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• 0. Then \\e~zfl° . • 0 and \]Ho • 1 for all jc e M^. Then we can write 0, there is a positive constant b—depending only on q,d and a—such that 4, they have a similar asymptotic behavior. (See [SilO, p. 70].) (b) For z 6 C+, [/ + zHo]~l is also an integral operator on L2 (Rd) and its kernel can be computed via entirely analogous formulas. (See Exercise 10.2.16 below.) (c) When t = 0, [/ + it HQ]~I does not have a representation of the form (10.2.52) but [I + it//o]~' = I~l = I is extremely simple in this case. (d) In [Lai, 2, La6-13] and [ReSi2], for example, the free Hamiltonian is taken to be two times HO; that is, it is chosen to be equal to —A instead of—A A. In this case, as was mentioned earlier, t on the right-hand side of (10.2.53) must be replaced by 2t. Exercise 10.2.16 (i) Show that for every z e C+, the complex resolvent [I + zHo]~l is a bounded integral operator on L2 (Rd) and calculate its kernel (which can be viewed as a complex Green's function). In particular, for z = t (resp., z = it), with t ^ 0, deduce the expression of the (standard) Green's function, the kernel of [I +1 HQ] ~l, and recover the expression obtained in (10.2.53) for the kernel of [I + itHo]~l. [Advice: In part (i), you may assume that d = 1 or 3, for simplicity. Further, for d = 3, for instance, you should find that the kernel of [—A + f 2 ]~' is given by (4jr||jt — y\\)-ie-S\\x-y\\, for all f e C+. By contrast, for d = D + 2 (D > 1), this same kernel is given for £ € C+ by C is sesquilinear and satisfies \q( 0 and all tp, ty 6 H, then there is a unique operator A 6 £(W) such that q — qA- The converse result depends on Riesz's lemma from which it follows quite easily [ReSil, p. 44]. Our main goal is a much deeper result which is in the spirit of the converse result above but involves semibounded forms and self-adjoint operators. Specifically, we will see that "if q is a closed semibounded sesquilinear form, then it is the form associated with a unique self-adjoint operator". This result has proved to be important in the mathematical treatment of quantum theory, as well as in various other subjects, including Dirichlet forms, Markov processes and the Feynman integral. Basic definitions and properties We will let Ti be a Hilbert space over C throughout our discussion. Definition 10.3.1 A sesquilinear form is a map q : Q(q) x Q(q) —»• C, where Q(q) is a dense linear subspace of H, called the form domain of q, such that q is linear in the first variable and conjugate linear in the second variable. If q(tf>, if) = q(ifr, 9) for every 0 such that 0 and q( 0 and q( • 0 as n —* oo, as we wished to show. Next we establish the converse. Let { () such that \\ 0. Certainly also(c+l)|| Oasn —> oo. Since we already saw that || • Oasn —> oo. Hence, to finish showing that \\ . Example 10.3.13 Define q : V x T> -*• C by . , At/0 for all . Then q immediately has an extension to a semibounded quadratic form on D(A) x D(A). By Propositions 10.3.8 and 10.3.9, the quadratic form qA associated with A as in Definition 10.3.5 is a further extension of q which is closed. But no such closed extension of q exists as we have shown above. Hence (ii) follows. D Representation theorems for quadratic forms -c\\ —c\\ -c|| 0 for every ip e Q(q). (In the more general case when q is semibounded, we simply add c((p, VO to the right-hand side of (10.3.28) below.) Since the form q is closed as well as nonnegative, we know (see (10.3.3) and Definition 10.3.3 with c = 0) that the domain Q(q) of the form is a Hilbert space under the inner product 6 Tt+i, and the fact that H+i = Q(q) is norm dense in H allow us to show easily that D(B) is norm dense in H: Given h e H and e > 0, first, take h+i e T-L+\ such that \\h - h+\ \\ < e/2. Next take k e D(A) such that H/.+,-*||+l < e/2. Then \\h - k\\ < \\h - h+i || + \\h+1 - k\\ < \\h - h+l || + H/r+i - ftll+i < e/2 + e/2 = e as desired. Next we show that B is a symmetric operator. Let e ft : f+(a>) > 0}r\{ca € ft : /_(«) > 0} = 0, we see that (R)) whereas the latter insures that D(A) n D(B) = {0}. Consequently, the operator sum A + B is trivial (being the zero operator with domain {0}) whereas, by Theorem 10.3.19, the form sum A+B is a well-defined (and hence densely defined) self-adjoint operator in H. To obtain a function V satisfying the above hypotheses, take K > 0, K e L 2 (R), but with K4 not integrable near x = 0. Then, if {rn}'^_l is an enumeration of the rational numbers, set (Md), equation (10.4.3) follows from Green's formula (and the fact that qH0( (M.d)inHl(E.d).] (b) Deduce from (a) that under the assumptions of Theorem 10.4.8, we have for H = HO+V, the form sum of HO and V, c((p, (p) for all ; then • T(ip) for all (#o) and that Z> c D(Ho). Thus —A is a symmetric operator on V. It is easy to show that V + 1 is a symmetric operator on T>, and so -A + V + 1 is itself a symmetric operator on T>. Recall also (Definition 9.6.4(i)) that the adjoint of a symmetric operator extends the operator. Thus 0 implies that un -> u in the distributional sense. Fix and let K — supp(^)). Then Oasn —> oo.But d/2for d > 4. [If V_ e Lp(Rd) + L°°(R.d), as in Theorem 7.5.1, then the conclusion of Proposition 11.2.14 holds even if we take p = d/2ford > 5 (i.e. d > 4)J (b) Theorems 7.5.1 and 11.2.19 are closely related, but the latter was not only proved but also is considerably more detailed. In addition, the conditions on V- are more general in Theorem 11.2.19; we assumed in Theorem 7.5.1 that V- e Lp(Rd) + Z.°°(Rd), with p as in (a) above, and avoided discussion of relative operator boundedness and the classes Sd. (c) The conditions on the function V in Theorem 11.2.19 permit a much broader class of potentials than were allowed in Nelson's 1964 paper [Nel], Nelson required that V : Rd -> R be a finite sum of functions in Lp (W*)for values of p satisfying p > 2 and p > d/2 and functions which "after an inhomogeneous linear change of variables are of theform W ( x 1 , . . . , Xj)where j < d and W is in Lp (Rd) for values of p satisfy ing p > 2 and p > d/2." It is Section 1 and Appendix B, especially Theorem 8, of [Nel] that are related to our present considerations and which the reader may wish to consult. The increased generality of the assumptions in Theorem 11.2.19 is particularly striking for the positive pan of the potential since it is only required that V+ e L^oc(Rd). However, we should note that Nelson's paper was already sufficiently general to accommodate the case of a finite number of particles with Coulomb interactions. The Feynman integral via the Trotter product formula (briefly, TPF) is probably the most widely known of the many approaches to this subject. Perhaps because of this and because of the close conceptual connection with the approximation approach in Feynman's original paper (see Section 7.4), this version of the Feynman integral is often referred to as simply "the Feynman integral". We will refer to it, however, as the Feynman integral via TPF and write f'TP(V) for this Feynman integral associated with the potential V. Definition 11.2.21 Let V : Rd ->• R be Lebesgue measurable and finite Leb.-a.e. Note that the operator (e~'"H°e~'"V)n on L2(K.d) makes sense for every positive integer n and that for every (p € L 2 (R d ) and Leb.-a.e. v, [(e~'"Hoe^'nV)n(p](v) equals the expression on the right-hand side of (11.2.38) with the integrals interpreted in the mean when necessary. The Feynman integral via TPF associated with the potential V, denoted 0, 0, J-'M(V) ^ / IIV 1. Then by (12.2.32), Vi,m < m and so Vi,m M2 < m\ip\2 e Ll(K.d); since in addition, V n > m |^| 2 | Vm\ oo and hence, in light of (12.2.33) and (12.2.34), we conclude that (12.2.35) holds. Let Vm, _ be the negative part of Vm. Since by (12.2.36), V m, - = V_ is Ho-form bounded with bound less than 1, the nonnegative form qoo,m is closed (see Theorem 10.3.19). Hence, in the notation of Theorem 12.2.4, (qoo,m)r = qoo,m. Let Tm be the (nonnegative) self-adjoint operator associated with q00,m. Note that by Theorem 10.3.19, Tm = H0 + (S + Vm), the form sum of the operators HO and S + Vm. By Theorem 12.2.4, the monotone convergence theorem for nonincreasing forms, [I + Tn,m]~} -> U + Tm]-] strongly as n -> oo. By Theorem 9.7.11 (or 8.1.12), a key approximation theorem for semigroups, it follows that for all t > 0, e-tTn,m -> e-tTm strongly as n —> oo; namely for every \fr e L 2 (R d ), 0 Leb.-a.e. with 0 (< 6 D(H)) is nonnegative for all times: u(•, t) > 0 for all t > 0. Proof This is an immediate consequence of the Feynman-Kac formula (12.1.4) and of Corollary 12.1.2. (C_;mo)), is solved at time t by T-tmo(t) 2 or for (a = 2 and ft > 1/4)—not only is the solution of "the" Schrodinger equation |+,/'(./+ !)• Further, let (T(J}(t)} (resp., [U$(t)}) denote the restriction of {T(t)) (resp., (Up(t)}) to the eigenspace "Hj. Then there exists a probability measure P(j) on the unit circle (or 1-torus) T(j) := S1 such that the counterpart of equation (13.6.32) holds; namely,
as desired. The case z = t > 0 is especially important.
APPLICATIONS OF THE SPECTRAL THEOREM
167
Theorem 10.2.6 (Free heat semigroup) For w e L2 (Rd) and t > 0, the formula
holds for Leb.-a.e. x e Rd. Further, forwp e D(H0) (see (10.2.7) and (10.2.10)) and t > 0, the function
gives the solution, in the sense discussed in Section 9.1, to the heat equation
with initial data w(.) (and potential V = 0). Proof The assertion regarding (10.2.26) is just a special case of Theorem 10.2.5. The claim regarding the heat equation follows from Theorems 9.1.4 and 9.1.14 and the fact that e t ( - H o ) is a (Co) semigroup on L2 (Rd) with generator -H0. The semigroup e-tHo, t > 0, is called the (free) heat semigroup. Later, we will also be interested in "heat semigroups" e-t(H°+v) t > 0, which involve a nonzero potential (or interaction term) V. Next, we wish to establish formula (10.2.25) for w e L1 (Rd) n L2 ( R d ) and z = it purely imaginary but not equal to 0. Recall from (10.2.14) that the operator e-zho is strongly continuous in C+. Now let {zn} be a sequence in C+ such that zn -> it, t = 0. By the strong continuity just noted above we have that ||e- ZnH ° w — e-it H °w||2 ->• 0 as n -> oo. By dropping to a subsequence if necessary (which we will not indicate in our notation), we can also insure that (e~ z n H °W)(x) —> (e-itHo W ( x ) for Leb.-a.e. x e_R d . On the other hand, since w e L1 and|g-| *-«x-u| 2/2z|<1foranx,u e Rd and z e C+ \ {0}, we see from the dominated convergence theorem that
Now Theorem 10.2.5 along with the facts discussed above allow us to finish this case (namely, z = it and w e L1 n L2) by taking limits:
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SELF-ADJOINT OPERATORS AND QUADRATIC FORMS
Finally, we wish to establish (10.2.25) for z = it and
First note that I k - M | | 2 = (x - u) • (x - u) = \\x\\2 + \\u\\2 -2x • u. Thus,
'ii-ir , Now since
7
r
'!M|2i\ ( x )
has the L -function F\
->
as its limit in L -norm as r —> oo. Thus, by
i \\x ii2 r i||.||2i (10.2.31), the L -function(it)~ e 2t F\
d/2
r —> oo of the integral
But this is precisely what is meant when one writes
and says that the integral on the right-hand side of (10.2.32) is to be interpreted in the mean. Now if we let
then \\(p - <prh ->• 0 as r -+ oo. Hence \\e~itH°(p - e-itH0lVrll2 -* 0 as r -* oo. Also (10.2.30) holds for the function (pr since <pr 6 L1 n L2. Putting the facts above together, we can write the following where the limits involved are in L2-norm and the last integral is interpreted in the mean:
APPLICATIONS OF THE SPECTRAL THEOREM
169
In the following theorem, we (i) summarize what we have proved above regarding the explicit representation of the unitary group e-it H0, and (ii) observe that e-it H 0 (p, (p 6 D(H0), provides the solution to the Schrodinger equation with initial data
where l.i.m. indicates that the integral on the right-hand side of (10.2.33) is to be interpreted in the mean; i.e. it is the L2-limit as r —> oo of the integral over the r-ball of the same expression. When tp e L1 (R d ) D L2 (R d ), we can write
where the right-hand side of (10.2.34) is an ordinary Lebesgue integral. Also, fort 6 Randy e D(H0) (see (10.2.7) and (10.2.10)) with \\p\\2 = 1, the function
gives the solution in the sense discussed in Section 9.1 (see also Definition 9.6.9(ii)) to the Schrodinger equation
with initial probability amplitude (p for the free particle (i.e. potential energy V = 0) in Rd. The function ( 2 n i t ) - d / 2 e 2 r appearing in (10.2.33) and (10.2.34) is often called the free propagator in the physics literature. (Mathematically, it can be viewed as the fundamental solution—based at the point x e Rd and evaluated at the time t € R—of the (distributional) Schrodinger equation (10.2.36), with initial condition
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SELF-ADJOINT OPERATORS AND QUADRATIC FORMS
Theorem 10.2.8 LettpeL2 (Mrf). Then
Proof We know that the Fourier transform T is a unitary operator on L2. Using this, it is not hard to show that the scaled transform t~d/2(F
is also unitary. Now we know that e "H° is unitary as well, and so it is not hard to see that it suffices to establish (10.2.37) for
Hence, by (10.2.38) and (10.2.39),
and so
where the inequality in (10.2.41) follows from the estimate
Hence the theorem is established. Theorem 10.2.8 is given in [ReSi2, Theorem IX. 31, pp. 60-61] where its physical interpretation is also discussed.
APPLICATIONS OF THE SPECTRAL THEOREM
171
Standard cores for the free Hamiltonian Next we wish to show that the operator Ho|S and H0|D are essentially self-adjoint and that f/0|S = //o|X> = HQ. (Here, //o|5 and HQ\T> denote the restriction of HO to S and T>, respectively.) We will make use of two results found in [ReSil, Theorem VIII.3 and Corollary, pp. 256-257]. We now state both the theorem and its corollary. (At this point, the reader may wish to review Section 9.6, especially Definition 9.6.4(iii), (v), where the notions of self-adjointness and essential self-adjointness were introduced.) Theorem 10.2.9 (Criteria for self-adjointness) Let T be a symmetric operator on a Hilbert space H. Then the following three statements are equivalent: (i) T is self-adjoint. (ii) T is closed and Ker(T* ± i) = {0}. (iii) Ran(T ±i) = H. Here, Ker(T* ± z) denotes the kernel of T* ± i and Ran(r ± i) denotes the range of T±i. In the corollary, we replace "self-adjoint" in (i) with "essentially self-adjoint". The corresponding changes in (ii) and (iii) turn out to be what one might guess. Corollary 10.2.10 (Criteria for essential self-adjointness) Let T be a symmetric operator on a Hilbert space Ti.. Then the following are equivalent: (i) T is essentially self-adjoint; that is, it has a unique self-adjoint (necessarily its closure T). (ii) Ker(r* ± i) = {0}. (iii) Ran(T ± j) are both dense in "H.
extension
We first show that HQ\$ is essentially self-adjoint and then use that fact as a key element in the proof that Hop has the same property. Theorem 10.2.11 (i) The operator H0|S " essentially self-adjoint. (ii) <S is a core for HQ. Proof (i) By Corollary 10.2.10, it suffices to show that Ran(Ho|S i 0 are both dense in L2 (Md). Since the argument that we will make does not really depend on the sign of i, we will just show that Ran(H0|S + 0 is dense in L2. Now, from (10.2.7) and (10.2.8) we know that
where
Clearly, S c D(//o), and so what we need to show is that the set of functions {F~l [\ INI 2 'F(f)\ + // : / 6 5} is dense in L2. Since S is dense in L2, it suffices to show that for any g e S, there is an / e S such that J""1 [| || • ||2 F(f)] -\-iF~1 [F(f)] = g or, equivalently, that (||| • ||2 + i ) F ( f ) = f(g). By Theorem 10.2.4, h :- f(g) is
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SELF-ADJOINT OPERATORS AND QUADRATIC FORMS
just another element in S. Hence, we need to know that there is f e S such that
Since F maps S onto S (Theorem 10.2.4), it is enough to show that the function
belongs to S. However, since h e S, it is not difficult to convince oneself that the function in (10.2.43) is infinitely differentiable and satisfies (10.2.21) for all multi-indices a and ft. [First note that |(||y||2 + 2i)-1 <1/2for all y € Rd. Next, observe that the derivatives of the function (10.2.43) involve higher and higher powers of ||y ||2 + 2i in the denominator and polynomials multiplied by h and its derivatives in the numerator. Thus 2f+2i e S as claimed.] Hence, there exists / e S such that (10.2.42) holds. Thus Ran(//Q|5 + i) is dense in L2. Since, similarly, Ran(//o|5 — 0 is dense, we see that //ojS is essentially self-adjoint as claimed. (ii) Since //o|5 is essentially self-adjoint by (i), we know that #o|S has a unique self-adjoint extension, necessarily the closure of HQ\S (see Definition 9.6.4(v)). But HQ is a self-adjoint extension of HQ\<; and so HQ = //o|S- It now follows from the fact that HQ = HQ is closed (see Theorem 10.2.9 or Proposition 9.6.2) and from Propositon 9.1.20(ii), that S is a core for HQ. D Next we will prove a result like Theorem 10.2.11 but for //o|D rather than for HQ\$. We cannot give the same proof as in Theorem 10.2.11 basically because the Fourier transform does not map D onto D. We will however use Theorem 10.2.11 in the proof to follow. Theorem 10.2.12 (i) The operator H0|D is essentially self-adjoint. (ii) D is a core for H0. Proof The key to the proof is to show that HQ\TJ 2 HQ\$. Let
Note that q>n e T> for all n. Also, it is not hard to show that \\ipn —
APPLICATIONS OF THE SPECTRAL THEOREM
173
where the next to last equality follows from the dominated convergence theorem with 4|
Applying (10.2.45) to
Now A
Further, since the partial derivatives f j r , . . . , fjr- are all in D, there is a uniform bound, say N, for all of these functions. Hence, we have
and so
Finally, since Aw e D and so is bounded, one has
Putting together (10.2.47)—(10.2.49), we see from (10.2.46) that || A
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SELF-ADJOINT OPERATORS AND QUADRATIC FORMS
//op and show that B = HQ. Such a self-adjoint extension B of //op is certainly closed since it is self-adjoint (see Theorem 10.2.9 or Proposition 9.6.2). However, since B is a closed extension of //op, it must certainly contain the smallest closed extension; namely, B l> (//op)** — HQ\T> = HO. From this and from the self-adjointness of B and HO, we deduce that
Thus we have both B ^ HO and B c HO; hence B — HO. Our proof of Theorem 10.2.12 is now complete. D The functions in D(//o) need not possess a classical Laplacian; on the other hand, D(Ho) falls well short of including all functions in L2 (R rf ). What can be said about the functions in D(//o)? We simply state one result along these lines [ReSi2, Theorem IX.28, p. 55] which is closely related to the Sobolev imbedding theorem [Ad, Bre2, Kes]. Theorem 10.2.13 Let y e L2 (Rd) be in D(//0), the domain of the free Hamiltonian HO. (i) If d < 3, then (p is bounded and continuous and for any a > 0, there is a positive constant b, independent of
(ii) Ifd >4andqe [2, 2d/(d - 4)), then
Imaginary resolvents We have discussed the operator HO and the exponential functions e~zH° (z e C+) at some length. In applying the "product formula for imaginary resolvents", Theorem 11.3.1, to the "modified Feynman integral" ([Lai, 2], [La6-13], [BivLa]) in Sections 11.4-11.6, we will also need to consider the resolvents [/ + itHo]~l, t e M. The central result that is needed expresses [/ + it//o]"1 as an integral operator. We will omit the proof, which can be found in [ReSi2, Theorem IX.29 and Examples 1,2, pp. 57-59]. Theorem 10.2.14 (Free imaginary resolvent) For all t e M, [7 + itHo]~l e C(L2 (Rd)). Further, for all t ^ 0, [/ + itH0]~l is the integral operator (with kernel the free "Green's function" G(x, j; t)) given for all
where the integral on the right-hand side o/(10.2.52) exists as a Lebesgue integral. The function G(x,y\ t) is known explicitly for all nonzero t andd = 1,2, In particular,
APPLICATIONS OF THE SPECTRAL THEOREM
175
for t > 0, we have for d = I
and for d = 3,
In (10.2.53a), z1/2 denotes the standard analytic determination of the square root which is positive for z > 0. Moreover, for t < 0 (instead of t > 0), the expression of G(x, y; t) in (10.2.53) would be the same as above except for (1 — i)t~1/2 replaced by (1 + i)\t\~1/2 inside the exponentials. Finally, ford = 1 or 3 (resp., otherwise), the integral in (10.2.52) converges for all x (resp., Leb.-almost every x) in Rd. Remark 10.2.15 (a) Theorem 10.2.14 is in much the same spirit as our earlier Theorem 10.2.7 which dealt with the free unitary group; indeed, there are common elements in the proofs which both make use of the Fourier transform. There are also differences between Theorems 10.2.7 and 10.2.14. The integrals in (10.2.33) are oscillatory and hence are interpreted as integrals in the mean when
176
SELF-ADJOINT OPERATORS AND QUADRATIC FORMS
where wrj denotes the solid angle in D dimensions, F is the gamma function, and HDn denotes the Bessel function of the third kind (Hankel function) of order D/2. Hence, in (10.2.52), G(x, y; t) is equal to (2it) -1 times this expression, where f := (1 — z)A/f or f := (1 + i)/ J\T\, for t > 0 or t < 0, respectively.)] (ii) Show that the operator-valued function [I + Z//Q]~' is analytic for z € C+ and strongly continuous for z € C+. [Hint: The proof of part (ii) parallels that of Theorem 13.3. L] 10.3 Representation theorems for unbounded quadratic forms Our interest here is in unbounded operators and forms, but let us begin by recalling the situation with bounded operators. Let H be a Hilbert space over C. If A e £(7i) and if we let qA '• T~i x H —»• C be defined by q& (
for
REPRESENTATION THEOREMS FOR UNBOUNDED QUADRATIC FORMS
177
Partly with this in mind, q acting on all of the product space Q(q) x Q(q) is often itself referred to as a quadratic form. (b) If q is semibounded with (10.3.1) holding, then q' defined for
is clearly a nonnegative sesquilinear (or quadratic) form. Because of the simple relationship between q and q', results which can be established for nonnegative quadratic forms usually carry over easily to semibounded quadraticforms. Hence proofs for semibounded forms are often given just for nonnegative forms. (c) One can show [BachN, Theorem 20.13, p. 369] that a sesquilinear form q is symmetric if and only if q((f>) e R/or all (p 6 Q(q)- It follows from (10.3.1) that a semibounded sesquilinear form is necessarily symmetric. (7f is implicit in the definition of semiboundedness that q((p, (p) is real and (10.3.1) holds.) Recall that an operator A : D(A) —>• H is said to be closed if and only if its graph is closed in H x H; further, recall that this is equivalent to requiring that D(A) is complete under the norm \\tp\\A := \\Aq>\\ + \\tp\\ (see Propositions 9.1.7 and 9.1.8). In order to study the relationship between self-adjointness and semibounded quadratic forms, we will need to extend the notion of "closed" from operators to semibounded forms. Let q be a semibounded quadratic (or sesquilinear) form and let c > 0 be such that (10.3.1) holds for all tp e Q(q). It is not hard to show that the formula
defines an inner product on Q(q). The associated norm is then given by
Definition 10.3.3 A semibounded quadratic form q as above is called closed if and only if Q(q) is complete under the norm \\ • \\+\ given by (10.3.4). If q is closed and D is a subspace of Q(q) which is dense in Q(q) in the norm || • ||+i, then D is called a form core for q. Our first proposition gives a necessary and sufficient condition for q to be closed. We will include the simple proof. Proposition 10.3.4 Let q be a semibounded (and hence symmetric) quadratic form satisfying (10.3.1) for all
as n, m -* oo. But (Q(q), \\ • ||+i) is complete and so there exists (po e Q(q) such that
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SELF-ADJOINT OPERATORS AND QUADRATIC FORMS
Hence \\
as n, m -» oo. Since the sum of the first two terms on the right-hand side of (10.3.6) is nonnegative,itfollowsthat||
and, for ( < p i , . . . ,
Recall that when N = oo, ( < p \ , . . . , (pu) should be interpreted as the infinite sequence (
REPRESENTATION THEOREMS FOR UNBOUNDED QUADRATIC FORMS
179
Definition 10.3.5 With the setting as described in the last two paragraphs, we let
and, for (
Then the form q^ is called the quadratic form associated with A, and we sometimes write Q(qA) = Q(A). The subspace Q(A) is called the form domain of the operator A. It follows from this definition that q& (
All the remaining proofs in this chapter could be rewritten with more concise notation using this formulation. However, the concreteness of (10.3.7)-(10.3.10) has its own appeal. Definition 10.3.7 A symmetric operator A : D(A) -> H is said to be semibounded (or bounded below) if and only if there exists a number c > 0 such that
for all
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SELF-ADJOINT OPERATORS AND QUADRATIC FORMS
Proposition 10.3.8 Let Abe a semibounded self-adjoint operator on the Hilbert space Ti. and let q& be the quadratic form associated with A. Then qA is semibounded and closed. Proof Let c > 0 be such that (10.3.15) is satisfied. It follows from Exercise 10.1.13 that
Hence we have, for p = 1 , 2 , . . . , Support(/u,p) c a(A) c [—c, oo). Therefore, (10.3.9) and (10.3.10) of Definition 10.3.5 can be rewritten in this situation as
and, for (
It follows readily from (10.3.17) and (10.3.15) that qA is semibounded. Now let (
By (10.3.16), x + c > Oona(A) andsox+ c+ 1 > 1 oner (A), Hence, it follows from (10.3.20) that ((")} is a Cauchy sequence in 0^° =| L 2 (cT(A), (x + c + \)d^p). Thus there exists (
REPRESENTATION THEOREMS FOR UNBOUNDED QUADRATIC FORMS
181
on [—c, oo) D o(A). Hence (
Proof As mentioned earlier, we will limit our attention in this proof to the case N = \ of the spectral representation. Let
But then certainly,
Since the function x -> \x\ is bounded on [—1, 1], u. is a finite measure and y e L 2 (R, u), we also have
It follows from (10.3.23) and (10.3.24) that /R \x\\
as claimed in (10.3.22). Note that A was not required to be semibounded in Proposition 10.3.9. Proposition 10.3.10 Let A be self-adjoint and semibounded. Then D (A) is \\ • \\ +1 -dense in Q(qA).
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Proof We again limit our attention to the case N = 1 of the spectral representation. Recall from Proposition 10.3.8 that qA is semibounded, so that || • ||+1 defined by (10.3.4) is a norm. Now let p e Q(qA) so that we have /r |^>(x)| 2 du < oo and, by (10.3.9), /R \x\\
Also \\
as n -*• oo, as desired. Exercise 10.3.11 Prove Propositions 10.3.9 and 10.3.10 for N = oo. Proposition 10.3.10 tells us, in the language of Definition 10.3.3, that the domain of a self-adjoint semibounded operator A is a form core for qA. Our next proposition says that more is true: Any operator core for such an A is a form core for qA. Proposition 10.3.12 Let A be self-adjoint and semibounded. Then any operator core for A is a form core for qA. Proof First we note that a self-adjoint operator is necessarily closed (see Proposition 9.6.2(v)). Now let DO be an operator core for A (Definition 9.1.16). By Proposition 9.1.17, DO is dense in D(A) under the graph norm |h|| := ||h|| + \\Ah\\ on D(A). We wish to show that DO is dense in Q(qA) under the || • ||+1 norm. Since D(A) is dense in Q(qA) under the || • ||+1 norm by Proposition 10.3.10, it suffices to show that D0 is dense in (D(A), || • ||+1). Let h e D(A). Since DO is dense in D(A) in the graph norm, there exists a sequence {hn} in DO such that
Now by Proposition 10.3.9, the Cauchy-Schwarz inequality, and (10.3.25a), we see that
as n —> oo. Using (10.3.25), we see immediately that
Thus DO is a form core for qA, as desired.
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Let D = D(R), the space of infinitely differentiable functions of compact support on R. Next, we give an example of a quadratic form q on D x D which is positive and symmetric but for which there is no semibounded, self-adjoint operator A such that q((p, ifr) = (
Then q is easily seen to be & positive, symmetric quadratic form. However, we claim that (i) q has no closed extension, and (ii) there is no semibounded, self-adjoint operator A such that q(
Proof Consider a sequence {$„} from T> such that
We are ready for the main theorem in this development. Theorem 10.3.14 (First representation theorem) Let q be a closed semibounded quadratic form. Then: (a) There exists a unique self-adjoint operator A such that q — qA (i-e. such that q is the quadratic form associated with the self-adjoint operator A, in the sense of Definition 10.3.5). Hence, by Proposition 10.3.9, D(A) C Q(q) =; Q(A) and
In particular,
(b)Ifc>0 is such that q satisfies q(
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give the proof under the additional assumption that q is nonnegative; i.e. q(
We will denote this Hilbert space H+1 and we let H-1 denote the space of bounded conjugate linear functions on H+1. Let j : H ->• H-1 be the linear mapping of H into H+1 defined by
Each j (VO is clearly conjugate linear. Also, it follows easily from the Cauchy-Schwarz inequality and the fact that \\
Hence j (\jf) is a bounded conjugate linear functional on H+\, as claimed. Next, we show that j is injective. Let TJt\, fa e H and suppose that j ( f a ) = j ( f a ) . Then for every (p e H+1, we have
But H+1 — Q(q) is dense in H and so ^r\ = fa as claimed. Finally, it is easy to see from (10.3.31) that j is continuous. In summary, j is a linear, continuous injective map from H to H-1. Clearly, the identity map / : H+1 -> H is also linear, continuous and injective. Since H+1 is a Hilbert space, every element * in H.+1 defines an element 54* of H-i via the formula
In fact, by the Riesz lemma [ReSil, Theorem II.4, p. 43], B is an isometric isomorphism ofH+i ontoW-i. We are now ready to define an operator B which will turn out to be related in a simple way to the operator A which we seek. Let R(j) denote the range of the imbedding j : H -»• H-i discussed earlier, let
and define B : D(B) -» H by
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The following diagram, along with (10.3.32}-(10.3.34), will help us to keep track of some of the relationships discussed above:
We will need to know that B is densely defined on "H. In this connection, we begin by showing that R(j) is dense in ~H-\. If this were not so, then there would be A € H*_\ such that X ^ 0 but for all V 6 H
Now T~i+i is a Hilbert space and H*+i = T~t-\. It follows that ti*_^ = ~H+\ and so there exists
for all T/f e 71., where the second equality in (10.3.37) comes from the definition of j. Combining (10.3.36) and (10.3.37), we see that
But (10.3.38) is not possible since
Now from (10.3.39) and the symmetric nature of the hypotheses on
and so we see that B is symmetric as claimed. Thus B is a densely defined symmetric operator.
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Next we show that B is actually self-adjoint. Let C = B~l o j. Since j maps H into Ji-\ and fi"1 is an isometric isomorphism of "H-\ ontoW+i c 7, it follows that C takes H into H. Note that C is every where defined and that C — B~loj — (j~loB)~l = B~l. We claim that C is symmetric. Let h1,h2 e W.SinceC = B ~l, there exist k1, k2 e D(B) such that h1= Bk1 and hi = Rki- Then since B is symmetric, we have
as claimed. The Hellinger-Toeplitz theorem [ReSil, p. 84], a simple consequence of the closed graph theorem, assures us that the everywhere defined symmetric operator C is bounded and self-adjoint. Also, C is injective since both j and B"1 are injective. Hence by Theorem 10.1.8, the spectral theorem in multiplication operator form, there is a finite measure space (£2, A, M) and an R-valued function / e L°°(£2, A, M) such that C is unitarily equivalent to the operator M/ on L2(£2, A, M) of multiplication by /. Since C is injective, M/ is also injective and so / ^ 0 u-a.e. Thus 1/f is R-valued u-a.e. and M1/f is a self-adjoint operator (not necessarily bounded, of course) on L 2 (Q, A, u.). The spectral theorem applied this time to an unbounded—rather than a bounded—function of a self-adjoint operator assures us that C-1 is unitarily equivalent to M\/f and so C"1 = B is self-adjoint. We define A := B - I with D(A) = D(B). Then A is self-adjoint. Also we see from (10.3.40) that (
But, by Proposition 10.3.9, we have that (
In order to finish the proof of the existence part of the assertion of this theorem, we need to show that Q(q) = Q(qA) and that q = q& on this common domain. Q(q) ^ Q(qA)'- Let h € H+\ = Q(q). Recall that in the process of showing above that D(B) is dense in H, we saw that D(B) is || • ||+1 -dense in H+\. Since D(A) - D(B), D(A) is then also || • ||+i-dense in H+I. Using this fact, we can take a sequence {an} from D(A) such that \\an - h\\+i -+ 0. We see from (10.3.4) that the self-adjoint operator A is nonnegative. For any k e Q(9A), let
Since qA = q on D(A) x D(A), we see that || • |U,+i = II • ll+i on D(A). Hence we have \\an - am |U,+i = \\an - am ||+i ->• 0. Since qA is closed by Proposition 10.3.8,
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there exists /JQ e HA,+\ '•= Q(QA) such that \\an — AolU.i -*• 0. The proof of the containment will be finished if we show that ho = h. However, q(an -h) + \\an — h\\2 = \\an - h\\2+l -> 0 and q(an - h0) + \\an - h0\\2 = \\an - h0\\\v -»• 0. It follows that ll^n — h|| ->• 0 and \\an — ho\\ ->• 0, and so h — ho as we wished to show. Q (QA ) £ Q (l) : The proof of this containment is essentially the same as that of the containment above and we leave it as an exercise for the reader except to remark that the || • || A,i-density of D(A) in Q(qA) comes from Proposition 10.3.10. The two containments assure us that Q(q) — Q(qA) and so it remains to show that q — qA on their common domain. Let h & Q(q). Take a sequence {an} from D(A) just as in the first part of the proof so that \\an — /Mil -> 0. From the earlier argument and the fact that ho above turned out to be h, we see that \\an — A |U, i -*• 0- It follows that ||an||i -» ||h||i and \\an\\A.\ -* \\h\\A,i. But || • |U = || - ||Ail on D(A) (as noted previously) and so ||&||i = ||/z|U,i- But then, by the polarization identity (equation (10.3.2)), q(., •) + (-, •) = qA(-, •) + (-, •) on Q(q) = Q(qA). Hence q = qA as desired. We will finish the proof of Theorem 10.3.14 by establishing the uniqueness that is claimed in part (a). Accordingly, let A and B be self-adjoint operators such that qA = qB = q. Then Q(qA) = Q(qB) and qA = qs on Q(qA) x Q(qA) = Q(qB) x Q(qB). We need to show that D(A) = D(B) and that A and B agree on this common domain. We show below that D(A) c D(B*) and B* = A on D(A). Since B is self-adjoint, it will follow that D(A) c D(B) and B - A on D(A). Further, from the symmetric nature of the assumptions, we will then have that D(B) c D(A) and A = B on D(B). Consequently, it will follow that D(A) = D(B) with agreement of A and B on their common domain. Thus, it remains only to show that D(A) c D(B*) with B* = A on D(A). Let (p e D(A). It suffices, by definition of B*, to show that (B^jr,
Hence,
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q = qA. (See Theorem 10.3.14.) Then Q(A), the form domain of A (see Definition 10.3.5), Is given by Q(A) := Q(q) = D(Al/2). Moreover,
and so, in particular,
Proof We first note that the existence and uniqueness of the nonnegative self-adjoint operator A such that q = qA is guaranteed by Theorem 10.3.14, the first representation theorem. Now, Theorem 10.3.15 is easy to prove using either the definition of the quadratic form qA associated with the self-adjoint operator A (see Definition 10.3.5 and (10.3.7)(10.3.10)) or the alternative described in Remark 10.3.6 (see (10.3.11)-(10.3.14)). We will use the latter. The operator A is (unitarily equivalent to) the operator Mg on L 2 (/z) = L 2 (£2, A, /J.) of multiplication by the .A-measurable, M-valued function g, where u(£2) < oo and g is finite /i,-a.e. As earlier, g will denote both the function itself and the operator Mg of multiplication by g. In the present case, g > 0 since A > 0. We see from (10.3.13) and (10.3.14) that
and
But since A 1 / 2 = g 1 / 2 , it follows that
and
Comparing (10.3.44) and (10.3.46), we see that Q(A) = D(Al/2). Similarly, from (10.3.45) and (10.3.47) we obtain (10.3.43). Exercise 10.3.16 Prove Theorem 10.3.15 using Definition 10.3.5 and formulas (10.3.7)(10.3.10). Given a self-adjoint operator A : D(A) —* 7, we discuss next the operators \A\, A+, A-, \A\1/2, A+ , A_ , and the relationships between them. Our first use of
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the results of this discussion will give information about the form domain of A in the case where A is not necessarily positive, but we will make use of these operators in several other places as we continue. (See especially Sections 11.3-11.6, as well as Chapters 12 and 13.) According to the spectral theorem in the multiplication operator form, Theorem 10.1.8, the self-adjoint operator A is unitarily equivalent to the operator Mf on L2(/u.) = L 2 (fit, A, n) of multiplication by an R-valued .4-measurable function /, where /K, is a finite measure. We will frequently suppress reference to the unitary operator U : Ti. -> L 2 (/z) and just regard A as equal to A//. We will be more formal here and write
where D(A) is related to D(Mf) as follows:
The operators | A \, A+ and A -—called respectively the absolute value, the positive part and the negative part of A— are then defined uniquely via the spectral theorem by
and
These three self-adjoint operators are all nonnegative since the functions |/|, /+, /_ are nonnegative. (Here, the functions /+ := max(/, 0) and /_ := max(—/, 0) denote the positive part and the negative part of /, respectively.) Further, from the relationships / = /+ — /- and | /| = /+ + /_, the spectral theorem assures us that
and
Now D(Mf) = D(M|/|) = {y & L2(,u) : ftp e L2(/j,)}, and D(Mf±) = {
and so
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In similar fashion, we define the following nonnegative self-adjoint operators, the square root of \A\, A+, and A_, respectively:
and
Since /+/_ = 0, we have |/|1/2 - /+/2 + /1/2, and so we see that
Arguing much as we did in connection with (10.3.53), we have
Now from the spectral theorem (Theorem 10.1.8), the second representation theorem for forms (Theorem 10.3.15), and (10.3.56), we deduce that Q(A), the form domain of A, is given by
Finally, from (10.3.9) and (10.3.10), the equation x = x+ - x-, and the second representation theorem, we obtain that for
In particular,
The form sum of operators The concept of the "form sum" of two self-adjoint operators will play an important role in two of our approaches to the Feynman integral (Sections 11.3-11.5 and Sections 13.113.2, respectively), as well as in our dicussion of the Feynman-Kac formula (Chapter 12). The ordinary operator sum (or algebraic sum) of two self-adjoint operators A and B will also be used and has a straightforward definition:
and
It can happen, however, that D(A) f~l D(B) is not dense in H. Indeed, the extreme case where A and B are densely defined self-adjoint operators but D(A) n D(B) = {0}
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is possible (see Example 10.3.21 below). The idea is to consider the sum qA + qs of the forms associated with A and B. Under appropriate hypotheses to be discussed below, qA + qa will be densely defined, semibounded and closed. Theorem 10.3.14, the first representation theorem, then assures us that there exists a unique densely defined, semibounded, self-adjoint operator associated with the form qA + qa- This self-adjoint operator will be denoted A+B and called the/orm sum of A and B. It can happen in cases of interest that A+B makes perfectly good sense even though A + B has a domain which is too small to be useful. Our concern with form sums comes from our desire to have the Feynman integral defined even when highly singular potential energy functions are involved as indeed they sometimes are in physical problems. In most of the concrete applications of form sums, including those which are our primary concern, the self-adjoint operator A is nonnegative. Hence, we restrict our attention to that case. Following (10.3.51a), we will write the self-adjoint operator B as B = o_j- — B~. Definition 10.3.17 We say that B_ is relatively form bounded with respect to A (briefly, B- is A-form bounded,) with bound less than 1 if and only if
and there exist positive constants y < 1 and S such that
The infimum of all such positive numbers y is called the A-form bound of B. (Hence it is strictly less than one.) Remark 10.3.18 (a) Definition 10.3.17 can easily be extended to the case where A is a semibounded, self-adjoint operator and B is a symmetric (densely defined) operator. We simply replace Q(B-) by Q(B) in (10.3.62a) and (10.3.62b) by
(b) When the number Y in (10.3.62b) can be made smaller, it may well be necessary to take S larger. We stress that the infimum of the numbers y which can be used in (10.3.62b) need not be a bound itself. From our earlier discussion, especially the second representation theorem (Theorem 10.3.15), we see that the expressions in Definition 10.3.17 make sense and that (10.3.62b) can equivalently be written as
We add to the hypothesis of relative boundedness the assumption that Q(A) n Q(B+) is dense in H. Now the map
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We leave it as an exercise for the reader to show that the form qA + qs is also closed. We can now apply the first representation theorem (Theorem 10.3.14) to produce the self-adjoint operator A+B. We summarize in the next theorem the central facts coming from the discussion above. Theorem 10.3.19 (Form sum of operators) Let A and B be self-adjoint operators on Ti, with A nonnegative. Suppose further that B_ is A-form bounded with bound less than 1 and that Q(A) D Q(B+) is dense in U. Then the quadratic form QA+QB given by (10.3.63) is closed and semibounded and so there exists a unique semibounded self-adjoint operator A+B, called the form sum of A and B, such that qA+B = qA+qB; further Q(A+B) = Q(A)r\Q(B) = Q(A)r\Q(B+). The operator A+B is a self-adjoint extension of the algebraic sum A + B. (Recall that D(A + B) = D(A) n D(B).) Remark 10.3.20 The density assumption could be eliminated from Theorem 10.3.19 at the price of working on the Hilbert subspace of H obtained by taking the closure of Q(A) n Q(B+) in H. A similar comment could be made at several other places in this book. When we apply Theorem 10.3.19 below, U will be L1 (Rd), A will be the free Hamiltonian HO — —1/2 A and B = B+ — B- will be the operator My of multiplication by the potential energy function (or briefly, potential) V. Thus
and we will need conditions on V+ and V_ which assure us that Q(Ho) n Q(My+) is dense in L2 (Ed) and that Mv_ is //o-form bounded with bound less than 1. We will provide such conditions in Section 10.4. (See especially Theorem 10.4.8.) We close this section by providing a somewhat pathological example ([Cher2], adapted to quadratic forms in [La6, footnote to pp. 1705-1706]) which shows that in general, caution must be observed when comparing the operator sum with the form sum. Example 10.3.21 Let H = L 2 (R).Let A (- H0) be the differential operator -1/2d2/dx2 with domain D(A) = {
CONDITIONS ON THE POTENTIAL V FOR #0-FORM BOUNDEDNESS
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(So that || A^vf = \ /+* \
where the series in (10.3.66) is convergent in L2-norm. We leave the verifications as an exercise for the reader. 10.4
Conditions on the potential V for Ho-form boundedness
Let HO be the free Hamiltonian as discussed in Section 10.2. Suppose that V : E.d -> R is Lebesgue measurable and let V = V+ — V- be the usual decomposition of V into its positive and negative parts: V+ = max(V, 0) and V_ = max(—V, 0). We will have in some key theorems to follow (in Chapters 11-13) the assumption that V_ is Ho-form bounded with bound less than 1. We state in this section various concrete conditions on V which guarantee its form boundedness. This will enable us to use the results obtained at the end of Section 10.3, particularly Theorem 10.3.19, to define the form sum H = H0+V, a natural realization of the quantum-mechanical Hamiltonian, under very general hypotheses. The classes Kd in the following definition were introduced by Kato [Kat2] and are often referred to as the Kato classes. Our main references for this material are [CyFKSi, §1.2] and Simon's survey article [Sil 1]. (Also, see [Kat8] and [AiSi], as well as the more recent work [GulKon] quoted in Remark 10.4.9(b) below, along with [Gull].) Definition 10.4.1 (The Kato class Kd) Let W : W1 -» R be Lebesgue measurable. The function W belongs to the Kato class K^ if and only if
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We will also need the following classes of functions in connection with the present dicussion as well as further on. Definition 10.4.2 Let W : Rd ->• E be Lebesgue measurable, and let p e [1, oo). (i) We say that W is in L^c(Rd)-uniformly and write W e tfoc(Rd)u (or simply ^^(L^J if and only if
(ii) We say that W is in (L^K^-strongly uniformly and write W e L^^R4)(or simply W € (i-j^.)-) if and only if
Remark 10.4.3 (a) Observe that K\ = L^R),,. (b) Simple compactness arguments yield the first two containments below:
where L^oc(Wi) is the space of functions W such that fc \W\pdy < oo for every compact subset G ofRd. (c) The class (•£-£,<.)- was first introduced by Simader in his work on Schrodinger operators with singular potentials [Sd]. (d) It is apparent that the classes (££,<.)» and (L^oc)u are related. See (Hi) and (iv) of Proposition 13.4.5 for some precise information on this point. The next proposition accounts for our interest in Kd • (At this point, the reader may wish to review Definition 10.3.17 and Remark 10.3.18 where the notion of Wo-form boundedness is introduced.) Proposition 10.4.4 If W e Kd, then W is Ho-form bounded with bound less than 1. (Actually, W has Ho-form bound 0 in this case.) A proof of Proposition 10.4.4—based on a characterization of the Kato class Kd in terms of Brownian motion—can be found in [AiSi] and is sketched in [Sill, pp. 458-459]. For smaller classes—such as (L\oc)~, for example—Proposition 10.4.4 easily follows from the Sobolev inequalities (e.g., [Ad, Bre2, Kat8, Kes]); see, for example, [BreKat, pp. 139-140]. We now give some information about Kd and its contents. Proposition 10.4.5 (a) Kd c L,1^ (E.d)u for all positive integers d. (b) If we take p > d/2 ford>2 and p = 1 for d = 1, we have
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We now briefly comment on the first containment in (10.4.1) above. Recall to begin with that, by definition, W e LP + L°° if and only if W = Wi + W2, with Wi e LP and V/2 € L°°. Next, since V in Lp implies that the measure | V(y)\pdy is absolutely continuous with respect to Lebesgue measure, it is easy to see that Lp c (L^c)~; further, since L°° is also contained in (L^- and (L^.)- is closed under addition, it follows that Lp + L°° is contained in (L^oc)~, as desired. Example 10.4.6 ([CyFKSi, p. 8]) If d > 3 and & € (0, 1], then
is //o-form bounded with bound less than one but is not in Kj. The function V does belong to Kd if and only if S > 1. We note that the first two inclusions in (10.4.1) hold for any p e [1, oo). Moreover, an example showing that the containment (££,<.)« £ (L^u is strict can be found in [Jo6, pp. 15-16]. Further, Example 2.6 of [JoKim] shows that the first containment in (10.4.1) is strict. Finally, by Remark 10.4.3(a), the last containment in (10.4.1) is actually an equality when p = 1. We state one further pleasant property which is possessed by the classes K^, d — 1,2, . . . . Proposition 10.4.7 Let d and d' be positive integers with d < d'. Suppose that the function W : E.d ->• R is in Kd and that T : R.d' ->• Rd is linear and surjective. Then V(x) := W(Tx) is in Kd<. The linear map T in Proposition 10.4.7 could be, for example, a projection onto a subspace or, in connection with an N-particle quantum system (see Section 6.4), we could have
where p, q £ ( I , . . . , N] with p ^ q. We close this section by applying the above results to Theorem 10.3.19. Here, H = L 2 ^) and the operators A and B from Theorem 10.3.19becomef/0 and V = V+-V-, respectively. Theorem 10.4.8 Let V : Rd ->• R be Lebesgue measurable. We assume that V+ e .LI'OC (Rd} and that V- is Ho-form bounded with bound less than 1. (We know from Propositions 10.4.4 and 10.4.5 that this condition on V- is satisfied if V- is in any of the classes Lp (Rd) + L°° (Rd) c L£C (Rd)~ c L£C (Rd)u c Kd, where p > d/2 if d>2andp = lifd=l.) Then the form sum H := Ho+V is well defined and is a semibounded, self-adjoint operator on L2 (Rd). Remark 10.4.9 (a) If we assume that V_ belongs to (L^z ^ Lp + L°°, then the conclusion of Proposition 10.4.4 holds even when p = d/2 for d > 3 (i.e. d > 2); see,
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for example, [BreKat, Lemma 2.1 and Remark 2.1, pp. 139-140]. Hence the condition on p in Theorem 10.4.8 can be relaxed accordingly. (b) Recently, A. Gulisashvili and M.A. Kon [GulKon] have obtained, in particular, an elegant potential-theoretic characterization of the Kato class Kd, building upon [Sill, Proposition A.2.3, p. 454] (obtained in [AiSi]) and comments in [Sill, e.g. p. 458]. Briefly, their result [GulKon, Lemma 2.1] can be stated as follows: Let V € L^ (Rd). Then V e K,j = Kj (Md) if and only i f \ V \ — k — Ak, where k is a bounded uniformly continuous function on Rd and where Ak denotes the distributional Laplacian (closely related to the second Bessel potential) ofk. (Actually, the main theorem in [GulKon] solves a conjecture ofB. Simon [Sil2] concerning the smoothness of the heat semigroup e~'H, where H = HQ+V and the potential V is a suitable function in the Kato class Kj. We note that very recently, this result has been extended to a certain class of singular, time-dependent potentials in [Gul2].) Although the definition of the Kato class (given in Definition 10.4.1) appears somewhat technical at first, it may now seem more natural in the light of the above characterization ofKdThe following exercise will be useful to us later on; it requires from the reader some knowledge of the theory of Sobolev spaces [Ad, Bre2, Kat8, Kes]. Exercise 10.4.10 (Form domain of the Hamiltonian H = Ho+V) (a) Show that
and that Q(H0) = H1 (W1), the Sobolev space of functions
for all (p € Q(H). We see from (10.4.4) that Q(H), theform domain of the quantum-mechanical Hamiltonian H, is precisely equal to the set of "quantum states"
11 PRODUCT FORMULAS WITH APPLICATIONS TO THE FEYNMAN INTEGRAL Now that we have assembled the necessary tools, we are going to obtain in the next three chapters (especially in Chapters 11 and 13) some very general existence theorems for the (operator-valued) Feynman integral. We begin in Chapter 11 by considering two approaches to the Feynman integral via product formulas associated with semigroups (or other suitable functions) of operators; namely, the "Feynman integral via the Trotter product formula" (for unitary groups) and the "modified Feynman integral" (via a product formula for imaginary resolvents). The Trotter product formula, you will recall, was introduced in Chapter 7 (Section 7.5). We now prove in Section 11.1 a more abstract form (by means of the Chernoff product formula [Cherl,2]) and then obtain in Section 11.2 a general existence theorem for the Feynman integral (via the Trotter product formula for unitary groups) F'TP(V) introduced by Nelson in [Nel]. The beautiful distributional inequality of Kato [Kat2] permits us to establish the essential self-adjointness of the Schrodinger operator HQ + V under more generality (on the potential V) than was known at the time of Nelson's paper and therefore leads to broader results. (See Section 11.2.) Interest in the Feynman integral led Lapidus ([Lai 1], [Lal3, Part I]) to obtain a "product formula for imaginary resolvents" (rather than for unitary groups, as used in [Nel]). This formula is established in Section 11.3. Its proof rests in part on quadratic form techniques and the spectral theorem (as discussed in Chapter 10), as well as on a result of Chernoff in [Cher2]. When specialized, the product formula for imaginary resolvents yields in turn the existence of the "modified Feynman integral", also introduced by the second author in [La6, 11]. (See Section 11.4.) Thus the existence of the "modified Feynman integral" F'M(V) associated with the potential V is established under significantly more general assumptions on V than for the Feynman integral via the Trotter product formula ^ P (V). In particular, essential self-adjointness of the Schrodinger operator HO + V is no longer required. This enables us to show the convergence of T1M(V) (essentially) whenever the Hamiltonian (or "energy operator", defined by means of quadratic forms) is bounded from below, and hence the corresponding Schrodinger equation (with potential V) is defined without ambiguity. It is natural to ask whether the Feynman integral is stable under small perturbations in the potential V. This question seemed not to have been directly addressed until the paper by Johnson [Jo3] which gives a "bounded-type" convergence theorem for the analytic operator-valued Feynman integral (to be studied in Chapters 15-16). The paper of Lapidus [La12] gives a "dominated-type" convergence theorem for the modified Feynman integral under far less restrictive conditions. (See Section 11.5.)
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Finally, in Section 11.6, we discuss extensions of the modified Feynman integral to the case of complex-valued potentials, following work in [BivLa]. This is of interest in the study of dissipative (rather than isolated) quantum systems. In closing this brief overview of Chapter 11, we point out the survey article [FrLdSp] which gives a number of examples where singular potentials (which play a central role in large parts of this chapter and in Chapters 12 and 13 below) are used as mathematical models in phenomenological high energy physics and quantum chemistry. Moreover, we mention the book [Ex] where complex-valued potentials are discussed extensively as mathematical models for dissipative quantum-mechanical systems. 11.1
Trotter and Chernoff product formulas
We turn back to the mainstream of our development and proceed to a proof of the Trotter product formula. Our interest is primarily in the Hilbert space setting; however, we have assembled the tools necessary to give the proof in the Banach space setting, and hence we will do so. In fact, we will prove a more general result, the Chernoff product formula, from which Trotter's theorem will follow quite easily. Given a positive integer n, it will be helpful to us below to know that
Formula (11.1.1) is an easy consequence of the calculation, found in introductory probability texts such as [Ros, pp. 54-55], of the mean and variance of a Poisson random variable Z with rate A. (A > 0). The Poisson distribution (or measure) is concentrated on the nonnegative integers 0,1, 2 , . . . and
It is shown that the mean and variance of Z are both A.. In particular,
We obtain (11.1.1) by applying (11.1.2) to the case A. = n. The proof of the Chernoff product formula is based on the following inequality from [Cherl]. Lemma 11.1.1 (Chernoff's lemma) If L is a bounded linear operator on the Banach space X such that \\L\\ < 1, then for all f € X and for each nonnegative integer n,
In the following, an operator L in L.(X) such that \\L\\ < 1, as in Lemma 11.1.1, will be called a linear contraction or, briefly, a contraction on X.
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Proof of Lemma 11.1.1. We will use the following fact. For any B e C(X) and any a eR,
Formula (11.1.4) is easily justified:
Now using (11.1.4) with a = n we can write
Using the assumption that \\L\\ < 1, we have for k > n,
Similarly, for k < n, we can write
Using (11.1.6) and (11.1.7) we obtain for all fc = 0, 1, 2 , . . . ,
Using again the fact that \\L\\ < 1, we have for m = 0, 1, 2 , . . . ,
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Now from (11.1.5), (11.1.8) and (11.1.9) and then the Cauchy-Schwarz inequality, we deduce
where the next to last equality comes from (11.1.1). Hence formula (11.1.3) is established. D
We are now prepared to prove the Chernoff product formula [Cher 1,2]. Theorem 11.1.2 (Chernoff product formula) Let {F(0}r>o be a family of contractions on the Banach space X with F(0) = /. Suppose that the derivative F'(Q)f exists for all f in a subspace D such that the closure A of F'(0)\o generates a (Co) contraction semigroup {T(t) : t > 0}. Then for each f e X,
uniformly for t in compact subsets of [0, +oc). [Of course, we denote by F'(G)\D the restriction to D of the (strong) derivative F'(Q).] Proof We will carry out the proof except for the uniformity which is claimed in connection with the limit in (11.1.11). The key steps will be justified by Theorem 9.7.10 and Lemma 11.1.1. Let AQ = A and, for t > 0, let
The operators F ( t ) are contractions by assumption. The bounded operators An generate norm continuous (see Proposition 9.5.2), and so certainly strongly continuous, semigroups Tn(t) := e'A". We now show that ||r w (OH < 1 for every n and for every t > 0. Formula (11.1.4) will again be useful to us.
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From (11.1.13) we see that the stability condition involved in the hypotheses of Theorem 9.7.10 is satisfied with M - 1 and w = 0. Applying Theorem 9.7.10 (or Theorem 8.1.12) we obtain
uniformly for t in compact subsets of [0, +co). Now Lemma 11.1.1 and the existence of F'(0)f for / e D allow us to write
as n -»• oo, for all / e D. Combining (11.1.14) and (11.1.15), we see that for every / e D, \\Fn(t/n)f - T ( t ) f \ \ ->• 0 as n -> oo. Since D is dense in X, it follows that for all / e X
as we promised to show.
D
Remark 11.1.3 The reader should note that the family of contractions {F(t)} involved in the Chernoff product formula is not required to form a semigroup. As mentioned earlier, the Trotter product formula [Tro2] follows easily from the Chernoff product formula. Theorem 11.1.4 (Trotter product formula) Suppose that A, B and A + B generate (Co) contraction semigroups {S/i(0}f>o- {Se(/)}(>o, [T(t)}t>o on the Banach space X. Then
for each f e X, where the convergence (is in the norm on "H and) is uniform for t in compact subsets of [0, +00). The formula (11.1.17) is often written instead as
We caution the reader that the exponential notation in (11.1.17')—although frequently used—can be misleading because, as we have seen in Chapters 8 and 9, semigroups do not possess all of the properties of the numerical exponential function. When the generators of the semigroups are self-adjoint (or skew-adjoint as in Corollary 11.1.6 below), the richness of the associated functional calculus (Section 10.1) can be used to more fully justify the exponential notation. (See especially Remark 10.1.12(a), (b).)
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We note that the limit in (11.1.17') (or (11.1.17)) is also sometimes denoted by
and referred to as the "strong operator limit" or "limit in the strong operator topology of £(H)". Proof of Theorem 11.1.4 We will apply Theorem 11.1.2 with D := D(A + B) = D(A) n D(B) and
Since SA(t) and SB(t) are contractions, F(t) is certainly a contraction. Also F(0) = SA(0)SB(0) = I2 = I. Now, for / € D, we can write
as t -*• 0+. (To justify the first limit in the last line of (11.1.19), we use the fact that IISxWII < !•) Thus the strong derivative F'(0) exists and equals A + B on D = D(A + B). Hence, from this and one of our assumptions, we have that the closure of F'(0)|o, that is, A + B, generates the (Co) contraction semigroup {T(t)}. Now applying the Chernoff product formula (11.1.11) in the present situation, we obtain
as desired. Remark 11.1.5 (a) In view of the Chernoff product formula (Theorem 11.1.2) the Trotter product formula (Theorem 11.1.4) extends to the case where there exists a subspace D of D(A) n D(B) such that (A + B)\D generates a (Co) contraction semigroup. The proof is the same; it suffices to replace A + B by (A + B)\D. (b) Formula (11.1.17') is sometimes referred to as the Trotter-Lie Product Formula in the literature; see, e.g., ([Cher2], [Kat4], [La1, 5,6], [La13, Part I]). Indeed, in about 1875, Sophus Lie established the finite-dimensional case of(11.1.17'), which corresponds to smooth vector fields generating one-parameter groups of matrices associated with the solutions of differential equations. (Of course, in this situation, there is no need to distinguish between A + B and A + B.)
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Product formula for unitary groups Corollary 11.1.6 (Trotter product formula for unitary groups) Let A and B be selfadjoint operators on a complex Hilbert space H, and suppose that there exists a subspace D of D(A) n D(B) such that (A + B)\D is essentially self-adjoint; i.e. C = (A + B)\D is self-adjoint. Then for every u e H,
uniformly in t on all bounded subsets of R. Proof This is an immediate consequence of Theorem 11.1.4, Remark 11.1.5(a), and the relationship between self-adjoint and skew-adjoint operators. Note, in particular, that the skew-adjoint operators ±i A, ±iB and ±i'C = ±i(A + B)\D generate (Co) contraction semigroups. (See Section 9.6, especially Definition 9.6.4 and Theorem 9.6.11, Stone's theorem, according to which e"(±A), e"(±B) and e't(±C)(f > 0) are (Co) semigroups of unitary operators.) We discussed the relevance of the Trotter product formula to the Feynman integral in Section 7.5. The reader should review that material at this point and might also wish to consult the book of Goldstein [Gol, §8.13, pp. 54-55]. We remark that the Trotter (or Chernoff) product formula is also useful in other contexts. For example, one can derive from it the Central limit theorem (see [Tro3] and [Gol, pp. 51-53]). We will discuss several additional applications of product formulas to diffusion theory or quantum physics throughout the rest of this book. Further, the Trotter (or Chernoff) product formula—along with its nonlinear extensions (see Remark 11.1.9(b) below) and various generalizations—provides useful numerical algorithms for solving certain (linear or nonlinear) partial differential equations and variational inequalities; see, e.g., [ChorHMM], [KatMasu], [La5], [Lal3, Part I]) and the references therein. Exercise 11.1.7 Let F(t) be given by (11.1.18). In cases of interest, [F(t)},>o is not a semigroup. Give a condition under which {F(t)} does form a semigroup. We shall also need (for example, in Step 2 of the proof of Theorem 11.3.1) a more general version of Theorem 11.1.2, also due to Chernoff [Cher2, Theorem 1.1, p. 4], which we now state without proof. Theorem 11.1.8 (Chernoff's theorem) Let {F(t)}t>o be a family of contractions on the Banach space X such that F(0) = /. Suppose that there is a (Co) contraction semigroup {T(t) : t > 0} generated by the m-dissipative operator — C, such that for one (and hence for all) A. > 0, we have
for all f
eX.
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Then for all f € X,
uniformly for t in compact subsets of [0, +00). (Note that following the convention in Exercise 9.3.8, we can write T(t) = e~1c.) Remark 11.1.9 (a) In the statement of Theorem 11.1.8, the word "m-dissipative" can be omitted in view of Theorem 9.4.7; moreover, under suitable assumptions, the word "semigroup" can be omitted as well (see [Cher2, Theorem 2.5.3, p. 28]). Finally, under the hypotheses of the theorem, the conclusion (11.1.23) not only follows from (11.1.22) but is equivalent to it (this is essentially a consequence of Theorem 9.7.11 or of [Kat8, Theorem IX.3.6, p. 513]); see [Cher2, Theorem 1.1]. (b) Chernoff's theorem has been extended in a number of ways; in particular, it was generalized to nonlinear semigroups by Brezis and Pazy [BrePaz]. (See also Brezis' book [Brel, Theorem 4.3, p. 124], as well as [Reil,2] and the relevant references therein.) Exercise 11.1.10 Show that the Chernoff product formula (Theorem 11.1.2) follows from Theorem 11.1.8. 11.2 Feynman integral via the Trotter product formula Feynman's approximation formula was discussed in Section 7.4, and we saw in Section 7.5 how this leads to Nelson's first approach to the Feynman integral via the Trotter product formula. Consistent with the tone of Chapter 7, proofs were not given. However, considerable background information in operator theory has now been provided in Chapters 8-10, and a quite general version of the Trotter product formula, Theorem 11.1.4, has been proved; further, its application to unitary groups has been provided in Corollary 11.1.6. Hence, we are now in a position to give a much more detailed and precise mathematical discussion. As mentioned in the introduction to Section 7.5, Nelson's result depended not only on the Trotter product formula but also on the 1951 result of Kato [Katl] insuring the essential self-adjointness of the (formal) energy operator H, where
(In (11.2.1), we have simplified (7.5.1) by taking the physical constants h and m equal to 1.) The energy operator (also called the "Schrodinger operator" or "Hamiltonian") is now known to be essentially self-adjoint under far more general conditions than were known at the time of Nelson's paper. We will discuss these matters below and will carry out the part of the proof which seems to us to be the most novel and interesting. The argument will revolve around a clever distributional inequality due to Kato [Kat2]. Criteria for essential self-adjointness of positive operators We begin by stating an important abstract criterion for essential self-adjointness [ReSi2, Theorem X.26, p. 182] which will be helpful to us in applying "Kato's inequality"
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(Theorem 11.2.3 below) to establish the essential self-adjointness of H, under suitable hypotheses on the "potential" V. (See the proof of Theorem 11.2.5 below.) In the result which we are about to state, it is only the fact that (ii) implies (i) that will be needed in this section. (For the notion of symmetric and essentially self-adjoint operators, we refer back to Definition 9.6.4.) Theorem 11.2.1 (Criteria/or essential self-adjointness of positive operators) Let Ti. be a Hilbert space and let Abe a symmetric operator on H which is also strictly positive; i.e. there exists c > 0 such that (A
extension
Remark 11.2.2 (a) The above theorem obviously applies to semibounded operators (see Section 10.3) A by replacing A by A+c, where c is a sufficiently large positive constant, (b) Theorem 11.2.1 is related to an earlier criterion for essential self-adjointness, Corollary 10.2.10, where semiboundedness of A was not assumed. Note, however, the differences between the statements of Theorem 11.2.1 and Corollary 10.2.10. In particular, in part (iv) of Theorem 11.2.1, we assume that A has a unique semibounded self-adjoint extension. [Contrast this with part (i) of Corollary 10.2.10 (or of Theorem 11.2.1).] This is not immediately equivalent to the essential self-adjointness of A because, in general, a symmetric semibounded operator can have a self-adjoint extension that is not semibounded. (See the comment following the proposition on page 179 of [ReSi2], as well as Problem 26, page 341 in [ReSi2J.) A brief outline of distribution theory We now give an extremely brief summary of the parts of distribution theory which will be helpful to us, especially in connection with Kato's inequality. A detailed treatment of distribution theory can be found in many places, for example, [Schw2, ReSil,2, H6r3]. The space of test functions will be T>(R.d), which we will denote by V throughout this section. (Recall that T>, also sometimes denoted C^(K rf ) in earlier chapters, is the space of C-valued infinitely differentiable functions with compact support in Rd.) The appropriate topology on Rrf is an inductive limit topology under which Z> is not metrizable [ReSil, p. 147]. However, the convergence of sequences in T) is all that we will need, and that is easily described. Let (<£>„} and
•••dXj
\a\ =«[ + ---+ad.) The space of distributions T>' is the space of all continuous linear functionals on T>. Given {Tn} and T from D', Tn converges to T if and only if Tn(
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A key fact about distributions is that they possess partial derivatives of all orders which remain in the space D'. The am derivative of T where a is a multi-index acts on tp e D as follows:
Further, differentiation is a continuous operation on V. Our concern here will be with the distributional Laplacian A = -^r + • • • + 3x -^-? acting on functions u e L1 which have 3jc, d
the property that the distribution AM is again in L2. (See the earlier related discussion in Section 10.2 as well as in the next paragraph.) The space L^ = Lj^R^) of C-valued functions u on Rd which are integrable over every compact subset of R* (or "locally integrable", in short) is continuously imbedded as a subspace of V via the formula
In particular, if Tv = Tw, then v = w Leb.-a.e. [H6r3, Theorem 1.2.5, p. 15], and so v = w as elements of L'OC. Further, it is easy to show that if vn -» v in L|OC (i.e., for every compact K c R rf , we have fK \vn — v\dx —> 0), then TVn —> Tv in V'. In the following, we will write i; instead of Tv when no ambiguity may arise. Various easily established facts about L'OC and L2OC will be used below, most often without comment. For example, if « e L2 and V e L2OC, then uV e L,1^ (by the Cauchy-Schwarz inequality). We say that T e V is positive and write T > 0 if and only if Tip > 0 whenever
for all (p e £>. Kato 's distributional inequality We come now to a key step, the beautiful distributional inequality of Kato [Kat2, Lemma A, p. 138]. Theorem 11.2.3 (Kato's inequality) Let u e L^ be such that its distributional Laplacian Aw e L'^. Define
Of course, sgn u e L°°, so that sgn u, (sgn u) AM, and Re[(sgn M)A«] are all in L^ and hence are all distributions.
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Then we have the following distributional inequality: Proof We will proceed in two steps. Step 1: We suppose for now that u e C°°(Ed), the space of C-valued infinitely differentiable functions on R rf , and we let Taking the gradient on both sides of the equality we obtain
Hence Using the fact that \u\ < \uf\ and (11.2.6), we obtain
It follows that (One may worry about the case u(x) — 0; but then it suffices to replace ubyu + l.) Now Applying (11.2.8) to both sides of (11.2.6), we have
Hence Next from (11.2.10) and (11.2.7), we deduce that Thus pointwise and so in a distributional sense. Now letting
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and using (11.2.12), we obtain (for all u e C°°(Rd))
In Step 2, we will use the inequality (11.2.13) and suitable approximation procedures to finish the proof of Kato's inequality. Step 2: Now we assume that u and AM are in L,1^. Let jg be an approximate identity (or "mollifier"); i.e. for S > 0, let js(x) := j(x/S)S~d where ; > 0, j € V, fRd j(x)dx = 1, and the support of j is the closed unit ball of W1. Let
where the convolution * is defined as in (10.2.23). Since us € C°°(lRrf) [H6r3, Stel, SteWe], we deduce from (11.2.13) that
It is well known and not difficult to check that us -> u in L11oc(IRrf). (See, for example, the proof of Theorem 1.18 in [Ste We, p. 10].)Since AM € Z,,1^ and A(« 6 ) = (AM)*/? (Aw) 5 (by [H6r3, Theorem 4.1.1, p. 88]), we also have Aw"5 -> AM in L^oc. By passing to a subsequence {Sp} (using a diagonalization argument) we also get convergence a.e. of AUSP to Aw and of USP to u; in this connection, we will suppress reference to the subsequence and simply write AM^ —> AM and us —> u a.e. (Here and thereafter, we write a.e. instead of Leb.-a.e. when no ambiguity may arise.) Now, for a fixed e > 0, we have (in view of (11.2.5) and (11.2.14))
where the convergence in (11.2.16) holds almost everywhere. Note also that both |sgn e (M (S )| and ]sgn e (M)| are bounded by 1 for all S and €. Now take y € V and let K = supp(^). Of course, K is compact. Using the fact that \\Aus - AM|| L I ( A -) -»• 0 as well as (11.2.16) and the dominated convergence theorem, we can write
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Thus
in the sense of distributions. It then follows easily from (11.2.17) that
in the sense of distributions. We have just shown above that for u such that both u and AM are in L1loc, the righthand side of (11.2.15) converges in the sense of distributions to the right-hand side of (11.2.13) as 8 ->• 0+. Our next goal is to show that the left-hand side of (11.2.15) converges in the same sense to the left-hand side of (11.2.13). As noted earlier, us —»• u in L\0(:. Using this, we can write, for any compact subset K of R rf ,
(Note that in (11.2.19), we have used the norm \ \ ( z \ , 12) \\p(c2) '•- (kl I2 + Iz2l 2 ) 1 / 2 , for
Hence (us)e -> uf in £>''. Finally, since differentiation is continuous in T>', we have
Now for u € L'OC such that AM e L^, the distributional inequality
follows from (11.2.15) via a limiting argument using (11.2.18) and (11.2.21). Finally we let € -» 0+. Clearly sgn€(u) -> sgn(w) pointwise with a uniform bound of 1, so that the right-hand side of (11.2.22) converges in the sense of distributions to the
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right-hand side of (11.2.4). Further, we clearly have uf -»• \u\ a.e. Since we also have u€ < (|«|2 + 1)1/2 < |M| + 1 e L,1^, we get uf -> \u\ in L^. Therefore «t ->• |w| in the sense of distributions and so AM€ —>• A|w| in the same sense. The distributional inequality (11.2.4) is now established and the proof is complete. D Although we shall not use this fact here, we note that more abstract (operatortheoretic) versions of Kato's inequality have been obtained, especially for the generators of positive (i.e., positivity preserving) semigroups. See [Si4, 7], as well as, for example, ([Are], [La5, §5.3], [Na], [NaUh]) and the references therein. We will use, however, a variant and extension of Kato's inequality in Section 13.6 and would also use it in the proof (not provided there) of one result in Section 13.5. (See Lemma 13.6.6.) Essential self-adjointness of the Hamiltonian H = HQ + V We will use the fact that (ii) implies (i) in Theorem 11.2.1 to show in Theorem 11.2.5 below that — A + V is essentially self-adjoint when V is nonnegative and in L2loc. In that connection, we will show that if u e £>[(- A + V + !)?£,] and [(- A + V + I)\T>]*U = 0, then u — 0. In order to carry this out we will need the following lemma. Lemma 11.2.4 Let V e L2oc(Rd) be such that V > 0 Leb.-a.e. Ifu e D[(-A + V + l)|p]*. then both [(—A + V + 1)|£>]*« and (—A + V + 1)« make sense as distributions and
as elements ofT>'. Proof We show to begin with that -A + V + 1 is defined on £> with values in L 2 (R d ). Since — A + 1 maps Z> into T>, it suffices to prove that V
Next we claim that (— A + V + \)u makes sense as a distribution for u e L2 and so certainly for u 6 D[(- A + V + !)*£>] c L2. First note that any u e L2 is in L2OC c LJ^ and so u e T>'. Since — A + 1 maps T>' into T>', it suffices to show that Vu 6 L^. and so can be regarded as a distribution. But V e L2OC and u € L2loc implies that Vu e L*oc, as observed earlier. Now[(—A+V4-l)| / z?]*MmakessenseasadistributionforM e D[(— A + V + l)\x>*] simply because [(-A + V + l)p]*w 6 L2 c L2oc c L,1^.
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We are now prepared to establish (11.2.23), the main point of this lemma. Let u e D[(-A + V + 1)|£>*] and take a sequence [un] from D> such that \\un - «||2 -*• 0. We claim that
and Since (-A + V + 1) and [(- A + V + l)|i>]* agree on T>, it will follow immediately from (i) and (ii) that the L2-function [(—A + V + l)|i>]*w (which is also a distribution) equals the distribution (—A + V + I)H in the distributional sense and so the proof of this lemma will be complete. (i): Fix tp e D. Then
where the second step follows from the symmetry of (—A + V + l ) o n £ > and the fourth comes from the definition of the adjoint (see Definition 9.6.1) and the fact that M € D[(-A + V + l)|u*] and
as needed. It follows immediately that (-A + l)un -»• ((—A + \)u in a distributional sense. It remains only to show that for
where the last assertion in (11.2.24) follows from the fact that \\un — u\\2 ->• 0 and fK V2\y>\2dx < ll^ll^j/jf V2dx < co. This finishes the proof of (ii) and so, as noted above, the proof of the lemma. D Note that it follows from the preceding lemma that if u e £>[(—A + V + l)|i>*], then the distribution (-A + V + l)u lies in L2(E.d). In fact, it can be shown [Kat2, p. 136] that D[(-A + V + l)|i>*] consists exactly of those functions u e L2(Efl!) such that (-A + V + 1)« 6 L2.
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We are now ready to show that if V e L2locwith V > 0, then (—A+V)|£> is essentially self-adjoint. Following this, we will state a result—also due to Kato in [Kat2,3]—proving the essential self-adjointness of (—A + V)\-p, where V — V+ — V- is an R-valued potential with V+ e L2loC and with V- "relatively operator bounded with respect to HQ with relative operator bound less than 1." Theorem 11.2.5 (Kato) Let V e £ 2 oc (R rf ) with V > 0 pointwise. Then -A + V is essentially self-adjoint on T>. Proof It is easy to see that - A + V is essentially self-adjoint on V if and only if — A + V + 1 is essentially self-adjoint on £>. We will work with the operator (-A + V + 1)|£> since it has the advantage of being strictly positive. [In fact, ((—A + V + l)(f>, ip) > \\ip\\2 for all (p e V.] According to the implication (ii) =>• (i) in Theorem 11.2.1, it suffices to show that if u e Z)[(-A + V + l)jx>*] and
thenw = 0. Now by Lemma 11.2.4, it suffices to show that if u e D[(-A + V + 1)|£>*] and
then H = 0. Since u e L2 c L2oc and V e L2OC, both u and Vu are in L,1^. Thus, by (11.2.26),
Hence, the hypotheses of Theorem 11.2.3, Kato's inequality, are satisfied and so we have A|M| > Re [(sgn u) AM] in the distributional sense. Thus, we can write
in the distributional sense. In particular, we see that
Let js be an approximate identity as in Step 2 of the proof of Kato's inequality. [For S > 0, take js(x) := j(x/S)S~d where j > 0, j e £>, /R(/ j(x)dx = 1, and the support of j is the closed unit ball of Md.] Let v := \u\ and i/ := v * 7'^. Now v e L2 since u e L2. Also vs e L2 since it is the convolution of the L1 function js with the L2 function v [Fol2, (8.7), p. 232]. Further, vs is infinitely differentiate (since it is the convolution of the locally integrable function, and hence distribution, v with j$ e T>) and Aw"5 = v * Aj5 [H6r3, Theorem 4.1.1, p. 88]. In addition, At) 6 e L2 since v e L2 and Ay's e Z> c L1.
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We have seen above that both vs and Ai/ are in L2. Recalling from (10.2.10) that D(Ho) = [w € L2 : Aw € L2 in the sense of distributions}, we see that vs e D(Ho). (The fact that HO equals — ^A instead of -A clearly does not matter for the present discussion.) Since HQ is a nonnegative operator, we have
Now since A |« | is by (11.2.28) a positive distribution, there is (see (11.2.3)) a measure fj. : B(R.d) -> [0, +00] which is finite on compact sets such that the action of the distribution A|«| on any (p e T> is given by
Further, At/ = A(|w| * y',$) = A(|w|) * jg and At/5 is infinitely differentiable [H6r3, Theorem 4.1.1, p. 88]. Hence, by (11.2.30),
so that At;5 is pointwise nonnegative. (Clearly, whether A \u \ is viewed as a distribution or as a measure does not affect the value of the convolution product here.) Since vs = \ u \ * js is nonnegative, we see that (vs, At/) > 0, and so
Combining (11.2.29) and (11.2.31), we have that
Now vs = \u\*js is infinitely differentiable with all of its derivatives in L2 (Davs — \u\ * Dajs is the convolution of the function Daj$ e T> c L1 with \u\ 6 L 2 ). Hence we have the formula
and so we see from (11.2.32) and (11.2.33) that Vvs = 0 a.e. In fact, since the first partials of v5 are continuous, (Vvs)(x) = 0 for all x e K rf . Thus vs is a constant function, and, since the only constant function in L2 is the zero function, t/ = 0. Finally, vs -» v a.e. as S —> 0+ and so v = u \ = 0 a.e. Thus u = 0 as we wished to show. D Remark 11.2.6 It can be shown ([Kat2] or [BreKat]) that the exact domain of the unique self-adjoint extension of(— A + V)\v where V > 0 and V e L^OC(R'') is {u € L 2 (M d ) : Vu is in Lloc(Rd) and the distribution (-A + V)u is in L2(R.d)}. We assumed in the preceding theorem that V > 0 pointwise. It suffices to assume that V is bounded below.
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Corollary 11.2.7 Let V e L^oc(Rd) and let c be a nonnegative constant such that V > — c pointwise a.e. Then — A + V is essentially self-adjoint on T>. Proof The preceding theorem shows that — A + (V + c) is essentially self-adjoint on T>. One can then show easily that - A + V = — A + ( V + c ) - c has a unique self-adjoint extension and so is also essentially self-adjoint. a It is easy to give many examples of functions V which satisfy the conditions of the corollary. One class of examples follows. Example 11.2.8 The operator - A + P ( x \ , . . . , xj) is essentially self-adjoint on T>(Rd), where P is any polynomial with coefficients in R which is bounded below. If d = 1, P can be any polynomial of even degree whose highest powered term has a positive coefficient. We next turn to the statement and some discussion of an essential self-adjointness result where a certain class of potentials having an unbounded negative part is permitted. Definition 11.2.9 Let A be self-adjoint and Bbea (densely defined) symmetric operator on a complex Hilbert space Ji. We say that B is relatively operator bounded with respect to A (briefly, B is A-operator bounded) with bound less than 1 if and only if
and there exist positive constants a < 1 and b such that
The infimum of all such numbers a is called the A-operator bound of B. We note that the analogue of Remark 10.3.18(b) holds. In particular, the infimum need not be a bound itself. The proposition to follow will eventually permit us to compare different approaches to the Feynman integral; particularly, the one discussed at the end of the present section and the "modified Feynman integral" [Lai 1] defined in Section 11.4 below. (We use here some of the notation from Sections 10.2 and 10.3.) Proposition 11.2.10 Let A and B be self-adjoint operators on H, with A nonnegative and Q(A) H Q(B+) dense in H. Then: (i)IfB- is A-operator bounded with bound less than 1 (as in Definition 11.2.9), then it is also A-form bounded with bound less than 1 (in the sense of Definition 10.3.17). (ii) Assume that B- is A-form bounded with bound less than 1, so that (by Theorem 10.3.19) A+B is well-defined. If A + B, restricted to a subspace D, is essentially selfadjoint , then its unique self-adjoint extension (A + B)\D, coincides with A + B, the form sum of A and B. We note that part (ii) follows from Theorem 10.3.19 since the latter implies that A+B is a self-adjoint extension of (A + B)\D; hence A+B must necessarily coincide with (A + B)\D, the unique self-adjoint extension of (A + fi)|o- The proof of (i) is left as an exercise for the reader.
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The following extension (also given in [Kat2]) of Kato's essential self-adjointness theorem (Theorem 11.2.5) enables us to deal with certain potentials V which are unbounded below. (Compare Corollary 11.2.7.) Theorem 11.2.11 (Kato) Let V = V+ — V- be an R-valued Lebesgue measurable function on Rd such that V+ 6 £j!oc(Rd) and V- is Ho-operator bounded with bound less than 1. Then HO + V is essentially self-adjoint on T> = T>(Rd). Remark 11.2.12 (a) The conclusion of Theorem 11.2.11 still holds if V - V\ + Vz, with V\ > 0, Vi e LJ!oc(R
Note that for d < 3, Sd = L^ oc (R d ) H . Recall that the space L£c(Rd)H has been introduced in Definition 10.4.2(i). The next proposition accounts for our interest in Sd.
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Proposition 11.2.14 If W € Sj, then W is Ho-operator bounded with bound less than 1. (In fact, W has Ho-operator bound 0.) We give some further information about Sj. and its contents. Proposition 11.2.15 (a) We have 84 c L^ oc (M d ) u /or all positive integers d. (b) If we take p > d/2for d > 4 and p = 2 for d < 3, then
Proof (a) For d < 3, the containment is trivially true and is actually an equality. We will work out the case d > 5', the remaining case, d = 4, can be done in a similar manner. Pick any number r > 0. It is easy to show that W e L^ oc (E d ) H if and only if
Suppose now that W is not in L^ oc (M d ) M . Let r e (0,1] be fixed (for now). We see from (11.2.36) that there exists a sequence {x^} in Rd such that
Since 0 < r < 1, it follows that
Hence,
We see then that
Thus W is certainly not in Sj. We will omit the proof of (b) except for the observation that the first containment is trivial. Example 11.2.16 ([CyFKSi, p. 8]) If d > 5 and S e (0, 1/2], the function
is #o-operator bounded with bound less than one but is not in Sj. The function W does belong to Sd if and only if S > 1/2. If we compare this example with Example 10.4.6 taking d > 5 and S e (1/2, 1], we see that Sj is not a subset of K^.
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Remark 11.2.17 (a) Note that Definition 11.2.13 makes perfectly good sense for C-valued functions. However, our main interest in these definitions is in connection with self-adjointness results and so we have kept W real-valued. (b) The analogue of Proposition 10.4.7 still holds for the class Sd. Feynman integral via the Trotter product formula for unitary groups We are now prepared to state and prove the key theorem establishing the existence of the Feynman integral via the Trotter product formula. We will actually postpone formal reference to "the Feynman integral" until (Definition 11.2.21 and) Corollary 11.2.22. However, the reader will recall (perhaps after reviewing Sections 7.4 and 7.5) that the following result is related in a rather straightforward way to Feynman's original paper [Fey2]. The proof below will not be difficult since it is simply a matter of applying results which we now have at hand. Unlike our preliminary discussion in Chapter 7 (see especially Theorem 7.5.1), we will simplify notation by taking the physical constants m and h equal to one. We begin with a lemma which is concerned just with the nth Trotter product and not with its limit. Note that the assumptions on V in the lemma are minimal. Lemma 11.2.18 Let V : R.d —>• K be Lebesgue measurable and finite Leb.-a.e. Then for every t e R and y> 6 L 2 (R rf ) and for Leb.-a.e. v, we have the equality
where xn := v and where the iterated integrals on the right-hand side of (11.2.38) exist for Leb.-a.e. v e Kd when interpreted in the mean (in the sense of Theorem 10.2.7) and define a function which is in L 2 (M rf ) as a function of v. Proof The operator e~'"v is the operator of multiplication by the function e~'«v, and this is the first operator that acts on (p on the right-hand side of (11.2.38). According to the left-hand side of (11.2.38), the operator e~~'*H° should now act on e~'"V( 'V(0- But we know from Theorem 10.2.7 that e~'nHa is precisely the second operator that acts on the right-hand side of (11.2.38), where the integral involved should be interpreted in the mean (at least when the function it is acting on is in Z,2 \ L 1 ). (See especially formula (10.2.33).) The product e~'nH°e~'"v should now be iterated n times according to the
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left-hand side of (11.2.38). In summary, the right-hand side of (11.2.38) is exactly the nth Trotter product (e~l«H°e~'«v)" acting on
defines a function which is in L 2 (R rf ) and is given for almost every v e E.d by the integrals (interpreted in the mean) on the right-hand side of (11.2.38). Further, there is a function \jf : (-00, +00) x E.d -> C such that ijr(t, •) e L 2 (R d ) for every t e R and
where the limit is uniform in t on all bounded subsets o/R. In fact, the function ^r(t, •) is given by the action of the unitary group e~"H on
where H := (Ho + V)\x> is the unique self-adjoint extension of the essentially selfadjoint operator (Ho + V}\x>- (We know from Proposition 11.2.10 that we must also have H = HQ + V, the form sum of HQ and V.) Finally, for
with initial state (p. Proof We will apply Corollary 11.1.6, the Trotter product formula for unitary groups, with H = L 2 (R d ), D = T> = V(Rd), A = H0 and B = V, the operator of multiplication by the R-valued function V. Now, HQ is self-adjoint as we saw in Section 10.2 (just below Equation (10.2.8)), and, of course, V is self-adjoint as well (by Proposition 10.1.3). Under the present hypotheses on V, (Ho + V)|£> = ( — j A + V)\x> is essentially selfadjoint by Theorem 11.2.11. (See Remark 11.2.12(d).) We let C = H = (H0 + V) |X >. Thus we see that the hypotheses of Corollary 11.1.6 are satisfied in our present setting. It follows from (11.1.21) that for all
where the limit is in the norm on L 2 (R d ) and is uniform in t on all bounded subsets of M.
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Now we simply rewrite the nth Trotter product on the right-hand side of (11.2.43) by means of Lemma 11.2.18. It follows (for Leb.-a.e. v) that i/rn(t, v) defined by (11.2.39) is given by the right-hand side of (11.2.38). Further, the fact that the limit of irn(t, v) exists as n -> oo in the sense claimed in the theorem and equals the unitary operator e~'tH acting on
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by FjpW), is the strong operator limit
in £(L2(Rd)) whenever this limit exists. (That is,
for all ip eL 2 (R J ).j The following corollary restates much of Theorem 11.2.19 in the language just introduced. Corollary 11.2.22 (Existence of the Feynman integral via the Trotter product formula) Let V : Kd -* R be Lebesgue measurable and such that V+ e £2oc(lRd) and V- is relatively operator bounded with respect to HQ with relative bound less than 1. Then F'Tp(V) exists for all t &R, where the limit in (U.2.44) is, for all
that is, f'TP(V) agrees with the unitary group which specifies the dynamics in the standard approach to quantum mechanics. Finally, for (p 6 D(H), ^^(V)^ is the unique solution (in the semigroup sense) of the Schrodinger equation (11.2.42) with initial state (p. 11.3 Product formula for imaginary resolvents We present in this section the results of Lapidus [Lai 1]—building upon his earlier work in [La1, 2, La6-9]—which establish a product formula for the "imaginary resolvents" of (suitable noncommuting and unbounded) self-adjoint operators. The main results (Theorem 11.3.1, as well as its consequences, Corollaries 11.3.5 and 11.3.7) will be key to defining and proving the convergence of the "modified Feynman integral" (also introduced by the second author in [La1, 2, La6-ll,Lal3]) for very general singular potentials. (See Section 11.4.) They will also be essential to establish "dominated-type convergence theorems" [Lal2] for this approach to the Feynman integral. (See Section 11.5.) Hypotheses and statement of the main result We next state the assumptions made in much of this section. Let H be a complex Hilbert space, with inner product denoted by (•, •) and corresponding norm || • ||. Let A, B be (typically unbounded) self-adjoint operators on H, with A nonnegative. Let B+ and B- be the positive and negative parts of B defined through the spectral theorem as in Section 10.3; then B+ and B- are nonnegative self-adjoint operators and B — B+ — B-. Assume that B_ is relatively/orm bounded with respect to A (in short, A-form bounded)
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with relative bound less than 1 (see Definition 10.3.17); i.e.
and there exist positive constants y < 1 and S such that
Assume, for simplicity, that Q(A) n Q(B+) (= Q(A) n Q(B) by (11.3.la)) is dense in H. (See part (i) of Corollary 11.3.7 below for the general case.) Recall then from Section 10.3 that the quadratic form
is bounded from below and closed, and so there exists a unique (bounded from below) self-adjoint operator A+B associated with this quadratic form. (See Theorem 10.3.19.) The operator A+B is called the form sum of A and B. We are now ready to state the main result obtained by Lapidus in [Lai 1]. (See [Lai 1, Theorem l,p. 263].) Theorem 11.3.1 (Product formula for imaginary resolvents) Under the above hypotheses, we have for all u e "H,
uniformly in t on compact (or equivalently, bounded) subsets of R. We remark that with the notation introduced in (11.1.17"), we can restate (11.3.2) more briefly as follows:
We will sometimes find it convenient to use (11.3.2') rather than (11.3.2). (See especially Section 11.5 below.) The following comment may help a reader (who is not familiar with operator theory) understand the statement (as well as the proof given below) of Theorem 11.3.1. [Our remark deals with the "imaginary resolvent" [/ + i t B ] ~ ] (in the terminology of [Lal,2, La6-13]) but, of course, an entirely analogous comment applies to justify the existence of the "imaginary resolvent" [7 + itA]~l of the unbounded self-adjoint operator A.] Note that since the operator B is self-adjoint, its spectrum cr(B) is contained in R (see Remark 10.1.7), and so its resolvent set p(B) := C \ a(B) contains the imaginary axis j'R; hence, by Definition 9.3.1, the imaginary resolvent [1 + itB]~^ exists (as an element of C(H)~) for all t e R. (Clearly, for t = 0, it is equal to /, the identity operator on H, while for t ^ 0, it is equal to (i/t)R((i/t); B), in the notation of Definition 9.3.4.) An important fact to keep in mind (in the proof of Theorem 11.3.1 below) is that it can also be obtained via the functional calculus associated with the unbounded self-adjoint
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operatorB(Theoreml0.1.11);namely,[I+itB] -1 = g(B),where g(x) :— (1+itx) - 1 for x e a(B) c R. (At this point, it may be helpful to some readers to briefly review the various forms of the spectral theorem for unbounded self-adjoint operators discussed in Section 10.1. Indeed, they will be used extensively in the proofs given below; see especially Theorems 10.1.8, 10.1.11 and 10.1.14.) Proof of the product formula We shall need two preparatory lemmas before proving Theorem 11.3.1. Lemma 11.3.2 Let B be a self-adjoint operator on H with positive part B+and negative part B-. Then, for all t > 0, we have
where
are nonnegative, bounded self-adjoint (i.e. Hermitian) operators on H. Moreover, as t -> 0+,
and
Proof A straightforward computation yields The first part follows since B = B+ — B-. Now by Theorem 10.1.8, the multiplication operator form of the spectral theorem, we may assume that B = Mg, the operator on L 2 (u,) of multiplication by the R-valued function g. Following standard notation, we use g to denote the operator Mg as well as the function itself. Let f e L 2 (u). Note that by (10.1.2) (or (10.3.11) and (10.3.13)), / € Q(B) = D(|B11/2) if and only if |g|1/2|f| e L 2 (u). Similarly, f e Q(B±) = D(Bi /2 )ifandonlyifgi /2 |/| E= L 2 (u,). Thus, in particular, Q(B) = Q(B+)n Q(B-). (Also see equation (10.3.57).) Further, one easily sees that
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(After some elementary manipulations, the first inequality in (11.3.4) follows from the fact that 0 < x2 + t2x4 + t4x6 for all x, t in R. The last two inequalities are immediate.) [Actually, all three inequalities can be obtained, up to a multiplicative constant, by observing that the function x H-> x-1 (1 — r(x)) is bounded on all of R, where r(x) := (1 + i x ) - 1 ; note that |T| < 1, r(0) = 1 and r'(0) = -i.] Taking f e Q(B) in the first inequality in (11.3.4), (11.3.3a) now follows from the Lebesgue dominated convergence theorem (applied in L 2 (u)). Similarly, (11.3.3b) [resp., (11.3.3c)] follows from the second (resp., third) inequality in (11.3.4) by taking f € Q(B+) [resp., f e Q(B_)]. O Lemma 11.3.3 Let {Ut}t>o, {Vt]t>o be families of bounded operators on a Banach space X such that U, exists as a possibly unbounded operator. Assume that || Vt || < 1 and that there is a constant k > 0 such that
Then, for sufficiently small t, [U,
— V t ] - 1 is a bounded operator on all of X and
Moreover, if Ut v —> v as t —> 0+, then
Proof Since ||U,V,|| < ||Ut|| ||V,|| < 1 - kt + o(t) < 1 for small t, it is well-known [Ru2, Theorem 18.3, p. 357] that [/ — Ut V t ] - 1 is a bounded operator defined on all of X and has a Neumann series expansion in £(X):
The operator Ut-1 may be unbounded and hence U-1 — V, is only defined on the subspace D ( U - 1 ) . Let us show that U, — Vt is invertible, Clearly
since the equality U, — V, = U, (I — U1Vt) between unbounded operators holds. Similarly, if J denotes the inclusion mapping of D ( U - 1 ) into X,
Hence U-1 - V, is invertible and (11.3.5) holds.
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Observe that
[Here, 0(1) (resp., o(l)) denotes a function which is bounded (resp., vanishes) as t -> 0+.] This follows since
Now, by (11.3.5),
Thus (11.3.6) follows immediately from (11.3.8).
D
Remark 11.3.4 Lemma 11.3.3 is due to Kato [Kat7, Lemma 4.1, p. 112]. Originally, Kato stated his lemma in a slightly different form, assuming instead that || Ut || < e-1, since he was primarily interested in the behavior of semigroups and not of resolvents. We are now ready for the proof of Theorem 11.3.1. Proof of Theorem 11.3.1 Fix A > 0. For t > 0, set
and
Note that Lemma 11.3.3 applies to the families {U t } t>0 and {V,} t>0 since
Clearly,
since, by (11.3.5) and (11.3.7),
Note further that U-1 = I +1 (A + i A) and that Ran(W t,y ) = D ( U - 1 ) = D(A). Hence Wt,y E D(A)for all u E H.
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The remainder of the proof is divided into two steps. In the first step, we shall hold the parameter A fixed; we shall let it tend to zero at the end of the second step. Step 1: Let A > 0. We will show that for every v € H, Wt,v -> [A + i C ] - 1 v as t -> 0+, where we have set C = A+B. To see this, fix v e H and put wt = Wt,y v. As noted above, wt e D(A). Thus w, e Q(B-) since, by (11.3.la), D(A) c Q(A) c g(B_). Moreover, from the definition of Wa, Thus
Also, we have
and
where RB, , Ig+ and IB- are the nonnegative, bounded self-adjoint operators defined in Lemma 11.3.2. Taking the inner product of v against wt and combining (11.3.10) and (11.3.11), we see that
Therefore
By (11.3.9),
Hence by the Cauchy-Schwarz inequality, Recall that IB- = B-[I+ t 2 B 2 ] - 1 . Thus by the spectral theorem (Theorem 10.1.8), we see much as in the proof of Lemma 11.3.2 that
(Indeed, (11.3.15) follows by using the third inequality in (11.3.4) but for f 0 Q(B-).)
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Applying the key observation (11.3.15) to £ = wt and recalling (11.3.1b), we obtain
(As was noted earlier, we have E = wt e Q(A) c Q(B-).) Consequently
and, by (11.3.12),
Using (11.3.13) and (11.3.14), we deduce that
Since y < 1, it follows that Al/2wt and I 1B/ 2 +w, are bounded as t -> 0+. By (11.3.16) t (and (11.3.13)), I1/2 J w is bounded as well. In view of (11.3.12) and (11.3.14), it is t B t immediate that wt and RB' wt are also bounded. Thus there exists a sequence of positive numbers [tn] (tn -> 0+) along which
(for some vectors w, a, a, B+ in H), where the symbol -^ denotes weak convergence in the Hilbert space H. (Here and in the next few lines, it is implicitly understood that the limits are taken along the sequence {tn}. Further, recall that, for example, wt —> w means that for every fixed y e H, ( w t , y) -> (w, y) in C.) Let y e Q(A). Equation (11.3.17) implies that
[Here and hereafter, we also use the following easily verified fact. If wt -> w weakly in H and pt —> p strongly in (the norm of) W, then (wt, pt) -> (w, p) in C.] Since A 1 / 2 is self-adjoint, it follows that w € Q(A) and a = Al/2w. Similarly, let y e Q(B) - Q(B+) n Q(B-). Then
where we have used (11.3.3a) in Lemma 11.3.2 and (11.3.17). Since Q(B) is dense in H, it follows that a = 0. Lety e Q(B+). By (11.3.3b) in Lemma 11.3.2 and (11.3.17),
Hence w e Q(B+) and 0+ = Bl+/2w.
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In the same manner, we see by using (11.3.3c) in Lemma 11.3.2 that ft- = B_i w. Since Q(C) = Q(A) n <2(B+), we may summarize the above discussion as follows:
Now let y e D(C). Note that D(C) c. Q (C) C Q(B-) n Q(B+), since Q(A) C Q(B-)by(11.3.1a). Then
where we have used successively (11.3.10), (11.3.11) and, after passing to the limit along [tn], (11.3.17), (11.3.18) and (11.3.3) in Lemma 11.3.2. Hence we have ( i - 1 ( v Xw), y) = (w, Cy) for every y e D(C). Since C is self-adjoint, it follows that w € D(C),
and so
The limits in (11.3.17) are, therefore, independent of the sequence {tn}. By a standard compactness argument (namely, the existence of a weak limit point for a sequence in a weakly compact set implies the existence of its weak limit), it follows that weak convergence holds as t -> 0+ in (11.3.17). In particular, w, -> w and RB' w, -^ 0, where w is given by (11.3.19b). Passing to the limit as t -»• 0+ in the real part of (11.3.12) and using (11.3.19a), we see that
Consequently,
Since X > 0, it follows in particular that w, -»• w (in the norm of Ti.), as required. This completes the proof of Step 1.
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Step 2: By Step 1 and equation (11.3.6) of Lemma 11.3.3, we have for every A. > 0 (with A. still fixed) and v e H,
Set F(t) = UtVt. Note that ||(1 + Xt)F(t)\\ < 1. (This follows since ||F(r)|| < IIU, || || V, || < (1 + X t ) - 1 ) Furthermore,
and (11.3.20) yield for every v e H,
Now we apply Theorem 11.1.8, Chernoff's theorem, to the family of contractions {(1 + kt)F(t)}t>o and to the m-dissipative operator — iC (see Definition 9.4.5 and Theorem 9.4.7) which generates the contraction semigroup {e-itc
uniformly for t € [0, T], for all T > 0. Fix T > 0 and t with 0 < t < T. Clearly,
At this stage, we recall that Ut, and hence also F(t), depends on A.. Now
In the next to last inequality, we have used the resolvent equation (see Proposition 9.3.5):
In view of (11.3.23) and (11.3.24),
Now an-bn = En-1 a n - 1 - k (a-b)b k for arbitrary a ,b in L(H).Hence' if and || b|| < 1, we have
||a||<1
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With the obvious notation, (11.3.25) and (11.3.26) yield
It now suffices to let X -> 0+ in (11.3.27) to deduce from (11.3.22) that (11.3.2) holds, uniformly for t e [0, T] for all T > 0. (Recall that C = A + B.) By replacing A, B with their negatives, we would reach the same conclusion for t € [-T, 0]. This completes the proof of Step 2 and of Theorem 11.3.1. D Consequences, extensions and open problems We now state an immediate—but important (see Remark 11.3.6)—corollary of Theorem 11.3.1. Corollary 11.3.5 (Arbitrary nonnegative self-adjoint operators) Let A and B be arbitrary nonnegative self-adjoint operators on the complex Hilbert space H. Assume, for simplicity, that their common form domain, Q(A) n Q(B), is dense in H. (See Corollary 11.3.7(ii) below for the general case.) Then, the form sum A + B is a well-defined (and hence densely defined) operator on H, and we have (with the notation introduced in (11.1.17"))
self-adjoint
uniformly in t on compact subsets of R. Remark 11.3.6 (a) Corollary 11.3.5 (as well as part (ii) of Corollary 11.3.7 below) was first obtained by the second author in [La6, Theorem5.1,p. 1720]. (See also [LaL,2 ].) Its original proof, however, was more involved than the proof of the more general "product formula for imaginary resolvents" statedin Theorem 11.3.1. It was using, in particular, techniques from papers by Kato [Kat5] and Ichinose [Icl]; the latter paper was also motivated by problems connected with the Feynman integral. (b) Under the assumptions of Corollary 11.3.5 (and, in fact, under the more general hypotheses of part (ii) of Corollary 11.3.7 below), Kato [Kat4,5] has obtained a "product formula for an arbitrary pair of self-adjoint semigroups"; namely,
as well as
uniformly in t on compact subsets of [0, +00). [Note that we now deal with self-adjoint (contraction) semigroups (by Theorem 9.6.7, since A and B are nonnegative) or real (rather than imaginary) resolvents in (11.3.29) or (11.3.30) respectively.] More precisely [Kat5], under the hypotheses of Corollary 11.3.7(ii) below, the right-hand side of
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(11.3.29) or (11.3.30) must be replaced by e~t ( - A+B) n, where U denotes the projection onto the closed subspace H1.— Q(A) n Q(B). These results have been extended to nonlinear semigroups generated by the subdifferentials of convex functionals by Kato and Masuda [KatMasu], except that the conclusion of [KatMasu] in the linear case is somewhat weaker when Q(A) n Q(B) is not assumed to be densely defined. [In the nonlinear setting (proper, lower semicontinuous) convex functionals are a natural analogue of (nonnegative) quadratic forms.] Both the results in [Kat5] and [KatMasu] have been further extended by the second author [La5] (and [Lal,3]) to "average formulas " (coined by analogy with the arithmetic mean and the geometric mean, in particular). Hence, for instance, (11.3.29) becomes
Similarly, the conclusion of the counterpart of Corollary 11.3.5—obtained in [La6, §4, esp. Theorem 4.1 and Corollary 4.1, pp. 1710 and 1717]—states that
(Actually, it also extends to an arbitrary finite number of nonnegative self-adjoint operators; see [La6, Corollary 4.1, p. 1717].) We note that none of formulas (11.3.29)-(11.3.32) holds (or even makes sense) when B has an unbounded negative part, as in Theorem 11.3.1 (or Corollary 11.3.7(i) below). We will further discuss this point in Remark 11.7.5. (c) Even when A and B are arbitrary nonnegative self-adjoint operators (with a dense common form domain)—and let alone under the more general assumptions of Theorem 11.3.1 (or Corollary 11.3.7 below)-the counterpart for unitary groups of the product formula (11.3.28) is not known. Recall that in Corollary 11.1.6, the "product formula for unitary groups", we had assumed that A + B, the algebraic sum of A and B defined on D(A) n D(B), is essentially self-adjoint; that is, A + B has a unique self-adjoint extension, necessarily its closure A + B; further, by Proposition 11.2.10(ii), we must have A + B = A + B, the form sum of A and B, under the assumptions of Theorem 11.3.1 (or Corollary 11.3.5). The problem of extending the product formula for unitary groups (Corollary 11.1.6) to more general situations (such as those considered in the present section) is an extremely difficult one which has been intensively investigated in the 1970s and the early 1980s; see, in particular, the works of Paris [Far 1], Friedman [Fri], Chemoff[Cher2], Ichinose [Icl], Lapidus [Lal,2, La4, La6—11], and the references therein. To our knowledge, it still remains an open problem (which we will formally state in Problem 11.3.9 below). It is, of course, motivated 2 by Nelson's approach to the Feynman integral via the Trotter product formula [Nel] described in Section 11.2 above. (d) Recall from Example 10.3.21 that there exist nonnegative self-adjoint operators A and B in H := L (R)suchthatQ(A)r\Q(B)isdensein'HwhereasD(A)r\D(B) = {0}. Consequently, their form sum A + B is well defined (and, in particular, densely defined
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in H), whereas their algebraic sum A + B (with domain D(A) D D(B) = {0}) is trivial (and hence certainly not essentially self-adjoint). In this situation, the product formula for imaginary resolvents (Theorem 11.3.1 or Corollary 11.3.5) still provides meaningful information although the product formula for unitary groups (Corollary 11.1.6)—at least in its current form (see (c) above)—is useless. Corollary 11.3.7 Let A be an arbitrary nonnegative self-adjoint operator and let B = B+ — B- be a self-adjoint operator with an arbitrary positive part B+ and a negative part B- which is A -form bounded with bound less than 1. (Note that we no longer require that Q(A) D Q(B+) is dense in W.) (i) Then the form sum A + B can be viewed as a semibounded self-adjoint operator on the closed subspace H1 := Q(A) n Q(B+) c H and, for all u E H1,we have
uniformly in t on compact subsets of R (ii) Hence, in particular, if in Corollary 11.3.5 above, we only suppose that A and B are arbitrary nonnegative self-adjoint operators on H (without assuming the density of Q(A) n Q(B)) then the conclusion (11.3.28) must be replaced by (11.3.33), with u in H1. Proof (i) This follows from the proof of Theorem 11.3.1 and from refinements of the classical approximation theorems which can be found in ([Kat5], [Kat8, §IX.3]). (ii) is an obvious consequence of (i); note that B+ = B in this case. D Remark 11.3.8 Observe that if Q(A) n Q(B+) (which equals Q(A) n Q(B) since Q(B+) c Q(A)) is dense in H, then H1 = Q(A) n Q(B) = H, and hence pan (i) (resp., (ii)) of Corollary 11.3.7 yields Theorem 11.3.1 (resp., Corollary 11.3.5) above. The following open problem is unusually challenging—as was noted in Remark 11.3.6(c)—and would have important consequences for Nelson's approach to the Feynman integral via the Trotter product formula (discussed in Section 11.2 above). In this form, it was stated in [La6, p. 1738] for the special case (ii), and then extended in [Lall] (or [Lal3, Part I]) to the more general situation considered in case (i). (The answer is conjectured to be affirmative in [La6, §7, Conjecture, pp. 1737-1738].) Despite attempts by many investigators, it does not seem to have yet been resolved. Problem 11.3.9 (General product formula for unitary groups?) (i) Is it true that for (noncommuting, unbounded) self-adjoint operators A and B satisfying the assumptions of Theorem 11.3.1, we have
where A + B denotes the form sum of A and B? (ii) Does (11.3.34) hold under the more restrictive assumptions of Corollary 11.3.5, that is, for an arbitrary pair of nonnegative self-adjoint operators (with Q(A) n Q(B) dense in H?
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Remark 11.3.10 (a) The point is, of course, that in contrast to the usual Trotter product formula for unitary groups (Corollary 11.1.6), we no longer assume that the algebraic sum A + B, defined on D(A) n D(B), is essentially self-adjoint. (b) It is shown in [La6, Proposition 3.1, p. 1706] (Proposition 11.7.3 below)—by means, in particular, of results of Chernoff[Cher2] and Kato [Kat5], aswell as of Vitali's theorem [HilPh, Theorem 3.14.1, p. 104] (Theorem 11.7.1(i) below) for sequences of analytic functions—that if the (strong operator) limit on the left-hand side of (11.3.34) exists (with A and B nonnegative, but otherwise arbitrary), then Q(A) D Q(B) is dense in H, the limit in (11.3.34) is necessarily a unitary group, say e~itc, and in addition, we must have C = A + B, the form sum of A and B. [See also [La6, Proposition 3.2, p. 1708] (Proposition 11.7.4 below) for some positive results.] We note that this partly justifies the conclusion (11.3.33) obtained in Corollary 11.3.7. Hence, the problem (in this situation) is not to identify the limit but rather to prove its existence. We refer to Appendix 11.7 at the end of this chapter for a more thorough discussion of the results of [La6] alluded to in Remark 11.3.10(b), as well as of the closely related issue of the "analytic continuation" of formulas such as (11.3.29) into (rather) weak forms of (11.3.34). (See, in particular, Proposition 11.7.4.)
11.4 Application to the modified Feynman integral In this section, we will apply the above product formula for imaginary resolvents (Theorem 11.3.1) from [Lall] in order to define the "modified Feynman integral", FM(V), introduced by Lapidus in [Lal,2, La6-ll, Lal3] (and especially [Lall] in the present generality). More precisely, we will apply, in particular, Theorem 11.3.1 to the self-adjoint operators A = HQ = —5 A and B = V, the operator of multiplication by a suitable (but possibly very singular) "potential" function V : R.d —> R, acting in the complex Hilbert space H = L2(Rd). This will enable us to establish the convergence of the "modified Feynman integral" F'M(V) in (essentially) the most general case when the energy operator (or Hamiltonian) admits a self-adjoint realization that is bounded from below. (In particular, in contrast to the Feynman integral via the Trotter product formula F t T p ( V ) studied in Section 11.2, we will no longer have to require that H0 + V is essentially self-adjoint. Compare Theorems 11.2.19 and 11.4.2.) Aside from a few exceptions, this covers all the situations of interest in the applications. We will address, however, in Chapter 13, the case when the energy operator is unbounded from below. (See Sections 13.4 and 13.5, which deal with the appropriate extensions of the definition of fTp(V) and fM(V), respectively.) Towards the end of this section, we will also briefly discuss the "modified Feynman integral" on a smooth Riemannian manifold (instead of Rd) and the case of the Schrodinger equation with singular scalar as well as magnetic vector potentials. (See Examples 11.4.10 and 11.4.12, respectively.) Finally, we mention that we will only consider here the case when the scalar potential V is real-valued—and hence the corresponding quantum system is conversative (or isolated), which is the situation that is traditionally considered in mathematical physics.
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The case when V is complex-valued—and hence the quantum system is dissipative (or "open" [Ex])—will be dealt with in Section 11.6 (as well as in Section 13.5). Modified Feynman integral and Schrodinger equation with singular potential Let HQ = —2A denote the free Hamiltonian acting in L 2 (R d ), as defined in Section 10.2. We first recall from Section 10.2 some facts concerning the imaginary resolvent [I + it H0]-1 of HO, with t € R. Let G = G(x, y; t) : Rd x Rd x R -> C denote the free Green's function defined in Theorem 10.2.14; that is, for t £ R, t = 0, G( . , .; t) is the integral kernel of the convolution operator [/ + it Ho]-1 acting in L 2 (R d ). Hence, for all
where the integral on the right-hand side is an ordinary Lebesgue integral (and hence converges absolutely). (Of course, for t = 0, [/ + it Ho]-1= I is simply the identity operator, which we continue to write in the form (11.4.1).) In the present case of Euclidean space, G(x, y; t) is (for fixed t) a function of x — y alone. Further, by Theorem 10.2.14, when t > 0, we have for d = 1
and for d = 3,
In (11.4.2a) and in the following, we use the analytic determination of ^/z which is positive for z > 0; so that, for example, (—i) 1 / 2 = (1 — i>/V2- (For t < 0, we must replace in (11.4.2) (1 — i ) / t by (1 + i)/^/\f\ inside the exponentials; see Theorem 10.2.14. Therefore, the formulas in (11.4.5) below will have an obvious counterpart for t <0.) We now proceed much as at the end of Section 11.2 (beginning with Lemma 11.2.18), except that unitary groups are replaced by imaginary resolvents. We will also point out differences with that situation along the way. Lemma 11.4.1 Let V : Rd -> R be Lebesgue measurable and finite Leb.-a.e. Then for every t e R and (p e L2(Rd) and for Leb.-a.e. v e Rd, we have the equalities
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where xn := v and where the n-fold iterated integrals in the first equality of (11.4.3) as well as the multiple integral in the second equality of (11.4.3) exist for Leb.-a.e. v € ~Rd as ordinary Lebesgue integrals, and define a function which is in L 2 (R d ) as a function of v. Here, G n (xo, x1, . . . , xn; t) is the nth iterated kernel of the convolution operator [I + i(t/n)H0] - 1 given by (11.4.1) (with t replaced by t/n):
In particular, it follows from (11.4.2)-(11.4.4) that when t > 0, we have ford — 1
and for d = 3 (the case of a single quantum particle moving in R3),
Moreover, for t < 0 (instead of t > 0), the expression of Qn(xo, x1, . . . , xn; t) in (11.4.5) is the same as above except for (1 — i)/V?/« replaced by (1 + i)/T/\t\Jn inside the exponentials. Proof Fix p € L 2 (R d ). The operator [/ + i(t/n) V]-1 is the operator of multiplication by the function (1 + i (t/n) V ) - 1 , and this is the first operator that acts on p on the righthand side of the first equality of (11.4.3). According to the left-hand side of (11.4.3), the integral operator [I+ i ( t / n ) H o ] - 1 must now act on [l+i(t/n)V(-)]~V(-)-By (11.4.1) (Theorem 10.2.14), we have for all f e L2(Rd) and Leb.-a.e. w € Rd,
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where the integral is an ordinary Lebesgue integral. (Note that since V is M-valued, \(l+i(t/n)V(y))-1\< 1.) Iterating (11.4.6) n times, we see that the n-fold iterated integral on the right-hand side of the first equality of (11.4.3) coincides precisely with the nth product ([/ + i(t/n)Ho]~l [I + i(t/n)V]~l)n acting on q and evaluated at v := xn. The second equality of (11.4.3) follows from the Fubini theorem and the rapid decay at infinity of G(x, y; t), viewed as a function of \\x — y\\ alone. D We can now state the main result of this section (see also Corollary 11.4.5 below), which establishes the existence of the "modified Feynman integral" (Definition 11.4.4 below) for a rather large class of potentials V; essentially, the largest class for which the Hamiltonian H (or rather, the associated "energy functional" or quadratic form) is bounded from below and hence the Schrodinger equation can be defined without ambiguity. (At this point, the reader may wish to review some of the material in Sections 10.3 and 10.4 concerning quadratic forms and their applications; see also Remark 11.4.3(d) below.) Theorem 11.4.2 Let V : Rd -> R be Lebesgue measurable and such that V+ is in L 1 ( R d ) and V_ is Ho-form bounded with bound less than 1 (in the sense of Definition 10.3.17). Also, let p € L2(Rd). Then, for every t e R,
defines a function which is in L 2 (R d ) and is given for almost every v e Rd by the finite-dimensional Lebesgue integrals on the right-hand side of (11.4.3). Further, there is a function ty : (-00, +00) x Rd -> C such that i/f(t, .) e L2(Rd) for every t 6 R and
where the limit is uniform in t on all bounded subsets of R. In fact, the function i(r(t,. ) is given by the action of the unitary group eith on p; that is,
where H := HO + V is the form sum of the self-adjoint operators HO and V. [Recall from Theorem 10.3.14 and Exercise 10.4.10 that H is the unique (boundedfrom below) self-adjoint operator acting in L2(Rd) and associated with the semibounded quadratic form (or "energy functional") q(u) := 1/2 fRd ||Vu||2 + fRd V|u| 2 .] Finally, for p e D(H), fy(t, v) is the unique solution (in the semigroup sense discussed in Chapter 9) of the Schrodinser equation
with initial state p.
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Proof We apply Theorem 11.3.1, the product formula for imaginary resolvents, with H = L2(Rd), A = H0 and B = V, the operator of multiplication by the R-valued function V. As we know, HQ is self-adjoint (see Section 10.2) and V is self-adjoint as well (by Proposition 10.1.3). Further, HO is nonnegative and, by assumption, (the multiplication operator by) V_ is Ho-form bounded with bound less than 1. Hence the hypotheses of Theorem 11.3.1 are satisfied and so for all p e L2(Rd)
where the limit is in the norm on L 2 (R d ) and is uniform in t on all bounded subsets of R. (Here, in the notation of Theorem 11.3.1, we have let C - A + B - HQ + V = H.) We now simply use Lemma 11.4.1 to rewrite the nth Trotter-like product on the right-hand side of (11.4.11). It follows that (for Leb.-a.e. v) on(t, v) defined by (11.4.7) is given by either one of the Lebesgue integrals on the right-hand side of (11.4.3). Further, the fact that the limit of 4 n ( f , v) as n -> oo exists in the sense claimed in the theorem and equals ( e - i t H p ) ( v ) also comes from (11.4.11). Finally, since H = HO + V is self-adjoint, the fact that (t, v) = ( e i t h p ) ( v ) yields the unique solution (in the semigroup sense) of the Schrodinger equation (11.4.10) when the initial state q is in D(H) follows from Theorems 9.1.4 and 9.1.14 (as well as from Section 9.6). D Remark 11.4.3 (a) Recall from Proposition 10.4.4 that if V- belongs to the Kato class Kd, then it satisfies the conditions of the preceding theorem. Also, by Proposition 10.4.5(b), LP(Rd) + L°°(R d ) c L p . ( R d ) y c L,poc(Rd)u c Kd, where p = 1 for d = 1 and p > d/2 for d > 2. Further, by Remark 10.4.9(a), we can allow p = d/2 when d > 3 if V_ belongs to the smaller class (Lf oc (R d ))u u Lp(Rd) + L°°(Rd). (We refer to Section 10.4 for the notation used here.) (b) The fact that V+ belongs to L1oc(Rd) can be greatly relaxed provided we do not require the quadratic form q and the operators involved to be densely defined in H. = L2(E.d). Indeed, if we assume that V+ : Rd -> [0, +00] is Lebesgue measurable (but otherwise arbitrary), then under the same hypotheses as above on V-, H = HO + V is a self-adjoint operator on H1:.= Q(q) (rather than H and the counterpart of (11.4.8)— (11.4.10) holds when restricted to (p in H1. (In the analogue of (11.4.10), we would still require that p E D(H) but now D(H) c H1.) This follows from Corollary 11.3.7. [Naturally, according to an observation made later in [Jo6] in a related context, we can allow V+ e L1loc(Rd\G), where G is an arbitrary closed subset of Rd of Lebesgue measure 0, without changing the fact that H is densely defined in H = L 2 (R. d ); this follows since then, L2(Rd) = L2(Rd\G).] (c) Recall that in Example 10.3.21, we have discussed a very singular positive potential V such that the form sum Ho+V is densely defined in L 2 (R d ) but the domain of the algebraic sum HO + V is reduced to {0}. In this case, the familiar space D = D ( R d ) is contained in the form core. It is possible, however, to have a form core that is much different than the usual ones but such that the form sum is still densely defined. In Example 13.7.20 and Remark 13.7.21, we will discuss a highly singular potential V such
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that Ho+V is densely defined but no nonzero continuous function can be in any of its form cores. (In fact, for every p > 0, V is nowhere p-th power integrable; in particular, V E Lfa. (R d ).) Theorem 5.7 of [AlMal] implies that there is a form corefor Ho+V such that all the functions in the form core are bounded, compactly supported and "quasicontinuous." This example and the theorem of Alheverio and Ma rest on fine properties of Brownian motion, some of which will be discussed in Section 13.7. (d) We will not discuss here the case of "potentials" (or "interactions") which are given by (suitable) measures u rather than by functions V (on Rd). This will be the object of Section 13.7 (based on the later work of Albeverio, Johnson and Ma in [AlJoMa]), in the context of the "analytic in time (operator-valued) Feynman integral". However, we mention that under the assumptions of the main result of Section 13.7, Theorem 13.7.16, the product formula for imaginary resolvents of [La] 1] (Theorem 11.3.1) can still be applied (since its hypotheses hold, by Remark 13.7.24(b)), and hence the "modified Feynman integral" still exists for "interaction measures" u, = u+ — u- as in Theorem 13.7.16. [Inparticular, u+ is a "smooth measure" (e.g. the area measure on the surface of a sphere in R3) and u_ is in the analogue of the Kato class for measures; see Section 13.7 for the relevant terminology.] We note that the Hamiltonian H = HQ+u—as well as the associated quadratic form (or "Dirichlet form")—is still bounded from below in that case. We point out, however, that in order to obtain a complete analogue of Theorem 11.4.2 (and of Corollary 11.4.5 below) in this broaderframework, there remains to obtain a concrete expression for the action in L 2 (R d ) of the resolvent operator [I + it u , ] - 1 ; see Remark 13.7.24(b). Definition 11.4.4 (Modified Feynman integral) Let V : Rd -> R be Lebesgue measurable. Note that the operator ([I+i(t/n)Ho]~l[I+i(t/n)V]~1)" on L 2 (R. d ) makes sense 2 d for every positive integer n and that for every cp e L (R ) and Leb.-a.e. v € Rd, the number [([I + i ( t / n ) H o ] - 1 [I + i ( t / n ) V ] - 1 ) n p ] ( u ) equals either one of the expressions on the right-hand side of (11.4.3), with the integrals interpreted as ordinary Lebesgue integrals. The modified Feynman integral associated with the potential V, denoted by Ft m(V), is the strong operator limit
in £(L2(Rd)) whenever this limit exists. (That is,
for all p € L2(Rd).) We next restate in the above language much ofTheorem 11.4.2. Corollary 11.4.5 (Existence of the modified Feynman integral for a semibounded Hamiltonian with highly singular potential) Let V : R.d —> R be Lebesgue measurable and such that V+ e L 1 o c (R d ) and V- is relatively form bounded with respect to HQ with relative bound less than 1. (This is the case, for example, if V_ belongs to any of the
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classes LP(E.d) + L°°(E.d) c Lfoc(Rd)z c Lf oc (R d ) u c Kd, where p = 1 f o r d = 1 and p > d/2ford > 2; see also Remark 11.4.3(a) above.) Then FM(V) exists for all t e R, where the limit in (11.4.12') is, for all p € L 2 (R. d ), uniform in t on bounded subsets of R Further, for all t € R,
that is, FM (V) agrees with the unitary group which specifies the dynamics in the standard approach to quantum mechanics. Finally, for p € D(H) and t e R, J-'M(V)p is the unique solution (in the semigroup sense) at time t of the Schrodinger equation (11.4.10) associated with the semibounded Hamiltonian H and with initial state (p. Remark 11.4.6 In Theorem 11.4.2 (and Corollary 11.4.5), our assumptions on both the positive part and the negative part of the potential V are significantly more general than those made in Theorem 11.2.19 (and Corollary 11.2.22) which guaranteed the existence of Ft (V), the Feynman integral (associated with V) via the Trotter product formula. Indeed, if V+ belongs to L^oc (Rd), then it obviously belongs to L1loc (Rd) (see also Remark 11.4.3(b) according to which V+ can be an arbitrary nonnegative measurable function); moreover, it follows from Proposition 11.2.10(i) that if V- is Ho-operator bounded with bound less than 1, then it is also Ho-form bounded with bound less than 1 (but, of course, the converse is not true in general). Hence, under the hypotheses of Theorem 11.2.19 (or Corollary 11.2.22), both F t T P ( V ) and f'M(V) exist (and coincide with the unitary group e - i t H , since then, by Proposition 11.2.10(ii) and Theorem 11.2.11, H = (Ho + V)\D = HO + V). We will pursue this comparison early in Section 13.4. [Recall that F'Tp(V) is only known to exist when the algebraic sum HO + V is essentially self-adjoint since it requires the convergence of the Trotter product formula for unitary groups (Corollary 11.1.6) rather than for imaginary resolvents (Theorem 11.3.1), as for F'M(V); see Problem 11.3.9.] The following example provides a simple situation where fM(V) exists although HQ + V has many self-adjoint extensions (and hence is not essentially self-adjoint). Example 11.4.7 (Strongly singular central potential) Assume that d = 3, for simplicity. Set
where \\x \\ denotes the length of AC e R3 \ (0) and a, B are positive constants. We are thus considering a single (nonrelativistic) quantum-mechanical particle moving in R3 under the influence of the attractive central potential V. (Here, V is said to be attractive if V is negative, and repulsive if V is positive.) The larger the exponent a, the more singular and attractive V is. (Note that for a = 1, V is just the standard Coulomb potential which plays a crucial role in quantum mechanics.) Then, for anyB,Ho+V is essentially self-adjoint on C^(R3\{0}) as long asa < 3/2. Hence, in region (I) of Figure 11.4.1 (0 < a < 3/2), both Fp(V) and f'M(V) exist (and coincide). However, for arbitrary ft, the hypothesis that V_(= \V\) is Ho-form
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bounded with bound less than 1 holds for all a < 2 and, in the limiting case where a = 2, for B < 1/4. (See [Far2, Example, pp. 96-97] and [ReSi2, pp. 169-170], [Kat8, Example VII.4.15, p. 402]; note for comparison purposes that in the present situation, we have HO = — 1 A rather than —A, which explains the factor 1/2 in the definition of the potential V in (11.4.14).) For 3/2 < a < 2 or a = 2 with ft < 1/4 (region (II) of Figure 11.4.1), HQ + V need not be essentially self-adjoint and hence one does not know whether it is possible to use Nelson's definition of the Feynman integral, F t T P ( V ) . (For a = 2 and ft > 0 arbitrary, for example, Ho + V has many self-adjoint extensions and hence is certainly not essentially self-adjoint; this follows, e.g., from [Far2, Cases 2 and 3, p. 97] or [Kat8, footnote 1, p. 403] by passing to spherical coordinates.) Nevertheless, the above modified Feynman integral F ( V ) converges because Theorem 11.4.2 (or Corollary 11.4.5) applies to this situation. (See Figure 11.4.1.) In region (II) of Figure 11.4.1, we have J^M(V) = e~itH, where H = H0 + V, the form sum of HQ and V (or the Friedrichs extension of HQ+V). It is natural, then, to wonder whether the modified Feynman integral selects the "correct" self-adjoint extension of H0 + V (restricted to C^(Rd) \ {0}). This is so, however, since H is the self-adjoint operator associated with the semibounded energy functional q, as was explained in Exercise 10.4.10 and is discussed at greater length in ([Nel,4], [Kat8], [Sil], [ReSil,2]). On the other hand, when a > 2 or a = 2 and B > 1 /4 (region (III) of Figure 11.4.1), q is no longer bounded from below and so the form sum HO + V is not well-defined; hence neither F' (V) nor ^(V) exists. We will discuss situations of this type in Chapter 13 where TP extensions of the definitions of J^p(-) and ^/(-) (based on a method due to Nelson [Nel] and adapted to the modified Feynman integral in [BivLa, §3]) will be considered. [See Sections 13.5 and 13.6, respectively. Further, for the situation of the present example corresponding to region (III) of Figure 11.4.1, we refer to Example 13.6.13 and—for the interesting special case of the attractive inverse-square potential (or = 2 and B = 1)—to Example 13.6.18 below.] This example extends to strongly repulsive (rather than attractive) potentials, where we now assume that V(x) = +p/2\\x\\a (rather than -B/2\\x\\ a , as in (11.4.14)), for x 6 R3 \ {0} and with a, B positive constants. Then, for instance, in the limiting case when a = 2, the symmetric operator HO + V is essentially self-adjoint (on C^(R3 \ {0})—and hence F ( V ) is presently known to exist—if and only if B > 3/4. (See, e.g., [Far2, pp. 96-97] or [Kat8, footnote 1, p. 403]; in the terminology of differential equations, the case B > 3/4 or B < 3/4 corresponds to the limit point or the limit circle case, respectively.) In contrast, f'M(V) exists for all values of B > 0. Actually, in view of Remark 11.4.3(b) (or 12.1.3(c)), f'M (V), the modified Feynman integral associated with the positive potential V (= V+), exists for all values of (a, B), with a > 0 and ft > 0. We remark that for notational simplicity, we have only taken into account the radial variable r := \\x\\, but ignored the angular variables in the present discussion. The minor adjustments needed for a more detailed discussion will be briefly explained in Example 13.6.18, especially in Remark 13.6.19. Finally, we note that Example 11.4.7 also extends to multiparticle Hamiltonians (see Section 6.4 above and [ReSi2, Theorem X. 19, p. 170]). Further, it is of interest in quantum chemistry, especially in the phenomenological study of certain large molecules. In this
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FIG. 11.4.1. The central potential V(x) = B/2\\x\\ a , x e R3 \ {0} regard, we point out the very interesting review article [FrLdSp] by Frank, Land, and Spector surveying (through the late 1960's), in particular, the various uses of singular potentials as mathematical models in quantum physics (e.g. Regge poles formalism), as well as their applications to molecular physics and high energy phenomenology. Remark 11.4.8 (a) The free unitary group e~ itH ° is a generalized integral operator in L2(Rd). (See, e.g., [Hor3,4, Ste2].) In fact, it acts as an ordinary integral operator, with kernel (2nit)~ d / 2 e'|| x - y | | 2 /2t, when restricted to a (dense) subspace of functions decreasing sufficiently fast at infinity, such as the Schwartz space S(Rd), for instance (or the larger space L1 (Rd) D L2(Rd)). It can then be extended by continuity to all of L 2 (R d ). (See Section 10.2, especially Theorem 10.2.7, as well as the beautiful discussion in [Kat8, §IX.1.8, pp. 495-496].) The free imaginary resolvent [I + it Ho]-1. however, is an ordinary integral operator in L2(Rd). Consequently, for any p 6 L 2 (R d ), the integrals in (11.4.3) are Lebesgue integrals and can be viewed either as n-fold iterated or multiple integrals; in particular, they do not have to be interpreted in the mean or some other special sense, as is the case in (11.2.38) with the definition of F t j p ( V ) . (See also Remark 10.2.15(a). On the other hand, an advantage of (11.2.38) over (11.4.3) is that the expression is given by the same familiar exponential function for all dimensions.) (b)As was pointed out by Mark Kac when he was presented by the second author with the above results, the definition of the modified Feynman integral (in Theorem 11.4.2 or Corollary 11.4.5) provides a "natural summation procedure " for the Feynman integral. (c) (The subsequent comments are purely heuristic and do not pertain to the domain of rigorous mathematics.) In some sense, from the point of view of the integral kernel,
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the imaginary resolvent [I + itHo]~l shares features with the unitary group e~ itH ° (see (a) above and (10.2.33)) as well as with the heat semigroup e~'H° (see (10.2.26)). When d = 1, for instance, the kernel G(x, y, t) of the operator [I + it Ho]~' is not only oscillatory but also decays exponentially fast for large values of \x — y|/\/7 (see (ll.4.2a)). Moreover, intuitively, in the last equality of (11.4.3), the factor ["]"=1 (1 + ;' (t/n)V (xj))~l can be interpreted as an approximation to the "exponential functional" exp{—i f0 V ( w ( s ) ) d s } , where w is a continuous path going from XQ to xn — v within the time interval [0, t]. Accordingly (and still assuming that d = 1 for simplicity), one can argue that the "important "paths in the finite-dimensional integrals in (11.4.3) are Holder continuous of order at most 1/2, as is the case of the "typical" Brownian paths. (Compare with Corollary 3.4.4 and Theorem 3.4.5.) Note that in order to reach a similar conclusion, Feynman had to use (formally) an argument related to the method of stationary phase; see [Fey2, pp. 375-376] and the comments following formula (7.3.5) in Section 7.3. The following exercise supplements the above remarks and may help the reader—as it did the second author—to motivate and understand intuitively the modified Feynman integral. Exercise 11.4.9 Go over the heuristic derivation of the Schrodinger equation (when d — 1) given in Section 7.3 above ([Fey2, pp. 374-376]) by using, instead of the function exp{'(*^y) }, the function exp{—(1 — i ) ^ j j ^ } [and hence, essentially, by replacing the kernel ofe~itH° given by (10.2.33) by that of [1 + itH0]~l (when d = 1 and t > 0) given by (11.4.2a)]. More specifically, determine in the process the analogue of the constant L in (7.3.3) and identify it with the pre-factor in (11.4.2a). [Advice: Recall that (—j)1/2 = (1 - OA/2 and calculate the Lebesgue integrals
This should enable you to carry out the analogue of the computation performed in Section 7.3 by means of ordinary Lebesgue integrals (and hence absolutely convergent integrals) over R rather than oscillatory "integrals " which need not be convergent in the sense of either Riemann or Lebesgue (see the comments following Equations (7.3.7)-(7.3.10)).] Extensions: Riemannian manifolds and magnetic vector potentials We close this section by considering two additional examples: (i) the more geometric setting of curved differentiable manifolds (Example 11.4.10), and (ii) the case of a quantum particle moving in a magnetic as well as electric fields (Example 11.4.12). (Nothing in the rest of this book—except, in the case of Example 11.4.12, part of Section 13.6 and the end of Section 13.5—depends on these examples, and so one or both of them can be skipped on a first reading. On the other hand, each of them identifies a direction in which further work related to other topics in this book might well be successfully pursued.) Example 11.4.10 (Modified Feynman integral on Riemannian manifolds) Let Md be a complete, smooth (d-dimensional) Riemannian manifold. (Recall that a Riemannian
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manifold is complete if, for example, it is compact or is a closed submanifold of Euclidean space Rd, such as Rd itself; see, e.g., [Chav2, §1.7, esp. Theorem 1.10, p. 26] or [Chob].) Let A — div(grad) be the Laplace-Beltrami operator acting in H = L2(Md); here, the inner product in H is taken with respect to the Riemannian volume of Md. Then, HO := — 2 A is a nonnegative self-adjoint operator on H and is (by definition) equal to the closure of its restriction to D ( M d ) = C^(Md) (the space of smooth functions with compact support in Md). It is to guarantee this latter (essential self-adjointness) property that the assumption of (geodesic) completeness is required. (We recommend Strichartz's nice article [Stril] as a simple introduction to the subject of the Laplacian on a Riemannian manifold; other references include [OsWe], [Chavl] and [Chob-dWm, Chapter V].) For real t = 0, [1 + itHo] - 1 is an integral operator acting on H, with kernel still denoted by G(x, y; t). Let V be a (measurable) real-valued function on Md with positive part V+ and negative part V- satisfying a condition analogous to that of Theorem 11.4.2. This will be the case, for instance, if V+ e L^M4) and V- e L?(Md) + L°°(Md), where p = 1 if d = 1, p > 1 if d = 2 and p > d/2 if d > 3, and if, in addition, the Sobolev inequalities hold in D ( M d ) . In fact, the proof of this statement is the same as in the Euclidean case since it relies on the Sobolev inequalities. (The Sobolev inequalities hold for functions in V(Md) if, for example, Md has nonnegative Ricci curvature and if the volume of the geodesic balls of radius r grows like rd at infinity. See, for example, [Yaul, p. 10] or [Au]; also see [Heb, esp. Proposition 3.6, p. 59] for a more detailed discussion and for further information.) Under these assumptions, Theorems 11.3.1 can be applied and so Theorem 11.4.2, Definition 11.4.4 (of the modified Feynman integral) and Corollary 11.4.5 extend easily to this situation provided, of course, that the integration in (11.4.3) is now performed with respect to the Riemannian volume of Md. (Naturally, the Hamiltonian H — HO + V is still defined as the form sum of HO and V.) Remark 11.4.11 (a) In general, however, G(x, y; t) does not have an explicit expression, in contrast to the Euclidean case. Hence, for example, formulas (11.4.2) and(11.4.5) do not have an exact counterpart in this setting. (An analogous comment would be true if we worked with the unitary group e~"H° instead of the imaginary resolvent [I +itHo\~l, as in the Feynman integral via TPF.) (b) In view of part (a) and Remark 11.4.8(c), it is noteworthy that precise kernel estimates are now available in a rather broad context. In particular, if the Ricci curvature of the Riemannian manifold Md is bounded from below, then the kernel G(x, y ; t ) exhibits an asymptotic behavior roughly similar to that of its Euclidean counterpart. (See [CheGT, §0.27 for a precise statement of this fact.) Example 11.4.12 (Schrodinger equation with singular magnetic vector potential) For simplicity, we again work in Rd (with d > 1), rather than in more general curved Riemannian manifolds as in the previous example. Let a = (a1,a2, . . . ,ad) € (LjJoc(Rd))d be a (locally square-integrable) vectorvalued function on Rd; that is, each of its components aj (1 < j < d) is real-valued and in L2 (Rd).
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Set H = L2(E.d), equipped with it usual norm || • \\2 = \\ . \\L2 Rd)- Consider the (nonnegative, closed) quadratic form
with domain
where V = (9i, . . . , 9j) = ( d / d x 1 , . . . , 3/dxd) denotes the distributional gradient on Rd and i = V=T. Let A = Ag = — j(V - ia)2 be the (nonnegative) self-adjoint operator associated with/I^ (viaTheorem 10.3.15). Now, let V be a measurable real-valued function on Rd such that V+ = max( V, 0) is in L]1oc(Rd) and V_ = max(— V, 0) is A-form bounded with bound less than 1. Further, let B = V be the operator of multiplication by V acting in L 2 (R d ). Then C — A + B, the form sum of A and B, is the semibounded self-adjoint operator associated with the (semibounded, closed) quadratic form
with domain
where we have used (11.4.15). (For more details, see ([Kat7], [ReSi2].) We write
Physically, H = H^y represents the Hamiltonian or energy operator with scalar potential V and magnetic vector potential a. In view of (11.4.16), H is a natural (bounded from below) self-adjoint realization of the symmetric operator — 1(V — ia)2 + V. (Of course, when a — 0, we have A — — 1 A = HO and so H = HQ + V as in the rest of this section.) We can now apply Theorem 11.3.1 and deduce from it that the unitary group e~"H is given by (11.3.2); which enables us to write the solutions of the Schrodinger equation -jj- = — iHty (t e R) in terms of expressions involving a and V separately.
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We note that we could work here on an arbitrary (nonempty) open subset Q of Rd instead of Rd itself. (We would then have to impose, for example, Dirichlet boundary conditions, corresponding to a particle constrained to remain in Q.) To avoid repetitions, however, we will postpone doing this until the latter part of Section 11.6, where the results and definitions of this section are extended to include Schrodinger operators with complex-valued scalar potentials. Remark 11.4.13 (a) If we wanted to apply Corollary 11.1.6 (the Trotter productformula for unitary groups) instead of Theorem 11.3.1, we would have to assume that — 1(V — ia)2 + V is essentially self-adjoint on D(R d ). This is known to be true under the more restrictive hypotheses a e (L4 oc (R d )) d and V-a :=diva = 0 (as well as V+ e L 2 Q C (R d ) and V_ is A-operator bounded with bound less than 1); see the work of Leinfelder and Simader [LeiSd] which extends Kato's essential self-adjointness results [Kat2,3] discussed in part in Section 11.2 (Theorem 11.2.11). (b) Some readers may wonder where the definition (11.4.17) of the Hamiltonian with magnetic vector potential comes from. Physically, when d = 3, a = a(x) represents the magnetic vector potential (associated classically with the magnetic field B = V x a := curia) and V represents the scalar potential (associated classically with the electric field £ = — VV := —grad V) acting upon a single quantum-mechanical particle moving in R3. The formal expression for the quantum-mechanical Hamiltonian,
can then be easily deduced from the "correspondence principle " between classical and quantum mechanics. (See, for example, [ReSi2, Example 5, p. 173].) Further, in physical units such that H = 1, H can be written formally as
where m and e denote the mass and the charge of the quantum particle, respectively, and where c is the speed of light. (c) Example 11.4.12 above extends naturally to an N-body problem corresponding to a system of N charged particles (of mass mk and charge ek, k = 1, . . . , N); with the obvious notation, the formal expression for the Hamiltonian is then given by
(In ordinary units, we must replace V and Vk by h V and HVk, in (11.4.18) and (11.4.19), respectively.) Exercise 11.4.14 Write down the quadratic form corresponding to (11.4.19) and give (under suitable hypotheses) a precise definition of the N-body Hamiltonian with magnetic vector potential discussed in Remark 11.4.13(c).
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11.5 Dominated convergence theorem for the modified Feynman integral The purpose of this section is to show that the "modified Feynman integral" from [Lai ,2, La6-ll, Lal3]—introduced and discussed in Section 11.4—presents the advantage of enjoying very satisfactory stability properties; in fact, close to the optimal ones that could be expected from any definition of the Feynman integral. The results of this section are due to Lapidus in [Lal2]. Our approach is operatortheoretic in nature. Following [La 12], we first obtain an abstract perturbation theorem for unitary groups generated by form sums of operators (Theorem 11.5.7). We then combine this result with certain analytic techniques in order to derive a dominated convergence theorem for Feynman-type path integrals (Theorem 11.5.13). In applying the latter to the modified Feynman integral, we use in an essential way the fact that the "product formula for imaginary resolvents" from [La1 1 ] (Theorem 11.3.1, established in Section 11.3 above) holds in a rather general context. This enables us, in particular, to consider perturbations with "arbitrary" positive part and with unbounded negative part, in the sense of quadratic forms. More precisely, in a special case, the main result of this section is the following "dominated-type convergence theorem" (see Theorems 11.5.13 and 11.5.19 for a more general statement): Theorem 11.5.1 (Dominated convergence theorem for the modified Feynman integral: special case) Let V,Vm,m — 1,2, . . . ,be Lebesgue measurable real-valued functions on Rd. Let Vm,+ — max(Vm,Q) and V m ,_ — max(-V m ,0) denote respectively the positive and negative part of Vm. Assume that " Vm converges to V dominatedly ", in the following sense: (a) Vm -> V Leb.-a.e. in Rd, (b) V m ,+ < U for some U e L1loc (c) Vm,- < W for some W e LP + L°°, where p = 1 if d = 1 and p > d/2 if d > 2. Let H = Ho + V and Hm = HO + Vm be the Hamiltonian [or Schrodinger operator, acting in L2 (Rd)] associated with V and Vm, respectively, where as before, HQ = — 5 A denotes the free Hamiltonian acting in L 2 (R d ). Then we have asm —^ oo:
and
where the limit in (11.5. la) [resp., (11.5.1i>)] holds in the strong operator sense in L(L 2 (R d )) and is uniform in t on compact subsets of R [resp., [0, +00)]. Moreover, the "modified Feynman integrals", f \ f ( V ) and F'M(Vm), associated with V and Vm, respectively, exist (i.e. are well defined, in the sense of Definition 11.4.4) and for all t 6 R
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thatis, for all p e L2(Rdl
in the norm of L2(Rd). More specifically, we have
Note that the last equality in (11.5.3) indicates that the limits in m and n, occurring in the definition of the modified Feynman integral (Definition 11.4.4) associated with the "potentials" Vm, can be interchanged. In the actual statement of Theorem 11.5.13, it is assumed that W belongs to the larger space (£loc)u, the set of all functions which are "locally strongly uniformly in Lp" (see Definition 10.4.2(ii)). [Naturally, in our present context, the "domination" refers to the potential and not to the corresponding "functional" in the integrand of a (formal) Feynman path integral.] This enables us to obtain a theorem in which no a priori assumption is made on the limiting function V, as should be the case with a "dominated convergence theorem". [More generally, we will slightly extend the result of [Lal2] by allowing W to be in the broader Kato class Kd for any d > 1 (see Definition 10.4.1 and Remark 11.5.15(d)).] The work in [La 12] was motivated in part by a private query of the first author concerning his "bounded convergence theorem" for the "analytic operator-valued Feynman integral" [Jo3, Theorem 2]. (As far as we know, the paper of Johnson [Jo3] was the first to directly address in any detail the question of stability of Feynman integrals under small changes in the potential.) The second author answered this query in the latter part of his article [Lal2, Corollary 4.2] and obtained some related results [Lal2, Proposition 4.1] pointing out the usefulness of operator-theoretic methods in this context. [We will take advantage of this in Chapters 15 and 16. Moreover, in Section 16.1 (Theorems 16.1.1 and 16.1.2), based on later work in [JoLal], we will significantly extend the stability results of [Jo3] to a broad class of Wiener functionals.] The rest of this section is organized as follows. We first recall some definitions and results from operator theory that will be useful to us here. We then present an abstract perturbation theorem for the form sum of self-adjoint operators (Theorem 11.5.7), and combine it with Theorem 11.3.1 above to justify an interchange of limits in the corresponding product formulas for imaginary resolvents. Finally, we establish a general dominated-type convergence theorem for Feynman integrals (Theorem 11.5.13) and discuss its applications to the modified Feynman integral (Theorem 11.5.19). We note that these results (or a special case thereof) will also be applied to other approaches to the Feynman integral towards the end of Section 13.4; see especially
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Corollaries 13.4.3 and 13.4.6 (from [Jo6] and [JoKim], respectively), dealing with dominated-type convergence theorems for the analytic in time Feynman integral and the Feynman integral via TPF. Preliminaries Our setting for the abstract results is much the same as that of Section 11.3. Let H. be a complex Hilbert space equipped with the inner product (., .) and the associated norm || • ||. The arrow -> (resp., ->) indicates strong (resp., weak) convergence in H. If T is an unbounded self-adjoint operator in H, we denote as before by T+ (resp., 71) its positive (resp., negative) part, defined through the spectral theorem (Theorem 10.1. 11), and we let Q(T) = Q(T+) n Q(T-) be its form domain. (See Section 10.3.) If T is nonnegative, Q(T) = D ( T 1 / 2 ) , while if T is bounded from below (say, T + al > 0, for some a s R), then Q(T) :- Q(T + al). We will need a counterpart for semibounded operators of the notion of form core for semibounded quadratic forms introduced in Definition 10.3.3. Definition 11.5.2 Let T be a self-adjoint operator on Ti. which is bounded from below (say, T + al > 0, for some a € R). Then a subspace D of Q(T) is said to be a form core for T if and only if it is dense in the space Q(T) equipped with the Hilbert norm
that is, D is a form core (in the sense of Definition 10.3.3) for the nonnegative quadratic formx H-> ||(T +aI) 1/2x || 2 , with domain Q(T). Exercise 11.5.3 Let T be as in Definition 11.5.2. Show that the linear operator (T + aI) 1 / 2 is continuous from (Q(T), ||| . |||1) to (H, \\ • ||). We will also need a suitable notion of convergence for unbounded self-adjoint operators. (See, for example, [ReSil, §VIII.7] for a number of additional properties.) This notion will be used again (in a slightly different form) in Section 12.2. Definition 11.5.4 Let B, Bm, m = 1,2, . . . , be (unbounded) self-adjoint operators in H. We say that {B m } m = 1 converges to B in the strong resolvent sense if and only if
that is, [I + i B m ] - 1 x -+ [I + iB] - 1 x, for all x e H. In this section, unless otherwise specified, all limits involving operators will be taken in the strong operator sense (that is, the topology of pointwise convergence on H) and will be denoted by -> or, equivalently, by s - lim. Finally, we reformulate and specify—in the special case of self-adjoint operators— Theorem 9.7.11 (or 8 .1.12), a basic result in the approximation theory of semigroups of operators. (For the present statement, see [Kat8, Theorem IX.2.16, p. 504] and [KuSe, Theorem 11.4, p. 312 and Corollary 11.4.1, p. 315] or [ReSil, Theorem VIII.21, p. 287].)
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Theorem 11.5.5 Let H,Hm, m = 1,2, . . . , be self-adjoint operators in H. Then the following statements are equivalent: (a) {H m } m = 1 converges to H in the strong resolvent sense. (b) e~itHm -> e~"H strongly, for all t € R. (c) [/ + iXHm]~l -> [I + i ^ H ] - 1 strongly, for all y e R. (d) The same as (b), uniformly in t on compact subsets of R. If, in addition, the operators Hm and H are uniformly bounded from below, then (a) implies: (e) e~'Hm -> e~'H strongly, uniformly in t on compact subsets of [0, +00). Perturbation of form sums of self-adjoint operators We establish here a simple abstract perturbation theorem for the form sum of (suitable) unbounded self-adjoint operators (see Theorem 11.5.7 and Corollary 11.5.8 below, obtained in [Lal2]), and then apply it (in conjunction with Theorem 11.3.1 above, from [Lai 1]) to justify the interchange of limits associated with the product formulas for the imaginary resolvents of these operators. We now present the aforementioned perturbation theorem that will be applied later on in this section. We do not strive here for maximal generality since we are mainly interested in the applications to convergence theorems for the modified Feynman integral. We first state our assumptions for this theorem. Hypotheses. Let A, B, Bm, m = 1, 2, . . . , be self-adjoint operators on H. Assume that A is nonnegative. Suppose for simplicity that Q(A) n Q(B+) is dense in H. Assume further that the following conditions (CO, ( C 2 ) are satisfied: (CO For each m > 1, Q(A) c Q(B-) n Q(Bm,-) and
for all x 6 Q(A), for some positive constants y < 1 and S independent of m and x. We then set H = A + B and Hm = A + Bm. (Note that by Theorem 10.3.19, H and Hm are semibounded self-adjoint operators.) ( C 2 ) There is a form core D for B+, B- and H (in the sense of Definition 11.5.2) such that for all x e D,
and
as m —> oo. Here and hereafter, we denote by 5m,+ (resp., 5 m ,_) the positive (resp., negative) part of Bm and by B^+ (resp., Bm'_) the square root of Bm,+ (resp., B m , _).
Remark 11.5.6 Condition (C\) implies that B- and Bm,- are A-form bounded (Definition 10.3.17) with uniform bound y less than one. Furthermore, conditions (CO
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and (C2) imply that D c Q(H) n Q(Hm); in particular, Q(H) n Q(Hm) is dense in "H. It thus follows from Theorem 10.3,19 that the form sums H = A + B and Hm — A + Bm (m = 1 , 2 , . . . ) are well-defined (and necessarily densely defined) semibounded self-adjoint operators on H, with uniform lower bound —8. We can now state the following perturbation theorem of the Kato-Rellich type, which is obtained in [Lai2, Theorem 3.1 (and Corollary 3.1), p. 43]. Theorem 11.5.7 (Perturbation theorem for form sums of self-adjoint operators) Under the hypotheses just stated: {/fm}^L] converges to H in the strong resolvent sense.
(11.5.4)
In view of Theorem 11.5.5, the next statement follows immediately from Theorem 11.5.7. Recall from Remark 11.5.6 that the stability condition (CO implies that the operators Hm and H are uniformly bounded from below, so that part (e) of Theorem 11.5.5 can also be used here. Corollary 11.5.8 Under the assumptions of Theorem 11.5.7, we have
and
strongly as m -> oo. Moreover, the convergence in (ll.S.Sa) [resp., (11.5.5b)] is uniform in t on compact subsets o/R [resp., [0, +00)]. Proof of Theorem 11.5.7 Fix v e «. Set wm = [1 + iHmr{v. Then wm e D(Hm) and
Clearly,
Consequently,
By (11.5.6) and the Cauchy-Schwarz inequality,
Since wm e D(Hm) c Q(Hm) = Q(A) n Q(Bm,+), condition (d) implies that
Taking the imaginary part of (11.5.8) and using (11.5.9), we see that
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Since y < 1, it follows that {Al/2wm} and {B^+wm} are bounded sequences. (Note that we have used here the fact that S is independent of m.) According to (11.5.6) and (11.5.10), so is {Bm' _wm}. Hence there exist vectors w, a, b+ and b- inH such that
along some subsequence {rrij} —> oo. (Here and in the next few lines, it is implicitly understood that the limits are taken along {mj}.) We claim (much as in Step 1 of the proof of Theorem 11.3.1) that
In fact, let y e Q(A). Then, by (11.5.11),
Hence w € Q(A) and a = Al/2w. (Recall that Q(A) = D(A1/2) and A1/2 is selfadjoint.) Now, for all y e D,
where we have used (11.5.11) and condition (€2). Since D is a form core for B+, it follows by continuity (see Exercise 11.5.3) that
Hence w € Q(B+) and b+ - B^w. Similarly, using (11.5.11) and (C2), we see that b- = B 1/2 _w. Since Q(H) = Q(A) n Q(B+), this concludes the proof of (11.5.12). Next, let y e D. Then
where we have used successively (11.5.7), and after passing to the limit along {mj}, (11.5.11), (11.5.12) and condition (C2). Here, we denote by q the sesquilinear form with which H is associated. Hence q(w, y) = ((v — w)/i, y), for all y 6 D. Since, by (C2), D is a form core for H, it follows from the first representation theorem for unbounded quadratic forms (Theorem 10.3.14) combined with a simple continuity argument that w e £>(//) and Hw = (v - w)/i. Thus
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and so
In view of (11.5.13b), the limit w in (11.5.11) does not depend on the subsequence {wmj}. By a standard compactness argument, it follows that wm —>• if as m —»• oo. Hence by (11.5.8) and (11.5.13a),
which in turn implies that || wm — w ||2 -> 0 and so wm -+ w,as required. This concludes the proof of Theorem 11.5.7. D It is natural to ask whether the limiting procedures occurring in the statements of Theorem 11.3.1 and Theorem 11.5.7 can be combined or even interchanged. The following result answers this question in the affirmative. It will be applied to the modified Feynman integral at the end of this section (see Theorem 11.5.19). Corollary 11.5.9 (Interchange of limits) Assume that the hypotheses of Theorem 11.5.7 hold and {Bm}™=} converges to B in the strong resolvent sense. Then, for all t e R, we have
Proof We shall show by repeated use of Theorem 11.3.1 (the product formula for imaginary resolvents) and Theorem 11.5.7 (the perturbation theorem for form sums) that both double limits exist and are equal to the same operator, namely e~lt(-A+B\ As before, we set H = A + B and Hm = A + Bm. Also, we fix t e R. By Theorem 11.3.1, we have for each fixed m,
Further, by Theorem 11.5.7 (or more precisely, by equation (11.5.5 a) of Corollary 11.5.8),
Hence the first double limit occurring in the statement of Corollary 11.5.9 exists and equals e~'tfi. On the other hand, for each fixed n, our additional assumption in Corollary 11.5.9 (about the strong resolvent convergence of {Bm}^=1 to B) implies that
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Since all the resolvents involved are contractions, it follows by an immediate induction that, for fixed n,
Now, applying Theorem 11.3.1 once more, we see that
Consequently, the second double limit in Corollary 11.5.9 exists and equals e "H. Hence the conclusion. D Remark 11.5.10 (a) It will be very easy In the applications to verify that {Bm}m=1 converges to B in the strong resolvent sense (see the end of the proof of Theorem 11.5.13). (b) The analogue of Corollary 11.5.9, obtained by replacing imaginary resolvents by unitary groups in its statement, would follow from Theorem 11.5.7 in a similar way if it were known that the Trotter product formula for unitary groups (Corollary 11.1.6) held under the assumptions of Theorem 11.3.1. Recall, however, that even for unitary groups generated by nonnegative self-adjoint operators, this problem remain open. (See Remark 11.3.6(c) and Problem 11.3.9, as well as [La6] and [Lai 1].) For the same reason, it is not known whether Theorem 11.5.13 below can be applied in its full generality to Nelson's definition of the Feynman integral (via TPF) introduced in [Nel] and discussed in detail in Section 11.2. Application to a general dominated convergence theorem for Feynman integrals It will be convenient to work with the class of functions (Lploc)u — (Lploc(Rd))u which was introduced in Definition 10.4.2(ii). This class was originally considered by Simader [Sd] in the context of Schrodinger operators. Loosely speaking, the elements of (Lpc)u are the functions which are "locally strongly uniformly in Lp". Indeed, recall from Definition 10.4.2(ii) that f E (Lploc)u if and only if f ||x _ y||
where Kd denotes the Kato class in dimension d (Definition 10.4.1). (For the containment of (11.5.14b), we assume that p > d/2 if d > 2 and p = 1 if d = 1.) Of course,
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(L^n
253
c Lploc. In particular, it follows that (as noted before)
The inequality 0 < f < g, with / Lebesgue measurable and
In the next two statements, (11.5.17) and (11.5.18), we assume that p > d/2 if d > 3, p > 1 if d = 2 and p= 1 if d= 1.
Set f+ = max(/, 0) and /_ = max(-f, 0). Let f+ 6 L^ and /_ e (ifoc)u. Then, for any r > 0, we have (unless otherwise specified, all integrations are taken over R d ):
where C (resp., Cr) is a universal constant depending only on d (resp., on d and r) but not on f_, and where Br(x) denotes the ball in Rd of center x and radius r. For a proof, based on the Sobolev inequalities, see, e.g., [BreKat, Lemma 2.1 and Remark 2.1, pp. 139-140]. Here, we denote by HQ (the Sobolev space) the completion of D = D(R d ) (= C^(Rd))—the space of infinitely differentiable functions with compact support in Rd—under the Hilbert norm |||M|||I = (||Vu||2;2 + || M ||2 2 ) 1/2 . Let HO = -5 A be the free Hamiltonian acting in H = L2(Rd) and let / be as in (11.5.18). In the following, we will (as before) use the same notation for the function / and the corresponding multiplication operator by / in H. The form sum HQ + f is a well-defined self-adjoint operator in "H. In fact, it suffices to choose r so small that C supxeRd ||f _||L p (B r (x)) < 1 for inequality (10.3.62b) to hold in Definition 10.3.17 (see, e.g., [BreKat, Remark 2.1, p. 140]). Naturally, here, A = H0, B = f, B+ - f+ and B_ = f _ . Remark 11.5.11 (a) The properties (11.5.15), (11.5.16) and (11.5.18) will enable us to state a "dominated convergence theorem" (Theorem 11.5.13) in which no a priori assumption is made on the limiting function. (b) In view of (11.5.17), Theorem 11.5.13 below will apply to most situations encountered in the applications. Further, (11.5.17) implies that Theorem 11.5.1 stated at the beginning of this section follows from Theorem 11.5.13. As a matter of fact, any space of functions enjoying properties similar to those of Simader's space (L^c)u (especially the counterpart of (11.5.15), (11.5.16) and (11.5.18)) could be used to formulate the statement of Theorem 11.5.13 presented below. (See, in particular, Remark 11.5.15(d) below which will enable us to replace (L^y (with p satisfying (11.5.19)) by the Kato class Kd in hypothesis (V 2 ).)
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We now state our assumptions: Hypotheses. Let V,Vm, m = 1, 2, . . . , be measurable functions on Rd. Set H = 2 L (Rd), A = Ho = -1 A, B = V and Bm = Vm. Assume that
Assume further that conditions (V1) and (V2) hold: (V1) Vm —> V Leb. almost everywhere (a.e.) in Rd. (V2) There exist U e L^ and W e (Lfoc)u such that for all m>\,
and
Remark 11.5.12 By (11.5.15) and (11.5.16), the above assumptions imply that V+, Vm,+ 6 L,1^ and V_, V m ,_ € (LJ^)jr; note that V+ < U and V_ < W. In view of the previous remark and Theorem 10.4.8, we may set H = HO + V and Hm = HO + Vm, and consider them as unbounded operators on H. We can now state the main result of this section [Lal2, Theorem 4.1, p. 52], which we will apply afterwards to the modified Feynman integral (see Theorem 11.5.19). Theorem 11.5.13 (Dominated-type convergence theorem for Feynman integrals) With the above notation and assumptions, the conclusion of Theorem 11.5.7, as well as of Corollaries 11.5.8 and 11.5.9, holds. In particular, for every fixed p e L 2 (R d ), e ~ i t H p converges to e~itH p in L2(Rd), uniformly in t on all bounded subsets of R. Proof of Theorem 11.5.13 We must check that Theorem 11.5.7 applies to this situation. For all u € H1 = Q(A) and r > 0, we have:
where we have applied inequality (11.5.18) to W 6 (Lpk,c)u By choosing r so small that y = C(sup xeRd ||W||Lp(B r (x))) < 1, we see that condition (C1) holds with S = CCr sup xeRd ||W||LP(Br(x)). Here and in the rest of this section, ||W||tp (instead of || W||p) often denotes the L^-norm of W on Rd; further, ||W||z,p(Br^)) denotes the Lpnorm of W on Br (x), the open Euclidean ball of radius r and center x in Rd. Next, we check that condition (€2) is satisfied with D := D = D(R d ). Since, for u e D, |V1/2+u|< U1/2\u\, withW 1/2 |u| e L2 and V1/2+u -> V1/2+u a.e., it follows from the Lebesgue dominated convergence theorem that
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Similarly, \V^u\ < \V^2\u\ with W'^lwl e L2 [in fact, W 6 L^ by (11.5.15)] and Vm _« —> V_ M a.e. Hence, again by dominated convergence,
Moreover, D = Z> is a form core for H by a theorem of Simon (see [Si8] and [Sil 1, Theorems A.2.8 or B. 1.5, p. 460 or 463]) which extends to form cores a theorem of Kato on operator cores (see [Kat2] and [Sil 1, Theorem A.2.9, p. 460]). [This is the only place where we have to require that p > d/2 for d > 3, Note that, in view of the remark at the top of page 464 in [Sil 1], Simon's theorem applies here since (L^z C Kj by (11.5.14b). (Further, since the latter theorem is valid if V+ € L1loc and V_ e Kj, the generalization of Theorem 11.5.13 provided in Remark 11.5.15(d) below will not pose any problem here.)] Finally, it remains to check that D = D is a form core for B+ = V+ and B- = V_; since V+ and V- belong to L^, this is an immediate consequence of the following simple lemma: Lemma 11.5.14 Let f € L^ with f > 0. Then V is a form core (in the sense of Definition 11.5.2) for the operator of multiplication by f in L2. Proof of Lemma 11.5.14 Recall that u e Q(f) if and only if u e L2 and f 1 / 2 u e L2. Observe that a subspace E of Q(f) is a form core for / if for all u 6 L2, there exists a sequence [un] in E such that un -> u in L2 and f 1 / 2 u n -> f l/2 u in L2 (see Definition 11.5.2). The result is now obtained by using a series of standard approximation procedures analogous to those involved in the proof of the Feynman-Kac formula in the general case (Theorem 12.1.1) to be discussed in Chapter 12 (Section 12.2). (Compare [Sill, Proof of Theorem A.2.8 (Theorem B.I.5), pp. 463-464].) Step 1: E\ := {u e Q(f) : u has a compact support) is a form core for /. In fact, let 0 be a continuous function with compact support in Rd and such that 0(0) — 1. For M € Q(f), define un by un(x) = 0(x/n)u(x). Clearly, un has compact support and |un| < ||0||ool"|. Hence |/ 1/2 w n ] < ||0||ool/ 1/2 «l- In particular, «„ e £1. Because wn -> M a.e., we also have that f 1/2 u n -> f1/2 u a.e. Since both M and f 1/2 u belong to L2, it follows from the Lebesgue dominated convergence theorem that un -> u and / 1/2 M n -» /'/ 2 M in L2. Step 2: EI := E\ n L°° is a form core for /. Let M e E1. By separating the real and imaginary parts of M, we may assume that u is real-valued. Define un by
Clearly, «„ is bounded and has compact support. Since \un\ < \u\ and un -> u a.e., it follows as above that un -> u and f 1/2 w n -> f 1/2u in L2. Since {un}lies in E2, the claim results from Step 1.
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Step 3:Disa form core for /. In fact, let u € £7- Let {j») be an approximate identity on Rd (i.e. jn e £>, jn > 0, Supp jn C Bn-\ (0) and / _/„ = 1, where Supp jn denotes the support of jn and Bn-\ (0) is the closed ball of center 0 and radius n~ ] ). Let un = jn * u be the convolution of jn with u. It is well known that since u € L2, un € V and un -H> u in L2. (See, e.g., [Stel, Theorem 2 (the p = 2 case), pp. 62-64].) By possibly considering a subsequence, we may assume that un -> u a,e. and hence that fl/2un -»• /^ 2 M a.e. Moreover, Supp «„ c Supp u + Supp ;'„ c K, where A' := Suppw + B\(0); note that K is compact because w has compact support and 61 (0) is closed. Since u e L°°, we see that
Thus \un\ < \\u\\ L°°XKi where XK denotes the characteristic function of AT. This implies that
Note-that f l / 2 X K e L2 since / e L,1^. By (11.5.22), it follows from the dominated convergence theorem that f^2un —> fl/2u. In view of Step 2, this concludes the proof of Lemma 11.5.14. n We have just shown that the assumptions of Theorem 11.5.7 are satisfied. To conclude, we note that the hypotheses of Corollary 11.5.9 hold since condition (Vi) alone implies that {Vm}^=1 converges to V in the strong resolvent sense. Indeed, [/ + j Vm]~l = (1 + iVm)~l (the multiplication operator), and similarly for V. Fix u e L2; then |(1 + iVm)~lu\ < l«l and (1 + iVm)~lu -» (1 + iV)~lu a.e. in Rd. Hence (I + iVm)~lu ->• (1 + iV)~lu in L2, by a final application of the dominated convergence theorem. The proof of Theorem 11.5.13 is now complete. n Remark 11.5.15 (a) It follows from the proof of Theorem 11.5.13 that, in condition (Vi), the assumption Vm,+ < U for some U € L\oc can be replaced by the following weaker hypothesis:
Indeed, for v 6 V, /(V m ,+ - V+)\v\2 -+ 0 since \v\2 = vv € T>. (b) The fact that, for nonnegative potentials Vm satisfying (11.5.23), Hm converges to H in the strong resolvent sense, also follows from the works of Simon [Si8, Theorem 4.1, p. 44] and Kato [Kat6]. We note, without going into the details, that Theorem 11.5.7 can be generalized in order for Theorem 11.5.13 to encompass the case of Schrodinger operators with magnetic vector potentials studied in Example 11.4.12. More precisely, let the "scalar potentials" V, Vm be as in Theorem 11.5.13 and let the "vector potentials" a, am be W1-valued functions on Rd such that am -> a in L2^. Then the counterpart of Theorem 11.5.13 holds with H := -|(V
- ia)2 + V and Hm := -i(V - iam)2 + Vm.
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(c) It is noteworthy that a "monotone convergence theorem" for quadratic forms is also available in this context. (See [Kat8, § V//7.3 and VIII.4, pp. 459-^62] and [ReSil, Supplement to VIII. 7, pp. 372-377], as well as Theorems 12.2.2 and 12.2.4 below.) This fact will be useful in the proof of the Feynman—Kac formula given in Chapter 12. (d) Close inspection of the beginning of the proof of Theorem 11.5.13 shows that in condition (V2) we could replace the hypothesis that W e (L^ l o c ) by the weaker assumption that W 6 Kd, where Kd denotes the Kato class in dimension d, with d > 1; see Definition 10.4.1. (This fact was not observed in [Lal2].) Indeed, it follows from Definition 10.4.1 and (the proof of) Proposition 10.4.4 that if for all m,Q< Vm_ < W, with W e Kd, then Vm_ e Kd and V m ,_ is H0-form bounded with uniform bound less than one (i.e. with uniform constants y < 1 and S > 0 in (CO). Hence condition (C\) of Theorem 11.5.7 still holds in this situation. (See the proof of Proposition 10.4.4 given in [AiSi] and sketched in [Sill, pp. 458—459]; alternatively, see the explicit estimates obtained in [Gull,2].) We now give examples of situations where Theorem 11.5.13 applies. Example 11.5.16 Let V be such that V+ e Llloc and V_ e (Lfoc)u (in view of (11.5.17), this is the case, for instance, if V_ e Lp + L°°), where p is given by (11.5.19). (More generally, by Remark 11.5.15(d) above, we could assume instead that V_ belongs to Kd.) Consider Vm obtained by truncation as follows:
Then Theorem 11.5.13 clearly applies to this setting. Example 11.5.17 Let d > 2. Consider the function p defined on (0, +00) by p(r) = P/2ra, where a, B are positive constants and a < 2. Define V, Vm, for m = 1,2, . . . , as follows. Assume that V+ e L^ and V-(x) < P(||x||), where |x| denotes the length of x e Rd. Set
Assume further that Vm+ —> V+ in L^oc. Let Vm := V m, + — V m ,-. By using the monotonicity of p, it is then easy to check that Theorem 11.5.13 applies (see Remark 11.5.15(a)); note that, for W(x) = p (||x||), we have W 6 L^ + L00 with p e (d/2,d/d) satisfying (11.5.19). We refer to Example 11.4.7 (as well as [ReSi2, pp. 161, 169-170] and [Far2, Example, pp. 96-97]) for a motivation of the present example. Note that one could not (at least in a straightforward way) make use of a monotone convergence theorem here. Remark 11.5.18 (a) By possibly passing to a subsequence, it is easy, in principle, to verify condition (V2) in Theorem 11.5.13. Indeed, recall thefollowing fact: For p e [1, +00),
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let {gm} be a sequence in Lp and g e Lp such that \\gm — g\\ip -> 0. Then there exists a subsequence {gmj} and a function h e Lp such that gmj —> g a.e. and \gmj \ < h for all j (see, e.g., [Bre2, Theorem IV.4, p. 58]). If, in addition, ||gm — g|| Lp tends to zero fast enough—in the sense that Em=1 ||8m — g\\L? < oo—then the subsequence {gmj} can be replaced by the entire sequence {gm} (see, e.g., [Des, Proposition 2.2, p. 94]). (b) An extension of Theorem 11.5.13 [Lal2, Theorem 4.1] (or rather, of the semigroup convergence (ll.S.lb) obtained in Theorem 11.5.1) was obtained by Voigt [Vo2] in the more general context of "absorption semigroups"; that is, of Schrodinger-type semigroups (generated by the infinitesimal generator of a positive (Co) semigroup [Na] perturbed by a suitable potential function) acting in L p (u), where 1 < p < oo and H is a measure on some measurable space (X, A). (See Theorem 3.5 and Corollary 3.6, pp. 126-127, as well as Example 4.3 in [Vo2].) The counterpart in [Vo2, Corollary 3.6] of condition (V\) and (V%) above is still assumed; in particular, in the analogue of (V2), different domination hypotheses are made on the positive and negative parts of the approximating potentials Vm (namely, "admissibility" and "regularity", in the sense of [Vo2]). The work of Voigt in [Vo2] was motivated by his earlier work [Vol] on "absorption semigroups", as well as by that of Lapidus in [Lal2] on "dominated-type convergence theorems ". We now conclude this section by applying the previous results to the modified Feynman integral f M (.). We briefly recall from Section 11.4 the definition of the latter. (We use more concise notation than in Definition 11.4.4 and the corresponding Theorem 11.4.2 or Corollary 11.4.5.) Let V, Vm, m = 1, 2, . . . , be as in Theorem 11.5.13 and let (p € L2 = L2(Rd). Then, for every t e R and a.e. v =: xn 6 Rd,
where the limits as n -> oo hold in L2. Here, Gn (XQ, . . . ,xn,t; V) denotes the integrand of (11.4.3); namely,
with Gn given by (11.4.4). In this manner, the solution at time t, e ~ i t H p , of the Schrodinger equation
with initial condition fy(., 0) = (p € D(H) c L2, is represented by a sequential limit of (Lebesgue-convergent) finite-dimensional integrals. Naturally, f'M(Vm) is defined in the same way, except for V replaced by Vm.
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We can now combine Theorem 11.5.13 and Corollary 11.5.9, as well as Theorem 11.4.2 (or Corollary 11.4.5), in order to obtain the desired stability result, and thereby complete the proof of Theorem 11.5.1 as well as of the more general theorem below. Theorem 11.5.19 (Dominated convergence theorem far the modified Feynman integral) If "Vm converges to V dominatedly"(i.e. in the sense of Theorem 11.5.13 or, more generally, of Remarks 11.5.15(a)) and 11.5,15(d)), then the "modified Feynman integrals" FM(V), F'M(Vm) associated with V, Vm are well-defined and for all p € L2 and all t e R:
where the above limits hold in the norm of L2 = Z,2 (Rd) and are uniform in t on compact subsets 0f R. Here, Gn is given as in (11.5.25). Exercise 11.5.20 Show that if pm —> p in L2, then
and so formula (11.5.26) holds, except for p replaced by pm on the right-hand side of the five equalities. Remark 11.5.21 We close this section by recalling from Remark 11.5.10(b) that we cannot deduce from Theorem 11.5.13 an entirely analogous dominated convergence theorem for Nelson's definition of the Feynman integral f T P ( . ) (Section 11.2, especially Definition 11.2.21 and Corollary 11.2.22) since we do not have at our disposal a sufficiently general existence theorem for this definition. (See Remark 11.5.10(b) and Problem 11.3.9 for a more thorough discussion of this point.) However, we note that the present dominated-type convergence theorem (Theorem 11.5.1 or 11.5.19 above, from [Lal 2])—along with the Trotter product formula for unitary groups (Corollary 11.1.6)— will be used in Section 13.4 to deduce a weaker result for ft-p(.); see Corollary 13.4.6 (from [JoKim]), 11.6 The modified Feynman integral for complex potentials In this section, we present results of Bivar-Weinholtz and Lapidus [BivLa] extending the earlier results of Lapidus [Lall] discussed in Sections 11.3 and 11.4 above. More precisely, we obtain an abstract formula for the imaginary resolvents of normal (rather
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than self-adjoint) operators. We then deduce from it the convergence of the modified Feynman integral with a highly singular complex (rather than real) potential V. This is of current interest in the study of the Schrodinger equation associated with dissipative (or "open") quantum systems [Ex]. Specifically, the assumptions on the real part of the potential, Re V, are exactly the same as in Section 11.4. Furthermore, we assume for the imaginary part that Im V e L11oc(Rd) and Im V < 0 (a mathematical condition akin to "dissipativity of the energy")- Quantum-mechanically, this will imply for the nonisolated system a "loss ofunitarity" and hence a "loss of conservation of probability" (compare with the discussion in Section 6.1). (As in Section 11.4, singular magnetic vector potentials are also allowed.) Product formula for imaginary resolvents of normal operators Let T be a (not necessarily bounded) normal operator in a (complex) Hilbert space H: that is, T is a densely defined closed operator on U such that T*T - TT*, where T* denotes the (Hilbert) adjoint of T, as given by Definition 9.6.1. (See [Rul, p. 348].) Then we can write
with T1, T2 self-adjoint operators on H. Here, T1, TI are respectively the real and minus the imaginary parts of T; they are defined by means of the operational calculus given by the spectral theorem for normal operators, with the functions Re z and Im z defined on the spectrum a(T) c C. (We note that so far, the spectral theorem and the associated functional calculus have been discussed only for (unbounded) self-adjoint operators; see Section 10.1. However, they can be extended from self-adjoint to normal operators; see, e.g., pages 348-355 in Rudin's book [Rul]. We will use this fact throughout this section.) Since D(T) = D(T*), it easily follows from Definition 9.6.1 that
If, in addition, we suppose the operators T1, T2 to be nonnegative, then —iT is seen to be m-dissipative (or equivalently, by Remark 9.4.6(b), iT is m-accretive) and, hence, by the Lumer-Phillips theorem (Theorem 9.4.7), it generates a (Co) contraction semigroup {e~l'T}t>o on H. In view of [Paz, Corollary 4.4, p. 15], this follows from the obvious fact that both — iT and — (iT)* are dissipative (see Remark 9.4.6(a)). (We refer to Section 9.4 for the basic facts concerning dissipative operators.) We shall denote by
the form domain of T; of course, | T |1/2 is also defined by means of the spectral theorem. If now A = A] — i A2, B = B\ — iB2 are normal (possibly unbounded) operators on H with Aj, Bj nonnegative self-adjoint (j = I, 2) and
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we can define the form sum A + B of A and B as the operator in H associated with the sesquilinear form
for u, v e Q(s) := Q. Naturally, the quadratic form associated with s is no longer semibounded (since, for instance, s(u) = s(u, u) may not be real); nevertheless, (part of) the first representation theorem (Theorem 10.3.14) can be adapted to the present situation, as we now briefly explain. By definition, u e D(A + B) if and only if the map v -> s(u, v) is continuous on Q c H with respect to the norm topology on H. Then, (A + B)u is the unique vector v in H (given by the Riesz representation theorem) such that
By (11.6.1) and (11.6.2), — i(A + B) is obviously dissipative; further, one can easily show by using the Lax-Milgram lemma (e.g., [Bre2, Corollary V.8, p. 84]) that it is in fact m-dissipative. We denote by {e~"^A+B)}t>o the (Co) semigroup which it generates. The same conclusions still hold if one replaces the hypothesis B\ nonnegative by a weaker assumption on its negative part B_. (Henceforth, we write B1 = B+ — B-, where B+, B- are respectively the positive and negative parts of the self-adjoint operator B1, given by the spectral theorem.) Namely, we shall assume, by analogy with Theorem 11.3.1, that B- is relatively form bounded with respect to the nonnegative operator A1 with relative bound less than 1; i.e. the counterpart of (11.3.1) holds, with A replaced by A1. In this case, in the definition (11.6.1) of the sesquilinear form 5, one has of course to replace the term ( B 1 / 2 u , B1 1/2 v) by (B+1/2u, 5+1/2u) - (Bl_/2u, Bl_/2v). We can now state the following abstract result, which extends Theorem 11.3.1 from self-adjoint to normal operators. (In summary, in this extended product formula, we assume suitable sign conditions on the real and imaginary parts of the normal operators, as well as relative form-boundedness of the negative real part of one of the operators with respect to the real part of the other.) Theorem 11.6.1 (Productformula for imaginary resolvents of normal operators) Under the above hypotheses,
for all u € H, uniformly in t on bounded subsets 0f [0, +00). Proof The proof parallels that of Theorem 11.3.1 and we shall only indicate the changes to be made in the argument. Fix A > 0, v e W; for t > 0, set
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and
We will show that wt -> [A. 4- iC]~lv as f -> 0+, where C := A + B. (This will establish the counterpart of Step 1 in the proof of Theorem 11.3.1.) We have wt & D(A) C Q(A) C Q(B-), and, by definition,
By using the spectral theorem for normal operators [Rul, pp. 348-355], we obtain
where /?g,, IB>, IB+ and IB- are nonnegative bounded self-adjoint operators given by
and
(Note that B\ and 82 commute since they are functions of the same operator B.) Taking the inner product of (11.6.3) against wt, we then easily check that
hence, also ||A2/2iu»||, ||/?^2if/|| and \\Ilg,2w,\\ are bounded independently of t. This, along with the observation (already made in equation (11.3.15), but now a consequence of the spectral theorem for normal rather than self-adjoint operators) that
allows us to apply the counterpart of (11.3.1) (with A replaced by AI) to deducemuch as was done following (11.3.15)—that ||7B4.w,|| and ||A,' w<|| are also bounded r independently of t. We can then pass to the limit, weakly in H, along a sequence tn -> 0; now, wt tends to a certain w, and to identify the limits of the other bounded sequences, we can use the spectral and Lebesgue theorems We first obtain, in particular, that w e Q(A) n Q(B) = Q. Note also that by the m-dissipativity of — i B , we have
so that, for z = x + iy e a(B), we obtain
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Hence, in fact, for every fixed e € Q(B),
from which we easily deduce now that, along [tn],
Next, let y e Q = Q(A) D Q(B); the inner product of (11.6.3) with y gives, upon passing to the limit in the second member,
So that, by definition of C = A + B, w e D(C) and
It follows that Wt tends weakly to w = (A + i C ) - 1 v . Norm convergence can be deduced much as in the lines following (11.3.19). This completes the proof of the analogue of Step 1 in the proof of Theorem 11.3.1. The rest of the proof can be carried out just as in Step 2 of the proof of Theorem 11.3.1, and so we omit it. D Remark 11.6.2 (a) A corollary analogous to Corollary 11.3.7—in which the form domain Q = Q(C) is not necessarily densely defined in H—is also available. Then the limit in Theorem 11.6.1 only holds for all u e H := Q. (b) In contrast to what happens in Theorem 11.3.1, the limiting (one-parameter) semigroup in Theorem 11.6.1, [e~"c — e~"^A+B^}t>o, is not a group in general and the operators e~ltc are (usually) not unitary for t > 0. (c) Even in the case of nonegative self-adjoint operators [Lai,2, La6], let alone in the case studied in [Lall] (Section 11.3 above) or the present situation of normal operators, it is not known whether one can replace the "imaginary resolvents " by the corresponding semigroups in the statement of Theorem 11.6.1; see Problem 11.3.9. Application to dissipative quantum systems We specialize now to ~H = L2(£2), where £2 is a nonempty open subset of R^with d > 1. Let A = HO = — ^ A be the free Hamiltonian acting in L 2 (£2), where A denotes the Dirichlet Laplacian on S2. (When £2 — Rd, HQ is the usual free Hamiltonian on K d .) Further, let B be the multiplication operator by a complex-valued function V = q+ — <7_ — iq', where q+,q_, q' are real nonnegative functions in £^(£2) and q- is relatively Ho-form bounded with bound less than 1. In this case, the form sum
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A + B extends, hence coincides with, the natural realization H of — ^ A + V(x), with domain
(Here, as before, HQ(&) denotes the Sobolev space defined as the closure of £>(£2) with respect to the Hilbert norm ||«||#i(Q) := (\\u\\2 + IIVM\\ 2 1 ) l / 2 \ see, e.g., [Kes].) In fact, H -\r X is known to be m-accretive (or equivalently, —(H + A.) is known to be m-dissipative) for some A. > 0 (see, e.g., [BreKat]). In the case when fi = Rd, with some supplementary assumptions on q-, one can replace "M e H^(QY'by"u e L 2 (£2)" in the definition of the domain (see [BreKat]). (The hypothesis q' > 0 is not required for the results of Brezis and Kato [BreKat] to hold and a somewhat stronger hypothesis is assumed on q-, but then it is the closure of the operator H + A. that is known to be m-accretive. In the present case when q' has a sign, however, even with the stated hypothesis on q-, the methods of [BreKat] easily adapt to show that H + A. itself is m-accretive with D(H) C D(A + B).) We can then apply Theorem 11.6.1 to A and B, which gives us a representation of the solutions of the Schrodinger equation
corresponding to the "energy operator" H (also called the "Schrodinger operator" in [BreKat] or the "pseudo-Hamiltonian" in [Ex]). [We refer to Exner's book [Ex] for a description of the properties and physical relevance of such operators to the study of "open" (that is, nonisolated or dissipative) quantum systems.] When £2 = E.d, this gives us an explicit representation of e~ltH as a limit of iterated integral operators that generalizes the modified Feynman integral from [Lai,2, La6-13] (introduced for real potentials in Section 11.4 above) to the case of a complex potential with a highly singular (merely L,1^) negative imaginary part. We summarize the above discussion (when Q, = R rf ) in the following theorem (from [BivLa]). Theorem 11.6.3 (Modified Feynman integral with highly singular complex potential) Let V : Rd —> C be such that V = q+ — q- — iq', where q+, q- and q' are nonnegative functions in L^. (Rd). Assume further that q- is Ho-form bounded with bound less than 1. Then the analogue of Theorem 11.4.2 holds (for all t > 0 instead of for all t 6 K). Hence the "modified Feynman integral" J-'M(V) (given as in Definition 11.4.4 except for V complex-valued and t > 0) exists for all t > 0, where the limit in (11.4.12') is, for all
where H is the "Schrodinger operator" (or "pseudo-Hamiltonian") associated with the complex potential V, as defined above. Finally, for
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unique solution (in the semigroup sense) at time t of the Schrodinger equation (11.6.6) with initial state ^r(0, •) =
where [bjk}dsk=\ is a symmetric, uniformly elliptic matrix of real-valued bounded measurable functions on £l and a = (a\, . . . , a^) is a d-dimensional vector of L^oc real-valued functions on Q (see Example 11.4.12). More precisely, L is the nonnegative self-adjoint operator associated with the closure of the minimal energy form corresponding to (11.6.8) (i.e. the closure h of the form defined on T>(&) = Cgg(ft)). The form domain of L is and L is the maximal restriction, as an operator on H, := L2(£i), of the operator
We refer to the work of Bivar-Weinholtz [Bivl-3] for the definition of such operators and for the properties that enable us to apply Theorem 11.6.1 in this case.
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Recall from Example 11.4.12 that when £2 = E.d and {bjk} = ^ I, then L = - ~ (V ia) ; further, if, in addition, V is real-valued, then H = — j(V — ia)2+V, the form sum of L and V, as in Example 11.4.12. 2
Remark 11.6.5 It is natural to wonder whether the stability results from [Lal2] discussed in the previous section can be extended to complex potentials. (See the query in [BivLa, Remark 2, p. 455].) Recently, such a "dominated-type convergence theorem" extending part of [Lai2] and [Vo2] (along the lines of Remark 11.5.18(b)) has been obtained in [LisM]. Combined with Theorem 11.6.1 (or Theorem 11.6.3) from [BivLa], and (an easy generalization of the proof of) Corollary 11.5.9, it would provide an analogue of Theorems 11.5.13 and 11.5.19 above (from [Lal2]) to the present context of ("absorption ") semigroups generated by Schrodinger operators with singular C-valued scalar potentials (and magnetic vector potentials). 11.7 Appendix: Extended Vitali's theorem with application to unitary groups Extension of Vitali's theorem for sequences of analytic functions The following theorem [La6, Theorem, p. 1739] extends Vitali's classical theorem [HilPh, Theorem 3.14.1, p. 104] and is abstracted from ideas of Babbitt [Babl-3] further developed by Feldman [Pel, p. 261] in the context of the analytic (in time) Feynman integral (Sections 13.1 and 13.2), and later used by Chernoff [Cher2, Remark, pp. 9091] and then the second author [La6, §3] in the context of product formulas for unitary groups of operators. (See the latter part of this appendix.) We will also use Theorem 11.7.1 in Chapter 16 (Proposition 16.2.4) when discussing a result from [JoLal]. The classical theorem of Vitali (part (i) of Theorem 11.7.1) will be used in several places in this book, especially in Chapter 13 and Chapters 16-17. [In all these cases, we will use the more general situation where the functions /„ are operator-valued (rather than C-valued) analytic functions, as discussed in Remark 11.7.2(c).] We will follow here the exposition given in the appendix, pages 1738-1741, of [La6]. Theorem 11.7.1 (i) (Vitali's classical theorem) Let D be a domain ofC, with closure D and boundary F. Let {/n}^l0 ^e a sequence of analytic functions on D, continuous on D, and uniformly bounded on D (say, \fn(z)\ < M for all n > 0 and all z in D). Assume that limn_,.oo fn (z) exists for all z in a subset of D having a limit point in D. Then limn_.oo fn (z) exists for all z in D, uniformly on compact subsets of D. Further, the limit, denoted by f ( z ) , is analytic in D. (ii) (Extended Vitali theorem) Suppose, for example, that D = C+ = {z € C : Rez > 0), the open right half plane, and so D = C+ = {z e C : Rez > 0} and F = {z 6 C : Rez = 0), the imaginary axis equipped with its natural Lebesgue measure. Assume, in addition to the hypotheses of part (i), that the limit f extends continuously to D. Then
for all p e L1 (F).
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Proof (i) This is just Vitali's theorem [HilPh, Theorem 3.14.1, p. 104], the proof of which can be found in [HilPh]. (Actually, in part (i), the functions fn need not be defined on the boundary F; see Remark 11.7.2(b) below.) (ii) It now remains to establish the extended Vitali theorem. For notational simplicity, we may assume (by means of a conformal equivalence) that D is the open unit disk, and hence F is the unit circle equipped with its natural (Haar) measure. The proof may be summarized as follows. Formula (11.7.1) holds for ip = Pr, the Poisson kernel of parameter r (with 0 < r < 1). It is then shown to hold for all
Given tp e L 1 ( F ) , we set
It will be convenient to use the notation p(9) or p(e i 0 ) interchangeably. The following properties will be useful: (a)||Pr||Li = l. (b)||P r *<0-«0|| L i ( r ) ->0,asr-> 1, p e L 1 (T). (c) f(Pr * p)(eW(9)d9 = fp(0)(P r * rlf)(6)de, In the proof of (c), the point is that Pr is even:
If g is a continuous function on D that is analytic in D, then it is the Poisson integral of its restriction to the boundary F (see, e.g., [Ru2, Chapter 1]):
Thus we can write for any
where
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and
where we have used property (c) in the last equality. Clearly,
Hence, in view of property (b), we can choose r close enough to 1 so that An
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Analytic continuation and product formula for unitary groups Our aim here is to present some results from [La6, §3]—extending results of Chernoffin [Cher2] and completing our discussion at the end of Section 11.3—which point out the limitations of the analogy between the "real" case, corresponding to self-adjoint semigroups, and the "purely imaginary" case, corresponding to unitary groups. (In physics, these would be called the "imaginary time" case and the "real time" case, respectively.) By analytic continuation (Theorem 11.7.1 (i)), we draw some preliminary information from results of Kato [Kat5], who has obtained a product formula for an arbitrary pair of self-adjoint contraction semigroups. (See Proposition 11.7.3.) We then deduce from the extended Vitali theorem (Theorem 11.7.1(ii)) a very weak form of the product formula for unitary groups in this context. (See Proposition 11.7.4.) Let H be a complex Hilbert space. Recall that if A is a nonnegative self-adjoint operator in H, the semigroup [e~zA : z e C+} is analytic for Rez > 0 and strongly continuous for Rez > 0. (See, for example, [Kat8, §IX.1.6] and [Gol, HilPh, ReSi2, Yo]. This fact can be proved much like Theorem 13.3.1) Here, e~zA is defined by means of the functional calculus (Theorem 10.1.11) since the function e~zx is bounded for x in a (A) c [0,+00). The next proposition ([La6, Proposition 3.1, p. 1706]) supplements [Cher2, Theorem 7.2, p. 84]. Proposition 11.7.3 Let A, B be arbitrary nonnegative self-adjoint operators on H. Assume that
exists for all t e R. Then Q(A)n Q(B) (= D ( A 1 / 2 r ) D ( B 1 / 2 ) ) is dense in H and U(t) coincides with the unitary group generated by A-}-B, the form sum of A and B; namely, U(t) = e-"
where fl denotes the orthogonal projection of ft onto Ti.\ := Q(A) n Q(B). Now, the (L(H)-valued) function Un(z) := ( e - (z/n)A e -W B ) n is analytic for Rez > 0 and strongly continuous for Rez > 0. Moreover, the sequence {t/n}£L0 is uniformly bounded for Re z > 0 (since || Un (z) || < 1 for Re z > 0). By applying Vitali's
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classical theorem (Theorem 11.7.1(1) and Remark 11.7.2(c)) as well as the principle of analytic continuation (on the right-hand side), we deduce from (11.7.4) that
The functions U(z) and e zC are analytic for Rez > 0 and bounded, strongly continuous for Re z > 0. Since, in addition, they coincide on the boundary, Re z = 0, they must also coincide in the entire right half-plane Re z > 0. For z — t (with t > 0), this fact combined with (11.7.5) (or (11.7.4)) yields
In particular,
Thus Q(A) n Q(B) is dense in H, A + B is a self-adjoint operator on U (= tC
U\)
t(A+B)
and e~ — e~ , for all t > 0. These two semigroups must then have the same infinitesimal generators. Whence C = A + B, as required. D As we pointed out at the end of Section 11.3, the above proposition shows that the difficulty with Problem 11.3.9 (concerning a general product formula for unitary groups) is no longer to identify the limit in (11.3.34), but rather to justify its existence. Finally, we close this appendix by providing some additional information regarding (part (ii) of) Problem 11.3.9. (See [La6, Proposition 3.2, p. 1708] which extends the corresponding result in [Cher2, p. 90].) Proposition 11.7.4 Let A, B be arbitrary nonnegative self-adjoint operators in H. Let n be the orthogonal projection of H onto its closed subspace H1 := Q(A) n Q(B). Then
uniformly on compact subsets of the open right half-plane Re z > 0. Moreover, we have
for all p e L l (R). Of course, if Q(A + B) — Q(A) n Q(B) is dense in H., then we may omit n on the right-hand side of formulas (11.7.7) and (11.7.8). Proof In view of (Remark 11.7.2(c) and) the main result in [Kat5] recalled in (11.7.4) above, (11.7.7) follows from the classical Vitali theorem (Theorem 11.7.1(i)), while (11.7.8) follows from the extended Vitali theorem (Theorem 11.7.l(ii)). n
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Aside from the very weak type of convergence obtained in (11.7.8) above, it appears to be difficult to obtain more precise information in this way about Problem 11.3.9. In conclusion, it does not seem possible to deduce directly by analytic continuation the existence of
from that of
where t > 0. We will see, however, in Chapter 13 as well as in Chapters 15-18, that various kinds of analytic continuation procedures are used in defining (and then studying) certain approaches to the Feynman integral. Our final comment points out another difference between self-adjoint semigroups and unitary groups in this context. In hindsight, it is quite natural since, in concrete settings, they correspond to very different physical situations (such as the diffusion case and the quantum-mechanical case, respectively). Remark 11.7.5 Note that we assume throughout that the operators A and B are both nonnegative. This corresponds to the assumptions of Problem 11.3.9(ii) (or Corollaries 11.3.5 and 11.3.7(ii)) rather than of Problem 113.9(1) (or Theorem 11.3.1 and Corollary 11.3.7(i)), where B- is allowed to be unbounded. Indeed, if we assume instead that B = B+ — B- is unbounded from below (or equivalently, that its negative part B- is unbounded), then the product formula for self-adjoint semigroups given in (11.7.4) no longer makes sense. This is so because then, for each t > 0, the operator e-tB = e -tB+ e' B - is unbounded (as can be seen, for example, by applying the multiplication operator form of the spectral theorem (Theorem 10.1.8) in conjunction with Exercise 10.1.2(b)). Thus, obviously, in this situation, we cannot deduce (after analytic continuation) results about a (possible) product formula for unitary groups from a corresponding formula for self-adjoint semigroups.
12
THE FEYNMAN-KAC FORMULA Be this as it may, the formal analogy between [formula (7.4.2) above] and integrals appearing in Wiener's theory is striking and since Wiener's theory is rigorously founded Feynman's heuristic connection between the Schrodinger equation and the path integral can be made into an unassailable theorem [namely, the Feynman-Kac formula]. Mark Kac, 1980 [Kac4, p. 51] In its various guises the F-K [i.e., Feynman-Kac] formula is ubiquitous throughout much of quantum physics on the one hand and probability theory on the other. It is probably safe to say that I am better and more widely known for being the K in the F-K formula than for anything else I have done during my scientific career.... The differential equation my formula solves is nineteenth-century, while the Wiener integral is a creation of the twentieth. It turned out that many properties of the solution which were difficult and even opaque from the nineteenth-century point of view became simple and transparent when looked upon from the point of view of the twentieth. The formula and its many close and distant relatives have proven to be useful and sometimes powerful tools in a variety of problems. I got tremendous mileage out of them and so did many others. Hardly a month passes without someone discovering yet another application. Mark Kac, 1984 [Kac5, pp. 115-116] It is not true that these problems all require functional integration for their solution (although, at the present moment, some of them have only been solved with such methods), but they all share the property of being problems with "obvious'-' answers and with elegant, conceptually "simple" solutions in terms of the tools we shall develop here. Once the reader has understood these methods and solutions, he will probably have little trouble giving a "word-by-word translation" into a solution that never makes mention of functional integration but rather exploits the Trotter product formula and the fact that e'A is an integral operator with a positive kernel. That is, there is a sense, somewhat analogous to the sense in which the Riemann integral is a systematized limit of sums, in which the Feynman-Kac formula is a systematic expression of the Trotter product formula and positivity of ' A . In part, the point of functional integration is a less cumbersome notation, but there is a larger point: like any other successful language, its existence tends to lead us to different and very special ways of thinking. Barry Simon, 1979 [Si9, p. 4]
The Feynman-Kac formula, which expresses the heat semigroup e~t(H°+ V) in terms of a Wiener integral, is one of the most influential results in the theory of path integration. Kac's formula does not deal directly with quantum mechanics although it is related to that subject and indeed was inspired by Feynman's path "integral". These points were discussed in some detail in Section 7.6, where an informal introduction to the
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Feynman-Kac Formula via the Trotter product formula as well as a brief account of the beginnings of the subject were also provided. Applications of Feynman-Kac formulas in various settings have been given in a broad range of areas such as, for example, diffusion equations, the spectral theory of Schrodinger operators, quantum mechanics, statistical physics and quantum field theory [BraRo, Car, Frei, GliJa, ReSi2, Si9, Va]. In addition to these references, Kac's original papers [Kacl,2] and his later expository writings, notably his paper in the Wiener Memorial Volume [Kac3] and his 1980 Fermi lectures Integration in Function Spaces and Some of its Applications [Kac4], make very interesting reading. 12.1 The Feynman-Kac formula, the heat equation and the Wiener integral We first state the hypotheses under which we will establish the Feynman-Kac formula. Let V : W* —> M be a measurable function, with positive part V+ = max(V, 0) and negative part V_ = max(— V, 0), and let //o = —5 A. We will assume that V+ e Lioc(Rd) and that V_ is Ho-form bounded with bound less than 1 (see Definition 10.3.17); that is,
and there exist constants 0 < y < 1 and S > 0 such that
[In this chapter, as before, we often abbreviate /Rrf \q>(y)\^dy by f |^|2; we also write ||V>|| 2 for the square of the magnitude of the (distributional) gradient of
with domain Q(q) := Q(H0) f~l Q(V+), is semibounded and closed. Consequently, by Theorem 10.4.8 (or 10.3.19), there exists a unique semibounded self-adjoint operator, denoted by H := Ho + V (and called the form sum of HQ and V), associated with this quadratic form. In the mathematical physics literature dealing with quantum theory, H = HO + V is often referred to as the Hamiltonian with potential (or interaction term) V. As we will see below, in the Feynman-Kac formula (12.1.4), the contribution of the free Hamiltonian HO will be absorbed in the term involving Wiener measure m. Of course, if V belongs to L°°(Rd), then we simply have H = H0 + V, the ordinary sum of the operators HO and V, with domain D(H) = D(Ho); further, H is essentially self-adjoint on V(Rd) = C^(Rd) in that case.
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We saw in Section 10.4 (Proposition 10.4.4) that a (nonnegative) function V_ which is in any of the classes
satisfies the condition (12.1.1) required in Theorem 12.1.1 below; in (12.1.3), we assume that p > d/2 if d > 2 or p = 1 if d - 1. (Also see Remark 10.4.9(a).) In that case, H is a natural realization of -± A + V(£) in L2(Rd). (Recall that the definition of the Kato class Kd was given in Definition 10.4.1.) We are now prepared to state the main result of this chapter, the Feynman-Kac formula. This formula, under the conditions given in Theorem 12.1.1 below, is stated and proved in Simon's book [Si9], Functional Integration and Quantum Physics. (See [Si9, Theorem 6.2, p. 51].) Our proof will fjllow the outline in [Si9] but is much more detailed. The result that we establish is slightly different than the theorem in [Si9], which will be given later in Corollary 12.3.3. Theorem 12.1.1 (The Feynman-Kac formula) Let V : Rd -+ M be such that V+ e L11oc(]Rd) and V- is Ho-form bounded with bound less than 1. Then the heat semigroup e~'H is given for every t > 0 by the formula
where iff e L2(l&d), £ 6 R rf , C'0 is the space of"K." -valued continuous functions x on [0, t] such that x(Q) =0 (i.e. Wiener space) and m is the associated Wiener measure (as defined in Chapter 3). The equality (12.1.4) is in the sense of L?-functions; in particular, for each 1/r 6 L 2 (R d ), the two sides o/(12.1.4) are equal for Leb.-a.e. % € Ed. Once Theorem 12.1.1 is established, it follows immediately from Theorem 9.1.4 (connecting a semigroup of operators to the evolution equation associated with its infinitesimal generator) that the solution to the heat equation associated with the Hamiltonian H — — j A - j - V = HQ + V can be represented via the Feynman-Kac formula (12.1.4), as desired. We formally state this key result as a corollary of Theorem 12.1.1. Corollary 12.1.2 (Heat equation and the Feynman-Kac formula) The solution
of the heat (or diffusion) equation
is given at time t > 0 and (a.e.) point t- € M.d by the Wiener integral on the right-hand side of '(12.1.4).
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Remark 12.1.3 (a) In probabilistic notation, (12.1.4) becomes
where E (resp., E%) denotes expectation with respect to Wiener paths starting at 0 (resp., £) at time t = 0. (b) The reader will note that the assumptions on the potential V = V+ — V_ in Theorem 12.1.1 (or Corollary 12.1.2) are the same as in Theorem 11.4.2 (or Corollary 11.4.5) which guarantees the existence of the modified Feynman integral T1M (V) in this setting. Recall, however, that Theorem 11.4.2—which follows from a product formula for imaginary resolvents (Theorem 11.3.1)—expresses as a limit of suitable finite-dimensional integrals the solution to the Schrodinger (rather than the heat) equation associated with the quantum-mechanical Hamiltonian H = Ho + V. (That is, in physicists" terminology, it deals with the "real time" case rather than the "imaginary time" case.) (c) The hypothesis V+ e L$oc(R.d) can be weakened to V+ € L/^K^G), where G is a closed subset of Rd of Lebesgue measure 0; indeed, it is not hard to show that q is still densely defined in that case, with D(Rd \ G) dense in L2(Rd) and satisfying D(Rd \ G) C Q(Ho)r\Q(V+) = Q(q). Actually, onecanallow V+ to be any (nonnegative) measurable function on Rd, provided that the semigroup e~tH is regarded as acting in the Hilbert space Q(q), the closure of Q(q) in L2(Rd). (We note that a direct probabilistic proof of the Feynman-Kac formula for arbitrary nonnegative, measurable potentials was given by McKean in [McK].) (d) Under the general hypotheses of the theorem, it is not true that the integral f0 V(x(s) + $-)ds exists for all % and x. However, it follows from Theorem 12.1.1 that for Leb.-a.e. f, this integral is defined for m-a.e. x. (See Corollary 12.3.1 below for a more precise statement.) Now we turn to a measure-theoretic lemma which will be useful in the proof of the Feynman-Kac formula as well as later on in the book. Let E : [0, t] x C'0 -> Rd be the evaluation map
Lemma 12.1.4 Let N be a subset of Rd of Lebesgue measure 0. Then for m-a.e. x € C'0, we have Leb.({s e [0, t] : x(s) € N}) = 0. In other words, m-almost surely, the "occupancy time" of the Wiener path x in N is zero. Proof Since every set of Lebesgue measure 0 is contained in a Borel set of Lebesgue measure 0 (as follows from [Roy, Proposition 12.7, p. 294]), it suffices to establish this result for a Lebesgue null Borel set N. By Exercise 3.2.6, E is continuous and so E ~ 1 ( N ) is a Borel subset of [0, t] x CQ. Now if XF denotes the characteristic function of the set F,
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By (12.1.7) and the Fubini theorem,
Hence, for m-a.e. x in C'0, f0' X E ~ I ( N ) ( S ^ X ^ S ~ 0- Thus, for m-a.e. x in C'0, 0 = XE~I(N)(S> x) = X{se.[Q,t\.x(s)eN}(s) for Leb.-a.e. s e [0, t]; that is, for m-a.e. x in C'Q, Leb.fs e [0, t] : x(s) e N] = 0 as we wished to show. . a Corollary 12.1.5 Let N e B(Rd) be such that N has Lebesgue measure 0. Then for every £ e IRrf it is the case that for m-a.e. x e CQ, Leb.({s € [0, f] : *(s) +1 € N}) = 0. d
Proof Fix f e M . Then A? - £ e S(lRrf) and Leb.(N - ?) = 0. Now simply apply Lemma 12.1.4 with Af replaced by N — t-. D 12.2
Proof of the Feynman-Kac formula
We will prove Theorem 12.1.1 in four steps: First, for V continuous (with compact support) in Step 1, and then V bounded in Step 2; second, for a general potential satisfying the hypotheses of Theorem 12.1.1, in Steps 3 and 4. Bounded potentials Step I: V is a continuousfunction of compact support. (One could begin with an infinitely differentiable function of compact support, but the extra assumptions would not simplify the proof of this step.) Recall that the operator My of multiplication by V is a self-adjoint operator, and that the operator e~tMv = e~'v. Further recall that HQ is a nonnegative self-adjoint operator and that the contraction semigroup {e~'H° : t > 0} is given on L2(Rd) by (see formula (10.2.26) of Theorem 10.2.6)
Thus
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and so the nth Trotter product is given by
where in the last expression of (12.2.3) we take e = UQ and we identify (Rd)" with Rdn. Letting Vj = Uj + e, j — 1, . . . , n, we obtain for every integer n > 1:
where MO = 0 in (12.2.4). Now from (12.2.4) and Theorem 3.3.5 (a basic Wiener integration formula for finitely based functionals) we can write
We would now like to apply Theorem 11.1.4, the Trotter product formula, to ( e -(t/n)Hoe e- (t/n)V)n However, Theorem 11.1.4 is not quite directly applicable since (for s > 0) the operator e~sV need not be a contraction. But, by Proposition 10.1.1, the norm of this operator equals ||e~sV/(^ ||oo which is less than or equal to e~ s " v "°°. Thus the semigroup e~s(v+]l\v^ = e^s^v^aoe~sV is a contraction semigroup. Using this, the fact that the operators of multiplication by the constant function e~~ s N V|l °° commutes with every operator in L(L 2 (R d )), and (a simple case of) the Trotter product formula
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THE FEYNMAN-KAC FORMULA
(Theorem 11.1.4), we can write
for every i/f 6 L 2 (R d ), where the limit (as n -> oo) is in the norm on L 2 (R d ). Of course, it follows immediately from (12.2.6) that
with the limit holding in the norm || • \\2. [Note that the Trotter product formula (Theorem 11.1.4) obviously applies here. Indeed, let W := V + ||V||oo; then, since W is bounded and nonnegative, so is the multiplication operator by W. Hence the algebraic sum HQ + W (with domain D(Ho)) is itself self-adjoint and nonnegative. It follows that —(Ho + W) is w-dissipative (see also Remark 10.1.12(b)) and thus Theorem 11.1.4 can be applied to yield (12.2.6). Finally, observe that H0 + W = H0+ W = H0 + W = H + || V||oo(The above argument could be simplified somewhat by observing that if the FeynmanKac formula (12.1.4) holds for a given potential V, then it also holds for its translates V + c,for any constant c. Hence we could have assumed without loss of generality that V > 0 and deduced directly (12.2.7) from Theorem 11.1.4.).] It follows from the convergence just established that there exists a subsequence { n p } for which the convergence in (12.2.7) holds for Leb.-a.e. e e R.d. Since the paths x are continuous and since V is continuous under our present assumptions,
for all x e C'0 and f e Rd. (This is the Riemann sum approximation procedure alluded to in Section 7.6, just below Equation (7.6.8).) Thus, for every f e W1, we have as p —»• oo:
for m-a.e. x (specifically, for every x such that ty(x(t} + f) is defined). Now (12.2.7) and (12.2.9) will allow us to finish the proof of the Feynman-Kac formula under the assumptions of Step 1 by taking the limit on both sides of (12.2.5)
PROOF OF THE FEYNMAN-KAC FORMULA
279
(with n replaced by np) as p -> oo provided that we can justify an application of the dominated convergence theorem on the right-hand side. But
and by Theorem 3.3.5 again (or, more specifically, by formula (9.2.15) with f := \ifr\ e L2(Rd)),
Step 2: V e L°°(Rd). We will use Step 1 and a limiting argument to establish the Feynman-Kac formula for V & L°°(Rd). With this in mind, we want a sequence {Vn} from T>(Rd) such that | Vn (it) \ < || V || oo for every positive integer and every u € Rd, and, in addition, Vn(u) —> V(u) for Leb.-a.e. u e Rd. We indicate in some detail how such a sequence can be produced by means of an approximate identity although we will not give a complete proof. Take a function J e D(R d ) such that J > 0, J(0) > 0, J ( u ) = J ( v ) when ||u|| = ||v||, J ( u ) > J ( v ) for ||u|| < ||u||, and /R J(u)du = 1. (The graph of a 1-dimensional version of such a function /* appears in Figure 12.2.1. Let K(u) := y*(||w||2) and then take J = cK, with the constant c chosen so that fRd J(u)du = 1.) Now for n = 1 , 2 , . . . , let J\(u) = ndJ(nu), Note that J\_ e T>(Wi) and that « « fyi.d JL (")^M = 1 for n = 1, 2, — Next we let n
FIG. 12.2.1. The graph of the function J* (when d = \)
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THE FEYNMAN-KAC FORMULA
Clearly, for every positive integer n and every u e Md,
Also, Wn is infinitely differentiable ([H6r3, Theorem 4.1.1, p. 88] or [Stel, Theorem 2, part (b) (the p = oo case), pp. 62-64]) and Wn(u) -> V(u) for Leb.-a.e. u. It need not be the case, however, that Wn is compactly supported. In order to remedy this, we begin by taking 0 e D(R d ) such that 0 < 0(w) < 1 for all u, 0(0) = 1, and fRd 0(u)du = 1. Now let
It is easy to see that Vn has the pleasant properties possessed by Wn and, in addition, Vn is compactly supported. Summarizing, we now have a sequence {Vn}£lj from T>(Rd) such that |V n (w)| < || V||oo for every u e M.d and for every n, and, in addition, Vn(u) -» V(w) for Leb.-a.e. u eR d . LetHn = Ho + Vn,n = 1,2, We wish to apply Theorem 9.7.10, an earlier result which dealt with the continuous dependence of a semigroup on its generator. From the third paragraph preceding the statement of the present theorem, we know that each Hn as well as H — HO + V is self-adjoint on D(Ho) and is essentially self-adjoint on P(Md). Further, all of these operators are uniformly bounded below; specifically, we have in the sense of self-adjoint operators,
as we now explain. Since by construction of the function Vn, we have Vn(u) > — ||V||oc pointwise, we can write for any 4> £ D(Ho):
and similarly for H. Hence (12.2.14) holds. Now(12.2.14) [and the spectral theorem (Theorem 10.1.ll(ii) to be precise)] easily imply (see Exercise 10.1.13(b)) that
Inequality (12.2.15) shows that the "stability condition" involved in the hypotheses of Theorem 9.7.10 is satisfied in our present setting.
PROOF OF THE FEYNMAN-KAC FORMULA
281
In order to finish showing that Theorem 9.7.10 can be applied, it suffices to prove that for every <j> eV(Rd),
(Recall from Theorem 10.2.12 that Z>(Rd) is an operator core for H since H is essentially self-adjoint on P(Erf).) But \\(H0 + Vn)4> - (Ho + V)(p\\2 = \\Vn<j> - V<j>\\2, and so it suffices to show that, for > e P(R rf ),
In fact, we will show that (12.2.17) holds for any <j> e L 2 (E rf ). Accordingly,
where the second equality in (12.2.18) comes from the dominated convergence theorem (since \Vn(u) - V(w)| 2 |>(w)l 2 < 4||V|| 2 0 i^(M)| 2 € L 1 ^)) and the last equality from the fact that Vn(u) -> V(u) for Leb.-a.e. u e Rd. We now have the hypotheses of Theorem 9.7.10 satisfied and so, for every t/r e Lr 2fmd\ (M. ),
as n -> oo. [In fact, Theorem 9.7.10 assures us that the convergence in (12.2.19) is uniform for t in compact subsets of [0, +00). We do not need this uniformity at present but it is sometimes a useful fact.] Now let N be the Borel subset of W1 of Lebesgue measure 0 such that for u e N, Vn(u) -» V(u). By Corollary 12.1.5, for every £ e E.d it is the case that for m-a.e. x e CQ, x(s) +%eNforat most a subset of [0, t] of Lebesgue measure 0. Hence for a.e. s in [0, t],
Since \Vn(u)\ < ||V||oo for all n and all M, (12.2.20) and the bounded convergence theorem imply that for every £ e Rd it is the case that for m-a.e. x e CQ,
Thus for every E and m-a.e. x, we have
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THE FEYNMAN-KAC FORMULA
and so for Leb.-a.e. £ and m-a.e. x,
as n —> oo. Since the function x -> e t||V||00 |i/r(jt(0 + £)| is a dominating function for the left-hand side of (12.2.23) which is integrable with respect to m as we saw earlier in (12.2.11), we deduce from the dominated convergence theorem that for Leb.-a.e. f e Rd,
Now by (12.2.19), we know that we can pass to a subsequence {np} such that for Leb.-a.e. £,
Thus by combining (12.2.25), Step 1 of the proof, and (12.2.24), we can write
Hence the Feynman-Kac formula is established for the case involved in Step 2. Monotone convergence theorems for forms and integrals We next wish to consider the general case when V+ e L,1^ and V- is H0-form bounded with bound less than 1. In order to deal with this case, we provide some operator-theoretic background regarding monotone-type convergence theorems for quadratic forms. We restrict our attention to a Hilbert space H. (In Steps 3 and 4 below, H will be the space L 2 (R d ), equipped with its natural norm || • ||2 = || • IIz, 2 ^) (Rd) inner product (.,.) = (•, •)L 2 (Rd).) At this point, some readers may want to review the introductory material on unbounded quadratic forms provided in Section 10.3. Recall that we denote by Q(q1) the domain of the quadratic form q\ and write q\ (p) instead of q\ (p, p). Our treatment will follow [ReSil, Supplement to §VIII.7, pp. 372-377], where the proof of Theorems 12.2.2 and 12.2.4 below can be found. (Also see [Kat8, §VIII.3 and §VIII.4, pp. 459-462], as well as [Si5,6] and the references therein.) Definition 12.2.1 Let q\ and q2 be quadratic forms on a (complex) Hilbert space H. We say that q\
PROOF OF THE FEYNMAN-KAC FORMULA
283
A moment's reflection will convince the reader that the inclusion Gfe) =2 (q1) is natural in the previous definition. We can now state the first convergence theorem, which will be used in Step 4. Theorem 12.2.2 (Monotone convergence theorem for nondecreasing forms) Let q1, q2,... , q n , . . . , be a nondecreasing sequence of closed, nonnegative forms:
Let
Then, if 2(qoo) is dense in H, the quadratic form qoo with domain Q(qoo) and defined by
is closed, Moreover, if Tn (resp., T) is the nonnegative self-adjoint operator associated with the form qn (resp., qoo), then as n -> oo, [/ + T n ]-1 —> [I +
T]
in the strong operator topology (or strongly, in short); (12.2.29a)
that is, for all p e H,
To state the second result, we need some additional preliminaries. We say that the form q2 is closable if it admits a closed extension. In that case, q2 has a smallest (in the sense of Definition 12.2.1) closed extension q2, called the closure of q2. Proposition 12.2.3 Let q2 be a nonnegative quadratic form on H. Then q2 admits a largest closable form (q2)r, that is smaller than q2_. We are now prepared to state the second convergence result, which will be used in Step 3 below. (In that case, the limiting form qoo will be closed and hence we shall simply have qoo = (qoo)r = (qoo)r.)
Theorem 12.2.4 (Monotone convergence theorem for nonincreasing forms) Let q1, q2,..., qn,..., be a nonincreasing sequence of closed, nonnegative quadratic forms:
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THE FEYNMAN-KAC FORMULA
Let Q(qoo) = Uoo n=1 Q(qn) and consider the form qoo with domain Q(qoo) and defined by
Let Tn be the (nonnegative self-adjoint) operator associated with the form qn and let T be the (nonnegative self-adjoint) operator associated with the closure of (q00)r (where (qoo)r is defined in Proposition 12.2.3). Then as n —> oo,
Remark 12.2.5 (a) In the literature, one often refers to the convergence in (12.2.29) and (12.2.31) as "strong resolvent convergence" of Tn to T. (b) To avoid unnecessary complications, we have assumed implicitly that the quadratic forms involved are densely defined. Finally, we shall need the following simple extension of the ordinary monotone convergence theorem for integrals; it is proved much like the standard theorem. (See, for example, Exercise 6, page 89 and the remark at the bottom of page 92 in [Roy].) Theorem 12.2.6 (Generalized monotone convergence theorem for integrals) Let (X, A, u) be a measure space with u a positive measure. Let {fn} be a sequence of measurable real-valued functions that is bounded below (resp., above) by an integrable function g. Suppose further that fn —> f u-a.e.as n -» oo and that for each n, fn < f (resp., fn > /) u,-a.e. Then
Remark 12.2.7 Note that the sequence {fn} in Theorem 12.2.6 is not assumed to be monotonic. When we apply Theorem 12.2.6 in Steps 3 and 4 below, however, we shall havefn| f (resp., f n f f). Unbounded potentials Let V satisfy the hypotheses of Theorem 12.1.1; hence V+ e L,1loc(Rd) and V- is Ho-form bounded with bound less than 1. Given positive integers n and m, we define d)
bu Vn,m ee LLoo(R (K ) by
\7
We will now complete the proof of Theorem 12.1.1 in two additional steps. In Step 3, we will keep m fixed and let n -» oo (after having applied Step 2 to the bounded potential V n,m ). Finally, in Step 4, we will let m —> oo and conclude that the Feynman-Kac formula (12.1.4) holds for the potential V.
PROOF OF THE FEYNMAN-KAC FORMULA
285
Step 3: Define the quadratic form qn,m by
Note that qn,m > 0. Indeed, since V n , m — V n , m + — Vn,m, - > — V-, where V n,m, ± denotes the positive (negative) part of Vn m, we have for
where we have used hypothesis (12.1.1b) in the second inequality. Clearly, the form qn,m is closed as the sum of a closed form (
we have
in the sense of quadratic forms (see Definition 12.2.1). [Note that Q(qn,m) — Q(Ho) for all n, so that the inclusion between domains required by Definition 12.2.1 is trivially satisfied.] Let Tn,m be the nonnegative self-adjoint operator associated with the closed, nonnegative form qn,m (see Theorem 10.3.15). Clearly, Tn,m = HO + (8 + Vn<m), with domain D(Tn,m) = D(//o); in other words, Tn,m is simply the algebraic sum of HO and of the bounded operator 8 + Vn^m. Let q00,m be the quadratic form defined in Theorem 12.2.4; that is,
We now show that
where
Indeed, since Vn,m ], Vm as n -> oc, (12.2.35) follows from (12.2.34) and (12.2.33) by use of Theorem 12.2.6, the generalized monotone convergence theorem for integrals.
286
THE FEYNMAN-KAC FORMULA
More precisely, fix
Let € L°°(Rd) and t > 0. Since F n,m = HO + (S + Vn,m), with Vw,m e L°°, we may apply Step 2, the Feynman-Kac formula for bounded potentials, to deduce that for Leb.-a.e. $ € Rd,
Next we turn to an examination, under the assumption of Step 3, of the Wiener integral side of the Feynman-Kac formula (12.2.38). Since V n,m Vm Leb.-a.e. as n -> oo, —Vn,m t ~Vm Leb.-a.e. as n -» oo. Hence by Corollary 12.1.5, we deduce as in Step 2 that for every £ and m-a.e. x
for Leb.-a.e. .s e [0, t]. Thus by a second application of Theorem 12.2.6,
for every £ and m-a.e. jc. (Note that —m < — V l , m (x(s)+f) and Q |— m\ds — mt < oo; so that, in view of (12.2.39), Theorem 12.2.6 can be applied to yield (12.2.40).) It follows immediately from (12.2.40) that for every f and m-a.e. x,
PROOF OF THE FEYNMAN-KAC FORMULA
287
Hence for <£ e L2(Wt) with tj> > 0, we have, for Leb.-a.e. £,
(To deduce (12.2.42) from (12.2.41), it suffices to apply the ordinary monotone convergence theorem since all the functions involved are nonnegative.) Now, by (12.2.37), we can pass to a subsequence {np} such that for Leb.-a.e. £,
Then by combining (12.2.42), the Feynman-Kac formula (12.2.38) obtained in Step 2, and (12.2.43), we see that for Leb.-a.e. f,
We next make some comments which will serve to both help clarify equation (12.2.44) and be useful to us in Step 4 (as well as to establish Corollary 12.3.1 below). The last expression in (12.2.44) is in L2(Ed) as a function of £ and so the same is true of the first expression. Hence for Leb.-a.e. £, the integral (in the first expression) over Ct exists. Also, the integrand (which is > 0 here since <j> > 0) must be finite for m-a.e. x. Thus (taking the case (p > 0), we see that
Note that the symbol /0'(<5 + V m (jf(.s)+ £))$• is defined since by (12.2.36), Vm < m and so /Q'(S + Vm,+(x(s) + $))ds <St + mt
We now complete the proof of Step 3. For an arbitrary (complex-valued) ^ e L2 (E.d), we can write V = (
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THE FEYNMAN-KAC FORMULA
function in L2(Rrd); here, <j>\ — 02 is the decomposition of Re rjf into its positive and negative parts and cfe -$4 is the decomposition of Im i/s into its positive and negative parts. By (12.2.44), the Feynman-Kac formula holds for each of the nonnegative functions (j>j and so, by linearity, also holds for arbitrary iff € L 2 (R d ); that is for every m > 1 and ir e L2(Ed), we have for Leb.-a.e. £,
This concludes Step 3. Step 4: In this step, we let m -> oo and complete the proof of Theorem 12.1.1. Since clearly Vm f V by (12.2.36), it follows from (12.2.35) that
in the sense of quadratic forms. Let q00 be the limiting nonnegative form as in Theorem 12.2.2; that is,
In view of definitions (12.2.35) and (12.2.46a), it follows from the generalized monotone convergence theorem for integrals (Theorem 12.2.6) that for tp e Q(H0),
where the quadratic form q is defined by (12.1.2). The use of Theorem 12.2.6 can be justified as we now briefly explain. Since Vm, __ = V_, (<5-V_)|
Thus (12.2.46a) yields (12.2.47).
PROOF OF THE FEYNMAN-KAC FORMULA
289
if and only if
Therefore, (12.2.46b) yields
It thus follows from (12.2.47) and (12.2.48) that
Consequently, if T is the nonnegative self-adjoint operator associated with q00,
where H = H0 + V is the Hamiltonian appearing in the statement of this theorem. We can now apply the second part of Theorem 12.2.2, the monotone convergence theorem for a nondecreasing sequence of closed forms {qoo,m}oom=1, and conclude that [I + T m ]-1 -» [/ + 7T1 strongly as m -> oo. (Note that by (12.2.48), Q(qoo) is dense in L 2 (R d ) as required by the hypotheses of Theorem 12.2.2.) By Theorem 9.7.11 (or 8.1.12), the Trotter-Kato-Neveu-Chernoff theorem, we thus deduce (as in Step 3) that for all t > 0, e-tTm
->
e-tT
in the strong operator topology as m -> oo.
(12.2.49)
Fix <j) > 0, (j> e L2(Rd). By (12.2.49), there exists a subsequence {mp} such that for Leb.-a.e. f in Rd,
We next apply the Feynman-Kac formula (12.2.45) obtained in Step 3; namely, for each m > 1 and 0 e L 2 (R d ),
Assume for now that
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THE FEYNMAN-KAC FORMULA
Rd such that (12.2.50) holds (pointwise) and both sides of (12.2.51) exist and are finite, for all m > 1. Then clearly, R d \Z has Lebesgue measure 0. Fix f e Z. According to the comment following equation (12.2.44), we then know that for m-a.e. .tin Ct, the functions H» Vm(x(s)+%) is integrable on the interval [0, t]. Therefore, since — Vm —V, we may apply Theorem 12.2.6, the generalized monotone convergence theorem for integrals, to deduce that
Note that for m-a.e. x e Ct 0, the integral on the right-hand side of (12.2.52) is either —oo or finite and thus, its counterpart in (12.2.53) is either 0 or > 0, but in any case finite. (One minor point: In view of Corollary 12.1.5, we may assume that £ € Z implies that
[Observe that by our choice of Z, the integrand on the right-hand side of (12.2.51) is a (nonnegative) integrable function of x over Ct.] Thus by passing to the limit along the subsequence {mp} on the left-hand side of (12.2.51), we deduce from (12.2.50) that
as desired. By the linearity of both sides of (12.2.54), the Feynman-Kac formula (12.1.4) follows for any ^ e L 2 (R d ). This concludes the proof of Theorem 12.1.1. Exercise 12.2.8 Try to prove Theorem 12.1.1 by interchanging Steps 3 and 4; that is, by first letting m —» oo and then letting n —» oo. 12.3 Consequences We close this chapter by giving a few simple corollaries of the Feynman-Kac formula stated in Theorem 12.1.1 (or Corollary 12.1.2). In each of these corollaries, we will implicitly assume that the hypotheses of Theorem 12.1.1 are satisfied.
CONSEQUENCES
291
Corollary 12.3.1 For Leb.-a.e. % in Rd, the integral /0' V(x(s) + f )ds exists and is either finite or oo for m-a.e. x in Ct0; i.e.
In particular, if V is bounded above, then for Leb.-a.e. £ in Rd, the function s -» V(x(s) + £) is integrable on the interval [0, t] for m-a.e. x in C'0. Exercise 12.3.2 Prove Corollary 12.3.1. [Hint: The proof is in the spirit of the comment following formula (12.2.44) in Step 3, except that the potential need not be bounded above here.] The next corollary yields the version of the Feynman-Kac formula that is given in [Si9, Theorems 6.1 and 6.2, pp. 49 and 51]. Corollary 12.3.3 Let p, $ e L 2 (R d ). Then
where (•, •) denotes the inner product in L 2 (R d ). Proof By the Feynman-Kac formula (12.1.4),
then by the Fubini theorem, this last iterated integral is equal to the right-hand side of (12.3.2), as desired. We next briefly justify the above use of the Fubini theorem in (12.3.3). Assume both (p, $ > 0. Hence the first integral in (12.3.3) is finite and thus so is the iterated integral in (12.3.3). Therefore the Fubini-Tonelli theorem yields formula (12.3.2) in this case. For general complex-valued (p, fy in L 2 (R V ), we complete the argument by use of linearity. Remark 12.3.4 The measure Leb. x m on Rd x C'0 appearing in (12.3.2) is referred to by Simon [Si9, p. 38] as "Wiener measure". For us, Wiener measure is the probability measure m acting on C'0. (See Chapter 3.)
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THE FEYNMAN-KAC FORMULA
Corollary 12.3.5 The heat semigroup [e-tH = e - t ( H o + V ) ; t > 0} is positivity preserving; that is, for all t > 0, e-tH t(> > 0 Leb.-a.e. whenever
when
13 ANALYTIC-IN-TIME OR -MASS OPERATOR-VALUED FEYNMAN INTEGRALS 13.1 Introduction In this chapter, we define and study the existence of analytic-in-time or -mass operatorvalued Feynman integrals. Sections 13.2, 13.3 and 13.7 will deal with analytic continuation in time, whereas Sections 13.5 and 13.6 will be concerned with analytic continuation in mass. In 13.2, 13.3, 13.7 and much of 13.5, the analytic continuation will be from an appropriate Wiener integral (as is generally understood when "the" analytic Feynman integral is discussed). However, in the last part of 13.5 and in 13.6, the analytic continuation in mass will start from an operator-valued function of mass (as well as time) which is obtained from a product formula. The product formula is of the "Trotter type" in Section 13.5 but does not follow from the Trotter product formula in Section 11.1. In Section 13.6, the product formulas are of the "resolvent type", but again do not follow from the product formulas for the resolvents of self-adjoint or normal operators given in Sections 11.3 and 11.6, respectively. We will refer to the integrals obtained by analytic continuation in mass from appropriate product formulas as analytic-in-mass Feynman integrals. This seems to us to be reasonable terminology even though it has been restricted in the past to cases where the starting point was a true integral with respect to Wiener measure. Our notation in Section 13.6 and in the last part of Section 13.5 distinguishes between the different starting points. The former deals with the "analytic-in-mass modified Feynman integral", ^^(V), from [BivLa], while the latter deals with extensions .?Y'£ian(V), based on [Kat7, BivPi, Biv3], of Nelson's work [Nel] on the "analytic-in-mass operatorvalued Feynman integral", K^(F-iv)The realization that there is a connection between Feynman's path integral and the Wiener integral, a true Lebesgue integral with respect to the countably additive Wiener measure, is due to Mark Kac [Kacl,2]. Kac's insight not only produced the FeynmanKac formula which was discussed in a very general setting in the previous chapter, but has also suggested approaches to "the" Feynman integral via analytic continuation from the Wiener integral. These matters were discussed informally in Section 7.6 where some background material and related references were also provided. The discussion in Section 7.6 focused primarily on operator-valued analytic continuation in mass from the Wiener integral. This approach will play a prominent role in Section 13.5. It will also be used throughout Chapters 15-18, but the goal of those chapters is quite different from the goal of Section 13.5 and so requires different techniques and a more stringent definition of the analytic in mass operator-valued Feynman integral (compare Definition 15.2.1 with Definitions 13.5.1 and 13.5.1')-
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We will show in Theorem 13.3.1 that the analytic-in-time operator-valued Feynman integral Jit ( F v ) (see (13.2.1) and Definition 13.2.1) exists and agrees with the unitary group e-"(Ho + V) for all t in K whenever V = V+ — V_ satisfies the following two assumptions: (i) V+ 6 L,1loc(Rd). (In fact, it is sufficient that V+ e L/^OR^G), where G is a closed subset of Rd of Lebesgue measure 0.) (ii) V_ is H0-form bounded with relative form bound less than 1. We note that the existence of the modified Feynman integral and its agreement with the unitary group was established in Corollary 11.4.5 under precisely the same assumption on V as in (i) and (ii) above. This permits us to conclude easily (Corollary 13.4.1) that under these same very general assumptions, the modified Feynman integral, the analyticin-time operator-valued Feynman integral, and the unitary group associated with the standard Hamiltonian approach to quantum dynamics all exist and agree. Further, under the still quite general assumptions used in Theorem 11.2.19 to establish the existence of the Feynman integral defined via the Trotter product formula (namely, the assumptions that (iii) V+ 6 L2loc(Rd) and (iv) V- is H0-operator bounded with relative operator bound less than 1), we conclude (in Corollary 13.4.2) that all three of the approaches to the Feynman integral via the Trotter product formula, the modified Feynman integral, and the analytic-in-time operator-valued Feynman integral exist and agree with one another and with the unitary group. Still further, these consistency results can be combined with the "dominated-type" convergence theorem of Lapidus for the modified Feynman integral, Theorem 11.5.19, arid some additional considerations to establish dominated-type convergence theorems for both the analytic-in-time operator-valued Feynman integral and the Feynman integral defined via the Trotter product formula (see Corollaries 13.4.3 and 13.4.6, respectively). Section 13.4 is particularly informative even though the proofs found there are not especially difficult. Indeed, the combination of new results and the review of earlier results provides an answer to some of the common objections made to mathematical theories of "the" Feynman integral (see items 1-3 along with the related discussion in Section 1.1 of Chapter 1, the introduction to this book). Theorem 13.3.1—which establishes the existence of the analytic-in-time operatorvalued Feynman integral—rests on some key results discussed earlier in this book: Theorem 10.3.14, the first representation theorem for quadratic forms; Theorem 10.3.19, an abstract result which gives conditions under which the form sum of self-adjoint operators is self-adjoint; Theorem 10.4.8, which insures that HO + V is self-adjoint and bounded below under the conditions on V discussed earlier; and Theorem 12.1.1, the Feynman-Kac formula. With these results in hand, the proof of Theorem 13.3.1 is not difficult. It depends primarily on consequences of the spectral theorem. The 1963 papers of Babbitt [Bab2] and Feldman [Fel] seem to have been the first mathematically rigorous papers on the Feynman integral which used analytic continuation in time. (A 1960 paper of Cameron [Cal] used scalar-valued analytic continuation in mass.) Theorem 13.3.1 is due to Johnson [Jo6]. Before [Jo6], there seems to have been a rather widespread opinion among "Feynman integrators" that the class of potentials V for which the analytic in time Feynman integral exists is extremely limited. The work
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on [Jo6] started with the idea of understanding and explaining this opinion. Instead a positive result, Theorem 13.3.1, was proved (Theorem 6.1 of [Jo6]). The 1963 paper of Feldman [Fel] which was mentioned above dealt with very general nonnegative potentials. In retrospect, this paper should have had more influence than it did. However, [Fel] is quite technical and does not apply to negative potentials such as the attractive Coulomb potential, the most basic potential of atomic physics. A number of variations of "the" Feynman integral defined via operator-valued analytic continuation in mass are discussed in Sections 13.5 and 13.6. Before commenting on the individual results, we emphasize some features that these approaches have in common and mention a few differences. The reader can consult the sections themselves for further discussion of the differences. A major strength shared by all of these approaches is that the potentials are allowed to have arbitrarily strong singularities independent of sign. A serious weakness is that the existence of the strong operator limits depends on a nontangential approach to the imaginary axis and, even then, these limits are shown to exist only for Leb.-a.e. value on that axis. The proofs of the results (some of which are just outlined or remarked on, or even omitted) have major differences but also have some common elements. The concept of (Newtonian) capacity plays a key role in all of these approaches. Throughout Sections 13.5 and 13.6, with the exception of Haugsby's extension (Theorem 13.5.9) of Nelson's results, the arbitrary singularities can occur on a closed set of capacity 0. In the approach of Nelson [Nel], this set of capacity 0 is the only place where finite range singularities are allowed to occur. In contrast, in Section 13.6 and near the end of 13.5, other strong (but not arbitrary) finite range singularities are permitted. Product formulas play a role in each of these approaches. A link is made with the Wiener integral in [Ne 1 ] and [Hau]. This is used to give a rather easy proof of the product formula in the case of [Nel]. The "product formula" in [Hau] is not discussed below but we remark that it is harder to prove both because of more general hypotheses and because arbitrary partitions of the time interval are considered and the limit is taken as the norm of the partition goes to zero. The product formula (13.5.41) used in the last part of Section 13.5 is due to Kato [Kat7, p. 107]; its proof depends in part on the deep distributional inequality (11.2.4) which is also due to Kato. A product formula due to Bivar-Weinholtz and Lapidus [BivLa] (see (13.6.1)) is key to Section 13.6. It is used, in particular, to establish the existence of the "analytic-in-mass modified Feynman integral" under very general hypotheses. All of the approaches in Sections 13.5 and 13.6 make crucial use of operator-valued versions of three beautiful results from complex analysis: Vitali's convergence theorem (discussed earlier in Theorem 11.7.1 and reviewed in the proof of Theorem 13.5.6), Poisson's representation theorem and the Fatou-Privaloff (or edge of the wedge) theorem (both reviewed in the proof of Theorem 13.5.7). Another common thread that runs through Sections 13.5 and 13.6 (and contrasts with the approaches in Chapter 11 and in Sections 13.2-13.4 and 13.7) is that there is no guarantee that the evolving system is described by a unitary group. (It is
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always described by a contraction semigroup, but in general—for instance, for highly singular attractive central potentials—the time evolution is not reversible; see Example 13.6.18.) We turn next to a brief discussion of some of the results in 13.5 and 13.6. The earliest mathematically rigorous papers on "the" Feynman integral defined via analytic continuation in mass were those of Cameron [Cal, 1960] and Nelson [Nel, 1964]. It is the beautiful results of Nelson that will be treated most completely in Section 13.5. (See the beginning of Section 13.5 for additional comments on the early papers [Cal, Nel] and the even earlier 1956 paper [GelYag] of Gelfand and Yaglom.) The hypothesis on the potential V in Nelson's results, Theorems 13.5.4, 13.5.6 and 13.5.7, is easily described: There is a closed subset J of Rd of capacity 0 such that V is continuous and R-valued on Kd \ J. Note that there is no separate mention of the positive and negative parts of V as there was in the results on the Feynman integral in Chapter 11 and as there will be in Sections 13.3, 13.4 and 13.7 of this chapter. The hypothesis just stated above (which is the same as (13.5.5) below) permits V to have arbitrarily strong singularities at oo and on J but requires continuity and so permits no finite range singularities on the open set R d \J. Haugsby's extension [Hau], Theorem 13.5.9, of Nelson's results comes next in Section 13.5. Arbitrarily strong singularities of the potential V are still allowed much as in Nelson's theorems. Further, V is permitted to be complex-valued and time-dependent. In addition, Nelson's continuity assumption is considerably weakened (although not removed entirely). Theorem 13.5.9 is the only result in this book in which the potential can be time-dependent and also have strong spatial singularities. Haugsby's theorem does not extend all aspects of Nelson's work. For example, one cannot expect the semigroup property to be satisfied when V is time-dependent. Thus the information about the semigroups from Nelson's results does not carry over to Haugsby's setting. As part of his work on the analytic in mass Feynman integral in [Nel], Nelson established some interesting operator-theoretic results for the Schrodinger operator with complex potential. The Wiener integral was naturally involved in this work (see Definitions 13.5.1 and 13.5.1') and the Lebesgue integral with respect to Wiener measure was used in a crucial way to establish a Trotter-type product formula (see (13.5.11) below). Kato showed in [Kat7] that the operator-theoretic aspects of Nelson's work including the product formula can be obtained and significantly extended by means of operatortheoretic methods and without the use of the Wiener integral. It is worth noting that Kato precisely identified the domain of the generator of the semigroup which gives the evolution. Nelson's result contained only partial information in this direction; see Remark 13.5.5(a). (In fact, Nelson did not need to know the precise domain of the generator to accomplish his main purpose.) Starting from the Trotter-type product formula of Kato (13.5.41), one can analytically continue in mass much as Nelson did and obtain "the" operator-valued Feynman integral in Theorem 13.5.16 below. Kato did not do this although it certainly seems that he could have. (Of course, operator theory was his main area of interest and expertise. It is also possible that he did not feel entirely comfortable with the elusive Feynman integral.)
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What are the conditions on the potential V in Theorem 13.5.16 (the theorem which stems from [Kat7]) and how do they compare with the assumptions (13.5.5) in Nelson's results? Recall that Nelson assumed the existence of a closed subset J of Rd of capacity 0 such that V is continuous and R-valued on Wd\J. Theorem 13.5.16 allows V to be C-valued but assumes the dissipativity condition, Im V < 0. Also, much less than continuity is required of Von Rd\J, namely that V e L r l o c (R d \J) where r = 2d/(d+2) for d > 3, r > 1 for d = 2 and r = 1 for d = 1. Corollary 13.5.18 is a consistency result. It tells us that when the hypothesis (13.5.5) of Nelson's results are satisfied, then the assumptions of Theorem 13.5.16 are also satisfied and the two Feynman integrals agree. Moreover, this corollary takes advantage of both results by identifying precisely the domain of the generator and making the connection with the Wiener integral. Kato's results in [Kat7] have been extended in various ways by Brezis and Kato [BreKat], Bivar-Weinholtz [Bivl-3], Bivar-Weinholtz and Piraux [BivPi], and BivarWeinholtz and Lapidus [BivLa]. Theorem 13.5.22 summarizes consequences (of all but the last) of these papers for the analytic in mass operator-valued Feynman integral where the analytic continuation starts from a Trotter product type formula. Part (a) of Theorem 13.5.22 ([Biv3], generalizing [BivPi]) extends Theorem 13.5.16 in that the potential V is only required to be in L1 loc (R d \J) instead of in Lrloc(Rd\7), where J is again a closed subset of Rd of capacity 0. Both a scalar and a magnetic vector potential are considered in part (b) of Theorem 13.5.22. (However, from a mathematical point of view, the assumptions made on the magnetic potential are not as natural as those made in Section 13.6.) The analytic-in-mass modified Feynman integral is considered in Section 13.6. The existence of such integrals is established in Theorem 13.6.4 and relies on a product formula for resolvents given in Theorem 13.6.1. The product formula is itself a special case of a more general result, Theorem 13.6.7. These results on the analytic in mass Feynman integral are based on work of Bivar-Weinholtz and Lapidus in [BivLa, §3]. The potential V is assumed to be in L1 loc (R d \J), where J is a closed subset of Rd of capacity 0. Also, V is allowed to be C-valued provided the dissipativity condition Im V < 0 is satisfied. Further, the Schrodinger operator may involve a magnetic vector potential a whose components are required to be in L2loc(Rd\J). The proof of Theorem 13.6.7, on which the results just mentioned depend, involves techniques from the theory of elliptic partial differential equations which are not used elsewhere in this book. It also relies on a suitable version and extension of Kato's distributional inequality (see Lemma 13.6.6). Theorems 13.6.10 and 13.6.11 are comparison results for analytic-in-mass, operatorvalued Feynman integrals which are in some respects analogous to the earlier comparison results, Corollaries 13.4.2 and 13.4.1. These theorems stress the unity of these analyticin-mass Feynman integrals within the intersection of their domains of validity. We conclude Section 13.6 by considering highly singular central potentials with the emphasis on potentials of the form ^j-, where r is the distance from the origin in R3; see Example 13.6.13. Such potentials are used, for instance, as phenomenological models for molecular interactions and aspects of quantum field theory. The attractive
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inverse-square potentials ^f- are studied in Example 13.6.18 and have some particularly interesting features. In Section 13.7, we will discuss recent results of Albeverio, Johnson and Ma [AUoMa] which extend substantially the already very general work in Section 13.3 on the analytic-in-time operator-valued Feynman integral. We have not provided in this book the necessary prerequisites for a detailed and mathematically rigorous treatment of [ AUoMa] and so we will settle in Section 13.7 for a discussion intended to give the reader some understanding of the results. The missing prerequisites include smooth measures /A on Rd (see Definition 13.7.8) and the associated positive continuous additive functionals Aul of Brownian motion (see Definition 13.7.11) as well as operators Hu := H o + u , where the usual potential V can be replaced by the measure u (see 3' and 4' in Section 13.7). These ingredients come together in a Feynman-Kac formula
which extends the Feynman-Kac formula (12.1.4) from Chapter 12. The measure u. plays the role of a "potential" but u need not be absolutely continuous with respect to Lebesgue measure in Rd. As a consequence, there need not be a potential V in the usual sense of the word. Area measure on the surface of a sphere in R3 is an example which is of physical interest and satisfies the assumptions of the results in Section 13.7 but not the assumptions of Theorem 13.3.1. The results in Section 13.7 provide additional information about the analytic in time operator-valued Feynman integral even when the "potentials" u are absolutely continuous withrespect to Lebesgue measure in Rd;i.e., u ( d x ) — Vu(x)dx as in Section 13.3. For instance, the positive potential V = Vu in Example 13.7.20 is nowhere locally integrable. This implies that the form domain of HO + V does not contain any nonzero continuous function. In spite of this, Ho + V is densely defined on L2(Rd) and J i t ( F y ) agrees with the unitary group e-it (H0 + V) for every t in R. The reader should compare the results in Section 13.7 with the analytic continuation in mass results in Sections 13.5 and 13.6 where the limits as the imaginary axis is approached are taken nontangentially and still may only exist for Leb.-a.e. value on the imaginary axis. On the other hand, it is only in Sections 13.5 and 13.6 that arbitrarily strong singularities are permitted which are independent of sign. In all of the other approaches to the Feynman integral in Chapters 11 and 13, the assumptions on V_ are more restrictive than those on V+. (It may be useful here to recall that physically, V_ and V+ represent, respectively, the attractive and repulsive part of the potential V.) Section 13.7 is the only place in Chapters 11 and 13 where potentials V = V+ — Vare allowed to be replaced by certain measures u = u+ — u-. However, it seems likely that this could be done for the modified Feynman integral as well (with the aid of the results of Section 11.4 and of parts of Section 13.7, see Remark 13.7.24(b)). 13.2 The analytic-in-time operator-valued Feynman integral Recall that Ct = C([0, t], Rd) denotes the space of continuous Revalued functions on [0, t]. Let V : Rd -> R be Lebesgue measurable. (In Sections 13.5 and 13.6, the
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299
"potential" V will be allowed to be C-valued.) Our primary interest in this section (as in the following one) is in functions F = FV on C' which are of the form
(Without further assumptions on V, the subset of C' on which FV is actually denned may be very small.) However, as we will see further on, there are reasons to be interested in functions which are not necessarily of the form (13.2.1), and so we will give the definition of the analytic in time operator-valued Feynman integral without the assumption that F has the special form above. [Recall from Section 10.2 that C+ (resp., C+ ) denotes the set of complex numbers z with Re z > 0 (resp., Re z > 0).]
The operator-valued function space integral J t ( F ) exists if and only if (13.2.2) defines J t ( F ) as an element of L(L 2 (R d )). If J t ( F ) exists for every t > 0 and, in addition, has an extension (necessarily unique) as a function of t to an operator-valued analytic function on C+ and a strongly continuous function on C+, we say that J t ( F ) exists for all t E C+. When t is purely imaginary, Jt (F) is called the analytic in time operator-valued Feynman integral. We now make several remarks concerning the definition just given. The comments in (a) below are most important since they describe our present concerns more fully and indicate how they are related to the last chapter and to previous operator-theoretic matters. Remark 13.2.2 (a) Note that when F — FV is given by (13.2.1), the integrand on the right-hand side of (13.2.2) has exactly the same form as in the Feynman—Kac formula, Theorem 12.1.1. Hence, if the potential V satisfies the hypotheses of Theorem 12.1.1, the Wiener integral in (13.2.2) is equal for Leb.-a.e. E to the operator e-tH from the heat semigroup { e - t H : t > 0} acting on the function W and evaluated at E. Thus, J t ( F v ) is certainly defined for all t > 0 under these circumstances. Using the comments just made as a starting point, we will show that J t ( F y ) exists for every t E C+ and equals, for t purely imaginary (the Feynman integral case), the unitary group given by the usual operator-theoretic approach to dynamics. Hence, writing the parameter in the purely imaginary case as t — ito, to € R, we will have J i t o ( F ) — e~it0H or, with a more complete notation,
(b) We caution the reader that what is seen as imaginary time above when we start with the heat semigroup, the Wiener integral and t > 0 but end up in the quantum setting with t = —ito, to E R, is seen as real time by the quantum physicists. Indeed, from their perspective, it is the methods that use the heat semigroup and the Wiener integral to study quantum problems that are referred to as "imaginary time techniques".
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(c) Definition 13.2.1 involves the concept of analyticity of an L(L 2 (Rd))-valued function. But what type of analyticity is meant? Fortunately, as was briefly discussed in Appendix 11.7, the three concepts of analyticity with respect to the operator norm, strong operator or weak operator topologies on L(L 2 (Rd)) are all equivalent [HilPh, §3.70, esp. Theorem 3.10.1, p. 93] and so any of these can be used. (See Remark 11.7.2(c) above.) (d) As indicated earlier, our main interest in Sections 13.2-13.4 and in 13.7 is in functions of the form (13.2.1) when V : Rd —> R. Potentials V which are C-valued and/or time-dependent are of physical interest (see [Ex], for example) and will be discussed further on, primarily in connection with the analytic-in-mass operator-valued Feynman integral. (See especially Section 13.5 and Chapters 15-18. See also Section 11.6 and Section 13.6 where highly singular complex potentials are discussed in connection with the modified Feynman integral.) We note that Definition 13.2.1 above does make sense for such functions (as well as for certain functions which are very different from the "Feynman-Kac functional" in (13.2.1)). (e) Definition 13.2.1 was phrased in such a way that it has an all or nothing character off the positive real axis. (It is in fact very analogous to the definition of the analytic-inmass operator-valued Feynman integral adopted in [JoLal, Definition 0.1, p. 10] and to be discussed later on in the book; see Definition 15.2.1.) This fits the theorem that we are working towards since the existence of J t ( F ) will be established in this strong sense. However, one may phrase the definition so that, for example, the question of the existence of strong limits as the imaginary axis is approached is examined at each point. (f) There are many possible variations in the definition of "the" analytic Feynman integral. We have already mentioned that we will consider the analytic-in-mass operatorvalued Feynman integral later on. We gave a limited discussion earlier in Section 4.5 of a scalar-valued analytic Feynman integral (see Definition 4.5.1). Even within the operator-valued approach, the limits as the imaginary axis is approached can be taken in the weak operator topology on L(L 2 (R d )) rather than in the strong operator topology; further, the analytic continuation can be taken through some subset of C which is more restrictive than C+. 13.3 Proof of existence We are now ready to state and prove the theorem which insures the existence of the analytic-in-time operator-valued Feynman integral for a large class of potentials V. We will also see that this Feynman integral agrees for the same class of potentials with the unitary group which specifies the dynamics in the standard operator-theoretic approach to quantum mechanics. Theorem 13.3.1 Let V : Rd -> R be such that V+ E Llloc(Rd) and V- is H0-form bounded with bound less than 1. Also, for any t > 0, let Fv : Ct —>• R be given by (13.2.1). Then J t ( F v ) (as in Definition 13.2.1) exists for every t E C+ and is analytic in C+. For t > 0, J t ( F v ) is given by the right-hand side of (12.1.4), the Feynman-Kac formula. For all t E C+,
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where e-t(H0+V) is given meaning via the spectral theorem applied to the self-adjoint operator H = H0 + V. In particular, the analytic-in-time operator-valued Feynman integral J i t o ( F v ) exists for every to E R and we have
where {e-itoH : t0 E R} is the group of unitary operators given by the usual operatortheoretic approach to quantum dynamics. Finally, for every y E D(Ho + V), W(to, E) := ( J i t o ( F v ) ( p ) ( % ) is the unique solution (in the semigroup sense as discussed in Chapter 9) of the Schrodinger equation
with initial state y. Remark 13.3.2 We remind the reader that we have previously discussed in Section 10.4 conditions which insure that the function V- satisfies the hypothesis imposed on it above. The least restrictive of these sufficient conditions is that V- belongs to the Kato class Kd (see Definition 10.4.1). This particular class of functions is mentioned here for the purpose of a later comparison of Theorem 13.7.16 with the theorem immediately above. We further recall that under the assumptions of Theorem 13.3.1, the operator H0 + V is self-adjoint and semibounded below. We forego any further review since the salient facts were discussed briefly at the beginning of Section 12.1. Proof of Theorem 13.3.1 In light of Definition 13.2.1 and the Feynman-Kac formula, Theorem 12.1.1, the key is to show that the operator-valued function T(t) := e-tH is defined and strongly continuous for all t E C+. (See Remark 13.2.2(a).) It will follow immediately that J t ( F v ) exists for all t E C+ and (13.3.1) holds. Restricting / to the imaginary axis, say t — ito, to 6 R, we then have (13.3.2); so that J i t o ( F V ) = e-itoH is the unitary group from the usual Hamiltonian approach to quantum dynamics. From this it follows that W(to, E) := (•/""(FV)<")(£) is, for every (y E D(H), the unique solution (in the semigroup sense) of the Schrodinger equation (13.3.3) with initial state (p. Establishing that T(t) = e-tH is strongly continuous in C+ and analytic in C+ is simply a matter of combining some consequences of the spectral theorem with standard arguments as we will now show. Since H is bounded below, —E:= inf o (H) is greater than —oo. Because of this, it is easy to check that the function/, : a(H) —»• Cdefinedby f t ( u ) :— e-tu is a bounded function of u for any t E C+. Using the functional calculus which arises from the spectral theorem, Theorem 10.1.1 l(ii), we see that T(t) = e~t H is defined and belongs to L ( L 2 ( R d ) ) for all t E C+. Next, let {tn} be a sequence in C+ such that tn —> t. Then ftn(u) -> f t ( u ) for all u E a ( H ) . Further, the sequence {11ftn1100}is bounded, where 11ftn11 oo = sup{|ftn(u)| : u E a(H)}. Using part (v) of Theorem 10.1.11, we have that ftn(H) -> f t ( H ) strongly; that is, T(tn) = e~tnH —> e~t H = T(t) in the strong operator topology as n - oo. It remains only to show that T(t) = e - t H is analytic in C+. In light of Remark 13.2.2(c), it is enough to show that(e~tH W, p) is analytic in C+ for every W, y E L 2 (R d ).
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In fact, by polarization, it suffices to show that ( c ~ t H p, p) is analytic for every p E L - ( R d ) . For the rest of the argument, we fix p E L 2 ( R d ) ; we may as well even assume that \ \ p \ \ 2 — 1. and we do so since it is mildly convenient. Let P be the projection-valued measure associated with the sell-adjoint operator H. See Definition 10.1.5, Theorem 10.1.16 and the associated discussion for the information about P which we w i l l need. The measure up.p defined for any Borel subset B of R by u p . p ( B ) : = ( p . P ( B ) p ) i s a probability measure such that u p . p ( a ( H ) ) = I . Further, by the projection-valued measure form of the spectral theorem (Theorem 10.1.16).
We know from an earlier part of the proof that e-tH is strongly continuous on C + . Certainly, then ( e ~ t H p.p) is continuous on C_|.. From Morera's theorem of complex analysis [Ru2, Theorem 10.17, p. 208], the desired analyticity w i l l follow if we can show that
for any triangle F in C+. We now give the argument which establishes (13.3.5) and then comment on the steps:
The first equality in (13.3.6) follows from (13.3.4): the third equality comes from the Cauchy integral theorem since the scalar-valued function e-tH is certainly an analytic function of t for t E C+ . The second equality is a consequence of the Fubini theorem. In order to see that the Fubini theorem is applicable, observe that the now fixed triangular contour F is bounded and is bounded away from the left half-plane. Hence, there is a bound for e=tu, where t runs over P and u runs over [ —E. oo). Since up.p is a probability measure and the integration over the contour is, alter parameterization. integration with respect to a measure of f i n i t e total variation, the use of the Fubini theorem is justified and the proof is now complete. D Remark 13.3.3 (a) Recall from Remark 12.1.3(c) that the hypothesis V+ E L l l o c ( R d ) can he weakened to V+ E L l l o c ( R d \ G ) , where C is a closed subset of Rd of Lebesgue measure 0. The same change can he made in the theorem which we just finished proving (ax was done in /Jo6/). Further, since extremelv singular potentials are of interest in quantum physics, this change has some significance. While C is a small set in terms of d-dimensional Lehesgne measure, it could he, for example, a (d — 1)-diniensional subspace of Rd , a rather large set on which to permit arbitrary singularities. (This comment
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303
also applies lo the modified Feynman integral studied in Sections 11.4-11.6 us well as to the Feynmun-Kac formula discussed in Chapter 12.) (b) We have proved directly that under the assumptions of Theorem 13.3.1, the semigroup {e~t H : t E C+ } is a hounded holomorphic semigroup (of angle n/2), and hence is analytic in C+, the open right-half plane. We could have used instead the general theory of holomorphic semigroups to establish this fact. /See / KalH, §IX.1.6] or [ReSi2, pp. 248-2571 ax well as jGol, HilPh, Yo] for the theory of holomorphic (or analytic) semigroups. I However, the more general results do not by themselves yield strong continuity on the imaginarv axis, the Fevmnan case. 13.4
The Feynman integrals compared with one another and with the unitary group. Application to stability theorems
This section interrelates and expands upon the most central results in this book regarding existence and stability theorems for "the" Feynman integral. More specifically, we are now in a position to show that the approaches to the Fey nrnan integral studied so far in this hook (see Corollary 1 1.2.22 of Theorem 1 1.2.19, Corollary I 1.4.5 of Theorem 1 1.4.2, and Theorem 13.3.1) are closely related to one another and lo the unitary group from Ihe standard approach to quantum mechanics. Further, these relationships along with Lapidus' dominated convergence theorem for the modified Feynman integral, Theorem 1 1.5.19, allow us to establish closely related convergence (or stability) results for the analytic-in-time operator-valued Feynman integral and for the Feynman integral defined via the Trotter product formula. The reader may have noted that the assumptions on V in Theorem 13.3.1 coincide exactly with the assumptions in Theorem 11.4.2 (or Corollary 1 1.4.5) which guaranteed the existence of the modified Feynman integral discussed in Section I 1.4. These two theorems tell us that the modified Feynman integral and the analytic-in-time operatorvalued Feynman integral both agree with the unitary group under the same assumptions. It follows immediately that these two versions of the Feynman integral must agree with each other. We state this formally as a corollary of Theorems 13.3.1 and 1 1.4.2. Corollary 13.4.1 Let V : Rd -> R he such that V+ E Llloc(Rd) and V- is Ho-form hounded with hound less than I. Also, for any t > 0, let Fv : Ct —> R he given by (13.2.1). Then the unitary group e ~ i t ( H o + V). the modified Feynman integral F l M ( V ) and the analytic-in-time operator-valued Feynman integral .1" ( F v ) all exist for every t E R, and we have
We can give a somewhat less general related result which also involves the Feynman integral defined via the Trotter product formula (TPF), obtained in Theorem 11.2.19 and Corollary 1 1.2.22. The conditions will be such that all three versions of the Feynman integral exist and agree with the unitary group. Corollary 13.4.2 Lei V : Rd -> R he such that V, E L2loc(Rd) and V- is H0-operalor hounded with hound less than I.
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Then (H0 + V )iD is essentially self-adjoint on D = D ( R d ) . Denoting the closure (H0, + V )ID by H. we have that the unitary group e itH, the modified Feynman integral .F1M(V). the Feynman integral F 1 T P , ( V ) via TPF, and the analytic-in-time operatorvaliied Feynman integral Jit ( F v ) all exist for every t E R, and
Proof Under the hypotheses of" Corollary 13.4.2. V_ is M0-form bounded with hound less than 1 (see Proposition 1 1.2.10(1)); further, by Proposition 1 1 . 2 . 1 0 ( i i ) . we also have H = HO + V, the form sum of M) and V. Therefore. Corollary 13.4.2 follows from Corollary 1 1.2.22 and Corollary 13.4.1. Recall from Remark 11.4.3(a) that (by Propositions 10.4.4 and 10.4.5(b)) V satisfies the condition of Corollary 13.4.1 if it belongs to any of the elasses
where p = 1 for d — I and p > d/2 for d > 2. Also, if V. belongs to either of the classes L p ( R d ) + L~ (Rd) cLoloc( R d ) u . then we can allow p = d/2 when d > 3. Further recall that (by Theorem 11.2.1 I. Propositions 11.2.14 and 1 1.2.15(b)) V satisfies the condition of Corollary I 3.4.2 if it belongs to any of the classes
where p — 2 for d < 3 and p > d/2 for d > 4. (Here and in the following, the spaces Lploc (Rd)u and Lploc are given as in D e f i n i t i o n 10.4.2.) Once Theorem 13.3.1 was established (Theorem 6.1 of | Jo6|), it was observed (Corollary 6.2 of |Jo6|) that the connection between F 1 M ( V ) and Jit(Fv) (Corollary 1 3 . 4 . 1 1 and Lapidus' dominated convergence theorem. Theorem 1 1.5.19-, easily yield the following result. [ F r o m now on. as in Section 1 1 . 5 .Vm,+ (resp.. Vm, ) denotes the positive (resp., negative) part of V m . 1 Corollary 13.4.3 (Dominated convergence theorem for the analytic-in-time Feynman integral) Let V. Vm. m = 1,2 he Lebesgue measurable real-valued functions on Rd . Assume that "Vm converges to V dominatedly" in the following sense: (a) Vm -^ V Leb.-a.c. in Rd. (b) Vm.+ < U for some U E L l l o c ( R d ) , (c) Vm.- < W for some W E Lploc(Rd)u. where p — 1 if d = I and p is any number in the interval ( d / 2 . oo) for d > 2. Then Jit (Fv) and J i t ( F v m ) . m =
1.2
all exist and
in the strong operator topology, uniformly in t on compact subsets of R.
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Proof Assumptions (a) and (b) imply that all of V+ and V m, +, m = 1, 2 , . . . , are dominated by U. Also, (a) and (c) imply that all of V_ and Vm, _, m = 1, 2 , . . . , are dominated by W. Then Corollary 13.4.1 along with Propositions 10.4.4 and 10.4.5 imply that all of F t M ( V ) , J i t ( F v ) , F t M (V m ) and Jit(Vm), m = 1 , 2 , . . . exist, and we have the equalities Ft M (V) = J i t ( F v ) and Ft M (V m ) = Jit(Fvm), = 1, 2 , . . . . Further, by Theorem 11.5.19 (and Theorem 11.5.13), Ft M (V m ) -> Ft M (V) in the strong operator topology, uniformly in t on compact subsets of R. The assertion (13.4.4) now follows immediately. D
Remark 13.4.4 (a) Recall from Section 11.5 [La 12] that an entirely analogous dominated convergence theorem holds for the modified Feynman integral. (See, in particular, Theorems 11.5.19 and 11.5.13.) (b) After the publication of [Lal2], the second author realized that the class of functions LPloc(Rd);; could be replaced in his dominated convergence theorem by the larger class Kd (see Remark 11.5.15(d)). A corresponding improvement can be made in Corollary 13.4.3; the only adjustment in the proof above is that Proposition 10.4.5 is no longer needed. The final result in this section depends mainly on Lapidus' convergence theorem and on Corollary 13.4.2. We begin with a proposition which will help us with the proof and will also clarify some related issues. Proposition 13.4.5 and Corollary 13.4.6 can be found in [JoKim]. Items (i), (ii) and the first containment in (13.4.8) of the proposition are known. The second containment in (13.4.8) and (iv) may not be known. Proposition 13.4.5 Let d be any positive integer and let p and q be real numbers satisfying 1 < q < p < oo. Further, let W be any R-valued, Lebesgue measurable function on Rd. Then we have the following: (i) For any x E Rd and r > 0,
where Br(x) is the ball of radius r centered at x in Rd and F is the gamma function. It follows immediately that
(ii) Further,
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Proof (i) The Lp-norm on a probability space is an increasing function of p. Hence,
It follows that
Thus (13.4.5) follows since
(ii) Let W e L^CM^)?. By definition, this means that
Hence, by (13.4.5),
Thus W E Lqloc(Rd)u as claimed. (iii) The first containment in (13.4.8) has already been noted in (10.4.1). See also the paragraph immediately following Example 10.4.6 where it is noted that the first containment in (13.4.8) holds for all p E [1, oo). It is the second containment in (13.4.8) which mainly concerns us. Let W E Lploc(Rd)u.Then supxERd \\W\\iP(Br(x)) < °°- Also, since q < p,
Hence, under the present assumptions, the (adjusted) argument in (13.4.10) still shows that WELqloc3(Rd)u. (iv) The fact that the left-hand side of (13.4.9) is contained in the right-hand side follows immediately from the first containment in (13.4.8). Finally, let W E Lploc(Rd)u for some p > c > 1. Choose q such that p > q > c. By the second containment in (13.4.8), we see that W e Lqloc(Rd);r and so W E U p>c Lploc(Rd)u, as we wished to show. D
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Corollary 13.4.6 (Dominated convergence theorem for the Feynman integral defined via the Trotter product formula) Let V,Vm,m = 1,2,..., be Lebesgue measurable real-valued functions on Rd. Assume that "Vm converges to V dominatedly" in the following sense:
where p — 2 for d = 1, 2, 3 and p is any number in the interval (d/2, oo)for d > 4. Then F'TP(V) and Ft T p (V m ),m = 1 , 2 , . . . , all exist and
in the strong operator topology, uniformly in t on compact subsets of R. Proof The domination assumption (b) insures that each V m ,+ belongs to L2loc(Rd). Combining (b) and the pointwise convergence from (a), we see that V+ is also in Lloc (Rd). Combining (c) and (a), we see that each V m , -,m = 1,2,..., and V- are dominated by W and so all belong to Lploc(Rd)u, where the number p depends on the dimension d as indicated in (c) above. It then follows from Corollary 11.2.22 and Propositions 11.2.14 and 11.2.15 that the Feynman integrals Ft TP (V) and Trp(V m ), m = 1,2,..., all exist and agree with the unitary groups e~itH and e~it Hm , m — 1, 2 , . . . , respectively. We also recall for use below that H and Hm,m = 1, 2,..., are all the closures of the essentially self-adjoint operators (Ho + V)ID and (Ho + Vm)ID, m = 1 , 2 , . . . , respectively (see Theorem 11.2.11); i.e.
Next, we recall that L21oc(Rd) c L11OC(Rd). Hencb, our dominating function U satisfies the hypothesis (11.5.20) of (V2) of the dominated convergence theorem, Theorem 11.5.13, for the modified Feynman integral. We also claim that W satisfies the hypothesis (11.5.21) of (V2) of the same theorem. Comparing the hypothesis just referred to with our assumption (c), we see that it suffices to show that for d = 1, d = 2, d = 3 and d > 4, respectively, we have:
But Proposition 13.4.5(iii) tells us that (i) holds and that (ii) holds with, say, p = 7/4. The same part of Proposition 13.4.5 yields (iii) above where we can, for example, again take p = 7/4. Part (iv) of Proposition 13.4.5 yields (iv) above where c is taken as d/2. Thus W satisfies the desired hypothesis.
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Now by the second author's dominated convergence theorem (see Theorems 11.5.13 and 11.5.19), it follows that F t M ( V m ) ->• Ft M (V); in fact,
in the strong operator topology as m ->• oo, uniformly in t on compact subsets of R. But since (Ho + V)ID is essentially self-adjoint, we know by Proposition 11.2.10(ii) that H := (Ho + V)ID (see (13.4.12)) agrees with the form sum HO + V. In the same way, Hm := (Ho + V m )ID = H0 + Vm, for m = 1, 2 , . . . . Hence, it follows that
in the strong operator topology as m —> oo, uniformly in; on compact subsets of R. But by Corollary 13A.2, F t T p ( V ) = F t M (V) and F t T p ( V m ) = F t M ( V m ) , and so by (13.4.14), we have
in the strong operator topology as m ->• oo, uniformly in t on compact subsets of R, as we wished to show. Q Remark 13.4.7 (a) Since L P ( R d ) + L°°(Rd) c Lploc(Rd)u for 1 < p < oo (see Proposition 10.4.5(b) and the paragraph following Example 10.4.6), we see that W E LP(Rd) + Loo (Rd) implies that the assumption on W in (c) of Corollary 13.4.6 holds. Here, p depends on the dimension d exactly as in (c) of Corollary 13.4.6. (b) As noted in Remark 13.4.4 above, the second author's dominated convergence theorem [Lal2] and so Corollary 13.4.3 can be strengthened by replacing the class Lploc(Rd)u by the larger class Kd. A related improvement in Corollary 13.4.6 can be made by replacingLploc(Rd)u by Sd n Kd (see Definition 11.2.13 and recall from Propositions 10.4.5(b) and 11.2.15(b) that LPloc(Rd)u c Sd n Kd for appropriately chosen p.) One might guess that Sd is a subset of Kd so that Kd could be replaced by Sd rather than Sd n Kd. However, Example 11.2.16 shows that Sd is not in general a subset of Kd. 13.5 The analytic-in-mass operator-valued Feynman integral The earliest mathematical papers on "the" Feynman integral defined this elusive "integral" via the analytic continuation in mass of a Wiener integral. The paper of Gelfand and Yaglom [GelYag] was published in Russian in 1956 and an English translation appeared in 1960. Their paper is, as we mentioned in Remark 4.6.2(d), well worth reading for its positive aspects but is best known for an error involving the analytic continuation process. (See the discussion in Section 4.6.) The first mathematically rigorous work using analytic continuation was, as far as we know, done by Cameron [Cal] in 1960 where the analytic continuation was scalar-valued and in the mass parameter. In that paper, Cameron pointed out the error in [GelYag] and also gave a positive result under what now seems like extremely restrictive hypotheses. (Two early papers [Bab2, Pel] on the analytic-in-time operator-valued Feynman integral appeared in 1963.)
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Our main concern in this section is the beautiful theorem of Nelson on the analyticin-mass operator-valued Feynman integral. This result was the focal point of the paper [Nel] in which additionally the approach to the Feynman integral via the Trotter product formula was first treated with mathematical rigor (see Sections 11.1 and 11.2). We will not establish all of Nelson's assertions but we will give a complete proof of the existence of the analytic-in-mass operator-valued Feynman integral (in the sense of Definition 13.5.1' below) for almost every value of the mass parameter. Recall that in the approaches to the Feynman integral in Sections 11.2, 11.4, 11.6 and 13.3, the assumptions on V_, the negative part of the potential, were (although quite reasonable) always stronger than the assumptions on V+. A strength of Nelson's result is that it allows certain potentials with arbitrarily strong singularities regardless of sign. (The results to be discussed in Section 13.6 also have this feature.) Haugsby extended the work of Nelson in his 1972 thesis [Hau] by weakening the continuity requirement on V and by allowing V to be time-dependent and complexvalued. Both of [Nel, Hau] have a serious shortcoming which we will discuss below. (See, in particular, Remarks 13.5.8(a) and 13.5.5(b).) This shortcoming is not shared by the approaches to the Feynman integral which were considered earlier. We will just state Haugsby's theorem and give a brief discussion of it. Unfortunately, Haugsby's thesis has never been published. However, the first author has stated and briefly discussed the main theorem of [Hau] in [Jo6, pp. 38-50] and in [Jo7, pp. 15-20]. Part of the present section is based on [Nel, Hau]. In the last part of this section, titled "Further extensions via a product formula for semigroups", we will state (essentially without proof) later results by Kato [Kat7] and then discuss their consequences for a suitable version of the analytic-in-mass Feynman integral. These consequences were not considered in [Kat7], Kato's results (and their later generalization by Bivar-Weinholtz and Piraux [BivPi, Biv3] also stated at the end of this section) reprove and extend the operator-theoretic part of Nelson's results without using Wiener integrals. Instead, they are based on a suitable product formula for semigroups which is used as a substitute for a generalized Feynman-Kac formula with imaginary potential. Definition of the analytic-in-mass operator-valued Feynman integral Let C and C+ denote, respectively, the complex numbers and the complex numbers with positive real part; further, let C+ denote C+ with the origin of the complex plane removed. We fix a positive integer d and also fix t > 0. Our definition will be given for rather general functions F : Ct = C([0, t], Rd) —>• C, but our interest in this section will be in functions of the form
where V : [0, t] x Rd -> C. In fact, most of our discussion will concern Nelson's theorem in which the "potential" V is time independent so that F has the form indicated in (13.2.1). We remark that in Chapters 15-18, the functions F that will be considered
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will not be restricted to the form (13.5.1). In fact, certain Banach algebras of functions At,t > 0, will play an important role in those chapters. Definition 13.5.1 Fix t > 0 and let F be a function from Ct to C. Given n > 0, W E L2(Rd) and E E Rd, consider the expression
The operator-valued function space integral exists for A. > 0 if (13.5.2) defines Ktn(F) as an element of L(L 2 (R d )). If Ktn(F) exists for every A > 0 and, in addition, has an extension (necessarily unique) to an L(L 2 (R d ))-valued function of n which is analytic in C+, we say that Ktn (F) exists for all n E C+. Finally, for no = —iqo purely imaginary (qo = 0), we say that Ktno (F) exists provided that the limit, limn->n0 Ktn(F), where A approaches A.Q through C+, taken in the strong operator topology, exists and equals the bounded linear operator Ktno (F). We write briefly,
and we callKtno(F) = Kt-iqo (F) the analytic-in-mass operator- valued Feynman integral of F with parameter n0. A number of variations of the preceding definition can be given. In fact, one of these will be the definition that is actually used in the theorems of Nelson and Haugsby. There will be no change from Definition 13.5.1 for n E C+ but, for n0 = —iqo E C+\C+, we will require only that
in the strong operator topology as n approaches n0 "nontangentially" through C+. We will see in the following paragraph and Definition 13.5.1' what "nontangentially" refers to. Let a e (0, n/2) and let n0 = — iqo purely imaginary (qo / 0) be given. The symbol Wa (A-o) will denote the wedge in C+ with vertex at no and with both sides making the angle a with the imaginary axis. Definition 13.5.1' Suppose that Ktn(F) exists in the sense of Definition 13.5.1 far all n E C+. Let n0 = — iqo be purely imaginary with qo = 0. We say that Ktn0 (F) exists provided that for every a 6 (0, n/2),
as A — n0 through the wedge Wa (n0) and where the limit is taken in the strong operator topology on L(L 2 (R d )). We will again refer to Ktn0 (F) as the analytic-in-mass operatorvalued Feynman integral with parameter n0, but, when confusion might arise, we will add the phrase "in the sense of Definition 13.5.1".
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Clearly, the existence of the analytic-in-mass operator-valued Feynman integral with parameter n0 = — iqo in the sense of Definition 13.5.1 implies its existence in the sense of Definition 13.5.1'. There will be a third definition, Definition 15.2.1, in Chapter 15. In that case, the Wiener integral on the right-hand side of (13.5.2) will be required to extend to an L(L2(Rd))-valued function which is strongly continuous in C+ and analytic in C+. The existence of the Feynman integral in the sense of Definition 15.2.1 certainly implies its existence for all n0 = -iqo in the sense of both Definitions 13.5.1 and 13.5.1'. Since Definition 15.2.1 will be used throughout Chapters 15-18, we will refer to it there as simply the analytic operator-valued Feynman integral. (We may sometimes even drop the words "operator-valued".) Definition 13.5.1 has been stated (as has often been done in the literature) so as to minimize reference to the somewhat subtle issues connected with scaling in Wiener space. This topic and its implications for the scalar-valued analytic Feynman integral has been discussed in detail in Chapter 4; Sections 4.5 and 4.6 are especially relevant to our present considerations. Analogous issues for the analytic-in-mass operator-valued Feynman integral will be treated in part I of Section 15.2. With that in mind, we will settle here for a few remarks. Remark 13.5.2 (a) Using the change of variable theorem, Theorem 3.3.2, we can rewrite the integral in (13.5.2) as
(The reader may wish to review the definition of ma, Definition 4.2.3, and to compare (4.6.1) with (4.6.2).) Thus we see thatjust the existence of the integral in (13.5.2) for every n > 0 means that for every one of the mutually singular measures on Ct0 in the collection {ma : a > 0} and for Leb.-a.e. E, F(x + E) must be measurable and integrable with respect to ma; in particular, for Leb.-a.e. E, F ( ( . ) + E) is scale-invariant measurable. Further, much as in Section 4.5, the proper equivalence relation for functions F(x + E) and G(x + E) is not simply equality mx Leb.-a.e. but rather equality ma x Leb.-a.e. for every a > 0. (b) Our starting point as well as Haugsby 's is the integral (13.5.2) for X > 0. Nelson's starting point is essentially the same although that is not apparent at first glance. Nelson begins in [Nel, §27 with a diffusion constant D — ^, where m is purely imaginary with Im m > 0. Next, Nelson integrates with respect to the probability measure which corresponds to the diffusion constant D > 0 and to continuous paths which start at E at time 0. The vector E appears in Nelson's notation but the diffusion constant D does not. No mention is made of equivalence classes of functions, but since D will (for m purely imaginary and Im m > 0) run over all positive numbers, the appropriate equivalence is equality ma x Leb.-a.e. for every a > 0. (c) Nelson eventually analytically continues to real m, with m nonzero. Throughout, his m and our A. are related by the simple equation n = — im. Comparing formula (2.1.2) for the diffusion constant D, we see that (when m > 0) Nelson absorbs the mass m of the quantum-mechanical particle into, say, the coefficient of viscosity in Einstein's formula.
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(d) Recall from Chapter 6 (especially Equation (6.4.1)) that for a single quantummechanical panicle of mass m moving in Rd, the Hamiltonian is given formally by
where A denotes the Laplacian in Rd. Note that the mass is involved with the Laplacian, but not with the potential V. The reader may wish to keep this in mind when following the rest of this section. We add two further comments unrelated to the issue of scaling. Remark 13.5.3 (a) We remind the reader that the concepts of analyticity of L ( L 2 ( R d ) ) valuedfunctions are equivalentfor the operator norm, strong operator and weak operator topologies on L ( L 2 ( R d ) ) (see Remark 13.2.2(c)). (b) What accounts for the fact that we will be able to establish the existence of the analytic in mass operator-valued Feynman integral under more stringent requirements in Chapters 15-18? Briefly, our goals in those chapters are different than in Chapters 10-13, and we will not attempt there to deal with highly singular "potentials". Nelson's results We will analytically continue in mass rather than in time. This change from Sections 13.2 and 13.3 will have both advantages and disadvantages (see Remarks 13.5.5(b) and 13.5.8(a) below). It is the potential V on Rd that will vary from problem to problem and so it is on V that we must place our hypothesis. Hypothesis on V: There is a closed subset J of Rd of capacity 0 such that V is continuous and real-valued on Rd\ J. (13.5.5) The concept of (Newtonian) capacity will be defined (see formula (13.7.1) and Definition 13.7.5) and briefly discussed in Section 13.7. For now, we just list the properties of capacity that will be helpful to us in the present section. The capacity of a subset A of Rd will be denoted Cap(A). (13.5.6) (1) Cap(W) = 0 implies that Leb.(AO = 0, but not conversely. (2) If Cap(N) = 0, then for every n > 0 (equivalently, for every diffusion constant D > 0) and for every E E Rd\N, the path n - 1 / 2 x ( s ) + E, 0 < s < oo, misses N for m-a.e. x [Nel, Theorem 5 and following paragraph]. This implies that the following is true for any V satisfying (13.5.5): For every t > 0, n. > 0 and for every E E Rd\J, the function V ( n - l / 2 x ( s ) + E), 0 < s < t, is continuous and real-valued for m-a.e. x € Cto. (3) The only subset of R of capacity 0 is the empty set, so that this concept will not be helpful to us for d = 1. On the other hand, every countable set has capacity 0 for d > 2. Thus, for example, a potential V : R.d ->• M satisfying (13.5.5) can have arbitrarily strong singularities independent of sign at oo and at any finite subset of Rd, d > 2. Larger sets of singularities are possible as d increases. Any line or line segment has capacity 0 in Rd for d > 3. In fact, any subset of a (d — 2)-dimensional affine subspace has capacity 0 in
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Rd. Also, such facts are by no means limited to flat subsets; any compact Riemannian submanifold of Rd with dimension less than or equal to d - 2 is a set of capacity 0 in Rd. (See, e.g., [RauTay, Example 3, p. 35].) Item (3) just above was included in order to give the reader a better feeling for the implications of the theorems in this section as well as in Sections 13.6 and 13.7. It will be convenient for us in this section and throughout Chapters 15-18 to restate our formulas for the operators [ e - z H 0 : Re z > 0} (see Theorems 10.2.5-10.2.7 and especially formulas (10.2.25), (10.2.26), and (10.2.33)) in a slightly different form. For s > 0 and n E C+, we write
with the convention that e -0(H0/n) = /. A brief review of the basic facts about these operators cast in the notation of (13.5.7) will be given in part £ of Section 15.2. The reader may find it helpful to consult that material. We settle for recalling here that when in the mean (see (10.2.33)). We now make further comments which will help us with the proof of Theorem 13.5.4 below. We will assume that our potential V is time independent and satisfies (13.5.5). It follows from (1) of (13.5.6) that V is R-valued Leb.-a.e. on Rd. Thus the operator V = MV of multiplication by V is self-adjoint (see Proposition 10.1.3) and the operator e~isV is unitary for all s e R. In the notation of (13.5.1), it is the function F_iiv that will concern us. Note that for y E Ct,
The formula (13.5.2) becomes in this case for any W E L 2 (R d )
Taking advantage of Remark 13.5.2(a), we can rewrite (13.5.9) as
We will give Nelson's results in three parts, beginning with the case n, > 0. (See Theorems 13.5.4, 13.5.6 and 13.5.7.) The theorems will be stated in full but not all the assertions
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will be proved. However, the three results taken together will provide a complete and rather detailed proof that under the assumption that V satisfies (13.5.5), the analytic in mass operator-valued Feynman integral Kt-iqo (F-iv) exists in the sense of Definition 13.5.1' for Leb.-a.e. qo in M. Theorem 13.5.4 (Imaginary mass) Let n > 0 be given and suppose that the potential V satisfies (13.5.5). Then, for all t > 0 and W E L2(Rd), we have
where the limit is in the norm on L 2 (Rd). The sense of (13.5.11) is that the operators (e~"(-H°^'le~l nv)n in £(L2(Rd)) converge in the strong operator topology to the operator K'^(F-iv), where the action of this last operator is given for all TJS in L 2 (Rd) Leb.-a.e. by the right-hand side of (13.5.9). Finally, for each A. > 0, the operators [K[(F-iv) : t > 0}form a (Co) contraction semigroup on H = L2(Rd). Proof Let A. > 0, t > 0 and E E Rd\ J be given. By (2) of (13.5.6) (which uses in an essential way the fact that J has capacity 0), the function V ( n - 1 / 2 x ( s ) + E)> 0 < 5 < t, is continuous and R-valued for m-a.e. x € C0. Thus the integral in the argument of the exponential in (13.5.9) makes sense for m-a.e. x. Further, the integrand in (13.5.9) is bounded by |W(n-1/2x (t) + E)|. By applying the Wiener integration formula in Theorem 3.3.5 and then the explicit formula for the free heat semigroup in Theorem 10.2.6, we obtain
But the right-hand side of (13.5.12) belongs to L2(Rd) as a function of E (see (10.2.12)) and so is finite for Leb.-a.e. E. (In fact, more is true. The right-hand side of (13.5.12) is the convolution of two L2-functions (see (10.2.26)) and so is easily seen to be bounded and continuous on Rd.) Because of the continuity of V(n-1/2x(s) + E) as a function of s for m-a.e. x, the integral in the equality to follow can be regarded as a Riemann integral and so can be written as a limit of Riemann sums for m-a.e. jc:
Using (13.5.13) and the Lebesgue dominated convergence theorem with |W ( n - 1 / 2 x ( t ) + E)| serving as a dominating integrable function, we obtain (for E E J)
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Now, the integrand on the right-hand side of (13.5.14) depends only on the value of the Wiener path x at the n times ^ . ^ . • • - . ^ = ^ Hence the integral can be calculated using Theorem 3.3.5, Wiener's integration formula. We have done closely related computations earlier (see (7.6.1)-{7.6.9) as well as the explicit formula (10.2.26) for the free heat semigroup) and so we will just use the result here. Doing so, (13.5.14) becomes for E E J,
Since (by (1) of (13.5.6)) the set J of capacity 0 necessarily has Lebesgue measure 0, (13.5.15) tells us that the sequence of Trotter-like products
each of which belongs to L2(Rd), converges Leb.-a.e. to the L2-function Ktn(F-iv)W We want to show that
First, it is easy to deduce from (13.5.7) that for n > 0, t > 0 and any p E L2(Rd)
We now wish to show that for all integers n > 1,
We will write out the case n = 2. (The same ideas are involved in the proof for general n.)
where the inequalities follow from (13.5.17). Now from (13.5.18), we have |Wn - Wn| < 2e-t(H0/n) |W| and so
But Wn — Wni -> 0 for Leb.-a.e. E since both Wn and Wni converge Leb.-a.e. to Ktn(F-iv)W. Thus by the Lebesgue dominated convergence theorem, we see that {W n } is a Cauchy sequence in L 2 (R d ). Hence {W n } converges in L2-norm to some element in L 2 (R d ) and, since we have seen that Wn -> Ktn(F- iV )W Leb.-a.e., (13.5.16) follows. But (13.5.16) is just the desired equality (13.5.11) in different notation.
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Since the operators e-s(H0/n), s > 0, N > 0, are all contractions (see (10,2.12)) and since the operators of multiplication by e - i s V , s > 0, are clearly unitary, it follows that the operators
are all contractions. Further, it is easy to show that the strong operator limit of contraction operators is also a contraction. Therefore, it follows from (13.5.16) that the operators Ktn(F-iv), t > 0, n > 0, are contractions. Nelson goes on to show that Ktn(F_iv) is strong operator continuous as a function of t for t E [0, +00) and that this family of operators satisfies the semigroup property
(In view of (13.5.11), the latter property is easy to establish.) Thus Theorem 13.5.4 follows. D Remark 13.5.5 (a) Since {Ktn(F-iv) : t > 0} is a (Co) semigroup, it has an infinitesimal generator. (Recall that much of Chapters 8 and 9 revolved in one way or another around this concept.) Nelson proves that every function p in the domain of the generator has finite gradient. More precise information is now known about this domain, as will be discussed further on. (See formula (13.5.42b), Corollary 13.5.18 and Remark 13.5. J9(a) below.) (b) Some of the differences in techniques between analytically continuing in mass as opposed to time have an elementary source. Let the potential V be R-valued and time independent. In the method of analytic continuation in time, we start with the semigroups H0 and e-sH0 and e-sV and analytically continue through C+ to the unitary groups e~is isV e~ . We begin with the pair of semigroups for diffusion (or heat) problems and end with the pair of unitary groups for quantum-mechanical problems. In contrast, when we analytically continue in mass, we begin with e~ s(H0 /n) (n > 0) and e~'sV and end with g-isH0 and e-isV Analytically continuing in A. has no effect on e~isv. If we want to end with e~is V , the right thing for (standard) quantum problems, e~isV has to be there to begin with. Of course, what we start with is then not the right thing for diffusion problems. This will not concern us much in this chapter as our focus is on the quantum-mechanical setting. However, we will use analytic continuation in mass throughout Chapters 15-18, and in those chapters we will often wish to draw conclusions about both the diffusion and quantum-mechanical situations. In order to cover the diffusion setting, we will have to start with V purely imaginary so that —isV will be real. The presence of e~isV with V real right from the start in the analytic continuation in mass framework has advantages and disadvantages. The function e~isV—and thus also the functional exp[—i ft0 V(x(s) +E)ds]—has constant absolute value 1, independently of the sign of V and no matter how large | V \ becomes. This makes the limiting arguments involving the Wiener integral in the proof of Theorem 13.5.4 rather elementary. In particular, it is easy to find dominating integrable functions. Also, the special care with the negative part of the potential that was needed in the analytic continuation in time approach was not needed above. On the negative side, HO + iV is not self-adjoint and
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this means that arguments that were based on the self-adjointness of H0 + V in Section 13.3 are not available here. (c) The reader may wonder why Nelson did not simply apply the Trotter product formula (Theorem 11.1.4) to deal with the limit of the functions Wn in (13.5.16). In fact, this formula does not apply here under the present assumptions because in general, HO + i V need not be "essentially m-accretive" on D(Ho) n D ( i V ) ; that is, —(Ho + iV) need not generate a (Co) contraction semigroup. We now proceed to the second case of Nelson's results, namely, A E C+. Theorem 13.5.6 below says, roughly, that Theorem 13.5.4 continues to hold for all A E C+. There is a crucial difference, however; the limit on the right-hand side of (13.5.11) will again be shown to exist but, in this case, will be used to define the operator on the left-hand side of (13.5.11). No claim will be made that the formula (13.5.9), or equivalently (13.5.10), holds for A E C+; in fact, there can be no such formula (in any straightforward sense) for n E C+\(0, +00) since there is no countably additive, complex-valued Wiener measure with such a variance parameter (see Section 4.6 and especially Theorem 4.6.1). Theorem 13.5.6 (Complex mass) Let A E C+ and suppose that V satisfies (13.5.5). Then, for all t > 0, the sequence (e~n ( H ° / n ) e~il/n) n from L ( L 2 ( R d ) ) converges in the strong operator topology to an operator in £(L 2 (R d )) which we will denote K^(F-iv). We write, for all W E L 2 (R d ),
where the limit in (13.5.19) is in the norm topology on L2(Rd). Further, the operatorvalued function Ktn(F-iv) of n is analytic in C+ and agrees with the operator Ktn(F-iv) in (13.5.9) for n > 0. Hence Ktn(F- iv ) exists in the sense of Definition 13.5.1 (or 13.5.1') for all n E C+. Finally, for each n E C+, the operators {Ktn(F-iv) : t > 0} form a (Co) contraction semigroup on H = L2(Rd). Proof We will prove all but one of the claims made in the statement of the theorem. Note to begin with that e~tH° is analytic in C+ as an operator-valued function of t. This is a consequence of Theorem 13.3.1 in the simple case where V = 0. (The reader should consult the proof as well as the statement of Theorem 13.3.1.) It follows that H e -s(H 0 /n) is anaiytic as a function of A in C+. Thus the products (e~n( o/We-' ^ V)« are analytic operator-valued functions of A throughout C+. Further, the operators e~« ( H °/n ) are all contractions for n E C+ (see (10.2.12)) and so the above-mentioned products are contractions as well. Summarizing, the sequence of operator-valued functions of A,
is analytic in C+ and bounded in norm by 1. Further, we know from Theorem 13.5.4 that for all n > 0, f n (A) -> Ktn(F-iv) in the strong operator topology as n -» oo. All but one of the conclusions of our theorem now follow immediately or at least easily from the Vitali convergence theorem for operator-valued analytic functions [HilPh, Theorem 3.14.1,
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p. 104] (see also Theorem 11.7.1 (i) and Remarks 11.7.2(b), (c) in Appendix 11.7 above for the corresponding classical Vitali theorem). The version of Vitali's theorem which we will apply says the following: Let {fn ( n ) } be a sequence of analytic operator-valued functions defined on a domain D in the complex plane and suppose that ||fn(n)|| < M for all n and all n E D. Finally, suppose that limn-oo fn (n) exists in the strong operator topology for all A in a subset DO of D, where DO has a limit point in D. Then limn_»oo fn (z) exists in the strong operator topology for all n E D, uniformly on all compact subsets of D. Further, the limit, denoted by f(n), is analytic in D and satisfies ||f(n)|| < M for all X 6 D. For us, /„(>.) is given by (13.5.20), D = C+, M = 1 and D0 = (0, +00). The existence of the limit (13.5.19) in the strong operator topology for all A, 6 C+ and the analyticity of the limit function Ktn(F-iv) throughout C+ follow immediately as does the existence of Ktn(F-iv) in the sense of Definition 13.5.1 for all n E C+. The inequality
also comes immediately from Vitali's theorem. Now, by Theorem 13.5.4, we have the semigroup identity
for all n > 0. It then extends to all n e C+ by analytically continuing (for fixed s, r > 0) all three expressions in this identity. The only assertion from the theorem that we will not prove is the strong operator continuity for all n E C+ of the semigroup {Ktn((F_iv) : 0 < t < 00}. O We are now prepared to establish the existence in the sense of Definition 13.5.1' of the analytic in mass operator-valued Feynman integral with parameter b0 = —iqo for Leb.-a.e. qo 6 R. As before, we will not give a complete proof of the theorem to follow. Our presentation of this third case, n purely imaginary, of Nelson's results is influenced somewhat by the proof of Theorem 3 in [JoSkl]. [The reader may wish to review Remarks 13.5.2(b),(c) to see why X purely imaginary corresponds to a real mass parameter.] Theorem 13.5.7 (Real mass) We suppose that V satisfies (13.5.5). Then for Leb.-a.e. qo in R., qo = 0, the analytic-in-mass operator-valued Feynman integral with parameter n0 = — iqo exists in the sense of Definition 13.5.1'; that is, for every t > 0 and a e (0,7T/2), Ktn(F-iv) converges in the strong operator topology as n —> n0 through the wedge Wa(no) to an operator in L(L 2 (R d )) which we will denote Ktn0(F-iv) or Kt-iqo (F-iv). When this happens, we write, for every W E L 2 ( R d ) ,
and understand that A. can approach n0 = — iqo through any of the wedges Wa(A.o). Finally, for every A-o — —iqo for which K'_. (F_,-v) exists, the operators (K^j (F-iv) : t > 0}form a (Co) contraction semigroup on L2^).
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Proof Fix t > 0. One of the keys to the proof is the use of the Fatou-Privaloff theorem [Ru2, Theorem 11.23, p. 244] which says (for the domain that concerns us) that a bounded analytic function on C+ has a nontangential limit almost everywhere on the real axis. Since Ktn[(F-iv) is analytic in C+ as a function of A. by Theorem 13.5.6, we know that (K[(F-iv) W, W) is a scalar-valued analytic function on C+ for every W, p in L 2 (R d ). Further, this function is bounded by ||W|| ||p|| since each of the operators Ktn(F-iv) is a contraction. Hence, by the Fatou-Privaloff theorem, there exists a Lebesgue null set N(W, p) of real numbers such that if qo E N ( W , p) then (Ktn(F-iv)W, p) converges as n —> — iqo nontangentially through C+. (During the rest of the proof, limits as n - — iqo are to be interpreted as taken nontangentially through C+.) Now the separability of L2(Rd) and the boundedness of {Ktn((F- iV ) : n E C+} allows us to find a null set N independent of W and p such that if qo E N, then (Ktn(F-iv)W, p) converges as n -> -iqo, for all W, p E. L2(Rd). The completeness of £(L2(Rd)) in the weak operator topology insures us that for each qo E N, there is an operator in L(L 2 (R d )) which we shall denote Kt_. (F_iv) such that as n -> — iqo
It follows from (13.5.23) that the operators Kt_iqo (F_,iv), qo E N, are contractions since all the operators Ktn(F-iv) are contractions and
We wish to show that for qo E N, Ktn[(F-iV) -» Kli^F-iv)
(strong operator topology),
(13.5.24)
Now for A. € C+, Ktn(F-iv)W is weakly measurable in n since it is analytic in A.. Hence, Kt-iq (F-iv)W is a weakly measurable function of q. But L 2 (Rd) is separable and so Kt_iq (F-iv) W is strongly measurable [HilPh, Corollary 2, p. 73]. Then it also follows that||Kt-iq(F-iv)WII is measurable as a function of q, by [HilPh, Theorem 3.5.2]. Next, for each A = u - vi E C+, the L2(Rd)-valued function
is Bochner integrable (by [HilPh, Theorem 3.7.4]) since
Note that the integrand in the middle expression of (13.5.27) is the Poisson kernel (with parameters u and v) for the region C+ (see [Hil]). This integrand is nonnegative and has
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L1(1K) norm 1 as a function of q. [The Poisson kernel and the Poisson representation theorem (used below) are for C+ and its boundary. We used the (conformally equivalent) Poisson kernel and Poisson representation theorem for the open unit disk and its boundary intheproof of Theorem 11.7.1(ii).] Now that we know from (13.5.27) that g(q) is Bochner integrable, we are ready to establish the (operator-valued) Poisson formula
which relates the "values" of the function on the boundary to its values on C+. Since both sides of (13.5.28) are in L 2 (R rf ), it suffices to show that we get equality when we take the inner product of the two sides with an arbitrary p € L2. But closed linear operators and so certainly bounded linear functionals can be taken inside Bochner integrals [HilPh, Theorem 3.7.12 and following comment], and thus it suffices to show that
However, (13.5.29) follows immediately from the classical scalar-valued Poisson representation formula [Hil, p. 455]. Hence, we have now proved (13.5.28). Using (13.5.28) and a simple inequality for Bochner integrals [HilPh, Theorem 3.7.6], we have for every X = u — vi e C+,
Now using the fact that || Kt_. (F-iv)W || is a bounded, real-valued measurable function of q and [Hil, Lemma 19.2.1], we have for Leb.-a.e. qo E R,
Combining (13.5.30) and (13.5.31), we see that for Leb.-a.e. go.
Enlarging if necessary the null set associated with (13.5.23) to reflect the step just above, we can also insure that for qo E N,
as A = u — vi —> — iqo. But when weak convergence is present in a Hilbert space, one always has the norm of the limit less than or equal to the lower limit of the norms [Kat8, Eq. (1.26), p. 137]. Thus, in our case,
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Combining (13.5.32) and (13.5.34), we see that for q0 $ N,
But this last equality and the weak convergence in (13.5.33) implies the convergence in the norm of L 2 (R d ) (see [BkExH, Proposition 2.1.5, p. 43]) that we wished to establish (see (13.5.24) and (13.5.22)). This will finish the proof for us, but Nelson goes on to show that the exceptional set N of qs can be chosen independently of t and that {K'_f (F_,-y) '• t > 0} is a (Co) semigroup for every qo £ N. D Remark 13.5.8 (a) The existence of K'^ (F-iv) only for Leb.-a.e. q in Theorem 13.5.7 is a serious weakness of Nelson's beautiful results. (Actually, all the approaches to the Feynman integral discussed in this section and the next depend on the Fatou-Privaloff theorem and so have this same shortcoming.) Given a particular A.Q = —iqo, one does not know whether K'^ (F_/v) exists or not. Further, even if it does exist, the possibility that a minute change in XQ may cause it not to exist is certainly not satisfying. (b) Can the existence of K t _ . ( F - i v ) only for almost every (nonzero real) q be used to perform the basic computation of quantum mechanics? Perhaps. We engage in some speculation which might provide a way of avoiding the difficulties. Suppose, for simplicity, that we are interested in a single quantum particle moving in R3 under the influence of the potential V. Let Abe a measurable subset o/R3. The probability that the particle is in A at time t is ordinarily calculated by
where W is the initial probability amplitude and q is the mass parameter. The trouble is that for a fixed qo, we have no guarantee that the expression in (13.5.36) makes sense. However, for any e > 0, it does make sense to average (13.5.36) in some way over the interval (qo — e, qo + e). For example, we may consider
The expression in (13.5.37) seems to have some heuristic appeal especially if E is taken so small that q is not known with better than e-accuracy anyway. Haugsby's result for time-dependent, complex-valued potentials Nelson considers potentials V : Rd -> M which take finite real values except possibly on a closed set J of capacity 0 (see (13.5.5)). It is such time independent R-valued potentials that are appropriate for the standard problems of quantum mechanics. However, potentials that are time-dependent or complex-valued are also of interest. Time-dependent potentials arise naturally when the force which the potential represents varies with time. One might have, for example, V = V\ + V2, where Vi is a time-dependent potential such that the associated force is under the control of an experimenter. Complex-valued
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potentials were discussed at some length in Section 11.6. We just mention here that such potentials are used to provide phenomenological models for "open quantum systems" such as decay problems where attention is restricted to the material that remains and the decay products being emitted are ignored (see [Ex, BkExH] for example). Haugsby's theorem permits potentials that are time-dependent and/or C-valued and can also have the same kind of strong singularities allowed in Nelson's results. We are now prepared to state the theorem of Haugsby [Hau]. Theorem 13.5.9 Let V : [0, t] x Rd -* C be Lebesgue measurable and let T be a subset of Rd of capacity 0. Further let V satisfy the following: (i) Re V(s, u) < Bfor all (s, u) € [0, t] x Rd, where B is a real constant. (ii) V(s, u) is bounded on every compact set disjoint from [0, t ] x P. (iii) V(s, u) is continuous almost everywhere on [0, t] x A, where A is an open subset of Rd satisfying at least one of the conditions (c1) P c A or (c2) P c A and A\A has Lebesgue measure 0. Then, for F = Fv given by (13.5.1), Ktn(FV) exists for all X € C+ and Ktn0(F v ) = Kt-iq0 (Fv) exists in the sense of Definition 13.5.1' for Leb.-a.e. qo € R. [We remark that our present notation involving the "potential" V and the associated functional F = FV are somewhat different than in the rest of this section and in Section 13.6.] A check of the conclusions of Theorems 13.5.4, 13.5.6, and 13.5.7 will show that Nelson's results include several assertions that have no counterpart in the theorem just above. However, the following simple proposition shows that if attention is restricted to the existence of Ktn[(Fv), n € C+, then Haugsby's theorem implies Nelson's theorem. Proposition 13.5.10 If the potential V\ satisfies the hypothesis (13.5.5) of Nelson's theorem, then the corresponding potential V(s, u) = —iV\(u) satisfies the hypotheses of Theorem 13.5.9. Proof The real part of V(s, u) — —iV\(u) is identically 0 and so (i) of Theorem 13.5.9 is certainly satisfied. The set of capacity 0, call it F here, in (13.5.5) is required to be closed and V\ is required to be continuous on R d \F. Clearly then V(s, u) — —i V\ (u) is bounded on every compact set disjoint from [0, t] x F; that is, on every compact subset of [0, t] x (R d \F). Thus (ii) is established. If V\ (u) satisfies (13.5.5), then V(s, u) = -i V\ (w) is continuous on [0, t] x (R
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almost everywhere on [0, t] x Q, is considerably less restrictive than the assumption in (13.5.5) of continuity (of the time independent potential V\) on E.d\J. (c) Haugsby's proof in [Hau] is clearly inspired by that of Nelson in [Nel] in a number of key places such as the use of (i) the Wiener integral, a true integral with respect to a countably additive measure, for X > 0, as well as (ii) three powerful theorems of complex analysis, the theorems of Vitali, Fatou-Privaloff, and the Poisson representation theorem. (d) The function F-w given by (13.5.8) has constant absolute value 1. This fact was quite useful in the proof of Nelson's results, but no longer holds in Haugsby 's setting for the function Fv(y) given by (13.5.1). This makes less difference in the proof than one might think since we have from assumption (i) of Theorem 13.5.9,
(e) In the proof of Nelson's theorem we had, for every t > 0, A > 0 and f E Rd\ J, the continuity of V ( n ~ l / 2 x ( s ) + E) as a function of s, 0 < 5 < t, for m-a.e. x E Ct0. This does not hold in Haugsby's setting and this fact adds technical difficulties to the first part (A > 0) of Haugsby's proof. (f) The "potentials" treated in Chapters 15-18 will be allowed to be time-dependent and/or complex-valued as in Haugsby's theorem, but they will not be permitted to have spatial singularities. It should be added that these latter chapters focus on perturbation series and Feynman 's operational calculus for noncommuting operators and have goals which are quite different from those of Chapters 11-13. Further extensions via a product formula for semigroups Let W : Rd —> C be a suitable (measurable) complex-valued function. Further, let C be a suitable m-accretive "realization" of the "Schrodinger operator" — 1 / 2 A + W ( y ) (y E Rd). As we have seen in the first part of this section, a key step in the proof of Nelson's result in [Nel] consisted in establishing the following Trotter-type product formula:
where { T ( t ) := e- t C : t > 0} denotes the (Co) contraction semigroup generated by —C. The convergence in (13.5.38) is assumed to hold in the strong operator topology for all t > 0. (See Theorem 13.5.4 and its proof, with the obvious changes of notation; in particular, with our previous notation, we have W = —i V, where V denotes the "potential".) We stress that (13.5.38) does not follow from the classical Trotter product formula (Theorem 11.1.4) or from Chernoff's generalization (Theorem 11.1.8) because the hypotheses of these theorems are clearly not satisfied under the hypotheses to be made on W. See (13.5.39) below. Recall that Nelson [Nel] established (13.5.38) under the assumption that Re W = 0 and W is continuous on Rd\J, where J c Rd is a closed set of capacity 0, but without any further restriction on the behavior of W near J. The proof of (13.5.38) given in
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[Nel] (as well as that of Theorem 13.5.4 above) uses the Wiener path integral in an essential manner and in fact, deduces the existence of the product formula in (13.5.38) from that of a corresponding Feynman-Kac formula with potential W (assumed to be purely imaginary, in [Nel]). We note that even though Nelson showed that the limiting semigroup T(t) in (13.5.38) is a (Co) contraction semigroup, he did not precisely identify the domain of its generator — C. (See Remark 13.5.5(a) above.) In a remarkable paper [Kat7], Kato has extended the operator-theoretic aspects of Nelson's results in several directions and has exactly determined the domain of the generator -C. (See (13.5.40) below, as well as (13.5.42) in Remark 13.5.15(b).) His extension—obtained by means of operator-theoretic methods and without the use of the Wiener integral—had the following goals: (i) Relax the conditions on the complex-valued potential W (with Re W < 0, say). (ii) Allow for stronger singularities on the imaginary part of W (and hence on the real part of the potential V, if we set W = —iV). [In view of Lemma 13.5.14 below, this is achieved by working with a suitable Schrodinger operator on the open set £2 — Rd\J, where J is as above.] (iii) Allow for the free Hamiltonian HQ = —1/2Ato be replaced by a suitable second order elliptic differential operator, with variable and possibly discontinuous coefficients. It should be noted that Kato did not apply his results to the Feynman integral, as we will do further on in Theorem 13.5.16. We will now discuss (essentially without proof) some of these extensions, and then mention some of their later generalizations obtained in [BivPi, Bivl-3]. [The corresponding (but more general) results for the analytic (in mass) modified Feynman integral— obtained by Bivar-Weinholtz and the second author in [BivLa] via a suitable product formula for imaginary resolvents—will be discussed in Section 13.6. The interested reader may wish to consult that section; especially the proof of Theorem 13.6.7, to have a more precise idea of the techniques used in proving the results below.] Let A C Rd be an arbitrary open set and put H = L 2 (A). Denote by HO =-1/2A the (normalized) Dirichlet Laplacian on A, acting in H. (Recall that we have already considered this choice of free Hamiltonian towards the end of Section 11.6.) Further, let W : A —»• C satisfy the following conditions. (These assumptions have later been relaxed in [BivPi,Biv3], as will be briefly discussed at the end of this section.) Hypotheses on W:
and
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Define the (linear, unbounded) operator C in H = L2(A) by
with domain
Remark 13.5.12 In (13.5.40b), much as in our discussion of Kato's inequality in Section 11.2, —1/2Au+ Wu is considered a priori as a distribution in D'(A). Further, as in Section 11.6, the Hilbert space H10 (ft) is defined as the closure in H1 (A) of D(A) = Coo00(A). Here, the Sobolev space H 1 ( A ) (used in Lemma 13.5.14 below) denotes the set of functions u in L 2 (A) with distributional gradient VM in L2(A); i.e. each of the d components of Vu lies in L 2 (A). Recall that H1(A) is a Hilbert space when equipped with the norm
(See, for example, [Ad, Bre2, Kes].) We can now state Kato's main result [Kat7, Theorems I and II, p. 107]: Theorem 13.5.13 Assume that W satisfies hypotheses (H1) and(Hi) statedin (13.5.39). Then the operator C defined by (13.5.40) is m-accretive (i.e. — C is m-dissipative by Remark 9.4.6(b)) and therefore, by Theorem 9.4.7, defines a (Co) contraction semigroup (T(t) — e~tc : t > 0}. Further, this semigroup is given by the Trotter-like product formula (13.5.38); namely, for all t > 0 and W E L 2 (A), we have
where the limit is in the norm on L2(A) and holds uniformly for t in bounded subsets of [0, +00). The proof of Theorem 13.5.13 makes essential use of Kato's distributional inequality (Theorem 11.2.3), as well as of operator-theoretic and semigroup methods. It also involves a well known lemma of Stampacchia [Stam] concerning the Sobolev space H1 (ft), and a lemma of Kato [Kat7, Lemma 4.1, p. 112] which we have already used in Section 11.3; see Lemma 11.3.3 above. We refer to Sections 3 and 4 of [Kat7] for the complete proof. Next, in order to connect the present situation to our earlier setting, we recall a very useful potential-theoretic lemma [Kat7, Lemma 2.6, p. 106], that seems to have first been established in this form in [HorLio]. (See also [RauTay, Lemma 2.1 and Proposition 2.2, pp. 34 and 36] for a closely related statement; the latter result relies on Carleson's results in [Carl].) Lemma 13.5.14 Let J be a closed subset of Rd and set A = R d \J. Then Hl0 (A) = H 1 ( R d ) if (and only if) J has (Newtonian) capacity 0.
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Remark 13.5.15 (a) The property expressed by Lemma 13.5.14 can be thought of as a potential-theoretic counterpart of the fact (used in the proof of Theorem 13.5.4) that almost surely, Brownian paths do not "hit'' a set of capacity 0; see (2) of (13.5.6) for a precise statement. (b) It follows from (13.5.40) (where we have set W :— -i V) and Lemma 13.5.14 that for each n > 0, the operator Cn in part (I) of Theorem 13.5.16 below is also defined by
with domain
Note that with our earlier notation in (13.5.40), we have C\ — C. We can now deduce the following result from Theorem 13.5.13 and Lemma 13.5.14 (for part (I)), as well as from the methods of proof of Theorems 13.5.6 or 13.5.7 (for part (II) or (III), respectively). We note that neither Theorem 13.5.16 nor Corollary 13.5.18 below were stated in [Kat7] because Kato was primarily interested in establishing the product formula (13.5.38) and did not discuss its consequences for the analytic (in mass) operator-valued Feynman integral. Theorem 13.5.16 Let J C Rd be a closed set of capacity 0. Further, let V : Rd ->• C be a measurable function such that V € Lrloc(Rd\J) (i.e. V € L r ( X ) for every compact set X C R d \ J ) , where the exponent r is as in hypothesis (H2) of (13.5.39). Assume, in addition, that V satisfies the "dissipativity condition" Im V < 0. [Of course, the case where V is real-valued—and hence Im V = 0—is that of most interest in ordinary quantum mechanics.] Then: (I) (Imaginary mass) Let n > 0 be given. Then, for every t > 0 and W 6 L2(Rd), we have
where Cn is the m-accretive operator defined just as in (13.5.40), except with —iV instead of W and with — ^ instead of — j. (Equivalently, by Remark 13.5.15(b) above, d is defined by (13.5.42).) (II) (Complex mass) Let A. 6 C+. Then the limit on the right-hand side o/(13.5.43) continues to hold and defines a (Co) contraction semigroup, denoted T^(t) (t > 0). Further, for all t > 0, the function X M- 7\(r) is analytic from C+ to £(L2(Rd)). (III) (Real mass) For Leb.-a.e. q^'« R> - 7X(t) has a (strong) nontangential limit (in the sense of Definition 13.5.1') as n approaches the parameter n0 = — iqo; that is, for every t > 0 and a E (0, n/2), Tn(t) converges in the strong operator topology as n -»• n0 through the wedge W a (no) to an operator in L(L2(Rd)), which we will still denote 7n0(t) or T-iqo(t).
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Finally, for every A.Q = —iqoforwhich T-iq0(t) exists, the operators {T-iq0(t) : t > 0}form a (Co) contraction semigroup on L 2 (R d ). We will say that the "analytic in mass Feynman integral" associated with V exists (see Remark 13.5.17(a) below) and denote it byF t , n , T p a n ( V ) ;so thatF t , n T p a n ( V )= Tn(t) for all n € C+ and for Leb.-a.e. nonzero purely imaginary n. Proof (I) In view of Lemma 13.5.14 and Remark 13.5.15(b), part (I) (with n = 1) follows from Theorem 13.5.13, specialized to A := Rd\J and W := -iV;so that, in particular, Im V = Re W < 0. (Naturally, the case where X > 0 is arbitrary amounts to nothing more than replacing — y by ^ in the proof of Theorem 13.5.13.) Note that by (1) of (13.5.6), J has Lebesgue measure 0 and hence H = L 2 (A) = 2 L (Rd). (II) Part (II) follows from part (I) exactly as Theorem 13.5.6 was deduced from Theorem 13.5.4, by analytic continuation via an application of Vitali's convergence theorem (see Theorem 11.7.1(1) and Remarks 11.7.2(b),(c)). Note that for all nonzero n E C+ and t > 0, we have||e-t(H0/n)||< 1 and ||e~ itv || < 1, because Im V < 0. (III) Finally, part (III) follows from part (II) much in the same way as Theorem 13.5.7 was deduced from Theorem 13.5.6 by using, in particular, the classical FatouPrivaloff theorem [Ru2, p. 244] regarding the boundary value of a bounded analytic function. D Remark 13.5.17 (a) Let V : Rd -> C be a measurable function. Then, by analogy with Definitions 13.5.1 and 13.5.1', the "analytic-in-mass Feynman integral via TPF" associated with V is said to exist and is denoted byF t , n T p , a n ( V )provided that the following conditions are satisfied: (i) The product formula for semigroups (13.5.43) holds for all n > 0. (ii) It continues to hold for all n E C+ and defines Tn(t) E L(L2(Rd)) which depends analytically on n € C+. (Hi) Finally, for Leb.-a.e. nonzero purely imaginary n0, Tn(t) admits a nontangential limit (as in part (HI) of Theorem 13.5.16 and denoted Tn0 (t)), as A, approaches n0 through C+. (The reader may wish to compare the definition of Ft,n T P,an(.) with that of the "Feynman integral via TPF" given in Section 11.2; see Definition 11.2.21.) We note that even though we use the term "TPF" in the above definition, the product formula (13.5.43) does not follow from the classical Trotter product formula (Theorem 11.1.4). (b) Of course—except for the claim (madeinTheoreml3.5.4)that for each n > 0, the operator Tn (t) is given by the Wiener path integral Ktn(F-iv) in (13.5.9) (see (c) below)— Nelson's original result (given in Theorems 13.5.4, 13.5.6 and 13.5.7 above) follows at once from Theorem 13.5.16 by taking V real-valued and continuous on R d \J. Clearly, in the case of time independent potentials and uniform partitions (along with minor notational changes), the same is true of Haugsby's generalization [Hau] of Nelson's result (see Theorem 13.5.9 and Proposition 13.5.10). (c) Note that—unlike in Theorem 13.5.4—we did not claim in pan (I) of Theorem 13.5.16 above that for n > 0, Tn(t) is given by an extended Feynman-Kac formula, with (complex) potential W := -iV, as in (13.5.9) and (13.5.11) with Tn(t) := Ktn(F- iV ). (See, however, Corollary 13.5.18 below.) Actually, it would be interesting to determine
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whether a counterpart of the extended " Feynman-Kac formula" (13.5.9) holds for all X > 0 under the hypotheses made in (13.5.39)—and, more generally, under the assumptions made at the end of this section as well as in Section 13.6 below. [Perturbation theorems and interchange of limits arguments of the type of [Lai2]—as discussed in Section 11.5 (and recently extended to C-valued potentials in [LisM])—may be useful to tackle this question; see Remark 11.6.5.] (d) Strictly speaking, according to the above comment made in (c), we cannot refer to Theorem 13.5.16 as establishing the existence of the analytic in mass operator-valued Feynman integral (in the sense of Definition 13.5.1'). (Compare Theorem 13.5.7.) However, in view of the product formula (13.5.43)—which is shown in Theorem 13.5.16 to holdfor all A. 6 C+ as well as for Leb.-a.e. A = —iq (q E R, q = 0)—it is natural in the present context to extend Definition 13.5.1' and to think of Tn(t) as the counterpart of Ktn(F-iv), the analytic-in-mass operator-valued Feynman integral associated with the potential V. (e) Note that in Theorems 13.5.4, 13.5.6 and 13.5.7 ([Nel]), the potential V was assumed to be real-valued, whereas it is now allowed to be complex-valued provided it satisfies (13.5.39), and particularly the "dissipativity condition" (H\). We will further relax these conditions at the end of this section—and even more so, in Section 13.6 below in the context of the analytic (in mass) modified Feynman integral [BivLa]. For now, we simply note that, obviously, we can replace the dissipativity condition Im V < 0 by Im V < B, for any real constant ft (as in Theorem 13.5.9). The only changes in the statement of Theorem 13.5.16 are then that a suitable translate of C is m-accretive and that the (Co) semigroup {Tn(t)}t>0 is no longer composed of contractions. (f) Just as in Theorem 13.5.7, in part (HI) of Theorem 13.5.16, the semigroup {T_iqo (t)}t>0 is in general not a group, let alone a unitary group, and can only exist (in the case of very singular potentials) for Leb.-a.e. nonzero real qo- In particular, when it exists, the m-accretive operator C_((?0 is usually not skew-adjoint—and hence the operator |C_iqo is not self-adjoint. Nevertheless, it follows from semigroup theory (see Theorem 9.1.14) that for p E D(C_iqo), W(t,.) := T-iqo(t)
with initial condition iff(0, .) = p; here, formally, —C-iqo W can be interpreted as "—i(—A\l//2qo + V (x)W)". (It can be checked that—with this interpretation—equation (13.5.44) also holds in the distributional sense.) Note that if qo is positive, it can be thought of as the "mass" of the associated particle in the "formal realization" of the "Schrodinger operator" — A/2go + V(x). The following corollary of Theorem 13.5.16 specifies Nelson's result (discussed in Theorems 13.5.4,13.5.6 and 13.5.7 above), as well as Theorem 13.5.16 itself; see Remark 13.5.19 below. It also shows the unity between the approach via a product formula for semigroups and the analytic (in mass) Feynman integral.
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Corollary 13.5.18 Suppose that V satisfies hypothesis (13.5.5) of Theorems 13.5.4, 13.5.6 and 13.5.7. Then the assumptions of Theorem 13.5.16 are also satisfied and for each A E C+,
In particular, for every X > 0, the generator of the semigroup [K^(F-iv) '• t > 0} is equalto —Cx, where C\ isthem-accretiveoperator given by (13.5.42a) and with domain as in (13.5.42b). Further, still for A > 0, 7X(t) = e~tCx is given by the generalized Feynman-Kac formula (13.5.9), with potential -iV. More precisely, for all k > 0 and W E L 2 (Rd), Tn(t)W = e~tcnW is given by the Wiener path integral on the right-hand side of (13.5.9). Moreover, for Leb.-a.e. nonzero real qo, the analytic in mass operator-valued Feynman integral (with parameter no = —iqo) %*_• (F-iv), exists (in the sense of Definition 13.5.1') and coincides with TL,-iqo(t). More precisely, for a given nonzero qo in R, Kt-iqo(F-iv) exists if and only if T-iqo(t) exists, and then we have Kt-iqo (F_iv) = T-iqo(t). In particular, under the above assumptions, the analytic-in-mass operator-valued Feynman integral, Ktn(F-iv), and the operator Tn(t) coincide whenever they exist for n E C+, n = 0. Proof For n > 0, (13.5.42) holds because by Theorem 13.5.4 and part (I) of Theorem 13.5.16 both Ktn(F-iv) and Tn(t) are given by the same product formula (13.5.38), and hence are the limit of the same sequence of operators. In view of Theorem 13.5.16 along with Theorems 13.5.4, 13.5.6 and 13.5.7, the result then follows by the uniqueness of the analytic continuation and of the boundary value of a bounded analytic function. Note that the claim about the generator of {Ktn[(F_,- v )}t>o for A > 0 follows from part (I) of Theorem 13.5.16, while that about the extended Feynman-Kac formula (13.5.9) follows from Theorem 13.5.4. D Remark 13.5.19 (a) Corollary 13.5.18 specifies Nelson's result in [Nel], especially Theorem 13.5.4 above, because for A > 0 (that is, for "purely imaginary mass" in the terminology of [Nel]), it identifies the infinitesimal generator —C\ of the semigroup {Tn(y) = e~itCn : t > 0). In particular, in view of (13.5.42b), it provides a precise description—in the language of distributions or generalized (Sobolev) functions—of the domain D(Cn) — D(—Cx) of this generator. Recall from Remark 13.5.5(a) that according to [Nel], we only knew that D(Cn) c H1(R d ); that is, given u in D(Cx), then its distributional gradient VM belongs to L2(Rd) (compare (13.5.42b)). (b) Further, Corollary 13.5.18 supplements part (I) of Theorem 13.5.16 (or equivalently, for A = 1, Theorem 13.5.13 above from [Kat7]) because for A > 0, it provides a Wiener integral representation o f T ^ ( t ) W = e~tCn W for each W in L 2 (E rf ), at least under the assumption made in [Nel]; see (13.5.10). This fact does not seem to have been noted in [Kat7] or in later papers on this subject.
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Kato's results in [Kat7] have been extended in several directions: (i) First, the study of Schrodinger operators with highly singular complex potentials begun in [Kat7] has been pursued by Brezis and Kato in [BreKat], where such operators -5 A + W—or rather, their m -accretive realization C in H = L 2 (A) := L 2 (A; C), with Dirichlet boundary conditions—have been defined and their domain D(C) analyzed under very general assumptions on the "potential" W : A —> C, with A an arbitrary open subset of Rd. (As in the proof of Theorem 13.5.13 given in [Kat7], a version of Kato's inequality, Theorem 11.2.3, also plays a crucial role in [BreKat].) We will implicitly use these results (and further extensions obtained in [Bivl-3, BivPi, BivLa]) in the statement of Theorem 13.5.22 below, as well as in the next section where the results of [BivLa] will be discussed. (ii) Second, the product formula for semigroups (13.5.38) has been established under progressively more general conditions on the potential W by Bivar-Weinholtz and Piraux in ([BivPi], [Bivl,3]). (We note that no such product formula was derived in [BreKat].) In particular, (13.5.38)—and, more precisely, the analogue of Theorem 13.5.13—has been obtained for the C-valued function W. Here, we assume that —W = q' — p + iq, with q', p, q : A -> R, q', p > 0, W E Llloc(A) and Re W = -q' + p < p; further, we suppose that either p E L°°(A) or else that the following condition (13.5.46) holds:
and
Remark 13.5.20 (a) For simplicity, the reader may well wish to assume in the following that p = 0, as in most applications. Then, in that case, condition (13.5.46b) is unnecessary. (b) When p = 0 (or, more generally, when p E L°°(&)), we must replace HO by H0-p := HO + (-p) and -W by q' + iq in the left-hand side of (13.5.38) and (13.5.41). Indeed, otherwise, the semigroup etw would consist of unbounded operators for all t > 0. In [Bivl-3], a magnetic vector potential a — (a1,. . . , ad) € (L/oc(A; R))d with divergence V.a E L2 loc(A) is also allowed, with suitable modifications. [In particular, we must replace the free Hamiltonian H0 by H0,a = — i (V — ia)2 as defined (when A = Rd) in Example 11.4.12 (see (11.4.17) with V = 0) and, at the end of Section 11.6, in (11.6.8) under more general conditions than here.] The case when a = 0 was obtained earlier in [BivPi]. Remark 13.5.21 We will see in Section 13.6 that—according to the work in [BivLa]— "the" natural conditions both on the complex-valued scalar potential W (exactly the
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same as in [BreKat]) and the vector potential a (for example, a in (L2loc(A))d ratherthan in (L4loc(A)) d ) can be assumed provided we replace the semigroups by the corresponding resolvents on the left-hand side of (13.5.38). By specializing to the case when A = Rd \ J, where / is a closed set of capacity 0, and by letting W — -iV, with V : Rd -> C (and with p = 0, for simplicity), we obtain the following extension of Theorems 13.5.13 and 13.5.16, which was not explicitly stated in the abovementioned references [BivPi, Bivl-3]. (To obtain Theorem 13.5.22, we use—just as in deducing Theorem 13.5.16 from Theorem 13.5.13—Lemma 13.5.14 and the fact that H := L2(Rd) = L2 (Rd\J), since J has Lebesgue measure 0.) Theorem 13.5.22 Let V : Rd -»• C be a measurable function and let J c Rd be a closed set of capacity 0. (a) (Scalar potential) Then the counterpart of Theorems 13.5.13 and 13.5.16 holds under the weaker assumption that V E L1loc\ J) and Im V < 0. It follows that Ft,nTP,an(V), the analytic in mass Feynman integral viaTPF associated with the potential V (as defined in Remark 13.5.17(a) above), exists under these hypotheses for every n E C+ and for Leb.-a.e. nonzero n0 = — iqo. (b) (Scalar and magnetic vector potentials) In addition, we can replace the free Hamiltonian HO = — 3 A by Ho,5 = — j(V — ia)2, where the "magnetic vector potential" a = (ai,..., ad) : Rd -» Rd satisfies both a € (L4loc(Rd \ J; R)) d and V • a E L2loc(Rd\J). The domain of C (or more generally Cn, with n > 0) is then defined appropriately, in the same way as in part (b) of Theorem 13.6.4 below. We close this section by posing a problem which does not seem to have been addressed in the literature. (A possible way of tackling it was suggested in Remark 13.5.17(c) above.) Problem 13.5.23 (Generalized Feynman-Kac formula with complex, singular potential) Determine for which complex-valued potentials V the "generalized Feynman-Kac formula" (13.5.9) holds. Specifically, does it hold under the assumptions of Theorem 13.5.13 and more generally, of Theorem 13.5.22 (with a = 0)? In particular, does it hold for V real-valued on Rd and V € Llloc(Rd \ J), with J as abovel Looking ahead at Section 13.6, does it hold under the more general assumptions of Theorem 13.6.1 below"? A reader familiar with the Brownian bridge stochastic process (see, for example, [Si9, pp. 40-41 and §V. 15]) may try to answer the same questions as in Problem 13.5.23 in the case when a = 0; that is, in the presence of a magnetic field.
13.6 The analytic-in-mass modified Feynman integral In this section, we discuss joint results of Bivar-Weinholtz and Lapidus—obtained in [BivLa, §3] and concerning an extension of the "modified Feynman integral" F t M ( . ) ([Lal,2, La6-13], [BivLa, §1-2]) discussed in Sections 11.4-11.6 of this book. (See, in
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particular, Definition 11.4.4 where Ft M (-) was formally defined.) We will call here this extension the "analytic-in-mass modified Feynman integral" and denote it by F t,n M,an (-)Recall that in Section 11.4 (based on the results of [Lall] presented in Section 11.3), the existence of the "modified Feynman integral" F t M ( V ) was established under (essentially) the most general assumptions for which the quantum-mechanical Hamiltonian H = — j A + V = HO+V can be defined without ambiguity (and hence the associated energy functional, 5 f ||Vu||2 + f V|u| 2 , is a bounded from below quadratic form). Namely, V : Rd -> R is measurable, V = V+ - V_, with V+, V_ > 0, V+ E Llloc(Rd) (or more generally, V+ is an arbitrary, nonnegative measurable function) and—most importantly from the present point of view—V_ is H0-form bounded with bound less than 1 (for example, V_ E Kd(Rd) or, in particular, by Remark 11.4.3, V_ € Lr(R d ) + Loo(Rd), with r > d/2 if d > 2, r > d/2 if d = 2 and r = 1 if d = 1); see Theorem 11.4.2 and Corollary 11.4.5. Of course, under these conditions, the Hamiltonian H is a self-adjoint operator and is bounded from below. In addition, in Section 11.4 (Example 11.4.12), a magnetic vector potential a = ( a 1 , . . . , ad) was also allowed under the minimal hypothesis that a belongs to (L2loc(Rd; R ) ) d . [Naturally, the "free Hamiltonian" H0 must then be replaced by H0,2 = -1/2(V — ia)2 (called Aa in Example 11.4.12); see (11.4.17) with V := 0.] Moreover, recall that in Section 11.6—based on the results of [BivLa, § 1 and §2] and motivated in part by a mathematical model of dissipative quantum systems—we have allowed V to be complex-valued; in particular, the real part of V satisfied the same assumptions as in Section 11.4 while its imaginary part satisfied the "dissipativity condition" Irn V < 0; see Theorem 11.6.3. In addition, a vector potential a was also allowed in Section 11.6 under the same hypothesis on a as in Section 11.4; see the end of Section 11.6. The purpose of the present section is twofold: (i) Firstly, we discuss the consequences of the main results of [BivLa, §3] for the "analytic in mass modified Feynman integral" F t , n M , a n ( V ) associated with a potential V; see Theorems 13.6.1 and 13.6.4 below. Our hypotheses will be even less stringent than those made in Sections 11.4 and 11.6. In particular, the negative part of Re V will no longer be restricted significantly. Actually, our assumptions will be completely symmetric in the positive and the negative parts of Re V. Specifically, Re V will be assumed to be an arbitrary locally integrable function off a closed set of capacity 0 in Rd,without any sign restriction, and to have arbitrarily strong singularities on the "exceptional set" J or at infinity; see Theorem 13.6.4. It will follow that when V is real-valued, as is the case in standard quantum mechanics, the aforementioned energy functional, j / ||V«||2 + f V\u\2, can take arbitrarily large negative values and hence is usually no longer bounded from below. The price to pay for this increased generality will be —much as in Nelson's second approach to the Feynman integral discussed in the previous section—that Ft,nM,an (V) will only exist for almost every value of the parameter A. on the imaginary axis—and hence, for Leb.-a.e. value of the "mass" of the corresponding quantum particle; see part (III) of Theorem 13.6.4 below. Further, as in the previous section or in the study of complex potentials made in Section 11.6, the corresponding
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time evolution will no longer be reversible. Mathematically, this is translated into the fact that the associated (Co) semigroup Tn(t) = e~tCn(t > 0) is usually not a group and hence that a suitable translate of Cn is an w-accretive operator but is no longer skew-adjoint, even for A purely imaginary. Here again, a vector potential a will also be allowed under significantly weaker hypotheses on a than in Sections 11.4 or 11.6. Specifically, the components of a will be allowed to have arbitrarily strong singularities on the exceptional set J or at infinity, and to be any locally square-integrable functions off 7; see part (b) of Theorems 13.6.1 and 13.6.4. (ii) Secondly, we represent the semigroup associated with a Schrodinger operator with highly singular complex potential via a product formula for the resolvent of the free Hamiltonian (or a suitable substitute thereof) and the imaginary resolvent of (the multiplication operator by) V; see Theorem 13.6.7 below, from [BivLa, §3]. [Recall that in Sections 11.4, where V was assumed to be real-valued, we have used instead (with A = HO and B = V) a product formula for the imaginary resolvents of two self-adjoint operators A and B obtained in [Lall]; see Theorem 11.3.1. Similarly, in Section 11.6, where V was assumed to be complex-valued, we have used Theorem 11.6.1 (from [BivLa, §1] ) which extends from self-adjoint to normal operators the product formula for imaginary resolvents established in Section 11.3.] We will be able to deal in Theorem 13.6.7 with the same class of singular complex-valued scalar potentials V as that considered by Brezis and Kato in [BreKat]. Further, as before, we will allow for a singular magnetic vector potential a of the same type as in (i). More generally, we will show (as in Section 11.6) that the "free Hamiltonian" can be replaced by a uniformly elliptic, second order operator with variable and possibly discontinuous coefficients. A special case of the product formula obtained in Theorem 13.6.7 yields Theorem 13.6.1 discussed in (i), and therefore establishes (when a := 0) the existence of Ft,nM,an( V), the "analytic-in-rnass modified Feynman integral" associated with V. The proof (and even the statement) of Theorem 13.6.7 is somewhat technical, and therefore some readers may wish to omit it on a first reading. However, we do include it here because—given the material discussed in Sections 11.6 and 11.3—it can be presented reasonably concisely. Moreover, it makes use of (a suitable version and extension of) Kato's distributional inequality (Theorem 11.2.3), as well as of techniques from the theory of elliptic partial differential equations not previously encountered in this book. It also gives the flavor of the proof of Theorems 13.5.13 and 13.5.22 (regarding the existence of the "analytic in mass Feynman integral via TPF") that was omitted in Section 13.5. We will continue Section 13.6 by comparing the various approaches to the Feynman integral via analytic continuation in mass considered in this and the previous section. We will stress, in particular, the unity between these approaches, within the intersection of their domains of validity; see Theorems 13.6.10 and 13.6.11. This material was not included in [BivLa]. Finally, we will illustrate these results by discussing the case of highly singular central potentials, and in particular, the inverse-square potential; see Examples 13.6.13 and 13.6.18. Even though such potentials are so singular that they possess certain unphysical features (such as unbounded from below total energy), they present some very interesting
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mathematical challenges (see, for example, [Nel, Case, R]) and also provide useful phenomenological models for various situations in molecular chemistry and in quantum physics. (See especially the survey article [FrLdSp] as well as the more recent physical references [MarPari, PariZi, GupRaj, HenRajl,2].) Existence of the analytic-in-mass modified Feynman integral We first state in Theorem 13.6.1 a product formula for resolvents (one of which is imaginary) that is the key to deduce the existence of F t,n M,an (V); see Theorem 13.6.4. As was mentioned above, in view of Lemma 13.5.14, it is a special case of the main result of [BivLa, §3], Theorem 13.6.7, which will be stated and established below; see Remark 13.6.9. Theorem 13.6.1 Let J C. Rd be a closed set of (Newtonian) capacity 0 in Rd. Let V : Rd ->• C be a measurable function such that V E Llloc (Rd\ J); i.e. V is integrable on every compact subset of Rd\J. Further, assume that the "dissipativity condition " Im V < 0 is satisfied. [The case where V is real-valued—and so Im V = 0—is that of most interest in quantum mechanics.] (a) (Scalar potential) Then there is a (Co) contraction semigroup ( T ( t ) = e~tc : t > 0} such that the following product formula for resolvents (one of which is imaginary) holds: For all t > 0 and every W E L2(Rd), we have
where the limit is in the norm on L2(Rd) and holds uniformly for t in bounded subsets of [0, +00). Here, H0 = — jA denotes the standard free Hamiltonian acting in L 2 (R d ). Further, the m-accretive operator C is a suitable m-accretive realization in H = L 2 (R d ) of the Schwdinger operator -^A + V(x), x E Rd; more specifically, C is given by Theorem 13.6.2 below. (b) (Scalar and magnetic vector potentials) Assume, in addition, that a = ( a 1 , . . . ,ad) is a measurable, vector-valued function on Rd belonging to the space(L2loc(Rd \J; R)) d . Then the analogue of the product formula (13.6.1) holds provided we replace HO = — A by H0, a = -1/2(V — ia)2, the free Hamiltonian with magnetic potential a. (Here, H0,a is acting in L2(Rd) and is defined as in Example 11.4.12, by (11.4.17) with V := 0, or equivalently, in view of Lemma 13.5.14, as at the end of Section 11.6 on the open set A := Rd \ J.) Moreover, the definition of C — C% is modified accordingly. (See Remark 13.6.9 and the discussion preceding the statement of Theorem 13.6.7.) We next specify the m-accretive operator C occurring in Theorem 13.6.1 (a) or equivalently, the m-dissipative operator —C which generates the (Co) contraction semigroup ( T ( t ) = e-tc : t > 0} in (13.6.1). Define the (linear, unbounded) operator E in H. := L2(Rd) by
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with domain
Theorem 13.6.2 Under the hypotheses of Theorem 13.6.l(a), the operator E defined by (13.6.2) is closable in H = L 2 (Rd) and its closure E coincides with the m-accretive operator C introduced in Theorem 13.6.1(a); in short, C = E. Further, for u € D(C), we have u E Hl(Rd) and
where (., .)L2(Rd) denotes the inner product in L 2 (R d ). Proof This follows from [BreKat, Theorem 3.1, p. 146] and from Lemma 13.5.14 applied to the open set A := Rd \ /. D Remark 13.6.3 In order to state Theorem 13.6.4 below, we will also need the following variant of Theorem 13.6.2. Given n > 0, let En be defined by (13.6.2), except with — ^ instead of — j in (13.6.2a) and (13.6.2b). Then En is closable and Cn := En is an m-accretive operator. (Observe that with the notation of Theorems 13.6.2 and 13.6.1, we have E\ = E and C\ = C.) Naturally, the quadratic form (or "energy functional") \ f \\Vu\\2 + f V\u\2 is then replaced by ^ / \\Vu\\2 + f V\u\2, so that the displayed equation in Theorem 13.6.2 must be modified accordingly. We are now ready to state (more explicitly than in [BivLa]) the main existence theorem (Theorem 13.6.4) for Ft,nM,an (.)- It is the counterpart in our present context of Theorems 13.5.16 and 13.5.22, which dealt with the existence of Ft,nM,an(.). Note, however, that the hypotheses of Theorem 13.6.4 on the scalar potential V are weaker than in Theorem 13.5.16 (where a was set equal to 0), while those on the vector potential a are weaker than in Theorem 13.5.22, where a was assumed, in particular, to be in (L4loc(Rd \ J))d rather than in the smaller and more natural space (L2oc(Rd \ J ) ) d . [The former condition on a is related to essential self-adjointness results in [LeiSd] (see Remark 11.4.13(a)) for Hamiltonians with magnetic vector potentials, whereas the latter is that expected from the more general point of view of quadratic forms (see Example 11.4.12).] Moreover, even though the hypotheses of Theorem 13.6.4 are exactly the same as in Theorem 13.6.1 above, we will repeat them for clarity of exposition. Theorem 13.6.4 (Existence of the analytic-in-mass modified Feynman integral, for a highly singular complex potential with "unrestricted" real part) Let V : Rd —»• C be a measurable function such that V E L1loc(Rd\J), where J C Rd is a closed set of (Newtonian) capacity 0 in Rd. Assume further that V satisfies the "dissipativity condition" Im V < 0. [Recall that the case where V is real-valued—and hence Im V = 0--corresponds to the situation of ordinary quantum mechanics. In that case, V is allowed to be any real-valued locally integrable function off J, without any sign restriction, and can have arbitrary singularities on the "exceptional set" J.]
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(a) (Scalar potential) Then: (I) (Imaginary mass) Let A. > 0 be given. Then, for every t > 0 and W E L2(Rd), we have
where Cn is the m-accretive operator defined just as in Theorem 13.6.2, except with — 2^ instead of — y. (See Remark 13.6.3 above.) Here, H0 denotes the standard free Hamiltonian acting in L 2 (Rd). (II) (Complex mass) Let A. 6 C+. Then the limit on the right-hand side of (13.6.3) continues to hold and defines a (Co) contraction semigroup, still denoted Tn(t) (t > 0). Further, for all t > 0, the function n Tn(t) is analytic from C+ to L(L 2 (R d )). (III) (Real mass) For Lebesgue almost every nonzero purely imaginary number no, the operator-valued function n > Tn(t) has a (strong) nontangential limit (in the sense of Definition 13.5.1'), when X approaches n0 through C+; that is, for every t > 0 and a 6 (0, n/2), 7n(t) converges in the strong operator topology as n —>n0through the wedge Wa(n0) to an operator in L ( L 2 ( R d ) ) , which we still denote Tn 0 (t). We will say that the "analytic-in-mass modified Feynman integral" associated with the (scalar) potential V exists (see Remark 13.6.5(a) below) and denote it by F t , n M , a n (V); so that F t , n M , a n ( V ) = Tn(t) for all n E C+ and for Leb.-a.e. nonzero purely imaginary A.. Finally, for every nonzero purely imaginary number A.Q such that Tn 0 (t) exists, the operators (T^Q(t) : t > 0} form a (Co) contraction semigroup on H = L 2 (R d ), with infinitesimal generator denoted —Cn 0 . Moreover, if we write A.Q = —imo (with mo real and nonzero), then for every fixed
with initial state W(0, •) = (p. (b) (Scalar and magnetic vector potentials) Assume, in addition, that a = ( a 1 , . . . , ad) is a vector-valued function on Rd belonging to (L 2 l o c (Rd\J; R))d. Then the counterpart of (I) through (III) holds, provided that, in particular, we replace the free Hamiltonian H0 = — 5 A by the nonnegative self-adjoint operator H0,a — — j(V — ia)2 (acting in L 2 (R d ) and defined as in Example 11.4.12 by (11.14.17) with V :— 0 or equivalently, in view of Lemma 13.5.14, as at the end of Section 11.6 with A := Rd \ J). It follows that the "analytic-in-mass modified Feynman integral", associated with the scalar potential V and the magnetic vector potential a, exists under these hypotheses (for every n E C+ and for Leb.-a.e. nonzero n0 = —iqo). Proof This follows from Theorem 13.6.1 (and Theorem 13.6.2) in the same way as Theorem 13.5.16 followed from Theorem 13.5.13.
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(I) More precisely, part (I) (with X = 1) follows from Theorems 13.6.1 and 13.6.2. [Of course, for X > 0, it suffices to replace — y by — ^ in case (a), and more generally, -j(V - ia)2 by -i(V - ia)2 in case (b).] (II) Part (II) follows from part (I) exactly as Theorem 13.5.6 followed from Theorem 13.5.4 via analytic continuation by an application of an operator-valued version of Vitali's convergence theorem (see Theorem 11.7.1(1) and Remarks 11.7.2(b),(c)). Note that for all nonzero A. eC+and all f > 0,wehave||[/+f(#oA)r 1 || < 1 and||[/-MWr'|| < 1 because Im V < 0. (III) Finally, part (III) is a consequence of part (II). It is obtained in much the same way as Theorem 13.5.7 was deduced from Theorem 13.5.6, by using an operator-valued version of the classical Fatou-Privaloff theorem [Ru2, Theorem 11.23, p. 244] for the boundary value of a bounded analytic function. (We leave it as an exercise for the reader to prove the statement regarding the distributional Schrodinger equation (13.6.4).) D Remark 13.6.5 (a) Let V : Rd ->• C be a measurable function. Then ^^(V), the "analytic-in-mass modified Feynman integral" associated with the potential V, is defined exactly like F t,n M,an (^) in Remark 13.5.17(a) above, except with the product formula for semigroups (13.5.43) replaced by (13.6.3). Hence, the product formula for resolvents (13.6.3) must hold for all X > 0 (as in part (1) of Theorem 13.6,4); further, it must continue to hold for all A e C+ and define Tn(t) e £(L2(Rd)), which depends analytically on n e C+ (as in part (II)). Finally, for Leb.-a.e. nonzero purely imaginary n0, Tn(t) is required (as in part (III)) to have a nontangential limit as n — n0 through C+. (b) In part (I) of Theorem 13.6.4, the fact thatfor fixed X > 0, {7x(0 = e~'c^ : t > 0} is a (Co) contraction semigroup on L2(Wi) follows from Theorem 13.6.1 (and thus from Theorem 13.6.7 below) since by Remark 13.6.3, — CA. is m-dissipative on L2(E.d) and hence generates a (Co) contraction semigroup, by the Lumer-Phillips theorem reviewed in Section 9.4. Moreover, the corresponding statement regarding {71(0 '• t > 0} in parts (II) and (III) of Theorem 13.6.4 follows much as in [Nel], as was explained in the proof of Theorems 13.5.6 and 13.5.7, respectively. (We note that a similar comment could be made about Theorems 13.5.16 and 13.5.22.) Product formula for resolvents: The case of imaginary mass We will now precisely state and then establish the product formula for resolvents obtained in [BivLa, §3]. It corresponds to the (generalized) Schrodinger equation with singular magnetic and complex scalar potentials (denoted a and V, respectively), as well as with "imaginary mass" (in the above sense). When specialized in the obvious way, it yields the key part of Theorem 13.6.4 above; namely, part (I) corresponding to the case of "imaginary mass". (See Remark 13.6.9 below.) At first, it will be convenient to contrast our hypotheses and notation with those of Section 11.6 [BivLa, §1 and §2] which dealt with the "modified Feynman integral" with a singular complex potential (as well as with a magnetic vector potential), defined via a product formula for imaginary resolvents (Theorem 11.6.1 [BivLa, §1], extending Theorem 11.3.1 [Lai 1 ] from self-adjoint to normal operators).
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With the notation of Section 11.6 (see especially Theorems 11.6.1 and 11.6.3), we can take A = —iAi = i^ (where A denotes the Laplacian with Dirichlet boundary conditions on an open subset Q of Rd) and B — V (the multiplication operator with the scalar potential V — q+ — q- — iq'). But then, the hypotheses of Theorem 11.6.1 would imply that ^_ is essentially bounded, since now A\ = 0. In this case, however, we can modify the proof of Theorem 13.6.1 to obtain the analogous result by assuming only that #_ belongs to L]1OC(S2); i.e. the real part of the potential can be any locally summable function on £2. [As the reader will see in the proof of Theorem 13.6.7 below, a number of very significant changes are required, the most interesting of which involve a suitable version of Kato's distributional inequality in this context (Lemma 13.6.6) and the use of results concerning the weak (or distributional) solutions of an elliptic equation. (For notational simplicity, we consider here the case when A. = 1; otherwise, it suffices, of course, to replace A by A."1 A, with X > 0.)] We can also add a nonnegative imaginary part p to the potential, with some boundedness hypothesis relatively to — ^A, provided that, in the product formula, p appears in the same resolvent factor as — 5 A; see the left-hand side of the displayed equation in part (i) of Theorem 13.6.7 below. (Otherwise, the resolvent associated with the potential would, in general, be unbounded.) Finally, as was done at the end of Section 11.6, we can replace — j A by an arbitrary second order, uniformly elliptic operator L. Under these general hypotheses, we can no longer define "A + B" as a form sum in a suitable way, but we can consider an m-accretive realization of "i(A + B)", defined in [BreKat] for the case A = '-£• and in [Bivl,3] for A = —iL. We shall make the same assumptions on the operator L = L (a) as at the end of Section 11.6. Namely, let [bjk }d.k=l be a symmetric, uniformly elliptic matrix of real-valued Loofunctions on A, and let a = ( a 1 , . . . , ad) e (L 2 O C (A; R)) d denote the magnetic vector potential. Then, formally, L is defined by equation (11.6.8):
More precisely, L is defined as the maximal restriction of the operator L (viewed as taking values in the space of distributions D'(£2)) defined just after equation (11.6.8). [Assume that [bjk] = j/, where / denotes the identity matrix of order d. Then we simply have L = L(a) = — j(V — ia)2 = //o,3> with Dirichlet boundary conditions on A, much as in Example 11.4.12 where we had assumed that A = Rd. Of course, if in addition, £2 = Rd and a = 6, then L = L(0) = -^A = HO, the standard free Hamiltonian acting on L2(Rd). More generally, although this may seem less obvious to some readers, it follows from Lemma 13.5.14 that if A = Rd\J, with J a closed set of capacity 0 in Rd, then L = L(a) = -j(V - ia) acting in all of L2(Rd); in particular, L = L(0) coincides with the free Hamiltonian HQ acting in L2(R.d).]
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Moreover, we will make the following hypotheses on the complex-valued scalar potential, which are exactly the same as in [BreKat]:
where q,p,q' are real-valued functions, with
p, q'>Q,
(13.6.6b)
and
when d > 2, for some arbitrarily small e > 0.
(13.6.6d)
[Caution: In (13.6.6a), (13.6.6b) and in the following, the function denoted by q' is not the derivative of q.] We note that hypothesis (13.6.6c) can be replaced by the weaker assumption that e P ^loc^-* 's infinitesimally form bounded with respect to — ^ A; i.e. for every fixed y > 0, there exists 8(y) > 0 such that
Since \\V\u\\\ < ||(V - ia)u\\, for all u e Q(L) (cf. [LeiSd] or [Biv3]), this hypothesis implies that we can define the form sum L + (—p + 8) = L — p + 5 as a positive selfadjoint operator with form domain Q(L), where we have set S = S(a) (in the notation of (13.6.6c')) and where a denotes the ellipticity constant of {bj k } d j , k = 1 . (The operator L — p + S is the maximal restriction to an operator on H := L 2 (Q) of L — p + S e C(Q(L), Q(L)'); see [Biv3, II, §2].) Then it is known that the operator iG such that
is closable, its closure iG being such that iG+8 is m-accretive', see [BreKat] for the case L = —5 A, and [Biv3] for the general case. In the adaptation of the proof in [BreKat] to a general L, the following lemma —which is an operator-theoretic version ofKato's inequality (Theorem 11.2.3), extended to generalized Schrodinger operators with scalar as well as magnetic vector potentials—plays a crucial role, and it will also be used later on in the proof of Theorem 13.6.7.
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Lemma 13.6.6 Under the above hypotheses, let g e L°°(£2) with a nonnegative real pan, and let u e Ji = L2(tt). Then, for all K > S :
Proof of Lemma 13.6.6 This lemma is closely related to Lemma 6 of [LeiSd], the proof of which can be easily adapted to this situation (see [Biv3, Lemma 2.1, p. 28]). We just note that in [LeiSd], although g is (real-valued and) nonnegative and p = 0, g is only assumed to be in Llloc(A). Since we suppose here that g E Loo(A), we do not need to start with u in Loo(A) and hence the proof does not require the use of Lemma 4 in [LeiSd]. D The solution of the generalized Schrodinger equation with imaginary mass and singular complex scalar (or electric) potential V, along with singular magnetic vector potential a,
is given by the semigroup e l'c, for which one has the following product formula ([BivLa, Theorem 2, p. 457]). [For the reader's convenience, in the statement of Theorem 13.6.7 and in Remark 13.6.9 following its proof, we will explain in much greater detail than in [BivLa] how to deduce Theorem 13.6.1—and thus also Theorem 13.6.4 establishing the existence of the analytic in mass modified Feynman integral—from this product formula.] Theorem 13.6.7 (Product formula for resolvents: the case of imaginary mass) (i) Under the above hypotheses, we have as n —*• oo :
for all u e Ti. = L2(Q), uniformly in t on bounded subsets of [0, +00). (U) Assume in addition that p = 0, so that (by hypotheses (13.6.6a) and (13.6.6b)) V satisfies the "dissipativity condition" Im V < 0. Then, when (by letting {bjk} = \1) the uniformly elliptic operator L = L(a) is replaced by //o,3 — —\(^~ ia)2 and in particular (when a = 0) by HQ = —\&—as is the case in Theorem 13.6.1—the conclusion of part (i) of Theorem 13.6.7 still holds, except that we can let S = 0 in the above statement of the product formula. Moreover, if we further specialize to the situation of Theorem 13.6.1, where V is a complex-valued function on R1 which is locally integrable off a closed set J c Rd of capacity 0, then the above product formula (with S — 0) yields the product formula of Theorem 13.6.1; namely, formula (13.6.1) in the case when a = 0, as in Theorem 13.6. l(a), or its counterpart (with H0 replaced by H0,a = — ^(^ — ia)2) in the presence of a magnetic field, as in Theorem 13.6. l(b). Consequently, Theorem 13.6.1 (and hence also Theorem 13.6.4) follows from this special case of the present theorem. (See Remark 13.6.9 below, along with the comment preceding the statement of (l3.6.6a) above.)
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Remark 13.6.8 (a) Since iG + 8 is m-accretive, the limiting semigroup appearing in Theorem 13.6.7,
is well-defined as a (Co) semigroup of contractions on H = L2(A). (b) We have adopted notation that facilitates the comparison of the statement of Theorems 13.6.7 and 11.6.1. (c) In the proof of Theorem 13.6.7 given below, we will also (in so far as possible) use notation that is similar to that of Theorem 11.6.1 (as well as of Theorem 11.3.1). Two notable exceptions are the following: (i) We denote by G the operator which used to be called C in the proof of Theorems 11.6.1 and 11.3.1. (ii) In order to avoid possible confusion with the "mass parameter", what used to be called X in the proof of Theorems 11.6.1 and 11.3.1 is now denoted K. Proof of Theorem 13.6.7 We will prove part (i) of the theorem since as will be explained in Remark 13.6.9, part (ii) is a corollary of Lemma 13.5.14, Theorem 13.6.2 and the proof of part (i). We will stress the new features of this proof [based—as was alluded to above—on a suitable version in this context of Kato's distributional inequality (Lemma 13.6.6) and on regularity properties of the weak solutions of elliptic partial differential equations] and will omit the parts that are similar to the proof of Theorems 11.3.1 and 11.6.1. Fix K > 0 and v € H = L 2 (fi). We shall use the notation of Theorem 11.6.1 with
As will be indicated at the end of the proof, a density argument enables us to assume that v E L 2 (A) n L°°(A). For such a fixed v in L 2 (A) n Loo(A), we now show that
as t ->• 0+, weakly in H. Here, wt is defined as in the proof of Theorem 11.6.1 (just above equation (11.6.3)), except with n replaced by K. [More precisely, as in the proof of Theorem 11.3.1 or 11.6.1, we first show that the limit in (13.6.7) holds along a sequence {tn}—> 0+, and then deduce from a compactness argument that (13.6.7) holds as t ->• 0+.] Fix t > 0. We have
but now wt is the solution of the following elliptic equation:
with
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and
[Note that like their counterpart in the proof of Theorem 11.6.1, Rt, Iq', and Iq, are bounded self-adjoint (i.e. Hermitian) and nonnegative operators on W.] As in the proof of Theorem 11.6.1, we can easily bound
independently of t; and so we can justify that along a sequence tn -> 0+, w, converges weakly in H to
with
In particular, q'w e £^(£2). (Recall that q' does not stand for the derivative of q.) If we multiply (13.6.8) by a test function
and integrate on £2, we can pass to the limit on the right-hand side of the equation obtained in this fashion, much as in the proof of Theorem 11.6.1. We point out the differences in the argument. For the term involving Rt := RB,, one has to use
a consequence of the Lebesgue dominated convergence theorem and of (the counterpart of) estimate (11.6.4), combined with the hypothesis V e £^(£2). For the last term,
we observe that by an additional application of Lebesgue's dominated convergence theorem, we have
either in L 1+£ (Q) or in L'(£2), according to the hypotheses on q or p (cf. (13.6.6d)). To pass to the limit in this term, it is then sufficient to obtain a bound on w, either in L f l + e ) (Supp tp) or in L°°(Supp (p), independently of t. (Here, Supp p denotes the support of (p and (1 + e)' = 1 + e~' denotes the conjugate exponent of 1 + e.) This can be achieved by using Lemma 13.6.6 with g :— Rt + Ig> + ilqt.
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[Since we are now in a concrete situation, we can think of Rt, Iq>, and Iqi as multiplication operators by (essentially) bounded functions. With the usual identification between a multiplication operator and the associated function, it is then immediate to check from the definitions above that g belongs to L°°(£2).] More precisely, a simple algebraic manipulation based on (13.6.8) enables us to apply Lemma 13.6.6 with the above choice of g to obtain (since L = L(a)):
Thus, we deduce that
where ^ is the unique solution in Hl0 (A) of the elliptic equation
But now, hypothesis (13.6.6c') and an easy generalization of [BreKat, Theorem 2.3, p. 143] ensure (by a "bootstrap argument") that
[In order to apply this extension of the above-mentioned theorem in [BreKat], we use here the assumption that v belongs to L2(A) n L°°(fi).] Hence, in particular, w, is uniformly bounded in L(1+£)'(£2), which allows us to conclude in the case when q e L110^e(fi). In the alternative hypothesis p e Lloc e(£2), we can apply standard elliptic local regularity (see, for example, [GigTr]) to equation (13.6.12) to conclude that W E Looloc(ft). Hence w, is uniformly bounded in L°°(supp (p); so, once again we can pass to the limit in (13.6.11), which guarantees,
From (13.6.6a), (13.6.6c'), (13.6.10) and (13.6.13), we deduce that
We can then pass to the limit in (13.6.8) in the sense of distributions, which gives for all
where h denotes the quadratic form associated with the nonnegative self-adjoint operator L (see the paragraph following equation (11.6.8)). [Here, (•, •} denotes the "duality
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bracket" between the space of distributions, D'(A), and the space of "test functions", 2?(A) = Coo00(A).] By definition of L (see the comments following equation (11.6.8)), we deduce from (13.6.15) the following equality between distributions (i.e. in D'(A)):
(Recall from (13.6.6a) that V = q + i(p — q').) We thus have, in particular,
In view of the definition of the operator G given just before Lemma 13.6.6, equations (13.6.14) and (13.6.16) now imply that
It thus follows that along a sequence {tn} of positive numbers tending to zero, wt tends weakly (in H = L 2 (A)) to
[Note that since iG + 8 — i(G — iS) is w-accretive (see Remark 13.6.8(a) above), its resolvent [K + i(G — iS)]~l is a well-defined bounded (contraction) operator on H, for every k > 0.] By (weak) compactness, we deduce that wt tends weakly (in H) to w as t —>• 0+, as claimed. Moreover, using the same argument as at the end of Step 1 of the proof of Theorem 11.3.1, we can easily conclude that in fact, wt tends strongly (in Ti.) to w as t —>• 0+; i.e. ||wt — w||2 —> 0, as t —>• 0+. Recall that we have assumed so far that v belongs to L 2 (A) n L°°(J2). However, the density of L 2 (A) n Loo(A) in L 2 (A) = H and the boundedness of the operators involved now enable us to reach the same conclusion for an arbitrary v E L 2 (A), as required. We thus obtain the exact counterpart of Step 1 of Theorem 11.3.1 (or 11.6.1). The rest of the proof of part (i) of Theorem 13.6.7 can be carried out exactly as in Step 2 of Theorem 11.3.1 above (from [Lall]). [Recall that this part of the proof of Theorem 11.6.1 (from [BivLa]) was also identical with that of Step 2 of Theorem 11.3.1.] D Our next remark explains how to deduce part (ii) of Theorem 13.6.7 from part (i), and consequently how to view Theorem 13.6.1 (and hence also Theorem 13.6.4) as a corollary of Theorem 13.6.7. (This reduction relies on Lemma 13.5.14 and Theorem 13.6.2.) It may be helpful to begin by reviewing the comment preceding the statement of hypothesis (13.6.6a). Remark 13.6.9 First, assume that {bjk} —1/2Iin the definition of the operator L, where I is the identity matrix, and let p = 0 in the definition of the potential V, as in part (ii) of Theorem 13.6.7. Then, a brief inspection of the above proof of part (i) of Theorem 13.6.7
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345
reveals that we can let S =0 throughout the proof, and hence also in the statement of the product formula. Indeed, under the present hypotheses, we no longer need to introduce the constant of uniform ellipticity of L and thus we can put S = 0 in the counterpart of (13.6.6c'). (It is only the existence of a positive imaginary part p that forces us in general to introduce a nonzero constant S on the left-hand side of the product formula in Theorem 13.6.7.) Next, assume in addition that the hypotheses of Theorem 13.6.1 are satisfied, as in the secondpartofthestatementofTheoreml3.6.7(ii);sothatwehave V : Rd — C, Im V < 0, and V E L1loc (Rd\J), where J C Rd is a closed set of capacity 0. Then we can set A = R d \J and apply Lemma 13.5.14 to deduce that L(a) = -1/2(V - ia)2 = H0,a (and, in particular, L(0) = —1/2A= H0) acting in all of L 2 (Rd); see the last comment preceding the statement of (13.6.6a). (Note that V can be written in the form V = q + i (—q')> with q' > 0, as in hypotheses (13.6.6a) and (13.6.6b), because it satisfies the "dissipativity condition" Im V = —q' < 0.) Consequently, in light of Theorem 13.6.2, we deduce that Theorem 13.6.1 holds. In particular, when a = 0, as in part (a) of Theorem 13.6.1, the product formula (13.6.1) holds, while when a ^ 0, as in part (b) of Theorem 13.6.1, the counterpart of (13.6.1) holds with HQ replaced by HQJ = — j(V — ia)2. Comparison with other analytic-in-mass Feynman integrals We now discuss material not included in [BivLa] but which is helpful in clarifying further the relationships between the different notions of analytic in mass (operator-valued) Feynman integrals considered in this and the previous section. As we have seen above, the analytic-in-mass modified Feynman integral is (so far, at least) the most general approach in this context, especially if we allow for singular C-valued scalar potentials V and/or for singular magnetic vector potentials a. However, for our present purpose^, we will now focus our attention on R-valued scalar potentials (as well as let 3 = 0) and make more restrictive assumptions on the function V. Our first comparison theorem supplements Corollary 13.5.18 and shows the unity— in the intersection of their domains of validity—between the various approaches to the Feynman integral considered in Sections 13.5 and 13.6. It follows from the results of [Nel], [Kat7], and [BivLa, §3] discussed in those sections. [Recall that the relevant definition of K t ( F - i v ) , with F-JV as in (13.5.8), is given in Definition 13.5.1', while that of Ft,P, an (V) (resp., F t , a n ( V ) ) is given in Theorem 13.5.16 and Remark 13.5.17(a) (resp., Theorem 13.6.4(a) and Remark 13.6.5(a)).] Theorem 13.6.10 Let V be a real-valued function on Rd that is continuous off a closed set of capacity 0 in Md. Then the analytic-in-mass operator-valued Feynman integral, K t ( F - i v ) , associated with the functional F-iv, the analytic-in-mass Feynman integral via TPF, Ft,p an (V), and the analytic-in-mass modified Feynman integral, Ft, an (V), associated with the potential V, all exist and coincide. More precisely, for all t > 0,
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for every A e C+ and for Lebesgue almost every nonzero purely imaginary A (say, for Leb.-a.e. A — —iqo, with go e R and 0} is a (Co) contraction semigroup. Finally, for A, > 0, each of the operators in (13.6.17) is also represented by a "generalized Feynman-Kac formula"; that is, when applied to an arbitrary vector W in L 2 (R d ), it is equal to the Wienerintegral onthe right-hand side of (13.5.9). Further, still for A. > 0, we have T\(t) — e~t Ck for all t > 0, where Cx is the m-accretive operator defined by (13.5.42) and, in particular, with domain D(C\) given by (13.5.42b). Proof This follows from Corollary 13.5.18 (itself a consequence of Theorems 13.5.4, 13.5.6, 13.5.7 and 13.5.16) along with part (a) of Theorem 13.6.4. D More generally, as our second comparison theorem shows, the analytic-in-mass Feynman integral via TPF and the analytic-in-mass modified Feynman integral agree with each other under minimal assumptions on V. Theorem 13.6.11 Let V be a real-valued function on Rd that is locally integrable off a closed set of capacity 0 in Rd. Then for all t >0,
for every A. e C+ and for Lebesgue almost every nonzero purely imaginary A,. Moreover, the equality (13.6.18) holds for every nonzero purely imaginary A such that one (and hence all) of the operators in (13.6.18) exists. Proof This follows from Theorem 13.5.22(a) (from [BivPi, Biv3]) and Theorem 13.6.4(a) (from [BivLa, §3]). D Remark 13.6.12 Note that, in Theorem 13.6.11, there is no longer the claim that for A > 0, the generalized Feynman-Kac formula holds; see Remark 13.5.17(c) and Problem 13.5.23. Instead, for A > 0, F^p ^(V) (resp., ^ X a n (V)) is defined as a limit of Trotter-like products involving, when V is real-valued, a semigroup and a unitary group (respectively, a resolvent and an imaginary resolvent); see equations (13.5.43) and (13.6.3), respectively. Highly singular central potentials—the attractive inverse-square potential This example suggests that, for highly attractive singular potentials, the solution of the Schrb'dinger equation obtained by [analytic in mass] Feynman integrals as developed here is physically relevant even though the operators may not be unitary. If the energy is not bounded below, the potential may produce collisions as well as scattering. Edward Nelson, 1964 [Nel, p. 338] Because of the problems concerning the physical interpretation of attractive singular potentials, physicists concluded for a long time that no significance could be given to any singular potential as regards the singularity at the center of force. Furthermore the mathematical difficulties made the
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problem of understanding singular potentials even more formidable. But it was finally realized that many aspects of singular potentials are physically meaningful, and that certain of their properties are relevant. In addition, mathematical techniques were developed for handling the calculational problems of singular potentials. W. M. Frank, D. J. Land and R. M. Spector, 1971 [FrLdSp, p. 37] Predazzi and Regge ([PreReg], 1962) were among the first to realize the usefulness of singular potentials as a formal "laboratory" for investigating many physical and mathematical ideas. They argued that physical interactions among particles in the real world are very likely highly singular in character. The study of regular potentials is unlikely to reflect this situation. Rather, the singular potentials, with their strong repulsion and lack of analyticity in the coupling constant, would be more likely to shed some reliable light on the physics of strong interactions. Others, as we shall see in this section, have used this argument in one variation or another to justify the examination of various features of singular potentials. W. M. Frank, D. J. Land and R. M. Spector, 1971 [FrLdSp, p. 74]
Let us assume that d = 3, for simplicity. Physically, this corresponds to a single nonrelativistic quantum particle moving under the influence of the scalar potential V. (Analogous examples can be given in every dimension d > 3 as well as for multiparticle systems.) Since a point has capacity 0 in R3 (see (3) of (13.5.6) above), in order for Theorem 13.6.10 (resp., 13.6.11) to apply, a potential V can have an arbitrarily strong singularity at the origin, say, provided V is continuous (resp., locally integrable) except at the origin in R3. We will now complete from our present perspective the study of highly singular (attractive) central potentials ( V ( x ) = -fi/ra, r = \\x\\, x € R3\{0}) made earlier in Example 11.4.7 from the point of view of the (standard) modified Feynman integral. (See Example 13.6.13.) We will then focus our attention on the very interesting special case of the attractive inverse-square potential (V(x) = —1/r 2 ). (See Example 13.6.18.) Example 13.6.13 (Highly singular attractive central potentials) Let us assume, as in Example 11.4.7, that V is an attractive inverse-power potential:
where r := \\x\\ denotes the length of x e R3\{0} and a, ft are given positive constants. Schematically, one can think of an electron rotating around (or spiraling down to) the nucleus of an atom (located at the origin of R3). The parameter ft is then proportional to the electric charge of the electron while a represents the strength of the interaction. Clearly, the larger a, the stronger the attraction between the electron and the positively charged proton of the nucleus, and hence the greater the singularity of the potential at the origin. Even though such situations are often described as nonphysical in the literature, they are used to model interesting physical problems. The survey article [FrLdSp] gives a lengthy discussion of the role played by highly singular potentials in the physics literature.
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Both attractive and repulsive potentials are considered there, with the emphasis on the attractive case. Sections I, V and VI of [FrLdSp] will give the reader a quick introduction to the ideas of that paper along with some of the many specific examples to be found in [FrLdSp]. We note that the inverse-square potential (a = 2 and ft = 1) is discussed in physical terms in [LL] and [Nel]. (The quote that begins this subsection is from [Nel].) This potential will be discussed further in Example 13.6.18. Of course, the standard interaction between an electron and a nucleus is that due to an attractive Coulomb potential (a = 1 : V(x) — —ft/r, with ft > 0). However, more complicated physical situations naturally involve highly singular central potentials of the type described here (as well as other central potentials that can be dealt with by means of the theorems mentioned above). As a first example, we mention that the electrostatic interaction between a charge Z and an induced dipole of polarizability p is described by the potential V(x) = -^Zpe2r~4, where e denotes the electron charge; see [FrLdSp, p. 36]. (In contrast, recall that the electrostatic interaction between two charges Z\ and Z2 is described by the classical Coulomb potential V(x) = Z1Z2e 2 r~ l .) Moreover, in physical chemistry, intramolecular interactions are often described phenomenologically by means of singular potentials; see, for example, [FrLdSp, pp. 37-38 and esp. § V. A] and the references therein for a discussion of several models of interactions between (polar and/or nonpolar) molecules, including the Lennard-Jones potential, V(x) — ar~12 — br~6 (a,b > 0) and its generalizations. (Note that this potential can be written as the sum of a repulsive and of an attractive potential.) See also [MarPari] and the references therein for a recent discussion of singular potentials in connection with the study of long-range interactions between polymers (or macromolecules), of interest in biophysics. As we have seen in Example 11.4.7, in the usual situations considered in the mathematical physics literature, the Hamiltonian is always bounded from below and thus we can always use the (standard) modified Feynman integral FtM (V) defined in Section 11.4 or the analytic-in-time operator-valued Feynman integral from Sections 13.2 and 13.3. Further, it is frequently the case that - ^ A -I- V is essentially self-adjoint, and hence we often can also use the (standard) Feynman integral via TPF F t p ( V ) defined in Section 11.2. More precisely, in the present situation, if a < 3/2 (for example, for the Coulomb potential where a = 1), then, by Corollary 13.4.2, all of Tt M (V), J i t ( F v ) and F t T P ( V ) exist and coincide with e~it H , where the Hamiltonian H is the unique self-adjoint extension of — j A + V; see region (I) of Figure 11.4.1. (Recall that the symmetric operator - j A + V, defined on C£°(R3\{0}), is essentially self-adjoint in that situation.) On the other hand, for stronger interactions (region (II) of Figure 11.4.1)— specifically,when3/2 < a < 2or(a =-2and/3 < 1/4)—only Ft M (V)and Jit(F v )can be used since, as was discussed in Example 11.4.7, we do not know whether F t p ( V ) exists. Then (in both regions (I) and (II) of Figure 11.4.1), by Corollary 13.4.1, we have Ft M (V) = Jit(Fv) = e-itH, where H = H0 + V denotes the form sum of H0 = -\ A and V—which, in this case, coincides with the Friedrichs extension [Kat8, §VI.3] of the symmetric operator — j A + V. Recall from Example 11.4.7 that the energy functional
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2 /R3 II^ M II 2 + /RS ^M2 is bounded from below in this situation. (We stress that in a notation compatible with that introduced in Theorem 13.6.14 below, we should replace 4A by -2^A and 5 /R3 IIV«il 2 by ^ /R3 y v w f , form0 > 0.) We now consider the only case not treated in Example 11.4.7; that of even more singular potentials, for which none of F t M ( V ) , Jit(Fv) or Ft P (V) is defined. (See Remark 13.6.15 below.) Namely, let us assume that either a > 2 or (a = 2 and ft > |); see region (III) of Figure 11.4.1. Then, leaving aside the border-line case when (a = 2 and ft — |), the above-mentioned energy functional is no longer bounded from below. This is so, in particular, for the attractive inverse-square potential (a = 2 and ft = 1), to be considered in Example 13.6.18 below. Since the central potential V is continuous away from the origin and—as was noted earlier—a single point has capacity zero in R3, the following result is just a corollary of Theorem 13.6.10 (as well as of Corollary 13.5.18 and Theorem 13.6.4 that led to it). Hence we will not repeat the entire statement of Theorem 13.6.10 but we will recall some of the main points and emphasize some new ones. Also, we will introduce a dimensional (mass) parameter mo (but continue to set h = 1). Theorem 13.6.14 (Highly singular attractive central potentials) Let the attractive central potential V be given by (13.6.19), where a and ft are arbitrary positive constants. Then the conclusion of Theorem 13.6.10 (and, a fortiori, of Theorem 13.6.11) holds for this potential. In particular, the analytic-in-tnass operator-valued Feynman integral K[(F-iv), the analytic-in-mass Feynman integral via TPF Ftp,an(V), and the analy'ticin-mass modified Feynman integral Ft,M (V) associated with V, all exist and coincide (for every A. e C+ and for Leb.-a.e. nonzero purely imaginary A.):
Further, set X = —imo, where mo > 0 can be thought of as the "mass" of the quantum particle. Then there exists a subset N C (0, +00) of (one-dimensional) Lebesgue measure 0 such that for all fixed mo € R\N, the family of bounded operators {T-imo(t) : t > 0} is a (Co) contraction semigroup on H = L2(R3), with generator an m-dissipative operator denoted C-imo. (We write T-imo(t) — e~ imo for t > 0.) Moreover, still for fixed mo 6 (0,+oo)\N, "the' formal "Schrodinger equation" with potential V
with initial condition W(0, •) =
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(13.6.21) provided by Theorem 13.6.14, but the "Schrodinger operator" with potential V (and so the "Schrodinger equation" itself) is given meaning via the use of any one of the above versions of the analytic-in-mass Feynman integral. (This fact will be illustrated in a striking manner in Example 13.6.18 below.) (b) Of course, as was already pointed out earlier (for instance, in Remark 13.5.8(a)), a significant drawback of the method of analytic continuation in mass is that in general, we do not know a priori for which values of the mass parameter mo the Schrodinger equation (with mass mo) is defined, whereas a quantum particle (like an electron, for example) has a well defined mass. (c) As was also noted earlier, the "Schrodinger equation" (13.6.21) is in general no longer time-reversible and the associated time-evolution is no longer unitary. Further, if we want to define the Schrodinger equation (13.6.21) for t < 0 (rather than for t > 0), we must first define K[(F-iv), .7>'p an^) and ^'M ^V^> f°r K < ° (rather than for A. > 0), then analytically continue to the left half-plane Re A. < 0 (instead of the right half-plane Re A. > OJ, and finally take the boundary value along the punctured imaginary axis (Re A = 0, A. ^ OJ. (Recall that A. and mo are connected throughout by the relation A = — imo-) It can be checked that for mo ^ (0, +oo)\N, we would then obtain another (Co) contraction semigroup, {7}mo(0 : t > 0), the adjoint semigroup of {T-imo(t) : t > 0} [inthe sense of'Theorem 9.6.14, except for Hilbert rather than Banach adjoints (see Remark 9.6.15(b))], and thus with generator —C*_im , the (Hilbert) adjoint of—C-imo. (An analogous comment applies to all the versions of the analytic-in-mass Feynman integral discussed in Sections 13.5. and 13.6; see, for example, the comment preceding Section 5 in [Nel, p. 336]for the case of K[(F-iv).) Exercise 13.6.16 Let V be given by (13.6.19), as above, and let N be the Lebesgue null subset o/(0, +00) defined in Theorem 13.6.14. (a) (Regions (I) and (11) of Figure 11.4.1.) Assume that a < 2 or that (a = 2 and ft < 1/4), so that by Example 11.4.7, the (standard) modified Feynman integral ^^(V) exists for all values of mo > 0. Then show that for all positive mo E N,
It follows that in this case, the semigroup {T_,-mo(?) = e tc-'mo : t > 0} is actually a unitary group and that Hmo := |C_,-mo is self-adjoint and equal to — ^- + V, the form sum of—^- and V. Further, it follows from Corollary 13.4.1 that the operators in (13.6.22) are equal to the analytic-in-time operator-valued Feynman integral J i t ( F v ) . (b) (Region (I) of Figure 11.4.1.) Assume in addition that a < 3/2, so that by Example 11.4.7, the (standard) Feynman integral via T PF FTp(V) exists for all values of mo > 0. Then show that for all positive mo £ N,
It follows that in this case, Hmo(= — ^- + V) = — ^- + V, the unique self-adjoint extension of the symmetric operator ~^- + V.
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Remark 13.6.17 (a) Of course, in (13.6.22) and (13.6.23), ^(V), J^M(V) and J"(Fv) correspond to the positive mass parameter mo (that is, to the operator — ^instead of — y in Definitions 11.2.21 and 11.4.4, respectively). [Earlier, there was no need to introduce a new notation because, as can be easily checked, each of these notions exists either for every mo > 0 or else not at all.] (b) (Repulsive central potentials) Some readers may wonder why we have not considered here the repulsive (rather than attractive) central potential V(x) = +ft/ra (rather than —f}/ra), with a, ft > 0. In fact, in that case—as was discussed at the end of Example 11.4.7—the (standard) modified Feynman integral exists for all values of mo > 0 and for every pair (a, /?) with or, /8 > 0, since V is a positive potential. Further, the same assertion holds for the analytic-in-time operator-valued Feynman integral J''(Fv); see Theorem 13.3.1 and Remark 13.3.3(a). Hence, there is no need to consider analytic-in-mass Feynman integrals in this situation. However, Theorem 13.6.10 can still be applied, and it can be checked (much as in Exercise 13.6.16 above) that for Leb-a.e. m0 € (0, +00), Ft,an(V) exists and coincides with both Ft M (V) and Jit(Fv). We point out, however, that—even though they are mathematically easier to deal with from our present perspective—such singular repulsive potentials are quite useful in various applications, particularly in molecular chemistry and aspects of phenomenological quantum field theory; see, for example, [FdLdSp, §IV.A and §V], [MarPari], [PariZi], and the relevant references therein. Example 13.6.18 (Attractive inverse-square potential) Let us now specialize the discussion in Example 13.6.13 by assuming that V is the attractive inverse-square potential (a = 2) with positive parameter ft; thus
In addition to the properties common to all attractive inverse-power central potentials considered in Example 13.6.13, this example possesses specific features of great interest, as we will soon see, especially in Remark 13.6.19 (3) and Theorem 13.6.21 below. (1) As we know from either Example 11.4.7 or Example 13.6.13, if ft < 1/4 (or rather, since the positive mass parameter WQ is now taken into account, if fmo < 1 /4), the total energy functional ^- / IIv «ll 2 + / V|w| 2 is bounded from below and there is a distinguished self-adjoint extension of the symmetric operator S := — j~—h V; namely, the form sum H = ~^~ 4- V. As was discussed previously, the Hamiltonian H—defined as a form sum (or equivalently here, as a Friedrichs extension of S)—is the physically natural self-adjoint realization of S in this case, even though S is not essentially self-adjoint. (Recall that this situation corresponds to region (II) of Figure 11.4.1.) Then, for all values of the mass parameter mo > 0, the Schrodinger equation
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with initial state (p e D(H) c L 2 (K 3 ), is defined unambiguously and its unique solution at time t e R is given by
where F'M(V) is the (standard) modified Feynman integral and J"(Fy) is the analyticin-time operator-valued Feynman integral associated with the potential V. (Recall that FjpCV) is not known to exist in this case.) Remark 13.6.19 (a) It may be helpful for some readers to provide a bit more technical information about the standard differential equation approach to such problems. (For more details, see, for example, [Case], [Nel,§5], [R], [FrLdSp], and the relevant references therein.) First, by passing to spherical coordinates, we write
where "®" denotes the tensor product of Hilbert spaces and where S2 = {x e R3 : \\x\\ = 1} is the unit sphere in M3. Then, we have the "spherical harmonics decomposition ":
where Bj is the jth eigenspace of the square of the "angular momentum operator" (which has eigenvalues j (j + 1), for j = 0, 1 , . . . ) ; see, for example, [Han, Chapter 8], [LL, §32 and §357, or [Nel, §57. (The integer j is often referred to as the azimuthal quantum number or spin, in more modem terminology; for simplicity, we do not discuss here half-integer spins, which would require enlarging the Hilbert space of wave functions. In any case, in the following discussion, the specific value of j is irrelevant.) Putting together (13.6.27) and (13.6.28), we obtain the orthogonal sum decomposition:
where
Next, after making the change of dependent variable u(r) = rWrad(r), where Wrad denotes the radial component of a separable solution W of the (formal) Schrodinger equation decomposed according to (13.6.29), we are led to solving (for each fixed j e N = {0,1,...}):
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where K2 := | + (j(j + 1) —ftmo)(K€ C) andu(t, •) is required to be in L2(0, +00) for each fixed time t. [Here, in the second equality of (13.6.30), we have used the fact that 2m0V = -ftm0/r2, by (13.6.24).] Case (1) above and (2) or (3) below corresponds, respectively, to K2 > 0, K2 = 0 or K2 <0.
We can then consider the time independent differential equation corresponding to (13.6.30), along with the associated indicial equation. (One recognizes a form of Bessel 's equation with parameter K.) At least in the present case (I), the necessary argument is standard and can be supplied easily. (For cases (1)-(3), see [Case, §11],[Mee], [Nel, p. 338] or [R].) (b) According to the discussion in (a), for a general azimuthal quantum number (or spin) j E N, the "critical value" of ft is given by ftmo = \ + j(j + I) (instead of ^). Hence, the "subcritical", "critical" or "supercritical" case (case (1), (2), or (3), respectively) corresponds to ftmo being less than, equal to or greater than, respectively,
l + yO' + l)-
For notational simplicity, we assume implicitly throughout much of this discussion that j — 0, so that ft = ^— is the critical value. [Alternatively, one does not lose any essential information by working instead in the one-dimensional situation; namely, the half-line (0, +00) with a suitable boundary condition at r — QJ The interested reader should not have any difficulty in restoring the correct value of the parameter; see, for example, Remark 13.6.22(a) below. (2) For ft = 1/4 (or rather, for ftmo = 1/4), the potential energy is no longer bounded from below in terms of the total energy [Nel, p. 338] and so the form sum of HO and V is not defined; also, neither are F'M(V), J"(FV) nor FTP(V). However, a suitable "cut-off regularization" (see [R, §4] and Remark 13.6.20 below) provides a reasonable definition of the Schrodinger operator in this situation. By [Nel] and [R], the resulting operator still gives rise to a unitary evolution and coincides (for Leb.-a.e. value of mo) with the operator }C_,-mo defined in Theorem 13.6.14. Remark 13.6.20 Briefly, the above regularization can be described as follows. Given R > 0, one considers the cut-off potential defined by VK(X) = —ft/2r2 for r > R and VK(X) = —ft/2R2for 0 < r < R. One then shows that in the limit when R -»• 0+, one recovers the operator provided by the standard differential equation approach discussed in Remark 13.6.19 above; see [R, p. 545]. This regularization procedure also works (for j ^ OJ in the previous case (1), but fails in the highly singular case (3) considered below; see [R, %4] or [Mee, §57. (3) We now come to the most interesting case from the present perspective; namely, that when ft > 1/4 (or rather, ftmo > 1/4). Then, it is known that the above total energy functional is unbounded from below; so that there exists a sequence of normalized L2eigenfunctions [pn}'^=^ (or bound states) with energy tending to — oc as n —> oo. As in case (2), neither the form sum of HO and V nor the Friedrichs extension of — ^- + V is defined; also, neither Ft M (V), J i t ( F v ) , nor F t p ^ V ) exists. However, it has been shown
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by Case that for every fixed mo > 0, the symmetric operator
has a one-parameter family of self-adjoint extensions {Sp}peT, see [Case, §11]. [Here, T = (Sl)®(= [0, 2;r)N) can be viewed as an infinite dimensional torus (while Sl is the unit circle or 1-torus) and the parameter p = (Pj)jeli corresponds in each component indexed by j e N to a choice of boundary condition at the singular point r = 0 [R, p. 545] or to a phase factor PJ common to all the eigenfunctions of a particular self-adjoint extension [Case, pp. 798-799]. (See also Meetz's work [Mee] which makes explicit use in this situation of von Neumann's method [Sto, Theorem 10.20] for determining the self-adjoint extensions of a symmetric operator.)] A remarkable fact also due to Case is that these operators [Sp : p e T) constitute precisely all the self-adjoint extensions of 5 (see [Case, §11] or [R]). However, there is a priori no physically natural way to choose between these various extensions. In the following, given p e T, we will denote by {Up(t)}t£K the unitary group {Up(t) = e~"s» : t e R} associated with the self-adjoint operator Sp. On the other hand, we know from Theorem 13.6.14 that in this case, the various forms of the analytic-in-mass Feynman integral exist and coincide for Leb.-a.e. mo > 0. More precisely, there exists a Lebesgue null set N c (0, +00) such that for each fixed mo e (0, +oo)\N, a suitable "Schrodinger operator" jC_/ mo is well defined and the unique solution at time t > 0 of the corresponding "Schrodinger equation" (13.6.25) with initial state p 6 D(C_,-imo) c L2(R3) is given by (13.6.26) (or, equivalently, by (13.6.31) below). Recall that C_imo is an m-dissipative operator and hence generates a contraction semigroup [e~ imo }(>Q, but that ]-C_j mo is not self-adjoint and hence does not give rise to a unitary evolution. To summarize the situation, "the most striking difference [between the above two approaches] is that Case ([Case]) finds the [evolution] operators to be unitary but not unique whereas Nelson ([Nel]) finds them to be unique but not unitary" [R, p. 544]. It is then natural to wonder whether these two very different points of view can be reconciled. In fact, in an intriguing work, Radin [R] has established the precise relationship between them. In a nutshell, he has shown that "Nelson's nonunitary solution is a simple (time independent) average over Case's family of unitary solutions " [R, p. 544]. Without going into the details, we will now give a somewhat more precise statement of Radin's main results. The following theorem is a corollary of Theorem 13.6.14 above and of [R, Proposition, p. 547]. Theorem 13.6.21 Fixmo in(0, +oo)\N such that m0 ft > 1/4, where N is the Lebesgue null set given by Theorem 13.6.14. Then the nonunitary evolution T(t) — e~~'c-'mo (t > 0) on H = L2(R3) for the "Schrodinger equation"
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defined in Theorem 13.6.14 ([Nel], [Kat7], [BivPi], [BivLa]) is a time-independent average of the unitary evolution operators Up(t) = e~ltSf> obtained by the classical method discussed by Case in [Case]. More specifically, there is a (time-independent, regular Borel) probability measure P on T = (51)N such that for each fixed t > 0,
for all
for all < p e H , and that for (p e £>(C_,-mo) c H, T(t)
for all (p e Hj (that is, physically, for all states of azimuthal quantum number j or equivalently, of angular momentum ^/j(j + I)). [The construction of [R] is valid for every value of ft > 0 (whether critical, subcritical or supercritical). Thus, it is not necessary to specify the value of ft in Theorem 13.6.21 or in the above comments. However, for ft subcritical (i.e. niQft < \ + j(j + l)j, all the self-adjoint extensions {Sp}p coincide with each other and hence so do all the unitary groups {Up(-)}p; in that case, it follows that, as expected, we recover the known conclusion T(t) = Up(t) and thus T^(t) == U(£(t) for all t.] (b) Although it was first discovered in the present situation [R], the above "averaging phenomenon " is not unique to the Schrodinger equation with inverse-square potential. Indeed, an abstract counterpart of Equation (13.6.32) has later been obtained by Powers and Rodin in [PowR] for a large class of boundary value and Cauchy problems. It makes use, in particular, of Choquet's beautiful integral representation theory, a far-reaching generalization of the Krein-Milman Theorem ([Cho3, Vol. II, Theorem 25.12, p. 105]) for the extreme points of a weakly compact convex set in an infinite dimensional topological vector space; see Chapter 6 in Volume II of [Cho3], as well as [ChoMey] and [Phe]. In the following difficult problem, the operator T(t) is defined by K'imo(Fiv) for t > 0 and by K~'imo(FiV) for t < 0, as indicated in Remark 13.6.15(c). We caution the
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reader that to our knowledge, parts of this problem are still open. (We are grateful to Brian Jefferies for pointing out an inconsistency in the original formulation of Problem 13.6.23.) Problem 13.6.23 Assume that V(x) = -fir^2, with ft > 0 and /3|mol > 1/4 (as in case (3)). (a) Show that the time evolution T(t) contracts distances in L 2 (R 3 ) but usually does not preserve them. Conclude that (T(t) : t e R} cannot form a unitary group. (b) Give a plausible physical interpretation for the results obtained in (a). [Hint: With regard to questions (a) and (b), you may wish to consult [Nel]. We caution the reader, however, that there may be more that one possible answer to question (b) and that there does not seem to be a consensus among mathematical physicists as to which one is "correct".] (c) Investigate whether the analytic-in-mass Feynman integral of V can exist for all (nonzero) real values of the mass parameter mo. (d) Answer the analogue of (a)-(c) for the analytic-in-mass modified Feynman integral. (e) Finally, answer the counterpart of(a)-(d)for other attractive central potentials (including those considered in Example 13.6.13), as well as for more general highly singular potentials V (as discussed in Section 13.5 and in Section 13.6, respectively). We close this section by making a few additional comments of either a physical or a mathematical nature. The last two of those are rather open-ended and speculative but may point to some interesting future research directions in this area. Remark 13.6.24 (a) Naturally, a careful physicist or theoretical chemist—when confronted with the dilemma of choosing between the Hamiltonian approach ([Case, Mee]) or the analytic-in-mass Feynman integral approach discussed in the present example (and the previous one)—will not decide on purely mathematical or aesthetical grounds. Instead, the choice will be dictated by more pragmatic and physical considerations based on the particular features of the physical problem at hand. (b) Actually, there is a "third" approach to highly singular potentials more recently discussed in the physics literature. It is based on renormalization techniques and seems to strike a middle ground between the standard Hamiltonian approach (where "uniqueness" is lost) and the analytic-in-mass Feynman integral approaches (where unitarity of the time evolution is lost). See the paper by Gupta and Rajeev [GupRaj] and the relevant references therein, including [PariZi]; see also the papers by Henderson and Rajeev [HenRajl, 2] for a closely related work. (The second author is grateful to Teaman Tiirgut for pointing out these papers and for related conversations on the material of the present remark and on that of its continuation in Section 13.7, Remark 13.7.18(d) below.) This third approach begins with the observation that in the above-mentioned approaches, the ground state energy is physically meaningless (or equal to — ooj. In a nutshell, it consists in introducing a cut-off a > 0 (so that r = \\x\\ > a instead ofr > 0) and in letting /8 = fi(a) vary with a in the definition (13.6.24) of the inverse-square potential.
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The suitable dependence of ft on a (or "renormalization" of the parameter ft) is then determined by requiring that the ground state energy be finite and independent of a (as well as by imposing Dirichlet boundary conditions on the sphere r = ||jt|| = a); this amounts to implementing Wilson's renormalization scheme [Wits] in the context of quantum mechanics [GupRaj]. For the moment, this intriguing approach is far from being as mathematically developed as the other ones. It would be very interesting to investigate it further and, if possible, to relate it with the approaches discussed in the present example. (The authors of [GupRaj, HenRajl,2]—who are motivated in part by the study of the strong force in quantum field theory, namely quantum chromodynamics (QCD)—do not seem to be aware of Nelson's approach via the analytic-in-mass Feynman integral or of its later extensions presented in Sections 13.5 and 13.6.) Along similar lines, it is natural to wonder whether the stochastic process constructed in [HenRajl,2] in association with the Dirac measure in two dimensions (i.e., the potential V(x) = S(x) in R2) has a suitable counterpart in the present more difficult case when V(x) = —l/r2 in R3. We will return to this and related questions in Remark 13.7.18(d) below. (c) To put it a bit dramatically, a quantum particle submitted to a very singular attractive inverse-square potential (V(x) = —ft/r2, with ft > 1/4 as in case (3) above) is confronted with the following rather unpleasant alternative: (i) Either to "commit suicide " by "choosing " a nonunitary time evolution eventually leading to a collision with the center of attraction [LL, Nel], if one of the analytic-in-mass Feynman approaches is used, (ii) Or else to choose a privileged direction and hence "break the symmetry " of the system, if the Hamiltonian approach is used (which consists in selecting a particular self-adjoint extension HO + V, and hence a suitable boundary condition at the origin). However, no such choice is necessary if the potential is less singular (namely, if ft < 1/4, as in case (1) above) since then the total energy is bounded from below and so the natural Hamiltonian Ho + V (the form sum of HO and V) can be used unambiguously. [Similarly, when ft = 1/4 (as in case (2)), there is also a unique choice of Schrodinger operator.] Note that the analytic-in-mass Feynman integral approaches have the advantage of not breaking the symmetry of the quantum system (whatever the value of ft is) since by Theorem 13.6.21 [R], they amount to a suitable averaging over all possible self-adjoint extensions of HO + V. On the other hand, they lead to the eventual destruction of the existing system. It would be interesting to investigate the relationships between the present situation and the "spontaneous symmetry breaking phenomena" studied by Bost and Connes in [BosCon 1,2] (see also [Con2, § V. 11 ]), in connection with the Riemann zeta function and following in part earlier suggestions made by the physicist B. Julia in [Ju]. In the above example of the inverse-square potential, the "spontaneous symmetry breaking" occurs for values of ft greater than the critical value fto := 1/4; then, the natural rotational symmetry of the quantum system (that is, the invariance under the orthogonal group 50(3)) is broken. Correspondingly, in [BosCon] (and with an appropriate choice of notation), for ft < fto, there is a unique KMS^ state (and the associated factor is of type IIIi), whereas for ft > fto, there is a continuous family of KMSp states, indexed by the symmetry group of the system (and the associated factors are then of type IQO). The spontaneous symmetry breaking occurs at "low temperatures", corresponding to ft > /Jo-
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(We refer, for example, to [Con 1,2] for the terminology from the theory of operator algebras adopted here. Physically, in quantum statistical mechanics, given ft > 0, the "KMS0 states" can be thought of as the possible phases of the system at "temperature" T = f ) ~ l , and a "phase transition" occurs at the critical temperature TO — fi^1.) To summarize the above discussion, we conjecture (in a work in preparation by the second author [La21]) that a "phase transition with spontaneous symmetry breaking" (in the sense of [BosCon]) does occur at the "critical value" ft = PQ. In particular, for ft > PQ (i.e. at low temperature), the singularity of the interacting potential V(x) = —p/r2 is sufficiently strong to maintain the system in "thermodynamic equilibrium" in an asymmetric state, whereas for ft < PQ (i-e. at high temperature), it is mild enough for "the disorder associated with high temperatures [to cause] a unique homogeneous phase" ([Lac, p. 332]). (Naturally, since the models studied in [BosCon] and in the present example are rather different, the detailed structure of the KMS/j states cannot be expected to be the same.) Further, partly in light of [PowR] (see Remark 13.6.22(b) above), we expect that a similar conjecture can be made about many other highly singular attractive potentials. Pursuing this analogy one step further, one may then wonder whether, in the setting of [BosCon], a suitable analytic continuation—as in the approaches via the analytic in mass Feynman integral studied in Sections 13.5 and 13.6—would provide an appropriate average of all the KMSy3 states when P > fa. In addition, one may ask what the mathematical or physical meaning of this "average state" might be in this situation. (It should be noted that, in general, this "average state" will no longer be a "state" in the mathematical sense of the term used, for example, in the theory of operator algebras.) It may also be worthwhile to investigate the possible connections between these questions and aspects of the theory of "complex dimensions of fractals" and the study of geometric, spectral and arithmetic zeta functions conducted by Lapidus and his collaborators, Carl Pomerance, Helmut Maier, Christina He, and Machiel van Frankenhuysen, in [La22-26, LaPoml-3, LaMail,2, HeLal,2] and especially in [La-vFl-4]. See, in particular, the comments regarding mathematical "phase transitions", "complex dimensions" and the Riemann hypothesis, made in [La23, p. 176] and [La24, pp. 146-147, esp. Question 2.6, p. 147], in reference to the work of [LaMail,2] which provides a geometric characterization of the Riemann hypothesis in terms of a natural inverse spectral problem for fractal strings. We note that recently, the latter work has been significantly extended and put in a more conceptual framework in the research monograph [La-vF2] (announced in [La-vFl]). (See also [La27].) We leave the possible investigation of these and related problems to future work. 13.7
The analytic-in-time operator-valued Feynman integral via additive functionals of Brownian motion Our goal in this section is to give the reader some idea about recent results of Albeverio, Johnson and Ma [AlJoMa], even though the background information necessary for a precise understanding has not been supplied in this book. Most of the missing material involves the Wiener (or Brownian motion) process and includes smooth measures, positive continuous additive functionals of Brownian motion (in the sense of Fukushima
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[Fuk, FukOT]), a generalized Feynman-Kac formula, and the connection between these three subjects. Introductory remarks The results in [AUoMa] are considerably more general than those in Section 13.3, as we will see in Examples 13.7.17, 13.7.19, 13.7.20 and 13.7.23 below. Nevertheless, if we restrict V_, the negative part of the potential V, to be in the Kato class Kj (see Section 10.4) and make a corresponding restriction in the setting of [AUoMa] to the "generalized Kato class" GKj (see Definition 13.7.3 below), then there is a close parallel between the two sets of ideas which should help the reader gain some insight into the more recent work. We will begin by pointing out this parallel below—initially with minimal explanation of the new concepts and facts. After that, we will discuss briefly several of the ideas used in [AUoMa] and follow with the examples mentioned above which will illustrate contrasts and connections between Section 13.3 and [AUoMa]. Finally, we will use a more general result from [AUoMa] and give a related example, Example 13.7.23 below, where the positive part of a highly oscillatory and singular potential helps to "control" the negative part. As indicated in [AlJoMa, p. 292], it is likely that even more general existence theorems for the analytic (in time) operatorvalued Feynman integral can be obtained by using Theorem 3.4.4 of that paper. However, we will not pursue that topic here. The paper [AUoMa] is based on deep results of Albeverio and Ma [AlMal,2] and Blanchard and Ma [BlMal,2], the influential book of Fukushima [Fuk], and ideas of Johnson [Jo6]. While reading this section, one might find it helpful to consult Chapter 3 (basic information about the Wiener process), Section 10.3 (quadratic forms), Section 10.4 (the Kato class), Section 12.1 (the statement of the Feynman-Kac formula) and Sections 13.2 and 13.3 on the analytic-in-time operator-valued Feynman integral. The reader will note the symbol C for the complex numbers in several places below. We will not concern ourselves with this, but we mention that the results in [AlMal,2, BIMal ,2, Fuk] are all obtained for Hilbert spaces over K; however, [AUoMa] was written with quantum mechanics and the Feynman integral in mind, and so it was necessary to have C as the scalar field. This required complexifying (roughly speaking) everything in sight. The parallel with Section 13.3 We now begin drawing the parallel between the "objects" involved in our earlier results in Sections 13.2 and 13.3 (numbers 1-6 below) and their replacements (numbers 1'-6'). The reader should keep in mind that in both 1 and 1', the conditions on the negative part of the "potential" do not cover all that is known. 1. [Earlier potential] V or Vdu; V = V+ - V-, where V+ 6L1loc(Rd/G)and V- 6 Kd- (Here, G is a closed subset of Md of Lebesgue measure 0.) 1'. [Replacement] /A, a "generalized signed measure"; ^ = /A+ — /u_, where /JL+ 6 5, the class of "smooth measures", and /u— e GKj C S. (Here, /^ is not necessarily a true signed measure since both ^+ and ,u,_ are permitted to be infinite measures. A key part
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of what is required to be a smooth measure is that 0 measure be assigned to all sets of capacity 0.) 2. [Earlier additive functional] A v ( y ) = ft 0 V(y(s))ds = ftV+(y(s))ds /0' V-(y(s))ds, where y € C([0, +00), Rd). [Note that the integral involving V+ is positive, possesses continuity properties as a function of y (because of the convergence theorems of the Lebesgue theory), and is additive as a function of t. Of course, the same observations hold for the integral involving V-. The integrals involving V+ and V- are not the only "positive continuous additive functionals" (PCAFs); see Definition 13.7.11 below as well as [Fuk, FukOT].] 2'. [Replacement] A? (y) = A?+ (y) - A?' (y), where Au+ and Au~ are the PCAFs associated with the smooth measures u+ and u_, respectively. 3. [Earlier Hamiltonian] H^ = HQ+V, the form sum of HO = -\ A and V. Here, H£ is the self-adjoint operator associated with the quadratic form £y = £c + q^. (The quadratic forms £c and q^ have been used earlier and are denned in item 4 below.) 3'. [Replacement] H£ = HQ + M» the form sum of HQ and /z. Here, H£ is the selfadjoint operator associated with the quadratic form €^ = £c +q^. (The quadratic from qc is denned in (b) of 4' below.) 4. [Earlier quadratic forms] (a)£c(i/r,
where Ef denotes integration with respect to the standard Wiener measure m* associated with continuous paths y which start at | e Rd at time 0. 5'. [Replacement] extended Feynman-Kac formula:
This formula can be found in [BIMal, §4] under closely related hypotheses. 6. [Earlier] Theorem (briefly stated). The analy'tic-in-time operator-valued Feynman integral J l t ( F y ) exists for every t € R and we have
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6'. [Replacement] Theorem (briefly stated). The analytic-in-time operator-valued Feynman integral Jit (Fu) exists for every t € R and we have
Recall that the analytic-in-time operator-valued Feynman integral was defined by starting with the expression on the right-hand side of the equation in 5 above and analytically continuing to an operator-valued function of t which is analytic in C+ and strongly continuous in C+. (The precise definition is given in Section 13.2.) The same procedure is used in the new situation except that one begins with the right-hand side of the equation in 5'. (For more precision, see Definition 2.2.3 of [AlJoMa].) We turn now to the definition of the terms used in l'-6' above, beginning with the concept of a "generalized signed measure" (see 1'). Our definition here is equivalent to—but different from—the one given in [AlJoMa, p. 271]. Generalized signed measures Definition 13.7.1 Let fj,+ and /z_ be a-finite measures on B = B(Rd), the a-algebra of Borel subsets o/Rd, and suppose that there exists B e B such that /^+ (Bc) = /u,_ (B) = 0, where Bc denotes the complement of B in R rf . Then /j. — /z+ — ^i_ is a generalized signed measure on B. Remark 13.7.2 (a) The set function n, is not necessarily defined on all ofB; it is defined precisely for those E e B such that at least one of the inequalities At+(6 n E) < oo, M_(0 C D E) < oo holds. (b) The total variation |/LI| := ju+ + /^_ is a a-finite measure defined on all ofB. The generalized Kato class The Kato class discussed earlier (see Definition 10.4.1) consists of functions on R.d. The "generalized Kato class" (see 1') is a straightforward extension of this concept to measures. Definition 13.7.3 A positive measure /u, on B(Rd) is said to be in GKj, the generalized Kato class, if and only if
Remark 13.7.4 (a) It is not hard to show that /j, € GKj implies that ^ is locally finite; that is, n(K) < oo/or all compact subsets K ofRd.
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(b) The Junction V on Rd belongs to Kj, the Kato class of functions on E.d, if and only if the measure \ V \du belongs to GKj. Thus Kj may be regarded in a natural way as a subset of GKd. However, Kd is not all of GKj, as we now show by giving an example of a measure K which is in GK3 but which is not absolutely continuous with respect to three-dimensional Lebesgue measure. Let K be two-dimensional Lebesgue measure on the (u1, u2)-plane in R3 = {u1, u2, u3) : uj e R for j = 1, 2, 3} and extend K to all of R3 by letting K be the zero measure on the complement of the (u1, u2)-plane. Since the support of K is the (u1, u2)-plane, a set of three-dimensional Lebesgue measure 0, K is certainly not absolutely continuous with respect to three-dimensional Lebesgue measure. We claim however that K e GK3. Let a > 0 be fixed for now. Thus we can write:
where the next to last equality comes from changing to polar coordinates. Letting a —» 0+, we see from Definition 13.7.3 and the argument just above that K e GKj,, as claimed. Capacity on W1 We used the concept of "capacity 0" in the preceding two sections even though we had not yet given the definition of the (Newtonian) capacity of a subset A of Rd. We present this definition now. First, for ^,
where (•, •) denotes the inner product in L2(Rd, C). Definition 13.7.5 Given an open subset U of Rd, we define Cap(U), the capacity of U, by the formula
Then, for any subset A of Rd, we define the capacity of A as Cap(A) := inf{Cap(C7) : A C U with U open}.
(13.7.3)
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Certain facts about capacity that were needed earlier were stated in (1 )-(3) of (13.5.6). The reader may wish to review those items at this point. Some properties of the set function Cap(-) are given in the following theorem, the first three parts of which are proved, for example, in [Fuk, Theorem 3.1.1]. Theorem 13.7.6 The set function Cap(-) is a Choquet capacity; that is: (i) If A c B, then Cap(A) < Cap(B). (ii) If {An}oo is an increasing sequence of subsets of Rd, then
(iii) If {An}oo is a decreasing sequence of compact subsets of Rd, then
Further: (iv) If{An}^_^ is a sequence of subsets of W*, then
Remark 13.7.7 (a) For all dimensions d, if A is the surface of a ball of positive radius, then Leb.(A) = 0 whereas Cap(A) > 0. Thus we see that strict inequality can hold in (v) above. Also we see that Leb.(A) = 0 does not imply that Cap(A) = 0, a fact that was already noted in (1) of (13.5.6). (b) Physically, the Newtonian capacity of a set ("conductor") A C Rd corresponds to its "electrostatic capacity". Further, in (13.7.1) or (13.7.2), £f(^, VO can be thought of as an "electrostatic energy". Naturally, the analogy with electrostatics has played a key role in the development of potential theory and, in particular, of capacity theory. (c) Further information about the theory of Choquet capacities can be found in [Cho2] and in Section 9 of Chapter 3 of [Cho3, Vol. I]. The relationships between the theory of (Newtonian or more generally, Choquet) capacities, potential theory and Brownian motion are explored at length in Doob's treatise [Doo2]. Smooth measures In each of the theorems in [AlJoMa] which insure the existence of the analytic-in-time operator-valued Feynman integral J i t ( F u ) , we require at least that both parts of the "potential" u = u+ — u- are "smooth measures". [In particular, the theorem in 6'
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above assumes that u+ is smooth and that u- belongs to GKd, a subset of the set S of smooth measures (see 1').] Thus, "smooth measures" are at the heart of the main results of [AlJoMa]. Definition 13.7.8 A positive measure u on B ( R d ) is said to be smooth if and only if C € B(R d ) and Cap(C) = 0 implies that u-(C) — 0 and if there exists an increasing sequence {Kn}oo=l of compact sets (in general depending on /u) such that (i) (i(Kn) < oo for n = 1 , 2 , . . . , and (ii) limn_,.oo Czp(K\Kn) = 0, for all compact subsets K of Rd. Remark 13.7.9 (a) The assertion that C e B(Rd) and Cap(C) = Q implies that ^(C) = 0 is often described by saying that fj, does not charge sets of capacity 0. (b) Any locally finite measure /u. on B(R d ) which does not charge sets of capacity 0 is smooth; in fact, if we take Kn to be the closed ball of radius n centered at 0, this one sequence {Kn} satisfies (i) and (ii) of Definition 13.7.8 for all such measures p,. (c) A measure JJL which does not charge sets of capacity 0 may fail to be locally finite and still be smooth. Example 13.7.19 gives a simple example of this (when d > 2 and a > d). The measure there is smooth although it is not locally finite since u ( U ) = +00 for all nonempty open subsets U of Rd which contain the origin. Example 13.7.20 is a more extreme illustration; the measure there is nowhere locally finite but is still smooth. (d) Every fj, e GKj is smooth, as was already noted in 1'; i.e. GKd c S. Definition 13.7.10 A generalized signed measure (Ji = n+ — /i_ (see Definition 13.7.1) will be called a generalized signed smooth measure if and only if both /u,+ and ^~ are smooth measures. The family of all generalized signed smooth measures will be denoted S — S. Positive continuous additive functionals of Brownian motion We turn now to the topic of "positive continuous additive functionals of Brownian motion". We have not provided the background necessary for a thorough discussion of this subject and its relationship with smooth measures, and so we will settle here for a brief description of the most relevant facts. More information can be found in [AlJoMa, Fuk, FukOT] as well as in [AlBlMa, AlMal,2, BlMal,2]. Let C([0, +00), Rd) denote the space of continuous, Rd-valued functions on the interval [0, +00). Given s e [0, +00), we define (much as in (3.3.2)) the operator
by
Now for any t e [0, +00), we let
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that is, F0 is the smallest a-algebra of subsets of C([0, +00), Rd) containing all of the sets P~l(B), where s varies over the interval [0, t] and B ranges over B(R d ). The a -algebra F0 can also be described as the smallest a-algebra making all of the functions Ps,0 < s
We have the following containments: For 0 < s < t < oo,
where C([0, +.00), Rd) is equipped with the topology of uniform convergence on compact subsets of [0, +00) and, as before, B(C([0, +00), Rd)) denotes the Borel class of C([0, +00), Rd). The first two containments in (13.7.7) follow immediately from the definitions in (13.7.5) and (13.7.6), and the third follows from the fact that each of the functions Ps in (13.7.4) is continuous. We actually enlarge the a-algebras F0 and F° by a type of "universal completion" that is appropriate to this setting. (See, for example, [AUoMa, p. 279].) These completed a-algebras Ft and T, respectively, satisfy
We are now ready to define a positive continuous additive functional (in the sense ofFukushima [Fuk]) of Brownian motion. Given f e Rd, we let m^ denote the Wiener measure whose support is the set of y 6 C([0, +00), Rd) such that >>(0) = E. As before, the expectation (or integral) with respect to the probability measure mE will be denoted E|. Definition 13.7.11 A function (or process) A : [0,+00) x C([0, +00), Rd) -+ R is called a positive continuous additive functional (PCAF) if and only if At(-) is Ttmeasurable for each t > 0 and there exists A € F (called a defining set for A) and N e B(Rd) (called an exceptional set for A) satisfying the following properties: (i) Cap(N) = 0 and m(A) = 1 for all t- e Rd\N. (ii) T,y e A for all y 6 A, where (Tty)(s) := y(t +s) for 0 < s < oo. (iii) For each y e A, the function t-> At(y) is continuous, increasing, and vanishes at 0 (i.e. A0(y) = 0); furthermore, it is additive in the sense that
Remark 13.7.12 The measurability assumption that is made on A in Definition 13.7.11 is usually referred to in the literature on stochastic processes by saying that "(At) is (f,)-adapted." The relationship between smooth measures and PCAFs There is, as we will see in Theorem 13.7.15 below, a one-to-one correspondence between smooth measures and certain equivalence classes of PCAFs (in the sense of Fukushima). The next definition specifies the nature of the equivalence.
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Definition 13.7.13 Two PCAFs A and B (of Brownian motion) are equivalent if and only if they share a common defining set A on which they agree. The correspondence between a smooth measure and its associated equivalence class of PCAFs is given (in general) by the implicit equation (13.7.15) in Theorem 13.7.15 below. We try to give the reader some insight into this relationship by showing that the formula holds for the familiar PCAFs of the form
where V is nonnegative, Borel measurable and (for simplicity) bounded. With Av as just described, the inner integral on the left-hand side of (13.7.15) below satisfies (see Problem 14, page 50 of [Fol2]) for y e C([0, +00), R rf ):
where/ : [0, +oo)xR d —* R is bounded, nonnegative and Borel measurable as required in the statement of Theorem 13.7.15. By first using (13.7.11) and then the Fubini theorem and the Wiener integration formula (Theorem 3.3.5), respectively in the third and fifth equalities below, we can write
where /z (or ^v) is given by the formula
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Hence, the relationship (13.7.15) below is established for all functions / as described above in the special case where the additive functional and the measure u, are given, respectively, by (13.7.10) and (13.7.13). Remark 13.7.14 (a) Since V is bounded, it is clear that the measure /j, — Vdv in (13.7.13) is locally finite. Hence, by Remark 13.7.9(b), to show that IJL is smooth, itsuffices to show that fj. does not charge sets of capacity 0; that is, B e B(E.d) and Cap(fi) = 0 implies thatfj,(B) = 0. But this is true since Cap(fi) = 0 implies (see Theorem 13.7.6(v)) that Leb. (B) = 0, which implies in turn (by (13.7.13)) that /j.(B) = 0. Thus /z is smooth. It is also easy to see that A^ is a PCAF under the present assumptions on V. Indeed, taking the defining set A and the exceptional set N from Definition 13.7.11 to be C([0, +00), Rd) and 0 (the empty set), respectively, the reader can easily verify this. (b) The relationship between V and /u that is established in (13.7.12), namely,
actually holds for an arbitrary nonnegative Borel measurable function V : Rd —»• [0, +00] and an associated measure given by (13.7.13). In fact, the argument changes very little. Equation (13.7.11) continues to hold and the Tonelli theorem (seepage 65 of [Fol2]) rather than the Fubini theorem needs to be used to justify the third equality in (13.7.12). However, we do not claim for Vs this general that Avt given by (13.7.10) is necessarily a PCAF nor that n given by (13.7.13) is necessarily a smooth measure. We are now ready to state the general result giving the connection between smooth measures and positive continuous additive functionals. Theorem 13.7.15 For each n e 5, there is a PCAF AM with exceptional set N such that the formula
holds for every bounded, nonnegative Borel measurable function f on [0, +00) x Rd and for every f e R d \N. Moreover, A.^ is unique in the sense that if another PCAF B satisfies (13.7.15), then A^ and B are equivalent (see Definition 13.7.13). Finally, given any PCAF A, there is a measure jj.eS such that A = AM. The analytic-in-time operator-valued Feynman integral exists for V = M+ - n- e S - GKd Now we restate (with a little more detail) the theorem from 6' which assures us of the existence of the analytic-in-time operator-valued Feynman integral Jl'(F^) for certain generalized signed measures /n = /j>+ — /z_. Theorem 13.7.16 Let n = /z+ — /^_ be a generalized signed measure belonging to S - GKd (i-e. M+ € 5 and /z_ e GKd), and let F M (v) = e~A?^\ where
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A? = A?+—A?~ and A?+ and A?~ are the PCAFs associated with the smooth measures jU,+ and /n_, respectively. Then the analytic-in-time operator-valued Feynman integral J"(Fn) exists and we have
for all t e R. Examples Near the end of this section, we will make brief comments about results from [AlJoMa] which are more general than Theorem 13.7.16. In particular, we will give an example where Jlt(Fll) exists even though /u_ ^ GKd and, in fact, is not even locally finite. However, for now, we give examples (or families of examples) of positive measures which are in GKj or are at least in S. Any choice then of a generalized signed measure /A = /i+ — /i_ such that fi+ e 5 and /j.- e GKd produces an example where Theorem 13.7.16 is applicable. We remind the reader that Kd c GKd c S (see Remarks 13.7.4(b) and 13.7.9(d), respectively), where Kd is the ordinary Kato class (see Definition 10.4.1) of R-valued functions on Rd. Example 13.7.17 Take d > 3 and let M be a (d — 1)-dimensional (smooth) submanifold of Rd. We take K = KM to be the Riemannian volume measure on M and extend K to R d \M by letting K(R d \M) = 0. [In the literature on smooth measures, somewhat by abuse of language, K = KM is often called the "^-function" (or "Dirac measure") supported on M, and is denoted by SM.] One can show that K e GKd. In fact, we established a very special case of this in Remark 13.7.4(b) where M was a plane in R3. The case of a (d - 1)-dimensional affine subspace in Rd (d > 3) is not much different. The submanifold M need not be flat but can be, for example, the surface of a sphere in Rd (still with d > 3); for instance, KS2 = SS2, the "^-function" on S2 (the unit sphere in R3)—which, mathematically, is simply the area (or Hausdorff) measure on S2—is a smooth measure on R3. Remark 13.7.18 (a) Let K — KM be as in Example 13.7.17 and for any bounded, Borel measurable function g, g : Rd —> M, let
Then for any B e B(Rd), we have
andboth of these measures are in GKd. Hence, fj, = /z+ — /z_ e 5 — GKj and Theorem 13.7.16 is applicable. (In fact, the comments made so far in this remark are not limited to the special measures K in Example 13.7.17 but hold for any K € GKd and any bounded Borel measurable function g.)
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(b) A submanifold oflHd(d > 3) of codimension 1 is the focus of the discussion in Example 13.7.17 and in part (a) of this remark. It is natural to ask if similar assertions hold for submanifolds of Rd of codimension 2 or more. The answer is "No"—at least for our present approach to this subject. The key is that such submanifalds have capacity 0 (see (3) of (13.5.6)) and so Wiener paths that start at a point f which is outside of the submanifold almost surely never visit it (see (2) of (13.5.6)). The result is that the Wiener integral which is involved in the Feynman-Kac formula (see 5'), namely
does not distinguish the measure K from the 0 measure or, equivalently, the 0 potential, and so does not distinguish the additive functional AKt from the additive functional that is identically 0. (c) In Example 13.7.17 and in (a) of this remark, the interaction is localized to a set which has Lebesgue measure 0 but positive capacity. (An example of such a set is S2, the unit sphere in R3, as was seen above.) A considerable amount of work has been done using Hamiltonian methods with interactions that are localized even further to such sets as points, curved wires or line segments that share a common end point. A good brief treatment of this subject including many references and a discussion of circumstances under which the resulting models are physically reasonable can be found in [BkExH] (see especially Section 14.6 in that book). An additional interesting feature of the work just referred to is that in a number of cases, the associated generalized Schrodinger equations are exactly solvable. (d) The origin in R2, (0), is a set of capacity zero in R2. (This follows from property (3) recalled in (13.5.6) because a point is of codimension 2 in M 2 .j Consequently, the interaction £t(= /Cjoj) = <5{0), the Dime measure at the origin in R2, is too singular to be a smooth measure (or a perturbation thereof) on #(R2). Hence, neither the extended Feynman-Kac formula 5' nor Theorem 13.7.16 (about the analytic-in-time Feynman integral) can be applied to this situation, when the measure is taken to be /it = S(o). However, recent physical work of Henderson and Rajeev [HenRajl,2]—based on Wilson's renormalization scheme applied to quantum mechanics (see Remark 13.6.24(b) above)— provides a (Euclidean or "imaginary time ")path integral representation for the solution of the corresponding (suitably defined) "Schrodinger equation ". (See also the relevant references in [HenRajl,2].) In fact, the path space measure involved is no longer Wiener measure, and relative to this new measure, the "potential" can be taken to be zero when \JL =• <5(0) in M2 in the counterpart of the Feynman-Kac formula. Moreover, it is expected—but, to our knowledge, not yet proved—that the new path space measure is singular with respect to Wiener measure m (in the usual measure-theoretic sense). Note that by contrast, all the results considered in the present section can be reinterpreted in terms of functional measures that are absolutely continuous with respect to m. // would be very interesting to rigorously investigate this example and to eventually develop a unified theory that would include interaction measures JJL which are of the type considered in Theorem 13.7.16 (and hence correspond to suitable perturbations of smooth measures), as well as singular interactions such as fj, = 5{o) in R2. (For the latter example, the earlier mathematical work in [Hugl,2] and the relevant references therein may be useful.)
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In particular, it would be important to be able to include in this extended framework the model case of the attractive inverse-square potential (say, V(x) = — 1/r 2 in R3, with r = \\u\\) discussed in some detail in case (3) of Example 13.6.18 from the point of view of the various analytic-in-mass Feynman integrals considered in Sections 13.5 and 13.6. The latter interaction, n(du) = V(u)du, has been studied at the physical level of rigor in [GupRaj] from the point of view of renormalization theory (see Remark 13.6.24(b) above) but, as far as we know, has not yet been treated successfully from the point of view of the "imaginary time" path integals considered in [HenRajl,2]. The measures involved in the rest of our examples are all of the form V(u)du; that is, the interactions are all given by R-valued potential energy functions V. The next example is a simple one. It shows that a smooth measure need not be locally finite and also illustrates how the compact sets { K n } from the definition of a smooth measure (see Definition 13.7.8) can be adjusted to the particular case at hand. Example 13.7.19 Let V(u) = \\u\\~a. Fix d > 2. We will see that the measure Vdu is smooth for any a e R. Note that Vdu does not charge sets of capacity 0 for any such a. Now, let a < d. Then V e Ljoc(R'/) and so Vdu is locally finite and hence smooth. However, when a > d, V £ Lj oc (R rf ) and so Vdu is not locally finite; specifically, fv V(u)du = +00 for any open set U containing the origin. Nevertheless, Vdu is still smooth. In order to see this, take the set Kn from Definition 13.7.8 to be the closed ball of radius n centered at 0, with the open ball of radius l/n centered at the origin removed. Note that the set Kn is constructed with the location of the singularity of V in mind. Theorem 13.7.16 can, of course, be applied where the measure IJL is Vdu from Example 13.7.19. In fact, Theorem 13.3.1 can be applied if it is supplemented by Remark 13.3.3(a) where the closed set G is just the singleton set {0}. Our next example shows that it is possible to have extremely singular positive potentials V such that the measure Vdu is smooth and so Theorem 13.7.16 is applicable. This example was discussed in connection with the Feynman integral on page 286 of [AlJoMa] but the potential appeared earlier in [AlMal]. A closely related example appeared still earlier in [StlVo]. Example 10.3.21 above is somewhat similar in character and illuminates the distinction between algebraic and form sums of operators. Example 13.7.20 Fix d > 2. Let {uj} be a countable dense subset of Rd and let { a j } be (for now) an arbitrary sequence of real numbers. It was shown in Proposition 1.3 of [AlMal] that there always exists a sequence [cj] of strictly positive numbers such that if we define
then Vdu is a smooth measure on Rd. Now if we choose [ctj} in such a way that ctj < —d for all j greater than some jo, then Vdu is nowhere locally finite; i.e. for any nonempty
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open subset U of Kd, we have
If, even further, we let a;- = — j for j — 1,2,..., then the function V defined by (13.7.18) is nowhere Lp-integrable for any p > 0; i.e. for any nonempty open subset U of Rd and any p > 0,
Remark 13.7.21 (a) It is unlikely that the example just given is of any physical interest; but it does tell us something about the boundaries of the mathematical theory. The potential V is everywhere (i.e. on every nonempty open set) extremely singular and yet the Hamiltonian H = Ho + V is defined on a dense subset of L2(R d ) and the Feynman integral Jit (Fy) exists for every value it on the imaginary axis. This situation should be contrasted with earlier results for positive potentials where arbitrary singularities were allowed either on closed sets of Lebesgue measure 0 (see Remarks 11.4.3(b) and 13.3.3(a)) or on closed sets of capacity 0 [see Theorem 13.5.7 (along with (13.5.5)) as well as Theorem 13.6.4] and where, in the case of analytic continuation in mass, the existence of the Feynman integral was established only for Leb.-a.e. value of the mass parameter. (b) When a positive potential V belongs to L,1oc(Rd), the familiar space D — D(Rd) (also denoted Coo00(Rd)) of infinitely differentiable functions with compact support, is a form core (see Definition 10.3.3) for the quadratic form associated with the operator HO + V. It is easy to see that when V is nowhere locally integrable as in Example 13.7.20, then the only function in TJ which belongs to the form domain of V, that is, which belongs to
is the function that is identically 0. In fact, Q(V) cannot contain any nonzero continuous function. It is natural to ask in this situation if it is possible to find a form core consisting of functions with at least some nice properties. Theorem 5.7 of[AlMa2] implies that there is a form core for Ho + V (with V as in Example 13.7.20) such that all thefunctions in the form core are bounded, compactly supported and quasi-continuous. /Note: A function f on Rrf is said to be quasi-continuous (seepage 64 of[Fuk]) if and only if f is defined except on a set of capacity 0 in Kd and for every e > 0, there exists an open subset U of Rd such that Cap(t/) < e and the restriction of f to K d \t7 is continuous.] We remark that one not only needs Theorem 5.7 of [AlMa2] for the above but also the fact that our quadratic form is positive (and so certainly semibounded) and closed, from which it follows that the quadratic form in [AlMa2, Theorem 5.7] coincides with ours. We note that the existence of a dense form core for H0+V in the case of extremely singular potentials, such as in Example 13.7.20, is relevant not only to the analytic-intime (operator-valued) Feynman integral but also, for example, to the modified Feynman integral; see Remark 11.4.3(c).
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Items l'-6' gave a theorem (restated later as Theorem 13.7.16) which insured the existence of the analytic-in-time operator-valued Feynman integral Jit(F l i ) under the assumptions that /u,+ e S and /u_ 6 GKj. As mentioned earlier in the section, this hypothesis on fj,-, while convenient for comparison with the work in Section 13.3 (see Theorem 13.3.1), is not the best that can be made. We illustrate this by simply stating an example where J" (FM) exists even though /*_ is not locally finite and so cannot be in GKd (see Remark 13.7.4(a)). We begin by defining a concept that plays a role in the example below. Definition 13.7.22 Let K and v be positive smooth measures with associated PCAFs AKS and Avs, respectively. We say that K is compatible with respect to v if and only if there exists t > 0 such that
where N is a set of capacity 0 containing the exceptional sets of both AK and Av. Example 13.7.23 Let d > 2. We define for any k > 2,
If we let /LI = Vdu, then /z is a generalized signed smooth measure (see Definition 13.7.10) and we have
Neither fi+ nor /u_ is locally finite and so neither belongs to GKd- Since //,_ £ GKd, Theorem 13.7.16 is not applicable. However, it was proved in [Sturl] that /z_ is compatible with n+ and so it follows from Theorem 3.4.8, page 293 of [AUoMa] that 7"(FM) exists and equals the unitary group e~l!Hc for every t e M. The rough idea is that HO and V+ rather than just HO are used to control the singularities of the negative part V-. [In the language of unbounded quadratic forms, one can say that V- is relatively form bounded with respect to HO + V+. With this in mind, we see that Example 13.7.23 is also relevant to the study of the modified Feynman integral (from Section 11.4).] In the example just given, the rapid oscillations of V help us to deal with the strong singularity at the origin. A more extensive discussion of such highly singular oscillatory potentials can be found in [Stur2]. In addition to Theorem 3.4.8 of [AUoMa] which was referred to just above, Theorems 3.4.4 and 3.4.5 from the same paper are directed towards extensions of Theorem 13.7.16. The consequences of Theorem 3.4.4 for the Feynman integral have not been explored and may well yield further interesting results. We recommend the paper [AlMa2], especially Theorems 4.1 and 5.5, to the reader who may wish to consider this possibility. Finally, we remark that q^ should be replaced by q^_ in (iv) of Theorem 3.4.4 of [AUoMa]. (In the notation of [AUoMa], one must replace Q^ by QI]L_.) We close this section with two comments.
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Remark 13.7.24 (a) Recently, Chang, Lim and Ryu [ChanLimRy] have established a stability theoremfor the analytic-in-time operator-valued Feynman integral in the setting of the present section. Part of their proof adapts techniques from Lapidus' dominated convergence theorem [Lal2] (see Theorem 11.5.19 along with Theorem 11.5.7). However, the assumption that Vm -> V Leb.-a.e. in Theorem 11.5.19 is replaced by the assumption that, for every B e B(M.d), fJ^m(B) and f j , m ^ ( B ) converge monotonically downward to /i(fi) and n-(B), respectively, as m —> oo. In addition, the sequence of measures {/^m}^=1 is assumed to "converge dominatedly" to fj. in the following sense: Mm,+ < v and fJ-mi- < r\, where v e Sa and rj e GKj, with Sa denoting the space of a-finite smooth measureson Rd. (Here, Mm,+ and nm,- denote the positive and negative part of n,m, respectively. Further, the order relation involved is the usual one between measures.) The changes noted above necessitate quite different techniques of proof in parts of the paper [ChanLimRy]. We note that the assumption in [ChanLimRy] was that v e G Kj, but that it was more recently shown in [Lim] that v e Sa suffices. This is in closer analogy with Theorem 11.5.19. However, to improve this result still further, it would still remain to eliminate the "monotonicity assumptions " made in both [ChanLimRy] and [Lim], but not in Theorem 11.5.19 or in [Lal2J. (b) If ^ = ^i+ — [A- € S — GKd (i.e. (f/u.+ is a smooth measure and /z_ is in the generalized Kato class), as was assumed in the statement of Theorem 13.7.16, then the self-adjoint operator associated with [i- is Ho-form bounded with relative bound less than 1. It follows that the abstract product formula for imaginary resolvents (Theorem 11.3.1) can be applied to this situation, thereby potentially enlarging significantly the class of examples which can be dealt with via the modified Feynman integral (of Section 11.4). However, the concreteform of the modified Feynman integralfor potentials belonging to this enlarged class remains to be investigated. If the problem suggested at the end of the last paragraph can be resolved, then the stability results discussed in part (a) of this remark can be fully applied to the modified Feynman integral. This would be a further example where (in the spirit of Section 13.4) one of our approaches to the Feynman integral yields information about another.
14 FEYNMAN'S OPERATIONAL CALCULUS FOR NONCOMMUTING OPERATORS: AN INTRODUCTION ... just as a poet often has license from the rules of grammar and pronunciation, we should like to ask for "physicists' license" from the rules of mathematics in order to express what we wish to say in as simple a manner as possible. Richard P. Feynman, 1951 [Fey8, p. 124] The mathematics is not completely satisfactory. No attempt has been made to maintain mathematical rigor. The excuse is not that it is expected that rigorous demonstrations can be easily supplied. Quite the contrary, it is believed that to put the present methods on a rigorous basis may be quite a difficult task, beyond the abilities of the author. Richard P. Feynman, 1951 [Fey8, p. 108] In my paper, the fact that XY was not equal to YX was very disagreeable to me. I felt that this was the only point of difficulty with the whole scheme. Werner Heisenberg, reminiscing about his key 1925 paper on quantum mechanics ([Hei]). (Quoted in [Han, p. 81].)
In this chapter, we give an elementary introduction to the general ideas of Feynman's operational calculus for noncommuting operators and indicate the connection between this calculus and the Wiener and Feynman integrals. It may appear at first that these path integrals have little to do with the operational calculus of Feynman, but the topics are intimately related as we will begin to see in Section 14.3 below. The Wiener and Feynman integrals will be used in Chapters 15-18 to study Feynman's operational calculus in the diffusion (or probabilistic) and quantummechanical settings, respectively. We will return to a more general setting for the operational calculus in Chapter 19. The last part of this introduction will concentrate on Chapters 15-18 and will stress those aspects of the path integrals that are connected with Feynman's operational calculus. However, many of the results in those chapters have an interest as contributions to the Wiener or Feynman integrals apart from that connection. Three final notes here: (1) Further introductory material on our general approach to Feynman's operational calculus can be found at the beginning of Chapter 19. (2) When the words "operational calculus" are used in Chapters 14-19, they will refer to Feynman's operational calculus for noncommuting operators unless it is said or is clear from the context that we have some other operational calculus in mind. (3) Our main purpose in this chapter is to introduce the heuristic ideas of the operational calculus and to describe in broad terms some of the highlights of Chapters 15-19. We leave the mathematically precise discussion for these later chapters. We settle here for mentioning that our standard setting will be a separable Hilbert space Ji over C with
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the operators either being (i) bounded, or else being (ii) unbounded operators which are the generators of (Co) semigroups (see Chapter 9). At the end of this chapter, we will give a list of references connected to Feynman's operational calculus. 14.1 Functions of operators It is useful in many areas of mathematics and its applications to form functions of operators. One sees this even in elementary settings. Consider the initial-value problem x' — Ax, x(0) = jco, where A is an n by n matrix of constants and x is an n vector whose components are functions of time. The elegant solution to this problem is given by*(0 = [exp(fA)]C*o)Since exponential functions play a central role in solving evolution equations, it is especially important to be able to exponentiate operators. The theory of semigroups of operators outlined in this book in Chapters 8 and 9 may be regarded as the theory of exponentiating operators. (It is necessary to exercise some caution in adopting this point of view; in fact, as was seen in Chapter 9, (Co) semigroups have some but not all of the properties of the numerical exponential function.) In quantum mechanics, where the basic observables are (possibly unbounded) selfadjoint operators on an infinite dimensional Hilbert space, one often deals with functions of such operators. Indeed, this is one of the recurring themes in Chapters 6-19 of this book. Let H be the self-adjoint energy operator or Hamiltonian for a quantum system and let HO be the free Hamiltonian for the same system. One of the following three functions of H or HO has either been directly involved in, or has been the main motivation for, the central results in the last four chapters: e~"H (the unitary group), e~'H (the heat semigroup), and [/ + i(t/n)Ho]~~] (the free resolvent). Various other functions of selfadjoint operators have arisen as well; see, for example, the proof of Theorem 11.3.1 where a number of such functions are used. The functional calculus for a single self-adjoint operator (bounded or unbounded) is extremely rich, as we have seen in Chapters 10-13. The situation is unchanged for functions of any finite number of self-adjoint operators provided these operators commute with one another. However, as soon as commutativity fails, a functional calculus becomes much more difficult, even in the presence of self-adjointness. The question of defining functions of noncommuting self-adjoint operators is not the only difficulty. Even when this can be accomplished, the process may not be unique. The lack of uniqueness gives rise to the "ambiguity of quantization." Before explaining how this ambiguity arises, we describe briefly and in a special case the canonical quantization of a classical mechanical system. We refer to [Mac, §2.4] for a much more complete discussion of the quantization procedure. Suppose that we have a single classical particle moving in K. A classical observable is an R-valued function f(q,p) of the position q and momentum p of the particle. The position observable q itself is quantized by replacing it by the position operator Q on L2(K) defined by
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The momentum operator,
quantizes the classical momentum p. (Here, as in Chapter 6,h = h/2n denotes Planck's constant divided by 2n.) Both Q and P are unbounded, self-adjoint operators. Note that Q and P do not commute with one another; in fact,
The idea is to carry out the general quantization procedure by replacing the classical observable f(q, p) by the quantum observable f(Q, P). If f(q, p) — q2 + p2, this can be done unambiguously by taking f(Q, P) = Q2 + P2. However, ambiguities arise in the procedure as soon as products of Q and P are involved. Consider the simple example f(q, p) := qp. Now qp = pq, and thus both products give the same classical observable. But QP / PQ by (14.1.3), and so QP and PQ are distinct quantum observables. Which operator should be associated with f(Q, P)? Should it be QP or PQ, or perhaps something else, say, \(QP + PQ)1 Remark 14.1.1 (a) Much of the focus so far in this section has been on self-adjoint operators since these are the observables in quantum mechanics. However, the difficulties and ambiguities involved in forming functions of operators are present whenever the operators involved fail to commute. (b) As we continue, we will see that the ambiguity "problem " can actually be an asset insofar as it will allow us to model a wider variety of physical phenomena. This positive aspect of the ambiguity will be involved in all of Chapters 15-19 but will perhaps be seen most clearly in Chapters 17 and 19. (c) The richness of the functional calculus for commuting self-adjoint operators carries over to commuting normal operators. Indeed, we made use of thisfact in Sections 11.6 and 13.6. (d) 'While it is true that a functional calculus for noncommuting self-adjoint operators is limited and difficult, interesting work along these lines has been done. See, for example, the papers [And], [Tayl] and, for recent work andfurther references, thepaper [KisRal,2]. 14.2 The rules for Feynman's operational calculus Motivated by his work on path integration in nonrelativistic quantum mechanics [Fey2] and on quantum electrodynamics [Fey 5-7], Feynman gave in his 1951 paper An operator calculus having applications in quantum electrodynamics [Fey8], a heuristic formulation for an operational calculus for noncommuting operators. Our purpose in this section and in the first part of Section 14.3 is not to discuss rigorous mathematics but to illustrate and discuss Feynman's heuristic ideas. Talking about the unconventional use of rules and formulas in his time-ordered operator calculus, Feynman writes [Fey8, p. 124]: ... The physicist is very familiar with such a situation and satisfied with it, especially since he is confident that he can tell if the answer is physically reasonable. But mathematicians may be
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completely repelled by the liberties taken here. The liberties are taken not because the mathematical problems are considered unimportant. On the contrary, this appendix is written to encourage the study of these forms from a mathematical standpoint. In the meantime, just as a poet often has license from the rules of grammar and pronunciation, we should like to ask for "physicists' license" from the rules of mathematics in order to express what we wish to say in as simple a manner as possible.
The position on the page is the standard way of keeping track of the order in which products of noncommuting operators act. Feynman had a different way of doing this, which is one of the keys to his operational calculus. Feynman's time-ordering convention Feynman used time indices to specify the order of operators in products, where it is understood that operators with earlier time indices always act before operators with later time indices. For example, given operators P and y,
Feynman's heuristic rules Some of the "rules", roughly described, for the operational calculus are as follows: (1) Attach time indices to the operators to specify the order of operators in products. (2) With time indices attached, form functions of these operators by treating them as though they were commuting. (3) Finally, "disentangle " the resulting expressions; that is, restore the conventional ordering of the operators. Feynman says of the disentangling process [Fey8, p. 110], "The process is not always easy to perform and, in fact, is the central problem of this operator calculus." Feynman did not attempt to prove his results mathematically, and it is not always clear, even heuristically, how his rules are to be applied. How should one attach time indices to the operators as mentioned in rule (1)? First of all, the operators may come with indices naturally attached. This happens with the operators of multiplication by time-dependent potentials, for example, and also in connection with the Heisenberg (or interaction) representation in quantum mechanics (see [Sud, Sections 3.4 and 3.5]). However, the operators that appear most frequently in the literature of quantum mechanics, perhaps especially in the mathematical literature, are time independent. Given such an operator a, almost without exception, Feynman attaches time indices according to Lebesgue measure as follows:
where a(s) := a for 0 < s < t. Despite its artificial appearance, we will eventually see that this device is extremely useful in many situations. However, it is not always the right
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thing to do. In the following chapters, a variety of measures will be used to assign time indices. Physical or mathematical considerations may determine the appropriate choice. We present two situations which are easily described and are motivated by physical considerations. An experimenter may wish to study the reaction of a system to a force which is being turned on or off at certain times. In this case, the natural measure to use would be H(ds) = g(s)ds, where g(s) alternates between 1 and 0 at the appropriate times. (Of course, in a case such as this, one can equally well continue to think of the measure as Lebesgue measure and regard g(s) as multiplying the operator. Indeed, Feynman adopts this point of view in two places in his paper [Fey8, pp. 113 and 114].) A system, perhaps again under the control of an experimenter, is subjected to significant forces which act over certain very short time intervals. Such a situation can often be usefully modeled by a linear combination of Dirac measures (or, heuristically, "<5-functions"). We will frequently encounter such situations in the following chapters. We now turn to two simple examples which illustrate Feynman's heuristic rules as well as the roles played by the time-ordering convention and by the measures. We note that systematic use of measures in connection with Feynman's operational calculus was introduced by the authors in [JoLal] and will play a prominent role in Chapters 15-19. Two elementary examples Let a and /3 be operators on the separable Hilbert space Ti.. We let a(s) := a and P(s) := f> for all s in the time interval in question. Example 14.2.1 (Disentangling aft) Given numbers a and b, let g(a,b)=ab.
(14.2.3)
Also let t = 1 for convenience. We follow the most common procedure with a. and assign time indices according to Lebesgue measure; i.e. a = f0 a(s)ds. On the other hand, we take /6 = /0 f}(s)8\ (ds), where S\ is the Dirac measure with total mass 1 concentrated on {!}. We first write down the string of "equalities" below and then discuss them:
The first equality in (14.2.4) just restates the assignment of time indices, while the second reflects the nature of the measures involved. The third "equality" time-orders the expression; that is, the time indices involved in the second factor on the right-hand side of the third "equality" are all less than the time index involved in the first factor. We are
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done disentangling when the expression is time-ordered; that is, when the order of the operators on the page corresponds to the order of the time indices. We are then free to calculate insofar as possible any remaining integrals. Doing this, we obtain the fourth equality in (14.2.4). // is typical ofFeynman's use of his rules that not all of the "equalities" in a calculation are valid if they are interpreted in the usual way. If a and ft do not commute, the third equality and also the final conclusion of (14.2.4) are certainly false if interpreted literally. How then should (14.2.4) be interpreted? The idea is that the product aft may have more than one meaning if a and ft do not commute. (Compare (14.2.4) with (14.2.6) and (14.2.8) below.) The sense of (14.2.4) is that with Lebesgue measure / and S\ associated with a and ft, respectively, the product is defined as fta. To state it another way,
We did not use Feynman's second rule in the calculation in (14.2.4). This is not typical. The second rule usually plays a crucial role. We will see an illustration of this in Example 14.2.3 below. Next, we replace &\ by So in (14.2.4) and see what changes result:
The first and second equality in (14.2.6) are much the same as before, but here the righthand side of the second equality is already time-ordered, and so we can go directly to calculating the integrals. It happens in this case that all three equalities in (14.2.6) are valid if interpreted in the usual way. The outcome of (14.2.6) can be summarized by writing What do we get if Lebesgue measure / is used to assign time indices to both a and ftl
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FIG. 14.2.1. The unit square [0,1]2 in the heuristic derivation of formula (14.2.9) In the second equality, we wrote the product of one-dimensional integrals as an integral over the unit square [0, I]2. Now, there is no definite time-ordering in [0, I]2, and so in the third equality we expressed the integral over [0, I]2 as the sum of integrals over the upper (s\ < $2) and lower (52 < si) triangles, respectively. (See Figure 14.2.1.) In the fourth equality, we made the ordering of the operators in each term consistent with the time-ordering. (More specifically, following Feynman's time-ordering convention (14.2.1), we wrote fi(s2)ot(s\) = flat for si < $2 and a(si)p(s2) = aft for $2 < s\. See Remark 14.2.2(b) below for the case when s\ = si.) Note that the upper and lower triangles in Figure 14.2.1 have the same area, equal to 1/2. As before, we could then calculate the remaining integrals. We see from the calculation in (14.2.8) that
We have an additional comment about the computation in (14.2.8), but first we make a remark that comes from comparing the three calculations above. Remark 14.2.2 (a) The three functions gi,S[, gi,s0 and gij all agree when a and ft commute, even though they are all distinct without this commutativity. What was one function in the commutative case has become threefunctions. (In fact, there are additional possible functions as well.) The three functions represent three different disentanglings of the simple product function; the distinct disentanglings are associated with the three pairs of measures (I, &\), (/, So] and (I, /). Is it useful to have different definitions for functions which somehow involve products ofnoncommuting operators? We will see that it is so in Chapters 15 through 19—perhaps most strikingly in Chapters 17 and 19 where exponential functions and associated evolution equations play a prominent role.
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(b) In the third equality of our calculation in (14.2.8), we have ignored the diagonal of the unit square. This seems reasonable since the diagonal C$i = $2) has measure 0 with respect to Lebesgue measure I x/ on [0, I]2v see Figure 14.2.1. Related steps will be involved in two or more dimensions in many places as we continue. However, when the measure we are dealing with has nonzero discrete part, ignoring the diagonal will not be possible and complicated combinatorial issues will frequently arise. [See especially Chapter 15 (Sections 15.3-15.6), as well as Chapters 17 and 19.] (c) There are circumstances in which it is desirable (or at least customary) to write the product of two noncommuting operators in one or another specific order as in (14.2.5) or (14.2.7). It is standard to write differential operators so that each term is a kth derivative (where k is a nonnegative integer) followed by a multiplication operator. However, in the functional calculus ofKohn andNirenberg [KoNi] for pseudodifferential operators, this order is exactly reversed; i.e. the multiplication operator acts first, Similarly, if a and ft are, respectively, the annihilation and creation operators from quantum field theory, then the ordering that results from (14.2.5) corresponds to Wick normal ordering [GliJa, p. 16] whereas the reverse ordering (resulting from (14.2.7)) corresponds to Wick anti-normal ordering. [Recall that the Wick (or normal) ordering is often used (possibly without explicit mention) in many calculations occurring in quantum field theory; see, for instance, ([Berl], [GliJa], [Wei,2]).] It is somewhat harder to explain, but we mention that if a and ft are the momentum and positions operators, respectively, then the disentangling associated with (14.2.9) is connected with the Weyl functional calculus ([And], [Foil, p. 79], [Hdrl], [Ne3, §6/j. We are now ready for our second elementary example. Let a and ft be operators on the separable Hilbert space H, and let a(s) = a and ft(s) = ft for all s in the time interval in question. Example 14.2.3 (Disentangling e-'a+<^) Given parameters t > 0 and a> e C, as well as numbers a and b, let
[Here, / (resp., w) can be thought of as a "time" (resp., "weight") parameter.] Let T e [0, t]. We attach time indices to a and ft as follows: a = j /Qa(s)ds, ft = ^ ft(s)&r(ds). Letting r e (0, t) for now and using rules, we write
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The first equality in (14.2.11) restates the assignment of time indices. The second equality is crucial; it follows Feynman's second rule by using the commutative exponential law even though the operators a and B need not commute. The fourth equality time-orders the product; that is, the order of the operators in the product is made to coincide with the time-ordering. This is the situation which one works towards in the disentangling process. Once this is done, we are free to calculate (wherever possible) any remaining integrals. Note that as in Example 14.2. 1, the "equalities" in (14.2.1 1) cannot all be interpreted in the usual way if a and B do not commute; specifically, the second and fourth equalities need not hold if interpreted literally. The calculation above is more typical of the disentangling process than the one in Example 14.2.1 insofar as the second rule is invoked. However, it is still much simpler than many of the calculations that will appear in later chapters. We can summarize the calculation in (14.2.11) by the following continuum of formulas: With / given by (14.2.10), we have for every r € (0, t),
Assume for simplicity that a and ft are bounded operators. If a and B ft commute, then the continuum of functions {fi,&r : 0 < rT < t) of a and B ft reduce to a single function. Remark 14.2.4 (a) When the operator —sa, s > 0, is being exponentiated, we will often want to permit a to be unbounded. In that case, our usual assumption will be that —a is the generator of a (Co) semigroup. We will continue to use the suggestive exponential notation in that case (as we frequently did in Chapters 8-13) even though what we really have mathematically is the semigroup generated by —a. (b) Note that it is normalized Lebesgue measure 1/tds that is used when we write a —1/t j /Q /Qa(s)ds. a(s)ds. Obviously, Obviously, one one can can equivalently equivalently write write ta ta == f^f^ a(s)ds, a(s)ds, and and itit isis this this that will usually be done as we continue. (c) Naturally, if instead of choosing f as in (14.2.10), we let
and set
(but keep a as before), then we obtain similarly that for every T e (0, t),
This choice of discrete measure (v = wSr, a Dime measure with total mass u> concentrated at the instant r) turns out to be better suited to our later developments in Chapters 15-19.
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Exercise 14.2.5 (The r = 0 and r = t cases of (14.2.12)) Using Feynman's heuristic rules much as in (14.2.11), show that we arrive at the formulas
and
[Observe that the r = 0 and r — t cases of (14.2.12) agree with (14.2.13) and (14.2.14), respectively.] Remark 14.2.6 Although we are not prepared to discuss the details now, we mention that formula (14.2.12) can be used to model the evolution of certain physical systems. In fact, this formula represents related but distinct evolutions for distinct values of T. The expression u>ft(s)ST(ds) is associated with an impulsive force acting sharply at time r. These and related matters will come up many times as we continue, especially in Chapters 15-17 and 19. (See, for instance, Section 15.3, formulas (15.5.18) and (15.5.24), Corollary 17.2.9, and Example 16.2.9.) We will continue to illustrate the use of Feynman's rules in the first part of the next section by using them to calculate a perturbation series. 14.3 Time-ordered perturbation series Time-ordered perturbation series will be one of the central themes of Chapters 15-19. Perturbation series via Feynman 's operational calculus We begin by using Feynman's heuristic ideas to calculate in a simple case the first few terms of a perturbation series. Our purpose in this is both to further illustrate and explain the use of Feynman's ideas and to see the connection between these ideas and perturbation series. Let H be a separable Hilbert space over C and let a be a fixed operator on H. It will be convenient to define «(» := a for every s > 0. In the sequel we will typically either have (Chapters 15-18) or require (Chapter 19) that —a is the generator of a (Co) semigroup on H. We attach time indices to a in the following way: For every / > 0, write
We allow for the possibility that B be time-dependent, B : [0, +00) ->• C(ri). If ft is constant, ft e £-(H), we write here tft = /0' ft(s)ds, where ft(s) := ft for every 5 > 0. We will frequently associate measures other than Lebesgue measure with ft, but note that, for now, we are using Lebesgue measure / on [0, t]. We will usually require (in Chapter 19) or have (in Chapters 15-18) that the operators {ft(s) : 0 < s < 00} all commute with one another, but we will not require that the Bs commute with the operators in the semigroup (T(s) = e~sa : 0 < s < 00} associated with a. We will
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need a measurability assumption on the function B, but we omit that discussion here. Finally, when Lebesgue measure is used, we will assume that for every t > 0,
This last condition is of course satisfied when B is a constant function. We now wish to discuss the disentangling of the exponential expression
Exponential functions are central to the study of evolving physical systems and they played a key role in [Fey8]. Their role is expanded still further in the remaining chapters of this book. We write
where the second "equality" comes from applying the commutative exponential law, exp(A + B) = [exp(A)][exp(5)], in a noncommutative setting as we did in the second "equality" in ( 14.2. 1 1 ). Just as in the earlier situation, the "equality" cannot be interpreted in the usual way. The second step in (14.3.3) is, of course, inspired by Feynman's second rule. A further point concerning the third expression in (14.3.3): It is counterproductive to calculate the integral — f0 a(s)ds at this stage. Leaving it as it is will facilitate the disentangling process. Continuing, we expand exp(/0 f)(s)ds) from (14.3.3) in a series and obtain
Next we work out the m = 0,1,2 terms of the series. m = 0: This term is just exp (- /Q a(s)ds\. Here, there are no operators present which fail to commute with a. Thus no disentangling needs to be done, and hence nothing is lost in calculating the integral. Doing so, we obtain the semigroup generated by —a:
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m = 1: We begin with the formal calculations suggested by Feynman's heuristic ideas and then comment on the steps involved.
In the first equality above, we relabel the dummy variable in the left-hand factor and move the expression inside the other integral. The second equality is obtained by writing the integral with respect to •$•] as the integral from 0 to s plus the integral from s to t. This step looks ahead to the third "equality", where the time-ordering is carried out. Once the time-ordering is complete, we can calculate the integrals in the exponentials and obtain the final expression in (14.3.6). Note that the third equality in (14.3.6) cannot be interpreted in the standard manner. m = 2: We begin as before with the formal calculations.
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Some of the steps in (14.3.7) are simple or are similar to earlier steps, and we will not need to comment on them here. The sixth 'equality' is one key; the time-ordering is completed there. This 'equality' cannot be interpreted in the usual way. Two issues are involved in the third equality. The first is that the integrand is unchanged by a permutation of the variables. The other is that the diagonal (i.e. {(s\, $2) € [0, r]2 : s\ = $2}) is a set of Lebesgue measure zero in [0, t]2 and so can be ignored. (See Remark 14.2.2(b) above.) Putting these two comments together, the integral over [0, t]2 can be reduced to 2! times the integral over the open triangle 0 < s\ < $2 < t. Using (14.3.4) through (14.3.7), we now write
Remark 14.3.1 (a) The mth term of the series (14.3.4) will be disentangled in Chapter 19 via direct use of Feynman 's ideas as above. Further, in the next chapter, the disentangling will be carried out in a setting where the Wiener and Feynman integrals can be employed. (See the special case of Corollary 15.3.6 where u is taken to be Lebesgue measure I.) (b) If a = — | A = i HQ and /6 (s) = —iV, the operator of multiplication by —i times the potential energy function V, then (14.3.8) becomes
which is the Dyson series expansion [Dysl] for the quantum-mechanical unitary group exp(— it(Ho + V)). (See formula (15.3.22) below.) On the other hand, if we take
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a = — 2 A = HO and B ( s ) = —V, we are in the diffusion (or probabilistic) setting, and (14.3.8) yields
Formulas (14.3.9) and (14.3. 10) are time-ordered perturbation series expansions for the unitary group e ~"(#o+V) and the heat semigroup e~t(-Ho+v), respectively. [Note that by using the integral expressions obtained in Section 10.2 (Theorems 10.2.7 and 10.2.6) for the free unitary group e~"H° and the free heat semigroup e~t H °, we can rewrite (14.3.9) and (14.3.10) in a more concrete form.] We stress that in order to insure that the perturbing operator B is bounded, we will assume in both cases immediately above that the potential energy function V is essentially bounded (see Proposition 10.1.1). (c) The series that is most often referred to as the "Dyson series " in the literature ([Dysl,2,4], [ReSi2, pp. 282-283], [Schwel, §11. f], [Sud, p. 119]) is more compact and slightly different from the series in (14.3.9). Let V(t) := U ( t ) - l VU(t), where U(t) = exp(-it H0). (The time-dependent operator V is the "Heisenberg representation" of the operator V.) Note that \\V(t)\\ = \\ V\\ for every t since U(t) and U(t)~l = U(-t)are unitary for all t. Form the series
This is the Dyson series in the "Interaction "interaction representation " (or "interaction picture ") of quantum mechanics; see, e.g., [Han, §11.1] or [Sud, §3.4 and §3.5]. It is not equal to the series in (14.3.9), but the two series are related in a simple way:
[In a much more general setting, we will use a closely related form of the "interaction (orDirac) representation" ([Han, §11.1]) of such series in Section 17.6; this will enable us, in particular, to obtain a product integrator "time-ordered chronological product") representation of the solutions to the associated evolution equation (see [La 16]).] Since exp(-t Ho) is a semigroup but not a group, the series (14.3.10) has no interaction representation. However, in Chapters 15-18, it will frequently be useful for us to compare (14.3.9) with (14.3.10). Hence (except in Section 17.6), formula (14.3.9) will suit our needs better than the series in the Heisenberg or interaction representation.
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(As the reader will see in Chapter 15, it is usually generalizations of (14.3.9) and (14.3.10) that will be compared.) We indicated in Section 14.2 that a variety of measures can be used to assign time indices to operators. Let u be a (Borel) measure on [0, +00), which is finite on compact subsets of [0, +00). What happens to the perturbation series (14.3.8) if /Q P(s)ds is replaced with /J B(s)[i(ds)? If we make the additional assumption that u is a continuous measure, that is, every single point set is u-null, then the appearance of (14.3.8) is not changed much. We obtain
where u x u denotes the measure on [0, +00) x [0, +00) obtained by taking the product of u with itself. The largely heuristic 'derivation' that led to (14.3.8) needs to be changed significantly at only one place to give (14.3.11) instead: In the third equality of (14.3.7), we used the fact that the diagonal D of the square [0, t]2 has Lebesgue measure zero and so can be ignored in the integral over [0, t]2. Similarly, it is true that (u x u ) ( D ) = 0. We will examine this issue carefully for m > 2 in Lemma 15.2.7; we simply mention for now that the equality (u x u ) ( D ) = 0 follows from Fubini's theorem and the continuity of the measure u (that is, the fact that u({s}) = 0 for every s in [0, t]). In spite of the similarity between formulas (14.3.11) and (14.3.8), different choices for u can represent very different disentanglings and very different time evolutions. We will examine various possibilities in later chapters, but the reader might wish to consider briefly even now the following cases: (a) The measure u is the zero measure until some time to and then is Lebesgue measure beyond to(b) The measure u, alternates between the zero measure and Lebesgue measure on successive time intervals. (c) The measure u(ds) = g(s)ds is approximately the Dirac measure at some time to (see Example 16.2.9). (d) The measure u is a continuous singular measure, such as the Cantor-Lebesgue measure (see Example 15.5.3). Suppose that the continuous measure u in (14.3.11) is replaced by a measure n which is not continuous; that is, n({r}) > 0 for at least one instant T in the time interval [0, t]. One can still obtain a perturbation series for exp ( — f a + /0' fi(s)r)(ds) J, but
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the series will no longer have the simple form indicated in (14.3.1 1). See Section 15.3 for a discussion of the case where rj = n + 0)ST with u continuous (and r in (0, t)). [As will be explained further on, the measure v :— a)Sr, with (constant) "weight" co and <5r the Dirac measure concentrated at the instant r, is called the discrete part of r;; see especially Section 15.2.F.] The situation can become extremely complicated as the reader will see in Section 15.4. We can get some idea why complications arise for discontinuous n [that is, for rj with a nonzero discrete part v, necessarily a (possibly infinite) linear combination of Dirac measures concentrated at various instants] by considering the m = 2 terms of the series and realizing that the diagonal cannot be ignored in integrals over [0, t]2 with respect to n x n. (See Figure 14.2.1 above, with [0, I]2 replaced with [0, t]2.) Perturbation series via a path integral We turn now to the calculation of the first three terms of a perturbation series using the path integral of Wiener (introduced and discussed in Chapters 3 and 4). We will compare this mathematically rigorous computation with the earlier one (14.3.8) which made use of the heuristic ideas of Feynman's operational calculus. In particular, we will compare our final result here with (14.3.10), the special case of (14.3.8) which involves the heat semigroup. We work in one space dimension for simplicity. Also, we assume that the potential V : R -+ R belongs to L°°(R); that is, V is essentially bounded on R. [Actually, in our work presented in Chapters 15-18, we will allow V to be time-dependent (and hence to be defined on [0, t] x R), but we will leave such issues aside for now.] This assumption makes it easy to verify the existence of the integrals involved as well as to justify the interchange of limit processes. Using the relatively simple case of the Feynman-Kac formula (Theorem 12.1.1) where V is essentially bounded, we can write for every
[Recall from Chapter 3 that Wiener measure m is a probability measure acting on Wiener space C'0 (the space of continuous paths x = x(s) : [0, t] -» M such that x(0) = 0).] Note that there is some similarity between (14.3.12) and (14.3.4). While the semigroup e~ t H 0 does not appear explicitly in the last three expressions in (14.3.12), it is there implicitly through the presence of Wiener measure m.
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Next we calculate the m = 0, 1, 2 terms of the series (14.3.12). It is the m = 2 term that will be the most instructive in relation to our later work. m — 0: By applying the Feynman-Kac Formula, Theorem 12.1.1, with the potential identically equal to zero, we obtain
The reader should compare this fact with (14.3.5) and with the m = 0 term on the righthand side of (14.3.10). We remark that (14.3.13) could also be calculated by using the Wiener integration formula in Theorem 3.3.5 and then the explicit formula for the free heat semigroup found in Theorem 10.2.6. m = 1: By Fubini's theorem, we have
We will typically need to work at ordering the time indices (see the case m = 2 below) and that will sometimes be combinatorially complicated. However, in (14.3.14) we have 0 < s\ < t , and since the set {s\ e [0, t] : s\ = 0 or si = t } has Lebesgue measure zero, we can limit attention to the set where 0 < s\ < t. Now applying Wiener's integration formula, Theorem 3.3.5, to (14.3.14) and then making the simple substitution i>o = MO + £» v\ = u\ + £, we can write
Finally, writing the inner integral in the last expression in (14.3.15) as an iterated integral and then using the formula for the heat semigroup given in Theorem 10.2.6, we have
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The reader should compare the last expression in (14.3.16) with the last expression in (14.3.6) and especially with the second term of the right-hand side of (14.3.10). In (14.3.16), the integrand has the operators acting on
where the equality immediately above uses the fact that the diagonal D of [0, t]2 has l x l measure zero, with / denoting Lebesgue measure on [0, t]. Note that in the first part of the last expression in (14.3.17), we have 0 < s1 < s2 < t whereas we have 0 < s2 < s1 < t in the second part. Actually, the two parts are easily seen to be equal. Using that fact, Fubini's theorem and the Wiener integration formula (Theorem 3.3.5), we can write
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We finish finding the desired formula for I2 by making the substitution ii0 = MO + £, DI = MI + £, vz = U2 + £, writing the resulting integral over R3 as an iterated integral and, finally, using formula (10.2.26) for the free heat semigroup e~sli° given in Theorem 10.2.6:
The reader should compare the third term on the right-hand side of (14.3.10) with the last expression above. As operators, they are the same except for the time-reversal. The calculation in (14.3.7) should also be compared with the calculation of I2 above, and the strong similarities should be noted. The computation of the Wiener integral I2 is, however, mathematically rigorous and can be done without ever explicitly invoking Feynman 's rules. Finally, we use the calculations of I0, I1, and I2 above to write the first three terms of the series in (14.3. 12):
[Eventually, we will want to analytically continue the expression (14.3.20) in a suitable (mass or diffusion) parameter A. When we do this (see, for example, Corollary 15.3.6 with n = 1), the form of the resulting perturbation series will essentially remain the same.] We have seen above that there is a connection between path integrals and Feynman's operational calculus. Feynman was aware of this; indeed, it was a desire to extend his
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work on the path integral [Fey2] to quantum electrodynamics (QED) which led him to his operational calculus. The origins of Feynman's operational calculus We give here a brief historical sketch of Feynman's invention of his "rules" for forming functions of noncommuting operators. As a graduate student, Feynman was fascinated by quantum electrodynamics (QED). The theoretical and computational difficulties with QED were regarded as a central problem of the physics of the time. Recall from Chapter7 that Feynman's 1942 thesis [Fey 1] and the resulting 1948 paper [Fey2] developed a path integral in the setting of nonrelativistic quantum mechanics. Feynman also wanted a path integral for relativistic QED. He did not succeed in this, but he did invent methods of calculation similar in certain situations to those for path integrals which allowed him to compute perturbation (or Dyson) series for QED. These calculations were in the spirit of (14.3.3)-(14.3.8) above but were more complicated. The famous Feynman diagrams were a way of keeping track of the terms of the resulting perturbation series. In Feynman's first two papers [Fey5,6] on this subject, he discussed the physics involved but did not explain his methods of calculation. These methods as applied to QED were explained in his 1950 paper [Fey7]. Finally, in 1951, he gave a largely mathematical, but far from mathematically rigorous, explanation and extension of his formal methods of calculation which he called the operator calculus (for noncommuting operators) [Fey8]. Although [Fey8] contained examples from quantum mechanics and QED, it is clear that Feynman thought of it as primarily a general mathematical method for forming functions of noncommuting operators. The papers [Fey 2-8]—all of which appeared in the period 1948-51 —are the components of a single extraordinarily rich and influential research development. This development is traced in much more detail in Jagdish Mehra's fine biography of Richard Feynman [Me, esp. Chapters 6 and 10-15] and in the valuable monograph [Schwe2] and book [Schwe3] of Silvan Schweber. Remark 14.3.2 (a) The Dyson series for nonrelativistic quantum mechanics (see (14.3.9)) is not difficult to compute from Feynman's path integral [Fey2j; somewhat surprisingly, it does not appear in [Fey2]. However, such Dyson series play a major role in the later book of Feynman and Hibbs [FeyHi]. (b) Dyson's role was, in particular, to explain in [Dysl] the connections between the approaches to QED put forth by Tomonaga, Schwinger, and Feynman. (The latter were jointly awarded the Nobel prize in physics in 1965 for their work on quantum electrodynamics.) In his paper [Dysl], Dyson introduced the time-ordered perturbation series that now bears his name. (We also refer to [Dys2,4], especially to Dyson's own comments in [Dys4, pp. 9-16], as well as to Schweber's description of Dyson's work on QED in [Schwe3, Chapter 9].) Feynman's work on the operational calculus for noncommuting operators is not mathematically rigorous. We will briefly discuss some ways of achieving rigor in the next section and then will pursue this topic in Chapters 15-19.
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14.4 Making Feynman's operational calculus rigorous The work of Feynman referenced at the end of this book has had a fundamental impact on physics and a significant impact on a number of developments in mathematics. However, this work is not rigorous in the sense in which mathematicians commonly understand this word. How can it be made mathematically rigorous? We have given some answers to this question earlier with regard to the Feynman integral, especially in Chapters 11 and 13. In this section, we will indicate three ways of making Feynman's operational calculus rigorous. Feynman himself recognized that lending rigor to his operational calculus was not likely to be easy. The reader might be surprised to read what he had to say about this. (See the second quote from [Fey8] given at the beginning of this chapter.) We note that in spite of our emphasis in this section on the problem of making Feynman's operational calculus rigorous, this is not our only concern in the remaining chapters of this book. Our goal in Chapters 15-19 is also to interpret and extend Feynman's ideas. I. Rigor via path integrals We indicated in Section 14.3 that Feynman's operational calculus can be regarded as generalizing some aspects of path integrals. We quote Schweber [Schwe2, p. 502] on this point of view (but the remarks in parentheses are ours): "In any case Feynman never felt order-by-order (the time-ordered integrals involved in the perturbation expansions obtained from applying Feynman's operational calculus) was anything but an approximation to the 'thing' and the 'thing' was the path integral." In the following paragraph [Schwe2, p. 502] Schweber quotes Feynman (taped interview, November 1980): "A way of saying what quantum electrodynamics was, was to say what the rule was for the diagram (or Feynman graph)—although I really thought behind it was my action form" (and so, presumably, an associated path integral of some sort). Since the nonrigorous operational calculus of Feynman is a kind of generalization of Feynman's nonrigorous path integral, it seems plausible that the rigorous Feynman and Wiener integrals could be used to obtain a rigorous version of the operational calculus in the quantum-mechanical and diffusion (or probabilistic) settings appropriate to the Feynman and Wiener integrals, respectively. Indeed, this is exactly what is done in Chapters 15-18. Those chapters—which are based on work by Johnson and Lapidus in [JoLal-4] (for Chapters 15, 16 and 18) as well as of Lapidus in [Lal3-18] (for Chapter 17)—will be discussed briefly in Section 14.5. We refer the reader to that section for further introductory material concerning this approach and, in particular, for a discussion of the noncommutative operations * and + on "disentangling algebras" of Wiener functional. Remark 14.4.1 (a) We have discussed in rather simple cases the use of measures other than Lebesgue measure in connection with Feynman's operational calculus in Examples 14.2.1 and 14.2.3, Exercise 14.2.5, Remark 14.2.2 and formula (14.3.11). The flexibility provided by such measures will allow us to substantially extend the operational calculus as the reader will see repeatedly in connection with approach I in Chapters 15-18
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and approach II (described below) in Chapter 19. A similar statement holds for approach III, but that approach will be discussed only briefly in this book; namely, in III of the present section. (b) We have been emphasizing above that the Wiener and Feynman path integrals will be used in Chapters 15-18 to make the operational calculus rigorous. However, as was already mentioned, many of the results in those chapters have an interest as contributions to the Wiener and Feynman integrals apart from their connection with the operational calculus. (c) Both the Wiener and Feynman integrals are intimately associated with the Laplacian, the "generator" of Brownian motion. (See Example 9.2.3, Remark 9.2.4, and Theorems 12.1.1 and 13.3.1 for earlier material related to this statement.) It seems clear that other path integrals associated with different generators and Markov processes (see [Fuk, FukOT, L2, MaRo, Or, Sh, StroVa, Va, Wil]) could be used in appropriate settings to make Feynman's operational calculus rigorous. The only work that we know of which is explicitly in this direction is due to Riggs [Rig]; it extends to the Dirac operator (in one space dimension) some of the results of [JoLal,2] and [Lal5,16] discussed in Chapters 15 and 17, respectively. The stochastic process involved in [Rig] is the Poisson process with generator and semigroup as indicated in Example 9.2.5. The related partial differential equations are the telegrapher's equation and the Dirac equation (both in one space dimension) rather than the heat and Schrodinger equations as in [JoLal,2] and [Lal4-18]. (See Remark 17.6.32(c).) II. Well-defined and useful formulas arrived at via Feynman's heuristic rules A common mathematical strategy is to use heuristic ideas to arrive at mathematical "objects" which can then be rigorously defined and are useful in themselves. This is the point of view adopted in Chapter 19, based on the work of DeFacio, Johnson and Lapidus in [dFJoLal,2]. Much of the motivation for [Fey8] as well as for the papers [Fey5-7] which led up to it was to find a method of calculation akin to path integration which allowed Feynman to "derive" formulas for evolving physical systems. As we indicated earlier, he was especially interested in the perturbation series for quantum electrodynamics where there was no path integral available. Although Feynman's operational calculus extends well beyond exponential functions, exponentials of sums of noncommuting operators will play a central role in Chapter 19 as they did in [Fey8]. In fact, in Chapter 19 we will be considering exponentials of sums of integrals of operators as in (19.1.2). The special emphasis on exponentiation comes from its connection with evolution equations. A major part of the idea in Chapter 19 is to develop mathematical models for complicated physical systems which combine features of simpler systems but are not readily treatable by familiar methods. In mathematical terms, the rigorously defined disentangled formulas for (19.1.2) will provide the unique solution to the evolution equation (in integral form) (19.5.1). As will be further discussed in Chapter 19, one advantage of this abstract approach [dFJoLa2] is that it allows us, for example, to deal with nonlocal interactions—such as nonlocal potentials given by suitable integral operators (instead of the traditional local
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potentials given by multiplication operators). This is of interest in studying models used in phenomenological nuclear physics [Tab, ChSa, Mc]. III. A general theory of Feynman 's operational calculus with computations which are rigorous at each stage We describe briefly here a part of recent work of Jefferies and Johnson [JeJo]. Fix bounded linear operators A1, . . . , An on a Banach space X over the complex numbers C. It is not assumed that A1, . . . , An commute. We form a commutative Banach algebra ID>(A1 , . . . , An) which consists of certain analytic functions f(A\, . . . , An), where AI , . . . , An are regarded as purely formal commuting objects. Within O(Ai , . . . , A n ), called the disentangling algebra, we can, in the spirit of Feynman's second rule (see (2) of Section 14.2) form functions of the 'operators' as though they were commuting. In fact, the actual elements of D(A1 , . . . , An) do commute. Now let u1, . . . , un be (for convenience) Borel probability measures on the time interval [0, 1]. The measures u1 , . . . , un will be used to assign time indices to specify the order of the operators in products (see (1) of Feynman's rules in Section 14.2). Saying it another way, u1, . . . , un will specify directions for disentangling functions f ( A 1 , . . , , An) of A1 , . . . , An (see rule (3) in Section 14.2). Disentangling maps
are defined and the resulting theory may be thought of as providing a family of functional calculi for the operators A1 , . . . , An . A simple example of a theorem from [JeJo] is that if AI, ... , An commute, then the functional calculi specified byTu1,...,^,with u1, ..., un as above, all agree with each other and with the usual analytic functional calculus for commuting operators. The discussion so far gives some idea of the content of the first several sections of [JeJo]. Those sections study the algebraic properties of Feynman's operational calculi and it is only the norm and not the spectrum of the individual operators that comes into play. In later sections, with additional information about the n operators involved, a "joint spectrum" can be defined and the functional calculus can be enriched. The operator-valued version of the Paley-Wiener theorem is one of the keys to this. Also, a connection with the Riesz-Dunford functional calculus for a single operator is made by appealing to the Cauchy integral formula from Clifford analysis for functions of n + 1 real variables taking values in a Clifford algebra. In [JeJo], the emphasis is on the algebraic properties of Feynman's operational calculi as well as on forming rather general functions of the not necessarily commuting operators A1, . . . , An. On the other hand, in order to encompass evolution problems within the theory and, in the process, to make connections with II above and Chapter 19 below, it is necessary to adjust the framework somewhat and to emphasize exponential functions and operator- valued functions of time. Such work is in progress in a paper by Jefferies, Johnson and Nielsen [JeJoN]. We finish here by mentioning the thesis of Lance Nielsen [Nie] which is nearing completion. Nielsen proves several stability theorems in the settings of either [JeJo] or [dFJoLa2]. For example, the first theorem in [Nie] says that if a sequence
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{(u1 m , . . . , un,m) : m = 1, 2, . . . } of n-tuples of probability measures on [0, 1] converges weakly to the n-tuple (u1, . . . , u n ) , then the sequence of disentangling maps {Tu1m,...,unm :m = 1,2, . . . } converges to the disentangling mapTu1,...,unin the strong operator topology on £(D(A1, . . . , An), £(X)). This conclusion can also be rephrased as asserting that the sequence of functional calculi associated with (u1 m , . . . , u n m ) converges to the functional calculus associated with (u1, . . . , u n ) . 14.5
Feynman's operational calculus via Wiener and Feynman integrals: Comments on Chapters 15-18 We will give here a brief description of some key parts of Chapters 15-18. The precise statement of definitions and theorems will be postponed, but we will try to convey the spirit of some of the results. Our discussion of Chapter 15 will be the fullest both because it is the essential starting point for the later chapters but also so that it can serve as part of the introduction to the next chapter. Chapter 75 Recall that Ct = C([0, t], Rd), for t > 0, and let F : C' -> C. The operator-valued function space integrals A"|(F), A e C+, which we will use throughout the next four chapters will be defined for A e C+ just as in Section 13.5. (Here, as before, C+ = {A e C : Re A > 0} and C+ = {X e C : X ^ 0 and Re X. > 0}.) In particular, K[(F) will be defined for A > 0 by the Wiener integral (13.5.2) in Definition 13.5.1 (or see (15.2.8) below) and then for A. e
where n is a C-valued measure of finite total variation on (0, t) and 6 : (0, t) x Rrf —> C belongs to Looi;>; (see parts F and G of Section 15.2). For our present purposes, one does not lose much by thinking of n as a finite (nonnegative, Borel) measure on (0, t) and regarding 6 as bounded. (In order to insure convergence in an appropriate sense of the infinite sums mentioned above, we will require (15.4.19).) The collection At of functions (equivalence classes, actually) which we have just described is larger than we will need for any one of the concrete problems that we will consider. However, At has the advantage of being an algebra and of being complete under an appropriate norm. In fact, At is a commutative Banach algebra with identity which we call the disentangling algebra (or the disentangling algebra associated with time t). (See Section 15.7.) The family of disentangling algebras (A, : t > 0} will play a prominent role in Chapter 18, as we will discuss further on. Given F e At, the function of operators that is to be disentangled can be identified from F; we will give an example of this below. The disentangling is carried out by
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calculating the path integral K[(F). [Recall that K[(F) is a Wiener integral for X > 0 and a Feynman integral for A, € C+ \ C+ (that is, for A ^ 0 purely imaginary).] The disentangled operator K[(F) is expressed as a generalized Dyson series (GDS). If F(x) = e\p[F-iv,i(x)] (see (14.5.1)), where / is Lebesgue measure on (0, t) and V is R-valued and time independent, then—except for an exact reversal of the time-ordering (see Remark 15.3.7)—the GDS forKt-1i(F)is just the Dyson series (14.3.9). We will see that the GDS can be quite simple or extremely complicated (as in Section 15.4). Generalized Feynman diagrams (see Section 15.6) are a visual aid in keeping track of the individual terms of a GDS, and they help in giving physical interpretations where that is appropriate. Next we take a simple example of F e At and identify in the notation of Sections 14.2 and 14.3 the function of operators being formed when K[(F) is calculated. Let
where / is a complex-valued function of one complex variable. We assume for convenience that / is entire. When the GDS for K'^(F) is calculated, the function of operators being computed is in fact
where 0(s) denotes here the operator of multiplication by the function &(s, •). In the notation of Sections 14.2 and 14.3, a(s) = Ho/A for every s € [0, t], and so t ( H 0 / X ) = /Q a(s)ds, while B(s) = 0(s) for 0 < s < t. Hence, (14.5.3) can be written as
One may well ask how exp[-t (Ho/A)] is obtained from (14.5.2). It actually comes out of the path integral which starts for A. > 0 with a Wiener integral. Indeed, the exponential factor exp[—t(Ho/A)] for some A e C- will be involved in all the disentanglings in Chapters 15-18. (If some appropriate stochastic process other than the Wiener process were used, then the infinitesimal generator of that process would replace the infinitesimal generator — HO of the Wiener process.) Various examples discussed in Section 15.5 are helpful in understanding the theory and its relationship to other subjects; we mention in particular Examples 15.5.1, 15.5.3, 15.5.5, and 15.5.9. Chapter 16 Every F e At is defined in terms of "potentials" and measures. It is natural to ask for conditions under which the operators K[(F) depend continuously on these objects. We consider such questions with respect to potentials in Section 16.1 and with respect to measures in Section 16.2.
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The results of Section 16.1 are more satisfying but those of Section 16.2 are perhaps more interesting. Example 16.2.9 is especially illuminating; it illustrates how the GDS for a sequence {nm } of absolutely continuous measures can pass to the GDS for a discrete measure as nm converges "weakly" to the Dirac measure ST, where r e (0, t). Chapter 1 7 The focus of Chapter 17 is on the work of Lapidus [La 14- 18] in which he established a "Feynman-Kac formula with a Lebesgue-Stieltjes measure" (in short, FKLS) and related it to the solution of various associated integral and differential (evolution) equations. In these same papers, Lapidus made connections with the operational calculus of Feynman through a variety of topics that are discussed in Chapter 17 and which are related to material in Chapters 15, 16, and 18. The classical Feynman-Kac formula was the subject of Chapter 12. Recall that the functionals dealt with there had the form
where 6 = - V with V time independent and R-valued R- valued (see Theorem 12.1.1). 1.2.1.1 ).In Incontrast, contrast, the functional functionals that will interest us in Chapter 17 will be of the form
where 90 : [0, t) x W Rd1 -> C is allowed to be time-dependent and C-valued and 77 is a C-valued Borel measure on [0, t] t ] (see (17.1.5)). ( 1 7. 1 .5)).InInterms termsofofthe therestrictions restrictionsmentioned mentionedsoso far, the functionals in (14.5.6) form a more general class than those in (14.5.5). However, the potentials V (= —6) in (14.5.5) were considerably more general in one important sense; they were permitted to have strong singularities (see Theorem 12.1.1). In contrast, the functions 6 in (14.5.6) are required to be in Looi-^ (see (17.1.3)) which implies, in particular, that 0(s, 9(s, •) is essentially bounded for n-a.e.s. It will be necessary in Chapters 17 and 18 to treat time t as a variable whereas it will suffice to fix one value of t in Chapters 15 and 16. Actually, it is not quite the functions t (-*• -> Kt/a(FK,n) K'^FK,^) that will solve the evolution equations (in integral or differential form) in Chapter 17, but rather (14.5.7) where FK,n is the time-reversal of the functional FK,n (see Definition 15.7.5 and Theorem 15.7.6). The functional FK,n belongs to the disentangling algebra At for every t > 0 by Theorem 15.4.1 (with m = 1) and Corollary 15.7.4. Hence, K [ ( F K n ) exists and can be disentangled via a generalized Dyson series (GDS) for every >. e C+ (see Theorem 15.7.1). The presence of the Lebesgue-Stieltjes measure n will allow us to consider a variety of interesting special cases and to blend continuous and discrete phenomena. [In the
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following, we let n = u + v denote the unique decomposition of 77 into its continuous part u. and its discrete part v (see Section 15.2.F).] The case n — u, with u absolutely continuous with respect to Lebesgue measure /, is not much different mathematically from the case n = l itself. However, even within the absolutely continuous case, the physical interpretations of the evolving operator u(t) (see (14.5.7)) can be drastically different. The fact that the measure n = u+v is allowed to have a nonzero discrete part v affects the physical interpretation of the evolving operators in both the diffusion and quantummechanical cases (see especially Section 17.5). The presence of v = 0 introduces phenomena that play an important role throughout all of the sections in Chapter 17. The emphasis is on finitely supported v, say v = X)p=i Mp^rp with 0 < TI < • • • < TJ, < ? and Wp e C, since that case is more tractable and seems to be the most reasonable physically. (See Sections 17.2-17.5, which discuss the results of [Lal5].) However, the general case, v — Y^Li (OP&TP with £^Li cop\ < oo, is treated in Section 17.6. [Actually, some of the results from [Lal8,16] discussed in Section 17.6 will provide new information even in the special case (treated in Sections 17.2-17.5) when v is finitely supported. In the quantum-mechanical case (i.e. 9 = —iV and k = —i), this is especially true regarding the distributional form of the differential equation and a product integral representation of the solution to the evolution equation, which are obtained for an arbitrary measure 77; see Theorems 17.6.15 and 17.6.10, respectively.] Equation (17.5.12) (or (17.2.11)) gives a striking formula for K[(F), where F = FK,T} is the Feynman-Kac functional (14.5.6). In addition, a comparison of the timeordered expression for the functional F in (17.5.11) with the last expression in (17.5.12) will eventually show us that the noncommutative operations which will be introduced in Chapter 18 fit well with both the operational calculus and the Feynman-Kac formula with a Lebesgue-Stieltjes measure. The study, in conjunction with the Feynman integral, of exponential (and other analytic) functions of functionals of the form
where n is an arbitrary C-valued Borel measure on [0, t], seems to have begun with [ChanJoSk2, Jo4]. However, in those papers the approach to the Feynman integral and the assumptions on 9 were different than in Chapters 15-18. The Feynman integral was scalar-valued rather than operator-valued and transform assumptions were placed on 6. The main assumption on 9 was that for every s in [0, t], 0(s, •) was equal to the Fourier transform of a C-valued Borel measure on Rd. The "Fresnel integral" [AlHol] also involved transform assumptions but otherwise appeared quite different. However, the two theories were shown to be essentially equivalent in [Jol] and a little later but independently and in greater generality in [KalBr]. It had been shown [JoSk10] under transform assumptions that the scalar-valued analytic Feynman integral of a functional much like exp(F1), except with n in (14.5.6) equal to Lebesgue measure l, satisfies the Schrodinger equation in integral form. Johnson asked a former graduate student if a similar result holds for a general C-valued Borel
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measure n. He also observed that the result follows easily from [JoSkl0] if n is absolutely continuous with respect to l since, in that case, the Radon-Nikodym derivativedn/dlcan be used to absorb n into the potential. The same issue came up while Johnson and Lapidus were working on [JoLal]]. Although the setting was different, earlier work [JoSk6, p — 2 case] again easily took care of the absolutely continuous situation. Lapidus went on to establish the much more complicated case of a general C-valued Borel measure n and to ask and resolve many further related questions. The operator-valued setting of these papers of Lapidus [Lal4-18] [and of the paper [JoLal] (discussed in Chapters 15 and 16) that helped lead up to them] provided a natural framework in which to study the relationship between the Feynman-Kac formula with a Lebesgue-Stieltjes measure and the operational calculus of Feynman. Chapter 18 In Chapter 18 — which discusses joint work of the authors in [JoLa3,4] — we study two noncommutative operations on Wiener functionals and their relationship with the disentangling algebras {At : t > 0} and the operational calculus of Feynman. We begin with a description of the noncommutative operations avoiding the technical details; a more precise treatment will be postponed until Sections 18.2 and 18.3. The reader may wish to review our brief comments on the disentangling algebra At at the beginning of this section (or else see Section 15.7). Let F e At1 and G 6 At2. Then F and G are C-valued functions on Ct1 and C'2, respectively. The noncommutative product F * G is to be a C-valued function on C't1+t2. Given x e C t1+t2 , (F * G)(x) := F(x1)G(X2), where x\ is the restriction of x to [0, t\] and X2 is obtained by first restricting x to [t1 , t1 + t2] and then translating the restriction to [0, t2]. In a similar manner, the noncommutative addition + is defined by the formula (F+G)(x) := F(x\) + G(x2), for x e C t1+t2 . We will show in Theorem 18.5.3 that F * G and F+G belong to Att+t2. Note that the operations * and + are not internal to At but rather map At1 x At2 to At1 +t2 . As the reader may have noticed, these operations are not limited to the disentangling algebras but make sense for any functions F and G on Cl1 and Ct2 , respectively. The operations have various pleasant algebraic properties as we will see in Sections 18.3 and 18.5. In order to illustrate how the operation * is related to Feynman's operational calculus, we will describe the problem which initially led the authors (in [JoLa3,4]) to the definition of *. Let F and G belong to At1 and At2, respectively. We know from our earlier discussion of Chapter 15 that for every y e C+~, both Kt1y (F) and Kt2y(G) are disentangled by their generalized Dyson series expansions (GDS). It is natural to ask if the operator product Kt1y (F)Kt2y(G) can also be disentangled. The answer is "yes". In fact, Kt1y (F)Kt2y(G) has a GDS since
(See Theorem 18.5.6.) Note that the definition of F * G and the formula (14.5.9) both involve time-ordering. We will see in Theorem 18.4.1 that the relationship (14.5.9) is not restricted to the algebras [At : t > 0} but extends to rather general functionals F and G on Ct1 and Ct2,
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respectively. However, the interpretation of (14.5.9) is different in that case since there is no reason to think that any of the three operators involved can be disentangled via a GDS. It is then reasonable to regard the right-hand side of (14.5.9) as providing a partial disentangling of Kt1 +t2 (F * G). The formula
is used in [Fey8] even when A and B do not commute. Following Feynman, we used essentially the same relationship in the second "equality" in (14.2.11) and also in (14.3.3); of course, these "equations" cannot be interpreted in the usual way. Such steps are in the spirit of rule (2) of Feynman (see Section 14.2) and are used to disentangle expressions which involve the exponential of sums of operators. Is there a useful way to rigorously interpret (14.5.10) without changing its form? In fact, given F E At1 and G E At2, we will show in Theorem 18.5.9 that
Note that (14.5.11) is a relationship between functionals whereas (14.5.12) gives the corresponding operator equation. The formula (14.5.11) looks like the standard exponential formula but involves the noncommutative operations + and * introduced in [JoLa3,4] (and in Section 18.3). The proof of the operator equation (14.5.12) follows easily from (14.5.11) and (14.5.9). Several other "paradoxical formulas" found in Feynman's paper [Fey8]—as well as new ones—can be given a rigorous interpretation in this framework. (See, in particular, Example 18.5.12.) Remark 14.5.1 An attempt to capture the essence of the algebraic and analytical structures underlying the construction [JoLal-4] (carried out in Chapters 15 and 18) of ({At}t>0, +, *)—the family of "disentangling algebras" equipped with the noncommutative operations * and + defined above—was made by the second author in [Lal9], where a possible set of axioms for (parts of) Feynman's operational calculus was proposed. (See Appendix 18.6.) The counterpart of the map F H» Kt y (F) was then viewed as a "quantization map" defined via a kind of generalized (Feynman) path integral. The difficulty in thisframework is, of course, to construct such a map in each concrete situation of interest in the applications [as was done in [JoLal—4] (see Chapters 15 and 18) in a setting corresponding to ordinary quantum mechanics]. In addition, it is likely (but has not yet been established mathematically) that the work in [dFJoLa2] (see Chapter 19) augmented by that in [JeJo] (described respectively in parts II and III of Section 14.4 above) provides further illustrations of the axioms proposed in [Lal 9] for Feynman's operational calculus via noncommutative operations acting on suitable "disentangling algebras".
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We close this chapter by giving a list (certainly not complete) of references connected with Feynman's operational calculus. These references are divided into four categories but the boundary between these categories is not always clear cut. The reader may be surprised at the diversity of this work. (1) References using path integrals to make Feynman's operational calculus rigorous: [AhJo, ChaJo, JoLal-4, JoP, Lal3-18, 20, Rig]. (2) References of a more abstract nature: [Alb, And, Ar, dFJoLal,2, Gil1,2, GilZa, JeJo, JeJoN, JoKal5, KisRal,2, Lal9, Masll, NazSS, Ne3, Nie, Rey, Tay1]. (3) References in the physical literature: [AgWo, dWmMN2, Fuji, Klal, KouN, Lo, Ya]. (4) A sample of further work in some way related to the operator calculus: [BayFFLS1,2, Ber, Chen1-3, Con1,2, DoFri1-3, DouRotS, FlSter1,2, Fli, Foil, GelKLLRT, GelRet1,2, GrsLouSte, Je3, JeMcIPic, H6rl-4, KoNi, Kon2,4,8, LM1,2, Moy, Pins, Pry 1,2, PrySol, Ric, Riel,2, Ster, Stri2, Tay2, Un1,2, Wei]. In particular, the book Operational Methods by Maslov [Mas11] and the papers by Nelson [Ne3] and Araki [Ar] were early and influential works. Maslov's work had a strong impact on and was further developed in the recent book Methods of Noncommutative Analysis by Nazaikinskii, Shatalov and Sternin [NazSS]. Finally, we mention that time-ordered integrals appeared in the literature well before Feynman's paper [Fey8]. Examples can be found in the work of Vito Volterra (see [Vol, VolHos] and the earlier references therein). Jagdish Mehra [Me, §14.4(a), p. 192] gives a brief description of Volterra's work and points out its relevance to both the functional calculus and functional integration.
15 GENERALIZED DYSON SERIES, THE FEYNMAN INTEGRAL AND FEYNMAN'S OPERATIONAL CALCULUS 15.1 Introduction The reader will recall that the first part of Section 14.5 introduced some of the ideas of this chapter. The present section continues this introduction but with an emphasis on the role played by the discrete part of the measures n that are involved. The present chapter (and the next one) discusses work of Johnson and Lapidus in their memoir [JoLal]. One of its main goals is to lay some of the foundations for Feynman's operational calculus, and particularly for the "disentangling process" discussed in Section 14.2. As was already briefly explained in Section 14.5, we will achieve this in a concrete setting by using the Wiener and Feynman path integrals of suitable functionals to represent the associated operators via time-ordered perturbation expansions, called generalized Dyson series (GDSs), valid in the diffusion (or probabilistic) as well as in the quantum-mechanical case. Further, we will show that the set of Wiener functionals giving rise to these GDS— modulo a suitable equivalence relation and a suitable choice of norm—forms a commutative Banach algebra, At, called the disentangling algebra. As will be seen throughout this chapter, these perturbation series possess a rich combinatorial structure, owing to the systematic use (introduced in [JoLa1,2]) of both discrete and continuous measures in the time-integration involved in the definition of the appropriate Wiener functionals. Fix t > 0. Recall from Section 14.5 that the commutative Banach algebra At (which will be constructed in Section 15.7 below) consists of infinite sums of finite products of functions of the form
where n is a C-valued Borel measure on (0, t). Recall further from our discussion of Chapter 17 in Section 14.5 that the function (14.5.3), that is, the function x i->\-> ex.p(F0^(x)), exp(F0,n(x)), is a particularly useful example of an element of AtAt. The measures nr\ as above can be quite diverse. (See Section 15.2.F below for basic definitions and facts regarding these measures.) Every such nr\ can be uniquely written as r\ = IJL n u + v, where nrj is continuous and and v is discrete. The The additional flexibility flexibility provided by the use of Lebesgue-Stieltjes measures nrj has many implications which allowed us in [JoLal^4, [JoLal-4, Lal4-18] to broaden and unify known concepts and to introduce new ones having an interest in their own right. When r\ n = un is a continuous measure, the generalized Dyson series (GDS) associated with the functional exp(Fe exp(Fo,u(x)) tM (x)) has the
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same formal appearance as in the classical case (see (14.3.9)). However, even when 77 = u is absolutely continuous with respect to Lebesgue measure, very different interpretations may be suggested. Cases with 77 = u continuous but singular with respect to Lebesgue measure / are interesting and quite different (see Example 15.5.3). When n has a nonzero discrete part, the form of the GDS changes markedly and strikingly different phenomena occur. The combinatorial structure of the series is much more complicated even when v is finitely supported. For instance, additional summations appear as well as powers of the potential 9 evaluated at fixed times. Some of the combinatorial complications and nearly all of the analytic difficulties are found in the (relatively) simple case n = u + wdT, where dT is the Dirac measure at r. Accordingly, we discuss this prototypical example in detail in Section 15.3 (and later in Section 17.3) and use it as a conceptual aid to the general development. Even the case when n = v is a purely discrete measure with finite support, i.e.
is of interest. By considering exp(F0,n) with F0,n Fo,n given by (15.1.1) and further specializing, we obtain apartial a partial product (see (15.5.19)) as in the theory of the product integral. Still more specializing yields the hth Trotter product. Connections with the (time-ordered) product integral and the Trotter product formula will be discussed in Section 17.6 (especially in Theorems 17.6.10 and 17.6.12) and Example 16.2.7, respectively. Our GDSs can be represented graphically by generalized Feynman diagrams. The nth term of the classical Dyson series (see (14.3.9) and (15.3.22)) corresponds to a single connected Feynman diagram. The same assertion is true if Lebesgue measure / in the classical Dyson series is replaced by any continuous measure n = u. (See Figure 15.6.1 and the discussion which precedes it.) However, when n = u + v with the discrete part v different from 0, then the nth term of the CDS GDS gives rise to many disconnected components, one for each summand. Figures 15.6.2-15.6.4 and the related discussion all deal with the case n = u + wdt, wdt, whereas Figure 15.6.5 deals with the case n=m u +Eh Ehp= 1wpdtp.The complex combinatorial structure of the GDS is accurately p=1 reflected in the generalized Feynman graphs, and the reader may find it helpful, after a brief look at Section 15.6, to draw such graphs while following the proofs and examples in Sections 15.3-15.5. A great variety of Feynman diagrams and perturbation expansions appear in the physics literature. We should make it clear that we do not claim to be generalizing all of these. The combinatorial complications that are introduced when n has a nonzero discrete part are associated with the inappropriateness of the third part of Feynman's timeordering convention (14.2.1). (See equation (14.2.11) for such measures.) For example, if n ( { T } ) = 0 and 0 < r < t, then (n x n ) ( { ( Tr , r)}) = = n ® 22 ( { r } ) = 0 and so the diagonal cannot be ignored in integrals over (0, t) x (0, t) with respect to n x n]. n. (See Figure 14.2.1. Recall that the diagonal could be omitted in the earlier calculations (14.3.7) and (14.3.17).)
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A striking instance of the distinct roles played by the continuous and discrete parts of the measure n will be found in Chapter 17 which deals with the Feynman-Kac formula with a Lebesgue-Stieltjes measure [Lal4-18]. See, for example, Theorem 17.2.7 and part (a) of the remark which follows it. In our experience, Feynman's paper on the operational calculus [Fey8] is, at least initially, difficult to read; perhaps more difficult for mathematicians than his celebrated paper [Fey2] on the "Feynman path integral". The operational calculus paper is rich in ideas, but is also in need of clarification and mathematical development. Indeed, Feynman himself wrote [Fey8, p. 108] about his operational calculus: "The mathematics is not completely satisfactory. No attempt has been made to maintain mathematical rigor. The excuse is not that it is expected that rigorous demonstrations can be easily supplied. Quite the contrary, it is believed that to put the present methods on a rigorous basis may be quite a difficult task, beyond the abilities of the author." The above quote may be surprising to the reader familiar with some of Feynman's negative statements regarding mathematics as reported in the press or printed in some of his own books or articles. However, it is in complete accord with many statements made by Feynman during private conversations with the second author on the need for the development of a mathematical theory of his operational calculus. It is our hope that Chapters 14-19 of this book will contribute to the understanding and rigorous development of Feynman's operational calculus for noncommuting operators. We now briefly describe the organization of the remainder of this chapter: The next section provides notation, facts and definitions that will be needed in Chapters 15-18, with an emphasis on the operator-valued analytic (in mass) Feynman integral. The section will close with three lemmas which will be used frequently in Chapters 15-18. In Section 15.3, we discuss the prototypical example n = u + wdt mentioned above; our most detailed analytic proofs are given in this case. Generalized Dyson series for the full class of functionals treated in this chapter are obtained in Section 15.4. The reader should note the main results of this section but may wish to skip the proofs, at least on a first reading. Section 15.5 may be particularly helpful to the reader as it deals with a variety of concrete examples of perturbation expansions. Much of the emphasis in this section is on the combinatorics. We present in Section 15.6 a graphical representation of our generalized Dyson series in terms of generalized Feynman diagrams. In Section 15.7, we show that the general class of functionals treated in Section 15.4 forms a commutative Banach algebra, and we discuss the related functional calculus. We also define the "time-reversal map" and study it in the context of this "disentangling algebra". The disentangling provided by the path integrals precisely reverses the natural physical ordering (of the GDS), which the time-reversal map restores. Finally, the last part of this section discusses some of the connections with Feynman's operational calculus. Possible physical interpretations are provided in various places throughout the chapter.
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We point out that some of the material in this chapter is new (and, in particular, was not contained in [JoLal]). The most essential addition is the definition of the time-reversal map T (Definition 15.7.5) and a careful discussion of its properties (Theorem 15.7.6 and Corollary 15.7.8). (See also Remarks 15.3.7 and 15.4.4, as well as Example 15.3.8, leading to the definition of T.) We also mention that an interesting explicit example of the computation of the operator Kty(F), in the case where the measure 77 = u is purely continuous but singular, is provided in Example 15.5.3. 15.2 The analytic operator-valued Feynman integral In A-I below, we recall some facts and introduce much of the notation which we will need in this chapter as well as in Chapters 16-18. With the exception of G and I, we suggest that the reader go over the material quickly and then return to it if and when it is necessary. First we mention some general references: For the theory of the Wiener process and applications of path integration that supplement our earlier discussion in Chapters 2-5 as well as 12, the reader may wish to consult [GliJa, Si9, Va]. For semigroup theory, we refer to Chapters 8 and 9 for the basic developments, as well as to Chapters 10 and 11 for related facts. For the theory of the Bochner integral, we refer to the treatise of Hille and Phillips [HilPh, Chapter III] where the essential facts about this analogue of the Lebesgue integral for Banach space-valued functions are given in relatively few pages. Finally, the basic facts of measure theory used in this chapter can be found in [ReSil, §1.3 and §1.4, pp. 12-26] and [Cho3, Coh, Fol2, Roy, Ru2, WhZy], Notation and definitions A. C, C+, C+~ : These denote, respectively, the complex numbers, the complex numbers with positive real part, and the nonzero complex numbers with nonnegative real part. B. L2(Rd): The space of Borel measurable, C-valued functions u on Rd such that 2 |u| is integrable with respect to Lebesgue (abbreviated Leb.) measure on Rd. C. L°°(Rd): The space of Borel measurable, C-valued functions on Ra which are essentially bounded. [It is the presence of a wide variety of Borel measures on (0, t) that leads us to work in B and C with Borel (rather than Lebesgue, as in the previous chapters) measurable functions on Rd. Further, as usual, the elements of L2(Rd) and L°°(Rd) are equivalence classes of functions, with u1 and u2 said to be equivalent if they are equal almost everywhere (a.e.) with respect to Lebesgue measure on R d .] D. L(L 2 (R d )): The space of bounded linear operators from L2(Rd) to itself. The notation || • || will be used both for the norm of vectors and for the norm of operators; the meaning will be clear from the context. E. The semigroup exp(—ZH0): We recall from Section 10.2, especially Theorems 10.2.5-10.2.7, some facts which we will use frequently concerning the holomorphic semigroup {exp(—zH0)}zEc~+ generated by the "free Hamiltonian" H0 = — (1/2)A = -(l/2)Eda=162/9x2aacting on L2(Rd). (Also, see [Kat8, §IX.1.8, pp. 495-97] or [ReSi2, Example 5, p. 254].) We use notation convenient for our purpose. The operators
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{exp[-s(H0/y)] : s > 0, A E C~+} are all in £(L2(Rd)) and satisfy:
In fact, when y E C~+is purely imaginary, exp[—s(H0/y)l is a unitary operator. As a function of y, exp[—s(Ho/y)] is analytic in C+ and continuous in the strong operator topology (or strongly continuous) in C~+. (This follows, for example, from the trivial case V = 0 of Theorem 13.3.1.) Recall from Remark 11.7.2(c) (or 13.2.2(c)) that for operator-valued (or for vector-valued) functions, the natural notions of analyticity coincide. See [HilPh, §3.10, esp. Theorem 3.10.1, p. 93].) Next, we state a familiar explicit formula for the operator exp[-s(Ho/y)]. (See formulas (10.2.25), (10.2.26) and (10.2.33).) Given V € L 2 (R d ) and X e C~+,
The integral in (15.2.2) exists as an ordinary Lebesgue integral for X e C+, but when y is purely imaginary and U is not integrable, the integral should be interpreted in the mean (equation (10.2.33)) just as in the theory of the Fourier-Plancherel transform. (See Theorems 10.2.5-10.2.7.) As is well known, the (negative) normalized Laplacian HO is the generator of the Brownian motion on Rd: It follows, in particular, that the semigroup {exp(-s H0) : s >: 0} is intimately connected with Wiener measure m defined in Chapter 3 and used in I below. (See, e.g., Chapters 3 and 12, as well as Example 9.2.3.) F. The measure space M ( 0 , t): Let t > 0 be fixed. In the following, M(0, t) will denote the space of complex Borel measures n on the open interval (0, t). For information on such spaces of measures, see, for example, [Coh, Chapter 4] or [Ru2, Chapter 6]. Given a Borel subset B of (0, t), the total variation measure \n\ is defined by \n\(B) = SUP{Enj=1 |n(Bj)|)' where the supremum is taken over all finite partitions of B by Borel sets (see [Coh, p. 126] or [Ru2, p. 116]); of course, \n\ is a positive measure in M(0, t). In fact, if n is complex-valued, the measure |n| is a finite measure. (See [Ru2, Theorems 6.2 and 6.4, pp. 117 and 118].) Under the natural operations, M(0, t) is a Banach space when equipped with the norm
the total variation of n. A measure u in M ( 0 , t) is said to be continuous if u ( { r } ) = 0 for every r in (0, t). In contrast, v in M(0, t) is discrete (or is a "pure point measure" in the terminology of Reed and Simon [ReSi1]) if and only if there is an at most countable subset {TP}8p=1 of (0, t) and a summable sequence {w p }8p=1 from C such that
where dTp is the Dirac measure with total mass one concentrated at rp [Coh, p. 22]. Every measure n e M(0, t) has a unique decomposition, n = u + v, into a continuous part u
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and a discrete part v [ReSi1, Theorem I.13, p. 22]. We will make frequent use of such decompositions. Results for complex measures can usually be reduced to the case of positive measures by using the formula ([Ru2, Theorem 6.12, p. 124]) n(dt) = e i g n ( t ) | n | ( d t ) , where gn(t) is a real-valued measurable function of t and |n| is the positive total variation measure defined above. The key to the proof is the fact that n is absolutely continuous with respect to |n | so that a Radon-Nikodym theorem can be applied. The formula above can often be used directly, but it is sometimes helpful to write
and then to decompose each of the cosine and sine functions into the difference of their positive and negative parts. The result is that 77 is expressed as a (simple) linear combination of four positive measures. Unless otherwise specified, we work with the space M(0, t) throughout, but M[0, t) could be treated without any essential complications. However, allowing n to have nonzero mass at 0 introduces additional alternatives which, with only a few exceptions, we have chosen to avoid. Finally, we remark that the product of measures m1, . . . ,uk (whether in M(0, t) or not) will be denoted u1 x . . . x uk or xku=1 uu. If u1 = . . . = uk = u, we will write uxk instead. (See Remark 15.2.10 below for more details.) G. The mixed norm space (Lool;n, || . ||oo1;n): Let n e M(0, t). A C-valued, Borel measurable function 8 on (0, t) x Rd is said to belong to Loo1;n if
Note that if 9 e L-oo1;n, then 0(s, .) must be in L°°(Rd) for n-a.e. s in (0, t). If one makes the usual identification of functions which are equal n x Leb — a.e., the mixed norm space Loo1;n, equipped with the norm || . ||oo1;n, becomes a Banach space. Note that all bounded, everywhere defined, Borel measurable functions on (0, t) x Rd are in Loo1;n for every n in M(0, t). (Here, as before, "Leb." denotes Lebesgue measure on R d .) The reader will see further on that the norm (15.2.5) appears in our estimates in a natural way. The functions 9 will be interpreted physically as potentials. The condition that 6 be in Loo1;n is rather minimal in most respects. No smoothness is required, and 9 is allowed to be time-dependent and C-valued. The use of C-valued functions 0 will enable us, in particular, to treat simultaneously the diffusion case and the quantum-mechanical case; that is, in the language often used by physicists, the "imaginary time" and "real time" case, respectively. (See Remark 15.2.6(b).) The importance of C-valued potentials in the study of decay systems in quantum-mechanics is discussed thoroughly in the book of Exner [Ex]. (See also Sections 11.6 as well as Sections 13.5 and 13.6.) Certainly, the most serious restriction in our assumptions is that 0(s, .) be essentially bounded for n-a.e. s. (See Remark 15.2.6(a).) However, even this condition seems quite reasonable in light of our goal of obtaining rigorously justified perturbation series valid in the quantum-mechanical case.
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If 9 e Loo1;n and if n — u + v is decomposed into its continuous and discrete parts, then it is not difficult to show that 0 e Loo1;u n Loo1;v and
H. The multiplication operators 0(s): We remind the reader that the operator of multiplication by a (C-valued) function in L°°(Rd) belongs to L(L2(Rd)) and has operator norm equal to the essential supremum of the function. (See the extension of Propositions 10.1.1 and 10. 1.3 stated in Remark 10.1.4; see also, e.g., [Kat8, Example 2.11, p. 146].) For us, the L°° -functions that arise will be of the form 0(s, .), where 0 E Loo1;n. It will be convenient to let 0(s) denote the operator of multiplication by 0(s, .), acting on L 2 (R d ). According to Remark 10.1.4, 9(s) is then a bounded (normal) operator and its operator norm ||0(.s)|| satisfies
The analytic (in mass) operator-valued Feynman integral Kty(.) Recall from Section 3.1 (and, e.g., Chapters 12 and 13) that Ct = C([0, t ] , R d ) denotes the space of Rd-valued continuous functions x on [0, t]. Further recall that Ct0 = Co([0, t], Rd) consists of those functions x in Ct such that x(0) = 0. The space Ct0is, as earlier, equipped with d-dimensional Wiener measure m, a probability measure which is just the product of d one-dimensional Wiener measures (see Chapter 3). I. The operator-valued function space integrals Kty(F), y e C~+: Definition 15.2.1 Fix t > 0. Let F be a function from Ct to C. Given X > 0, u e and E e Rd, we consider the expression
The operator- valued function space integral Kty(F) exists for y > 0 if (15.2.8) defines Kty((F) as an element of L(L 2 (R d )). If, in addition, Kty(F), as a function of y, has an extension (necessarily unique) to an analytic function on C+ and a strongly continuous function on C~+, we say that Kty(F) exists for y e C~+.When X is purely imaginary, Kty(F) is called the analytic (in mass) operator-valued Feynman integral of F. Remark 15.2.2 (a) Our notation for Kty(F) reflects the fact that later on, t will be allowed to vary (see Chapters 17 and 18). Throughout the present chapter (and Chapter 16), the time parameter t will be fixed. (b) The analytic Feynman integrals considered throughout the rest of this book (specifically, in Chapters 15-18) will be given as in Definition 15.2.1 and so we will always omit the words "in mass" when referring to Kty(F). Note that the parameter y can be expressed in terms of either the mass parameter or the diffusion constant. (See Section 13.5, especially Remark 13.5.2(c).) We stress that the definition adopted here is much stronger than that used in Section 13.5, where only a "nontangential limit" of y i-> Kty(F) was assumed to exist
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along the imaginary axis. (Compare Definitions 13.5.1' and 15.2.1.) Hence, when Kty (F) exists (for all y e C~+) in the present sense of Definition 15.2.1, then it will certainly exist in the weaker sense of Definition 13.5.1'. (See the comments following Definition 13.5.1'.) (c) The function F in Definition 15.2.1 (often referred to as a "functional" in the physics literature), need not be everywhere defined; however, in order to have Kty(F) defined for all A > 0, it must be the case that, for every y > 0, F (y - 1 / 2 x + £ ) is defined for m x Leb.-a.e. (x, £) € Ct0 x Rd. Definition 15.2.3 Given functions F and G defined (in the sense of Remark 15.2.2(c) above) on Ct, we say that F is equivalent to G and write F ~ G if, for every y > 0, F(y- l / 2 x + E) = G(y- 1 / 2 x+E) for m x Leb.-a.e. (x, £) e Ct0 x Rd. We leave it to the reader to check the following two simple facts: (i) The relation ~ just introduced above is an equivalence relation. (ii) The algebraic operations of pointwise addition, pointwise multiplication, and scalar multiplication are compatible with the relation ~. For example, if F1 ~ F and G1 ~ G, then F1 + G1 ~ F + G. This equivalence is necessitated by the pathology of Wiener measure under scale change and the fact that infinitely many scale changes (corresponding to all y > 0) are involved here. These matters were discussed in detail in Chapter 4 for the scalar-valued analytic Feynman integral. Although the setting here is somewhat more complicated, the essential difficulties are not increased by the presence of £ e Rd. We illustrate this by converting Example 4.5.3 to the present operator- valued setting. Some of the notation and facts from Chapter 4 will be useful to us here. Example 15.2.4 Let G : Ct -> C be identically 0. Then for every y > 0 and u E ), we have
Thus Kty (G) is the 0 operator for every y > 0. Certainly thenKty(G) exists in the sense of Definition 15.2.1 and equals the 0 operator for every y E C~+. Next fix d0 > 0 such that d0 = 1 . Let F : Ct —> C be given by
where £la is given by (4.2.1) for every a > 0 (with b — a = t in this case) and xna denotes its characteristic function. The function F is Borel measurable since Qdo is a Borel set by Proposition 4.2. l(i). For x € Ct0 and £ E Rd, we have F(x + £) = xndo (x). Hence F(x + £) is a Borel measurable function of the two variables (x,E) € Ct0 x Rd. (In fact, F ( x + E ) is independent of E.) Since m ( £ 1 ) = m1 ( £ 1 ) = 1 and Q1 n $d0 = 0 by Proposition 4.2.1, we see that F(x + i- ) = Xfi do (x) = 0 m1 x Leb.-a.e. (In fact, more is true than we need here; for every E e Rd, F(x + E) = 0 m1-a.e.) Thus F(x + E )
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and G(x + £) are equivalent functions on Ct0 x Rd, in the sense of equality m1 x Leb.a.e. Nevertheless, we claim that F fails to have an operator-valued analytic Feynman integral. For (x, £) E Ct0 x Rd, we have by Proposition 4.2.1(ii) that y-1/2x e fid0 if and only if x e y1/2od0 = £y1/2 d 0 . Thus
Therefore, F(y- 1/2 x + £) = 1 for m1 x Leb.-a.e. (x, £) if and only if y1/2d0 = 1 or = d 0 -2 .For y = d0-2, F(y- l/2 x + E) = 0 for m1 x Leb.-a.e. (x, £). Hence, for y > 0,
Recall from (9.2.15), a simple consequence of Wiener's integration formula (3.3.9), that
for every y > 0 and every U e L 2 (R d ). Hence, from (15.2.10) and (15.2.11), we see that for all y > 0,
Clearly then Kty(F) cannot be analytically continued to C+. In summary, the function (x,E) i-»- F(x + £) is Borel measurable and equivalent to 0 in the sense that it is equal to 0 m1 x Leb.-a.e. on Ct0 x Rd. However, Kty(F) fails to exist at every A 6 C~+ \ (0, oo). This concludes Example 15.2.4. A family of examples much like the above is described in Corollary 31, page 172 of [JoSk7]. Further related information can be found on pages 171-174 of the same reference. It is easy to show that the more refined equivalence relation ~ given in Definition 15.2.3 has the properties which we need. The simple theorem which we are about to state is the analogue for the operator-valued analytic Feynman integral Kty(F) of our earlier -1/2x analytic Feynman integral. result, Theorem 4.5.7, 1/2 for the scalar-valued Theorem 15.2.5 Let F and G be C-valued functions on Ct such that F ~ G; that is, for every A > 0, F(y- x + £) = G(y + E) for m x Leb.-a.e. (x, E) E Ct0 x Rd. Suppose that the operator-valued integral Kty(G) exists for every y E C~+. Then Kty (F) exists for every y e C~+ and we have
for every y E C~+.
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The simple proof is much like that of Theorem 4.5.7 and is left to the reader. Definition 15.2.1, given in [JoLa 1,2], is a variation of a definition given by Cameron and Storvick [CaSt1] and earlier, in the exponential case which has traditionally been of most interest, by Nelson [Ne1]. For A purely imaginary (the Feynman case), the requirements in Definition 15.2.1 for the existence of Kty (F) are more stringent than the requirements in either of [CaSt1] or [Ne1]. (See Section 13.5.) Hence, when Kty (F) exists in our sense, it will certainly exist in the sense of [CaSt1]. The "integral" introduced in [CaSt1] has been studied in several later papers including, for example, [JoSk1,2, JoSk6]. The interested readers may wish to check some of the references in [CaSt1, JoSk2, JoSk6]. They should also note the differences in notation between this book, [JoLa1,2], and earlier papers such as [CaSt1, JoSk2, JoSk6]. A reader not familiar with the difficulties involved in defining the Feynman integral may wonder why the Wiener integral in (15.2.8) is not used to define Kty (F) for all y E C~+. We do not wish to go into this in detail but remark that formula (15.2.8) can be rewritten with A appearing as the scaling m o y1/2 of the measure m rather than in the argument of the functions in the integrand. (See the first equality in (4.3.1).) The problem associated with using (15.2.8) to define Kty (F) for y E C~+\ R are then due to the fact that scaled Wiener "measure" is not countably additive for nonreal scalings, a fact discussed in some detail in Section 4.6 above that dealt with the nonexistence of Feynman's "measure". (See especially Theorem 4.6.1.) Remark 15.2.6 (a) In order to avoid possible misunderstandings, we mention that the theory in Chapters 15-19 is different in spirit and in purpose from the approaches to the Feynman integral discussed earlier in this book, especially in Chapter 11 (Sections 11.2 and 11.3-11.6) and Chapter 13. In particular, as observed in G above, no attempt is made here to treat very singular potentials as was done in those earlier chapters. On several occasions, however, we shall take advantage of some of the earlier results and techniques. (b) The physical interpretations that we will emphasize refer to the quantummechanical case, i.e. to y purely imaginary. We stress that the standard quantum mechanical case corresponds to 0 = —iV, with V real-valued, as well as A = —i (i := \/-1). In contrast, the diffusion (or probabilistic) case corresponds to 9 = —V and y = 1. This flexibility will enable us to treat the quantum-mechanical case and the diffusion case in parallel.
Preliminary results We finish this section with three lemmas. We begin with a simple lemma that we will frequently need. Lemma 15.2.7 Let J be an interval in R and let K1, . . . , Kk be continuous d-finite measures on (J, B(J)). Then the following sets have K1 x . . . x KK measure 0: (i) The subset of Jk where two or more coordinates are equal. (ii) The subset of Jk where one or more coordinates has a fixed value.
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Proof We just prove (i) since the proof of (ii) is similar and slightly easier. Clearly it suffices to show that
If we section the set involved in (15.2.13) at an arbitrary point (S2, . . . , Sk) E J k - 1 , we obtain the single point set s\ = s2 in J. But this set has K1-measure 0 since K1 is continuous. Now if we integrate the K1-measure of such sections over Jk-1 with respect to K2 x . . . x Kk, we will get 0. Thus (15.2.13) follows from the Fubini theorem. We continue with a somewhat technical measure-theoretic result which will be often used throughout Chapters 15-18, most of the time without explicit mention. The reader should note the result but may wish to skip the proof at least initially. Lemma 15.2.8 Let n E M(0, t) and suppose that 9 e Loo1;n. Let
for any y e C' for which the integral exists. Then, for every y > 0, F1 (y -1/2 x + E) is defined and satisfies
for m x Leb.-a.e. (x, E) e Ct0 x Rd. Proof We first show: (*) For every y > 0 and m x Leb.-a.e. (x, E), 0(s, y - 1 / 2 x ( s ) + E) is defined and satisfies |0(s, y-1/2x (s) + E)| < ||0(s, .)||oo for |n|-a.e.s. Let Hy : (0, t) x Ct0x Rd -> (0, t) x Rd be defined by Hy (s, x, E) = (s, y - 1 / 2 x(s) + E). The map Hy is everywhere defined and continuous and so 0 o Hy is certainly Borel measurable though it need not be everywhere defined. Let N :={(s, v) € (0, t) x Rd : 6(5, v) fails to be defined or£(s, u)isdefinedbut|0(s, u)| > ||0(s, .)||oo}. Since 0 E Loo1;n, a part of the Fubini theorem [Coh, Chapter 5] assures us that 9 is defined and satisfies 0 ( s , v)| < ||0(s, .)||oo for |n| x Leb.-a.e. (s, v); i.e. N is |n| x Leb.null. Let y > 0 be given. Note that to establish (*), it suffices to show that Hy-1 (N) is |n| x mx Leb.-null. Accordingly, we section H-1y(N) at (s, v) € (0, t) x Rd:
where N(s) := [u e Rd : (s, u) e N}. Now, since N is |n| x Leb.-null, it follows that for |n|-a.e. s and every E, and so certainly for |n| x Leb.-a.e. (x, e) e Ct0 x Rd, the set y1/2[N(s) - E] is Leb.-null. But
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it is well known and easily follows from Wiener's integration formula (3.3.9) that the set of Wiener paths x whose value at a particular time s lies in a Leb.-null set is a set of m-measure zero. (See Lemma 12.1.4.) Hence, by the Fubini theorem [Coh, Theorem 5.2.2, pp. 159-160], H-1y (N) is |n| x mx Leb.-null. Thus (*) is established. It now follows from (*) that for every y > 0 and mx Leb.-a.e. (x, E),
Hence, for every y > 0 and m x Leb.-a.e. (x, E),
is defined and we have
This concludes the proof of Lemma 15.2.8. We remark that even if 9 is everywhere defined, there may be continuous functions y in C' for which F1 (y) fails to be defined. If 9 is everywhere defined and bounded, then F1 (y) is defined for all y in C1. Note that even when 0 is essentially bounded, the graph of the function s i-» y - 1 / 2 x ( s ) + E could lie in {(s, v) € (0, t) x Rd : |0(s, v)| > ||0(s, .)||oo) for some y > 0, x and E. In this case, (15.2.15) could fail since the second inequality in (15.2.16) could fail. Our last lemma does not seem to be explicitly stated in the standard references on the Bochner integral. We include its easy proof. Lemma 15.2.9 (Bochner integrals depending on a parameter) Let X be a complex Banach space. Let (E, y) be a a-finite measure space and let T be a metric space. Consider the function g : T x E —> X [or L ( X ) ] . Assume that/or all y in T, g ( y , y) is a strongly measurable function of y in E. Suppose further that there exists h in L1 (E, y) such that ||g(y, y)|| < h(y) for y-a.e. y € E and all y e T. Set
(Note that G is well-defined by the basic Bochner integrability criterion; that is, Bochner measurabitity of a function and integrability of its norm implies its Bochner integrability [HilPh, Theorem 3.7.4, p. 80].) (1) Assume that for y-a.e. y € E, g(y, y) is a strongly continuous function of y e T. Then G is strongly continuous on T.
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(2) Assume that T is an open subset of C and that for y-a.e. y E E, g ( y , y) is an analytic function of y on T. Then G is analytic on T. Proof If g(y, y) is operator- valued, we consider the vector-valued function for fixed U in X; so that we may assume that g is X-valued. Part (1) is a consequence of the dominated convergence theorem for Bochner integrals [HilPh, Theorem 3.7.9, p. 83] which is the exact counterpart of the usual theorem for scalar-valued functions. Let X* denote the dual space of X and {•, •) the duality bracket between X and X* (see, e.g., Definition 9.4.5). Fix U* in X*. Recalling our earlier remark in Section 15.2.E about the equivalence of all the natural notions of analyticity, we see that (2) will follow if we show that G1(y) := {U*, G(y)} is analytic on T. Let g1(y, y) = (U*, g(y, y)}. Clearly, under our assumptions, g1(y, y) is an analytic function of y for y-a.e. y E E. Further, by [HilPh, Equation (3.7.5), p. 80], G1(y) = fE g1(y, y)y(dy). Moreover, for y-a.e. y, we have
By hypothesis, the dominating function in (15.2.17) lies in L 1 ( E , y). Part (2) now follows from the corresponding result for Lebesgue integrals of scalar-valued functions depending on a parameter. When we apply Lemma 15.2.9 in Chapters 15-18, we shall always choose T — C~+ in part (1) and T = C+ in part (2). Remark 15.2.10 (Notation for product measures) Product measures will appear frequently in Chapters 15-19. When the product of the measures u1, . . . , uk appears under the integral sign, we prefer to write (u1 x . . . x uk)(ds1, . . . , dsk) or xku=1 uu (ds1, . . . , dsk), but, for the sake of brevity, we will sometimes write xku=1 uu(dsu), When the measures are all the same, say u1 = . . . = uk = u, we will use the briefer (tensor product) notation u® k (ds1, . . . , dsk) to denote the product of k copies of u.
15.3
A simple generalized Dyson series (n = u + wdr)
We begin by treating a prototypical example and deriving the corresponding generalized Dyson series. The difficulties that we will encounter in this chapter are of two kinds, analytic and combinatorial. Most of the analytic obstacles involved in Sections 15.315.5 occur in the special case that is our concern in Theorem 15.3.1 and Corollary 15.3.4 below and will be dealt with in detail in this section. The combinatorial problems encountered here provide a good introduction to Sections 15.4 and 15.5 where we will concentrate on the combinatorial aspects. This division makes the exposition easier to follow, and, once the reader has understood the present section, it will be quite clear how to supply the analytical details omitted in Sections 15.4 and 15.5. (However, in Section 15.4—where the "generalized Dyson series" obtained is of much greater combinatorial complexity—some of the main analytic arguments will also be provided.)
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Let n = u + wdr, where u e M(0, t) is continuous, dT is the Dirac mass at r € (0, t) and w e C. Let 0 € Loo1:n. Set
We will be interested in analytic functions of F1 (y), but we postpone that until Corollary 15.3.4. Most of the problems arise in dealing with the nth power Fn(y), as in (15.3.1), and that is the subject of our first theorem. Theorem 15.3.1 The operator Kty (Fn) exists for all y e C~+ and
where, for 0 < j < k < n,
The integral over Ak;j in (15.3.2) is a Bochner integral in the strong operator sense. (That is, for every U E L 2 (R d ), Lj U is an L 2 (R d )-valuedBochner integrable function; see [HilPh, Chapter III] and Remark 15.3.2(e) below.) Moreover, for all E 6 C~+,we have
Before beginning the proof, we make some remarks intended to help the reader understand the notation and follow the arguments. Remark 15.3.2 (a) Note that in the expression (15.3.3) for A k ; j , we have 0 < r < s1 < • • • < Sk < t when j = 0 and 0 < s1 < . . . < Sk < T < t when j = k. Thus L0 begins with e - r ( H 0 / y ) [ 0 ( r ) ] n - k and L k ends with [0(r)]n-k e- ( t - r ) ( H 0 / y ) . (b) Figure 15.3.1 shows the regions Ak;j for k = 2, where
(c) The integrand Lj of (15.3.2) depends on k as well as j.
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(d) The expression u®k in (15.3.2) denotes the product of k copies of the measure u, in agreement with our conventions in Section 15.2.F (and Remark 15.2.10); alternatively, we sometimes write the integral in iterated form. (e) The sense of the equality (15.3.2) is that for every U € L2(Rd),
We caution the reader that the integral appearing in (15.3.2) may not make sense in the operator norm topology (often called the uniform operator topology). (f) The reader may now find it useful to consult Section 15.6 on Feynman diagrams. In particular, Figures 15.6.1-15.6.3 are relevant to the present theorem. Proof of Theorem 15.3.1 We will first emphasize the formal calculations leading to (15.3.2). Later we will provide the justification for the numbered steps. Let V 6 L2(Rd), E e Rd and y > 0 be given. Set
Since u is a continuous measure, one can see by a sectioning argument (analogous to the one used in the proof of Lemma 15.2.7) that
except for a set of mxk-measure zero. Note for later use in the proof that, by (15.3.3), the sets Ak;j in (15.3.7) are pairwise disjoint. The reader may find it helpful to consult
FIG. 15.3.1. The regions Ak;j, k = 2
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Figure 15.3.1 above. Now, in view of (15.2.8),
Hence, (15.3.2), with Lj given by (15.3.4), is established (formally, at this point) for all y > 0. (Of course, formula (15.3.8) only holds for a.e. E E Rd.)
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Before proceeding to the justification for the numbered steps, we note that Lj = L j ( s 1 , . . . , Sk) as given by (15.3.4) is measurable in the strong operator topology [HilPh, §3.5, esp. Corollary 2, p. 73]. Step (I) results from writing n as u + wdr and carrying out the integral with respect to wdT . In (II), we separate the continuous part of n from the discrete part of n by means of the binomial theorem. Step (III) follows from the "simplex trick" which works just as well for an arbitrary continuous measure as for Lebesgue measure I as we now explain briefly: the set (0, t)k is, except for boundaries, the union of k! simplexes, one being Ak, the others differing from Ak only by permutations of the s-variables. Since m is a continuous measure, Lemma 15.2.7 implies that these boundaries have uxk-measure zero. Furthermore, the integrand is invariant under permutations of the s-variables. Hence, the integrals over the k! simplexes are all equal and (III) follows. Equation (15.3.7) implies (IV). In Ak;j, the time indices are ordered in anticipation of carrying out the Wiener integral. Step (V) follows from Fubini's theorem which will be justified below in conjunction with the proof of the norm estimate (15.3.5). Finally, (VI) is obtained by application of Wiener's integration formula (Theorem 3.3.5) for finitely based functionals, followed by a simple change of variables for finite-dimensional Lebesgue integrals. For more detail on this last step in a simpler situation, see the m = 1 and m =2 cases of the calculations which begin with equation (14.3.14) and end with equation (14.3.19). Replacing Lj U by ||LjU||, w by |w| and u by |u|, we obtain the norm estimate (15.3.5) for y > 0 essentially by reversing the steps above. First,
Using this, the norm inequality (15.2.1) and the equalities (15.3.4), (15.3.7), (15.2.5), (15.2.7) and (15.2.6), we can write:
Consequently (15.3.5) follows.
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As previously noted, Lj is strongly measurable; thus, in view of (15.3.9) and [HilPh, Theorem 3.7.4, p. 80], Lj is Bochner integrable in the strong operator topology as a function of (s1, . . . , sk). Doing the calculation in (15.3.8) with U, 9 and w replaced by their absolute values and u replaced by its total variation \u\ leads to the Wiener integral of the function
Since \ U ( y - 1 / 2 x ( t ) + E)| is Wiener integrable, by (15.2.11) with U replaced by |U| E L2(Rd), the use of the Fubini theorem in Step (V) above is justified. We assumed throughout that Kty (F n ) exists for A > 0. However, the strict logical order would be as follows: First, establish the norm estimates (15.3.9) and deduce that the right-hand side of (15.3.2) defines a bounded linear operator on L 2 (R d ); secondly, reverse the steps in (15.3.8) and conclude that Kty (Fn) exists for A > 0 and is given by (15.3.2). We finish this proof by showing that the right-hand side of (15.3.2) is strongly continuous for A in C~+ and analytic for A in C+. This will establish the existence of Kty (Fn) (in the sense of Definition 15.2.1) as well as the equality (15.3.2) for all A in C~+. (We continue to write Kty (Fn) for the extension to C~+ of the function initially defined for A > 0. Note that, for nonreal y e C~+, there is no claim that Kty (Fn) is given by a Wiener integral.) The operator Lj given by (15.3.4) depends on A; we indicate this dependence now by writing Lj = Lj(y; s1, . . . , Sk). Using the norm inequality (15.2.1) and the norm equality (15.2.7), we see that for all y e C~+ and u®k-a.e. (s1, . . . , sk),
Next, noting that the multiplication operators involved in (15.3.4) are independent of A and recalling (from Section 15.2.E) the strong continuity and analyticity properties of exp[—^(H0/y)], we see that, for uxk-a.e. (s1, . . . , sk), Lj(y; s1, . . . , sk) is strongly continuous for A in C~+ and analytic for y e C+. Moreover, the right-hand side of (15.3.11) is M®k-integrable since 0 e Loo1;n. (See (15.2.5) and (15.2,7).) It thus follows from Lemma 15.2.9 on Bochner integrals depending on a parameter that
is strongly continuous on C~+ and analytic on C+. We conclude that Kty (Fn) exists and is given by (15.3.2) for all A e C~+. Further, the norm inequality (15.3.5) follows from (15.3.9) which clearly continues to hold for A e C~+. Remark 15.3.3 (a) The existence of Kty (Fn) and the norm estimate (15.3.5) can be obtained more easily and more directly for A > 0, as someone familiar with both the Wiener and Feynman integrals might guess. Simply use the bound (15.2.15) from Lemma 15.2.8 and replace F n ( y 1 / 2 x + E) by (||0||oo1;n)n.
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(b) Note that, although we have not explicitly mentioned it, Lemma 15.2.8 has been needed all along to assure us that, for every y > 0, F n (y-1/ 2x + E) is defined for m x Leb.-a.e. (x, E) € Ct0 x Rd.
Now, let
be an analytic function with radius of convergence strictly greater than Corollary 15.3.4 Consider the functional
with f as in (15.3.13). Then Kty (F) exists for all X e C~+and is given by the "generalized Dyson series" (or "time-ordered" perturbation expansion)
where Fn is the functional defined by (15.3.1) and Kty (Fn) is given by (15.3.2). Moreover, for y e C~+, the series in (15.3. 15) converges in operator norm and we have
Proof We could show that the convergence of the series in (15.3.15) is uniform in y e C~+ and argue from there. Instead, we regard the series as an additional integral; then the proof parallels that of Theorem 15.3.1. The application of the Fubini theorem, the Bochner integrability, the analyticity and the strong continuity all follow from the analogue of the norm estimate (15.3.9). Remark 15.3.5 (a) A slight variation of the argument yielding (15.3.16) gives the following estimate for the norm of the tail of the series for Kty (F):
for all nonnegative integers p. We leave the easy verification as an exercise for the reader. (b) Even when the "potential" 9 is time independent, the family of operators {Kty (F)} is not in general a semigroup in the time parameter t when f ( z ) is different from exp(z) or n is different from Lebesgue measure l. (See Chapter 17, especially Sections 17.2, 17.3 and 17.6.) Nevertheless, the use of results connected with the theory of semigroups (Chapters 8 and 9, as well as Section 10.2) is still helpful and allows us (as was the case in
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wdT)
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[JoLa1]) to simplify considerably earlier arguments for measurability, strong continuity and analyticity found in [CaSt1, JoSk2, JoSk6] and elsewhere for n = l, In addition, the systematic use of operator-theoretic notation (also made in [JoLa1]) substantially shortens all the expressions. These comments apply equally well to later sections in this chapter (and to Chapters 16-18). The advantage of operator-theoretic methods in this context was pointed out in [La12, pp. 58-60] in reference to [Jo3]. (See the introduction to Section 11.5.)
Next, we examine the case w = 0; further specialization yields the classical Dyson series (see Equation (15.3.22) below). We refer to Figure 15.6.1 for the Feynman diagrams associated with the series in formula (15.3.17) below. Corollary 15.3.6 (Purely continuous measure: n = u) Let F be given by (15.3.14). When CD = 0, the perturbation expansion for Kty (F) becomes
where
and, for (s1 , . . . , sn) € An,
Proof It suffices to consider Kty (Fn) as given by (15.3.2). When w = 0, the only nonzero term on the right-hand side of (15.3.2) is obtained when k = n; thus
We observe that when k = n, Lj—as given by (15.3.4)—does not depend on j; in fact, since (15.3.21)
Lj equals the integrand £ of (15.3.17). In view of (15.3.7), we have
as desired.
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The classical Dyson series The classical Dyson series—except for an exact reversal of the time-ordering which will be discussed specifically in Remark 15.3.7 below—corresponds to the special case of (15.3.17) obtained by letting (u = l = Lebesgue measure, f(z) = exp(z) (so that n!an = 1), as well as A = —i and 6 = -i V (the "quantum-mechanical case", as in Remark 15.2.6(b)), where the potential V belongs to Loo1;n. In fact, it is given by
where the simplex An is as in (15.3.18). (Usually, V is assumed to be time independent and so for each j = 1, . . . , n, V(sj) is replaced by V in (15.3.22).) It will be helpful to recall briefly the physical interpretation of the nth term of the classical Dyson series (15.3.22) ([Dys 1,2,4], [Schu, pp. 67-68], [Schwel, § ll.f]). For now, we think of a single quantum-mechanical particle moving in the potential V. The integrand of the nth term may be described as follows: A free evolution between times 0 and s1, interaction with the potential V at time s1, free evolution between s1 and S2, and so on up to an nth interaction with V at time sn followed by a free evolution between sn and t. We now integrate (or "sum") over the simplex An to take into account all the possible times of interaction; in the present case, the times receive equal weight since n = l. The form of the series in (15.3.17) is essentially the same for any continuous measure n = u. However, even with n = u absolutely continuous, different physical interpretations are suggested. For example, the weighting may force all the interactions to take place in some short time interval. The absolutely continuous case amounts to multiplying 0(s, .) by a function of time which equals the density of u with respect to l. (With this in mind, one can see that this situation is included in the p = p' = 2 case of [JoSk6].) In general, n = u has a singular part. (See, for example, [Coh, p. 141], [ReSi1, pp. 20-23] or [WhZy, pp. 35, 116 and 180].) This could be, for instance, the LebesgueStieltjes measure associated with the Cantor function. Cantor-like functions have been used, for example, in connection with the notion of "fractal time" in the theory of errors in data transmission lines. (See [Mand, Chapter 8, esp. pp. 74-75 and 78-83].) The Cantor function and its associated measure will be used in a quantum-mechanical setting in Example 15.5.3 below, as well as in several places in Chapter 17. When n has a discrete part, as in Corollary 15.3.4, even the formal appearance of the generalized Dyson series changes markedly. This is most striking in the general case to be examined in the following section, but can already be seen in formulas (15.3.2)(15.3.4) of Theorem 15.3.1 where n = u + wdr and w = 0. The single integral over An in the nth term of (15.3.17) is replaced in (15.3.2) by a double sum of integrals over the sets Ak;j. This double sum accounts for the various placements of the variables Sj with respect to r. Note the special role played by the fixed time r; indeed, r always appears and 0 ( r ) appears to powers ranging from 0 to n. In the time intervals [0, r) and (r, t],
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there is a succession of free evolutions and interactions weighted by u®k just as in the case w = 0. The reader can easily imagine that there will be substantial additional complications in the next section where many different 9s and ns are involved and where the discrete parts of the ns may have countable support. Remark 15.3.7 Each summand in the generalized Dyson series (15.3.15) (with Kty (Fn) as in (15.3.2) and Lj as in (15.3.4)) should, in the natural physical interpretation, be read from time t to time 0 rather than from 0 to t, as you go from left to right on the page. The change to the natural physical ordering can be brought about mathematically by replacing (in the definition (15.3.14) or (15.3.1) of F or of Fn, respectively) the "potential" 9 and the measure n by their "time-reversals" 9 and n~, respectively, where
and
the image measure of n under the "time-reversal" map s -> t — s from [0, t] to itself. (We will in fact need to substitute the pair (0, n~) for the pair (0, n) in Chapter 17 where we will be interested in the time evolution of Kty (F).) Observe that our assumptions (in this section) are still satisfied by the new pair (0, n~) since 0 E Loo1;n~ if and only if 0 e L001;n. Indeed,
since \n~\ = \n\ and by the abstract change of variable theorem (Theorem 3.3.2),
An alternative (which we will not use) would be to start Brownian paths at time t and let them flow backwards to time 0. The above comments in Remark 15.3.7 also apply to the simpler generalized Dyson series (15.3.17) (with n = u and £ as in (15.3.19)) and will be extended in Remark 15.4.4 to the general situation considered in Section 15.4 below as well as throughout Chapters 15-18. For now, we illustrate it by a very simple example. (We leave the easy verification to the reader.) Example 15.3.8 Let u be a continuous measure and let 0 e Loo1;u (so that, by (15.3.24), 0 E Loo1-u~).
(a) Let
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Then we have
[Note that (15.3.25) is just the n = 2 term of the series (15.3.17) in Corollary 15.3.6 where a2 = 1/2! and an = 0 for all n = 2.] (b) Next, let ~F be the "time reversal" of F; namely,
where 9 and u~ are given as in (15.3.23). Then, in contrast to (15.3.26), we have
Note that, in agreement with Remark 15.3.7, (15.3.28) follows from (15.3.26) (and Corollary 15.3.6) by substituting the pair (~0, u~) for (6, u~). Indeed, it can be deduced from (15.3.26) (and the abstract change of variable formula, Theorem 3.3.2) by means of the substitution s' = t — s. 15.4 Generalized Dyson series: The general case We now extend the results of Section 15.3 to a much broader setting. The main results of this section, Theorem 15.4.1 and Corollary 15.4.3, show that for a large class of functions F, the operator Kty (F) exists and can be disentangled by a (rather involved) time-ordered perturbation expansion or generalized Dyson series (in short, GDS). This will be the key to the construction in Section 15.7 of a commutative Banach algebra of functionals, called the disentangling algebra, which will play a central role in later developments (see Chapter 18). By contrast to Section 15.3, many measures and potentials may be involved; moreover, the discrete part of each measure is unrestricted. The new complications are mainly combinatorial in nature; otherwise, the proofs proceed much as in Section 15.3. For this reason, we shall emphasize the combinatorial aspects. Eventually, we wish to consider series of functions of the type we are about to introduce, but for now, we let
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where nu e M(0, t) and 0u e Loo1;nu for u = 1, . . . , m. Note that by Lemma 15.2.8, for every X > 0, F ( y - 1 / 2 x + E) is defined for m x Leb.-a.e. (x, £) in Ct0 x Rd. Let
be the unique decomposition of nu into its continuous part nu and its discrete part vu (see Section 15.2.F). For each u = 1, . . . , m, we write
where {r p ; u }oop=1 is a sequence from (0, t) and {wp;u}?li is a sequence from C such that
We will write each of the factors in (15.4.1) as an absolutely convergent infinite series. We will then find it advantageous to multiply these series together. With this is mind, we introduce the following notation: Given k between 0 and m, [k; m] will denote the collection of all subsets of size k (or k-sets) of the set of integers {1, . . . , m}. If {a1, . . . , ak} e [k; m], we shall always write
Now, in view of (15.4.2) and (15.4.3),
Note that when k = 0, {ak+1, . . . , am] = [1, . . . , m] and the integral involving the continuous measures does not appear in the corresponding term of the final expression. On the other hand, when k = m, then {a1, . . . , ak} = {1, . . . , m} and only the continuous measures appear.
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Soon we will wish to calculate the Wiener integral defining Kty (F). For this purpose, we will need to order the time variables. We begin by ordering the rs that appear within a given term of the series in (15.4.4). For fixed k,{a1, . . . , ak} in [k; m] and pk+1, . . . , pm, let a be a permutation of {k + 1, . . . , m} such that
(If the rS involved in (15.4.5) are distinct, then the permutation a is unique.) We can now state the key result of this section ([JoLa1, Theorem 2.1, pp. 27-29]). (See also Corollary 15.4.3 below [JoLa1, Corollary 2.1, pp. 33-34].) Theorem 15.4.1 Let F be defined by (15.4.1). Then Kty (F) exists for y € C~+. In particular, for y purely imaginary, the analytic operator-valued Feynman integral of F is well defined. (See Definition 15.2.1.) Moreover, for all y e C~+,
where, for each fixed k E {0, . . . , m}, p ranges through the group Sk of permutations of [1, . . . , k]and
In addition, for (s1, . . . , Sk) e Ak;j1, . . .,j m _ k+1 (p) and r = k, . . . ,m,
Here, a is a permutation of {k + 1, . . . , m} as defined in (15.4.5).
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The integrals in (15.4.6) are to be interpreted as Bochner integrals in the strong operator topology and the series converges in the operator norm. Further, for all y € C~+, we have
Remark 15.4.2 (a) In (15.4.7) and (15.4.8) above, we adopt the conventions tpd(k);ad(k) = 0; tpa(m+1);ad(m+1) = t and 0(T pa(k);aa(k) ) = 1, the identity operator. Further, we take jo = 0; then, when r = k, it is reasonable to interpret j\ + . . . + jr-k + 1 as 1, and we also get jr-k = j0 = 0. The s-values between two equal rs are omitted in (15.4.7). (b) If Tpa(r);ad(r) = rpd(r+1);aa(r+1), then jr-k+1 = 0, Lr in (15.4.8) reduces to 0ad(r) ( r AV(r);ad(r)) and the term L r L r +1 involves the product
In fact, more than two successive TS may be equal in (15.4.5), giving rise to a corresponding product of 0s evaluated at the same time and without intermediate semigroups. (c) Note the simplicity of the norm estimate (15.4.9) in spite of the complexity of the formula (15.4.6) for Kty (F). Proof of Theorem 15.4.1. Except for the combinatorics, the proof parallels that of Theorem 15.3.1. We first show that Kty (F) is given by (15.4.6) for y > 0. Fix U € L2(Rd). For notational simplicity, we let A = 1 and E = 0. (Actually, the calculations hold for any y > 0 and almost every £ € Rd.) Also, we write 6(x(s)) instead of 0(s, x(s)). From (15.2.8), (15.4.4) and (15.4.5), it follows that
For p E Sk, define the simplex
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The continuity of the measures u au , u = 1, . . . , k, and a sectioning argument as in Lemma 15.2.7 give the equality (0, t)k = Upesk Ak(p),up to a set of xku=1 uau -measure zero. Consequently, since the above union is disjoint,
Again using Lemma 15.2.7 and the fact that the measures u1, . . . , um are continuous, we see that
where the pairwise disjoint subsets Ak;j1, . . .,j m-k+1(p) are defined in (15.4.7) and the equality holds up to a set of x ku=1 uap(u) -measure zero. In view of (15.4.12) and (15.4.13), we thus have
By application of Fubini's theorem, we deduce from (15.4.10) and (15.4.14) that
Observe that, within the integrand of the Wiener integral in (15.4.15), the time variables are explicitly ordered according to (15.4.7). By Wiener's integration formula (3.3.9), it thus follows that this Wiener integral is equal to ((Lk . . . Lr . . . L m )U)(0), with Lr given by (15.4.8). We conclude that (15.4.6) holds for all y > 0.
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It will be helpful to keep in mind that, in the above derivation, the string of inequalities
corresponds to the operator Lr. The additional combinatorial complications of our present setting make the proof of the norm estimates somewhat more involved. Once we have the norm estimates, however, the rest of the proof proceeds much like that of Theorem 15.3.1.
Hence,
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where (15.2.7) and (15.2.6) are used to obtain the last two equalities. By replacing U, 0u, w p;u by their absolute values and uu by |uu| in the calculation leading to (15.4.15), we obtain, exactly as in the proof of (15.4.17), the Wiener integral of the function
Since by (15.2.11), U ( x ( t ) ) is Wiener integrable, this justifies the use of Fubini's theorem in the derivation of (15.4.15). Analyticity and strong continuity of Kty (F) are argued just as in the proof of Theorem 15.3.1 by using Lemma 15.2.9 which deals with Bochner integrals depending on a parameter. Corollary 15.4.3 (Generalized Dyson series) Let{Fn}oon=0be a sequence of functionals each given by an expression of the type (15.4.1):
where nn,u e M(0, t) and 0n,u € Loo1;nn,u. (Note that if mn = 0, we take Fn = 1.) Assume that
Then for all y > 0, the individual terms of the seriesEoon=0F n (y-1/ 2 x + E) are defined and the series converges absolutely for m x Leb.-a.e. (x, E) e Ct0 x Rd. Let
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Then for all y e C~+, Kty (F) exists and is given by the following time-ordered perturbation expansion (GDS):
where Kty (Fn) is defined by (15.4.6) with the functional F from Theorem 15.4.1 replaced by Fn as in (15.4.18). The series in (15.4.21) converges in operator norm; furthermore, for all A 6 C~+, we have the estimate
Proof The fact that for every A > 0 the terms of the series in (15.4.20) are defined and that the series converges absolutely for m x Leb.-a.e. (x, E ) follows from Lemma 15.2.8 and the assumptions on the sequence {bn}, where
(See (15.4.19) according to whichEoon=0bn < oo.) Let y > 0. A use of the integrable (by (15.2.11)), dominating function
allows us to interchange the order of the integral and sum and write
Now, the inequality||kty(Fn)|| < bn from (15.4.9) in Theorem 15.4.1 assures us that the series Eoon=0 Kty (Fn) converges in operator norm, uniformly for y e C~+. The analyticity and strong continuity ofEoon=0Kty(Fn) follow from this and the corresponding assertions about Kty (Fn) which were given in Theorem 15.4.1. In the light of Definition 15.2.1, the existence of Kty (F) for y € C~+ and the formula (15.4.21) now follow from (15.4.23).
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Remark 15.4.4 (Time-reversal and natural physical ordering) The GDS in (15.4.21) (with each Kty (Fn) as in (15.4.6)-(15.4.8)) is not written in the natural physical order. In that order, consistent with Feynman's time-ordering convention (14.2.1), the operators involving smaller time indices always act before those involving larger time indices. (This same idea was expressed differently in Remark 15.3.7.) We can obtain the natural physical order by replacing (in the definition (15.4.18) of Fn) each pair (0 n,u , n n , u ) by its "time-reversal" (0n,u, nn,u), where the "potential" 0n,u and the measure n~ are defined as in (15.3.23); namely,
and
the image measure of nn,u under the "time-reversal" map s |-> t — s of the interval [0, t]. Observe that our assumptions (in Corollary 15.4.3) are still satisfied since by (15.3.24), we have ||0n,u||oo1;n~n,u = ||0n,u||oo1;nn,u and so, by (15.4.19),
is also finite. Hence our claim follows by simply applying Corollary 15.4.3 to the new functional
the "time-reversal" of F, and then using the abstract change of variable theorem (Theorem 3.3.2) by making the substitution s' = t — s in each term of the series (15.4.21) (with Kty (Fn) given by (15.4.6)), where (0n,u,n~n,u)has been substituted for (0n,u nn,u). (For a simple illustration of this, we refer back to Example 15.3.8.) The above comments in Remark 15.4.4 (which will be expanded upon in Section 15.7, especially in Theorem 15.7.6) apply to the generalized Dyson series occurring throughout Chapters 15-18 and are particularly relevant to the development in Chapter 17. We note that in Chapter 19, where we address Feynman's time-ordered calculus in a more general setting, our perturbation expansions will be obtained directly in the natural physical order. 15.5 Disentangling via perturbation expansions: Examples We now wish to consider various special cases. In each of these cases, the existence of the operator Kty (F) for y E C~+ and corresponding norm estimates follow from Theorem 15.4.1 and Corollary 15.4.3. In each example, we shall give the time-ordered perturbation expansion (GDS) of Kty (F) for y e C~+. The series appearing in these formulas all converge in operator norm and the integrals are taken as Bochner integrals with respect to the strong operator topology.
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A single measure and potential Let n e M (0, t) and 9 € Loo1;n. As usual, n = u + v will be the decomposition of n into its continuous and discrete parts, and we will write
where {T P }oop=1 is a sequence from (0, t) and {w P }oop=1 is a sequence from C such that
be an analytic function with radius of convergence strictly greater than ||0 ||oo1;n. Consider the functional
The case where v has finite support is perhaps most likely to be of physical interest; this is the object of our first example. Example 15.5.1 (Finitely supported v) Let u be a continuous measure in M(0, t) and let
where we may as well assume that 0 < r1 < . . . < Th < t. Then
where q0, . . . , qh, j1, . . . , jh+1 are nonnegative integers,
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and, for (s1, . . . , sq0)EAq0;j1,. . .jh+1 and r E {0, . . . . h},
(We use the conventions r0 = 0, rh+1 = t and [ 0 ( t 0 ) ] q 0 = 1.) Moreover,
We briefly explain the structure of the GDS (15.5.5). Let
We abbreviate 0(s, x ( s ) ) to 9(x(s)). Using the multinomial formula, we have
where Aq0 is given as in (15.3.18). We now order the s-variables with respect to the TS, thereby giving rise to the sum over the js. Finally, we calculate the Wiener integral to obtain the nth term of the series (15.5.5). While formula (15.5.5) is not especially simple, note that the arguments which led to it are much simpler than in the general case discussed in Theorem 15.4.1 and Corollary 15.4.3. The Feynman diagram corresponding to the nth term of the generalized Dyson series (15.5.5) is given in Figure 15.6.5. Observe that we recover formula (15.3.15) of Corollary 15.3.4 by letting h = I , T 1 = T and w1 = w in Example 15.5.1 . By specializing even further (w1 = 0), we obtain the case of a purely continuous n. Example 15.5.2 (n = u purely continuous) The generalized Dyson series corresponding to this case is given in Corollary 15.3.6. It is similar in form to the classical Dyson series given in (15.3.22). The latter is recovered by letting f ( z ) = exp(z), n = u = l and 0 = -iV.
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In our next example, we consider a special case of Example 15.5.2 where u is the Lebesgue-Stieltjes measure associated with the Cantor function C = C(t) defined on [0, 1]. (See [Coh, p. 55 and pp. 22-24], [ReSi1, pp. 20-23] and [Mand, Chapter 8, esp. Plate 83, pp. 82-83]. Recall that C is a nonnegative, nondecreasing continuous function on [0, 1] such that C(0) = 0 and C(l) = 1; further, C is constant on each interval in the complement of the (ternary) Cantor set in [0, 1].) It is natural in this setting to extend C periodically to [0, +00), but we will (mostly) restrict our attention to the usual Cantor function and to t between 0 and 1. The calculation is not complicated but the resulting formula for Kty (F) is interesting, especially in the quantum-mechanical case, and seems to be new. Example 15.5.3 (n = u purely continuous and singular) The basic relationship between the Cantor function C and the Cantor-Lebesgue measure u is given by the formula
The measure u is a continuous probability measure which is supported by the Cantor set. Although the cardinality of the Cantor set is the same as the cardinality of [0, 1], it has Lebesgue measure 0. Thus u is singular with respect to Lebesgue measure. We now take 9 constant (say, 0 = K), and f(z) = exp(z). The function F is independent of the path x in this case and is given by
Since an =1/n!,we see from (15.3.17) of Corollary 15.3.6 that for t E [0, 1],
Hence, for all y E C~+,
In particular, taking K = — 1 and y = 1, we have in the diffusion or probabilistic case,
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On the other hand, with K = —i and y = —i, we obtain in the quantum-mechanical case,
Whether (15.5.15) models some real physical phenomenon is not clear to us; nevertheless, we make some comments about the situation described there. Note that for any initial state U,
also; hence the computation of probabilities associated with the evolution of this isolated "particle" is exactly the same as for the free evolution of that particle. The phase changes induced by the factor e -ic(t) will have an influence on the calculation of probabilities if the particle interacts with other particles since the changing phases will alter interference effects. Because the Cantor function is constant on the countable union of intervals which make up the complement of the Cantor set, the phase remains constant across those intervals. The changes take place at the fractal set of times in the Cantor set itself. Finally, we note that our results (15.5.13)-(15.5.15) extend in an obvious manner if instead of the standard Cantor function C on [0, 1] as above, we consider its periodic extension to [0, +00) or to R (as is done in a different context in [Mand, Chapter 8, esp. pp. 74-75 and 78-79]). Exercise 15.5.4 (a) Calculate the final expression for Kty (F) in (15.5.13) by beginning for X > 0 with (15.2.8) in Definition 15.2.1 and then extending your result first to C+ by analytic continuation and then to C~+by strong continuity. (b) Pick some other positive Borel measure u on [0, +00) which has finite total variation on [0, t]for every t > 0, and let Du (t) be the associated function of bounded variation [Coh, pp. 22-24]. Do a calculation for this situation as in the proof of (15.5.13) and interpret your result in the quantum-mechanical case. We will return to Example 15.5.3 in several places in Chapter 17 where we will discuss the integral and differential equations associated with this evolution. Next we single out the case when n from Example 15.5.1 is purely discrete. We will use this result in Sections 16.2 and 17.3. Example 15.5.5 (n purely discrete and finitely supported) Let n = v be given by(15.5.4) with the rs ordered. We write Kty (F) in two ways; the first emphasizes the connection with formula (15.5.5), the second, the similarity of the inner sum to the multinomial
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formula:
If f ( z ) — z, (15.5.16) simply becomes
Alternatively, (15.5.16) and (15.5.17) can be obtained with relative ease in this case by returning to Wiener's integration formula (3.3.9) followed by analytic continuation and strong continuity. Note that the formulas for Kty (F) in (15.5.16) do not involve integrals; it is the continuous part u of the measure n which leads to the integrals in (15.5.5) as well as to the sum over the js. If f(z) = exp(z), then Wiener's integration formula (3.3.9) and the multiplicative property of the exponential function yield
(See Exercise 15.5.6 below for a simple special case.) Of course, one can also recover (15.5.18) from (15.5.16); indeed, since n!an = 1 in (15.5.16), the multinomial formula can be applied to yield (15.5.18). Specializing further, we begin to see connections with the product integral ([DoFri3], [Mas], [dWmMN2, §2]) and the Trotter product formula (Chapter 11, especially Section 11.1). Let w1 = T1, w2 = t2 — T1, . . . ,wh = th — th-1. Then (15.5.18) becomes
a "partial product" which plays a role in the theory of the product integral analogous to that played by partial sums in Riemann integration. Product integration in this context was pursued by Lapidus in ([La14,15, Lal8] and especially [Lal6]). It is not one of the main themes of this book, however, although we will return to it briefly in Section 17.6.
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If the Tp's are equally spaced (i.e. if rp = p(t/h) for p = 1, . . . , h), and if 9 is time independent, we obtain
which is a Trotter product. Observe that if y = 1 and 6 = — V, then
on the other hand, if y = —i and 9 = -iV, then
We shall use this link with the Trotter product formula in Example 16.2.7 of Chapter 16, in connection with our discussion of "stability in the measures". Recall from Section 11.1 that this formula involves taking the limit as h -> oo in expressions such as (15.5.20)(15.5.22). (Caution: The integer h in (15.5.22) does not stand for Planck's constant.) For convenience, our assumption all along has been that the rs were in (0, t). Consistent with our remark at the end of Section 15.2.F, it is quite possible to let rh = t, and we have taken advantage of that in (15.5.20)-(15.5.22). Exercise 15.5.6 Given w E C, y E C~+, T € (0, t) and 9 essentially bounded, let
The results of the next example will be used in Section 16.2 (Example 16.2.11) and in Section 17.6. Example 15.5.7 (n an arbitrary Borel measure) We treat here the case of a single arbitrary n e M(0, t) and a single 0 e Loo1;n. Let v, the discrete part of n, be defined by
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(15.5.1). Then
where, for each h, a is the permutation of {1, . . . , h} such that
and
further, for (s1, . . . ,s q 0 ) e Aq0;j1,...,jh+1 and r e {0, . . . , h},
(We use the conventions rd(o) = 0, rd(h+1) = t and [0(td(0))] qd(0) = 1.) Finally, the norm estimate (15.5.8) holds. In particular, if f(z) = exp(z),
Note that if [rp] is an increasing sequence, then the permutation a that appears in (15.5.25)-(15.5.28) can be omitted. The key to the combinatorial structure of Equation (15.5.25) is the following version of the "No-nomial formula":
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Equation (15.5.30) can be obtained as follows:
which equals the right-hand side of (15.5.30). To use (15.5.30) in order to deduce (15.5.25), take bo — f ( 0 , t ) 0 ( x ( s ) ) u ( d s ) and bp = w p 0 ( x ( T p ) ) for p > 1. For a given h, introduce the permutation a of {1, . . . , h] to order T1, . . . , rh, as in (15.5.26). One may now finish much as in Example 15.5.1. Several measures and potentials We now proceed somewhat less formally than before. Our main intention is to suggest the rich variety of possible examples within our framework. The measures denoted by u, u1, u2, . . . , will always be continuous measures in M(0, t) and T, T1, T2, . . . will lie in the interval (0, t). Example 15.5.8 Let
where 0 e Loo1;u and 01 e L001;dr. (Note that 01(r, x(r)) = f(0,t)01(s, x(s))d T (ds).) Then
where An and An;j are as in (15.3.6) and (15.3.3), respectively. Hence
where
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If
it then follows from (15.5.33) and Corollary 15.4.3 that
One can think of 01(t) appearing in (15.5.36) and (15.5.34) as corresponding to the first order term of an external disturbance, at the fixed time T, of the physical system determined by 6 and u. Recall from Remark 15.2.6(b) that taking X = -i and 0 = — i V puts us in the quantum-mechanical setting. Such "instantaneous interactions" might occur at a finite number of fixed times T1 , . . . , rh. We consider this situation in the next example where we shall also highlight some of the connections with Feynman's operational calculus. Example 15.5.9 Let 0 < r1 < . . . < rh < t. Put
Then
where An;j1,...,jh+1 is as in (15.5.6). We write explicitly the case when n = 1:
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and so
The passage from the functional F1 written in the form (15.5.39) to the operator Kty (F1) respects the ordering of the time indices. (Mathematically, this is due to Wiener's integration formula (3.3.9).) Further, in (15.5.40), we have taken the operators not involving the variable s out from under the integral sign. We see here an explicit connection with Feynman's time-ordering convention [Fey8, p. 109] recalled in formula (14.2.1). This convention is central to the "disentangling" process which, as Feynman notes [Fey8, p. 110], lies at the core of his operational calculus. Next we consider the functional
Then, in view of (15.5.38),
where, for r = 0, . . . , h,
here, r0 = 0, rh+1 = t and 00(ro) = 1. The Feynman graph corresponding to the nth term of the generalized Dyson series (15.5.42) is given in Figure 15.6.6. Example 15.5.10 For 7 = 1, 2, let 8j e Loo1;uj. Put
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Then
Note that within Am, r1 < . . . < rm, whereas within An, s1 < . . . < sn. In order to obtain the formula for Kty (F m , n ), one needs to order the r and .s variables with respect to each other much as in some earlier examples. (See also Example 19.7.2 and related material in Chapter 19.) However, we will simply give the formula for Kty (F1,1):
By considering the functional
where u2 is a probability measure on (0, t), we would obtain a continuous analogue of the perturbation series (15.5.36) in Example 15.5.8. Exercise 15.5.11 Compute Kty (F) for F given by (15.5.47). More generally, by using methods similar to those of the previous examples, we can treat functionals of the form
where nj e M(0, t), 0j € L0o1;nj for j = 1, . . . , q and where / is a function of q complex variables which is analytic in a region containing
We have discussed several rather diverse examples, but we should emphasize that much more is possible. Actually, not only any finite number, but even an infinite number of distinct ns and 9s may appear. We finish this section by giving, as an exercise, a simple example of this type.
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Exercise 15.5.12 (Infinitely many ns and 8s) Suppose that nn E M(0, t) and 0n e Loo1;nn for n = 1, 2, . . . . Further assume thatE8n=1||0n||0nllool;nn< oo. Let
Then show that
15.6
Generalized Feynman diagrams
At Cornell I was learning Richard Feynman's quite different way of calculating atomic processes. . . . He had his own private way of doing calculations. His was based on things that he called "Propagators", which were probability amplitudes for particles to propagate themselves from one space-time point to another. He calculated the probabilities of physical processes by adding up the propagators. He had rules for calculating the propagators. Each propagator was represented graphically by a collection of diagrams. Each diagram gave a pictorial view of particles moving along straight lines and colliding with one another at points where the straight lines met. When I learned this technique of drawing diagrams and calculating propagators from Feynman, I found it completely baffling, because it always gave the right answers but did not seem to be based on any solid mathematical foundation. Feynman called his way of calculating physical processes "the space-time approach", because his diagrams represented events as occurring at particular places and at particular times. . . . Freeman J. Dyson, 1996 [Dys4, p. 12]
In this section, we introduce generalized Feynman diagrams corresponding to our generalized Dyson series. The diagrams contain most of the information conveyed by the series and help to understand and visualize its various terms. A great variety of Feynman diagrams associated with perturbation expansions appear in the physics literature. Although our diagrams can be complicated in their own right, they generalize the simple diagrams of nonrelativistic quantum mechanics but not those of, for example, quantum electrodynamics. We begin by considering the Feynman diagrams associated with the Dyson series (15.3.17) of Corollary 15.3.6 where n = u and f ( z ) — exp(z). When n = u = l, we shall recover the classical Feynman diagrams associated with the classical Dyson series. Figure 15.6.1 shows the graph corresponding to the nth term
of the series (15.3.17). This diagram describes a succession of free evolutions during the time intervals (0, s1), (s1, S2), . . . , (sn, t) alternating with interactions which take place at the times s1, S2, . . . , sn. The symbol u®n appearing at the bottom of the diagram indicates integration over An with respect to u®n. In the classical Dyson series (15.3.22), we have u = l, but in general, the continuous measure u might be quite different from
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FIG. 15.6.1. The graph corresponding to the nth term of the GDS for n = u, continuous Lebesgue measure. (See the discussion following the proof of Corollary 15.3.6, as well as Example 15.5.3 above.) Note that in the setting of Example 15.5.3, where u is the Cantor-Lebesgue measure, interactions could occur only at the points of the Cantor set. A simple introduction to the concept of Feynman diagrams, as they appear in the physics literature, can be found in [Mat, esp. §3.2.]; for the connections between Feynman diagrams and Dyson series, we also mention ([Mi, Chapter 4], [Schu, Chapter 10], [Si9, §20]). We now consider generalized Feynman graphs corresponding to the nth term of the generalized Dyson series (15.3.15) from Corollary 15.3.4. Recall that n = u + wdr in this case. We assume for simplicity that f ( z ) = exp(z). Figure 15.6.2 gives three representations for the graph corresponding to the term
To avoid the extreme cases for now, we assume that 0 < k < n and 0 < j < k. We have n interactions just as in the example above but n - k of them are now occurring at the fixed time T. We use a large dot at r to stress its special role. From the graph-theoretic point of view, the number of vertices has been reduced by n — k — 1 from n + 2 to k + 3. What we denoted in (a) by [ 0 ( r ) ] n - k is expressed in (b) and (c) by n—k—1 loops attached to the vertex r. The segments connecting successive times represent edges of the graph. The loops can be thought of as "infinitesimal edges" connecting r to itself as we will discuss further on. The number of edges is reduced by n — k — \ without the addition of the loops, but, with them, it remains unchanged and equal to n + 1. In place of u®n in our earlier example, we now have u®k and wn-k /(n— k)! Here, u®k represents integration with respect to the k variables s1, . . . , sk, whereas w n - k / ( n — k)! arises from integration with respect to wdT of the remaining n — k variables. (We will see eventually in Example 16.2.9 of the next chapter that the factorial comes from the ordering of the variables.) In the classical case, the term of the Dyson series corresponding to n interactions is represented by a single Feynman diagram (see Figure 15.6.1). Here, however,
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FIG. 15.6.2. Three representations of the graph corresponding to n = u + wdr for n, k and j fixed the generalized Feynman graph associated with n interactions has many disconnected components, one corresponding to each term of the type (15.6.2) in the sum
(15.6.3)
Figure 15.6.3 shows these disconnected components for the case n = 3. Remark 15.6.1 Clearly, there are k + 1 disconnected components associated with a fixed k, k < n. If k = n, however, we have only one component rather than n + 1; in fact, T does not appear, and we obtain the same graph as in Figure 15.6.1 where n = u. (See Figure 15.6.3.) Note that the total number of disconnected components for a given n equals (n (n + We wish to indicate an alternative way of thinking of Figure 15.6.2(b) drawing on the intuition of infinitesimals. We adopt the language of [Ke]. In Figure 15.6.4, we use an "infinitesimal microscope" to magnify part of the monad of T. The loops of Figure 15.6.2(b) are replaced by the edges connecting n — k distinct vertices within the monad of T. (Compare the methods of Example 16.2.9.) We now consider generalized Feynman diagrams associated with the generalized Dyson series (15.5.5) obtained in Example 15.5.1. Recall that n = u + Ehp=1 wpd T p with 0 < T1 < . . . < Th < t. Figure 15.6.5 gives the graph corresponding to the
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FIG. 15.6.3. The four graphs corresponding to n = u + wdr with n = 3
FIG. 15.6.4. The instant T in Figure 15.6.2 under an infinitesimal microscope
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FIG. 15.6.5. A generic graph corresponding to n = u +
(generic) term
We close this section by drawing the graph associated with the term
of the Dyson series (15.5.42) obtained in Example 15.5.9. This is given in Figure 15.6.6 and provides one illustration of a generalized Feynman graph involving several 9s. The reader may have noticed that our Feynman diagrams are drawn to correspond to the "natural physical order" as discussed in Remark 15.3.7, Example 15.3.8, Remark
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FIG. 15.6.6. An example with multiple 9s 15.4.4, and below in Section 15.7; to obtain the order resulting directly from calculating the Wiener integral of functionals F as in Section 15.4, simply read our diagrams from top to bottom.
15.7 Commutative Banach algebras of functionals: The disentangling algebras The class of functionals considered in Section 15.4— equipped with a natural norm that we shall soon define—forms, for every t > 0, a commutative Banach algebra At under pointwise multiplication. This Banach algebra structure along with our earlier results will help us to make rigorous certain parts of Feynman's operational calculus [Fey8]. Indeed, we have made related comments in Section 15.1 and we shall make additional remarks in this section. We will develop this topic much further in Chapter 18—where two auxiliary noncommutative operations will play a key role. As we will see in Chapter 18, the fact that we have at our disposal a one-parameter family of algebras (At : t > 0}, indexed by time t, will be crucial to our discussion of these noncommutative operations. In the present section, however, t will be fixed.
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The disentangling algebras At Given t > 0, the Banach algebra At will consist of equivalence classes of functions, where the equivalence is that given in Definition 15.2.3. (At this point, the reader may wish to review this definition as well as Example 15.2.4 and Theorem 15.2.5.) The equivalence relation ~ is compatible with the algebraic operations as we observed just after Definition 15.2.3. Following the common convention, we will often blur the distinction between equivalence classes and representatives. However, there will be places in this section and in Chapter 18, specifically, Sections 18.2 and 18.3, where we will need to keep the distinction in mind. For the reader's convenience, we begin by recalling some facts and notation from Section 15.4. Let {Fn}8n=0 be a sequence of functionals each of which is given by an expression of the following form:
with mn a nonnegative integer, nn,u e M(0, t) and 0n,u e L001;nn,u. Assume that
Then define a functional F as in Corollary 15.4.3 by
where y > 0 and (x, E) e Ct0 x Rd (so that y-1/2x + £ e Ct). Recall that for every X > 0, the series in (15.7.3) converges absolutely for m x Leb.a.e. (x, £) e Ct0 x Rd. Denote by At the set of all equivalence classes of functionals F obtained in this manner. It is important to realize that the representation of functionals in (15.7.3) and even in (15.7.1) is not unique. For F in At, we let ||F||t be the infimum of the left-hand side of (15.7.2) over all representations of F of the form (15.7.3). We make two observations concerning the equivalence relation ~ and representations of functionals as in (15.7.1)-(15.7.3). These facts follow trivially from the transitivity property of ~; nevertheless, it is helpful to keep them in mind: (i) If F ~ G, then F and G have exactly the same set of representatives as described in (15.7.1)-(15.7.3). (ii) Let the equivalence classes [F] and [G] belong to At. If there exist F1 6 [F] and G1 e [G] such that F\ and G\ have in common even one representation as described in (15.7.1)-(15.7.3), then [F] = [G]. Now we come to the main result of this section ([JoLa1, Theorem 6.1, p. 70]). (Our proof of the completeness will be more detailed than its counterpart in [JoLa1].)
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Theorem 15.7.1 (The disentangling algebras {At}t>o) For each t > 0, the space (At, || . ||t) is a commutative Banach algebra with identity. Moreover, given F in At, Kty (F) exists for all y e C~+and satisfies the norm estimate
Further, for every choice of a representation of F, the operator Kty (F) is given (or disentangled) by the corresponding generalized Dyson series (15.4.21). Proof We begin by showing that if ||F||t = 0, then F is equivalent to 0. Let p be any positive integer. Since \\F\\t = 0, there exists a representation for F given by (15.7.1) and (15.7.3) such that the left-hand side of (15.7.2) is less than 1 / p . We see from the inequality (15.2.15) in Lemma 15.2.8 that, for every y > 0, \ F ( y - 1 / 2 x + £)| < 1/P for a.e. (x, £). Since p is arbitrary, it follows that, for every A > 0, F ( y - 1 / 2 x + £) = 0 for a.e. (x, £). Hence F is equivalent to 0. Next suppose that F and G are in At. We claim that FG is in At and that
Given e > 0, take a representation for F defined by (15.7.1) and (15.7.3) such that the left-hand side of (15.7.2) is less than ||F||t + e. Choose a similar representation for G. Then, for every y > 0 and a.e. (x, E), we have
and
where the convergence is absolute. It follows that, for every A > 0 and a.e. (x, E), the series, written by using the diagonalization procedure,
converges absolutely and has the sum F(y - 1 / 2 x + £ ) G ( y - 1 / 2 x + E). Further, each term F j ( y - 1 / 2 x + E ) G k ( y - 1 / 2 x + E) in (15.7.6) is of the type (15.7.1). Consequently, FG e At and
Inequality (15.7.5) now follows since € was arbitrary.
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Using arguments similar to some of those above, it is now easy to show that (At,||.||t) is a normed linear space. To show completeness, it is convenient to use the fact that a normed linear space is complete if and only if every absolutely summable series is summable in the norm of the space [Roy, Proposition 6.5, p. 124]. Let {Fj} be a sequence from At such that
Then for each j, we can choose representations of the form (15.7.3) and (15.7.1),
such that for each j, the left-hand side of (15.7.2) is less than || Fj||t + 2-j. The terms of the series
are, of course, of the form (15.7.1), and the corresponding series of the form (15.7.2) converges to a number less than £8j=1||Fj||t+ 1. Thus, by Corollary 15.4.3, the series (15.7.8) with its terms evaluated at y-1/2x + § is absolutely convergent for every y > 0 and m x Leb.-a.e. (x, £) in Ct0 x Rd. Let
We see from the discussion above that F e At. What remains is to show that ||F - Enj=1 Fj||t -> 0 as N -> oo. Given € > 0, it follows from (15.7.7) that we can choose N0 large enough so that E8j=N0+1 ||Fj||t + 2-j) < €. Using the definition of the norm || . ||t, it is now not difficult to show that for N > N0, we have
We leave this last step to the reader.
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The fact that Kty (F) exists for all y e C~+ and is given by a generalized Dyson series has already been shown in Corollary 15.4.3. (Also, recall from Theorem 15.2.5 that Kty (F) does not depend on the choice of representative in the equivalence class of F.) Finally, the estimate (15.7.4) follows from the fact that the norm estimate (15.4.22) in Corollary 15.4.3 holds for any representation for F. This concludes the proof of Theorem 15.7.1. Remark 15.7.2 Banach algebras smaller than but related to At were studied by Johnson and Skoug in [JoSk2,6]. It is the p = 2 case of [JoSk6] which compares most directly to [JoLaI, §67 and to our present discussion. There was an error made in [JoSk6] in connection with the definition of the norm on the Banach algebra. This difficulty is easily corrected as we pointed out in Remark 6.1(a) of [JoLa1], and it is then easy to see that the Banach algebra At = A from [JoSk6] is a subalgebra of At. (More specifically, A, is the subalgebra obtained by taking all measures nn,u equal to Lebesgue measure l in the definition of At; see equations (15.7.1)-(15.7.3).) Various formulas from [JoSk2,6] are, from the perspective of our present work, formulas for disentangling. The authors of [JoSk2] and [JoSk6] did not think of their work in these terms although they did suspect that it had some connection with Feynman's time-ordering ideas. The 1 < p < 2 case of [JoSk6] as well as [JoSkS] show that mathematically rigorous meaning can be given via functional integration to parts of Feynman's operational calculus even when the space of functionals involved is not an algebra. However, the presence of a Banach algebra structure permits a fuller development of the ideas from [Fey8], Work of Cameron and Storvick beginning with [CaSt1] influenced ([JoSk2,6], [JoSk5]) and so had an influence on the memoir by Johnson and Lapidus [JoLa1] and hence on the present chapter. Our first corollary follows immediately from Theorem 15.7.1. Corollary 15.7.3 For each fixed y 6 C~+, the mapping Kty : At -> L ( L 2 ( R d ) ) which associates Kty (F) with F e At, is a bounded linear operator of norm at most 1, and equal to 1 if y is purely imaginary. Our second corollary follows from well known facts about the holomorphic functional calculus in Banach algebras ([KadRi, §3.3] or [Nai, pp. 202-205]). Corollary 15.7.4 Let F E At and let f be a complex-valued function of a complex variable which is analytic in a disk about the origin with radius strictly greater than ||F||t. Then the function f o F is also in At. The time-reversal map on At and the natural physical ordering In the following, we further address—within the context of the Banach algebra At—the twin issues of the time-reversal operation and of the natural physical ordering of operators that were already discussed in a closely related context in Remark 15.4.4 above (and, in a much simpler situation, in Remark 15.3.7 and Example 15.3.8). We note that this material was not included in [JoLal].
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Let F be an arbitrary element of At. Then the "time-reversal" F of F is defined as follows. In each representation of F of the form (15.7.3), we substitute the pair (0n,u, ~nn,u) for (0n,u nn,u), where 0n,u and~nn,uare the time-reversals of 0n,u and nn,u, respectively, as defined by (15.4.24). More precisely, if F is given by (15.7.1)-(15.7.3), then F is given by (15.4.26); namely,
where A > 0 and (x, £) e Ct0 x Rd. (This definition is compatible with the equivalence relation ~, as we will soon see.) Moreover, by (15.4.25), the counterpart of (15.7.2) still holds, so that F also belongs to At and ||F||t = ||F||t. We can now formally define the "time-reversal" operation T as a map from the Banach algebra At onto itself. Definition 15.7.5 The time-reversal map T: At -> At is defined by
where F is given as in (15.7.10) above. The first assertion in Theorem 15.7.6 below will assure us that T is well defined on the set of equivalence classes that make up At. The following result explains our interest in the map T. Theorem 15.7.6 (i) The time-reversal map T is well defined on At. Further, it is an involution on At (i.e. ~F = F for all F e At) and is an isometric isomorphism of the Banach algebra (At, || . ||t) onto itself. (ii) For every F e At and X € C~+, we have
the Hilbert adjoint of the bounded linear operator Kt/y (F), where F (e At) denotes the complex conjugate of F. Moreover, by Remark 15.4.4, Kty (F) can be disentangled (as in Theorem 15.7.1) via a GDS of the type (15.4.21), except that each term in the time-ordered perturbation series (15.4.21) is now written in the natural physical ordering; that is, in agreement with Feynman's time-ordering convention (14.2.1), the operators involving smaller time indices always act before those involving larger time indices. Proof It follows easily from observation (ii) just before Theorem 15.7.1 that the map T respects the equivalence relation ~; that is, F ~ G implies that F ~ G. In other words, given an equivalence class [F] e At, it makes sense to define [F] as [F]. Thus T is a well defined map on the set of equivalence classes that make up At • The equation F = F follows from (15.7.10) and the fact that 0n,u = 0n,u and nn,u = nn,u for each n and u. It is clear that the map T is linear and onto, and it follows from (15.4.25) and (15.3.24) that it is an isometry.
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The last claim made in (ii) of the theorem was established in Remark 15.4.4. (The combinatorial complications involved in (15.4.21) through the associated formula (15.4.6) tend to obscure the simplicity of the idea. The reader may find it helpful to compare formulas (15.3.26) and (15.3.28) of Example 15.3.8.) The proof of the theorem will be complete if we establish the equality (15.7.12); to do this, it suffices to show that
Recall from (15.4.21) that Kty (F) = E8n=0 Kty (Fn), where the series converges in operator norm. Since the adjoint operation on L(L 2 (R d )) is continuous (because it is a conjugate linear isometry by Proposition 9.6.2), it suffices to establish the desired formula for F = Fn. We will do this by using the formula (which will be discussed at the end of this proof)
where n is a complex measure on the measurable space (Y, y) and g : Y -> L ( H ) is Bochner integrable in the strong operator sense with respect to n. [Here, H is an abstract complex separable Hilbert space (rather than L2 (Rd)).] What happens when the formula above for the adjoint of the integral is applied along with the conjugate linearity of the adjoint operation to the formula for Kty (Fn) in (15.4.6) with the Lrs given by (15.4.8)? The (as and the us are conjugated and the product Lk . . . Lm is replaced by L*m . . . L*k. Further, each L*r is the product in reverse order of the adjoints of the operators appearing on the right-hand side of (15.4.8). Since the adjoints of the multiplication operators 0(s, .) are given by 0(s, .)* = 0(s, .) (by Theorem 10.1.11(i) or Remark 10.1.4) and since ( e - s ( H 0 / y ) * =e-s(H0/y)for y e C~+ and s > 0 (by Theorem 10.1.11(i) with g(x) := e - s x / y for x e d(H0) C R, and so g(x) = e-sx/y), we obtain the formula Kty (F)* = Kt/y (F) and hence (15.7.12). (The reader may find it helpful to calculate Kty (F)* in the simple but instructive case where F is given by (15.3.25) in Example 15.3.8.) We now discuss briefly the proof of the formula given above for the adjoint of the integral. Since the integral is in the strong operator sense but the adjoint operation fails to be continuous in the strong operator topology [DunSc1, p. 513], the limiting argument that may first come to mind fails. However, one can quite easily prove that the operator fy g * ( y ) n ( d y ) acts as the adjoint of fY g ( y ) n ( d y ) should; that is, for every o, U e H,
where (., .)H denotes the inner product on H. Actually, it is enough to show this last equality when n is a positive measure since the general case then follows from the linearity (in the measure) of the integral, the conjugate linearity of *, and the fact that every complex measure n can be written as n = n1 — n2 + i(n1 - n4), where nj is a
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positive measure for j = 1, . . . , 4 [Coh, p. 126]. The equality of the two inner products above follows from the theorem [HilPh, Theorem 3.7.12, p. 83] allowing one to take a closed linear operator, and so certainly a continuous linear functional, under the integral sign. Remark 15.7.7 (a) When Kty (F) from Theorem 15.7.6 is expressed in terms of the pairs (0n,u, n n , u ), the natural physical ordering is not achieved. Returning once again to Example 15.3.8 for an illustration, we have
In contrast, when Kty (F) is written in terms of the pair (0, u), we do have the natural physical order as we saw in equation (15.3.28) of Example 15.3.8:
(b) While we were writing [JoLa1], we knew from Feynman's ideas, from earlier work such as [CaSt1, JoSk2, JoSk6], and especially from work then in progress [La14-16, La18] that the time-ordering in the GDS for Kty (F) is exactly the reverse of the natural physical order. We also realized that taking the adjoint would produce the desired order. However, we thought mistakenly that taking the Banach space adjoint rather than the Hilbert space adjoint would yield the right order (see [JoLa1, Remark 1.5, p. 25]) without introducing complex conjugates on 0, n] and y. This is correct when 0, n and y are all real-valued, but is not so otherwise. This error was carried on in [La14-18], but since the GDS actually being used was the physically natural one, no serious harm was done. We will study the material from [La14-18] in Chapter 17 below, but we will of course use Kty (F) and not the Banach space adjoint. Corollary 15.7.8 Let R : At -> A, be defined by
[In other words, R is defined as T above except that each pair (0n,u nn,u) is replaced by (0n,u, nn,u) instead of (O n,u , n n , u ).] Then: (i) (At, || . | | t ) , equipped with R, is a (commutative) complex Banach algebra with involution, in the sense of [KadRi, §4.1, (i)-(iii), p. 236] (that is, F## = F, ( F G ) # = G#F#(= F#G# here) and (aF + BG) # = aF# + BG#, for all F, G e At and a, B € C). Further, R is an isometric anti-isomorphism from (At, || . ||t) onto itself.
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459
(ii) For every F e At, we have
the (Hilbert) adjoint of Kty (F). (In particular, for y > 0, the adjoint of Kty (F) is given by Kty (F#).) Consequently, Kty (F)* = Kty ( F # ) can be disentangled via a GDS of the form (15.4.21), except with the operators in their natural physical ordering (and the potentials 0n,u and measures nn,u replaced by their complex conjugates). Proof (i) follows easily from part (i) of Theorem 15.7.6. Note that by (15.7.13) and the definition of the norm || . ||t,
for all F e At. (ii) follows immediately from (15.7.13) and (15.7.12), as well as from Theorem 15.7.1. Remark 15.7.9 Clearly, (At, || . ||t, R) is not a C*-algebra since, in general, ||FF#||t does not equal ||F||2t for F e At, as the reader will easily verify. (See, for example, [KadRi, § 4.1, (i)-(iv), p. 236] for the definition of a C* -algebra.) Exercise 15.7.10 Show that the algebra At from [JoSk6] (and defined in Remark 15.7.2 above) is a subalgebra of At which is stable under both of the maps T and R. Connections with Feynman's operational calculus We close this section by comments that may help put our work in this chapter in a broader context and connect it with material discussed in later chapters. One of the purposes of a functional calculus is to allow one to form a rich class of functions of the objects of interest. In the traditional cases, the Feynman-Kac formula (y = 1) and the Feynman integral (y = —i), one considers the exponential of
where / is Lebesgue measure. The present theory is more general in many respects. In [JoSk6, pp. 121-123], one could already form rather general analytic functions of Gl. Here, however, we are able to treat analytic functions of
where n e M(0, t) is arbitrary. The significance of this additional flexibility is seen throughout the present chapter and much of the rest of the book. We note that we can actually deal with analytic functions of a general element of the Banach algebra At as in Corollary 15.7.4. Further, it follows from the spectral theory of
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Banach algebras—more specifically, from the holomorphic functional calculus in that context ([KadRi, §3.3] or [Nai, pp. 202-205])—that it suffices for the function / of Corollary 15.7.4 to be analytic in a neighborhood of the spectrum of F. We can also take functions of infinitely many variables as was done in Exercise 15.5.12. As commented on earlier, Feynman [Fey8] observed that disentangling is the key to his operational calculus. Within the setting of nonrelativistic quantum mechanics (alternatively, the diffusion equation (12.1.5)) and multiplication operators as in the present chapter, our generalized Dyson series (GDS) provide a method of disentangling in a rather broad context. The general theory to be presented in Chapter 19 is in many respects broader still but does not have the analytic power connected with the Wiener integral. (See, in particular, Theorems 16.2.3 and 16.2.6 as well as Examples 16.2.7 and 16.2.11 in Chapter 16 below which illustrate the use of the dominated convergence for the Wiener integral in this context.) In the present setting, even when X is not real, the explicit nature of the GDS for all y e C~+ will have its advantages. We will see illustrations of this in Chapters 16, 17 and 18. It will allow us, in particular, to make limiting arguments valid for both the diffusion and quantum-mechanical case. [See Example 16.2.9 as well as the "Feynman-Kac formula with a Lebesgue-Stieltjes measure" (FKLS) in the general case (Section 17.6 and [Lal8]).] The generalized Dyson series will also be used to obtain the results in Chapter 17 on the FKLS from [Lal4-16, Lal8] (see especially Sections 17.3 and 17.6) and further used in Chapter 18 in relation with the disentangling algebras At [JoLa3,4] (see Section 18.5). It is important to note that the process of disentangling is not unique. Example 15.5.5 provides a simple illustration of this. Compare formula (15.5.16) in the case where f ( z ) = exp(z) (so that n!an = 1) with formula (15.5.18). These two timeordered expressions for the same operators Kty (F) are quite different from one another. Examples 16.2.7 and 16.2.11 in the next chapter, as well as the results from [Lal4-16, La18] discussed in Chapter 17, will provide further illustrations of nonuniqueness. When there is more than one disentangling of the operator Kty (F), the most useful choice may depend on the goal being pursued. Even though F e At does not uniquely determine the disentangling of Kty (F), the operator Kty (F) itself is unambiguously associated with F. This follows from the fact that Kty (F) is defined by the Wiener integral (15.2.8) for y > 0, followed by analytic continuation for X € C+ and strong continuity for y e C~+. It is clear that the process of disentanglement, which might be viewed as the main theme of the present chapter, is intimately related to the noncommutativity of the operators involved. Since the algebra At is commutative whereas £(L 2 (R d )) is not, we would not expect the linear mapping Kty defined in Corollary 15.7.3 to preserve multiplication. Indeed, it does not as the following simple example shows. Example 15.7.11 Take F = 1. Then F2 = F and, for all y e C~+,
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where the last equality comes from (15.2.11). On the other hand, by the semigroup property of e-tH0/y,
Thus, we see that (for t > 0)
We will introduce in Chapter 18 an auxiliary noncommutative operation * : At1 x At2 -> At1+t2 which will fit well with Feynman's operational calculus and which will allow us to write (for t1 , t2 > 0 and y e C~+)
The operation * is a type of multiplication but is rather different from the pointwise product of functions involved in Example 15.7.11. Nevertheless, it is interesting to note the formal contrast between (15.7.20) and Example 15.7.1 1. As we will see in Chapter 18, this "noncommutative multiplication" * —along with its companion operation +, a kind of "noncommutative addition"—will enable us to take into account in a more complete way the noncommutativity underlying Feynman's operational calculus.
16
STABILITY RESULTS The functionals introduced in the previous chapter and studied in Chapters 15-18 are defined in terms of measures and potentials. It is natural to ask if the corresponding operators are stable under perturbations of either of these objects. First, we consider in Section 16.1 stability with respect to the potentials. We then consider in Section 16.2 stability with respect to the measures. Apart from some extensions discussed at the end of Section 16.2, the results of this chapter are due to Johnson and Lapidus in [JoLa1, §4]. Those in Section 16.1 are motivated by and extend to a broader class of Wiener functionals the stability theorem (for analytic operator-valued Feynman integrals) of Johnson in [Jo3] and [JoSk2, p = 2 case]. Further, the stability theorems with respect to the measures obtained in Section 16.2 will enable us to reinterpret in a new light and unify various phenomena connected with discrete and continuous measures. (This theme will be pursued in parts of Chapter 17.) 16.1 Stability in the potentials For much of the notation in our first theorem, we refer the reader to Corollary 15.4.3. Theorem 16.1.1 Let nn,u e M(0, t) and let 0n,u e L 0 0 1 n , u . Assume that the hypotheses of Corollary 15.4.3 are satisfied for the family of potentials ®n,u. Let 0n,u(m), m = 1 , 2 , . . . , be Borel measurable functions on (0, t) x Rd such that, n x Leb. almost everywhere,
and
Then0n,u,0(m)n,uall belong to Further, let Fn (resp.,Fn(m)) be the functional associated with the potentials &n,u (resp., 0n,u(m)) and the measure nn,u as in equation (15.4.18) of Corollary 15.4.3. Moreover, let F (resp., F(m)) be the functional defined in terms of the sequence {F n }8 n=0 (resp., {Fn(m)}£n=0) via equation (15.4.20). Then, for all y € C~+, Kty (F), Kty (F(m)) exist and
(That is, the convergence in (16.1.1) holds when applied to any vector U in L2(Rd).)
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Further, the form of the generalized Dyson series is preserved. More precisely, for each n, the combinatorial form of the nth term can be taken to be the same for all of the GDSs involved. We omit the proof of Theorem 16.1.1 since it is, except for notational complications, essentially like the proof of our next result. In the following theorem, we treat the special case considered in Theorem 15.3.1 and Corollary 15.3.4 and we use our earlier notation. Theorem 16.1.2 Let n = u + wdT as in Theorem 15.3.1 and let ® € Loo1;nLet e (m) , m = 1, 2, . . . , be Borel measurable functions on (0, t) x Rd such that, n x Leb.-a.e.,
and
Then 0, 0 (m) belong to Loo1;n. Moreover, let
where f is given by (15.3.13) and has radius of convergence strictly greater than ||®||oo1;n. Let F(m) be defined as in (16.1.3) except with 0 replaced by 8 (m) . Then, for all X e C~+, Kty (F), Kty (F(m)) exist and
Further, theform of the generalized Dyson series is preserved in thefollowing precise sense:
strongly as m -» oo. Here, Lj is defined by (15.3.4) and Cj(m) is given as in (15.3.4) except with 0 replaced by 0(m). Proof By (16.1.2), ||0(m)||oo1;n < ||O||oo1;n, form = 1, 2, . . . and 0(m), 0 lie in Loo1;n. Using (16.1.2) and the Lebesgue dominated convergence theorem, we see that 0(m)(s) -> 0 ( s ) strongly as m -» oo, for n-a.e. s in (0, t). It follows that
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Now, by (16.1.6) and the norm estimate (15.3.16), we have
Hence the conclusion follows from equations (15.3.1), (15.3.2), (15.3.15) and the dominated convergence theorem for Bochner integrals [HilPh, Theorem 3.7.9, p. 83]. Remark 16.1.3 (a) The above proof shows that the form of each summand in (16.1.5) is preserved. We should stress that, in all cases, for instance, in the examples of Section 15.5, the form of the generalized Dyson series is preserved just as in Theorem 16.1.2. (b) In both of Theorems 16.1.1 and 16.1.2, if U (m) -> V in L2(Rd), then in It has been notoriously difficult to establish pleasant analytic properties of the Feynman integral. As far as we know, the first reasonably satisfactory stability theorem for the Feynman integral appeared in Johnson's paper [Jo3]. The results in [Jo3] and [JoSk11, p = 2 case] are special cases of Theorem 16.1.2. The paper [JoSk11] also includes results where the "Feynman integral" was interpreted as a bounded linear operator from L P ( R d ) to Lp ( R d ) , 1 < p < 2, where1/p+ 1/p' = 1. In [JoSk11], restrictions are placed on the dimension d which become more and more serious as p gets closer and closer to 1. The p = 1 case was covered in [Cha]. Recall that Theorems 11.5.13 and 11.5.19 above gave a stability theorem for the modified Feynman integral which permitted highly singular potentials. (See Section 11.5.) This result is due to Lapidus [La 12]. In [Lal2, pp. 58-59] (or [La9]), one can also find an operator-theoretic proof of the result of [Jo3]. 16.2 Stability in the measures We give a simple theorem establishing stability in the measures and use it, in particular, to make connections with the Trotter product formula (Example 16.2.7) and to explain the link between the absolutely continuous case and the discrete case (Example 16.2.9). We do not strive here for maximum generality; indeed, additional study of the topics in this section might make a good subject for further research. Our next theorem involves the concept of weak convergence of measures. Definition 16.2.1 Let n, nm, m — 1, 2, . . . be in M(0, t). We say that nm converges weakly to n and write nm -> n, provided that
for every bounded continuous function b on (0, t ). Remark 16.2.2 The term "vague convergence" is sometimes used instead of weak convergence. Weak convergence is weak* convergence from the functional analysis point of view since the space M(0, t) is a closed subspace of the dual of the Banach space of all bounded continuous functions on (0, t). The concept of weak convergence of measures
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makes sense well beyond the real line; a common setting for its study is a separable metric space. This notion of convergence is important in the probability literature where it is restricted to probability measures and often referred to as "convergence in law". Most of the results in the probability literature carry over easily to finite, positive measures. Hence, many things carry over as well to complex measures of finite total variation since each such measure is a linear combination of four positive measures [Coh, p. 126]. Two good references for weak convergence of measures are the books by Billingsley [Bil] and Dudley [Du, esp. Chapters 9 and 11]. Theorem 16.2.3 Let 0 be a continuous function bounded by a finite constant c on all of (0, t) x Rd. Let n, nm, m = 1, 2, . . . , be in M(0, t). Assume that
Put
where f is an entire function given by (15.3.13). Let Fm be defined as in (16.2.3) except with n replaced by nm. Then
uniformly in y on all compact subsets of C+. Proof Fix U E L 2 (R d ) and E e Rd. Let X > 0. In view of (15.2.8),
and similarly for F except with nm replaced by n. Given x e Ct0, the function 0 ( s , y - 1 / 2 x ( s ) + £) is bounded by c and is continuous as a function of s. Hence, by hypothesis (16.2.2),
Since / is continuous,
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Note that
where c1 = c supp ||np||. Since a weakly convergent sequence of measures is bounded in total variation norm, we have c1 < oo. Thus,
where c2 = sup |z| < Cl |f(z)| < oo. Recall from (15.2.11) that |U(y 1 / 2 ;t(f) + £)| is Wiener integrable and
In view of (16.2.5), (16.2.7) and (16.2.9), the dominated convergence theorem yields
as m ^ oo for a.e. £ e Rd. Moreover, by (16.2.9) and (16.2.10),
A second application of the dominated convergence theorem yields
in L2(Rd) as m ->• oo for all y > 0. Now, Kty (Fm) U is analytic for y e C+ and, by (16,2.12) and (15.2.1), we have
Hence, by (16.2.13) and (16.2.14), Vitali's classical theorem (Theorem 11.7.l(i) and Remark 11.7.2(c)) gives the result for y e C+. Our most interesting application of Theorem 16.2.3 will come in Example 16.2.9. There we will see how the Dyson series associated with certain absolutely continuous measures nm converges to the combinatorially very different Dyson series associated with the Dirac measure dr as nm converges weakly to dT. It may well be that deeper facts about weak convergence of measures can be used to extend Theorem 16.2.3 substantially. At the end of this section we will state a slight adaptation of part of a result in [Bil, Theorem 5.2, p. 31] which will allow us to make a modest start in that direction. Observe that the conclusion of Theorem 16.2.3 is stated only for X 6 C+. The next proposition gives some information for y purely imaginary (much as was done in Appendix 11.7, especially in the proof of Proposition 11.7.4, in the related context of unitary groups of operators). For notational convenience, we write y = iy with y e R. In the statement of Proposition 16.2.4, (., .) denotes the inner product in L 2 (R d ).
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Proposition 16.2.4 Assume that the hypotheses of Theorem 16.2.3 hold. Let U1, U2 be in L2(Rd). Then for all o e L1(R.),
Proof The functions Kty (Fm) and Kty (F) are analytic in C+ and strongly continuous in C~+. Moreover, by (16.2.14), \\Kty (F m )U\\ < c2. By the extended Vitali theorem discussed in Appendix 11.7 (see Theorem 11.7.l(ii) and Remark 11.7.2(c)), the conclusion follows. (Note that, even though Kty (F) is not defined for y = 0, it is in H°°, the space of (essentially) bounded analytic functions on the right half-plane. In view of [Dure, Theorem 1.1, p. 2], the argument in the proof of Theorem 11.7.1(ii) still applies to this situation.) The convergence in (16.2.15) is in too weak a sense to be widely useful. However, Proposition 16.2.4 does yield the following corollary via standard functional analysis arguments. We omit the proof. Corollary 16.2.5 Let the assumptions of Theorem 16.2.3 hold. If the strong limit of K t i y (F m ) exists as m ->• oo and equals A(iy) (in £(L2(Rd)), then A(iy) = Ktiy (F) for Leb.-a.e. y in R. If we further assume that the limit function A(i y) is strongly continuous on R as a function o f y , then A(iy) — Ktiy (F) for every nonzero y in R. If we make the much stronger assumption that n,n —» n in norm, we can show without any continuity assumption on 9 that Kty (Fm) —»• Kty (F) in operator norm. Theorem 16.2.6 Let 9 : (0, t) x Rd -> C be everywhere defined, Borel measurable and bounded by B and suppose that f is analytic at least in a circle of center at 0 with radius greater than B \\n\\. Finally, suppose that \\nm — n\\ -> 0 as m —>• oo. Then
uniformly in A on all compact subsets 0f C+. Moreover, for all X > 0, we have the norm estimate
where
with
Proof Since || nm — n || -» 0 as m -»• oo, we have that, at least for large m, f is analytic in a circle centered at 0 with radius greater than B \\ nm \\. We will assume throughout the
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rest of the proof that m is large enough so that this is so. Now for any x 6 Ct, we clearly have
where Mm is given by (16.2.19); thus, for U e L 2 (R d ), a.e. £ € Rd and X > 0,
where cm is defined by (16.2.18). Now by (16.2.5), (16.2.21), and (16.2.10), we have, for X > 0,
and thus, since \\e~ t(Ho/y) || < 1,
Since U is arbitrary in L2(Rd), (16.2.17) follows for A > 0. Now since Mm -» 0 as m ->• oo and / is uniformly continuous on the closed disk \z\ < B\\n\\, we see that cm -» 0 as m —»• oo and so (16.2.16) follows for y > 0. Since strong analyticity is equivalent to analyticity with respect to the operator norm [HilPh, Theorem 3.10.1, p. 93] and since ||Kty((Fm)|| < B||nm|| < B[\\n\\ + 1] (at least for larger m), Vitali's theorem (Theorem 11.7.1(1)) yields (16.2.16) for all X 6 C+. We now present some applications of Theorem 16.2.3. We assume the conditions of that theorem and use the corresponding notation insofar as possible. Our first example relates our point of view to the Trotter product formula and the classical Feynman-Kac formula. Example 16.2.7 (Trotter product formula and classical Feynman-Kac formula) Let /(z) = exp(z), y = 1, and suppose that 6 — —V, with V time independent. Further, let
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Observe that
since, for every bounded continuous function b on (0, t), we have the Riemann sum approximation:
as m -» oo. Thus, by Theorem 16.2.3,
strongly, where F(x) := exp(- f(0,t) V(x(s))ds). If we assume the Trotter product formula (Theorem 11.1.4), that is,
strongly, then from (16.2.27) we have
We thus recover the classical Feynman-Kac formula (Theorem 12.1.1)
where U € L2(Rd) and £ e Rd. Actually, we do not need the full strength of the Trotter product formula to obtain (16.2.29); in fact, we know from Theorem 16.2.3 that the limit in (16.2.29) exists. By [Cher2, Theorem 3.1, p. 34], the limit in (16.2.27) is a semigroup and is generated by Ho + V; thus (16.2.29) follows. As an alternative to the above, we can assume the Feynman-Kac formula (16.2.29) and recover this concrete case of the Trotter product formula from (16.2.27). Of course, we know from our work in Chapters 11 and 12 that both the Trotter product formula and the Feynman-Kac formula are valid under conditions on V that are far more general than in this example. Further, in Chapter 12 we saw the Trotter product formula used in the proof of the Feynman-Kac formula. (References to some of the earlier work on these formulas and their relationship can be found in Chapters 11 and 12.) In spite of all this, there is something to be gained from the present example. Our framework provides a unifying point of view and a broader perspective. Both the approximating product and the limit in (16.2.27) are Wiener integrals of expressions involving Lebesgue-Stieltjes measures. Moreover, the approximating procedure—that is, essentially, the role played by Lebesgue-Stieltjes measures in our theory—is reminiscent in some ways of that played by such measures in the context of one-dimensional integrals.
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Remark 16.2.8 (a) We have restricted attention to X = 1 so that our formulas coincide with the Feynman-Kac and Trotter product formulas as they are most often stated. Actually, the argument works just as wellfor X € C+. For X purely imaginary, our method gives only the rather weak result obtained from Proposition 16.2.4. Note, however, that there is no problem in applying the Trotter product formula in this case since V is bounded. The difficulty of going beyond Proposition 16.2.4 in this manner is closely connected with the problem of passing from the Wiener integral to the Feynman integral and of establishing the Trotter product formula for unitary groups in a general setting (see Problem 11.3.9 above and [La6], as well as the discussion at the end of Appendix 11.7). (b) A " Feynman-Kac formula with a Lebesgue-Stieltjes measure" [La14-18] will be treated in detail in the next chapter. There the functional will have the form
The traditional Feynman-Kac formula is, of course, the special case where n = l. One of the interesting phenomena that will be discussed in Chapter 17 is the distinct roles played by the discrete and continuous parts of the measure n. It is known that, for bounded time independent potentials, the Trotter product formula is related to the product integral (see, for example, [DoFri3, §4.3, pp. 135-137]). A product integral representation for Kty (F) is given in [La16] and will be discussed briefly in Section 17.6; see especially Theorems 17.6.10 and 17.6.12. Here, the functional F is the time-reversal of F (see Definition 15.7.5 and part (b) of Remark 15.7.7) and F is given by (16.2.31). Example 16.2.9 (Absolutely continuous measures approximating dT) Let [pm] be a 8sequence [Sta, pp. 106-117] centered at the point T in (0, t). For instance, let
Let nm be the absolutely continuous measure on (0, t) with density pm. Then, it is not hard to show that
Let y e C+ and f(z) = exp(z). Then Theorem 16.2.3 and formula (15.3.17) from Corollary 15.3.6 yield:
strongly as m ->• oo, where An and £ are given by (15.3.18) and (15.3.19), respectively.
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471
Next, with pm given by (16.2.32), we discuss a direct derivation of (16.2.34) using the generalized Dyson series for Kty (Fm). One purpose for this is to show that (16.2.34) holds even for imaginary values of y. The (rough) principle seems to be that, when one can work directly with the Dyson series, all y e C+ can be included. Note that, in the proof of Theorem 16.2.3, Dyson series do not appear; the key step in that proof is the use of the dominated convergence theorem for the Wiener integral. A second reason for including the discussion to follow is that it may help the reader, as it did the writers, to develop further intuition, not only for such limiting arguments, but also for some of the new phenomena arising in this theory, such as the appearance of powers of the potential evaluated at fixed times. Because of the norm estimate given in Remark 15.3.5 for the tail of a Dyson series, it suffices to show that the nth term of the Dyson series for Kty (Fm) converges strongly to the nth term of the Dyson series for Kty (F). Let U e L2(Rd). In view of (16.2.34), we must verify that, as m —»• oo,
Now,
as (s1 , . . . , sn) -»• (T, . . . , T), with (s1, . . . ,sn) e An. Hence, given e > 0, for large enough m,(s1, . . . , sn) E An n [T — ( 1 / m ) , r + ( l / m ) ] n implies that
For m so large that (16.2.37) holds, we can now use properties of pm and write:
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Hence (16.2.35) is established and (16.2.34) holds for all y e C~+. Remark 16.2.10 The proof above reveals rather nicely some of the intuitive ideas underlying the present theory. Taking m large forces all n interactions represented in the expression for LU in (16.2.36) to take place very close to T and thus leads to [0(T)]n in the limit. All the intermediate free evolutions are forced close to the identity operator. The n! in the first expression of (16.2.38) may appear at first glance to be troublesome. However, using the fact that the Lebesgue measure of [r — 1/m,r + 1/m]n is n! times the Lebesgue measure of [T —1/m,r+1/m]nn An, one sees further on in (16.2.38) that the initial n! is just what is needed. By considering more elaborate examples in a like manner, we could presumably acquire a deeper understanding of the expressions giving our generalized Dyson series in the previous chapter and, in particular, of the appearance of powers of the potentials as well as of the values of the combinatorial coefficients. We next indicate how Theorem 16.2.6 can be used to obtain a new expression for Kty (F) in the case when n is an arbitrary Borel measure (compare Example 15.5.7). Example 16.2.11 (Approximation of an arbitrary n by measures with finitely supported discrete part) Let n e M(0, t) and let 6 satisfy the hypotheses of Theorem 16.2.6; thus 8 is everywhere defined, Borel measurable and bounded by B. As in Example 15.5.7, we may write n in the form
Note that the times rp in (16.2.39) are not necessarily ordered. Let nm in M(0, t) be defined by
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473
For our purpose, it is convenient to rewrite vm in the form vm = where for a given m, ii is the permutation of {1, . . . , m} such that rpi(o) := 0 < Observe that if {tp}8p=1 is an increasing sequence, then TT is the identity for any positive integer m. Clearly, nm converges to n in norm since || nm — n || =£8p=m+1|wp| —> O as m -*• oo. It thus follows from Theorem 16.2.6 and from (15.5.5)-(15.5.7) in Example 15.5.1 (with h replaced by m) that
as m -»• oo, uniformly in y on compact subsets of C+. Here,
with
and, for (s1, . . . , sqo) e A go; j 1 ,...,j m+1 and r = 0, . . . , m,
Moreover, for all y > 0, (16.2.17)-(16.2.19) of Theorem 16.2.6 as well as the computation of \\nm - n|| above give the norm estimate where we have set
with
We can see directly from the above or alternately, from the proof of Theorem 16.2.6, that cm -> 0 as m —> oo.
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Remark 16.2.12 (a) If we add the assumption that 9 in Example 16.2.11 is continuous, then equation (16.2.15) of Proposition 16.2.4 holds since ||nm — n|| —> 0 certainly implies that nm —*• n. (b) The nonuniqueness of the disentangling process was discussed in Section 15.7, and examples were given or referred to. (Compare, for example, formulas (15.5.16) and (15.5.18).) Example 16.2.11 in conjunction with Example 15.5.7 illustrates this nonuniqueness once again. Example 16.2.11 above was based on Theorem 16.2.16. However, it is possible to treat that example under weaker assumptions than are made in Theorem 16.2.6 as the exercise below shows. Exercise 16.2.13 Let n and nm be as in Example 16.2.11 and suppose that 9 e L001;n. Further assume that / is analytic in a circle centered at 0 with radius greater than (a) Show that for X > 0, (16.2.45) and (16.2.46) hold but with Mm given by (b) Show that \\Kty (F m ) - Kty ( F ) | | -> 0 as m -*• oo uniformly in y on all compact subsets of C+. [Hint: Review the proof of Theorem 16.2.6 and then begin by showing thatfor any x e Ct,
Actually, we will see in Theorem 17.6.27 ([La18, Theorem 4.2, p. 180]) that the stability result of Example 16.2.11 can be considerably improved, not only by merely assuming that 9 e Loo1;n (as in Exercise 16.2.13 above), but more significantly, by also obtaining an estimate of the type of (16.2.45) (with cm —> 0) which is valid for all X in C~+and hence, in particular, in the diffusion as well as in the quantum-mechanical case. (See Theorem 17.6.27 and Remark 17.6.29(a).) We finish this section by discussing how our weak convergence result can be strengthened in ways which are relevant to the two examples which made use of it. In Theorem 16.2.3 and its applications, namely Examples 16.2.7 and 16.2.9, we have used the fact that nm -± n implies that f(0,t) b(s)nm(ds) -> f(0,t) b(s)n(ds) for every bounded continuous function b on (0, t). This implication comes immediately from the definition of weak (or vague) convergence (see Definition 16.2.1). However, stronger implications of this kind are known. We adapt one part of [Bil, Theorem 5.2, p. 31] to our setting: If nm -^ n and b is a C-valued, bounded, Borel measurable function on (0, t) with \n\(D(b)) — 0, then f(0,t) b(s)nm(ds) ->• f(0,t) b(s)n(ds), where D(b) denotes the discontinuity set of b. We now give a result which is more restrictive in some respects than Theorem 16.2.3 but which makes a less restrictive continuity assumption. The proposition to follow is
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475
motivated by Example 16.2.7 where the potential was time independent and the limiting measure was Lebesgue measure. Proposition 16.2.14 Let 0 : Rd ->• C be Borel measurable, bounded by c and continuous Leb.-a.e. on Rd. Let n, nm, m — 1, 2, . . . , be in M ( 0 , t) and suppose that |n| is absolutely continuous with respect to Lebesgue measure l on (0, t). (We write|n| « /.) Assume that
Put
where f is an entirefunction given by (15.3.13). Let Fm be defined as in (16.2.49) except with n replaced by nm. Then
uniformly in A on all compact subsets of C+. Proof The only step in the proof of Theorem 16.2.3 which does not carry over immediately to the present situation is the limit taken in (16.2.6). There the definition of weak convergence was invoked and the limit existed for every x e Ct0. We will use the weak convergence result adapted from [Bil] and stated above and establish convergence for m-a.e. x e Ct0. This slightly weaker form of the limit causes no problems in finishing the proof as before. We will show that for every y > 0 and every £ € Rd,
for m-a.e. x e Ct0.This will follow immediately from the adapted weak convergence result if we can show that for m-a.e. x 6 Ct0, the function s >- 0 ( y - 1 / 2 x ( s ) + E) is continuous |n|-a.e. But this last function is continuous except at values of s such that y - 1 / 2 x ( s ) + E € D(0), where D(0) denotes the discontinuity set of 9 (a subset of Rd). But y-1/2x(s) + £ e D(9) if and only if x(s) e X 1 / 2 [ D ( 6 ) - £]. Now D(0) has Lebesgue measure 0 by assumption, and since every translate and every scaling of a set of Lebesgue measure 0 has the same measure, we see that the set y1/ 2 [D(0) — E] also has Lebesgue measure 0. It follows then from Corollary 3.3.4 that for every 5 6 (0, t), m({x e Ct0 : x(s) E y1/ 2 [D(0) - £]}) = 0. Using this and applying the Fubini theorem to the l x m-measure of the set {(s, x) e (0, t) x Cto : x(s) E y 1 / 2 [ D ( 0 ) - E ]}, we see that for m-a.e. x e Ct0, l([s e (0, t) : x(s) e y 1 / 2 [D(0) -£]})= 0. (Also see Corollary 12.1.5.) But |n| «l by assumption and so for m-a.e. x e Ct0, |n|({s e (0, t : x(s) € A 1/2 [D(0) - £]}) = 0. Hence, for m-a.e. x e Ct0> the function s K> 0(y-1/2x(s) + £) is continuous |n|-a.e. This concludes the proof as we observed above.
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We will not state a formal proposition related to Example 16.2.9, but we will indicate briefly how Theorem 16.2.3 can be adjusted so that much weaker continuity assumptions on 9 are sufficient. The key as in Proposition 16.2.14 is to find a replacement for the limiting argument justifying (16.2.6). Recall that the limiting measure in Example 16.2.9 is the Dirac measure dT. Accordingly, for every y > 0 and £ e Rd, we need to have the function s i-> 0(s, y - 1 / 2 x ( s ) + E) continuous at r for m-a.e. x € Ct0. A condition which is sufficient for making this argument in a manner similar to that in the proof of Proposition 16.2.14 is that the r-section of the discontinuity set of 0(., .) has Lebesgue measure 0 in Rd.
17 THE FEYNMAN-KAC FORMULA WITH A LEBESGUE-STIELTJES MEASURE AND FEYNMAN'S OPERATIONAL CALCULUS 17.1 Introduction Our goal in this chapter is to describe (and partially prove) results of Lapidus in [Lai 4-18] in which he obtained a natural extension of the Feynman-Kac formula (from Chapter 12) to the framework of the authors" memoir [JoLa1] studied in Chapters 15 and 16; namely, a "Feynman-Kac formula with a Lebesgue-Stieltjes measure" (FKLS, in short). As was discussed in Section 7.6 and in much greater detail in Chapter 12, the classical Feynman-Kac formula expresses the solution of the heat (or diffusion) equation (with potential V) as a functional integral of
with respect to the Wiener process; here, 6(x(s)) stands for the potential 6 (= —V, in the notation of Chapter 12) evaluated along the path x = x(s) between times 0 and t. Following [Lal4-18], we now investigate what happens if in the Feynman-Kac functional (17.1.1), one performs the time integration with respect to a Lebesgue-Stieltjes measure n(ds) rather than ordinary Lebesgue measure l(ds) — ds. More precisely, we study the time evolution of the operator u(t) associated through path integration (followed by analytic continuation and passage to the limit) with the exponential functional
where n = u+ v is an arbitrary (C-valued and thus bounded) Borel measure on a bounded interval containing [0, t], with continuous part u and discrete part v. (In our present context, FK,n, can be called the "Feynman-Kac functional with Lebesgue-Stieltjes measure n".) Reversing the original procedure, we look for an evolution equation (in integrated or differential form) satisfied by the function u = u(t). [For mathematical reasons, the integrated form of the evolution equation will be the most suitable for our purpose. Further, the assumptions on 6 will be of the same nature as in Chapters 15 and 16. Consequently, they will be far more restrictive than in Chapter 12; on the other hand, 6 will be allowed to be time-dependent and C-valued. In addition, (most of) the results of [Lal4-l 8] presented here will be valid both in the diffusion and the quantum-mechanical case.]
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Actually, we show that u obeys a certain Volterra-Stieltjes integral equation (of which u = u(t) is the unique bounded solution). We then deduce from this key fact a number of related results, some of which will serve as a bridge between Chapters 15 and 16, on the one hand, and Chapter 18 on the other hand, which will take into account more explicitly the noncommutative aspects of Feynman's operational calculus within the context of path integration. [In Chapter 19 we will investigate in a more abstract setting the relationships between Feynman's operational calculus and certain evolution equations. Part of the latter work is similar in spirit but quite different in detail from that presented in the present chapter. (In particular, compare Sections 19.5 and Section 17.3.)] New phenomena will appear in this setting, connected in particular with the discrete part v of n; for example, u = u(t) will usually have a (multiplicative) time-discontinuity at every point in the support of v. [Of course, when n = u = Lebesgue measure = l and thus v = 0, the (operator-valued) function t \-* u(t) will be (strongly) continuous and the above integral equation yields—or rather, is "formally equivalent to" (in the sense of [CaSt1,2] and [JoSk2,6])—the heat or the Schrodinger equation in the diffusion or quantum-mechanical case, respectively.] We note that when specialized to the quantummechanical case, this will enable us, in particular, to study and write in terms of simpler expressions (as in formula (17.2.11) below) the [analytic (in mass) operator-valued] Feynman integral associated with a broader class of exponential functionals than in the standard situation where n = l. For the simplicity of exposition, we will present these results (and their possible physical interpretation) in two steps: First, in Sections 17.2-17.5, in the case when v is finitely supported (say, v = Ehp=1 wPdT P , with wp e C and a < r1 < . . . < rh, < b), as in [Lal4,15]. Then, in Section 17.6, in the general case when n = u + v is arbitrary (say, v = E8p=1 WpdTp, with£8p=1|wp|< oo), as in [Lal6,18]. More specifically, we will first obtain in Section 17.6 a form of the integral equation that is valid for an arbitrary measure n, as in [La 18]. Then, as in [La 16], we will obtain (in the quantum-mechanical case) a product integral (or "time-ordered exponential product") representation of the solution to this evolution equation, as well as a corresponding distributional differential equation valid for an arbitrary n (possibly with a nontrivial singular part). We will derive (in Section 17.3) the integral equation in the prototypical case when n = u + wdr (as in Section 15.3) and refer to [Lal5] for the general case when v is finitely supported, studied in Section 17.2. Furthermore, we will (mostly) refer to [Lal6, 18] for the proof of the results stated in Section 17.6. We stress, however, that both when v is finitely or infinitely supported, the generalized Dyson series (GDSs) from [JoLa1] obtained in Chapter 15 play a crucial role in deriving the results of [Lal4-l 8], especially in the quantum-mechanical case. Notation and hypotheses Let a, b be real numbers such that a < b. [In the following, we adapt in the obvious manner the notation from Chapter 15 (particularly, Sections 15.2.F-15.2.I).] Let M([a, b]) denote the space of complex Borel measures on [a, b]. [Note that in contrast to Section 15.2.F (where a — 0), we allow the measures to have mass at the endpoints of the interval.]
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Let 0 = 0(s, £) be a complex-valued Borel measurable function on [a, b] x Rd. Given n in M = M([a, b]), we say that 9 belongs to Lool;n = Lool;n(a, b)) if
where |n| denotes the total variation measure of n. It follows from (17.1.3) that for n-a.e. (almost every) sin[a, b], 0(s, •) is in L°°(Rd), and that 0(s), the multiplication operator with the function 6(s, .), belongs to £(L 2 (R d )) and has norm ||0(s)|| = ||0(s, .)||oo. Recall that if 0 = 0(s, £) is bounded or if 6 — d(s) is strongly continuous on [a, b], then 0 € Lool;n. Finally, C+ and C~+ denote, respectively, the complex numbers with positive real part and the nonzero complex numbers with nonnegative real part. Given t e [a, b], let Ct = Ca,t = C([a, t], Rd) be the space of continuous functions from [a, t] to Rd. Then the Wiener space Ct0 = C 0 ( [ a , t], Rd) is the set of paths x e Ct such that x(a) = 0, equipped with Wiener measure m. (See Chapter 3.) Fix t € (a, b). Given a function F : Ct -> C, U e L 2 (R d ), and y > 0, we consider (as in formula (15.2.8)) the expression
In Chapter 15 (Theorem 15.4.1), we have established that the operator-valued function space integral Kty (F) exists for all y e C~+, in the sense of Definition 15.2.1, and is "disentangled" by a time-ordered perturbation expansion (or GDS). [Recall from Definition 15.2.1 that Kty (F) e L(L 2 (R d )) is given by (17.1.4) for y > 0, defined by analytic continuation to y e C+ and then extended by strong continuity to A e C~+. Further, when y is purely imaginary, Kty (F) is called the analytic (in mass) operator-valued Feynman integral of F.] These results (from [JoLa1]) are valid for a broad class of functionals, including those we are about to introduce (in (17.1.5) and (17.1.7)). Let n e M([a, b]) be an arbitrary complex Borel measure on [a, b] and let 9 6 £oo1;n([a, b)), as above. We consider the exponential functional
associated with the "potential" 0 and the "Lebesgue-Stieltjes measure" n. It is mathematically convenient and physically natural to work with the Hilbert adjoint Kt/y (F)* ofKt/y(F) , rather than with Kty (F) itself (see Remark 17.3.5 below); here, I (resp., F) denotes the complex conjugate of y (resp., F). Note that Kt/y (F)* depends on time t both through the definition of F in (17.1.5) and that ofKt/y(F) in (17.1.4). We thus set, for y e C~+ and t € (a, b],
It is understood that u(a) = I, the identity operator.
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Equivalently, sinceKt/y(F)* = Kty (F) by Theorem 15.7.6 (formula (15.7.12)), we can also define u(t) in a simpler manner by
where F denotes the "time-reversal" of F (see Definition 15.7.5); namely, for x e C',
where 9 and n denote the "time-reversal" of the "potential" 9 and the measure n, respectively. Recall from (15.4.24) (with [0, t] replaced by [a, t]) that 0(s) = 0(t - s + a) for s € [a, t] and that n is the image measure of n under the time-reversal map s i-> t — s + a from the interval [a, t] onto itself. (We know from Remark 15.3.7 that 9 € Loo1;n implies that 9 € Loo1;n Further, note that in (17.1.6), we have slightly modified the definition of u(t) given in [Lal4-18], for reasons discussed in Remarks 17.3.5(b) and 15.7.7(b). However, this will change neither the proofs nor the statements of the results of [Lal4-18].) Finally, as in Chapters 15 and 16, we will refer to the case when y = 1 and 9 = — V (resp., y = —(' and 6 = —iV), with V e Loo1;n, as the probabilistic or diffusion (resp., quantum-mechanical) case. 17.2
The Feynman-Kac formula with a Lebesgue-Stieltjes measure: Finitely supported discrete part v Whereas the study in Chapters 15 and 16 [JoLa1] was conducted for fixed time t, we are interested here in the behavior of u(t) as time varies. One of our main goals is to extend the Feynman-Kac formula to the present setting, as well as to obtain results that are valid both in the diffusion and quantum-mechanical cases. Let n = u + v be the unique decomposition of n into its continuous part u and its discrete part v. This decomposition arises naturally in the study of the generalized Dyson series (Chapter 15). We assume in the present section and in Sections 17.3-17.5 that v is finitely supported; specifically, we suppose that
Integral equation (integrated form of the evolution equation) We first state our central result in this case ([Lal5, Theorem 2.1, p. 98]). [In the following, unless otherwise specified, we always assume that y e C~+ is arbitrary. Hence, for example, the integral equation (17.2.2) and the differential equation (17.2.3) below hold for all y e C~+.]
MEASURE WITH FINITELY SUPPORTED DISCRETE PART v
481
Theorem 17.2.1 For each fixed p = 0, . . . , h, the operator-valued function u satisfies the Volterra-Stieltjes integral equation
for all t € (Tp,T p+1 ]. The integral in (17.2.2) converges in the strong operator topology (or is a strong Bochner integral ). Moreover, there is a unique bounded (strongly measurable) function u : [a, b] —> £(L2(Rd)) which is a solution of the integral equation (17.2,2) on each interval (rp, Tp+1], with p = 0, . . . , h. In Section 17.3, we will establish this integral equation in a simple but prototypical case (77 = u + wdr) and refer to [Lal5, §4] for its proof in the general case stated above. [The last statement of Theorem 17.2.1 (concerning the uniqueness of the solution) will be proved in Section 17.5.] Differential
equation (differential form of the evolution equation)
The next two statements follow easily from Theorem 17.2.1. However, the formulation of Theorem 17.2.2 and Corollary 17.2.3 will perhaps seem more familiar to the reader than that of Theorem 17.2.1. Theorem 17.2.2 Let p e {0, . . . , h}. Then for Leb.-a.e. t E (TP, T P +1),
wheredu/ds(t) denotes the Radon-Nikodym derivative of u with respect to Lebesgue measure l. (That is, for each U in D(H0), the domain of HO, u(t)U is differentiable for Leb-a.e. t and the differential equation (17.2.3) holds when applied to U.) Consequently, in the diffusion or quantum-mechanical case, respectively, u satisfies the heat or Schrodinger equation with potential (du/ds)(t)V in each open interval (TP, Tp+1), p = 0, . . . , h, as is stated in the following corollary. Corollary 17.2.3 In the probabilistic case (i.e. y = 1 and 6 — —V with V e Loo1;n)> (17.2.3) becomes the heat equation
with potential V\ := ( d u / d s ) ( t ) V . On the other hand, in the quantum-mechanical case (i.e. A = — i and & = —iV with V e Loo1;n), (17.2.3) yields the Schrodinger equation
with potential V\ defined as above; note that V1 e
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If, in addition, /z = / and 9 = 0(s) is strongly continuous (for instance, if V is time independent and bounded), then V[ = V and the differential equation(17. 2.4) or (17. 2.5) holds everywhere (not just l-almost everywhere) in each open i n t e r v a l ( r p , r p interval. Warning: The logically correct form of the differential equation is of a distributional nature; for simplicity, however, we postpone to Section 17.6 (specifically, to Theorem 17.6.15) a precise discussion of this point (in the quantum-mechanical case) and refer to [Lal6, §2.B] for the details. (See also Remark 17.2.8(d) below.) For this reason, the integrated form of the evolution equation (Theorem 17.2.1)— although possibly less intuitive to the reader— is to be preferred in the mathematical exposition of the theory. Discontinuities (in time) of the solution We next see that the solution of the integral equation (17.2.2) (or of the differential equation (17.2.3)) will usually have a discontinuity at a point rp of the support of v. (This will be further discussed in Section 17.4.) Theorem 17.2.4 The function u is strongly left continuous in (a, b]. Moreover, for p = 0, . . . ,h, u is strongly continuous in (rp, t p +1] and
where U(TP+) denotes the strong limit of u(t) as t —> rp, t > TP. Remark 17.2.5 In (17.2.2), (17.2.6), and thererafter, we set a)0 = Qifr0:=a< r\ and (DO = a>\ if TO = T\. It is worth while singling out the special case where r] is a continuous measure (i.e. Corollary 17.2.6 (Purely continuous measure: rj = /j.) (i) Assume that r] = /u, is a continuous measure. Then, for all t 6 [a, b],
Moreover, u is strongly continuous in all of [a, b] and satisfies the differential equation (17.2.3) l-almost everywhere in (a, b). (ii) Suppose further that r/ — (j. = I (so that d/j,/ds)(t) = 1). Then, in the probabilistic case, u satisfies the heat equation
while in the quantum-mechanical case, u obeys the Schrodinger equation
MEASURE WITH FINITELY SUPPORTED DISCRETE PART v
483
When 77 = [i — I as in part (ii) of Corollary 17.2.6—or, more generally, as commented upon in Section 14.5, in the special case of part (i) when r/ = /x is absolutely continuous with respect to Lebesgue measure /—the integral equation (17.2.7) follows from that obtained in the paper by Johnson and Skoug [JoSk6] (the p — p1 = 2 case). (Also see the earlier related work by Cameron and Storvick, beginning with [CaStl], as well as the relevant comments in Section 14.5 above.) Equation (17.2.8) yields the classical Feynman-Kac formula (Chapter 12, especially Theorem 12.1.1 and Corollary 12.1.2), which expresses the solution of the diffusion equation in terms of a Wiener integral. (Note that in Chapter 12, the Feynman-Kac formula was stated for a time independent potential V, but under much more general assumptions on V.) On the other hand, equation (17.2.9) leads to one interpretation of the Feynman path integral (see Chapter 15, especially Definition 15.2.1 and the comments following it). Strictly speaking, as was previously stressed, it would be more appropriate in the present context to refer to the probabilistic (resp., quantum-mechanical) case, of the evolution equation in its integrated form (17.2.7) rather than in its differential form (17.2.8) (resp., (17.2.9)). In view of (17.2.2)-(17.2.9), it is thus justified to refer to the above results from [Lal5]—and, strictly speaking, to the probabilistic case of Theorem 17.2.1—as a "Feynman-Kac formula with a Lebesgue-Stieltjes measure" (in short, FKLS). We emphasize that the procedure used in [Lal5] (or [Lal4]) is the reverse of the usual one, since, starting with a Wiener functional [given here by formula (17.1.7)], we find the corresponding differential or integral equation. We note that the approach to evolution equations (via Feynman's operational calculus [Fey8]) which we will use in Chapter 19 (based on [dFJoLal,2]) is in the same spirit as in the present chapter. One begins in Chapter 19 with an exponential function of sums of noncommuting operators. After "disentangling", one then obtains an expression which can be shown (see Section 19.5) to satisfy an evolution equation. Propagator and explicit solution The fact—stated at the end of Theorem 17.2.1 and established in Section 17.5—that the integral equation (17.2.2), and a fortiori (17.2.7), has a unique bounded solution, enables us to introduce the following operator. Given a < t\ < ?2 £ b, let P f a , t\) be the propagator for the integral equation (17.2.7); that is, P(t2,t1) is the operator in £(L 2 (R d )) that maps the solution u(t\) at time t\ of (17.2.7) onto the solution u(t2) at time ti. Clearly, for a < t\ < 12 < (3 < b,
Note that, by definition, u(t) = P(t, a) is the solution at time t of (17.2.7) when rj = /j.; further, we stress that by definition of (17.2.7), P(ti, t\) depends on IJL but not on v. We are now able to describe the combined effects of the continuous part n, and the discrete part v of n. (See [Lal5, Theorem 2.4, p. 99].)
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FEYNMAN-KAC FORMULAS WITH STIELTJES MEASURES
Theorem 17.2.7 Let rj be as above. Then
Remark 17.2.8 (a) Equations (17.2.2), (17.2.6) and (17.2.11) suggest that the Dirac mass opSrp in the discrete part of n corresponds to a "shock" or "instantaneous interaction" occurring at time rp. We will elaborate this interpretation in Section 17.5. For now, we simply mention that it is physically reasonable to assume that v is finitely supported, since only a finite number of events (for instance, shocks or scatterings) should occur within the time interval [a, b]. Moreover, from a mathematical point of view, we shall find it useful in Section 17.6 to approximate —as was done in [La 18] —an arbitrary measure r\ by measures with finitely supported discrete part. (See Theorem 17.6.27 below.) (b) The adjective "strong" in the statements of Theorems 17.2.1 and 17.2.4 means that the integrability, or limit, holds when the corresponding expression is applied to an arbitrary vector i/r e L2(Rd). (c) When u — l, (17.2.7) is nothing but the standard Duhamel integral equation (see, for example, [JoSk6, §57 in conjunction with (Lal2, pp. 58-60] or [La9]). If, in addition, the potential is time independent, then
in the diffusion or quantum-mechanical case, respectively. Note that since V must then be (essentially) bounded, the Hamiltonian HQ + V has the same domain as HQ; namely, (d) We stress that the measure /u, need not be absolutely continuous. It could be, for instance, the singular Lebesgue-Stieltjes measure associated with the Cantor function, as was considered in Example 15.5.3; we will return to this situation in Section 17.4. As we will see, the integral equation (17.2.2) is then more informative than the differential equation (17.2.3) [since dfi/dl = 0 l-a.e.]. Alternatively, one can replace (17.2.3) by a distributional differential equation, established in [La16, Theorem 2.5, p. 286] and discussed in Section 17.6 below. (See Theorem 17.6.15.) We conclude by considering the special case when r\ is discrete (i.e. n = 0). Corollary 17.2.9 (Purely discrete measure: n = v) (i) Assume that r\ = v is a discrete measure given by (17.2.1). Then, for p € {0, . . . , h] and t € (TP, Tp+i],
MEASURE WITH FINITELY SUPPORTED DISCRETE PART v
485
(ii) Suppose in addition that a = 0, b = t, rp = p(t/h), and u>p = TP — TP+\ = t/h, for p = 1,..., h; so that r\ = (t/h) Yjp=i &p(t/h)- Then, for time independent potential, we have in the probabilistic case (i.e. A. = 1 and 9 = — V)
and in the quantum-mechanical case (i.e. X = —i and 6 = —i V)
We recognize in ( 1 7.2. 14) or ( 1 7.2. 1 5) the familiar hth Trotter product for semigroups or unitary groups, respectively. The link with the Trotter product formula (Section 11.1) is discussed in Example 15.5.5 and especially in Example 16.2.7 above. More generally, the connections with product (or time-ordered) integrals (pointed out in [La 14]) are developed in [Lal6]. We will briefly describe them in Section 17.6. See especially Theorems 17.6.10, 17.6.12 and 17.6.20, which deal with the quantum-mechanical case and refer to the counterpart of the operator u (t) in the so-called Dirac (or interaction) representation. Remark 17.2.10 Note that in (17.2.13)— and in (17.2.11)— the expression for u(t) = K ( ( F ) appears in the "natural physical ordering" (as discussed in Remarks 15.3.7 and 15.4.4, as well as in Section 15.7). Of course, (17. 2.1 3) follows from (17. 2.11) [by letting H = 0 and hence P(ti, t\) = e -('2-n)flbWjf] while (17,2.14) and (17.2.15) follow from (17.2.13) and Theorem 15.7.6 [or more simply, by replacing 9 and r\ = u by their "time-reversals" 6 and rj, as defined by (15.3.23)]. Between the two extreme cases of a purely continuous and a purely discrete measure treated in Corollaries 17.2.6 and 17.2.9 lies a whole range of possibilities described by Theorems 17.2.1, 17.2.2, and 17.2.7. Exercise 17.2.11 Rewrite the (classical) Feynman-Kac formula obtained in Chapter 12 (Theorem 12.1.1 and Corollary 12.1.2) in the form of an integral equation (rather than of a differential equation, as in (12.1.5)). [Hint: Your answer should be analogous to (17.2.7). Recall that in Chapter 12, Y\ = //, = /, A. = 1, and 0 = — V, with V time independent and satisfying the hypotheses of Theorem 12.1.1.] We now outline the contents of the remainder of this chapter. We begin in Section 17.3 by deriving the integral equation (Theorem 17.2.1) in the relatively simple case when v = a>8r. We then sketch (in Problem 17.3.6) the proof of Theorem 17.2.1 for an arbitrary finitely supported v. [This general proof of Theorem 17.2.1 (detailed in [Lal5, §4]) is interesting in its own right because of its rich combinatorial structure and because it shows that the generalized Dyson series introduced in Chapter 15 [JoLal] can be very useful computational tools. We have chosen, however, to omit it here.] In Section 17.4, we analyze the discontinuities due to the discrete part v (Theorem 17.2.4) and explain heuristically how they are connected with a change of initial condition in the differential equation. In Section 17.5, we establish the "explicit" expression obtained in Theorem 17.2.7.
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FEYNMAN-KAC FORMULAS WITH STIELTJES MEASURES
We also provide physical interpretations of our results in both the quantum-mechanical and probabilistic cases as well as suggest further connections between Chapters 15-16 [JoLal], the present work [Lal5], and Feynman's operational calculus. Finally, in Section 17.6, we consider the general case of the FKLS (derived in [Lal8] and [La 16]) when r\ is an arbitrary Borel measure [and hence u may have countably many (possibly unordered) points in its support]. We state, in particular, an integral equation [Lal8], a corresponding distributional differential equation [Lal6], as well as (in the quantum-mechanical case) a product integral representation [La 16] of the solution u(t) = K[(F) (and of the associated "scattering operator"). We will refer the reader to [La 18, La1 6] for the detailed proofs of these results. The basic idea of the proof of the integral equation, however, consists in applying the FKLS when v is finitely supported (Theorem 17.2.1), rewriting it in a way that does not depend on the ordering of the times TP, and then passing to the limit in a suitable way; a sketch of this proof will be provided at the end of this chapter. 17.3 Derivation of the integral equation in a simple case (n = n + w5 r ) We give a detailed proof of Theorem 17.2.1 (apart from the uniqueness statement which will be established in Section 17.5) in the prototypical case n = u + co8T, as in [Lal5, §3]. Hence we assume that
where u is a continuous measure, w E C, and r e (a, b]. At the end of this section, we shall then briefly explain how to deal with the additional combinatorial complications encountered in the case when the discrete part of n is finitely supported (see Problem 17.3.6). (The complete proof in that situation can be found in [Lal5, §4].) We stress that whether in the case treated here or in the general case when v is finitely supported, the results from [JoLal] on generalized Dyson series (GDSs) obtained in Chapter 15 are crucial to the proof of the integral equation (see Theorem 17.3.1 below and part (a) of Problem 17.3.6). Our starting point is the following result, which in view of (17.1.6), (17.1.7) and Theorem 15.7.6, is a consequence of Theorem 15.3.1 as well as Corollaries 15.3.4 and 15.3.6. We purposely use more precise notation than in Chapter 15. [Observe that each CDS below is written according to the "natural physical ordering", as will be further explained in Remark 17.3.5 below. This is precisely where we use the fact that u(t) = K[(F) (rather than K [ ( F ) ) , where F denotes the time-reversal of F. See Remarks 15.3.7 and 15.4.4, as well as Theorem 15.7.6.] Theorem 17.3.1 For all A e C+, u(t) = K[(F) is disentangled (time-ordered) by the following perturbation expansion or CDS: For t e (T, b],
DERIVATION OF THE INTEGRAL EQUATION WHEN n = fjt + toS,
487
where for 0 < j < k < n,
Here,
and for (si,...,Sk) € &k;j(t),
On the other hand, for t e [a, T],
where
Here,
and for(si,...,sn) € A n (f),
In (17.3.2) or (17.3.6), the series is summable in £(L 2 (R d )) and converges in operator norm, uniformly for t in (T, b] or (a, T], respectively; further, the integrals in (17.3.3) or (17.3.7) are strong Bochner integrals. Moreover, for all t € [a, b], we have
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FEYNMAN-KAC FORMULAS WITH STIELTJES MEASURES
Remark 17.3.2 (a) For j = k, Cn,k-j in (17.3.5) begins with e-(t-T)(Ho^[e(T)]n-k, whereas for j = 0 it ends with [e(T)]n-ke~(r~a)(Ho/^; in fact, in (17.3.4), we have by convention a < s\ < •• • < Sk < T or T < s\ < • • • < SK < t for j = k or 0, respectively. Further, in (17.3.3), the notation /u,®* stands for the product measure of k copies of n, as in Chapters 15 and 16; of course, the integral in (17.3.3) could equivalently be written in iterated form. (See Remark 15.2.10.) (b) Assume that r\ = ^ = I. Then, in the quantum-mechanical case, (17.3.6) yields the classical Dyson series (see formula (15.3.22)). Further, if 9 is time independent, then K [ ( F ) = K[(F)[=KL(F)*, by (15.7.12)]. (Actually, u(t) =
(As before, unless otherwise specified, we assume throughout that A. 6 C+ is fixed, but arbitrary.) In the course of the proof of (17.3.11), we shall use the following key result ([Lal5, Proposition 3.1, p. 103]):
DERIVATION OF THE INTEGRAL EQUATION WHEN r, = /x + euSr
489
Clearly,
Now, for sk+i 6 (r, t) and (s\,..., sk) e A* ; ;(s k +i), we have
In fact, by (17.3.5) with t replaced by SK+I,
Hence, if we observe that Q<j
equations (17.3.16) and (17.3.5) yield successively
This proves (17.3.15). (See (17.3.19) below.) Next, for Sk+i € (T, t), we have
In the first equality of (17.3.18), we have used (17.3.3) with f replaced by sjt+i. In the second one, we have taken the given bounded linear operator inside the integral, according to a special case of [HilPh, Theorem 3.7.12, p. 83].
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We note that, by (17.3.4) and Remark 17.3.2(a),
for 0 < j < k. (Observe that j ^ k + 1 implies that sk+\ > r if (s\,..., sk, s^+i) e Aft+i ; y(0-) Consequently,
by (17.3.3). In view of (17.3.13), this yields (17.3.12), as desired. (For notational simplicity, we write jU®*+1 rather than jtt®(*+1) for the product of k + 1 copies of the measure ^.) We now explain the main steps leading to (17.3.20). In the first two equalities of (17.3.20), we have used (17.3.14) and (17.3.18), respectively. In the third one, we have combined Fubini's theorem and (17.3.19). Note that the Fubini theorem for (strong) Bochner integrals [HilPh, Theorem 3.7.13, p. 84] applies here; in fact, for V € L2(R.d),
Further, by (Lemma 15.2.7 and) the "simplex trick" for continuous measures discussed shortly after formula (15.3.8), we have (with A^+j (t) defined as in (17.3.8) above),
since 6 e Lcci;?; implies that 6 e £001-,;* (see equation (15.2.6)). This completes the proof of Proposition 17.3.3.
DERIVATION OF THE INTEGRAL EQUATION WHEN r\ = fj. + (o8T
491
We can now give the proof of Theorem 17.2.1 in the case when rj = & + co8T. (We focus here on the derivation of the integral equation; the fact—stated at the end of Theorem 17.2.1—that this integral equation has a unique bounded solution will be established towards the beginning of Section 17.5.) Proof of Theorem 17.2.1 For t e (T, b], set
As was noted previously, it suffices to show that (17.3.11) holds for t e (T, b], or equivalently,
We first see that
where wn,k\j is defined as in (17.3.13). Indeed, by (17.3.2),
in view of (17.3.13), this yields (17.3.24). The interchange of the order of integral and sum in the last equality of (17.3.25) will be justified at the end of the proof. By Proposition 17.3.3,
Now, by making the change of indices m = n+l and A" = fc + 1 and noting once again that m — K = n — k, we obtain
Observe that we may as well assume that the indices m and K start at 0 instead of 1 in (17.3.27). Hence
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It follows from (17.3.2) and (17.3.28) that
Further, we claim that for t e (T, b],
with uk given as in (17.3.7). In fact, by (17.3.4) and (17.3.8),
for all t e (T, b], and by (17.3.5), (17.3.9) and Remark 17.3.2(a), we have for (si,...,sk) e Ajt ; jfc(r),
Thus by (17.3.3), (17.3.31) and (17.3.7),
This proves (17.3.30). In view of (17.3.29) and (17.3.30),
where we have used (17.3.6) in the last equality; moreover, in the second equality of (17.3.32), we have recognized the Cauchy product of two absolutely summable series
DERIVATION OF THE INTEGRAL EQUATION WHEN rj = n + a>Sr
493
in £(L2(Rd)). Since (17.3.32) yields (17.3.23), we obtain (the integral equation stated in) Theorem 17.2.1 in the present case where rj = n, + a>ST. We conclude this proof by justifying the interchange of sum and integral in (17.3.25). Indeed, for 5 e (r, f),
since A^ = IJ 7=0 &k;j UP to a set of I Ml®* -measure zero (see equation (15.3.7)),
Hence, for ^ € L2(Rd),
and so
In the last equality of (17.3.33), we have used the fact that (see equation (15.2.6))
We may therefore apply Fubini's theorem in (17.3.25). From a strict logical point of view, the following comment should have been made at the beginning of this section.
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Remark 17.3.4 We justify the existence of the strong Bochner integral in (17.3.11) or (17.3.12) by appealing to Bochner's integrability criterion [HilPh, Theorem 3.7.4, p. 80], The (strong) Bochner measurability of the integrand in (17.3.11) or (17.3.12) follows from a simple measurability lemma for Bochner integrals depending on a parameter, analogous to Lemma 15.2.9. (It can also be obtained by applying the recent abstract measurability results in [Jo8, BadJoY] which will be briefly discussed at the beginning of Section 19.4 or else by showing directly, as in [Lal4, §2.2, pp. 11-12], that u or un k-j is continuous on (a, b], except at T, where it has one-sided limits.) Moreover, let us denote by Aa(s) the integrand in (17.3.11) or (17.3.12), with a = 1 or 2, respectively. Then, fora given iff & L2(Wi), the integrability of\\J\fa(s)i/\\ with respect to \/j.\(ds) follows from the norm estimate (17.3.10) or (17.3.21), respectively. We next explain why it is advantageous in this context to work with K[(F) rather than with K[(F) itself. Remark 17.3.5 (a) The GDS for K[(F) and K[(F) differ only by the time-ordering of each summand. (Compare equations (17.3.2) and (17.3.3) with equations (15.3.2) and (15.3.15).) More precisely, according to Theorem 15.7.6, each summand in the series for K[(F) (resp., K[(F)) is written according to increasing (resp., decreasing) time indices as you read from right to left on the page. (Compare equation (17.3.5) with equation (15.3.4).) (This distinction is blurred in the classical case where rj = /u, = I because of the invariance of Lebesgue measure under time-inversion; see Remark 17.3.2(b).) Clearly, the ordering according to increasing time indices in the series for K^(F) corresponds to the natural physical interpretation (see Remarks 15.3.7 and 15.4.4). Further, it is in agreement with Feynman 's time-ordering convention for noncommuting operators, according to which the operator with the smallest time index acts first (see formula (14.2.1)). As we know from Section 15.7, this comment applies to all the generalized Dyson series (and associated Feynman diagrams) treated in Chapters 15 and 16, and in particular to that considered in Problem 17.3.6(a) below. Moreover, in the present case of an exponential functional, the use of F, the time-reversal of F, also enables us to obtain for u(t) = K[(F) a differential or integral equation in a convenient form. (b) In [Lai4-18], the second author had erroneously defined u(t) by K'^(F)', the (Banach) adjoint of K((F), rather than by K[(F) [ = KL(F)*, the (Hilbert) adjoint of Ki-(F), by(15.7.12)] as in (17.1.6). (This is correct in case the parameterX, the potential 6 and the measure r\ are all real-valued, but not in general; see formula (17.1.6a) and Remark 15.7.7(b).) The reason for introducing the adjoint was to handle the situation associated with the "time-reversal" and hence to obtain the GDS in the "natural physical ordering ", as explained in (a) above. [We have already commented in Remark 15.7.7(b) on this problem which led us to introduce (in Definition 15.7.5) the "time-reversal" map F M» F in order to formalize the notion of "time-reversal" for the GDS disentangling K[(F).] However, since u(t) was then precisely given by the physically natural GDS [for example, in the setting of the present section (where r) = ^ + a)Sr), by the GDS in Theorem 17.3.1], neither the statement of the results in [Lal4~18] nor their proof need
DERIVATION OF THE INTEGRAL EQUATION WHEN r\ = n + a>8r
495
to be changed in any way [once, of course, the original definition of u(t) is modified according to (17.1.6)]. Sketch of the proof when v is finitely supported We close this section by briefly commenting on the proof of the integral equation (17.2.2) from Theorem 17.2.1 in the case when the discrete part v is finitely supported. Because of the greater combinatorial complexity of the GDS for u (t ) , the proof of Theorem 17.2.1 is technically much more involved than in the special case studied above; the previous proof should serve as a helpful guide, however. The following problem — which should take longer to solve than most exercises in this book —provides an outline of the proof of Theorem 17.2.1 for finitely supported v. [The interested reader can find a complete proof of the integral equation in [Lal5, §4]; more precisely, the proof of part (b) (resp., (c)) below is given in [Lai 5, Proof of Proposition 4.1, pp. 113-114] (resp., [Lal5, Proof of Theorem 2.1, pp. 114-118]).] Problem 17.3.6 Let n = n + v, with discrete part v given by (17.2.1). (a) Deduce from Corollary 15.4.3 and Example 15.5.1 (as well as Theorem 15.7.6) that for all A. € C+ and all t e (r/,, b], the operator u(t) = K[(F) is disentangled by the following GDS:
where for nonnegative integers qo, q\,..., qh and j\,..., jh+i such that q^-i n and j\ + . . . + jn + jh+i = qo,
Vqh =
(17.3.35) Here, &q0;j\,...,jt,+\ — Aqo'Ji.—Jh+i^ is given by (15.5.6) (with 0 replaced by a), while for (s\ Sq0) e A^,,...,^^), we have £n,go;j\,...,JH+\ = £n,90;h JH+\ (s\,..., sqo; t) := Lh ... L\L$, with Lr defined by (15.5.7) for r =0,... ,h. Further, show that the series (17.3.34) is summable in £(L 2 (R d )) and converges in operator norm, uniformly in t e (r/,, b), while the integral in (17.3.35) is a strong Bochner integral. Moreover, prove that estimate (17.3.10) holds for all t € [a, b]. (b) For t e (T/J, b] and nonnegative integers qo, q\,..., qh and j\,..., jh+\ such that qo H \-qh = n and j\ H h jh + jh+i — qo, set
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Then establish the exact counterpart of Proposition 17.3.3; namely, show that for all t 6 (r/,, b], we have (17.3.37) [In order to understand (17.3.37), it is useful to note that j\ + - • • + jh+ Jh+\ = <1Q and qo-\-----\-qh = n implies that ji-\-----K/ft+C/A+l + l) = qoand(q0+l)-\ ----- \-qh - n+l. Further, if we were to indicate explicitly the dependence on q\, . . . , qh in the left-hand side of (17.3.37), then the same h-tuple (q\, ... , qh) would appear on both sides of (17.3.37).] (c) Use (b) and (a) to show that for p := h, the integral equation (17.2.2) holds for allt € (rh,b]. [Warning: The proof of part (c) is combinatorially more involved and requires several additional steps than its counterpart in the simpler case when r\ = IL + u>Sr.] (d) Conclude from (c) that for every p e {0, . . . . A}, the integral equation (17.2.2) holds for all t e (TP, rp+i], as desired. [Hint: If I < p < h and t € (TP, Tp+i], simply replace the discrete part of r\ by We close this section with a very simple exercise that may, however, be helpful to some readers. Exercise 17.3.7 (a) Write the integral equation (17.2.2) in a more concrete form (valid for any X € C+ and when applied to an arbitrary \jf e L2(R.d) ), not involving explicitly the operator notation. Specialize your result to the diffusion case and then to the quantummechanical case. [Hint: Use formula (15.2.2) or, more precisely, use the integral expressions for the free semigroup (A e C+) given in Theorems 10.2.5-10.2.7.] (b) Looking ahead to Section 17.6, answer the same question as in part (a) for the more general integral equation (17.6.2), valid for an arbitrary Borel measure r]. [Advice: Use in addition formula (17.6.5').] 17.4 Discontinuities of the solution to the evolution equation We establish here Theorem 17.2.4 and discuss, in particular, its relationships with the associated evolution equation. Unless otherwise specified, we assume throughout this Section and the next that rj = ^ + v has finitely supported discrete part v given by (17.2.1), just as in Section 17.2. The time discontinuities We first show that the discrete part v gives rise to discontinuities in the solution u = u(t) of the evolution equation (17.2.2). Proof of Theorem 17.2.4. Fix X € C+ and p with 0 < p < h. By (17.2.2), we have for all t e (TP,-CP+\\,
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497
where we have set
For i < t, we also set
so that by (17.4.2),
Note that because ^ is a continuous measure, we may integrate over (TP, t) instead of [TP, t). Further, since g-O-^XtfoA) js strongly continuous in t, so is f ( s ; t). Clearly, by (17.3.10),
Since the right-hand side of (17.4.3) is |?j|-integrable and a fortiori |/i|-integrable, the Lebesgue dominated convergence theorem for Bochner integrals [HilPh, Theorem 3.7.9, p. 83] easily yields that w is strongly continuous on (rp, TP+\]. By (17.4.1), it follows that u is strongly continuous on (TP, rp+1 ] and, since W(TP+) — w(rp) = 0,that u(r p +) exists and is given by
This establishes Theorem 17.2.4. Differential equation and change of initial condition It is easy to deduce (at least formally) the differential equation (17.2.3) in Theorem 17.2.2 from the integral equation (17.2.2) in Theorem 17.2.1. We now briefly explain how, from the point of view of the differential equation, the discontinuities of u = u(t) established above can be viewed as a change of initial condition occurring at every point TP in the support of v. Our discussion will be more informal than in the rest of this section. First we observe that Theorems 17.2.2 and 17.2.4 can be restated as follows. Fix p e {0,..., h} and \jf e D(HQ); then for /-a.e. t e (TP, rp+\), the function u(t)\fr is differentiable in the norm topology of L 2 (JR rf ) and satisfies the differential equation
with initial condition
(Note that (17.4.5b) holds for all w e L 2 (R d ).)
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Actually, it is of interest to interpret this change of initial condition from ^.probabilistic point of view. Consider the Wiener integral (17.1.4) that defines K[(F)\{s for A. > 0 and iff 6 L2(Rd); here, the functional F is given by (17.1.5). For notational simplicity, we let | = 0 and A, = 1 in (17.1.4); further, we use F rather than its time-reversal F given by (17.1.7). For t e (TP, TP+I] and x e C', we write
As t —»• TP+ and under appropriate assumptions, the expression between brackets tends to
Consequently, as t -*• TP+, F(x)$(x(t)) converges to
where we have set
We thus see that at time rp, the "initial condition" 1/r is replaced by the new initial condition <j) given by (17.4.6). We conclude that under suitable hypotheses and for A, > 0, the analogue of (17.4.4) for K[(F) (or rather, for u(t) = K^(F)) can be obtained in this manner. However, technical difficulties—only partly overcome by use of Vitali's theorem for the convergence of a sequence of analytic functions (Theorem 11.7.1 and Remark 11.7.2(c))—arise when we attempt to establish this result for A. purely imaginary. (This is analogous to the difficulties described at the end of Appendix 11.7.) Recall that the Wiener integral representation of K'^(F) given in (17.1.4) no longer holds for nonreal A. (See, for example, the discussion preceding Remark 15.2.6.) In contrast, the following exercise illustrates the fact that our perturbation expansions (or GDSs) in Chapters 15 and 16 allow us to work directly for any A. in C+ and, in particular, to obtain results valid both in the probabilistic and quantum-mechanical cases. (Of course, we already know that Theorem 17.2.4 is valid for all A e C^ since we have deduced it from Theorem 17.2.1 which was proved by means of these same GDSs.) Exercise 17.4.1 Fix an arbitrary A. in C+. Assume for simplicity that r/ = /u, + a>ST, as in Section 17.3. Then establish directly—by using the GDSs in (17.3.2) and (17.3.6)— Theorem 17.2.4, and in particular, the time discontinuities expressed by (17.2.6) (or (17.4.4)).
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We now comment on the relationships between Theorem 17.2.4 and the various forms of the evolution equation. As we know, the integral equation (17.2.2) obtained in Theorem 17.2.1 provides the mathematically correct form of the evolution equation. For instance, u, could be a singular measure, in which case (dp,/ds)(t) = 0 for /-a.e. t e [a, b]', then, clearly, a great deal of information is lost (in (17.2.3)) and hence (17.2.3) combined with (17.2.6) can not imply (17.2.2). (Of course, this is so even if v = 0.) In order to take into account, in particular, both the singular part of /j, and the time discontinuities caused by v, we would have to replace equations (17.2.3) and (17.2.6) by a single distributional differential equation on [ a , b ] . We refrain from doing this in a rigorous manner at this stage and instead refer to [Lal6, Theorem 2.5, p. 286] which will be stated and briefly discussed in Section 17.6 (see Theorem 17.6.15). However, we mention that we can approximate v by absolutely continuous measures /% with density pm := ]S£/>=i u>pPm;p, where (Pm:p}^=i is an appropriate <5-sequence centered at TP. If we let um(t) = K[(Fm), where Fm is defined as in (17.1.5) except with n replaced by r\m := \L + (j,m, then methods similar to those of Example 16.2.9 show that um(t) —>• u(t) as m —> oo, uniformly for all t € [a, b]. Since by Theorem 17.2.2 (or Corollary 17.2.6), we have for /-a.e. t € (a, b)
one may expect to obtain in the limit a distributional differential equation for u. The method just outlined would probably seem quite natural to a physicist having to deal with a similar situation. Actually, when presented by the second-named author with some of these results, Richard P. Feynman pointed out that (in the quantum-mechanical case) strong interactions between elementary particles could be modeled in a similar manner, although, in practice, it would usually not be necessary to pass to the limit. Exercise 17.4.2 (a) Make the above limiting argument more precise and guess the form of the distributional differential equation obtained in the limit. (b) Verify your guess by taking for u the Cantor-Lebesgue (singular) measure ([Coh, p. 55 and pp. 22—24] and [Mand, pp. 82-83]) and by approximating it by absolutely continuous measures associated with (suitable) step functions. (c) In the quantum-mechanical case, compare your results with Theorem 17.6.15 below ([Lal6, Theorem 2.5, p. 286]), where is provided a suitable distributional differential equation. Further, compare directly your answer with the result obtained in Example 15.5.3; see especially formula (15.5.15). 17.5 Explicit solution and physical interpretations Let u = u(t) = K[(F) be given by (17.1.6), with the functional F as in (17.1.7). In this section, we show that the integral equation of Theorem 17.2.1 has a unique bounded solution, and we establish Theorem 17.2.7, which yields an "explicit" expression for u (t) in terms of simpler operators. We also provide physical interpretations for our results both in the quantum-mechanical and probabilistic case, as well as suggest further connections with Feynman's operational calculus. (The latter subject will be pursued in Section 17.6
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during our brief discussion of the relationships [La 16] between the present work and product integration.) We begin by proving Theorem 17.2.7 for the case of a continuous measure and then treat the more general case where v is finitely supported. Continuous measure: Uniqueness of the solution Recall from Corollary 17.2.6 that when q = /* is a continuous measure, u = u(t) satisfies the integral equation
for all? e [ a , b ] . It is easy to see that (17.5.1) has a unique bounded (strongly measurable) solution, which must therefore equal u in [a, b]. In fact, let v be a solution of (17.5.1) such that ||u(OII < M for all? e [a,£]. Aftern- 1 iterations of (17.5.1), we obtain for t e [a,b]:
where Aa and Ca are given as in (17.3.8) and (17.3.9), respectively. It follows that
Hence
and, by (17.3.6) (and (17.3.10)), v = u on [a, b]. For a < t\ < t% < b, we may therefore consider, as was done in Section 17.2, the propagator P(ti,t\) associated with the integral equation (17.5.1). Recall that P(t2, t\)u(t\) = ufo) and that, in particular,
is the unique bounded (strongly measurable) solution of (17.5.1) at time t e [a, b].
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Measure with finitely supported discrete part: Propagator and explicit solution Let r] = n + v, with v given by ( 17.2. 1 ). With the obvious change of notation, it follows from the above discussion that the integral equation
has a unique bounded solution on the interval (rp, TP+I], given by (We use here implicitly the fact that equation (17.5. 1) holds when applied to an arbitrary vector \lr e L 2 (R rf ).) By applying this result successively on the intervals [TO, r\], [r\, TZ], . . • , [r/,, r/,+i], we conclude that there is a unique bounded function v from [a , b] to C (L 2 (M.d ) ) satisfying (17.5.2) on (TP, Tp+i] for p = 0, . . . , h. (Recall that by convention, TO = a and T/,+I = b.) Moreover, v is given by (17.5.3) in (rp, rp+\] Since, by (17.2.2) in Theorem 17.2.1 and by (17.3.10), u is bounded and satisfies (17.5.2) on each interval (TP, TP+\], it follows that u = u on [a, b]. In view of (17.5.3) and (17.5.4), we thus obtain for A. 6 C+ and p e {0, . . . , h},
for all t e (TP, TP+I]', here, we use for O>Q the same convention as in Remark 17.2.5. Since (17.5.5) is clearly equivalent to (17.2.11), we have established Theorem 17.2.7. The following comment may help the reader gain some further intuitive understanding of (17.5.5) from a probabilistic viewpoint. Remark 17.5.1 Note that the Dirac mass a)p8Tp in the discrete part of ij acts in the immediate present and, according to (17.2.2), (17.2.6) and (17.5.5), influences only the future, whereas, due to the Markov property [Hid, §2.47, the increments of the Wiener process forget the past. (See also Chapter 3.) Recall that for X > 0, K^(F) is given by the Wiener integral (17.1.4) and the exponential function in the definition (17.1.7) of F is multiplicative. Since we have proved that the integral equation (17.5.2) had a unique bounded solution, we may now introduce the associated propagator U. Recall that by definition, U(t2,ti)u(t\) = «(?2) for a < t] < t2 < b. It follows from the above results that U(ti,t\) = I and fora < t\ < ti < (3 < b,
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Moreover, for
where Ct (resp. C'0 ) is replaced by Ct2, t1 = C([t2, t1], Rd) (resp., Cgt2,t1) and F (resp., K t 2 , t 1 ( F ) ) ) is given as in (17.1.7) (resp., (17.1.4)), except with a replaced by t\ and t replaced by t2. In particular, for all t e [a,b],
is the solution of the integral equation (17.5.2) at time t. We point out a special case of interest. Proposition 17.5.2 (Unitary propagator) Assume that u = l, v is real (i.e. wp is real for all p) and V is real-valued. Then in the quantum-mechanical case, U(t2, t1) is a unitary operator for a - t1 - t2 - b. Proof Since ul — l and in view of Corollary 17.2.6, P(t2,t1) is the standard unitary propagator for the Schrodinger equation (17.2.9) with potential V (or rather, for the corresponding integral equation (17.2.2)); see, for example, [Schwel]. Further, our assumptions imply that w p V ( t p ) is self-adjoint and hence that e®?o(tp) = e-iwPv(Tp) is unitary. Since a finite product of unitary operators is unitary, the conclusion follows from Theorem 17.2.7. Remark 17.5.3 Recall that for time independent potentials, the condition 9 € Lool;^ implies that 0 is (essentially) bounded. In this respect, the spirit of the present work is quite different from the previous work in [La6, Lall, BivLa] presented in Sections 11.311.6 and which focused on defining the "modified Feynman integral" for very general potentials. Nevertheless, it would be interesting to know whether the results from [La12] discussed in Section 11.5—or related perturbation results in Section 12.2—can be used in conjunction with Theorem 17.2.7 above (in the case where u = l) to extend some of the present results to unbounded potentials. The following exercise asks the reader to investigate this open-ended question. Exercise 17.5.4 Suppose that u — l and v is given by (17.2.1). Attempt to determine a class of unbounded (time independent) potentials to which some of the results of Section 17.2 (including Theorems 17.2.1 and 17.2.7) can be extended in the diffusion or in the quantum-mechanical case. Remark 17.5.5 For the discussion that follows, it is important to observe that since
0 € Lool;n if and only 9 e Lool;u and 6 ( t p , •) is well defined and bounded for each p e {0,..., h}. Consequently, the values of the potentials w p 6 ( t p ) are completely independent of one another and of that of 0(s), for (u-a.e.) s e [a, b] \ {t1, ..., Th}.
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In fact, we could just as well have given different names to the potentials ( o p 0 ( t p ) and e(s)forse[a,b]\{Tl,...,Th}. We now provide physical interpretations of our results in the quantum-mechanical and probabilistic cases. For simplicity, we assume throughout our discussion that u = l and v = Ehn= w) n S T n , with to = a < TJ < • • • < T/, < b = TA+I . Physical interpretations in the quantum-mechanical case Suppose that A = —i and 9 = — i V with V € Lcci;^- Then P(t2, ?i) is the propagator for the Schrodinger equation (17.2.9) with potential V (or rather, for the corresponding integral equation (17.2.2)). Recall from Proposition 17.5.2 that if V is R-valued, then P (t\, t-i) is a unitary propagator. In particular, if V is also assumed to be time independent, then P(t\, t\ + T) = e~'T<-H°+V) is the unitary group generated by the Hamiltonian H := Ho + V.
Consider a single nonrelativistic quantum-mechanical particle. We propose to interpret (17.5.8) as follows: The particle moves in the potential V between times a and TJ, interacts instantaneously with the potential w V(T\) at time t1, moves (or propagates) in the potential V between times t1 and T2, interacts instantaneously with (or is scattered by) the potential w2 V(t2) at time T2, and so on until a final instantaneous interaction at time tp followed by a propagation between TP and t. If the particle is in the initial stated e L 2 (K d )attimea, we interpret \jf(s) := U(s)tjf as representing its state at time s e [a,b]. According to Theorems 17.2.1, 17.2.2, 17.2.4 and 17.2.7, \js(s) obeys the Schrodinger equation (17.2.9) with potential V during each time interval (r ? , Tq+\), 0 - q - p; for t\, t-i inside this interval, we have i/r(t2) = P(t2,t\)\jr(t\). However, immediately after time rq (in reality, a very short time afterwards), the particle is in the state
As was seen above, the factor e~ lwq v(tq) can be interpreted as the result of a change of phase due to an instantaneous interaction with (or a scattering by) the potential wqV(rq) at time Tq. Similarly, it can also be thought of as being created by a "shock" or a "hit" of some kind. (Strictly speaking, we should use the expression "change of phase" only in the most common case where wq and V(tq) are real; see Proposition 17.5.2 above which then implies that U(t2, t1) is a unitary propagator.) The reason why we may use the term "instantaneous interaction" should be clear by now; we refer, for instance, to the discussion at the end of Section 17.4 (as well as in Example 16.2.9), where we approximated the Dirac mass (wq&rq by absolutely continuous measures with density sharply concentrated about time rq .
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[Caution: The expression "change of phase " used above solely refers to the operators involved, and not to the wave functions (considered as functions of the space variable). Hence it corresponds to an observable physical effect.] The results from [Lal5] described in this chapter discussed here could possibly provide a simplified model for multiple scattering, as we next further illustrate. (We caution the reader that there are several defects with this model, one of which is mentioned below.) Imagine that a particle is projected onto a thick target; it is then subjected to multiple scattering when it collides with the atoms of the target, say at times T\ ,..., TH • When the projectile leaves the target, it is in the state ty(t) = u(t)\lf, with u(t) as in (17.5.8) and with p = h. An obvious problem here is that one does not know how many scatterings occur or at what times they take place; in order to obtain a more realistic model, it may therefore be necessary to use some kind of averaging (or randomization) process. (See Problem 17.6.31(c) and Remark 17.6.32(a) below.) Actually, because of the difficulties involved, experimental physicists try to avoid multiple scattering by using a very thin target; ideally, a single scattering occurs in that case, and from our point of view, we are in the simple situation where n — u + w S T :
Remark 17.5.6 A special case of interest is obtained when V = 0 in [a, b ] \ { t 1 , . . . , th). Then P (t2, t1) = e~l<-t2~'^H°, and the particle propagates freely during the time interval (rq, Tq+\), 0 - q - h. Since V = 0 l-a.e. in [a, b] in this case, everything happens as if n] were purely discrete and equal to v. (See Corollary 17.2.9.) The analogue of this remark holds in the probabilistic case, where P(t2,t1) = e ~ (t2 ~t1 )Ho represents free diffusion. We now turn to the probabilistic or diffusion case; we shall be more concise in our explanations, since, once the proper identifications have been made, the discussion parallels that of the quantum-mechanical case. Physical interpretations in the diffusion case Assume that A = 1 and 9 = —V, with V e Looi.ij- Then P(t2, t1) is the propagator for the heat equation (17.2.8) with potential V (or rather, for the corresponding integral equation (17.2.2)). In particular, if V is time independent, P(t1, t1 + T) = e - T ( H ° + V ) is the heat (or diffusion) semigroup generated by H = HO + V. Since it may not be as well known as in the quantum-mechanical case, we briefly recall the physical meaning of P (t2, t1). (See [Klul, pp. 161-163 and p. 177] for an elegant exposition.) Imagine a solvent which fills the entire space Rd. We suppose that "a substance reacting with [this] solvent spreads through it according to the laws of diffusion" [Klul, p. 16]. The rate of reaction and diffusion is proportional to the amount of substance present, the coefficient of proportionality at time s and location E being equal to V (s, E); with our sign convention, V(s,%) - 0 or - 0 means that the substance is being destroyed or created, respectively. Now, the propagator P(t2, t1) corresponds to the phenomena of reaction and diffusion taking place simultaneously.
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If the substance has T]S e Ll(E.d) n L2(E.d) (ijr > 0) as its initial mass distribution at time a, we interpret ^(*) := M(*)^ as being its mass distribution at time £ e [a, b]. (Note that since V(rg, •) is bounded, by Remark 17.5.5, it follows from (17.5.10) that V^(s) e L1 n L2 and yf(s) > 0; a more natural assumption to obtain this result would bethato> 9 V(r 9 , •) > 0.) According to Theorems 17.2.1, 17.2.2,17.2.4and 17.2.7, VCO satisfies the heat equation (17.2.8) with potential V during each time interval (rp, r p+ i); however, immediately after tq, the mass distribution of the substance is given by
This suggests that at time r?, an "instantaneous reaction" with rate characterized by a>qV(Tg, •), occurs throughout space. No diffusion is then taking place. Somehow, the diffusion is "frozen" at time rq. In the following discussion, we assume that V > 0 and coqV(Tq, •) > 0 for all q e ( l , . . . , / z ) . We now adopt an alternative interpretation of Pfo, t\) in probabilistic terms (see, e.g., [Klul, p. 177]). Consider a Brownian particle which is exposed to a risk of destruction characterized by V. (If at time s the particle is located at £ e Rd, the probability that it will be destroyed within the short time period AT is approximately equal to V(s, £)AT.) Now, in view of (17.5.10), we may interpret the contribution of the discrete part v of of as creating an additional risk of destruction, characterized by wq wq, •) and occurring throughout space at precisely each instant rq. Further connections with Feynman 's operational calculus An interesting aspect of the above results and of the theory developed in [JoLal] (Chapters 15 and 16 above) is that they allow us to blend discrete and continuous structures by use of Lebesgue-Stieltjes measures, as well as to unify concepts which were not previously thought to be directly related. This work can be extended in various directions. In particular, by making more explicit use of the rules of Feynman's operational calculus in place of path integral methods, some of the results of this chapter will be extended to a more general setting in Chapter 19 (see Theorems 19.5.1 and 19.6.1). The techniques of Chapter 19 have a good deal in common with some of those of the present chapter and of Chapter 15. This is not so surprising (after the fact) if one recalls from Sections 14.3 and 14.4 that Feynman's operational calculus for noncommuting operators can be thought of as generalizing some aspects of path integration. There doubtless exist other derivations of the "Feynman-Kac formula with a Lebesgue-Stieltjes measure". This is especially true in the probabilistic case, where we are aware of several alternative proofs. (The quantum-mechanical case could then be handled by analytic continuation followed by a suitable limiting argument.) However,
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we have chosen here to emphasize an approach that is very close to the one that led the second author to discover this formula in [Lal4,15]. It possesses a rich combinatorial structure and makes explicit connections with the disentangling process, which—as was discussed in Chapter 14—is central to Feynman's operational calculus [Fey8] and is one of the main themes of Chapter 15. In fact, it yields a very concrete example of the usefulness of our generalized Dyson series in [JoLal] (Chapters 15 and 16) which, as we recall, provide a way of disentangling the operator u (t). It is natural to wonder whether the "miraculous cancellations" that occur in the course of the proof of Theorem 17.2.1 (especially in the general case omitted here but studied in detail in [Lal5, §4]) do not suggest the existence of an underlying mathematical structure. This is indeed the case. Actually, motivated by the present work and that in [JoLal] (Chapters 15 and 16 above), one can define a new noncommutative and timeordered multiplication on the space of Wiener functionals. It explains, in particular, the striking similarity between the expression for the functional F in (17.1.5) (with 1 = V- + Ej=i WA, as in (17.2.1) and t € (rp, r p +i]),
and the formula (17.5.5) obtained in the threorem 17.27 for he operotor as in (17.1.6):
In the quantum-mechanical case, the passage from (17.5.11) to (17.5.12) could be considered as a kind of quantization procedure. In Chapter 18—which discusses the authors' work in [JoLa4]—we will develop this deeper interpretation of a rigorous and extended form of Feynman's operational calculus for noncommuting operators. Remark 17.5.7 In his 1948paper [Fey2, pp. 381-382], Feynman noted somewhat as we did just above the correspondence between time-ordered functional and time-ordered operators obtained by path integrating the functional. It seems likely that such observations played a key role in his later formulation of the perturbation series in quantum electrodynamics [Fey5—7] and in his development of the operational calculus for noncommuting operators [Fey8].
THE GENERAL CASE (ARBITRARY MEASURE n)
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17.6
The Feynman-Kac formula with a Lebesgue-Stieltjes measure: The general case (arbitrary measure 77) In this section, we give an overview of the main results obtained by Lapidus in ([Lal8], [Lal6]). These results extend to an arbitrary measure rj the corresponding ones from [Lal5] discussed in Section 17.2, where v was assumed to be finitely supported. (See [Lal8].) In the quantum-mechanical case, they also provide information—such as a product integral representation of the solution and a distributional differential equation—that are new even in the setting of Sections 17.2-17.5. (See [Lal6].) We now briefly recall our assumptions and notation from Section 17.1. Let n € M([a, b]) be an arbitrary (complex-valued and hence bounded) Borel measure on [a, b]. Let n = n. + v be the unique decomposition of n into its continuous part u, and its discrete part v. We write
where {rp}™=l is a sequence from [a, b} and {cap}^=l is a sequence from C such that Y^L\ \U>P\ < °°- (Without loss of generality, we may assume that all the rp are distinct.) Note that in (17.6.1), the times rp need not be naturally ordered; they could be, for instance, all the rational numbers in the interval [a, b]. This is, of course, in sharp contrast with the situation of Sections 17.2-17.5 where the definition of v involved only finitely many times TP. (Compare (17.6.1) with (17.2.1).) Finally, given 9 e Looi;r/ = Looi;ij([a, b)), t e [a, b] and A. e C+, we let u(t) = K((F) = KL(F)*, as in (17.1.6), where F and its time-reversal F are given by (17.1.5) and (17.1.7), respectively. Integral equation (integrated form of the evolution equation) The first main result ([Lal8, Theorem 4.3, p. 185]) provides, in this general setting, the integrated form of the evolution equation. We will refer to it as the "Feynman-Kac formula with a Lebesgue-Stieltjes measure" (briefly, FKLS) in the general case. Theorem 17.6.1 (FKLS in the general case) For all X e C+, the operator-valued function u satisfies the following Volterra-Stieltjes integral equation on [a, b]:
for all t 6 [a, b], where 0M is given by (17.6.5) below. The integral in (17.6.2) is a strong Bochner integral; so that, for all \{r e L2(K.d) and t 6 [a, b],
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Further, for all t 6 [a, b],
Here, we have set
where rj(s) := n([s]) and
for all
that is,
Note that #M is defined by means of the holomorphic functional calculus in the (complex) Banach algebra £(L2(E.d)). (See, e.g., [KadRi, §3.3].) We will see in Lemma 17.6.24 ([Lal8, Lemma 2.1, p. 170]) that since 0 e Looi;n, so does 9v^-t hence, 0
As will be further discussed below, the replacement of 9 by 0M in formula (17.6.3) was motivated in part by the work of Dollard and Friedman ([DoFri2], [DoFri3, Chapter 5]) in the related context of product integrals of matrices with respect to measures. Of course, if s is not a "pure point" of n (i.e. if n(s) = 0), then Ov,n(s) = 0(s). (In view of (17.6.5), this follows since 6(0) = 1. Here, we use the term "pure point of or "atom of 77" to mean any point 5 in [a, b] such that n(s) = 0, or equivalently, any point rp in the support of v; that is, such that a)p ± 0.) In particular, if /? = u is a purely continuous measure, as in Corollary 17.2.6, then (17.6.2) becomes (17.2.7). Recall that, in general, u has both an absolutely continuous and a singular part. If we assume in addition that n = u = I, the ordinary Lebesgue measure on [a, b], then as was discussed in Section 17.2, we obtain the classical Feynman-Kac formula in the probabilistic case (A = 1 and 6 = — V) and an interpretation of the Feynman path integral in the quantum-mechanical case (A = — i and 0 = —iV). (See, e.g., [CaStl,2] and [JoSk6].) We stress that if v is finitely supported, as in Section 17.2, then the integral equation (17.6.2) in Theorem 17.6.1 is equivalent to that provided by (17.2.2) (fbr p = 0 , . . . , h)in Theorem 17.2.1. Actually, as we shall see at the end of this section, this basic fact ([Lai 8, Theorem 3.2 (and Proposition 5.1), pp. 173 and 198])—which may not seem entirely
THE GENERAL CASE (ARBITRARY MEASURE rf)
509
obvious —constitutes the first step of the proof of Theorem 17.6.1. (See Theorem 17.6.23 below.) The second step of the proof of Theorem 17.6.1 consists in approximating an arbitrary Borel measure rj (with discrete part given by (17.6.1)) by measures ijm with finitely supported discrete part of the form: rjm = n + Y^"^={ CDpSrp,farm = 1,2 ..... (See Theorem 17.6.27 below [Lai 8, Theorem 4.2, p. 180].) In the next result ([La 1 8, Theorem 5. 1 , p. 1 92]), we establish in particular the uniqueness of the solution to the evolution equation (17.6.2). Basic properties of the solution to the integral equation Theorem 17.6.2 Fix A. e C+. Then, under the same assumptions as Theorem 17.6.1, the integral equation (17.6.2) (with u replaced by v) has a unique bounded (strongly measurable) solution v : [a, b] —> £(L 2 (R d )), necessarily equal to the function u of Theorem 17.6.1. Moreover, for all t e [a, b], u(t) = K[(F) is given by the following time-ordered exponential series:
where
and for
with
defined as in (17.65).
In (17.6.9), the series is summable in £(L 2 (R )) and the integral is a strong Bochner integral. Remark 17.6.3 It follows from Theorems 17.6.1 and 17.6.2 that for each t e [a, b], the time-ordered exponential series in (17.6.7) is equal to the time-reversal (in the sense of Theorem 15.7.6(ii)), of the CDS given by formulas (15.5.25)-( 15.5.28) (withn\an = lj of Example 15.5.7. (Indeed, by Example 15.5.7 and Theorem 15.7.6, u(t) = K[(F) is equal to the time-reversal of the series in (15.5.25); further, by the uniqueness statement in Theorem 17.6.2, u(t) is also equal to the series in (17.6.7).) This fact was certainly not obvious a priori. Note that the "disentangling process" (in the sense of Feynman's operational calculus as developed in Chapter 15) is carried out much further in the generalized Dyson series in (15.5.25) than in the series in (17.6.7). However, the latter series is written in a rather concise form. We next give further properties of the solution to the integral equation (17.6.2). (See [Lal8, Theorems 5.2 and 5.3, pp. 204-205].)
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Theorem 17.6.4 (a) (Time discontinuities) Fix A. e C+ and fef u(f) = K((F) be the unique solution of the evolution equation (17.6.2), as in Theorem 17.6.1. Then u = u(t) is strongly left-continuous on (a, b] and is strongly continuous at every point of [a, b] which is not a pure point of n. Moreover, for every r 6 [a, b) such that w := n ( r ) 7^ 0, we have
where M(T+) denotes the strong limit of u(t) as t —» r, t > i (i.e. for every i/r 6 L 2 (R d ), limt-t,t > u(r)tfr = e w0(r) u(r)VO. ffcj The function z defined by z(t) - u(t) - e -(t-a)(H0) /5 strongly of bounded variation on [a, b]. Exercise 17.6.5 (a) Prove part (a) of Theorem 17.6.4 in two different ways: (i) By using the integral equation (17.6.3) obtained in Theorem 17.6.1. (ii) By using the corresponding result (Theorem 17.2.4) obtained in Section 17.2 for measures with a finitely supported discrete part, and then by approximating 77 by rim •= M + Ep=i WPSV [Hint: In case (i), use the identity
which follows from the definitions (17.6.6) and (17.6.5) of w andOy^, respectively. (See formula (17.6.38) in Lemma 17.6.24(b) below.) In case (ii), use Theorem 17.6.27, the (strengthened) stability theorem relative to measures ([Lai 8, Theorem 4.2, p. 180]) stated below; see [La 8, pp. 204-205] for mo re details.] (b) Show directly (that is, without using Theorem 17.6.2) that the series in (17.6.9) and the time-reversals of the GDSs in (15.5.25) coincide. [Advice: For combinatorial simplicity, you may assume at first that v is finitely supported, as in Section 17.2; then a direct computation shows that the series in (17.6.7) and the GDS in (17.3.34) are equal.] Remark 17.6.6 (a) Theorems 17.6.1, 17.6.2, as well as (when v is finitely supported) Theorem 17.2.7 and Corollary 17.2.9, all suggest the existence of a (strong) product integral in this context. For instance, equation (17.6.2) looks like Duhamel's formula or the "sum rule " [DoFriS, pp. 33, 106 and 1 72]; further, formula (1 7.6. 7) yields the corresponding time-ordered perturbation expansion, which is often taken to be the definition of the product integral (see the discussion in the introduction of [DoFril]). Finally, formula (17.2.11) shows that, when v is finitely supported, it suffices to establish the existence of the product integral for a continuous measure. For the sake of brevity and simplicity, however, we choose not to consider the general case when A. e C^ here but to focus instead on the quantum-mechanical case in the rest of this chapter. Actually, the corresponding abstract theory of product integration (developed in Section 3 of [Lal6]) is more elegant in this setting.
THE GENERAL CASE (ARBITRARY MEASURE t))
511
(b) (Discreteness in space vs. discreteness in time.) We have previously discussed how to reinterpret a simple case of the Trotter product formula in our context by letting weighted sums of Dirac measures (in time) converge to Lebesgue measure I. (See, e.g., Corollary 17.2.9 and especially Example 16.2.7.) When we assume that /u = /, it would be interesting (but probably difficult) to investigate whether the work in [Hug 1,2]— which was partially motivated by [La6,ll]—could be adapted to this situation in order to show the existence of the product integral for more singular potentials than those considered below, such as, for example a Dirac measure in the space (rather than in the time) variable. We close this subsection by considering the "propagator" {/(•, •) associated with the integral equation (17.6.2); that is, for each t\,ti e [a, b] with t1 - 12, Ufa, t1) e £(L 2 (Rd)) sends the solution of (17.6.2) at time t\ onto the solution at time t2- (In particular, with this notation, we have u(t) = U(t, a) for all t in [a, b].) Clearly, for A. > 0, U (t2,t1) is also given by the right-hand side of (17.1.6), but with the time interval [a, t) replaced by [t\, t2) in the definition of the functional F given by (17.1.5). (Note that the operator Ufa, t\) was denoted P (t2, t1) in Section 17.2.) Theorem 17.6.7 (a) (Propagator) For all t1,t2, t3 in [a, b] with t\
(b) (Unitary propagator) Assume that we are in the quantum-mechanical case (A, = —i and 9 = —iV, with V e LOO,;/^), as we did in (17.5.6). If in addition, n = l, the ordinary Lebesgue measure on [a, b], v is real (i.e. a>p is real for all p > 1) and V is real-valued, then Ufa, t\) is a unitary propagator (with inverse identified with U(t\, 12)) for all t1, t2 in [a,b]). (Note that this implies in particular that u(t) = U(t,a) is unitary for all t in [a,b].) Theorem 17.6.7 is proved much as is suggested in part (ii) of Exercise 17.6.5(a), by deducing the stated property from the corresponding fact (equation (17.2.10) and Proposition 17.5.2, for part (a) and (b), respectively) for measures with finitely supported discrete part. (Note that a norm limit of unitary operators is still unitary.) The remainder of this chapter will be largely devoted to the study of the unitary propagator Ufa, t\) (and, in particular, of u(t) = U(t, a)) in the quantum-mechanical case (but under more general assumptions and with different notation, to be introduced below.) Quantum-mechanical case: Reformulation in the interaction (or Dirac) picture We now focus exclusively on the quantum-mechanical case. This is the situation of greatest physical interest from our present perspective since it leads to a suitable interpretation of the Feynman path integral in this context. Thus we assume throughout that A. = —/ and 6 = —iV, with V e Loo1;n, b)). We continue to use our earlier notation; in particular, u(t) = K'^F) is defined as in (17.1.6), with F as in (17.1.5). It will be convenient to begin by reformulating our basic results in the "interaction (or Dirac) representation". (For an introduction to this subject, closely related to the
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"Dirac or interaction picture" of quantum mechanics, we refer, for instance, to [Han, §11.1], [Sud, §3.5], or [Berl, Appendix on Wick's theorem, pp. 203-222].) Fix V e L2(Rd). For s € [a, b], set
and for w
Since u(a) = I, we have Q(a) = I and ^(a) = T/T. Clearly, Q(s) € £(L 2 (M d )) and $(s) e L2(R.d) for all s e [a, b]; further, W(s) € £(L 2 (K d )) for n-a.e. s € [a, b]. [Note how the "observables" (operators) W(s) as well as the "states" (vectors) ur evolve with time s in the "interaction (or Dirac) representation"; see (17.6.12c) and (17.6.12b). (On the other hand, in the closely related "Heisenberg picture" of quantum mechanics— which can be viewed as a special case of the "interaction picture" [Han, §11.1 ]—the states remain constant while the observables evolve with time according to (17.6.12c); see, e.g., [Han, §11.1], [Sa, § 1.3, pp. 4-5], or [Sud, §3.4 and §3.5]. We remark that in the literature, the time-dependent operator W is often called the "Heisenberg representation" of the operator W. Of course, the symbol ^ does no^stand here for the Fourier transform.)] We shall simply write WM instead of (W) M . Indeed, in view of (17.6.5) and (17.6.6), it is easy to check that
(Even though W does not necessarily belong to LQO^, it is still possible to define (W) M by (17.6.5); see also [La 16, §3].) We can now reformulate Theorems 17.6.1 and 17.6.2 in the quantum-mechanical case in (the analogue in our context of) the "interaction representation". Theorem 17.6.8 The function Q = Q(t) satisfies the following Volterra-Stieltjes integral equation on [a, b]:
Further, the integral equation (17.6.14) (with Q replacedby R) has a unique bounded (strongly measurable) solution R : [a, b] -> C(L2(E.d)), necessarily equal to Q. Moreover, for all t e [ a , b ] , Q(t) is given by the following time-ordered exponential series:
THE GENERAL CASE (ARBITRARY MEASURE n)
513
where A^(0 is as in (17.6.8) and for (s\,... , sn) e Aj,(f),
The series in (17.6.15) is (absolutely) summable in £(L2(Kd)) and the integrals in (17.6.14) and (17.6.15) are strong Bochner integrals; so that, in particular, we have for all r 6 [a, b],
In addition, for all t e [a, b],
Proof This follows easily from Theorems 17.6.1 and 17.6.2. We provide the proof, however, in order to explain how the expressions of the operators involved are significantly simplified in the interaction representation. (Compare, for example, (17.6.16) and (17.6.9).) Note that since by (17.6.2) (applied with 6 := -iV),
we have, in view of [HilPh, Theorem 3.7.12, p. 83] and (17.6.12)-(17.6.13),
this establishes (17.6.14). We deduce similarly (17.6.15) from (17.6.7). Remark 17.6.9 (a) If ^, the continuous part of r), is equal to Lebesgue measure I, we have seen in pan (b) of Theorem 17.6.7 that u(t) is unitary, for n and V real; in view of (17.6.12a), Q(t) is also a unitary operator in this case. In particular, we have \\Q(t)\\ = 1 and not merely \\Q(t)\\ -exp(|| VHoo,-,,) as statedin (17.6.18). (b) In general, (—i V)v^ is not equal to — i (V)v
The formulas (17.6.14)—(17.6.17) can thus be simplified accordingly in this case.
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FEYNMAN-KAC FORMULAS WITH STIELTJES MEASURES
(c) It is noteworthy that, in the quantum-mechanical case, all the time-ordered perturbation expansions (GDSs) obtained (or discussed) in Chapters 15-16 (as well as in the previous sections of the present chapter) can be rewritten in the interaction representation. This provides a much more concise (and possibly more natural) expression/or our generalized Dyson series. As an exercise, some readers may find it helpful to check this fact concretely in a few examples; for instance, for the GDS defined by (17. 3. 2)-(17.3. 5) (or (17.3.6)-(17.3.9)) or, more generally, for the GDS given by formulas (17.3.34)(17.3.35) (and corresponding to the "time-reversal" of that in (15.5.5)-(15.5.7) from Example 15.5.1). Product integral representation of the solution We still suppose that we are in the quantum-mechanical case and continue to discuss the results from [Lal6]. We stress that all the results provided through to the end of this chapter (with the exception of Theorem 17.6.14) will be new and of interest even in the context of Sections 17.2-17.5, where the discrete part of n was supposed to be finitely supported. The following Theorem ([Lal6, Theorem 2.2, pp. 282-283])— which is obtained by combining Theorem 17.6.8 above ([Lal6, Theorem 2.1]) with Theorems 3.1, 3.3, and 3.4, pages 293, 299, and 304 in [Lal6] —provides the key link between the general "Feynman-Kac formula with a Lebesgue-Stieltjes measure" (as presented in Theorem 17.6.8) and the theory of strong product integration of measures (as elaborated in [Lal6, §3]). The theory of (strong) product integrals (for general Borel measures)— also called "time-ordered multiplicative (or chronological product) integrals" — has been developed in the present context by Lapidus in [La 16], building on earlier work by Dollard and Friedman [DoFril-3] and others (and going all the way back to early work (on the composition of) integral operators by Volterra ([VolHos], [Vol]) which can be thought of in some respects as a precursor of some of Feynman's ideas on his operational calculus [Fey8]); see Remark 17.6. ll(b) below. For conciseness, however, we will not provide here all the necessary definitions and details of the theory, for which the interested reader is referred to [Lal6, esp. §3] (as well as to the book [DoFri3] for the general framework in more particular situations). We note that for many readers, Theorem 17.6.12 below ([Lal6, Theorem 2.3, p. 285]) may provide a suitable substitute for the general definition ([La 16, Theorem 3. 1, p. 293]) of the product integral appearing in formula (17.6.19). (Later on, we will use indifferently n[a,t) or nj, in the expression of the product integral.) Theorem 17.6.10 Let n be an arbitrary Borel measure on [a, b] and let V € LOO,;,,. Then the strong product integral of—iV with respect to the measure n exists over [a, b) [and hence over every subinterval of [a, b)]. Moreover, for all t € [a , b], we have
where V is defined by (17.6.1 2c). (In particular, Q(t) € C(L2 (Rd), for every t E [a, b].)
THE GENERAL CASE (ARBITRARY MEASURE n)
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The product integral in (17.6. 19a) converges in the strong operator topology; so that, for every faced w e L 2 (R d ),
wherep(t)isgivenby(17.6.12b).(Recallthatp(a)=p.)
Here is a sketch of the proof of Theorem 17.6.10: By [Lal6, Theorem 3.1, p. 293], the product integral on the right-hand side of (17.6.19) exists; thus, for every fixed t e [a, b], formula (17.6.19a) defines a bounded linear operator in £(L 2 (R af )), denoted by w(t). Next, by [Lal6, Theorem 3.3, p. 299], w = w(t) is a bounded solution of the integral equation (17.6.14) on the entire time interval [a, b]. Since—according to Theorem 17.6.8—such a solution is unique and necessarily equal to Q(t), as defined by formula (17.6.12a) and by the time-ordered exponential series in (17.6.15), we conclude that (17.6.19a) holds; that is, for all t e [a, b], w(t) = Q(t) and hence Q(t) is given by the product integral in (17.6.19a), as desired. For a detailed proof of Theorem 17.6.10 ([Lal6, Theorem 2.2, pp. 282-283]), we refer the interested reader to [Lal6, pp. 283-284]. Remark 17.6.11 (a) Roughly speaking, product integrals are multiplicative and noncommutative analogues of the usual (additive) integrals. In the physics literature (and in the classical case when n = u = I), the product integral in (17.6.19a) is often called a "chronological" or "time-ordered" (exponential,) integral and is denoted by
(with n(ds) = ds), where T stands for the time-ordering symbol ([Dyl,2,4], [Fey8]). For instance,
(See, for example, [Sa, p. 5]. Note the complete agreement between (17.6.20b) and Feynman's time-ordering convention (14.2.1) discussed in Section 14.2.) With a similar notation and in the general case of Theorem 17.6.10, Equation (17.6.19b) yields
(We use here Dirac's "bra and ket" notation [Dir2].) Of course, the product integral used in Theorem 17.6.10 and throughout the end of this Section are not merely formal expressions but are precisely defined mathematical objects, as is explained at some length in Section 3 of [La16]. (b) A rigorous treatment of strong product integrals, in the case when n — I—as well as of product integrals of measures in the case of finite-dimensional matrices—is given in [DoFriS], Chapters 3 and 5, respectively. (See also the research articles [DoFril,2] and
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FEYNMAN-KAC FORMULAS WITH STIELTJES MEASURES
the relevant references therein.) In [Lal6], Section 3, the second author has unified and (partially) extended both approaches to strong product integrals of measures, in order to make sense, in particular, of the product integral appearing in (17.6.19) and to establish some of its main properties [such as the associated evolution equation in integrated or (distributional) differential form]. Note that in (17.6.19), an arbitrary Borel measure occurs and that, in addition to V, the unbounded operator HQ acting in the infinite dimensional space L 2 (R d ) appears in disguised form (see (17.6.12c) above). (c) The theory of (strong) product integration of measures developed in [Lai 6] is sufficiently general to handle considerably more abstract situations than the present one. Although this has not yet been carried out explicitly, it should help provide a product integral representation of the solution to the evolution equation later obtained in our (joint) paper [dFJoLa2J and discussed in Chapter 19 below. [In [dFJoLal,2], the setting for "Feynman 's operational calculus" involves suitable (noncommuting) abstract bounded linear operators acting in a Hilbert space, as well as (unbounded) generators of (Co) contraction semigroups; see Chapter 19.] Moreover, it could possibly help establish connections between aspects of the present work and the notion of "expansionals" in Banach algebras introduced by Araki in [Ar]. (d) An informal introduction to the notion of product integrals, primarily directed towards physicists, is provided in [dWmMN2, §2, pp. 284-293], in the simple case of finite-dimensional matrices and of ordinary Lebesgue measure. We have to disagree, however, with the statement in [dWmMN2, p. 284] according to which "product integrals are a simple and rigorous vehicle for Feynman's operator calculus." In fact, both on the basis of his paper [Fey8] and of the second author's private conversations with him, we think that Feynman had a much more ambitious program in mind, extending well beyond the mere use of the (admittedly very important) exponential function. It would perhaps be more accurate to state that close relationships exist between product integration and Feynman's operational calculus. Fix t e [a,b] and let P be an arbitrary partition of [ a , t ) . HP is of the form S0 := a < s1 < • • • < t =: sn, we set Ik = [sk-1,Sk)> for k = 1,... ,n; further, as usual, we define the mesh of P by mesh(P) = max1
THE GENERAL CASE (ARBITRARY MEASURE n)
517
The integrals in (17.6.21) are strong Bochner integrals and the limit holds in the strong operator topology, uniformly for t € [a,b]; so that, in particular, for every ^ e L2(E.d), w ( t ) = Q(t)t/f ^(appearing in (17.6.19b)) is obtained by applying the right-hand side of'(17'.6.21) to w ( a ) = w, Remark 17.6.13 We briefly comment on the rigorous definition—given in [Lal6, §37— of the productintegral occurring in (17.6.19a) and (17.6.21). Fort e [a,b], the(strong) product integral, ^ \ ^ a t ) e x p { B ( s ) n ( d s ) } , of an operator-valued function B : [a,b] £(L 2 (R d ) (with respect to the Borel measure n on[a, b]) is first defined for a step function (and then denoted Y[g (t)), much as in the theory of Riemann-Stieltjes integrals, but in a multiplicative rather than additive way (and also by taking into account the noncommutativity of the exponential function in a Banach algebra). (See [Lal6, Definition 3.4, p. 292].) Then, for a general (suitable) function B, it is defined as the (strong) limit of the operators \\Bn(t) (uniformly in t e [a, b]) associated with a sequence {Bn}^1 of step functions converging (in an appropriate sense) to B. (See [Lal6, Theorem 3.1, p. 293].) Of course, it can be shown that for the class of functions B considered in [La}6], there always exists a sequence of step functions converging to B, and that, in addition, the (strong) limit of{\\B (t)}%Li always exists and is independent of the choice of the sequence of step functions (suitably) converging to B. (See [Lai6, Proposition 3.1 and Corollary 3.1, p. 293].) [We note that the setting of [Lal6, §37 is more general than indicated here, as it allows for functions B with values in a strongly closed Banach subalgebra of C(X), where X is an arbitrary Banach space.] The next result is just a restatement of Theorem 17.6.4 in the interaction representation. (In the light of Theorem 17.6.10, it also follows directly from the properties of the product integral obtained in [Lal6, Theorem 3.2(d), p. 297]. Note that e«?(*) = eisH0eaV(s)e-isH0 for a e C and ^.a.e. s e [a< b),) Theorem 17.6.14 The (operator-valued) function Q = Q(t) is of bounded variation on [a, b]. Moreover, it is left-continuous on all of (a, b] and is continuous at every point of [a, b] which is not a pure point of n. Further, if r is a pure point of rj, then Q(r+) exists and
(i.e. u(r+) = e "°V^U(T)), where w := n(T) ^ 0. These statements must be understood in the strong operator topology. Distributional differential equation (true differential form of the evolution equation) Heuristically, one may expect that the integral equation (17.6.14) can be rewritten in the form of a distributional differential equation. (Refer to the comments concluding Section 17.4 above.) We now see that this intuition is correct, as was shown in [Lal6, Theorem 2.5, p. 286].
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Theorem 17.6.15 The following distributional differential equation holds on (a,b):
in the sense of distributions on the open interval (a, b). (Recall that the function 1/r satisfies the initial condition w ( a ) = w.) Specifically, this means that for all infinitely differentiable complex-valued functions p with compact support contained in (a, b), we have
where p' denotes the (ordinary) derivative of the "test function" p.
Theorem 17.6.15 follows by combining Theorem 17.6.10 above and [Lal6, Theorem 3.5, p. 305]. The interest of equation (17.6.23) is twofold: (i) Firstly, the measure n (or rather, its continuous part u) may have a nontrivial singular part. (See, for instance, Example 15.5.3 and Exercise 17.4.2, as well as Exercise 17.6.16 below.) (ii) Secondly, the measure n may have a nonzero discrete part v, the action of which (as seen, for example, in formula (17.6.22) of Theorem 17.6.14) is taken into account by the (modified) "potential" (—z V) . (The comments in (i) and (ii) are of interest even in the context of Sections 17.2-17.5, where v was assumed to be finitely supported.) Note that in the special case when n = u is a continuous measure, (17.6.23b) becomes, in view of Remark 17.6.9(b):
Exercise 17.6.16 Let n — ul be the Cantor-Lebesgue measure, as in Example 15.5.3. Hence n is a singular continuous measure and so ( d n / d s ) = 0 l-a.e. Rewrite the integral equation (17.6.14) and the distributional differential equation (17.6.23) or formula (17.6.24) in this case. Compare these equations with each other. Finally, check that their solution is consistent with the result obtained in formula (15.5.15) of Example 15.5.3 (where a = 0). [Hint: You may use (17.6.25) since n = u. Further, recall that Q (t) is given by(l 7.6.12a), withu(t) = K'_i(F).]
THE GENERAL CASE (ARBITRARY MEASURE
n)
519
Unitary propagators For a - t1 - t2 - b, we will frequently use the symbol f|^ instead of Yl[tt t2y ^ follows from results of [Lal6, §3] that ["[^ exp{— i V (s)n(ds)} is invertible in the Banach algebra £(L2(Md)); further, its inverse, denoted by f]^ exp{-/ V(s)n(ds)}, is defined just like the above product integral, except with a time-ordering with respect to increasing instead of decreasing time indices as you read from left to right on the page. (See [La 16, Lemma 3.2, p. 295] and the discussion preceding it.) For any t1 , 12 6 [a, b], we now set
Note that, in particular, Q(t ) = Q(t , a) for t e [a, b]. Our next result follows from Theorems 17.6.10 combined with [Lal6, Theorems 3.2(c), 3.3, and 3.4, pp. 296, 299, and 304], Theorem 17.6.17 For any t\ , ti, t2 e [a, b], we have Q(t1 , t1) = I and Q(t3, t1) = Q(h, t2)Q(t2, ti).
(17.6.27)
Moreover, for a < t\ < t2 < b,Q(t2,t1) is the "propagator" for the integral equation (17.6.14) (i.e. tyfa) = Q (t2, t 1 ) w ( t } ) ) and is given by (17.6.15), with &'n(t) and Ln^ as in (17.6.8) and (17.6.16) except with a, t replaced by t\, ti, respectively. [For ti < ?2, Q(ti,t\) = Q(t\,t2)~l is the "propagator" for the "time-reversal" of'(17.6.14) (or equivalently, of(17.6.14') below); that is, for the integral equation (17.6.14') stated below.] We refer the reader to [Lal6, Corollaries 3.2 and 3.4, pp. 303-304] for a more complete statementofTheorem 17.6.17.There,itisshownthatforanya, B e [ a , b ] , Q(a, ft) and its inverse, Q(B, a) = Q(a, B ) ~ l , are given by a time-ordered perturbation expansion analogous to (17.6.15), and satisfy a Volterra-Stieltjes integral equation similar to (17.6.14). In particular, for fixed ft (resp., a), Q(a, ft) satisfies a kind of "forward" (resp., "backward") integral equation; see [Lal6, equations (3.23) and (3.24), p. 303] with A := —iV. More precisely, for arbitrary a, ft 6 [a, b], Q(a, ft) is the unique bounded solution of each of the Volterra-Stieltjes integral equations
and
where, for z e C, o(z) is given by (17.6.6) and p (z) := p(-z) = e-2p(z).
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FEYNMAN-KAC FORMULAS WITH STIELTJES MEASURES
We note that given Theorem 17.6.10, the remaining results (Theorems 17.6.12, 17.6.14,17.6.15 and 17.6.17) follow easily from the corresponding results about abstract product integrals established in [La 16, §3]. Remark 17.6.18 It follows from Remark 17.6.9(b) that for any a, B in [a, b], Q(a, B) is a unitary operator when u, = l and the potential V and the measure v are both realvalued, with n = u + v as before. (For a - ft, this is obvious, while for a > B, we simply use the fact that Q ( a , ft) is the inverse of the unitary operator Q(B, a).) By abuse of language, we still refer to Q(a, ft) as a "unitary propagator" even when this assumption is not satisfied. Scattering matrix and improper product integral In order to state our last result, we shall need to slightly modify our hypotheses. Assume that n is a (complex-valued) set-function defined on (the bounded Borel subsets of) R and whose restriction to every compact subinterval [a, b] c R lies in M([a, b]) (the set of complex Borel measures on [a, b]). (See Remark 17.6.19 below.) Suppose further that V : R x Rd -> C is a Borel measurable function which belongs to Looj^R) (i.e. b\\V(s,-)\\00\rl\(ds)
THE GENERAL CASE (ARBITRARY MEASURE
n)
521
where u(b) = Kb_t (F) is given by (17.1.6), with F as in (17.1.7). The limits in (17.6.28) hold in the strong operator topology. If r) = I, then S is often called the "scattering operator" or "S'-matrix" and is denoted by
in the physics literature. (See [Sa, p. 5] and Remark 17.6.1 l(a) above.) Therefore, we may think of Theorem 17.6.20 as establishing the existence of the scattering operator in our context. When v is finitely supported, this is of particular interest in view of Theorem 17.2.7 [see equation (17,5.8) with t := b, which provides (in the quantummechanical case) an "explicit expression" for u(b) as a function of b (and a) in R] and of the physical interpretations in terms of multiple scatterings provided in Section 17.5 above [seethe discussion following equation (17.5.8), which also deals with the quantummechanical case]. We close our exposition of the results of [La 16, 18] by giving a natural sufficient condition under which S is unitary. (We write /z instead of the continuous part of the restriction of n to an arbitrary compact interval [a, b] C R.) Corollary 17.6.21 Suppose further that u = l and that v and V are real-valued but otherwise arbitrary. Then S, as defined by (17.6.28), is a unitary operator. Exercise 17.6.22 Deduce Corollary 17.6.21 from Theorem 17.6.20and Remark 17.6.18. [Hint: With the obvious notation, it follows from the comment preceding equation Sketch of the proof of the integral equation We conclude this section by providing a sketch of the proof of the integral equation (17.6.2) stated in Theorem 17.6.1 ([Lal8, Theorem 4.3, p. 185]) for an arbitrary Borel measure n on [a, b]. We will closely follow the outline of the proof given in [La 18], and (except for the main part of the proof of Step 1) we will refer to [Lai 8] for most of the details. More precisely, in Step 1 below, we assume that v, the discrete part of r;, is finitely supported (as in Section 17.2) and then reformulate the integral equation obtained in Section 17.2 (Theorem 17.2.1, recalled in (17.6.29) below) in the more invariant form of (17.6.2) (or (17.6.30) below). (Compare equation (17.6.29) with equation (17.6.30), which no longer depends on the specific definition of v.) Then we treat the case of an arbitrary Borel measure r\ = u + v in Steps 2 and 3. In Step 2, we use in a crucial way the time-ordered perturbation expansions (GDS) derived in Chapter 15 to approximate the operator u(t) = K((F) associated with 77 = u. + v by operators um(t) = K[(Fm) associated with the measures r)m = u + vm (m — 1, 2, . . . ), with vm finitely supported. Finally, in Step 3, we use Step 1 along with the approximation theorem from Step 2 to deduce that u = u(t) satisfies the integral equation (17.6.2), as desired.
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FEYNMAN-KAC FORMULAS WITH STIELTJES MEASURES
We now proceed to give a more detailed outline of the proof of Theorem 17.6.1. Step 1: Reformulation of the integral equation when v is finitely supported. Fix A 6 C+. Assume that v is finitely supported; namely, v = £p=i oP5rp, with (Op e C and TO := a < T\ < • • • < rm < b =: tm+\ (exactly as in (17.2.1), except with the subscript h replaced by m). Then by Theorem 17.2.1, u = u(t) satisfies the integral equation (17.2.2) on each interval (TP, TP+\]; namely, for p = 0 , . . . , m,
The next result ([Lal8, Theorem 3.2, p. 173]) shows that u satisfies the integral equation (17.6.2) on the entire interval [a, b]. (Some of the necessary technical facts will be provided in Lemma 17.6.24 below.) Theorem 17.6.23 (Finitely supported v) Under the above assumptions,
Proof of Theorem 17.6.23 . Fix t e [a, b] and A e C+. We will show that u(t) is given by the right-hand side of (17.6.30). If t = a, this is obvious since u(a) = I. We now assume that t e (a, b]; let p be the unique integer in {0,... , m} such that t e (rp, TP+i ]. According to Theorem 17.2.1 (see equation (17.6.29) above),
Hence, since u is continuous and since in shorthand notation,
where a = 1 if n(a) = 0 and a = 2 otherwise (recall from Remark 17.2.5 that &>o = (a\ and TO = TI = a if n(a) ^ 0), we have:
THE GENERAL CASE (ARBITRARY MEASURE 77)
523
By inserting a telescoping sum, we obtain
[Here, we have again used the convention specified in Remark 17.2.5 and the fact that W(TO) = u(d) = I to deduce that (for a — 1,2)
Now, by Theorem 17.2.1—more precisely, by equation (17.2.2) or (17.6.31) applied to Tq e (Tq_I , T?] for # e {1,... , m]—we see that
Thus, by [HilPh, Theorem 3.7.12, p. 83] and by the semigroup property of >o, it follows that for q e {1,... , p],
Combining (17.6.32) and (17.6.33), we obtain
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FEYNMAN-KAC FORMULAS WITH STIELTJES MEASURES
By Lemma 17.6.24(a) below, 0w,n(5) = 6(s)
|u|-a.e. in (a, t),
(17.6.35)
and since for q e {0, . . . , m), n(T?) = a>q, it follows from formula (17.6.38) in Lemma 17.6.24(b) that wq0
(17.6.36)
and hence, in particular,
Consequently, in view of Remark 17.2.5, equation (17.6.34) becomes successively:
This proves equation (17.6.30), as desired. Let 9 € Looi;i? and define as before Qv
THE GENERAL CASE (ARBITRARY MEASURE 17)
525
(c) Under the above hypotheses. 0n,n e loo;n and
Exercise 17.6.25 Prove Lemma 17.6.24. Exercise 17.6.26 Establish the converse of Theorem 17.6.23; namely (still for v finitely supported), show that if u satisfies (17.6.30) on [a, b], then it also satisfies (17.6.29) on (Tp,Tp+\]forp = 0 , . . . , m. [Hint: The proof—which can be found in [Lal8, Proposition 5 1, pp. 198-199]—is shorter than that of Theorem 17.6.23 and makes use of Lemma 17.6.24.] Step 2: Approximation by measures with finitely supported discrete part. Let n = u + v be an arbitrary Borel measure on [a, b], with v = ELi w^-ip an^ Xi^Li \o>p\ < °o (as in (17.6.1)). Without loss of generality, we may assume that the times tp are pairwise distinct (but we may not assume that they are linearly ordered). For m > 1, set nm = u + vm, where vm := £)p=i w p & r p . (Note that the measures 7?m and n have the same continuous part u.) Let F be defined by (17.1.5), and let Fm be defined similarly, except with n replaced by nm. Given t € [a, b) and A. e C+, let u(t) = K((F) (as in (17.1.6)) and let um(t) = K [ ( F m ) f o r m > 1. The following result ([Lai8, Theorem 4.2, p. 180]) provides a strengthening in this specific context of some of the stability theorems (from [JoLa 1 ]) with respect to measures discussed in Chapter 16 (Section 16.2). (See Remark 17.6.29(a) below.) Theorem 17.6.27 (Stability theorem: approximation ofv by vm) With the above notation, we have:
so that, in particular, {um}^_^ converges to u in (the norm operator topology of) £(L2(E.d)), uniformly in t on all of [a, b]. We next introduce some cumbersome but convenient notation for use in the solution of Problem 17.6.28 below. Given t e [a, b], we let (for p > 1)
so that the restriction of r\ to [a, t) is equal to the measure n, := u + Yl^Li Op.t&Tp (where u, now denotes the restriction of the continuous part of r] to [a, t)).
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FEYNMAN-KAC FORMULAS WITH STIELTJES MEASURES
so that the restriction of r)m to [a, t) is equal to the measure n m > , := /j, + £!^Li
where cm —> 0 as m —> oo. This proves Theorem 17.6.27. [Note: More precisely, the positive number cm in (17.6.44) is explicitly computable and can be chosen to be the tail of a specific convergent series; see [Lai 8], equation (4.22), page 183.] Remark 17.6.29 (a) In Example 16.2.11, we used our earlier stability Theorem (Theorem 16.2.6) to obtain a conclusion analogous to that of Theorem 17.6.27. We point out, however, a significant difference with the present result: Estimate (16.2.45)—the counterpart in Example 16.2.11 of estimate (17.6.41) or (17.6.44)—was valid only for A > 0 (rather than for all A. e C+, as in Theorem 17.6.27). This is so because the proof of Theorem 16.2.6 relied on the dominated convergence theorem for functional integrals whereas that of Theorem 17.6.27 makes use of our generalized Dyson series, which are well defined and have the sameform for all A. € C+. This provides one more illustration of the usefulness of these time-ordered perturbation expansions (GDSs) to establish results which are valid in the diffusion as well as in the quantum-mechanical case. [See also, e.g., Example 16.2.9 or the proof of the integral equation (Theorem 17.2.1) discussed in Section 17.3.]
THE GENERAL CASE (ARBITRARY MEASURE
n)
527
(b) It is the uniformity in t e [a,b] of the limit in (17.6.41) that permits us to interchange the limits involved in a variety of situations and to deduce results for general Borel measures/mm the corresponding onesfor measures with finitely supported discrete part. (See, for example, part (ii) of Exercise 17.6.5(a) as well as Step 3 below.) Step 3: Passage to the limit in the integral equation. Fix A. e C+. With the notation and hypotheses of Step 2, we know from Step 1 (Theorem 17.6.23) that for each fixed m > 1 , um = um(t) satisfies the integral equation (17.6.30) on [a, b]; namely, for all t e [a, b],
The present step consists in justifying the passage to the limit (as m —> oo) in (17.6.45); namely, we show that, for all t e [a, b],
Fix t e [a, b]. Clearly, the left-hand side of (17.6.45) converges to that of (17.6.46), according to the stability result from Step 2 (Theorem 17.6.27). Moreover, by means of Lemma 17.6.24 and Theorem 17.6.27 (including the uniformity of the convergence of um(s) to u(s) for s e [a, b], as stressed in Remark 17.6.29(b)), we deduce after somewhat lengthy computations that the right-hand side of (17.6.45) also converges to that of (17.6.46). This proves that the integral equation (17.6.46) (or equivalently, (17.6.2)) holds for all t e [a, b] and thus completes the proof of Theorem 17.6.1, as required. The following problem asks the reader to provide the missing details in the above argument. (The full proof can be found in [Lal8, §4.C, pp. 185-191].) Problem 17.6.30 Complete the proof of Step 3. [Hint: It should be useful, in particular, to note that &
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FEYNMAN-KAC FORMULAS WITH STIELTJES MEASURES
(17.6.2). Then (possibly under a different set of hypotheses), deduce in two steps that (17.6.2) holds for all A, e C+; namely, by analytic continuation (A, e C+) followed by passage to the limit along the imaginary axis (A. e C+). [Advice: For simplicity, you may wish at first to assume that v is finitely supported. Further, only when A, > 0, you may look for "optimal" hypotheses on the potential V under which the FKLS holds.] (b) (Stochastic version of the FKLS?) Suppose again that we are in the diffusion case (say A. = 1 and 0 — —V). Try to generalize the "Feynman-Kac formula with a Lebesgue-Stieltjes measure" (FKLS) to the case when the measure n (as well as, possibly, the potential V = V(s, •)) is allowed to be "random" (hence, for instance, the atoms of n are no longer located at predetermined times TP, but at "random " times, which can be interpreted probabilistically by using the notion of "stopping time"). That is, obtain a suitable stochastic version of th- integral equation (17.6.2) (or (17.6.46)). (c) Can you extend to the quantum-mechanical case (when A. ^ 0 is purely imaginary and 9 = —iV) the probabilistic results which you have obtained in part (b)? More specifically, can you find—by means, for example, of a suitable analytic continuation procedure followed by a passage to the limit—a counterpart in the quantum-mechanical case of the stochastic (integral) equation obtained in part (b) in the probabilistic case? Under which assumptions does your extension hold? [Advice: In part (c), you may have to restrict considerably your hypotheses on the potential V.] Remark 17.6.32 (a) We note that a stochastic version of the FKLS in the diffusion (part (b)) or, especially, in the quantum-mechanical case (part (c)) would broaden the scope of the physical interpretations and possible applications discussed in Section 17.5 above. For instance, in the quantum-mechanical case, it might improve the simplified model of "multiple scattering" (within a target) discussed just before equation (17.5.9), by allowing the impacts (or scatterings) of the particles to occur at random (rather than at fixed predetermined) times. (For example, the corresponding probability distribution could be that associated with a Poisson process or with a more complicated "jump process".) (b) In part (a) of this remark, we have discussed a possible "randomization" of the discrete part of the measure. In some sense, recent work [JoKalS] of Johnson and Kallianpur on Stochastic Dyson series and the solution to stochastic evolution equations can be thought of as providing a suitable "randomization" of the continuous part of the measure; namely, heuristically, u,(ds) is replaced by %(s)ds, where t-(s) is "white noise" (roughly speaking, the "derivative" of Brownian motion). (c) Another type of extension of some of the results of this chapter would consist in replacing Brownian motion by a more general (Markov) stochastic process; that is, in allowing HQ to denote the generator of (the heat semigroup associated with) a Markov process other than the Wiener process. (See Remark 14.4. l(c).) For instance, for d = 1 and under suitable assumptions, the FKLS of Theorem 17.2.1 was extended by Riggs [Rig] to the case when HQ denotes the generator of the Poisson process (as in Example 9.2.5). Then, in the (analogue of the) diffusion or quantum-mechanical case,
THE GENERAL CASE (ARBITRARY MEASURE n)
529
respectively, we obtain the (generalized) telegrapher's equation or the Dirac equation (both in one space dimension) rather than the (generalized) heat or Schrodinger equation, respectively. In this context, the (counterpart of the) Dirac equation describes the time evolution of a relativistic quantum-mechanical particle subjected to "shocks " or "scatterings " occurring at deterministic times. Naturally, in the spirit of Problem 17.6.31 and of part (a) of this remark, one may still wonder what happens if those times are randomized. We leave the investigation of these and related questions to the interested reader.
18 NONCOMMUTATIVE OPERATIONS ON WIENER FUNCTIONALS, DISENTANGLING ALGEBRAS AND FEYNMAN'S OPERATIONAL CALCULUS 18.1 Introduction A family {At '• t > 0} of commutative Banach algebras of functionals on Wiener space was introduced in Chapter 15, and it was shown (see Theorem 15.7.1) that, for every F e At, the functional integrals K[ (F) exist and are given by a time-ordered perturbation expansion which serves to disentangle, in the sense of Feynman's operational calculus, the operator K[(F). In this chapter, noncommutative operations * and + on Wiener functionals are defined which will enable us to provide a precise and rigorous interpretation of certain aspects of Feynman's operational calculus for noncommuting operators. The noncommutative operations, the "disentangling algebras" {At '• t > 0} and the functional integrals provide a rich interlocking structure. The present chapter is based on the joint paper [JoLa4] of Johnson and Lapidus (announced in part in [JoLa3]). The paper [JoLa4] in turn relied heavily on [JoLal], which was the main subject of Chapter 15. The last part of Section 14.5 discussed Chapter 18 and can serve as part of the introduction to this chapter. For the convenience of the reader, however, we recall some of the most important elements and make a few additional comments. In Section 14.5, we discussed the fact that if F 6 Atl and G 6 At2, then both F * G and F+G are in Att+t2. Now that we have defined the norms || • ||(, t > 0 (see Section 15.7), we can state associated norm inequalities (see Theorem 18.5.3):
Recall from Section 15.7 that for a fixed t > 0, the Banach algebra (At, II • ||/) is a suitable set of (equivalence classes of) functions [often called "Wiener functionals"] defined on C' - C([0, t], Rd), the space of continuous paths x : [0, t] - Rd. (Also, At is equipped with the natural pointwise addition and multiplication.) Further, given F e Atl and G e At-, with t1, t2 > 0, F * G e At1 +t2 is defined by
INTRODUCTION
531
where the paths x\ e C'1 and X2 € C'2 are given by
and
Similarly, with x\ and xi as in (18.1.4), F+G £ A,+r 2 1 S defined by
We leave aside for now the measure-theoretical technicalities associated with the definition of the algebras At (due mainly to the properties of scale-change in Wiener space), and hence of the operations * and +. We refer the interested reader to the earlier material in Sections 15.2.1 and 15.7. The precise definitions of the "noncommutative multiplication" * and the "noncommutative addition" + will be given in Section 18.2 below. Our central result (Theorem 18.5.6)—established below in the broader context of the analytic operator-valued Wiener and Feynman integrals (see Theorem 18.4.1)—can be formulated as follows in the present setting of the disentangling algebras {A, : t > 0}. If F 6 At} and G e At2, then for all A e C+, we have the operator identity (in r i 1 2/in>\\. L\L (IK )).
[Observe that if we replace F * G by G * F on the left-hand side of (18.1.6), then the order of the operators on the right-hand side must be reversed.] We deduce in particular from (18.1.6) that the operator product K^(F)K^(G) can be disentangled via a generalized Dyson series (GDS) because (by Theorem 15.7.1) the operator K'^+'2 (F * G) itself can be disentangled in this manner. (See Corollary 18.5.7.) This answers a very natural question motivated in part by our earlier work presented in Chapter 15 ([JoLal]) and in Chapter 17 ([Lal4-18]). (See, in particular, the comments at the end of Sections 15.7 and 17.5 above.) [Note that although the time parameter t was kept fixed in Chapter 15, we now allow / to vary in Chapter 18, much as in the study of the evolution equations in Chapter 17. This fact is crucial in interpreting and deriving formula (18.1.6) and its corollaries discussed below.] There are a variety of equalities relating * and + which have corresponding operator equalities. Formulas (14.5.11) and (14.5.12) are one such pair. More generally (see Example 18.5.14), given arbitrary tj > 0 and Fj e Atj (for j = 1 , . . . , h), we have the identity between elements of Atl+...+th'•
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NONCOMMUTATIVE OPERATIONS ON WIENER FUNCTIONALS
and os, by (18.1.6) above (theorem 18.5.6),
[We stress that in (18.1.8), for instance, exp(FH ----- i-F/,) is defined by means of the analytic functional calculus in the Banach algebra At1-\_____\-th', further, for j = 1, ...,/!, exp(F;-) is similarly defined within the Banach algebra Atj.] Under the same assumptions as for (18.1.7) and (18.1.8), another example is given by a type of multinomial formula and a related operator equation (see (18.5.35) and (18.5.36)):
and hence
Finally, it makes sense to consider the commutator [F, G] := F * G - G * F of F € Atl and G 6 At2 and to ask if the functional integrals K[ preserve commutators. They do (see Corollary 18.5.8), and we have
The rest of Chapter 18 is organized as follows. The main part of the chapter (Sections 18.1-18.5) deals with the authors' work ([JoLa3,4]) discussed above. Moreover, in the appendix (Section 18.6), we discuss some work of the second author ([Lai 9]) in which was proposed an axiomatic approach to Feynman's operational calculus, motivated in large part by the results of [JoLa4]. 18.2 Preliminaries: maps, measures and measurability Throughout this chapter, we will need various maps between spaces of continuous functions and related results involving measures and measurability. Proposition 18.2.1, which concludes this section, will help us to show in Section 3 that the operations * and + are well defined. At least on first reading, one may wish to skip the proof of Proposition 18.2.1. Let Ca
In the following, ma,b will denote Wiener measure on CQ' & . Frequently, we will have a = 0 and then we will write Cb, CQ and m^ rather than C°!b, CQ 'b and mo, b, respectively.
PRELIMINARIES: MAPS, MEASURES AND MEASURABILITY
533
In particular, with this notation, we have m, = m, Wiener measure on C'0 = C0'r, as defined in Chapter 3. (Caution: Throughout the present chapter, mt is defined as above and hence does not denote scaled Wiener measure, as defined in Chapter 4.) We begin by considering two restriction maps and a translation map all of which will be involved in the definitions of the operations * and + • Suppose that a < b < c and let R\ : Ca'c ->• Ca'b be the map of restriction to the first part of the interval. If we want to emphasize that R\ depends on a, b and c, we will write R"' 'c rather than simply 7?i. So
Similarly, /?2 : Ca'c -> C b,c will denote the map of restriction to the second part of the interval; that is,
In this chapter, we will need certain variations on the maps and results which we are describing. When these differ only slightly from items being discussed here, we will usually treat them briefly. For example, if [a, b] is partitioned into n subintervals, Rj will denote the map which restricts x in Ca-b to the jth subinterval, j = 1, 2 , . . . , n. Let T : Ca'b ->• Cb~a be the translation map
The restriction TO of T to CQ ' has range CQ a. Further, the stationarity of the increments of the Wiener process (see Proposition 3.3.17) yields
Equation (18.2.5) along with (18.2.7) below will be instrumental in the proof of formula (18.1.6); the latter equation will be used in turn in the proof of formula (18.1.8). There are three bijective maps, p\, p2, p3, onto product spaces which we will find useful. Given a < b < c, the map pi : CQ'C -> CQ' x C0'c is defined by
Since, by Proposition 3.3.18, the Wiener process has independent increments, we have
We will often regard p\, pi and p$ as identifying the spaces involved. For example, given x in CQ'C we will often write (y, z) in place of x, where y = R\x and z = RZX — x(b). It is then natural to write m a]C = ma,b x Tnj,,c rather than (18.2.7). The map p^ : Ca'b ->• Rd x C^b is defined by
We will frequently think of R4 x CQ 'b or, under the "identification" pi, Ca-b, as equipped with the measure Leb. xm a ,ft, where as before, Leb. denotes Lebesgue measure on Rd.
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NONCOMMUTATIVE OPERATIONS ON WIENER FUNCTIONALS
Finally, Given a < b < c, the map p3:
is defined by
We will sometimes think of Rd x Ca,b x CQ'C or, under the "identification" p3, Ca-c, as equipped with the measure Leb.xm fli £ x rrifoiC. Given (£, y, z) in W* x CQ' x C0'c, we will often write x = (£,y, z) rather than the more precisely correct equality x = pj1 (£, y, z). Similarly, given (£, y) in E.d x Ca b ', we will often write x = (£, y) rather than* = /^~ 1 (£,y). The spaces of continuous functions above are equipped with the sup norm topology. Under these topologies, R\ and R2 are continuous maps and T, p\, p2, and p^ are all homeomorphisms. Let F and F1 be C-valued Borel measurable functions on the space Ca'b which is identified, via the map p2 above, with Rd x CQ'*. Experience with standard measuretheoretic settings suggests that we require functions to be defined Leb. x ma,t-a.e. and that we regard F and F\ as equivalent provided that F(*+£) = Fi(jt+f)forLeb.xm a ,fo-a.e. (|, x) in Rd x CQ'*. However, due to the pathology of Wiener measure under change of scale (see Chapter 4) and the fact that a continuum of positive scalings enter into the definition of the analytic operator-valued Feynman integral (Definition 15.2.1), we need as in several places before to use more refined equivalence classes. (See Sections 14.2-14.4 and especially Sections 15.2.I and 15.7.) We will say that F is equivalent to F1, and write F ~ F1, provided that, for every p > 0, F(px + £) = F\(px + £) (or F(£ + px) = FI(£, px)) for Leb. x ma,b-a.e. (£, x) in Rd x C^b. We conclude this section with a rather technical proposition which will be needed as we continue; in particular, it will enable us to show that the operations * and +, which will be introduced in the next section, are well defined when thought of as acting on the appropriate equivalence classes of functions. Proposition 18.2.1 Let a < b < c. (i) Suppose F, F\ : Ca'b —> C are Borel measurable and that F ~ FI. Then F o RI, F, o #1 : Ca-c ->- C and F o R\ ~ FI o RI. Similarly, if H, H\ : Cb-c ->• C are Borel measurable and H ~ H\,thenHoR2, H\oR2 : Ca'c -»• CandHoR2 ~ H\oR2. (ii) Suppose G, G\ : Cc~b ->• C are Borel measurable and that G ~ G\. Then G o Tb~c o R2, G\ o Tb'c o R2 : Ca-c -^C and Go Tb>c oR2~G\o Tb*c o R2. Proof (i) Given x = (|, y, z) € C a ' c = Rd x CQ'* x Cb,c R1x = R\($,y,z) = (£,y) e R d xCo' f c = C a 'b.Letp 0 befixed.By assumption,F(py+§) = F{(py+%) or F(|, py) = Fi(f, py) for Leb.xma,fc-a.e. (§, y). Then (F o /?!)(£, py, pz) = (F1 o R)(t, py, pz); that is, F(£, py) = F,(|, py) for Leb.xma,b-a.e. (§, y) e Rd x C^'fo and every z 6 CQ' C . Certainly then (F o fli)(£, py, pz) = (Fj o R\)(%, py, pz) for Leb.xma,fe x mfc,c-a.e. (|, y, z) 6 Rd x CQ'* x CQ'C. In the proof just given, after R\ is applied to (£, py, pz), pz is no longer present. The corresponding statement is not quite true when R2 is involved and this makes the next proof more complicated.
THE NONCOMMUTATIVE OPERATIONS * AND
+
535
Given x = (£, y, z) s Ca'c = Rd x Ca,b x C0b'c, R2* = R2(£, y, z) = (I + Y(b), z) eR d xC0' c = Cb, c .Let/o > 0 befixed.Since H ~ Hi,H($,pz) = H1(£, pz) for Leb.xm&iC-a.e. (£, z) e Rd x C0b'C. Let M C Rd x Cg'c be the Leb.xmb,c,-null exceptional set; that is, M = {(E, z) e Rd x Cp'e : H(f, pz) #,(£, pz)}. We wish to show that (H o R2) Ay, pz) = (H1 ° R2)(f, Py, pz), that is, H(£ + py(b), pz) = H1(? + py(b), pz) for Leb.xma,b x mb,c-a.e. (£, y, z) e Rd x Ca,b x Cb'C. In fact, we will show more. We will prove that for every y e CQ' , H(£ + py(b), pz) = H1(f + py(b), pz) for Leb.xmb,c-a.e. (c, z) € Rd x Cp'c. Fix y 6 Ca,b. To complete the proof, we will show that M' := {(c, z) e Rd x Cg'c : (f + Py(b), z) € M) is a Leb.xmfo>c-null set. Since .M is Leb-xm^^-null, the section M.(z) is Leb.-null formb,c-a.e.z € CQ'C. Take any z such that M(z) is Leb. null. Sectioning M' at z, we obtain
But ./M^' is null and Lebesgue measure is translation invariant on Rd and so (M')^ = M{z) - py(b) is null. Since (M')(z) is Leb.-null for mb,c-a.e. z e Cb'c, M' is null as we wished to show. (ii) Let T = Tb'c : Cb
(18.2.10)
where the last equality follows from (18.2.5). Because of the second part of (i), we can establish (ii) by showing that G o T ~ G1 o T. Let p > 0 be fixed. We wish to show that (G o T)(%, pz) = (G| o T)(E, pz) or G(£,p7bz) = Gi(f,p7bz) for Leb.xmi>c-a.e. (^,z) 6 Rrf x CQ' C . Since G ~ G1, G(E, Px) = GI(§, pjr) except for a Leb.xmc_6-null exceptional set M. Thus .M = {(£,*) e Erf x Co~b : G($,px) ± GI(|,PJC)} is Leb. x m c _ fe -null. Let M' := {(I, z) € Rd x CQ'C : (£, T0z) e >f}. Note that X' = T- 1(M) . From this, (18.2.10), and the fact that M. is Leb. x mc_(,-null we can write (Leb. x mb,c)(M) = (Leb. x m b , C ) ( T - l ( M ) ) = (Leb. x mc_b)(M) = 0. It follows that G(E, pTbz) = G1(E, pF0z) for Leb. x mb,C-a.e. (§, z) e Rd x Cb,C. 18.3 The noncommutative operations * and + In this section, we introduce the noncommutative multiplication * and the noncommutative addition + and present some of their algebraic properties.
536
NONCOMMUTATIVE OPERATIONS ON WIENER FUNCTIONALS
Throughout the rest of this chapter, we will assume, usually without explicit mention, that thefunctions introduced are Borel measurable. (Actually, except for Theorem 18.3.2, the results of the present section can be thought of purely algebraically.) Also, as we continue, t , t 1 , t 2 , . . . will denote positive real numbers. Let F and G be functions on C'1 and C'2, respectively. Both F * G and F + G are to act on Ct1+t2. Given x e Ct1+t2, we define x1 e Ct1 and x2 € Ct2 by
and
Note that x1 is the restriction of x to [0, t1 ] and X2 is the restriction of x to [t1,t\ + t2] followed by translation to [0, t2]. In terms of the brief form of the notation from Section 18.2,
We now define F * G and F + G as C-valued functions on C'l+'2 via the formulas
and
Alternatively, we can write formulas (18.3.2) and (18.3.3) as
and
Remark 18.3.1 (a) In the definitions of * and +, one can begin to see connections with Feynman 's time-ordering ideas [Fey8] discussed in Chapter 14. Both equations (18.3.2) and (18.3.3) involve time-ordering. Indeed, note that F(x1) depends on the part of the path x over the interval [0, t\], whereas G(x2) depends on the part of x over the later time interval [t1, t1 + t2]. (b) It may well be natural for some purposes (such as the study of an evolving physical system) to work with the abutting intervals [0,t1 ] and [t1, t1 + t2] rather than with [0,t1] and [0, t2\. If this alternative were followed, given x 6 Ct1l+t2, we would define x\ as before but would take x2 := R2X as there would be no need for the translation T. Given functions F and G on Ct1 and Ct1,t1+t2,respectively, we would define F * G and F + G by (18.3.2) and (18.3.3) as before except that x2 would now be simpler. We will not pursue this line of thought in the present work even though it would be easy to make the necessary adjustments throughout the rest of the chapter.
THE NONCOMMUTATIVE OPERATIONS * AND +
537
We will need to work with equivalence classes of functions rather than with the functions themselves. Our first result shows that the equivalence relation ~, introduced in Definition 15.2.3 and discussed further in Section 18.2, is compatible with the operations * and +. Once we have this result in hand, we will usually follow the standard measure-theoretic convention and blur the distinction between equivalence classes and their representatives. Theorem 18.3.2 Let F, F] : Ct1 -> C and let G, G1 : Ct2 -» C. Further suppose that F ~ F1 and G ~ G1. Then
Proof Proposition 18.2.1 assures us that F o R1 ~ FI o R1 and that G o T o R2 ~ GI o T o R2. The result now follows from the compatibility of ~ with the usual product and sum (see Section 15.7) and from the definition of * and + as Equations (18.3.2') and (18.3.3') make particularly clear. D It is rather evident that * and + are noncommutative operations (even if t\ = t2); however, these operations do have a variety of pleasant algebraic properties as the next three theorems show. Theorem 18.3.3 (Algebraic properties of*) (1) Let F, FI : C" -> C and let G, G1 : Ct2 -> C; further, let a, B e C. Then
and
(2) Let F : C"1 -» C, G : C'2 -+ C, and H : C'3 -> C. Then (F * G) * H and F * (G * //) are C-valuedjunctions on C'l+'2+'3 and
(3) Lef 1 fee the Junction identically equal to one on C[0, 0] = C({0}). Then for all F : C' -»• C,
Proof The bilinearity (18.3.5) of * is easily verified as is (18.3.7). The key to establishing the "associativity" of * is to show that, for any x e Ct1+t2+t3, both sides of (18.3.6), when applied to x, yield F(x\)G(x2)H(x3,), where x\ and x2 are given by (18.3.1) and
The two arguments are similar. We carry out the proof that
538
NONCOMMUTATIVE OPERATIONS ON WIENER FUNCTIONALS
By definition of *,
where x'1(s) = x(s), s e [0, t\ + t2], and x'2(s) — x(t\ + t2 + s), s e [0, t3]. Note that x'2 — XT, so that the second factor on the right-hand side of (18.3.10) is just H(XI). Hence, it remains to show that (F * G)(x() = F(x\)G(x2). But, by definition,
where x'{(s) = x [ ( s ) , s e [0, ti], and x^(s) = x[(t\ + s), s e [0, t2\- Hence, it remains only to show that x" = x\ and x£ — X2But t\ + s e [t\,t\ + t2\ since ^ € [0, ^1 and so x((t\ + s) = x(t\ + s). Hence x%(s) = x\ (t\ + s) = x(t\ + s) for ^ e [0, t2\; that is, x% = X2 as desired. Finally, x\(s) — x(s) since ^ e [0, t\] and so *"(s) = x(s), s e [0,t\]. Hence x" = x\ as desired. Theorem 18.3.4 (Algebraic properties of +) (1) Let F, FI : C"1 -> C and let G, G{ : C'2 -* C; let a, ft 6 C. Then
(3) Let 0 be the function identically equal to zero on C[0,0] = C({0}). Then for all F : C' ->• C,
Proof The "associativity" of + is obtained just as in the proof of (18.3.6) by showing thatbothsides of (18.3.13), when applied to* € C'[+t2+'\ yield F(xi) + G(x2) + H(x3). As for the linearity of +, we note that (18.3.11), when applied to x e ct1+t2+t}, is equivalent to the equality
as required. We conclude this section by giving two properties relating * and 4-. Two further examples of such properties will be provided by equations (18.5.30), (18.5.32), (18.5.35) and (18.5.37) of Section 18.5.
THE NONCOMMUTATIVE OPERATIONS * AND +
539
Theorem 18.3.5 Let F : Cf' -*• C and G : C* -> C. (1) Then F + G, exp (F + G) and exp(F) * exp(G) all map C"1 +'2 to C and we have
(2) Let n be a positive integer. With the conventions indicated in Remark 18.3.6(b) below, we have
Remark 18.3.6 (a) By (18.3.15), we also have exp(G-j-F) = exp(G) * exp(F), but due to the noncommutativity involved, these quantities are not equal to exp (F + G). An analogous comment applies to (18.3.16). (b) In (18.3.16), we let F° = l t ] , where lf| is the function which is identically one on [0, t}]; further, G° is interpreted as It2, with It2 similarly defined. Proof of Theorem 18.3.5 We first establish the exponential formula (18.3.15). For* e C" 1+ ' 2 ,wehave
We next derive the binomial formula (18.3.16). Let x e C'1+'2. Then
Note that in the last equality of (18.3.17) we have made use of Remark 18.3.6(b) according to which
and
Hence the result is established. The reader should keep in mind that the results of this section hold for equivalence classes of functionals; however, except for Theorem 18.3.2, they clearly hold for pointwise operations as well.
540
NONCOMMUTATIVE OPERATIONS ON WIENER FUNCTIONALS
18.4 The functional integrals K[(-) and the operations * and + In this section, we present results relating the noncommutative operations introduced in Section 18.3 and the functional integrals K[(F) defined in Section 15.2.1. Let F and G be C-valued functions on C'] and C'2, respectively. We originally conjectured the formula
for F € Ati and G e At2 • (The definition of the Banach algebras At, t > 0, introduced in Section 15.7, will be briefly reviewed in Section 18.5.) It is still in the framework of the Banach algebras that we will provide in the next section an affirmative answer to our question, discussed in the Section 18.1, about the possibility of disentangling K^ (F)K'^(G). However, it turns out that equation (18.4.1) itself is true in much greater generality; we establish it here for the functional integrals K [ ( F ) , t > 0, A. e C+ and, in particular, for the Feynman integral (A. e C^ \ C+). The case where A > 0 is treated in Theorem 18.4.1, whereas the case A e C+ is dealt with in Corollary 18.4.4. (See [JoLa4, Theorem 4.1 and Corollary 4.1, pp. 85 and 88].) Theorem 18.4.1 Let X > 0 be given and suppose that F : C'1 -» C and G : C'2 -> C are such that K'^(F) and K^(\G\) exist. Then K[I +'2(F*G) exists and formula (18.4.1) holds. Remark 18.4.2 Let A. > 0 be given. If we assumed in Theorem 18.4.1 only that K'£ (G) exists, this would mean that, for any
exists as an absolutely convergent integral and defines a function of E which is in L2(Rd). [See equation (15.2.8) from Definition 15.2.1; further, recall that with our present notation, mt2 (= m) denotes Wiener measure on CQ.] It is not required by the definition that
is an L2-function of. However, our use of the Fubini theorem in the proof below does require this. It is in order to justify the use of the Fubini theorem that we have assumed the existence of K£ (\ G \). Proof of Theorem 18.4.1 The arguments will depend heavily on the notation and results given in Section 18.2. LetA. > Oandi/f e L 2 (R d )be given. The first equality in(18.4.4)belowfollows from the definition of *. (See (18.3.2').) The second, third and fourth equalities in (18.4.4) are the main steps in the proof. The second equality follows from (18.2.7), which relies on the independent increment property of the Wiener process; the fourth results from the application of the translation map given in (18.2.4) and from (18.2.5), which rests on the
THE FUNCTIONAL INTEGRALS K((-)
541
stationarity of the increments of the Wiener process; the third equality follows from the Fubini theorem. The use of the Fubini theorem will be justified below. In (18.4.4), we make implicit use of the maps p1, pi and p3 given in (18.2.6), (18.2.8) and (18.2.9), respectively, which identify appropriate spaces of continuous functions. We remind the reader that we write C'0 rather than C®''.
Note that in (1), (5) and (6) we have made use of the definition of K'^(F) given in (15.2.8). To justify the use of the Fubini theorem, it will be convenient to work backward beginning with expression (5) above. That expression exists with F, G and ty replaced by | F|, \G\ and |u | as we next explain. Since we have assumed that K'£ (\G\) exists, the function
542
NONCOMMUTATIVE OPERATIONS ON WIENER FUNCTIONALS
belongs to L2(Wd) as a function of f. Then, since K^(F) exists, we know that for Leb.-a.e. £ e Rd,
By combining (18.4.5) and (18.4.6), we obtain the existence (for Leb.-a.e. £) of expressions (5) and (4) in (18.4.4) with absolute values on F, G and i/r. Finally, to go from (4) to (3), we use (18.2.5) and the abstract change of variable theorem (Theorem 3.3.2), but, this time, with \F\, \G\ and |^| involved. This shows that the Fubini theorem can be applied to (3) (without absolute values) to yield (2). Hence, the proof of Theorem 18.4.1 is now complete. d Remark 18.4.3 (a) It is not our present concern, but it appears that the proof given above carries over to the setting of a certain class of stochastic processes which includes the Wiener process as one example. (b) Assume that GO : C'2 -*• [0, oo) dominates G in the sense that, for X > 0 and forLeb.xmt2-a.e. (f, w) e Rd x C%, \G(X~^2w + |)| < GD(l.-l/2w+$). Then, the existence of K£(GD) for X > 0 implies that of K^(\G\) (and of K'*(G))for A > 0. This simple observation will be of use to us in Section 18.5 (see, in particular, Lemma 18.5.5 and Theorem 18.5.6) and should be kept in mind by the reader throughout the rest of this section. Next we proceed to Corollary 18.4.4 and the case A. e C+. We wish to make assumptions on F and G which will insure the existence of ^1+/2(f * G) and the validity of formula (18.4.1) for all X e C+. A minimal set of assumptions would seem to be that K^(F) and K^(G) exist for all A. e C+. In fact, we will make one mild additional assumption: the existence, for all A. > 0, of A^ 2 (|G|). We remind the reader that the case A 6 C+ \ C+ (i.e. A. purely imaginary and nonzero) corresponds to the analytic operator-valued Feynman integral. (See Definition 15.2.1.) Corollary 18.4.4 Suppose that F : C'[ -+ C and G : C'2 -+ C are such that K^ (F) and K%(G) exist for all \ € C+ and that K%(|G|) exists for all A > 0. Then K1J+'2(F * G) exists for all A € C+ and
Proof According to Theorem 18.4.1, equation (18.4.7) holds for all A > 0; the cases A e C+ and A 6 C^ can now be dealt with by analytic continuation and strong continuity, respectively. Q Remark 18.4.5 Recall that, in Remark 18.3.1, we pointed out the connection between time-ordering and the operation * and +. Equation (18.4.7) is intimately related to this time-ordering. The function F in the formula (F * G)(x) = F(x\)G(x2) depends only on x\, the pan of the path x over [0, t\], whereas G depends only on X2, the part of x over [t\, t\ + t2\; correspondingly, the operators K'^ (F) and K'^ (G) appear first and
THE FUNCTIONAL INTEGRALS K{(-)
543
second, respectively, on the right-hand side of (18.4.7). [Since the operator K£ (G) acts first, formula (18.4.7) actually correponds to precisely reversing the time-ordering. This can be rectified by working with the "time-reversal" of the functionals involved, as is discussed carefully in Remark 15.4.4 and Section 15.7 (as well as in Remark 15.3.7 and Example 15.3.8).] Note that in order for equation (18.4.7) to hold, it is important that the functional integrals K[(F), defined in (15.2.8), act at the appropriate times. There is a natural definition of "commutator" relative to the operation *. Given F : Ct1 ->• C and G : C'2 -»• C, the commutator [F, G] of F and G is a C-valued function on C'1 +'2 which we define by
The next corollary shows that the functional integrals K'^(-) preserve commutators, a simple but possibly useful fact. Corollary 18.4.6 Let F : C" ->• C and G : C'2 -» C be given. (1) FixK>Q. Suppose that K^ ( \ F \ ) and K**(\G\) exist. Then K[l+'2([F, G]) exists and
where the bracket on the right-hand side of (18.4.9) denotes the usual commutators of bounded operators on L2(Rd); namely, [K'^(F), K'2(G)] := K[l(F)K^(G) K'2(G)K^(F). (2) Suppose that K^ (F) and K%(G) exist for all A. e C+ and that K^ ( \ F \ ) and K%(\G\) exist for all A > 0. Then, for all A e C~, Kt1 + t 2 ([F, G]) exists and (18.4.9) holds. Proof The existence of K!X[+'2(F * G) and K[l+'2(G * F)—and hence of the operator K[[+t2([F, G])—is insured by Theorem 18.4.1 and Corollary 18.4.4. A straightforward calculation now yields (18.4.9):
as desired.
The following result is easily obtained by combining Theorem 18.3.5 with Theorem 18.4.1 and Corollary 18.4.4.
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NONCOMMUTATIVE OPERATIONS ON WIENER FUNCTIONALS
Theorem 18.4.7 Let F : Ctl -> C and G : C 2 t 2 - C be given. (1) Fix A > 0. Suppose that A^1 (exp(F)) and K^2 (exp(Re G)) exist. Then Kl+'2(exp(F + G)) exists and
(2) Suppose that Kt1(exp(F)) and Kt2(exp(G)) exist for all A. 6 C+ and that Kt (exp(Re G)) exist for all X > 0. Then, for all A e C+, ^1+'2(exp(F + G)) exists and (18.4.10) holds. 2
Proof According to Theorem 18.4.1 (respectively, Corollary 18.4.4), we know that for A. > 0 (respectively, A e C+), K'^+'2 (exp(F) * exp(G)) exists and equals the right-hand side of (18.4.10). Moreover, by equation (18.3.15) and Theorem 18.3.5, exp(F + G) = exp(F) * exp(G). The present theorem now follows by combining these facts. Just as the exponential formula (18.3.15) was used to obtain equation (18.4.10), we could use the binomial formula (18.3.16) to obtain a related equation. We forego this here, but we will give such a formula in the Banach algebra setting, namely, equation (18.5.34) of Section 18.5. In fact, it is possible to give a variety of further formulas. (See, for instance, Examples 18.5.12-18.5.14 below.) However, we will limit ourselves in this section to one additional result; although it is a simple consequence of Corollary 18.4.6 and Theorem 18.4.7, the nature of the formula is perhaps slightly surprising. Corollary 18.4.8 Let F : C" -» C and G : C'2 -» C be given. (1) Fix X > 0. Suppose that /s^'(exp(Re F)) and /sT^(exp(Re G)) emf. TTzew K^ +t2 (exp(F + G)) and Kt +'2 (exp(G + F)) exist and
(2) Suppose that K1(exp(F)) and KT(2(exp(G)) exist for all X <= C+ and that AT('(exp(Re F)) and A^2(exp(Re G)) exist for all A. > 0. Then, for all X e C+, ^'+'2 (exp(F + G)) anrf ^'+tl (exp(G + F)) exwr and (18.4.11) toWs. Proof The existence of the expression on the right-hand side of (18.4.11) is guaranteed by Theorem 18.4.7. Further, by (18.4.8) and (18.3.15),
Equation (18.4.11) now follows by appling K^+'2 to both sides of (18.4.12) and using Corollary 18.4.6. D 18.5
The disentangling algebras At, the operations * and +, and the disentangling process We begin this section by reviewing facts about the Banach algebra At introduced in Section 15.7. We show, in Theorem 18.5.3, that the family of Banach algebra (A '• t > 0}
THE DISENTANGLING ALGEBRAS A,
545
is closed under * and 4- by proving that, if F € Atl and G e At2, then F * G and F + G are in Atl+tr Several formulas given in the functional integration framework of the last section are seen to hold in the Banach algebra setting; in particular, we show that, if F € AH and G 6 A,2, then (without further hypotheses) K^+'2(F * G) = Kt^(F)Kt)2(G) for all X e C+. An immediate corollary of this result and of our earlier work in [JoLal] (see Chapter 15, especially Theorem 15.4.1 and Corollary 15.4.3) is that K^ (F)K^(G) can be disentangled via a generalized Dyson series, thus resolving in the affirmative the question raised in Section 18.1 and which initially motivated our work in [JoLa3,4]. We will see in this section that the noncommutative operations * and + combine with the Banach algebra framework and functional integration to provide a rich interlocking structure. Let t > 0 be fixed for now. We recall briefly some definitions and notation from Chapter 15. More explanation and related references can be found in Sections 15.2,15.4 and 15.7. Let M[0, t) denote the space of C-valued Borel measures on [0, t). Given n e M[0, t), a C-valued Borel measurable function 9 on [0, t) x Rd is said to belong to Looi;n - £<x>i;>n[0, t)if
where |n| in (18.5.1) denotes the usual total variation measure associated with n) (see Section 15.2.G). Recall from Section 15.7 that functions (often called functionals) in At are formed as follows: take a sequence {Fn}^L0 of functions on C' each of which is given by an expression of the form
with mn a nonnegative integer, Nn,u e M[0, t), and 0n,u e Loo1;Nn,u• Assume that
Define a functional F by
It is shown in Corollary 15.4.3 that for every k > 0, the series in (18.5.4) converges absolutely for Leb.xm/-a.e. (|, x) € Rd x C'0; then At is defined as the set of all equivalence classes of functionals F obtained in this manner. The representation of functionals in (18.5.4) and even in (18.5.2) is not unique. For F € At, we let \\F\\t be the infimum of the left-hand side of (18.5.3) for all representations of F of the form (18.5.4). In Theorem 15.7.1, we have shown that (At, \\ • \\t) is a commutative Banach
546
NONCOMMUTATIVE OPERATIONS ON WIENER FUNCTIONALS
algebra under pointwise multiplication and addition such that, given F € At, exists for all A. e C+ and satisfies
K^(F)
The inequality (18.5.5) shows that the linear operator K( from At to £(L 2 (R d )) is bounded. (See Corollary 15.7.3.) One of the main results of Chapter 15 shows that each operator K[(F), F e At, is given in disentangled form by a generalized Dyson series (GDS). See Corollary 15.4.3 (in conjunction with Theorem 15.4.1). As in Section 15.7, we will often refer to the Banach algebra At as the "disentangling algebra". Roughly speaking, a GDS is a timeordered perturbation expansion of operators. One operator K[(F) may have many GDSs correponding, in particular, to various representations of the form (18.5.4), (18.5.2). Remark 18.5.1 (a) As was explained in various places in Chapters 15-17, the use of Lebesgue-Stieltjes measures in the definition of At enabled us to blend continuous and discrete structures. We used the space Af (0, t) throughout Chapter 15 with only a few exceptions; however, the use of the open interval (0, t) was merely a matter of convenience as was essentially noted in Section 15.2.F. (Also see Chapter 17.) (b) If the measures nn:U in (18.5.2) are all taken to be Lebesgue measure I, we obtain a Banach subalgebra At of the disentangling algebra At- (See Remark 15.7.2.) This Banach subalgebra was introduced and studied in [JoSk6, esp. pp. 121-124] except that, in [JoSk6], the functions on,u were taken to be Lebesgue measurable rather than Borel measurable. A Banach algebra closely related to At was introduced earlier in [JoSk2]. In [JoSk6], a Dyson series was given for each F e At- These perturbation series are rather general in certain respects but possess a much less complicated combinatorial structure than the GDSs from [JoLal] and studied in Chapter 15. The papers [JoSk2, 6] are relevant to Feynman 's operational calculus. However, the connection, although suspected by the authors of [JoSk2,6], was not understood at the time. Our first theorem provides the key step in showing that if F is in At, and G E At2, then F * G and F + G are in Atl+t2Theorem 18.5.2 Let t > 0 and suppose that H e As-r, where 0 - < s < t. Let R : C' —* Cr's be the restriction map (i.e. the map that restrictsx in C' to the subinterval [r, s]) and let T : Cr's -> Cs~r be the translation map. Then H o T o R e At and
Remark. Note that T = Tr-s as in (18.2.4) but that R is slightly different than R\ and R2 from (18.2.2) and (18.2.3), respectively, since the restriction is allowed to be to any subinterval. Proof of Theorem 18.5.2. Let H e As-r- We take an arbitrary representation of H in terms of measures <$„,„ e M[0, 5 - r) and functions finiU e £ooi;/3 nu [0, s - r),
THE DISENTANGLING ALGEBRAS A,
547
n = 0, 1, 2 , . . . , u = 1,2, ...,mn, such that
where
Translate £2n,u and fin,u to [r, s). Call the resulting functions and measures &n\ and Pn,u> respectively. Mure precisely,
and, given B 6 B([r, s)), the Borel class of [r, s), we have
In view of (18.5.9), we have
Next we extend Q^u and fi^l to [0, t) by letting both be zero off the interval [r, s). Call the resulting functions and measures £2J^ and /3^, respectively. In particular, Pn2l(B) = Pn]l(B n [r, s)), for B e B([0, t ) ) . Note that A® € M[0, t) and that
where we have used (18.5.10) in the last equality. In view of (18.5.11), £2^, € L v[0,t). °°l;Bn,u
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(18.5.12) It now follows from (18.5.1), (18.5.11), (18.5.12) and the definition of the norm as the infimum of the left-hand side of (18.5.3) taken over all representations that
Since the representation of H in (18.5.7) was arbitrary, it follows from (18.5.13) that H oT oR e At and \\HoToR\\t < \\H\\s-r, as desired. The following theorem will enable us to study the operations * and 4- in the context of the Banach algebras [At '• t > 0}. It will also serve as a connecting link with the results of the previous sections. Theorem 18.5.3 If F e Atl and G e At2, then F * G and F + G are in At1 +t2- Further, we have
and (18.5.15) Proof Let x 6 C'1+'2. By definition of the operations * and + (see (18.3.2') and (18.3.3')),
where R1 : C'1+'2 -> Cfl and R2 : C'1+f2 ->• Ct1,t1+'2 are restriction maps and T : C'i,ti+t2 _^ c'2 is the translation map. The map R of Theorem 18.5.2 corresponds to RI and R2, respectively, for the special cases (i) r = 0, s = t1 and t = t\ + t1 and
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549
(ii) r = t1 , s = t = t\ + t2- Further, the translation map of Theorem 18.5.2 corresponds to T in case (ii), whereas it is equal to the identity in case (i). We deduce from Theorem 18.5.2 that F o R1 and G o T o R2 belong to At1+t2 and that
It now follows from (18.5. 16) and (18.5.17) that F * G and F 4- G are in A,t+,2 and that, by (18.5.18),
and
as desired. Note that we have used here the fact that Atl+t2 is a normed algebra (Theorem 15.7.1). We emphasize to the reader that each disentangling algebra At has its own operations, pointwise addition and commutative multiplication. The operations * and + are additional operations which act on a pair of algebras Atl x A,2 and produce an element of the algebra At1 +t2 . The next result shows that * and + are compatible with the structure of the Banach algebras (A, II • II/), ' > 0. Proposition 18.5.4 The operation * (resp., + ) is C-bilinear (resp., C-linear) and continuous from At{ x At2 to A^+tr Further, * and + are "associative" in the sense that if F € Ati, G € -^2 an^ H 6 At3, then the following equalities hold in Atl+t2+t3'-
and
Proof The algebraic properties follow from their counterparts in Theorems 18.3.3 and 18.3.4 as well as from Theorem 18.5.3. The continuity of* (resp., + ) is a consequence of the bilinearity (resp., linearity) and the norm estimate (18.5.14) (resp. (18.5.15)). Lemma 18.5.5, to follow, will be useful in the proof of Theorem 18.5.6 below and is of some independent interest. Lemma 18.5.5 Let G e At . Then there exists GD € At which dominates G. (Compare (18.5.21) and (18.5.22) below.) Further, K((\G\) exists for all X > 0.
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Proof Consider a representation of G of the form
where
Define
The functional GQ dominates G in the sense that
for all X > 0 and Leb.xm,-a.e. (£,*) € Rd x CQ. Clearly, GD € A- Consequently, by Theorem 15.7.1, K t ( G D ) exists for all A e C^ and, in particular, for all A, > 0 as we recalled just preceding (18.5.5). It now follows from Remark 18.4.3(b) that K[(\G\) exists for all A. > 0. Note that Lemma 18.5.5 does not assert that |G| € At but only that Kt(\G\) exists for all X > 0. The nexttheorem ([JoLa4, Theorem 5.3, p. 96]), which is now an easy consequence of our earlier results, is one of the main theorems of this chapter. As an immediate corollary, we obtain a solution to the problem concerning disentangling which was stated in the introduction and that initially motivated this work. Theorem 18.5.6 Let F e At1 and G e At2- Then, for all X e C+, the analytic operatorvalued Feynman integrals K[{ (F), K[2 (G) and Kt1 +'2 (F * G) exist and we have
Proof Recall again that if H € At, K[(H) exists for all A > 0. The result now follows from Theorem 18.5.3 according to which F * G 6 Atl+t2, Corollary 18.4.4 on functional integrals, as well as Lemma 18.5.5. Note that this lemma justifies the fact that the hypothesis of Corollary 18.4.4, stating that K^(\G\) exists for all A > 0, is satisfied. D
Corollary 18.5.7 If F e A, and G e A,2, then, for all X e C+, K^ (F)K^(G) can be disentangled via a generalized Dyson series (GDS) for Kt1 +t2 (F * G). Proof Recall that if H 6 At, then, for all X € C^, K{(H) can be disentangled via a GDS (Theorem 15.7.1). But F * G is in the disentangling algebra At1 +t2 and so the result follows from equation (18.5.23) of Theorem 18.5.6.
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551
The next result is the counterpart of Corollary 18.4.6 and is an immediate consequence of Theorem 18.5.6. Corollary 18.5.8 Let F e Atl and G e At2. Then [F, G] := F * G - G * F is in A
In reference to the next result, the counterpart of Theorem 18.4.7, we mention that the holomorphic functional calculus for Banach algebras ([Nai, pp. 202-205], [KadRi, §3.3]) assures us that exp(H) e At if H e At. Since the Banach algebra operations are just pointwise addition and multiplication, exp(H), in the sense of the Banach algebra At, coincides with exp(H), which was defined pointwise in Section 18.3. Theorem 18.5.9 Let F e Atl and G e At2. Then the following equality holds in A,l+t2:
Proof Equation (18.5.25) is just equation (18.3.15) of Theorem 18.3.5, interpreted in At1+t2- Equation (18.5.26) now follows from Theorem 18.5.6. Note that the fact that H e At implies that K^ (exp(H)) exists for all A. e C+. Remark 18.5.10 (a) The exponential formula (18.5.26) is the counterpart of Theorem 18.4.7. Observe that this result, along with Theorem 18.5.6, Corollary 18.5.8, and Corollary 18.5.11 below, takes a simpler form in the Banach algebra setting than in Section 18.4. (b) It is well known that if A and B are noncommuting operators, then the formula
may fail. Nevertheless, Feynman gives examples which show that equation (18.5.27), properly understood using his time-ordering convention and "disentangling ", may also "hold" in a certain sense and be useful when A and B are noncommuting [Fey8, equation (4), p. 110 and equation (8), p. 111]. There are various ways of making Feynman's ideas precise in certain cases [Ar, Gil, GilZa, JoLal, Masll, NazSS, Ne3, ...]. In our present setting, the form of the paradoxical formula (18.5.27) is preserved in formula (18.5.25), but of course we have changed the ordinary algebraic operations + and • to the noncommutative operations + and *, respectively. Note that we are not working directly with operators but with functionals; we first establish (18.5.25) for functionals and then derive the corresponding formula (18.5.26) for the operator-valued functional integrals. (c) In formulas (18.5.25) and(18.5.26), one sees particularly clearly the relationships between aspects of the present work and that of Lapidus on the "Feynman-Kac formula
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with a Lebesgue-Stieltjes measure" [Lal4-18] discussed in Chapter 17. (See especially Theorem 1 7.2. 7 [Lai 5, Theorem 2.4, p. 99] and the comments preceding Remark 1 7.5. 7.) (d) In [Lal9], an axiomatic formulation of key aspects of Feynman's operational calculus is developed in which, for instance, the abstract analogue of formula (18.5.25) holds, as will be discussed in Appendix 18.6 below. The last corollary which we will formally state is the counterpart of Corollary 1 8.4.8 in the setting of the Banach algebras [At : t > 0}. Corollary 18.5.11 Let F e At1 and G € At2. Then the following equality holds in
further, for all h e C=,
Proof The result follows immediately from Corollary 18.4.8 with the aid of Lemma 18.5.5. Examples: Trigonometric, binomial and exponential formulas Many further formulas hold in the setting of the Banach algebras (At : t > 0}. We conclude this section by simply stating three examples. Example 18.5.12 (Trigonometric formulas) (i) Let F € At] and G € At2, then the following equality holds in Atl+t2'.
(ii) Similarly, under the same assumptions as in (i), we have the following equality in Atl+t2-
fu rther
,foralh
e C+,
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553
Example 18.5.13 (Binomial and multinomial formulas) (i) Given F e A, and G € At2, the "binomial formula" (18.3.16) holds in Atl+t2, and, for all A. e C+,
(ii) More generally, let F/ € Atj, j: = 1,..., h. Then the "multinomial formula"
holds in A,l+...+tk and, for all A 6 C+, the following equality holds in £(L2(Rd)):
Recall fromRemarkl8.3.6(b)that F° = 1t1, and G° = 1t2. It follows that A^(F 0 ) = exp[-?i(//oA)] and K^(G°) = exp[-f2(#oA)L where HO is the free Hamiltonian acting in L2(E.d). Some basic facts about the holomorphic semigroup exp(-z//o) are reviewed in Sections 10.2 and 15.2.E. Out last example extends Theorem 18.5.9 to a finite number of terms. Example 18.5.14 (Exponential formula) Let Fj• e Atj, j' = 1 , . . . , h. The equality
holds in Att-+...+th and so, for all A. e C+,
A careful discussion of formulas (18.5.35)-(18.5.38) should involve the maps RJ, j = 1,... ,h, which were mentioned in Sections 18.2 just before equation (18.2.4). Exercise 18.5.15 Establish the facts stated in Examples 18.5.12-18.5.14 above. [Hint: In view of Theorem 18.5.6, it suffices to prove the formulas at the level of the algebras At-] 18.6
Appendix: Quantization, axiomatic Feynman's operational calculus, and generalized functional integral We have developed in Chapters 15 and 16 a rigorous theory of Feynman's operational calculus in the context of quantum mechanics and based on the use of Wiener integrals and (analytic-in-mass) Feynman integrals. Partly motivated by the results of Chapters 15-16 ([JoLal]) and 17 ([Lal4-18]), we have then defined in the present chapter ([JoLa4]) noncommutative operations * and + acting on certain Banach algebras of Wiener functionals
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(called the "disentangling algebras" and denoted At = (At, \\ • \\t)- Further, we have used these operations—in conjunction with the path (or "functional") integrals constructed in Chapter 15—to "disentangle" (in the sense of Feynman's operational calculus [Fey8]) the operators involved. In this appendix, we discuss a general axiomatic framework for (part of) Feynman's operational calculus, proposed by Lapidus in [La 19]. It puts in a broader and more conceptual framework the theory developed in this chapter as well as could significantly enlarge its potential range of applications. In a nutshell, it is expressed in terms of a family of Banach algebras {At}t€G parametrized by a semigroup G (that plays the role of "time" and is denoted additively), two noncommutative operations * ("multiplication") and 4- ("addition") which let these algebras "interact" at different "instants" t e G, and of a "generalized functional integral" K,'(t e G), from At to a fixed Banach algebra B. Intuitively, this integral enables us to "quantize" the elements of the algebra of "symbols" At. We refer to (18.6.1)-(18.6.8) below for a precise statement of the axioms proposed in [Lal9]. We simply mention here that for t1, t2 in G, * and + are maps from At1 x At2 to At1+t2 and that it follows from the axioms (see Theorem 18.6.10) that if a e Atl and b e At2, then the equality exp(a+b) = exp(a) * exp(b) holds in At1+t2, much as in Theorem 18.5.9, except that the underlying commutativity making such an identity possible is now more apparent (see Theorem 18.6.8 and Remark 18.6.17(a)). It should be made clear from the outset that considerable difficulties may have to be overcome in order to obtain, in each concrete situation of physical or of mathematical interest, the rich interlocking structure (including the "quantization map" K,t) postulated in [La 19] and discussed below. Having stated the axioms proposed in [La 19], we will deduce some of their main consequences and then see, in particular, that they are satisfied by the structure (({At}t>o, +, *), (K.' '•— K[)t>o) constructed in Chapters 15 and 18; that is, by the family of "disentangling algebras" At equipped with the noncommutative operations * and +, together with the (operator-valued) functional integrals K[ (if A. > 0) or with the analytic-in-mass Feynman integrals K'_- (if A. = —iq ^ 0 is purely imaginary). We will close this appendix by some comments of a more speculative nature concerning possible connections with other mathematical or physical theories. Algebraic and analytic axioms Let G be a semigroup, denoted additively. (In contrast with the rest of this book, the word "semigroup" refers here to an algebraic semigroup (or monoid), and not to a semigroup of operators as in Chapter 8 or 9. The main examples that we have in mind for the applications are G = (0, +00), the additive semigroup of positive real numbers, and G = N* = {1, 2, 3,...}, the additive semigroup of positive integers.) (18.6.1) For each t € G, let At be a (real or complex) unital algebra, with unit element denoted 1t and with zero denoted Or. Given any t1,t2 e G, we assume that there exist two operations (usually noncommutative) * and + such that * (resp., +): At} x At2 -> Ati+t2 is bilinear (resp., linear) and "associative" in the following sense: iftj e G and
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Remark 18.6.1 Heuristically, as was alluded to above, * (resp., +) can be thought of as a kind of noncommutative "multiplication " (resp., "addition ") that enables us to let the algebras Att and At2 "interact" at the "times" t\ and t2 in G. In addition, we assume that the following algebraic axioms, (18.6.2)-(18.6.5), are verified:
Remark 18.6.2 (a) In (18.6.2)~(18.6.5), the stated equalities hold between elements of the algebra Att+t2. Moreover, on the right-hand side of (18.6.3), for example, a1b1 indicates the product of a\ and b1 in the algebra At( whereas the sign "+" indicates the addition in the algebra •4(1+t2- (Of course, an entirely analogous comment can be made about each of the algebraic axioms (18.6.2)-(18.6.5).) (b) We can complete the above axioms in the case when G has a unit element or is a group. However, this extension will not be needed here. (c) The semigroup G (except in axiom (18.6.8) below) or the algebras A[(t e G) are not necessarily assumed to be commutative (i.e. Abelian). (d) Naturally, when we say that the operations * and + are "noncommutative ", as in Remark 18.6.1 above, we mean that even when G is Abelian (so that Atl+t2 = At2+t1), we usually have a\ *a2 ^ a2*a\ and a1+a2 = a2+a1, in the algebra Atl+t2 = Ai2+tl, fora\ e AH and 02 e A,2. We assume, in addition, that the following analytic axioms, (18.6.6) and (18.6.7), are satisfied:
Remark 18.6.3 It follows at once from (18.6.1) and (18.6.6) that the mappings * and + are continuous from At1 x At2 to At1+t2. ( 1 8.6.7) (Density axiom) For all t1,t 2 e G, the set V := {a1 * a2 : a1 e At1 , a2 e At2 } is total in At1 +t2 : that is, the space of (finite) linear combinations of elements of V is dense in (At1+t2, \\ • ||<1+(2). In order to state our last axiom, we introduce a Banach algebra B (independent of t e G) and also assume that G is Abelian (so that Atl+t2 — At2+tl , for all t\, t2 e G).
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(1 8.6.8) (Integration or "quantization" axiom) For every t e G, there exists a continuous linear map K,' : At —> B such that for all tj € G and aj e Atj (j = 1 , 2), we have
where [a1 , 02] := a\ * 02 — a2 * a1 (s -A tl+t2 ) and where the second bracket in (18.6.9) denotes the commutator in the algebra B; that is,
. Moreover, we assume that supt€G |k' || < oo, where \\ • \\ denotes the norm in £(At, B). Actually, we will be mostly interested in the following "multiplicativity property", which is stronger than that of preserving commutators. Definition 18.6.4 We will say that K,' is multiplicative if instead of requiring (18.6.9) in axiom (18.6,8), we assume that
Remark 18.6.5 (a) Property (18.6.9') does not require G to be Abelian; however, when G is commutative (as will always be the case below), (18.6.91) clearly implies (18.6.9). (b) In the applications [as in Theorem 18.6.16 below where B = C(L2(Rd))], B will frequently be an algebra of operators. Thus, intuitively, the "integral" ACf enables us to associate an "operator" K}(d) with the "symbol" a e At, or in other words, to "quantize" the elements of the algebra At. (c) It is immediate to check that [-, •]—defined as in (18.6.8)—is a "Lie bracket", in the sense that it is skew-symmetric and bilinear from At{ x A,2 to Att+t2 and satisfies the obvious analogue of the "Jacobi identity" in this context; namely, if aj e Atj(j = 1,2,3,), then
Consequences of the axioms We now assume that ({At}teG> +, *) satisfies axioms (18.6.1)-(18.6.7) and that ({Kt}teG, B) satisfies axiom (18.6.8). The following results—obtained in [La 19]—are simple consequences of those axioms but may provide a better conceptual understanding of the approach to Feynman's operational calculus discussed in the present chapter, as well as broaden its potential range of applications. Theorem 18.6.6 We have the following equalities:
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(iii) The map L (resp., R), defined by L (a) = a* \t2fora e At} (resp., R(b) = 1?1 *b for b e At2), is a continuous algebra homomorphism from A^ (resp., At2) to Remark 18.6.7 Intuitively, the elements of the subalgebra L(Ati) (resp., R(At2)) can be thought of as the "left" (resp., "right") elements ofAt}+i2. Theorem 18.6.8 (Canonical decomposition of * and +) Let L and R be defined as in Theorem 18.6.6(iii). Then, for all a € At( andb e At2, wehave the following equalities inAtl+t2: Moreover, for all a e At\ and b € At2, we have Remark 18.6.9 In the terminology of Remark 18.6.7 above, equation (18.6.11) implies that all the "left" elements of the (not necessarily Abelian) algebra Atl+t7 automatically commute with its "right" elements. Then, equation (18.6. JO)provides a canonical decomposition of a * b (resp., a +b) as the product (resp., sum) of a "left" and a "right" element of At\ +t2 • We have replaced the order relation (between the time parameters) and "Feynman's time-ordering convention" (14.2.1) by the translation in the semigroup G and the noncommutative operations * and 4- acting on the algebras At(t e G). In this way, we can justify in a general framework the "paradoxical formulas" suggested by [Fey8] and obtain many identities between expressions involving noncommuting variables, as we did at the end of Section 18.5 in the concrete setting of the "disentangling algebras". We give two such identities in the following theorem. Theorem 18.6.10 Fora 6 Atl andb e At2, we have the following equalities in At[+t2:
and
[Here, the exponential ofu 6 At is defined by means of the analytic functional calculus in the Banach algebra A,.] So far, we have only used axioms (18.6.!)-(18.6.7), but the following corollary of Theorem 18.6.10 also makes use of axiom (18.6.8). Corollary 18.6.11 Let a 6 Atl and b e A,2. Then If, in addition, K? is multiplicative (in the sense of Definition 18.6.4), then
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Remark 18.6.12 Of course, all the identities stated at the end of Section 18.5 have an exact counterpart here. In particular, thanks to the "associativity " of + and * (see axiom (18.6.1)), we can generalize formulas (18.6.12) and (18.6.13) to an arbitrary (finite) number of elements, much as in (18.5.34) and (18.5.37), respectively. We now briefly explain how to establish the above results (Theorems 18.6.6, 18.6.8 and 18.6.10). We shall use the following simple lemma, which is an immediate consequence of axiom (1 8.6. 1) and of the "density axiom" (18.6.7). Lemma 18.6.13 If u, v e At]+t2 are such that u * (a1 * ai) = v * (a1 * a2), for all aj €. Atj (j = 1, 2), then u = v. Proof of Theorem 18.6.6 (i) According to axioms (18.6.5) and (1 8.6.4), we have for all a, a1 e At1 and b € At2'- (a * If 2 )( a i * a2) = (aa\) * 02 = (a-j-0/2)(ai * 02)', hence the conclusion, in view of Lemma 18.6.13. The second equality in (i) is proved in exactly the same way. (ii) According to (i), 0(,+0r2 = 0(, * 0/2 = Ot\+t2 while lf, * l,2 = Itl+t2 follows from (18.6.5) and Lemma 18.6.13. (iii) is now easily proved. Proof of Theorem 18.6.8 We deduce from axiom (18.6.5) that (since L(a) = a * It2 and R(b) = 1,, * b)
indeed, by (two applications of) (18.6.5), we have successively:
Moreover, by (18.6.3) and Theorem 18.6.6(ii),
Hence the esult.
Proof of Theorem 18.6.10 Since L(a) and R(b) commute with each other and the maps L and R are continuous algebra homomorphisms (by Theorems 18.6.8 and 18.6.6(iii)), we have in view of (18.6.10):
thus (18.6.13) holds. Formula (18.6.12) is proved in much the same way, by using Theorems 18.6.6(iii) and 18.6.8, except that it does not require the continuity of L and R. D
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559
We next briefly indicate how to embed the family of Banach algebras (•At, || • \\t)tsG into a single Banach algebra (A, \\ • ||) and to extend the operations * and + to all of A. (See Remark 18.6.15 below.) Let
equipped with the norm ||a|| := J^teG \\at\\t anc* the natural vector space operations. For a = (at)teG and b = (bt)tec in A, we define a * b and a+b in A by
and
Finally, we define K : A-- B by
Then we can easily obtain the following theorem, whose verification we leave to the reader. Theorem 18.6.14 The space (A, \ \ - \ \ ) is a Banach space. Moreoever, the operations + and* given by (18. 6. 16) and the "integral" K. given by (18.6.17) are well defined and satisfy the analogue of axioms (18.6.1) and (18.6,8). Remark 18.6.15 (a) The above construction is mostly of interest when G = N* = {1,2,...}. When G = (0, +00), the additive semigroup of positive real numbers, the notion of direct integral (rather than that of summable family) would be more appropriate, but we will not develop this subject here. (b) Theorem 18.6.14 can be used in the concrete setting studied in this chapter (except for (0, +00) replaced with N*) and revisited below. Examples: the disentangling algebras and analytic Feynman integrals We now illustrate the above axioms by considering the concrete situation studied in the rest of this chapter (see especially Sections 18.4 and 18.5). Let G = (0, +00), the additive semigroup of positive real numbers. For t > 0, let At be the "disentangling algebra" of (equivalence classes of) Wiener functionals, as in Sections 15.7 and 18.5. Further, let + and * be the noncommutative multiplication and addition of functionals defined by (18.3.2) and (18.3.3), respectively. Finally, given A e C+ and t > 0, let 1C' := Kt : At ->• C(L2(Rd)) be the (operator- valued) "functional integral" defined in Section 15.2.1 and used, in particular, in Sections 18.4-1 8.5. Recall that for A > 0, Kt is given by a bonafide Wiener path integral, while it is obtained by analytic continuation for A € C+\(0, +00) and passage to the limit for A = —iq (q real, q = 0). In particular, for A = 0 purely imaginary, Kt = Kl_iq is given by the (operator-valued) analytic (in mass) Feynman integral.
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Theorem 18.6.16 For every fixed A € C+, the system of "disentangling algebras" ({A}r>0> +, *) and of "integration maps" (K' =: K[)t>o —constructed in Chapters 15 and 18 (see especially Section 18.5) —satisfies axioms (18.6.1)-(18.6.8), with G := (0, +00) and B := £(L2(lRd)). Moreover, the integrals K.' = Kt are multiplicative (in the sense of Definition 18.6.4). Proof One checks easily (18.6.2)-(18.6.5). Further, one deduces (18.6.1) and (18.6.6) from Proposition 18.5.4 and Theorem 18.5.3, respectively. Also, the multiplicativity of 1C' = Kt is established in Theorem 18.5.6. Finally, the "density axiom" (18.6.7) can be obtained by analyzing an appropriate set of generators of the algebra At (see Section 15.7 and equations (18.5.2)-(18.5.3) above). The verification of this last statement is left as an exercise for the reader. Remark 18.6.17 (a) Note that even in the present case, the above axiomatic approach ([La] 9]) enables us to better understand the algebraic and analytic structures underlying the theory developed in this chapter ([JoLa4]). For instance, since the "disentangling algebras " At are commutative (cf. Theorem 15.7.1) it was not obvious a priori that the "left" and "right" elements had to commute automatically. (See Theorem 18.6.8 as well as Remarks 18.6.9 and 18.6.7.) (b) More precisely, given F 6 Atl and G € At2, the associated "left", L(F), and "right", R(G), elements (in the sense of Theorem 18.6.8 and Remark 18.6.7) are the elements of At1+t2 given respectively by
where x\ e C'1 , X2 e C'2 and (just as in equations (18.3.1) and (18.3.1')) x\ (s) = x(s) fors e [0, t\], while x2(s) = x(t\ +s)fors e [0, t2]. (Formula (18. 6. 18) follows easily from the definition of the maps L and R given in Theorem 18.6.6(iii).) Then, of course, in this concrete situation, the "canonical decomposition"
of* and + obtained in Theorem 18.6.8 (see equation (18.6.10)) is precisely equivalent here to the original definition of these operations given by formulas (18.3.2) and (18.3.3); namely, in view of (18.6.18) and (18.6.19), we have
We close this appendix by some very brief comments of a more speculative nature. Remark 18.6.18 Although this did not constitute our initial motivation and, in particular, the theory of [JoLal-4] was developed independently and with different objectives in mind — it would be interesting to extend the present axiomatic framework (from [Lal9]) to supersymmetric algebras and to establish connections between aspects of our work in this chapter ([JoLa4]) and aspects of string theory [Gre-SWit, Kak] and Cannes' noncommutative geometry [Conl,2J. Specifically, in Witten's "string field theory" [Wit5,6] —which combines the use of formal path integrals and of noncommutative
APPENDIX: AXIOMATIC FEYNMAN' S OPERATIONAL CALCULUS
561
geometry—an operation resembling our *-multiplication is defined, which describes physically the interaction between two (open) strings. [There are, of course, a number of technical differences between the two settings; in particular, in [Wit5,6], the endpoints of the interval parametrizing an open string are fixed, whereas in our case (see Sections 18.1-18.5 and Remark 18.6.17 above), the right endpoint of the interval parametrizing a path is allowed to vary.] We hope that some of those analogies—which were pointed out in this context in [Lal9]—will be developed and turn out to be useful in future work.
19 FEYNMAN'S OPERATIONAL CALCULUS AND EVOLUTION EQUATIONS 19.1 Introduction and hypotheses In Chapters 15-18, much of the focus has been on making Feynman's operational calculus mathematically rigorous via the use of rigorous path integrals. Specifically, it was the path integrals of Wiener and Feynman that were used for this purpose; accordingly, the theory was restricted to the diffusion or quantum-mechanical settings, respectively. The present chapter is based on joint work [dFJoLal ,2] of the authors with Brian DeFacio. (See especially [dFJoLa2], announced in part in [dFJoLal]). Here, we take a more general approach suggested in II of Section 14.4 and use Feynman's heuristic rules to disentangle the exponential of sums of integrals of operators and arrive at a formula which can be shown to make sense (see Theorem 19.4.1) and to give the unique solution (see Theorems 19.5.1 and 19.6.1) to an associated evolution equation in integral form. These results provide one resolution of a mathematical problem which is suggested implicitly but naturally by Feynman's paper [Fey8]. In [Fey8] and [Fey7], Feynman used his rules to disentangle the exponential of certain sums of integrals of operators. He was confident that the resulting expression would describe the evolution of an associated physical system. The main goal of this chapter is to justify Feynman's confidence in a mathematical sense under a precisely identified set of assumptions. Whatever Feynman's paper may lack in precision is more than offset by the rich supply of ideas found there. Several lines of research are suggested by this paper [Fey 8]. A partial list of references on this topic was given at the end of Chapter 14. As in Chapters 14—18, measures other than Lebesgue measure will often be used to give directions for disentangling. This is in contrast to [Fey8] which focused on Lebesgue measure. However, except for Examples 19.7.1, 19.7.3, and 19.7.4, we will limit our attention to the case of continuous measures u; that is, measures which assign zero mass to all single point sets. Feynman's operational calculus as a generalized path integral We pointed out earlier, especially near the end of Section 14.3, that Feynman's operational calculus is closely related to and, indeed, may be regarded as an extension of certain aspects of path integration. We emphasize that point again by quoting part of the next to last paragraph of the book of Feynman and Hibbs [FeyHi] on path integration: "Nevertheless, many of the results and formulations of path integrals can be reexpressed by another mathematical system, a kind of ordered operator calculus [Fey8]. In this
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563
form many of the results of the preceding chapters find an analogous but more general representation . . . involving noncommuting variables." Before proceeding further, the reader should review the discussion of Feynman's rules in Section 14.2 and may also find it helpful to review the simple examples 14.2.1 and 14.2.3. Exponentials of sums of noncommuting operators We have seen in Chapters 14-18 and especially in Chapter 15 that a broad range of functions of noncommuting operators can be formed using Feynman's ideas; see also [AhJo, Alb, Fey8, JeJo, JoLal, Lal3-16], as well as Remark 19.4.5 and Example 19.7.2 below. However, throughout most of this chapter, we will concentrate, as Feynman did in [Fey 8], on exponentials of sums of operators or sums of integrals of operators. We focus on such a special class of functions simply because of their connection with evolution equations. This connection is developed much further here than in earlier work, but it is hardly new that exponential functions are important in connection with evolving physical systems (see Section 14.1). Not only will our emphasis be on the exponential functions discussed above but, more particularly (as in [Fey 8]), on the perturbation series related to them. One can argue plausibly that the most important development for physics from both the theoretical and computational point of view of Feynman's path integral [Fey2] and his related operational calculus [Fey8] was the perturbation series that came from them. Disentangling exponentials of sums via perturbation series We will use the ideas of Feynman's operational calculus in Sections 19.2 and 19.3 to disentangle the formal expression
This calculation will be employed as a vehicle for further illustrating Feynman's key ideas. Sections 19.2 and 19.3 will be partly heuristic; however, the combinatorial arguments—which are an important part of this subject—will be done rigorously in both sections. The heuristic aspects will be more dominant in Section 19.2 than in Section 19.3 where the combinatorics becomes more difficult. The final result of Sections 19.2 and 19.3 will provide the definition of(19.1.1) on which the mathematical results in Sections 19.4-19.6 are based. A precise discussion of the hypotheses that are involved in this chapter will be given at the end of this introduction. Let us mention briefly for now that —a in (19.1.1) will be the generator of a (Co) contraction semigroup, each uP will be a continuous, locally finite measure on [0, +00), and for each p = I , . . . ,n, the operators { B p ( s ) } will be a commuting family of bounded linear operators on a separable Hilbert space H. We will not assume that the operators Bp and Bq commute with one another for p = q nor that the operators Bp commute with the operators e~sa in the semigroup. The formula for the expression in (19.1.1) that will be obtained in Section 19.3 is perturbative with respect to the operators Bp and permits one to take into account the
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strength of various forces at different times and locations. For example, in a certain time interval and region of space, one may want to truncate the perturbation series to consider second order effects in B1 and first order effects in B2 but to ignore the negligible influence of B 3 , . . . , Bn. At other times and in other regions of space, different operators Bp may provide the dominant effects. One of the strengths of Feynman's operational calculus and of the present chapter is its generality. The formula (19.3.14) that results from the calculations in Sections 19.2 and 19.3 may be thought of as giving the evolution of complicated physical systems involving several forces. Feynman's operator calculus as applied to exponentials of sums as in (19.1.1) tells us how to blend together the features of simpler systems. Cases where one (or more) of the operators Bp is something other than the standard operator of multiplication by an R-valued, time independent, local potential V are discussed briefly further on in this introduction and in more detail in Example 19.7.5. The generator —a of the semigroup will typically be an unbounded operator in applications. Further, it may well happen that one or more of the perturbing operators is also unbounded. However, our hypotheses require that all of the operators B p ( s ) , p = l , . . . , n , o < s
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One can write V = ^p-\ Vp, The individual parts of the potential V each have an effect on the physics involved, in particular, on the constants of motion, and it may be helpful to study these effects alone or in various combinations. In Section 14.2, we discussed the question: How do we attach time indices to operators (in a natural way) to specify the order of operators in products? In other words, how do we follow rule (1) of Feynman? We mentioned that operators sometimes come with time indices naturally attached (time-dependent potentials or the operators in the Heisenberg or interaction representations in quantum mechanics, for example) and that a physical problem at hand (for example, a potential being turned on and off at certain times) might well suggest an attachment of time indices. Further, we have seen the usefulness of varied assignments of time indices in several places, especially in Chapters 15 and 17. The main results of this chapter will provide additional evidence of this usefulness. Sometimes, it is a combination of physical and mathematical reasons which suggest an appropriate measure. We mentioned above the problem of adding one or more additional (that is, in addition to a) unbounded operator to our analysis. One approach is to approximate Lebesgue measure on [0, T] by a discrete measure, sayTNYlg=i ^T/N, where SqT/N is the Dirac measure with unit mass atqT/N. If the additional unbounded operatorai is such that—ai is the generator of a (Co) semigroup, then in this model, — a\ appears only in the semigroup exp(—sa\) (see (19.7.14)). But exp(-sai) is a bounded operator and will not cause problems in our formulas. See Example 19.7.4 for more detail. We discussed briefly above two strategies for dealing with operators other than the generator —a which are unbounded. But what are some examples of a commutative family of operators {/#(•*) • 0 < s < 00} which is of physical interest and is such that is bounded for each si Local and nonlocal potentials The standard operator of multiplication by a real-valued potential energy function V is bounded if and only if V is essentially bounded. (See Proposition 10.1.1 and Exercise 10.1.2.) Further, as was noted in Remark 10.1.4, this property continues to hold for complex-valued potentials. (Recall that complex-valued potentials are used in phenomenological models of dissipative quantum systems [Ex].) Of course, for any fixed bounded operator A and any scalar-valued Borel measurable function of time g(s), j3(s) := g(s)A is an example. The usual potentials V(x) are local in the sense that they depend only on a single point x in configuration space. (So far in this book, as in most of the mathematical physics literature, we have only used such local potentials to illustrate our results in applying them to situations relevant to quantum physics.) However, nonlocal potentials V(x, y) provide an interesting class of examples. They are used in many-body problems in several areas of physics, especially in nuclear theory. (See, for example [ChSa, Me, Tab].) The operator associated with the nonlocal potential V(x, y) is, in this case, the integral operator having V as its kernel:
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where the integral is over the configuration space, say Md. If V is time independent, the "family" of operators (19.1.2) is clearly commutative. Even with V = V(s; x, y) time-dependent, the family may be commutative, and it definitely is if V(s; x, v) = Vi(s; x — y) is of convolution type. When convolution operators act on L 2 (M d ), it is easy to show that V\ e L '(R d ) implies that || A v\\ < ||Vilh (see, e.g., [DunSc2,p.951]). Two good references in the physics literature regarding nonlocal potentials as well as sources for further references are [ChSa, esp. Chapter 8] and [Me, esp. pp. 73-78 and 381-384]. Nonlocal separable potentials of finite rank, that is, having the form
are used frequently in phenomenological nuclear physics. It was Tabakin's work [Tab] which justified from a physical point of view the use of such potentials in this context. [It showed, in particular, that they can be viewed as "effective potentials" (or interactions) in the so-called "Hartree-Fock approximation".] Example 19.7.5 will give some of the terms of a perturbation expansion involving a generator, a time-dependent local potential, and also a nonlocal potential as in (19.1.3). We now give a precise discussion of the assumptions that will be made throughout this chapter. Hypotheses (H.I) The operator —a will be the infinitesimal generator of a (Co) contraction semigroup of operators on a separable Hilbert space H (over C or R). Such an operator is always closed (Proposition 9.1.10) but is unbounded in most of the interesting cases. We remind the reader that the operators in the (Co) semigroup T(s) = e~sa, 0 < s < oo, associated with a are all in jC(H), the space of bounded linear operators on H. The assumption that the operators T(s) in the semigroup are all contractions is not actually needed but is made here because it simplifies the estimates and also because it is satisfied anyway in most of the potential applications. (H.2a) For each p = 1, . . . , n, fj,p will be a Borel measure on [0, +00). It is perhaps most natural from the point of view of applications to think of the up as positive measures, but they can also be signed or even complex-valued measures. [Strictly speaking, these are local (positive or complex) measures on (0, +00), in the sense of [DoFri3, Definition 6.1, p. 179] or of Remark 17.6.19(b).] In any case, we require that (H.2b) the associated (necessarily positive) total variation measures \ u p \ satisfy lMpl([0, T}) < oo for every T > 0 and p = 1, . . . , n. Except in Section 19.7, we assume that (H.2c) the Up are continuous measures; that is, u p ( { t } ) = o for every t > 0. In addition to many familiar measures, the class of continuous measures includes those that are continuous but singular with respect to Lebesgue measure. Recall that
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567
such measures have been discussed to some extent earlier in this book (see Example 15.5.3 as well as Remark 17.2.8(d) following Theorem 17.2.7). In Section 19.7, we will consider cases where the measures are allowed to have nonzero mass at a finite number of times. We remind the reader that even measures with arbitrary discrete part were permitted earlier in this book (see Sections 15.4 and 17.6). (H.3a) For each p = 1, ...,n, f}p : [0, +00) -*• C(H) is assumed to have the property that B - p l ( E ) is in 6[0, +00)), the Borel class of [0, +00), for every strong operator open subset E of £(H). It follows that each of the nonnegative functions ||Bp(-)ll is |uP [-measurable [HilPh, Theorem 3.76.4, p. 80]. Remark 19.1.1 Certainly, functions Bp : [0, +00) -> £(H) which are continuous in the strong operator topology satisfy hypothesis (H3. a) (see Exercise 3.2.10(b)); of course, this is the case when Bp is time independent. Further, for each p = 1 , . . . , n, (H.3b) we require that $ Wp(s)\\\iJLp\(ds) < oo for every T > 0. (H.3c) We assume that for each p, the family of operators {/3p(s): 0 < s < 00} is a commuting family in C(H). We do not assume for p=q that fip(s) commutes with fiq(t) (s, t in [0, +00)), nor do we assume that any of the operators ftp(s) commute with the operators in the semigroup T(t) — e~'a. The fact that Bp need not commute with Bq for p=q will complicate the combinatorics considerably as we will see in Sections 19.3-19.6 below. In Chapters 15-18, all of the operators involved other than the generator are multiplication operators and so they all commute with one another. However, different combinatorial complications arise in Chapters 15-18 from the systematic treatment of measures with nonzero discrete parts. The only consideration (beyond remarks) of such measures in this chapter appears in Section 19.7, where further illustrations of disentangling are given. Remark 19.1.2 (a) (Unitary groups of operators) We are interested in unitary groups as well as in contraction semigroups. For notational convenience, we work primarily with the semigroup case, but all of our results in this chapter can be rephrased in terms of unitary groups, which is the case most relevant to quantum physics. More specifically, if H is a complex Hilbert space and a is assumed to be the infinitesimal generator of a unitary group of operators (see Definition 9.6.9 and Remark 10.1.12(a)) rather than of a semigroup, with Bp defined on R and otherwise suitably adjusted, our results hold without change except that the time parameter s lies in R rather than in [0, +00). (b) Some of our work here may be somewhat related to [Klal] and may fill out one aspect of Klauder's "exponential representations" [Klal, p. 610], an extension of the idea of an exponential Hilbert space (or Fock space). (c) The work in this chapter was partly motivated by applications in quantum theory and so the setting of a separable Hilbert space is appropriate. In his thesis [Rey], Tristan
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Reyes—building on a draft of the paper [dFJoLa2] on which this chapter is based— worked with a separable Banach space and, further, did not assume that the operators ( B p ( s ) : 0 < s < 00} are commuting for fixed p. Thus, his work enlarged substantially the range of potential applications of [dFJoLa2], and hence of the present chapter. (d) Let a = i HQ and take n = 1. Further, let B1 — V, the operator of multiplication by the time independent potential V. Recall that in the Heisenberg or in the interaction representation (see Remark 14.3.l(c)), one takes V(t) — e~"H°Hel'H°. It is interesting to note that the operators [f$\(t) = V(t) : t € K} form a noncommuting family even though the initial family {B1(s) = V} consists of a single operator and so is certainly commuting. Hence Reyes' results can be used to disentangle exp{—it Ho + f^ fi\ (s)ds] whereas the results of this chapter cannot be applied directly. However, by disentangling exp{—itHQ + f0 f t ( s ) d s } and then rewriting the terms of the resulting perturbation series in the Heisenberg or in the Dirac representation (much as was done, e.g., in Section 17.6 above), we obtain essentially the "same" final series; more precisely,
(Analogous comments can be made about the case n >2.) Of course, the results of [Rey] also apply to more complicated situations involving noncommuting families. We close this introduction with a brief summary of the contents of Sections 19.2-19.7. Feynman's heuristic rules are applied in Sections 19.2 and 19.3 to disentangle exponentials of sums of operators. (Care is taken that the combinatorial arguments involved are done with mathematical rigor.) In Section 19.4, it is proved that the time-ordered perturbation series (19.3.14) which result from the calculations in Section 19.3 make sense and converge (see Theorem 19.4.1). We prove in Section 19.5 (see Theorem 19.5.1) that the operator-valued function defined by the perturbation series satisfies an evolution equation (in integral form), and then go on to show in Section 19.6 that the solution to this evolution equation is unique (Theorem 19.6.1). Further examples of the disentangling process are given in Section 19.7, our last section. We particularly call the reader's attention to Example 19.7.5 involving nonlocal potentials. 19.2 Disentangling exp { - f a + /0' /6(s)^(ds)} Feynman's heuristic ideas were used in the first part of Section 14.3 to calculate the m =0,1,2 terms of the perturbation expansion of exp{-ta + /0' B(s)ds}. (The reader may find it helpful to review that earlier material, especially the calculation of the m = 2 term.) Here, we will calculate the general term of the expansion. One purpose of this section is to further illustrate and clarify some of Feynman's ideas in an important but simple setting. Also, our calculations will be useful in the more complicated setting of the next section where an arbitrary finite number of (ft, u) pairs are treated. The combinatorial aspects of our arguments will be done carefully in this section and the next, but much of the emphasis will be on the heuristic ideas. The formula (9.3.14) arrived at in Section 19.3 will be the starting point for the mathematically rigorous results established in Sections 19.4-19.6.
DISENTANGLING exp{-ta + /„' P(s)n(ds)}
569
We will work out in detail below the case where ^ is Lebesgue measure and then state the formula for general u as described in hypotheses (H.2a) - (H.2c) at the end of the introduction. Although different measures u suggest very different physical interpretations, the explanation for the calculations differs only at one point, and we will explain that point carefully. Rule (1) stated in Section 14.2 says that we should "attach time indices to the operators to specify the order of operators in products". How should we attach time indices when they do not come naturally with the problem? There are many ways to do this as we have seen throughout Chapters 14-18 but, in this section, except at the very end, the indices will be distributed uniformly on [0, f], that is, according to normalized Lebesgue measure. We write
and, if b is time independant
where a(s) = a and B(s) = B for all s. e [0, t]. This attachment of time indices is appropriate for many of the problems arising from quantum theory and follows the method used throughout much of [Fey8]. We let a be given by (19.2.1) but we allow for the possibility that B(s) is timedependent. The heuristic calculation that we are about to give was discussed by Feynman [Fey8, p. 109] for B time independent and t = I. We will give much more detail in order to clarify the heuristic ideas. The calculation that begins with (19.2.3) below and ends with (19.2.9) will repeat some steps from the corresponding calculation ((14.3.3) through (14.3.8)) in Section 14.3. We write
where the second "equality" in (19.2.3) comes from an important special case of "rule" (2) which said, "with the indices attached, form functions of operators by treating them as if they were commuting". We did this in (19.2.3) by using the commutative law, exp(A + B) = [exp(A)][exp(B)J, in a noncommutative setting. Once "rule" (2) has been applied, it should be understood that the resulting expression is not to be (and, indeed, cannot be unless the operators involved are commuting) interpreted in the usual way until the "disentangling" called for in the rather vague "rule" (3) is complete. In particular, (19.2.3) above is not true in general if the first and third expressions are interpreted in the usual manner.
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A further point concerning the third expression in (19.2.3): It is counterproductive to calculate the integral — /J a(s)ds at this stage. Leaving it as it is will facilitate the disentangling process. Continuing, we expand exp (/0' fi(s)ds J from (19.2.3) in a series and obtain
Now we write
where
The last equality in (19.2.5) follows from the fact that the function ( s 1 , . . . , sm) B(Sm) • • • B ( S I ) is invariant under permutations of the integers {1,..., m} and the fact that subsets of [0, t]m where two or more of O , s 1 , . . . , s m , t are equal have Lebesgue measure 0. We make two remarks concerning (19.2.5): (i) The ordering of the operators in the last expression in (19.2.5) corresponds to the time-ordering in (19.2.6); this is irrelevant when (19.2.5) stands by itself since the /8s commute, but it will facilitate disentangling when (19.2.5) is substituted into (19.2.4). (ii) If Lebesgue measure in the integral f^ B(s)ds is replaced by a measure which charges points, ignoring the sets where two or more of 0, s1,..., sm, t are equal is no longer possible. One consequence of this is that the third line of Feynman's timeordering convention (14.2.1) is inappropriate. Section 15.4 illustrates how involved the combinatorics can become when the measure has nonzero mass at a countably infinite number of times.
DISENTANGLING exp{-ta + /„' P(s)n(ds)}
571
Now combining (19.2.4) and (19.2.5) and then doing some simple calculations, we obtain
where so = 0 and im+i := ?. The next step is simple but crucial to the disentangling; we finish time-ordering the integrands in the last expression in (19.2.7), thereby obtaining
Note that each term in the sum on the right-hand side of (19.2.8) is now completely time-ordered; i.e., the conventional ordering indicated on the page corresponds to Feynman's time-ordering convention (14.2.1). It is at this point, following Feynman's heuristic ideas, that the disentangling is complete and that we can interpret our expression in the conventional way. We are now free to evaluate if possible whatever integrals remain. In this case, the integrals of a are easily calculated, and we finally obtain
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We now rewrite (19.2.9) with Lebesgue measure replaced by a measure fj, satisfying the hypotheses (H.2a) - (H.2c) from the end of Section 19.1:
where, as before, u®m is the measure obtained by taking the product of u with itself m times. The same steps that led to (19.2.9) yield (19.2.10) as well. In fact, it is only the last equality in (19.2.5) that requires a different explanation. In that equality, we were able to ignore the m tuples in [0, t ]m where two or more coordinates are equal because of the familiar fact that such sets have m-dimensional Lebesgue measure 0. The /u®m measure of such sets is also 0 by Lemma 15.2.7 and the continuity of u (see (H.2c)). Even though the situation in (19.2.10) is not all that complicated, it is helpful to draw a Feynman graph to keep track of the mth term on the right-hand side of (19.2.10). The operator a might be iH, for example, where H is the Hamiltonian of some quantum system. The operator H need not be a free Hamiltonian but could be the sum of a free Hamiltonian and some other rather singular operator. The (is should be thought of as some possibly time-dependent perturbing operators. These might be the familiar operators of multiplication by a potential as in (14.3.9) or (15.3.22) but could also be one of the types of operators discussed in Section 19.1 such as an integral operator (as in (19.1.2)) determined by a nonlocal potential V(x, y). Looking at Figure 19.2.1, we think of the quantum system evolving according to the Hamiltonian H over the time interval [0, si). At time s, the system is scattered by the perturbing operator /3(si). Between times s\ and $2, the system again evolves according to H and then is scattered at $2 by /Jfe)- This process continues until a final scattering at time sm followed by evolution according to H over the time interval (sm, t]. One then integrates (or "adds up") over all possible locations in A m (r) of the m scatterings. The distribution of the scatterings is given by [L®m, the product of m copies of the measure fj.. Usually in such diagrams no measure is included, because it is understood that the integration is with respect to m-dimensional Lebesgue measure l®m (or ds\ • • • dsm). We include /u®m in Figure 19.2.1—much as we did in Figure 15.6.1 (and in all of Section 15.6)—to emphasize to the reader that various distributions of the scatterings are possible. If at = i/fo. where HQ = — j A is the usual free Hamiltonian from nonrelativistic quantum mechanics acting on "H = A 2 (R rf ) (the space of C-valued square-integrable functions on Rd), /6(s) = —iV where V : Rd ->• E is a time independent essentially bounded potential, and fj, = I, then the perturbation series (19.2.10) becomes the classical
DISENTANGLING
573
FIG. 19.2.1. A Feynman diagram for the pair (ft, Dyson series from nonrelativistic quantum mechanics [Dysl.Schu]:
19.3 Disentangling exp{-ra + /0' )3i(j)Aii(dj) + • • • + /0' pn(s)nn(ds)} In this section, we use Feynman's heuristic ideas to disentangle the exponential appearing in the title above. Our purpose is both to further clarify Feynman's ideas and to arrive at an expression which will be used throughout Sections 19.4-19.6. As before, only the combinatorial arguments will be treated carefully here, but they will be more complicated than in Section 19.2 and so will require more of our attention. We will state our final formula for measures f j , \ , . . . , fj,n satisfying (H.2a) — (H.2c), but for notational simplicity, we do the calculations where /A \ , . . . , / / , „ are all Lebesgue measure /. When the "explanation" of a step below is similar to one in Section 19.2, any comment that we make will be brief.
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Using formula (19.2.1), Feynman's first and second rules (see Section 14.2) as well as (19.2.5), we write
where for p = 1 , . . . , n,
In order to shorten the expressions to follow, we also let
Now continuing from (19.3.1) and making use of (19.3.2) and (19.3.3), we write
where m\,..., mn are summed over all nonnegative integers.
DISENTANGLING exp{-?« + /„' p\(s)tn(ds) + ••• + /„' pn(s)nn(ds)}
575
Note that in each summand on the right-hand side of (19.3.4), the second through (n + l)st factors are time-ordered when considered by themselves. However, they are not time-ordered with respect to one another and that is our next concern. Temporarily fix m\,..., mn and let
We want to consider certain permutations n of the integers ( 1 , . . . , m}. We do not wish, however, to consider all such permutations since for each p , s p , 1 , . . . , sp,mp are already in order with respect to each other and we do not want to disturb that order. Let Pmi mn be the set of all permutations Ti of [l, ...,m] which leave the components of each Sp,mp, for p = 1 , . . . , n, in the same relative order, where
(Recall that m = m\ + • • • + mn, by (19.3.3).) Thus, it is required of n that for each p = 1 , . . . , n, the integers m1 + . . . + mp-1 + 1 , . . . , m\ + . . . + mp-\ +mp appear in order (but not necessarily consecutively) in the list ;r(l), 7t(2),..., n(m). Another way of saying this is that for each p = 1 , . . . , n, n should satisfy
Following our present general discussion, we will give a concrete example which may be helpful to the reader. (See Example 19.3.1.) Since the discussion immediately above is crucial, it seems worth while to give a part of that example here. Let n = m \ = m^ = 2 so that
where
The permutation n, with TT(!) = 3, n(2) — 1, jr(3) = 2, n(4) = 4, belongs to Pz,2 since 1 appears before 2 and 3 appears before 4 in the list n(l), n(2), jr(3), ;r(4). In contrast, the permutation no, with jro(l) = 2, jro(2) = 1,7To(3) = 3,3To(4) = 4, is not in p2,2 since 2 appears before 1 in the list no(l), no(2), jro(3), no(4). We will soon sum over all n e Pmi mn and so it is of interest to know the cardinality of Pm\,...,mn- Among the (m\ + • • • mn)\ = ml permutations of {1,..., m}, how many have the property that they leave the components of each of the Sp<mp in the same relative order? Once m\ (out of the available m) places have been chosen for the components of S\,mi, mi places have been chosen for the components of S2,m2, • • • ,mn places have been chosen for the components of Sn,mn, the permutation is completely determined by
576
FEYNMAN' S OPERATIONAL CALCULUS AND EVOLUTION EQUATIONS
the requirement (19.3.7) of keeping the components of each Sp,mp in the same relative order. Thus
Given n s Pmi,...,mn, we let (with Amp defined by (19.3.2))
and
Next, we claim that up to a set of Lebesgue measure 0,
It is clear from (19.3.9) that the union in (19.3.11) is disjoint and is a subset of Am, x • • • x A m n . Now the elements of Ami x • • • x Amn for which two or more tq are equal form a subset of Lebesgue measure 0. Thus, to establish the claim it suffices to take (t\,..., tm) € Am, x • • • x Amn where the tq are all distinct and show that (t\, ...,tm) € (A m , x • • • x A m J(7r) for some n e Pmi,...,mn. But given such a ( * i , . . . , ?m), its components have a definite order, and so there is a permutation n (which is unique in fact) such that tn(\) < • • • < tn(m). Further, this permutation it must belong to Pm] mn since for each p = 1 , . . . , n, the components of (tmi+...+mp_l + \,... ,tm[+...+mp_t+mp) are already in order. Now making use of Feynman's second "rule" (see Section 14.2), the preceding paragraph and (19.3.10), we can write
where we remind the reader that m := m1 + • • • + mn.
DISENTANGLINGexp{-ta+ /„'B1(s)u1(ds) + • • • + /„'
B n (s)/*„(ds)}
577
Using (19.3.12), we can now continue from (19.3.4) and write
where the calculation of the integrals yielding the last equality is "permitted" since each summand on the left-hand side of that equality is completely time-ordered. We have now used Feynman's "rules" to "disentangle" the expression which we began with:
The same discussion which yielded (19.3.13) leads us to the next equality, where the product of m — m\ + • • • + mn copies of Lebesgue measure in (19.3.13) is replaced by
578
FEYNMAN' S OPERATIONAL CALCULUS AND EVOLUTION EQUATIONS
the product measure /if"1 x /if2
x • • • x v®m":
where Pmt,...,mn is defined just above (19.3.6) and (A m , x ••• x A m J(7r) andy^fe(p)) are given in (19.3.9) and (19.3.10), respectively. Here, as before, on the right-hand side of (19.3.14), the summation over m\,..., mn is taken over all n-tuples of nonnegative integers. Some of the main difficulties in the mathematically rigorous part of this chapter are combinatorial in nature and revolve around the series in (19.3.14) (or (19.3.13)) and the associated concepts and notation. Consideration of a special case may help the reader to understand the preceding discussion as well as arguments to follow, especially in Sections 19.5 and 19.6. We return to the case of Lebesgue measure for the example. Example 19.3.1 Let n = m\ = m2 = 2 and start with the expression
which forms part of the right-hand side of (19.3.4) in this case. Here,
and
According to (19.3.8), there should be (2 + 2) !/(2!2!) = 6 permutations n of {1, 2, 3,4} such that
(Note that by (19.3.3), m — m1 + m2 = 4 here.) In fact, the following six permutations have that property as the reader can easily check:
DISENTANGLING exp{-fa + /„' B1(s)u1 (ds) + • • • + /„' Bn (S)//„(ds)}
579
All that is needed is to observe that in the second column under each of the permutations, the number 1 appears before 2 and 3 appears before 4. None of the remaining 18 permutations of (1,2, 3, 4} has this property. Consider, for example, the permutation
While 3 comes before 4 in the list
and
In this case, (19.3.12) then becomes
580
FEYNMAN' S OPERATIONAL CALCULUS AND EVOLUTION EQUATIONS
With n fixed at 2, the m1 = m2 = 2 term of the series in (19.3.13) then becomes
One can draw Feynman graphs representing each of the six integrals in (19.3.24). We limit ourselves to the graph corresponding to the permutation nj,. The operator a could be iH, where H is the Hamiltonian of some quantum system. The B1s andB2s should be thought of as time-dependent perturbing operators. Consulting Figure 19.3.1, we see that the system evolves according to a over the time interval [0, t3). At time t3, the system is scattered by the perturbing operator B2(t3). Between times t3 and t\, the system again evolves according to a and is scattered by B1\(t1) at t\. This is followed by another a-evolution over (t1,12) and a B1 (12) scattering at time t2- Next comes an a-evolution over (?2, U) and a B2 scattering at t4. There is a final a-evolution over (?4,t). One then integrates over all possible locations in (A2 x A2)(7T3) of the four scatterings, two by B1 and two by ft. The distribution of the scatterings is uniform over (A2 x A2)(JT3); that is, it is governed by l®4, four-dimensional Lebesgue measure. The situation summarized in Figure 19.3.1 is a fourth order scattering which is second order in each of B1 and ft. We remind the reader that there are five more such graphs corresponding to the remaining five permutations in P2,2-
CONVERGENCE OF THE DISENTANGLED SERIES
581
FIG. 19.3.1. A fourth order diagram associated with Lebesgue measure l and two Bs If uP is associated with Bp, for p = 1,2, then the measure in Figure 19.3.1 will be ux u2® instead of/® 4 . The graph itself will not change for this general situation, but in specific cases it can have very different interpretations. We mention some possibilities: Case 1. Suppose that u-2 is associated with a force that is not "turned on" until time t/2, so that we can take u2 = 0 on the Borel class of [0, t/2]. Since the first interaction or "scattering" is by ft for the permutation nj,, all four interactions will occur in the interval [t/2, t]. Case 2. Suppose that the force associated with B1 and u1 is "on" over the interval [0, t/2] and "off" over [t/2, t] and that the situation is reversed for the pair (B2, u2)Then all of the ft scatterings will precede all of the ft scatterings. This eliminates five of the six permutations in P2,2 including TIT, and leaves only the identity permutation. A less extreme but related situation occurs if u1 and u2 are both spread over all of [0, t ] but u1 has nearly all of its mass on [0, t/2] and u2 has nearly all of its mass on [ t / 2 , t]. Case 3. One or both of /u,j, /u-2 is continuous but singular with respect to Lebesgue measure with mass concentrated for example on a Cantor set. 19.4 Convergence of the disentangled series We have used Feynman's ideas in Sections 19.2 and 19.3 to "disentangle" the formal exponential expression in (19.3.14) and have arrived at the series on the right-hand side of the same equation. Except for the combinatorics, we have not worried about mathematical rigor up to this point. However, we now wish to use the series to define the left-hand side of (19.3.14). Accordingly, we will show in this section that the terms of the series make sense and that the series converges. We will also obtain a norm bound for the series which is simple despite the fact that the series itself is rather complicated.
582
FEYNMAN' S OPERATIONAL CALCULUS AND EVOLUTION EQUATIONS
The first issues that arise in connection with the terms of the series in (19.3.14) have to do with the nature of the integrals involved. Recall that although we are using the suggestive exponential notation exp(—sa), what we really have is the (Co) semigroup with generator a (see (H.1) of Section 19.1). Such semigroups are continuous in operator norm if and only if a is a bounded operator (see Propositions 9.5.2 and 9.5.3). However, we do not wish to make an assumption that forces a to be bounded since that rules out many of the most interesting generators. Now a (Co) semigroup, even with an unbounded generator, is strongly continuous; i.e. is a continuous function from [0, +00) to C,(H), where £(H) is equipped with the strong operator topology. Certainly then such a (Co) semigroup is a measurable function from ([0, +00), B([0, +00))) to (£(H), Bso), where B([0, +00)) denotes the Borel aalgebra (or Borel class) of [0, +00) and Bso denotes the a -algebra generated by the subsets of C(H) which are open in the strong operator topology. We are assuming that the ftp are measurable in the same sense; see hypothesis (H.3a) from Section 19.1. But does this make the integrands in (19.3.14) measurable with respect to the strong operator topology on C(H)1 It does, but to see this, the key is to show that the product of C(H)valued functions measurable in the sense just described is also measurable in the same sense. Surprisingly, this has only recently been proved, and the result was motivated by the paper [dFJoLa2] on which this chapter is largely based. What we need is an easy corollary of the result of Badrikian, Johnson and Yoo [BadJoY, Theorem 2]. (In fact, Johnson's paper [Jo8] came first, and Theorem 5 of that paper is sufficient to handle the case where fi\ = • • • = / £ „ . ) We note that Schliichtermann [Schl1,2] has more recently proved results closely related to those in [BadJoY, Jo8] but using different techniques and without the separability assumptions required in both of those papers. We are now ready to state the main result of this section ([dFJoLa2, Theorem 4.1, p. 180]). Theorem 19.4.1 Suppose that hypotheses (H.1)-(H3) from Section 19.1 are satisfied. Then the terms of the series in (19.3.14) make sense, the series converges absolutely in operator norm, and we have the following bound for the operator norm of the series:
where the integrals in (19.3.14) and (19.4.1) are to be thought of as Bochner integrals in the strong operator sense.
CONVERGENCE OF THE DISENTANGLED SERES
583
Remark 19.4.2 (a) More precisely, the integrals in (19.3.14) and (19.4.1) need to be regarded as acting on a fixed vector
Further, let
and let
(Of course, it is only as we finish the proof below that we will be assured that (19.4.4) makes sense.)
584
FEYNMAN'S OPERATIONAL CALCULUS AND EVOLUTION EQUATIONS
Now since the semigroup involved in (19.4.2) is a contraction semigroup,
Recalling the notation (19.3.3), (19.3.5), (19.3.9), (19.3.10) and using (19.4.5) as well as the fact (see (19.3.11)) that, up to a set of measure 0, Ami x • • • x Amn is the disjoint union of the sets (A m , x • • • x A mn )(jr), we can now write
where the first two equalities in (19.4.6) reverse heuristic calculations that were made in Section 19.3; here, however, scalar-valued functions are being integrated, so that noncommutativity is not an issue and the steps are rigorous. It follows from (19.4.6) that
and so the proof of Theorem 19.4.1 is complete.
CONVERGENCE OF THE DISENTANGLED SERIES
585
We are now prepared to give a rigorous definition of the previously purely formal expression
Definition 19.4.3 The exponential expression in (19.4.8) is defined by (or "disentangled by") the series in (19.3.14) whose absolute convergence (and hence convergence) in operator norm is guaranteed by Theorem 19.4.1. We will see in Section 19.5 that the "exponential" in (19.4.8) satisfies an evolution equation determined by the infinitesimal generator a and the n pairs (£1, Mi ) • • • • • (Pn, Mn)- We finish the present section by giving some error bounds which follow easily from the bound (19.4.1). Proposition 19.4.4 Suppose that hypotheses (H.1)- (H. 3) are satisfied. Then, given any nonnegative integers M 1 , . . . , Mn, we have the following error bounds for the indicated approximation to the exponential expression (19.4.8):
586
FEYNMAN' S OPERATIONAL CALCULUS AND EVOLUTION EQUATIONS
Proof The second inequality follows easily from the first. The first comes from (19.4.7) and the fact that
where the sets in the union (19.4.10) are pairwise disjoint. Remark 19.4.5 We disentangled the exponential expression (19.4.8) in Sections 19.2 and 19.3 and then proved in this section that the resulting series makes sense and converges. A careful look at our work will show that most of the difficulties came in disentangling the (m1,..., mn)-term of
where f was the entire function
Given a more general analytic function
we can, by taking a to be the 0 operator and writing
use the formula (19.3.14) to formally disentangle
Convergence of the series giving (19.4.15) can then be proved for 0 < t < T provided that the series (19.4.13) converges absolutely at least on the closed polydisk
Thus, although our work in this chapter focuses almost entirely on the expression (19.4.8), the reader can see that it is not far removed from rather general expressions like (19.4.15)
THE EVOLUTION EQUATION
587
with g given by (19.4.13). (Of course, functions other than exponential functions of noncommuting operators have been involved in Chapters 15, 16, and 18.) In Example 19.7.2 below, the disentangled series for (19.4.15) will be written out in the case n = 2. 19.5 The evolution equation Feynman was concerned in [Fey8] with the general problem of forming functions of noncommuting operators. However, as we discussed earlier, his particular focus was on exponentiating sums of noncommuting operator expressions. The idea was that these "exponentials" should be defined so as to give the time evolution of the associated physical systems. This suggests that the now rigorously defined expression in (19.4.8) might satisfy an evolution equation corresponding to a system governed by the infinitesimal generator a and the time-dependent (in general) forces represented by the pairs (/?!. Mi). • • • » ( & i , M«)- The theorem below ([dFJoLa2, Theorem 5.1, p. 185]) will show that this is indeed the case. As the reader will see, the arguments involved are considerably more complicated than the familiar arguments in related situations. Theorem 19.5.1 Suppose that the hypotheses (H.1)-(H.3)from Section 19.1 are satisfied. Then Lt defined by (19.4.2)—(19.4.4) satisfies the integral equation
where, as in Remark 19.4.2(a), the integrals are Bochner integrals in the strong operator sense. Remark 19.5.2 (a) We will work with the interval 0 < t < T throughout and show that Lt satisfies (19.5.1) on [0, T]for every T > 0. Since T > 0 is arbitrary, it will follow immediately that Lt satisfies (19.5.1) for 0 < t < oo. (b) Some readers may wonder if it is reasonable to refer to (19.5.1) as an evolution equation. Taking n = 1 and n\ = / (Lebesgue measure), we get for different choices of a and fi\ the integrated form of many familiar evolution equations. For example, if H = L2(Rd),a = — |A (equivalently, a = iHo) and f$\ is the operator of multiplication by —iV (briefly, we write ()\ = —iV), (19.5.1) is the integrated form of the Schrodinger equation. If |A and —iV are replaced by — ^ A and —V, respectively, (19.5.1) corresponds to the heat equation rather than the Schrodinger equation. For certain other choices of a and fl1, (19.5.1) becomes the integrated form of various evolution equations corresponding to an assortment of Markov processes. (c) Forn = l,H~ L2(E.d), and with a - -|A (= iHo), ft1 = -iV (resp., a = j A, ft1 = —V), the integral equation (19.5.1) reduces to the special case when IJL is a continuous measure of the integral equation obtained by the second author in [Lal4, 15] in the "quantum-mechanical" (resp., "diffusion" or "probabilistic") case. (See especially Theorem 17.2.1 and Corollary 17.2.6, which were taken from [LalS].) It would be interesting to extend the present setting to the case when /J,p is replaced by r/p = Up + vp (p = !,...,«), where vp is a finitely supported discrete measure (on each finite interval [0, T]), as in [Lal4,15] and Sections 17.2-17.5 above, or even
588
FEYNMAN'S OPERATIONAL CALCULUS AND EVOLUTION EQUATIONS
to the case of a general (locally bounded) Borel measure r\p = fj,p + vp (i.e. vp has a (possibly infinite) countable support on each interval [0, T]), as in [Lai 8,16] and Section 17.6 above. We suspect that such generalizations are possible in the present setting but that their justification would be very involved combinatorially. We will go no further in this direction in the present chapter. We will, however, discuss in Section 19.7 the disentangling of various examples involving measures with nonzero discrete part. (See Examples 19.7.1, 19.7.3, and 19.7.4 as well as Examples 14.2.1 and 14.2.3.) Before beginning the proof of Theorem 19.5.1, we state a lemma which we will prove further on. This lemma is actually the key to the proof of Theorem 19.5.1. Lemma 19.5.3 Suppose that the hypotheses of Theorem 19.5.1 are satisfied and that m i , . . . , mn are nonnegative integers not all of which are 0. Then for 0 < t < T, we have
Here, L™1 mn isdefined by(19.4.3) and (19.4.2); further, the integrals on the right-hand side of (19.5.2) are Bochner integrals in the strong operator sense. Proof of Theorem 19.5.1 In order to justify the existence of the appropriate integrals and sums below and the permissibility of the necessary interchanges, one argues in essentially the same way as in Section 19.4 that the integrals and sums involved are absolutely convergent. We will omit these aspects of the proof. The validity of each equality in the following argument will be discussed below.
The first and last equalities in (19.5.3) come from (19.4.4) and the second from the fact that L®'""° = e~ta. Lemma 19.5.3 gives the third equality, and the fifth is an
THE EVOLUTION EQUATION
589
interchange of order of a sum and integral. In order to see that the fourth equality in (19.5.3) holds, it suffices to fix k out of the finite set {1,...,«} and show that the sum of the terms involving that particular fa is exactly the same on both sides of the fourth equality. Of course, the sum of such terms on the right-hand side of that equality is just
Now fa appears on the left-hand side of the fourth equality only for those (m1 ,..., mk,..., mn) such that mk > 1. But, for this fixed k, the sum
reduces to
However, (19.5.4) and (19.5.5) are the same series differing only in appearance by a shift of 1 in the kth index. Hence, subject to the proof of Lemma 19.5.3 which will be carried out as we continue this section, the proof of Theorem 19.5.1 is complete. O The expressions Lt,m1 mn and Lsm1 m k - 1 . . . m n (for k such that mk > 1) which appear in equality (19.5.2) of Lemma 19.5.3 are defined as sums over appropriate sets of permutations. Understanding how the permutations on the two sides of (19.5.2) are related to one another is crucial to the proof of Lemma 19.5.3. Definition 19.5.4 Given k e {!,...,«} and nonnegative integers m1,... ,mn with mk > 1. we let
The set of permutations just defined can also be described as follows:
Thus P%m1 mn consists of those permutations it in Pm1,...,mn such that the last "scattering" in Lrm1...,mn(n) is by fa. We wish to define a map $* from Pm1,...,mt-1 mn into the set of all permutations of the integers 1, ...,m1 + • • • +mn = m. We will see below that 4>* is in fact a bijective map from Pm, mk-1,...,mn onto Pkm, ,...,mt,...,mn -
590
FEYNMAN' S OPERATIONAL CALCULUS AND EVOLUTION EQUATIONS
Definition 19.5.5 Let k and m1,...,m n be as in Definition 19.5.4. Given p e £mi,...,m t -i,...,».„, the permutation n = 4>^ m / t _,,. > m n P = $*/> of the integers 1,... m, is defined by n(m) — m1 + • • • + mk, and otherwise
Examples 19.5.7 and 19.5.8 which come after the proof of the next lemma may help the reader follow parts of that proof. Lemma 19.5.6 The function $k from Definition 19.5.5 is a bijective map from r m 1 ,— ,mk-1,...,m n
onto
l^m1,...,mk
mn-
Proof A close look at (19.5.8) shows that n is a well-defined function from the set {1,..., m} onto itself. Hence n is one-to-one and so is a permutation of the integers 1 , . . . , m. We wish to show that n e Pk m1..., mk.....,mn. It suffices to show that n e Pm1 mk,...,mn since it certainly then follows from the first part of (19.5.8) that n 6
"m1 mk,...,mn.
As we continue, we will refer several times to lists of integers as appearing in order in another list. We do not mean to imply by this that the integers in the shorter list appear consecutively in the longer list. Since p e /Pm1,...,mt-1,...,mn (where n = 3>kp), we have that in the list
each of the following lists of integers appears in order:
Thus, forq < k, them ? integers mi + • • • +mq_1 + 1 , . . . , m1 +... + mq-1 +mq, appear in order in the list
since from (19.5.8) they are in exactly the same place (counting from the left) in the list (19.5.11) as in the list (19.5.9). In fact, the assertion just made is also true for the
THE EVOLUTION EQUATION
591
same reason for q = k and the m^ — 1 integers m1 + • • • + mk-1 + l , . . . , m 1 + - - - + m1+...+ (mk — 1). The assertion then follows immediately for the list of m^ integers m\ + ••• + mk-\ + 1, • • • , m1 + • • • + m^-i + mk since by the first part of (19.5.8), m1 + • • • + mk is the last integer in the list (19.5.11). Now let q be such that k < q < n. We must show that the mq integers
or, written in more detail,
appear in order in the list (19.5.11). But the mq integers in (19.5.12) (equivalently, (19.5.13)) appear in (19.5.11) in exactly the same places where the mq integers in the list
(each of which is exactly one less than the corresponding integer in (19.5.13)) appear in (19.5.9). (See the last part of Definition 19.5.5.) However, since p e Pm1 m t _1 „,„, the integers in (19.5.14) appear in order in (19.5.9). Hence the integers in (19.5.12) appear in order in (19.5.11). Therefore n 6 Pm1,...,mn and so, as observed earlier, n € ?£,....,«„• ThuS
We See that the
map
Pkm1,...,mn.
** = *i,,...,m t -l,... l B f B °" Pm,....,i» t -l
m, is
into Pkm1,.....,mn. Let n e In order to show that <&k is onto, we will identify AT e Pm mk-1,...,mn and prove that */>* = n. Accordingly, let
The reader should note that m1+ + m^-1 + mk does not belong to either of the two sets in (19.5.15). It may be helpful to check Example 19.5.8 at this point. We show first that px is a permutation of the integers {1,...,m — 1}. Now, Px(q) is defined for every positive integer q such that n(q) is in either one of the two sets in (19.5.15). This is the case for all of the integers 1,..., m — 1 except for the integer which TT maps into m1 + • • • + mk; that is, except for m1 + + mk + + mn = m. Thus p^ is defined on 1 , . . . , m — 1 = mi + •• • + (mk — 1) + • • • + mn, as it should be. Further, the range of pn consists o f { l , . . . , m 1 + " - + mk-1 + (mk — 1)} and (m1+....+mk + 1, ...,m1+....+mk+....+mn}-{!} = {m1+.....+mk,..., m-1], and so the range of pn is {1,..., m — 1} as desired. Finally, since px is a map of the finite set (1,..., m — 1} onto itself, it must be one-to-one. Hence pn is a permutation.
592
FEYNMAN'S OPERATIONAL CALCULUS AND EVOLUTION EQUATIONS
We omit the proof that pn e Pm1,...,mk-1 mn since it is much the same as the proof above that TT e Pm1, . . , mn< where n is given by (19.5.8) of Definition 19.5.5. We finish the proof that ok is onto Pkm1,...,mk,...,mn by showing that ok (pn) = n.By definition of ok acting on Pm1 mk-1,... ,mn (see (19.5.8)), we have
and
Using (19.5.16), the definition of pn given n e Pm1,...,mn (see (19.5.15)) and noting that ( o k p n ) ( m 1 +...+mk +...+m n ) = m1 + ...+mk = n(m1 +. . . + m k + . . . + mn), we have
that is, ok pn = n as claimed. It remains to show that ok is one-to-one. Since we have seen that ok is onto and Pm1 mk-1,...,mn and Pkm1,...,mk,...,mn are finite sets, it suffices to show that card(Pm1 mk -1,...,m n ) - card(Pkm1, mk,...,mn). Now we know that
We claim that card (Pkm1,...mk,...mn) is also equal to the right-hand side of (19.5.18). In Pm1 mk,...,mn we pick m1 (out of m) places to put the integers 1, . . . , m1; . . . ; mk places to put m1 + • • • + mk-1 + 1, . . . , m1 + . . . + mk-1 + m k ; . . . , and mn places to put m1 + . . . + mn-1 + 1, . . . , m1 + . . . + mn-1 + mn In contrast, in Pkmt, . . . , mk, . . . ,mn the last of the m places is reserved for m1 + . . . + m*, and so we pick m1 (out of m — 1) places for 1, . . . , m1; . . . , mk — 1 places for m1 + . . . + mk-1 + 1, . . . ,m1 + . . . + mk-1 + (mk — 1); . . . , and mn places for m1 + . . . + mk + . . . + mn-1 + 1, . . . , m1 + . . . + mk + . . . + mn-1 + mn. Hence, as claimed above,
This completes the proof of Lemma 19.5.6.
THE EVOLUTION EQUATION
593
We turn next to the two examples mentioned just before the statement of Lemma 19.5.6. Example 19.5.7 We take n — 2, k = 1 and m1 = m2 = 3. We want to fix a particular p e Pm1-1,m2 — P2,3. Let p be the permutation (13452); that is, p ( 1 ) — 3, p(2) — 1, p(3) = 4, p(4) = 5, p(5) = 2. This permutation does belong to P2,3 since both 1, 2 and 3, 4, 5 appear in order in the list p(l), p(2), p(3), p(4), p(5). Following Definition 19.5.5, we wish to write down the corresponding permutation n = O1 (p) e P13,3. We know that n(6) = n(m1 + m2) = m1 = 3. To continue, we need to identify those qs in {1, 2, 3, 4, 5} for which p(q) € {1, 2} as well as those qs for which p(q) e {3,4, 5}. The first of these sets is {2, 5} and the second is {1, 3, 4}. Thus n (2) = p(2) = 1 and n (5) = ,0(5) = 2 and n(\) = p ( 1 ) + 1 = 4, n (3) = p(3) + 1 = 5, and n (4) = p(4) + 1 = 6. Hence, we can write n = (146352). Note that 1, 2, 3 as well as 4, 5, 6 appear in order in the list n (l) = 4, n (2) = 1, n (3) = 5, n (4) = 6, n (5) = 2, n (6) = 3, and so n e P3,3. Since n(6) = 3, we have n e P1 3. Example 19.5.8 Here, we take n = 3,k = 2, m1 = 1 and m2 = m3 = 2. Then m = m1 +m2 + m3 = 5 and m — 1 = m1 + (m2 — 1) + m3 = 4. Let n = (124)(35); that is, n (l) = 2, n (2) = 4, n (3) = 5, n (4) = l, n (5) = 3. Note that the lists 2, 3 as well as 4, 5 appear in order in the list n (l), n (2), n (3), n(4), n(5). Since also n (m1 + m2 + m3) = n (5) = 3 = m1 + m2, we have that n € P21,2,2. Next we compute pn following (19.5.15). Now {q: n(q) e {1, 2}} = {4,1} and {q: n(q) e {4, 5}} = {2, 3}. Thus pn(4) = n(4) = 1, p n (1) = 7r(l) = 2 and Pn(2) = n (2) — 1 = 3, pn (3) = n(3) — 1 = 4. Note that pn e P1,1,2 since 3, 4 appear in order in the list p n (1) = 2, pn(2) = 3, pn(3) = 4, pn(4) = 1. Finally, we wish to calculate n* := o2 pn and observe that n* = n. Now {q: n(q) € {1, 2}} = {4, 1} and also {q: n(q) 6 {3,4}} = {2,3}. Thus, using (19.5.8), n*(4) = p n ( 4 ) = 1 and n*(l) = P n (1) = 2 whereas n*(2) = pn(2) + 1 = 4 and n*(3) = pn(3) + 1 = 5. Also n* (5) = 3. In summary, n*(1) = 2, n*(2) = 4, n*(3) = 5, n*(4) = 1 and n*(5) == 3. Hence n* = (124)(35), and so n* = o2Pn = n as we claimed. We are now prepared to give the proof of Lemma 19.5.3 and so to complete the proof of Theorem 19.5.1. Proof of Lemma 19.5.3 Suppose that mk > 1 and that p e Pm1,...,mk-1,...,mn. Then, by Lemma 19.5.6, the associated n = ok p is in Pkm1,...,mk,...,mn and we claim that
Why does (19.5.20) hold? Both n and p determine a finite sequence of perturbing Bs ordered in the expressions forLtm1,...,mk,...,mn(n) andLsm1,...,mk-1,...,mn(p), respectively, according to increasing times. The key is that n = ok p has been defined in terms of p precisely so that the first (m — l)-perturbing Bs determined by n are exactly the same sequence of Bs determined by p. Of course, n also determines a final Bk, but the presence of the extra Bk in the integrand on the left-hand side of (19.5.20) compensates for that. In fact, it is the multiplication of L m1,...,mk-1,...,mn (p) by e-(t-s)a Bk(s) and the integral
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over [0, t] with respect to uk on the left-hand side of (19.5.20) which makes the equality (19.5.20) true. (Further comments about (19.5.20) can be found in Remark 19.5.9 below.) It follows from (19.5.20) that
Hence,
where the second equality in (19.5.22) follows from the fact that Pm1,....mn is the disjoint union of the sets {Pkm1,...,mk,...,mn : mk > 1} Finally, moving the sum over p in the first expression in (19.5.22) inside the integral, we obtain the desired equality (19.5.2) from (19.5.22); that is, we have
This concludes the proof of Lemma 19.5.3. Remark 19.5.9 (a) We wish to make some comments about (19.5.20) which may be helpful to the reader. Equality (19.5.20) is rather simple but the somewhat involved combinatorial notation obscures this to some extent. Perhaps the quickest way to gain insight into (19.5.20) is to check it for a particular choice of k, m1, . , . , mn (with mk > 1), p € 7>mi,...,mk-1,...,mn and n = ok p. Also, it may help the reader to recall that tm1+...+mk = Sk,m k and that tm1+...+mt = tn(m1+...+mn) since n e Pkm1,...,mk,...,mn• Using this notation, the left-hand side of (19.5.20) becomes
(b) Lemma 19.5.3 is the counterpart in the present context of [Lal 5, Proposition 4.1, p. 112] (see Problem 17.3.6(b) and, for a special case, Proposition 17.3.3 above) which
THE EVOLUTION EQUATION
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appears in the proof of the integral equation obtained in [Lal5] (see Theorem 17.2.1 above). However, the combinatorial arguments that were involved in the proof of Lemma 19.5.3 (and of the associated Lemma 19.5.6) are of a rather different nature and are also more complicated (due to the fact that Bp does not commute with Bq for p = q). Next we show that Lt, the solution of our integral equation, is strong operator continuous (i.e. continuous in the strong operator topology) as a function of t. Theorem 19.5.10 Suppose that hypotheses (H.1)-(H.3)from Section 19.1 are satisfied. Then the function L, defined by (19.4.2)-(19.4.4) is strong operator continuous as a function of t on [0, +00). Proof We know from Theorems 19.4.1 and 19.5.1 that L, has the following properties: (i) It is measurable as a function of t in the sense described in (H.3a). (Note in this connection that the series defining Lt converges for every t.) (ii) It is bounded in operator norm on [0, T] for every T > 0. (iii) It satisfies the integral equation (19.5.1) on [0, T] for every T > 0. Using (iii), the fact that t |-» e-ta is strong operator continuous and the fact that the sum on the right-hand side of (19.5.1) is a finite sum, it suffices to show that
is strong operator continuous on [0, T], where (B, u) could be any of the pairs (B1, u1), . . . , (Bn, un). We can rewrite h(t) as
where X[0,t] denotes the characteristic function of the interval [0, t]. Now let w e H be fixed and let tj —> to. We will use the dominated convergence theorem for the Bochner integral [HilPh, Theorem 3.7.9, p. 83] to show that \\h(tj)p h(to)o|| -» 0 as j -> oo. The function s |-> ||B(s)||B||p||, where B = B(T) is a bound as in (ii) on [0, T], dominates the integrand of
for every j, and
where (19.5.27) follows from (H.3b). Hence, to apply the dominated convergence theorem, it remains only to show that as j -> oo for |u||-a.e. s e [0, T]. We will show in fact that (19.5.28) holds for every s e [0, T] \ {to}. But |u|({t0}) = 0 by (H.2c) since u, is a continuous measure, and so we will have convergence for u-a.e. s € [0, T].
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Let s e [0, T] \ {to} befixed.Then w := B ( s ) L s p is a fixed element of H, and we wish to show that
Now, X[0,tj](s) -» X[0,t0](s) as j -» oo since s = to. Also, since the semigroup is strongly continuous, we have ||e-(tj-s)a w — e-(to-s)a W || —» 0 as j —> oo. Hence we can write
as j -» oo; so that (19.5.29) holds as we wished to show. The preceding proof establishes a little more than was claimed in Theorem 19.5.10. We state this additional information as a formal corollary since it will shed some light on the uniqueness result in the next section. Corollary 19.5.11 Suppose that hypotheses (H.1)-(H.3) are satisfied. Then any L(H)valued function Mt, 0 < t < oo, which satisfies (i)-(iii) from the proof of Theorem 19.5.10 must be strong operator continuous. Proof A close look at the proof of Theorem 19.5.10 will show that it does not depend on the special nature of Lt but only on the fact that Lt satisfies (i)-(iii). Hence any function Mt as above which also satisfies (i)-(iii) must itself be strong operator continuous. D How does Corollary 19.5.11 shed light on the uniqueness result? We will see in the next section that any function Mt which satisfies properties (i)-(iii) must equal Lt for every t. One might think that instead, there could be some exceptional set, say, at least a finite set. The strong operator continuity of Lt and Mt helps to explain why this cannot happen. 19.6 Uniqueness of the solution to the evolution equation We show in this section that the solution to the integral equation is unique ([dFJoLa2, Theorem 6.1, p. 196]). Much of the argument follows standard lines rather closely, but equation (19.6.6) below is more involved. Theorem 19.6.1 Suppose that hypotheses (H.1)-(H.3) in Section 19.1 are satisfied. If M : [0, +00) —> £(H) is any function which satisfies (i)-(iii) from the proof of Theorem 19.5.10, then M, = Lt for every t e [0, +00), where L, is defined by (19A.2)-(19AA). Hence, Lt is the unique solution to the evolution equation (19.5.1) satisfying conditions (i)-(iii).
UNIQUENESS OF THE SOLUTION
597
Proof It suffices to fix T > 0 and show that Mt = Lt on [0, T]. Since Lt and Mt both satisfy (i)-(iii) on [0, T], we see that Lt — Mt satisfies the integral equation
Letting B = B(T) be a bound for ||Lt - Mt\\ on [0, T], we see from (19.6.1) that
Using (19.6.1) to justify the first two equalities below, we can write
Proceeding inductively, we obtain for any positive integer m,
Using (19.6.4), one easily obtains the inequality
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for any m. Further, we will show that for any m and 0 < t < T, we have
Indeed, (19.6.6) is the key to the proof and is the main place where the argument deviates from standard lines. Using (19.6.5) and temporarily assuming (19.6.6), we have
Since (19.6.7) holds for every t e [0, T] and any positive integer m, we see that Lt = Mt for all t e [0, T] as we wished to show. Thus it remains to establish the equality (19.6.6). While we discuss the general case, the reader might find it helpful initially to think in terms of the case n = 2, m = 3. This special case is the simplest one that contains the essential ideas of the general case. We begin by using the multinomial formula to rewrite the right-hand side of (19.6.6) as
Now we focus our attention on one of the terms of the series in (19.6.8); that is, we fix nonnegative integers m 1 , . . . , mn such that m1 + . . . + mn = m and write
The proof will be finished by showing that the expression in (19.6.9) equals the sum of those terms on the left-hand side of (19.6.6) that involve m1 B1s, . . . , mn B n s. (Remark: Here and further on, we refer just to the operators Bk. Recall, however, that each Bk comes with an associated measure uk, so that we are actually considering the mk — (Bk, uk) pairs.) In order to do this, we start by writing the right-hand side of (19.6.9) as the constantm1!...mn!times the sum of m! integrals over the simplices corresponding to all m! orderings of the variables s1,... ,sm. Note that the assumption of continuity on the measures uk is involved in this.
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Now these m! integrals break up into m! groups of m1! . . . mn! equal integrals. The m1! . . . mn! equalities hold because any of the m1! . . . mn! permutations that reorder the variables appearing as arguments of Bk for each k = 1, . . . , n leave both the integrand and the measure |u1|®m1 x • • • x |un|®mn unchanged. Collecting the equal integrals and paying attention to the cancellation of constants, we are left with the sum of m! integrals each with a constant of 1 in front of it. Now we use the Fubini theorem to write each of those m! integrals as an iterated integral corresponding to the ordering of the variables. Next, rename the dummy variables as needed so that ds1, . . . , dsm appear in each integrand with their subscripts in increasing order. Finally, note that this sum of m! integrals is precisely the same as the sum of those m! integrals on the n! left-hand side of (19.6.6) that involve exactly m1 B1s, ... ,mn Bns. Thus (19.6.6) is established and so our proof of Theorem 19.6.1 is complete. Formula (19.6.6), while probably not new, is new to the authors and may be useful in related situations. With this in mind, we restate it in simplified notation as a corollary of the preceding proof. Clearly, further analogous formulas could be obtained in a similar way. We stay close to the relevant assumptions of Theorem 19.6.1 although they could probably be varied or relaxed further. Corollary 19.6.2 For k = 1, . . . , n, let uk be a finite, positive, continuous Borel measure on [0, t ] and let fk : [0, t] -> R be uk-integrable. Then for all integers m > 1,
19.7 Further examples of the disentangling process Our purpose in this section is to give additional examples of the use of Feynman's rules and/or to illustrate remarks made earlier in the chapter. Discrete measures are missing from this chapter so far, except for brief comments. In contrast, three of the five examples here will involve such measures. Examples 14.2.1 and 14.2.3 also involve discrete measures; the reader may wish to review these two simple examples at this point. (Of course, measures with a nonzero discrete part were involved throughout much of Chapters 15-18 and were treated rigorously there.) What is the level of mathematical rigor in this section? All five examples are rigorous at least insofar as assumptions are indicated which insure that the expressions resulting from the disentangling process make sense. (This is the level of rigor that was reached at the end of Section 19.4 in the main body of this chapter.) Examples 19.7.2 and 19.7.5 are entirely rigorous. Example 19.7.2 expands on Remark 19.4.5 and deals briefly but rigorously with the problem of forming analytic functions of noncommuting operators. Example 19.7.5 illustrates the theory developed earlier in this chapter; it involves nonlocal potentials and has a more explicitly physical character than the others. We conjecture but will not prove here that the disentangled expressions in Examples 19.7.1, 19.7.3 and
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19.7.4 give the unique solution to an evolution (integral) equation involving a measure with a finitely supported discrete part. (Example 19.7.3 has been rigorously dealt with earlier in the diffusion and quantum-mechanical cases via path integral methods; see Theorems 17.2.1 and 17.2.7.) Example 19.7.1 involves two (not necessarily bounded) operators and Trotter products. Example 19.7.1 (The nth Trotter products) We assume that both -A and —B are (Co) semigroup generators. Given t > 0, attach time indices to — t A and — tB, respectively, as follows: — tA = — ft A(s)ds and — tB = — ft B(s)v(ds), where v is the discrete measure v =1/nEn=1dqr/n• Following Feynman's ideas, we write
In the second equality in (19.7.1), we followed Feynman's second rule (see Section 14.2) and in the fourth, we time-ordered the expression. Thus when e~t(A+B) is disentangled according to the given measures, we obtain the nth Trotter product which is the last expression in (19.7.1). Recall that Trotter products and especially the related Trotter product formula, Theorem 11.1.4 (which involves the strong operator limit of the last expression in (19.7.1)), have—along with the product formula for imaginary resolvents, Theorem 11.3.1— played a key role in this book. Indeed, these product formulas and their relationship with "the" Feynman integral are the subject of Chapter 11; they also played a role in several other places in this book, including Examples 15.5.5 and 16.2.7, as well as Corollary 17.2.9. A point that may be confusing to the reader in the derivation of (19.7.1) above is that defined by (19.7.1), need not denote the (Co) semigroup generated by the operator A + B; and, in fact, A + B itself need not be the generator of a (Co) semigroup. We caution the reader once again that the individual steps in calculations using Feynman's rules as in (19.7.1) above should not be interpreted literally; this applies in particular to the second and fourth equalities in (19.7.1). The interested reader may wish to do a calculation similar to that performed in (19.7.1) with v := En=1 wqdTq, where 0 < T1 < . . . < Tn < t and wq e C for q = 1, . . . ,n. e -t(A+B), as
FURTHER EXAMPLES OF THE DISENTANGLING PROCESS
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We remark that Example 19.7.4 below is somewhat related to the example just given. Example 19.7.2 (Analytic functions of noncommuting operators) Let T > 0 be given and suppose that u1, u2 and B1, B2 satisfy hypotheses (H.2) and (H.3). Further, let
where the series (19.7.2) converges absolutely at least on the closed polydisk
From Remark 19.4.5, in particular, taking a = 0 and using (19.4.14), we can disentangle g (ft B 1 ( s ) u 1 ( d s ) , ft B2(s) u2(ds)) for 0 < f < T by means of (19.3.14):
where the notation is just as in Section 19.4 but for n = 2. We saw in Section 19.2 how to disentangle the exponential expression
where u, is a continuous measure. The final formula in this case is given by (19.2.10). Not much changes in (19.2.10) if we start the evolution at time a rather than at time 0. We now give the adjusted formula and also introduce notation which will be useful to us in the next two examples. Let / denote Lebesgue measure on R and suppose that t > a. Then
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where
As in Chapter 17, it is natural to take P(a, a) .= /, the identity operator. Example 19.7.3 (Adding on a discrete measure) We now wish to consider the consequences of adding to u in (19.7.5) a finitely supported discrete measure; that is, replace u by
where wq e C is the "weight" atTqand
In addition to the usual requirements on a, B, u,, we insist that B(Tq) be defined for q = 1, . . . , p. Now for 0 < t < T1,
On the other hand, for T1 < t < T2, the use of Feynman's heuristic ideas yields
The third equality above comes from Feynman's second rule (working temporarily in the "commutative world"), the fourth from time-ordering and Feynman's second rule,
FURTHER EXAMPLES OF THE DISENTANGLING PROCESS
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and the fifth from our calculation in Section 19.2 and the notation in (19.7.5). Continuing along this line, we obtain
Recall that when H = L2(Rd) and a =-i/2A = i H0, B = -iV (resp., a = 1/2A, B = V), the right-hand side of (19.7.10) was shown in Chapter 17 [Lal5] to give the unique solution to an integral equation associated with the Feynman-Kac formula with Lebesgue-Stieltjes measure n in the "quantum-mechanical" (resp., "diffusion" or "probabilistic") case. (See especially Theorems 17.2.1 and 17.2.7.) Example 19.7.4 (Two unbounded operators) Our starting point as in the previous example is the expression exp (— f0a(s)ds + ft B(s) u ( d s ) ) , where a, B, and u satisfy the usual conditions; i.e. (H.1)-(H.3) in Section 19.1. As discussed in the introduction to this chapter, we may well want to add another unbounded operator to the argument of the exponential. Suppose that — a1 is such an operator which is also a (Co) semigroup generator. We would like to disentangle
but we do not know how to do this with a and a1 both unbounded except when — (a+a1) is itself a (Co) semigroup generator. The results which guarantee this (see Theorems 9.7.1, 9.7.3, Corollary 9.7.8 and Theorems 10.3.19 and 10.4.8) may not be applicable. Even if they are, it is not always desirable to proceed in this way. For one thing, the individual effects of a and a1 get lost in the process. Further, —a might well be such that a concrete and tractable formula for e-sa is known but no such useful formula is known for e-s(a+a1).
Suppose that we are interested in the evolution over the interval [0, T], where T > 0 will be fixed throughout the rest of this discussion. Motivated by the fact that the sequence of measures
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converges weakly (or "vaguely") to Lebesgue measure l on [0, T) (i.e. for every bounded, continuous function g on [0, T),
as N —> oo), we pick an integer N and replace Lebesgue measure in the third integral in (19.7.11) by vN. The revised model will result from disentangling
The motivation for the calculation yielding (19.7.14) below is quite different than in the preceding example, but the calculation itself differs very little. Hence, we simply state the result, again making use of the notation from (19.7.5):
We close this example by remarking that we could treat finitely many (B, u) pairs and finitely many additional unbounded operators much as in the manner above. We would begin by regarding (19.3.14), the final result of our calculations in Section 19.3, as P(0, t). Nonlocal potentials relevant to phenomenological nuclear theory Example 19.7.5 (Nonlocal potentials) Our final example has a different purpose than those above. We will write out and comment on some of the terms of a perturbation expansion involving a nonlocal potential B2 as well as a time-dependent, complexvalued local potential B1(s) = V(s, •) and a generator —a. We will associate Lebesgue measure with both B1 and B2 although any measures u1 and u2 satisfying (H.2) of Section 19.1 could be used. It is of course no surprise that any operators and measures satisfying our hypotheses can be substituted into our formulas to give a variety of concrete examples. We give this example nevertheless because nonlocal potentials do not seem to be discussed in the mathematical physics literature and also because the associated perturbation series have features which are quite different from those arising from the usual local potentials.
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Physically, these nonlocal interactions arise in many-body problems, especially in the theory of nuclear structure [Mc, Tab]. (See also [ChSa, Chapter VIII].) Even if the two-body interactions are local (which seems to be the case), systems with three or more particles require nonlocal potentials. These potentials are also momentum-dependent. The compactness of the associated integral operators acting on L2(R3) guarantees that the nonlocal potential can be written as an infinite separable sum. Empirically, the infinite sum can be replaced by a finite sum so that the associated operators have finite rank; in fact, a few terms suffice in practice. Tabakin [Tab] first showed this by using canonical transformations on Hartree-Fock models of nuclei. He not only showed that the momentum-dependence and nonlocality were accurately treated by low rank separable potentials, but he also found that the highly singular "hard cores" in the realistic potentials could be replaced by very smooth functions. Hence, finite rank, separable potentials are physically useful and so elucidating their structure as we are about to do in the simple example below is of strong physical interest as well. Let p € L2(R3) = H and suppose that
where the nonlocal potential W is separable and of finite rank as on the right-hand side of (19.1.3). As discussed above, it is reasonable physically to assume that the rank is low. We will take the rank to be one for brevity and because, once this case is understood, it is not hard to work out the general case. Thus
where «, v e L2(R3). Hence, from (19.7.15),
where (•, .)2 denotes the inner product in L2(R3). Note that we have taken W and hence B2 to be time independent. We remark that, in the light of the assumption (H.3b), we require f0t || V(s, •) ||oods to make sense and be finite. The full perturbation series involves a sum over m1 and m2, both going from 0 to oo. In the (m1,m2) term, B1 and B2 appear m1 and m2 times, respectively. We now examine various terms of the series. The terms of the form (m1, 0) do not involve the nonlocal potential and so look familiar. We consider their action on an initial state p € L2(R3).
Next, we consider terms involving only the generator and the nonlocal potential. First we use (19.7.17) to calculate the third order term (0,3) to get the idea and then just write down the general term (0, m2).
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(0, 3) term:
Now we make some comments about the last expression in (19.7.18) which will also be valid for the expression in (19.7.19) below. First, all the factors except the first and the last are almost exactly the same, the only difference being in the length of the time intervals in the exponentials. Second, the effect of the initial state p is felt only in the first factor (on the right). Third, the only remaining vector is in the last factor; the other factors are all scalars. Hence, all the factors can be permuted at will. We just write the final result for the (0, m2) term. (0, m2) term:
We finish this discussion by considering the (1, 1) and (1, 2) cases. Formula (19.3.8) tells us that these involve 2! = 2 and 3! = 3 terms, respectively. (1, 1) term:
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Observing the integrands on the right-hand side of the last equality in (19.7.20), we see that the effect of the initial state p dies off after the first appearance of B2. The same comment is valid for any m1 and m2. We will skip the calculations in the remaining case and just give the first expression and the answer. (1, 2) term:
We close this example by noting that in this case, the k = 2 term on the right-hand side of the integral equation (19.5.1) acting on p becomes
Exercise 19.7.6 (a) Work out a few additional terms of the perturbation series obtained in Example 19.7.5; for instance, write down the (2,1), (2,2), and (1,3) terms. (b) Extend the above example to the case where fa is a rank two operator; more specifically, assume that the kernel W(x, y) of B2—instead of being given by (19.7.16) above—is defined by
where uj, Vj € L 2 (R 3 ) for j = 1, 2. As was alluded to earlier, integral operators other than those with finite rank also arise in applications to many-body problems. In particular, convolution operators are used in this context.
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Exercise 19.7.7 Let B1 be the multiplication operator by V(s, .), as in Example 19.7.5, but replace B2 in (19.7.17) by a convolution operator with kernel W (x, y) = K(x — y), where the function K belongs to L1 (R3). Write out a few terms of the resulting perturbation series. The perturbation series that arose from disentangling the exponential expression (19.1.1) have been the main focus of this chapter. However, Feynman's ideas extend well beyond such series. We saw this in the setting of the Wiener and Feynman path integrals in Chapters 15-18 and we hope that we have given the reader some indication that this continues to be the case in the present setting.
20 FURTHER WORK ON OR RELATED TO THE FEYNMAN INTEGRAL Our work in this book has focused on a number of operator-valued approaches to the Feynman integral in which product formulas and analytic continuation played a prominent role. We also studied Feynman's operational calculus for noncommuting operators and have seen that this subject is closely related to the Feynman and Wiener integrals and can, in fact, be regarded as extending certain aspects of path integration. One of our purposes in this chapter (see Section 20.1) is to remark on or at least to provide references for some of the many further rigorous approaches to the Feynman integral. Since the appearance of the influential monograph of Albeverio and Hoegh-Krohn [AlHol] in 1976, much of the rigorous work has involved "transform assumptions". We will concentrate on these developments in Section 20.1.A but will simply provide references for other work in the very short Section 20.1.B. Heuristic versions of the Feynman integral have had a considerable influence on many subjects in physics over the years and, more recently, on several topics in mathematics as well. The work of Edward Witten [Witl-15] has been central to a number of exciting developments. We begin Section 20.2 by identifying ingredients that are often present when heuristic Feynman integrals are used in quantum physics. Section 20.2. A provides a rather lengthy but far from complete discussion of the mathematical results in low-dimensional topology and knot invariants which have been heavily influenced by quantum field theory and heuristic Feynman integrals. Section 20.2.B begins with a discussion of the "proof" of the celebrated Atiyah-Singer index theorem via supersymmetric Feynman integrals. It continues with a brief discussion of the use of Feynman-type path integrals in connection with a few further topics in physics and mathematics. These topics are deformation quantization, gauge field theory and string theory. The reader should be aware that many of the concepts involved have not been rigorously formalized in a mathematical sense. Further, the authors are far from being experts on the subjects involved in Section 20.2. We now turn to Section 20.1 and to the topic of Feynman integrals via transform assumptions.
20.1
Transform approaches to the Feynman integral. References to further approaches Many of the remarks that we will make in this section are intended only to give a rough idea of the nature of various developments. Readers seeking more detail will need to check the relevant references for precise definitions and statements of results. It is the
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monograph of Albeverio and Hoegh-Krohn [AlHo1] and the paper of Kallianpur, Kannan and Karandikar [KalKanK] which will receive the most attention in this section. 20.1.A. The Fresnel integral and other transform approaches to the Feynman integral The idea of focusing on potentials which are Fourier (or Fourier-Stieltjes) transforms of bounded, C-valued Borel measures has its roots in work of Ito (see [Ito2], 1967). (Note also the earlier 1961 paper [Itol].) Actually, Ito considered potentials of the form
Albeverio and Hoegh-Krohn extended this idea and introduced and developed many further ideas in their influential monograph [AlHol] and in further work [AlHo2,3] that followed soon after. Much of the work that will be commented on in this subsection will be described in reference to [AlHol-3], and so we briefly outline the basic framework from [AlHol]. In the case of a reader who has no previous knowledge of work on or related to the Fresnel integral, it will probably not be clear at first what the discussion through formula (20.1.18) has to do with "the Feynman integral". The Fresnel integral Let H be a separable Hilbert space over R and let M(H) denote the space of C-valued Borel measures on H. A measure u e M(H) necessarily has a finite total variation measure |u|; see, for example, [Ru2, Chapter 6]. We let
Addition and scalar multiplication are defined in a straightforward way in M(H), and multiplication is taken to be convolution. Specifically, given u and v in M(H),
for any B e B(H), the Borel class of H. Under these operations and the norm defined in (20.1.2), M(H) is a commutative Banach algebra with identity. (The identity in M(H) is the Dirac measure 60 with mass one concentrated at 0 e H.) The interest is primarily in functions f(= fu= u) on H which are Fourier-Stieltjes transforms of measures u € M(H). Given u, € M(H), we let
where (, ) denotes the inner product in H. The set of all such functions fu is denoted F(H) and is called the Fresnel class of H. Looking ahead to formula (20.1.9), F(H) is also referred to as the class of Fresnel integrable functions. Now the familiar relationship
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between convolution and the Fourier transform holds in this setting; that is, for u, v in M(H), we have
In alternative notation,
The correspondence u |-> fu from M(H) to F(H) is onto (by definition of F(H)) and can be shown to be one-to-one. Further, this correspondence preserves algebraic operations, where the multiplication in F(H) is ordinary pointwise multiplication of functions (see (20.1.6)). In light of the discussion just given, if we carry the norm on M(H) over to F(H), that is, if we define
then F(H) is a commutative Banach algebra with identity. Clearly, F(H) with this norm is, via the correspondence u |-> fu, isometrically isomorphic as a Banach algebra to M(H). It is easy to see from (20.1.4) that fu is everywhere defined on H and that
for every h e H. Using (20.1.4) again as well as the dominated convergence theorem, it easily follows that each fu is continuous on H. In fact, one can show that fu is uniformly continuous on H. The Fresnel integral F(fu) of fu e F(H) is defined by
Clearly, the definition just given makes sense for all fu e F(H) and we have
Since also F(1) = F(fd 0 ) = 1 and ||f S o || = ||d0|| = 1, we see that the Fresnel integral is a continuous linear function on F ( H ) of norm 1; i.e.
Properties of the Fresnel integral The space F(H) and the Fresnel integral F(.) have a variety of further pleasant properties. We now list some of these: (1) We already know that the product of Fresnel-integrable functions is Fresnelintegrable (see (20.1.5) or (20.1.6)). Also, since F(H) is a Banach algebra with identity, any entire function of a Fresnel-integrable function is Fresnel-integrable. Of particular
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interest in applications is the fact that the exponential of a Fresnel-integrable function is Fresnel-integrable. (2) If fu e F(H) and U is an orthogonal transformation on H, then fu o U € F(H), fn o U = u ° U and
(3) The Fresnel integral is sometimes called the "normalized integral" and denoted
"Properly" interpreted, the normalized integral is translation invariant. (See [AlHol, pp. 2.6-2.8] and [AlHo2, pp. 8, 9 and especially the formula on the bottom of p. 11].) However, F ( f ( a ) ) , where F(a) is the translate by a e H of f e F(H), is not in general equal to F ( f ) as Example 20.1.1 below shows. (4) There is an analogue of the Fubini theorem for the Fresnel integral. (See [AlHol, pp. 2.8-2.10].) Next we give the simple example referred to in (3) above. Example 20.1.1 Let b e U with b = 0. Take / = db. Then
Also,
Next, choose a e U such that (a, b) e {0, ±2n, ± 4n, . . .}. Then by (20.1.14),
Hence, by (20.1.16) and (20.1.15), we have
Comparing (20.1.15) and (20.1.17), we see that
It is natural to wonder at this point what the Fresnel integral has to do with the Feynman integral.
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An approach to the Feynman integral via the Fresnel integral We will indicate here choices that can be made for the Hilbert space H and the function / so that the Fresnel integral of / provides a solution to the Schrodinger equation. We consider a quantum-mechanical particle moving in R under the influence of a potential V. The restriction to a single space dimension is made simply for convenience. The Hilbert space H1 consists of those R-valued absolutely continuous functions y, with Y : [0, t] -> R, such that y (t) = 0 anddy/dse L2[0, t]. The inner product on H1 is given by
Note that all the paths in H1 vanish at t whereas the paths in Wiener space C0 vanish at 0. This gives rise to some minor technical points which we will usually ignore when we make comparisons further on which involve these spaces. We remark that H 1 ( 0 , t) can be viewed as the space of real-valued functions y in the Sobolev space H1(0, t) which satisfy a Dirichlet boundary condition at t; namely, y(t) = 0. In order to insure that the function / in (20.1.20) below is in F(H1), Albeverio and Hoegh-Krohn [AlHol, §3] assume that the potential V and the initial state p are Fourier-Stieltjes transforms of C-valued measures on K; i.e. V and p are in F(R). Then the Fresnel integral F ( f ) , where
satisfies the Schrodinger equation with initial state p [AlHol, Theorem 3.1]. (We have assumed throughout the preceding discussion that units are chosen so that H and the mass m are equal to 1.) In [AlHol], the authors went on to view scattering theory and wave operators from the perspective of the Fresnel integral. Following this, they added to their potential a quadratic form and defined a Fresnel integral associated with the form. Various cases were considered including quadratic forms that were not necessarily positive definite. Applications were given. The simplest corresponds to the case where the quadratic form is strictly positive definite; this amounts to adding an anharmonic oscillator to the potential V, where V is the Fourier-Stieltjes transform of a C-valued measure on Rd. We will postpone a fuller discussion of the role of quadratic forms in transform approaches to the Feynman integral until we come to the paper of Kallianpur, Kannan and Karandikar. At that point, we will discuss quadratic forms as presented in [KalKanK] as well as recent and related nice work of Albeverio and Brzezniak [AlBr3]. Advantages and disadvantages of Fresnel integral approaches to the Feynman integral The Fresnel integral approach to the Feynman integral has both major advantages and disadvantages as we will now discuss. [Some of these points apply also to the other
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transform approaches that will be discussed or at least referred to in this subsection (i.e. 20.1.A).] Advantages: (1) The Fresnel integral has several nice properties and the part of the theory that was sketched above is mathematically appealing and relatively simple. (The part having to do with quadratic forms is less appealing and not so simple and the developments connected with (2) and (3) below are far from simple.) (2) One of the most appealing features of Feynman's approach to mechanics on the quantum level is the insight that it provides into how quantum mechanics "approaches" classical mechanics as h —» 0. (Recall that the heuristic "argument" for this was discussed in Section 7.2 of this book but that a rigorous proof was never presented.) In spite of the great appeal of this method of stationary phase and further related developments, it has been extremely difficult to turn the heuristic ideas into rigorous mathematics. Perhaps this is not surprising; what is needed is an infinite dimensional version of a subject which is difficult in the finite-dimensional case. Rigorous results have been given, however, in the (infinite dimensional) framework of the Fresnel integral of Albeverio and Hoegh-Krohn (see [AlHo2,3, AlBlHo, Rez, AlBrl]). Note that finite-dimensional oscillatory integrals have been studied in many places; see for example, ([Coiv2], [Dui], [DuiHor], [Fed], [H6r3, esp. Chapter VII], [Hor2,4], [GuiSbg, esp. Chapter I], and [Ste2, esp. Chapter VIII]), along with the much more elementary textbook [Er]. We point out that many of these references discuss, in particular, oscillatory integrals that occur naturally in the study of geometric optics or of the "classical limit" associated with the Schrodinger equation. (3) The Fresnel integral is rather broad with respect to the range of topics to which it can be applied, as can be seen to some extent in the original monograph [AlHol]. The 1995 paper of Albeverio and Schafer [AlSca] is an interesting recent illustration of this; it looks at the Abelian Chern-Simons theory and linking numbers as in [Witl4] in terms of the Fresnel integral in an appropriately chosen setting. (See case (i) of Remark 20.2.5(b) below for further related information. We note that although the "Gauss linking number" has played an important role in the history of knot theory, it is the non-Abelian Chern-Simons theory which is most directly relevant to the contemporary study of knot invariants; see [Witl4] and Section 20.2.A.) The paper [Witl4], the non-Abelian Chern-Simons theory and linking numbers, along with a variety of related topics, will be discussed in Section 20.2. A below. (We remark that the work on the Feynman integral in this book is essentially restricted to nonrelativistic quantum mechanics but, within that setting, is far broader than the Fresnel integral.) Disadvantages: (1) The Fresnel integral permits no singularities in the potential other than the singularity at oo possessed by a quadratic form. This is a serious shortcoming since many of the potentials of physical interest including such fundamental ones as the attractive and repulsive Coulomb potentials have such singularities. (Our results in Chapters 11 and 13 clearly go far further in this respect.)
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(2) Recall that to get, for example, the function / from (20.1.20) in F(H), one requires that the potential V and the initial state p be Fourier-Stieltjes transforms of complex measures on Rd. This limits V and p to bounded uniformly continuous functions, as noted earlier. But even if V is bounded and uniformly continuous, it is often not easy to tell whether or not V = u for some complex measure u. Of course, a similar comment holds for (p. (No such issue arises for our work in this book.) (3) Although the Banach algebra F ( H ) has some mathematically pleasing limiting properties, the limiting process seems to be neither physically natural nor easy to verify. Given f and fn, n = 1, 2, . . . , from F(H), to show that \\fn - f\\ -> 0, one needs to find the measures u and un, n = 1, 2, . . . , such that f = u and fn = un, n = 1, 2, . . . , and then show that un —> u. in total variation norm. The uniform convergence of fn to / on H is a necessary but not sufficient condition to have ||un — u|| —> 0. (The limiting processes in our stability theorems are far more straightforward to deal with and are more natural physically.) The work of Albeverio and Hoegh-Krohn inspired many others directly or indirectly to approach the Feynman integral using transform assumptions. We will now give references for and, in some cases, comments on, a portion of this work. Many of the comments that we make will be brief and often they will be rather vague. We will leave it to the interested reader to check the references for the definition of terms and the precise statement of results. The Feynman map Aubrey Truman [Tru2-5] quickly took up ideas from [AlHol] and combined them with ideas from Feynman's 1948 paper [Fey2] to define the "Feynman maps." The space H1 of paths used by Truman is the same Hilbert space as in [AlHol] in the case where the Schrodinger equation for nonrelativistic quantum mechanics is solved (see the paragraph containing (20.1.19) above). Polygonal paths are used as in [Fey2] and the Feynman map is defined as the limit of what might be called "scalar-valued Trotter products". Transform conditions come into play when a sufficient condition is established for the existence of the Feynman map. Quadratic forms fit smoothly into the Feynman map approach and they do not require an adjusted theory as is the case of the Fresnel integral. On the other hand, the Fresnel integral seems to be more readily adaptable to settings other than nonrelativistic quantum mechanics (see especially the recent paper of Albeverio and Schafer [AlSca]). In 1945, Cameron and Martin [CaMa2] established a linear change of variables formula in Wiener space. Truman gave and made good use of a parallel formula for the Feynman map. (A Cameron-Martin formula for the Feynman integral will be stated in (20.1.74) below.) A quasiclassical representation is developed by Truman and combined with the Cameron-Martin type formula to study various applications to quantum mechanics. The 1979 paper [Trul] includes at least some discussion of all of the topics mentioned above. The later paper of Elworthy and Truman [ElTr] further develops the Feynman map especially in regard to the Cameron-Martin type formula, and also gives simpler proofs
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of key facts. Results of Hormander [H6r2] on finite-dimensional oscillatory integrals are the basis for the revised proofs. The paper [ElTr] seems to have had a significant impact on later work on infinite dimensional oscillatory integrals (see, for example, the papers of Albeverio and Brzezniak [AlBrl-3]). The Poisson process and transforms Further, significant work has been done in which transform assumptions are made on the initial state and on the potential and in which the Poisson process plays a crucial role. In these developments, the setting is "momentum space" rather than position (or configuration) space. This work began with Chebotarev and Maslov [ChebMasll,2]. Additional progress along these and related lines was centered at CNRS, Luminy (in the Centre de Physique Theorique). Blanchard, Combe, Hoegh-Krohn, Rodriguez, Sirugue and Sirugue-Collin were the people most consistently associated with this effort. The papers [ComHRSS, BlSir] are examples of this work. Many further references can be found in the expository paper [B1CSS]. We also refer the reader to [ChebMasll,2] as well as to the survey paper [ChebKMasl] and to the recent contribution [ChebQu]. The Poisson process also plays a central role in work related to the Dirac rather than to the Schrodinger equation. Here, however, one can work in position rather than momentum space and it is not necessary to assume that the initial state and the potential are transforms. (Hence, strictly speaking, this work does not quite fit into this section.) The solution to the Dirac equation (or sometimes to a closely related equation) is written as a path integral or expectation with respect to the Poisson process. The telegrapher's equation plays a role in these developments that is in some ways analogous to the role played by the heat equation with respect to the Schrodinger equation. Except in the case where strong spatial symmetry is assumed, this work is limited to one space (plus one time) dimension. The work of Zastawniak [Zas2] is informative with regard to this restriction. See the papers [B1CSS] and [ChebKMasl] for much more information and further references regarding the Poisson process and the Dirac equation. We also call to the reader's attention the work of Chargoy [Charg], Gaveau, Jacobson, Kac, and Schulman [GaJKacS] (and the relevant references therein, including an early paper by Mark Kac), Ichinose [Ic2,3], Ichinose and Tamura [IcTaml-4], Zastawniak [Zasl], Jefferies [Je2, Sects. 6.4 and 6.5] and Riggs [Rig]. The thesis of Riggs is not only concerned with the Poisson process and the Dirac equation but also with appropriately adjusted generalized Dyson series and Feynman-Kac type formulas with a LebesgueStieltjes measure much as in Chapters 15 and 17 of this book ([JoLal], [Lal5-16, 18]). Our references above to the work of Ichinose and Ichinose and Tamura have been brief, but, in fact, their work has played a significant role in the developing understanding of the relationship between the Dirac equation and path integrals. A "Fresnel integral" on classical Wiener space Inspired in large part by the monograph of Albeverio and Hoegh-Krohn [AlHol], Cameron and Storvick began in 1980 [CaSt4,5] the study of a related theory. In their approach, however, the Feynman integral was interpreted via scalar-valued analytic continuation in mass (see Definition 4.5.1) rather than by the Fresnel integral (see
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(20.1.9)). Further, the space F(H) of Fresnel-integrable functions (see (20.1.4)) was replaced by a space S of functions a = Fa on Wiener space Ct = C([0, t], Rd) which are a type of stochastic Fourier-Stieltjes transform of C-valued Borel measures a on L2[0, t] = L 2 ( [ 0 , t ] , R d ) . For notational convenience, we will assume that d = 1. Given a in the Banach algebra M(L2[0, t]) and x e Ct,
where the integral in the argument of the exponential in (20.1.21) can be interpreted either as the Paley-Wiener-Zygmund (see [CaSt4]) or Ito (see [Kal1]) stochastic integral of the deterministic function v with respect to the stochastic variable x. Heuristically, one can think of f0 v(s)dx(s) as the "inner product" of v with "white noise"dx/dx.Of course, dx/ds does not exist as a legitimate function as we know from Theorem 3.4,7, but, for each v e L2[0, t], the stochastic integral f0f v(s)dx(s) does exist for s-a.e. x e Ct. The elements of the space S referred to above are actually equivalence classes of functions where the appropriate equivalence is equality s-a.e. (see Definitions 4.2.3 and 4.5.5). We note at this point that some key parts of [CaSt4] were substantially simplified by Johnson and Skoug in [JoSk8]. In particular, the essential results concerning S were obtained rather briefly in [JoSk8] without the use of several intermediate spaces that played a role in [CaSt4], Elements of [JoSk8] will be incorporated into our continuing discussion of [CaSt4]. It was shown that the map carrying a to [Fa ], the equivalence class of Fa, is bijective, preserves addition and scalar multiplication and turns the convolution of measures into the pointwise multiplication of "functions". Letting
and using the fact that M(L2[0, t]) is a commutative Banach algebra with identity, one sees then that (S, || • ||) is also a commutative Banach algebra with identity. (The reader should compare our earlier discussion of the Banach algebra F(H) with the discussion above and, in particular, should compare (20.1.22) with (20.1.7).) It is then easy to show ([CaSt4]) that for every F e S, the scalar-valued analytic Feynman integral of F with mass parameter q exists and equals
where F = Fa is given by (20.1.21). The reader should compare (20.1.23) with the right-hand side of (20.1.9). In order to show that the Banach algebra S is relevant to nonrelativistic quantum mechanics, one needs to find sufficient conditions on a potential V to insure that an appropriate function F on Wiener space belongs to the Banach algebra 5. Cameron and Storvick treated time-dependent and complex-valued potentials V : [0, t] x R —> C
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and, in that setting, appropriate conditions turned out to be that (a) V(s, •) = Vs, where vs e M(R) for.y e [0, t], (b) vs(B) is a Borel measurable function of s for every B e B(R), (c) ft||vs||ds < oo. Under these assumptions, one can show that the function
belongs to 5. Since S is a Banach algebra, it follows immediately that exp(F) is also in 5. It is a slight variation of exp(F) that is used to make the connection with quantum mechanics and the Schrodinger equation. Given E € R, one replaces F in (20.1.25) with
(The change from s in (20.1.25) to t — s in (20.1.26) is related to the time-reversal map which was discussed in Remarks 15.3.7, 15.4.4 and 17.3.5. See also Definition 15.7.5 and Theorem 15.7.6.) Then the functions G and exp(G) are in S. One also needs to know conditions on the initial state p such that for every E € R, the function
belongs to S. For this, it suffices that p = u0 where uo € M(R). Cameron and Storvick [CaSt5] showed that for d = 1, p = uo as just described above and under somewhat more restrictive assumptions on V than in (20.1.24), the scalar-valued analytic-in-mass Feynman integral of the function
satisfies the integrated form of the Schrodinger equation with initial state (p. The techniques from [CaSt5] do not extend to dimension d > 2. By using a different summation procedure which permitted exact calculations in some places rather than estimates, Johnson and Skoug [JoSk10] obtained the results of [CaSt5] for an arbitrary space dimension d. Also, the assumptions on the potential in [JoSk10] are given by (20.1.24) and are somewhat milder than those in [CaSt5]. Cameron and Storvick were aware from the beginning that their work [CaSt4] shared common elements with the monograph [AlHo1]. They believed, however, that there were functions in S which had no counterpart in the Banach algebra treated in [AlHol]. They proposed such an example in the introduction to [CaSt4, p. 20]. A special case of the example for d = 1 is the function
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They expressed doubt that such a function could be Fresnel-integrable because of the unboundedness of the integrand in (20.1.29). (Being unbounded, the "potential" V(s, u) :— —u2 associated with (20.1.29) certainly could not be the Fourier-Stieltjes transform of u e M(R).) However, they showed in Example 3 of Section 7, pages 63-67, that functions "like" Q do belong to S. As a consequence of a later result of Johnson [Jol], one can see that, in fact, the quadratic functions from [CaSt4] and further quadratic functions do have Fresnelintegrable counterparts. Here, we should clarify a matter that could be a point of confusion by emphasizing that such quadratic potentials are not the same as those for which Albeverio and Hoegh-Krohn extended their theory of the Fresnel integral [AlHo1, Sects. 4-6]. The quadratic potentials in [AlHol] start out R-valued (see (4.2) in [AlHol]) but are multiplied by —i when one passes to the extended Fresnel integral (see (4.3) in [AlHol]). Those in [CaSt4] (and in later related papers [JoSk9, ChanJoSk2, ParSk]) start out and remain R-valued since the analytic continuation is in the mass (rather than in the time) parameter. The quadratic potentials in [AlHol] appear frequently in standard quantum-mechanical problems. This is not the case for the quadratic potentials without the factor of i although these do seem to be of some physical interest having arisen in connection with the theory of the nucleus. See [Gla, Sects. 21-24, 35, 60, 66] for a brief discussion, some related mathematics and further references. We will return soon to a discussion of [Jol]; but first we describe briefly a nice result of Park and Skoug [ParSk] which had its beginning with the quadratic examples from [CaSt4]. Johnson and Skoug [JoSk9] extended the examples given by Cameron and Storvick to certain time-dependent quadratic forms and the same authors and Chang [ChanJoSk2] dealt with the case where there was a double dependence on time. The most interesting feature of [ChanJoSk2] was the natural appearance of two-parameter Wiener space (sometimes referred to as Yeh-Wiener space) in a problem stated entirely in terms of ordinary one-parameter Wiener space. One then needed to prove a stochastic integration by parts formula involving a mix of one- and two-parameter Wiener spaces. Park and Skoug extended the result to n time and d space parameters. In their case, a stochastic parts formula involving 1 and n parameters needed to be proved; this parts formula rested in turn on a stochastic Fubini theorem for multi-parameter Paley-WienerZygmund integrals. In the earlier papers, it was assumed that the square-roots of the eigenvalues of the time-dependent matrices were of bounded variation. Park and Skoug found a simpler and more natural condition; namely, that the eigenvalues belong to L 1 ([0,t] n ). As mentioned above, Cameron and Storvick knew that their space S was closely related to the space F(H) from [AlHol]. However, they believed as did Johnson and Skoug that the Banach algebras S and F(H) were quite different objects. This turned out not to be the case, as Johnson discovered in [Jo 1 ]. Actually, the connection made in [Jo 1 ] is between S and F(H1), where H1 is the particular Hilbert space used by Albeverio and Hoegh-Krohn in connection with nonrelativistic quantum mechanics. Recall that H1 consists of the absolutely continuous functions y on [0, t] such that y ( t ) — 0 and dy/ds e L2[0, t]. (As usual, for simplicity, we are limiting our notation to the case of one space dimension even though the results hold for any space dimension d.) The inner
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product of y1 and Y2 H1 is given by calculating the inner product in L2[0, t] of dy1/ds anddy2/ds;i.e. the inner product is given by (20.1.19). Among other things, it was shown in [Jol] that S and F(H1) are isometrically isomorphic as Banach algebras. Slightly later and working independently, Kallianpur and Bromley [KalBr] obtained the same kind of results for general H. Wiener space C' was replaced by abstract Wiener space B in [KalBr] and S was replaced by F(B), a space of (equivalence classes of) functions on B which are stochastic Fourier-Stieltjes transforms of measures from M(H). These ideas were explored significantly further by Kallianpur, Kannan and Karandikar in [KalKanK]. We will return for additional discussion of [KalBr, KalKanK], but first we give further information about [Jol] and some consequences that came out of it. The Banach algebras S and F(H1) are the same The following observations and results from [Jol, pp. 2091-2092] are the key to the connections between the Banach algebras F(H1) and S and to the Fresnel and scalarvalued analytic Feynman integrals acting, respectively, on these spaces. (1) We begin by describing a relationship between the inner product in L2[0, t] and the stochastic integral appearing in (20.1.21). Given v in L2[0, t] and y in H1, the Paley-Wiener-Zygmund (or Ito) stochastic integral f0 v(s)dy(s) exists and we have
where D is the differentiation map acting on H1. (2) The map D is an isometric isomorphism of H1 onto L2[0, t]. The integration map
is the inverse of D. (3) The maps
send M(H1) into M(L2[0, t]) and M(L2[0, t]) into M(H1), respectively. In fact, P is a Banach algebra isometric isomorphism of M(H1) onto M(L2[0, t]) and T = D-1. (4) If a = Du, where u e M(H1), then
(5) The map o : S -> F(H1) defined by
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identifies S and F(H1) isometrically and isomorphically as Banach algebras. Further,
(6) If u belongs to F(H1) and [ D u ] is the corresponding element of S, then the Fresnel integral of u equals the analytic Feynman integral with parameter q = 1 of Du; that is,
In fact, for any F in the equivalence class [Dp],
Consequences of the close relationship between S and F (H1) Some consequences of the close connection between .F(H1) and S have been given already in [Jol]. There are functions on Wiener space which are of some physical interest which had been shown to be in S but were not known to be in F(H1). These are functions arising from certain time-dependent (see (20.1.24) and (20.1.25)) and quadratic potentials. (Formula (20.1.29) gives a simple example of such a quadratic potential.) Corollaries 1 and 2 of [Jol] showed that the restrictions of these functions in S to H1 do belong to F(H1). Recall that the quadratic potentials had initially been thought to be examples of functions in S with no counterparts in F(H1). In Theorem 2 of [Trul], Truman explores the connection between his Feynman map acting on F(H1) and the Wiener integral. In the last two of the three assertions of his theorem, Truman considers f = u e F(H1) and assumes that / has an extension which is continuous on the space of continuous functions C[0, t]. Corollary 3 of [Jol] shows that there is no need to assume the existence of such an extension. One simply uses any F in the equivalence class [ D u ] . Truman's result and its improvement are stated precisely in [Jol, pp. 2093-2094]. Corollary 4 of [Jol] gives a Cameron-Martin type translation theorem for S. This result was not new. It had been proved a short time before [Jol] by Cameron and Storvick [CaSt6]. Also, Albeverio and Hoegh-Krohn had established such a translation theorem for F(H) in [AlHol, pp. 19-20]. The purpose of Corollary 4 in [Jol] was to illustrate how results in F(H1) could be transferred to 5. The proof of Corollary 4 is not difficult, but it does suggest that carrying results from F(H1) over to S may give rise to measuretheoretic issues which will not always be easily resolved. We turn next to brief remarks about work of Chang, Johnson and Skoug [ChanJoSkl, 3-5] some parts of which were inspired by [Jol]. The paper [ChanJoSkl] was motivated by a specific question dealing with nonrelativistic quantum mechanics and the Hilbert space H1. Before stating and answering that question, we give a simple related result,
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Theorem 13 of [ChanJoSkl], for a general separable Hilbert space H over R: Let K be a closed subspace of H and let P be the orthogonal projection of H onto K. Also, let p : K. -» C and define f : H -> C by
Then f is in f(H) if and only if p is in F(K.). Now we describe the question which motivated [ChanJoSkl ]. We know from a simple result in [AlHol, p. 29] that if the initial probability amplitude p is the Fourier-Stieltjes transform of some measure in M(R), i.e. if p e F(R), then the function f : H1 -> C defined by
(alternatively, f ( y ) := (p(y(0) + x), x e K) belongs to F(H1). Question: Are these the only functions of the form (20.1.39) that belong to F(H1)? The answer is "yes", as was shown in [ChanJoSkl, Corollary 10]; that is, if f as in (20.1.39) belongs to F(Hi), then the initial probability amplitude
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Theorem 1 of [ChanJoSk3] is a general result which identifies functions belonging to F(H). One of the elements of the proof is a simple unsymmetric Fubini theorem [Jo2]. Eight corollaries follow which include most of the results that were already known as well as some new results. Corollary 5 of [ChanJoSk3] showed that functions of the form
belong to F(H1), where the key assumption on V is that V(s, •) is, for each s, the Fourier-Stieltjes transform of a measure vs e M(K). Since F(H1) is a Banach algebra, it follows as usual that exp(g(.)) is also in F(H1). The reader should note that under less restrictive assumptions on V, functions on C[0, t] of the form exp(g(-)) are central to the generalized Feynman-Kac formulas (from [Lal4-18]) discussed in Chapter 17 of this book. In fact, functions in the Banach algebra A, (see Theorems 15.7.1 and 15.4.1 as well as Corollary 15.4.3) form a large class of functions on C[0, t] which are related to functions of the type (20.1.40). Corollary 6 in [ChanJoSk3] is similar to Corollary 5 but deals with potentials that have a double dependence on time and Borel measures n on [0, t]2. Hilbert spaces in addition to H1 are considered in [ChanJoSk3]; these are the two reproducing kernel Hilbert spaces associated with, respectively, the Brownian bridge and the Brownian sheet stochastic processes. (Some facts about these processes are discussed in the next to last subsection of the present section.) The results of [Jol] suggest that there ought to be results for the Banach algebra S that correspond to those results in [ChanJoSk3] which involve F(H1). This was shown to be true in [ChanJoSk5]. A unified theory of Fresnel integrals: Introductory remarks We turn next to a discussion of the beautiful paper [KalKanK] of Kallianpur, Kannan, and Karandikar which unified and substantially extended most of the earlier work involving transform assumptions. Some key aspects of [KalKanK] had previously appeared in the paper [KalBr] of Kallianpur and Bromley. In spite of this, our comments will be based primarily on [KalKanK]. We mention that [KalBr] explores some issues which are not discussed in [KalKanK]. As we have seen, the setting for the work of Albeverio and Hoegh-Krohn [AlHol] involves a general separable Hilbert space H. over E, with the particular Hilbert space H1 playing an especially significant role in connection with applications to nonrelativistic quantum mechanics. One of the key features of [KalKanK] (and already present in [KalBr]) is that an analogue for the close connection between the Banach algebras F(H1) and S from [Jol] is established for general H between the Banach algebra F(H) and another Banach algebra F(B), where B is a separable Banach space over R which is related to W in a certain way. In order to discuss [KalKanK], we will need to provide some information about B and the relationship between H and B. The space B or, more precisely, the triple (B, H, v), is called an abstract Wiener space; here, v is a Gaussian probability measure on the Borel class B(B) of B. Before turning to a brief but more systematic presentation of the necessary background material, we mention
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the monograph of Kuo [Kuol] and the book of Kallianpur and Karandikar [KalKar, Chapters 1 and 3] where much more detailed information can be found. The background material is somewhat technical, which probably explains in part why [KalKanK] is not more widely appreciated. Background material We begin by describing the canonical Gauss measure on the cylinder sets in an infinite dimensional Hilbert space H. Let P be the set of all orthogonal projections on H with finite-dimensional range. For each P e P, let
The collection Cp for fixed P is a a -algebra of subsets of H. In contrast, the cylinder sets defined by
form an algebra but not a a -algebra of subsets of H. The canonical Gauss measure m acting on the cylinder set P-1 (E) is given by
where n is the dimension of PH and the integration is with respect to Lebesgue measure on PH. The canonical Gauss "measure" is only finitely additive on C although it is rotation invariant on C and countably additive on Cp for each fixed P e P. The measure m o P-1 given by (20.1.43) is the joint distribution of n independent random variables each of which is normally distributed with mean 0 and variance 1. A norm || • ||0 on H is said to be measurable if for every e > 0, there is PO e P such that m({h e H: \\Ph\\0 > c}) < e whenever P in P is orthogonal to P0. We note that H cannot be complete with respect to || • \\0 when H is infinite dimensional (see [Kuol, p. 62]). Let B denote the completion of H with respect to || • ||o and let t be the natural injection of H into B. The adjoint operator i* is one-to-one and maps B*, the dual of B, continuously onto a dense subset of H*. By identifying H* with H and B* with i*B*, we have the triple B* c H* = H c B. Further, if (•, •) denotes the natural dual pairing between B and B*, we have (h, b*) = (h, b*)n for all h e H and b* € B*. A central fact—due to Gross (see [Gros] and [Kuol, Theorems 4.1 and 4.2 of Chapter 1])—is that the set function m o l-1 has a countably additive extension v which is a probability measure on B(B). The triple (B, H, v) is called an abstract Wiener space (abbreviated AWS) and H is called the generator of (B, H, v). The concepts of a measurable norm and of an AWS are also due to Gross [Gros]. The Hilbert space H\ along with classical Wiener space and Wiener measure, that is, the triple (Ct0, H1, m), provides an example of an AWS. (Note: The spaces H\ and Ct0 are not quite lined up right since the paths in these two spaces vanish at different endpoints. However, this is a minor technical matter which we will ignore as we have
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previously.) A few more examples of AWSs will be mentioned further on in connection with our later discussion of [AhJoSk], and additional examples can be found in [Kuol]. Also, it can be shown that for any real separable Banach space B with norm || • ||0 and for any countably additive Gaussian measure v on (B, B(B)) which is positive on all nonempty open subsets of B, there is a Hilbert space H. such that H c B, \\ • \\0 is a measurable norm on H, B is the completion of H under || • \\0 and v — m o i-1, where m is the canonical Gauss measure on H. Saying it briefly, any real separable Banach space B equipped with a Gaussian probability measure with support B is an AWS. (See [Kuol, Theorem 1.1 of Chapter 3] for a discussion of these matters.) Next we briefly describe an integration theory for functions on H, with respect to the canonical Gauss measure m. See [KalKar, Chapter 3] for a much more detailed treatment of this subject. We begin by defining a stochastic bilinear functional (•, -)~ on H x B. Fix a complete orthonormal set {ej} oo j=1 for H. such that ej e B* for all j. For h e H and b e B, let when the limit exists, otherwise. One can show that for each h e H, the limit in (20.1.44) exists for v-a.e. b e B; in fact, the limit exists for j-a.e. b e B (see [Chul]). Further, the random variable (h, -)~ on (B, B(B), v) is normally distributed with mean 0 and variance \\h\\2. Now let / : H -» C be a cylinder function. Such a function has the form
f(h) = p((h1, h), . . . , (hk, h)),
(20.1.45)
where k is a positive integer, h1, . . . , hk belong to H and p : Rk -> C is Borel measurable. We lift / to a function Rf on B by letting
(Rf)(b):=
. . , (h k ,b)~).
(20.1.46)
Next we extend the lifting map R beyond the cylinder functions. The limit involved in the extension will be taken with respect to the net P of orthogonal projections on H with finite-dimensional range. If P1, P2 are in P, we write P1 < P2 when Ran(P1) c Ran(P2). We say that a Borel measurable function f : H -> C belongs to L(H, C, m) provided that the net of lifted cylinder functions {R(f o P) : P e P} is Cauchy in v-probability. In this case, R(f o P) converges in v-probability to a function (or random variable) on (B, B(B), v) which we denote Rf and call the lifting of /. Usually, one is interested in functions / on H which cannot only be lifted to B but which are integrable over B. Let
For / 6 £ 1 ( H , C, m) and C 6 C, define
where xc denotes the characteristic function of C.
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There is already in the literature a theory of integration with respect to finitely additive measures. This subject is treated in detail in [DunScl]. A comparison of that theory with the approach via lifting is made in Section 5 of Chapter 3 in [KalKar], An example is given in that section of a function which belongs to £1 (H, C, m) but which is not integrable with respect to the theory in [DunScl]. However, it is shown that the finitely additive measure m on C can be extended to a finitely additive measure m on a larger algebra C such that £ l (H, C, m) coincides with the space of functions on H that are integrable with respect to m (see Theorem 5.9 in Chapter 3 of [KalKar]). We remark that it is possible to lift the integration theory associated with (H, C, m) to more than one space (which need not even be an AWS). One consequence of Theorem 5.9 in [KalKar] just discussed above is that the integration theory is independent of the particular lifting chosen. Thus, although it is convenient for many purposes to work with the countably additive measure v, it is the Hilbert space and the canonical Gauss measure that are primary. A unified theory of Fresnel integrals (continued) We are now preparecrto discuss some of the key ideas and results from [KalKanK]. A review of the definition of the scalar-valued analytic Feynman integral for functions on classical Wiener space (see Definition 4.5.1) immediately suggests how the definition should be made for functions F on B, where (B, H., v) is our abstract Wiener space. We begin by considering
for all A. > 0. If JF(A) exists for all X > 0 and has an analytic continuation to C+, the right half-plane, we denote this analytic continuation by I a ( F ) . Finally, for q e R, we write
provided the limit exists. For any q e M for which the limit does exist, we call Ia -iq (F) the (scalar-valued) analytic Feynman integral of F over B with parameter q. The appropriate equivalence relation for functions F and G on B is, much as in the case of classical Wiener space, not simply equality v-a.e. but rather equality s-a.e. with respect to v. The class of functions on B that are shown in [KalKanK, KalBr] to be analytic Feynman-integrable includes the Banach algebra F(B) of "Fresnel-integrable functions". These functions are a type of "stochastic Fourier-Stieltjes transform" of measures u in M(H); more specifically, they have the form
where (-, -)~ is the stochastic bilinear functional defined in (20.1.44). (There is a larger class of functions Qq (B) all of which are shown in [KalKanK] to be analytic Feynmanintegrable; we will discuss this class further on.)
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It was shown in [KalBr] that F ( B ) and F ( H ) are isomorphic as Banach algebras. Also, the analytic Feynman integral of FM is given for all q e R, q = 0, by
Comparing (20.1.52) with (20.1.9), we see that the analytic Feynman integral of Fu in F ( B ) in the case q = 1 equals the Fresnel integral of the corresponding function fn e T(H). The analytic Feynman integral was defined for the first time in [KalKanK] for functions on the Hilbert space H. Here, the role traditionally played by Wiener measure m is played instead by the canonical Gauss measure m. We begin by considering an expression similar to (20.1.49), namely
for all A > 0. We proceed as before by analytically continuing if possible to C+, obtaining ia(f), and finally, for q e R, by letting
provided the limit exists. For any q € R for which the limit in (20.1.54) exists, we call ia -lq (f) me (scalar-valued) analytic Feynman integral of f over H with parameter q. It is shown in [KalKanK] that for every q e R, q = 0, i a - i q ( f ) exists for every / in F(H.) (in fact, for all / in a larger class Qq (H) to be discussed later). Further, for f = fn € F(H.), we have
The facts just referred to are established by showing that f — fu in F(H) given by (20.1.4) lifts to Rfu, = Fu,given by (20.1.51). Certainly then f e L l ( H , C, m) since Fu is bounded by ||u||. The fact that ia-iq(fu) is given by (20.1.55) and agrees with Ia-lq(Fu) given by (20.1.52) follows from the relationship
We remark that the preceding outline has been oversimplified in that one needs to relate fu(A -1/2 (-)) and Fu(A--1/2(-)) for every A > 0 rather than just fu(-) and Fu(-). Kallianpur, Kannan and Karandikar go on to discuss sequential Feynman integrals in the setting of the Hilbert space H and also for the related Banach space B. We begin with H.
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Let f : H -» C be such that for all P e "P and all A > 0,
For such an / and P and for any X € C+, we define
where n is the dimension of PH. Since (20.1.57) holds for all X > 0, the integral in (20.1.58) exists as an ordinary Lebesgue integral for all A e C+. Now let q e R with q = 0 and suppose that lim jf(Pl, Al) exists for all sequences {Al}, {Pl} where Ae e C+ and Pl e P for all l and with Al -> —iq and Pl -> I (the identity operator) in the strong operator topology on L(H). Then we define the limit—which is easily seen to be independent of the sequences {Al}, {Pl}—to be the sequential Feynman integral of f over "H with parameter q and denote it by is -iq(f). We summarize this situation briefly by writing
It is then shown in [KalKanK] that for every f = fu 6 F ( H ) and for every q e R, q = 0, id -iq (fu) exists and we have
(As in situations discussed before, there is a larger class of functions on H for which the sequential and analytic Feynman integrals over H with parameter q both exist and agree.) So far, we have considered three definitions from [KalKanK] of Feynman integrals: analytic Feynman integrals over B and over H, and sequential Feynman integrals over H. The lifting map R was not involved in these definitions but did play a prominent role in relating the definitions to one another on the spaces F(H) and F(B). We turn now to the definition from [KalKanK] of the sequential Feynman integral over B. In this case, the definition itself makes use of the lifting map. Before presenting this definition, we remark that sequential Feynman integrals can be defined independently of the lifting map, at least for special choices of B. For example, such a definition is given for classical Wiener space in [CaSt7]. Suppose that F : B -» C is such that for some f 6 L ( H , C, m),
Then for P 6 P, define FP by
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By definition of the lifting map R for cylinder functions (see (20.1.46)), one can check that
Now F p ( A - 1 / 2 ( - ) ) converges in v-probability to F(A. - 1 / 2 (-))- Therefore, for each A., Fp (A-1/2 (•)) is a finite-dimensional approximation to F(A-1/2 (•)). We further assume JF P (A) given by (20.1.49) exists for all A > 0 and has an analytic continuation I a ( F p ) to C+. Next let q e R with q = 0 and suppose that lim Ial ( F p l ) exists for all sequences {Al}, {Pl} with Al e C+ and Pl e P for all t and with Al -» -iq and Pl -*• / in the strong operator topology on £(H). Then we define the limit—which is easily seen to be independent of the sequences {Al}, {Pl}—to be the sequential Feynman integral of F over B with parameter q. We summarize briefly by writing
Kallianpur, Kannan and Karandikar go on to show that for every F = Fu in F(B) and for every q 6 R, q = 0, Is - i q (Fu) exists and we have
The Fresnel classes along with quadratic forms We have alluded above to classes of functions on H and B which are larger than the Fresnel classes F(H) and F(B), respectively, and are such that the types of Feynman integral discussed above exist. We will state these more general results in Theorem 20.1.2 below after some preliminaries. The first preliminary provides specific information about the lifting map R [KalKanK, Proposition 2.4]: Let u e M(H) and let A be a self-adjoint trace class operator on H. Further, let g : H -» C and G : B -» C be defined, respectively, by
and
Then, we have
and, in fact, for every A > 0,
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Finally, if Pm -> I in the strong operator topology on £(H), then for every A > 0
The reader should note that if A — 0, then the functions of the form (20.1.66) or (20.1.67) are just the functions in F(H) or F(B), respectively. The classes Qq (U) and Qq(B) Let q e R, q = 0, be given. We denote by Qq(U) (alternatively, Qq(B)) the class of functions g (or G) of the form (20.1.66) (or (20.1.67)) for some u e M(H) and some self-adjoint, trace class operator A on H such that I +1/qA has a bounded inverse. For a self-adjoint trace class operator A with eigenvalues {aj}, the Fredholm determinant of / + A, denoted det(1 + A), is defined by
Further, the Maslov index (see [Masl2]) of / + A, denoted ind(I + A), is the number of negative eigenvalues of / + A; that is,
We have not stated theorems formally in this section. We will, however, do so for the next result since it not only contains most of the theoretical developments in [KalKanK] but also extends or is at least closely related to many of the earlier developments that make use of transform approaches to the Feynman integral. The result is called a CameronMartin formula for the Feynman integral because it plays a role similar to that of the classical Cameron-Martin formula [CaMa2]. The classical result is in some respects very different in that the integral involved is a true Lebesgue integral with respect to Wiener measure whereas the Feynman integral is far from that. Theorem 20.1.2 Let q e R, q = 0, be given and suppose that
belong to Qq(H) and Qq(B), respectively. Then the Feynman integrals i a - i q (g), ia -iq (g), I-iq (G) and Ia-iq (G) all exist and we have
The reader should note that when either of the sequential Feynman integrals i s -iq (/) or I s - i q ( F ) exists, there is great freedom in how it is computed. First of all, there is
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no restriction as to how the sequence {A n } approaches — iq through C+ as there is in some other definitions of the Feynman integral (see, for example, Definition 13.5.1')More importantly, the sequence {Pn} of finite-dimensional projections such that Pn —> I strongly can be chosen arbitrarily. For example, in the case where H = H1, it is not necessary to take the familiar polygonal projections which were used in Feynman's original paper [Fey2] and in many subsequent papers. Indeed, the idea of replacing the approximating polygonal paths with trigonometric polynomials was mentioned in the book of Feynman and Hibbs [FeyHi, p. 71]. Theorem 20.1.2 tells us that if f e Cq(H) or F e Qq(B), the limit is independent of how the sequence {Pn} is chosen. Of course, there is no guarantee that the independence persists for more general functions. In fact, it need not, as follows from a result of Elworthy and Truman [ElTr, Corollary 3C] where the trace class operator A from Theorem 20.1.2 is replaced by a compact operator which is in the Hilbert-Schmidt class but not in the trace class. One of the truly striking features of "the" Feynman integral acting on the Fresnel classes F(H) or F(B) or, more generally, on Qq(H) or Qq(B), is the agreement both between different approaches to the Feynman integral and also for different sequences {Pn} of finite-dimensional projections such that Pn -> / strongly. Parts of this story had appeared in earlier work, but it was the paper of Kallianpur, Kannan and Karandikar which provided the fullest development of these issues. Quadratic forms extended The recent paper of Albeverio and Brzezniak [AlBr3] which was mentioned earlier in this section extends the work in [ElTr] as well as some aspects of the work in [KalKanK], Recall that the operator A in Theorem 20.1.2 is required to be in the trace class Q1. In contrast, in the corresponding Cameron-Martin type formula, Theorem 2.1 of [AlBr3], A is permitted to be in any of the Schatten p-classes Qp, 1 < p < oo. If 1 < p1 < P2 < oo, we recall that
As in [KalKanK] but different from [ElTr], the Hilbert space H is not restricted to be H1, nor are the projections from P which converge strongly to / restricted to be of any special kind. At first glance, the freedom in choosing the approximating projections in Theorem 2.1 of [AlBr3] seems to contradict what we have said above about Corollary 3C of [ElTr]; namely, for A e Q2, the Hilbert-Schmidt class, the limit can change when the sequence of projections converging strongly to / is changed. Actually, there is no contradiction. Albeverio and Brzezniak give a new and extended definition of a Fresnel (or "oscillating" or "Feynman") integral Fp,h -A which, when restricted to H1 and p = 2, is related to but not the same as the definition used in Corollary 3C of [ElTr]. The AlbeverioBrzezniak definition involves a renormalization which depends on p of the quadratic form h |-> (h, Ah), as well as a regularized Fredholm determinant which also depends on p. As an application of the results in [AlBr3], the authors express a solution of the Schrodinger equation with a scalar and a magnetic potential in terms of their oscillatory
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integral. They remark that it was this application that motivated the study which resulted in [AlBr3]. (Recall that the Schrodinger equation with a highly singular scalar and magnetic potential was treated in Example 11.4.12, at the end of Sections 11.6 and 13.5, as well as in Section 13.6.) Functions in the Fresnel class of an abstract Wiener space: Examples of abstract Wiener spaces We turn next to a paper of Ahn, Johnson and Skoug [AhJoSk]. The main result of that paper, Theorem 2, identifies functions in the Fresnel class F ( B ) of an abstract Wiener space. Theorem 2 is followed by eleven corollaries, all of which assert that some class of functions belongs to F(B) for some B. Corollaries 3-8 are concerned with classical Wiener space. Part of our purpose here is to give some examples of AWSs that are different from classical Wiener space. Theorem 2 of [AhJoSk], even when restricted to classical Wiener space, is somewhat more general than the corresponding theorem in [ChanJoSk5]. Hence, all of the eight corollaries in [ChanJoSk5] follow from Theorem 2 of [AhJoSk], although this may not be immediately apparent from a quick glance at both papers. For one thing, the corollaries in [ChanJoSk5] permit Wiener paths with values in Rd. These corollaries could all have been repeated in [AhJoSk] but were not. Corollary 8 in [AhJoSk] is for classical Wiener space but has a somewhat different character than the corollaries in either of the earlier papers [ChanJoSk3,5]. The evaluation of the path x at a sharp time s as in the integrand of
is replaced in Corollary 8 of [AhJoSk] by the integral operator with kernel g(s, y) in the expression
Note that we can take Y = [0, t] in (20.1.77), in which case we obtain
If, further, g(s, T) — g1(s)X[0,T](s), then (20.1.78) becomes
Corollary 5 of [AhJoSk] treats functions on Wiener space of the form (20.1.76), where V satisfies the conditions in (20.1.24) except that the integral in (c) of (20.1.24) is with respect to the measure |n|. A multiple dependence on time is allowed in Corollary 6 of [AhJoSk], but otherwise Corollaries 5 and 6 involve similar assumptions.
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Before finishing with the specific examples of further AWSs which are involved in the last three corollaries of [AhJoSk], we remind the reader of the fact noted earlier that any real separable Banach space B equipped with a Gaussian probability measure with support B is an AWS. Thus there is no shortage of AWSs. The multiparameter Wiener spaces Bn,0 := Co([0, t]n) equipped with the sup norm and with the n-parameter Wiener measures vn,0 on B(B n , 0 ), n = 2, 3, . . . , are abstract Wiener spaces. We will give most of the pertinent facts but will not attempt to explain this assertion entirely. We remark that two-parameter Wiener space B2,0, often called Yeh-Wiener space, has been studied most. The sample functions x in B2,0 are sometimes referred to as Brownian surfaces or Brownian sheets. The covariance function for the one-parameter (or ordinary) Wiener process was calculated in Chapter 3 (see Proposition 3.3.9 and Remark 3.3.11). The formula is similar in the n-parameter case: the covariance function maps the pair ((T1, . . . , T n ), (s1, . . . , .Sn)) from [0, t]n x [0, t]n into
(Note that the material in Chapter 3 just referred to corresponds to the case n = 1; that is, ordinary Wiener space.) We now describe the reproducing kernel Hilbert space (RKHS) Hn,0 = H n , 0 ([0, t]n) of this process. The space Hn,o consists of all functions y : [0, t]n —> R for which there exists g in L2([0, t]n) such that for all (s1, . . . , sn) e [0, t]n,
It is apparent that y(s1, , . . ,sn) =0 whenever at least one of the numbers Sj equals 0. Further, it is not hard to show that for a.e. (s1,... ,sn) e [0, t] n ,
The inner product on Hn,o is defined by
The space Hn,o, equipped with this inner product, is a separable Hilbert space over R. The family of functions {y(T1, . . . ,Tn) : (T1, Tn) e [0, t]n} from Hn,Q defined for (s1, . . . ,sn) 6 [0, t]n by the expression in (20.1.80) has the reproducing property In fact, the triple (Bn,o, Hn,o, vn,0) is an AWS [Chul, Kuell]. Further, the reproducing property carries over to Hn,o x Bn,0 in the sense that for s-a.e. x € Bn:o, where (•, -)~ in (20.1.85) denotes the stochastic bilinear functional defined by formula (20.1.44).
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With the above facts in hand, a variety of functions F e F(Bn,o) could now be identified. We will limit ourselves to the type of function considered in Corollary 9 of [AhJoSk]. Let n be a Borel measure on [0, t]n which either belongs to M([0, t]n) or else is positive and a -finite. Let V : [0, f]n x R -» C be given by V((T1, . . ., T n ), •) = u(T1, . . . . , Tn )(-), where u(T1,...,Tn) € M(R) for every (T1,. . . ,T n ) € [0, t]n and where the family {u( T1 Tn) : (T1, ... , Tn) e [0, t]n} satisfies (i) u(T1, ,. . . ,T n )(E) is a Borel measurable function of (T1, . . . , Tn) for every £ € B(R), and (ii) ||u(T1, . . . ,Tn)|| e L1([0, t]n, |n|). Then
belongs to F(B n , o ). (As usual, exp(F(x)) also belongs to F(B n , o ).) We turn next to the Brownian bridge stochastic process and the associated abstract Wiener space. This process has been used to express the fundamental solution (or Green's function) for the Schrodinger equation as a Feynman integral. In [KalKanK, §5b], for example, the integration is taken over a space of paths that deviate from the classical path. Since the focus is on the deviating paths, these paths should vanish at the endpoints of the time intervals which we will take to be [0, t]. (If the paths are instead tied at nonzero values at one or both endpoints, then the spaces of functions are not linear and an adjustment is needed as in [Ito2].) The AWS is closely related to classical Wiener space in this case. In fact, the Brownian bridge stochastic process can be obtained from the classical Brownian motion (or Wiener) process via conditional expectation. The Banach space is just Bo,O = Coo([0,t]) = (x e C([0, t]) : x ( 0 ) = x(t) = 0} equipped with the sup norm, and the appropriate measure V0,0 can be obtained from classical Wiener measure on Co([0, t]) by conditioning. The covariance function of the Brownian bridge process maps (T, s) e [0, t] x [0, t] into
The Hilbert space Ho,o associated with the covariance function (20.1.87) is a subspace of the Hilbert space H1 which has appeared frequently in this section. Specifically, HO,O consists of all absolutely continuous functions y on [0, t] which have square-integrable derivatives and which satisfy y (0) = y (t) = 0. The inner product on Ho,O x H0,0 is just the restriction to that space of the inner product on H1 x H1 given by formula (20.1.19). Note that Ho,o is certainly a Hilbert space since it is a closed subspace of H1. The family of functions {yT : T e [0, t]} from Ho,o given for s e [0, t] by the expression in (20.1.87) has the reproducing property
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Further, the triple (Bo,o, Ho,o, v0,0) is an abstract Wiener space [Kuel2]. Finally, the reproducing property extends to Ho,o x B0,0 in the sense that for s-a.e. x e BO,O,
where (•, -)~ is the stochastic bilinear functional as defined in (20.1.44). Functions of a variety of forms belonging to F(.Bo,o) could now be identified. We settle for mentioning that a function F of the form (20.1.86), with n — 1 and V satisfying the assumptions discussed just above (20.1.86), is an example. Our last example of an abstract Wiener space is actually a rather large family of examples which is given in Watanabe's book [Wat, pp. 7-9]. Watanabe considers the case of an arbitrary finite number of space dimensions and an arbitrary finite number of "time" parameters. We follow here the presentation in [AhJoSk] which simplifies matters by taking both of these numbers equal to one. Let I = [0, t] and let p be a positive integer. We assume that K is an R-valued, symmetric, nonnegative definite function (or kernel) on I x I having 2p continuous derivatives and satisfying the following condition: There exists d e (0, 1] and c > 0 such that
for all (s, T) e I x I. Now, given 0 < e < 1 and any function x on / possessing p continuous derivatives, let
and let
Then (C p , e , ||-|| p,e ) is a Banach space. Also, it is known [Wat, p. 8] that for any e e [0, d), there exists a mean 0 Gaussian measure vp,e on (Cp-£, B(C P , e )) having K(s, T) as its covariance function. Further, the reproducing kernel Hilbert space Hp,e = H p , e ( I ) associated with K is such that (C p , e , H p , e , v p,e ) is an abstract Wiener space [Wat, p. 8]. We remark that to obtain the space HP'E, one begins by defining an inner product on all finite linear combinations of functions of the form K(s, •). Given two such functions, En1k akK(sk, •) and En2 blK(T t , •) =En2l=1,blK(-, Tl), we take their inner product to be
The Hilbert space Hp'e is the completion of this inner product space. The family of functions {yr = K(r, •) : r e I} has the reproducing property; i.e.
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Theorem 2 of [AhJoSk] can now be used to show that functions of various forms belong to F(B p ' e ). Corollary 11 in [AhJoSk] is one such example. There is a technicality that should be mentioned. It may well be true here as in (20.1.89), (20.1.85) and in the setting of classical Wiener space, that the reproducing property carries over to the stochastic bilinear function (•, •)~ on HP'E x B p , e . However, if (y r , x)~ = x(T) fails, then the point evaluation involved in, for example, functions of the form
needs to be replaced by (yT, x)~, obtaining instead an adjusted function
This concludes our discussion of abstract Wiener spaces and of their associated Fresnel classes. Fourier-Feynman transforms, convolution, and the first variation for functions in S We finish Section 20.1.A by calling attention to interesting recent work of Park, Skoug, Storvick and Huffman that is connected with but is also rather different from the rest of this section. Several formulas are developed which relate (i) the "Fourier-Feynman transform" defined via a Feynman integral, (ii) "convolution", and (iii) the "first variation" for functions on Wiener space belonging to the Banach algebra S. The paper [ParSkSt2] gives the fullest treatment of the subjects just mentioned but builds on earlier work in [ParSkStl] and [HufParSk]. Integration by parts formulas for analytic Feynman integrals and for Fourier-Feynman transforms can also be found in [ParSkStl] and Parseval relations are established in [HufParSk]. Not all of the work found in these last two papers is restricted to the Banach algebra S. It seems likely that some of the work just described is connected with the distributional theories found in the books of Hida, Kuo, Potthof and Streit [HidKPS] and Kuo [Kuo2]. We now go to Section 20.1.B where we will give some further references to the Feynman integral and related topics. 20.1 .B. References to further approaches to the Feynman integral In this very brief section, we give a sample of additional references on or related to the Feynman integral. With only a few exceptions, these references do not repeat those given elsewhere in the book (including Sections 20.1.A and 20.2). Even with the addition of these supplementary references, the bibliography of this book is far from complete, especially with regard to references of a physical nature. We now list these additional references in alphabetical order: [AlRezZam], [AlSen], [Az], [AzDos], [BalBlo], [benACas], [Berr], [BrBh], [CaSt3,8,9], [Cart-dWml-3], [ChanAKKR], [ChanRy2], [Chaz], [Chu2], [ChuSk], [Coivl,2], [DauKlal,2], [DauKlaPau], [dWml-4], [dWmMNl.2], [Dk], [Dos], [Dui], [DuiGui], [DuiHor], [Dynl,2], [Fed], [Frd], [Fuj1,2], [Ga], [GaV], [GuiSbg], [Gutl,2],
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[HidKPS], [Hor2-4], [HuMey], [Jel-3], [JoKall-4], [JoSk3,4], [Kal2], [KalUs], [Kla2,3], [Klul-3], [KtSW], [Kuo2], [Masll,2], [MaslFed], [MeyYan], [Mor], [Mul], [Pop], [Rob], [Schu], [Tarl,2], [Zam]. Much of the work represented in the list above involves approaches to the Feynman integral not discussed elsewhere in this book. Most of these approaches can be divided into the following rough categories: (1) Phase space path integrals. (2) Infinite dimensional stochastic analysis including, in particular, various types of white noise analysis. (3) Semiclassical expansions. (4) Feynman integral via vector measures. We caution the reader that the boundaries between the above categories are not always clear. 20.2
The influence of heuristic Feynman integrals on contemporary mathematics and physics: Some examples
Still the intuitive appeal of Feynman's definition is enormous, and it gave birth to a dream that perhaps all of Physics could be rewritten in terms of "sums over histories". MarkKac, 1980 [Kac4, p. 51]
Our concern in this section is with relationships between "the" heuristic Feynman integral and a variety of further mathematical and physical topics. Our emphasis will be on Section 20.2.A where connections between Witten's heuristic Feynman integral and the related subjects of knot theory and low-dimensional topology are discussed. The mathematical theory of these latter two subjects has been revolutionized in recent years by the work of Edward Witten [Wit 12-14]. Witten's paper [Wit 14] on the Jones polynomial [Jone4] will be especially relevant to our discussion. In Section 20.2.B, we will limit ourselves to references and brief remarks (with one exception) regarding the use of Feynman-type path integrals in a few additional areas. The first of these, described in more detail than the others, is an approach to the famous Atiyah-Singer index theorem via supersymmetric Feynman integrals. The remaining ones are deformation quantization, gauge field theory, and string theory. We note that some aspects of these subjects as well as of those treated in Section 20.2.A are related to Feynman's operational calculus, where perturbation series played a significant role. As a general rule, in Section 20.2, we will stress the mathematical or physical concepts and intuitions motivated by—or closely related to—the use of heuristic Feynman path integrals. Throughout Section 20.2—and much more so than in the previous chapters of this book (or than in Section 20.1)—we will make use of mathematical concepts or physical topics that have not been precisely defined or previously introduced in the text. Given the nature and the scope of the material being discussed, as well as the relative brevity of the present section, we have found it impossible to do otherwise. When appropriate, however, we will try to provide suitable general references where the interested reader can find further information on the subject at hand.
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We caution the reader that the authors have not been participants in the developments described in this section and are far from being experts on these subjects. It should also be made clear to the reader that the nature of the present section is quite different from that of Section 20.1 as well as from that of much of the rest of this book, from several additional points of view. In particular, except in a few instances which will usually be pointed out, it frequently discusses material or draws on "facts" which—in our present state of knowledge—are far from having been precisely formulated, let alone established in a rigorous manner. The heuristic Feynman path integral In recent years, heuristic Feynman path integrals have played a remarkable role in various aspects of quantum physics and in related areas of mathematics. Before providing examples of these developments, we identify some ingredients that are typically present in these heuristic path integrals: (1) A set P of virtual "paths", to be integrated over in (4) below. (2) An action functional S = S(p), p e P. (3) A "measure" Dp, whose existence as a true measure is typically doubtful. (4) The "Feynman integral"
(5) The limit as k —» oo in (20.2.1), i.e. the "classical limit" or the method of stationary phase, remains important. (6) Perturbation series can be formally calculated from (20.2.1) and have provided, at least in some cases, the most useful result of the heuristics. In Chapter 7, where the standard Feynman integral associated with quantum mechanics was discussed, the virtual "paths" were true paths in the usual sense of the word. As we continue, the set P from (1) will have various interpretations, for example, as a space of gauge fields (or connections) in the applications to knot theory and low-dimensional topology to be discussed in Section 20.2.A. In quantum field theory, P usually consists of a space of fields satisfying suitable boundary conditions; in general relativity, it can be taken to be a set of space-time metrics, while in string theory, it usually consists of a suitable moduli space of Riemann surfaces. (See Section 20.2.B for a few examples and some further discussion and precision.) The integrand in (4) above may well involve additional functions, as we will see in Sections 20.2.A and 20.2.B. (See, for example, formula (20.2.8) below as well as Remark 20.2.20.) More specifically, if F : P -> R is a suitable "functional" (that is, an appropriate function on the "generalized path space" P)—frequently interpreted physically as an "observable" of the associated "quantum theory"—then the heuristic Feynman path integral (20.2.1) appearing in (4) is replaced by the following more general one:
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which is often referred to in the physics literature as the (unnormalized) "expectation value" of the "observable" F. (The "normalized expectation value" of F would be given by the ratio of the functional integral in (20.2.1') by that appearing in (20.2.1), with the understanding that the "constant" K occurring in both integrals would automatically be cancelled out.) Furthermore, the real parameter k in (4) and (5) plays the role of l/h in (7.2.1) for the "traditional" Feynman path integral associated with quantum mechanics. (This is why in (5), the limit as k —> oo corresponds to the "classical limit" as h -> 0. In the physics literature, k is often referred to as the "coupling constant"; see, for example, [Witl3,14].) In addition, in (20.2.1) (or in (20.2.1')), K is intended to be an appropriate normalizing constant which is typically infinite if interpreted straightforwardly. Recall that in the quantum-mechanical context, the constant K was not identified by Feynman in (7.2.1), but that suitable approximating constants were explicitly given in the discretized version (7.4.4) of (7.2.1). In more complicated settings, K can sometimes be defined via an appropriate renormalization (or regularization) scheme. Feynman-type diagrams are frequently used to keep track of the terms of the perturbation series referred to in (6). When rigorous mathematics has come out of these heuristic ideas, it has often been the case that the terms of the perturbation expansion have been replaced by (or interpreted in terms of) suitable combinatorial (and/or algebraic) objects associated with the geometry of the underlying problem. We note that—even though the settings in which they arise are quite different from those discussed in Chapters 14-19—the approaches via perturbation series alluded to in (6) above are more in the spirit of Feynman's operational calculus for noncornmuting operators than of the Feynman path integral itself.
20.2.A. Knot invariants and low-dimensional topology Since just before and ever since the Jones polynomial ([Jone2-4]) it has been possible to express "polynomial" invariants of links [or knots] as state summations analogous to partition functions in statistical mechanics. After the landmark paper of Witten in 1988 ([Wit 13,14]) it became possible to express these invariants as functional integrals in a quantum field theoretic context. These integrals, void of traditional mathematical existence, have cast an enchanted spell on the subject of lowdimensional topology from which it may hope to wake through either the means of combinatorics or perhaps in the ultimate justification of these integrals. . . . Louis H. Kauffman, 1995 [Kau9, p. 2402]
The Jones polynomial invariant for knots and links Motivated by certain problems in the theory of operator algebras (including his earlier work on the index of subfactors of von Neumann algebras [Jonel]), Vaughan Jones [Jone2-4] discovered in 1984 a new polynomial invariant for knots (and, more generally, links), now called the Jones polynomial and denoted by VK(t). Unlike previous knot invariants, notably the Alexander-Conway polynomial [Conw], the Jones polynomial was often able to distinguish between a knot K and its mirror image K; namely, in the
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notation of [Jone2-4], we have the following "chirality property":
Remark 20.2.1 Actually, given a knot K, VK(t) is a Laurent polynomial (with integer coefficients) in the indeterminate t; that is, a polynomial in t and t-1. More generally, the same is true if K is a link with an odd number of components. Further, if K is a link with an even number of components, then VK(t) is equal to Vt times a Laurent polynomial. (We will recall the formal definition of knots and links towards the beginning of the next italicized subsection.) We now briefly discuss some of the key properties of the Jones polynomial VK. (For more information, see, for example, [Jone2-4], [Kau3,9] or [Kau8, Part I, esp. §I.7 and §I.9].) The Jones polynomial is a topological invariant for knots (or links), in the sense that VK1 (t) = VK2(t) if K1 is "ambient isotopic" to K2; that is, if K\ can be continuously deformed into K2 (via smooth embeddings of S1) in the ambient three-dimensional space—or equivalently, according to Alexander's theorem ([Kau8, p. 18], [BuZi]), if one can pass from the planar diagram of K1 to that of K2 via a finite number of basic operations, called the "Reidemeister moves" ([BuZi], [Kau8, §I.2, esp. Fig. 8 on p. 16]). Moreover, the Jones polynomial satisfies the following normalization condition (20.2.3) and "skein relation" (20.2.4) (see [Jone2-4] or [Kau8, §I.5]):
where 0 denotes the "unknot" (or unknotted circle), and
where K+ (a positive crossing), K- (a negative crossing) and K° (a smoothed or "spliced" crossing) are links related as in Figure 20.2.1, the rest of the links being identical. Here, we use the standard representation of knots and links by means of planar diagrams, as described, for instance, in [Kau8, §I.2] or in [BuZi]. Thus, the meaning of Figure 20.2.1 is that, at the level of planar diagrams, K+ has a positive crossing which is replaced by a negative crossing in Kr and a smoothed (or "spliced") crossing in K°, all other crossings remaining unchanged; besides [Jone2-4], see, for instance, [Kau8, §I.7] or [Kau9, pp. 2406-2407].
FIG. 20.2.1. The links K+, K-, and K° in the skein relation (20.2.4)
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As can be easily checked, the polynomial invariant VK. is determined uniquely— and hence characterized—by the skein relation (20.2.4), along with the normalization condition (20,2.3). Remark 20.2.2 For example, the fact that the "trefoil knot" K (see [Kau8, Fig. 6, p. 9]) is inequivalent to its mirror image K. (i.e. that K, is "chiral") can be deduced from a simple computation based on relations (20.2.4) and (20.2.2). Indeed, if K is the righthanded trefoil, then VK(t) = t + t3 — t4 whereas for the left-handed trefoil K,, we have VK(t) = V K ( t - 1 ) = t-1 + t-3 - t-4; so that VK = VK and hence, by the topological invariance of VK, K is not equivalent (i.e. ambient isotopic) to 1C. (See, for example, [Jone6, p. 129] or [Kau8, §I.5], where different conventions are used.) Moreover, since the same computation shows that VK = 1, itfollows from (20.2.3) that K is not equivalent to the unknot O; that is, the trefoil knot is "knotted". Of course, in this simple situation, such statements can also be verified directly (see, for instance, [Kau8, pp. 21-25]). Finally, we mention another key property of VK, to which we will return in the next subsection; namely, the Jones polynomial invariant is multiplicative under connected sums. More precisely,
where K1#K2 denotes the (suitably defined) connected sum of the links K1 and K2. The multiplicative property (20.2.5) is also satisfied by the two-variable knot invariants extending both the Jones polynomial and the Alexander-Conway polynomial (namely, the homfly and Kauffman polynomials). We will see in the next italicized subsection (specifically, in Remark 20.2.7(a)) that it can be derived from the corresponding property of Witten's topological invariants defined by means of Feynman-type path integrals. The latter property is intimately related to the functional integrals themselves. Witten's topological invariants via Feynman path integrals In recent years there has been a remarkable renaissance in the relation between Geometry and Physics. This relation involves the most advanced and sophisticated ideas on each side and appears to be extremely deep. The traditional links between the two subjects, as embodied for example in Einstein's Theory of General Relativity or in Maxwell's Equations for Electro-Magnetism, are concerned essentially with classical fields of force, governed by differential equations, and their geometrical interpretation. The new feature of present developments is that links are being established between quantum physics and topology. It is no longer the purely local aspects that are involved but their global counterparts. In a very general sense this should not be too surprising. Both quantum theory and topology are characterized by discrete phenomena emerging from a continuous background. However, the realization that this vague philosophical view-point could be translated into reasonably precise and significant mathematical statements is mainly due to the efforts of Edward Witten who, in a variety of directions, has shown the insight that can be derived by examining the topological aspects of quantum field theories. Michael Atiyah, 1989 [At6, p. 175] As for the Jones polynomial and its generalizations [Jonel,2, Jone4, FYHLMO, Kaul,2, Kau6, Turl, PrzT, Birl], these deal with the mysteries of knots in three dimensional space. The puzzle on
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the mathematical side was that these objects are invariants of a three dimensional situation, but one did not have an intrinsically three dimensional definition. There were many elegant definitions of the knot polynomials, but they all involved looking in some way at a two dimensional projection or slicing of the knot, giving a two dimensional algorithm for computation, and proving that the result is independent of the chosen projection. This is analogous to studying a physical theory that is in fact relativistic but in which one does not know of a manifestly relativistic formulation—like quantum electrodynamics in the 1930s. Edward Witten, 1989 [Witl4, p. 352]
Jones' discovery [Jone2-4] led to a revival of the subject of knot theory and to rather surprising connections between low-dimensional topology and several areas of physics, including statistical mechanics, quantum field theory and conformal field theory. (See, for example, [Jone5,6, Kaul,2, Kau4,5, Kau8, Seg].) In 1987, during the Hermann Weyl Symposium, Michael Atiyah [At5] posed the problem of finding a so-called threedimensional "topological quantum field theory" that would provide a suitable topological, geometric and physical interpretation of the Jones polynomial. (See especially [At5, pp. 298-299].) Shortly afterwards, an answer to his query was provided in remarkable fashion by Edward Witten ([Witl3], [Witl4]) in the form of a heuristic Feynman-type path integral associated with a non-Abelian Chern-Simons gauge field theory. (An early Abelian version of such a quantum field theory had been considered, with different motivations, in a simpler geometric context by Albert Schwarz in [Schwal]; see Remark 20.2.5(a) below as well as [Schwa3, pp. 199-200] for further relevant information.) Witten's formulation provided, in particular, an extension of the Jones invariant—and of its later (two-variable) generalizations, namely, the homfly ([FYHLMO], [PrzT]) and Kauffman polynomials ([Kau6], [Kau8, §I.5])—to knots and links in arbitrary compact (smooth, oriented) 3-manifolds M (rather than just in the unit sphere S3). Remark 20.2.3 The second author was fortunate to be able to attend the 1987 Hermann Weyl Symposium at Duke University during which Michael Atiyah presented his conjecture as a "challenge" to mathematicians and physicists, as well as the 1988 International Congress on Mathematical Physics in Swansea, Wales, at which Edward Witten announced his resolution of that conjecture ([Will 3], expanded upon in [Wit 14]). He remembers it vividly as an inspiring intellectual experience. We will now describe somewhat more precisely Witten's heuristic formula for his topological invariant in terms of a functional integral [Witl4]. First, recall that a knot K is an embedding of a circle S1 in three-dimensional Euclidean space R3 (or, for convenience, in S3, the unit sphere in R4, viewed as the one-point compactification of R3). (We note that all embeddings involved here are assumed to be smooth.) More generally, a link (still denoted by K) is an embedding of a finite product of circles, S1 x . . . x S1, into S3. Hence a link K can be thought of as consisting of finitely many nonintersecting knots, called the components of K and denoted {K a }. Clearly, if M = M3 is a (compact, oriented) three-dimensional manifold, one can define similarly a knot or a link in M by replacing S3 by M in the above definitions. (In particular, a knot in M is a simple closed curve in M.) In the following, all the knots and links in M will be assumed to be oriented.
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Next, we indicate what the "Feynman integral" (20.2.1) looks like in the situation which we have just been discussing. Our comments on the objects involved will be limited. In addition to [Witl4] (or to [Witl3]) itself, the reader who wants more information about this subject can consult, for example ([At5-8], [Fa2], [Kau8, esp. §I.17], [Kau9], [Roz3]). We begin with the case corresponding to the empty knot, so that only the manifold is involved. In this case, the form of the Feynman integral is essentially the same as in (20.2.1):
where k e Z. (The constant 1/K which appeared in (20.2.1) is thought of as being absorbed into the "measure" DA.) Here and from now on, M is a compact, smooth and oriented three-dimensional manifold (with or without boundary), and G is a gauge group, a compact simple Lie group. The "integral" in (20.2.6) is over the set P of all gauge fields A on M modulo gauge equivalence; these gauge fields (or rather, gauge potentials) take values in the Lie algebra g of G. Mathematically, a gauge potential A is a connection on a (trivial) G-bundle over M; in particular, it is represented by a g-valued 1-form on M. Further, the (fictitious) "measure" DA (as in (3) above) in (20.2.6) (or in (20.2.8) below) should be thought of as a measure on a suitable "moduli space" P of connections modulo the group of gauge transformations. (This moduli space P is the set of virtual "paths" referred to in (1) above (20.2.1).) Following the usual convention, however, we will blur the distinction between a connection (or gauge potential) A and its equivalence class modulo gauge transformations. (For a good introduction to gauge theory from the present viewpoint, we refer, for example, to [Atl] or [EgGH]; see also Remark 20.2.20 below for some brief comments.) We can now identify the action functional S = S(A) (see (2) just above (20.2.1)) appearing on the right-hand side of (20.2.6): It is called the Chem-Simons action (or functional) and is given by the formula
where A denotes the wedge product of differential forms on M. (We note that this functional was introduced in 1974 in [ChernSim] for geometric rather than for physical reasons.) The expression A A dA +2/3A A A A A is a 3-form since A is a 1-form. The integrand of (20.2.7) is called the Chem-Simons Lagrangian. (Caution: The terminology used in this context in the literature is not always as uniform or consistent as one might like.) Since the Lie algebra g in which A takes its values is finite-dimensional, there is no question about the existence of the trace that appears in (20.2.7). Therefore, the integrand of (20.2.7) becomes a scalar-valued 3-form, which we now integrate over the 3-manifold M to obtain a real number; namely, the action S(A) of A. The integral W(M) defined by (20.2.6)—with the action S = 5(A) given by (20.2.7)—M Witten's invariant for the three-manifold M.
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We next turn to the case where a nontrivial link K, made up of a necessarily finite number of knots {Ka} (namely, its components), is embedded in M. This will add another functional to the integrand of (20.2.6):
The knot Ka is a simple, closed oriented curve in M and the expression
denotes the product integral of the matrix-valued (more precisely, g-valued) gauge field A. The integral W ( M ; K ) defined by (20.2.8) is Witten's knot (or link) invariant for the manifold M with embedded link K. Clearly, in the earlier case when K is the "empty knot" 0 (a link with zero component), then (20.2.8) reduces to (20.2.6):
so that W(M) is a topological invariant of the three-manifold M itself. We stress that both W(M) and W(M; K) depend on the choice of the parameter k e Z and of the Lie group G; further, W(M; K) (but not W(M)) also depends on suitable choices of representations of the associated Lie algebra g, as is noted in Remark 20.2.4(d) below. For simplicity, however, we have chosen not to indicate this explicitly in our notation. When the context will require it, we will simply specify the choice of G or point out the dependence on k. Clearly, the path (or rather, functional) integral in (20.2.8) provides one example of the kind of heuristic Feynman integral introduced in (20.2.1') above. Also, as will be further discussed in Remarks 20.2.4(a), (c) and (f), it can be interpreted physically as the (unnormalized) "expectation value" of the "observable" associated with the product of the "Wilson loops" (corresponding to the monodromy along the loops Ka). Recall that in part of Section 17.6 ([Lal6,18]), we have worked with time-ordered (rather than path-ordered) product integrals in an infinite dimensional context. Observe that in (20.2.9) and (20.2.8) above, we have used the physicist's notation for product integrals rather than that of [DoFri3] or of Section 17.6. (We refer to Remarks 20.2.4(a) and (c), respectively, for geometric and physical interpretations of these product integrals in the present situation.) A perturbation series for W(M; K) can be obtained formally from the right-hand side of (20.2.8) by regarding the second factor in the integrand as perturbing the first. We assume for simplicity that the link K consists of a single knot, which we still denote by K. In this case, the second factor in the integrand is just
Now, a familiar theorem from the theory of product integration [DoFri3, Appendix II, esp. §A.II.9] permits one to write the product integral in (20.2.11) as a "time-ordered"
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exponential series. Actually, time is not the natural variable here, but it is enough for product integration to have a totally ordered set. The oriented, simple closed curve K plays that role here. The variables involved in the nth term of the exponential series are points x1, . . . , xn on the knot which have the order induced by the choice of an initial point on K and by the orientation of K. (Of course, one can parametrize K by means of a one-to-one map x : [0, 1] -» K, with x(0) = x(l), and think of t, where 0 < t < 1, as time.) We have just been discussing what Kauffman [Kau8, Remark, pp. 290-291] calls the "level zero heuristics" of perturbation series associated with the Feynman-Witten integral (20.2.8). (See also [barNl,2] and [GuMMl].) Kauffman goes on to discuss the heuristics at a higher level [Kau8, pp. 291-313]. For a still more penetrating discussion of the integral (20.2.8) and associated physics or perturbation expansion as an infinite sum over Feynman-type diagrams [Witl4, Fig. 3, p. 364], we refer to Sections 2-4 of Witten's paper [Wit14], which will be briefly discussed below. (See also the second part of Rozansky's survey article [Roz3].) We caution the reader, however, that the prerequisites for large parts of [Wit14] are rather formidable. We now comment on some of the steps leading to the perturbation series obtained in [Wit14]. In summary, in [Wit14], the perturbation expansion associated with the Feynman-type integral (20.2.8) is first obtained in [Witl4, §2] for the model case when M = S3, the three-dimensional sphere; see, in particular, [Witl4, Fig. 3, p. 364], representing "the first in an infinite series of Feynman diagrams, with gauge fields [continually] emitted and absorbed by the same knot." At this stage in [Witl4], there is already a beautiful interplay between two of the key tools used in the physics literature to deal (at least heuristically) with Feynman-type path integrals; namely, perturbation expansions as infinite sums over terms represented by Feynman diagrams (as was alluded to in (6) below (20.2.1)), and the method of stationary phase (interpreted naturally as the classical limit as k —> oo, as in (5) just below (20.2.1)). This interplay is crucial to much of [Witl4] (as well as to a number of Witten's related papers [Wit7-15]) and in one form or another, has become a recurring theme in many of the later mathematical or physical works related to Witten's topological invariants. (See, for example, [Roz3] and the relevant references therein.) Then, in [Wit14, §3 (and §4)], the case of a general 3-manifold M is dealt with. The corresponding perturbation expansion is now obtained by performing "surgery" (in the sense of differential topology) on M; very roughly, by "cutting" M along a compact Riemann surface E; see [Witl4, Fig. 4, p. 366]. In this context, the model case is that in which M is globally a product manifold; say, M = E x R1. (Instead, one may wish to work with the compact 3-manifold E x S 1 .) In general, however, this only corresponds to the local picture; that is, M only "looks like" E x K1 near the cut along the twodimensional compact surface E. [When knots or links are added, as in [Wit14, §3.3 and §3.4], the associated "Wilson lines" (see Remark 20.2.4(c) below) pierce the surface E at finitely many points, called "marked points". This leads to perturbation expansions involving a "moduli space of Riemann surfaces with marked points". Such perturbation expansions appear in several mathematical works motivated in part by conformal field theory [Seg, PresSeg] or by string theory [GreSWit, Wit7], including the recent work of Kontsevich [Konl-8] leading, in particular, to the so-called "Kontsevich integral"
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(or rather, perturbation series) representation of the Vassiliev invariants [Konl]. These invariants will be briefly discussed at the very end of the present section.] The model case when M = E x R1 is treated via "canonical quantization" [Witl4, §3] and leads to the construction of the (necessarily finite-dimensional) "physical Hilbert space of [physical states of] the Chern-Simons theory quantized on E" [Witl4]. (From a mathematical point of view, this finite-dimensional vector space is associated with the Teichmiiller space of complex structures associated with E.) This construction relies in part on the physical work of Belavin, Polyakov, and Zamolodchikov on "rational conformal field theory" via "conformal blocks" [BelPolyZ], as well as on G. Segal's axiomatization of two-dimensional conformal field theory in terms of "modular functors" [Seg]. As Witten points out [Wit14, p. 366]: "The key observation in the present work was really the observation that precisely those [modular] functors can be obtained by quantization of a three dimensional quantum field theory, and that this three dimensional aspect of conformal field theory gives the key to understanding the Jones polynomial." In other words, there is a natural and deep connection between the three-dimensional ChernSimons "topological quantum field theory" (three = two space + one time dimensions)— constructed by Witten in [Witl4] in answer to the question raised by Atiyah in [At5]— and a suitable underlying two-dimensional (two = one space + one time dimensions) conformal field theory. More precisely, [Wit14, p. 396]: "The basic connection that we have so far stated between general covariance [see Remark 20.2.4(g) below] in 2 + 1 dimensions and conformally invariant theories in 1 + 1 dimensions is that the physical Hilbert space obtained by quantization in 2 + 1 dimensions can be interpreted as the space of conformal blocks in 1 + 1 dimensions." Witten goes on in [Will 4, §4] to explain how to perform "surgery on knots or links" in order to reduce the general case when M is a compact, oriented, three-dimensional manifold to the original model case mentioned above; namely, that when M = S3. We will not discuss this fascinating topic any further but instead refer the interested reader to [Witl4, §4] and to [Roz3] where the connections between surgery theory and perturbation expansions for the Feynman-type integral (20.2.8) are discussed in detail. (See also Section 3 of the recent paper [RozWit] for perturbative calculations of invariants of 3-manifolds in a closely related context.) We further refer to [At6] and [At7, esp. Sects. 2-6] for more mathematical precision about several aspects of the above discussion. Remark 20.2.4 (a) Geometrically, the product integral (or path-ordered exponential) given by (20.2.9) represents the holonomy ([Chob-dWm], [EgGH, pp. 276-277]) of the connection A around the closed curve Ka; see, for example, [Kau8, pp. 294-299]. It is associated with the parallel transport of A along the "Wilson loop" Ka and is closely related to the curvature of A. (We refer the interested reader to Section A.II.9 of Masani's appendix in [DoFri3, pp. 240-245] for a discussion of holonomy in terms of product integrals in a somewhat simpler situation than the present one.) (b) The fact that the parameter k must belong to 1 is a "quantization condition" which guarantees the gauge-invariance of the expression e x p { i k S ( A ) } in the integrand of (20.2.6) or of (20.2.8), much as in Dirac's theory of magnetic monopoles; see [Wit14, p. 354].
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(c) Physically, the gauge-invariant local observables
in the integrand of (20.2.8) are associated with "Wilson lines", much as in standard Yang-Mills gauge theory, particularly quantum chromodynamics (QCD); see [Wit14, pp. 354-355]. Furthermore, the ath factor in (20.2.11') is often referred to in the literature as the "Wilson loop observable" associated with the knot Ka. (d) The traces involved in (20.2.8), (20.2.11) and (20.2.11') depend on the choice of a representation for the Lie algebra g, which is usually chosen to be the "fundamental" or "standard" representation of g in the concrete applications to knot polynomials; however, we omit explicit reference to this fact in our notation. We note that in general, one can choose different representations of g to define the trace occurring in (20.2.11) as well as for defining in (20.2.8) or (20.2.11') the trace of the product integral associated with each component Ka of the link K; see [Wit14]. [We point out that this comment does not apply to the invariants of 3-manifolds defined by (20.2.6). Indeed, whereas there are many "coloured" Jones polynomials, for example, the definition of the 3-manifold invariants does not require the choice of a representation (i.e. of a "colour").] (e) As was noted earlier, the method of stationary phase—applied to the heuristic Feynman integral (20.2.6) or (20.2.8)—plays an important role in [Wit14]; see especially [Witl4, §27 and [At7, §7.2]. As in (5) above (which appears just after (20.2.1)), it corresponds to the classical limit as k —> oo, also called "the weak coupling limit" in [Witl4, p. 356] or else the "semiclassical approximation." We will briefly discuss one concrete consequence of this method in Remark 20.2.7(b). For now, we mention that from a mathematical point of view, the results deduced from the method of stationary phase provide "the first indication that topological invariants really can be obtained from the Chem-Simons theory" [Wit14, p. 358]. [This fact—which makes use of earlier work of Albert Schwartz in [Schwal]—relies on the close analogy (and actual equality in the present situation [At7, pp. 60-677) between the so-called Reidemeister torsion (itself a topological invariant) and the Ray-Singer analytic torsion [RaySin] (defined for elliptic operators via "regularized determinants" in spectral geometry); see, for example, [BurFKM] and the relevant references therein, including the original papers by Cheeger and by Muller; see also [AtPatSin].] We close this comment by providing some additional information regarding the application of the method of stationary phase in this context. To avoid unnecessary complications, we assume that M is a manifold without embedded links, so that the infinite dimensional oscillatory integral to be considered is given by (20.2.6). The stationary phase approximation to (20.2.6) involves a sum overall critical (i.e., stationary) points of the Chern-Simons action functional S = S(A) given by (20.2.7); that is, the (equivalence classes of) connections A such that dS(A) = 0, where dS = dS/dA denotes the functional derivative of S with respect to A. But, as is well-known (see, for example, [At7, §7.27, [Kau8, Proposition 17.3, pp. 300-301] or [Wit14])—although by no means obvious—these critical points are precisely the "flat connections"; in other words, the connections with curvature zero. In summary, the "classical virtual paths" associated
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with the "classical limit" k -> oo in the Feynman path integral (20.2.6) are exactly the (gauge equivalence classes of) flat connections on M. This key fact is at the basis of the computation alluded to above (as well as in Remark 20.2.7(b) below). (f) By analogy with quantum statistical mechanics, the Feynman "path integral" W(M; K) in (20.2.8) is often referred to in [Witl4] (or in later physics papers, such as [Roz3]) as the "partition function" of the link K or, in agreement with the comments following (20.2.1'), as the "expectation value" of the observable (20.2.11'). (g) Witten [Wit14, pp. 352-353] stresses the fact that his three (= two space + one time) dimensional quantum field theory is "generally covariant", in the sense that all observables must be topological invariants (i.e. independent of the choice of a metric on M). The key point here is that neither the Chern-Simons action S(A) in (20.2.7) nor the functional (20.2.11') in the integrand of (20.2.8) depends on the choice of a metric. As noted in [Wit14], this yields a new kind of "generally covariant" physical theory, different from Einstein's general relativity, where in the Feynman path integral formalism, the observables are obtained by integrating over all (equivalence classes of) metrics of space-time (rather than over all gauge fields as in the present situation). In spite of the fact that, in general, there is no mathematical proof of the existence of Witten's version of the Feynman integral (see, however, Remark 20.2.5(b) below), Witten makes frequent use of techniques of integration in [Wit14] and elsewhere (see, e.g., [Wit5-15]) to do calculations. Further, these calculations have led to significant results which have in several cases been verified mathematically by other means. The integration techniques employed by Witten include changes of variables which take advantage of the symmetries present in the associated physical and/or mathematical problems, integration by parts, and the method of stationary phase (see Remarks 20.2.4(e) and 20.2.7(b)). Further, a multiplicative property of the heuristic Feynman integral (20.2.8) plays a crucial role in [Witl4] (as well as, for example, in [Witl2]). (See [Witl4, Eq. (4.1), p. 374] and especially, equation (20.2.13) in Remark 20.2.7(a) below [Witl4, Eq. (4.56), p. 394].) This property rests on the quantum nature of the underlying theory and is related to the basic formula (6.2.2) discussed in Chapter 6. It is also in the spirit of our noncommutative multiplication * in Chapter 18 [JoLa3,4]. (See Section 18.3 for the definition of * and of the related noncommutative addition +; also see Corollary 18.4.4, Theorems 18.4.7 and 18.5.6 for various expressions of the multiplicative property of the Feynman path integral in the context of Chapter 18.) The reader should note that the multiplicative property in [Wit14] is associated with a splitting up of the manifold. (See [Witl4, Fig. 5, p. 375].) In contrast, formula (6.2.2) comes from splitting the time interval. The noncommutative multiplication * starts with a splitting of the time interval which then induces a corresponding splitting of the space Ct1+t2 - C([0, t1 + t2], Rd) of Rd-valued continuous functions on [0, t1 + t2\. Despite the analogies with aspects of Chapter 18 pointed out above, it goes without saying, however, that to our knowledge, there are still many obstacles, in general, to the construction of a similar multiplication * (on a suitable space of functionals) in the context of low-dimensional topology and knot invariants, via a rigorously defined path (or rather, functional) integral. Among these difficulties, we mention the fact that the
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functional to be integrated in (20.2.6) and (20.2.8) no longer depend on standard paths in ]Rd but on (equivalence classes of) connections on the manifold M. [In some sense, however, Kontsevich's beautiful work [Konl-8] to be briefly discussed at the end of the present section and in Section 20.2.B—in which, in particular, the universal Vassiliev invariants are defined via suitable perturbation series—can be thought of as realizing a similar objective.] Remark 20.2.5 (a) We should point out—as was alluded to earlier—that an important example of (topological) quantum field theory in the form of an Abelian Chem-Simons theory defined by a heuristic Feynman path integral of the simpler type (20.2.6) has been provided by Albert Schwarz in 1978 in [Schwal]. Schwarz also later conjectured in [Schwa2]—at about the same time as but independently of Michael Atiyah in [At5]— that Jones' polynomial invariant could be expressed as a path integral by using an extension of the quantum field theory discussed in [Schwal]. (See [Schwa3, pp. 199200].) In [Schwal], the so-called Ray-Singer (or analytic) torsion [RaySin]—an analytical counterpart of the Reidemeister torsion, a well known topological (or combinatorial) invariant—was expressed in terms of a (heuristic) path integral associated with an Abelian Chern-Simons theory for which G = SU(1) — U(1), the (commutative) group of planar rotations. (See Remarks 20.2.4(e) and 20.2.7(b).) Further, Schwarz's conjecture in [Schwal] was motivated by a query of Vladimir Turaev regarding the possible connections between [Schwal] and the Jones polynomial. Moreover, as is also pointed out in [Schwa3], Witten knew about [Schwal] (and actually used it in [Wit14], see Remarks 20.2.4(e) and 20.2.7(b)) but was unaware of the abstract [Schwa2] at the time of writing his seminal paper [Witl4]. (Naturally, Atiyah was also unaware of [Schwa2].) In this regard, Schwarz writes elegantly in [Schwa3, p. 199]: "Of course, Witten went much further than I [in [Wit 13, 14]]. (I consider the contribution of [Schwa2] as negligible in comparison with [Witl4] or [Schwal].) The constructions of his paper [Witl4] (in particular, the connection with two-dimensional conformal field theory) were exploited later in hundreds of papers and led not only to heuristic, but also to important rigorous results." (Unfortunately, the above information—which can be found with some more detail in [Schwa3]—does not seem to be widely available. The second author is grateful to Masoud Khalkhali for bringing references [Schwa2] and [Schwa3] to his attention.) (b) It is noteworthy that recently, the Feynman-type functional integral (20.2.8) has been made sense of rigorously—within the context of the "Fresnel integral" (discussed in Section 20.1.A) [Sea, AlSca] or in terms of "white noise (distributional) analysis" [LeuSca, AlSen]—in the following two cases: (i) In the Abelian case, in [Sca, AlSca, LeuSca]; i.e. when the group G is commutative. (See also [Nil] where a rather different approach is taken.) (ii) In the non-Abelian case, in [AlSen], but only in the flat case (i.e. for G-principal fiber bundles over R3). Unfortunately, the Abelian hypothesis in case (i) excludes most (but not all) situations relevant to Witten's link invariants in the context considered in [Witl4]. Indeed, as we will see below, the group G will typically be nonAbelian. Moreover, the flatness hypothesis in case (ii) requires the links (or knots) to lie
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in S3 (or equivalently, in R3), and hence excludes the general case of links in arbitrary compact 3-manifolds. Nevertheless, the results of [AlSen] in case (ii) are relevant to the study of the Jones polynomial (or of its two-variable extensions). (At this point, however, it is not clear to the authors whether in this special case, all the aforementioned integration techniques used in [Wil13,14] can be applied within the framework of [AlSen].) In addition, the results of [Sca, AlSca, LeuSca, Nil] in case (i) can be applied to the simplest type of knot invariant, namely, the Gauss linking number (see, for example, [AlSca]) which corresponds to G — U(l) = S1. Also, after further specialization, they can be used to justify rigorously the use of the heuristic Feynman integral made in the early example of (Abelian) Chem-Simons theory studied by Schwarz in [Schwal]; see part (a) of the present remark. We now explain the relationships between the heuristic Feynman integral (20.2.8), Witten's link invariants and the Jones polynomial. In his paper [Witl4], Witten showed formally that the Jones polynomial VK corresponded to the choices M = S3 (the 3sphere) and G = SU(2) (the special unitary group in two dimensions). More precisely, with the "coupling constant" k e Z as in (20.2.8), we have for the gauge group G = SU(2):
for all knots K (in S3 or R3) and all values of the parameter k e Z, where W(S3; K) is given by the Feynman-type path integral (20.2.8). (Observe that since k e Z is arbitrary, relation (20.2.12) determines uniquely the Jones polynomial VK = VK(t). Further, we point out that a more accurate statement of relation (20.2.12) will be given in equation (20.2.14) below; see Remark 20.2.7(b).) Thus, in more physical terminology (see Remarks 20.2.4(c) and (f) above), the Jones polynomial—evaluated at suitable roots of unity (which play a key role in Jones' original operator-algebraic interpretation [Jone2-6] of VK)—is nothing but the expectation value of the Wilson loop along the knot K in SU(2) Chern-Simons gauge field theory. (An entirely analogous statement holds for links as well.) We note that the latter gauge theory is an example of a 2 + 1 (= two space + one time)-dimensional "topological quantum field theory", as will be further discussed below. Hence, in light of (20.2.8), (20.2.7) and (20.2.12) (or rather, (20.2.14)), Witten's work in [Witl4] provides an intrinsically three-dimensional interpretation of the Jones polynomial invariant VK, thereby answering Atiyah's original question in [At5, pp. 298-299]. (See Witten's quote [Wit 14, p. 352] at the very beginning of the present italicized subsection.) Remark 20.2.6 (a) In an equally remarkable development, Witten [Witl2] had constructed in 1988 a four (= three space + one time)-dimensional "topological quantum field theory" (TQFT) for Donaldson's theory ([Don1,2], [DonKr]), thereby providing a physical interpretation of Donaldson's polynomial invariants of 4-manifolds ([Don1,2], [DonKr]) and answering the other main question raised by Atiyah in [At5]. We note that the path integral formalism also played an essential role in [Witl2]; in addition to [Witl2], see for example, [Wit13] and [At6-8].
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(b) In [Wit9], Witten has also proposed a 1 + 1 topological quantum field theorycalled a "topological sigma model"—naturally associated with Gromov's beautiful theory of "pseudo-holomorphic curves" [Gro], The action (or Lagrangian) in the corresponding Feynman path integral involves a supersymmetric version of quantum physics discussed earlier in [Wit4]. (See also, for example, the brief overview given in [Benn].) A closely related theory—also inspired in part by [Wit9J and by Witten's approach to Morse theory [Wit4] on which it is based—has been developed by Floer [Flo1-3] (see [At6, Example 3.1(a), p. 183]) and has now flourished into a new mathematical subject, called "quantum cohomology". (See, for example, [Run, Hofl~2, Tau5] and the relevant references therein.) (c) As is explained in [At6,7], all topological quantum field theories—and, in particular, those used in (a) and (b) above—are dynamically trivial: The associated Hamiltonian is equal to zero. Regarding this point, Atiyah writes in [At6, p. 182]: "The reader may wonder how a theory with zero Hamiltonian can be sensibly formulated. The answer lies in the Feynman path-integral approach to QFT [quantum field theory]. This incorporates relativistic invariance . . . [see Remark 20.2.20 below] and the theory isformally defined by writing down a suitable Lagrangian [or action]—a functional of the classical fields of the theory. A Lagrangian which involves only first derivatives in time formally leads to a zero Hamiltonian, but the Lagrangian itself may have non-trivial features which relate it to the topology." Similarly, the specialization M — S3 and G = SU(N) (the special unitary group in N dimensions) or G = SO(N) (the special orthogonal group in N dimensions), respectively, leads to a relation analogous to (20.2.12) between Witten's knot invariant W(S3; K) and the homfly polynomial ([FYHLMO], [PrzT]) or the Kauffman polynomial ([Kau6], [Kau8, §I.5]). Again, the Feynman path integral formulation (20.2.8) of Witten's invariant is key to establishing these relationships, and, in particular, equation (20.2.12) above (or more correctly, equation (20.2.14) in Remark 20.2.7(b) below). In short, using the formal properties (mentioned earlier) of the functional integral (20.2.8), Witten [Wit 14, §4.1 ] derives a certain "skein relation" satisfied by his topological invariant W(M; K). When specialized to the aforementioned situation, this relation yields the skein relation (20.2.4) for the Jones polynomial (or its analogue for the homfly and Kauffman polynomials). (See [Witl4, Eqs. (4.22) and (4.21), p. 382], applied with N = 2.) Relation (20.2.12) follows from this fact because, as was noted earlier, the skein relation (20.2.4), along with the normalization (20.2.3) (see Remark 20.2.7(b) below), determines the Jones polynomial. It is noteworthy that once (20.2.12) is established, then the multiplicative property of the Jones polynomial under connected sums (see (20.2.5) above), as well as the chirality property (20.2.2), all follow from the corresponding properties of the Witten link invariants, which are themselves formal consequences of the path integral formalism. Exactly the same comment applies to both the homfly and Kauffman polynomials. (In view of relation (20.2.12) or (20.2.14), it is helpful to recall that two polynomials which agree at infinitely many points must coincide.) Remark 20.2.7 (a) More precisely, the multiplicative property of the Jones polynomial under connected sums, equation (20.2.5), follows from (20.2.12) and from the special
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case M1 = M2 = S3 (and G = SU(2)) of the aforementioned multiplicative property of WK; namely [Wit14, Eq. (4.56), p. 394]:
where M1#M2 (resp., K1#K2) denotes the connected sum of the manifolds M1 and M2 (resp., of the links K1 and K2). The appropriateness of the word "multiplicative" is more obvious when (20.2.13a) is rewritten in the following form:
In light of (20.2.13b), W(S 3 , O), suitably interpreted, plays the role of a normalization factor, not unlike that played by the normalization factor K occurring in the analogue (20.2.1') of the Feynman integral (20.2.8). We note that in the same manner, the multiplicative property of the homfly (resp., Kauffman) polynomial follows from the special case M1 — M2 = S3 and G = SU(N) (resp., G = SO(N)) of equation (20.2.13b). (b) In the literature—and, in particular, in later rigorous interpretations of the Witten invariants—the "normalization factor" W(S3; O) has frequently been omitted. In writing the identity (20.2.12), we have implicitly followed this convention. However, in light of the previous discussion, relation (20.2.12) should actually be written as follows:
for all k 6 Z. (See, in particular, Witten's comments following equation (4.2) in [Witl4, p. 374]; also see [Wit13].) Witten stresses in [Witl4, p. 375] that (at least in his Feynman path integral formulation of the knot invariants) the normalization factor W(S3; O) in (20.2.13) and (20.2.14) cannot be dispensed with and set equal to one. Indeed, in his words [Witl4, p. 395], "the axioms of quantum field theory are strong enough that the value of W (S3; O) is uniquely determined and cannot be postulated arbitrarily." In fact, in [Witl4], Witten has computed a closely related normalization factor (namely, W(S3), defined as in (20.2.6), rather than W(S3; O) itself); see [Wit14, Eq. (2.26), p. 362], which is obtained by means of perturbation expansions and is consistent with the results obtained by applying the method of stationary phase to the Feynman integral (20.2.6). [The latter method yields a formula involving the Ray-Singer analytic torsion [RaySin], defined in terms of a ratio of zeta-regularized determinants, and which is familiar to spectral geometers when they evaluate a "semiclassical limit" (or equivalently, when they compute the "stationary phase approximation" to a suitable infinite dimensional oscillatory integral); see [Witl4, §27 and for more detailed information, [At7, §7.27, along with Remark 20.2.4(e) above. This particular aspect of [Wit14] is an easy generalization to non-Abelian groups of Albert Schwarz's earlier work on Abelian Chern-Simons theory [Schwal], corresponding to the case when G = S U ( 1 ) = S1. We note that formally, the term involving the analytic torsion is equal to the coefficient of the first term in the infinite
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perturbation expansion for W(S3) obtained in [Wit14].] Further, more recently, the first two coefficients of W(S3; O) in the perturbation expansion associated with the Feynman integral (20.2.8) have been computed explicitly; from which it follows, in particular, that W(S 3 ; O) is finite and nonzero. (See [GuMM2], later simplified in [LinWa] by means of methods from integral geometry.) Observe that the normalization condition (20.2.3), V0(f) = 1, follows at once from equation (20.2.14) applied to K = 0 and to all k € Z. We note that similar comments apply to the generalizations of the Jones polynomial. More precisely, in the counterpart of (20.2.14) for the homfly (resp., Kauffman) polynomial, the denominator k + 2 (appearing in the exponential) must be replaced by k + N, and, as before, the gauge group G is taken to be SU(N) (resp., SO(N)). (See [Witl4, Eq. (4.21), p. 382].) (c) We briefly comment on the relationships between Witten's path integral formulation of the Jones polynomial and the axioms of topological quantum field theory (TQFT), as formulated by Atiyah in [At6] or [At7, Chapter 2]. The multiplicative property (20.2.5) of VK (itself a consequence of (20.2.13) above) is the analogue of the "multiplicative axiom" of TQFT (see [At6, Axiom (3), p. 178 and Eq. (3), pp. 178-179]), while the chirality property (20.2.2) of VK parallels the "involution axiom" of TQFT (see [At6, Axiom (2), p. 178 and esp. Eqs. (5) and (5'), p. 181]). It is noteworthy that at least on a formal level, the multiplicative property (20.2.5) had a counterpart in Chapter 18 [JoLa4] (see, for example, equation (18.4.7) above), as was mentioned earlier, while the chirality property had an analogue towards the end of Chapter 15 (see, for instance, equation (15.7.12) or (15.7.14) above). In particular, the "gluing" of the manifolds M1 and M2 along their common boundary in the formulation of the multiplicative axiom (as given in [At6, Eq. (3c), p. 179]) resembles the "concatenation" of paths involved in the definition of the noncommutative operation * of Chapter 18; see equation (18.1.3), along with equation (18.1.4) above. Of course, the context of Chapters 15-18 as well as the motivation for these chapters are quite different from the present geometrically more complicated situation. For a much more detailed discussion of topological quantum field theory in this context, we refer to [At6] and to Atiyah's book [At7], entitled The Geometry and Physics of Knots. (See also, for example, the survey article [Saw] for a more recent and somewhat different perspective on this subject.) As noted in [At6, p. 177], early motivation for axiomatic TQFT came not only from Witten's work ([Witl2-14], along with [Wit4, 8, 9, 11]) but also in part from work of Graeme Segal [Seg] on two-dimensional conformal field theories. (d) With the benefit of hindsight, one might add that early examples of "topological quantum field theories" (well before the very notion of TQFT was formalized) include, for example, the joint work of Moshe Flato and Chris Fronsdal in the context of de Sitter theory or "singleton theory" [FlFrl], as well as the aforementioned work of Albert Schwarz [Schwal] on Abelian Chern-Simons theory. We note that the paper [FlFrl]— and later work arising from it (see, e.g., [Ang FFS, BiFlFrSo, BiFrH1-3, F12, FlFr2])— has recently been used in nonperturbative string theory, and especially in "M-theory"; see, for instance, [FerFr, FerKPZ, Wit23, 24].
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(e) We close this discussion by briefly mentioning a rather subtle issue concerning the connections between path integrals and topological quantum field theory. In order for the Feynman path integral (20.2,8) to satisfy the axioms of topological quantum field theory (TQFT), and hence to be eventually identified with the Jones polynomial via formula (20.2.14), one must really work on the tangent bundle T(M) of the given 3-manifold M. (This is so even though the corresponding "bare 3-manifold" is defined by considering a trivial principal G-bundle over M.) Now, although the tangent bundle is trivial, there is no preferred trivialization (i.e. no canonical framing). This mathematical difficulty was originally overlooked in the first version of Witten's paper [Wit14]. Atiyah has later recognized this "framing problem" and resolved it in [At9] by defining a canonical framing (trivialization) for the "double" (T(M) ® T(M)) of the tangent bundle T(M) of an arbitrary 3-manifold. Fortunately, this subtle modification turns out to be sufficient to guarantee that the axioms of TQFT are satisfied. (We refer to [At9] for a detailed treatment of the above framing problem; see also [At7].) Further developments: Vassiliev invariants and the Kontsevich integral The Witten integral [W(M) in (20.2.6) or W(M; K) in (20.2.8)] is, in its form, a typical integral in quantum field theory. In its content, [it] is highly unusual. The formalism and internal logic of Witten's integral supports the existence of a large class of topological invariants of 3-manifolds and associated invariants of knots and links in three manifolds. The invariants associated with this integral have been given rigorous combinatorial descriptions [ResTur2, TurW, KirM, Lick, Wa, KauLins], but questions and conjectures arising from the integral formulation are still outstanding. (See, for example, Refs. [At7, G, FrdGom, Jefl, Rozl].) Specific conjectures about this integral take the form of just how it implicates invariants of links and 3manifolds, and how these invariants behave in certain limits of the coupling constant k in the integral [the stationary phase approximation corresponds to the limit k -> oo]. Many conjectures of this sort can be verified through the combinatorial models. On the other hand, the really outstanding conjecture about the integral is that it exists! [Italics added.] At the present time there is no measure theory or generalization of measure theory that supports it. It is a fascinating exercise to take the speculation seriously, suppose that it does really work like an integral and explore the formal consequences. . . . Here is a formal structure of great beauty. [Italics added.] It is also a structure whose consequences can be verified by a remarkable variety of alternative means. Perhaps in the course of the exploration there will appear a hint of the true nature of this form of integration. Louis H, Kauffman, 1995 [Kau9, p. 2413] Witten's results described in the previous subsection—along with Jones' earlier work and its various generalizations—have stimulated a tremendous amount of activity in knot theory or in low-dimensional topology and have given birth to a subject now called "quantum topology". The latter consists in the "study of quantum invariants of knots and three manifolds [and] can be seen as a program to give mathematical definition to the invariants and structure promised by the Chern-Simons path integral [(20.2.6) or (20.2.8)], using the apparatus of quantum groups [Dr] (otherwise known as quantized universal enveloping algebras)" [Kri, p. 4]. A sample of the recent literature related to this subject includes [AlSca, AlSen, AltFrie, At6,7, Ax, AxPW, AxSinl,2, barNl-3, barNG, barNGRT, barNWit, Bir2,
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BirLin, BlaHMV, BotTau, FrdGom, Free, G, GetKap, GuMMl,2, Jefl,2, Kapl,2, Kau7-l 1, KauLins, KirM, Konl-8, Kri, LeuSca, Lick, Lin, LinWa, MurOh, Nil, Ohl-4, ResTurl,2, Rozl-9, RozWit, Saw, Sca, Sm, Tur2, TurVi, TurW, Vas, Wa]. This list is far from exhaustive and many further references as well as much more detailed information from several different perspectives can be found in the survey articles [Bir2, Kau9, Lin, Saw], the paper [BirLin], the books [Kas, Tur2, KasRTur], and in the recent thesis [Kri]. Some of the references above are directly influenced by (and make heuristic use of) the Feynman path integral formalism. (See, for example, [AlSca, AlSen, At6,7, Ax, AxPW, AxSinl,2, barNl,2, barNG, barNGRT, barNWit, BotTau, FrdGom, Free, GuMMl,2, Jefl,2, Kau7, Kau9-11, LeuSca, Nil, Rozl-9, RozWit, Sca, Sm] and [Kau8, §I.17].) In other cases, the influence is indirect and mathematical rigor is achieved by means of combinatorial (and/or algebraic) constructions that are often strongly suggested by— but avoid the use of—the path integral. (See, for example, [AltFrie, BlaHMV, GetKap, Kapl,2, KasRTur, KauLins, KirM, Konl-8, Kri, Lick, Lin, MurOh, Ohl-4, ResTur2, TurVi, TurW, Wa].) We note, however, that this dichotomy is not clear cut and that as a consequence, several of these references could be put in either category. We close Section 20.2.A by briefly commenting on some recent significant developments: Vassiliev invariants and the Kontsevich integral. We begin by mentioning a paper [ResTur2] which helped lead to these developments. In [ResTur2], Reshetikhin and Turaev provided a rigorous but purely combinatorial (and algebraic) description of Witten's link invariants (when M = S3) for a large class of (semi-simple) Lie algebras g and associated representations. In particular, in the case when g = su(N) (resp., 0 = so(N)), the Lie algebra of the Lie group G = SU(N) (resp., G = SO(N)), they recovered the homfly (resp., Kauffman) polynomial. (Recall from our earlier discussion that the su(2) case corresponds to the Lie group G = SU(2), and hence to the Jones polynomial.) They used, in particular, representations of the associated "quantum groups" [Dr, ChariPres, Kas]; in this case, Uq(su(N)) or Uq(so(N)), respectively, specialized to complex roots of unity q. They also used in an essential way connections between such representations and the solutions of the YangBaxter equation, occurring in the study of exactly solvable models in statistical physics [Bax] as well as in the study of the Jones polynomial [Jone5,6, Kaul, 2, Kau4,5]. (See also, for example, the earlier papers [Turl, ResTurl].) We refer to the books [Tur2, Kas, KasRTur] and the review article [Saw] for a much more detailed account of the connections between quantum groups and knot invariants in this context. A key question remained, however, after the work in [ResTur2] was completed; namely, to find a clear topological interpretation of the new "quantum invariants" constructed in [ResTur2] and associated with the given representation of the quantum group involved. Vassiliev's later work in [Vas] has provided a framework in which, in particular, such a question can be formulated, as we now briefly explain. Vassiliev's theory [Vas] has given topologists a very novel way in which to think of knot invariants by introducing the "space of all knots", A, and proposing a method for computing the associated cohomology. In short, £, the space of all knots (in M = S3), can be defined as the complement of the set of singular immersions from S1 to S3 in
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the space of all (isotopy classes of) such immersions. More precisely, in the notation of [BirLin], let M denote the space of all smooth maps from S1 to S3, and let £ (traditionally called the "discriminant locus" of M) denote the set of all maps in M which fail to be embeddings of S1 into S3. Then, by definition, A = M\E, the complement of £ in M, is the "space of all knots" (in M = S3). Strictly speaking, in the definition of .8, one must work with the isotopy classes of knots, also called "knot types". Hence, thanks to Vassiliev's insight, the point of view has shifted from the study of a single knot to the study of all knot types simultaneously. Using singularity theory and cohomology theory, Vassiliev [Vas] has defined a graded space of knot invariants (now called "Vassiliev invariants"), where the degree (or "type") is provided by the degree of the singularities on the discriminant locus E. We refer the interested reader to the paper by Birman and Lin [BirLin] for a very clear description and a useful axiomatizati~n of the theory of Vassiliev invariants. We also refer to Arnold's paper [Arno] for an exposition of Vassiliev's theory that is closer to singularity theory. Moreover, we point out the paper by Bar-Natan [barN3] for a comprehensive treatment of the combinatorics of the Vassiliev invariants. Finally, a short nontechnical exposition of the theory can be found in [Sos]. We now provide an alternative and perhaps more concrete description of k, viewed as a graded vector space, and define the associated notion of Vassiliev invariant. (We closely follow here the exposition given in [Oh4, pp. 475-476].) Let k denote the complex vector space (freely) spanned by the isotopy classes of (framed) knots. Further, let { k d } d > 0 be the associated filtration of k in vector subspaces of degree d (to be defined below). Then, given a nonnegative integer d, a linear functional p : R -> C is called a Vassiliev invariant (or a "finite type invariant") of degree d if its restriction to kd+1 is trivial; namely, p1kd+1 = 0. Here, for each d > 0, we denote by kd the linear space of knot types of degree d, which can be defined as follows. Given a knot K and a set C of crossings of K, we let
where C' runs over all subsets of C (including the empty set), #C1 denotes the cardinality of C1, and Kc1 is the knot obtained from K by crossing changes occurring precisely at the crossings of C'. Then Ad is defined as being the vector subspace of R spanned by the (isotopy classes of) "brackets" [K, C], where K is a knot and C is a set of d crossings of K. (Naturally, one can check that this definition is compatible with the equivalence relation induced by the isotopy of framed knots.) We refer, for example, to [Oh3,4] for further information regarding this discussion. The central conjecture in [Vas] is that the "Vassiliev knot invariants "form a complete set of invariants; that is, if two knots are different (i.e. not isotopic), then there exists a Vassiliev invariant (or "knot invariant of finite type") that takes different values on the corresponding knot types. So far, Vassiliev's conjecture has been neither established nor disproved. However, there seems to be a consensus among knot theorists that it is likely to be correct. If it is proved some day, then an entire chapter in knot theory and low-dimensional topology will have been completed. (It should be mentioned that it is
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known that the homfly and Kauffman polynomials together do not form a complete set of invariants. Moreover, there exist infinitely many pairwise nonisotopic knots with the same Jones polynomial; see, for example, [Kau8] and the relevant references therein. Finally, the Vassiliev invariants can distinguish certain knots which neither the Jones nor the homfly polynomials can differentiate; see, for instance, [Sos] and [CLq].) Beginning around 1992, the work of Kontsevich [Konl-8] (especially, [Konl-3]) has provided a very powerful way of understanding and constructing the (universal) Vassiliev invariants. It has also provided a new rigorous framework within which to study Witten's topological invariants discussed in the previous subsection. We will limit ourselves here to a few comments about this subject since Kontsevich's work is so recent that it is not yet fully understood (at least by the authors), although it has already given rise to many further papers, including, for example, [barN3, barNGRT, GetKap, Kapl,2, Kau9-l 1, Kri, Lin, MurOh, Ohl-4, RozWit]. In short, Kontsevich associates some kind of integral—now known as the "Kontsevich integral"—to the entire space of Vassiliev invariants. This "integral" is defined rigorously by a perturbation expansion, each term of which is represented by a suitable Feynman diagram. A bit more precisely, with each invariant of a given (necessarily finite) type (called "degree" in our above discussion), Kontsevich associates a well-defined finitedimensional integral, itself also given perturbatively by a sum involving finitely many Feynman diagrams (and hence corresponding to finitely many terms in the perturbation series). At this point, it seems that the full "Kontsevich integral" can only be defined perturbatively via the above-mentioned perturbation expansion. However, at least heuristically, it appears that it should also be given by a suitable Feynman-type path integral. Indeed, many of Kontsevich's own talks on the subject—and several of his own papers, for example [Kon3-5]—begin (or end) with some heuristic comments regarding such a Feynman integral, (We note that some very recent work has been done in this direction from several points of view; see Remarks 20.2.8(a) and (c) below.) A lot more mathematics than is alluded to above is involved in Kontsevich's work [Konl-8]. In particular, deep aspects of algebraic geometry are used to define suitable "moduli space of configurations" [FulMcp]. In addition, the coefficients of the perturbation expansion are defined via an appropriate use of dilogarithms in the hyperbolic plane and are conjectured to have special arithmetic properties. Finally, we stress that the Feynman-type diagrams used to describe the various terms of the perturbation series have a very rich and interesting combinatorial structure, now often referred to in the literature as "the Kontsevich graph complex". We refer the interested reader to Clifford Taubes' recent article [Tau6, esp. §2, pp. 121-123]—written on the occasion of the award of the Fields medal to Maxim Kontsevich—for a brief introduction for the nonexpert to Kontsevich's work on Vassiliev invariants. In particular, it makes transparent how Kontsevich's notion of "graph cohomology" [Kon2,3] "succinctly summarizes the algebraic side of the [Vassiliev and the Chern-Simons perturbative] invariants" and "how his constructions also vastly simplified the analytic aspects of the definitions" [Tau6, p. 119]. Kontsevich's work has already been fruitful in other areas of mathematics or of mathematical physics. In particular, very recently, a closely related construction—also
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based on a perturbation expansion for the "Kontsevich integral" and on the so-called "Kontsevich functor" [Cart, Kri]—has enabled Kontsevich [Kon8] to resolve in the affirmative a long-standing conjecture in the theory of "deformation quantization" [BayFFLSl,2, FlSterl,2, Wei, Ster]. We will briefly return to this last subject in Section 20.2.B (towards the end of the second subsection). Remark 20.2.8 (a) Very recently, in the flat case (i.e. for knots in R3 or equivalently, in M — S3) and with a particular choice of gauge, Kauffman [Kau10] has provided a (semi-heuristic) bridge between Witten's heuristic Feynman integrals and Kontsevich's rigorous diagrammatic description of the Vassiliev invariants. It seems that combined with (a suitable interpretation of) the work in [AlSen] (see case (ii) of Remark 20.2.5(b) above), Kauffman's work could be used to connect several of the main subjects discussed in this section. It would be interesting—but. would seem to require a rather different method—to extend the results of [Kau10] to arbitrary compact 3-manifolds. (We note that the paper [Kaul 0] has a sequel, [Kaul 1 ], which contains a detailed exposition of the relationship between Kontsevich integrals and Witten's heuristic Feynman path integral. Further, it is pointed out in [Kau11] that conclusions similar to those of [Kau10] were reached independently by Labastida and Perez in [LPz].) (b) Motivated by Witten's theory of knot and 3-manifold invariants given by Feynman path integrals, in conjunction with Vassiliev's theory of universal knot invariants and its later interpretation by Kontsevich in terms of suitable perturbation expansions, a rich theory of "finite-type invariants " of certain 3-manifolds (namely, the "integral homology 3-spheres")—introduced by Ohtsuki in [Oh2]—has been developed over the last few years; see, for example, [Kri, Lin, Ohl—4, MurOh] and the relevant references therein. There are subtle differences and intriguing analogies with the existing theory of knot and link invariants of 3-manifolds. We refer the interested reader to Lin's and Ohtsuki's recent survey articles [Lin, Oh3,4] for further information and references regarding this subject. (We note that the first part of [Oh4] also provides a concise introduction to Vassiliev's and Kontsevich's theories of knot invariants.) (c) An algebraic description and extension of Kontsevich's graph complex has recently been given by Getzler and Kapranov in [GetKap]. It is formulated within the framework of the theory of operads and involves the so-called "Feynman transform", which can be thought of heuristically as a purely algebraic (and topological) interpretation of an infinite dimensional Fourier transform. See, for example, Kapranov's recent survey article [Kap2] and the relevant references therein (including [GetKap] and [Kap1]) for further information on this and related subjects. We note that at least on a formal level and despite the obvious differences in motivations and approaches, interesting connections may exist between the above-mentioned algebraic formulation in terms of operads and aspects of the theory of Feynman's operational calculus developed in the second part of this book. (d) We point out that the relationship between the "space integrals" and the "graph complex" have also been studied from different points of view by Axelrod and Singer [AxSin1,2], Bar-Natan [barN1-3], Bott and Taubes [BotTau], Altschuler and Friedel [AltFrie], among others. (The "space integrals" are a suitable generalization of the
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Gauss linking integral, and are defined for arbitrary differentiable embeddings of knots in R3.) (e) The second author is very grateful to his colleague, Xiao-Song Lin, for useful conversations, comments and references regarding the subject of this last subsection, as well as about various aspects of the remainder of Section 20.2.A. He would also like to thank Louis H. Kauffman for his helpful comments on this section. 20.2.B. Further comments and references on subjects related to the Feynman integral . . . The mathematical theory of Feynman's magnificent integrals, which physicists write in vast numbers, is not really far removed from the stereometry of wine barrels [in the spirit of Kepler's work on the latter subject in which volumes of solids of revolution were computed before the general definition of an integral had been introduced]. From the viewpoint of a mathematician, each such calculation of a Feynman integral simultaneously defines what is calculated, i.e. it constructs a text in a formal language whose grammar has not previously been described. In the process of such computations a physicist calmly multiplies and divides by infinity (more precisely, by something which, if it were defined, would probably turn out to be infinite); sums infinite series of infinities, assuming here that two or three terms of the series give a good approximation to the whole series; and generally lives in a realm of freedom unencumbered by all "moral standards". Yu I. Manin, 1979 [Manil, p. 93] The Feynman path integral is the mathematicians' pans asinorum. Attempts to put it on a sound footing have generated more mathematics than any subject in physics since the hydrogen atom. To no avail. The mystery remains, and it will stay with us for a long time. Gian-Carlo Rota, 1977 [Rot, p. 229] In this last section, we wish to give a sample of additional mathematical or physical subjects related to (or partly motivated by) the heuristic Feynman path integral. Some of these subjects have already been briefly mentioned before, for example in Chapter 7 or Chapter 14; others appear here for the first time. We will content ourselves with providing a sample of relevant references (favoring introductory or survey articles, as well as textbooks, when they are available). In a few cases, we give a short summary of some of the main features of the subject at hand, stressing the relationships with Feynman's functional integrals. The first of our topics, related to the Atiyah-Singer Index Theorem and the use of supersymmetric Feynman-type integrals, will be treated with somewhat more detail (but not mathematical precision) than the remaining ones, which are connected with deformation quantization, and with the use of heuristic Feynman integrals in gauge field theory and in string theory.
Supersymmetric Feynman path integrals and the Atiyah-Singer index theorem To summarize, we have shown that the elliptic operators defined by the classical complexes are given either by the Hamiltonian or certain other symmetry operators of a 0 + 1 dimensional
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supersymmetric [quantum] field theory [i.e. a supersymmetric quantum-mechanical model]. This allows us to identify and interpret the index of the operators in field theoretic terms, namely, in terms of the ground states of the theory. Once this identification has been made, we can represent the index of the operator in a very compact form as a functional integral for the given field theory. This can be understood as a systematic and compact way of rewriting the computation of the index using the heat kernel of the relevant operator [AtBotPat, Gik]. The fact that the field theory we have to use is supersymmetric is crucial in identifying the operators of interest in terms of a field theoretic Hamiltonian, in proving that the index will be invariant under continuous deformations, and in furnishing a systematic way of evaluating the index densities using standard methods in perturbation theory. Luis Alvarez-Gaume, 1983 [Alv, p. 166] Stationary phase approximation applies to the computation of oscillatory integrals of the form f eitf(x) dx and it asserts that for large t the dominant terms come from the critical points of the phase function f ( x ) . Many interesting examples are known where this approximation actually yields the exact answer. Recently a simple general symmetry principle has been found by Duistermaat and Heckman [DuiHec] which gives a geometrical explanation for such exactness. In the first part of this lecture I will explain the result of Duistermaat-Heckman. I will go on to outline a brilliant observation of the physicist E. Witten suggesting that an infinite-dimensional version of this result [applied to a suitable supersymmetric Feynman (or Wiener)-type integral] should lead rather directly to the index theorem for the Dirac operator. Such interactions between mathematics and physics played a prominent part in the work of Laurent Schwartz. Michael F. Atiyah, Lecture in Honor of L. Schwartz, 1985 [At2, p. 43] Supersymmetry is one of the boldest, most original and most fruitful ideas to appear in physics in a very long time. In common with nonabelian gauge theories and spontaneous symmetry breaking, it has great depth, and just like those ideas it has traveled quite a way on its own momentum, without carving out for itself a rock bed of supporting experimental evidence. Guided by these analogies, nobody doubts that when discarding some blinding prejudices, or coming by some new data, supersymmetry will come into its own, experimentally as well. Unlike nonabelian gauge theories and spontaneous symmetry breaking [sic], supersymmetry does not build on well-understood mathematics. Rather, it has created its own, rich, truly new mathematics. Yes, we are faced here with one of those rare instances when the mathematicians, in all their wisdom, have overlooked a beautiful and most useful structure, and come to appreciate it only at the demand of physicists. We are living in an era in which the contacts between mathematicians and physicists are being vigorously renewed (particularly through work in supersymmetry and gauge theories). This is a good omen, since such contacts have historically always led to great advances both in mathematics and physics. Peter G. O. Freund, 1986, in the preface of [Freu, p. ix]
New proofs and various extensions of the classical Atiyah-Singer index theorem for elliptic operators [AtSin, AtSe, At3] have recently been given in the mathematical literature, beginning in the early to mid-1980s. These were based on (partly unpublished) ideas of Witten which were implemented at the physical level of rigor by Alvarez-Gaume in [Alv] and which relied in part on [Witl-4] and on an application of the method of stationary phase to a suitable supersymmetric Feynman-type path integral.
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The first rigorous "supersymmetric" proof of the Atiyah-Singer index theorem was given by Getzler in 1983 in [Getl] (and later on, in a somewhat different form in [Get2]). Other important contributions to this subject include those of Bismut [Bis1,2, 4-9] (see also [BisFrd]), as well as of Berline and Vergne [BerlVe] (see also [BerlGetVe]) and Melrose [Mel], among others. A semi-expository article by Atiyah [At2] on Witten's proposed Feynman integral heuristic proof of the index theorem for the Dirac operator (later expounded upon in [Wit10]) and entitled "Circular symmetry and stationary-phase approximation", has also played a significant role. Part of the aforementioned work of Bismut (especially [Bis5-7]) has provided a rigorous (infinite dimensional) stochastic version of this approach, where "Euclidean" supersymmetric functional integrals appear prominently. (We note that in the physics literature, a "Euclidean" path integral is one where the imaginary exponential of the action, e i S ( p ) , is replaced by a real exponential, e-S(p).) Recall that the Atiyah-Singer index theorem for a given elliptic pseudodifferential operator L (acting on a vector bundle over a finite-dimensional differentiable manifold M) asserts the equality of the Fredholm index of L (also referred to as the "analytic index") with its so-called "topological index". Hence, its power lies in its ability to directly connect a purely analytical quantity with a purely topological one. (For simplicity, we will assume M to be compact, Riemannian and without boundary. Also, in the appropriate cases, M will be implicitly assumed to have a spin or a spinc structure.) A classical proof of this theorem—commonly referred to as "the heat equation proof" of the index theorem—consists (roughly) in expressing the Fredholm index of L in terms of a suitable spectral function of L (namely, z(t), the trace of a corresponding heat semigroup) and then exploiting our knowledge of the short-time (t -> 0+) and longtime (heat kernel) asymptotics (t —> +00) of z(t). In some sense, due to appropriate cancellations, the long-time asymptotics yields the "analytic index", while the short-time asymptotics yields the "topological index". More precisely, let P1 (resp., Pr) be the nonnegative, self-adjoint elliptic operator defined by Pl = L* L (resp., Pr = LL*). Then the Fredholm index of L can be expressed as follows:
Now, as can be easily checked, the positive (i.e. nonzero) eigenvalues of Pl and Pr are the same (with identical multiplicities). Therefore, since dim(Ker Pl) (allowed to vanish) is just the multiplicity of the zero eigenvalue of Pl, and similarly for Pr, we have for all t > 0:
where we have set
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and similarly for Pr. (Since the present situation is different from that considered in Section 20.2.A, we use here a different symbol to denote the trace of an operator; compare, for example, equation (20.2.11') along with Remark 20.2.4(d) above.) We note that z pt (t), the trace of the heat semigroup generated by — Pl, is also called the "partition function" in quantum statistical physics, where t then plays the role of an inverse temperature, measured in suitable units, and the eigenvalues of Pl (now viewed as a "Hamiltonian") represent the various energy levels of the associated quantum system; in particular, the zero eigenvalue corresponds to the "ground states" of the quantum system. While the first equality in (20.2.16) follows from (20.2.15), the second one can be justified as follows. Let {Aj,l}oo j=1 (resp., {Aj,r}ooj=1)denote the sequence of eigenvalues of P1 (resp., Pr), written in nondecreasing order and repeated according to their multiplicity, such that Aj,l = 0, for j = 1, . . . , dim(KerPl) (resp., Aj,r = 0, for j = 1, . . . , dim(KerPr)). Then, according to definition (20.2.17) and the comments preceding equation (20.2.16), we have successively:
as desired. Note the huge cancellations occurring in the derivation of (20.2.18), and hence of (20.2; 16). (See, for example, [AndsLa2, § 1 .c] and the relevant references therein and in [AndsLal], or for more details, [LawM, §III.6].) Loosely speaking, the rest of the proof of the index theorem consists of first establishing the topological invariance of the index of L (based on the homotopy invariance of the Fredholm index). Next, one appeals to the short-time asymptotics of the heat kernel of a nonnegative pseudodifferential operator; namely, zpl is given by the asymptotic series
where d denotes the dimension of M and m the order of Pl, with the coefficients aj,l computed in terms of the symbol of Pl, and similarly for zpr- Hence, by means of (20.2.19), one expresses explicitly in terms of topologically invariant quantities the analytic index of L given by (20.2.16), thereby concluding the equality of the analytic and topological indices. This latter step of the "heat equation proof" is crucial and can be combinatorially very involved, although it has been somewhat simplified since its inception.
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An excellent exposition of the heat equation proof of the Atiyah-Singer index theorem can be found in the paper by Atiyah, Bott and Patodi [AtBotPat]. Moreover, a detailed account of this approach, along with the necessary background and various examples, can be found in Gilkey's book [Gik], entitled Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem. (See also, for instance, Chapter III of the book on Spin Geometry by Lawson and Michelsohn [LawM] for this and other approaches to the index theorem, as well as the Physics Report by Eguchi, Gilkey and Hanson [EgGH] for an introduction to this and related subjects that uses both mathematical and physical language. For a physically inclined reader, we mention the book by Nakahara [Naka], entitled Geometry, Topology and Physics, where are discussed, in particular, several physical applications of the index theorem to the understanding of anomalies in gauge field theories; see especially [Naka, Chapters 12 and 13]. We also point out the nice book by Booss and Bleecker [BooBle], entitled Topology and Analysis, where are discussed various applications to gauge theory.) We now discuss some elements of the derivation of the Atiyah-Singer index theorem, as originally sketched in the aforementioned paper by Luis Alvarez-Gaume [Alv]. We will stress the physical aspects, based on the use of supersymmetry and of functional integrals. Loosely speaking, supersymmetry is a symmetry that is forbidden in standard quantum physics and consists in exchanging fermions (particles which obey Pauli's exclusion principle, such as electrons, protons, neutrons and other constituents of matter) and bosons (particles which mediate interactions, such as photons and gravitons). It was introduced in physics in the 1970s towards the beginning of "string theory". It was later used, in particular, in the so-called "grand unification theories" and has played since then a key role in theoretical physics, including quantum field theory and modern string theory. (We note, however, that the existence of supersymmetry has never been verified experimentally, although many theoreticians hope that this will occur with the next generation of particle accelerators, perhaps during the first decade of the twenty-first century.) A good physical introduction to the notion of supersymmetry and its applications is provided in the monograph by Freund [Freu]. (Also see [Alva].) Other (more mathematical) references will be given further on. Paraphrasing [Alv, p. 163], we can summarize the "proof as follows (see also the quote from [Alv, p. 166] heading the present italicized subsection): The outline of [the supersymmetric] rederivation of the Atiyah-Singer results for classical complexes is as follows: We first find a supersymmetric quantum mechanical system whose conserved supercharge Q is the operator whose index we want to calculate, and then we compute the t-independent term of the functional integral [(20.2.28), for example], which automatically yields the index density for the operator Q. [Here, Q plays the role of the elliptic operator L above.] It is remarkable that the index theorem for the classical complexes (de Rham, Dirac, signature and Dolbeault complexes) can be obtained from a single supersymmetric system, namely the supersymmetric non-linear a -model. Furthermore, by a slight modification of the non-linear a -model, it is possible to find as well the G-index theorem [i.e. the equivariant index theorem] for the classical complexes.
Next, we begin by recalling some basic facts concerning supersymmetry in the present context. The supersymmetry algebra (with supersymmetry N) is defined by
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the anticommutation relations;
for a, B = 1, . . . , N, where daB denotes the Kronecker delta and, for example, the anticommutator of Qa and QB,is given by {Qa, QB} = Q a QB + QBQa- Here, H is the Hamiltonian of the supersymmetric quantum-mechanical system, Qa (a = 1, . . . , N) are the supersymmetric charges (also called "supercharges"), viewed as (unbounded) operators on the underlying Hilbert space H, and Q*a (a = 1, . . . , N) are the "conjugate supercharges" (that is, Qa is the adjoint of Qa with respect to the inner product of H). Moreover, the "fermion number operator", usually denoted by (— 1)F in the physics literature (see, e.g., [Wit1-3] and [Alv]), is required to anticommute with the supercharges:
for a = 1, . . . , N. Without real loss of generality, we can assume that N = 1 for the discussion that follows. Thus, we now have a single supercharge Q and its conjugate supercharge Q*, satisfying the corresponding special case of (20.2.20):
Consequently, if S = (Q + Q*)/V2, it follows that
From (20.2.22) and from relation (20.2.21) (with N = 1), it is easy to deduce that nonzero energy states (i.e. eigenfunctions u = O of H belonging to a nonzero eigenvalue X) come in "boson-fermion pairs"; specifically, in pairs (u, Su), with u and Su both eigenfunctions of H with the same nonzero eigenvalue X but also having different parity: If (-1)Fu = ± u, then (-1) F Su(= - S ( - 1 ) F u , by (20.2.21)) = +u. In general, the above reasoning does not apply to eigenstates with zero energy (i.e. to ground states of the system); see equations (20.2.25) and (20.2.26) below. At this point, it may be helpful to mention that the Hilbert space H admits the following Z2-grading (where Z2 = {0, 1} is the standard Abelian group of 2 elements):
where HO (resp., H1) is the +1 (resp., —1) eigenspace of the fermion number operator ( — 1 ) F . Then the elements of Ho (resp., H1) are said to have even (resp., odd) parity; that is, they represent bosonic (resp., fermionic) states. Further, HO (resp., H1) is called the bosonic (resp., fermionic) subspace of the theory.
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With respect to the orthogonal decomposition of H given by (20.2.24), 5 is represented by an off-diagonal operator
and so the bosonic (resp., fermionic) ground states u of H are given by the equation Qu = 0 (resp., Q*u = 0). (We leave the easy algebraic verification of these facts to the interested reader.) Consequently [Wit3], the "trace of (—1) F ", often denoted by Tr(-1) F in the physics literature and called the "supertrace" in the mathematics literature, can be expressed as follows:
where nA=0 (resp., nA=0) denotes the number of linearly independent bosonic (resp., fermionic) eigenstates of H with zero energy (i.e. ground states). More precisely, this formula should be interpreted as follows after a suitable regularization involving the "Euclidean propagator":
for all t > 0. Note the remarkable cancellations involved in this formula, much as in the derivation of (20.2.16) given above. One then rewrites the left-hand side of (20.2.26) as a (heuristic) supersymmetric functional integral (by means of [CeGir], see [Alv, §III]), and uses the homotopy invariance of the Fredholm index (and hence the invariance of the path integral under suitable continuous deformations of the Hamiltonian, much as in [Wit4]) to establish the topological invariance of Tr(— 1)F and to facilitate its computation. [We note that if instead of the "Euclidean propagator", we had taken the associated "Feynman propagator", Trace((— 1 ) F e - i tH) (sometimes called the "propagator of the fermions" in the physics literature), then the corresponding functional integral would have been a genuine supersymmetric Feynman integral, the existence of which would naturally be even more difficult to justify mathematically.] The last step of the (heuristic) supersymmetric proof of the index theorem consists in applying the method of stationary phase (or some other standard perturbation method) to the path integral in order to collect, in particular, the t-independent terms. In light of formula (20.2.26), we know a priori that these terms (corresponding to the leading contribution of the critical points of the action functional) will yield the desired result, namely, the topological index; see [Alv, §IV]. As a simple illustration, we briefly discuss the case of the de Rham complex when the supercharge is given by the exterior derivative. (See [Alv, pp. 163-164] for more details.)
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Then, the corresponding supersymmetric nonlinear a -model can be described as follows. The Hilbert space H is the space of square-integrable sections of A*(M) = A(T*(M)), the exterior power of the cotangent bundle of M. Further, the bosonic subspace Ho (resp., the fermionic subspace H1) is the space of even (resp., odd) differential forms on M, while the supercharge Q (resp., the conjugate supercharge Q*) corresponds to the exterior derivative d (resp., its adjoint d*). (From a physical point of view, a suitable version of Noether's theorem for the conservation of observables associated with supersymmetries plays a key role in carrying out the identification of the conserved supercharge Q with d, and hence of the adjoint Q* with d*.) Therefore, the Hamiltonian H is simply equal to (half) the Hodge-de Rham Laplacian on forms: 2H = dd* + d*d. (Note that equation (20.2.22) is obviously satisfied.) In addition, by means of standard algebraic manipulations (see, e.g., [Gik]), one can check directly that KerQ (resp., Ker Q*) corresponds to the bosonic (resp., fermionic) ground states; that is, in the present situation, to the even (resp., odd) harmonic forms on M. Also, formula (20.2.26) becomes for all t > 0:
where x (M) e Z denotes the Euler characteristic of M, d the dimension of M, and for j = 0, . . . , d, the integer bj stands for the jth Betti number of M (that is, the dimension of the space of harmonic j-forms on M). (As is well-known, if d is odd, then we have X(M)=0.) We now express the "supertrace" occurring in (20.2.27) as the following heuristic path integral (using [CeGi]):
where the functional integration is carried out over all field configurations satisfying periodic boundary conditions with period t; namely, $(s + t) = $ ( s ) and similarly for (p. Here, the Euclidean action S = S ( $ , p) is determined by the Lagrangian given by equation (3.2) of [Alv, p. 163]. Further, $ is a bosonic field while p is a fermionic field, specifically, a two-component real spinor field. [We note that instead of Q = d and Q* = d*, one sometimes considers as "generators of supersymmetry" the (formally) self-adjoint operators d + d* and i (d — d*); see, for example, [Benn, p. 71]. These operators also exchange forms of even degree ("bosons") and forms of odd degree ("fermions").] Remark 20.2.9 (a) Observe that because we have taken the (super) trace of the heat semigroup e-tH, the corresponding (supersymmetric) "Feynman-Kac formula" in (20.2.28) includes a double functional integral, unlike the single functional integral
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encountered in Feynman's original formulation discussed at the beginning of Chapter 7. (Compare also with equation (12.1.4), the version of the Feynman-Kac formula established in Chapter 12.) (b) In the present case of the Euler characteristic (and shortly after Getzler's work in [Get 1,2]), John Lott [Lot] has provided a rigorous definition of the necessary supersymmetric path integration and established the corresponding supersymmetric version of the Feynman-Kac formula. Hence, he has justified mathematically the above formulas (20.2.28) and (20.2.27), as well as the derivation of the index theorem in this special case. (For the expert reader, we mention that the formulas in [Lot] exhibit the correct scaling required by physical supersymmetry whereas this is not the case of those in [Getl,2].) Combining (20.2.27) and (20.2.28), one can use, in particular, the aforementioned analogies with quantum statistical physics (where t-1 is the "temperature" and ( - 1 ) F e - t H represents the "density matrix" of the system) along with standard computations of supersymmetric Gaussian integrals (based on "Berezin determinants" [Berl]) to carry out the analogue of the method of stationary phase and recover the classical Gauss-Bonnet formula (as extended to any dimension d by S.-S. Chern in [Chern1,2]) expressing the Euler characteristic x ( M ) in terms of the integral of the curvature of M; see [Alv, p. 167]. Remark 20.2.10 (a) Assume that M is even-dimensional. Then, to obtain the "Hirzebruch signature " T (M), we still work with the de Rham complex, but now identify the fermion number operator (— \)F with the Hodge-* operator which implements the Poincare duality (that is, for p = 0, . . . ,d, it sends a given p-form w to the (d-p)-form *w, called its Hodge dual). Then, and the corresponding path integral in the counterpart of (20.2.28) now involves both fields satisfying periodic and antiperiodic boundary conditions; see [Alv, p. 164 and pp. 167-169]. [Recall that T(M) is defined as the difference between the dimension of the space of self-dual harmonic forms (necessarily of even degree) and the dimension of the space of antiself-dual harmonic forms (also of even degree); the form w is said to be self-dual (resp., antiself-dual) if *w = w (resp., = —w).] (b) From the point of view of the general Atiyah-Singer index theorem, the most interesting case is that of the spin complex on a spin, even-dimensional Riemannian manifold M (or rather, a slight variation thereof [AtBotPat]). Then the supercharge Q is identified with (a suitable multiple of) the Dirac operator D, viewed as an elliptic operator acting on H, the Hilbert space of square-integrable sections of the spin bundle of M. Further, Ho (resp., H1) the bosonic (resp., fermionic) subspace of H corresponds to the space of spinors of positive (resp., negative) chirality. Finally, Index(Q) = A(M), the "Dirac (or A-) genus" of M. [All the geometrical and topological notions used here are described in detail in [Gik], [EgGH], [Naka], [BooBle], and [LawM], for example.] As was mentioned earlier, the first rigorous version of (Witten-) Alvarez-Gaume's supersymmetric "proof of the Atiyah-Singer index theorem [Alv] was given in 1983
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by Ezra Getzler in [Getl]. His method consisted in developing a symbol calculus for pseudodifferential operators on supermanifolds M, and then mimicking the standard (symplectic) proof (see, for example, [Wl,2] or [H6r4, §29.3]) of the extension of Weyl's classical theorem for the asymptotics of the eigenvalue distribution of elliptic (pseudo)differential operators on a compact, Riemannian manifold M. (Very roughly, a "supermanifold" is modeled locally on Rp/q, rather than on Rd, where R p / q denotes the real (super) vector space with p commuting and q anticommuting coordinates; see, for instance, B. DeWitt's book on this subject [dWi] as well as Leites' very helpful survey article [Leit] for an extensive treatment. Moreover, intuitively, instead of working with classical "phase space"—that is, with T*(M), the cotangent bundle of M, viewed as the natural symplectic manifold associated with M—one works with the "cotangent superbundle" of the supermanifold M; see [BerMv] and [Getl, esp. pp. 164-165].) In Getzler's own words [Getl, p. 167], "[his] proof may be considered to be an improved version of those of Patodi [Pat] and Atiyah, Bott and Patodi [AtBotPat] in two senses: the use of Clifford algebras and their symbols eliminates the combinatorics of the earlier proofs, and shows that the cancellations which occur are not all that remarkable; also this proof leads to the explicit formula for the A-genus [see Remark 20.2.10(b) above], while the earlier proofs had to appeal to various topological characterizations of the A-genus." We note that a second proof—motivated in part by the kind of deformation of a supersymmetric Hamiltonian introduced in [Wit4] but perhaps less directly connected than the first one to a heuristic supersymmetric path integral—was later given by Getzler in [Get2]; there too, a suitable pseudodifferential operator calculus played a key role. "Getzler's calculus" has also been very useful in later mathematical developments. (See, for example, [Con1,2], [LawM], [Mel, §8.4], and the relevant references therein.) Remark 20.2.11 (a) Recently, a different (physically motivated) supersymmetric derivation of the Atiyah-Singer index theorem has been given by A. Mostafazadeh in [Mot1,2]. It also makes a key use of heuristic supersymmetric Feynman path integrals but relies on early ideas of Bryce DeWitt (rather than of Edward Witten) presented in part in his aforementioned book on Supermanifolds [dWi]. (Unfortunately, the computations in [Mot] are rather involved.) One interesting aspect of the work in [Mot] is that it seems to provide a "resolution" of the long-standing controversy regarding (the numerical multiplicative factor of) the scalar curvature term in the Schrodinger equation on a curved Riemannian manifold; see [Mot2]. (b) Recall that we are assuming throughout that M is a closed manifold (that is, a smooth manifold without boundary). A supersymmetric and entirely rigorous proof of a very general extension of the index theorem to manifolds with boundary (and possible singularities) can be found in Melrose's recent book [Mel], entitled The AtiyahPatodi-Singer Index Theorem. (The original extension of the index theorem to nonclosed manifolds was obtained in [AtPatSin].) We note that a suitable version of Getzler's approach in [Get1,2] plays an important role in [Mel]. We close this subsection by discussing Witten's original idea for establishing the Atiyah-Singer index theorem for the Dirac operator via a heuristic supersymmetric (or
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"fermionic") Feynman-type integral, as it was presented by Atiyah in his aforementioned 1985 paper [At2] (and later expounded upon by Witten in 1988 in [Wit10]). (This second idea of Witten is different in several respects from the one developed by Alvarez-Gaume [Alv] and discussed above, although it also makes use of several key results from the papers [Wit1,3,4], in particular.) We think that despite—or perhaps because of—its conceptual simplicity, this idea has led to one of the most beautiful uses of the heuristic Feynman path integral in contemporary mathematics. In short, it consists in extending from the finite-dimensional to the infinite dimensional situation a general fixed-point formula due to Duistermaat and Heckman [DuiHec] which explains by means of a suitable symmetry principle why in certain situations, the stationary phase (or semiclassical) approximation to a Feynman (or Wiener)-type integral is exact. (See the quote from [At2, p. 43] given at the beginning of the present italicized subsection.) Similarly, when the potential is quadratic (that is, for imaginary or real Gaussian integrals)—and, in particular, for the harmonic oscillator—the method of stationary phase for oscillatory integrals is well-known to yield an exact result rather than a mere approximation. At least for finite-dimensional integrals, the DuistermaatHeckman "symmetry principle" implies that this will be the case whenever the "moment map" associated with "reduced symplectic space" is involved. (Although this terminology is now standard in symplectic geometry [AbMars, §4.2 and §4.3], it will not be needed to understand the following statements; see, however, [DuiHec, AtBot2] and Remark 20.2.12(a). The concepts from symplectic geometry and classical mechanics used below can be found, for example, in [AbMars] and [GuiSbg].) Let N be a compact, symplectic 2d-dimensional manifold, with symplectic form the (closed, nondegenerate) 2-form w and associated Liouville form w d / d ! (defining the analogue of the conserved "phase space volume" via the "Liouville measure" on N). Assume that N is acted upon by the multiplicative group S1, in a way compatible with the symplectic structure of N; that is, with the symplectic form w and the associated Hamiltonian function H. (Recall that in classical mechanics, the Hamiltonian H is a smooth real-valued function on N rather than a self-adjoint operator; see, for instance, [AbMars].) More precisely, this means that the vector field v associated with the S1 -action (i.e. the infinitesimal generator of the "circular symmetry") is mapped onto the 1-form dH; in other words, v is a periodic Hamiltonian vector field (see, for example, [GuiSbg, §IV.6], where a slightly different sign convention is used). (For more detailed information concerning "Hamiltonian systems with symmetry" and their geometric treatment, we also refer the interested reader to Chapter IV of [AbMars].) For the sake of simplicity, we suppose, in addition, that the S 1-action has only isolated fixed-points p, necessarily characterized by integers n j, p (j = 1, . . . , d) corresponding to the d planar rotations describing the circle action on TP(N), the tangent space of N at p. (The sign of these integers has been chosen in a way compatible with the orientation of N.) Then, the Duistermaat—Heckman fixed-point formula [DuiHec] (or exact integration formula) can be stated as follows:
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where the summation is taken over all fixed points of the S1-action. In (20.2.29), the number t can be taken to be in C+ (i.e. Re t > 0, t = 0); the two most important cases being those when t is purely imaginary or when t > 0. When t is purely imaginary, the integral over N appearing on the left-hand side of (20.2.29) is oscillatory and the fixedpoints p are just the critical (or stationary) points of H; i.e. those at which dH(p) = 0. Hence [At2, p. 44], "[equation (20.2.29)] asserts that when the phase function H is the Hamiltonian of a circular symmetry, stationary-phase approximation is exact." Remark 20.2.12 (a) If A e R is not a critical value of H, denote by wA the symplectic form induced by w on the "reduced symplectic space" (or "reduced phase space") NA = H-1 (A)/S 1 , where H-1 (A) is the level set of H of energy A. (See, for example, [AbMars, §4.3].) Then the proof of the "localization formula" (20.2.29) given in [DuiHec] consists essentially in showing that the cohomology class of wA varies linearly in X. Hence, formula (20.2.29) is of a cohomological nature, much in the same way as Cauchy's integral formula from complex analysis. This point is made more transparent in the paper by Atiyah and Bott [AtBot2] where a new derivation of (20.2.29) is given. (b) Several generalizations of (20.2.29) which are of significance for the infinite dimensional analogue discussed below can be considered, as we now briefly explain (see [At2, pp. 44—46]). For example, if we consider a torus (rather than a circle) action on N, then the fixed-point set (typically a union of submanifolds of N) no longer consists of isolated points and formula (20.2.29) becomes (see [AtBot2] or the addendum in [DuiHec]):
where M denotes the components of the fixed-point set, 2k is equal to the codimension of M in N, the integers nj are the rotation numbers normal to M, and the total Chern class of the normal bundle to M is given by Hj=1 (1 + P j ) , where pj is a suitable cohomology class (symbolic 2-form) for j = 1, . . . , k; finally,
Recall that w is a 2-form and hence that wq = Ofor q > d+1, since M is of dimension 2d, (c) Also, w is allowed to be degenerate; for instance, one can assume that wd = 0 on a hypersurface of N. Then the issue of orientability of N becomes important. It turns out [At2, p. 46] that once we have chosen a Riemannian metric g on N, N is orientable if and only if det(wg) has a well-defined and smooth square root, where at a point x e N, wg is the skew-adjoint endomorphism of the tangent space to N at x, TX(N), corresponding to the 2-form w via the isomorphism defined by g. Then, the "Liouville measure" (or "symplectic volume") on N, wd /d!, is related to the Riemannian volume measure on N, Volg, by the following formula:
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where P f ( w g ) denotes the Pfaffian of the skew-symmetric endomorphism wg (expressed in terms of the above square root). We now briefly indicate how (at least heuristically) the Duistermaat-Heckman formula can be adapted and extended to an infinite dimensional context in order to derive the Atiyah-Singer index theorem. Our presentation will be rather sketchy and will be limited to providing some of the main entries of a dictionary between the finite and infinite dimensional situations. We refer to [At2] and [Benn, esp. §1 and §2] for a more detailed exposition of the ideas (and of the mathematical difficulties) involved. We also refer the interested reader to the impressive work of Bismut ([Bisl-9], especially [Bis4-8]) for a rigorous mathematical formalization of several of the heuristic arguments described below, based on infinite dimensional stochastic analysis ([Bis3], [Mal2]); also see [BisFrd]. We note that Bismut makes use of "imaginary time" (rather than "real time") path integrals. Part of Bismut's results also make use of Quillen's work in [Quil,2], for example. We remark that analysis on "loop spaces" has since been developed in several directions by a number of researchers, including Getzler and his collaborators. Further information and references about various aspects of these topics can be found, for instance, in ([At2], [Benn], [Con1,2], [LawM]) and—in the special case when the manifold M introduced below is a Lie group—in [PresSeg]. Let M be a suitable (Riemannian) finite-dimensional manifold, and let N := L(M) denote the loop space of M; that is, the space of all smooth maps y from S1 to M. The (infinite dimensional) "manifold" N is equipped with a Riemannian structure as follows. A "tangent vector" to N at a point y of N is a variation of y, that is to say, a lifting of the loop y to the tangent bundle T(M). Moreover, the inner product of two tangent vectors is given by the integral over S1 of the corresponding scalar product in T(M). Let v be the "vector field" on N generated by the action of S1 on N; i.e. v is generated by the rotations of loops (say, y(0) |-> y(0 + O0), with 6$ fixed). The fixed-points of the action of S1 on N (i.e. the fixed-points of v) are the constant loops, and thus the fixed-point set of v can be identified with the underlying finite dimensional manifold
M C N.
As is suggested by the "nonlinear a-model" from quantum field theory, the Hamiltonian (or "energy operator") H is the infinite dimensional analogue of the Hodgede Rham Laplacian dd* + d*d (suitably deformed) acting on forms. In short, it is the (positive) "Laplacian" on N = L(M). (A precise definition of H can be obtained by combining, in particular, the work of Witten in [Wit4] and the aforementioned techniques from infinite dimensional stochastic analysis; see, for example, [Bis6].) Given a loopy in N = L(M), the "covariant derivative" Vy alongy is a skew-adjoint endomorphism of the "tangent space" TY (N) to N at y. Moreover, it can be thought of as giving rise to a (closed) 2-form w, and hence to a (mildly degenerate) "symplectic structure" on N; see [At2, Lemma 1, p. 47]. (Since the kernel of Vy is finite-dimensional, w is only degenerate along a submanifold of N of finite codimension; compare with Remark 20.2.12(c) above.)
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The Hamiltonian function H on N = L(M) is given by the energy functional (or Dirichlet integral)
which measures the energy of a loop y € N. We assume from now on that the Riemannian manifold M = M2d is oriented, evendimensional (say, of dimension 2d), and equipped with a spin structure (that is, with a lifting of the spin group Spin(2d) to the structure group SO(2d) of M). Then the "manifold" N = L(M) is naturally oriented. Further, let Ty denote the holonomy (or parallel transport) along the loop y. Then the Pfaffian of w (defined as a regularized \ determinant of VK, and equal to (det(I — Ty)) 2, by [At2, Lemma 2, p. 48]) is given by
where S+ and S denote the half-spin representations (i.e. the even and odd representations of Spin(2d), see [LawM]). Thus, by analogy with Remark 20.2.12(c) above, the counterpart in this context of "Liouville measure" on N is provided by
with P f y ( w ) given by (20.2.34) and dy given by an appropriate version of Wiener measure (or rather, of the "Brownian bridge", since we are working here with closed paths on M). Now that we have described some of the relevant entries of a suitable dictionary, we are ready to translate the Duistermaat-Heckman localization formula (20.2.29) into this infinite dimensional context. According to the previous discussion, its left-hand side should be given (in "imaginary time") by
with P f Y ( w ) given by (20.2.34) and H(y) defined by (20.2.33). Much as in the first proof of the index theorem sketched above, this can be viewed as a supersymmetric (or "fermionic") path integral and (after a simple time reparametrization) is given by the supertrace of the heat semigroup e-tH (denoted Tr((— 1 ) F e - t H ) in the physics literature). Actually, in light of Remark 20.2.12(b) above, we must work with the counterpart in this context of formula (20.2.30) (rather than of (20.2.29) itself), an extension of the Duistermaat-Heckman fixed-point formula. Indeed, as we have seen, the fixed-point set of the action of S1 on N = L(M) is given by M C N, a submanifold of N of finite dimension 2d. Thus, in the analogue of (20.2.30), we must replace "Liouville measure" w d / d ! by the expression (20.2.35), with P f y ( w ) given by (20.2.34). Then, a short but "miraculous computation" (see [At2, pp. 54-55], [Benn, p. 78], or Remark 20.2.13(a) below), along with the determination of the "normal bundle" of M in N = L(M), shows
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that the right-hand side of the counterpart of formula (20.2.30) is equal to the Dirac genus of M (also called the Hirzebruch form of M):
from which the Atiyah-Singer index theorem
follows by standard arguments referred to earlier in this subsection. [See especially equation (20.2.26) and Remark 20.2.10(b) above. In (20.2.38), as in the latter remark, D denotes the (elliptic) Dirac operator on M. Moreover, in (20.2.37), the pJ are such that ch(M) = Hj=1 (1 + Pj) is the total Chern class (or Chern character) of M.] Remark 20.2.13 (a) For the intrigued reader, we provide here the intuitive basis for the "miraculous computation" alluded to above and leading to (20.2.37) and hence to (20.2.38). First of all, it is at least plausible that the constant functions on S1 (with values in T(M)) represent the vectors tangent to M in N = L(M). Thus, N(M), the "normal bundle" to M in N, can be identified with the space of Fourier series (with values in T(M) and) with vanishing constant coefficient. Therefore, N ( M ) is given by the following infinite direct sum:
where for each n = 1, 2, . . . , Tn(M) is a copy of TC (M), the complexified tangent bundle of M, on which S1 acts with rotation number n. Next, one notes that the Chern class of Tc(M) is given formally by ITj=i(l + py)(l - Pj). It follows that in the counterpart of (20.2.30), the denominator (appearing on the right-hand side) can be computed heuristically as follows (when t = 1, say, since the left-hand side is independent of t):
where the last "equality" is obtained after regularization (or "renormalization") of a suitable determinant, using the fact that f '(0) = — 5 log(2n), where f denotes the classic Riemann zeta function; see [At2, pp. 48—49 and pp. 54-55]. The end of the argument (see [At2, p. 55]) presents no serious difficulty and one then concludes that the right-hand side of the counterpart of (20.2.30) is equal to A(M), as given by (20.2.37).
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(b) The reader familiar with the computations involved in applying the method of stationary phase to the Feynman integral associated with a quadratic potential (say, for example, in the case of a harmonic oscillator) will perhaps not be surprised by the outcome of the computation referred to above; see the right-hand side of equation (20.2.37) and recall that we have worked above in "imaginary" (rather than in "real") time. (c) The above "proof" is particularly well suited to obtain the analogue of the Atiyah-Singer index theorem for families of elliptic operators. (See [At2], [Bis5,6] and [BisFrd].) (d) The second author (M. L. L.) is indebted to Moshe Flato for proposing as the second subject (necessarily oral) of his These d'Etat es Sciences ([Lal3], June 1986) an exposition of the "supersymmetric proof of the Atiyah-Singer index theorem." At the time of the defense, Getzler's proofs and the beginning of Bismut's work on this topic were still rather recent. (The supervisor of the second author's These d'Etat itself [La13] was Haim Brezis, at the Universite Pierre et Marie Curie, Paris VI.) Deformation quantization: Star products and perturbation series [Here], we briefly present the fertile ground which was needed in order for deformation quantization to develop, even if from an abstract point of view one could have imagined it on the basis of Hamiltonian classical mechanics. Indeed, there are two sides to "deformation quantization." The philosophy underlying the role of deformations in physics has been consistently put forward by Flato since more than 30 years and was eventually expressed by him in [F11] (see also [Fal], [F12]). In short, the passage from one level of physical theory to another, more refined, can be understood (and might even have been predicted) using what mathematicians call deformation theory. For instance, one passes from Newtonian physics to special relativity by deforming the invariance group (the Galilei group SO(3) • R3 • R4). to the Poincare group SO(3, 1) • R4 with deformation parameter c-1, where c is the velocity of light. There are many other examples, among which quantization is perhaps the most seminal. Daniel Sternheimer, 1998, from the introduction to his review article [Ster, §I]
Deformation theory of algebraic structures has proved itself extremely efficient and very much so in the last two decades. . . . [Let us now briefly] present the essence of our deformation philosophy [F11]. Physical theories have their domain of applicability mainly depending on the velocities and distances concerned. But the passage from one domain (of velocities and distances) to another one does not appear in an uncontrolled way. Rather, a new fundamental constant enters the modified formalism and the attached structures (symmetries, observables, states, etc.) deform the initial structure; namely, we have a new structure which in the limit when the new parameter goes to zero coincides with the old formalism. In other words, to detect new formalisms, we have to study deformations of the algebraic structures attached to a given formalism. The only question is in which category we perform this research of deformations. Usually, physics is rather conservative and if we start e.g. with the category of associative or Lie algebras, we tend to deform in this category. However, recently, examples were given in the literature of generalizations of this principle. For instance, quantum groups [Dr] are in fact deformations of Hopf algebras. [See e.g. [GerSck2,3, BnFGP, BnFP].] Other examples include more general
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deformations like in the quantization of Nambu mechanics [see e.g. [DitFST, F1DS]] or in non Abelian deformations [Pinc]. . . . Moshe Flato, 1998 [F12] Quantum mechanics is often distinguished from classical mechanics by a statement to the effect that the observables in quantum mechanics, unlike those in classical mechanics, do not commute with one another. Yet classical mechanics is meant to give a description (with less precision) of the same physical world as is described by quantum mechanics. One mathematical transcription of this correspondence principle is the fact that there should be a family of (associative) algebras Ah depending nicely in some sense upon a real parameter h such that A0 is the algebra of observables for classical mechanics, while AH is the algebra of observables for quantum mechanics. Here, h is the numerical value of Planck's constant when it is expressed in a unit of action characteristic of a class of systems under consideration. (This formulation avoids the paradox that we consider the limit H -> 0 even though Planck's constant is a fixed physical magnitude.) The first order (in h) deviation of the quantum multiplication from the classical one is to be given by the Poisson bracket of classical observables. This idea goes back to Dirac [Dir2], who emphasized the analogies between classical Poisson brackets and quantum commutators. It played an important role in much of Berezin's work ([Berl-5]) on quantization. Although the terminology and much of the inspiration comes from physics, noncommutative deformations of commutative algebras have also played a role of increasing importance in mathematics itself, especially since the advent of quantum groups about 15 years ago. Alan Weinstein, 1995, from the introduction to his June 1994 Bourbaki Seminar on "deformation quantization" [Wei, p. 389] A fundamental question remains [motivated essentially by the founding papers [BayFFLSl,2]]. Is every Poisson manifold deformation quantizable? This question may be broken into the following two parts. Is every Poisson manifold locally deformation quantizable? Is every locally deformation quantizable Poisson manifold globally deformation quantizable? Alan Weinstein, 1995 [Wei, p. 405] In this paper, it is proven that any finite-dimensional Poisson manifold can be canonically quantized (in the sense of deformation quantization). Informally, it means that the set of equivalence classes of associative algebras close to algebras of functions on manifolds is in one-to-one correspondence with the set of equivalence classes of Poisson manifolds modulo diffeomorphisms. This is a corollary of a more general statement, which I proposed around 1993-1994 ("Formality conjecture"); see ([Kon6], [Vor]). For a long time, the Formality conjecture has resisted all approaches. The solution presented here uses in an essential way ideas of string theory. Our formulas can be viewed as a perturbation series for a topological two-dimensional quantum field theory coupled with gravity. Maxim Kontsevich, 1997 [Kon8, §0]
What has now come to be known as "deformation quantization" is a subject lying at the intersection of physics and mathematics, with a number of ramifications in each of those disciplines. It has received a lot of attention over the last few years, culminating with the recent work of Kontsevich [Kon8] on the "quantization of Poisson manifolds". Motivated in part by the ideas and the work of Dirac [Dir2-4], Weyl [Wey], Moyal [Moy] and Berezin [Berl-5] (see also [BerSu]) on quantization, as well as on the more
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purely mathematical side, by the seminal work of Gerstenhaber [Ger1,2] on algebraic deformations and by the paper of Vey [Ve], among others, it has really originated in the late 1970s with the seminal papers by Bayen, Flato, Fronsdal, Lichnerowicz and Sternheimer (see [BayFFLSl] and especially [BayFFLS2]). It is the brainchild of Moshe Flato who has consistently stressed the role played by deformations in physics (and, as a consequence, in mathematics as well) and has articulated it into a 'deformation philosophy' that permeates much of his work and of that of his collaborators or of other researchers on the subject. (See [F11] and the first two quotes heading this subsection.) A vast literature on the subject is now available and still growing, a sample of which can be found in the 1994 Bourbaki Seminar by Alan Weinstein [Wei] as well as in the 1998 survey article by Daniel Sternheimer [Ster], entitled "Deformation quantization: twenty years after." (See also, for example, the earlier review articles [FlSter1,2] for developments of the theory through the 1980s.) A sample of works on deformation quantization since the founding papers [BayFFLSl,2] includes [AngFFS, BiFlFrSo, BiFrHl-3, BnFGP, BnFP, Bou, CGR, ConFlSter, dWLel,2, Del, Dit1-3, DitFST, Feds1,2, Fl1,2, FIDS, FlFr1,2, FlPincSim, FlSimTaf, FlSterl,2, GqrVy, Gui, Kon8, Lich3, NesTsl,2, Oe, OmMY, Pan, Pinc, Riel,2, Ster, Vor, Wei]. We will limit ourselves here to a brief discussion of some of the key concepts involved in "deformation quantization", namely, star products and the associated perturbation series. The interested reader is invited to consult the aforementioned survey articles for a much more complete treatment of this subject and of its physical and mathematical applications, as well as for many additional references. Remark 20.2.14 Actually, it seems that Vey's paper was itself motivated in part by some early articles [FlLichSter1,2] on aspects of the theory that was later developed in [BayFFLS2] (see also [Lich1,2]). In turn, it has played an important role in [BayFFLS2] and in later papers on the subject. Further, Moyal's work in [Moy] only came to the attention of the authors of [BayFFLS2] while they were writing their papers; see, e.g., [FlLichSter3] and [Ster]. However, the results of [Moy] have also played a significant role in the theory. Moreover, another important and closely related motivation for deformation quantization is provided by the symbolic calculi of pseudodifferential operators, including the so-called Weyl calculus; see, for example, [AgWo, And, GrsLouSte, KoNi, Hor1,2, Un],2] and the treatise [Hor3,4]. The latter topic is also related to aspects of Feynman's operational calculus, as was briefly discussed in Chapter 14. (See, for instance, [Masl1], [NazSS] and [Ne3].) Finally, we note that there are several points of contact between deformation quantization and the Kostant-Souriau "geometric quantization" program ([GuiSbg, Chapter V], [Sou], [Wo]). (See [GqrVy, KarMasl1,2, Ster, Wei] and the relevant references therein.) Let M be a finite-dimensional manifold and let A = C°°(M) denote the (commutative) algebra of smooth functions on M, equipped with its natural pointwise product. The type of deformation of A considered in "deformation quantization" involves a family of algebras {Ah}, where for each value of the real parameter h, Ah is an associative algebra such that as a set, An = A, but the pointwise multiplication in A has been replaced by
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a different multiplication, denoted *h and called a star product. (See Weinstein's first quote above.) More specifically, if R[h] (resp., A[[h]]) denotes the ring of formal power series in h with values in K (resp., A), then a star product on A is an associative R[[h]]-linear product on A[[h]] given by the following formula for f, g € A C A[[h]]:
where the scalar h is a formal parameter and for each j > 1, Bj is a "bidifferential operator"; that is, Bj : A x A -> A is a bilinear map which is a differential operator with respect to each argument, of globally bounded order, and vanishing on the constants. The product on general elements of A[[h]] is defined by
Two star products on A are said to be gauge equivalent if they can be deduced from each other via an element of the (gauge) group of automorphisms of A[[h]] of the form
for f e A c A[[h]], where for each j > 1, the linear map Dj : A -> A is a differential operator. (This definition is then extended to all of A[[h]] much as above.) Remark 20.2.15 (a) In certain applications, the deformation parameter H should be replaced by a different one, denoted by v, say. For example, one lets v = 1/c, where c is the speed of light, if one wants to describe the passage from classical mechanics to special relativity. On the other hand, one may choose, for instance, v = R, the scalar curvature of space-time, if one wants to describe the passage from special relativity to general relativity. Naturally, the traditional choice v = h = h/2n, normalized Planck's constant, is used to describe the passage from classical mechanics to quantum mechanics (as well as its converse, when it is possible); see the first three quotes heading this italicized subsection. (We note that one often sets v = ih/2 instead of v = h in this context.) (b) For the simplicity of exposition, we have made above a specific choice for the algebra Ao = A that is being deformed; namely, A = C°°(M), where M is a suitable finite-dimensional manifold. Indeed, in the applications to quantum mechanics, M is often chosen to be a symplectic manifold that plays the role of "phase space" in classical mechanics. (More generally, it can also be a "Poisson manifold", as will be discussed after the end of this remark.) However, in different applications of deformation quantization, various other choices are made for the (typically infinite dimensional) algebra A. (For example, in the study of nonlinear evolution equations, the role of "phase space" is played by an "infinite dimensional symplectic manifold of initial conditions" [Ster, §II.1]; see, e.g., [FlSimTaf] and the relevant references therein.) We note that in
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the algebraic theory originally developed by Murray Gerstenhaber in [Gerl,2], considerable freedom is allowed in the choice of the abstract algebra (or ring) A. (c) A fundamental example of a star product on "phase space" M — R2d is provided by the so-called Moyal star product (defined by means of the Moyal bracket [Moy]), which is associated with a kind of "symplectic Fourier transform." This topic is intimately connected with the notion of Weyl quantization and that of Weyl ordering as extended to pseudodifferential operators. (In addition to [Moy], see, e.g., [FlLichSter4], [Kon8, §1.4.1], [Ster, §I.1] and [Wei].) (d) A key requirement of the above definition of star product is that *h (as given by (20.2.39)) be associative; i.e. f *& (g *h k) = (f *h g) *h k, for all f, g, kin A. In turn, this imposes severe constraints on the bidifferential operators Bj in (20.2.39)-(20.2.40), which play a crucial role in the theory; see, e.g., [Wei, p. 390]. (We note that, in practice, the associativity equation is often interpreted asymptotically, say to begin with to second order in h, and then fully by a "bootstrap" argument. This is the case, for instance, in the construction of star products given in [Feds1] and [Kon8].) (e) If the manifold M is equipped with a Poisson structure (that is, with a Poisson bracket{., •}), we not only require that Bo(f, g) — f . g, but also that the skew-symmetric part of B1 coincides with its Poisson bracket; i.e. for all f, g 6 A = C°°(M),
(Recall that if M = R2d, with coordinates (q1, . . . , qd, p1, . . . , pd) and equipped with its standard symplectic form w = dp A dq, then the ordinary Poisson bracket of two functions f, g in C°°(M) is defined by
More generally, on a symplectic manifold (M, w), a Poisson bracket can be defined similarly in (local) symplectic coordinates; alternatively, it can be defined intrinsically via the symplectic 2-form w; see, for example, [AbMars, Definition 3.3.11 and Corollary 3.3.14, pp. 192-193].) If condition (20.2.42) is satisfied, we let the "commutator" be given heuristically by [f. g]h = (f *h g — g *h f)/2h (interpreted as a formal "limit" as h —> 0). Then we obtain the counterpart of (20.2.39)):
where for each j > 1 , C j is a suitable bidifferential operator. Clearly, formula (20.2.44) can be extended to all of A [ [ h ] ] , much as in (20.2.40). Further, it is R[[h]]-bilinear and satisfies the analogue of the Jacobi identity: [[f, g]h, k]h+ cyclic permutations = 0. (Caution: The notation *h and [•, -]h have a rather different meaning here than in Chapter 18 above.) (f) Following-up on the previous comments (especially in (d) and (e) above), we very briefly mention the cohomological interpretation of deformation quantization.
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The bidifferential operators Bj in (20.2.39) (or in (20.2.40)) can be viewed as Hochschild 2-cochains, whereas their counterpart Cj in (20.2.44) can be viewed as Chevalley 2-cochains. The interplay between these two cohomology theories—namely, Hochschild cohomology of associative algebras and Chevalley (-Eilenberg) cohomology of Lie algebras—plays an important role in this subject; see, for example, [Ster, §I.4]. (See also [FlSter2] and, for the algebraic deformation theory, [Gerl,2] and [GerSck1,2].) (g) We stress that the perturbation series defining the star product *h in (20.2.39) and (20.2.40) (or the Lie bracket [•, .]h in (20.2.44)) is a purely formal power series. In particular, no claim of convergence, in any sense, is implied here; this is why the resulting quantization is sometimes referred to as a "formal quantization." (In contrast, the perturbation expansions—also called "generalized Dyson series"—occurring in Chapters 15—19 above were shown to be convergent in a precise analytical sense.) To a large extent, this may be required by the nature and the breadth of the physical applications being pursued in the theory of deformation quantization. (See, for example, [Ster, esp. §II.3.1.3], where a suitable notion of "cohomological renormalization" is also referred to in this context.) We note that recently, in certain situations relevant to quantum mechanics, a more analytical approach to deformation quantization was studied, in particular, by Rieffel in [Rie1,2]. The work on deformation quantization—a sample of which was given towards the beginning of this subsection—can roughly be divided into the following categories (several of which have overlapping boundaries): (i) Existence and classification of star products, (ii) Quantization and pseudodifferential calculus. Analytic and algebraic index theorems, (iii) Applications to statistical physics, quantum mechanics and field theory. Path integrals and "star exponentials." (See [Ster, §II.3.1. and §II.3.2].) (iv) Applications to the study of nonlinear evolution equations occurring in mathematical physics, by taking into account their underlying symmetries. (v) Quantization of "Nambu mechanics." (The latter topic requires an extension of the notion of deformation quantization; see, e.g., [F12].) (vi) "Singleton theory" and its applications to quantum electrodynamics or, more recently, to string theory. (See Remark 20.2.7(d) above.) We refer the interested reader to the aforementioned survey article [Ster] for an overview of these topics (along with related ones) and for appropriate references. (See also [Wei] and the quotes heading the present subsection.) For now, we will close this short discussion of deformation quantization by briefly describing the work of Maxim Kontsevich [Kon8] on the quantization of Poisson manifolds. As was alluded to above in Remark 20.2.15(e), a Poisson manifold is a natural extension of the notion of symplectic manifold, in which the Poisson bracket is allowed to be degenerate. Geometrically, it can be thought of as a differentiable manifold M that is foliated with symplectic leaves (not necessarily of constant even dimension).
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Analytically, this means that M is equipped with a Poisson structure {•, •}, in the following sense (with A := C°°(M), as above):
is a bilinear map which turns A = C°°(M) into a Lie algebra. Moreover, we must have
for all /, g € A, where a is a nondegenerate section of the second exterior power A 2 (T(M)) which satisfies certain (quadratic differential) constraints; see, e.g., [Kon8] or [Tau6, p. 124]. (We note that conversely, a suitable such a determines a Poisson structure on M.) Then, the problem of quantizing a Poisson manifold (M, {.,.}) consists in determining whether there always exists a star product *h on the algebra A — C°°(M) that is compatible with the given Poisson structure in the sense of Remark 20.2.15(e); i.e. such that equation (20.2.42) holds, with B1 given by (20.2.39). In a beautiful (and very recent) work [Kon8], Kontsevich answers this long-standing question in the affirmative. (See the last two quotes heading the present italicized subsection.) Moreover, he also classifies (up to gauge equivalence, as defined in (20.2.41) above) the set of all star products on a given smooth manifold M (that is, the set of all "differentiable deformations" of the associative algebra A = C°°(M)) by showing that it is in one-to-one correspondence with the set of (equivalence classes modulo diffeomorphisms) of the (null) Poisson structures on M. In particular, he shows that any Poisson structure {•, •} comes from a canonically defined (up to gauge equivalence) star product, and thereby resolves the afore-mentioned "quantization problem." In the model case when the manifold M is equal to R2d (or is an open subset of R 2d ), Kontsevich's answer takes the following striking form:
where a is the section of A 2 (T(M)) associated with the given Poisson structure on M. Here, for n > 1, Gr[n] is a set of (n(n + 1))n labeled oriented graphs F (each with n + 2 vertices and 2n edges). (For n = 0, Gr[n] is composed of a single graph.) Finally, for each F e Gr[n], BF,a is a specific bidifferential operator acting on A = C°°(M) (and with total degree 2n); further, the coefficient cor of the perturbation expansion in (20.2.45) is given by an extremely intriguing formula. The latter involves the integral of a suitable differential form of order 2n over the "moduli space of configurations" [FulMcp] of n distinct points in the Poincare disk. (A suitable version of Stokes' theorem plays an important role in this context. In addition, the formulas defining the numbers wr may have interesting arithmetic interpretations, as is hinted at by Kontsevich in his talks and in various places in his work on this and related subjects [Konl-8].) When M is a general (finite-dimensional) differentiable manifold, then "the" star product *h associated with a given Poisson structure {., •} on M is deduced from the
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above formula by suitable patching techniques; that is, roughly speaking, by "gluing" star products of the type (20.2.45) which are defined on local charts of M. Remark 20.2.16 (a) As is stated in the last quote (from [Kon8, §0]) heading this subsection, the perturbation series appearing on the right-hand side of (20.2.45) is inspired by string theory and can be viewed as being associated with a "topological two-dimensional quantum field theory [TQFT] coupled with gravity." However, it seems that aspects of Kontsevich's formulas were a surprise even to many theoretical physicists working in those areas (and certainly, to most mathematicians as well). (b) As was alluded to at the end of Section 20.2.A in discussing Kontsevich's work on Vassiliev invariants, the perturbation expansion in (20.2.45) can also be thought of as representing a suitable heuristic "Feynman integral". Similarly, the corresponding "Kontsevich graph complex" can be viewed as representing the "Feynman diagrams" of the physically motivated TQFT mentioned in (a) above. (c) The reader may wonder why it is important to be able to quantize an arbitrary Poisson manifold rather than merely an arbitrary symplectic manifold. In short, from a physical point of view, it is because one wants to be able to quantize nonautonomous systems. [See, for example, [Ster, §I.2], where are briefly discussed Dirac's so-called first (resp., second) constraints [Dir4, FlLichSter3], associated with Poisson (resp., symplectic) manifolds. Also see [Wei] for natural mathematical motivations.] (d) We note that the existence of a star product on an arbitrary symplectic (rather than Poisson) manifold was established earlier by De Wilde and Lecomte (see, e.g., [dWLe1,2]). Another quite different construction was provided in that setting by Fedosov (see, e.g., [Feds1,2] and earlier works quoted therein). (See also [OmMY].) A bridge between the approaches used in [dWLel,2] and [Feds1,2] has recently been provided by Deligne in [Del]. (Other references and further information concerning the existence and the classification of star products on symplectic manifolds can be found in [Kon8], [Ster, §II.2.1], as well as in [Wei].) (e) The second author would like to thank Maxim Zyskin—who was then visiting him at the University of California, Riverside—for giving in his Mathematical Physics and Dynamical Systems Seminar in April-May 1998 a stimulating series of lectures on Kontsevich's work on deformation quantization. We refer the interested reader to Kontsevich's own paper [Kon8] (along with its forthcoming sequels) for a precise statement of the results briefly discussed above, as well as for further relevant information and extensions. For an exposition of some of these results from different points of view, we also refer to [Ster, §II.2.2], [Tau6, §4, pp. 124-125], and to the recent Bourbaki seminar by Oesterle [Oe] about Kontsevich's results on this subject. As is pointed out in [Ster], the latter results are referred to in [Oe] as "the cherry on the cake" (or perhaps "the frosting on the cake") of deformation quantization. Remark 20.2.17 Sadly, part of the present subsection was written while the second author (M. L. L.) was in Tel Aviv, Israel, in order to attend Moshe Flato's funeral. Indeed, tragically, Moshe Flato passed away suddenly towards the end of November
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1998. (Unfortunately, a few weeks later, his mentor, Andre Lichnerowicz—who considered Moshe as his "spiritual son"—also passed away.) In the early 1980s, at UCLA, a long friendship and intellectual kinship began when Andre Lichnerowicz introduced the second author to Moshe Flato and to his long-time collaborator and friend, Daniel Sternheimer. Over the years, it has resulted in numerous passionate and fruitful conversations about a wide variety of topics, including, of course, mathematics and physics along with what Moshe liked to call "mathematique physique." In closing, the second author would like to dedicate this brief introduction to deformation quantization to the memory of Professor Moshe Flato, a brilliant scholar as well as an outstanding, generous and compassionate human being. Gauge field theory and Feynman path integrals The final form of the principle of least action [given by Hamilton] is extremely general and finds a fundamental place in a large number of physical theories. Moreover, Hamilton's principle changed the style of doing calculations in theoretical physics. All modern field theories, like classical and quantum electrodynamics, theory of gravitation, quantum chromodynamics, theory of electroweak interactions, and theories of supergravity and superstrings, were derived from the corresponding straightforward generalization of Hamilton's principle for systems with infinitely many degrees of freedom. Today the principal problem of every new fundamental theory is how to find the corresponding Lagrangian. This principle is very useful also for calculations in many practical problems. Thus we see that Hamilton's principle of least action looks like a fundamental law of nature and there exists the great fundamental question as to what lies behind this principle, what its meaning is, and what makes its large number of applications possible. The physicists before P. A. M. Dirac and Richard Feynman were not able to answer this question because the principle of least action is perhaps the most basic principle of classical physics, but its full significance lies in quantum physics; one can find it only in Feynman's path-integral formulation of quantum mechanics. Jagdish Mehra, 1994 [Me, p. 129] The path-integral formulation is ideally suited to discuss symmetry. If classical physics possesses a symmetry, then the action is invariant under certain symmetry transformations, as we learned [earlier]. It follows that since the same action controls quantum physics, quantum physics possesses the same symmetries as classical physics. For this and other reasons, over the last ten to fifteen years the path-integral formulation has largely supplanted the older wave-mechanical and matrix formulations in discussions of fundamental physics. In the path-integral formulation, the essence of quantum physics may be summarized with two fundamental rules: (1). The classical action determines the probability amplitude for a specific chain of events to occur, and (2) the probability that either one or the other chain of events occurs is determined by the probability amplitudes corresponding to the two chains of events. Finding these rules represents a stunning achievement by the founders of quantum physics. The mental processes involved can only be described as quantum leaps of genius. The law of the quantum is not so much a theory in itself, but a prescription to obtain a theory relevant in the realm of the quantum. One obtains quantum mechanics by applying the prescription to Newton's theory of mechanics, quantum electrodynamics by applying it to Maxwell's theory of electromagnetism, quantum gravity by applying it to Einstein's theory of gravity. But the action of quantum electrodynamics is still Maxwell's action, with all its symmetries. A. Zee, 1989 [Zee, pp. 143-144]
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Gauge field theories now provide the most commonly accepted theoretical basis for elementary particle physics. The quantization procedure in these theories is usually carried out via Feynman-type "path" integrals. Roughly, the integration is over all gauge fields (or rather, potentials) appropriate to the particles under consideration. More precisely, one integrates over equivalence classes of gauge potentials. This is handled in practice by choosing a representative which satisfies an appropriate gauge condition from each equivalence class. For example, in quantum electrodynamics (QED), the most fully developed (quantum) gauge field theory, it is the Coulomb gauge which is used and one chooses from each equivalence class the vector field with divergence zero [Pop, p. 48]. Remark 20.2.18 (a) In the more geometric language [Atl, EgGh, Law] used in Section 20.2.A, the gauge potentials are represented by connections on a G-principal bundle over a manifold M (a model of "space-time" equipped with a Lorentzian metric in the relativistic version, or else with a Riemannian metric in the Euclidean version), where G (the "gauge group") is a suitable compact Lie group. Further, the Feynman "integration" is performed over the "moduli space" of connections modulo gauge transformations. (Both the geometric formulation of Yang-Mills gauge theories and the corresponding Feynman integral approach will be discussed in more detail in Remark 20.2.20 below.) (b) In addition, in the Yang-Mills gauge field theories that form the basis for the so-called "standard model" of elementary particle physics, the gauge group G is taken to be U ( 1 ) = S U ( 1 ) = S1 (the multiplicative group of complex numbers of modulus one) for describing Maxwell's electromagnetism at the classical level [as well as quantum electrodynamics (QED) at the quantum level], SU(2) for describing the weak force, U(1) x SU(2) for describing the electro-weak force, and SU(3) to represent the strong (nuclear) force in quantum chromodynamics (QCD). Finally, in the standard model of particle physics, one chooses G = U ( 1 ) x SU(2) x SU(3) to describe the unification of the electromagnetic, weak, and strong forces—which are the three basic known forces apart from the gravitational force (described by Einstein's general relativity). (See, for example, [ChengLi, EgGH, HennTe, We1,2].) We note that the electro-weak interaction, the unification of electromagnetism and of the weak force in the Salam-Weinberg model, was definitely confirmed experimentally in 1983 and 1984 with the discovery of the bosons carrying the weak force, namely the W+, W- and Z° particles, within the theoretically predicted range of masses. (c) The path integral in "phase space" (i.e. involving position and momentum coordinates) is sometimes considered in the physics literature on gauge field theories; see, for example, [FaSl, Pop]. In contrast, we have worked throughout this book in "configuration space" (i.e. involving just position coordinates); see especially Chapters 6, 7, 11, 13, and 15-19. It is usually perturbation series and associated Feynman diagrams that are used in the most fully developed part of these gauge field theories. The first term of the action is associated with the free evolution and is typically assumed to be a quadratic form. Such a form gives rise to Gaussian integrals but with an i in the exponent instead of a real number. The remaining part of the action is taken to be the perturber. Whether
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or not the theory is "renormalizable" becomes an important issue. (For instance, the phenomenal theoretical and experimental success of quantum electrodynamics [Fey 10, Feyl5, Schwe3, Schwi] is often attributed to the fact that QED is "renormalizable", due to the relative smallness of its "coupling constant". On the other hand, the use of perturbative methods in quantum chromodynamics is much more problematic due to the larger size of the coupling constant involved.) Of course, in the present situation, one should not expect the perturbation series or even the individual terms of the series to make sense in a straightforward way. (Recall Manin's quote given at the beginning of Section 20.2.B. As was alluded to above, however, the situation is much better for QED than for QCD.) We should mention that even when Feynman-type integrals are not explicitly used in quantum field theories, Feynman's operational calculus, especially time-ordered operators and perturbation series, play a central role (see, for example, [MandlSha]). The Feynman diagrams in these quantum field theories (whether or not they are gauge theories) are significantly more complicated than in our book in that they come from a much greater variety of physical phenomena. However, from another standpoint, they are less complicated in that, for example, only very simple discrete measures are involved. Remark 20.2.19 One key advantage of the Feynman path integral approach to gauge field theories (and other physical theories) is the fact that it naturally enables us to take into account the underlying symmetries (here, the space-time and especially the gauge or "internal" symmetries) and, in particular, to relate the symmetries of the classical and the quantum theories (when they both exist) via the use of the action functional. We have already discussed in a more general context one aspect of this point in Chapter 7 (at the very end of Section 7.2), and, in the present situation, we will further elaborate on it in Remark 20.2.20. [For a more physical discussion, see also Zee's quote given at the beginning of the present italicized subsection and excerpted from his interesting book for a general public entitled Fearful Symmetry, as well as (for a more elaborate discussion and sometimes contrasting views) [Mani3] and [Mani1], especially Chapter 5.] The main references used in connection with the brief remarks above are [FaSl, MandlSha, Pop, Ram]. Some further relevant physical references are [ChengLi, Col, Fey 10, HennTe, ItzZu, Mi, Poly3, We 1,2]. For a more geometric and mathematical approach to gauge field theory from various points of view, see, for example, [Atl, At4, AtBotl, AtHit, Chob, Chob-dWm, EgGH, FrdUh, JaTau, Law, Mani2, Naka, RaSinW]. Papers of a probabilistic nature relevant to aspects of relativistic gauge field theory include [ArnPay, ChebKMasl, DriHa, GaSchu, GaV, SchrTayl,2]; see also the relevant references in Albeverio's survey article [A1] and in Malliavin's treatise on infinite dimensional stochastic analysis [Mal2]. An excellent nontechnical introduction to the variety of Feynman diagrams occurring in quantum field theory and in gauge field theory, in particular (especially in quantum electrodynamics), is provided in Mattuck's book [Mat] (now reprinted as a Dover Classic). A very readable (and completely nonmathematical) introduction to quantum electrodynamics is given in Feynman's aforementioned UCLA lectures for a general scientific public, published in 1985 under the title QED: The Strange Theory of Light and
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Matter [Fey15].Although the phrase "path integral" is probably never used explicitly in the text of [Fey15],the method of the Feynman path integral and its underlying basic physical principles permeate much of the book. Finally, in addition to Schweber's book [Schwe3] on QED already mentioned in Chapter 14, we refer to Weinberg's beautiful book for a general audience, entitled Dreams of a Final Theory [We3], where a fascinating and nontechnical account of the genesis of quantum electrodynamics and of the more general electroweak gauge field theory can be found. (See especially pages 107-116 and 116-131 in Chapter V of [We3].) Remark 20.2.20 It may be helpful for some readers to summarize (albeit succinctly) some of the basic elements of the Feynman path integral approach to (Yang-Mills) gauge field theory, as formulated in a geometric language, for example, in [At7, §1.2]. As was mentioned earlier, in the geometric formulation of Yang-Mills gauge field theory (see, for example, [Atl, At 4,5, EgGH, Law]), we are given a gauge group G (a compact Lie group) and a suitable (finite-dimensional, oriented smooth) manifold M. [In the relativistic theory, M represents "space-time", a Lorentzian manifold. In particular, in the physical applications to quantum electrodynamics, electroweak theory and quantum chromodynamics, M is a four-dimensional (oriented) manifold, usually taken to be fourdimensional Minkowski space-time; that is, R4 equipped with the standard Lorentzian metric of signature+, —, —, —, and given by ds2 = (dx°)2 — (dx1)2 — (dx2)2 — (dx3)2, where{xu}3u=0denote the canonical coordinates. We refer back to Remark 20.2.18(b) above for the choice of the gauge group G in the aforementioned physical gauge field theories.] Then, a gauge potential is represented by a connection A for a G-principal bundle over M (assumed here to be trivial); thus A is a g-valued 1-form over M, where g is the Lie algebra of G. Moreover, the associated gauge field (also called "field strength" in the literature) is given by the curvature of A, denoted F = FA; namely, F = FA is the g-valued 2-form over M given (in local coordinates) by its components:
where the subscripts u and v run from 0 to 3 and [ A u , Av] denotes the Lie bracket (commutator) of the g-valued (matrix-valued, say) components of A. [More intrinsically, FA can be written as the sum of the covariant derivative of A and of the Lie bracket of A with itself, [A, A]. Note that in (20.2.46), the bracket is usually nonzero (except when the gauge group is abelian, as in Maxwell's theory or in QED).] In Yang-Mills' gauge field theory, one assumes in addition that F = FA satisfies the so-called Yang-Mills equations (in vacuum):
where d denotes exterior differentiation and d* denotes the (formal) adjoint of d. These equations reduce to Maxwell's equations (without sources) written in geometric form, in the case of Maxwell's theory of electromagnetism; recall that we then have G — U(1), the Abelian group of complex numbers of absolute value 1.
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All physical quantities (i.e. "observables") are invariant under the group of gauge transformations (roughly speaking, a group of suitable maps from M to G). This is the case, in particular, of the (Yang-Mills) action functional S = S(A), given by
where "dx" denotes the natural volume measure on M and the norm of FA is defined by means of an invariant metric on G. Then, the (unnormalized) partition function of the theory is given by the (heuristic) Feynman path (or rather, functional) integral
where "DA" now denotes the (fictitious) "Feynman measure" and "integration" is performed over the moduli space of all connections modulo the group of gauge transformations. More generally, given any (suitable) functional F = F(A), the (unnormalized) expectation value of the "observable" F is given by the Feynman path integral
(Naturally, formula (20.2.49) above is the analogue of (20.2.6) in the present context, while (20.2.50) is the counterpart of formula (20.2.8) from Section 20.2.A. One striking difference, however, consists in the choice of the action in Yang-Mills and ChernSimons gauge field theories, respectively; compare equations (20.2.48) and (20.2.7), respectively.) We note that the normalized expectation value of' F would be defined by Z-1 feiS(A) F(A)DA, where Z is given by (20.2.49). By construction, the Feynman path integral approach to gauge field theory is relativistically invariant (and respects the symmetries of the underlying theory, as was recalled in Remark 20.2.19). This is one of the key advantages of this approach. We note that in the (mathematically more tractable) nonrelativistic (or Hamiltonian) approach to gauge field theories or to more general quantum field theories, M is chosen to be a (three-dimensional) Riemannian manifold (say M = R3, the ordinary Euclidean 3-space), and the time evolution governing the system is given by a unitary group {e-itH: t E R}, the infinitesimal generator of which is called the Hamiltonian of the theory and is traditionally denoted by H. One can then relate Feynman's path integral formulation and the Hamiltonian approach via formulas closely analogous to those encountered in nonrelativistic quantum mechanics discussed in Chapters 6 and 7 (as well as in much of the rest of this book); see, for instance, [At7, p. 4]. (The only formal difference now is that we are dealing with a theory with an infinite number of degrees of freedom, and hence with "fields" on R3 rather than with ordinary geometric paths in R3. This seemingly innocent change, however, is one of the main reasons why it is so difficult to provide a rigorous formulation of Yang-Mills gauge field theory—based on a true
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bona fide Feynman functional integral—in situations of physical interest, even in the case of quantum electrodynamics.) We now close this subsection with some comments relating, in particular, aspects of Section 20.2.A with the present material on gauge theory and Feynman integrals. Remark 20.2.21 (a) As is well known, Donaldson's theory of topological invariants for four-manifolds relies in part on concepts and techniques from (non-Abelian) YangMills gauge field theory. (See [Don1-2, At5] and especially the book by Donaldson and Kronheimer [DonKr], entitled The Geometry of Four Manifolds, along with the very interesting CBMS Lectures by Lawson [Law], which also discusses the beautiful analytical and geometrical contributions by Clifford Taubes and by Karen Uhlenbeck, among others. Although Donaldson's original theory did not make explicit use of functional integrals, Widen—thereby answering a question raised by Atiyah in [At5]—constructed in [Wit12] a 3 + 1 (= three space + one time dimensions) topological quantum field theory expressed in terms of a heuristic Feynman path integral, as was briefly discussed in Section 20.2.A (see Remark 20.2.6(a) above). (b) Even more recently, an entirely new perspective on these aspects of Donaldson's theory based on gauge theory has been provided by the physically motivated work of Seiberg and Witten ([SeiWit1,2], [Wit18]) on "N = 2 supersymmetric (fourdimensional) Yang-Mills theory". The latter has spurred a number of mathematical developments (see, for example, [Taul-5, Kon3—4, Kot]) on this and other geometrical and topological subjects (including the so-called "Gromov-Witten invariants", see [Gro, Wit9] along with Remark 20.2.6(b) above) and is closely related to one of the topics to be briefly discussed in the next italicized subsection; namely, dualities in string theory. [As is noted, for example, in [Witl9, p. 157], the "N = 4 super Yang-Mills theory can arise as a low energy limit of string theory (via toroidal compactification of the heterotic string)", and the so-called "electric-magnetic dualities" that play a crucial role in [SeiWit1,2, Wit18,19] can be viewed as an analogue of "T-duality" from string theory, which itself includes at least formally, "mirror symmetry"; see, for example, [AspMorr, Giv1-4, Kon3,5, KonMani, LiaLYau, LiaYau, Morr, Witl6, Yau2] and the relevant references therein.] (c) In Section 20.2.A, we have discussed at some length Witten's definition [Witl4] (and generalization) of the Jones polynomial invariant via a heuristic Feynman path integral of the type (20.2.6) and more generally (20.2.8), in the case of knots and links, respectively. (See equations (20.2.49) and (20.2.50), respectively, with the action functional S = S(A) given by (20.2.7) rather than by (20.2.48); see also (20.2.1) and (20.2.1.').) The gauge theory used in Witten's formulation of his topological invariant is a "Chern-Simons gauge field theory", associated with a 2 + 1 (= two space + one time) topological quantum field theory. [Recall from Remark 20.2.6(c) that in contrast to the Yang-Mills gauge theories, the topological quantum field theories referred to here or in (a) just above have trivial dynamics; that is, the corresponding Hamiltonian (see the end of Remark 20.2.20 above) is identically zero: H = O.] (d) In certain recent approaches to "quantum gravity"—motivated in part by physical work of Ashtekar and his collaborators, Rovelli, Smolin and others, but quite different in nature and scope from those based on string theory discussed in the next
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subsection—aspects of Chern-Simons gauge theory and the associated Feynman functional integral also play an important role. (Recall from (c) above and Section 20.2.A that intriguing and intimate connections exist between knot theory and Chern-Simons theory.) In particular, polynomial invariants (or rather, the closely related Kauffman bracket or polynomial [Kau6,8]) can be viewed as a "state" of this version of quantum gravity. (Just as in Section 20.2.A, however, despite several recent attempts, the rigorous definition of the Feynman integrals involved remain problematic in this context.) See, for example, [As, AsLwMMT, Bae, BaeSaw, RovSm1,2, Sm] and the relevant references therein for a sample of physical or mathematical works on this subject. (e) The earlier and more traditional approaches to "quantum gravity" have most commonly been based on certain types of Feynman functional integrals. See, for example, [GibHaw, Haw] and the relevant references therein. Based on ideas of Feynman and Wheeler [Whe] pursued in particular by Hawking, in the Feynman "path" integral formulation to traditional quantum gravity, the "integration" is carried out over all metrics on space-time M (or rather, over the moduli space of all equivalence classes of metrics modulo the group of diffeomorphisms of M). As was alluded to in Remark 20.2.4(g), this "averaging over all metrics" guarantees the "general covariance" of the underlying theory. We stress that the corresponding "classical virtual paths" are then the solutions to Einstein's equations for general relativity. In the Feynman "path" integral involved, the action functional is usually chosen to be the so-called "Einstein-Hilbert action". Further, in the Euclidean version of quantum gravity, integration is carried over Riemannian (rather than Lorentzian) metrics on M. Also, a suitable analytic continuation of Feynman integrals in a physical parameter (including "time") plays an important role in this approach to quantum gravity. (See, for example, [Haw, GibHaw].) (f) The authors would like to thank Sylvie Paycha for her comments on certain aspects of this material. String theory, Feynman-Polyakov integrals, and dualities I have tried to make it plausible that path integrals on Riemann surfaces can be used to formulate a generalization of general relativity. What is more, the resulting generalization is (especially in its supersymmetric forms) free of the ailments that plague quantum general relativity. If the logic has seemed a bit thin, it is at least in part because almost all we know in string theory is a trial and error construction of a perturbative expansion. [The Feynman-Polyakov path integrals over moduli space of Riemann surfaces with a given genus and a given number of marked points] are probably the most beautiful formulas that we now know of in string theory, yet these formulas are merely a perturbative expansion . . . of some underlying structure. Uncovering that structure is a vital problem if ever there was one. Edward Witten, 1987, concluding his lecture at the 1986 International Congress of Mathematicians in Berkeley [Wit7, p. 302] Let me first try in one paragraph to summarize the state of knowledge of physics. (For a more extensive account, see the beginning of my article [Wit7].) Gravitation is described at the classical level by general relativity, which is based on Riemannian geometry. Straightforward attempts at extending general relativity to a quantum theory have always led to extremely severe difficulties.
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Other observed forces are described by a quantum gauge theory (the "standard model"), whose construction involves (in addition to the machinery of quantum field theory) the choice of a Yang-Mills gauge group (SU(3) x SU(2) x U ( 1 ) encompasses the known interactions); a representation of that group for charged fermions . . . ; and a relatively little understood mechanism of symmetry breaking. The main unsolved problems are generally considered to be to overcome the inconsistency between gravity and quantum mechanics; to unify the various other forces with each other and with gravity; and to understand symmetry breaking and the vanishing of the cosmological constant. In the early 1980s, it became clear—through the work of M. B. Green, J. H. Schwarz, and L. Brink, building on pioneering contributions of others from the 1970s—that string theory offered a framework (in my view the only promising framework known) for overcoming the inconsistency between gravity and quantum mechanics. Actually, that is a serious understatement. It is not just that in string theory, unlike previous frameworks of physical theory, quantum gravity is possible; rather, the existence of gravity is an unavoidable prediction of string theory. . . . Then in 1984, . . . the gauge group and fermion representation of the standard model suddenly emerged rather naturally from the theory. On a more theoretical side, supersymmetry (or bose-fermi symmetry; supergeometry) is another general prediction of string theory. World-sheet supersymmetry was invented by P. Ramond in 1970 to incorporate fermions in string theory; fermions exist in nature, so this was necessary to make string theory more realistic. Space-time supersymmetry was invented by J. Wess and B. Zumino in 1974 based on an analogy with world-sheet supersymmetry. Ever since then, supersymmetry has fascinated physicists, especially in connection to the little-understood symmetrybreaking mechanism of the standard model. Supersymmetry is not an established experimental fact, though a possible partial explanation for the measured values of the strong, weak, and electromagnetic coupling constants based on supersymmetry has attracted much interest. There is an active search for more direct experimental confirmation of supersymmetry at high-energy accelerators . . . . The main immediate obstacle to progress in extracting more detailed experimental predictions from string theory (beyond generalities such as the existence of gravity) would appear to be that the vanishing of the cosmological constant is not understood theoretically. More fundamentally, I believe that the main obstacle is that the core geometrical ideas—which must underlie string theory the way Riemannian geometry underlies general relativity—have not yet been unearthed. [Italics added.] At best we have been able to scratch the surface and uncover things that will most probably eventually be seen as spinoffs of the more central ideas. The search for these more central ideas is a "mathematical" problem which at present preoccupies primarily physicists. Some of the spinoffs have, however, attracted mathematical interest in different areas. Edward Witten, 1994, excerpts from [Witl7, pp. 205-206] Because string theories incorporate gravitons and a host of other particles, they provided for the first time the basis for a possible final theory. Indeed, because a graviton seems to be an unavoidable feature of any string theory, one can say that string theory explains why gravitation exists. Edward Witten, who later became a leading string theorist, had learned of this aspect of string theories in 1982 from a review article by the CalTech theorist John Schwarz and called this insight the "greatest intellectual thrill of my life." String theories also seem to have solved the problem of infinities that had plagued all earlier quantum theories of gravitation. Although a string may look like a point particle, the most important thing about them is that they are not points but extended objects. The infinities in ordinary quantum
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field theories can be traced to the fact that the fields describe point particles. (E.g., the inversesquare law gives an infinite force when we put two electrons at the same position.) On the other hand, properly formulated string theories seem to be free of any infinities. Steven Weinberg, 1992 [We3, p. 216] Each string theory also has its own internal symmetries, of the same general sort as the internal symmetries that underlie our present standard model of weak, electromagnetic, and strong forces. But a major difference between the string theories and all earlier theories is that the space-time and internal symmetries are not put in by hand; they are mathematical consequences of the particular way that the rules of quantum mechanics (and the conformal symmetries thus required) are satisfied in each particular string theory. String theories therefore potentially represent a major step toward a rational explanation of nature. They may also be the richest mathematically consistent theories compatible with the principles of quantum mechanics and in particular the only such theories that include anything like gravitation. Steven Weinberg, 1992, [We3, p. 218]
In string theory, a point particle is replaced by a string—moving in a d-dimensional manifold M, viewed as "space-time". Mathematically, a "string" can be thought of as a parameterized curve in M; more precisely, in the standard terminology adopted in the theory, an "open string" has (possibly) different endpoints whereas a "closed string" is a closed curve (or loop) in M. [In the bosonic open string theory, the dimension d is equal to 26, while in the physically more interesting supersymmetric closed string theory (involving both bosonic and fermionic variables), d is equal to 10. Mixed versions of these string theories—involving both closed and open strings (heterotic strings), for example—exist and are also very useful. Further, in the simplest models, M is taken to be Rd (in the Euclidean version) or d-dimensional Minkowski space-time (in the relativistic theory).] Hence, what was the path (or "world-line") of a point particle in ordinary quantum mechanics now becomes a "world-sheet"; that is, the two-dimensional surface swept by the string as time evolves. Instead of "integrating over all paths", as in the usual Feynman path integral discussed in Chapter 7 (and treated in much of the rest of this book), one must now integrate over all "world-sheets" (most often interpreted mathematically as Riemann surfaces). Consequently, the space P of "virtual paths", in (1) of the introduction to Section 20.2, just above (20.2.1), must now be interpreted mathematically as a "moduli space of Riemann surfaces". [See, for example, [GreSWit], [Wit7] or [Mani3]. Further see the paper by Beilinson and Manin [BeMani] for rigorous work along these lines, and in particular, for the construction of a suitable measure on this moduli space, called the "Polyakov measure". The paper [BeMani] deals with the free case (i.e. with noninteracting strings). When strings are allowed to interact, the corresponding moduli space involves marked Riemann surfaces and so "vertex operators" [Kac-V] must be inserted at the marked points in order to describe the resulting interactions; see, for example, [Wit7] or [GreSWit] for more information.] Actually, the situation is somewhat more complicated than that. Indeed, in the notation of (2) (just above formula (20.2.1)), the action functional S = S(A) is not only a function of the world-sheet £ (a compact Riemann surface), but also depends on the choice of a suitable metric on E as well as on
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the space-time M; in general, it may also depend on various (bosonic and/or fermionic) fields. Then, in the corresponding Feynman-type path integral (often called a " Polyakov integral", a "Feynman integral", or else a "Feynman-Polyakov integral" in the literature), the "integration" must also be carried out over several of these additional data. A common choice for the action functional S is the so-called "Polyakov action" [Poly 1,2]. A detailed discussion of this action and of the associated Feynman-Polyakov integral would take us too far afield and so we prefer instead to refer the interested reader to the relevant physical literature on string theory. (Some aspects of the Feynman-type integral in this context are briefly discussed in Remark 20.2.22 below, however.) See, for example, [Wit7, §4], the treatise [GreSWit], the survey article [Polcl], [Polc3, Vol. I, Chapter 3], or Polyakov's own book [Poly3], entitled Gauge Fields and Strings. (Also, see [Kak, Part I].) Remark 20.2.22 We mention that the Feynman-Polyakov integrals involved can be viewed as path integrals over moduli spaces of Riemann surfaces (of a given genus) with a finite number of marked points, say n. (In this remark, we will closely follow aspects of the discussion provided in [Wit7, §4].) More precisely, according to [Poly1], the probability of scattering of n interacting particles of a given type (specified by the embedding of a Riemann surface £ in space-time) is given by a suitable integral over the "moduli space of configurations" of n (distinct) points on a suitable compactification £ of E, assumed to be of genus 0 and hence identified with the Riemann sphere. See formula (131) in [Wit7, p. 301]), which is one of the two "most beautiful formulas" of (perturbative) string theory referred to in the first quote from [Wit7] heading this italicized subsection. (Writing down the precise form of the integrand would require introducing some background from conformal field theory, which is provided, for instance, in [Wit7, §3].) Moreover, when quantum corrections to the above formula are taken into account, say corrections of order H8 (with h = h/2n, the normalized Planck constant, and g a nonnegative integer), then the corresponding formula involves an integral over the moduli space Mn,g of Riemann surfaces of genus g with n marked points; see formula (132) in [Wit7, p. 302], which is the second "beautiful formula" referred to in the abovementioned quote. (The closed Riemann surface E—which was assumed to be of genus 0—has now been replaced by a Riemann surface of genus g.) In summary, as is explained in [Wit7], the "classical answer"—corresponding to H = 0—for the "scattering amplitude" of n particles is given by the first formula (involving an integral over the moduli space Mn of n marked points on the Riemann sphere), while its "quantum corrections" of order hg (g = 1, 2 , . . . ) are given by the second formula (involving an integral over the moduli space M n , g ). We therefore obtain a perturbation expansion indexed by the nonnegative integer g. Finally, we note that in the above discussion, the nonnegative integer n was kept fixed, but that if we also let it vary, then we obtain a perturbation expansion indexed by g (the "genus") as well as by n (the "number of interactions"). Each "(n, g)-term" of the resulting perturbation series can be kept track of by a Feynman-type diagram, as is alluded to below, and corresponds to a path integral over the moduli space M n ,g.
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The method of stationary phase (i.e. "the" classical limit) and the perturbation expansions, as in (5) and (6) (just below formula (20.2.1)), still play a crucial role in this context (at least in the so-called "perturbative string theory" of the 1980s and the early 1990s). The "classical limit" takes various forms depending on which physical theory (for example, general relativity, quantum mechanics, or gauge field theory) one wishes to recover from string theory; see, for instance, [Wit7], [GreSWit] or [Witl9-21]. Moreover, the perturbation expansions associated with the heuristic "Feynman integral" now involve Feynman-type diagrams (often called "string diagrams"). (As was briefly discussed in Remark 20.2.22 above, these diagrams are keeping track of suitable path integrals over the moduli spaces Mn,g of Riemann surfaces of genus g and with n marked points, for n, g = 0, 1, 2 See, for example, [GreSWit] and [Wit7].) Surprisingly at first, the form and the combinatorial structure of these Feynman diagrams are significantly simpler than their counterpart in ordinary quantum field theory. [See [Wit7] and the first few chapters of Volume I of [GreSWit] for a detailed explanation of this key fact as well as for many examples of "string diagrams" describing the interactions between several strings; see also, for instance, [Kak, esp. §1.1 and §1.2] and the second paragraph of the quote from [We3] at the beginning of this subsection. In a nutshell, we simply mention here that because strings are "extended objects" rather than point-like, the associated Feynman (or string) diagrams keeping track of their interactions do not have singularities, unlike their counterpart in ordinary quantum field theories which do have singularities concentrated at certain points (rather than being smeared along "tubes").] Due in large part to the special structure of the Feynman diagrams, the divergences that plagued the previous attempts at quantizing general relativity miraculously disappear in (supersymmetric) string theory. In short, it is widely believed that string theory is "renormalizable" (more precisely, "finite to all orders in perturbative theory"), whereas earlier attempts at quantizing gravity (that is, general relativity) were definitely not. (See, for example, the first three quotes heading this italicized subsection for even stronger statements.) This fact—fully realized in the early 1980s and often referred to as having given birth to the first "superstring revolution"—gave hope to a number of theoretical physicists that they had finally caught a glimpse of a theory potentially able to successfully reconcile and even unify quantum mechanics and general relativity. (Needless to say, even if such a theory of fundamental interactions is ever completed, many basic problems will remain to be solved in other areas of physics.) Whether or not this vastly ambitious attempt at unifying the four known forces of nature—namely, electromagnetism, the weak force, the strong force, all described by Yang-Mills gauge field theories, and the gravitational force, described by Einstein's general relativity—will be successful remains to be seen. [In particular, it has been notoriously difficult to make predictions in string theory that could be experimentally verifiable with our present-day technologies or at least in the foreseeable future. Moreover, on a more theoretical level, one has yet to discover the basic geometry underlying string theory; see, for example, [Wit7, We3, GreSWit, Wit20-22], as well as the end of the second quote (from [Witl7]) heading this subsection.]
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However, skipping a number of intermediate steps, we wish to close this discussion by mentioning the recent spectacular developments of nonperturbative string theory. As its name indicates, the latter no longer relies on perturbation series as suitable substitutes for (or complements to) the path integrals involved. Nevertheless, it still makes an essential use of Feynman-type functional integrals. The various physical observables can be "computed" in several situations by exploiting new "dualities" (including, for example, the so-called 'T-duality" and "S-duality") relating the basic types of string theories which have emerged over the last twenty years. (The discovery of these dualities, along with their afore-mentioned applications, is now referred to as the "second superstring revolution" [Schwc2].) It is believed by some theoretical physicists that the resulting theory—often called "M-theory" in the literature—may be the long sought after "unified field theory". [It does seem to possess certain fascinating features, which are not yet fully understood. In particular, strings (or one-dimensional submanifolds) are no longer the only replacements for point particles; indeed, nonperturbative string theory involves "p-branes" (as well as "D-branes" or "Dirichlet branes"), which correspond, in particular, to (possibly) higher-dimensional submanifolds moving in M; for instance, a 2-brane (a two-dimensional "membrane") sweeps a "world-volume". For a simple introduction to this subject and the associated notions of duality, see [Duf] or [Wit20-22], and for a more sophisticated but still accessible introduction, see, for example, the survey articles [GivePR, Polc 1,2, Schwc2-4, V1,2]. Also see the groundbreaking papers [SeiWitl,2] and [Wit18-19, 21] for the significance of a closely related notion of duality, called "electric-magnetic duality."] Only time will tell whether this dream will eventually be concretely realized. What is already clear, however, is that these developments, both in perturbative and nonperturbative string theories—in which the use of heuristic Feynman-type path integrals has played a significant role—have begun to strongly influence large parts of mathematics and will surely continue to do so for a number of years into the twenty-first century. With some confidence, the same can be said about many of the other mathematical or physical developments (connected to the heuristic use of Feynman integrals) discussed in this section (i.e. in Section 20.2). Remark 20.2.23 (a) The language of Feynman-Polyakov path integrals still plays a prominent role in nonperturbative string theory. (See, for example, [Polc3, Vol. II], [Quev], and the relevant references therein.) However, it seems that to understand at a more fundamental level the dualities occurring in the resulting "M-theory", one may have to go beyond the usual concept of functional integral [V3]. (b) It is interesting and instructive to read what a famous physicist like Steven Weinberg (the co-discoverer with Sheldon Glashow and Abdus Salam of the electroweak theory) writes about the status of quantum mechanics in a hypothetical future "final theory" of the laws of nature. In [We3, pp. 88-89], he states, in particular: ... This theoretical failure to find a plausible alternative to quantum mechanics, even more than the precise experimental verification of linearity, suggests to me that quantum mechanics is the way it is because any small change in quantum mechanics would lead to logical absurdities. If
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this is true, quantum mechanics may be a permanent part of physics. Indeed, quantum mechanics may survive not merely as an approximation to a deeper truth, in the way that Newton's theory of gravitation survives as an approximation to Einstein's general theory of relativity, but as a precisely valid feature of the final theory.
In sharp contrast, Weinberg discusses as follows in [We3, pp. 215-216] the status of the quantum field theories most directly relevant to theoretical physics, namely the electroweak gauge field theory and quantum chromodynamics which comprise the standard model of elementary particles: ... From this point of view a quantum field theory like the standard model is a low-energy approximation to a fundamental theory that is not a theory of fields at all, but a theory of strings. We now think that such quantum field theories work as well as they do at the energies accessible in modem accelerators not because nature is ultimately described by a quantum field theory but because any theory that satisfies the requirements of quantum mechanics and special relativity looks like a quantum field theory at sufficiently low energy. Increasingly we regard the standard model as an effective field theory, the adjective "effective" serving to remind us that such theories are only low-energy approximations to a very different theory, perhaps a string theory. The standard model has been at the core of modern physics, but this shift in attitude toward quantum field theory may mark the beginning of a new, postmodern, era in physics.
This is not to say, of course, that quantum field theories do not and will not continue to play a crucial role in physics or in mathematics (see, for example, Weinberg's own treatise on the subject [We1,2], as well as Witten's Gibbs lecture [Wit22] and much of Witten's work briefly discussed in this chapter), but the above quotes suggest that at least on a fundamental level, quantum mechanics may occupy a more "permanent" place than its more elaborate counterpart, quantum field theory. (For completeness, we mention that Weinberg's book Dreams of a Final Theory [We3] was written before the advent of "M-theory ", although it is not clear whether these recent developments would have affected his two statements cited just above.) In our last comment, we provide some additional information and references regarding string theory. Remark 20.2.24 A brief introduction to string theory from a mathematician's viewpoint is provided in Manin's Mathematical Intelligencer article [Mani3], previously referred to in Chapter 7. A more physical approach is given in Witten's 1986 International Congress address [Wit7]. In addition, a vast amount of information on the subject can be found in the 1987 treatise by M. B. Green, J. H. Schwarz and E. Witten [GreSWit], entitled Superstring Theory, as well as in the more elementary textbook by Kaku [Kak], entitled Introduction to Superstrings, and in the edited volumes [GreG, Schwcl]. (In particular, the introductory chapters of Volume I of [GreSWit] are still relatively accessible to a nonexpert or to a mathematician and provide an excellent account of some of the main features of string theory through the mid-1980s.) We also refer the interested reader to the aforementioned book by Polyakov [Poly3] for a somewhat different but enlightening perspective on the subject.
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Other (more recent) references include, for example, [Wit20-22, Duf] as well as the survey articles on dualities and M-theory [GivePR, Polc2, Schwc2—4, V1,2]—which all stress the new developments in string theory that occurred in the 1990s (and that were very briefly discussed above). We point out that a very clear exposition of string theory through the mid-1990s is provided in Polchinski's article, entitled What is string theory? [Polc1]. It includes a review of perturbative as well as of nonperturbative string theory, and ends in particular with an introduction to "matrix models" and "M-theory." Moreover, we mention that Polchinski [Polc3] completed in 1998 a new two-volume treatise, entitled String Theory, which should be a very welcome addition to the existing literature and may in fact become an instant classic. In most of these references, (heuristic) Feynman-type path (or functional) integrals play a prominent role. Actually, it may be worth pointing out that the first part of Kaku's graduate textbook on superstrings [Kak] is entitled First Quantization and Path Integrals, and that it contains, in particular, an introduction to the Feynman path integral approach to ordinary quantum mechanics [Kak, Chapter 1]. Interestingly, Chapter 1 of [Kak] ends with the following statement (see [Kak, p. 47]): "We will shortly see the advantage of carefully working out the details of point particle [Feynman] path integrals. We will find that almost all of this formalism carries over directly into the string formalism!" Finally, we note that a mathematical (and especially, probabilistic) introduction to some aspects of string theory is given in [AlJosPS], while a course for mathematicians on Quantum Fields and Strings has just appeared in [Del-Et]. In addition, we point out that a nontechnical account of string theory directed to a general scientific public has appeared in early 1999. It is entitled The Elegant Universe [Gr] and is written by Brian Greene, an active contributor to string theory. What lies ahead? Towards a geometrization of Feynman path integrals? String theory carries the seeds of a basic change in our ideas about spacetime and in other fundamental notions of physics. Edward Witten, 1996 [Wit20, p. 24] It would hardly be possible to construct a consistent and applicable mathematical theory of Feynman integrals without progress in the understanding of physics. The very idea of "quantization" belongs not to physics, but to the history and psychology of science. There can only be a substantive meaning to "dequantization", i.e., the passage from a quantum description to a classical one, when the latter is meaningful; the converse, however, can never have substantive meaning. The classical fields occurring in the Lagrangians of the weak and strong interactions, for example, are physical phantoms: we do not know their meaning outside of quantization, and it is unlikely that they would describe the virtual classical histories of anything whatsoever. (It is believed that the situation is better with the quantization of electromagnetism.) Yu I. Manin, 1979 [Manil, p. 94]
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We certainly do not pretend to know the answer to the above questions. We only wish to offer in conclusion of this section some possible scenarios, proposed by researchers who have devoted a great deal of thought to these or related subjects. At a fundamental level, it can be safely ascertained that we are very far from having reached a mathematical understanding of ordinary quantum field theories (let alone the more involved string theories briefly discussed above); indeed, this was one of the main themes of the 1998 Gibbs lecture given by Edward Witten [Wit22]. As far as quantum field theories are concerned, our lack of understanding is manifested technically by the tremendous difficulties associated with the divergences occasioned by the coincidence of points in space-time. There seems to be an emerging consensus among theoretical and mathematical physicists that the actual remedy to such problems is to significantly revisit the notion of space-time, both from a geometrical and physical point of view. (Perhaps a suitable extension or modification of Connes' powerful "noncommutative geometry" [Conl,2] along with the aforementioned perturbative and nonperturbative approaches to string theory will someday provide a clue as to how to proceed.) In its most extreme form, this has led to the even more radical suggestion to do away with the notion of space-time altogether (at least as a primary concept). (See, for example, Witten's article [Wit20]— entitled "Reflections on the fate of spacetime", and from which the first quote heading this italicized subsection is excerpted—as well as the two citations from [Manil] given just above and below.) A closely related—but less commonly stated—belief is that the formidable difficulties encountered in attempting to make rigorous the heuristic Feynman path integrals occurring in all aspects of theoretical physics (and in the present section) are also connected to our lack of geometrical and physical understanding of the notion of "spacetime". The mathematician Yuri Manin is perhaps the most articulate proponent of this view. He wrote beautiful texts on this theme (and related subjects) in his book [Manil], entitled Mathematics and Physics. (See especially Chapter 5 and, in particular, pages 92-94, in [Manil].) We close by quoting one additional excerpt from Manin's book [Manil, p. 94]: Finite dimensional quantum models allow us to guess which features of the Feynman formulation are essential, and which are atavistic. As was explained [earlier], the evolution operator for a closed localized quantum system at its local time t has the form eis(t), where S(t) is now an operator with the dimension of action. If we imagine the different world-lines of the system with different local times, we see that quantum action is a connection in the space of internal degrees of freedom of the system, determining physically admissible histories as parallel translations. The recipes for quantization are a primitive manifestation of the fact that the space of internal degrees of freedom "at a single point" in vacuo is already infinite dimensional because of the virtual generation of particles. Further understanding is blocked until we relinquish the idea of space-time as the basis for all of physics.
Whatever lies ahead in the future of this subject, if the recent developments briefly discussed throughout this section are any indication, it seems likely that a suitable geometrization (yet to be discovered) of some notion of Feynman path integral will play a significant role.
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[W2] [Wi1] [Wi2] [Wi3] [Wig] [Wi1] [Wils] [Wit1] [Wit2] [Wit3] [Wit4] [Wit5] [Wit6] [Wit7]
[Wit8] [Wit9] [Wit10]
[Wit11] [Witl2] [Witl3]
[Witl4] [Wit15]
743
Widom, H., A complete symbolic calculus for pseudodifferential operators. Bull. Soc. Math. 104 (1980), 19-63. Wiener, N., Differential space. J. of Math. and Phys. (of MIT) 2 (1923), 131-174. (Reprinted in [Wi3], pp. 455-498.) Wiener, N., The average value of a functional. Proc. London Math. Soc. 22 (1924), 454-467. (Reprinted in [Wi3], pp. 499-512.) Wiener, N., Norbert Wiener: Collected Works. Vol. I (ed. P. Masani). MIT Press, Cambridge, 1976. Wiegel, F. W., Introduction to Path-Integral Methods in Physics and Polymer Science. World Scientific, Singapore, 1986. Williams, D., Diffusions, Markov Processes and Martingales. Vol. I. Wiley, New York, 1979. Wilson, K. G., Renormalization group and critical phenomena, I and II. Phys. Rev. B 4 (1971), 3174-3183 and 3184-3205. Witten, E., Supersymmetric form of the non-linear a -model in two dimensions. Phys. Rev. D 16 (1977), 2991-2994. Witten, E., An SU(2) anomaly. Phys. Lett. B 117 (1982), 324-328. Witten, E., Constraints on supersymmetry breaking. Nuclear Phys. B 202 (1982), 253-316. Witten, E., Supersymmetry and Morse Theory. J Differential Geom. 17 (1982), 661-692. Witten, E., Non-commutative geometry and string field theory. Nuclear Phys. B 268 (1986), 253-294. Witten, E., Interacting field theory of open superstrings. Nuclear Phys. B 276 (1986), 291-324. Witten, E., Physics and geometry. In: Proc. Internat. Congress Math., Berkeley, 1986(ed. A. M. Gleason). Vol. 1,pp. 267-303. Amer. Math. Soc., Providence, 1987. Witten, E., Elliptic genera and quantum field theory. Commun. Math. Phys. 109 (1987), 525-536. Witten, E., Topological sigma models. Commun. Math. Phys. 118 (1988), 411-449. Witten, E., The index of the Dirac operator in loop space. In: Elliptic Curves and Modular Forms in Algebraic Topology (eds. Landweber, et al.). Lecture Notes in Math., Vol. 1326, pp. 161-180. Springer-Verlag, Berlin and New York, 1988. Witten, E., 2 + 1 dimensional gravity as an exactly soluble system. Nuclear Phys. B 311 (1988/89), 46-78. Witten, E., Topological quantum field theory. Commun. Math. Phys. 117 (1988), 353-386. Witten, E., Some geometrical applications of quantum field theory. In: Proc. IXth Internat. Congress Math. Phys., Swansea, Wales, July 1988 (eds. B. Simon, A. Truman and I. M. Davies), pp. 77-116. Adam Hilger, New York, 1989. Witten, E., Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121 (1989), 351-399. Witten, E., Gauge theories, vertex models, and quantum groups. Nuclear Phys. B 330 (1990), 285-346.
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[Wit16]
[Witl7] [Wit18] [Witl9]
[Wit20] [Wit21 ] [Wit22]
[Wit23] [Wit24]
[Wo]
[Ya] [Yau1]
[Yau2] [Yeh] [Yo] [Zag] [Zam] [Zas1]
[Zas2] [Zee] [Zum]
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INDEX OF SYMBOLS Entries in boldface type refer to definitions
A A* (Hilbert adjoint of the (possibly unbounded) operator A) 144 A' (Banach adjoint of the operator A) 147 A (closure of the operator A) 133 A+, A_ (positive, negative part of the self-adjoint operator A) 189 \A\ (absolute value of the self-adjoint (or normal) operator A) 189 A 1 / 2 (square root of the nonnegative self-adjoint operator A) 190 A^2 = (A±) l/2 190, 222 A IP (restriction of the operator A to the subspace D) 171 Af (•) (positive continuous additive functional or PCAF [in Section 13.7 only]) 360, 365 A(M) (Dirac genus (or Hirzebruch form) of a spin manifold M) 667, 673 a = ( a 1 , . . . , ad) (magnetic vector potential on Rd)242 Ai (the disentangling algebra at time t > 0) 452-53 {At}t>0 (the family of disentangling algebras) 530 [a, b] (closed interval from a to b) (a, b) (open interval) [a, b), (a, b] (half-open intervals) A C B (the operator B is an extension of A) 144 A + B (the algebraic sum (or operator sum) of the operators A and B) 190 A+B (the form sum of the operators A and B) 192 No(-nomial formula) 441 B Br(x) (open ball of center x and radius r) 33, 41, _ 253 Br(x) (closed ball of center x and radius r) 41 BUC(R) (bounded uniformly continuous functions on R) 134 B(T) (Borel class of the topological space T (o-algebra of the Borel subsets of T)) 32
C card (cardinality of a finite set) 592 C(X) (continuous functions from X to R (or C))
C([a, b], X) (continuous functions from [a, b]
to X) C a,b = C([a, b], Rd) 32 Cb = C0,b = C([0, b], Rd) 32 Ca,b (continuous paths from [a, b] to Rd which vanish at a) 32 C0 = Cp'' Wiener space (t > 0) 274, 410 C0(R) (continuous functions on R vanishing at infinity) 134 C°°(Q) (infinitely differentiable functions on the open set Q C Rd) C00(Q) = D(Q) (infinitely differentiable functions with compact support on the open set Q C Rd (test functions)) 265, 344 C°° = C°°(Rd) (infinitely differentiable functions on Rd) C00 = D = D ( R d ) 205 C1 ([a, b], X) (continuously differentiable functions from [a, b] to X) 127 (Co) ((Co) (or strongly continuous) semigroup) 124 Cap(A) (capacity of the set A C Rd) 312, 362 Cov (X, Y) (covariance of two random variables X and Y) 47 C (complex numbers) 79, 407 C+ (complex numbers with positive real part) _ 79, 163, 407 C+ (complex numbers with nonnegative real _ part) 163 C+ = C+\{0} (nonzero complex numbers with nonnegative real part) 407 D
dF, d*F (differential and co-differential of the differential form F) 685 det(T) (Fredholm determinant of the operator T) 630
D+, D~ (right, left derivative) 130 Da
746
INDEX OF SYMBOLS
D = D(R d )(= C00(R d )) 205 V = D(Rd) 205 D (Dirac operator) 667 Sr (Dirac measure with unit mass concentrated at {T}) 405 d
A = E 32/3xt (Laplacian (or Laplace k=1 operator)) 162-63, 206 An = A n (r) (simplex) 423, 487 Ak;j = Ak;j (t)417, 487 A k (p) 429 Aq0;j1 jh+l 435, 495 SQ (boundary of the open set Q) 123 V (operator del) 207 V/ (gradient of the (scalar-valued) function or distribution /) 207, 243 V.a = div a (divergence of the vector-valued function a) 207, 330 V x a = curl a (rotational of the vector-valued function a) 244 (V - i a)2 243-44 E E(X) (expectation (or average) value of a random variable X) 46–7, 275 n (measure on the time-interval [0, t]) 408-9 (and Chapters 14-18) 1n1 (total variation measure associated with n) 408 \\n\\ (total variation of n) 408 F ¥ = ¥A (curvature of the connection A [in Section 20.2.B] only) 685 F (Wiener functional) 80, 410, 452 £ (time-reversal of F (F e A)) 456 F (complex conjugate of F) 456 F* (time-reversal of the complex conjugate of F)458-59 F+G (noncommutative sum of the Wiener functionals F and G) 536, 548 F * G (noncommutative product of F and G) 536, 548 FX (distribution function of the random variable X) 49 Fx1 xn (joint distribution function of the n random variables X 1 , . . . , Xn) 49 F (resp., .F-1) (Fourier (resp., inverse Fourier) transform operator) 161, 162 Ff (resp., F - 1 f ) (Fourier (resp., inverse Fourier) transform of the function or of the tempered distribution /) 161, 165
F(.) (Fresnel integral [in Section 20.1.A only]) 611 T (functional [or "observable"] [in Section 20.2 only]) 638-39, 686 (F) (expectation value of the observable T [in Section 20.2 only]) 638, 686 FTP(V) (Feynman integral via TPF (Trotter product formula) associated with the potential V) 219-20 P^(V) (modified Feynman integral associated with the potential V) 237 FTP, an(V) (analytic-in-mass Feynman integral via TPF associated with V) 326-27 Fm, an (V) (analytic-in-mass modified Feynman integral associated with V) 335-37 f+ = max(f, 0) (positive part of the (R-valued) function /) 112, 189, 193 f_ = max(-f, 0) (negative part of the (R-valued) function /) 112, 189, 193 |/| (absolute value of the (R- or C-valued) function f) 189 / (complex conjugate of the (C-valued) function /) 456 ||f|| (norm of f) f * g (convolution of the functions f and g) 165 G GKd (generalized Kato class (of Rd)) 361 G(x, y; t) (free Green's function) 174-75 G n (x0,x1,... ,xn,t; V) 258 Q(A) (graph of the operator A) 131 Q n (X0, X 1 , . . . , x n ; t) (a certain convolution kernel) 234 H h (Planck's constant (sometimes also used as an index without this meaning)) 97 h (Planck's constant divided by 2n (quantum of action)) 97 H (Hamiltonian (or energy) operator (much of the time, H = H0 + V, HO + V or H0+V))95, 111, 195, 218, 235, 243-44, 665 H (Hamiltonian function (or energy functional) in classical mechanics [in Section 20.2.B only]) 669-72 H-> (Hamiltonian with electrostatic scalar potential V and magnetic vector potential a ) 243-44 H0 free Hamiltonian (H0 = -1 A) 162-63 H1 (Q) (first Sobolev space on the open set Q C Rd) 264, 325
INDEX OF SYMBOLS
H0 (Q) (first Sobolev space on Q c Rd, with "zero boundary conditions" (H0 (Rd) = H1 ( R d ) ) ) 264, 325 2 H (Q) (second Sobolev space on Q) 163 H1, H0,H 2 (Q = Rd) K (a Hilbert space) 144, 176, 247 I i = V-1 (the square-root of -1 (never used for any other purpose)) 2 / (identity operator) 124 [I + itH0]-1 (free imaginary resolvent) 174-75 [I + itA]~[ (imaginary resolvent of the self-adjoint (or normal) operator A) 221-22, 261 /B+, Ig- (the positive, negative part of a certain / ' Hermitian operator) 222 'JU (the square-root of IBr±) 222 I (t1,...,tn; (a1,B1]x . . . x (an, Bn]) = I->((a1, B l ] x . . . x (an,Bn]) (an interval (or cylinder set) in Wiener space) 36 X (semi-algebra of intervals in Wiener space) 36 Im (A) (or Im A) (imaginary part of the complex number A) 226 Im (/) (or Im /) (imaginary part of the C-valued function /) 335 ind(T) (Maslov index of the operator T) 630 Index(L) (index of the Fredholm operator L) 661 J /-»(£) (a certain cylinder set in Wiener space) ' 42 /' (F) (analytic-in-time operator-valued Feynman integral of the (Feynman-Kac functional) F) 299 J (duality map (of a Banach space)) 141 K Kd(Kato class (of Rd)) 193 K[(F) (A > 0) (operator-valued analyticin-mass function space integral (of the Wiener functional F)) 310, 410 K[(F) (A e C~) (operator-valued analytic-in-mass function space integral (of the Wiener functional F)) 310, 410 Kt(F) (A = 0 purely imaginary) (analytic(in-mass) operator-valued Feynman integral of F); in Section 13.5: 310-11 in Chapters 14-18: 410 Ker(A) (kernel of the operator A) 171 K (knot (or link) [in Section 20.2.A only]) 639-40, 642
747
L LP(Q) (p-th power integrable functions on Q (Lebesgue space, 1 < p < oo); norm of Zf(Q) denoted either \ \ - \ \ p or ||.||lp (Q) )339 L 2 (Q) (square-integrable functions on Q (in a quantum-mechanical context, L 2 (Q) always consists of complex-valued functions)) 263, 325, 339 d LP = L f ( R ) 112, 219, 236, 245 L L = L foc(Rd) (functions locally in Lp) 194 (Lloc)u (functions locally uniformly in Lp) 193-94 (££c)u (functions strongly uniformly in Lp) 193-94 L? + L°° = Lp(R d ) + L°°(Rd) (space of sums of a function in Lp and of a function in L00) 112 (L^x.(Q))d (Rd-valued functions on Q, with components in L p x (Q)) 338 2 L (u) (square-integrable functions on a measure space (Q, A, u)) 153 £<x>i;u (mixed norm space (with norm denoted ||.||o0];/,))409
£(X, Y) (bounded linear operators from X to Y) 133 L ( X ) = £(X, X) (bounded linear operators on X) 124 Leb. (Lebesgue measure on Rd) 44 Leb.-a.e. (Lebesgue almost everywhere) 274 Leb (Rn) (a-algebra of Lebesgue measurable sets on Rn) l.i.m. (limit in the mean (in the sense of the Fourier-Plancherel theory)) 112, 169 M m (Wiener measure [on Wiener space (C°-b, B(Cg-h)) and, in particular, on (C'0, B(CJ))]) 40, 274 ma (scaled Wiener measure (with parameter a > 0) (m = mi))68 M f (operator of multiplication by the function /) 153 M(0, t) (space of measures on (0, f)) 408 mesh(P) (the mesh of a partition P) 516 N NA (Avogadro's number) 28 N(0, t) (normal distribution (with mean 0 and variance t)) 29 N = {0, 1,2,...} (natural numbers (nonnegative integers)) N* = {1, 2, 3,... ((positive integers) 554
748
INDEX OF SYMBOLS
n! (factorial) 423 n!!85 vf, v* (inner, outer measure associated with v) 92
Qa67
O
p Prob(-) probability of an event) 28 0 e-'v^i(ds) ((strong) time-ordered product '"•" integral of the (time-dependent) operator -i V(s) with respect to the oo _ measure n) 514-17 f[ e-iV(s)r,(di) ((strong) improper time-ordered ~°° product integral of — i V(s) with respect to n) 520-21
Q qn (quadratic form associated with an operator A) 179 G(A)(= Q ( q A ) ) (form domain of the (self-adjoint) operator A) 179 Quad.Var.(x) (quadratic variation of the Wiener path x) 67 R R(\) = R(\\ A) = (A. - A)'1 (resolvent of the operator A) 125, 139 KB, (a certain (nonnegative) Hermitian operator) 222
Rgt (the square root of RB, ) 222 R (real numbers) 36 Rd (d-dimensional Euclidean space (norm of £ e Rd denoted ||e||)) 32 R([a, b], X) (Riemann integrable functions from [a, b] to X) 127 p(A) (resolvent set of the operator A) 138 Ran(A) (range of the operator A) 171, 224 Re (A) (or Re A) (real part of the complex number A.) 79, 227 Re (/) (or Re /) (real part of the C-valued function /) 322 S S = S(-) (action functional (or integral)) 2, 101, 102, 106, 638 S = S(A) (action of the gauge potential A [in Section 20.2 only]) Chem-Simons 643 Yang-Mills 686 SA(J) ((Co) semigroup with generator the m-accretive operator A) 201 Sd (Stummel class of functions on Rd ) 215
S1 (= SU(1)) (unit circle (group of complex numbers of modulus one)) 354-55, 649-50 S" (unit sphere of R"+1) 352-55, 649-50 SO(N) (special orthogonal group (in N dimensions)) 651, 655 SU(N) (special unitary group (in N dimensions)) 651, 655 so(N) (Lie algebra of SO(N)) 655 su(N) (Lie algebra of SU(N)) 655 5 = 5(Rd) (Schwartz space (of rapidly decreasing functions on R d )) 165 5' = S'(Rd) (space of tempered distributions on Rd (dual of 5)) 165 s-a.e. (scale-invariant almost everywhere) 77 s-lim (strong (operator) limit) 202, 220 Support(u) (support of the measure u) 180 Support(/) or Supp(/) (support of the function /) 256 sgn u (signum of the function u) 206 o(A) (spectrum of the operator A) 139 a (I) (a -algebra generated by the semi-algebra I) 41 T Tv (distribution associated with the locally integrable function u) 206 {T(0)}, {T(t)},>0, {T(t) :t>0) (semigroup of (bounded linear) operators) 124, 127, 200 T(V(si)V(si)) (time-ordering symbol) 515 Te~'f" VMiW (time-ordered product integral) 515 tr(') (trace of a matrix or of an operator) 644 Tr((-l) f )) or Trace((-l)F-) (supertrace [in Section 20.2.B only]) 665 T (the time-reversal map) 456 8 (can be thought of as a "potential" in Chapters 14-18) 409 9 = — V and A = 1: (diffusion (or probabilistic) case) 409, 413 8 — —iV and A. = — i: (quantum-mechanical (or Feynman) case) 409, 413 9(s) (operator of multiplication by the function 6(s, 0)410 p^>,tj 508
u M(T+) ((strong) right-hand limit at r of the operator-valued function u = u(t)) 482 V V (potential) 95 Vm,+ , Vm,- (positive, negative part of Vm) 254
vJ,± = (V m ,±)2 254-55
INDEX OF SYMBOLS
V (interaction (or Dirac) representation of the time-dependent potential V) 512 Var(X) (variance of the random variable X) 29, 47 Vjc (Jones polynomial of the knot (or link) 1C [in Section 20.2.A only]) 639-41
W w (Wiener process) 91 W(M) (Witten's invariant for the 3-manifold M [in Section 20.2.A only]) 643 W(M: K.) (Witten's knot (or link) invariant [in Section 20.2.A only]) 644 W,,x 224
Wa(t,U)36 X X* (dual of the Banach space X) 141 xa (monomial (a = multi-index)) 165 XE (characteristic function of a subset £) 157, 159
Other notation: s-lim (strong operator limit) 202, 220 —> (convergence (strong convergence)) 249 —> (weak convergence) of vectors in a Hilbert space 226 of measures 464, 474-75
749
|| • || (norm of a vector in Euclidean space Rd and of a vector in a Banach or a Hilbert space) (•, •) (inner product on a Hilbert space) 220 (•, •> (duality bracket (or pairing)) 142 O (unknot (unknotted circle)) 640 O (end of a proof) Conventions used throughout the book: f f(u)t)(du) (the integral of a function / with respect to a measure n) 414 / F(x)dm(x) (the integral of a Wiener functional F with respect to Wiener measure m = m1) 80, 274, 410 (u\ x ••• x u k ) ( d s j , . . . , dsk) = x* = ,/i u (rfi u ) (product of the measures M l , . . . , Mi) (If/ii = ••• = /**=/*, we write ij.®k instead of fj. x • • • x £t.) 416 Acronyms:
AWS (abstract Wiener space) 624 FKLS (Feynman-Kac formula with a Lebesgue-Stieltjes measure) 477 GDS (generalized Dyson series) 404 homfly (knot polynomial invariant) 651, 655 PCAF (positive continuous additive functional) 365 QCD (quantum chromodynamics) 683 QED (quantum electrodynamics) 393, 683 TPF (Trotter product formula) 219-20 TQFT (topological quantum field theory) 653
AUTHOR INDEX A Ahn, J. M., Johnson, G. W. and Skoug, D. L. 632-36, 697 Albeverio, S. 684, 697 Albeverio, S. and Brzezniak, Z. 106, 613, 616, 631-32, 697 Albeverio, S. and Hoegh-Krohn, R. 8, 101, 106, 610,613-15, 619, 697-98 Albeverio, S., Johnson, G. W. and Ma, Z. M. 237, 298, 358-59, 367-68, 698 Albeverio, S. and Ma, Z. M. 359, 698 Albeverio, S. and Schafer, J. 614, 649-50, 698 Albeverio, S. and Sengupta, A. 649-50, 698 Alexander, J. W. 640, 641 Altschuler, D. and Friedel, L. 658, 698 Alvarez-Gaume, L. 659-60, 663-68, 698 Araki, H. 403, 699 Arnold, V. I. 656, 699 Ashtekar, A. 687-88, 699 Atiyah, M. F. 641-42, 646, 649-51, 653-54, 660, 668-73, 685-86, 699 Atiyah, M. F. and Bott, R. 670, 699 Atiyah, M. F., Bott, R. and Patodi, V. K. 663, 667-68, 699 Atiyah, M. F., Patodi, V. K. and Singer, I. M. 668, 700 Atiyah, M. F. and Singer, I. M. 659-74, 700 B Babbitt, D. G. 266, 294, 700 Bachelier, L. J. B. A. 28, 700 Badrikian, A., Johnson, G. W. and Yoo, II 582, 700 Bar-Natan, D. 656, 658, 701 Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A. and Sternheimer, D. 675-76, 701 Beilinson, A. A. and Manin, Yu I. 690, 701 Belavin, A. A., Polyakov, A. M. and Zamolodchikov, A. B. 646, 701 Berezin, F. A. 667-68, 675, 701-2 Berline, N., Getzler, E. and Vergne, M. 661, 702 Berline, N. and Vergne, M. 661, 702 Billingsley, P. 465, 702 Birman, J. and Lin, X.-S. 656, 702 Bismut, J.-M. 661, 671, 674, 702-3 Bivar-Weinholtz, A. 265, 297, 703 Bivar-Weinholtz, A. and Lapidus, M. L. 259-66, 295,297, 324, 331, 334-36, 340-41, 703
Bivar-Weinholtz, A. and Piraux, R. 297, 309, 330-31, 703 Blanchard, Ph., Combe, Ph., Sirugue, M. and Sirugue-Collin, M. 616, 703 Blanchard, Ph. and Ma, Z. M. 359, 360, 703 Blanchard, Ph. and Sirugue, M. 616, 703 Bohr, N. 1 Booss, B. and Bleecker, D. D. 663, 704 Bost, J.-B. and Connes, A. 357-58, 704 Bott, R. See Atiyah, M. F. See also Atiyah, M. F. (and Patodi, V. K.) Bott, R. and Taubes, C. 658, 704 Bratteli, O. and Robinson, D. W. 119, 704 Brezis, H. 204, 674, 704 Brezis, H. and Kato, T. 264, 297, 330, 333, 338-39, 704 Brezis, H. and Pazy, A. 204, 704 Brink, L. See Green, M. B. (and Schwarz, J. H.) Bromley, C. See Kallianpur. G. Brown, R. 17, 24-5
C Cameron, R. H. 12, 58, 67, 74-5, 83, 294, 308, 705 Cameron, R. H. and Martin, W. T. 45, 67, 72, 76, 615, 621, 630, 705 Cameron, R. H. and Storvick, D. A. 413, 455, 483, 616-21, 705-6 Carleson, L. 325, 706 Case, K. M. 354-55, 706 Cavalieri, B. 99, 111 Chang, K. S., Johnson, G. W. and Skoug, D. L. 621-23, 706-7 Chang, K. S., Lim, J. A. and Ryu, K. S. 373, 707 Chang, K. S. and Ryu, K. S. 60, 707 Chargoy, J. 616, 707 Chebotarev, A. M. 616, 707 Chebotarev, A. M., Konstantinov, A. A. and Maslov, V. P. 616, 707 Chebotarev, A. M. and Maslov, V. P. 616, 707 Chern, S.-S. and Simons, J. 642-43, 708 Chernoff, P. R. 125-26, 150-51, 197-204, 228, 230, 232, 269-70, 323, 707 Choquet, G. 355, 363, 708 Choquet, G. and Meyer, P. A. 355, 708 Combe, Ph. 616, 708-9
AUTHOR INDEX
See also Blanchard, Ph. (Sirugue, M. and Sirugue-Collin, M.) Combe, Ph., Hoegh-Krohn, R., Rodriguez, R., Sirugue, M. and Sirugue-Collin, M. 616, 708 Combe, Ph., Sirugue, M. and Sirugue-Collin, M. 616,709 Connes, A. 560, 696, 709 See also Bost, J.-B. Conway,J. H. 641, 709 D Davies, E. B. 130, 709 DeFacio, B., Johnson, G. W. and Lapidus, M. L. 7, 395-96, 562, 568, 582, 585, 587, 595-96, 604-5, 709 Deligne, P. 681, 709 De Wilde, M. and Lecomte, P. B. A. 681, 710 DeWitt, B. S. 668, 710 Dirac, P. A. M. 102, 104-5, 152, 675, 682, 710 Dollard, J. D. and Friedman, C. N. 514, 516, 710-11 Donaldson, S. K. 650, 687, 711 Donaldson, S. K. and Kronheimer, P. B. 650, 687, 711 Doob.J. L. 92, 363, 711 Dudley, R.M. 465, 711 Duistermaat, J. J. and Heckman, G. J. 660, 669-73, 711 Dunford, N. and Schwartz, J. Z. 153, 711 Dyson, F. J. 386-87, 393, 404, 424, 446, 712 E Eguchi, T, Gilkey, P. B. and Hanson, A. J., 663, 712 Einstein, A. 17, 24, 26-8, 91, 94, 641, 682, 688-89, 692, 694, 712 Elworthy, K. and Truman, A. 106, 615-16, 631, 712 Exner, P. 20, 85, 264, 409, 712 Exner, S. 25 F Fans, W. G. 230, 239, 712 Fedosov, B. V. 681, 713 Feldman, J. 266, 294-95, 713 Feynman, R. P. 1-2, 4-5, 7, 10, 13, 17-8, 32, 62, 95-6, 99-111, 115-16, 120, 217, 272, 374, 376-77, 379, 383-84, 386, 393-94, 405-6, 444, 446, 460, 499, 506, 562-63, 568-69, 573, 631, 637-39, 659, 680-86, 688, 713 Feynman, R. P. and Hibbs, A. R. 104, 393, 562-63, 631, 714 Flato, M. 674-76, 676-81, 681-82, 714
751
See also Bayen, F. (Fronsdal, C., Lichnerowicz, A. and Sternheimer, D.) Flato, M. and Fronsdal, C. 653, 679, 714 Floer, A. 651, 715 Frank, W. M., Land, D. J. and Spector, R. M. 198, 240, 346-47, 715 Freidlin, M. 119, 715 Fresnel, A. 87 Freund, P. G. O. 660, 663, 715 Friedel, L. See Altschuler, D. Friedman, C. N. 230, 715 See also Dollard, J. D. Fronsdal, C. See Bayen, F. (Flato, M., Lichnerowicz, A. and Sternheimer, D.) See also Flato, M. Fukushima, M. 359, 363-64, 715 Furth, R. 25, 28, 712 G Gaveau, B., Jacobson, T., Kac, M. and Schulman, L. S. 616, 716 Gelfand, I. M. and Vilenkin, N. Y. 33, 716 Gelfand, I. M. and Yaglom, A. M. 82, 84, 296, 308, 716 Gerstenhaber, M. 676, 679, 716-17 Getzler, E. 661, 667-68,717 See also Berline, N. (and Vergne, M.) Getzler, E. and Kapranov, M. 658, 717 Gibbs.J. W. 31 Gilkey, P. B. 663, 717 See also Eguchi, T. (and Hanson, A. J.) Glashow, S. 693 Glimm, J. and Jaffe, A. 119, 717 Goldstein, J. A. 121, 126, 717 Graf, S. 58 Greene, B. R. 695, 717 Green, M. B., Schwarz, J. H. and Brink, L. 689 Green, M. B., Schwarz, J. H. and Witten, E. 694, 718 Gromov.M. 651, 687, 718 Gross, L. 624, 718 Guilisashvili, A. and Kon, M. A. 196, 718 Gupta, K. S. and Rajeev, S. G. 356, 370, 718 H Hamilton, W. R. 104-5, 682 Haugsby, B. O. 19, 296, 309, 311, 321-23, 327, 718 Hawking, S. W. 688, 718 He, C. Q. and Lapidus, M. L. 358, 718 Heckman, G. J. See Duistermaat, J. J.
752
AUTHOR INDEX
Heisenberg, W. 96-7, 374, 719 Hellinger, E. 153 Henderson, R. J. and Rajeev, S. G. 356, 370, 719 Hibbs, A. R. See Feynman, R. P. Hida, T., Kuo, H., Potthof, J. and Streit, L. 637, 719 Hilbert, D. 153, 688 Hille, E. and Phillips, R. S. 407, 719 Hille, E. and Yosida, K. 140-41 Hoegh-Krohn, R. 616 See also Albeverio, S. See also Combe, Ph. (Rodriguez, R., Sirugue, M. and Sirugue-Collin, M.) Hormander, L. 614, 616, 719 Huffman, T., Park, C. and Skoug, D. L. 636, 719 Huygens, C. 103-4 I Ichinose, T. 229, 230, 616, 720 Ichinose, T. and Tamura, H. 616, 720 Ito, K. 610, 720 J Jacobson, T. See Gaveau, B. (Kac, M. and Schulman, L. S.) Jaffe, A. See Glimm, J. Jefferies, B. 616, 720 Jefferies, B. and Johnson, G. W. 23, 396, 720 Jefferies, B., Johnson, G. W. and Nielsen, L. 396, 721 Johnson, G. W. 11, 197, 246, 294-95, 300-1, 359, 462, 582, 619-22, 721 See also Ahn, J. M. (and Skoug, D. L.) See also Albeverio, S. (and Ma, Z. M.) See also Badrikian, A. (and Yoo II) See also Chang, K. S. (and Skoug, D. L.) See also DeFacio, B. (and Lapidus, M. L.) See also Jefferies, B. See also Jefferies, B. (and Nielsen, L.) Johnson, G. W. and Kallianpur, G. 528, 721 Johnson, G. W. and Kim, J. G. 305-8, 722 Johnson, G. W. and Lapidus, M. L. 5, 394, 397-98, 401-2, 404, 417, 422, 428-29, 432-33, 453, 455, 462-63, 465, 468, 470, 506, 530, 532, 540, 544-45, 548, 550-51, 560-61, 616, 648, 653, 679, 722 Johnson, G. W. and Skoug, D. L. 67-70, 74-5, 400-1, 413, 455, 483, 617-19, 722-23 Jones, V. F. R. 637, 639-42, 655, 657, 723
K Kac,M. 1, 10-1, 13, 31, 115-16, 119-20, 240, 272-73, 293, 469, 483, 616, 637, 723 See also Gaveau, B. (Jacobson, T. and Schulman, L. S.) Kaku, M. 694-95, 723 Kallianpur, G. See Johnson, G. W. Kallianpur, G. and Bromley, C. 620, 623, 723 Kallianpur, G., Kannan, D. and Karandikar, R. L. 610, 613,620, 623-31,723 Kannan, D, 527, 724 See also Kallianpur, G. (and Karandikar, R. L.) Kapranov, M. 658, 724 See also Getzler, E. Karandikar, R. L. See Kallianpur, G. (and Kannan, D.) Kato, T. 125-26, 147, 150-51, 187, 193, 197, 204, 206-7, 212, 215, 224, 229-30, 232, 244, 255, 256, 269, 295-97, 309, 324-26, 329-30, 724 See also Brezis, H. Kato, T. and Masuda, K. 230, 724 Kauffman, L. H. 639, 641, 645, 651, 654, 657-58, 688, 724-25 Kepler, J. 659 Khalkhali, M. 649 Kirchhoff, G. R. 103 Klauder, J. R. 567, 725 Kluvanek, I. 505, 725-26 Koehler, F. 58 Kohn, J. J. and Nirenberg, L. 381, 726 Kolmogorov, A. N. 36-7, 89-93, 726 Kon, M. A. See Gulisashvili, A. Konstantinov, A. A. 616 See also Chebotarev, A. M. (and Maslov, V. P.) Kontsevich, M. 645-46, 649, 655-59, 675, 679-81, 726 Kostant, B. 676 Kricker, A. 654, 726 Kuo, H. H. See Hida, T. (Potthof, J. and Streit, L.) L Labastida, J. M. F. and Perez, E. 658, 727 Laca, M. 358, 727 Lamperti, J. 30, 727 Land, D. J. See Frank, W. M. (and Spector, R. M.) Lapidus, M. L. 5, 7, 9-10, 18, 111-12, 115, 197, 220-21, 229-31, 232-38, 238-44, 245-46, 248-49, 251, 254, 258-59,
AUTHOR INDEX
259-60, 266, 269-71, 304-5, 307, 358, 394, 399-401, 406, 439, 464, 477, 480-86, 488, 495-96, 499, 503-6, 507-8, 510-13, 514-21, 525-27, 532, 551-52, 553-61, 587-88, 616, 642, 674, 681-82, 727-29 See also Bivar-Weinholtz, A. See also DeFacio, B. (and Johnson, G. W.) See also He, C. Q. See also Johnson, G. W. Lapidus, M. L. and Maier, H. 358, 729 Lapidus, M. L. and Pomerance, C. 358, 729 Lapidus, M. L. and van Frankenhuysen, M. 358, 729-30 Lawson, H. B. 687, 730 Lawson, H. B. and Michelsohn, M.-L. 663, 730 Lecomte, P. B. A. See De Wilde, M. Leibniz, G. W. 62 Leinfelder, H. and Simader, C. G. 244, 730 Leites, D. A. 668, 730 Levy, P. 17, 62-7, 730 Lichnerowicz, A. 676, 682, 730-31 See also Bayen, F. (Flato, M., Fronsdal, C. and Sternheimer, D.) Lie, S. 202 Lim, J. A. 373, 731 See also Chang, K. S. (and Ryu, K. S.) Lin, X.-S. 659, 731 See also Birman, J. Lott, J. 667, 731 Lumer, G. 141-43 M Ma, Z. M. See Albeverio, S. See also Albeverio, S. (and Johnson, G. W.) See also Blanchard, Ph. Maier, H. See Lapidus, M. L. Manin, Yu I. 97, 106, 659, 684, 694-96, 731 See also Beilinson, A. A. Martin, W. T. See Cameron, R. H. Maslov, V. P. 403, 616, 731 See also Chebotarev, A. M. See also Chebotarev, A. M. (and Konstantinov, A. A.) Masuda, K. See Kato, T. Mattuck, R. D. 447, 684, 732 McKean, H. P. 275, 732 Maxwell, J. C. 641, 682-83, 685 Meetz, K. 359, 732
753
Mehra, J. 104-5, 398, 403, 682, 732 Melrose, R. B. 661, 668, 732 Meyer, P. A. See Choquet, G. Michelsohn, M.-L. See Lawson, H. B. Mills, R. L. See Yangs, C. N. Mostafazadeh, A. 668, 732 Moyal, J. E. 675, 678, 732 N Nakahara, M. 663, 732 Nazaikinskii, V. E., Shatalov, V. E. and Stemin, B. Yu, 403, 732 Nelson, E. 9, 12, 19, 28, 111-13, 204, 217, 219-20, 231, 239, 252, 293, 295-97, 308-14, 317-18, 327-29, 332, 346, 354, 357, 403, 413, 733 Neveu, J. 125-26, 150-51 Nielsen, L. 398-99, 733 See also Jefferies, B. (and Johnson, G. W.) Nirenberg, L. See Kohn, J. J. O
Oesterle, J. 681, 733 Ohtsuki, T. 658, 733 Orey, S. 119, 733 P Park, C. and Skoug, D. L. 619, 733 Park, C., Skoug, D. L. and Storvick, D. A. 636, 734 See also Huffman, T. (and Skoug, D. L.) Patodi, V. K. 668, 734 See Atiyah, M. F. (and Bott, R.) See also Atiyah, M. F. (and Singer, I. M.) Paycha, S. 688 Pazy, A. See Brezis, H. Pauli, W. 94, 663 Perez, E. See Labastida, J. M. F. Perrin, J. 17, 24, 27, 51-2, 62, 734 Phillips, R. S. See Hille, E. See also Lumer, G. Piraux, R. see Bivar-Weinholtz, A. Polchinski, J. 694-95, 734 Polyakov, A. M. 688, 690-94, 734 See also Belavin, A. A. (and Zamolodchikov, A. B.)
754
AUTHOR INDEX
Pomerance, C. See Lapidus, M. L. Potthof, J. See Hida, T. (Kuo, H. and Streit, L.) Powers, R. T. and Radin, C. 355, 734
Q Quillen, D. 671, 735 R Radin, C. 354-55, 735 See also Powers, R. T. Rajeev, S. G. See Gupta, K. S. See also Henderson, R. 3. Ramond, P. 689, 735 Reed, M. and Simon, B. 153, 155, 735 Reshetikhin, N. Y. and Turaev, V. G. 655, 735 Reyes, J. T. 567-68, 735 Rezende, J. 106, 735 Rieffel, M. A. 679, 735 Riesz, F. 152 Riggs, T. D. 395, 528-29, 616, 735-36 Robinson, D. W. See Bratteli, O. Rodriguez, R. 616 See also Combe, Ph. (Hoegh-Krohn, R., Sirugue, M. and Sirugue-Collin, M.) Rota, G.-C. 659, 736 Rovelli, C. 687-88 Rovelli, C. and Smolin, L. 687-88, 736 Rozansky, L. 646, 736 Ryu, K. S. 76, 737 See also Chang, K. S. See also Chang, K. S. (and Lim, J.A.) S Salam, A. 683, 693 Schafer, J. See Albeverio, S. Schechter, M. 215, 737 Schluchtermann, G. 582, 737 Schmidt, E. 152 Schrodinger, E. 6, 95, 98 Schulman, L. S. See Gaveau, B. (Jacobson, T. and Kac, M.) Schwartz, L. 165, 660, 737 Schwarz, A. S. 642, 647, 649-50, 652, 653, 737 Schwarz, J. H. See Green, M. B. (and Brink, L.) See also Green, M. B. (and Witten, E.) Schweber, S. S. 120, 393, 394, 685, 738 Segal, G. B. 645-46, 653, 738
Seiberg, N. and Witten, E. 687, 693, 738 Sengupta, A. See Albeverio, S. Shatalov, V. E. See Nazaikinskii, V. E. (and Sternin, B. Yu) Simader, C. G. 194, 252, 738 See also Leinfelder, H. Simon, B. 10, 119, 193, 196, 255, 256, 272, 274, 291, 738 See also Reed, M. Simons, J. See Chern, S.-S. Singer, I. M. See Atiyah, M. F. See also Atiyah, M. F. (and Patodi, V. K.) See also Axelrod, S. Sirugue, M. 616 See also Blanchard, Ph. (Combe, Ph. and Sirugue-Collin, M.) See also Blanchard, Ph. See also Combe, Ph. (Hoegh-Krohn, R., Rodriguez, R. and Sirugue-Collin, M.) Sirugue-Collin, M. 616 See also Blanchard, Ph. (Combe, Ph., and Sirugue, M.) See also Combe, Ph. (Hoegh-Krohn, R., Rodriguez, R. and Sirugue, M.) Skoug, D. L. 60, 739 See also Ahn, J. M. (and Johnson, G. W.) See also Chang, K. S. (and Johnson, G. W.) See also Johnson, G. W. See also Huffman, T. (and Park, C.) See also Park, C. See also Park, C. (and Storvick, D. A.) Smolin, L. 688, 739 See also Rovelli, C. Smoluchowski, M. 28, 739 Souriau, J.-M. 676, 739 Spector, R. M. See Frank, W. M. (and Land, D. J.) Stampacchia, G. 325, 739 Sternheimer, D. 674-82, 739 See also Bayen, F. (Flato, M., Fronsdal, C. and Lichnerowicz, A.) Sternin, B. Yu See Nazaikinskii, V. E. (and Shatalov, V. E.) Stone, M. H. 144, 146, 739 Storvick, D. A. 636 See also Cameron, R. H. See also Park, C. (and Skoug, D. L.) Streit, L. See Hida, T. (Kuo, H. and Potthof, J.) Strichartz, R. S. 242, 739-40 Stummel, F. 215, 740
AUTHOR INDEX
T Tabakin, F. 605, 740 Taubes, C. H. 657, 687, 740 See also Bott, R. Toeplitz, O. 152 Tomonaga, S.-I. 393 Trotter, H. F. 8-9, 114, 125-26, 150-51, 197, 201-3, 204, 218, 468-69, 741 Truman, A. 106, 615, 621, 741 See also Elworthy, K. Turaev, V. G. 649, 741 See also Reshithikin, N. Y. Turgut, T. 356
755
Weinstein, A. 675-77, 742 Wess, J. and Zumino, B. 689 Weyl, H. 381, 668, 676, 742 Wheeler, J. A. 688, 742 Wien, W. 94 Wiener, N. 17, 26-7, 29-30, 31, 34-36, 40, 42, 44, 89, 91, 92, 272, 274, 410, 420, 430, 432, 743 Wilson, K. G. 357, 369-70, 744 Witten, E. 8, 20-1, 100, 560-61, 609, 637, 639, 641-54, 654-55, 658, 660-61, 668-74, 687, 688-96, 743-44 See also Green, M. B. (and Schwarz, J. H.) See also Seiberg, N.
U Uhlenbeck, K. 687
V Vafa, C. 693, 695, 741-42 van Frankenhuysen, M. See Lapidus, M. L. Varadhan, S. R. S. 119, 742 Vassiliev, V. A. 654-59, 742 Vergne, N. See Berline, N. (and Getzler, E.) See also Berline, N. Vey, J. 676, 742 Vilenkin, N. Y. See Gelfand, I. M. Vitali, G. 266, 467 Voigt, J. 258, 742 Volterra, V. 403, 481, 507, 514, 742 von Neumann, J. 152, 354, 742 W Weierstrass, K. 52 Weinberg, S. 683, 689-90, 693-94, 742
Y Yaglom, A. M. See Gelfand, I. M. Yang, C. N. and Mills, R. L. 685-87, 689, 692, 743 Yoo, II See Badrikian, A. (and Johnson, G. W.) Yosida, K. See Hille, E.
Z Zamolodchikov, A. B. See Belavin, A. A. (and Polyakov, A. M.) Zangger, H. 94 Zastawniak, T. 616, 744 Zee, A. 682, 684, 744 Zumino, B. See Wess, J. Zyskin, M. 681
SUBJECT INDEX Entries in boldface type refer to definitions
A absolute value (of a self-adjoint or of a normal operator) 188, 189 abstract Wiener space (AWS) 624, 623-36 accretive operator 142 See also m-accretive action See action functional action functional (or integral) 2, 99, 101-6, 638-39, 643-48, 682, 686, 696 Chem-Simons 643, 643-48, 686 classical (in quantum mechanics) 2, 99, 102, 101-6 Polyakov action 690-92 Yang-Mills 686 See also principle of least action (Hamilton's) adjoint (of an unbounded operator) Hilbert adjoint 144, 144-46, 171, 260, 456-57, 479-80, 486, 494-95 Banach adjoint 146, 147, 458, 494 adjoint (of a semigroup of operators) 147, 350 Alexander-Conway polynomial 641 algebra algebra of sets 626 o-algebra (of sets) 32, 36, 40, 42, 69-71, 89-91, 582 Banach algebra 397, 451-61, 530, 540, 544-46, 551-57, 610-11, 615, 620-21, 623 C*-algebra 459 disentangling algebra 6-7, 397, 402, 452, 451-61, 530, 544-53, 559-61 Lie algebra 647, 655 semi-algebra (of sets) 36, 89, 91 algebraic sum of operators (or operator sum) 9, 10, 12, 18-22, 190, 192-93, 204, 230, 238, 348-50, 354 analytic-in-mass modified Feynman integral CF^an(V)) 10-2, 20-2, 239, 293, 295-98, 329-31,335-37, 331-45, 345-46, 346-58 analytic-in-mass (operator-valued) Feynman integral (K[(V)) in the sense of Definition 13.5.1' (Section 13.5) 10-2, 18-22, 79, 293-98, 310, 308-18, 328-29, 345-46, 346-58 in the sense of Definition 15.2.1 (Chapters 14-18) 5-8, 12, 14-7, 18-20, 22,
79, 311, 410, 410-13, 416-76, 477-529, 530-32, 540-44, 559-60 analytic-in-mass (scalar-valued) Feynman integrals (in the sense of Section 4.5 and of Section 20.2.A) 8, 62, 79, 79-81, 311, 611, 609-36 analytic-in-mass Feynman integral via TPF (rfp an) 10-2, 14-5, 18-22, 293, 295-97, 310-11, 308-23, 327, 323-31, 345-46, 346-58 analytic-in-time (operator-valued) Feynman integral ( J ' ( F ) ) 3-41, 10-1, 246-47, 293-94, 298, 299, 298-303, 303-6, 348-51, 353, 358-73 approximate identity (or mollifier) 208, 208-9, 256, 279-80 approximation theorem (for semigroups) 125-26, 150-51, 248-49 Atiyah-Singer index theorem 8, 609, 637, 659-74 Avogadro's number 28, 27-8 axiomatic description of the Wiener process 51 axiomatic Feynman's operational calculus 402, 532, 553-61 B Banach algebra See algebra Banach space 32-3, 122, 124, 127, 133-34, 139, 141-42, 146-47, 408, 415-16, 454, 464, 559, 568 Berezin determinant 667 Bessel function 175-76 potential 196 binomial formula (or theorem) 420, 553, 557 Bochner integrability criterion 415, 422, 494 -integrable 415, 421, 432, 494 integral 415-16, 421, 481, 490, 494, 507, 582-83, 587-88 measurable 415, 420, 494, 582 Boltzman's constant 27 bootstrap argument 343, 678 Borel
SUBJECT INDEX
class (or a -algebra) 32, 32-3, 36, 40-4, 47, 60-1, 68, 71, 82-4, 91-3, 567, 582-83 measurable 42-3, 60, 80, 407-15, 462-63, 467, 478-79, 507-8, 534-35, 545, 582-83 measure 34, 35, 43, 388, 400-1, 404, 408-9, 436-38, 440-1, 464-5, 477-78, 507, 509, 517, 520-27, 545, 587 boson (or bosonic) 663, 665, 666, 683 See also fermion bounded operator 20, 124, 138-39, 143, 152-55, 176, 222-25, 261-62, 407-9, 508, 564, 567, 629-31 quadratic form 152, 176, 629-31 bounded from below (or semibounded) quadratic form 9-10, 176, 176-96, 197, 214, 220-71, 272-90, 300-3, 360-62, 367-73 self-adjoint operator 9-10, 146, 158, 179, 179-83, 192-93, 195, 197, 214, 220-21, 232, 237-38, 242-56, 269-71, 272-73, 359-61, 367-68 Brownian bridge (stochastic process) 326, 619, 634, 672 Brownian motion 17, 24-30, 31, 34-6, 51-2, 57, 107, 274-76, 358, 364-68, 408, 410, 504-5, 634 See also Wiener measure and Wiener process Brownian (or Wiener) path 24-6, 28-9, 34-5, 51-7, 62, 107, 275-76, 312, 325-26, 415, 504-5 particle 24-5, 27-9, 34, 504-5 surface (or sheet) 633
C (Co) (or strongly continuous) semigroup 13, 20, 124, 121-51, 200-4, 261, 314, 316-18, 323-24, 326-28, 336-37, 341, 346, 349-50, 516, 566 See also contraction (semigroup) See also semigroup (of linear operators) Cameron's theorem (nonexistence of Feynman's measure) 12, 82-5 Cameron-Martin translation theorem (or formula) 45, 76, 615, 630 for the Feynman integral 615-16, 630 cancellation (or interference) effects 2, 3, 85, 96, 103, 105-6 Cantor function 22, 437-38, 484 set 437 Cantor-Lebesgue measure 22, 437-38, 484, 499, 518
757
capacity Choquet 363 Newtonian 295-98, 312-14, 322-23, 325-26, 331, 334-35, 340-41, 344-45, 345-46, 349, 362, 363-72 Caratheodory's extension process (or theorem) 16, 36, 40, 42, 83-4, 90 Cauchy (or initial-value) problem 122-24, 151, 132-33 Cauchy's principal value (of an integral) 87, 108 Cauchy-Schwarz inequality 182, 184, 206, 225, 249 central limit theorem 26-7, 29-30 change of variable theorem (or formula) 43-51, 65, 311, 426, 434, 648 Chapman-Kolmogorov equation 37, 39, 135 Chernoff product formula 197-203 theorem 203-4, 228 Chern-Simons action (or functional) 643, 643-44, 647-48, 686
Lagrangian 643 path integral 643-44, 646-50, 654 Chern-Simons (gauge field) theory 21, 641-54, 686, 687-88 Abelian 614, 649-50, 653 non-Abelian 614, 649-53 chirality property 640, 641, 651, 653 Choquet capacity 363 Choquet's integral representation theory 355 classical action 2, 102, 101-6 classical Dyson series 386-88, 393, 424, 424-25, 447, 572-73 classical limit 614, 638-39, 645, 647, 654, 692 classical (or Newtonian) mechanics 1, 95, 97, 104-6, 244, 669-70, 674-75, 677, 682 classical path (or path of least action) 104-6 classical phase space 637, 668, 669 closed graph theorem 131, 139 closed operator 131, 130-34, 140-41, 171-74 quadratic form 177, 180, 183, 273, 283-86 closure of an operator 133, 144, 171-74, 200-2, 205, 214, 303-4, 307, 335, 339, 341, 344 of a quadratic form 283, 283-84, 286 cohomology 655, 670, 678-79 Chevalley (-Eilenberg) 679 Hochschild 679 commutative (Banach) algebra 451-53, 460, 530, 545-46 commutator 551-2, 556-57
758
SUBJECT INDEX
complete normed algebra 452-55 measure 42, 57-61, 70-3 measure space 42, 57-61, 70-3 complex dimensions (of fractals) 358 conditional convergence 2, 87, 108, 241 conformal field theory 646, 653 connection 647-48, 685-86 curvature of a connection 685 flat connection 647-48 continuous measure 22, 388-89, 400, 404-5, 408, 407-9, 413-14, 417, 424-25, 430, 435-38, 462-64, 480, 482-83, 486, 499, 518, 522, 524, 566-67, 577-78, 587-88, 595-96, 601-2 continuous singular measure 22, 436-38, 484, 499, 518 See also Cantor-Lebesgue measure contraction (linear) operator 124, 134, 198, 198-204, 228 semigroup 124, 125, 134, 140-42, 147-50, 157, 201-4, 228, 314, 316-17, 323-24, 334, 336-37, 341, 346, 349-50,516, 566, 584 convolution of functions 165, 165-66, 208-9, 213, 256, 267-68, 279-80, 636 of measures 610 convolution kernel 165-66, 114-15, 174-76, 217, 233-34, 258-59, 565-66 core for an operator (or "operator core") 133, 171-74, 181-82, 255 for a quadratic form (or "form core") 177, 181-82, 236-37, 255-56, 370-71 correspondence principle (of quantum mechanics) 244, 675 coupling constant 639, 684 countably additive (or o-additive) measure 34, 36, 40, 42, 62, 82-5, 91-2, 102, 317,413, 626 covariance function (of random variables) 47, 633 D deformation quantization 8, 609, 637, 658, 674-82 delta sequence 470-72 derivative (strong) 129-30, 200-2 differential equation 121, 130, 352-53, 399-401, 478, 481-85, 497-99, 507, 517-18 distributional 206, 478, 482, 484, 507, 516-18 See also evolution equation See also integral equation differential form (of an evolution equation) 130, 481-83, 486, 517-18
differential form (on a manifold) 643, 665-73 differential operator 126, 265-66, 338 diffusion (or probabilistic) case 5, 16, 271, 374, 387, 409, 413, 477-78, 480-83, 486, 496, 498-99, 503-5, 508, 526, 527, 528, 587, 600, 603 See also quantum-mechanical (or Feynman) case diffusion constant (or coefficient) 26-7, 311-12 diffusion (or heat) equation 10-2, 14, 26-7, 119, 121-22, 136-37, 272-75, 292, 299, 395, 478, 481-82, 504-5, 587 Dirac constraints 681 Dirac genus (or A-genus) 667-68, 673 Dirac equation 395, 528-29, 616 Dirac measure (distribution, or delta function) 357, 368-70, 382-83, 389, 400, 405, 408, 416-17, 424, 427, 435, 438-42, 449-50, 468, 470-72, 480, 484-86, 495-96, 501, 503, 505, 506, 507, 509-10, 511, 525, 600-4 Dirac operator 395, 662, 673 Dirac (or interaction) representation (of quantum mechanics) 387, 485, 512, 511-21, 568 Dirichlet boundary conditions 122, 265-66, 338 form 176, 196, 360 Laplacian 265-66, 338 discrete (atomic or pure-point) measure 400, 404-5, 408, 408-9, 427, 438-43, 448-50, 468-70, 470-72, 472-73, 480, 484-86, 501-5, 507, 510, 525-26, 587-88, 600-4 disentangle, disentangled, disentanglement, disentangling 6-7, 14, 16, 377-86, 395-97, 401-2, 417, 422-23, 428-29, 432-33, 434-46, 451-53, 459-61, 479, 495, 530-32, 540, 542-46, 550-53, 562, 568-83, 585, 586, 599-608 disentangling algebras (of Wiener functionals) 6-7, 397, 399, 402, 452, 451-61, 530, 544-53, 559-61 disentangling process 377, 382, 401-2, 404, 451, 460, 530-32, 544-53, 557, 559-61, 562, 608 dissipative operator with respect to a duality section (of a Banach space) 141-42 in a Hilbert space 142, 260 See also m -dissipative operator See also maximal dissipative operator dissipative (or open) quantum system 20, 198, 260, 263-66, 321-22, 332, 409
SUBJECT INDEX
dissipativity condition 328, 331, 334-35, 340, 345 distribution (or generalized function, in the sense of L. Schwartz) 165, 205-13, 265, 328-29, 343-44, 517-18 positive 206, 213 tempered (or Schwartz) 165 distribution (of a random variable) function 29, 49, 49-51, 82-3, 97 measure 50-51 (joint) distribution function (of n random variables) 51, 50-1 distributional derivative 206 differential equation 400, 478, 482, 484, 507, 516-18 gradient 196, 243-44, 325, 329, 360 inequality (Kato's inequality) 197, 206-15, 297, 325, 338-41 Laplacian 206, 206-15, 295, 297, 325, 338-41 domain of an unbounded operator 121, 127, 162-63, 178-79, 189-90, 192-93, 211, 329, 335, 339 of an unbounded quadratic form (or "form domain") 10, 179, 179-93, 196, 221-23, 229-32, 236-37, 248-49, 269-70, 273-75, 282-84, 339-42, 370-71 dominated-type convergence theorem for Feynman integrals 4, 10, 220, 245-59 (esp. 245-46, 254, 256, 258-59), 294, 303-5, 307-8, 373, 464 application to the modified Feynman integral 4, 10, 220, 245-47. 254, 256, 258-59, 266, 294, 303-5, 308, 464 application to the analytic-in-time Feynman integral 4, 246-47, 294, 304-5 application to the Feynman integral via TPF 4, 246-47, 259, 294, 307-8 further extensions 258, 266, 373 dominating function 245, 259, 282, 304, 307, 316, 540-42, 549-50 Donaldson's (polynomial) topological invariants 650, 687 dual (of a Banach space) 129-30, 141-42 dualities in string theory 688, 693, 695 duality bracket (or pairing) 130, 142 Duistermaat-Heckman fixed-point formula (or localization formula) 660, 669-72 Dyson series classical 386-88, 393, 424, 424-25, 447, 572-73
759
generalized (or GDS) 6, 15-6, 397-99, 401, 404-561 (Chapters 15-18), 679 See also perturbation series (or expansion) E Einstein-Hilbert action 688 Einstein's general relativity 641, 688-90, 692 Einstein's probabilistic formula 28-9, 34-5 Einstein's theory of gravitation 682, 690, 692 electric field 19, 244 electro-weak theory (or force) 682-83, 685, 692-94 elliptic boundary value problem 265-66, 338 (partial differential) equation 343 (partial differential) operator 265-66, 647, 661-62 pseudodifferential operator 381, 556, 647, 661-62, 667-68, 676 ellipticity constant 339-41, 345 energy functional 196, 235, 332, 351, 672 operator (or "Hamiltonian"); see Hamiltonian kinetic energy 2, 10, 196 potential energy 2, 10, 196, 354 total energy 10, 196, 265, 351, 357 equation Chapman-Kolmogorov 37, 39, 135 Dirac 395, 528-29, 616 differential; see differential equation distributional; see distributional equation elliptic 343 evolution; see evolution equation heat (or diffusion); see diffusion equation integral; see integral equation Schrodinger; see Schrodinger equation wave 123 essential range (of a measurable function) 155 essential self-adjointness criteria 171, 204-5 theorems 197, 210-15, 335 essentially bounded function 153-55 essentially self-adjoint operator 9, 144, 171-74, 197, 204-5, 210-15, 217-20, 230, 232, 238-39, 242, 244, 307-8, 348, 351 Euler characteristic 666-67 evolution equation 7, 13-5, 122, 129-30, 132-33, 328, 399-400, 478, 482-83, 496, 499-500, 517-18, 562-63, 568, 587-99 expectation value (of a functional or observable) 638-39, 686 expected (or average) value (of a random variable) 29, 46, 47, 275
760
SUBJECT INDEX
exponential formula 5, 7, 140, 402, 533, 551-52, 557 for semigroups of operators 140 Feynman's paradoxical formula 5, 7, 402, 551, 557 noncommutative 5, 7, 402, 533, 551-52, 557 exponential law(s) 15 exponential of sums of noncommuting operators 5, 7, 395-96, 558-87
F fermion (or fermionic) 663, 666, 669, 689 See also boson fermion number operator 666 Feynman amplitude See transition amplitude See also Feynman propagator Feynman's approximation formula 109-11 Feynman (or quantum-mechanical) case See quantum-mechanical case Feynman diagram (or graph) 6, 120, 393, 398, 405, 418, 446-51, 488, 572-73, 580-81, 639, 645-46, 657, 680-81, 683-84, 692 generalized 6, 398, 405, 418, 446-51, 488, 572-73, 580-81, 684 See also string diagram Feynman (path or functional) integral (heuristic) Feynman-Polyakov integral (in string theory) 638, 675, 680-81, 688-95 Feynman-Witten integral (in knot theory and low-dimensional topology) 8, 20-1, 100, 609, 637, 639, 641-59 heuristic Feynman integral (in mathematics or physics) 8, 20-1, 100, 106, 609, 637-39, 641-59, 659-74, 679, 681, 682-88, 688-93, 695, 695-96 in gauge field theory 21, 100, 106, 637, 682-88 in quantum field theory 21, 100, 638-39, 646, 686-87 in quantum gravity 100, 675, 680-81, 687-88, 695-96 supersymmetric 8, 609, 637, 659-61, 666-74 Feynman integral (rigorous) analytic-in-mass (operator-valued) Feynman integral (K[(F)) in the sense of Definition 13.5.1' (Section 13.5); see analytic-in- mass Feynman integral via TPF (esp., 310-11, 308-24) in the sense of Definition 15.2.1 (Chapters 14-18) 5-8, 12, 14-7, 18-20, 22, 79, 311, 410, 410-13, 416-76, 477-529, 530-32, 540-44, 559-60
analytic-in-mass (scalar-valued) Feynman integral, 62, 79, 79-81, 311, 616-20 analytic-in-mass Feynman integral via TPF (^P m(V)) 10-2, 14-5, 18-22, 293, 295-97, 310-11, 308-23, 327, 323-31, 345-46, 346-58 analytic-in-mass modified Feynman integral (^an(V)) 10-2, 20-2, 239, 293, 295-98, 329-31, 335-37, 331-45, 345-46, 346-58 analytic-in-time (operator-valued) Feynman integral (J'(F)) 3-4, 10-1, 246-47, 293-94, 298, 299, 298-303, 303-6, 348-51, 353, 358-73 Feynman integral via TPF (Trotter product formula) (.^(V)) 3-4, 8-9, 13, 111-15, 197-98, 219-20, 204-220, 231-32, 238-41, 246-47, 259, 270-71, 294, 303-4, 307-8, 309, 348-51, 353 modified Feynman integral (FM(V)) 3-4, 8-10, 13-5, 8-15, 18-23, 111-12, 114-15, 174-76, 197-98, 220-32, 237, 232-44, 245-59, 259-66, 271, 275, 293-94, 295-98, 302-3, 303-5, 307-8, 331-32, 335-37, 331-45, 345-46, 346-58, 373, 464 transform approaches to the Feynman integral (including the "Fresnel integral" and the "Feynman map") 4, 8, 101, 106, 611, 609-36 Feynman-Kac formula 10-11, 13, 115-20, 272-92, 293, 299, 301, 303, 314-17, 322-24, 328-29, 331, 345-46, 360, 367-70, 468-70, 477, 483, 485, 508 classical 10-11, 13, 115-20, 272-75, 276-79, 291-92, 293, 468-70, 477, 483, 485, 508 with a singular (real) potential 10-1, 13, 115-20, 272-92, 293, 298, 299, 301 with a singular complex potential 13, 314-17, 322-24, 328-29, 331, 345-46 with a measure (as potential) 13, 299, 360, 367-70 Feynman-Kac formula with a Lebesgue-Stieltjes measure (FKLS) 7, 13-4, 399-401, 460, 470, 477-529 (Chapter 17), 528, 531, 551-52, 587-88, 603 finitely supported discrete part 399-401, 477-506 (esp. Sections 17.2-17.5), 603 general case 399-401, 507-29 (Section 17.6), 603
SUBJECT INDEX
Feynman-Kac functional classical 14, 274-75, 300, 399, 477 with a Lebesgue-Stiehjes measure 6, 14, 399-400, 477 Feynman map 615-16 Feynman's "measure" 62-3, 82-5, 101-3, 106, 413, 638, 643-44 non existence of 62-3, 82-5, 413 Feynman's operational calculus (for noncommuting operators) 4-8, 15-7, 148-50, 374-608 (Chapters 14-19), 609, 637, 639, 648, 653, 679 Feynman's paradoxical formula(s) 5, 7, 402, 533, 551-52, 557 Feynman propagator 446, 665 Feynman's heuristic rules 4-5, 7-8, 377-89, 392, 568-78, 600, 602 Feynman's time-ordering convention 4—5, 377-89, 391, 434, 444, 455-58, 494, 506, 536, 542-43, 568-78, 551-52 finite-dimensional distribution (of a stochastic process) 91, 90-1, 93 finite rank potential (or operator) 566, 604-8 finitely additive measure 624—26 form bounded (or boundedness) 191, 191-96, 220-21, 231, 235-44, 248-49, 261, 263-64, 273-74, 300-1, 303, 372-73 form core (for a semibounded quadratic form) 177, 181-82, 236-37, 255-56, 370-71 form domain (for a semibounded operator) 10, 179, 179-93, 196, 221-23, 229-32, 236-37, 248-49, 269-70, 273-75, 282-84, 339-42, 370-71 form sum (of operators) 9-10, 13, 192, 192-93, 195-96, 221, 229-32, 245-52, 254, 259, 261, 263-66, 269-71, 273-75, 285-90, 301, 303-4, 339-42, 348-52, 360-61, 367-68, 370-73 Fourier inversion formula 161 Fourier (-Plancherel) transform 161, 161-71, 175, 658 on Schwartz space S and on its dual S' 165 Fourier-Stieltjes transform 610, 610-12, 626 fractal 358 fractal properties of Wiener paths 24, 51-7, 62 time 424, 437-38, 446-47, 484, 499, 518 fractal string 358 Fredholm determinant 630 index 661, 661-62, 665-67, 673 free
761
heat complex semigroup 163, 163-64, 407-8 Green's function 174-75, 174-76, 233-35, 242, 258-59 Hamiltonian 137, 162, 161-63, 171-74, 407-8 heat semigroup 137, 161-63, 167, 407-8 imaginary resolvent 174, 174-76, 232-38, 246, 258-59, 375 particle motion (or evolution) 424-25, 438, 446, 488, 504 propagator 504 unitary group 9, 114-15, 169, 169-70, 217-20, 386-88, 408-9, 424, 512 Fresnel class 610, 610-12, 620-23 -integrable 610, 610-12 integral (one-dimensional) 87 integral (infinite dimensional) 611, 610-36 approach to the Feynman integral 613-15 Fubini's theorem 55, 276, 291, 302, 390-91, 414, 420-22, 430, 432, 475, 490, 493, 541-42 functional (linear, continuous) 144, 205, 458 (in the sense of path integrals, including "Wiener functionals") 6-7, 14, 274-75, 300, 374-561 (Chapters 14-18), 638-39, 643-44, 666, 672, 686, 690-91 functional calculus associated to a self-adjoint (possibly unbounded) operator 156-59, 162-64, 166, 221-29, 260, 262-63 for commuting self-adjoint operators 15, 158, 375 for noncommuting self-adjoint operators 158, 375-76 associated to a normal (possibly unbounded) operator 260-63 associated to a Banach algebra 455, 459-60, 517, 532, 551-53, 557 functional calculus (for noncommuting operators) 4-5, 15-6, 374-76; see also Feynman's operational calculus functional calculus (for pseudodifferential operators) Kohn-Nirenberg functional calculus 381 Weyl functional calculus 381, 676 functional (or path) integral (see the entire book, esp. Chapters 3-5, 7, 12, 13, 14-18. and 20, along with Section 19.1) functions of rapid decrease (Schwartz test functions) 137, 165, 165-66 fundamental solution 165
762
SUBJECT INDEX
G gauge field (or "field strength") 643-44, 646-48, 683, 685-87 potential 643-44, 646-48, 683, 685-87 symmetry 682-84 gauge (field) theory 609, 641-54, 660, 682-88, 689-90, 692, 694-95 Chem-Simons 641-54, 686, 687-88 Yang-Mills 683, 685-87, 689, 692 See also Maxwell's electromagnetism See also electroweak theory See also quantum electrodynamics (QED) See also quantum chromodynamics (QCD) See also standard model Gauss-Bonnet formula 667 Gauss measure canonical 624 Gaussian (-type) integrals 85-7 general covariance in general relativity 648, 688 in topological quantum field theory 648 generalized Feynman diagrams (or graphs) See Feynman diagrams (generalized) generalized Dyson series (GDS) See Dyson series (generalized) generation theorems for semigroups 125, 140-43, 146-47 for unitary groups 143, 144, 146 generator of a semigroup 124 of a unitary group 146 graph (of an operator) 131, 131-32 graph norm 131 Green's function free 174-75, 174-76, 233-35, 242, 258-59 group (of operators) group of isometries 134 unitary group 9, 12, 14-5, 114-15, 134, 144, 146, 157, 169-70, 198, 203, 204, 240-41, 245-46, 248-49, 252, 254, 258-59, 263, 266, 269-71, 294, 299-301, 303-8, 316, 328, 346, 350, 356, 360, 367-68, 375, 386-87, 408, 424, 438, 440, 466, 485, 503, 512, 567, 686 group gauge 650-53, 655, 683, 685-86 quantum 655, 674-75 H Hahn-Banach theorem 130, 141 Hamilton's principle of least action 104-5, 682 Hamiltonian (or "energy operator") 9-11, 13, 18, 95, 97-8, 112, 114, 122, 137,
195-97, 218-20, 235-38, 239, 242-44, 245-46, 249, 259-60, 273-74, 300-1, 303-8, 348, 351, 360, 367-68, 503-4, 664-67, 686 See also free (Hamiltonian) Hamiltonian approach to quantum mechanics (or dynamics) 4, 14, 17, 20, 94-5, 97-8, 220, 238, 294, 303-4, 353-54 heat (or diffusion) equation See diffusion equation heat (or diffusion) semigroup 11, 121-22, 135, 135-38, 274-75, 292, 407-8, 504 See also free (heat semigroup) Heisenberg's representation (or picture) of quantum mechanics 387, 512, 512-21, 568 Hermitian form (= bounded, symmetric quadratic form) 152, 176, 629-31 operator (= bounded, symmetric or self-adjoint operator) 20, 124, 138-39, 143, 152-55, 176, 222-25, 261-62, 407-9, 508, 564, 567 heuristic Feynman integral See Feynman integral (heuristic) Hille-Yosida (generation) theorem 140-41 Holder continuity (of Wiener paths) 51-7 holomorphic (or analytic) functional calculus (in a Banach algebra) 455, 459-60, 517, 532, 551-53, 557 holomorphic (or analytic) semigroup 303, 407-8 holonomy 646, 656 homfly polynomial 642, 651, 653, 655 Huygens' principle in wave mechanics 103—4 Huygens' principle in wave optics 103-4 I image measure 43, 68, 82, 311, 313 imaginary time 11, 116, 119, 269, 275, 299, 671-72 imaginary resolvents 9, 111-12, 174-75, 197-98, 220-31, 233-38, 240-41, 246, 251, 258-59, 261-65, 275, 334-37, 340-44, 373 of self-adjoint operators 9, 111-12, 174-75, 197, 220-31, 233-38, 240-41, 246, 251, 258-59, 275, 294, 373, 375 of normal operators 198, 259-66 of highly singular complex potentials 297, 334-37, 340-44 independent random variables 50, 50-1 index analytic 661-62, 665 Fredholm 661, 661-62, 665-67, 673 Maslov 630
SUBJECT INDEX
topological 661-62, 665 index theorem 8, 21, 609, 659-74 infinitesimal generator 124, 146 infinitesimal (microscope) 447-49 initial-value (or Cauchy) problem 122-24, 151, 132-33 instantaneous interaction 443, 447, 449, 470-72, 484,488, 501, 503 integral equation for abstract semigroups 129 Volterra-Stieltjes integral equation 478, 480-85, 486, 495-96, 499-501, 507-9, 512-13, 517-19, 521-29 associated with the exponential of noncommuting operators 587-88, 595, 596 integral in the mean 110, 168-69 integrated form (of an evolution equation) 129, 478, 480-82, 486, 496, 499, 507-9, 512-13, 517-18, 521, 587-88, 596 interaction (or Dirac) representation (or picture) [of quantum mechanics] 387, 512, 511-21, 568 interference effects 2-3, 96, 105-6, 438 inverse spectral problem (and the Riemann hypothesis) 358 J Jones' polynomial (invariant) 21, 639-41, 650-53, 655, 687 and Witten's knot invariant 21, 650-53 joint distribution function of random variables 51, 50-1 measure of random variables 50-1 K Kato class (of functions) 193, 193-96, 216, 236, 237, 246, 257, 274, 301, 304, 361-62 generalized Kato class (of measures) 361, 361-62, 364, 367-68, 369 Kato's (distributional) inequality 197, 206-15, 297, 325, 338-41 Kato's essential self-adjointness theorem 197, 210-15, 335 Kato perturbation 150 Kauffman bracket 688 polynomial 642, 651, 653, 688 kernel (of an integral operator) 166-69, 174-75, 240-41, 565-66, 604-8 kinetic energy 2, 10, 196 knot 642 See also link knot (topological) invariant 8, 21, 610, 637, 639-44, 649-54, 654-59, 687
763
See also Jones, homfly, or Kauffman polynomial See also Vassiliev and Witten invariants knot theory 8, 21, 100, 609, 614, 637-39, 642, 656
knot type 656, 655-56 Kolmogorov's consistency theorem 89-93 Kontsevich graph complex 645-46, 657-58, 680-81 integral 654, 655, 657-58 star product (for Poisson manifolds) 675, 679-81 L Lagrangian 2, 102, 643 Laplace transform 124-25, 139 Laplacian (or Laplace operator) Dirichlet 265-66, 338 distributional 206, 206-15, 295, 297, 325, 338-41 Lebesgue's dominated convergence theorem 223, 254-56, 262, 268, 281-82, 315, 342, 463, 464-66, 471, 526 Lebesgue measurable 44, 58-60, 78, 219, 235 Lebesgue measure 13, 40, 43, 45, 57, 275-76, 291, 377-83, 400-1, 403, 407, 424, 436, 468-69, 477, 478, 482-83, 485, 494, 499, 502, 511, 515, 518, 520, 562, 569-70, 572-73, 576-78, 580 Lebesgue-Stieltjes measure 388, 400-1, 404, 408-9, 440-41, 464-65, 469, 477-78, 507, 509, 517, 520-27, 545, 587 Lebesgue's (abstract) theory of integration (or of integrals) 2, 21, 36, 42,44, 85, 100-1 Levy's theorem (or Levy's law of quadratic variation) 17, 62, 63-7 link 642 Liouville form 669-70 measure 669-72 loop space 671 low-dimensional topology 8, 21, 100, 609, 637-39, 641, 656 Lumer-Phillips theorem 141-42, 260, 337 M m-accretive 142, 260, 317, 325, 328, 330, 334-35, 338-39, 341, 346 m-dissipative 142, 203-4, 228, 260, 325, 334, 337, 349, 354 magnetic field 19, 244, 331 vector potential 232, 242-44, 256, 265-66, 330-31, 334-36, 338, 340, 344-45
764
SUBJECT INDEX
Markov (stochastic) process 96, 126, 395, 528, 587 Markov property 501 Maslov index 630 maximal dissipative 142 Maxwell equations 685 Maxwell's (theory of) electromagnetism 641, 682-83, 685, 692 measurable function Bochner 415, 420, 494, 582 Borel; see Borel (measurable) Lebesgue 44, 58-60, 78, 219, 235 scale-invariant measurable 17, 77, 77-8, 79-81, 311, 411-12 Wiener 44-5, 58-60, 77, 78, 311 measure complex (or C-valued) measure 62-3, 82–4, 111, 408-9 positive 409 finitely additive 624-26 a-additive (or countably additive); see countably additive measure o-finite 34 measure theory (or measure-theoretic) 21, 32, 414 measurement process (in quantum mechanics) 95-7 method of stationary phase 4, 105-6, 107, 241, 614, 638-39, 645, 647-48, 652, 654, 660-61, 665, 668-74, 692 mixed initial-boundary value problem 122 mixed norm 409 norm space 409 modified Feynman integral (fM(-)) 3-4, 8-15, 18-23, 111-12, 114-15, 174-76, 197-98, 220-32, 237, 232-44, 245-59, 259-66, 271, 275, 293-94, 295-98, 302-3, 303-5, 307-8, 329-31, 331-32, 335-37, 331-45, 345-46, 346-58, 373, 464 for a semibounded Hamiltonian with singular potential 3-4, 8-10, 13-5, 18-20, 23, 111-12, 114-15, 174-76, 197-98, 220-32, 237, 232-44, 245-59, 259-66, 271, 275, 293-94, 298, 302-3, 303-5, 307-8, 331-32, 348-51, 373, 464 on a Riemannian manifold 19, 220-32, 241-42 with a singular magnetic vector potential 19, 111-12, 220-32, 241, 242-44, 256, 259-60, 331-32 for a singular complex potential 10, 20, 112, 115, 175-76, 198, 259-66
analytic-in-mass modified Feynman integral 10-2, 20-2, 239, 293, 295-98, 329-31, 335-37, 331-45, 345, 345-46, 346-58 moduli space 638, 643, 680, 686, 688, 690-92 momentum operator (or observable) 375-76 Moyal star product 678 monotone convergence theorem for quadratic forms first 257, 283, 288-90 second 257, 283-84, 285-88 generalized monotone convergence theorem for integrals 284, 286, 288, 290 M-theory 653, 693-95 multi-index 165, 205-6 multinomial formula 438-39, 532, 553 Ko-nomial formula 441 multiplication operator (operator of multiplication) 153, 153-55 multiplication operator form of the spectral theorem 155-56, 159, 179, 222-23, 225 N negative part of a (real-valued) function 112, 189, 193 of a self-adjoint operator 189 Nelson's first approach to the Feynman integral (Feynman integral via TPF, Section 11.2) See Feynman integral via TPF Nelson's second approach to the Feynman integral (analytic-in-mass Feynman integral, Section 13.5) See analytic-in-mass Feynman integral (in the sense of Definition 13.5.1') nonncommutative exponential law 15 See also exponential formula noncommutative operations (on Wiener functionals) 6-7, 401-2, 466, 530-32, 536, 535-39, 540, 542, 544, 549, 551-53, 559-60, 648 noncommutative addition 6-7, 401-2, 466, 506, 530-32, 536, 535-39, 540, 542, 544, 549, 551-53, 559-60, 648 noncommutative multiplication 6-7, 401-2, 466, 506, 530-32, 536, 535-39, 540, 542, 544, 549-53, 559-60, 648-49, 653 noncommutative geometry 560-61, 696 noncommutative operational (functional, or operator) calculus See Feynman's operational calculus noncommuting operators 4-8, 9, 15, 114-15, 120, 149, 158, 201-3, 218, 220-21, 228, 229-32, 235, 245-46, 248-49,
SUBJECT INDEX
251, 260-61, 264-65, 269-71, 315, 323, 326-27, 335-36, 339-40, 374-76, and Chapters 14-19 nondifferentiability (of Wiener paths) 51-7, 107 nondifferentiable curve 25, 63 nowhere differentiable (Wiener paths, functions) 51-2 nonlinear sigma model 663, 666, 671 nonlocal potential (or interaction) 20, 395-96, 565-66, 604-8 nonrelativistic quantum-mechanical particle 94-5, 100, 424-25, 438, 488, 503-4 quantum mechanics (or dynamics) 2, 15, 99, 393 normal (probability) distribution 29, 34, 51 normal operator bounded 154-55 unbounded 259-66 O observable 375-76, 638-39, 647, 686 occupancy time (of a Wiener path) 275-76 operator algebra 358, 459, 556 operator bounded 214, 214-16, 218-20, 294, 303-4 operator of multiplication (by a function) 153, 153-55 operator-valued function space integral; see analytic-in-mass Feynman integral (in the sense of Definition 15.2.1) analytic (-in-mass) Feynman integral; see Feynman integral (rigorous) Bochner integral; see Bochner (integral) oscillatory integral finite-dimensional 87, 105-6, 107, 241, 614, 660, 668-70 infinite dimensional 2-4, 85, 100-1, 105-6, 107, 110, 241, 614, 638-39, 645, 647-48, 652, 660-61, 665, 668-74 P parallel translation 646 parallel transport 646, 656 partial differential operator 119, 126, 242-44, 265-66, 338-45 partition function 648, 661-62, 686 path Brownian (or Wiener); see Brownian path path of least action (or classical path) 104-6, 638-39, 647-48 path of unbounded variation 56
765
path (or functional) integral (see the entire book, esp. Chapters 3-5, 7, 12, 13, 14-18, and 20, along with Section 19.1) PCAF (positive continuous additive functional of Brownian motion) 365, 364-72 perturbation series (or expansion) time-ordered (Chapters 14—19); see Dyson series (generalized) other 393, 564, 637, 638-39, 645-46, 652-53, 657-58, 677-81, 683-84, 688-90, 691-92 perturbation theorem (for semigroups) 125-26, 147-51, 248-49 perturbation theorem (for form sums of self-adjoint operators) 248-49, 251 phase transition 357-58 phenomenological models 18-20, 198, 264, 265, 296-98, 321-22, 333-34, 346-48, 351, 355-58, 369-70, 409, 424, 437-38, 446-47, 484, 499, 503-5, 518, 527-29, 562, 564-66, 587, 604-8 physical ordering (natural) 425-26, 434, 450-51, 458, 479-80, 485, 486, 494, 542-43 Planck's constant 2, 95-8, 99, 101-2, 105-6, 675, 677, 691 Poisson bracket 678 integral (representation) formula 267-68, 295, 319-20 kernel 267, 319-20 manifold (quantization of a) 675, 679-81 (stochastic) process 137-38, 395, 528-29, 616 semigroup 137-38, 395 structure 678, 680 polarization identity 176 Polyakov (or Feynman-Polyakov) integral 638, 675, 680-81, 688-95 Polyakov measure 690 polynomial (topological) invariant (for knots or links) Alexander-Conway 641 homfly642, 651, 653, 655 Jones 21. 639-41, 650-53, 655, 687 Kauffman 642, 651, 653, 688 positive continuous additive functional (PCAF) 365, 364-72 positive distribution (= positive measure) 206, 212 positive part of a (real-valued) function 112, 189, 193 of a self-adjoint operator 189 positive (or positivity preserving) semigroup 210, 292
766
SUBJECT INDEX
position operator (or observable) 375-76 potential real-valued (or R-valued) 19-20, 95, 97-8, 102, 112-13, 118, 212-15, 218, 235, 237-38, 244, 245, 265, 274, 298, 300-1, 303-4, 307, 312, 314, 317-19, 345-46, 349 complex-valued (or C-valued) 19-20, 198, 259, 260, 263-66, 297, 322-23, 325-29, 331-45, 409-11, 604, 613, and Chapters 14-18 local 20, 95, 97-8, 102, 565-66, 604 measure-valued 367-68 nonlocal 20, 565-66, 604-8 time-dependent 19, 296, 321-23, 409-11, 604, and Chapters 14-18 potential scalar-valued (or electrostatic); see potential (R-valued, C-valued, local or nonlocal) (magnetic) vector; see magnetic vector potential See also electric or magnetic field potential energy 2, 10, 196, 354 potential (multiplication operator) 3, 12-3, 18-9, 95, 114-15, 192-96, 215-16, 232, 298, 303-4 See also multiplication operator probabilistic (or diffusion) case See diffusion case probability amplitude 94-6, 102-3, 107-9, 111, 113, 119, 321; see also wave function density (or distribution) 29, 30, 34, 97; see also distribution (of a random variable) measure 50-51 product formula (for contraction semigroups) 9, 13, 111-15, 198-203, 204, 217-20, 230-46, 251-52, 258-66, 271, 272, 277-78, 293-94, 297, 331-46, 373, 440, 468-70, 485 Trotter product formula 9, 13, 111-15, 197, 198, 201-3, 204, 218-20, 259, 271, 272, 277-78, 293-94, 440, 468-70, 485 Chernoff product formula (or theorem) 197-203, 204, 217-20, 230-32 product formula for imaginary resolvents of self-adjoint operators 9, 111-12, 115, 174, 197, 220-31, 231-44, 245-16, 251-52, 258-59, 275, 294, 373 of normal operators 115, 198, 259-66, 294
product formula for resolvents (involving a singular complex potential) 297, 331-46 product formula for self-adjoint (contraction) semigroups 229-30, 269-71 product formula for unitary groups 9, 111, 113-15, 197, 203, 204, 218-20, 230-32, 259, 440, 470, 485 product integral (or time-ordered exponential integral) 400, 478, 485, 486, 507, 510, 514-17, 519-21, 647 for unbounded operators and with respect to a (Lebesgue-Stieltjes) measure 400, 478, 485, 486, 507, 510, 514-17, 519-21 product integral representation 400, 486, 507, 514 product measure 416, 488 projection (orthogonal) 159, 159-61 projection-valued measure 160, 159-61 projection-valued measure form of the spectral theorem 161 propagator (Feynman propagator) 446, 665 propagator (for an evolution equation) 483-84, 501-6, 511, 519-20, 521 unitary propagator 502, 503, 511, 519-20, 521 pseudodifferential calculus 381, 403, 554, 668, 676, 679 operator 381, 556, 647, 661-62, 667-68, 676
Q quadratic form (on a Hilbert space) bounded 149, 172, 614-17 unbounded 176-77, 176-96 (Sections 10.3-10.4), as well as Sections 11.4-11.7, Sections 12.1-12.2, Sections 13.3 and 13.7 semibounded; see bounded from below (quadratic form) (symmetric) Dirichlet form 176, 196, 360 quadratic variation of Wiener paths 62-7 quantization 1-2, 4-8, 100, 375-76, 376-77, 394-97, 401-2, 403, 453, 455, 460-61, 505-6, 540, 542-44, 550-53, 553-54, 556, 560, 561-63, 578, 582, 609, 637, 658, 674-82, 695 deformation quantization 8, 609, 637, 658, 674-82 geometric quantization 676 map 453, 455, 506, 540, 542, 550-53, 556, 560 Weyl quantization 678 See also pseudodifferential (calculus)
SUBJECT INDEX
quantum dynamics 3, 4, 9-10, 94-8, 220, 238, 294, 303-4; see also Schrodinger equation (or unitary) evolution 20, 95, 353, 354, 355, 356, 425, 437-38, 446-51, 472, 481, 484, 488, 503-4, 507-8, 511-13, 514-21, 528-29, 562, 572, 580-81, 587, 596 observable 375-76, 638-39, 647, 686 (-mechanical) particle 94-95, 96-7, 97-8, 100-4, 105, 106-9, 238-40, 244, 265, 312, 321, 328, 346-48, 347-48, 348-56, 356-57, 425, 437-38, 443, 446-51, 484, 488, 503-4, 521, 528-29, 572, 580-81 state 10, 94-5, 97, 196, 357-58, 438, 688 quantum chromodynamics (QCD) 357, 647, 682-85, 693-94 quantum electrodynamics (QED) 4, 5, 6, 376, 392-93, 564, 679, 682-85, 687, 695 quantum field theory 21, 100, 273, 297, 351, 381,609,639,641, 647, 649, 651-52, 671, 684, 687, 689-90, 692, 693-94, 694-96 See also conformal field theory See also gauge field theory See also topological quantum field theory quantum gravity 100, 682, 687-88, 689, 692 quantum group 654-55, 674, 675 quantum mechanics; see esp. Chapters 6, 7, 11, 13, 14, 15-19, 20 (particularly, pp. 690, 693-95) quantum-mechanical (or Feynman) case 5, 16, 119, 271, 374, 386-87, 409, 413, 470, 477-78, 480-83,486, 488, 496, 498-99, 503-5, 508, 511-21, 526, 528, 600, 603 quantum-mechanical particle; see quantum (particle) (matter) wave 3 quantum physics 99, 108, 199-200, 272-73 See also quantum field theory See also quantum mechanics quantum statistical physics 100, 119, 273, 357-58, 665-68, 679 quantum topology 654 quasi-invariant measure 33 R regularized determinant 647, 652 renormalizable theory 684, 692 renormalization 356-57, 369-70, 564, 639 representation theorems (for unbounded quadratic forms) 152-53, 176-93
767
(Section 10.3), 193-96 (Section 10.4), 198, 220-21, 222-23, 224-28, 231, 260-61 first 183, 190-93 second 187-88, 190 See also form sum resolvent (of an operator) 124-25, 139, 138-40 See also imaginary resolvent See also product formula for imaginary resolvents See also product formula for resolvents resolvent convergence 247, 247-49, 251, 254, 284, 283-84, 286, 289 resolvent equation 139, 228 resolvent set (of an operator) 138, 138-39, 155 Riemann hypothesis 358 Riemann integral 87, 108, 241, 272, 314, 439 Riemannian manifold 19, 232, 241-42, 661, 672, 686 Riemann sum (approximation) 118, 272, 278, 314, 439, 469 Riemann surface 688, 690-92 Riemann zeta function 357-58, 673 S Salam-Weinberg model 683 sample path properties (of Wiener paths) 51-7, 62-7, 275-76, 281, 286, 289-90, 312-13, 314, 414-15, 421, 430 scalar-valued analytic (-in-mass) Feynman integrals, 62, 79, 79-81, 311, 616-20 scale-invariant almost everywhere (s-a.e.) 69, 78 scale-invariant measurable function 17, 77, 77-8, 79-81, 311, 411-12 measurable set 69, 69-74 scattering 484, 503-4, 528-29, 572, 580-81 scattering operator (or S-matrix) 486, 520-21 Schrodinger equation 9-10, 11, 13, 14, 19-20, 94-8, 106-9, 113, 122, 169, 197-98, 218-20, 235-38, 238-40, 241-42, 242-44, 245-46, 258-59, 259-60, 263-66, 271, 297-98, 300-1, 304, 307, 313-14, 317, 321-22, 326, 328-29, 330-31, 331-34, 334-37, 345-6, 346-57, 358-59, 361, 367-73, 386-89, 399-01,437-38, 477-78, 480-86, 502-4, 507-11, 511-21, 521-27,527-28,616 heuristic derivation 106-9, 241 on a Riemannian manifold 19, 232, 241-42 with a bounded (but not necessarily smooth, possibly complex-valued) potential 14, 19-20, 276-82, 399-400, 424,
768
SUBJECT INDEX
Schrodinger equation - Continued 478, 481-82, 485, 488, 503, 515, 521; see also Schrodinger equation (with a measure in time) with a highly singular (attractive or repulsive) central potential 19-20, 198, 238-41, 263-66, 297-98, 333-34, 370 with a highly singular (scalar or electrostatic) complex potential 19-20, 198, 259-60, 297-98, 313-14, 321-22, 326, 328-29, 330-31, 331-33, 334-37, 337-41, 345-46, 346-58 with a highly singular (scalar or electrostatic) real potential (and bounded from below Hamiltonian) 9-10, 11,13, 19, 197-98, 218-20,235-38, 238-40, 241-42, 242-44, 245-46, 258-59, 300-1, 304, 307, 348-49, 351-52 with a highly singular (scalar or electrostatic) real potential (irrespective of sign, and with unbounded from below energy functional) 297-98, 313, 317, 321-22, 326, 328-29, 330-31, 331-34, 334-37, 345-6, 347-48, 349,351,352-58 with a (singular) magnetic vector potential 19, 232, 241, 242-4, 265-66, 330-31, 331-33, 334-37, 337-41, 345 with a measure in space (viewed as a generalized potential) 11, 358-59, 361, 367-68, 369-70, 370-72, 373 with a measure in time (and a bounded, but not necessarily smooth scalar potential) 14, 386-89, 399-401,437-38, 480-86, 503-4, 507-11, 511-21, 521-27, 587-88, 595, 596, 604-8 with a (uniformly continuous) potential that is the Fourier-Stieltjes transform of a bounded measure (possibly added on to a bounded quadratic form) 610, 614-15, 616, 618, 629-31 with a time-dependent (scalar or electrostatic) potential 19, 321-23, 386-89, 399-t01, 437-38, 480-86, 503-4, 507-11,511-21,618 Schrodinger operator 114, 197, 264, 296, 324, 328, 339, 350, 353-54 Schwartz space (or space of functions of rapid decrease) 137, 165, 165-66 self-adjoint operator bounded 146, 152-55, 156, 157, 176 unbounded 8-11, 12-3,144, 144-47 (Section 9.6), 152-96 (Chapter 10), 197-98,
203; see also bounded from below (self-adjoint operator) See also functional calculus (associated to a self-adjoint operator) self-adjoint semigroup 144,146, 157, 229-30, 269-71,274-78 See also heat semigroup semi-algebra (of sets) 36, 89, 91 semibounded self-adjoint (or symmetric) operator; see bounded from below (self-adjoint operator) quadratic form; see bounded from below (quadratic form) semiclassical approximation (or expansion) 637, 647, 652, 660, 669 See also stationary phase approximation semigroup (of linear operators) adjoint semigroup 147, 350, 355-56 analytic (or holomorphic) semigroup 303, 407-8 (Co) contraction semigroup; see contraction (semigroup) (Co) semigroup (or strongly continuous semigroup); see (Co) semigroup self-adjoint semigroup 143,146, 229-30, 269-71, 274-78 uniformly continuous semigroup 143 unitary group (strongly continuous); see group (unitary) weakly continuous semigroup 143 semigroup heat; see heat semigroup Poisson 137-38, 395 translation 134 semigroup property 135, 316, 318, 523 separable (topological space) 33, 158 sesquilinear form 176, 176-77 o-T-additive (or countably additive) measure 34, 40, 42, 62, 82-5, 91-2, 102, 317, 413, 626 a -finite measure 34 o-algebra (of sets); see algebra (o-algebra) simplex trick 420, 490 singular continous measure 22, 436—38, 484, 499, 518 skein relation 640, 651 Sobolev embedding theorem 194, 242 inequalities 194, 242, 253 norm 254, 264, 325 space(s) 163, 196, 254, 264, 325 space-time 685, 690, 695-96 spectral measure 159-61
SUBJECT INDEX
spectral theorem (for unbounded self-adjoint operators) 13, 152-53, 153-61 (Section 10.1), 161-76 (Section 10.2), 179, 198, 221-23, 224-29, 269, 271-72,301 multiplication operator form 155-56, 159, 179, 222-23, 225 functional calculus form 156-58, 221-22, 269, 301; see also functional calculus (associated to a self-adjoint operator) projection-valued measure form 161, 302 spectral theorem (for unbounded normal operators) 260, 261-63 spectral theory 153 spectrum (of an unbounded operator) 139, 155-58, 160, 162, 180, 221-22, 239, 248, 262, 269, 301-2, 356-57 square root (of a nonnegative self-adjoint operator) 190, 190-93, 221-22, 225-27, 243, 247-50, 254-56, 260-63, 273, 342-43 stability condition 150, 248, 254, 257, 280 stability properties with respect to the potentials (in space) 245-46, 258-59, 266, 304, 307; 462-64 (quantum-mechanical case of Section 16.1, combined with the main results of Chapter 17). See also dominated-type convergence theorem for Feynman integrals stability properties with respect to the measures (in time) 467, 470-72, 474 (quantum-mechanical case of these results of Section 16.2, combined with the main results of Chapter 17); 525-27 (quantum-mechanical case) standard model (for elementary particles) 683, 689, 693-94 stationary phase approximation 654, 660, 669 See also semiclassical approximation stationary phase (method of) See method of stationary phase See also oscillatory integral star product 674, 677, 676-81 Moyal star product 678 stochastic analysis (infinite dimensional) 637, 671-73,684 stochastic integral (Ito or Paley-Wiener-Zygmund) 620 stochastic process 51, 89-93 (Chapter 5), 138, 326, 357, 365, 398, 528, 672 See also Brownian bridge See also Markov process See also Poisson process
769
See also Wiener process string diagram 692 string theory 100, 645,653, 675, 681, 688-96 nonperturbative 653, 693, 694-96 perturbative 645, 675, 681, 688-92, 694-96 strong continuity 124, 482 convergence (or limit) 201-2, 219-20 derivative 129-30, 200-2 operator topology 202, 481, 515, 517 strong (nuclear) force (or interaction) 357, 499, 683-84, 692 sum over histories 2, 13, 15, 488, 637 summation procedure 240 supermanifold 667-68 supercharges 664, 663-67 supersymmetric Feynman-Kac formula 666-67 nonlinear a -model 666, 671 (Feynman) path integral 609, 637, 659-61, 666, 667, 669, 672 proof of the Atiyah-Singer index theorem 609, 659-74 quantum mechanics 660, 662, 663-68, 671 quantum field theory 659-60, 687 string theory (or superstring theory) 689—90, 694-95 supersymmetry 659-60, 663, 663-64, 667,689 symmetric operator 144, 144-46, 179, 350, 351, 354-55 (quadratic) form 176, 176-96; see also bounded from below (quadratic form) symmetry breaking (spontaneous) 357-58, 660, 689 symplectic form 669-70 Fourier transform 678 manifold 669-71, 677, 679 T telegrapher's equation 395, 529, 616 time discontinuity 471-72, 478, 482, 484, 485, 488, 496-99, 503-4, 504-5, 510, 528-29 time evolution See quantum (evolution) Also see evolution equation time-ordered chronological product See product integral time-ordered perturbation expansion (or series) See Dyson series (generalized) time-ordering convention (Feynman's) See Feynman's time-ordering convention
770
SUBJECT INDEX
time-reversal 22, 399, 406, 425-26, 434, 450-51, 455-59, 479-80, 484-85, 494-95,519,542-43 of a functional 399, 456, 456-59, 479-80, 484-85, 488, 494-95, 507, 542-43 of a measure 425, 425-26, 434, 456-58, 480, 485 of a potential 425, 425-26, 434, 456-58, 480, 485 time-reversal map (on the disentangling algebra) 22, 406, 456, 455-59, 479-80, 488, 494-95, 542-3 See also physical ordering (natural) topological invariants (for knots (or links) and low-dimensional manifolds) See Donaldson See also Jones, homfly, or Kauffman polynomial See? also Vassiliev and Witten invariants topological quantum field theory (TQFT) 641-2, 646,649-54, 679, 681 torsion Reidemeister 647 Ray-Singer (or analytic) 647, 652 total energy 10, 196, 265, 351, 357 total variation measure 408, 408-9 total variation (of a measure) 82-3, 408,408-9 trace class operator 566, 604-8, 629-31 finite rank 566, 604-8 transform approach to the Feynman integral 609-36 and Poisson process 616 See also Fresnel integral transition amplitude 95-6, 101-2, 110-11 translation invariant measure 32-4, 102, 110, 488 translation pathology (in Wiener space) 74-6 translation semigroup 134 Trotter product 218, 277, 440, 485, 600 Trotter product formula (TPF) 9, 13, 111-15, 197, 198, 201-3, 204, 218-20, 259, 271, 272, 277-78, 293-94, 440, 468-70, 485
U unbounded operator; see operator (unbounded); see also self-adjoint operator (unbounded) quadratic form; see quadratic form (unbounded) uniform boundedness theorem (or principle) 131-32, 268 uniformly continuous semigroup 143 unitarily equivalence, equivalent 155, 155-59 unitary group; see group (unitary)
operator 134, 146, 219, 301, 316, 354-55, 502 evolution; see quantum (evolution) propagator 502, 503, 511, 519-20, 521 V variance parameter (of a normal distribution or of the Wiener process) 29-30, 68, 87-8 variance (of a random variable) 29-30, 47, 68, 87-8 Vassiliev (knot) invariants 646, 649, 656, 654-59 viscosity 25-6, 27 Volterra-Stieltjes integral equation; see integral equation (Volterra-Stieltjes equation) W wave equation 123 wave function (in Schrodinger's quantum mechanics) [also called probability amplitude] 9-10, 94-5, 102-3, 106-9, 111, 116, 122, 169, 218, 235, 258-59, 321, 352, 365 weak convergence of vectors in a Hilbert space 226 of measures 464,474-76 weak force 683 weak star convergence 464 weakly continuous semigroup 143 well posed (initial-value) problem 123-24 well posedness theorem 124 Weyl quantization 678 Weyl's functional calculus 381, 676 Weierstrass (nowhere differentiable) function 51-2 white noise 528, 637, 649 white noise (stochastic or distributional) analysis 528, 637, 649 Wick (or normal) ordering 381 Wiener functional; see functional (Wiener) integrable 44, 46, 47-50, 279, 288, 290, 311, 314-15, 421, 422, 432, 433, 466, 530-32 integral 5-6, 11, 16, 22, 79, 82, 115-16, 118-20, 272, 274-75, 279, 282, 286-91, 293, 296, 299, 308-9, 310, 314-15, 317, 322-23, 327, 329, 331, 360, 367-69, 374, 389, 391 measurable (function) 44-5, 58-60, 74-6, 77, 78,311 measurable (set) 36, 42, 58-60, 68-74 null (set) 69-71 Wiener's integration formula 44, 42-51, 53, 64-5, 83, 116-17, 136, 279, 315, 390-92, 420, 436,439
SUBJECT INDEX
Wiener measure 17, 31, 40, 42, 34-60, 62-3, 66-81, 82, 85, 92, 117, 274-76, 291, 299, 310-11, 313, 360, 369-70, 389, 410-13, 414-15, 469, 479, 533-35, 540 d-dimensional Wiener measure 57, 410 scaled Wiener measure (with variance parameter a2, i.e. "diffusion constant" a) 68, 68-81, 82, 311, 313, 317, 413 with complex variance parameter (non existence of) 62, 82-5, 317, 413 Wiener process 16, 27, 46, 51, 57, 68, 87-8, 92-3, 136, 138, 286, 289-90, 312-13, 314, 325-26, 359, 633-34 d-dimensional 57 multiparameter 60, 633-34 Wiener space 32, 34, 36, 44, 52, 66-81, 82-5, 87-8, 92-3, 274-76, 291, 389, 479, 506, 532-35
771
Wilson line 647 Wilson loop (observable) 644-47 Wilson's renormalization 357, 369 Witten's topological invariants 4, 21, 609, 641-54, 654-55, 658 for 3-manifolds 643 for knots (and links) 644 See also Jones (homfly, or Kauffman) polynomial See also Vassiliev invariants Witten (Feynman-type) integral (for topological invariants) 8, 20-1, 100, 609, 637, 639, 641-59 Y Yang-Mills action (functional) 686 equations 685 gauge (field theory) 100, 647, 683, 685-87, 689, 692