Geometry of Quantum Theory

~1(E)) is an observable associated with j£? whose range is obviously contained in j£?0. Suppose now that v is an observable associated with 3? whose range «5?vcjg?0. If we use the notations of the previous paragraph, and define £f~ by &>- = { / - i ( J p ) :Fe@{Rn)}, then h maps £f~ onto J5?0. Applying theorem 1.4 to £f~ and v9 we infer the existence of a real valued y ~ -measurable Borel function c o n l such that h(c~1(E)) = v(E) for all Borel sets E of R1. By lemma 1.5, since c is S?~ -measurable, there exists a real valued Borel function

(?{x)) for all x e X. If now 2£ is any Borel set on the line, we have:

u(
BOOLEAN ALGEBRAS

ON A CLASSICAL

PHASE SPACE

17

Remark. The uniqueness of u, guaranteed by (ii) of theorem 1.6 shows that u is independent of the constructs X, <9*, and h, used in its construction. Consequently, for any real Borel function cp on Rn, the map E -> u((p~1(E)) is uniquely determined by ul9 u2, • • •, un and cp. It is natural to denote this observable associated with ££ by
CHAPTER II PROJECTIVE GEOMETRIES 1. COMPLEMENTED MODULAR LATTICES The point of departure of our discussion of quantum phenomenology is the observation that the partially ordered set of experimentally verifiable statements associated with an atomic system cannot be expected to possess the distributivity properties characteristic of the Boolean algebras associated with classical systems. The simplest and most interesting of the mathematical structures that model such systems are the projective geometries, namely, the lattices of subspaces of vector spaces. More precisely, let D be a division ring, and let V be a vector space of finite dimension n ^ 2 over D; we shall always suppose our vector spaces to be left vector spaces (cf. Jacobson [1]). Then j£?=j£?(F,D), the lattice of linear subspaces of V partially ordered by inclusion, is called the projective geometry of V. This chapter will begin with a brief review of the basic properties of projective geometries. Although these are too simple to serve as models for realistic quantum systems, they already possess many of the fundamental features of the more complex systems. The first example of an abstract projective geometry that anyone comes across is the projective plane, described axiomatically in terms of its points, lines, and their incidence properties. To define higher dimensional projective spaces one replaces incidence by partial order and uses a dimension function to characterize the hierarchies of points, lines, planes, and so on; the properties of incidence that are characteristic of a projective geometry are then summarized by certain natural properties of the dimension function. We recall that a lattice 3? is complemented if for any aeJ? there is a b e S£ such that a Kb = 0 and a yb = 1. A complemented lattice JS? is modular if (1)

a,2>,CGJ^,

c < a => a A (bye) = (a A 6) Vc

Boolean algebras are modular; for any division ring D and any finite dimensional vector space V over D, ^(V,D) is a modular lattice which is not a Boolean algebra if dim(F) > 2. We next introduce the notion of a lattice of finite rank. A chain in a lattice is a sequence (a^i^i^n of elements at of the lattice such that ax < a2 < ... < an, with ax ^ 0 and a{ / a , for i <j; n is called the length of the chain, and the rank of the lattice is the supremum of 18

PROJECTIVE

GEOMETRIES

19

the lengths of all possible chains. An element of a lattice is called a point if it is nonzero and minimal. If i' 2' * * *' n are points, they are called independent if x{<£ V j V t ^ for * = 1,2,..., n. The lattice ^(VJ)) has rank dim(V), its points are the one-dimensional subspaces, and independence is just t h e usual linear independence. For any lattice 3? and a n y a / 0 in iff, let us write J?[0,a] for the set of all elements b of ££ with 0dim(a) has the following properties: (i) dim(0) = 0, dim(a) is an integer ^ 0 for all a; (ii) if a < 6, dim (a) ^ dim(6), with equality only for a = b; (2) (hi) dim(ayb) + dim(aAb) — dim(a) -f dim(&) for all a, b; (iv) if s = dim(a) > 0, then s is the maximum number of independent points contained in a; and a family of independent points contained in a has sum equal to a if and only if it has s elements. I n particular dim(l) = n and dim(#) = 1 for any point x. The converse is also true; if ££ is a complemented lattice of finite rank admitting a function d:J?->Jl satisfying (i)-(iii) of (2) and such t h a t d(x) = l for all points x, then ££ is modular and d(a) =dim(a) for all a. The lattice J£?(F,D) is modular, dim being the usual linear space dimension. At the other extreme is the Boolean algebra of all subsets of a finite set, dim being the cardinality. To single out the geometries we need the notion of irreducibility. An element of a complemented modular lattice j£P is called central if it has a unique complement in ££\ which is written as a'; the reason for this terminology is the easily proved fact t h a t a e j£? is central if and only if for all x e J*? (3)

x =

{x/\a)\f{x/\a').

The set of all central elements is called the center oi<£; it is a Boolean algebra, which coincides with 3? only when J$? itself is a Boolean algebra. & is called irreducible if 0 and 1 are its only central elements. A geometry is an irreducible complemented modular lattice of finite rank. The lattices JS?(F,D) (dim(F)^2) are geometries. Every complemented modular lattice of finite rank m a y be viewed in a canonical manner as a direct sum of geometries over its centre. More precisely, let ££ be a complemented modular lattice of finite rank and let cv...,ck be the minimal (nonzero) elements of the center of 3?. Let J^,- = JSf[0,cJ

and let

& = ££X x ... x Seh

20

GEOMETRY

OF QUANTUM

THEORY

where the partial order in ££ is denned by saying that (xl9...,xk)

<

(yl9...,yk)

if and only if xt < y{ for 1 ^ i^ k. Then each J^- is a geometry, and the map that takes x e££ to (xAcv xAc2, ..., xAck)eJ? is an isomorphism of ££ with J*?, the inverse map taking (x1,...,xk) to xx\/...\fxk. For arbitrary geometries S^i9 ££ denned as above is a complemented modular lattice whose center has (0, ...0,1,0, ...,0) (1 in the j t h place) as its minimal elements (l^j^k). The usual geometrical terminology with which we are familiar may be introduced as follows. Let JS? be a complemented modular lattice of finite rank. If aeS£ and dim(a) = l, a is a 'point', if dim(a) = 2, a is a line; if dim(a) = 3, a is a plane. If a = b vc where b and c are distinct points, a is called the line joining these points, and 6, c are said to be on a; if d is a point < a, 6, c, d are said to be collinear. It can be proved that ££ is a geometry if and only if every line has at least three points on it. The fundamental theorem of classical geometry is that every geometry of rank ^ 4 is isomorphic to some j£?( F,D) wher the division ring D is determined uniquely up to isomorphism; and that if the rank is 3, this is true if (and only if) the plane in question is Desarguesian. We shall discuss this question a little later. At this time we shall take a closer look at the geometries S£ (F,D). 2. ISOMORPHISMS OF PROJECTIVE GEOMETRIES. SEMILINEAR TRANSFORMATIONS. The first problem that arises naturally is to obtain a description of all possible isomorphisms between j£?(F,D) and J?(V',D'). We suppose the vector spaces to be always finite dimensional and that the scalars act from the left. To formulate the answer we need the concept of semilinear transformations. Let D, D' be two division rings which are isomorphic and let (4)

G:C-+C<>-

(ceD)

be an isomorphism of D onto D'. Let V (resp. V) be a vector space over D (resp. D'). A map L (v->Lv) of V into V is said to be semilinear relative to a or a-linear if L(vx + v2) = Lvx + Lv2 L(cv) = c^Lv

(vv

v2eV),

(c eD, v e F).

If L is in addition a bijection of V onto Y\ we say that L is a a-linear isomorphism,

PROJECTIVE

GEOMETRIES

21

If L is a <7-linear isomorphism of V with V, then for any linear subspace M <= 7, its image X[lf] under L is a linear subspace of V, and Jf ^ ^ [ i l f ] is an isomorphism of ^ ( F , D ) with j£?(F',D'), denoted by £z,. We can now formulate the main result. Theorem 2.1. There exists an isomorphism between if?(F,D) and ^(F'jD') if and only if D and D' are isomorphic and dim( V) = dim( 7'). / / £ ^s aw i'somorphism of J*?(F,D) wi£A J5?(F',D'), there is an isomorphism a(D ^ D ' ) awd a a-linear isomorphism L(V^> V) such that | = £L. If ^-(D^D') is another isomorphism and L'( V ^ V) a r-linear isomorphism, then £L = £L' if and only if for some d e D, d ^ 0, we have cr = dc*d-1 (ceD),

(6)

L'v = dLv

(veV).

For the proof we refer to Baer [1], If D = D' it makes sense to call an isomorphism £(j£?(F,D) ^ J^(F,D')) linear if there is a linear isomorphism L(V^> V) such that £ = £L. Theorem 2.1 leads to Theorem 2.2. In order that every isomorphism ofJ?(V,D) with j£?(F',D) be linear it is necessary and sufficient that every automorphism of D be inner. The division rings of greatest importance in physics are R, the field of real numbers, C, the field of complex numbers, and H, the division ring of "Hamiltonian" quaternions. (a) R. The identity map of R is its only automorphism, i.e., R is rigid. So every isomorphism between real projective geometries is linear and the linear map is determined up to a multiplicative constant. (b) C. One knows from the theory of algebraic fields that there are infinitely many automorphisms (cf. Bourbaki [1], pp. 114-115). However, the identity and complex conjugation (c^c*) are the only analytically wellbehaved ones (e.g., measurable, bounded, etc.). Since C is commutative, any isomorphism between two complex geometries determines uniquely the automorphism of C associated with it. (c) H, the quaternions. Let 1=

(o i ) '

jl=

(o

-if'

J2=

( - i o)'

h==

[i

o)'

where i is the usual square root of — 1 in the complex number field. We define H to be the R-linear span of 1 and j a (1 ^ a < 3 ) ; since j a 2 = —1 (1 ^ a ^ 3 ) and j«j&= — j&ja=jc whenever (abc) is an even permutation of (123), H is an (associative) algebra. It is easy to verify that it is the algebra generated by j a with the above relations. Let |q| = +det(q)*= +(?o2 + Qa e R) • Then | • | is a norm on H which is multiplicative, i.e., |qq'| = |q||q'|, and q is invertible in H if and only if

22

GEOMETRY OF QUANTUM

THEORY

q ^ 0. Thus H is a division ring. For q e H , say q = ^01 + 2i i t s conjugate q* is the quaternion g^ —2i
so that the unit quaternions constitute the group SU(2,C) of 2 x 2 unitary matrices of determinant 1. Two quaternions are said to be in the same class if they can be transformed into each other by an inner automorphism; for q, q' to be in the same class it is necessary and sufficient that tr(q) = tr(q') (tr = trace) and |q| = |q'|. All the automorphisms of H are inner so that isomorphisms of quaternionic projective geometries are linear. The map x-+x\ (#eR) imbeds R as the center of H. H may be viewed as a vector space over C, although in a noncanonical manner. We identify C as a subfield of H by the correspondence

H is then a vector space over C, scalar multiplication being from the left. Then dim c (H) = 2, and {l,j2} may be taken as a basis. For any q e H , R q :q'->q'q* is then an endomorphism of H over C, q->R q is a faithful representation, and the matrix of j 0 (1 ^ a < 3 ) is c a j a , where ea= — 1 for a = 1,3 and + 1 for a = 2. 3. DUALITIES AND POLARITIES We shall now examine the anti-automorphisms and orthocomplementations of the projective geometries j£?(F,D). It will turn out that orthocomplementations essentially arise only from Hilbert space structures, at least when D is one of R, C, and H. We begin with the concept of the dual of a division ring. Let D be a division ring. We define D° to be the ring with the same elements as D, with the same rule for addition, but reversing the order of multiplication. D° is a division ring, said to be dual to D. Isomorphisms of J)1 with D2° may be viewed as anti-isomorphisms of T>1 with D2. If V is a vector space over D, its dual, defined as the space of D-linear maps of V into D, becomes a vector space over D° if we define scalar multiplication by writing

(«'/)W=/Wa

(aeD°,veV9feV*);

addition in V* is defined in the usual way. F* is then a (left) vector space over D° and has the same dimension as V. Clearly (D°)° = D and so V can

PROJECTIVE

GEOMETRIES

23

be canonically identified with F**. If (vv ..., vn) is a basis of V, the corresponding dual basis (v±*,...,«„*) of F* is defined by v-*(vk) = Sjk. If i f <= F is a linear subspace, its annihilator is the linear subspace of F* defined by (7)

Jf° = {/:/eF*,/(v) = 0 for all veM}.

It is clear that (8)

dim( Jf) + dim( Jf °) = dim( V) 00

and so M = M . The map M->M° is obviously an inclusion-reversing bisection of J£?( F,D) with «J^( F*,D°). We call it a duality, and use the same term to refer to any inclusion-reversing bijection of one geometry onto another. To determine the most general duality we need the concept of a nonsingular semibilinear form. Let 6 be an anti-automorphism of D, F a vector space over D. A 0-bilinear form is a map <. , . ) : x, y ~> (x, y) of F x F into D with the following properties: (9)

(i) {x1 + x
,

m

2/l + 2/2> = <*#!> + < * # 2 >

(ii) = c{x,y)dd

(c,detof a;,yGF).

We call <. , . ) nonsingular if (lOr) and

= 0

VzeF=>*/ = 0

(101)

<s,y> = 0

VyeF=>a? = 0.

For any anti-automorphism 0 of D, the map (»!,...,rr n ), (yls ...,yn)->Xiyie + — +Xnyn0 is a nonsingular 0-bilinear form on D n x D n . Theorem 2.3. Let V be a vector space of dimension n over D; 6, an antiautomorphism of D; and (.,.), a 6-bilinear form on Vx F. Then (. , . ) is nonsingular if and only if it satisfies either of (101) or (lOr). For any linear subspace M of V let (11)

M' = {u:ue F, (x,u) = 0 for all zeilf}.

Then, if(.,.)is nonsingular, M->M' is a duality of\££(V ,D). If n^3, every duality of j£?(F,D) arises in this manner, for suitable 6 and (.,.). The pair 6', (.,.)' induces the same duality as 6,(.,.) if and only if there is a nonzero deD such that for all x, y e V, c e D, (12)

{x, yy = (x, y)d,

Co'

= d-Wd.

In particular, if D is commutative, 6' = 6. Proof. For y eV let tyeV* be defined by ty(u) = (u,y) (ueV). Then t (y->ty) is a 0-linear map of F into F*. Clearly (lOr) is equivalent to the

24

GEOMETRY

OF QUANTUM

THEORY

statement t h a t t is injective while (101) is equivalent to the statement t h a t (0) is the annihilator in V of the range of t; i.e., t h a t t is surjective. The first statement is now clear. Suppose now t h a t ( . , . ) is nonsingular. Then t is a 0-linear isomorphism and so, as M' = t~1(M°) we see t h a t M->M' is a duality of &(F,D). Conversely, if f is a duality of J^(F,D), v:M-> (^(M))0 is an isomorphism of J*?(F,D) onto ^ ( F * , D ° ) . This shows already t h a t (13)

dim(M) + dim(i(M))

= n.

Moreover, theorem 2.1 gives the existence of an anti-automorphism 6 of D and a 0-linear isomorphism t of V with V* such t h a t M° = t[£(M)] for all M in«S?(F,D). If we set {x,y) = (ty)(x) (x,ye V), it is immediate t h a t ( . , . ) is a nonsingular 0-bilinear form and (j(M) = Mf for all M. If 6' and ( . , . ) ' is another pair and ty'{x) = (x,y)' (x, y e V), 0', (.,.}' also give rise to £ if and only if t and t' induce the same isomorphism of j£?(F,D) with «^f(F*,D°). Theorem 2.1 now leads to the required result. This completes the proof of theorem 2.3. A duality £ of j£?(F,D) which is involutive is known as a polarity, this means i f = |(£(Jf)) for all M. A polarity is called isotropic if ilf <= | ( J f ) for all one-dimensional M. Theorem 2.4. Let V be a vector space of dimension n^3 over D. Then J?(V,D) admits an isotropic polarity if and only if D is commutative and dim(F) is even. In this case, if£is an isotropic polarity and 2N = dim(V), we canfind a nonsingular skew-symmetric bilinear form (.,.) such that i-(M) = M' for all M. Moreover, we can find a basis {x1,y1,...,xN,yN} for V such that (xi,xj) = (yi,yj) = 0, (xi,yj)=-(yj,xi) = Sij (l^i,j^N). In particular, if £ and £' are two isotropic polarities of3?{V',D), there is a linear automorphism a ofSf( F,D) such that £' = af cr 1 . Proof. Let f be an isotropic polarity of j£?(F,D). By theorem 2.3 there exists an anti-automorphism 6 of D and a nonsingular 0-bilinear form < . , . ) giving rise to f. Since f is isotropic (x,x) = 0 for all xe V. Replacing x b y x+y we see t h a t < . , . ) is skew symmetric. If now c e D, c^ = -c(y,x)

=

c(x,y);

choosing x, y such t h a t (x,y) = 1 we see t h a t 0 is the identity. This means t h a t D is commutative and < . , . > is bilinear. The remaining statements are standard. Theorem 2.4 is proved. We now consider nonisotropic polarities. A 0-bilinear form ( . , . } is called symmetric if (x,y)0 = (y,x) for all x,yeV. Notice t h a t this property is relative to 6. We observe t h a t if ( . , . > is not identically zero, 6 is necessarily involutive; then the set of values of (.,.) is all of D, and

(x,y)62 = (y,xy = (x,y).

PROJECTIVE

GEOMETRIES

25

Lemma 2.5. Let dim( V) ^ 2, ( . , . ) a nonsingular d-bilinear form such that (x,x) = (x,x)d^O for some xeV. Suppose (.,.) induces a polarity. Then this polarity is nonisotropic, 9 is involutive, and (.,.) is symmetric. VrooLIfu,veV9(u9v) = OoB'V^ (D-u)'o(D-u) g (D-v)'o(v,u) Let c = (x,x). Then x $(D -x)' and dim((D •#)') = % — 1, so that

= 0.

F = (D-z)0(D-a;)'. If u,ve(D'x)', ((u,v)x — u,x + cv) = 0, so that {x-\-cv,(u,v)x — u) = 0, proving that {u,v)e — (v,u). Now (.,.) is not identically zero on (D.z)'x(D.z)', and so this already proves that 9 is involutive. The symmetry of {.,.) V xV now follows from direct computation. Lemma 2.5 is proved.

on

Theorem 2.6. Let 9 be an involutive anti-automorphism of D, V a vector space of dimension n over D, and (.,.) a nonsingular symmetric 6-bilinear form on V xV. Then the duality corresponding to ( . , . ) is a nonisotropic polarity unless D is commutative and of characteristic 2. Conversely, let n^3 and let £bea nonisotropic polarity o/J§?(F,D). Then there exists an involutive anti-automorphism 9 of D and a nonsingular symmetric 9-bilinear form ( . , . ) on V x V such that £ is induced by {.,.). If 9' and {.,.)' is another pair, £ is also induced by them if and only if there is a d^O in D such that for all x,yeV,ceD (x,y)f = (x,y)d, c6' = d~xced. In this case de — d. Proof. Given 9 and ( . , . ) , the corresponding duality is a polarity. If this were isotropic we can argue as in theorem 2.4 to conclude that D is commutative, 9 = identity, and <. , . ) is skew symmetric. As (. , . ) is symmetric it follows that D has characteristic 2. Conversely, let n^3 and let | be a nonisotropic polarity of J>?(F,D). Then £ corresponds to a pair 90, ( . , . ) . Choose x e V such that (x,x)0 = c0^0. Writing (u,v)

= (U^QCQ-1,

ce =

C^CQ-1,

we see that £ is also induced by 9, ( . , . ) , and that (x,x) = l. Lemma 2.5 shows that 6 is involutive, and {.,.) is symmetric; clearly, 0, {.,.) induce £. Suppose 9' is an anti-automorphism of D and ( . , . ) ' a symmetric 0'-bilinear form. Then, by theorem 2.3, 9', (.,.)' induce £ if and only if for some d^O in D, (x,yY = (x,y)d, (x,yeV) and cd' = d-1ced for all ceD. If (x,y) = 1, then (y,x) — 1 and so {{x,yy)e' — {y,x)' = d while {{x,yyf' = dd' = d-1ded, also. So we have dd = d.

26

GEOMETRY

OF QUANTUM

Remark. It is useful to note that the form (.,.) erty that (w,w) = 1 for some weV.

THEORY

inducing £ has the prop-

4. ORTHOCOMPLEMENTATIONS AND HILBERT SPACE STRUCTURES An orthocomplementation ofJ?( F,D) is a polarity £ such that M n £(M) = 0 for all M, i.e., an inclusion reversing bijection £ of J^(F,D) with itself such that g(i;(M)) = i f and M n f (ilf) = 0 for all M. It is easy to see that a polarity is an orthocomplementation if and only if (D-x) n £(D-a:) = 0 for all xe F. The preceding discussion leads to Theorem 2.7 (Birkhoff-von Neumann [1]). Le£ w = dim(F)^3 and £ an orthocomplementation of JS?(F,D). TAew £Aere is aw- involutive antiautomorphism Oof J) and a nonsingular symmetric O-bilinear form (.,.) on V x V such that 6 and {.,.) induce £. We have (x,x) = 0ox = 0, and {.,.) can be chosen so that (w,w) = 1 for some weV. Let a symmetric 0-bilinear form < . , . ) be called definite if (14)

(i) (x,x) = 0 o x = 0, v \ >/ (ii) (w,w) = 1 for some

weV.

A definite form is necessarily nonsingular. Assume now that D is one of R, C, or H. (a) D = R. Then the definite forms are precisely the positive definite quadratic forms. So the definite form (.,.) of theorem 2.7 defines a scalar product and f is the orthocomplementation in the corresponding Hilbert space structure. (b) D = C. Here we must assume that 6 is continuous or measurable (analytically well behaved). Since complex bilinear forms are not definite, 6 must be the complex conjugation.The definite forms are thus the Hermitian positive definite forms, and in theorem 2.7, £ is the orthocomplementation defined bv the Hilbert space structure corresponding to the scalar product (c) D = H, the quaternions. A scalar product for a vector space V over H is a *-bilinear form on F x F such that: (i) for any x-+V, (x,x} is in the center of H and coincides with a nonnegative real number; (ii) (x)x) = 0 if and only if x = 0; if \\x\\ = -f- (x,x)%, || • || is a norm for V. If V is complete under ||-1|, it is called a Hilbert space; this is the case when dim(F)
PROJECTIVE

GEOMETRIES

27

ueVbe such that (u,u) is a unit quaternion, say a, and let ueg (H• u) be such that (u,u) is also a unit quaternion, say b; of course V = H • u © £(H *u) • From (u,u)6 = <w,^) we get, after a small calculation, <->«< = (_%

r

c)

where c e R , |c2| + |r| 2 = l. Hence tr(aq0) = 0. Similarly, tr(bq0) = 0. So aq0 and — bq0 are in the same class, proving that qiaq0q1*q0~1 + b = 0 for some unit quaternion qv But then (x,x) = 0 for x = qxu + v. So, q 0 2 = +1 is the only possibility. Since det(q0) = 1, we must have q0 = + 1. So 6 must be conjugation and (.,.) is *-bilinear. In particular, (u,u) is real for all ueV, and >0 for ue V\(0). V thus becomes a Hilbert space under < . , . ) , and £ is the associated orthocomplementation. We thus have Theorem 2.8. Let D = R or H and let V be a vector space of dimension n^3 over D. / / £ is any orthocomplementation in J?(V,D), there is a scalar product for (.,.) converting V into a Hilbert space such that £ is the corresponding orthocomplementation. If D = C and f is regular in the sense that the anti-automorphism of C defined by £ is continuous, then it is the complex conjugation, and £ is induced by a Hermitian scalar product {.,.) that converts V into a Hilbert space over C. Finally {.,.) is determined by £ up to multiplication by a positive real number. We shall now assume that D is one of R, C, or H, and that V is a Hilbert space of dimension n ^ 3 over D, with scalar product ( . , . ) . Theorem 2.9. Suppose £ is an automorphism of J?(V,D) which preserves the orthocomplementation of V. Then there exists a semilinear automorphism Lof V inducing £ such that: (i)

if D = R or H, L is linear, and for D = C, L is either linear or conjugate linear;

(ii)

for all

x,yeV,

(15)

(Lx,Ly) = (x,y) if L is linear, and

(16)

(Lx,Ly)=(y,x) ifJ) = C and L is conjugate linear. In particular, L is an isometry of V.

Proof. We begin with the following observation.Let 9? be an automorphism of D and Lx a ^-linear isomorphism of V onto itself such that the induced automorphism of ^(V,D) preserves the orthocomplementation of V; then, for some d^0 in D, we have, for all x,yeV, (17)

(L1x>L$) = {x,y)<ed

28

GEOMETRY OF QUANTUM

THEORY

Indeed, (x,y) = 0 if and only if {Lxx,Lxy) = 0; so, if we define

(x,y)~ = (LiXtLtfyv1) (x,yeV)

and 0 = y i ^

where 6 is the anti-automorphism corresponding to ( . , . ) , then the pair ifj, (.,.)~ also induces the orthocomplementation in J£( V,D). Theorem 2.3 now leads to (17). This said, let us first suppose that f is induced by a linear automorphism Lx of V. Then (17) becomes (L1xiL1y) = (x,y)d. This shows that d is a real positive number, and we get the required result on taking L — d~^Lv If £ is not linear, then D = C and there is an automorphism

5. COORDINATES IN PROJECTIVE AND GENERALIZED GEOMETRIES The study of the projective geometries =£?(F,D) gains significance in view of the fundamental theorem of classical geometry which asserts that these are the only geometries, at least when the rank is ^ 4. When the rank is 3, i.e., when we are dealing with a projective plane, the geometries j£?(F,D) are characterized by the property of being Desarguesian, namely, that the classical theorem of Desargues is true for them. Of course a projective plane imbedded in a geometry of higher rank is always Desarguesian; this can be proved directly and in a simple manner. However, as was mentioned at the beginning of the chapter, the geometries ££{ F,D) and, more generally, lattices of finite rank, are not adequate to serve as models for the proposition calculi of complex quantum systems. We shall therefore enlarge the scope of our discussion by including a class of partially ordered sets of infinite rank, and proving for them a coordinization theorem that will include the classical result. The objects that we shall study will be called generalized geometries; and the main result concerning them is that they are isomorphic to the partially ordered sets of finite dimensional subspaces of possibly infinite dimensional vector spaces over division rings.

PROJECTIVE

GEOMETRIES

29

Let ££ be a set with partial ordering <; we do not assume that ££ has a unit element. 3? is called a generalized geometry if the following conditions are satisfied: (i) for any finite subset F of J^, \/a&Fa, /\a£Fa, exist in <£?, (18) (ii) if a e 3? and a ^ 0, j£?[0,a] is a geometry. Thus J^7 is a geometry if and only if it has a unit element. If D is a division ring and V a not necessarily finite dimensional vector space over D, the partially ordered set J^(F,D) of finite dimensional subspaces of V is a generalized geometry, which is a geometry if and only if dim( V) < oo. If & is a generalized geometry, ££ has points and every element of ££ is a sum of points. The dimension functions of the J?[0,a] are mutually compatible and define a dimension function on all of J§P. We write d i m ^ ) for sup aGif (dim(a)). We can obviously speak of lines, planes, etc., in j£f. In the rest of this section we shall sketch a proof that any generalized geometry is isomorphic to some J2?( F,D), provided dim(oSf) ^ 4. Our proof is a straightforward variant of the classical one; but we have decided to include a sketch since it is not easy to locate it in the literature in the precise form we need. The reader who wants more details on the part dealing with Desarguesian planes should consult Seidenberg [1] and Heyting [1] (see the Notes at the end of this chapter). Let S£ be a generalized geometry with d i m ^ ) ^ 4. Then all the planes in ££ are Desarguesian. The classical method of constructing the division rings associated to 3? is as follows. We take a line t in ££ and a plane a in 5f containing t, and we choose three distinct points 0, E, W on t. To this data we shall associate a division ring D whose elements will be the points of the line t that are different from W. The operations of addition and multiplication in D will be defined with the help of geometrical constructions (in the plane a) that imitate the familiar constructions in the Euclidean plane. Addition. Let AGD; then we define the map aA'B-+B + A ofD with itself in the following manner. In the plane a we take distinct lines q, r through W9 different from t, and distinct points X, Y on q, different from W9 such that if n(X; r,t) (resp. n(Y; t,r) is the perspectivity from r to t with center X (resp. from t to r with center Y), then n(X; r,t) n(Y; t,r) takes 0 to A; then we define (Fig. 1) aA =

7r(X;r,t)7T(Y;t,r).

If we view q as the "line at infinity" then this construction resembles the usual Euclidean one (Fig. 1(a)).

30

GEOMETRY

OF QUANTUM

FIG.

1(a)

THEORY

PROJECTIVE

GEOMETRIES

31

Multiplication. Given A e D we define right multiplication TYIA by A; if A=0, MAB = 0 by definition for all B and so we take A^O. We take in the plane a a line g through W different from t, and a line p through 0, different from t. We choose distinct points X, Y on q different from W such that n(X; p,t)n{Y; t,p) takes E to A (see Fig. 2); then m^ = n(X;pft)7r(Y;

t,p).

If we view g as the "line at infinity", this resembles the usual construction in the Euclidean plane (Fig. 2(a)). Of course we could have equally well taken this as the definition of A -B; our choice is dictated by the requirement of getting isomorphisms with the geometry of subspaces of a left vector space. Although these definitions involve the choices of auxiliary lines and points, the Desarguesian nature of the plane can be used to prove that addition and multiplication are well defined and convert D into a division ring with 0 and E as its null and unit elements. If t, 0, E, W are fixed, this division ring is independent of the choice of the plane a that contains t; for, if 6 is a different plane containing t, and X is a point in the three-dimensional space avb which does not lie on a or 6, the perspectivity from X gives an

FIG.

2

32

GEOMETRY OF QUANTUM

FIG.

THEORY

2(a)

isomorphism of ^f[0,a] with J£[0,b] which is the identity on t — a/\b, and it is clear that addition and multiplication on D are the same whether we use a or b. The division ring is also unchanged up to isomorphism if we change t 0y E, W. More precisely, let t' be another line in the plane a, and O', E', W, three distinct points on t'. Then any projectivity in the plane that takes t to t' and 0, E, W, respectively to 0', E', W', is an isomorphism of the associated division rings; this follows from the well-known fact that any projectivity of t with t' extends to an automorphism of the plane j£?[0,a]. We remark finally that D is commutative if and only if the geometry of J&?[0,a] is Pappian for any plane a in jSf, i.e., the theorem of Pappus is valid in o£?[0,a] for any plane a. The first step in the coordinatization of ££ is to define" afnne " coordinates for all points of JSf that do not "lie at infinity". If S£ is a geometry we can choose an arbitrary hyperplane to be at infinity. For a generalized geometry we use a simple modification of this procedure. To obtain an intuitive picture of the definition of coordinates the reader is asked to picture everything in a Euclidean context, interpreting incidences at infinity in terms of parallellism. Let us call a set of points of j£? independent if every finite subset of the set is independent. If {P3} (j e J) is an independent set of points, then we have a map u (K->u(K)) from finite subsets of J into j£? defined by (19)

u(K)=

V Pi9

u(4>)=0.

PROJECTIVE

GEOMETRIES

33

I t is easy to verify t h a t (20)

u(Kt

u K2) = u{Kx) v u(K2),

u(K± n K2) = u{Kx) A tt(X2).

An independent set C of points of J§? is called a 6cms if it is maximal. Every independent set is contained in a basis in view of Zorn's lemma. If C is a basis of J£? and P is a point of if, it follows from the maximality of C t h a t there is a finite set F <= (7 such t h a t P < \/QeFQ; hence for any ae^, there is a finite set (r c 0 with a <\/Qe0Q. Let 0 be a point of if. By a frame a t 0 we mean a pair (#,{PJ ; . G t / ) such t h a t {0,{P,} yet/ } is a basis for if. There are frames at any point of i^7. From now on we fix a frame (0,{Pj}jeJ) a t 0. An element a e i f is said to lie at infinity if there is a finite set K^J such t h a t adim(u(K)) so t h a t avu(K)

=

Ovu(K);

if a^ = aAu(K), then dim(a 00 ) = dim(a) — 1, from which the characterizations of a^ foliow immediately. The uniqueness of a^ is obvious and a^ is called the hyperplane at inifinity in a. I n particular, if P is a point not in ifoo a n d K ^ J a finite set, then for a = P\/u(K), u(K) is the hyperplane at infinity. Every line not in ££^ has a unique point at infinity on it, every plane not in ^^ has a unique line at infinity on it, and so on. Let rrij be the line 0\fP3 (jsJ). For each j e J we choose a point Ej in m3distinct from O and P3. As explained earlier we convert the set of points of nij different from Fj into a division ring Dy with O and E3 as its null and unit elements, respectively. The division rings D^ are of course mutually isomorphic b u t we can do better and introduce a " c o m p a t i b l e " system of isomorphisms. Let j , keJ, j ^ h. The lines E3vEk and P3\/Pk, being in t h e plane m3\imk, meet at some point, say P3k; this is the point at infinity of EjVE^ and the perspectivity from P3k, of the line mk onto the line m3, fixes 0 and takes Ek (resp. Pk) to Ej (resp. P3). I t thus induces an isomorphism of Dfc onto D ; , which we denote by (21)

d3k:Dk^D3,

6jk = n(P3k;

mk,m3).

We define d33 to be the identity. Then the "compatibility" of the 6jk may be formulated as the following lemma. Lemma 2.10. Ifj, (22)

k,reJ,

then &rj ° Ojk = 6rk-

34

GEOMETRY

OF QUANTUM

THEORY

Proof. From (21) it is immediate t h a t 0kj = 6jk~ . Suppose now t h a t j , Jc, r are three distinct indices from J, XeBk, Y = Ojk(X), Z = 6rj(Y). W e must prove t h a t Z = 6rk(X). W e m a y assume X^O, X^Ek. Desargue's theorem applied t o the triangles X YZ and EkEjEr which are in perspective from 0, gives a line t on which t h e points (X V Y) A (Ek V Et) = Pik, and

(YvZ)A

(XvZ)A(EkvEr)

(E, V Er) = Pjr,

=Q

lie. Obviously t is a t infinity a n d so Q = P rfc , proving t h a t Z = 6rk(X). From (22) we see t h a t there is a division ring D a n d isomorphisms (23)

,-: D ^ > D ,

such t h a t t h e diagrams

&. D

(24)

^

D

'

are commutative. W e write 0 a n d E for t h e null a n d unit elements of D. Let ty be the set of points of «£? which do not lie a t infinity. Let us denote b y W the vector space of all functions with finite supports on J with values in D. W is a left vector space over D in t h e usual fashion. Suppose now P ety. We can choose a finite setK^J such t h a t P < Ovu{K). For any j e J, u(K-{j}) is t h e hyperplane a t infinity on Pyu(K-{j}) a n d so t h e latter does n o t contain P y ; thus MjKP =

mjA(Pvu(K-{j}))

is a n element of D,. If Kx <^K is such t h a t P
MjP = mj A ( P V !*(#-{j}))

( P < 0 V w(JC)).

p

Transferring i f , t o D now gives t h e coordinates of P , i.e., let fp be t h e function from J t o D defined b y (26)

tpU)=vr1(M,p)

(jeJ).

Lemma 2.11. For any P e ^ , fp is in W; fo = 0, the origin ofW; and P-> fp is a bijection ofty with W . Proof. If P
U(j'-{jr})

=0

PROJECTIVE

GEOMETRIES

35

by (20). So, P e ty and it is easy to verify that fP = g. If P' e $ is also such that fP, = g, it is clear that P' < Ovu(J'), and hence, for 1 < r < 8,

Qr=MJrr'
P
M^^MfJeK.

Fixj,keK,j^k.

The triangles PMjpMkp and QM^Mjfi

are in perspective from O and so have an axis of perspectivity. This is a line at infinity (cf. the remark above) and so (M,p V Mkp) A (Mfl vMkQ) = T is a point at infinity, which lies on the line Pj\/Pk because all four points are in the plane a = 0vP,vP fc - If n denotes the perspectivity, in the plane a = OvPj\/Pkiofmjto mk with center T, we can use the definition of multiplication in-D, to conclude that if Nj = (6jkon) (Ej), then for any UeDjt (6jko7T)(U)=U'Nj. Thus djk(MkQ) = MjQ-Nj and djk{Mkp) = Mjp>Nii proving that 0Jk(Mk) = Mj. For the converse, suppose Q ^ 0 is in ^ with fQ = M • ip. Assume as before, to avoid trivialities, that K has at least two elements. Choose a>j0eK and let Q' be the point on the line 0\jP such that the lines Q'vMJoQ and PyMJop meet at infinity. Obviously, Q'ety, Q'j-O, and Jf, o «' = Jfy G. Since

GEOMETRY

36

fQ/ = j|f'-f P for some M'eD, Le.,Q = Q'.

OF QUANTUM

THEORY

we have
Lemma 2.13 ("parallelogram law of addition"). Let 0' e ^S, 0 V 0, and let t' be a line through 0' not containing 0. Let U, R be the points at infinity on the lines OyO' and t', respectively, and let t be the line OyR. If PeSfi lies on tand (27)

P' =

then

(UyP)At',

fp/ = fp-ffo-.

Conversely, suppose P' e ^} lies ont'\ then fp' — fcr =fp for some Pont, P and P' are related by (27).

and

Proof. For the first part we must show that Mf^Mf + Mfi* for all jeK where X g J i s a finite set such that O', P, P' are all <0\lu(R). If U
FIG.

3

As 0" < 0\ju{K - {j}) we have U' < u(K - {j}) and so Mf

= Mjp",

Mf

=

M?K

It is clear from Fig. 3 that with respect to addition on the line t (minus R), Pm = P + 01\ so, if t = mh Mf'^Mf + MpK Ift^mj, let T be the common

PROJECTIVE

GEOMETRIES 0

37 P

point at infinity on the lines O^siM, *, PvMf, P"yMj ". Consider the plane tymj9 and in it, the perspectivity from t to m5 with center T. As this takes Ov P, P" to if,0*, M$p, and Mf", respectively, we get Mf^Mf

+ M,0*.

For the converse let g = fp'-fo-. Define P = tA(UvP'). Then P and P ' are related by (27) and so g = fp. Lemma 2.14. Let P x ', P 2 ', P 3 'feeJAree distinct points inSfi. Then they are collinear if and only if for some A^O inD we have (28)

tps^A-tps + iE-Aytp,.

Proof. Assume first that the P{ are on a line t'. Then (28) follows from lemma 2.12 if t' contains 0. If 0 <£ t\ we can apply lemma 2.13 with 0' = P 2 'If P x and P 3 are the points on the line 0\lE corresponding to PX' and P 3 ', we have fp3, = fp2, -|- fp3> fPi, = fp2, -f fPi) while lemma 2.12 gives a nonzero ^. eD with fp3=A -fp,. This implies (28). For the converse, assume (28) and let t' be the line P^yP^. We may suppose that Oj;tf, as the case 0
gp(i) = g ( «

(29) and (30)

y(P) = D-gp.

Suppose P is a point at infinity on ££. We choose a point Q on the line 0\lP such that # ^ P , Q ^ 0 and define g P , Q e \ , y(P) eS£' by (31)

g

3 eJ, 3 = 00,

and (32)

y(P) = D- &P.Q-

Lemma 2.12 shows that y(P) in (32) is independent of the choice of Q and so is well defined.

38

GEOMETRY

OF QUANTUM

THEORY

Lemma 2.15. The map P -> y(P) is a bijection, of the set of all points of J£? with the set of all points of j£?', which preserves collinearity. Proof. The bijective nature of the map P->y(P) is a straightforward consequence of the definitions and we sketch the argument for proving that y preserves collinearity. Fix three points of j£?, say Pf (i— 1,2,3); we must show that they are collinear in j£? if and only if y(Pi) are collinear in -Sf". Case 1. P{• G^} for all i. In this case lemma 2.14 gives what we want. Case 2. Exactly one of the Piy say P 3 , lies at infinity. Let Q be a point on OvP 3 , Q¥^0,Q^Pz. It is trivial that y{Px), y(P2), a n d 7(^3) a r e collinear if and only if for some M eD, £Q = M -(fp2 — fpj. Lemmas 2.12 and 2.13 show that this is precisely the condition for Q to lie on the line joining 0 to the point at infinity on Px\/P2, i.e., for P 3 to lie on P1yP2. Case 3. P 2 and P 3 are both at infinity. Suppose P l 5 P 2 , P 3 are collinear, so that P1 is also at infinity. Chooce Q*G^P> Qi^O (i = 1,2) such that: (i) Qi a n d 7(^3) are collinear. Conversely, if y(Pi), y(P2)> y(P3) are collinear, choose Q*e^p, Qi^O in OvP*, * = 1,2,3. Then for some

M,N^O

mD}

By lemma 2.12 we can find points Q/ ^ 0, such that Then *b3' = ^ ' —f^x'. If X is the point at infinity on Qi\zQ2', lemma 2.13 and this relation tell us that Q3 is on OyX. So X = P 3 , i.e., the P z (^ = 1,2,3) are collinear. This proves the lemma. These lemmas imply the following result which is the fundamental theorem on coordinatization of geometries. Theorem 2.16. Let & be a generalized geometry with dim(J^) ^ 4. Then there exists a division ring D and a left vector space V over D such that 3? is isomorphic to the generalized geometry of all finite dimensional subspaces of V. V is finite dimensional if and only if 3? is a geometry.

NOTES ON CHAPTER II 1. For more detail on quaternionic geometry see Chapter I of Che valley [1]. The identification of H with C2 in Section 2 gives rise to a bijection

PROJECTIVE n

GEOMETRIES

39

2n

H ^>C which is C-linear and associates q = (qv ...,#«) to x=^{x1,...ix2n) where qk = x]^-\-xn+k}i. We may regard H as a Hilbert space with scalar product (q,r)=21*/> =

S

(Zkyn+k-yk%n+k),

where x = (xv ...,#2«) and y = (yv ...,y2n). 2. The definitions in Section 5 are classical. For any line t in a Desarguesian plane n let G be the set of all projectivities of t with itself which are compositions of two perspectivities: from t to a second line s followed by a perspectivity from s to t. The line s, as well as the centers of perspectivity, are arbitrary. Assume that s^t and that the two centers, X, Y are distinct. Then the projectivity has one or two fixed points according as the line Xv Y passes through sAt or not. In the former case the projectivity is called special and the point W, which is on s, t and XV Y, is the unique fixed point and is called its canonical fixed point. If 0 = sAt and W = (XyY)At are distinct, they are the two fixed points and are respectively called the first and second canonical fixed points of the projectivity. For any W on t (resp. distinct 0, W, on £), GStw (resp. Gg>o, w) is the set consisting of the identity and the special (resp. general) projectivities of t with itself, with W as canonical fixed point (resp. 0 and W as the first and second canonical fixed points). The key fact, which follows from the Desarguesian nature of 77, is that if aeGStw (resp. aeGgio, w), a is determined by its action on one point oit other than W (resp. its action on one point of t other than 0 and W), independently of the choices of s, X, Y. It is immediate from this that the operators a A and rriA are well defined. The possibility of using different s, X, Y to define the same projectivity then allows one to prove that GSt w and Gg>0>w are groups, the former being abelian. This will prove that + converts D into an abelian group with O as its null element, and • converts D \ 0 into a group with E as unit element. Right multiplication is of course in Gg, o. w\ left multiplication is also a projectivity of t with itself fixing 0 and W, which however is a composition of three perspectivities. Distributivity laws are special cases of the more general result that if a is any projectivity of t fixing 0 and W, a is an automorphism of the additive group of D. This is an immediate consequence of the extension principle stating that any projectivity from one line of n to another can be extended to an automorphism of 7T. This principle, in the case of a perspectivity, is an easy application of the following lemma: Lemma. Let C be a point of n; t, t' distinct lines through C; and A, A' distinct points ont', different from C. Then there is a unique automorphism a

40

GEOMETRY

OF QUANTUM

THEORY

of the plane n such that: (i) a fixes each point of t\ (ii) a(A) = A'; (iii) for any point X not on t} a(X) is on the line joining C to X. Proof. Consider first a line u through C distinct from t and two points X, X' on u, distinct and different from C. Let P(u) be the set of points of the plane not on u. We define dx, x' to be the m a p P(u) -> P(u) such t h a t for a n y B in P(u), B' = dx,x'(B) is the point where the line KvX' meets the line CvB, K being the point where the line ByX meets t (draw a figure). I t is easy to see t h a t dx, x' is bijective and its fixed points are precisely those on t (except for C). If v is the line ByB' for some B in P(u), 6B, B' is well defined on P(v), and a suitable application of Desargues' theorem shows t h a t QB,B' = QX,X' on P(u) nP(v). We now define a as follows. For a n y point P<^,a(P) = P;ifPH:^andP^r,a(P) = ^,^(P);ifP
PROJECTIVE

GEOMETRIES

41

of principal right ideals to form a complemented partially ordered set; and finally, that in this case the set of principal right ideals is a complemented modular lattice (so that principal = finitely generated). Of course in all of this "right" can be replaced by "left". The lattice will be irreducible if and only if R is indecomposable in the sense that R does not admit any decomposition of the form R = 7^©R 2 where R^O are two-sided ideals; and this is equivalent to requiring that the center of R is a field. Finally, if a ring R has the property that its principal right (or left) ideals satisfy the ascending and descending chain conditions, then R is regular if and only if it is semisimple, thus tying in the classical coordinatization theory with the Wedderburn theory. It was also established by von Neumann that dualities of 3? correspond naturally to anti-automorphisms of R, and that orthocomplementations correspond to involutive anti-automorphisms x->x* which are definite in the sense that x*x=0oa; = 0. Rings of bounded operators in a complex Hilbert space are regular only when they are finite dimensional; however, the ring of not necessarily bounded operators affiliated (in the sense of Murray and von Neumann) to a type IIX factor is a regular ring, and is in fact the regular ring whose principal right ideals form the lattice isomorphic to the lattice of projections of the factor. Occupying a special place among the complemented modular lattices are the continuous geometries also discovered by von Neumann. He characterized completely the associated regular rings. The lattices of projections of type IIX factors are orthocomplemented continuous geometries. Roughly speaking the axioms for an irreducible complemented modular lattice to be a continuous geometry require the completeness of the lattice and the continuity of the lattice operations, and lead to the existence of a dimension function with values in the unit interval [0,1]. Let D be a division ring, Vk and Vmk (m an integer ^ 2) be left vector spaces of dimensions k and mk, respectively, over D. We then have a map M-+M<$>...®M (ra summands) of ^(VkiD) into J^(Fmfc,D). Using a succession of such mappings (with changing "ratios" m), von Neumann constructed continuous geometries associated with D which, when D = C, were not isomorphic to the projection lattices of type IIX factors. These scattered remarks should convey a glimpse of the beautiful generalization of classical projective geometry that von Neumann erected. The reader who wants to get a sharper perspective on these things should begin with a study of von Neumann's book (Continuous Geometry, Princeton University Press, Princeton, New Jersey, 1960, which has annotations by I. Halperin that illuminate many aspects of the theory and contain references to further literature), and his papers on the subject that appear in volume IV of his Collected Works (6 volumes, Pergamon, 1961).

CHAPTER III THE LOGIC OF A QUANTUM MECHANICAL SYSTEM 1. LOGICS Let © be any quantum mechanical system. We have seen that one can associate with <& the partially ordered set ££(!&) of all experimentally verifiable propositions concerning <&. The partial ordering is that induced by the implication relation. Moreover, the map, which associates with any element of ^ ( G ) its negation, behaves very much like an orthocomplementation. This leads one to introduce axiomatically a class of partially ordered sets and to study their properties. We call these systems logics. The basic assumption of modern quantum theory can then be described (if we anticipate some terminology) by saying that ££(<&) is a standard logic. Let j£? be a lattice under a partial ordering < . By an orthocomplementation of j£? we mean a mapping a-^a1

(_L) :

of & into itself such that (i) J_ is one-one and maps «£? onto itself, (ii) a < b implies b1 < a1, (1) (hi) a 1 1 = a for all a, (iv) a A a 1 = 0 for all a, (v) a V a1 = 1 for all a. We note that the relations 0 1 = 1 and 1 1 = 0 are easy consequences of (ii). Moreover it follows easily from (iii) that J_ is one-one and onto, so that (iii) => (i). Finally, we observe that (ii), (iii), and (iv) imply (v). In fact, let a e 3? and b = a\jaL. Since both a and a1 are
(i) for any countably Vn «n and / \ n an (ii) if al9 a2e J? and that b
infinite sequence al9 a2, - - - of elements of JSf, exist in &, a1
THE

LOGIC OF A QUANTUM

MECHANICAL

SYSTEM

43

Before proceeding further we make the observation t h a t if a1
SP[a,b] = {c : c G J^,

a < c < b}.

Then under the partial ordering inherited from <j£f, j£?[0,6] becomes a lattice, in which countable unions and intersections exist and whose zero element is 0 and unit element is b. If we define, for any x in ^ [ 0 , 6 ] , its orthogonal complement x' by x/ = x1Ab, then it can be shown t h a t j£?[0,6] equipped with the orthocomplementation x —> x' is a logic (cf. corollary 3.3). The central assumption t h a t one makes in any quantum mechanical application is t h a t the set of experimentally verifiable propositions is a logic. The only thing t h a t m a y be open to serious question in this is assumption (i) of (2) which forces any two elements of S£ to have a lattice sum; the others m a y be regarded as technical necessities. We can offer no really convincing phenomenological argument to support this (cf. BirkhofT-von Neumann [1]). If we omit the assumption t h a t 3? is a lattice the axioms become so weak t h a t it is very difficult to avoid pathology in any mathematical discussion. We point out also t h a t every calculation in quantum mechanics is based on assumptions which not only imply t h a t the set of experimentally verifiable propositions is a logic but in fact a very special one. We shall now make a few general remarks. If 3?^ and ££2 are two logics, an injection of 3?x into j£?2 is a m a p / ( a - > / ( « ) ) of 3?x into 3?2 such t h a t ( i ) / i s one-one,/(0) = 0 , / ( l ) = l ; (ii) if al9 a2, • • • is a n y at most countable sequence of elements of &l9 f(\/nan) = V n / K ) and f(/\an) = A / K ) ; (iii) f(a1)=f(a)1 for all a e S£x. An isomorphism of S£x on S£2 is an injection which maps <£x onto J^ 2 - If <^i = o^ 2 , isomorphisms are called automorphisms. The set of all automorphisms of a logic J£ is a group under composition; we denote it by Aut(J^). If ££x and ££ are two logics, we say t h a t ££x is a sublogic of S£ if (i) ^ c ^ and (ii) the identity m a p of =£?! into ££ is an injection.

GEOMETRY

u

OF QUANTUM

THEORY

Any Boolean cr-algebra is a logic provided we define, for any element a, a1 to be the complement of a. These logics are of course not very interesting from the point of view of quantum mechanics. To obtain a more typical example, let us consider a Hilbert space £F over the real, complex or quaternionic division rings. We denote by J^(Jf) the collection of all closed linear manifolds of 3tf. If we now define < to mean set inclusion, and J_ to mean the usual operation of orthogonal complementation in ^ ( ^ f 7 ) , then it can easily be verified t h a t ^f (Jf) is a logic. The isomorphism class of ^(Jti?) depends only on the field D of definition of 34? and the dimension of ^f \ A logic ££ is said to be standard if it is isomorphic to the logic j£?(c^) of a separable infinite dimensional Hilbert space over one of t h e three division rings R, C, or H. Modern quantum theory works with the assumption t h a t the logic of any atomic system is, if not standard, at least a sublogic of a standard logic. We shall make a deep study of standard logics in Chapter IV. We shall also describe there a number of logics t h a t are sublogics of standard logics, b u t are not standard themselves. We shall now proceed to derive a few simple consequences of the axioms denning a logic. Lemma 3.1. Let ££ be a logic and ava2, ...a sequence of elements of ££. Ifb e S£ and b\_anfor all n, then &JL.Vn an- Moreover we have the identities:

(V OnY = A «n\

Finally,

if ax
c
Proof. Since b
b
(a V 6) A c = (a A c) V b.

Proof. Write d = (aAc)vb

and e = (avb) Ac. I t is obvious t h a t

d<e.

THE

LOGIC OF A QUANTUM

MECHANICAL

SYSTEM

45

Hence, by (ii) of (2), there exists an element g of J? such t h a t d_\_g, and dvg = e. Since a±b and the element g satisfies the relations gx' defined above is a logic. Proof. We must verify (ii), (hi), (iv) of (1) and (ii) of (2). (ii), (iv) of (1) are obvious. We shall now prove (hi) of (1). Let x
2. OBSERVABLES We shall introduce in this section the concept of an observable associated with a logic. The formal definition is given in the next paragraph, but we shall first t r y to motivate the definition. Suppose t h a t (5 is a physical system, S£(&) its logic, and £ a physical quantity or observable. Then in any experiment which an observer performs, the statements t h a t can be made concerning £ are of the type which asserts t h a t the value of £ lies in some set E of real numbers. I t is natural and harmless to require

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that the sets E be Borel. If we denote by x(E) the statement that the value of £ lies in the Borel set E^R1, then one has a mapping x : E -> x(E) of ^(R1) into J^(©). We shall regard two observables as "identical" if and only if the corresponding mappings are the same. If / is a real valued Borel function of a real variable, then we mean by / o £ the observable whose value is f(r) whenever £ takes the value r; to this observable clearly corresponds the mapping E -> x(f~1(E)) (cf. Chapter I, sections 2 and 4). Motivated by these remarks we introduce the following definitions. Let j£? be a logic. An observable associated with ££ is a mapping x : E -> x(E) of the o--algebra ^(R1) of Borel sets of the real line into ££ such that (6)

(i) a.( 0 ) ==o,a?(J81) = 1, (ii) if E, F e ^(R1) and E n ^ = 0 , x(E) _[_ x(F), (hi) if 2£1? 2£ 2 >' ' ' i s a sequence of mutually disjoint Borel sets in R1,

x(y j?n) = y ^ j . We write 0 or 0(j£?) for the set of all observables associated with J£f. We remark that the properties (i) through (iii) are natural if we want to interpret x(E) as the statement that the value of the observable £ lies in E. If X is any set and j£? is an arbitrary cr-algebra of subsets of X, then j£? is obviously a logic provided we define < to be set inclusion and _[_ to be set complementation. Theorem 1.4 asserts that the set of observables is in canonical one-one correspondence with the algebra of all real valued J^-measurable functions on X; if x(E —> x(E)) is any observable, then the corresponding function / on X is the unique ^-measurable one such that f~1(E) = x(E) for all E e ^(R1). Suppose that x is an observable and / a real valued Borel function of a real variable. Then (7)

fox:E^x{f-\E))

is also an observable. Lemma 3.4. Let S£ be a logic and x an observable associated with £\ Then for any sequence El9 E2, • • • of Borel sets (8)

U

/

»

*(n En) = A *{En).

THE LOGIC OF A QUANTUM

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47

Suppose further that fx and f2 are two real valued Borel functions of a real variable t. Let f± of2 denote the function t ->/i(/ 2 W)- Then (9)

(/i°/2)°z=/i°(/2°*).

Proof. Since x(En)±x(R1-En) and x(En) vx(Rl-En) = l, it follows 1 1 that x(R — En)=x(En) . Consequently, in view of the equation (4), the second equation in (8) will follow if we prove the first. To prove the first, we begin by observing that if A c= B, then x(A) <x(B); in fact, x(B) = x(A) v

x(B-A).

This said, let E = \Jn En. Then x(En) < x(E) for all n so that \Jn x(En) < x(E). On the other hand, let F1 = E1 and for n>l, let Fn = En-\Jj
<

n

\/z(En). n

This completes the proof of (8). Equation (9) is trivial. Let x be an observable associated with ££. A real number A is said to be a strict value of x if x({\}) ^ 0 . x is said to be discrete if there exists a countable set C = {cu c2, • • •} of real numbers such that x(C) = 1. a; is said to be constant if there exists a real number a such that x({a}) = l; we shall then write x — a. x is said to be bounded if there exists a compact set K such that x(K) = 1. If we define a(x) by (10)

a{x)=

p|

0,

C closed, x(C)= 1

then G(X) is a closed set called the spectrum of x. Since the topology of R1 satisfies the second countability axiom, there exists a sequence of closed sets C1,C2i--' such that x(Cn) = l for all n and a(x) = P| n C n . Since x(Rx-Cn) = 0 for nil n, x(Ri-*(x))

=

x(\J(Ri-Cn)}

=

\/x(Ri-Cn) n

= o, which proves that x(o(x)) = 1. or(#) is thus the smallest closed set C such that x(C) = l. The numbers A GCT(#)are called the spectral values of #. A strict value is a spectral value but the converse is not true in general, x is bounded if and only if G(X) is a compact set. A e a(x) if and only if for any open set U containing A, x(U)^0. Since the defining axioms of a logic are somewhat weak, it is not reasonable to hope for any incisive description of the set of all observables associated with ££\ In the next chapter we shall examine this question

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for logics which are associated with Hilbert spaces. For the moment we confine ourselves to a remark on discrete observables. Suppose ££ is an arbitrary logic. Let # be a discrete observable and {cl5 c2, • • •} the set of its strict values. Then x{{c$) and #({c;}) are orthogonal whenever i^j and

V*({c,}) = l. i

Conversely, let {Ci}ieD be an at most countably infinite set of distinct real numbers and {at}ieD a family of mutually orthogonal elements of j£? (with the same indexing set D) such that

v «f = i. ieD

Then there exists a unique discrete observable x associated with £? such that x({ci})=ai for all i e D. In fact, we have only to define, for E e^(R1), x(E) =

V «iiiCieE

It is easy to verify that x is an observable. It is clear that x({ci}) = ai for all ieD and that x is discrete. The uniqueness of x is obvious. Even though many interesting and important observables are discrete, there are quantum observables which are not discrete. The construction of observables which are not discrete is, however, quite complicated. We shall indicate how it can be done in the next chapter for logics which are associated with Hilbert spaces. If j£? is an orthocomplemented geometry of finite dimension N, then it is easy to see that every observable associated with j£? is discrete and has at most N values. Clearly, such logics are not suitable models to represent the experimentally verifiable propositions of complicated atomic systems.

3. STATES If © is a classical mechanical system, the concept of a state of © is so defined (cf. Chapter I) that if © is in a state p, and £ is any observable, then £ has a value in p. This leads one to postulate that the states are points of the phase space and observables are real valued functions on it. In quantum theory this has been rejected as irreconcilable with the available experimental facts regarding the behavior of atomic systems when subjected to refined microscopic observations (cf. von Neumann [1], Heisenberg [1]). The modern approach to atomic physics is based on the principle that in a given state of the system, an observable has only a probability distribution of values and in general no sharply defined value; and no matter how carefully the state is prepared, there will be some observables whose values are distributed according to probability distributions

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having arbitrarily high variance. Accordingly, if (B is in a state or, one can associate, with each observable x, a probability distribution, say Px , on the line, which will be interpreted as the probability distribution of the values of x when the system is in the state a. If a; is a discrete observable, and cl9 c 2 , • • • are its possible values, then Px would have mass concentrated on the set {c±, c2, • • •}; the probability mass which is then assigned by Px to the point cn will be the probability t h a t an experiment on @, designed to yield an exact value of x, gives the value cn, when @ is in t h e state a. I f / is any real valued function of a real variable t, then, for t h e o b s e r v a b l e / o x, we would have the equation: P r o b { / o x = a, when (5 is in the state o-}

(ii)

P

= 2

**(K}).

n:f(cn) = a

With these remarks serving as our motivation we proceed to the formal definitions. Let 3? be any logic and 0 the set of all observables of ££\ A state function of ££ is a m a p (12)

P :x->Px

{xeO)

which assigns to each observable x e 0 a> probability distribution Px on ^(R1) such t h a t for any real valued Borel function / on R1 and a n y observable x, (13)

Pf0X(E)

=

Px(f-HE)).

Equation (13) has an interesting consequence. We claim first t h a t if 0 is the zero observable, then P0 is the probability measure whose entire mass is concentrated at the origin. I n fact, if g is the function identically zero, then g o 0 = 0; since g~1(E) = R1 or 0 according as 0 e E or 0 <£ E, we have, by (13), P0(E) = P0(g-1(E) = l or 0 according as 0 e E or 0 £ E. We next observe t h a t if x is a n y observable and E is a Borel set such t h a t x(E) = 0, then Px(E) = 0. I n fact, ifx{E) = 0, then, for the f u n c t i o n / whose value is 1 on E and 0 on R1 — E, it is obvious t h a t y —f o x is the zero observable and hence Py is the measure concentrated at the origin. As E = {t:f(t) = l}, we have Px(E) = Py({l}) = 0. I n other words, (14)

PX(E) = 0

(Ee ^(R1),

x(E) = 0).

(14) implies t h a t if x is discrete, the mass of Px is concentrated on t h e set of strict values of x. I n turns out to be possible to construct all the state functions of ££ in a particularly simple manner. In order to describe this construction we introduce, with Mackey [1], the notion of a question. An observable x

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is said to be a question if #({0,1}) = 1; x is then discrete. If a = x({\}), it is clear that x is the only question such that x({l}) = a. We shall call it the question associated with a and denote it by qa. Suppose now that x is an observable associated with the logic 3? and E any Borel set ^R1. If XE *S the function which takes the value 1 on E and 0 on R1 — E, then XE ° x *s the question associated with x(E), i.e., (15)

XE°x

= qx(Ey

In fact, since XE takes only the values 0 and 1, it is clear that XE ° x i s discrete and (xE ° #)({0,1}) = 1. XE ° # is therefore a question, and, as E is the set where XE takes the value 1, (XEOX)({1})=X(E).

The description of all state functions can now be given in terms of the concept of a measure on 3. A measure on S£ is a function (16)

j9:a-H^(a)

(a

eg)

such that (i) # is real valued and 0 < p(a) < 1 for all a e ^ , (H) 3>(0) = 0, i»(l) = 1, (17) (hi) if ax, a2, • • • is a sequence of mutually orthogonal elements of Se and a = Vn «n> then p(a) = 2n ^ K ) Notice that if al9 a2 e j£? and ax
p(a±) < p(a2)

(ax < a2).

The concept of a measure on ££ is a generalization of the well known concept when j£? is a Boolean cr-algebra. However ££ need not be distributive and hence the structure of a measure on & is much more complex than that of a measure on a Boolean cr-algebra. Theorem 3.5. Let 3 be a logic and 0 the set of all observables associated with ££. Let p be a measure on J&f. / / we define, for any observable xe(9 and any Borel set Eg^R1, (19)

P/(E)

= p(x(E)),

then Pxp is a probability measure on ^(R1) and (20)

Pp : x -> P /

is a state function of ££. Conversely, if P(x -> Px) is an arbitrary state

THE LOGIC OF A QUANTUM

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function of ££\ there exists one and only one measure p on 3? such that Px(E)=p(x(E))forallx. Proof. Suppose that p is a measure on ££\ If El9 E2, • • • are mutually disjoint Borel sets of E1 with union E, then x(Ex), x(E2), • • • are mutually orthogonal elements of ££ with x(E) = \Jn x(En); moreover,

PX»(E) = 2

p p

x {KY

n

Since Pxp(E1)=p(l) = l) we see that Pxp is a probability measure on ^(E1). If / i s a real valued Borel function of a real variable, then P?ox(E)=p((foX)(E)) =

p{x{f-\E)))

=

Px*(f-i(E)).

This proves that x ~> Pxp is a state function of ££\ Conversely, let P(x —> Px) be a state function of ££. If a e <£?, P a is a probability distribution on the real line and as qa is a question, (14) ensures that PQa({0,l}) = l. Define (21)

p(a) = P, o ({l}).

^)(a->^(a)) is clearly a well-defined, real valued function on ££ and 0 i be a sequence of mutually orthogonal elements of j£? and let a = \/nan. Let a; be a discrete observable such that ^({0})=^ and x({n}) = an, n = 1, 2, • • •. Let fn be the function on E1 whose value is 1 for the point n and 0 otherwise. Then/ n o x is the question qUn and we have, by (21) and (13), p(an) = Px({n}). Since Px is a probability measure, it follows that

J p(an) = PX(Z), n=l

where Z is the set {1, 2, • • •}. If/ denotes the function whose value is 1 on Z and 0 on E1 — Z, f o x is the question ga and hence, exactly as before, p(a) = Px(Z), so that

(22)

*>(a) = 2 ^ ^ ) . n= l

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p is thus a measure on 3?. If x is any observable and E e ^(R1), we have, on writing g for the function whose value is 1 on E and 0 on E1 — E, PX(E) = PgoX({l}) = p{x{E)). Finally, equation (21) implies easily that p is unique. This completes the proof of the theorem. In view of the theorem we proved just now, it is natural to define a state of a logic to be a measure on 3?. The set of all states of j£? will be denoted by £f. If p{a-^p{a)) is a state of S£ and x(E -> x(E)) any observable, E -> p(x(E)) is a probability measure on ^(R1). It is called the probability distribution of x in the state p and is denoted by Pxp. Using the probability distribution Pxp, we may define the concepts of moments of an observable. The (mean) expected value S(x\p) of an observable x in a state p is defined by

(23)

*(x\p) = r

tdP/(t).

J - 00

The variance of x in the state p is defined when (x2\p) < oo; in this case, (24)

var^)

= £{x2\p)-[£(x\ p)f.

Clearly va,T(x\p)>0 and is = 0 if and only if PXP is concentrated at the single point t0 = (o(x\p), and then it is natural to say that, in the statep, x has a sharply defined value, namely t0.

4. PURE STATES. SUPERPOSITION PRINCIPLE Let ££ be a logic and Sf the set of all states of ££. If pl9 p2, • • • is a sequence of elements of S? and cu c2, • • • is a sequence of constants such that c n >0 for n = l, 2, • • • and c1-j-c2+ • • • = ] . , then the function p(a->p(a)) denned by (25)

p(a) = ^ cnVn{a) n>l

is easily verified to be a state of ££\ We shall write (26)

p = ^ n>l

cnPn.

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Sf is therefore a convex set. If x is any observable, it follows at once from (26) t h a t (27)

c P Pn

PJ> = ^

«* -

n>l

The equation (27) may be given the following interpretation. Suppose t h a t one knows t h a t the state of a system is one of px, p2, • • • with probabilities cl9 c2, • • •, respectively. Then, for any bounded observable x, the expectation value of x is 2n;>i cn^(x\Pn) which is equal to ${x\p). I n other words, assuming t h a t the system is in the state p is, statistically speaking, equivalent to assuming t h a t its state is one of px, p2, • • • with respective probabilities c1, c 2 , • • •. I t is customary to say t h a t the state p is a classical superposition of the states pn(n> 1). I n view of the probability interpretation of (26) and (27), interest centers around states for which no representation of the form (26) is possible. A state p is said to be pure if the equation (28)

P = cp1 + {l-c)p2

(0
pl9p2eST)

implies t h a t p=p1=p2' We write SP for the set of all pure states of j£?. 3P is the set of extreme points of the convex set SP. We may now introduce the notion of superposition. Let 2 be a set of states and p0 an arbitrary state. We say t h a t p0 is a superposition of states in 2 if the following property is satisfied: (29)

a eg,

p(a) = 0

for all

p e 2 => p0(a) = 0.

If p = clp1 + c2p2+ - • • where cl9 c 2 - • • > 0 , and c1 + c2+ • • • = 1 , t h e n p is a superposition of the states in the set {px, p2, • • •}. As we shall presently see, if 3? is a Boolean a-algebra, this is, roughly speaking, the only kind of superposition possible, in particular, no pure state can be a superposition of other pure states distinct from it. This is in contradistinction to quantum mechanics, where the structure of J^, namely the fact t h a t it is standard, forces the concept of superposition to be nontrivial even among pure states. Theorem 3.6. Let ££ be a Boolean o-algebra of subsets of a space X such that (i) ££ is separable (cf. Chapter I) and (ii) {a} e ££ for allae X. For any aeX let Sa be the state defined by (\

if

[fi

if

as

A a$A

Then {Ba : a e X} is precisely the set of all pure states of ££. If 2 is any set of pure states, and p0 an arbitrary pure state, p0 is a superposition of states of 2) if and only if p0 e 2.

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Proof. Let {Al9 A2, • • •} generate J*?. T h a t 8a is pure is trivially verified. Suppose t h a t p is a pure state. If for some A0 E J?, 0
= (l-p(A0))-ip(A

n

(X-A0))9

we would obtain the decomposition p =

p(A0)'p1+(l-p(A0))'p2;

p1y£p2 as p1(A0) = l and p2(A0)=0. This is not possible as p is pure b y assumption. Therefore we m a y conclude for any A e J?, t h a t ^(^4) = 0 or 1. B y replacing An by X — An if necessary, we may assume t h a t p(An) = 1 for all n, where {Al9 A2, • • •} generates ££\ Let

B = C]An. n

Then p(B) — \. I n particular, B is nonempty. We assert t h a t B cannot contain more t h a n a single point. I n fact, the collection of all sets G e ££ with the property t h a t either B^C or B n C = 0 , is a a-algebra which contains all the An. Hence it coincides with j£f. Therefore, as {x} e J?, this is possible only if B = {a} for some a e X. But then p = Sa. Finally, let p0 be a superposition of the states of 2) where each element of 2 is pure. If p0 = 8aQ, but pQ<£2, then p({aQ}) = 0 for all p e 3i but Po({a0}) = l; a contradiction. The theorem is completely proved. Let ££ be an arbitrary logic and 0 the set of all pure states of 3?. For a n y subset Ss^&, we write S for the set of all p e 0* such t h a t p is a superposition of the states of S. Let us now consider Jt, the collection of all 8^0 with the property t h a t S=S. Under set inclusion, Jt becomes a partially ordered set. If J£? satisfies the assumptions of theorem 3.6 for example, then jft is the class of all subsets oi0 and is therefore a Boolean algebra. However, in the case of more complex j£f, Jt is not a Boolean algebra. I n general, the geometric structure of Jt seems somewhat hard to determine. For the so-called standard logics to be introduced in the next chapter, this can be done and it becomes possible to determine the structure of Jt completely. For standard ££\ Jt then becomes isomorphic to j£f. I t is this fact which confers on the set of pure states of a standard logic a geometric structure characteristic of q u a n t u m theory. I n the physical literature (cf. Dirac [1]) this is elevated to a physical principle, the so-called superposition principle. From our point of view, the geometry of pure states will appear as a consequence of the structure of the logic 3?. 5. SIMULTANEOUS O B S E R V A B I L I T Y One of the features which sets apart quantum mechanics from classical mechanics is the existence of observables whose values cannot always be

THE

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55

simultaneously measured. We shall now introduce this idea formally. We begin with the elements of ££ themselves. Let a9b e ££. We shall say t h a t a and b are simultaneously verifiable, a <—> b in symbols, if there are elements al9 bl9 and c e i f such t h a t *

(i) al9 bl9 and c are mutually orthogonal, (ii) a = a1 V c, b = b± V c.

'

If a: and y are two observables, they are said to be simultaneously observable if for any two Borel subsets E and F of R1, x(E) i/. If A and i? are two subsets of j£f, we write A <^> B when a<^b for all a e ^4 and b e B. The analysis of these concepts depends on a study of Boolean subalgebras of ££. A subset J / c: Jgf is a Boolean subalgebra of S£ if it is (a) a sublogic of J5? and (b) a Boolean algebra, i.e., if (i) 0, les/, (ii) if a, b e <s/9 then a v b9 a A b9 and a 1 e JS/, (iii) if a9b9 c e s/9 then a A (6 V c) = (a A b) V (a A c), a V (b A c) = (a V b) A (a V c). J2/ is said to be a Boolean sub-a-algebra of ££ if it is a Boolean subalgebra, and if, for any countable sequence al9 a2, • • • e <$/, \Jn ane^/ and

Lemma 3.7. Let a,b e ££, 3? being any logic. Then the following

statements

are equivalent: (a) a <-» b9 (b) a-(a A b) J_ b, ^C) 6 ~ ^ a A 6 ) - L a ' (d) ^ e r e erciste an observable x and two Borel sets A and B of the real line such that x(A) = a and x(B) = b, (e) there exists a Boolean subalgebra of ££ containing a and b. Proof. Suppose a <-> b. We m a y then write a = a1y c, b = b±vc, where al9 bl9 and c are mutually orthogonal. Clearly c
ax = a— (a A b) ±_ b± v c = b. This proves t h a t (a) => (b). By symmetry, (a) => (c). If a — (a A b)_\_b, then, on writing a x = a — (aA6), b1 = b — (aAb) and c = aAb9 we find t h a t a = a1v c and b = b1v c. Since %_[_&> w e have «i_L^i a n d a i_L c > while, by

56

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definition, b1_[_c. Therefore a<-> 6, proving that (b) => (a), (a), (b) and (c) have thus been proved equivalent. We shall complete the proof by showing (a) => (d) => (e) => (a). If a = a1vc, b = b1vc, where al9 bl9 and c are mutually orthogonal, we write d = a1vb1\/ c and define x to be the discrete observable such that x({0}) = a1, #({!}) = &!, x({2}) = c and x({3}) = d1. Then x({0,2}) = a, x({l,2}) = b. This proves that (a) => (d). Suppose that (d) is satisfied. Then x(A-Ar\B) = a~-aAb, x(B — AnB) = b — aAb. Writing a1 = a — aAb,a2 = aAb,a3 = b — aAb,a± = (avb)1iwe see that the a{ are mutually orthogonal and a± V a2 V a3 V a4 = 1. If sf is the collection of elements which are the lattice sums a

h V ai2 V • • • V aik

(k < 4, 1 < t\ < i2 < • • • < ik < 4),

then it is easy to verify that s/ is a Boolean subalgebra of ££. Since a,b e jtf, we see that (d) => (e). Finally, let (e) be satisfied, and let $0 be a Boolean subalgebra of ££ containing a and 6. Clearly [a — (a A b)] A 6 = 0. But, as a— (a A 6), 6 and 6 1 are in the Boolean algebra s>/, a-(a

A b) = {(a-(a

A 6)) A b} V {(a-(a A 6)) A &1}

= {a-(aA6)} A b1 (b) and hence that (e) * (a). Corollary 3.8. If a<-*b, the elements ax, bl9 and c of (31) are uniquely determined. In fact, c = a Ab, a±=a — aAb, and b1 — b — aAb. Lemma 3.7 shows that in the presence of simultaneous verifiability of two elements of ££\ one can operate with the statements corresponding to these elements as if they were classical. We shall now take up the question of generalizing this fact to observables. Our aim is to prove that simultaneous observability is essentially a characteristic of classical systems. We shall deduce, as a corollary, that, if x, y, z, • • • are observables, associated with a logic, they are mutually simultaneously observable if and only if they are all functions of a single observable. This was first proved by von Neumann (cf. [1]) when the logic in question was a standard one. For a general logic, it was proved by Varadarajan [1]. Theorem 3.9. | Let J? be any logic and {xA}AeD a family of observables. Suppose that A

—^ *^A'

for all A, X' e D. Then, there exists a space X, a f The definition of a logic given in Varadarajan [1] does not require it to be a lattice. I t was pointed out to us by Dr. S. P . Gudder [1] that theorem 3.9 does not remain valid for these more general classes of partially ordered sets considered in our paper [1].

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57

o-algebra & of subsets of X, real valued ^-measurable functions gK on X(X e D), and a a-homomorphism T of & into ££ such that (33)

r{gA-\E))

= xK(E)

1

for all A e D and E e^{R ). Suppose further that either ££ is separable in the sense that every Boolean sub-a-algebra of ££ is separable, or that D is countable. Then there exists an observable x and real valued Borel functions f\ of a real variable such that for all A e D, (34)

xK = fA o x.

The proof of this theorem (cf. Varadarajan [1]) depends on a series of lemmas. J? is a fixed logic in these lemmas. Lemma 3.10. Let 6, al9 a2, • • • be elements of <S£.Ifb an\

(35)

&<->V«n, n

b+-> /\an. n

Moreover, we then have the distributivity laws (36)

o A A / < | = V (h A an), U n ' o V (A «») = A (6 V an).

Proof. We use the criteria of (32). It follows from the equivalence of (a) and (d) of (32) that if c <-> d, then any two of the four elements c, c1, d, d1 are simultaneously verifiable. This proves the first of the three relations of (35). Now, we come to the proof of the rest of (35). Write a = N/m am. Since b A am a. Changing all the an into anL we deduce from the relation b <-> \/n an that b <-> An an- Next we come to (36). Once again it is enough to prove only the first of the two relations. Clearly \/n (b A an) < b A a, where a = \Jn an. Let d < b A a be such that d_|_Vn (b^an) a n d (Vn (bAan))yd = bAa. Since d_j_&Aan and 6? an, we conclude that dj_a n , from (c) of (32). Since this is true for all n, d\_a. As d
58

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P | a ££\ is also a sublogic of <£\ From this it follows that if {^a}aeA i s a n y family of sublogics of ££, there exists a smallest sublogic containing all the j£?a. We denote it by \Ja j£?a. 9ft is thus a complete lattice. If each j£?a is a Boolean subalgebra of ££\ it is however not true in general that \/a J?'a is a Boolean subalgebra. We shall now prove that this is so if and only if the relations ££\ <—> ££\> are satisfied for all a, a e A. We begin by considering the case when there are only two Boolean subalgebras involved. Lemma 3.11. Let Mx and 0t2 be two Boolean subalgebras of ££\ Then 8%x <—> ^ 2 if and onty if &i V ^ 2 is a Boolean subalgebra of ££. Proof. Suppose that ^t1 M 0t2 is a Boolean subalgebra of ££. Then Si1 <—» ^ 2 m view of (e) of (32). We shall now prove the converse. We shall first consider the case when the Boolean subalgebras are finite. Let F% (* = 1, 2) be finite Boolean subalgebras of 3? such that Fx <->#*2. Then, their Stone spaces (cf. Chapter I) are finite and hence there are elements al9 • • •, ar e Fx and bl9 • • •, bs e F2 s u c n that W ail.i'-> i^i\ &/_L&y> j ^ j ' ; (ii) ax V a 2 V • • • V ar = 6X V &2 V • • • V bs = 1; (hi) Fx is the collection of all lattice sums a ^ v a ^ v • • • V a{ (1 < ix < i2 < • • • < ip < r) and F2 *s ^ n e collection of all lattice sums bj± V &,-2 V • • • V &,- (1 < j i < j 2 ^ *' * ^ j a < ^). Write (37)

cw = af A 6,

(* = 1, 2, • • •, r, j = 1, 2, • • •, 5).

Obviously c{j±_cvr unless i = i' and j = j ' . Since ai^-^bj we infer from (36) that

for all i and j ,

s

ij = at>

c

V

(38)

V % = &/• i=l

Consequently,

(39)

V V % = Ii=l

;= 1

If ^" denotes the collection of all lattice sums of the ci;, then F is a Boolean subalgebra of ££ and (38) implies that J j C j fori = l, 2. The formulas (37) show that if IF' is any sublogic of ££ containing both Fx and F2, then J ^ c J ' . Clearly, therefore, F^Fxy^2, Note that •^"1 V ^"2 *s a ^ s o finite. Let ^ (i = l, 2) be Boolean subalgebras of ££ such that ^ <->^ 2 - Let ^ be defined by (40)

» = U ( * ' i V ^2),

where the union is over all pairs [FX,F2) of finite Boolean subalgebras of S£ such that F^M^ i — 1, 2. We assert that ^ is a Boolean subalgebra

THE LOGIC OF A QUANTUM

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59 1

of «£? containing 0kx and 0k2. If a e 0k, then it is obvious that a e 0k. Clearly, 0, 1 e &. Suppose now that a,b e0k. Then there are finite Boolean subalgebras J^ l5 J ^ ' , ^ 2 , J*Y of & such that J ^ , . F / are c # x , J*"2, J ^ ' are ^ ^ 2 , a e ^ V ^ a n d fte^/v^'Since ^ and «^Y are contained in the Boolean algebra 0kx, tFx <—> <^Y and hence ^ \ v ^ Y = ^ Y ' is finite and contained in ^ x . Similarly, ^ 2y ^ 2 =: ^ 2" is finite and contained in 0 2 . As mi0k2, J V <-» J 5 "/. Therefore J^Y'V^a" is a finite Boolean subalgebra of j£? contained in ^ . Let us write ^ = SF± V ^Y'As J ^ , J*Y> J^2, and J*Y are contained in J5", it follows that &x\j &^& and J*Y v ^ ' ^ ^ - Thus, a,be^. Therefore avfe and aA& are in ^*. This proves that a\fb and a A b are in ^ . We shall next establish the distributivity laws. Let al9 a2, a3 be three elements of 0? and (for j = l , 2, 3) let aj e &?> V &f\ where J^jfc) is a finite Boolean subalgebra of j£? contained in 0kk (j = 1, 2, 3, k — 1, 2). Since ^rf)^^k for all j , V?=i ^ f c ) i s a finite Boolean subalgebra of 0kk. Let it be denoted by J*™. Obviously J^(1) <-> J*~(2), and hence ^ = J^(1> v J^(2) is a finite Boolean subalgebra of ££. Since J ^ c ^ " for all j and A;, j r u ) v jr(2)c j r for all j so that ay 6 J5" for all j . As J5" is a Boolean subalgebra of J£f, we have: «i A («2 V a3) = (ax A a2) V (ax A a 3 ), «i V (a2 A a3) = (ax V a2) A (ax V a 3 ). This proves that 0k is a Boolean subalgebra of o£\ We observe finally that ^ and 0k2 are contained in 0k. If a e ^ , we have a e J ^ v J ^ , where ^ 1 ={0,a,a 1 ,l} and ^ 2 = {0,l}. Hence Stx^St. Similarly, 0k2<^0k. 0kxy @2 is thus a Boolean subalgebra of <£. The lemma is completely proved. Corollary 3.12. / / 0k is defined by (40), then 0k=0kx\j0k2. If <€ is any subset of & such that <€ <-> 0kk, k = 1, 2, ^en ^ ^ ^ v ^ 2 Proof. If ^*fc is a finite Boolean subalgebra of 0kk (k = l, 2) and if 0k' is a sublogic of j£P which contains both ^ and 0k2, &\\j&'2^0k''. Hence 0k<^0k'. This shows that 0kx V 0k2 coincides with 0k. To prove the second assertion, let c e ^ \ Let &rk<^0kk be finite Boolean subalgebras of j£\ From (37) and (35) we may conclude that c <—• !FX v 3F2. Equation (40) shows then that c f ^ J . This proves that ^ <-> 0k ± V ^ 2 Corollary 3.13. Ze£ ^ (i = 1, 2, • • •, s) be Boolean subalgebras of 3? such that 0ki
60

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valid for all s<s0 and let s = s 0 + l. Since Sti V i i V ^i- ^ n ^ s completes the induction and the proof of the corollary. Lemma 3.14. Let {^ A } AeD be a family of Boolean subalgebras of <£ such that Stx , for all A, A' e D. Then there exists a Boolean subalgebra St of Se such that StK<^St for all XeD. Proof. Let F be a, finite subset of D. By corollary 3.13, V A G F ^ A *S Boolean subalgebra of ££\ Let us write (41)

»{F)

a

= V #AAeF

Obviously, St{F)<^St{F')

(42)

if F^F'.

Define St by

St =

U

St(F).

F finite

Obviously, StK<^St for all A e D. We shall now complete the proof by showing t h a t SI is a Boolean subalgebra of j£?. I t is enough to prove t h a t any finite subset of St is contained in a Boolean subalgebra of ££ contained in St. If a1,a2,'-'iar are elements of St there are finite subsets Flt F2, • • •, Fr of D such t h a t a{ e St{F{). If F = ( J t .F t , then a t e ^ ( ^ ) for all *. This proves t h a t ^ is a Boolean subalgebra of J^. Corollary 3.15. Let {StK}KeD &e a family of Boolean sub-a-algebras of 3? such that StK <—> StK' for all A, A ' e D . Then, there exists a Boolean sub-aalgebra St of <£ such that StK<^St for all XED. Proof. By lemma 3.14 there are Boolean subalgebras St' such t h a t St^St' for all A. By Zorn's lemma we can choose a maximal such, say St. We claim t h a t St is a Boolean a-algebra. I t is enough to prove t h a t if al9 a2> * * * is a sequence of elements of St, then a=\/m am is also in S$. Since am eSt, b <-» a m for all 6 G ^ . Hence a < - > ^ by (35). If j / = {0,a,a 1 ,l} then j / is a Boolean subalgebra of «£? and stf ±-*St. Hence si M St is a Boolean subalgebra of j£? containing S%. Since ^ is maximal, stf'v St=St. Therefore a e i For any observable # associated with 3? we write

(43)

St(x) = {z(#) : # e ^(i? 1 )}.

We shall call S%(x) the ran^e of a:. Lemma 3.16. Let St be a Boolean sub-o-algebra of j£f'. Then, in order that St=St(x) for some observable x, it is necessary and sufficient that St be separable.

THE LOGIC OF A QUANTUM

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Proof. If 0k=0k(x) and @ = {x(E) : E an open interval with rational end points}, then ^ is a countable subset of 0k and 0k is the smallest sub-cralgebra of itself containing Q). 0k is thus separable. Conversely, let 0k be a separable Boolean sub -a- algebra of j£f\ By theorem 1.3, there exists a space X, a a-algebra 01 of subsets of X, and a a-homomorphism h of 01 onto ^ . Let {^4^ ^42> • • •} be a sequence of subsets of X such that An e 08 for all n and {^(^4n) : n = l, 2, • • •} generates ^ . Let «^0 be the sub-o-algebra of 9S generated by {An : w = l, 2, • • •}. By theorem 1.6(i), there exists a cr-homomorphism w of ^(R1) onto «^0. Define a; by (44)

x(E) = A(M(JS))

(# G ^(i? 1 )).

It is obvious that x is an observable and 0k=0k(x). Now to the proof of theorem 3.9. Suppose that x is an observable and xA=fA ox (A G D), where each /A is a real Borel function on R1. Then 0k(x^)<^0k(x) and hence &(xK) <->0k{xK>) for all A, A' e D. This proves that xx <-> # v , for all A, A' e D. Conversely, suppose that {xA} is a family of observables and that #A<->#A, for all A, X'eD. By corollary 3.15, there exists a Boolean sub -a- algebra 01' of ££ such that 0k(xK)^0k' for all A. By theorem 1.3, there exists a space X, a cr-algebra 01 of subsets of X, and a cr-homomorphism r of 0? onto ^ ' . It is now clear that for each A G D, there exists, by theorem 1.4, a real valued ^-measurable function gA on X such that, for all E e ^{R1), xx(E) = r[gK-\E)}

(XeD).

This proves the equation (33). For proving (34) let 0t be the smallest sub-cr-algebra of 0k' containing all the 0k(xK). If 3? is separable, so is 0k. On the other hand, if D is countable, and i^A is a countable set generating &(x\)> UA ®\ i s a countable set generating £%. Hence 0k is separable in either case. By lemma 3.16 there exists an observable x such that 0t(x) =0k. As 0k(x-^ <^0k(x)i we conclude from theorem I.6(ii) that there exists a real Borel function / A of a real variable such that xx =/ A o x. This proves (34) and completes the proof of the entire theorem. Remark. It is obvious that if the xK satisfy either (33) or (34), A ^ ^

A'

for all A, A' ED. Theorem 3.9 therefore tells us that the classical logics are precisely those for which any two observables are simultaneously observable. As soon as ££ is not a Boolean algebra, it follows that the properties of a physical system which are embodied in the logic ££ have too complex a structure to be exhausted by a single set of simultaneously observable quantities. In general, when certain propositions are experimentally verified, certain others cannot be so at the same time.

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6. FUNCTIONS O F S E V E R A L OBSERVABLES The characterization given in theorem 3.9 of simultaneous observability enables us t o construct a calculus of functions of several observables which are mutually simultaneously observable. Theorem 3.17, Let ££ be any logic and observables such that x{ <-> Xj for all i, j . Then there exists one and only one a-homomorphism r of @(Rn) into & such that for all E e ^{R1) and all i = 1, 2, • • •, n, (45)

xx(E) =

rfrc^E)),

where iri is the projection (tl9 t29 • • •, tn) - > tt of Rn on R1. If g is any real valued Borel function on Rn, (46)

g o (Xl,x29.

- .9zn) : E ->

rlg-HE))

is an observable. If gl9 g2, • • •, gk are real valued Borel functions on Rn and yi — gi o (xl9 - - -, xn), then yl9 • • •, yk are simultaneously observable, and for any real valued Borel function h on Rk, (47)

ho (yl9--,yk)

= (h(gl9 • • •, gk)) o (xl9 • • ., xn),

where h(gl9 • • •, gk) is the function t =

(t1,--,tn)->h(g1(t),...,gk(t)).

Proof. B y theorem 3.9, there exists a n observable x and real valued Borel functions fl9 • • - , / n of a real variable such t h a t xi=fi o x for all i. Let t h e Borel m a p f of R1 into Rn be defined by f : * - M / i ( 0 , •••,/»(*)). If we define, for M e &(Rn), (48)

r(M) = xtf-^M)),

t h e n r is a a-homomorphism of &(Rn) into j£? which satisfies (45). If r and T' are two a-homomorphisms of &(Rn) into ££ satisfying (45), then T(M)

=

T\M)

whenever M is of the form Ex x E2 x • • • x En9 the Ej being Borel subsets of R1. Since the class of such sets generates &(Rn), it follows t h a t T = T'. The fact t h a t g o (xl9 • • •, xn) is a n observable is trivial. Note t h a t its range is contained in t h e range 0t of r which is a Boolean sub-a-algebra oiX.

THE

LOGIC OF A QUANTUM

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63

Suppose now that yi — qi o (xx, • • •, xn). Then the range of y{ is contained in 0t and hence y1, • • •, yk are simultaneously observable. Let g be the map t -» (^(t), • • •, gk(t)) of Rn into Rk. Then

y. M-> r(g-\M)) is a <7-homomorphism of &(Rk) into J£ such that

y\^c\m

= AgrHE)) = yt(E),

TTX being the map (sl9 s2, • • •, sk) -> s{ of Rk on R1 (i = 1, 2, • • •, &). Therefore we conclude from the uniqueness of the map r in (45) that, for w=

h°(yi>y2>--,yk)>

u{E) = y{h-\E))

(E e ^{R1)).

But y(A~1 (E)) = T(g~1(A~1 (2£))), so that we may conclude that A o (2/i, • • •, yk) = ( % i , • • •, &)) ° (»i, • • •, s n ) .

This proves the theorem. Theorem 3.17 enables us to define the joint distributions of xlt ...,#„. Let ^(a-^^)(a)) be a state of J*? and xl9 • • •, xn observables such that X^ ^—^ »£y for all i, j . We may then clearly define the probability measure P* 1> ..., Xn on^( J R»)by (49)

P»l.....Xn(M) = p(r(M)). s

Pxlf-,xn * called the joint probability distribution of (xl9 • • •, xn) in the state p. It follows easily from (47) that for Borel Mg:Rk, (50)

P5lt...iiyfc(Jlf) =

P*lt...tXn{g-*{M)),

where g is the map t =

{h,--,tn)-+(g1(t),...,gk(t)).

We see from (50) that the rules for the calculation of the joint probability distributions are the standard ones of probability theory.

7. THE CENTER OF A LOGIC We begin by recalling the discussion of Chapter II in which we singled out the geometries among the modular lattices of finite rank by demanding that their centers be trivial. We shall now do the same for logics. The irreducible logics would then have properties very far removed from those of Boolean algebras and would consequently serve as plausible models for the logics of atomic systems.

GEOMETRY

u

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Let ££ be any logic. We define the center of 3? to be the set %>, where (51)

V = {a:ae&,

a^b,

for all b e &}.

*& is nonempty since 0 and 1 are in *€. The elements of ^ are thus simultaneously verifiable with every element of ££. If x is any observable with &(x) $=:<£, then for any observable y, x+-+y. Such observables x are called central. ££ is said to be irreducible if <€={0,1}- Clearly, ££ is irreducible if and only if the only central observables are the constants. Theorem 3.18. Let ££ be any logic and <€ its center. Then <$ is a Boolean sub-o-algebra of ££\ Proof. The equations (35) and (36) reveal immediately that ^ is a Boolean subalgebra of «£?. If ax, • • •, ar, • • • is a sequence of elements in < €, then bar for all r(b e J?) and so \/n an *~* b by (35). Hence \Jn ane^ also. This proves that ^ is a Boolean sub-a-algebra of ££. Since the Boolean (r-algebras are relatively familiar, one may ask whether there is a decomposition of J£? over # analogous to that for modular lattices of finite rank. There is no difficulty in doing this when # is discrete, i.e., when there exists an at most denumerable set {a n } neD of mutually orthogonal elements of ^ such that (i) \fnan = \\ (ii) ^ consists precisely of all the lattice sums \/neZ an, where Z is an arbitrary subset of D. The an are called the atoms of %\ If # is not discrete, then such decompositions seem much more difficult to obtain. On the other hand, let us define ^ to be continuous if, given any c e ^ which is nonzero, there exists a nonzero c1s(^ such that c1
p~(a) =p(a A an)

(ae&),

then p~ is a pure state of ££. If (53)

^ n = {p~ : p a pure state of J?n},

then the {0*n : ne D) are disjoint subsets of & and (54)

0> = U &„

If *& is continuous and separable, & is empty, i.e., J? has no pure states.

THE

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Proof. We have already noted that for any n, j£?[0,an] = «£?n is a logic (corollary 3.3). We prove that ££n is irreducible. In fact, let c a for all a < an. On the other hand, as an e %', an*->b for any b e ££, so that b = (b A an) V (6 A a / ) (cf. (36)). Now c 6 A a / since c J_6 A an1. Therefore (35) enables us to conclude that c <-> 6. This shows that c e ^ and hence that c = 0 or c = an. This proves the irreducibility of ££n. Suppose next that p is a pure state of j£?n. Equation (52) defines p~ as a function on j£f. From the equations (35) and (36) it follows that p~ is a state of ££. We claim that p ~ is a pure state of <£?. Suppose that ^ ~ = cpx + (1 — c)p2, where 0 < c < 1 and JJ1? p2 are states of ^ . Since p ~ ( a ^ ) = 0 , PliVn1) = P2K1) = 0. Hence Pi(an)=P2(an)==l> a n d hence the restrictions of p1 and ^ 2 to j£?n are states of j£?n. Since ^ is a pure state of «j£?n, it follows that p{a) = px(a) = ^ a (a) for all a~ is thus a pure state of JSf. If m, n e D and m^n, 0>m C\ 0^n— 0 , for, if g e ^ m and g' e ^ n , g(am) = 1 while g'(am) = 0. We show now that each element of £P lies in some 2Pn. Suppose that p is a pure state of ££. We claim that p(an) = 1 for some n e D. Otherwise there will be some m e D such that 0
qM = (*>(««)) " ^ ( a A an), ?2(a) = ( l - ^ ( a m ) ) - ^ ( a A amL).

Since am e # , it can be easily verified using (35) and (36) that qx and q2 are states of ££\ Since qi(am) = l, and g ^ m ) ^ * <7i#(?2- As 2> = ^ K ) • ?i + (1 -!>(««)) • q2> p is not pure, a contradiction. Therefore p(an) = 1 for some ne D. The argument given in the previous paragraph can now be repeated to enable us to conclude that the restriction of p to «j£?n, say q, is a pure state of 5£n and that p = q~. This proves that pe&n and completes the proof of (54). We now come to the proof of the last assertion. Assume that ^ is continuous and separable. Suppose that p is a state of ££'. We assert first that for some c e # , 0
GEOMETRY

66

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THEORY

L

cn if necessary, assume that p(cn) = l for all n. Let c0 = /\ncn. Clearly, p(c0) = l. In particular, c 0 ^ 0 . We assert that c0 is an atom of <€. In fact, the subset of all elements c in ^ such that either c_[_c0 or c> c0, is a sub-o-algebra of ^ containing all the cn and so coincides with # ; this shows that if a
g2(a) = (l-p{d)y1p(a

A d 1 ).

Then p=p(d)q1 + (1 — p(d))g2 and p is not pure. The proof of the theorem is complete. We shall end this section with a few remarks of a general nature regarding the Heisenberg uncertainty principle. We must point out that a considerable part of its meaning is physical and we do not go into this here (cf. the discussions of von Neumann [1] and Heisenberg [1]). We shall be concerned only with mathematical formulations. Suppose that (x2\p)
op*{x) =

£{x*\p)-{£{x\p)f.

If crp(x) = 0, then the probability distribution of x in the state p is concentrated at the point &(x\p). x has then a sharply defined value in the state p. One might ask the question whether there are states in which every observable has, if not a sharply defined value, at least a small variance. The Heisenberg principle of uncertainty asserts that even this is not possible. Roughly speaking, it asserts that in any state, no matter how prepared, there are observables whose distributions have large variances. Notice that this is not possible if the logic is a Boolean a-algebra of subsets of a space. In fact, let «j£? be the cr-algebra of subsets of a set X. Then, for any observable with corresponding function/, the observable has a sharply defined value f{x0) whenever the state p is a measure concentrated in a point x0 of X. On the other hand, let us assume that © is the physical system of some atomic particle and that x and y are the position and momentum observables along some direction in space. If we assume that the logic .if of © is the standard logic associated with the complex number field, then (57)

°p(x)*p(y) > P ,

where h is Planck's constant divided by 2TT (cf. H. Weyl [1] pp. 77 and 393-394 for a rigorous derivation). From (57) it follows quite simply that

THE LOGIC OF A QUANTUM MECHANICAL

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67

in any state in which the distribution of a: has small variance (i.e., x is "localized"), the distribution of y has very large variance. I t is obvious that this is a property of the structure of the logic JS? of <3. 8. AUTOMORPHISMS Let ££ be a logic, 0 the set of observables of ££, and Sf the set of all states of ££\ The set of all automorphisms of ££ forms a group in a natural fashion, denoted by Aut(j£f). We shall now introduce the notion of an automorphism of Sf. By a convex automorphism of S? we mean a map (58)

(:p-+p*

(peS?)

such that (i) $ is one-one and maps Sf onto itself, (ii) if cl5 c2, • • • are numbers > 0 with 2 n cn —1> and jpl5 p 2 , • • • e £?, fecnPnY V n

= 2 /

C

»^'

n

From (ii) of (59) it follows at once that convex automorphisms of £P send pure states of J£? into pure states of ££? and that the correspondences induced in the set of pure states are one-one and onto. If a : a -> aa

(60)

(ae^f)

is an automorphism of j£? and if, for p e Sf we define pa by pa(a)

(61)

=p(a^~%

a

then p -> p is a convex automorphism of t9^. We shall call it the automorphism induced by a. The set of all convex automorphisms of Sf also forms a group denoted by A u t ( ^ ) . Let ££ be a a-algebra of subsets of a space X, and let * : x -> <(a)

(a? e Z)

be any one-one map of X onto itself such that for any set A<=iX, A e J? if and only if t" X{A) e J^7. When X and «^f satisfy some technical regularity conditions, it can be shown that every sufficiently well-behaved automorphism of Sf is induced by a mapping t such as the one described above. Suppose X is the phase space of a classical mechanical system with a C00 manifold M as its configuration space. X thus appears (cf. Chapter I) as the cotangent bundle of M. In this case, the subgroup of those automorphisms of ££ induced by (both ways) differentiate homeomorphisms of X onto itself and which preserve the canonical 2-form on X, plays undoubtedly a very crucial role (contact transformations).

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The axioms satisfied by an arbitrary logic are too weak to allow any reasonable determination of its automorphisms. In the next chapter we shall determine the automorphisms associated with standard logics. In the meantime, we shall briefly illustrate with two (among many) examples the important role played by the group Aut(^) in physical problems. As a first example we consider the problem of describing the dynamics of a physical system <&. Let j£? be the logic of ©, £P the convex set of all its states. Then, for each real number t, there exists a unique one-one map of SP onto itself, say D(t), with the following physical interpretation; if p e Sf is the state of <5 at time t0, then D(t)p is the state of <S at time t+t0. The dynamical group Qj where (62)

9 : t -> D(t)

satisfies the conditions (3) of Chapter I. We now postulate that each D(t) is in fact an automorphism of SP, i.e., if plt p2i • • • is a sequence of elements of SP and clt c2, • • • a sequence of nonnegative numbers such that c i + c 2 + • • • = 1, then, for each t, (63)

D(t)(2ciPi)

=2^(*)3>i.

This postulate may be justified by the following argument. Suppose one does not know the state at time 0 but only that it is one ofp1,p2, • • • with probabilities c1? c2, • • •, respectively. Then from the fact that the state at time t is D(t)p{ if the state time 0 is pt, it follows that at time t the state of the system (3 is one of D(t)pu D{t)p2i • • • with the same probabilities c l5 c2, • • •. But the statement that the state of S£ (or (&) is one of qx, q2i • • • with probabilities cl5 c2, • • • is physically equivalent to the statement that the state of ££ is c1q1+c2q2+ • • •. This proves the validity of (63). It follows now that each D(t) leaves invariant the subset of pure states of j£? and induces a one-one mapping on it. Furthermore, if a e ££, then, for each p e £f, (64)

t^(D(t)p)(a)

is a real valued function of t; it is physically reasonable to assume that for any 3 , this function is Borel for each p e SP and a E &. If this condition is satisfied, we call 2 a Borel one-parameter group. Thus the dynamical group of (3 may be assumed to be a Borel one-parameter group of convex automorphisms of 6f. This leads us to introduce the concept of representations of groups. Let ( j b e a locally compact topological group satisfying the second axiom of countability. The sets of the smallest a algebra of subsets of G

THE

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69

containing the open sets are called the Borel sets in G. By a representation of G in Aut(y) we mean a map (65)

T:g^Tg

(geG)

of G such that

(66)

(i) each Tg e Aut(^), (ii) Tgig2 = TgiTg2 (gltg2e€f), (iii) for each a e ££ and p e Sf, the function g -> (Tgp)(a) is Borel on£.

With these definitions we may say that the dynamical group of a system © is determined by a representation of the additive group of real numbers into Aut(^), where £f is the set of states of the logic ££ of ©. As our second example, we consider the problem of describing the quantum mechanical theory of a free particle. The configuration space of the particle in classical mechanics is the three-dimensional space, denoted by M. Let ££ be the logic of the physical system under consideration and Sf the convex set of its states. Suppose now an observer 0 wants to describe the system. He would then select a point of M, an orthogonal frame at 0, and map M onto R3 by a map (67)

y0'-M->

R3.

Let 0' be another observer and let yQ> be the corresponding map. Then there exists a unique Euclidean motion D0t0, of R3 such that for all meM (68)

yo,{m)

=

D0t0,(y0(m)).

Suppose now 0 describes the state of the system © by an element p0 e £?. Then, 0', using his orientation, would describe the same physical state by an element p0, of SP. It is clear that the map pQ —> pQ, is an automorphism, say a0t0> of Sf. In other words, a00, has the following physical interpretation: if the physical state of © is described by 0 by the element ge£f, then a0t0\g) is the element of £f describing the same physical state of © from the point of view of 0'. It is obvious that a0tQ, is the identity for 0 — 0\ and (69)

a0i0» = a0,f0»a0t0>.

We now remark that the basic physical laws of © are independent of the choice of observers and their frames. Therefore, we may conclude that a o,o' depends only on the transformation D0,o'- Let G be the group of all

70

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Euclidean motions of R . Then, from (68), (69), and the above remarks on the a00,, we infer the existence of a homomorphism (70)

T:g->Ta

of G into A u t ( ^ ) with the property t h a t whenever 0 and 0' are two observers of
«o.o> =

TD0,0,-

We m a y reasonably make the assumption t h a t for each a e & and p e £f, g - > (Tgp)(a) is a Borel function on G. T is thus a representation of G in Aut(^). We see thus t h a t the description of <& gives rise to a representation of G in A u t ( ^ ) . The determination of such representations is therefore the basic problem to be solved in the mathematical description of the physical system of a free particle. The reader might notice t h a t the arguments given above are very general. I n fact, the same type of arguments lead to the remarkable fact t h a t with every relativistic free particle there is associated a representation of t h e inhomogeneous Lorentz group. These representations were first completely determined by Wigner under t h e assumption t h a t t h e logic of <3 was standard. This led in t u r n to his celebrated classification of the relativistic wave equations [1].

N O T E S ON C H A P T E R I I I 1. As a general reference on lattice theory we refer to the book of F . Maeda and S.Maeda, Theory of Symmetric Lattices, Springer-Verlag, Berlin, 1970. 2. There has been a good amount of work by m a n y people in recent years on the structure of logics and probability measures on them, motivated mostly b y the relevance of such problems to the foundations of Quantum Mechanics. The encyclopedia volume of E . G. Beltrametti and G. Cassinelli (The Logic of Quantum Mechanics, Volume 15, Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Reading, Mass., 1981) gives an excellent overview of this work (Chs. 10-20). I n addition, this book presents a well-organized, comprehensive discussion of the foundations of Quantum Mechanics from different points of view with m a n y references to current literature, examples, and exercises. Especially illuminating are the critiques of controversial topics such as hidden variables (Chs. 15, 25) consistency of physical interpretation (Chs. 7, 8), the process of measurement (Ch. 26), and so on. As additional references we mention the following: J . M. J a u c h , Foundations of Quantum Mechanics, Addison-Wesley, Reading, Mass., 1968.

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C. Piron, Foundations of Quantum Physics, Benjamin, Reading, Mass., 1976. M. Jammer, The Philosophy of Quantum Mechanics, Wiley, New York, 1974. C.A.Hooker (Ed.), The Logico-algebraic Approach to Quantum Mechanics, Vol. 1, Reidel, Dordrecht, 1975. E. G. Beltrametti and B. C. van Frassen (Eds.), Current Issues in Quantum Logic, Plenum, New York, 1981. 3. The structure of the states of orthocomplemented continuous geometries is still mysterious. One of the most striking results is due to von Neumann who showed that in the presence of a transition probability function with suitable invariance properties, such a lattice is nothing but the lattice of projections of a type 1^ factor (Continuous geometries with a transition probability, Mem. Amer. Math. Soc. No. 252, Providence, R.I., 1981). 4. It is possible to determine, in the standard model, the states for which the lower bound of the Heisenberg uncertainty relations (57) is attained, for the quantum mechanical system of a one-dimensional particle. Let je = L2 (-00,00),

q = operator of multiplication by xt p = - j - ; then ov(g)-M#)=2 for the family of states

0) given by 9(x) = ( ^ ) % x p { - ( y / 2 S ) (*-*)* + (p/h)ix}; then oM = («/2y)*,

a9(p) = («y/2)*

(cf. von Neumann [1], pp. 233-237). As we shall see in greater detail in Chapter IV, any vector

S-(p is called the wave function, and since the spectral measure of q is E-*qs where qs is multiplication by the characteristic function of E, the probability distribution of q in the state g> is the measure fi :E-^

\cp\ JE

dx.

Thus the wave function allows one to calculate, by this formula, for any region E^(~ao,co), the probability/*(!£) of finding the particle inside E. This is the simplest instance of the statistical interpretation of Quantum Mechanics which is due to Max Born (Z. Physik, 37 (1926), pp. 863-867).

CHAPTER IV LOGICS ASSOCIATED WITH HILBERT SPACES 1. T H E LATTICE O F SUBSPACES O F A BANACH SPACE The general theory t h a t was developed in Chapter I I I is concerned mainly with the broad principles of Quantum Theory. Further developm e n t of the theory, such as a discussion of simple quantum mechanical systems, leads to problems of a more technical nature. These problems are, however, difficult to answer in the context of abstract logics, and therefore, in dealing with them, it becomes necessary to restrict the class of logics under consideration. I n this chapter we shall examine some of these logics and describe, in a somewhat more concrete fashion t h a n in Chapter I I I , the observables, states, and symmetries associated with t h e m . We have seen in Chapter I I t h a t a large class of orthocomplemented geometries are obtained by considering Hilbert space structures on finite dimensional vector spaces. These logics are, however, not of much use in discussing even very simple atomic systems, since every observable associated with t h e m is finite. Consequently, if one wants to obtain realistic examples, one is forced to deal with infinite dimensional vector spaces. I t is known (cf. Baer [1]) t h a t the lattice of all linear manifolds of an infinite dimensional vector space does not admit any polarity. Therefore it becomes necessary to consider lattices j£? such t h a t the elements of <£? are certain linear manifolds of a given vector space V and, wherein, the partial ordering is the inclusion relation. I t must be noted t h a t the lattice sum of two elements of such a lattice contains, but in general does not coincide with, the algebraic sum in V. A simple and interesting class of such lattices may be obtained in the following way. Let D be a topological division ring, say, for example, one of R, C, or H. Let V be a topological vector space over D (cf. Bourbaki [2] for the definitions and elementary properties). Then, the set of all closed linear manifolds of V is a lattice—in fact, a complete lattice. One m a y then construct examples of logics whose underlying lattices are these lattices of closed manifolds. The description of such logics would provide us with a large class of examples of logics which are complex enough to serve as models for the logics of atomic systems. 72

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The main purpose of this section is to prove a theorem of Kakutani and Mackey ([1]), which asserts t h a t if V is a Banach space, the above procedure leads to no new examples of logics other t h a n the standard ones. I n particular, if the underlying lattice of a logic is isomorphic to the lattice of all closed linear manifolds of a separable infinite dimensional Banach space over one of R, C, or H, then the logic is standard. Let D be one of R, C, or H and V any vector space over D. We shall say t h a t a topology 3T on V is compatible with the vector space structure of F i f it is Hausdorff, and, if the map, a, b,x, y^ax+byof&x'Dx VxV into V, is continuous. A topological vector space over D is a pair (F,^"), where V is a vector space over D, and «^", a topology over V compatible with its vector space structure. By the usual abuse of language we shall say t h a t V is a topological vector space. We shall denote by «j£?(F,D) the lattice of all closed linear manifolds of V. This is a complete lattice. If we denote by V and A the lattice operations in j£?(F,D), then, for any family {Ma} of closed linear manifolds of V, we have:

(i)

A Ma = n Mtt> a

(2)

a

\JMa=^May,

where 2 denotes the algebraic sum, and the bar denotes closure. If N is a n y finite dimensional linear manifold of V, then N is necessarily closed and for any closed linear manifold M, N v M = N -\-M. Moreover, if M is a closed linear manifold whose codimension ( = dim VjM) is finite and N any algebraic complement of M, i.e., such t h a t M n N = 0, M +N= V, then N is necessarily closed and V = M y N (for these facts cf. Bourbaki [2], Ch. 1). We shall recall the notion of Banach spaces. Let D be one of R, C, or H and V a vector space over D. A norm over V is a function x - > ||a?|| of V into the real numbers such t h a t

(3)

(i)

||a || > 0, = 0

(h)

\\x +y\\ < \\x\\ + \\ y\\

(in)

||ca;|| = |c| • ||#||

if and only if for

for

x = 0,

x, y e V,

c e D, x e V.

A Banach space over D is a pair (V, \\ • ||), where V is a vector space over D and || • || a norm on V such t h a t V is complete under the metric d defined by d(x,y) =\\x — y\\. A Banach space over D is obviously a topological vector space over D. We are now in a position to formulate the theorem of Kakutani and Mackey. We recall the notion of Hilbert spaces and inner products associated with D (cf. Chapter I I , section 4).

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Theorem 4.1 (Kakutani-Mackey [1]). Let D be one of the three division rings R, C, or H, V an infinite dimensional Banach space over D, and jSf = o^(F,D) the lattice of all closed linear manifolds of V. Suppose that ±_{M-> M1) is an orthocomplementation of j£f. Then there exists a Dvalued inner product <. , . > on V xV such that (i) V becomes, under <. , . >, a Hilbert space over D; (ii) the topology of V, induced by the norm associated with <. , .>, coincides with its original topology; and (iii) the map M -> ML coincides with the orthocomplementation induced by <. , .>. In particular, if the underlying lattice of a logic is isomorphic to the lattice of closed linear manifolds of a separable Banach space of infinite dimension over D, then the logic is standard. The proof is rather long and depends on a series of lemmas. With later applications in mind we formulate our first lemma somewhat more generally than is necessary for our immediate purposes. Lemma 4.2. Let K be a division ring and W a vector space of infinite dimension over K. Let Jtbe a lattice of linear manifolds of W such that (i) Jt contains all finite dimensional linear manifolds of W, and (ii) if M,N e Jt and at least one of them is finite dimensional, then My N = M+N. Suppose Jt{M -> ML) is an orthocomplementation in Jt. Let w0e W be any nonzero vector. Then there exists an involutive anti-automorphism 6 of K and a symmetric 6-bilinear form <. , .) on W x W such that (i) (w0,woy = 1 and (ii) (x,y} = 0 if and only if x e (K • y)L • 6 and (. , . > are uniquely determined. The form <. , .) is definite, i.e., (x,x) = 0 if and only if x = 0. Proof. Let F^ W be a finite dimensional linear manifold. Then F e Jt'. For any linear manifold L^F we have L e J also. We now define (4)

(F(L) =

L^nF.

Note that if M is finite dimensional and iV" e Jt, then M AN = MnN. In fact, M AN^MOLN always while the finite dimensionality of Mc\N implies, by (i), that MnN e Jt, so that MnN^M A N. Let JtF be the projective geometry of all linear manifolds L^F. £F is a map of JtF into itself. We shall show first that £F is an orthocomplementation of J/F. Clearly ^(0) = ^ , £F(F) = F1nF = F1 A F = 0. Since M —> ML is order inverting, it is obvious that fF is order inverting. We observe next that for LeJtF, Ln£F(L)^LnL1 = L A L1 = 0. On the L L other hand, W = L v L = L + L so that F = L+£F(L). Thus L and £F(L) are complements of each other in JtF. To complete the proof that £F is an orthocomplementation, it remains to prove that £F is involutive. For LeJtF, £F{L)<^LL so that L<^ijF(L)L. This shows that L£gF(L)Ln F=€F(€F(L)) for all LeJtF. Let L' = t;F(L). Then, by what we proved just now, £F(L') is a complement of L' in JtF. But L is also a complement of L' = £F(L) and we have seen that L^£F(L'). This shows that L must

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coincide with £F(L'). We see thus t h a t (jF is an orthocomplementation ofJ?F. We shall fix a nonzero vector w0 e W. Let !F be the class of all finite dimensional linear manifolds Fs^W such t h a t (i) w0 e F, (ii) dim F>3. By theorem 2.7 there exists, for any Fe^, an involutive anti-automorphism dF of K and a nonsingular symmetric 0 F -bilinear form <. , . } F on Fx F such t h a t < • , - X F induces £ F ; i.e., for Z, (5)

&•(-£) = {u : ue F, <jx,u)F = 0

for

allXG!}

and (6)

Oo,w 0 > = 1.

Moreover, (6) determines 0 F and <. , . } F uniquely. Suppose now t h a t Fl9 F2e3r with Fx c JF2. Then, for any L e JtFi,

fc1(I) = ^ n f 1 = { 6 > ( i ) } n ^ , This shows t h a t for any Z e ^ # F l , (7)

£Fl (L) = {u:ue

Flt (x,u}Fz

= 0

for all

x e £}.

In other words, the restriction of <. , . } F 2 to F1 x Fxis a, 6F2 -bilinear form which also induces the orthocomplementation f F . I n view of the normalizing condition (6), we m a y conclude at once t h a t

(8)

*'•

=

°F-

<*,y>F2 = <*,y>Fl

(x> y e ^ i ) -

Since any two elements of & are contained in a third element of J^, we easily deduce from (8) t h a t there exists an involutive anti-automorphism 6 of K, and a symmetric 0-bilinear form <. , . > on W x W such t h a t (i) 0=6F for all F e J ^ and (ii) the restriction of <. , .> to Fx F is ( . , . )F for all F e^. Lemma 4.2 follows a t once from this. We now resume our discussion of the lattice ££ of all closed linear manifolds of the Banach space V over D. We take V= W in lemma 4.2 and obtain an involutive anti-automorphism 6 of D and a definite symmetric 0-bilinear form <. , . > on V x V such t h a t (x,y) = 0 if and only if x±_y. We fix a nonzero vector v0 e V; then 6 and <\ , .> are uniquely determined by the condition (v0,voy = l. Let V* be t h e set of all continuous linear maps of V into D. Then V* is a vector space over D°, the division ring dual to D. For ueV, let j8tt be the linear m a p of V into D defined by

(9)

pu(x) = (x,u)

(xeV).

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We shall also use the conjugate norm on F*; this is defined by (10)

||AII = sup \\{u)\

(A e V*).

\\u\\
I t is well known t h a t V*, under this norm, is a Banach space over the division ring D°. Lemma 4.3. pu e V* for any u e V. Moreover, u —>• fiu is an additive isomorphism of the abelian group V onto the abelian group V*. Proof. I t is a well known fact (cf. Bourbaki [2]) t h a t a linear m a p A of V into D is continuous if and only if the linear manifold (11)

ZA = {x:xeV,

X(x) = 0}

is closed in V. Let now A = j8u, and write Zu for ZA, if and only if = 0, i.e., if and only if y\_u. Hence

ueV.liyeV,yeZu

Zu = (D-tO 1 .

(12)

B u t (D-u)1 e ££ and is therefore closed by assumption. This proves t h a t fiu e V*. The m a p u-> flu is clearly additive. If, for ue V, fiu = 0, then (uius)=pu(u)=0; this implies t h a t u = 0. I n order to complete the proof of t h e lemma it remains to prove only t h a t u^/3u maps V onto F*. Suppose A e F * and # 0 . Let ZA be defined by (11). Then ZA e & and ZAT^ V. Therefore ZAL^0. Let u be a nonzero vector in ZKL. Since (x,uy = 0 for all x e ZA, f$u vanishes over ZA; and since u^O, j8 M ^0. Since ZA and Zu are both of codimension 1 and ZAciZu, we infer t h a t ZA = ZU. Let xQ e V be such t h a t X(x0) = 1. Since x0 <£ ZA, xQ $ Zu and hence Pu(xo) = <^o^> ¥* 0. Let c = (xQ,u)~1

and let us write u0 = ceu. Then for x e V, <X,UQ)

=

<X,U)C.

This shows t h a t PUo(xQ) = 1 = X(x0) and implies t h a t p = A. The main question at this stage is whether 6 is continuous. The nontrivial case is when D = C and we handle it first. I t is at this point t h a t essential use is made of the infinite dimensionality of V. Lemma 4.4. Let W be a complex Banach space of infinite dimension. Then one can find a sequence of vectors xl9 x2, • • • e W with the property that, for any bounded infinite sequence {cn} of complex numbers, there exists a continuous linear functional A on W such that (13)

X(xn)=cn

(W=l,2,.-.).

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Proof. Let || • || be the norm on W. Let W* be the space of all complex valued continuous linear functionals on W. We define x± e W to be an arbitrary vector of norm 1 and choose xx* e W* such that

The celebrated Hahn-Banach theorem ensures that such an xx* e W* exists. We shall now define by induction two sequences {xn} and {xn*} in W and W*, respectively. Suppose xlt x2, • • •, xn e W and xx*, • • •, xn* e W* have been already constructed so that xi*(xj) = 8 y (Kronecker delta) (14)

(i 1)

1^*11 = 2" "

(i,j = 1, 2, • • •, n),

(i = 1,2,...,»).

Since W is infinite dimensional, there is a nonzero vector xn + 1 in W linearly independent ofxlf---,xn such that

Using the Hahn-Banach theorem once again, we can find x*+ x e W* such that x*+ i{xn + 1)-=^0, x*+ 1(a;1) = 0, i <w. Replacing #n + x and x*+ ± by suitable multiples, we may assume that \\x*+ ± || = 2 ~ n, a;*+ 1(xn + x) = 1. The sequences xl9 x2, - - -9 xn + 1 and xx*, • •, x*+1 then satisfy (14) with n replaced by n + 1 . By induction it follows that there exist infinite sequences {xn} and {xn*} such that xne W for all n, xn* e W* for all n, and *i*(*y) = 8 tf

(*,j =

|*i*| = 2 " « - »

1,2,..-),

(* = 1,2, • • •).

We claim that the sequence {xn} has the properties asserted in the lemma. In fact, let {cn} be any bounded sequence of complex numbers. For x e W, let X(x) be defined by (16)

\(x) = 2

onxn*(x).

n= l

The inequality |a;n*(a?)| < ||rrn*|| • \\x\\ = 2-(n""1)||a?|| shows that A is well defined and that x -> X(x) is a continuous linear functional on W, i.e., A e If*. Clearly, \{xn) = cn for all w. We now resume our discussion of V and 0. Lemma 4.5. If D = R, 6 is the identity; if D = C, 6 is the conjugation; and ijD = H, 6 is the canonical conjugation. Proof. Let F be any linear manifold of finite dimension > 3 containing v0. The map L -^L^ n F is, as we saw in the proof of lemma 4.2, an orthocomplementation in the geometry of subspaces of F and is, moreover,

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induced by <. , . )F (the restriction of <. , . ) to F x F). From theorem 2.8 we deduce t h a t 0 is the identity if D = R and is the canonical conjugation if D = H. Suppose now t h a t D = C. Clearly, as (X,X)F = 0 for am XEF implies t h a t # = 0, 6 is not the identity. 6 will therefore have to be the conjugation in this case, provided we show t h a t it is continuous. Since 6 is additive, this reduces to showing t h a t if cn - > 0 in D, {cne} is a bounded sequence. Suppose then t h a t D = C and t h a t 6 is not the conjugation. Then there exists a sequence {cn} in D such t h a t

Since dim F = oo, we can select a sequence {xn} in V such t h a t it has the property described in lemma 4.4. Let j8n be the element x->{x,xn) of F * (cf. lemma 4.3). Since \cne\ ^ o o , we may, by passing to a subsequence if necessary, assume in addition t h a t for all n,

(18)

K'| >H|j8 n ||.

By lemma 4.4 we can find a A e F * such t h a t X(xn) = cn for all n. Evidently, A7^0. Moreover, it follows from lemma 4.3 t h a t there is a u^O in V such t h a t X = f3u. Then (u,u)^0 and (xn,u} = cn for all n. If (19)

Zn =

(UyUy-^-Cn'^,

then A(Zn) = <2n>W> =

0

for all n. But then, as <. , . ) is symmetric,

(u,zny = o for all n, i.e.,

(20)

«u,uye) - \u,uy - ( O - \u,xny = o.

Since K O " 1 ^ , ^ ) ! < \cn°\ - ^ I f t J \\u\\, (18) implies t h a t (cne)-\u,xn}->0

(n->co) e

and hence we infer from (20) t h a t (u,uy((u,uy )~1=0, which is impossible as u^0. This contradiction proves t h a t 0 must be the conjugation and completes the proof of lemma 4.5. Lemma 4.6. f$(u -> f$u) is a continuous map of V onto V*. Proof. Since we may obviously consider R as a subfield of D as well as D°, V and F * m a y both be regarded as Banach spaces over R, denoted by ^ R a n d ^ R * J respectively. From lemma 4.5 we conclude t h a t fi(u —> j8u) is an R-linear map. As /3 is denned on the whole of F R , we m a y use the closed graph, theorem to reduce the proof of the continuity of j3 to the proof t h a t j8 is a closed map. Let {xn} be a sequence in F R such t h a t

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xn -> x and px ~> A. We must prove that \ = f$x. Since /? is onto, we may assume that \=f$u for some ue V. From the formula (10) for the norm in V*, it follows that fi Xn(v)-> fiu(v) for all veV. Since, by lemma 4.5, 6 is continuous, we have, for v e V, (x,v) = lim (xn,v} n

= lim «v,xnye) n

= (v,u}e = (u,v). This proves that x = u or /3X = X. We have thus proved lemma 4.6. Lemma 4.5 shows that (. , .} is an inner product on V xV. Thus, on defining ||2J|02 = <(z,z>, we obtain a norm on V. V is a pre-Hilbert space with respect to <. , . >. Lemma 4.7. V is a Hilbert space under <. , . >. Proof. We must prove that V is complete under || • ||0. Let {xn} be a sequence in V such that \\xn — xm\\02 ~> 0, n, m -> oo. Then {y,xn — xm} -> 0 as n, m -> oo for each y e V. Let A(?/) = limn (y,xn}. Then A is a linear map of V into D. Since f$x e F R * for all n, we ma}^ apply the well known uniform boundedness principle to F R , a Banach space over R, to conclude that Ae V*. By lemma 4.3 there exists a ze V such that A=&. We claim that \\xn — z\\0 -> 0. Let e > 0 be arbitrary and N. be such that \\xn-xm\\0

<e

for n, m>N. Then, for y e V, \ ^n-^m>| < 4y\\o

(w>m ^

N

)

and hence \
(n> N),

which shows that \\xn — z\\0<e for n>N. Lemma 4.8. The topologies induced on V by || • || (the original norm) and || • || 0 coincide. Proof. Let y, xlf x2, • • • be elements in V. We must show that as n -> oo, \\xn~y\\o -** ^ ^ a n ( i o n ly ^ ll^n —2/|| -^ 0- Since F is a Banach space over R under both || • || and || • ||0, it is sufficient, in view of the open mapping theorem, to show the implication in one direction only. Let then \\xn — y\\ -> 0. Since the map j8 from V to F* is continuous, fiXn — f$y -> 0 in F* so that (21)

sup II 2 II £ 1

\(z,xn-y)\-+0

(n^co).

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Now, \xn — y\ < 1 for all sufficiently large n and hence we infer from (21) that

\\xn-y\\o2 =

zn-y>-+o>

which is the relation required. We can now complete the proof of theorem 4.1. V is a Hilbert space under <(.,.> and the topology induced by the corresponding norm has been shown to coincide with the original topology for V. Consequently, the same linear manifolds are closed in both topologies. Let |(Jf -> £(M)) be the orthocomplementation of the Hilbert space V. Now, (y,x} = 0 if and only if x _L y, and hence for any M e&,ye M1 if and only if y e i(M). This proves that JL and £ coincide. The proof of theorem 4.1 is complete. Theorem 4.1 shows that a logic may be asserted to be standard as soon as it is identified as a member of an apparently more general class of lattices. We have proved theorem 4.1 to emphasize our point of view that the important role played by standard logics is not due to accident but stems from deeper mathematical reasons, and that theorems such as the present one lead to a clarification of some of these reasons. This said, the stage is set for the detailed analysis of the standard logics and we proceed to do it now. 2. THE STANDARD LOGICS: OBSERVABLES AND STATES We now proceed to describe the structure of the set of observables and the set of states associated with a standard logic and examine the manner in which the general concepts of Chapter III specialize. Throughout this section we write D to denote one of R, C, or H. Let 3f be a separable infinite dimensional Hilbert space over D. We write ££ for the logic o$f(^f ,D) of all closed linear manifolds of J f (with its canonical orthocomplementation). For any closed linear manifold M of ^ , let us write PM for the orthogonal projection on M. If f(E ->f(E)) is any observable associated with the logic j£f, then the map E -^ PnE) is a projection valued measure^ based on ^(R 1 ). Conversely, it is obvious that to any projection valued measure P(E -> PE) based on J^R 1 ) there exists a unique observable/(2£ -+f(E)) such that PE = PnE) for all real Borel sets E. •f The concept of a projection valued measure is nothing new; it is a standard tool of spectral theory. For complex Hilbert spaces we refer the reader to sections 35-38 of Halmos [2]. For the real case very few changes have to be made. For the quaternionic case and for projection valued measures defined on the real line, the theory is essentially the same, since the real field is the center of H . We shall assume the basic facts concerning projection valued measures as known.

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Lemma 4.9. Let Ml9 M2 e J^. Then M1 <—> M2 if and only if the projections PMi and PM2 commute. Proof. Let Mlf M2 e j£f. Assume first that Mx <-» M2. Then there exists mutually orthogonal elements Nl9 N2, N of 3? such that Mi = Ni-\-N (i = l, 2). Clearly, PMi = PNi+PN, and one is led immediately to the equation PMiPM2 = PMiPMi. Conversely, let PMi and PM2 commute and let P = PMiPM2. Then P is a projection. If we write Q{ = PMi-P, then it is easily verified that Qt is a projection (£ = 1,2) and PQi = QiP = 0, Q1Q2 = #2$i = 0.1{N l9 N2, and N are the closed linear manifolds which are the ranges of Ql9 Q2, and P, respectively, then it follows easily that Nly N2, and N are mutually orthogonal and Mi^=NiyN (i = l, 2). This proves that M1 <-> M2. Lemma 4.10. Let X be a set and & a o-algebra of subsets of X. Let P(E -> PE) be a projection valued measure in £F based on & and !F the algebra (over R) of all real valued bounded measurable functions on X. For ge^ let ||^||=sup{|gr(a;)| :xeX}. Then, for any ge^, there exists a unique, bounded, self-adjoint operator Bg on #F such that
g(x)dvUtV(x)

(u, v e Jt?)9

(22) J Bg =

gdP,

in symbols I

where vUfV is the J)-valued measure on & defined by vUtV{E) = (PEu,vy. The spectral measure of Bg is the projection valued measure E -> Pg~ iiE). One has: (23)

\\Bg\\ < \\g\\

(gelF).

The correspondence g -> Bg is a homomorphism from the algebra 3F into the algebra (over R) of all bounded operators on Jf. In particular, B9l and Bg2 commute for all gl9 g2e &'. Proof. The proofs are similar to the ones in section 37 of Halmos [2] and we omit them. In the quaternionic case we have to use the fact that the real field is the center of H. Theorem 4.11. Let D be one of R, C, or H and ££ the logic of all closed linear manifolds of #f\ For any observable f(E->f(E)) associated with

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££\ let Af be the self-adjoint {not necessarily bounded) linear operator with spectral measure E - > P / ( E ) . Then (24)

f->At

is a one-one correspondence between the set of all observables associated with ££ and the set of all self-adjoint operators on Jf. The observable f is bounded if and only if the operator Af is bounded. Two bounded observables fx and f2 are simultaneously observable if and only if the (bounded) operators Afl and Af2 commute. Iff is any bounded observable and p is a polynomial with real coefficients (in one variable), then the operator corresponding under (24) to p o f is p(Af). More generally, if fl9 • • • , / r are bounded observables, any two of which are simultaneously observable, and if p is a real polynomial of r real variables, then the observable ^ ° (A, ...,/r) (cf. theorem 3.17) has the corresponding operator p(Afi,..., Afr). Proof. The first assertion concerning the correspondence / - > A f is an immediate consequence of the spectral theorem. Next, an observable / is bounded if and only if for some compact set K £ R 1 , f(K) = 3tfp. This is possible if and only if PnK) = / , which in t u r n is well known to be equivalent t o the requirement t h a t Af be bounded. Let now fx and / 2 be bounded observables. Then fx and f2 are simultaneously observable if and only if fi(Ei) *~* 12(^2) f ° r an* pairs of Borel sets E1 and E2. From lemma 4.9 and standard spectral theory, it now follows t h a t A <->/2 if and only if Af and Af2 commute. We now" come to the last pair of assertions. Since the first of these is a special case of the second for r = 1, we shall confine ourselves t o the proof of the second. Let A , • • • ,fr be bounded observables such t h a t fi<->ft for all i, j = l, 2, • • •, r. By theorem 3.17 there exists a a-homomorphism F ->f(F) of &(Rr) into 3? such t h a t for any i (l t{ of R r onto R 1 . Let PF = Pf{F). Then F -> PF is a projection valued measure on «^(R r ). Since the/,- are bounded observables, there exists a compact set K c Ri such t h a t fi(K) = Jt? for i=l, 2, • • -, r. We now apply lemma 4.10 by taking X = Kr, @ = &(Kr). For any E e @(K), Pni-\E) = PftE\ and so, writing VuAF)

=
vlv(E)

=

(P^u,v),

we get (Bniu,vy

=

TT^!, t2, • • ., tjdv^fa, JKr

= J tdvljt) =


•••,*,)

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Thus, Bni = Aft, i = 1, 2, • • •, r. Since g -> Bg is a homomorphism, we have, for any real polynomial p of ^, • • •, tr, Bp=p(BXl,...,BMr)=p(Afl,...,Afr). But the self-adjoint operator Bp has the spectral measure E -> Pp- i(E) by lemma 4.10. Bp therefore coincides with the operator corresponding to the observable E ->f(p~1(E)) which is precisely p o (flt • • -,/ r ). This proves the theorem completely. Remark. That f1 <—>/2 is equivalent to the equation AflAf2 =Af2Afl is a celebrated result of von Neumann. The fact that p o (fl9 • • -,/ r ) corresponds to the operator p{Afl, • • •, ^t /r ) is remarkable since it links the algebra of functions of fl9 • • • , / r with the algebra of operators on Jf. We now take up the problem of determining the states of the logic j£f. Our basic result is the theorem of Gleason [1] which asserts that every state of ££ can be described in a canonical way by what is usually known as a density matrix. We shall now proceed to a detailed discussion of this theorem. To any vector uetf? with \u\ = 1, we associate the mapping pu of J&? into the set of nonnegative real numbers by defining (25)

pu{M) = (PMu,u} = ||PMi*||2

(Me&).

2

If M=Jf, pu(M)=\\u\\ = l, while pu(0)=0. If Ml9 M2, • • • is a sequence of elements of ££ such that Mi\_Mj (i=£j) then, writing M = Vt Mu we have : pu{M) = \\PMu\\2

= 2 11****11* i

i

In other words, pu is a state of ££. It is clear that if c e D and \c\ = 1 , Pcu=Pu> The general description of states depends on the concept of an operator of trace class. On complex Hilbert spaces they are defined and treated in many books on Hilbert space theory (cf. Dunford and Schwartz [2], for example). The treatment in the real and quaternionic cases is similar. A bounded operator A of $f is of trace class if, for any orthonormal basis {ej, the series 2 |<4«f,«f>| < oo; i

then the sum tr(^) = ^ i

<MA>

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exists for any orthonormal basis {et} and is independent of the basis used; it is called the trace of A. If A is of trace class and B is any bounded operator, AB and BA are of trace class; and moreover, tr(AB) = tr(BA). If A and B are of trace class, so are a A +/3B where a, j8 are in the center of the division ring over which J f is defined. We shall assume the reader to be familiar with the basic properties of these operators. The states of the form pu(u e Jf, ||w||=l) can also be described in another way. Let U be the projection on the one-dimensional linear manifold D u. Then, for any bounded operator A, AU is a bounded operator, which is of trace class, since U is (trivially) of trace class. Let {en} be an orthonormal basis for Jf7 with u = e±. Then tr(AU) = tr(UA) (26)

=^(AUeiA> i

=

(Au,u).

In particular, (27)

pu(M) = tr(PMU).

Suppose that %, u2i • • • is any sequence of vectors in #F with H^H = 1 for all i. Let alf a2, • • • be a sequence of nonnegative numbers with «i +«2 + • * • = L Let Ut be the projection on the one-dimensional manifold D-iv Then

(28)

^ =

2 ^ i

is a well-defined, bounded, nonnegative, self-adjoint operator of trace class and (29)

tr(*7) = 1.

If M e <£?, we have tr(PMU) = ^at

tT(PMUt)

i

= 2a^twi

Thus the function pv defined by (30)

pu(M) = tr(PMU)

is a state of j£? and in fact is the state 2i ciPut' Conversely, let U be a bounded, self-adjoint operator such that (i) U is nonnegative, i.e., (Ux,x}>0 for all xeJf, and (ii) U is of trace class and tr(U) = l. Then,

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the function pv defined by (30) is a state of ££\ We call the operators U which are bounded, self-adjoint, nonnegative, and of trace class, von Neumann operators. The question naturally arises whether every state of J&? is of the form pv for a suitable von Neumann operator U of trace 1. It was Gleason [1] who first proved that this is indeed so. His result, which is a cornerstone of the mathematical foundations of quantum mechanics, is undoubtedly one of the most profound theorems in this subject. Gleason's theorem is quite complicated to prove and we need to develop a number of technical lemmas. We begin with the concept of a frame function. Let $ be the unit sphere of Jtf, i.e., (31)

g = {x '.xeJtf, ||*|| = 1}.

A real valued function/(x ->/(#)) defined on & is a frame junction if (i) J (ex) = f(x) for xei and c e D with \c\ = 1, (ii) there exists a constant W such that 2n/( e n) — W f° r a n v orthonormal basis {en} of 3tf (recall that J f is separable). The constant W is called the weight of/. Suppose that U is a bounded, self-adjoint operator of trace class. Let us define the function fv by (33)

fu(x) =

(xet).

Then/^ is obviously a frame function whose weight is tr U. A frame function / is called regular if there exists a bounded, self-adjoint operator U such that (34)

f=U

Clearly, if/ is regular, there exists only one bounded, self-adjoint operator U such that (34) holds; for, if U and V satisfy (34), <(*7- V)x, x} = 0 for SbllxeJf? and hence U— V = 0. Suppose that /x is a state of ££\ For any vector x e $ let us define (35)

Ux) = pQt-x).

Then/^ is a nonnegative frame function of weight 1. If U is a von Neumann operator of trace 1 and p=Pu t n e state defined by (30), a simple calculation shows (36)

f, = fu,

i.e., fu is regular. Conversely, if fi is a state of ££ and if we assume that

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fu =fu for a bounded, self-adjoint operator, then it can be shown t h a t U is a von Neumann operator of trace 1, and fi=pv. I n fact, as 0, U is nonnegative; then, if {en} is any orthonormal basis of Jf7,

2
n

so t h a t U is of trace class and has unit trace. This shows t h a t the states fM and pv coincide for all one-dimensional subspaces of Jt? and hence coincide for all elements of ££\ I n other words, Gleason's theorem is entirely equivalent to proving t h a t every nonnegative frame function is regular. We shall first examine t h e special case when D = R and when J f is finite dimensional. If #? is the real line, then the only frame functions are constants. If #? is t h e plane R 2 , then it is obvious t h a t a frame function can be arbitrarily defined on any quadrant of the unit circle and hence need have no special properties. Lemma 4.12. Let &? — R2, the plane (D = the real field) and let $ be described as (37)

£ = {(cos 0, sin 0) : 0 < 0 < 2n}.

Then the function f : (cos 6, sin 6) - ^ cos nQ, n being an integer, is a frame function if and only if either n = 0 or n = 2 (mod 4). Proof. I n order t h a t / be a frame function we must have, for some constant W, (38)

cos7i0 + cos ( ™ ( 0 + | ) ) = W

for all 6. Differentiating with respect to 6, we have: M sin for all 0, i.e., ...

-./_

n7r\

.

nir

.,

n\ (sin nd)[ 1 + c o s — I + s m — 2 cos ^ 0 ^ = 0. This is possible if and only if either n = 0 or cos(mr/2)= — 1 , i.e., if and only if n = 0 or n = 2 modulo 4. Lemma 4.13. Let J f = R 3 be the three dimensional real Euclidean space. Then every continuous frame function on the unit sphere
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Proof. The proof makes use of harmonic analysis on $ (cf. Weyl [1], pp. 60-63,142-146). Let # be the set of all real valued continuous functions on $ and G the group of all orthogonal transformations of R3 which have determinant + 1 . For any element h of ^ and any a e G, we define ha by ha(x) = hia-^)

(39)

(xeg).

A linear manifold Ji c <$ is said to be invariant if ha e J£ for all a e 6r whenever & / e ^#. Then, according to classical spherical harmonic analysis, one has the following description of <S\ (1) for each integer n = 0, 1, • • • there is a 2n + 1 dimensional subspace Fn of ^ which is invariant and no proper linear submanifold of which is invariant, i.e., Fn is irreducible under the action (39) of the group G; (2) an element h e <€ belongs to !Fn if and only if it is the restriction to $ of a polynomial p of x, y, z (the coordinates on R3) with the properties (i) p is homogeneous in x, y, z of degree n, (40)

,..v / a 2 a 2

(u)

d2\

W 2 + ^ 2 + ai 2 F = 0;

(3) every continuous function on $ is the uniform limit of a sequence of functions each of which is a finite linear combination of elements of the «^"n; and finally (4) if IF is any invariant linear manifold, closed under the uniform convergence topology of # , and nlt %, • • • (0
(41)

^= (2^)~>

where 2 denotes the algebraic sum, and the bar denotes the closure. Let now IF be the set of all continuous frame functions on $. Evidently G-transforms (cf. (39)) and uniform limits of continuous frame functions are continuous frame functions. Thus the set F is a closed linear manifold in ^ which is invariant. Therefore, there exist integers nx, n2, • • • with the properties described in (41). We now claim that if n is any integer other than 0 and 2, then, IF^IF. In fact, let n be any integer. We use cylindrical coordinates p, 6, z on $ denned by x = p cos 0, y = p sin 6, z = z. Then it is well known that d2

d2

dx^^df^d?

d2 _ d2

Id2

~

2+

Id 2

+

d2 +

Wp 'p d¥ }Tp d?'

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One can easily check that the functions n P

cos n6, [pn-2(n-l)z2Pn-2]

cos {n-2)d

are homogeneous polynomials of degree n which satisfy the equation /a2

a2

d2\

Thus if J*"ncJ*-, then pncosnd and [pn-2(n-l)z2pn-2]-cos(n-2) d are frame functions. For any 0, (cos 0, sin 0, 0), ( — sin 0, cos 0, 0) and (0,0,1) form a frame of B3 so that the functions obtained by putting z = 0 are frame functions on the unit circle of the x, y plane. Thus cos nd and cos(w — 2)6 are both frame functions. By lemma 4.12 this is not possible unless w = 0 or 2. Thus we must have ^ ^ ( J ^ + J ^ ) - i . e . , J ^ c J ^ o + J ^ , since ^0 and ^ 2 are finite dimensional. We now claim that JF = ^" 0 +^2 and that every element of ^0 +^2 *s a regular frame function. In fact, ^0 consists of the constants which are regular. On the other hand, J^2 consists of the quadratic forms q(x,y,z) which satisfy id2

2+

d2\

d2

2+

W W ^)q= ' i.e., q(x,y,z) = ax2 + by2+cz2+ 2fyz + 2gzx + 2hxy with a+b+c = 0. Thus a function of ^ lies in ^0 +^2

^

an(

* only if it is of the form

x —> (Ax,x} (x is a vector in
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which are orthogonal to q. If the latitude of q is 6, it can be shown that this great circle passes through only those points whose latitudes 9? satisfy — 6<(p<0. It is therefore appropriate to call this the EW great circle through q. The essence of our argument is contained in the next lemma. To motivate this lemma, let us consider in the usual coordinates x9 y, z the function (x,

y,z)->x2+y2.

This is a frame function which is nonnegative and vanishes at the north pole p = (0,0,1). It is constant on points with the same latitude. For any point q of the northern hemisphere, the points of Np other than q on the EW great circle through q, have latitudes strictly less than the latitude of qy and consequently this frame function reaches its minimum on the EW great circle through q, at q. The next lemma asserts that every frame function which is nonnegative has this property approximately. Lemma 4.14. Let p e $ and g be a nonnegative frame function which is constant on the equator opposite to p. Then for any q in Np we have: (42)

g(q) < g(x)+g(p)

for every point x on C, the EW great circle through q. Proof. We first assert that if r is any point of Np, then (43)

g(r) < k+g(p),

where k is the constant value which g takes on the equator. To see this, consider the EW great circle through r. If r' is one of the points at which this meets the equator, then r_[_r' and hence r, r' are two points of a possible frame. The frame function g being nonnegative, one has: g(r)+g(r') < W, where W is the weight of g. But if we take any two points a, a' on the equator which are orthogonal, a, a', p is a frame and, consequently, 2k+g(p) = W. Thus g(r) < W-g(r') < 2k+g(p)-k

< k+g(p).

Suppose now q is an arbitrary point in the northern hemisphere. The inequality just now established proves the lemma if x lies on the equator. For x below the equator, — x lies above the equator, and, since g(u) =zg( — u) for all u, it is enough to prove the inequality (42) when x lies on the EW great circle C through q and is also situated in the northern hemisphere.

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Let y be a point on C, orthogonal to x, and lying in the northern hemisphere. We then have, if q' is a point where C meets the equator, g{x)+g{y) = g(q)+g(q') = g(q)+k and, since g(y)
by (43), one has: g{*) =

g(q)+k-g(y)

> g(q)-g{p)> which establishes the lemma. Lemma 4.15. With p and g as in lemma 4.14, let z be a fixed point in Nv. Let Mz be the set of points x on Np which have the property that for some point y of Np the EW great circle through x contains y and the EW great circle through y contains z. Then the interior of Mz is nonempty. Moreover, for any x of MZi one has: (44)

g(x)


Proof. Since g(x) i?»£) = (£2+y2) sin a —££ cos a. This is greater than 0 on the equator. On the other hand, there are points of Np for which it is <0. Therefore, if we define A to be the set of points

A ={(f,i ? ,0:0<J
0(^,£) <0},

then A is a nonempty open subset of Np. Suppose x lies in A. Then ip(x)<0 while the EW great circle through x meets the equator at a point at which ijj is positive. Since 0 is continuous, there must be a point y on the EW great circle through x at which 0 vanishes; thus x e Mz. In other words, i c M z . This proves the lemma. Lemma 4.16. With p and g as in lemma 4:A4:iletrj>0bea positive number such that g(p) < r/. Then there exists a point q of Nv and an open set M around q such that (45)

0 < sup g(u)- inf g(u) < 3^. ueM

ueM

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Proof. Define b = miueN g(u). Then b>0. Choose zeNp such t h a t b
ueM

This proves t h e lemma. Lemma 4.17. Let f be any nonnegative frame function such that for some open neighborhood M of r we have: (46)

0 < s u p , / » - w£f(u) ueM

and r any point

< a.

ueM

Then for every point s of $ there is an open neighborhood Ms of s such that (47)

0 < s u p , / » - inf f{u) < 4a. ueMs

ueMs

Proof. Use coordinates so t h a t r appears as the north pole. Let t be any point on t h e equator opposite r. Since M is an open neighborhood of r, we can assume t h a t every point whose latitude is > (TTJ2) — d0 belongs to M, 60 being a suitable angle in ]0,7r/2[. Consider now a point u due south of t and latitude — J0 O . Then by continuity it is clear t h a t there exists an open set Mt around t with the following property: if k is a n y point of Mt, then, on t h e great circle through u and k, there are points orthogonal to u a n d k, respectively, which are situated in M. Let k, k' be two points of Mt, and C, C the respective great circles through u, k and u, k'. Let x, y and x', y' be pairs of points, respectively, on these two great circles such t h a t x, y, x\ y' e M, % A-U,

y _1_ &> x' _\_u,

y' J_ k'.

We then have: f(x)+f(u)=f(k)+f(y), f(x')+f(u)=f(k')+f(y'). Subtracting and rearranging, we have:

\f(k)-f{k')\ <

\f(x)-f(x')\+\f(y)-f(y')\

< 2a, so t h a t 0 < sup f{u)— inf f(u) < 2a. ueMt

ueMt

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lis is any point of $, there exists a point t of $ such that t is on the equator opposite r and s is on the equator opposite t. Applying the preceding argument twice, we see that there is an open set Ms around s such that 0 < sup f(u) — inf f(u) < 4a. ueMs

ueMs

This proves the lemma. Lemma 4.18. Let f be a nonnegative frame function on . Then f is continuous. Proof. Let e > 0 be a given positive number. We shall prove that every point of S has a neighborhood in which the oscillation of/ is at most e. Since a constant is a continuous frame function, we can, by subtracting the infimum of/ (over S) from /, assume without any loss of generality that

inf/fa) = 0. Let 7} > 0 and p a point such that f(p) < rj/2. Let or be the rotation around the z-axis with^) as north pole through an angle TT/2. Let the frame function g be denned by 9(x)=f(x)+f(ax). g is nonnegative, and at p we have: 9(P) < V> moreover, g is constant on the equator. By lemma 4.16 there exists a point q of the northern hemisphere and some neighborhood Mq of q such that 0 < sup g(u)— inf g(u) < 3^. ueMq

ueMq

By lemma 4.17 there exists an open set Mp around p such that 0 < sup g(u)— inf g(u) < I2rj. ueMp

ueMp

Since g(p)
13TJ.

ueMp

Since 0
for all u, we have: 0 < sup f(u) < 13^ ueMp

and hence we have: 0 < s u p / ( u ) - inf f(u) < 137?. ueMp

ueMp

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Applying lemma 4.17 again, we see that every point of $ has a neighborhood over which the oscillation of / is at most 52^. If we choose 7/ = e/52, we are through. Lemma 4.19. Let ^ be a real separable Hilbert space of dimension at least 3. Then every nonnegative frame function f on the unit sphere $ of 34? is regular, Proof. Since every two-dimensional subspace L of J f can be imbedded in a three-dimensional subspace, it follows from lemma 4.18 that the restriction of/ to the unit sphere S'(L) of L is regular, and consequently there is a unique symmetric bilinear form on Lx L, say BL, such that (48)

BL(u,u) = f(u)

for all u G ${L). We shall now define a real valued function B on 3f x ^ as follows. Let x and y be two vectors of #F and L a two-dimensional subspace containing x and y. We define (49)

B(z,y) = BL{x9y).

Clearly, B(x,y) is well defined if x and y are linearly independent, since in this case L is unique; or if one or both of them is zero, since we then have BL(x,y) = 0 for any such L. If x and y were dependent and neither is zero, then they span a one-dimensional subspace. If Lx and L2 are two two-dimensional subspaces containing x and y, then both Lx and L2 are contained in a three-dimensional subspace, say M. Let C be a symmetric bilinear form on M x M such that

G(a,a)=f(a) for all vectors a in $(M). The restrictions of C to Lx x L± and L2x L2 are symmetric bilinear forms whose quadratic forms coincide with / on the unit spheres of L1 and L2, respectively. Since BLl and BL2 also have this property, we conclude that BLl{x9y) = C(x9y), BL2{x,y) = C(x,y), which proves that B(x,y) is well defined. Moreover, we have (50)

B(x,x)

=f(x)

for all unit vectors x in 3tf. We now claim that B is bilinear. The homogeneity of B as well as its symmetry involves only two vectors and so follows from the homogeneity and symmetry of BL. To complete the proof of the bilinearity (in view of the symmetry of B)> it is enough to show that B(x,y+z)

=

B(x,y)+B(x,z)

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for any three vectors x, y, z. Let M be a three-dimensional subspace containing x, y and z. Let Lly L2, L3 be two-dimensional subspaces of M containing, respectively, x9 y; x, z; and x, y+z. Let C be a symmetric bilinear form on M x M such that C(u,u) =f(u) for all unit vectors u in M. Then we see at once that C(x,y) = .BLl(*,y),

O ( ^ ) = BL2(x,y)

and C(x,y+z)

=

BL3(x,y+z).

Since C is bilinear, we have: B(x,y+z)

= S L a (»,y+2) = C(x,y)+C(x,z) = =

BLl(xiy)+BL2(x,z) B{x,y)+B{x,z),

which proves the bilinearity of B. Since 0 < -B(a?,a;) < 1 for all x with ||a;|| = 1, x, y-> B(x,y) is a bounded bilinear form. Thus there exists a bounded, self-adjoint operator U such that for all x, y in J f -B(a,2/) = <JJx,y>, f(x) = <J7o?,a?>. Thus we have proved that / is regular. We now proceed to the case of Hilbert spaces over D, where D may either be the complex number field or the quaternion division ring. Let 3tf be a Hilbert space over D. A closed R-linear manifold S of Jtf* is called completely real if the inner product <(.,.> in J f x MP takes real values on S xS; equivalently, if and only if there is an orthonormal set {ej) such that 8 is the closure of the real linear combinations of the ey. Lemma 4.20. Let 01 be a real separable Hilbert space, and f0 a regular nonnegative frame function of weight W. Then for all a,be& with

HI = l*I=i, (51) |/o(«)-/o(6)| * 2JF||o-&|. Proof. There is an operator T, of trace class and self-adjoint such that (Tc,c} =f0(c) for all unit vectors c. Then |/o(«)-/o(6)| = \| < 2||T||.|a-6|.

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But 0<(Tc,c}< W for all unit vectors c so that \\T\\< W. Thus \fo(a)-f0(b)\

<

2W\\a-b\\.

Lemma 4.21. Let J#* be a Hilbert space over D of finite dimension n. If f is a nonnegative frame function which is regular on each completely real subspace, then f is regular. Proof. We prove this by induction on n. For n = 1, the lemma is trivial. Let n>\. Let d = suj)m = 1f(u) and {xn} a sequence of points on $, the unit sphere of ^ , such that f(xn) -> d. We may assume, in view of the compactness of the unit sphere, that xn->yy \y\ = 1. We first show that f{y) = d. Let An=||-1<2/,^n>. Since xn-*y, A„->1. The inner product is real and consequently Xnxn and y span, over R, a completely real subspace. We now have, if W is the weight of/,

\f(y)-d\ < \f(y)-AKxn)\ + \f(Kxn)-d\ <2W\\y-Xnxn\\+\f(xn)-d\, since /, by hypothesis, is regular when restricted to the completely real subspace spanned by \nxn, y, and f(xn)—f(Xnxn) (|An| = l). This shows that f(y) = d. Let us now extend the function / to a function F defined on all of 3^ by setting (52)

F(v) = 1

lH2/(IN"H

v*o

Clearly for c e D, (53)

F(cv) = \c\2F{v).

We now assert that if u is any vector of 3tf orthogonal to y, then (54) F(cy+u) = \c\2d + F(u) for all c E D. Let L be the two-dimensional real linear manifold spanned by y and u. This is completely real, / i s regular on its unit sphere so t h a t / i s the restriction to L of a nonnegative definite quadratic form Q. But y is the point at which the maximum of the quadratic form is reached and u is orthogonal to y in L. Thus the matrix of the quadratic form Q is diagonal with respect to the basis {y,u} of L, and consequently we have, whenever r and r' are two real numbers, F(ry+r'u) = \r\2d + \r'\2F(u). It is to be noted that u here is an arbitrary vector orthogonal to y.

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Suppose now that v is any vector orthogonal to y and c^O an element from D, not necessarily real. We have: F(cy+v) =

F^y+c-H)) a f

= |c| J (y+c- 1 v) = \c\2F{y +

\c\-i\c\c^v),

x

but u= \c\c~ v is a vector orthogonal to ?/. Thus j ^ + c " 1 * ; ) = d + \c\-2F(\c\c-1v) = d + \c\~2F(v), proving that

F(cy+v) = \c\2d + F(v).

The restriction of / to the orthogonal complement Jf* of y is a frame function, which, on Jf, has the same properties as /. Suppose we assume the present lemma to be true for all Hilbert spaces of dimension k < n. Then F restricted to J f is regular by the induction hypothesis and hence there is an orthonormal basis {el5 • • •, en_x} for J f and real numbers ^i> * * • 9 dn_! > 0 such that F(ciei + • • • +c tt _ 1 e n _ 1 ) = {c^d, + • • • + |cn_1|2<Jn_1 for all cl9 • • •, c n _! G D. By the proof given in the previous paragraph, this means that F{cy+c1e1 + > • • +c n _ 1 e n _ 1 ) = |c|2d + |c 1 | 2 ^ 1 + • • • +|c n _ 1 | 2 d n _ 1 , which shows that / is the restriction to the unit sphere of the regular frame function fUt where U is the operator denned by Uy = dy, Uej = dfi^ j = 1, 2, • • •, n — 1. This proves the lemma. Lemma 4.22. Let 3f be a separable Hilbert space over D which is one of R, C, or H of dimension at least 3 and f a nonnegative frame function on the unit sphere $ of 3f. Then f is regular. Proof. Since the dimension of 3tf is at least 3, it follows that the restriction of / to any completely real subspace is regular. Only the case dim «^ = oo remains. Lemma 4.21 is applicable to any finite dimensional subspace of 3tf\ Thus for any finite dimensional subspace 8 of Jf7 there is an operator Us which is an endomorphism of 89 bounded, self-adjoint, and > 0 such that f(x) = (TJsx,x)

(xe8,

|| a; || = 1).

The uniqueness of Us implies that if Sx S^82i( USi x,y} — < US2x,y} if x,y e #!. This implies in the usual way that there exists a symmetric semi-bilinear form on 3/F x 3tf\ say, B such that

B(z,x)=f(z)

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for all x with ||a|| = l. Since 0 < B(x,x) < 1, there exists a bounded, nonnegative, self-adjoint operator U such that <JJx& = B(x,x) = f(x). This proves the lemma. Theorem 4.23 (Gleason [1]). Let Wbe the convex set of all von Neumann operators of unit trace on a separable Hilbert space #F of dimension at least 3 (possibly infinite) over D. Then the map (55)

U-^pu

(UelT)

(where pv is defined by (30)) is a convex^ isomorphism of iV onto the convex set £f of all states of the logic 3?. A state p e £f is pure if and only if p=pu for a unit vector u of Jf, pu being defined as in (25). pu=pv if and only if for some c e D with \c\ = 1, v = cu. In particular, the pure states of 3? are in natural one-one correspondence with the rays of 3^. Proof. That U ->pv is an isomorphism of iV onto Sf is immediate from lemma 4.22. Next we shall obtain a description of the pure states of ££. For any vector uetff with \u\ — 1, let pu be the state of ££ denned by pu(M) = \\PMu\\2 (Meg). We assert that pu is a pure state. Let U0 be the projection on the onedimensional space D-w. Clearly pu(M) — tr(PMU0) so that Pu = Pu0To prove that pu is pure, let pl9 p2 be states of JSP such that (56)

pu = apl + (\-a)p2,

where 0 < a < l . Since U -> PV is a one-one convex map of W onto the set of all states, there are (unique) elements Ulf U2e^r such th.dX pj=pUj (j = l,2) and hence (57)

U0 =

aU1+(l-a)U2.

7

Since <£7ya;,a;>>0 for all x e Jf , j = 0, 1, 2, and since (Uox,xy = 0 if x is orthogonal to u> we conclude from (57) that for xj_u, (Ujx,x}=0 (j== 1, 2). Now Vj is > 0 and hence there is a bounded self-adjoint Yi such that Vj2=Uj and we have <^Ujx,xs)= ||F;a:||2. This implies that Fy# = 0 for all x orthogonal to u and, consequently, that UjX = 0 for all x orthogonal to u. Since Uj is self-adjoint, this implies that Uj = cjU0 for some constant t If A and B are convex subsets of two vector spaces over the real number field and if u is a map of A into B, we say that u is convex if u (ax + by) = au(x) -f bu(y) for x, y eA, 0 < a , 6 < 1 and a -f 6 = 1.

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c,(j = l , 2 ) , and c, = l as tr(C/i) = l . Therefore U1=U2=U0, which proves t h a t pu is a pure state. Conversely let UQ e iV be such t h a t ^C7o is a pure state. If al9 • • •, an, • • • ( > 0 , a x -fa 2 + • • • = ] . ) are the eigenvalues of U0, and ux, u2, • • •, un, • • • corresponding eigenvectors of norm 1, then (58)

3>c0 =

^anpUn. n

As pv is pure, (58) implies t h a t all but one of the an must be 1, the rest being 0. This shows t h a t pUo =pUj for some j . Remark. From theorem 4.23 we can obtain at once an important characterization of the principle of superposition of pure states. Let {ua} be a family of unit vectors of J f and let us write pa for pUa. Let p=pu be a pure state and let M be the smallest closed linear manifold containing all the ua. If p is a superposition of the collection {pa}, then p(M1) = 0 as pa(M1) = 0 for all a. Hence ue M. On the other hand, if u e M, then for any N e ££ having the property t h a t pa(]^) — ^ for all a, p(N) = 0, since a n y such N is contained in M1. In other words, pu is a superposition of the family {puJ if and only if u lies in the smallest closed linear manifold of J f containing all the ua. Thus, if we make the assumption t h a t the logic of the physical system © is standard, the pure states can be identified with the points of an infinite dimensional projective geometry with the principle of superposition translating into the usual concept of linear dependence of points. Our next object is.to discuss the nature of the statistics of one or more observables associated with the standard logic «£?. We begin with t h e following theorem: Theorem 4.24. Let x(E - > x(E)) be an observable associated with J?, U a von Neumann operator on 3f of trace 1 and pU9 the state M - > tr(P M U) of ££\ Let Ax be the self-adjoint operator on $F which corresponds to x. If f is any real Borel function bounded on the spectrum of x, f o x is a bounded observable and (59)

StfoX\Pu)=tT(f(Ax)U).

Ifx is nonnegative, so is A x and then $(x\ pv) exists if and only if B = AX12 U112 is an everywhere defined, bounded operator of Hilbert-Schmidt class; in this case (60)

*(z\Pu)

=

tr(B*B).

If x is arbitrary, then x has finite variance in the state pv if and only if AXU112 is an everywhere defined, bounded operator of Hilbert-Schmidt class. In this case, (60) reduces to (61)

*(x\pa)

=

tv{Vll2AxU112).

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In particular, if u is a unit vector in Jf and pu is the state M -> || P u ||2 of ££\ x has a variance in the state pu if and only if ue @(AX) ;*(* in this case, (62)

£(x\pu) = .

Proof. If S is the spectrum of x and K = (/[$]) ~, the closure of/[$], then (/ o x)(K) = 3f; and as K is compact, this proves that y=f o x is bounded. The spectral measure of the operator Ay is the measure E -> Qf- iiE) where Q is the spectral measure of Ax. Thus Ay=f(Ax) in the usual sense of the functional calculus of self-adjoint operator on 3tf. This proves (59) at once. We next take up the more delicate questions involving possibly unbounded x. Let x be any nonnegative observable i.e., a;([0,oo[)=Jf7, associated with ££\ Then Ax is a nonnegative, self-adjoint operator. We denote by Ax12 the unique, nonnegative, self-adjoint operator C such that C2 = AX. Let U be any von Neumann operator of unit trace on Jf such that S(x\pv) < oo. Let Q(E-+QE) be the spectral measure of Ax and k(E) = tr(QEU). Clearly i([0,oo[) = 1 and

*(*\Pu) = J" tdk(t). If cpetf is any unit vector, (QEUll2q>,Ull2(Py
= k(Eyi

(t^fdk^t) < oo, /Jo; where k^ is the measure E ~> . This implies that ifs lies in the domain of Ax12. In other words, AXI2U112 is everywhere defined. Since Ax12 is closed and £71/2 is bounded, it is easily seen that Al.l2U112 is closed. Consequently, B — A]}2XJ112 is a bounded operator. To prove that B is of Hilbert-Schmidt class it is enough to prove that 2n II-^Pn II2 < °° f° r some orthonormal basis of 3tf. Let {0, 2 n a n = l. Let kn(E) = (QEcPn,(pn). Then each &n is a probability measure and k = 2n a A- But

jn^nii^s^n^vir Jo

n

V' 2 ) 2 «M*)

= T an\ n

tdkn(t) JO

p oo

= < oo.

tdk(t)

\ &{L) is the domain of the linear transformation L.

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112

For the converse, if B = AX U is defined everywhere and is a bounded Hilbert-Schmidt operator, we have, using the above notation, tr(B*B) = ^ I I i W = 2 «n f" Mkn(t) < oo. n

n

Jo

But then, by Fatou's lemma,

i:

tdh(t) = g(x\pv)

< oo.

Jo

At the same time, the above argument proves that *(z\Pu) = tr(B*B). Next we consider the case of finite variance. We use the same notation. x has a finite variance in the state pv if and only if f "^ t2dk(t) < oo. Proceeding exactly as before, we conclude that if this condition is satisfied, D = AXU112 is an everywhere defined operator of Hilbert-Schmidt class. Note that, if a n > 0 , ne@(Ax). In this case, (Ax(pn,
a

n(Ax

2 n:an >0

n

J - oo

= f

tdk(t).

J — 00

This proves (61). For proving the converse, let AXU1I2 = D be everywhere defined and of Hilbert-Schmidt class. We want to prove that f°° t2dk(t)

< oo.

J — oo

Write, for any Borel set

E^R1,

K(E) =

{QEu^
Then tTiU^QsU1'2)

k(E) =

k

= 2

nW

n

and

if

' " t*dkn(t) = 2 IMx^'Vnl 2 n <

00.

I t follows once again by Fatou's lemma that P° t2dk(t) < oo.

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For the last part, let U be the projection on the one-dimensional space spanned by ,q>> as || Q(c; E) be the spectral measure of A(c). If cp G Jtf is a unit vector, and if we define ^ ( c ; E) by (63)

q*(c;E)=

\\Q(c; E)
<7p(c; •) is a probability measure on ^(R 1 ). We say that the observables {x±, • • •, xk} have joint statistical distributions if for each unit vector q> eJtf', there exists a probability measure q^ on &(Rk) such that for all vectors c = (c1? • • •, ck) of Rfc and all E e^(R 1 ), (64)

q9(E(c)) = q9(c; E),

where (65)

E(e) = {(«!, • • •, y : cA + • • • + c ^

G #}.

fc

The sets E(c) are the "half-spaces" in R . Since any D-valued measure on &(Rk) is uniquely determined by its values on the sets of the form E(c), it follows that q9, if it exists at all, is unique. Obviously, the above sense is a quite conservative one in which we may speak of the statistics of the observables {xly • • •, xk}\ the probability measure q0 will then be the joint distribution of {x±, • • •, xk} under the state of J? determined by 9. We shall now prove the following theorem (Varadarajan [1]). Theorem 4.25. With the notations described above, {xx, • • •, xk} have joint statistical distributions if and only if the operators Aj(l<j
(66)

&«r\m = *m 1

for allEe&iR ) andj = l, 2, • • •, k, TTJ being the projection (tl9 • • •, tk) -> tt of Rfc into R1. If we define for a unit vector yetf, and E e@(Rk), (67)

q9(E) = ||P'<*V||9,

then (63) and (64) are satisfied. Thus {xu • • •, xk} possess joint distributions.

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We now proceed to prove the converse. If {x±, • • •, xk} possess joint distributions, it is obvious that for i,j
(D = R)>

ft*.*- = Ifa+o'-q*-*') (68)

q9%9. = Ifa + v'-qv-v'+iq-iv

+ v'-iq-to-o')

(D = C),

Qw = i(2*+ *'-&>-*'+ 2 to-Jr
( D = Q)

We assert that for each M e^(R 2 ), the map q^^M) is bounded and *-bilinear on J f x«?f. To verify this, let 9^, 902, is a D-valued measure on <^(R2). However, if M = E(c), where EeSKJk1), then g(?)(if) = , and hence from (68) we infer that q9.AE(*)) = < « c ; E). Consequently, v(M) = 0 whenever M = E(c). Hence v = 0. Similar arguments complete the proof that 99,

q^^M) is *-bilinear. Since

< ||9||2,

0<^ w

it is clear that 99, q>' -> q^^M) is a bounded form. We can now use a well known theorem to construct a unique bounded self-adjoint operator PM such that fr^W

= ,9>'>.

Clearly, 0 q^^M) is a measure, P M is additive over disjoint sets M, i.e., if Mv M2, - • • are pairwise disjoint Borel sets of R2 and M={Jn Mn, 00

<^M9>9>'>

= 2 '>n= l

We note also, as qOt0(M)>O, that for M±^M2, PMI
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L

B vanishes on Y and hence B = 0 on (X n Y) -B, being self-adjoint, therefore leaves X C\ Y invariant and for x e X n Y, (Bx,x}<\\x\\2. X Y Thus B

From lemma 4.26 we m a y conclude t h a t PE1*E2

< G I A Q2.

Replacing E2 by R 1 —E 2 we obtain PEIXR'-E2

< Qi A ( / - & ) .

The last two relations yield, on addition, (69)

Q,
A Q2+e! A (/-&).

However, Qi AQ2 and Qi A (I — Q2) are orthogonal projections which are both
A

(I-Q2).

This last equation leads quite simply to the conclusion t h a t Q1 and Q2 commute (cf. lemma 3.7). This theorem, taken together with theorem 3.9 may be regarded as giving a complete description of the circumstances under which the statistics of several observables may be regarded as arising from observables with joint distributions. We end these remarks with a mention of the concept of the quasiprobability distributions, introduced by Wigner [2] and studied extensively by Moyal [1]. Let xl9 x2, • • •, xk be bounded observables with Al9 • • •, Ak as their corresponding operators. We use the same notation as in theorem 4.25. For fixed

Q(c; E) = q0(c; E) is a nonnegative measure on ^ ( R 1 ) . We define (70) 0(ci,
J R j f c exp{(-l) 1 / 2 (c 1 ^ + - - - + c A ) } ^ ( c ; - ) .

I t is easy to prove t h a t <$( •; 93) is a continuous function of c±, • • •, ck for each 99 e J f and |*(
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We may therefore introduce its Fourier transform, in the sense of distributions of Schwartz, say (-;
3. T H E S T A N D A R D LOGICS: S Y M M E T R I E S At the focal point of the axiomatic development of modern physics lies the concept of automorphism or symmetry. We have already pointed out t h a t the momentum observables of a physical system are intimately related to one-parameter groups of automorphisms of the configuration manifold of the system. Moreover, we had described in Chapter I I I how the dynamical group of a physical system can be described b y a one-parameter group of convex automorphisms of the set of states. I t is therefore extremely important to examine the concept of an automorphism when the logic in question is a standard one. The main purpose of this section is to obtain a complete description of the automorphisms associated with a standard logic. We shall introduce, for this purpose, the notion of a symmetry of a Hilbert space Jf. As always, J f is separable and the field of scalars D is one of R, C, or H. A mapping T(x-+Tx) of Jtf* into itself is said to be a symmetry (of 3tf) if (i) T is additive, one-one, and maps Jf7 onto itself, and (ii) there exists a continuous automorphism 0 of D such t h a t T is 0-linear and, for all x, y e Jf7, (71)



=

<x,yy.

If D = R, then 6 is the identity and the symmetries are none other t h a n t h e unitary operators of J f . If D = C, then 6 is either the identity or t h e conjugation; in the former case, T is unitary, whereas in the latter case T is anti-unitary. If D = H, then there exists a c e D such t h a t |c| = l and de = cdc ~1 for all d e D; in this case, T(dx) =

(cdc-^Tx

and (Tx,Ty}

=

c<x,yyc-\

Equation (71) implies t h a t

\(Tx,Ty)\

=

\(x,y}\

I n the general case of (71) we shall call T 6-unitary. The symmetries of 3f form a group; if T} is 0 ; -unitary( j = 1,2), TX T2 is dx 0 2 -unitary and T5"1 is 0 ; - 1 -unitary. The unitary (linear) operators of Jf7 form a normal subgroup

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%{£?) of the group of all symmetries. We shall write 8(Jf) for the group of symmetries. The map which associates with any 0-unitary T, the automorphism 6 of D, is a homomorphism of S(J4?) onto the group Aut(D) of all continuous automorphisms of D. Since the kernel of this map is ^ ( J f ) , we have: (72)

S(Jtr)l<%{Jf) ~ Aut(D).

Note finally that if T is a symmetry and c e D is such that \c\ = 1, cT is a symmetry. Theorem 4.27. Let J^f be a separable infinite dimensional Hilbert space over D. Let a(M -> Ma) be an automorphism of the logic ££\ Then, there exists a symmetry T of ^ such that (73)

Ma = T[M]

(M e &).

If T' is a symmetry such that (73) is satisfied for all M, then there exists a c e D with \c\ = 1 such that T' = cT. Finally if T is any symmetry of Jf7, M -> T[M] is an automorphism of the logic ££\ Proof. It is quite trivial to check that if T is a symmetry of ^f7, M -> T[M] is an automorphism of the logic j£f. Suppose now that a(M -> Ma) is an automorphism of the logic J£. Let F be a linear manifold, chosen once for all, such that dim F = 3. By theorem 2.1 there exists an automorphism 6 of D and a 0-linear isomorphism L0, ofF onto Fa, such that for any linear manifold M^F, Ma = L0[M]. Let us define & to be the collection of all finite dimensional linear manifolds G which contain F. We claim that for each G e & there exists a unique 0-linear isomorphism LG of G onto Ga such that (74)

LGu = L0u

(u e F)

and (75)

Ma = LG[M]

(M c G).

In fact, by theorem 2.1 there exists an automorphism 0' of D and a fl'-linear isomorphism L' of G onto Ga such that Ma = L'[M] for all linear manifolds M<^G. Since Ma = L'[M] = L0[M] for all linear manifolds M^F, we conclude from the same theorem that there exists a nonzero C G D such that (i) L,u = cL0u for all ueF, and (ii) ae'=ca9c~1 for all aeD. If we write LG = c~1L/, then LG is a 0-linear isomorphism of G onto Ga satisfying (74) and (75). The uniqueness of LG follows from (74) and (75).

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Since LG is uniquely determined by (74) and (75), it follows quite easily t h a t there exists a 0-linear m a p L of J$? into itself such t h a t (76)

Lu = LGu

(ueG,

GeF).

We claim t h a t L is one-one and maps 2tf onto itself. To prove t h a t L is one-one, let x e «*f and I># = 0. Evidently, x e G for some G e ^ and Z/# = LGx = 0. Therefore # = 0 as LG is an isomorphism. If y e 3f there is a 6r e IF such t h a t y e Ga and once again, as XG maps G onto 6ra, there will exist &n x e G such t h a t Lx — y. Since any M e £f with finite dimension is contained in some element of F, we see t h a t L is a 0-linear isomorphism of J f onto itself such t h a t (77)

Ma = £ [ i f ]

(dim I f < oo).

Let now G e JF. Since a preserves orthogonal complementation, it follows t h a t for x, y e G, (x,y} = 0 if and only if (Lx,Ly} = 0. Thus, the semi-bilinear forms x, y -> (x, y) and x, y -> (Lx,Ly}w~ 1} of GxG induce the same polarity of the geometry of subspaces of G, and hence there exists, by virtue of theorem 2.3, a unique dG^Q in D such t h a t (Lx,Ly>

= (x,y}ddG

(x, y e G).

Since G e

= 'd

(*, ^ G J f )

for a nonzero d e D . We now show t h a t 6 is continuous. This is obvious for D = R or H, since 6 is the identity in the former case and is an inner automorphism in the latter. Consider now the case D = C. If 9 is not continuous, there exists a sequence {cn} in D such t h a t cn —>• 0, but \cne\ - > oo. Choose an orthonormal infinite sequence {en : n — 1, 2, • • •} in 3tf and let xn = nen. By replacing cn by a subsequence if necessary, we may assume t h a t (79)

||Z*n||/|cn'|-*0

(n-*oo).

Write

Since c n * - > 0, z is well defined in J f . Clearly, z ^ O . Then, <#n,z> = c n and hence, for all n, un = (z,z}~1z — cn~1xn is orthogonal to z. Consequently, from (78) we obtain, for all n, (Lun,Lz>

= 0.

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This implies, on simplification, that (80)

Kz,z>9) - \Lz,Lz>

= (cne) -

\Lxn,Lz>

for all n. But

KO-'iKM,,!,*)! < ic/i-^inizzji, which tends to zero by virtue of our assumption (79). Hence (80) implies that (Lz,Lz} = 0 or 2 = 0. This contradiction shows that 6 is continuous. We are now ready to complete the proof of theorem 4.27. Since 6 is continuous, it follows that for any real number r, re = r. Hence we see, by putting x — y in (78), that d is a positive real number. Write T — d~ll2L. Then T is a 0-linear isomorphism of Jti? onto itself such that (81)

=

(x,yy

and Ma = T[M]

{M e &, dim M < oo).

Since ||T#|| = ||#||, T is an isometry and is hence continuous. It follows therefore that M* = T[M] (M E &). The remaining statements are easy consequences of theorem 2.3. Therefore, we omit their easy verifications. The proof of theorem 4.27 is complete. We shall say that T induces a if (71) is satisfied. Corollary 4.28. / / D = R or H, then any automorphism a of J£ is induced by a unitary operator of Jf. 7/D = C, then a2 is induced by a unitary operator. Proof. For D = R there is nothing to prove. If D = H and T± is a 0-unitary operator inducing a where 6 is the automorphism d-^cdc-Wc\

= 1),

T = c~1T1 is a unitary operator inducing a. If D = C and a is induced by a *-unitary Tlt Tx2 is unitary and induces a2. In the physical literature it is customary to prove the above theorem in a somewhat different formulation (cf. Wigner [3]). To motivate Wigner's formulation we must introduce the notion of transition probabilities. Let r, r' be rays in 3? and |2 evidently depends only on r and r' and not on cp and cp'. We denote it by [r,r']. It is called the transition probability between the states p and p' which correspond (cf. theorem 4.23) to the rays r and r'. If a physical system © is in the state p, [r,rr] is the probability that it will be actually found in the state p' when an experiment is performed to

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decide whether it is in the state p' or not. We shall define a physical symmetry associated with the logic J? to be any one-one m a p a(r -> r a ) of the set of all rays of Jf onto itself with the property t h a t [r a ,r' a ] = [r,r']

(82)

for all rays r, r'. The following theorem describes the natural connection between physical symmetries and symmetries of Jt?. Theorem 4.29. (Wigner [3]). Let $P be a separable infinite dimensional Hilbert space over D and T a symmetry of Jf\ Then r -> T[r] is a physical symmetry associated with ££'. Conversely, let a(r -> r a ) be any one-one map of the set 0k of all the rays of 3F onto itself with the property that r_[_r' if and only if r a JLr' a . Then, a is a physical symmetry associated with <&, and there exists a symmetry T of J f such that r a = T[r] for all r e@£. T is determined by a up to multiplication by a number c e D with \c\ = 1. Proof. If T is a symmetry of JtT, \(Tx,Ty}\ = \(x,y}\ for all x,yeJtf. From this we conclude easily t h a t r - > T[r] is a physical symmetry associated with ££. We now examine the converse. Let a(r —>• r a ) be any one-one map of @t onto itself which preserves orthogonality. For any subset A of 0t let (83)

Aa = {r01 :re

(84)

[A] = {x : x e Jf, x e r

A}, for some r in A},

and (85)

i ^ ^ r l r '

for all

r' e A}.

We shall first prove t h a t if i c j is such t h a t M = [A] is a closed linear manifold of ^f, i.e., if A is the set of all rays of a closed linear manifold M of Jf\ then [Aa] is also a closed linear manifold. To prove this, we note t h a t as M = [A] is a closed linear manifold, a ray r belongs to A if and only if r e (A1)1. But as a preserves orthogonality, this means t h a t a ray r' e Aa if and only if r'e (A1*)1, i.e., Acc = (Ala)1. But for any subset S^Zft, the set [S1] is a closed linear manifold. Consequently [Aa] is a closed linear manifold. We now set up a correspondence, also denoted by a, between elements of <£. For any M e 5£, let A be the set of all rays r such t h a t r c M; we set: (86)

Ma = [^«].

The argument of the preceding paragraph has shown t h a t Ma e j£?. By very definition it is clear t h a t a(M - > Ma) is an order-preserving m a p of ££. Since r c i f 1 if and only if r is orthogonal to all rays contained in M, it follows t h a t (Jf-L)a = ( i f a ) 1 . Thus a preserves orthogonality. These

LOGICS

ASSOCIATED

WITH

HILBERT

SPACES

1

109 ia x)

results applied to the m a p a " of 0t show t h a t the m a p M->M ~ defined b y (86) is the inverse of the map M - > Ma of J£. Consequently, we m a y conclude t h a t M - > Ma is a lattice automorphism of <£ which preserves orthogonality. B y theorem 4.27 there exists a symmetry T of J f such t h a t M<* = T[M] for all M e 3?. B u t then r a = T[r] for all rays r showing t h a t r -> r a is a physical symmetry associated with ££. Since any automorphism of ££ is determined by its restriction to the set of all one-dimensional subspaces of #F, it is clear t h a t T is determined by the m a p a of 3% u p to multiplication by a number c e D with \c\ = 1. This completes the proof of the theorem. I n Chapter I I I we introduced the concept of a convex automorphism of the set of all states. We shall describe all such automorphisms of the set of states of a standard logic. Let T be a symmetry of the Hilbert space Jf, and iV the convex set of all bounded, nonnegative, self-adjoint operators U of trace class with tr(Z7) = l. For any U e HT let aT(U) be defined by (87)

aT(U) =

TUT-1.

Lemma 4.30. Let U eiT and let T be a symmetry of J#*. Then aT( U) eW* and U -> aT(U) is a convex automorphism of W'. Moreover, T —> aT is a homomorphism of the group of symmetries of J4? into the group of convex automorphisms of # " . The proof is a routine verification. We leave it to the reader. If T is a symmetry of 3f and if we set, for U e W, (88)

<*T(pu) = p t t T ( U ) ,

where pu is the state M -> tr(PMU) of J&, then the map pv - > aT(Pu) is a convex automorphism of £f. We shall prove t h a t any convex automorphism of £f is of this form. Let iT + denote the set of all von Neumann r operators on 3tf. Clearly, iT^iT to + . If Tlt T2 e if + , we write T10. Lemma 4.31. Let a be a convex automorphism of iV. Then there exists a unique one-one map a ~ of # " + onto itself such that (i) (ii)

a(U) = a~(U) a~(c1A1+c2A2)

=

(UeiT), c1a~{A1)-{-c2a^{A2) (Al9 A2 e i^ + , cl9 c2 real numbers > 0 ) ;

a ~ has then the property: [

}

(hi) (iv)

tr(a~(,4)) = tv(A) (AeiT + ), TX
GEOMETRY

110

OF QUANTUM

Proof. Define a~(0) = 0. If T eiT t~1T eiT\ we then put (90)

a~(T) =

THEORY

and T^O,

+

t = tr(T)>0,

and

ta^T).

It is trivial to check that a~ has the properties described in the lemma. The uniqueness of a" follows from the fact that the property (ii) of a~ implies (90). Equation (90) implies that tr(a~(T))=tr(T), proving (iii). Finally, let Tx < T2. Then a~(T2)

= a-iTJ+a-iTt-TJ

> a~(T x )

as a~{T2-T1)eiT+ and is hence >0. Conversely, let a^(Tx) TX. Lemma 4.32. Let
(i)

^J-^2* (ii) For any Aeitr trA>2.

+

, the relations P1
imply that

Proof. Suppose that cp1±_cp2 and suppose that i e # + satisfies the relations P10, 00

tr(^) = ^ n= 1 > (Acp^cp^

+(Acp2,cp2)

^ <^l9l>9>l>+<^29>2>9>2>-

Consequently, tr(A) > 2. This shows that (i) implies (ii). Conversely, let (ii) be satisfied. Let us assume that cp1 is not orthogonal to
0 2 = lala-^&l"^-^'.

LOGICS ASSOCIATED

WITH HILBERT

SPACES

111

S an

Then {
P

1

(l °\ \0

p i

\<

2

OA

HI6I\ |6| 2 /

\|o||6|

Since |a| 2 + |6| 2 = l, the eigenvalues of the matrix of P2 — P1 are ±\b\, and hence there exists a unitary operator V such t h a t the operator V(P2 — P1)V~l has t h e matrix given by

/\b\ ° \ -\b\J

\0

1

Let M be the unique operator such t h a t VMV'

has the matrix given by

h

l\ \ °\ \0

0/

Obviously M is self-adjiont, > 0 and tr(Jf) = |6| < 1. Further, 7if7-i_7(P2_Pl)F-i > o and hence M>P2-P1. Let A = P±+M. tr(^4) = 1 + \b\ <2. This proves the lemma.

Then P1
P2
but

Theorem 4.33. Let Jf be a separable infinite dimensional Hilbert space over D and T a symmetry of ffl. For any U eW let pv be the state M -> tr(PMU) of & and UT=TUT~1. If (92)

<*T(PU) =

PUT,

s a

then aT{pu —>PuT) ^ convex automorphism of Sf. Conversely, let a be a convex automorphism of £f. Then there exists a symmetry T of Jt? such that (93)

a = aT.

T is determined up to multiplication

by a number c e D with \c\ = 1.

Proof. The first p a r t of the theorem is rssentially the content of lemma 4.30. To prove the converse, let a be a convex automorphism of £f. By theorem 4.23, a induces a convex automorphism of # " , denoted once again by a. Let a~ be the m a p of # " + , determined by a (cf. lemma 4.31). Since the extreme points of iT are the orthogonal projections on the onedimensional subspaces of $F and since a is convex, a induces a one-one correspondence of the set ^ , of all rays of Jf7, onto itself. Let a(r - > ra) be this correspondence. Since a~ preserves the trace and the partial ordering < of # " + (by (iv) of lemma 4.31), and since the orthogonality of two rays has been completely characterized in terms of the trace

112

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function and the partial ordering < in lemma 4.32, it follows a t once t h a t r - > r a is orthogonality-preserving. By theorem 4.29 there exists a symm e t r y T such t h a t r a = T[r] for all rays r. If aT is the convex automorphism of £f associated with Ty this means t h a t aT and a coincide on the set of extreme points of / #^. From this it follows t h a t a — aT. The fact t h a t T is determined up to multiplication by an element C G D with |c| = l now follows from theorem 4.29. The theorem is completely proved.

4. LOGICS ASSOCIATED W I T H VON N E U M A N N ALGEBRAS If the logic J^7 of a quantum mechanical system is not isomorphic to a standard one, then the question of describing its observables, states, and symmetries becomes a more difficult one. There are no general results in this connection. I t is our aim to give a few examples which illustrate the wide possibilities as well as the difficulties involved in the construction of a n y general theory. A large class of logics which are sublogics of standard logics m a y be constructed using the theory of von Neumann algebras. We confine ourselves to the case of complex scalars. Let D = C be the complex number field and 3^ a separable infinite dimensional Hilbert space over C. We write ty(J^) for the algebra of all operators on Jf. For u, v e Jtf, let XUtV : A-> (Au,v}

(Aety(Jf)).

Then the Xuv are linear functionals on ty(34f). The weak topology on ty{3^) is the smallest one with respect to which all the AMfU are continuous. A von Neumann algebra ty is a subalgebra (containing / ) of ty(3#*) such t h a t (i) if A e ty, then A* e ty, and (ii) ty is a closed subset of ty(J4f) in the weak topology. lity' is the set of all elements of ty(Jf) commuting with all t h e elements of ty, ty' is also a von Neumann algebra and (ty')' = ty (cf. Dixmier [1] for the general theory of von Neumann algebras). Let ty be a von Neumann algebra. We define (94)

Se% = {M : M e Jg%?f), PM e ty}.

I t is then easy to verify t h a t J ^ is a logic. We shall call it the logic associated with ty. A closed linear manifold M lies in S£^ if and only if every element of ty' leaves M invariant. Thus the logic ££% may also be introduced as the logic of closed invariant linear manifolds of some von Neumann algebra. The one-one correspondence x - > Ax between observables of ££% and self-adjoint operators on 34?, which was discussed in section 2 of this chapter, clearly persists even in this case. However, not every selfadjoint operator is the operator of an observable associated with «#V

LOGICS ASSOCIATED

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113

Let A be a self-adjoint operator, not necessarily bounded. Then A=AX for an observable x of jSf9 if and only if the spectral projections of A lie in 51. If A is bounded, t h e condition will then be t h a t A e 21. If A is unbounded, t h e condition will have to be stated more carefully. I n this case, A — Ax for an ^ - o b s e r v a b l e if and only if UAU~1 = A for all unitary operators U e 21' (cf. Dixmier [1]; such ^ ' s are said t o be affiliated to 2t). 3 for all n, we can describe all the pure states of J*?^. I n fact, from theorem 3.19 a n d Gleason's theorem we conclude immediately t h a t p is pure if and only if there exists an n and a unit vector cp in Mn such t h a t (95)

p(M) = \\PM
(Jfe-SP.).

I n other words, the pure states of S£% are still in one-one correspondence with t h e rays of Jf7; however, not every ray can be used. Only the rays which belong to some Mn correspond to pure states. Physicists describe this situation by saying t h a t no ray of ^f7, which is a nontrivial superposition of rays in distinct Mn, can describe a (pure) state of J ^ . Each Mn is

114

GEOMETRY

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THEORY

called a coherent subspace of (pure) states. The conditions which demand t h a t the rays describing the pure states should lie in Mn for some n are usually referred to as super selection rules, (cf. Wick, Wightman, and Wigner [1]).

5. ISOMORPHISM AND I M B E D D I N G T H E O R E M S I t is very important in the foundations of quantum mechanics to examine the extent of generality of the concept of an abstract logic. I n particular, the question arises whether every logic can be regarded as a sublogic of a standard logic. Definitive answers to such questions are not known. We shall prove in this section some theorems which analyze t h e structure of a somewhat restricted class of logics. The main theorem of this section was proved by Piron [1]. A related but somewhat weaker result was obtained by Zierler [1]. The main tool of our analysis is the coordinatization theorem of Chapter I I . I n order to be able to apply t h a t theorem we must impose a sharp restriction on the class of logics to be analyzed. We introduce accordingly t h e following definition. A logic ££ is said to be projective if the following conditions are satisfied:

(96)

(i) given a # 0 in iff, there is a point x
For any a e = ^ , let (97)

$ ( a ) = {x : x a point, x < a}.

Note t h a t if S£ is projective and its lattice is complete, then every element a of j£? is the lattice sum of the points it contains. I n fact, if b is the sum of these points and b^= a, c = 6 i A a ^ 0 and there would exist a point x < c by (i) of (96) which contradicts the definition of b. Lemma 4.34. Let ££ be a projective logic whose underlying lattice is complete. If a and bj (j eJ) are elements of ££ and if a\bjfor allj, a_J_V; ^'* We have the usual identities:

LOGICS ASSOCIATED If a,be^,

then a
WITH HILBERT

SPACES

115

if and only if $(a)c=*g(&); a\_b if and only if

Proof. The first two assertions are proved exactly like the corresponding assertions in lemma 3.1. We therefore come to the proof of the last assertion. If a be a 0-bilinear symmetric form on V x V. Let us assume t h a t <. , .> is definite; i.e., (x,x} = 0 if and only if x = 0. For any set M of vectors we write (98)

Mx = {u : (x,u}

= 0

for all

x e M}.

Obviously, M1 is a linear manifold in F, M n M1 = 0, and M^M11. If M is a linear manifold and M = M11, we shall say t h a t M is <. , . >-closed. Note that no topology is involved in this definition. The pair (V, <. , .>) is said to be Hilbertian if for any closed linear manifold M, one has: (99)

V =

M+ML

algebraically. For any set i f c f w e write M~ = i f 1 1

(100)

If A, B^ V we write A^_B whenever = 0 for all a e A and b e B. Clearly 0 " = 0 , V~ = F, and 0L= V, V1 = 0. If M and JV are linear manifolds and M^N, then NL<^ML. If {if ; } is a collection of linear manifolds, ilf their algebraic sum, and Nj_Mj for all j , then i^ J_if. All this is trivial. Lemma 4.35. The following statements on a set M^V (i) M is a <. , .^-closed linear manifold. (ii) i f = i f " . (iii) M = N1 for some set N c F. Moreover, for any set (101) (102)

M^V, if111

=

if1,

( i f - ) 1 = if 1 ,

and (103)

(if-)" =

if".

are equivalent.

116

GEOMETRY

OF QUANTUM

THEORY

Finally, if {M^ is a collection of <\ , . >-closed linear manifolds, also a <. , .>-closed linear manifold.

P | ; Mj is

Proof. Clearly for any set i f c V, M^MLL. Thus ML<^(ML)LL. On the LL 1L L L other hand, as M^M , (M ) <^M . Therefore M1 = M111, which is 11 (101). Since M~ = if , (102) follows from (101). Further, (M-)~

= ( i f 1 1 ) 1 1 = (M111)1

=

M11

b y (101), so t h a t ( i f - ) - = i f " . This proves (103). We now come to the proof of the equivalence of (i) through (iii). Suppose M is a <. , . >-closed linear manifold. Then M = M1L, so t h a t M = M~. Thus (i) => (ii). If M = M~, then M = i f 1 1 = ( i f 1 ) 1 , proving t h a t (ii) => (iii). Suppose now t h a t M = NL. Then, by (98), M11 = N111 = N1 = M, proving (iii) => (i). Finally, let M = f|y M,. As M^Mj9 M11^Mj11 = Mj. Thus

M11

c Q if. = M.

This proves t h a t M is a <. , . >-closed linear manifold. Corollary 4.36. If M^V, M~ is the smallest <. , . >-closed linear containing M. In particular, if M^N, then M~^N~.

manifold

Proof. M" is <. , . >-closed b y (103). I t is therefore enough to prove t h a t if iV is a <. , .>-closed linear manifold and M^N, then M~^N. But M-=MxlcN11 = N. Finally if J f c t f , M^N~, d as iV" is <. , .>an closed, M~ <^N~. Lemma 4.37. If M is a finite dimensional subspace of V, then M is <. , . >closed. Proof. I t is enough to prove t h a t M1L^M. Let x e if 1 1 . Let N be the LL subspace spanned by M and x. N^M . Since < . , . > - is definite, its restriction to NxN is definite and hence nonsingular. B u t N is finite dimensional and hence it follows from (13) of Chapter I I and the definiteness of <\ , .> t h a t N is the direct sum of M and (M1 n N). Therefore, x = x' +x" where x' e M and x" e M1 n N. Since N^M11, x" e i f 1 1 n ML and hence x"= 0. Thus x e M. Lemma 4.38. If {if ; } is a family of linear manifolds of V, and if 2 / Mj denotes the {algebraic) linear span of the Mj, (104)

g

if,)-

= (c\Mj-y.

Proof. Let M = %jMj. Since Mj^M, ML^MjL so t h a t i f 1 ^ . if/. 11 1 1 This shows t h a t M~ =M ^(f)j Mj ) = N, say. On the other hand, as H ; Mj1^Mj1, Mj^Mj- <^N for all j so t h a t M^N. Since N is <. , .>closed, it follows from corollary 4.36 t h a t M~ s^N. Therefore N = M~.

LOGICS ASSOCIATED

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117

Lemma 4.39. If (V, (. , . >) is Hilbertian and Mx, M2, • • •, Ms is a finite set of mutually orthogonal <. , .)-closed linear manifolds, then 2? = i -^j ^s also a <. , . >-closed linear manifold. Proof. I t is enough to consider the case 5 = 2 since the general case follows at once by induction. We want to prove t h a t (M1+M2)'

=

M1+M2.

Write M = M1+M2. Let x e V. Since (V, <. , . » is Hilbertian, we have the unique decompositions x = x±+y y = x2+z Now, as J f 2 c MXL, x2 e M^

(x^M^yeM^), (x2 e M2, z e M2L). so t h a t z e MXL also. Therefore, z e M^

n

M2L,

and we have: x = x1 +x2 +z

(x± e M±, x2

G

M2, z e MXL n M21).

Suppose now t h a t x e (M1 + M2)~. Then the above equation shows t h a t ZE (M1+M2)~ n Mi1 n M2L, which implies t h a t z = 0 on using (104). Therefore (M1 + M2)~=M1 +M2. Theorem 4.40. Let D be a division ring, V a vector space over D with 4: ) be the set of <. , . >-closed linear manifolds of V, partially ordered under inclusion. If (V, < . , . > ) is Hilbertian, then &{V, <. , . » is a complete projective logic, and for any collection {Mj} of <(. , .)-closed linear manifolds of V, the lattice operations in ££{V, <. , .>) are given by

y*, = £*,)-, (105)

A*, = nM,j

o

Conversely, let 3? be any complete projective logic. Then there exists a division ring D, an involutive anti-automorphism 6 of D, a vector space V over D, and a definite symmetric 6-bilinear form <\ , . > on V x V such that (V, <. , . ) ) is Hilbertian and ££ is isomorphic to <^{V, <\ , . ) ) . Proof. Let D, 6, V, <. , .> be as given in the first half of the theorem. Let j£? = j£?(F, <\ , . >) be the partially ordered set of <. , .)-closed linear manifolds of V. Assume t h a t (V, <. , . >) is Hilbertian. For any linear manifold M, M ~ is the smallest element of ££ containing it. I t is clear from lemmas 4.35 through 4.38 t h a t ££ is a complete lattice with the lattice

118

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operations given by (105). Next we observe t h a t if we define, for a n y i f e i f , M1 by (97), then j£? becomes a logic. To check this, the only thing not immediately obvious is the weak modularity (2) (ii) of Chapter I I I . Suppose then MvM2e^ and M±^M2. Let N=M1±nM2. By lemma 4.35, N is a ( . , .)-closed linear manifold so t h a t NeJ?. Clearly N^M2, N±MX. Since (V, <. , . » is Hilbertian, V = M1+M11 and for a n y x e V we have the decomposition x — x1 +x2, where x± e Mlf x2 e MXL. If now x e M2, then x1 and x2 e M2 also, so t h a t x2 e N. Therefore M2 = MX +N and a fortiori M2 = M1 v N. <£ is thus a complete logic. We claim t h a t it is projective. By lemma 4.37, i f contains all finite dimensional subspaces. As dim F > 4 , (i), (ii), and (iv) of (96) follow at once from this fact; (iii) follows from the algebraic decomposition V—M+M1 for any M e J*?. ££ is thus a complete projective logic. We now come to the converse. Let i f be a complete projective logic. Let us now define ££' by (106)

^ ' = { a : a e ^ ,

a finite}.

B y (ii) of (96), a e 3?' if and only if the underlying lattice of if[0,a] is a geometry. Under the induced partial ordering, JSP' is thus a generalized geometry. Therefore there exists by theorem 2.16, a division ring D, a vector space V of dimension at least 4 over D, and an isomorphism of ££' onto the generalized geometry of all finite dimensional subspaces of V. Let y denote any such isomorphism. Fix a nonzero vector v0 e V. Let W be any finite dimensional subspace of V containing v0 and of dimension at least 3. The isomorphism y transfers the orthocomplementation in o£P[0,a] into an orthocomplementation on the projective geometry of subspaces of W. Hence, by theorem 2.7 there exists an involutive anti-automorphism 9W of D and a symmetric, definite # w -bilinear form <. , . > w on W x W, inducing the orthocomplementation in question, with (v0,voyw = l. We now argue as in lemma 4.2 to conclude t h a t 6w=0 is independent of W and t h a t there exists a symmetric, definite 0-bilinear form ( . , . > on V x V such t h a t (i) . Let «£?~ =j£?(F, < - , . > ) be the lattice of all <. , .>-closed linear manifolds of V. We shall now extend y to an isomorphism y~ of i f onto j£?~. F o r any ae if, let (107)

y~(a) = {v : v e V

and

v e y(x)

for some point x < a}.

We shall first show t h a t y~(a) is a linear manifold in V. This is obvious for a = 0 or a = 1, as then y ~ ( 0 ) = 0 and y ~ ( l ) = V. Let then a ^ O , ^ 1 . If v i> v2 e y~( a )> there are points x1,x2
LOGICS ASSOCIATED

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119

dimensional subspaces of V, y{xx) and y(x2) are rays in y(x1 V x2). Thus if v = c1v1 + c2v2 where c1? c 2 e D, t; lies in y(x1vx2) and hence v e y(x) for some point a; < x± V # 2 < a' y ~ ( a ) is thus a linear manifold. We shall prove next t h a t y ~ (a) is <. , . >-closed. From lemma 4.34 it follows easily t h a t for a point x e J?, a; < a if and only if x_\_y for all 1/ 6 $ ( a x ) . Therefore, (108)

y ~(o)

= (y^a1))1,

which shows t h a t y~(a) is <(. , .>-closed. Moreover, as (108) is valid for all a (in particular, for a1), we have: (109)

y~(a±) = (y~(a))^.

(109) shows t h a t y~ preserves orthocomplementation. From (107) we conclude, on using lemma 4.34, t h a t for a, b e J^, y~(a)£y~(&) if and only if a), it remains only to prove t h a t y~ is surjective. Let ilf be any <(. , .>closed linear manifold of V and let the subsets A and B of j£? be denned as follows: ^4 = {x : # e «£?, x is a point such t h a t y(#) c= Jtf}5 .8 = {x : a; e ££, x is a point such t h a t y(x) c: jlf 1 }. Since M = M11, x e A if and only if #_[_i?. Let a and fc be the lattice sums of the points of A and B, respectively; these exist as «£? is complete. I t follows from lemma 4.34 rather easily t h a t a and b are orthogonal complements of each other and A =^S(a), B = ^(b). The equation A = ^(a) implies t h a t y~(a) = M. The proof t h a t y~ is surjective is complete. y~ is thus a lattice isomorphism of J ? onto «Sf(F,<. , . » . We observe finally t h a t (V, <. , .> is Hilbertian. I n fact, let M, # 0 , ^ F , be any <. , .)-closed linear manifold, and v e V, v^0. Let a e J£? be such t h a t y~(a) = Jf. Clearly, a ^ 0 , # 1 . Then y ~ ( a 1 ) = i f 1 . Choose x e & such t h a t v e y(#). By (hi) of (96), there are points y, z of ££ such t h a t y < a , 2 < a 1 and x
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assert that, given any complete projective logic £? associated with D which is one of R, C, or H, there exists a pre-Hilbert space V, with inner product <. , .>, such t h a t (i) (V, < . , . > ) is Hilbertian and (ii) 3? is isomorphic to the complete projective logic of all <. , . >-closed linear manifolds of V. I t is natural to expect t h a t V is then actually a Hilbert space. This essentially leads to Piron's theorem. We shall obtain it as a consequence of a few lemmas. I t is an interesting problem to examine what natural assumptions on a geometry imply t h a t the associated division ring is isomorphic to one of R, C, H. This is a classical question and is intimately connected with t h e topologies on a geometry (cf. Kolmogorov [1], Weiss and Zierler [1]). Lemma 4.41. Let V be a pre-Hilbert space over D (one of R, C, or H) and let Jf be its completion. If x0, x0' e Jti? are orthogonal, there exists a pair of sequences xn and xn' such that /110x

(i) xn _L xm' fa, m = 0, 1, 2 , . . .), (ii) #n> *^m ^ ^ and xn —> x0, xm —> x0 •

Proof. We shall prove first t h a t if z e Jf? is a nonzero vector, t h e n {z}1 n V is dense in {z}1. I n fact, let y_[_z, and let {yn}, {zn} be sequences of elements of V such t h a t yn->y, zn-> z. Write (HI)

Vn =

Then yn' e V and (yn',z}

Vn-iyn^X^n^}'1^

= 0 for all n. As yn - > y and zn -> z,

<2 n J 2>" 1 2n-^<^>" 1 » = 0,

showing t h a t yn' -> y. This proves our assertion. Suppose next t h a t y> Vii ' ' ' > Vsare mutually orthogonal nonzero vectors. Applying repeatedly t h e result proved just now, we conclude in succession t h a t { y j 1 n V is dense in {2/J 1 , {ys^s-i}1 n F is dense in {f/s^s-i} 1 and so on. I n other words, given any finite dimensional subspace Y of £? and a vector y\_ Y, there exists a sequence {zn} in V such t h a t zn_]_ Y for all n and zn - > y. This said, we come to the proof of the lemma. Let x0, x0' e J f be given with x0_\_x0'. We shall show by induction t h a t there are sequences {xn} and {xn'} in V such t h a t (U2)

K-*o|| ^2"n' \\*n'-*o\\ <2"», ^n J_ xm' (n,m = 0, 1,2, • • • ) .

Suppose ajj, z2> • • •, xk and x±', x2', • • •, xk' have been constructed so t h a t (112) is satisfied for all n, ra = 0, 1, 2, • • •, k. By the result proved in t h e previous paragraph, there is a vector x'k + 1e V such t h a t 4 + i _L*n

(n = 0, 1, •••,&),

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Applying the same argument again, we can find xk + 1e V such that (n = °> 1, • • • , * + !.),

a* + i ±xn'

Thus {xn}0<.n£k +1 and {xn'}0^n^k + 1 satisfy (112) for n and m = l. Let z0 = y — x0. Then y = x0+z0 and x0±_z0. By lemma 4.41 there are sequences {xn} and {zn} in V such that (113)

xn-+x0,

zn-+z0,

xn±zm

(n, m = 0, 1,2, • • •)•

Let M c 7 be the lattice sum in & (F, <. , . » ofthe rays D-a n (n = 1,2, • • •)• Then, as (F, <.,.>) is Hilbertian, V =

M+M1.

Now zm_]_#n for all n so that zm e ML for all m. Hence #0 e Mcl and z0 e (M1 ) c l where cl denotes closure in Jf. Moreover, it is obvious that (ML)cl^{Mcl)'. Therefore, (114)

y = x0 +z0,

xQ e Mc\

z0 e (Mcl)'.

On the other hand, as V = M + MX9 we can also write (115)

y = x0' +z0',

x0f e M,

z0' e M1.

As M-t-^iM01)', a comparison of (114) and (115) shows that x0' = x0 and z0' = 20. In particular, x0 e V. This completes the proof of the lemma. Lemma 4.43. Let ££ be a projective logic with the property that any family of mutually orthogonal points of ££ is at most denumerable. Then ££ is complete. Proof, It is enough to prove that arbitrary lattice unions exist in 3?. Let {aj- : j e F} be a family of elements of ££\ For any countable subset

D^F, let c(D) = V «y jeD

f The argument for completeness is essentially that of I. Amemiya and H . Araki, Publ. Res. Inst. Math. Sci. Kyoto Univ. A2 (1966), p . 423. Their theorem is t h a t if V is pre-Hilbertian and &(V, < . , . » is orthomodular, then V is complete.

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Let B be a family of mutually orthogonal points x such that (i) for each x e B there is a countable set D^F such that x

and for each n let Dn c: F be a countable set such that *n < c(Dn)

for all 7i. Write D = [Jn Dn. If we define x by a = V xn> n

then x
NOTES ON CHAPTER IV 1. Hidden variables: Is the quantum mechanical description of microphysics complete'1. The question of "hidden variables" has persisted in the foundations of Quantum Theory from its very birth. The basic assumption in Quantum Mechanics, namely that in any state of an atomic system a physical quantity can in general have no sharply defined value (even under idealized measuring conditions) but only a probability distribution of the possible values, appeared to many people to be in very sharp conflict with all classical experience, and hence unacceptable, unless it was interpreted as follows: one is not able to get a complete description of the physical state of the system; there are additional coordinates ("hidden variables") that cannot be measured; and therefore the statistical character of the results of the experiments is due to the averaging over these hidden variables. This view, which regards the Hilbert space description of quantum states as

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fundamentally incomplete, is reinforced by the example of statistical mechanics where the situation is essentially of this character. Von Neumann was among the earliest to analyze this question, and his conclusion was that no mechanism of hidden variables could account for the statistical content of Quantum Mechanics as currently formulated. That he regarded this question, as well as his resolution of it, to be of great importance is clear from the fact that he raises the issue already in the preface of his book (pp. ix, x), returns to it briefly (pp. 209-210) before devoting a substantial effort (pp. 295-328) to a detailed mathematical and physical treatment (all references are to the 1955 English translation by Robert T. Beyer of von Neumann's book). We shall briefly summarize his argument but suggest strongly that the reader study von Neumann's brilliantly presented analysis. Von Neumann's reasoning, roughly speaking, has two parts to it, one physical, the other mathematical; and they reinforce each other. The physical side of his argumentation uses the technique of ensembles and begins by considering a statistical ensemble consisting of a large number of models of the physical system under consideration. If we reject the view of Quantum Mechanics, and suppose that the values of physical quantities in the ensemble have positive dispersions only because the ensemble has not been resolved in a sufficiently fine manner, we must admit that we should be able to carry out such a resolution into subensembles in which the dispersions are zero, or at least diminished from those of the original ensemble. The method of doing this is to measure the physical quantities in succession, replacing the ensemble each time to one of the subensembles where the quantity has a sharply defined value. But then, the Heisenberg uncertainty relations between complementary physical quantities operate in such a way that the precision achieved for a quantity in any stage is destroyed at the next stage when a complementary quantity is measured; we thus do not get ahead. However, one may take exception to this line of reasoning by pointing out that there could conceivably be other methods of penetrating to the dispersion free ensembles. To settle this point decisively one must therefore answer the following question: Given an ensemble in which certain physical quantities have positive dispersion, is it possible to resolve this ensemble into a superposition of two subensembles, different from the original one and distinct from each other? Ensembles which cannot be so resolved are the pure (homogeneous in von Neumann's terminology) ones, and the above question becomes the following: Is every pure ensemble dispersion free? If the answer is no, then the hidden variables interpretation must be given up. To answer this question in the conventional model of Quantum Mechanics von Neumann begins with the very interesting observation that since an ensemble is characterized (statistically) by the expectation values of all the

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physical quantities in it, one may replace the ensemble in the mathematical argument by the corresponding expectation functional; we must remember that if A is any bounded observable, its probability distribution is completely determined by the knowledge of all the expected values Ex$(An),n

= 0,1,2,....

Thus, for von Neumann, an ensemble is the same as the corresponding real valued (expectation) functional Exp:J5R on the space B0(J^) of all bounded self-adjoint operators on the complex Hilbert space Jf; then von Neumann assumes that it satisfies the conditions: (a) Exp(l) = l; (b) E x p ( ^ 2 ) ^ 0 , (c)

V^G^(Jf);

Exp is linear (over R);

(d) Exp is continuous in the strong operator topology. Nowadays, one works with the space B(Jf) of all bounded operators on &?\ the conditions (a)-(c) define a state, (b) being equivalent to Exp(^4J.t) ^ 0, yAeB(J^); with (d), we have a normal state. He showed that the functionals Exp are in one-one correspondence with bounded operators U which are self-adjoint, ^ 0 , and of trace 1, the correspondence being defined by (2)

Exp(A)=tT(UA).

The correspondence preserves convexity and so matches the pure states with the extreme points of the convex set of the U's. He verified that the latter are the one-dimensional projections, and showed by explicit calculation that these are never dispersion free. This concluded his proof. One must emphasize that his reasoning explicitly assumes that Quantum Mechanics is described by the conventional model. It is possible to modify von Neumann's argument so as to allow for the presence of superselection rules, but the essence of the reasoning does not change. Many subsequent criticisms of von Neumann's proof are unjust because they do not take into account the above assumption in his proof, although von Neumann states it quite unmistakably (see the first sentence of the first paragraph on p. 210, and lines 17 -h on p. 324; see also the remarks of L. van Hove, Von Neumann's contributions to Quantum Theory, pp. 95-99, in John von Neumann, 1903-1957, Bull. Amer. Math. Soc. 64 (1958), No. 3, Part 2). The mathematical content of von Neumann's argument may thus be summarized as follows: B(3tf) does not admit any dispersion free normal state.

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The first question t h a t arises naturally now is whether von Neumann's definition of a state was unduly restrictive. Obviously the conditions (c), (d), of (1) need to be examined more closely. A conservative definition would replace (c) b y (3)

(c')

E x p ( ^ +B) = Exp(^4) + E x p ( £ )

when

AB =

BA.

Let us call maps A: BQ(J^) -> R satisfying (a), (b) of (1) and (c') of (3)physical states. I t then follows t h a t |A(-4)| <\\A\\ and t h a t the restriction of A to the projections defines a finitely additive probability measure on the logic J£?(e2f). Conversely, it follows from the spectral theorem t h a t any finitely additive probability measure on ^ ( J f 7 ) is the restriction to the projections of a unique physical state of B0(J^). I n other words, physical states m a y be identified with finitely additive probability measures on j £ ? ( ^ ) . Thus the strongest generalization (in this direction) of the von Neumann result would be to show t h a t there are no finitely additive dispersion free probability measures on J>?(Jf). The case when 3 ^ d i m p f ) 1 one can show t h a t v is a state which is not normal. Our final remark is to point out t h a t it was Mackey who first realized t h a t one should really prove von Neumann's result with physical states rather t h a n states, or at least with countably additive probability measures on J ^ p f ) ; Gleason's work which showed t h a t every countably additive probability measure on j£?(Jf) defines a normal state on B(Jf) a t least when 3 ^ dim(^f) < oo, was inspired by Mackey's investigations. I n addition to being a significant and aesthetically satisfying strengthening of the von Neumann result, this work and others t h a t were inspired by it have led to a deeper understanding of the questions involved. The fact t h a t physical states are the same as states was proved in

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full generality o n l y recently, and the main motivation for reaching this result had always been the desire to generalize Gleason's theorem to more general von Neumann algebras t h a n B(Jf). A second question t h a t arises now is whether one can relax the assumption in the von Neumann-Mackey-Gleason result t h a t the logic of the physical system is <&(<#?). There is an extensive literature on this question and I shall limit myself to a brief discussion of two results, those of Neal Zierler and Michael Schlessinger (Boolean imbeddings of orthomodular sets and Quantum Logic, Duke Math. J., 32 (1965), pp. 251-262), and of Simon Kochen and E . P . S p e c k e r (The problem of hidden variables in Quantum Mechanics, J . Math. Mech., 17 (1967), pp. 331-348); for more detail and other references t h e reader should consult the lucid discussion of these problems in the book of Beltrametti and Cassinelli (of notes to Chapter I I I ) . I n both of these articles, the interpretation of what is meant by hidden variables is such t h a t it implies the existence of some sort of imbedding of the proposition system into a Boolean algebra. So the mathematical issue in both cases comes down to showing t h a t under suitable assumptions on the proposition systems such imbeddings do not exist. I n Z-S it is assumed t h a t the set i ? of experimentally verifiable quantum propositions is partially ordered, has 0 and 1, and an orthocomplementation ± : S£->£ (0 X = 1, l x = 0, a ^b=>b^ < a - \ a-L± = a, avaA- = 1, aAa± = 0) which is weakly modular (a^b=>a1Ab exists and b = a\/(a-LAb)); it is not assumed t h a t j£? is a lattice. As usual if a ^ 6 X we say a and b are orthogonal; sums of mutually orthogonal elements exist in «j£f. If x, ye<£f they are said to commute if we can write x = xxyz, y — yx\/z where xv yv and z are mutually orthogonal; this is possible if and only if x and y are contained in a Boolean subalgebra of & (recall t h a t 5 c i f is a Boolean subalgebra if 0, leB; if c,a,beB ^C^GB, a\jb, a Kb exist in ££ and belong to B; and if B becomes a Boolean algebra with respect to v, A, _L). If B is a Boolean algebra, a n imbedding of S£ into B is a m a p h (<& ->B) such t h a t

(4)

(a) (b) (c) (d)

A(0) = 0, h(l) = l; x ^yoh(x)
Actually it is enough to require (b), (c) and the single set of relations h(x+y) = h(x)\/h(y) in (d). Z - S prove t h a t if ££ contains a copy of j£f(R 3 ), then i f has no imbedding into a Boolean algebra; indeed, it follows from Stone's theorem t h a t any Boolean algebra has {0,l}-valued probability measures, so t h a t such imbeddings would give rise to a {0,l}-valued probability measure on i f ( R 3 ) . Z-S also proved t h a t for arbitrary ££\ if one

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assumes t h a t there are imbeddings into Boolean algebras which satisfy (d) of (4) in the following stronger form (5)

(d')

h(x v y) = h(x) V h(y)

iixyy

exists in 3?,

then x\jy will exist in ££ only when x and y commute; thus if S£ is in addition a lattice (e.g., a logic), such an imbedding will exist only when J£ is already a Boolean algebra. The analysis of K - S is along similar lines but it is formulated in the language of partial Boolean algebras. A partial Boolean algebra is a set L with a family 93 ={B} of subsets of L with the following properties:

/m

(a)

each B e 93 has the structure of a Boolean algebra; if B, B' e 93 and B^Bf (as sets), B is a Boolean subalgebra of B';

(b)

given B , B ' e SB, S n B ' e S B ;

(c) (d)

each element of L is in some Be 35; if a l 5 ..., an e L and any two of them are contained in some member of 93, there is a member of SB t h a t contains all of them.

(6)

I t follows easily from these axioms t h a t L has the following properties: (a')

t h e unit and null elements of all members of 93 are the same; we write 0 and 1 for them;

(b')

if a e L there is a unique a±eL such t h a t a1- is the complement of a in any B e SB t h a t contains a;

(c')

write, for a, b e L, a ^ b to mean t h a t a and b are in some B e SB and satisfy this relation there; then ^ is a partial order on L;

(d')

with respect to (<,J_), L becomes orthocomplemented and weakly modular. Obviously, if we start with an orthocomplemented weakly modular L its Boolean subalgebras will give L the structure of a partial Boolean algebra if (d) of (6) is satisfied. A morphism of a partial Boolean algebra (L,SB) into another (2/,93') is a m a p h:L->Lf such t h a t for any 2?e93, there is a 22'eSB' such t h a t h(B)^B' and h:B->B' is a morphism of Boolean algebras. Imbeddings are injective morphisms. For K - S the basic assumption is t h a t the set 3? of experimental propositions is a partial Boolean algebra, and as we mentioned earlier, their interpretation of the existence of hidden variables will imply the existence of an imbedding of S£ into a Boolean algebra. They prove t h a t the existence of such an imbedding is equivalent requiring t h a t if a, b e ££ and a^b, there is a {0,1}-valued probability measure honJ? such t h a t h(a) ^h(b). They then proceed to construct explicitly a finite subset F of J^(R 3 ) with the property t h a t the partial Boolean subalgebra of J^(R 3 ) generated by F does not admit a single {0,1}-valued probability measure.

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The case of the logics of the two-dimensional spaces has been completely excluded in the above discussion. The spin observables of an electron are an example of such a system. Here the Hilbert space <#?= C2. If

(8)

*!-£ I), *,-(<

""Q),

*,= (J _J)

are the Pauli spin matrices, then for a n y a = (ava2>a3) in the unit sphere S2 of R 3 , the self-adjoint operator a • | ( 1 +a-a) is a bijection of S2 with the set of (Hermitian) one-dimensional projections of Jf, and gives rise to a bijection p->fP between the set of probability measures^? on JS?(JH and functions/: £ 2 - > [ 0 , l ] such t h a t

(9)

/(2)>0,

/(o)+/(-S) = l

(«e^2),

with_p(J(l+a-a))=/3,(a). The q u a n t u m states, i.e., the states in t h e conventional model of Quantum Mechanics are the functions ql (beS2) denned by (10)

qt(a) = i ( l + a -S)

(• = scalar product in R 3 ).

I t is clear t h a t the functions (10) form only a small part of the convex set of functions (9) (even if we assume they are Borel). The extreme points of the latter set are the functions t h a t take only the values 0 and 1, the dispersion free states; they are the characteristic functions 1A where A^S2 is a n y subset with Au ( — A) — 82i Ar\( — A)= 0. This suggests t h a t we can express the states (10) as mixtures of suitable families of dispersion free states and hence t h a t a classical imbedding may exist for t h e q u a n t u m system associated to Jf. This is in fact so, and the details were first given in articles of Kochen-Specker (loc. cit.) and J . S . B e l l (On the problem of hidden variables in Quantum Mechanics, Rev. Mod. Phys., 38 (1966), pp. 447-452). I n what follows we shall describe briefly the construction of Kochen-Specker; Bell's example is built along similar lines. The space of "hidden variables " is S2. I t s points w define dispersion free states frv\ for each self-adjoint operator AsB(J^), there corresponds a function FA on S2; and for each pure q u a n t u m state q, there corresponds a probability measure JJLQ on S2. The FA and /xfl are related by (i) (11) (ii)

FU(A) = U(FA) for a n y Borel function u, r q(A) = FAdnQ. Js*

The relation (i) asserts t h a t the functional relationships among q u a n t u m observables are preserved under the m a p A ->FA, and relation (ii) asserts

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that the probability distribution of A in the state q is that of FA with respect to /zff. If A = |(A7+?• a) (A e R) (having eigenvalues J(A ± 1)), (12)

^

= *(A+1)1«.T>0+*(A-1)1^
where 1^> 0 is the characteristic function of the set of w's with w -"f ^ 0, and so on. It is immediate that the assignment (12) possesses the property (i) of (11). If qt is the quantum state (10), the associated probability measure fit=Pot is defined by (13)

dfit = 4(6-5?) l%fe0dw,

where did is the surface measure on $ 2 , given by dw = (l/4;r) sin Odd d

1_HMW= f

4(S-tS)diS.

Finally, for each w let/^(a ->/^a)) be the function which is 1 when w • a > 0, 0 when w-a<0, and which is defined so satisfy (9) when w-1z = 0. Then/^ defines a dispersion free state on J^(J^), and the formula (14) becomes (15)

qt== fadix%{w) fadptii

exhibiting qs as a mixture of dispersion free states; the absence of explicit definition off$(r) when w-T=0 does not matter, since this set has measure 0 in w. We now turn our attention to the analysis from the hidden variables point of view, due to Bell, of the Einstein-Podolsky-Rosen example (J.S.Bell, On the Einstein-Podolsky-Rosen paradox, Physics, 1 (1964), pp. 195-200); actually Bell considers the variant, due to Bohm and Aharanov, of the EPR example. The system consists of a pair of spin J particles and only their spin observables are of interest. Thus the Hilbert space is (cf. the remarks below in the discussion of composite systems) j T = C2(g)C2. The particles are in the "singlet" state defined by the state vector (16)

® = -jz{(p+®(p--(p-®
where 9^ are the spin states of eigenvalues + 1, say, in the z-direction. For instance, the spin parts of the states of the two electrons in the Helium atom or the H2-molecule are in this state (which is the unique rotation invariant

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antisymmetric state); even if the electrons are stripped away by dipole radiation and become widely separated spatially, (16) describes to a good approximation the compound state of the two electrons (cf. the discussion of E.P. Wigner in Interpretation of Quantum Mechanics, in Quantum Theory and Measurement, edited by J.A.Wheeler and W.H.Zurek, Princeton University Press, Princeton, NJ, 1983, pp. 260-314, especially pp. 291-293). Let Xlt% and X2,t be the observables measuring the spin components of the first and second particle in the direction a. In Quantum Mechanics they are represented respectively by the operators a • 5 : P QM (a,6) = ((a-a(x)S-a)
Pw(a,b)

= -a-b=

-cos0,

where 6 is the angle between a and b. The question studied in loc. cit. by Bell is whether the correlated statistics of the two particles in the singlet state that is summarized by (17) can be obtained from a hidden variables model that is local in the sense introduced by Einstein, Podolsky, and Rosen in their famous paper, namely, measurements on either particle do not disturb the other. Bell's analysis showed that the correlations arising from any local hidden variable model must satisfy certain limitations that conflict with (17). For nonlocal models, however, these inequalities may be violated (cf. M.Flato et al, Helv. Phys. Acta, 48 (1975), pp. 219-225). The model consists of a probability space X of points A, and a probability measure dX on X; moreover, the assumption of locality enables us to say that there are random variables A^aiX) and A2(b:X) corresponding to Xltt and X2.%, respectively, for a, beS2. The Aj(a:X) are ± 1 valued. The correlations are defined by P HV (a,6) = f A1(a:X)A2(b:X)dX, where the suffix stands for hidden variables. If a, b, a', b' are four points of S2 then (suppressing A)

\PBv(*,ty-PKv&h\^\JAi(a)AM

+I

U^AS'Jil+A^AMdxl

<(1±PHVK^)) + (1+PHV(^).

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So (cf. M. Lamehi-Rachti and W. Mittig, Phys. Rev. D, 14 (1976), pp. 25432555) we obtain the inequality (18)

|P HV (a,S)-P H V (a,6')| + 1 i V («',&')+^Hv («'£)l < 2,

which must be satisfied by the correlations in any local hidden variables model. In Quantum Mechanics, P Q M ( ^ 3 ) = — 1; PHv satisfies this if and only if (19)

A^aiX) =

-A2(a:\).

If we assume (19), then (18) becomes (20)

|P HV (S,6) - P H V (a,c)| ^ 1+P H V (S,c).

This is PeW's inequality. It is trivial to check that (17) violates (20). Indeed, if 012, 023> #3i a r e the angles between a, b, b, c, and c, a, then for (17) to obey (20) we should have sin2 (0ia/2) + sin2 (023/2) ^ sin2(031/2), which is false if 612 = 023 — n/3, 631 = 2TT/Z. If we assume the model to have natural covariance properties with respect to rotations, then P H v (*W depends only on the angle 0 between a and b say P HV (a,b) = P H V (0). Obviously,

The inequalities £18) now imply bounds for Puy(6). Indeed, taking a', b' coplanar with a, b we get, writing P for P H V , (21)

\P(e)-P(d + a)\ + \P(0')+P(6' + a)\ < 2 ,

for all 9, 6', a from which one gets (cf. Lamehi-Rachi and Mittig, loc. cit.), (22)

|P(jr/6)|«f

\P(n/4)\
\3P(n/3) + P(0)\<2.

The article of Bell attracted the attention of physicists and led to experiments whether (22) or (17) is true (cf. Wigner loc. cit.). The experimental results so far support Quantum Mechanics (see some of the experimental evidence presented in the articles of the collection edited by Wheeler and Zurek, loc. cit.). Even though the basic Hilbert space has dimension 4 here, the above treatment of the hidden variables question cannot be subsumed under the von Neumann-Mackey-Gleason paradigm. The point is that for each A e X, the model gives only the values of Exp(Z1>^m-Z2,Sn) (m,n^Q); and hence each XeX can define a dispersion free state only on the maximal abelian subalgebras jtf(aj)) of J^ generated by a-a®l and l®b-<j. Hidden variable theories such as these are called contextual in the book of Beltrametti and Cassinelli (cf. notes on Chapter III). The reader who wants to

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know more about these and get a sharper perspective on the entire question of hidden variables cannot do better t h a n consult the book of Beltrametti and Cassinelli, for its thorough t r e a t m e n t of related issues as well as t h e references it contains to other literature. For a question as deep as the one t h a t asks whether quantum mechanical description of the atomic world is complete, a mere mathematical analysis will hardly provide a completely satisfying answer. The argument must also be based on analyzing possible measurements and the limitations imposed on t h e m by the uncertainty relations. The paper of Einstein, Podolsky, and Rosen, and Bohr's refutation of it, offer perhaps the most profound discussion of this question from the natural philosopher's point of view (cf. the relevant articles in the collection edited by Wheeler and Zurek, loc. cit.). 2. Algebraic formulation of Quantum Mechanics. I n some sense the algebraic method of viewing the foundations of Quantum Mechanics is the oldest of all approaches, and arguably, the most flexible one. The early papers of Born, J o r d a n , Heisenberg, and Dirac repeatedly emphasized t h a t q u a n t u m observables are represented by matrices and so form a noncomm u t a t i v e algebra, in sharp contrast to the commutative algebra of classical observables (for a historical account as well as translations of the early papers, see Sources of Quantum Mechanics, B. L. van der Waerden, Dover, New York, 1967). Eventually this came to be formalized as the principle t h a t the quantum observables are in one-one correspondence with t h e selfadjoint operators on a (separable, complex) Hilbert space. I n spite of t h e overwhelming success of the models built out of this assumption it was always understood t h a t this was a very ad hoc solution to the mathematical, physical, and philosophical problems raised by atomic physics. Especially troublesome was the fact t h a t both the operations of addition and multiplication involved possibly noncommuting (hence nonsimultaneously observable) observables, t h u s making it very difficult to give a n y convincing " e x p l a n a t i o n " for them. (Perhaps one should not p u t too much faith in such " e x p l a n a t i o n s " . Most explanations make implicit or explicit use of classical perceptions and are not too relevant in the atomic realm.) There was also the fact t h a t every self-adjoint operator was supposed to represent an observable; in the presence of superposition rules one should modify this suitably, b u t even then this was gradually seen as a far-reaching enlargement of what can be effectively observed in atomic systems—see Interpretation of Quantum Mechanics, E . P . Wigner, in Quantum Theory and Measurement, edited b y J . A . W h e e l e r and W . H . Z u r e k , p . 298. Multiplication was felt to present the greater problem since it could be argued t h a t if A and B are observables, A+B can be effectively defined by prescribing t h a t its expected value in any state is the sum of the expected values of A and B in t h a t state. This, of course, is a little deceptive, for, as we have

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already remarked above, the linearity of expectation values is a deep property of the propositional logic; its validity for the standard logics (in dimension ^ 3) is a consequence of Gleason's theorem while its truth for the more general logics of projections in von Neumann algebras has only recently been proved (see the remarks on Gleason's theorem, and also §14.5 of the book of Beltrametti and Cassinelli, cited in the notes to Chapter III). The first systematic investigations on the algebraic structure of the set of observables are due to Jordan, von Neumann, and Wigner; the initial papers are those of Jordan in Gott. Nachr. (1932), pp. 569-575, Gott. Nachr. (1933), pp. 209-217, and Z. Phys., 80 (1933), pp. 285-291, and that of Jordan, von Neumann, and Wigner in Ann. Math., 35 (1934), pp. 29-64. One of the motivations for these studies was the feeling that the generalizations of Quantum Mechanics to relativistic and nuclear physics might force such a more general view of observables. Jordan's basic observation was that although the set of bounded self-adjoint operators do not form an algebra under multiplication, it is nevertheless an algebra over R, even abelian, if we define a new product by A • B = \{AB + BA). This product is bilinear and satisfies the distributive laws A • (B + C) — A • B + A • C, although not the associative law; moreover, the identity A -B = \[{A +B)2 — A2 — B2] shows that this product can be obtained from the linear and power structures. Nowadays such algebras, emerging as they did out of the pioneering studies of Jordan, are known as Jordan algebras; see the book of Jacobson, The structure of representations of Jordan algebras, Amer. Math. Soc. Colloq. Publications, vol. XXXIX, 1968, Providence, R.I. Typically, one starts with an associative but not necessarily commutative algebra and define the new multiplication by a-b — \(ab + ba) to obtain a Jordan algebra. But it is not always possible to obtain a given Jordan algebra in this fashion. The fundamental result of Jordan, von Neumann, and Wigner in their 1934 paper is the complete classification of all finite dimensional Jordan algebras A over R which are irreducible and formally real, i.e., satisfy al9 ...,aneA,

ax2+...+ a2 = 0 => ax = ... =an = 0.

It follows from their classification that, with one exception (that of M38, the algebra of self-adjoint 3 x 3 matrices over the Cayley numbers), all of these are obtained in the above-mentioned manner from associative algebras; and further, that if some special cases are disregarded, one gets just the Jordan algebras of Hermitian matrices over R, C, or H. The finite dimensionality assumption is a very severe one and von Neumann began the study of infinite dimensional topological algebras (Mat. Sb., 1 (1936), pp. 415-484; Collected Works, Vol. I l l , No. 9, Pergamon, Oxford, 1961). His investigation in this paper contains a deep analysis of

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t h e spectral theory of individual elements and its relationship to simultaneous observability. Although he had, in connection with his analysis of t h e possible existence of hidden variables (cf. his book), introduced the notion of the state of a system as a positive definite linear form on t h e observable algebra, this paper does not contain a study of the states. With the appearance, in the early 1940s, of the path-breaking papers of Gel'fand and Naimark on Banach algebras, the essential aspects of this approach became clearer. This theme was taken up by I . E . S e g a l in 1947 (Postulates for general Quantum Mechanics, Ann. Math., 48 (1947), pp. 9 3 0 948). Segal's postulates on the observable algebra were simpler and more direct t h a n von Neumann's, and were motivated by what is operationally possible; in addition, he studied in detail the structure of the states. The case when the (bounded) observables are the self-adjoint elements of a complex C*-algebra (a complex Banach algebra with involution * such t h a t \\x*x\\ = ||#||2 = ||#*||2, ||1|| = 1) is a possible system in t h e sense of Segal. Although special, it is an important one since it includes the conventional model of Quantum Mechanics. If U is the C*-algebra whose self-adjoint elements are in correspondence with the bounded observables of the system, a state of the system is a linear function / o n U such t h a t /(1) = 1 and f(x*x)^0 for all xeVL. The Gel'fand-Naimark-Segal representation theorem associates t o / a n essentially unique triple (J^f,nf,ifjf) consisting of a Hilbert space J^f, a unit vector i/jf in 2tfu and a *-representation 777 of U in J f f with ifjf as cyclic vector such t h a t f(x) = (7rf(x)ijjf,ifjf) (x e U); / is pure if and only if nf is irreducible. I n this way state vectors are once again unit vectors in Hilbert spaces; b u t the Hilbert space is not fixed as in conventional Quantum Mechanics, b u t varies with the state. The importance and usefulness of these more general types of observable algebras become clear in the theory of Second Quantization where one works with systems of identical particles (photons, electrons, He 3 , He 4 , etc.), b u t where the number of particles m a y not remain fixed. I n this case it t u r n s out in contrast to the case of systems of a fixed number of particles, t h a t t h e canonical commutation rules (or anti-commutation rules) have m a n y essentially different representations. Nevertheless, it was proved by Segal (Mat. Fys. Medd. Dan. Vid. Selsk, 31 (1959), No. 2) t h a t the C*-algebra, which is the norm-closure of the observables depending only on finitely m a n y particles, is canonically determined; therefore it m a y be regarded as t h e algebra of field observables. The possibility t h a t one m a y have to v a r y t h e Hilbert space with the state has also been discussed by Dirac (Lectures on Quantum Field Theory, Belfer Graduate School of Science Monograph Series, No. 3, Belfer Graduate School of Science, Yeshiwa University, New York, 1966). The study of infinite dimensional observable algebras has been pursued in recent years in another direction, closer to the original theme of J o r d a n ,

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von Neumann, and Wigner. This is the study of what are called J B , J B W algebras, which are the J o r d a n algebra analogues of the self-adjoint parts of C*-algebras and von N e u m a n n algebras. The state spaces of these are compact convex sets with an interesting geometric structure, and they can be characterized in an intrinsic manner. This thus opens up another approach to the foundations in Quantum Mechanics. For a survey of many aspects of this theme we refer the reader to the article of E. M. Alfsen (On the state spaces of Jordan and C*-algebras, in Algebres d'operateurs et leurs applications en Physique Mathematique, CNRS Coll. Intern. No. 274, 1979, pp. 15-40). I n recent years people (Haag, Kastler, and others) have used algebras of operators for formulating some general properties of physical systems (for example, a quantized field) extended in space-time, t h a t are local in the sense t h a t the observables t h a t are localized in noncausally connected regions commute with each other. These ideas have clarified somewhat the nature of the high-energy scattering processes t h a t dominate elementary particle physics. The reader who wants to understand the algebraic approach m a y start with the book of G. G. E m c h (Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley, New York, 1972), and the references cited there. I t should be remarked, however, t h a t the mathematical questions arising in the a t t e m p t to construct rigorous models of scattering processes involving particle creation and annihilation (Quantum Electrodynamics, for example) are quite deep and are tied up with themes of an entirely different sort (cf. J . Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, Springer-Verlag, New York, 1981). The approach t o Quantum Mechanics via functional integrals was pioneered by R . P . F e y n m a n [2] (cf. also his book with A . R . H i b b s , Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965) and is known as the path integral formalism. I t is widely used by physicists although mathematically difficult to justify. The attempts to understand this formalism a t a rigorous level have proved very beneficial and effective in the mathematical theory of elementary particles and their interactions; see the book of Glimm and Jaffe, loc. cit. Finally, for a view of the foundations of Quantum Theory t h a t is very close to phenomenology one should refer to the series of papers of J . Schwinger which are a p a r t of his book Quantum Kinematics and Dynamics, W. A. Benjamin, New York, 1970. 3. Statistics of mixed states. Consider a system governed by the standard logic ££(ffl) where &? is a separable complex Hilbert space, and let it be in t h e state pu where U is a von Neumann operator on #F. By the spectral theorem we have U = YinKPLn where \v\2,... are the distinct eigenvalues > 0 of U, Ln the eigensubspace corresponding t o An, and PLH

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is the orthogonal projection ^->Ln. If dn = dim(Ln), wn = dn\n, Un=d^P^y then tr( C7W) = 1 , 2 nW„ = 1, and the formula (23)

Pu =

and

^wnpun n

exhibits pu as a canonical mixture of the states pun with weights wn, the supports of the p un being mutually orthogonal. If U has a simple spectrum, i.e., if dn = l for all n, t h e ^t/ n are all pure states; and (23) is the unique decomposition of p u as a convex combination of pure states defined by orthonormal vectors. If some of the dn are > 1, the right side of (23) can be split further but not uniquely; indeed, if (ttnx-)i^
PU^^Wn

2

dn^PunV

which exhibits p u as a convex combination of pure states defined by t h e orthonormal family of vectors (uni). Thus, as soon as some dn > 1 we cannot definitely say of which pure states p u is a mixture of; the phenomenon is clearly nonclassical and is another example of the sharp contrast between q u a n t u m and classical statistics. 4. Composite systems. One of the most widely used methods of analyzing complicated systems is t h a t of viewing them in terms of suitable subsystems and their interactions. For example, for m a n y purposes it is completely adequate to view an atom as a system of electrons moving in a centrally symmetric force field. We are thus led to the general problem of composition of systems. Let <5lf S 2 , • • • , S ^ be q u a n t u m mechanical systems; in conventional Quantum Mechanics we associate complex separable Hilbert spaces Jtf?1,...93tf,N9 to them with the usual prescriptions of states and observables. I t is then possible to form a composite system S containing S 3 as subsystems in a natural manner, and in fact to do this in a universal way. The logic of S is ££{#?) where Jf =

J$?1®...(x)3eN.

The m a p ia:A->l®...®A®...®l

(1 ^ a ^ N, A in the ath factor)

imbeds the algebra B(J^a) in B(3tf)\ by restricting the ^4's to the projection operators, each ia gives rise to an imbedding ia :&(Jt?a) c> jgf(jT)

(1 < a < N).

Thus the properties of the S a are faithfully reflected within S . If a is a state of S , i.e., a probability measure on j£Ppf), then (25)

aar=(TOia

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is a probability measure on ^(J^a) and so defines a state of S a , the state that S a is in when S is in the state a. The map (26)

CT^(CT 1 J

...,GN)

is clearly surjective; if The probability of having spin parallel to any given direction is | for each of the particles; but the spins are so correlated that, given that the first particle has spin along the direction a, the probability is 1 that the second particle has spin along — a. This example is a variant due to Bohm and Aharanov (Phys. Rev., 108 (1957), p. 1070) of the example used by Einstein, Podolsky, and Rosen in their famous paper (Phys. Rev., 47 (1935), pp. 777-780). By measuring the spin of the first particle in any direction a we can predict with certainty the spin of the particle along a without apparently disturbing its state, a paradoxical result since the spin components along different directions are not simultaneously observable. The paradox arises because of the assumption (which ultimately involves preconceptions about the structure of matter, among other things) that a measurement of the spin state of the first particle does not affect the state of the second when they are spatially far apart; the paradox disappears if one realizes that when the composite system is in a state such as (27), a measurement of the spin state of the first particle changes the state of the composite system, and consequently also the state of the second system (cf. the detailed analysis of the situation in Bohr's reply to the EPR paper, Phys. Rev., 48 (1935), pp. 696-702). The theory of composition of systems in contexts other than the standard ones is more involved. For instance, since there is no reasonable way to form tensor products of vector spaces over noncommutative division rings, (27)

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t h e theory of composite systems in quaternion Quantum Mechanics becomes more difficult. F o r quaternionic Quantum Mechanics, t h e impossibility of forming tensor products is interpreted by Finkelstein et al.[l] to mean t h a t there are no truly independent systems, i.e., t h a t there are complementarity relations between observables of a n y two systems in quaternionic Quantum Mechanics. The issue becomes even more complicated when we treat it in t h e framework of general logics. The reader is referred t o t h e discussion in Chapter 24 of t h e book of Beltrametti a n d Cassinelli (cf. Notes on Chapter I I I ) and t o t h e literature cited there. The above considerations have t o be modified substantially when one wants t o treat systems of identical particles, because, in Quantum Mechanics there are n o experiments t h a t can keep track of individual particles throughout t h e course of some physical process. Thus, in such assemblies of identical particles, only the physical quantities t h a t involve t h e various particles symmetrically are permitted t o be observables. Mathematically, this means t h e following: if 34? is t h e Hilbert space of a single particle, t h e n for a system of N such particles, the self-adjoint operators on tfW) = 34?® ... & jtr (N times) t h a t represent physical observables must commute with t h e natural action of t h e permutation group SN of { 1 , . . . , N} on 34?{N); here we recall t h a t a n y GSSN acts on 3^{N) by sending u± (x)... ® uN t o ^o-i(D® ••• ® ^ - W ^ e ^ ) . Let U be t h e algebra of all operators on #£{N) commuting with this action on SN- T O analyze t h e structure of U we proceed as follows. For a n y irreducible character r of SN let E(T) be t h e projection operator (N! Vdim(r))- 1 £ T(*)«

on j f w

seSjf

t h a t projects onto t h e space of " r - s y m m e t r i c " tensors; if j f W ( r ) = E(T) (JirW), t h e 34?{N)(T) are mutually orthogonal, stable under SN and U with (28)

JfW = @ j f W ( T ) ; T

moreover, we can factorize each Jf{N\r) (29)

(but noncanonically), as

#W(T)KK{T)®L{T),

where U (resp. SN) acts only on t h e first (resp. second) factor in such a w a y t h a t SN acts irreducibly on L(r) with character r while U is isomorphic t o B(K(T))®1 (see Weyl [1]). Thus, if J ? is t h e logic of ^ - s t a b l e closed subspaces of ^N), then the J f iN\r) are the atoms of the center of J*?, t h e logics £?(T)=£? njfN\r) are standard (being isomorphic t o JP(K(T))9 a n d

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J£? = @ T J > ? ( T ) is t h e central decomposition of «^f. Although (29) is not canonical, once we fix such a factorization, for each r, t h e pure states of S£ m a y be identified with the rays of Jf { N ) t h a t lie in some K(T), SO t h a t the K(T) are t h e coherent subspaces of pure states. The reader should view this as an example of noncommuting superselection rules. Two specific coherent spaces of pure states deserve special attention. These are t h e cases where T is one-dimensional so t h a t K(T) = ^N)(T). More precisely, we consider t h e two cases: (a) T = 1 , T(S) = 1 for all SESN; (b) T = sign, T(S) = signature of s for all s e # # . I n (a), 34?{N)(r) is t h e subspace of symmetric tensors of rank N over 34?, or the subspace of degree N, 8{N)(34?), in t h e symmetric algebra S(34?) over 34?. I n (b), 34?{N\T) is t h e subspace of skew-symmetric or alternating tensors of rank N over 34?, or t h e subspace of degree N, A{N)(34?), in t h e Grassmann or Exterior algebra A(34?) over 34?. I n atomic theory, in order to get agreement with experiment it is necessary to assume t h a t for systems of N electrons no states other t h a n those in Am{34?) occur in nature, i.e., the logic is the standard one associated with the Hilbert space Am(34?). If u{ (l^i^N) are orthonormal vectors in 34?, it is customary t o interpret U±A...AUN as " t h e state in which t h e electrons (in some order) occupy t h e states uv ...,M;v". The fact t h a t uiiA...AuiN = 0 if two of t h e ip's are identical is usually formulated as t h e Pauli exclusion principle, namely, " n o two electrons can be in t h e same state." I n radiation theory which studies photon assemblies it becomes necessary t o assume t h a t for a system of N photons only t h e symmetric tensor states occur in nature, i.e., t h e logic is t h e standard one associated with t h e Hilbert space S{N)(3f). I n either case it is necessary t o give additional prescriptions, typically by introducing specific operators, t o describe physical quantities a n d processes characteristic of the systems of particles in question. The systematic way of doing this is known as Second Quantization. I n particle physics it is assumed t h a t t h e above description in terms of A{N)(34?) or S{N)(34?) extends to assemblies of particles of a n y type whatsoever. Particles which require t h e exterior algebra description are known as Fermions a n d are said t o obey Fermi-Dirac statistics, while t h e others are known as Bosons and obey Bose-Einstein statistics. Experiment has shown t h a t particles with half integral spin (electrons, neutrons, protons, H e 3 atom, etc.) are Fermions while particles with integral spin (photons, H e 4 atom) are Bosons. All theoretical explanations of this remarkable connection between spin a n d statistics depend on t h e theory of relativistic quantum fields. 5. Measurement in Quantum Mechanics. The theory of measurement is a t the heart of Quantum Mechanics. Indeed it is the uncontrollable disturbance caused by measurements on t h e physical systems being observed t h a t leads to t h e statistical character of t h e quantum mechanical picture of t h e

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physical world. It is natural to try to determine the extent to which one can understand the nature of the measurement process within the framework of Quantum Mechanics itself. Of course Quantum Mechanics gives unambiguous prescriptions for calculating the probability distribution of any physical quantity at a given instant of time, if the state of the physical system is given at an earlier time (Schrodinger's time-development equation), so that, no theory of measurement is needed to ensure the logical consistency of the theory. But the state vectors are not physical objects; they are idealizations, and one has conceptual access to them only through results of experiments. Thus, if the theory is to be confronted with experience, and is to be used as a guide for understanding existing phenomena as well as for predicting new ones, it has to contain instructions that tell us how to determine the states from physical data, i.e., how measurements on a system change its state. The fundamental work on the general theory of measurement is that of von Neumann, most of which is contained in Chapters V and VI of his book [1]. Since then, the subject has received a lot of attention and new themes have been introduced. We shall limit ourselves to a brief discussion of von Neumann's work and some of the later developments inspired by it. The reader who wants to get a historical perspective and a deeper insight into these questions should refer to the volume edited by Wheeler and Zurek (Quantum Theory and Measurement, Princeton University Press, Princeton, NJ, 1983) that collects together most of the crucial papers on the subject. For simplicity we shall restrict ourselves to the conventional model of Quantum Mechanics. Let a quantum mechanical system be in a pure state represented by the normalized vector cp (of the Hilbert space Jf7 underlying the system), and let A be a self-adjoint operator with a purely discrete simple spectrum that represents some observable. If a1,a2,... are the eigenvalues of A and ••• • However, if a particular value, say ak, is obtained, and the measurement of A is again carried out immediately after, the value does not change. This is interpreted in the following way: the act of measurement has changed the state of the system from (p to cpk. Since the value ak is obtainable only with probability | (y,(pk)\2, one may then summarize this set of circumstances by saying that the statistical effect of the measurement of A consists in changing the state of the system from the pure state cp to the mixed state in which the pure states q>v cp2,... occur with probabilities K^,^)! 2 , |( ••• • If the initial state is not pure but mixed and U is the von Neumann operator representing it, the operator of the changed state is (30)

UA=Z

{U
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where P[
MA:U-+UA

on the convex set of states. I t is clear from (30) t h a t MA is a convex m a p ; but it is a convex endomorphism and not a convex automorphism since in general it changes pure states into mixed ones. The essential content of von Neumann's work is a profound analysis of the mathematical a n d physical nature of the endomorphisms of the type MA. The first question is to what extent MA fails to be invertible. Von Neumann viewed this from the more general point of view of deciding whether (the effect of) MA is irreversible. To this end, he introduced also the automorphisms of the convex set of states of the form ajyiU^DUD-1;

(32)

these are induced b y unitary operators D of 3tf t h a t represent the dynamical transformations. H e then formulated the irreversibility of MA as the fact t h a t it is impossible to go from MA(U) to U by repeated applications of transformations of the form MB, OLD. For proving this he introduced, for any U, its entropy H(U): (33)

H(U) = — Ktr(Uln

U),

K > 0 being Boltzmann's constant.

Then he proved t h a t (34)

H(U)^0,

H{U) = 0oU

=p(p

for some cpeJf with |M| = 1,

and further t h a t (35) (36)

# K ( £ 7 ) ) = #(*7), H(MA(U))

> H(U)

unless

MA(U)

= U.

The irreversibility of the transition U^UA follows from (35), (36). As remarked by von Neumann himself, there are m a n y unitary invariants besides H(U) t h a t do not decrease under the measurement transition U->UA. For instance, the function t h a t assigns to U the value —A where A is the largest eigenvalue of U, has this property. But as we shall explain briefly below, the entropy function H has thermodynamic significance; and the relations (35)-(36) show t h a t the measurement transition U->UA is actually irreversible even at the thermodynamic level. Since it is easy to construct examples of changes U->U' where the entropy stays the same but the largest eigenvalue decreases strictly, we see t h a t there are thermodynamically reversible changes which cannot be brought about by repeated

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applications of MB, az>. Thus the inequalities (36) go far beyond merely establishing the mathematical irreversibility of the transitions U->UA. Von Neumann was led to the definition (33) of entropy by his analysis of ensembles of noninteracting systems each of which is in the state represented by the (von Neumann) operator U. H e used the methods of phenomenological thermodynamics for his purpose. The only difference between t h e classical setting and the present one is t h a t the individual systems are now governed by the laws of Quantum Mechanics. The problem was to calculate, for an ensemble of N systems in the state U, the decrease in entropy t h a t should be achieved to bring the systems to a pure state p^. He found t h a t the pure states are transformable among themselves without any change of " h e a t energy" and so all ensembles where the individual systems are in a pure state p^ have the same entropy, which can be normalized to be 0. H e then showed, assuming the validity of the first and second laws of thermodynamics, t h a t the ensemble of N systems with state U requires a decrease in entropy equal to —Nf
-NKtr(UlnU).

The reversal of the processes used here shows t h a t two ensembles with the same entropy can be transformed into each other by thermodynamically admissible processes. This analysis, which led to the formula (37) for the entropy, also suggested the properties (33)-(36). The relation (35) asserts t h a t the entropy remains constant during time evolution if no measurements are carried out. This will appear paradoxical because classically the entropy always increases. The paradox disappears if we realize t h a t the entropy H(U) is the entropy of the microstate. The classical entropy is a macroscopic quantity and its time variations are due to the fact t h a t the observer does not know the microstate of the system. To define an analogue of the classical entropy one m a y introduce a Boolean cr-algebra 9ft of "macroscopic projections" t h a t form the logic of macroscopic quantities. If we assume t h a t 9JI is atomic and its atoms EVE2,... are finite dimensional projections where (38)

sn = dim(En)

> 1,

then it is reasonable to define the macroscopic entropy H^iJJ) £7 by (39)

-^tr(UEn)lntT{UEn).

H(U) = n

$n

Von N e u m a n n proved t h a t (40)

HniU)

> H(U),

of the state

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with equality if and only if (41)

U =

j;(tT(UEn)/sn)En. n

Unlike the previous case, the macroscopic entropy is always > 0 for any U. Moreover, it will change under time evolution. Von Neumann investigated this as well as the associated ergodic theorems in his paper in Z. Physik, 57 (1929), pp. 30-70, to which we refer the reader for further details. The second aspect of the work of von Neumann is the investigation of the maps MA from the point of view of the Quantum Mechanics of the composite system formed by the system S under investigation and the system S ' consisting of the "measuring a p p a r a t u s . " Let 34? and 3tf" be the Hilbert spaces associated with them and let ^ = ^f?®Jf7/. Let A be as before a self-adjoint operator with a purely discrete spectrum, ava2,..., (pv 9? 2 ,... being the corresponding eigenvectors (normalized). The act of measurement of A in £> m a y be regarded as a " t e m p o r a r y " insertion of an energy coupling in the composite system t h a t allows S ' to interact with S . The time development of the composite system in the (short) time interval of measurement is then given by the dynamical group t -> exp( — itE) where E is the energy operator in 2tf. If £ is a unit vector in 3tf" representing the initial state of the measuring apparatus, and q> is a unit vector in £? representing the initial state of S , the state of the composite system after measurement is exp( — irE)((p(x)£;) where r > 0 is the duration of measurement. This state induces a state of S which m a y then be regarded as the state of (5 after measuring A. To make sure t h a t this scheme captures at least some of the key aspects of measurement it is natural to require t h a t the following conditions be satisfied. (i) (ii)

There is an orthonormal set (| n )n^i in &" such t h a t the state £ n of S ' " c o r r e s p o n d s " to the state (pn of (3. Forah>eJ^, exp( - irE)(

with M ? ) | = |(?Wn)|. The pairing (pn <-> f n is the mechanism by which the observer, recording the state f w of £>', recognizes t h a t S is in the state ®^) is

2IW..)IX. = -aMP*)n

If for a given A we can find (£ w )n^i, f > ®> T s u c n t n a t W a n d (ii) are satisfied, then it would be justifiable to view the transitions U-+MA(U) of the states of S , which are irreversible within S , as the effects on S of the

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perfectly reversible and continuous transitions provided in the composite system by its time development. I n other words, the causal and continuous change of states arising out of the dynamical evolution gives one a unified quantum mechanical way of looking a t the physical world and its interactions with the observer. Von Neumann proved t h a t this is possible; t h a t is, for any A, and for fixed r > 0, £, (£n)w^i> there is a suitable E satisfying (i) and (ii) above. This result of von Neumann inspired much subsequent work whose focus was on the following issue: To what extent do the requirements (i) and (ii) above as well as the operator E constructed by von Neumann give a faithful picture of measurement processes encountered in the physical world ? I t could be argued t h a t the description of S ' by the Quantum Mechanics of the logic ££{2tf") does not take into account the fact t h a t very often Q' is " v e r y big " compared to S and so the logic of S ' has a rich supply of superselection rules. The requirement t h a t measurement is of very short duration is also troublesome if these questions are to be treated relativistically. Finally, if there is additional structure in Jf, Jf", and J ^ , it is not clear t h a t the E constructed would respect them. The first example of nonmeasurability of this kind was exhibited by E.Wigner (Z. Physik, 131 (1952), p . 101). He showed t h a t if there are additive conservation laws for E, then A cannot be measured unless it commutes with the conserved quantity; and further, t h a t for approximate measurements of A with arbitrary accuracy to be possible, the measuring apparatus should be large enough t o admit states which are superpositions of sufficiently m a n y states with different quantum numbers of t h e conserved quantity. Actually for Wigner, A was the z-composnent of the spin of a spin i particle, the conserved quantity being the z-component of the angular momentum. These results were proved in full generality by H . Araki and M.Yanase (Phys. R e v . , 120 (1960), pp. 622-626). They proved t h a t if Lx (resp. L2) is a bounded self-adjoint operator on J4? (resp. ^ f ' ) , and if r > 0, vectors (£n)n^i t h a t are orthonormal in ^f', and a self-adjoint operator E on J f can all be found such t h a t U = exp( — irE) satisfies the von Neumann criteria for measurement of A for some initial state £ in 3tf\ then for U to commute with L = LX®1 + 1 ® £ 2 *t is necessary t h a t A commutes with Lv Their work also showed t h a t when Jf7 and &" are finite dimensional and the cardinality of the spectrum of L2 is allowed to be sufficiently large, then approximate measurements of A (in a certain sense) can be made with any desired accuracy. This raises the question, when A, g, and J*f' are fixed, of explicitly calculating the lower bound of the accuracy of approximate measurements of A in terms of the variance of L2 in the state £. F o r spin measurements of spin i particles (when the conserved quantity is the z- component of the angular momentum), M. Yanase (Phys. Rev., 123 (1961), p p . 666-668) has investigated this question.

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6. Quantum Probability Theory. For a given logic j£? the notions of probability measures on ££ and observables associated to ££ define a noncommutative generalization of conventional probability theory. For standard logics one can develop such a generalization in great depth: see for instance the articles in Quantum Probability and Applications to the Quantum Theory of Irreversible Processes, Springer Lecture Notes in Mathematics, 1055 (1984), edited by L. Accardi, L. Frigerio and V. Gorini. One of the concepts that is needed for such a theory of Quantum Probability, and which goes beyond what we have treated in Chapters III and IV is that of conditional probability. Let J f be a complex separable Hilbert space and ££ = <££{3tf). Suppose £> is a probability measure on j£? and U the associated von Neumann operator. We identify elements of S£ with the orthogonal projections corresponding to them. If E is a projection and p(E) > 0, the conditional probability measure given that E has occurred is defined as the probability measure p (. \E) given by (42)

P(F\E) =

tr(EUEF)/tr(EUE).

It is not difficult to show that p(. \E) is the unique probability measure on j£f with the following property: to any F^E, it assigns the probability p(F)/p(E). Clearly p(. \E) is defined by the von Neumann operator V where (43)

(tr(UE))-1-EUE.

V=

If F is a projection commuting with E we have (44)

p(F\E)=p(EF)/p(E),

which is exactly as in the conventional probability theory; however, for arbitrary F we should use (42). Ifp=p(p where ye Jt and \\q>\\ = 1, we have (45)

P(-\E)=P+,

*P=\\E
Let E = E1 + E2+... be an orthogonal sum of projections EXiE2i... with p(En) > 0 for all n. If the projection F commutes with all the En, we have (46)

p(F\E) = 2

(p(F\En)p(En)/p(E)).

n

However, for general F, (46) is no longer true. If p =p(p (

(«, «m .? (g)>i*.>+j, f « K « . The first term in the right side of this formula is the same as that in the right side of (46). The additional terms in the right side of (47) may be viewed as arising out of the "interference effects " that are typical in quantum theory

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when F does not commute with the En. For a very interesting discussion of the well-known two-slit experiments of Quantum Mechanics from the point of view of conditional probability see Chapter 26 of the book by E. Beltrametti and G. Cassinelli (cf. Notes on Chapter III). We refer to the same place for a treatment of conditional probabilities relative to a Boolean sub<7-algebra of «£?, and, more generally, relative to a sublogic of ££. 7. Measures on the projections of a von Neumann algebra: Generalizations of Gleason's theorem. Gleason's theorem may be viewed as a complete description of all countably additive probability measures on the standard logic J^(^f) when dim(Jf) ^ 3. It is natural to ask whether similar results can be proved for probability measures on the logic of projections of general von Neumann algebras. This question was first raised by G. W. Mackey (Amer. Math. Monthly, 64 (1957), pp. 45-57), and it appears that it has now been settled completely. Let ^ b e a complex separable Hilbert space and s/ a von Neumann algebra of bounded operators on £F. We w r i t e r p for the logic of projections in s/ and s/sa for the real vector space of all self-adjoint elements of s/. By a finitely additive probability measure ons/ p we mean a map \x: si p -> [0,1] such that /u(0)=0, /x(l) = l, fi(P + Q)=ti(P)+fji(Q) for P, Qesip with PQ = 0. Given such a JJL, one can associate to it the corresponding expectation functional Eu:jtfsa->R r

in the following way: if A es/8a and PA(E->PE) is the spectral measure of A on the a-algebra £% of Borel subsets of R, then (48)

E„(A) = r

\dv(\),

J — 00

where v is the finitely additive probability measure E-*n(PE) on £8. I t is easy to see that the correspondence

is bijective from the set of all finitely additive probability measures to the set of all functional

ons/p

E:jtfsa->R with the following properties: (49)

(i) E is linear (over R) on any abelian subalgebra of s/sa; (ii) E(A2) > 0 for any A e
Such functionals are called physical states of s/. The concept of a physical state is in principle weaker than that of a state ofs/ which demands linearity

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in place of (i) while retaining (ii) and (iii). If <$/ = B(34?), Gleason's theorem implies that for any countably additive /z, E ^ is a state of B(J^). The problem of generalizing Gleason's theorem on s/p may then be split into two parts: (1) to determine for which s/ all physical states are states; (2) if a state of s# is normal, i.e., if it is countably additive on mutually orthogonal projections, to obtain representations of it in terms of the usual trace ons/ (whenever such a trace exists). The first problem focuses directly on the issues arising out of the noncommutativity of the algebra of observables, and was recognized from the beginning to have crucial significance from the foundational point of view. Since the linearity of physical states is no longer true when dim(^f) = 2 it is clear that we must assume that jrf has no central direct summand of type I 2 . Recent work seems to have now shown that for such an stf, all its physical states are actually states. This result, certainly one of the most fundamental and profound ones in the subject, is the culmination of the efforts and contributions of many people spread over several years. In addition to the famous paper of Gleason the following (partial) list of articles may be cited: J. F. Aaarnes, Trans. Amer. Math. Soc. 149, (1970), pp. 601-625; J. Gunson, Ann. Inst. H. Poincare, A 17 (1972), pp. 295-311; A.A.Lodkin, Funk. Anal. Priloz., 8 (1974), pp. 54-58; M.S.Matvieicuk, Funk. Anal. Priloz., 15 (1981), pp. 41-53, and Teor. Mat. Fiz., 48 (1981), pp. 261-265; E.Christensen, Comm. Math. Phys. 86 (1982), pp. 529-538; F. J. Yeadon, Bull. London Math. Soc, 15 (1983), pp. 139-145, and Bull. London Math. Soc, 16 (1984), pp. 145-150; and A. Paszkiewicz,.preprm£, to appear in J. Funct. Anal. For a survey of these and other relevant articles and a discussion of the ideas used in the proof we refer to the article of P. Kruszynski: Extensions of Gleason's theorem, in Quantum Probability and Applications to the Quantum Theory of Irreversible Processes, Springer Lecture Notes in Mathematics, 1055 (1984), pp. 210-227, edited by L. Accardi, A. Frigerio and V. Gorini. The second problem is really part of the study of normal states, and may be viewed as a type of Radon-Nikodym theorem. Such theorems go back to H. A. Dye (Trans. Amer. Math. Soc, 72 (1952), pp. 243-280), and the reader should consult the literature on von Neumann algebras for definitive formulations of such results.

CHAPTER V MEASURE THEORY ON G-SPACES 1. BOREL SPACES AND BOREL MAPS F r o m this point onwards we shall emphasize some of the more sophisticated and specialized aspects of q u a n t u m theory. We shall deal only with complex separable Hilbert spaces, referring to them as Hilbert spaces without a n y qualification. We pointed out in Chapter I I I the role played by representations of physical symmetry groups into the groups Aut(j£?) and A u t ( ^ ) , where =£? is the logic and £f is the state space of some q u a n t u m mechanical system. If we assume t h a t <£ is standard, then the results of Chapter I V describe rather completely the structure of 3? and Sf. I t would t h u s appear quite feasible to study the representations of the groups which are important from the physical point of view. Such a study would shed considerable light on various aspects of q u a n t u m theory. Now, t h e methods used in the theory of representations of groups are very sophisticated and depend heavily on analysis on homogeneous spaces. The object of this chapter is to present the basic mathematical theory of homogeneous spaces and the function spaces associated with them. The main reference to the theory of measure and integration on groups a n d homogeneous spaces is the book of A. Weil [1]. This however deals mostly with compact and abelian groups. The general theory which is presented in the following sections is to a very large extent the work of Mackey [2], [3], [4], [5], [6], [7]. We shall begin with the concept of a Borel space. A Borel structure on a set X is simply a o--algebra & of subsets of X\ t h e pair (X,$) will be referred to as a Borel space. Elements of & will be referred t o as t h e Borel subsets of X. We shall use this terminology only when there is no ambiguity about t h e CT-algebra involved. By the usual abuse of language we shall refer to X itself as a Borel space. If (X,3S) and (Y f€) are Borel spaces, and / i s a m a p of X into Y, / i s said to be Borel i f / _ 1 ( ^ ) ^ J*; if/ is one-one, maps X onto Y, and i f / _ 1 ( ^ ) = ^ , we shall say t h a t / is a Borel isomorphism, and speak of X and Y as being isomorphic. When X=Y and / is an isomorphism we shall call / an automorphism. 148

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The theory of Borel spaces is a very extensive one. We shall constantly use the main results of this theory in our work. For this, our references are to Halmos [1], Kuratowski [1], [2]. Articles of Mackey [5] and Blackwell [1] are also very close to our general point of view. I n the next few paragraphs we shall summarize what is essential for our purposes; proofs of these and other related results may be found in the sources mentioned above; cf. also Varadarajan [3], [4]. A Borel space (X,36) is said to be separable if (i) {x} e 36 for all x e X and (ii) there is a countable set @^3# such t h a t 36 is the smallest oalgebra of subsets of X containing Q)\ Q) is then said to be a generating class. A class Q)<^36 is said to be separating if, for each pair of points of X, there exists a set A EQ) which contains one b u t not both of them. I t is useful to note t h a t in any Borel space (X,38), if 3) generates 36 and {x} e 3$ for all x E X, then Ql is separating. I n fact, if this is not true, there will be a pair of points x, y e X {x^y) such t h a t for each A e 3, either {x,y}^A or {x,y} n A= 0. The set of all such A is a a-algebra containing 3d and hence every set in 36 has the above property. But this is impossible as {x} does not have the property. Let X be a topological space. The smallest Borel structure containing all the open subsets of X is called the natural Borel structure of X. When X is a metric space, the natural Borel structure coincides with the smallest Borel structure with respect to which all continuous functions on X are Borel. Suppose Xa (a E J) are Borel spaces and X any set. For each a e J let 7Ta be a m a p of X into Xa. Then there exists a unique smallest Borel structure on X with respect to which all the maps 7ra are Borel. If Y is a Borel space and / a mapping of Y into X, f is Borel if and only if 7ra o / is Borel for each a. If X is the product space of the Xa and ira the projection of X on Xa, X, equipped with t h e above mentioned Borel structure, will be called the product of the Borel spaces Xa. If each Xa is a topological space and its Borel structure is the natural one, then it is not in general true t h a t the product Borel structure coincides with the natural Borel structure associated with the product topology on X. This is so, however, if J is countable and each Xa has a second countable metrizable topology. If (X,36) is a Borel space and Y is a subset of X, then the class 3#Y defined by aY

= {BnY:

BeO}

is a Borel structure for Y. (7,36 Y ) is said to be the Borel subspace defined by Y. The natural injection of Y into X is Borel. Note t h a t Y itself need not be a Borel set in X. If X is a topological space with its natural Borel structure, then 36Y coincides with the natural Borel structure of Y when Y is considered as a topological space with the relative topology.

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An important class of Borel spaces may be singled out in the following way. Let T be a complete separable metric space and X a Borel subset of T. Then the Borel subspace defined by X is said to be standard. The motivation for this terminology rests on the remarkable theorem t h a t two standard Borel spaces are isomorphic if and only if they have the same cardinal number and t h a t a standard Borel space is finite, countable, or has the power of the continuum. Standard Borel spaces are separable. If (X,&) is a standard Borel space, (Y f€) is separable, and if/ is any one-one Borel m a p of X into Y, then (i) the range Yf of / is a Borel set in Y, (ii) the Borel subspace defined by Yf is standard, and (iii) / is a Borel isomorphism of X with Yf. If (X^&i) (i = l,2,- • •) is a sequence of standard Borel spaces, then their product is standard. A Borel subspace of a standard Borel space is standard if and only if it is defined by a Borel set. Any uncountable standard Borel space is isomorphic to the unit interval. If X is any locally compact Hausdorff space satisfying the second axiom of countability, the Borel space obtained by equipping X with its natural Borel structure is standard. A measure on a Borel space is any nonnegative set function which is countably additive on its Borel structure; +00 is an allowed value. A measure p on (X,&8) is finite if p(X) < 00; is a-finite if X = Un -^n> where each Xn is a Borel set and for each n, p{Xn) < 00. p is said to be standard if there exists a Borel set X 0 such t h a t (i) p(X — X0) = 0 and (ii) the Borel subspace defined b y X0 is standard. Let X be a locally compact Hausdorff space satisfying the second axiom of countability. By a Borel measure on X we shall understand a measure with the property t h a t the measure of a n y compact set is finite. Any Borel measure is cr-finite. Let CC(X) be the linear space of all complex valued continuous functions on X with compact support and let \L be any Borel measure on X. Then

£:/->£/^

(feCc(X))

is a linear functional on CC(X) such t h a t fl(f)>0 whenever / is real and > 0 . Conversely, if A(/—>• A(/)) is a linear functional on CC(X) such t h a t A(/) > 0 w h e n e v e r / i s real and > 0 , there exists a unique Borel measure /x on X such t h a t A = /x. This is the well known Riesz theorem (cf. Halmos [1], pp. 216-249). Any Borel measure on X is regular, i.e., for any Borel set E^X, (1) fi(E)= sup p(C). C compact

Given a Borel space (X,&) and a a-finite measure p on it, we call a set A^X p-measurable if there exist Borel sets Bx and B2 such t h a t B±^A^B2 and p(B2 — B1) = 0. When there is no ambiguity about the measure t h a t is involved, we speak simply of measurable sets. The

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collection of ^-measurable sets is a cr-algebra and hence defines a Borel structure on X, say &p, called the p-completion of &. The measure p has a unique extension to &v as a measure. The functions which are Borel with respect to the Borel structure &v are called ^-measurable. If / is any ^-measurable function on X, there exists a Borel function g such t h a t f=g almost everywhere, i.e., {x : f(x) y^g(x)} e &v and has ^-measure zero. Let (X,&) be a Borel space, let q be any cr-finite measure, and let p be a measure. We shall say t h a t p is absolutely continuous with respect to q, p«q in symbols, if p(E) = 0 for any Borel set E for which q(E) = 0. If / is a nonnegative Borel function, the function p \ E —> J fdq is a cr-finite measure which is « g ; p is finite if and only if/ is an element of ^?1(q). Conversely, if p is a cr-finite measure « g , there exists a nonnegative Borel function / on X such t h a t p(E)=\fdq for all Borel sets E. f is essentially uniquely determined by p in the sense t h a t if / ' is a Borel function and p(E)—\ fdq for all Borel sets E, then / = / ' g-almost everywhere. / is called a Radon-Nikodym derivative of p with respect to q and is denoted by dpjdq. If p, q, r are cr-finite measures, then p<^q and q«r implies p « r and the relation dp\dr = dpjdq. dqjdr holds r-almost everywhere. I n particular, if p<^q and q<£p, p and q have the same null sets and dp\dq-dq\dp = \ almost everywhere; both the derivatives are then positive almost everywhere. Let p be any cr-finite measure on X, and let us choose disjoint Borel sets X1,X2, • • • such t h a t X = U n Xn and p(Xn) 0 be such t h a t 2n cnP(^n)<°°' Then, the measure q defined by

q(E) =2CnP(EnXn)

{E*&)

n

is finite, and p and q are mutually absolutely continuous. The relation of mutual absolute continuity is obviously an equivalence relation in the set of all cr-finite measures on X. The corresponding equivalence classes are called measure classes on X. If £ is a Borel automorphism of X and p is a cr-finite measure on X, we define the cr-finite measure pl by (2)

p\E)

= p(t-\E))

(Ee&)

p is said to be invariant if pl=p; p is said to be quasi invariant if pl and p are mutually absolutely continuous. Clearly p is quasi invariant if and only if the class of null sets of p is invariant under t. A measure class is said to be t-invariant if it contains a cr-finite measure quasi invariant with respect to t. We shall now prove two lemmas of a technical nature. They are of importance in the sequel.

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Lemma 5.1 (Federer-Morse [1]). Let X and Y be two mectric spaces and let f be a map of X onto Y. Suppose that there exists a sequence {Kn} of compact subsets of X such that (i) K1^K2^ • • •, (ii) X = ^Jn Kn, and (iii) for each n, the restriction of f to Kn is a continuous map of Kn into Y. Then there exists a Borel set E^X such that (i) / maps E onto Y, (ii) for any yeY, E meets / _1 ({?/}) in exactly one point, and (iii) Enf-Hf[Kn])
{m(y):yeY}

and E = g[F]. B y its definition, F meets each set h'1^}) exactly once. Hence h is one-one and maps F onto Y. From this it follows immediately t h a t / is one-one on E and maps E onto Y. I n other words, E meets each set f'Hty}) ( y e Y) exactly once. I t remains to prove t h a t E is a Borel set. For any integer n>\, An<^C be defined by

let

An = {c : c e C, h(d) ^ h(c) for a n y d eC with d < c — l/n}. We claim t h a t An is open in C. Suppose this is not true. Then there exists a sequence cs in C such t h a t cs $ An for all s = 1,2, • • •, but cs - > c e ^4n as 5 - > oo. Since cs $ An there exists a c / < cs — (l/n) in (7 such t h a t h(cs') — h(cs). If s± < s2 < • • • is a sequence of integers such t h a t cSk' -> c' as k - > oo, then c' e C , c r < c —(1/TI), and A(c') = A(c), a contradiction. ^4n is therefore open. We show next t h a t F = (~}n An. If c e F, it follows from the definition of m(y) t h a t for any c'
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the right of c. Hence there exists a d > 0 such t h a t all points of T are at a distance >d away from c. If (l/n) 1, (3)

En = {Dn-Dn

n (f-VLKn-i)))}

u #„-!•

We claim t h a t the 2£n's have all the desired properties. Clearly E1s^E2— We claim t h a t for each n, / is one-one on En and / [ ^ n ] = = / [ J ^ n ] - This is proved by induction on n. Since E1 = D1, this is true for n = \. Suppose now t h a t n > 1 and t h a t / is one-one on En _ x with f[En _1] =f[Kn _ -J. Then the formula (3) implies easily t h a t / is one-one on En and maps En onto f[Kn]. Let E = {JnEn. Then i£ is a Borel set, / is one-one on E, and f[E]=Y. Since En^Kn, it is obvious t h a t i£ n / - ^ / ^ ] ) ^ ^ for all n. The lemma is completely proved. Corollary 5.2. Let X and Y be standard Borel spaces and f a Borel map of X into Y. Let p be a finite measure on X and q the measure E -> p(f ~ 1(E)) defined on Y. Then, the range f[X] of f is q-measurable and its complement has q-measure zero. Moreover, there exist Borel sets A and Z such that (i) i c l , Z^f[X], (ii) q(Y— Z) = 0, and (hi) A is a section for f over Z, i.e., f is one-one on A and maps A onto Z. Proof. The corollary is trivial if X is countable. Hence we consider the case when X is uncountable. We m a y assume t h a t X and Y are both identical with the Borel space associated with the unit interval [0,1]. B y a theorem of Lusin (Halmos [1], p. 243) we can find compact sets Kns^X such t h a t ( a j ^ c ^ c . . . , (b) p(X — {Jn Kn) = 0, and (c) / i s continuous on each Kn. Let Z = {Jn f[Kn]. Each f[Kn] is compact and hence Z is a Borel set. Since \JnKn<^f~1{Z), p{X-f~1(Z)) = 0 so t h a t q(Y-Z) = 0. This already shows t h a t / [ X ] is g-measurable and q(Y — f[X]) = 0. We can now apply the Federer-Morse lemma to [JnKn and construct a Borel set A^X such t h a t / is one-one on A and maps A onto Z. The proof of the corollary is complete. Remark. I t is actually true (though it is harder to prove) t h a t the r a n g e / [ X ] is a universally measurable set; i.e., for any finite measure a on Y, f[X] is an a-measurable set. Corollary 5.2 is valid even in this case,

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with q replaced by a. We shall not prove these results but refer the reader to the paper of von Neumann [2] and Kuratowski [1], [2]. We refer to this stronger version of corollary 5.2 as von Neumann's cross-section lemma. Suppose X and Y are Borel spaces. Let v be a measure on Y, and for each x e X let us be given a ^-equivalence class of Borel functions on Y. The problem we now want to examine is whether we can select for each x a representative f(x,.) in the corresponding class such that the function x, y ~^f(x,y) is Borel on Xx Y. We need two auxiliary lemmas. Lemma 5.3. Let X be a Borel space, 8 a separable metric space, and f a Borel map of X into 8. Then there exists a sequence {fn} of Borel maps of X into 8 such that (i) each fn takes only countably many values, and (ii) fn(x) ->/(#) uniformly for x e X (as n -> oo). Proof. Fix the integer n> 1. Since 8 is a separable metric space, we can cover 8 by a countable family {Bm} of balls of radius 1/2%. Let the sets C m be defined by C1 = B1 and Cm = Bm — {Jj<m B} (m>l). Then the Cm are disjoint and ^ = l J m C r Since Cm^Bm, diameter of Cm< \\n for all m. For each m such that Cm is nonempty, let sm be some point of Cm. We define fn as the map

/»(*) = *« if

xef'HCJ.

Then/ n is Borel, has countably many values at most, and dist(/ n (#), f(x)) < \\n for all xe X. Lemma 5.4. Let X be a Borel space and 8 a separable Banach space. Let K be a subset of S*, the dual of 8, with the property that the linear combinations of elements of K are dense in 8*. Iff is a map from X to 8 such that for each he K, the function x -> k(f(x)) is Borel on X, then f is a Borel map of X into 8. Proof. Let ^ * be the smallest Borel structure on 8 with the property that all the elements of S* are Borel functions with respect to it. From our assumption about K it follows easily that for each A e&*, f~1(A) is a Borel subset of X. To prove that / is Borel, it is enough to prove that &* coincides with the natural Borel structure & of 8. Obviously, &*^&. Since £% is generated by the open sets, it suffices to prove that any open set belongs to «^*. Now, as 8 is separable, every open set is a union of countably many balls. Thus it is enough to prove that «^* contains all balls. Further, as x -> ex (c e C) and x -> x + a (a e 8) are easily seen to be automorphisms of the Borel space (8,&*), we are reduced to proving that the unit ball B = {x : \\x\\ < 1} lies in ^ * . Now, it is well known that under the weak *-topology for 8* ( = the smallest topology for 8* such that k -> lc(x) is a continuous function on S* for each xeS), the unit ball B* of

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S* is a compact metric space (Dunford-Schwartz [1]), hence separable. Let { ^ ^ ^ ^ - - J b e a countable dense subset of t h e unit ball of S* in the weak *-topology. Then (by the Hahn-Banach theorem), B = {x : \x*(x)\ < 1 for all x* e B*} = 0 {x

:

\xn*(*)\

< 1 for all n).

n

This shows t h a t 5 e J * and completes t h e proof t h a t &* = &. This proves t h a t / is a Borel m a p . Now we are in a position to formulate a n d prove t h e lemma on the selection of representatives. Lemma 5.5. Let X and Y be Borel spaces, with Y separable. Let /x and v be a-finite measures on X and Y, respectively. Let f be a complex valued function on Xx Y and let Y be a union of Borel sets Y{ of finite v-measure such that (i) for each x e X, y ->f(x,y) is a Borel function on Y which is v-integrable on each Yt, (ii) for any i and any Borel set F^Y{, x -> J", f(x,y)dv(y) is a Borel function of x. Then, there exists a complex Borel function f* on XxY and a Borel set N^X of fx-measure zero such that for each xeX — N,

W

f*(*,y) = f(*,y)

for v-almost all y. Proof. We m a y clearly drop down on one of t h e Yt so t h a t we m a y assume t h a t v{ Y) < oo. Also since fi intervenes only in terms of its null sets, we m a y replace /x b y a finite measure mutually absolutely continuous with respect to JJL; hence we assume t h a t /x is also finite. Write S = J£>1(v), t h e Banach space of (equivalence classes of) v-integrable Borel functions on Y. 8 is a separable Banach space since the Borel structure of Y is countably generated. Since y->f(x,y) is a v-integrable Borel function on Y, it defines an element, s&y f(x), of 8. We claim t h a t x-+f(x) is a m a p of X into 8 which satisfies t h e conditions of lemma 5.4. Now the dual of ££x{v) is J£™{v) and every bounded Borel function on Y is a uniform limit of linear combinations of characteristic functions XF of Borel subsets F of Y. XF defines t h e linear functional tF : u-> [ u(y)dv{y) on ^(y). B y hypothesis, x -> tF(f(x)) is Borel for all F. We m a y therefore conclude from lemma 5.4 t h a t x->f(x) is a Borel m a p from X to 8. So X is the disjoint union of countably many Borel sets on each of which / is bounded. We m a y assume therefore t h a t / itself is bounded. By lemma 5.3 there exists a sequence {/„} of Borel maps of X into S such t h a t (i) each fn is bounded and takes only countably many values, a n d (ii)/ n (:r) ->f{x) uniformly in XEX. Since fn is bounded and takes

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only countably m a n y values, it is obvious t h a t there exists a /x x vintegrable Borel function fn* on I x Y such t h a t for each xeX, the function y->fn*(%,y) defines the element fn{x) of S. We then have:

sup xeX

\fn*(x>y)-fm*(z>y)\My)-+o JY

as n, m —> oo and hence

J Jl/,*(*.y)-/»*(*.y)|% x ")(*,») -* o XxY

as n,m-> oo. As ^(fxxv) is complete, there is a Borel function / * on X x Y such t h a t / * is (p x y)-integrable a n d

JJ|/»*(*,y)-/*(*.y)|^xv)(a;,y)-»«0 Xxy

as n->co. Select a subsequence {/nfc*} such t h a t fnk*(x,y) -+f*(%,y) for (//, x y)-almost all x, y. Then there exists a Borel set N of ^-measure zero such t h a t for each x e X — N, fn*(x,y) ->f*(%,y) for v-almost all y e Y. I t is clear t h a t for each x e X — N, f*(x,y) =f(x,y) for v-almost all y. This proves the lemma. Corollary 5.6. Let X, Y, p, and v satisfy the same restrictions as in lemma 5.5. Suppose that Q(x->qx) is a map from X into the space of all finite measures on Y such that (i) qx«.v for all x e X, and (ii) for each Borel set Fs= Y', x -> qx{F) is a Borel function on X. Then, there exists a Borel function f* on XxY and a Borel set N^X of fx-measure zero such that for each x e X — N, y ->f*(x,y) is a version of dqx\dv. Proof. For each x e X select a Borel function fx on Y such t h a t fx = dqxjdv, and define / on 1 x 7 by f(x,y)=fx(y). The corollary follows from the lemma at once.

2. LOCALLY COMPACT G R O U P S . HAAR M E A S U R E The groups which are commonly encountered in physical problems, such as the Lorentz group or the rotation group, are topological groups (cf. Pontrjagin [1] for the theory of topological groups). Moreover t h e underlying topological spaces are locally compact and satisfy the second axiom of countability. Because of this and other technical reasons, we shall restrict ourselves in the sequel to locally compact groups satisfying the second axiom of countability (lcsc). The Borel structures associated with the underlying topologies of these groups are standard. We shall

MEASURE

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157

describe in this section a few known facts about lcsc groups and H a a r measures. A detailed t r e a t m e n t can be found in Halmos [1] (pp. 266-289). Suppose t h a t G is a group which is at the same time a Borel space. We shall say t h a t G is a Borel group if the map x, y - > xy ~1 of G x G into G is Borel; G is called a separable (standard, etc.) Borel group if the underlying Borel structure is separable (standard, etc.). I n this book we shall deal only with separable Borel groups. If G is a separable Borel group, the maps x - > x'1, x —> xh, and x-^hx (h e G) are Borel automorphisms of the Borel space G. Any lcsc group is a standard Borel group. Suppose G is any lcsc group. I t is a classical fact t h a t there exists a nonzero cr-finite measure \L on G such t h a t fx is left invariant, i.e., p(E) = JJL(XE) for all Borel sets E and all x e G. /x is essentially unique in the sense t h a t if /x' is another cr-finite left invariant measure, then \x! = C-(JL for some constant c>0. /x is called a left Haar measure on G. /x( £7) > 0 for every open set U and fx is a Borel measure. Similarly there exists a nonzero Borel measure /xr which is right invariant, i.e., fjLr(E) = /jLr(Ex) for all Borel sets E and x e G. Again, the right invariance determines /xr u p to multiplication b y a positive constant. /xr is called a right Haar measure on G. I n general, it is not true t h a t a right H a a r measure is left invariant. When this is so we shall say t h a t G is a unimodular group. Suppose G is any lcsc group and t h a t /x and /xr are left and right H a a r measures on G. For a n y x e G, the measure E - > fi(Ex) is also a left H a a r measure and hence there exists a constant A(x) > 0 such t h a t (5)

^Ex)

=

A(xME)

for all E. Clearly A does not depend on which left H a a r measure we use in (5). The function A(x - > A(#)) is known to be a continuous homomorphism of G into the multiplicative group of positive real numbers, i.e., (i) (6)

A(e) = 1

(e the identity of G),

(ii)

A(xy) = A(x)A(y)

(iii)

A is continuous.

(x,y e G),

(i) and (ii) are easy consequences of (5); the third needs a little work (cf. Loomis [2], pp. 117-120). A is called the modular function on G. If/ is a continuous function vanishing outside a compact set of G, we have, from (5), (7)

f f(zx)dfi(z)

= A(a)-1 f

JG

f(zW(z).

JG

From this it follows t h a t (8)

f f(z)A(z)-^(z) JG

= f JG

f(zx)A(z)-^(z).

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The equation (8) shows that the measure /xr defined by (9)

^(E) = f A(2" i)dM(z)

is a right Haar measure. It follows from (9) that \xr and \i are mutually absolutely continuous; the Radon-Nikodym derivatives d^jd^ and d[L\d[Lr are, respectively, the functions z -> A(2) -1 and 2 -> A(z). A simple calculation using (9) shows that (10)

^(stf) =

A(x)~^r(E)

for all Borel sets E and all ^ e G . It is clear from (9) that G is unimodular if and only if A is the trivial homomorphism, i.e., A(z) = 1 for all ze G. Since x -> log A(#) (# e G) is a continuous homomorphism of G into the additive group of real numbers which does not have any nontrivial compact subgroups, it follows that A is trivial on any compact subgroup of G. Compact groups are therefore unimodular; in this case /x and /xr are finite. It is customary, when G is compact, to reserve the term Haar measure to the unique left and right invariant measure giving measure 1 to the whole of G. If G is not compact, fji and jjir are not finite. Abelian groups are trivially unimodular. Since A is a homomorphism, A(z) = l whenever z is of the form xyx~xy~x for x, y G G. Let G' be the smallest closed subgroup of G containing all elements of the form xyx~xy~x (x, y e G); A is then identically 1 on G'. Hence a sufficient condition for G to be unimodular is that G = G'. There exist groups which are not unimodular. Suppose that G is the group of all matrices g,

ly

x

\0

l)

\

(x, y real, y > 0).

We identify g with the upper half-plane by the mapping g —> (x,y). If

^E)=jyhdxdy'

^E)=\\\dxdy*

E

E

then /x and \xr are, respectively, left and right Haar measures on G.

3. ^-SPACES Suppose that X is a Borel space and G is a separable Borel group. We shall say that X is a G-space if for each g e G, there exists a Borel automorphism tg(x -> g • x) of X such that (i) te is the identity, (11) **i*a = **Aa

fai'

£2 e 0).

MEASURE

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G-SPACES

159

X is said t o be a Borel G-space if the map g,x->g-xofGxX into X is Borel. We shall say t h a t X is a standard Borel G-space if X is a Borel G-space and if the Borel structure of X is standard. For any G-space X and any x e X, the set (12)

Gx = {g:geO,

g-x = x)

is a subgroup of G. I t is called the stability subgroup a t x. The set (13)

G-x = {g-x: g e G}

is called the orbit of x. If y = h-x (he G), then Gy = h-Gx-h~1 as is easily verified. Thus the stability subgroups of the points on the orbit of x are conjugate to Gx and any subgroup conjugate to Gx is the stability subgroup of some point on the orbit of x. If X is a Borel G-space and Y is a subset of X, we shall say t h a t Y is G-invariant if Gx^Y for all x e Y; in this case, t h e Borel subspace defined by Y becomes in a natural and obvious fashion, a Borel G-space. I t is called the Borel G-subspace defined by Y. If Xx and X 2 are two Borel G-spaces, and / a Borel m a p of Xx into X 2 , then / is called a G-homomorphism if for all x±e Xl9 and all g e G, (14)

f(9-x1)

=

g-f(x1).

A G-isomorphism is a G-homomorphism / with the additional property t h a t / i s a Borel isomorphism of X1 onto X 2 . If there exists a G-isomorphism of X x onto X 2 , we shall call X x and X 2 isomorphic Borel G-spaces. Suppose t h a t X is a locally compact Hausdorff space satisfying t h e second axiom of countability and, further, t h a t G is a lcsc group. Let X be a G-space with the additional property t h a t the map g,x—>g-x of G x X into X is continuous. Then X is obviously a Borel G-space. We shall refer to it as a locally compact G-space. G is said to act continuously on X. I n this case, for each g e G, x -> g-x is & homeomorphism of X onto itself. Since G is a countable union of compact sets, G-x is also a countable union of compact sets for any x e X. Every orbit in a locally compact G-space is thus a Borel set. The stability subgroups are, moreover, closed subgroups of G. A m e a s u r e p o n a Borel G-space X is said to be invariant ifp(g- E) =p(E) for all Borel sets E^X and all g e G. I n m a n y problems it is necessary t o work with measures which are only quasi invariant. A measure p on X is said to be quasi invariant if p(E) = 0 if and only if p(g-E) = 0 for g G G, i.e., if the action of G on X transforms j9-null sets into ^-null sets. Notice t h a t if p is quasi invariant a,ndf1,f2 are two functions on X which are equal almost everywhere, their transforms f±9 and f29 also have this property for a n y g e G, where ft9(x) =fi{g~1-x) (x e X, £ = 1,2). For a n y measure p on X and any g e G, let us define the measure p9 b y (15)

p'(E)=p(g-1-E).

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9

p is quasi invariant if and only if p and p are mutually absolutely continuous for all g e G. Therefore, if p is quasi invariant, so is any measure mutually absolutely continuous with respect to p. If X is a locally compact (r-space and p is a Borel measure, then so is p9. A simple example of a (r-space (0 any lcsc group) is obtained when we take X — G and define, for x e G and geG, g-x = Lg(x) = gx. Obviously G — Ge and the stability subgroups are all trivial. If yu is a left H a a r measure, /x is invariant. If /xr is a right H a a r measure, \xr is not in general invariant; but it is quasi invariant since it is mutually absolutely continuous with respect to fx (cf. (9)). A Borel (r-space X is said to be transitive or homogeneous if X = G- x for some x e X. I n this case, X — G-y for each y e X. The above example is a simple example of a transitive (r-space. If X is a standard Borel (r-space which is transitive and if the stability groups associated with the points of X are all trivial, then we shall say t h a t X is an affine space associated with G. If we assume G is standard and choose a point x e X, the m a p g -> g • x is a Borel isomorphism of G onto X which is a (r-isomorphism of the (r-space G (G acting on itself by left translations) on the (r-space X. B u t this isomorphism depends on the choice of x. We shall conclude this section with a theorem due to Varadarajan [2] which imbeds a standard Borel G-space in a compact metric (r-space, G being any lcsc group. Theorem 5.7. Let G be a lcsc group and X be any standard Borel G-space. Then there exists a compact metric G-space Y, and an invariant Borel set E^Y such that X is G-isomorphic to the Borel G-subspace of Y defined byE. Proof. Let /x be a n y left H a a r measure on G. Denote by 3?1 the Banach space of (equivalence classes of) complex Borel functions f on G such t h a t (16)

\\f\l, = J \f(x)\dp(z)

< oo.

UhsQ and fh(x)=f(h~1x) for x e G, fh e J^ 1 and | | / 1 i = |l/lli- I t is h well known t h a t f,h^f is a continuous map from J^f1 x G into J^f1 (Loomis [2], p . 118). Let !F be the complex vector space of all complex valued bounded Borel functions on X, and for u e !F let (17)

Halloo = sup

\u(x)\.

xeX

Since the m a p g,x->g~1-x of GxX into X is Borel, the function g,x->u(g~1-x) is Borel for each ue^. If feJ?1, the function x - > u 1,x jG f(9) (9~ )dv<(g) is Borel on X; we shall denote it b y / * u. I t is obvious t h a t / * ue^F and | | / * ^||oo< \\fWx' |M|a>. u-*f* u is a linear m a p of J ^ into itself.

MEASURE

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161

For any nonempty set ^ c ^ w e write Y[Q)) for the set of all complex valued maps c{f,u->c(f,u)) on J ^ x i ^ such t h a t (i) for each ue@, / - > c(/,w) is linear on JS?1, and (ii) \c(f,u)\ < | | / | | r \\u\\«, for a l l / , u e &1 x 2. We equip Y(@) with the smallest topology with respect to which all the maps c->c(f,u)(fE^?1,ue^,ceY(2)) are continuous. From the defining properties (i) and (ii) it is obvious t h a t Y(@) is a compact Hausdorff space. If Q) is countable, it is clear t h a t Y(@) is even a compact metric space. We shall now convert Y(@) into a 6r-space. We define, for c e Y{Q)) and g e G, the element g • c by (18)

g-c(f,u)

c(f^1\u),

=

where, as usual, fh(x)=f(h-1-x) {x,heG). Since | | / f f _ 1 | | i = \\f\\lf it is clear t h a t g-ce Y(@), and a simple calculation shows t h a t Y(Q)) is a (r-space. We now claim t h a t the m a p g, c-> g-c of G x F(i^) into F ( S ) is continuous. I n view of the definition of the topology on Y(@) this reduces to proving t h a t for each (f,u) e ££x x 2, the map g, c - > c(/ g - 1 ,w) from ^ x 7 ( S ) into the complex numbers is continuous. However, an easy calculation shows t h a t the inequality

\c(f°,u)-c'(f\u)\

< l^p^-c'ip^l

+

Wp-f^WuW^

is valid for all c, c' e Y{Q)) and g,he G, and the right side tends to 0 when h->g and c' - > c. G thus acts continuously on Y(Q)). We shall now imbed X in F ( ^ ) for a suitably chosen 3i. Let AX,A2, • • • be a sequence of subsets of X which generate the Borel structure of X, and let un be the function which is 1 on An and 0 on X — An. Let 2 = {ul3u2,--'} and let Y—Y{2). Since 2 is countable, F is a compact metric space. For any x e X let us define cx by (19)

c x (/,») = ( / * « ) ( * )

( / e J2?1, « e # ) .

Obviously cxe Y. x -> cx is thus a map, say £, of X into F . Since the m a p # — > ( / * tt)(a;) is Borel for all fixed / , w, it follows t h a t f is a Borel m a p of X into F . Further, if <7 e G, we have: <W(./» = = =

(f*n)(g-x) jj(h)u((h-i.g).x)dn(h) jGf(gt)u(t-^x)dfM(t)

= (9'<>)(f,u). This proves t h a t £ is a Borel 6r-homomorphism from the 6r-space X into the 6r-space F . Let E be the range of £. 2£ is clearly invariant.

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We assert t h a t £ is one-one. Suppose in fact t h a t x, y are two points of X such t h a t (/ * u)(x) = (f * u)(y) for all / , u e &1 x Q). Then, for each ue@, u(g~l •x) = u(g~1 -y) for almost all geG. As Q) is countable, there exists a Borel null set N^G such t h a t for any g e G — N, u(g~1-x) = u(g~1-y) for all ueQ). Let <70 be any point of G — N, and let # 0 — #o~ 1 *x> y0 = g0~1 -y. As the Borel space X is separable and as Al9 A2i- • • is a generating family, it is also a separating family (cf. Section 1). Hence, u(x0) = u(y0) for all u e Q) implies t h a t x0 = y0. Therefore x = y. Therefore f is a one-one Borel m a p of X into Y. Since X and Y are standard, this implies t h a t the range E of f is a Borel subset of Y and t h a t | is a Borel isomorphism of the Borel space X onto the Borel subspace of Y defined by E. Since we have already proved t h a t E is an invariant subset of Y and t h a t £ is a (r-homomorphism, the proof of theorem 5.7 is complete. Corollary 5.8. Let Gbea Icsc group and let X be a standard Borel G-space. Then, for any x e X, the stability subgroup Gx (of G) at x is closed. Moreover, the orbit G-xof any x in X is a Borel subset of X. I n fact, it is enough, in view of theorem 5.7, to prove these facts when X is an invariant Borel subset of a compact metric space on which G acts continuously. I n this case, both assertions are obvious. We shall end this section with two technical results which describe the manner in which the quasi-invariant measures on a Borel 6r-space transform under the action of G. For any cr-finite measure p on X we define P9 by (15). Lemma 5.9. Let X be any Borel G-space, G, a Icsc group. Let p be a a-finite quasi-invariant measure on X and let fp(g,.) be a version of the Badon-Nikodym derivative dpjdp{g~1S). Then for fixed glfg2e G, (20)

fp(9ig2,x)

= fp(9i,g2 • x)fp{g2tx)

for p-almost all x. Proof. We begin the proof with a formula which describes how a Radon-Nikodym derivative transforms under a Borel automorphism. Let /x, v be a-finite measures on X such t h a t IJ,«V. Let t(x->t-x) be a Borel automorphism of X. Define the measures yf and vl by setting fjLt(E) = fM(t-1(E)) and vt(E) = v(t~1(E)), for all Borel sets E. Then / x i «v t . If E is any Borel set, li\E)

= f Jt 1(E) = f JE

(dfildv)(x)dv(x)

(dfjildv)(t-1'x)dvt(x),

MEASURE

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163

showing t h a t (21)

drf/diS = (dfji/dvY.

This said, we come to the actual proof. Fix gl9 g2 e G. Then p{929i)~ (p i) 2 and dpldp(929i)=:(dpldp92)-(dpldp9i)92 from what we said above. This implies easily t h a t 9 9

fP(9i92>x) =fp(9i>92'V)fp{92>v) for ^-almost all x. Our next result tells us t h a t we can choose these Radon-Nikodym derivatives so they are " a l m o s t " Borel on GxX. Theorem 5.10. Let G be a Icsc group, \x a left Hoar measure on G, and let X be a separable Borel G-space. Let p be a a-finite quasi-invariant measure on X. Then, there exists a Borel function Fp on GxX which is positive everywhere such that for ^-almost all g e G, x -> Fp(g,x) is a version of the Radon-Nikodym derivative dp/dp^'^. In particular, the identity (22)

Fp(gig2,x)

=

Fp(gi,g2-x)Fp(g2,x)

is satisfied for JUXJLIX p-almost all (gi,g2,x) e GxGx

X.

Proof. We s t a r t with the special case when p is finite. We first claim t h a t for each Borel set E^X, g~^p(g-E) is Borel. I n fact, the map t : g, x—>g, g-x is a Borel automorphism of GxX and hence A = t[GxE] is a Borel subset of GxX. Let ^ be a left H a a r measure on G. Then we know t h a t for each g e G, the set Ag = {x : (g,x) e A} is a Borel subset of X and g - > p(A9) is a Borel function on G. Since Ag = gE, our claim is proved. We are now in a position to apply corollary 5.6. I t follows from this corollary t h a t there exists a Borel function Fx on GxX such t h a t for almost all g e G, x-> F^g^x) is a version of dp{9~^\dp. Since p is quasi invariant, for each g, F-^g.x) is j9-almost everywhere positive and hence, by the Fubini theorem, F1 is positive for almost all points of GxX. We m a y thus assume t h a t F1 is positive everywhere without altering the conclusions. P u t Fp = \jF1. Then F is a positive Borel function on GxX and for /x-almost all g,x-^ Fp(g,x) is a version of dp/dp(9~1}. Finally, the identities (22) follow from (20). This completes the proof of the theorem when p is finite. If p were a-finite, we choose a finite measure q such t h a t q<&p and p«q. Let a(x -> a(x)) be a positive Borel function which is a version of the Radon-Nikodym derivative dp/dq. Choose Fq as a Borel function on GxX corresponding to q, and define (23)

Fp(g,x)

=

a{x)-Fq(g,x)-a-1{g^x).

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The identity dpjdp™ =

dpldq-dqldq™.(dqldp)™

implies t h a t Fp has all the required properties.

4. T R A N S I T I V E ^-SPACES We shall now examine the transitive G-spaces somewhat more closely. Let G be a lcsc group. Let GQ be a closed subgroup of G. Let X = GjGQ, the space of all left cosets gG0, g e G. Let ft be the canonical mapping of G onto X defined by (24)

p-.g-^gGo,

and for any element g of G and a; = hG0 of X let <7 • x be defined by (25)

g-x =

ghG0.

Clearly, g-x depends only on g and x and not on the element h used to define x. G t h u s acts on X and this action is transitive. Let X be equipped with the quotient topology, t h a t is, [ / c l is open if and only if ^~1{U) is open in G. A m a p c(x - > c(x)) of X into 6? is called a ItoreZ (continuous) section if it is a Borel (continuous) m a p and if ft(c(x)) — x for all x e X; t h e n its range meets each left G0 coset exactly once. I t is said to be regular if for each compact subset K of G, c[X] n /? _1 (j8[ir|) is a set with compact closure. Theorem 5.11. X, under the quotient topology, is a locally compact Hausdorff space satisfying the second axiom of countability and ft is an open continuous map. Moreover, X is a G-space and the map g,x->g-x of G x X into X is continuous. If K is any compact subset of X, there exists a compact subset Kx of G such that ft[K1] = K. Further, there exists a regular Borel section for G/G0. A map f of X into some Borel space Z is Borel if and only if the map f°=f o ft of G to Z is Borel. In particular, if E^X, E is a Borel set if and only if the set ft~1(E) is a Borel subset of G. Finally, let Y be a transitive standard Borel G-space, y e Y and let G0 be the stability subgroup at y. Then the map t : ft(g) —> g-y is a Borel isomorphism of X = GjGQ onto Y which is a G-isomorphism of the associated G-spaces. If Y is a locally compact G-space satisfying the second axiom of countability, t is a homeomorphism. Proof. That X is locally compact Hausdorff second countable and t h a t ft is open are well known (cf. Pontrjagin [1]). Now, G is clearly transitive on X. The m a p (g,gr) - > {g,fi(g')) of G x G onto G x X is continuous and it is easily shown t h a t the topology for GxX is the quotient topology relative to this m a p . The m a p g, x - > g-x is therefore continuous.

MEASURE

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165

Let Wlt W2i- - • be a sequence of compact sets in G with nonempty interiors such t h a t ( J n (inter ior (W n)) = G. If then K is a compact set c j , then for some n,

K £ U /TO, and hence # = £ [ 7 ^ ] , where K^p-^K) n (\JJ=1 Wj). Clearly Z x is compact. We shall next prove the existence of a regular Borel section for G/G0. Let {Kn} be a sequence of compact subsets of G such t h a t (a) K1^K2^ •• • and U n ^ « = ft and (b) any compact subset of G is contained in some Kn. Using lemma 5.1 we can construct a Borel set E such t h a t (i) i£ n /S _1 (j8[iL n ])cir n for all n, and (ii) ft is one-one on i£ and maps 2£ onto X . ft is thus a Borel isomorphism of i£ onto X as E and X are both standard. If c denotes the inverse map, c is a Borel isomorphism of X onto E. c is a Borel section. I t is obvious from (i) and (b) above t h a t c is regular. Suppose now the set A c X is such t h a t 0~ 1(A) is a Borel set in 6?. Then E n P~1(A) is also Borel. Since j8 is one-one on E, and since the Borel subspace defined by E is standard, A = fi[E n j8 _1 (J[)] is a Borel subset of X. If / is a Borel m a p of X into Z, where Z is some Borel space, then / ° = / © ($ is a Borel m a p of G into Z. Conversely, let / ° be a Borel m a p of G into Z such t h a t f° =f o p for some m a p / of X into Z. If F is any Borel subset of Z, p-1{f-1{F))=f°-1{F) is Borel in 0 and hence / ^ ( - P ) is Borel in X . / is thus a Borel map. We finally come to the uniqueness of transitive G-spaces. Let 7 , y be as in the statement of the last assertion of theorem 5.11. Clearly the map t:P(g)^9'V is one-one, maps X onto Y, and is moreover such t h a t t(g • x) = g • t(x) for all xeX and g e G. If M^Y is open, t~1{M)=p[A], where A = 1 {9 '-g-ytM} so t h a t t' (M) is open, t is thus continuous. A classical category argument (Pontrjagin [1]) now implies t h a t t is a homeomorphism. Suppose we assume in the above proof t h a t Y is only a standard Borel G-space. We then use corollary 5.8 to conclude t h a t Gy is a closed subgroup of G. Thus t is well defined and the same argument as t h a t given above now shows t h a t t is Borel. Since both X and Y are standard, t is a Borel isomorphism. This completes the proof of the theorem. Since G acts on X , we m a y speak of invariant measure classes on X . We shall now proceed to study these. We use a "lifting" technique to reduce the problem to G itself. Now G acts on itself by left and right translations. We have accordingly left and right invariant measure classes. We write /x for a left H a a r measure and define fxr by (26) fjir is a right H a a r measure.

^(E)

= /x^-1).

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Lemma 5.12. There exists a unique measure class on G which is left (or right) invariant, and it is the measure class determined by fx. In particular, it is both left and right invariant. Proof. Let a be a finite right quasi-invariant measure. We want to prove t h a t a(E) = 0 if and only if /JL(E) = 0. Let / be a Borel function on G such t h a t (i) 0 < f(x) < 1 for all xeG, (ii)

f(x)dfi(x)+ JG

f(x)dfir(x)

< oo.

JG

Such / are easily constructed. Suppose now for a Borel set A, a(A) = 0. Since a is right quasi invariant, a(Ax) = 0 for all x in G and hence a(Ax~1)f(x)dfji(x)

= 0,

/JG. i.e., writing XA f ° r the characteristic function of A, L ( 1 XA(yx)da(y)^f{x)d^x)

= 0.

By Fubini's theorem, this means

J (J XA(yx)fWdA*))My) = o and hence, as the inner integral is > 0 , XA(yx)f(x)Mx)

= o

JG

for a-almost all y. Thus, for some y0 e G, JG

XA(y
= 0

a n d hence, as f(x) > 0 for all x, XA(VOX) =

0

1

for /x-almost all a;, i.e., fM(y0' A) = 0 so t h a t /x(^4) = 0. Conversely, if fi(A) = 0, /x( y ~ XA) = 0 for each y and hence

J XA(yx)f{xW(x) = 0 for each i/, proving t h a t

Using Fubini's theorem, we get:

J.

a(Ax-1)f(x)dfx(x)

= 0

ME AS U BE THEORY

ON

G-SPACES

167

and hence, as the integrand is > 0 , a{AxQ-x)

= 0

for some x0, showing, as a is right quasi invariant, t h a t <x(A) = 0. Thus /xr, a, ft are all in the same measure class. If a is left quasi invariant, the measure a defined by a(A) = a(A~1) is right quasi invariant so t h a t /x, a, jjir are in the same measure class. This proves the lemma. Remark. Lemma 5.12 enables us to speak without ambiguity of null sets in G; a Borel set A c= 0 will be called null if it is a set of measure zero for the unique measure class singled out by lemma 5.12. We now return to the general transitive case. Let G0 be a closed subgroup of G and X = G/G0. We now observe t h a t , on X, quasi-invariant measures can be constructed easily. Let aQ be a finite measure on G which is quasi invariant, and let a be the measure on X defined by a(A) = aoip-^A)). Then a(E) = 0 if and only if p~1{E) is a null set in G. a is thus quasi invariant. Our aim is to show t h a t all quasi-invariant measures can be obtained in this manner, up to absolute continuity. Let CC(G) denote the class of all complex functions on G which are continuous and have compact support. Let CC(X) denote the analogously defined space of functions on X. For any / in CC(G) let us write Mf for the function on G defined by (27)

(Mf)(g)

= f

f(gh)dfi0(h);

J G0

here /x0 is a left H a a r measure on the lcsc group G0 chosen in some way. I t is then clear from the left invariance of/x 0 t h a t , for any h0 in G0, (28)

(Mf)(gh0)

=

(Mf)(g)

for all g e G. Hence there exists a function / ~ on X such t h a t (29)

f~W(9))

= (Mf)(g)

(geO).

We shall use the mapping / -> / ~ to lift measures from X to G. Lemma 5.13. Let P be the set of all real nonnegative elements of CC(X) and P' a subset of P having the following properties: (i) if fl9f2 e P' and c1? c2 are constants > 0 , c 1 / 1 + c 2 / 2 e P', (h) if f' £ P' and fx e P, fife P', and (iii) for any x e X, there is an f e P' with f(x)>0. Then P' = P . Proof. Let xeX. Then there exists an fxeP' such t h a t fx(x) = \. Therefore, for some open set Ux with compact closure and containing x, fx(y) ^ i f ° r a U V E Ux. Let fx be an element of P such t h a t f^y) =fx(y)~x E an< for all y e Ux. Then fx'=fifx P' ^ has the property t h a t fx'(y) — ^ for all y e Ux. Suppose now t h a t K is any compact subset of X. For each

168

GEOMETRY

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THEORY

x e K, let Ux be an open set containing x and fx an element of P ' such that fx = 1 on Ux. Choose points xl9 • • •, xk e K such that K^[Jf=1 UXj. Thenf'=f'Xl+• • +f'Xk belongs to P ' and f'{y)>\ for all y e K. Choose an f2 e P such that f2(y)=f'(y)~1 f° r a u y E %• Then f"=f2f belongs to P' and f"{y) = 1 for all y e K. Finally, let / e P and let K be a compact set such that f(y) = 0 for y $ K. Choose a n / " e P' such that f"(y) = 1 for all y e K. Then / = / / " e P'. This proves the lemma. We are now ready to formulate the main properties of the map / - * / - •

Lemma 5.14. For any feCc(G), f~ eCc(X). The map f->f~ has the following properties: (a) / > 0 implies f~ > 0 ; (b) / - » / ~ maps CC(G) onto CC(X) and maps the set of real nonnegative elements of CC(G) onto the set of real nonnegative elements of CC(X); (c) for any continuous function u on X, {u°f)~=uf~ (where u° = u op) (d)ifaeG, (fa)~ = (f~)a. Proof. We first prove that for a n y / in CC(G), / ~ eCc(X). Clearly if/ vanishes outside a compact set K^G,f~ vanishes outside f$[K]. Therefore there remains only the continuity of/". For this it suffices to prove that Mf is continuous on G since then the continuit}' of/" would follow from the fact that X has the quotient topology. Let g0 e G and W0 be a compact neighborhood of e. Now

\(Mf)(g)-(Mf)(g0)\ < f

\f(gh)-f(g0h)\d^0(h).

J G0

Since feCc(G), given e>0, there exists a compact neighborhood W^ W o around e such that \f(a)— f(a')\ <e whenever aa'~1eW. If/ vanishes outside the compact K, and gg0~x e W,

\{Mf){g)-{Mf)(g0)\ <

J

|/(^)-/(^)|^oW

(fifO -1 W-lK)nG 0

< e

dfM0(h), K'nG0

where K' = (g0 ~1W0 " XK) n G0. This proves that Mf is continuous at gr05 finishing the proof t h a t / " e CC(X). The map / - > / ~ is linear and it is obvious that / > 0 =>/~ >0. Next, let 16 be a continuous function on X and u° = u o p. Then

Jf(«°/)(0) = «•>((/) f f(gh)dp0(h). J G0

This shows that M(u°f) = u°Mf. Hence (u°f)~=uf~. Third, let aeff. Then (Mfa){g) = (Mf)(a~1g) from which we conclude that (fa)~=(f~)a. This proves that (a), (c), -and (d) are satisfied. It only remains to prove (b).

MEASURE

THEORY

169

ON G-SPACES

Let P' ={f~

:feCc{G),

/ r e a l and > 0 } ,

and let P be the set of all real nonnegative elements of CC(X). We claim t h a t P' satisfies the conditions of the lemma 5.13. (i) is obvious and (ii) follows from (c). To check t h a t (iii) is satisfied at each point of X, it is enough to do so at x0 = /3(e) because G acts transitively on X and P' satisfies (d). But then, if feCc(G)>0 and is 1 in a neighborhood of e, (Mf)(e) =f~(x0) > 0 obviously. This shows now t h a t P' = P . I n particular, CC(G) is mapped onto CC(X). The lemma is completely proved. For any Borel measure a on X, the map / - > [ f~da

is, by virtue of

lemma 5.14, a nonnegative linear functional on CC(G). Let us denote by a0 the Borel measure defined on G by this linear functional. We thus have a mapping a -> a0 from the set of Borel measures on X into the set of Borel measures on G. Since / - > / ~ has the whole of CC(X) for its range, it is obvious t h a t ax° = a2° implies a± = a 2 . a -> a0 is therefore a one-one map. Let us now consider the equation (30)

f /d«o

f f~da

=

JG

(feCc(G)).

JX

Our next aim is to extend (30) to certain Borel functions on G. Lemma 5.15, Let f be a Borel function on G such that

(31)

f

1/(^)1^0(4)

<

00

J Go G 0 '" ^

for each g e G. Then there exists a unique Borel function f~ on X such that for all g e G, f~(P(9))

= f

f(gh)dno(h)-

G00 JJ G

If f> 0, then f ~ > 0 <md we have (32)

f /rfa° == [ JG

f~da

Jx

in the sense that either both sides are infinite or both sides are finite and equal. Iff is bounded and vanishes outside a compact subset of G, thenf~ is bounded and vanishes outside a compact subset of X. Proof. Since g,h^f(gh)

is Borel on GxG0,

it is clear t h a t g ->

X

\G / ( ^ ) ^ / O ( ^ ) is Borel on G. Moreover, this function of g is constant on the left 6r 0 -cosets. Hence, by theorem 5.11, the existence and uniqueness of / ~ are immediate. A l s o , / > 0 i m p l i e s / ^ > 0 . Suppose /(g) = 0 for all g $ K,

170

GEOMETRY

OF QUANTUM

THEORY

K being a compact set, and let \f(g)\
= I f I < A

f(9h)di*o(h)

JG0

\ Jgg

<

dfji0(h)

1x

KnG KnG0

ApoiK-WnGo).

This proves t h a t / ~ is bounded. We now come to the proof of (32). We consider first the case when f=XK> the characteristic function of the compact set K^G. We can choose a sequence {/n} of nonnegative elements in CC(G) such t h a t fn j XK point wise on G. Then / n ~ \ f~ point wise on X and

f fda° = lim f fnda° JG

n^co JG

= lim

fn~da

n-*co Jx

= f da

L~ '

This proves (32) in this case. Next, for any Borel set A^K, J=XA is bounded and has compact support and hence %A~ *S bounded and has compact support. I t is therefore a-integrable and the m a p A - > j XA~dcc is easily verified to be a measure on the Borel sets of K. By what we have seen earlier, this measure coincides with a 0 on all compact A^K. Therefore it must coincide with a0 for all Borel sets A. But then this proves (32) for f=XA (A^K). At this stage we know t h a t (32) is true for all simple functions / vanishing outside compact sets. For the general case, l e t / > 0 and let {/n} be a sequence of nonnegative simple functions with compact supports such t h a t fn f / pointwise on G. T h e n / n ~ f / ~ pointwise on X. Hence by the Fatou-Lebesgue lemma, f fda* = lim f

JG

n->oo j

fnda°

G

= lim

fn~da

n->oo J x

L

f~d, a.

This completes the proof of the lemma. Corollary 5.16. For a ° ( / S - V ) ) = 0.

a Borel

set A^X,

a(^4) = 0, if

and

only

if

MEASURE

THEORY

171

ON G-SPACES

Proof. Suppose t h a t A is a Borel set c j and a(A) = 0. If K is a compact set ^P~1(A), then, XK~ = 0 outside ^4 and so, by (32),

«°(#) = j XK~d* = 0. 0

This proves, by regularity of a , t h a t a°(p~1(A)) = 0. Conversely, let a0(j3~1(A)) = 0. Let K^A be compact. Since ^~1(K) is closed, there exists an increasing sequence {Kn} of compact subsets of ^~1(K) whose union is ^~1(K). Then a°(Kn) = 0 for all n, so t h a t we can infer from (32) and the nonnegativity of XKU ~ t h a t X#n ~ = 0 almost everywhere for each n. Suppose now t h a t a(K) > 0. Then, there will exist an x1 e K such t h a t XKn~(x1) = 0 for all n. Choose a g± e ^~1(K) such t h a t fi(gi)=x1. Then /xo((^ri ~ 1Kn) n #o) = = 0 f ° r a U ^. This is absurd as (gx ~ 1Kn) n 6r0 f G0. Thus a(K) must vanish. By regularity of a, a(A) = 0. Corollary 5.17. / / a± and a2 are Borel measures on X, <x2<£cc1 if and only if a 2 ° « a 1 0 . In this case, (33)

dcc^/da^

Proof. Let a 2 « a i nonnegative ueCc(G), 8.15 and, moreover,

an

(da^dcc^0.

=

d ^ t = da2/da1. We m a y assume £ > 0 . For any the function ut° satisfies the conditions of lemma

(ut°)~ 1

1

=u~t.

0

Since u~t e J * ? ^ ) , ut° e J ^ ^ ) and we have: uPda^

=

JG

Jx

=

u~tda1 u~da2

=

uda2°. JG

This shows at once t h a t a 2 ° « a i ° a n d dcc2°lda1° = t0. On the other hand, if a 2 ° « a 1 ° , we obtain the relation a 2 « a 1 at once from the previous corollary. Corollary 5.18. / / G is compact, then for any Borel set A^X, cfi(p-*(A)).

a(A) =

Proof. I n this case, on normalizing fx and /x0 as usual, we see t h a t for any ueCc(X), corollary.

{uQ)~ =u.

Hence,

\ uda=\

u°da°.

This implies

the

172

GEOMETRY

OF QUANTUM

THEORY

From lemmas 5.14 and 5.15 the following theorem follows immediately. I t is a basic theorem in the theory of quasi-invariant measures on X. Theorem 5.19. There are quasi-invariant Borel measures on X = GjG0, where G is a Icsc group and G0 a closed subgroup. If A is a finite quasiinvariant measure on G, and A~ is the finite measure defined on X by the equation A~(^4) = X(^~1(A)), then A~ is quasi invariant. Any two quasiinvariant a-finite measures on X are mutually absolutely continuous. If E^X is any Borel set, the quasi-invariant a-finite measures on X vanish for E if and only if p~1(E) is a null set of G. If a is any Borel measure on X, a is quasi invariant (invariant) if and only if a0 is a quasi-invariant (left invariant) measure on G. Proof. If g E G and if a is a Borel measure on X , (a9)0 = (a0)9 as is easily seen from (d) of lemma 5.14. Moreover, any cr-finite measure is mutually absolutely continuous with respect to some finite measure. Lemmas 5.14 and 5.15 and corollaries 5.16 and 5.17 then lead quickly to the proofs of all the assertions of this theorem. Theorem 5.19 enables one to speak of null sets of X and G without specifying any quasi-invariant measure. If A is any quasi-invariant measure on X, the class of A-measurable sets is independent of A. I n other words, we m a y also speak of measurable sets and measurable functions on X without ambiguity. Corollary 5.20. Let A^X. Then A is measurable if and only if p~1(A) is measurable. If f is a map of X into some Borel space Z, f is measurable if and only iff0 =f o £ is a measurable map of G into Z. Proof. I t is enough to prove the first p a r t since the second p a r t follows immediately from the first. Suppose now t h a t A^X is measurable. Then there are Borel sets A1 and A2 such t h a t A±^A^A2 and A2 — A1 is a null set. Clearly p-1(A1)^p-1(A)^^1(A2) and p~1(A2)-p~1(A1) = ft~1(A2 — A1) is also null, ft'1 (A) is thus measurable. Conversely, let B = j3~1(A) be measurable. Let A be a finite quasiinvariant measure on G and let A~ be the induced measure on X. There exist sequences {Kn} and {Kn'} of compact subsets of G such t h a t (i) Kn^B and Kn'^G-B for all n, and (ii) X(B-\JnKn) = X{(0-B)U n Kn') = 0. Since j8 is continuous, the sets A1 and A2, where

A = U KKn] n n

are both Borel. I t is obvious t h a t AX^A^A2 A is t h u s measurable.

and A~(^42 — ^41) = 0.

MEASURE

THEORY

ON G-SPACES

173

Corollary 5.21. G acts ergodically on X, i.e., if f is a Borel map of X into some Borel space Z such that f(g-x)=f{x) for almost all (g,x) e GxX, then f is a constant almost everywhere. In particular, there exists, up to a constant multiplying factor, at most one invariant a-finite measure on X. Proof. By the Fubini theorem, there exists a point x± of X such that f(9'xi)=f(xi) f° r almost all g. We may clearly assume that x1 = x0 = f$(e). Select a Borel set A^G such that A is the union of countably many compact sets, G — A is of Haar measure zero, and f(g-x0)=f(x0) for geA. Let X0 = A-x0. Then X0 is a Borel set and ^~1(X — X0)^G-A is null, showing that X — X0 is null. Let a± and <x2 be nonzero invariant afinite measures. Then they are quasi invariant. Let f=da1jda2. Then, for each g e G, f(g-x)=f(x) for almost all x. f is thus a constant almost everywhere, proving that a± is a constant multiple of a2. Corollary 5.22. Let 8 be a standard Borel space and let f be a Borel map of G into 8. Suppose that for each h e G0 f(gh) =f(g) for almost all g. Then there exists a Borel map a of X into 8 such that f(g) = a{fi{g)) for almost all g. Proof. We may clearly assume that 8 is the unit interval. Let A be a quasi-invariant measure on G with A(6r) = l. Let A0 be a quasi-invariant measure on G0 with A0(6r0) = 1. Let A~ be the measure on X induced by A through the map /?. By the Fubini theorem, f(gh)=f(g) for almost all (g,h) GGXG0. Hence, for some Borel set N of G of measure zero, we can assert that for each g e G — N, f(gh) =f(g) for almost all h e G0. If we write fi(9) = f

f(9h)dXo(h)9

J G0

t h e n / x is Borel, and for geG — N, f1(g)=f{g)> Moreover, for any fixed g e G — N and h' e G0, the functions h ->f(gh'h) and h ^f(gh) are both equal to f(g) for almost all h. Consequently, f1{gh')=fx(g) for all {g,h')e{G-N)xGQ. Consider now the set X'=p[Q-N]. X' is a A~ measurable subset of X by corollary 5.2, and hence ^(X') is a A-measurable set which is a union of left 6r0-cosets, and, for all (g,h) e p~1(Xf) x G0, f1(gh)=f1(g). Moreover, X — X' is null and so G — ^~1(X') is also null. Define now the function f2 on G by

{M)

M9) =

\o

to^-W).

f2 is A-measurable and f2(g)=f(g) almost everywhere. Further, f2 is constant on the left 6r0-cosets, and therefore there exists, by corollary 5.20, a A~-measurable function/ 3 on X such t h a t / 2 = / 3 0 . Choose a Borel function a and X such that a(x)=f3(x) for almost all x. Then f(g) = a(f$(g)) for almost all g. This completes the proof.

174

GEOMETRY

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THEORY

5. COCYCLES A N D COHOMOLOGY The functional equations (22) are extremely significant in the theory of induced representations associated with 6r-spaces. We shall devote t h e present section to a study of these equations in an appropriately general context. Let X be a standard Borel 6r-space, G being a fixed lcsc group. We choose an invariant measure class ^ o n l . Let M be a standard Borel group, fixed throughout this section. We write 1 for the identity of M. Let / be a function on GxX with values in M. We shall say t h a t / is a (G,X,M)cocycle relative to ^ if the following properties are satisfied: (i) (35)

/ i s a Borel m a p of GxX

(ii)

f(e,x)

(iii)

= 1

for almost all

f(gi92>z) = f(gi,92'X)f{92>z)

into M, xel, for

almost all (9n92>x) E

GxGxX.

If there is no ambiguity concerning the group M, we shall refer to / simply as a (G,X)-cocycle relative to *$ or, even more briefly, as a cocycle relative to <€. Note t h a t the null set in (ii) is with respect to ^ while t h e null sets in (iii) are those of the measure class which is a product of t h e usual measure classes in G with %\ A c o c y c l e / is said to be a strict cocycle if it satisfies (35)(i) and further, (36) (ii') and (iii'), where (36)

(ii') (m')

f(e,x) = 1

J

for all

x e X,

f(9i92>x) = f(9i,92 • x)f(92,x)

for all

(91,92^)

eGxGxX.

Our decision to call the functions satisfying (35) and (36) cocycles is prompted by the fact t h a t these equations are generalizations of t h e identities which describe the cocycles in the cohomology theory of groups (cf. Eilenberg [1], pp. 3-37). We note here t h a t it was G. W. Mackey who first studied the cocycles in the context of an arbitrary transitive Borel G-space. I t was the detailed study and analysis of cocycles which enabled Mackey to formulate and prove the correct generalizations of the classical work of Frobenius on induced representations (of finite groups), to the case when the group in question was an arbitrary lcsc group. The aim of this section is to carry out a cohomology theory of (G,X,M)-cocycles. We begin by introducing equivalence relations in the set of all cocycles relative to a fixed invariant measure class # . Suppose t h a t f± and f2 are two cocycles relative to <€. We shall say t h a t fx and f2 are cohomologous, fi~f2 in symbols, if there exists a Borel m a p b(x -> b(x)) of X into M such t h a t

(37)

f2(g,x)

= %-a;)/i((7,a;)6(a;)

x

MEASURE

THEORY ON G-SPACES

175

for almost all (g,x) e GxX. B y taking b to be identically 1, we see t h a t if / i a n d / 2 are equal almost everywhere on Gx X, then fx~f2- I t is easy to verify t h a t ~ is a genuine equivalence relation in the set of all cocycles relative to %\ Each ~ equivalence class will be called a cohomology class relative to %>\ Suppose further t h a t f± and f2 are strict cocycles. We shall then say t h a t / x a n d / 2 are strictly cohomologous, fx xf2 in symbols, if there exists a Borel m a p b(x -> b(x)) of X into M such t h a t

for all (g,x) e GxX. # is also easily verified to be an equivalence relation in the set of all strict cocycles. Each #-equivalence class will be called a strict cohomology class. The m a p 1 : g,x -> 1 is a strict cocycle. Any cocycle cohomologous to it will be called a coboundary. If the cocycle is strictly cohomologous to 1, it will be called a strict coboundary. Obviously, a cocycle / is a coboundary (strict resp.) if and only if for some Borel function b on X with values in M, f(g,x) = b(g-x)b(x)~1 for almost all (all resp.) (g,x) eGxX. Suppose t h a t M is abelian. Then the set of Borel maps o f 6 r x I into M is an abelian group under pointwise multiplication; both the set of cocycles and the set of strict cocycles are subgroups of this group. The coboundaries form a subgroup of the group of cocycles and for two cocycles fi,f2,fi~f2 if and only if f±f2 ~x is a coboundary. Therefore the cohomology classes are in one-one correspondence with the elements of the quotient group of the group of cocycles mod the group of coboundaries. This quotient group is known as the (G,X,M)-cohomology group. From now on we shall assume that G is transitive on X. We choose a point x0 6 X arbitrarily and write G0 for the stability subgroup a t x0. G0 is closed, ft denotes the m a p g -> g • x0 of G onto X. /3 is a Borel map. For any m a p u of X into some space Z, u° denotes the m a p g - > u(P(g)) of G into Z. On the other hand, if v is a m a p of G into Z which is constant on the left 6r 0 -cosets, there exists a unique map u of X into Z such t h a t u° = v; we denote u by v~. Thus (v~)° = v and (u°)~ =u. We shall also extend this notation to two variables. Let Y, Z be Borel spaces and let u be a m a p of 7 x 1 into Z. We then write u° for the map y,9-+u(y£(g)) of Y x G into Z. We note t h a t u is Borel if and only if u° is Borel. At various points of our arguments in this section, we shall be concerned with functions on X, G, GxG, GxX, GxGxX, and so on. Since G and X possess unique invariant measure classes, there are canonically determined measure classes on each of the other spaces. For instance, if A and X are quasi-invariant a-finite measures on G and X, respectively, the measure Ax A' determines a measure class on GxX which depends only

176

GEOMETRY

OF QUANTUM

THEORY

on the measure classes of A and A'. By a (Borel) null subset of G x X we mean a set on which all the measures of this class vanish. Similar remarks apply to GxGxX, etc. In view of the uniqueness of the invariant measure classes, we are entitled to omit any reference to them and speak of cocycles and coboundaries. This we shall do. Suppose now t h a t / is a strict cocycle. Then it follows easily from (ii') and (iii') of (36) that h -^f(h,x) is a Borel homomorphism of the stability group Gx at x into M. We shall call it the homomorphism defined by fat the point x e X. If mx and m2 are two Borel homomorphisms of G0 into M, we shall say that mx and m2 are equivalent if there exists a k e M such that, for all heG0, (38)

m2(h) =

km^k-1.

The cocycles and their cohomology classes are difficult to handle because of the null sets involved. The strict cocycles and their classes are much better behaved in this respect since there are no null sets. Our main concern is to show that in every cohomology class of cocycles there is a strict cocycle and that the latter is determined uniquely up to strict cohomology. This means that the cohomology classes of cocycles are in canonical one-one correspondence with the strict cohomology classes of strict cocycles, which are easier to analyze. The simplest strict cocycles are the strict (G,G)-cocycles. Our method of examining strict (6r,X)-cocycles consists in "lifting" these to strict (6r,6r)-cocycles and analyzing the latter. Lemma 5.23. Let f be a strict (G,X)-cocycle. Then the function f° on GxG defined by

/ W ) = f(gMg')) is a strict (G,G)-cocycle. If F is any strict (G,G)-cocycle, there exists a unique Borel map b of G into M such that b(e) = l and (39)

F(g,g') = b(gg')b(gf)^

for all (g,gf) eGxG. Proof. The first assertion is trivial. For the second, we note that (36) gives F(g,g') = F(gg\e)F(g',e)-i for all (g,gf)eGxG. Equation (39) follows by putting b(g') = F(gf,e). If b± is another Borel map of G into M such that b1(e) = l and F(g,g') = biigg'Mg')'1, then HggT^igg'^Hg'r^g') for all (g9gf) EGXG. This 1 shows that for some keM, b(g)~ b1(g) = k for all g. Since b1(e) = b(e) = l, & = 1 and b = bx.

MEASURE

THEORY ON G-SPACES

177

In view of this lemma, with any strict (G,X) cocycle/, we can associate a Borel function on G. But f° is not the most general {G,G)~cocycle since it arises from a (G,X)-cocycle. This Borel function must therefore satisfy some special identities. The next lemma describes these identities. Lemma 5.24. Let m be a Borel homomorphism of G0 into M. Then there exists a Borel map b of G into M such that b(e) = 1 and (40)

b(gh) = b(g)m(h)

for all (g,h) eGxG0. Corresponding to any such map b, there is a unique strict (G,X)-cocycle f such that

f°(g,g') = Hgg'MgT1

(41)

for all (g,gf) e GxG; f defines m at x0. Conversely, if f is a strict (G,X)cocycle and b is the Borel map of G into M such that b(e) = 1 and b satisfies (41), then the restriction ofb to G0 coincides with the homomorphism m defined by f at x0 and b satisfies (40). Proof. Let m(h -> m(h)) be a Borel homomorphism of G0 into M. Let c(x -> c(x)) be a Borel section for G/G0. We may assume c so chosen that c(x0) = e. For any g e G, let a(g) = c(P(g))~1g. Since g and c(f$(g)) lie on the same left coset gGQ, a(g) e G0. We put (42)

b{g) = m(a(g)).

Since c and fi are Borel maps, g —> a(g) is a Borel map of G into G0 and hence g —> b(g) is a Borel map of G into M. Also b(e) = m(a(e)) = l. Moreover, for g e G and h e G0, a(gh) = a(g)h. This shows that b(gh) = b(g)m(h) for all (g,h) e GxG0. Let

* W ) = Hgg'W)-1. F is obviously a strict (G,G)-cocycle and (40) shows that F(g,g'h) = F(g,g') for all (g,gf,h) e GxGxG0. Hence, there exists a unique Borel map / of G x X into M such that

MP(g')) = * W ) for all (#, g') e Gx G. It is easily checked t h a t / i s a strict (6r,X)-cocycle, that f° = F, and that it is the only (6r,X)-cocycle with this property. Finally, for h e G0, f(h,x0) = F(h,e) = m(h). For the results in the converse direction, let / be a strict (G,X)-cocycle and let b be such that b(e) = 1 and (43) for all {g,g') eGxG.

f°(g,g') =

Hgg'MgT1

Then, for all (g,g\h) b(gg'h)b(g'h)-i =

eGxGxG0,

Hgg'MgV1,

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from which it follows easily that for each h e G0, g' -» 6(g,)"16(gr/^) is a constant, say m(h). Thus b(g'h) = b(g')m(h) for all (g,h) e GxG0. Since b(e) = l, b(h) = m(h), while (43) implies that b{h)=f(h,x0), for all h e G0. This proves everything. Lemma 5.25. Let ft be a strict (G,X)-cocycle and let mi be the homomorphism defined by ft at x0 (i = 1, 2). Then the following statements are equivalent: (a)

(44)

mY and w,2 are equivalent,

(b)

f^f2,

(c)

/i~/2-

Proof, (b) => (c) is trivial. We shall prove that (a) => (b) and (c) => (a). (a) => (b). Suppose that m1 and m2 are equivalent homomorphisms. Now there exists by lemma 5.24, a Boral map bt of G into M such that (i) mt is the restriction of bt to G0, (ii) b{{gh) — b^m^h) for all (g,h) e GxG0 and (iii) fi0{g,g') = bi(gg')bi(g')-1 for all (g,g')eGxG. Let k e M be such that (45)

km^fyk-1

m2(h) =

for all h eG0. From (45) we obtain the equation b2(gh)kb1(gh)-' =

bMkbM-1

for all (g,h) e GxG0. This shows that the Borel map g -> b2(g)kb1(g)~1 is constant on the left 6r0-cosets. Therefore there exists a Borel map b(x —> &(#)) of X into Jtf with the property that (46)

ba{g)^i(9)-1

= W ) ) = b°(g)

for all g e G. An easy computation using (46) and the relation of bt to / f now gives us the relation f2°(9,9') = from which we obtain

b°(gg')UO(g,g'mg')-\

f2(g,x) = b{g^x)f1{g,x)b{x)-1 for all (g,x) eGxX. (c) => (a). We assume / i ^ / 2 - Therefore there exists a Borel map b of X into M such that

f2(g,x) = Hg-rfMg^Mx)-1 for almost all (<7,#) eGxX. (47)

/2 W )

This implies that =

bOigg'Wfag'mg')"1

for almost all (g,gf) EGXG. Let fy be the Borel map of Ginto M such that (i) mt is the restriction of b{ to # 0 , (ii) fi°(g,g,) = bi(ggf)bi(g,)~1 for all

MEASURE

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(g,g')eGxG (lemma 5.24). We then obtain from (47), after an easy calculation, the equation (48)

^(gg'yWiggY'Ugg') = M ^ r w r 1 ^ ' )

for almost all (g,g')eGxG. This means (corollary 5.21 with X = G) that the function g ->b1(g)~1b°(g)~1b2(g) is almost everywhere equal to a constant, say k'1 e M. Thus (49)

b2(g) =

b^b^k-1

for almost all g e G. We now conclude from the connection between b{ and m{ that for each h e G0,

for almost all g. Thus (50)

m2(h) =

km^fyk-1.

This proves that (c) => (a) and completes the proof. Lemma 5.26. / / / is any (G,X)-cocycle, there exists a strict (G,X)-cocycle j 1 such that f(9>x) = fi(9>v) for almost all (g,x) s G x X; f1 is determined uniquely up to strict cohomology. Proof. L e t / b e a (6r,X)-cocycle and l e t / 0 be defined as usual, i.e., /W )

= f{0,P(9'))-

Then f°(gi92,93) = Z ^ i ^ s ) / 0 ^ , ^ ) for almost all (g1,g2>g3) eGxGxG.

Hence there exists a g0 e G such that

f°(gig2,go) = P(9i>9&o)fQ(92>9o) for almost all (g1,g2) £ GxG. If we put g=gi, g' = 9r20ro> w e that

Pigg'go-^goWg*-1^)-1

f°(9,g') = for almost all (g,gf) eGxG.

Let d0(9) =f°(99o~1>9o)>

Then d0 is a Borel map of G into M and

/ W ) = doigg'WoigV1

can

conclude

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f

for almost all (g,g ) e GxG. Changing d0(g) to d(g) = d0(g)d0(e) ~1, we secure d(e) = 1 and at the same time

(51)

/W)

= digg'W)-1

for almost all (g,g') e GxG. Fix now an arbitrary h e G0. Then it follows from (51) that d(gg'h)d(g'h)-i = d(gg')d(g')^ for almost all (g,gf) e GxG, i.e., d{gg')-1d{gg'h) = d(g')'H(g'h) for almost all (g,gf)eGxG. Corollary 5.21 (with X = G) implies now that the function g' -> d{g')~1d(g'h) is almost everywhere equal to a constant, say m(h). In other words, (52)

d(gh) = d(g)m(h)

for almost all g. Clearly m(h) is uniquely determined by the requirement that the condition (52) be satisfied for almost all g. It now follows from (52) and the uniqueness of m(h) that (i) m(e) = 1, and (ii) for hlf h2 e G0, m{h1h2) = rn(h1)m{li2). We assert that mis a Borel map of G into M. Since M is standard, we may assume, in this argument, that M is the unit interval. Let

As cp is Borel on G x G0, this shows that m is Borel on G0. In other words, m is a Borel homomorphism of G0 into M. By lemma 5.24 there exists a Borel map d' of G into M such that d'(e) = l and d'(gh) = d'(g)m(h) for all (g,h) eGxG0. From (52) it now follows that for each h e GQ, d(g)d'(g)-1

=

d(gh)d'(gh)-i

for almost all 0. By corollary 5.22 we can find a Borel map k of X into ilf such that d(g) = k(p(g))d'(g) for almost all g. Write d1(g) = k(^(g))d'(g). Then d1(gh) = d1(g)m(h) for aZZ (g,h) e GxG0 and d1(g) — d(g) for almost all 0. By lemma 5.24, there exists a unique strict (6r,Z)-cocycle / x such that

A W ) = dAggld^g')-1 for all (g,gf) e GxG. Since dx~d almost everywhere, the last equation implies that f° =fx° almost everywhere on GxG. Thus (53) for almost all (<7,#)

fi(g,*) = fig,*) EGXX.

MEASURE

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181

If f2 were another strict cocycle such t h a t (53) is satisfied with / 2 in place o f / j , then f±=f2 almost everywhere on Gx X. In particular, / i ~ / 2 . From lemma 5.25 we now conclude t h a t / ^ / g . This completes the proof. Theorem 5.27 (Mackey [2], [6]). Let G be a Icsc group, X a standard transitive Borel G-space, and M a standard Borel group, x0, an arbitrary point of X, and G0, the stability group at x0. If y is any cohomology class of (G,X,M)-cocycles, the set yst of all strict (G,X,M)-cocycles in y is nonempty and is a strict cohomology class. The set y~ of the homomorphisms of G0 defined by the elements of yst at x0 is an equivalence class of Borel homomorphisms of G0 into M. y - > y~ is a one-one correspondence of the set of all cohomology classes onto the set of all equivalence classes of Borel homomorphisms of G0 into M. If M is abelian, the (G,X,M)-cohomology group is isomorphic to the group of all Borel homomorphisms of G0 into M. Finally, if X is an affine G-space (i.e., G0 is trivial), all cocycles are coboundaries. Proof. The proofs of all the statements are contained in lemmas 5.23 through 5.27. Remark. The theorem just proved is the cornerstone of the theory of induced representations. I t reduces the problem of construction and classification of cocycles to the problem of constructing the equivalence classes of Borel homomorphisms of G0 into M. Given any Borel homomorphism of G0 into M, lemma 5.24 tells us the method of constructing the strict cocycles which define this homomorphism. This construction depends however on the choice of a Borel section for the cosets gGQ. In general, there are no canonical choices for these sections. We shall see later on how, in at least some special cases, these difficulties can be overcome. Remark. I n m a n y problems, M is already a topological group with a second countable topology. I t is easy to indicate a sharpened form of theorem 5.27 which can be formulated in this context. We shall say t h a t a map u of a topological space A into a topological space B is locally bounded if for any compact set K^A, u[K] has compact closure in B. A mild sharpening of theorem 5.27 is contained in lemma 5.29. To give this we need a preparatory lemma, which is classical. Lemma 5.28. Let G be a Icsc group, and M a topological group satisfying the second axiom of countability. Suppose that M is given its natural Borel structure and that m is a homomorphism of G into M which, as a map G into M, is measurable. Then m is continuous. Proof. Since we can replace M by the range of m, we shall assume t h a t m maps 0 onto M. Let U be an open subset of M containing 1 and W an open set such t h a t I e W and WW'1^!!. Since the open sets sW (s e M) cover M, there is a countable family {sx, s2,- • •, sn,• • •} such t h a t

182

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M = {JnsnW. Choose gneG such t h a t m{gn) — sn. If m~1(W) = A, then it is easily seen t h a t G = {Jn gnA. Let f t b e a left H a a r measure. By assumption, the sets gnA are all measurable and hence (JL(A) > 0. By a well known result, there exists an open set B such t h a t e e B^AA'1. Since W W'1^ U, it follows t h a t m[B]<^ U. This proves t h a t m is continuous at e. Since m is a homomorphism, m is continuous everywhere. Lemma 5.29, Let the notation be as in theorem 5.27. Suppose that there is a second countable topology on M which converts M into a topological group and whose associated Borel structure is the given one. Then, any cohomology class y contains a strict cocycle which is locally bounded as a map of GxX into M. Moreover, if there is a continuous section for GjG0, y contains a strict cocycle which is a continuous map of GxX into M. Proof. I t is enough to prove t h a t given a Borel homomorphism m of G0 into M, there are strict cocycles defining m at x0 and having the required properties. Now, ra, by lemma 5.28, is continuous. Let c(x->c(x)) be a regular Borel section for G/G0 (theorem 5.11) such t h a t c(x0) = e. For t e G, h(t) = c(P(t))'H. Then it follows from the regularity of c t h a t h is a locally bounded m a p from G into G0. If b(g) = m(h(g)) for all g e G, b is a locally bounded m a p of G into M. g, g' ~^b(gg')b(g')~1 defines a strict (6r,X)-cocycle/; / is clearly locally bounded and defines m at # 0 . If c is a continuous map of X into G, f is even continuous. Remark. The reader would have noticed t h a t the transitivity of the Borel G-space X has played a crucial role in the main theorems of this section. I t is a natural question to ask whether some of these results can be generalized to the case when X is not homogeneous. Suppose then t h a t X is an arbitrary standard Borel G-space and ^ an invariant measure class. The measure class will now have to play an explicit role since it is not unique unless X is transitive. With respect to a given measure class <€, the problem of describing the cohomology classes of cocycles is then well posed. When there is an orbit G-x0 such t h a t X — G • x0 is a null set for %\ this problem is essentially the problem for the transitive Borel 6r-space X1 = G-x0. Such measure classes on X we shall call transitive. However, not every invariant measure class is transitive. Let us call a measure class ^ which is invariant, an ergodic measure class if any Borel function / , having the property t h a t for each g e G, f(g-x)=f(x) for <€almost all x, is constant almost everywhere. Transitive measure classes are ergodic but not conversely. When X is not transitive under G, there will often exist ergodic invariant measure classes ^ which are not transitive. For any such ^ the problem of describing all the cohomology classes seems quite difficult.

MEASURE

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183

If ^ is transitive and G acts freely on X, i.e., for ^-almost all x e X the stability subgroups are trivial, i.e., consist only of {e}, then there exists obviously an xQ e X such t h a t X — G-x0 is ^ - n u l l and GXQ = e. I n this case, there are no cocycles other t h a n coboundaries. When ^ is assumed to be merely ergodic, this is not true any longer. We shall now construct examples where ^f is ergodic, G acts freely on X, and yet there are cocycles which are not coboundaries. Example 1: G is the discrete additive group of integers, X is the multiplicative group of complex numbers of modulus 1, and M = X. fx is the H a a r measure on X. We denote by z0 an element of X such t h a t the powers zQn (n=l,2,- •) are dense in X. For neG and zeX, n[z] = z0nz. The measure class of //, is ergodic, as m a y be verified easily. Suppose t h a t / is a Borel m a p of X into M. If we define, for (n,x) eGxX,

(m/M'-m-1*) rf(n,x)

(n> i),

= h

(n = 0),

l/^o-^)- 1 "-/^^)- 1

(*<<>),

then rf is a strict cocycle, as is easily verified, and

I t is obvious t h a t every cocycle is equal almost everywhere on G x X to some rf. We now ask when rf is a coboundary. Clearly, this is so if and only if there exists a Borel m a p b of X into M such t h a t rf(n,x)

b(z0nx)b(x)~1

=

for /x-almost all x. I n particular, b{x)f(x)

= b(z0x)

for almost all x. Now, not e v e r y / can satisfy this equation for suitable b. I n fact, we have, for/(#) = #, bjc + i

=

z

o

o fc ,

where & - > bk is the Fourier transform of b, i.e., Sfc = f b{x)xkdix{x),

(k = 0, ± 1 , ± 2 , - • •).

This implies t h a t |S fc | 2 is independent of k, and hence t h a t bk = 0 for all & since b e j£?2(X,/x). This means t h a t 6 = 0, a contradiction. Example 2: I n this example, G = R and ilf the multiplicative group of complex numbers of modulus 1; it is due to Helson and Lowdenslager [1].

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We make frequent use of the Pontrjagin theory for locally compact abelian groups (Pontrjagin [1], Kaplansky [2], pp. 54-56). We write R for the additive group of reals in the usual topology and Rd for the additive group of the reals in its discrete topology. Let D be a countable subgroup of Rd and X the multiplicatively written character group of D. I n the (usual) topology of pointwise convergence, X is a compact metrizable group. Since no element of D is of finite order, X is connected. For X E D and x e X we write for the value of x at A. For a n y teR, A - > eitx (A e D) is an element of X. We write i for this element, t -> £ is a homomorphism of R into X. We note t h a t the image of R under this homomorphism is dense in X. Otherwise, by the duality there would exist a A ^ 0 in D such t h a t eitx

=

Y

for all teR, which is impossible. If D contains at least two elements which are rationally independent, t -> i is injective, as m a y be easily verified. The injection t - > t gives rise to an action of J? on X if we set t-x = tx

(t e R, x e X).

I t is obvious t h a t t, x - > t -x is a continuous map of R x X into X and t h a t X becomes an R-space. The H a a r measure /x on X is invariant and ergodic under this action. Our object is to exhibit (R,X,M )-cocycles which are not coboundaries. Suppose XeD. Then the function 0A : t, x—> (1 — eitx) is continuous on RxX and ifjx(t + u,x) = i/)A(t,u-x) + i/jA(u,x) for all t, ueR and x e X. However, I/JA is not real valued. Hence we pass on to A(M0 = < A , x > ( l - e ^ ) + < - A , a : > ( l - e - ^ ) . Then
Cj-> +oo,

(b)

A; G D

(c)

and the A; are rationally independent,

V Xfj < oo. i=i

MEASURE

THEORY ON G-SPACES

185

Put c0 = 0, cj = c_j, X_j=— Xj. Then the series 2f= - »fy(l— ef'Ai) converges uniformly when £ varies over a bounded set and x varies over X. It therefore represents a real valued function 99 continuous on RxX. As 00


(*)

f(t,x) = bfaMx)-1

for almost all (t,x) e RxX. Write at for the map x -> if>(t,x), and for yeX, let ^ be the map a?-> fe^a;)^)-1. Then y ~> ay is a continuous map of JL into the Hilbert space J^2(/x). This implies that (*) is valid for each t for almost all x. Suppose now that y e X is such that = 1 for ally, then we claim that ay is a constant (of absolute value 1) for almost all x. To see this, let us consider, for fixed s e R and each y e X, the function cy, where cy(x) = e x p { ( - i )

£

cXV>(l-)(l-e«**)\.

Since s is fixed, the series under the exponential is uniformly convergent as x, y vary over X and hence x, y -» cy(#) is a continuous function on X x X. Hence y —> cy is a continuous map of X into J^2(/x). But if we put y = t for t e R, an easy calculation shows that

The continuity of the maps y -> cy and y -> ay shows that for each y e X, we must have for almost all x. If now y e X and (X^y) = 1 for all j , cy(x) = 1 and hence ay(s-x) = ay(x) for almost all x. As i? acts ergodically on X, ay must be a constant almost everywhere. This constant is obviously of modulus 1. The conclusion of the preceding paragraph can be sharpened. Suppose y e X is such that = l for j^k and (Xk,y)= — l(k, j>l). Then, the result of the previous paragraph, applied to the strict cocycle obtained by dropping the terms with indices + k, shows that for some complex number z of modulus 1, ay(x) ~ z exp{4icfc Re <(Afc,a;>} for almost all x.

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We are now in a position to give the decisive argument. Since the A; are rationally independent, there exists, for each & = 1 , 2 , « - & ykeX such t h a t (

I

j ± k,

(j > 1),

Write ak for aVk. Since X is compact, there exists a subsequence {ykn} and a y0 e X such t h a t ykn -> y0. Then akn' -> aVQ. As = 1 for all j , we know t h a t ayo(x) = r for almost all x, T being a constant with \T\ = 1. Hence

On the other hand, we claim t h a t , since ck -> +oo,

J aykdfx->0. I n fact, consider the integral JCfx =

exp{4ic Re}d/x(#)

for c > 0. If A ( # ) / 1 for some x e X, then x - » maps X onto a compact nontrivial subgroup of M, which is connected as X is connected, and so coincides with M. The measure JJL then goes over to H a a r measure on M and hence Jc,\ ==

exp{4ic Re m}dm JM

1 = — ^

f2*

exp(4ic cos 0)d0.

Jo

J c A is t h u s independent of A and - > 0 as c -> oo. This proves, as we asserted, t h a t

J aykdii-+0. This contradiction shows t h a t our cocycle I/J is not a coboundary. We shall end this section with an application to t h e theory of quasiinvariant a-finite measures on standard transitive Borel 0-spaces. For a direct derivation of the results which are about to be obtained now, t h e reader m a y consult Mackey's paper [3]. We write B+ for the multiplicative group of positive reals. Let G be a lcsc group, G0 a closed subgroup, X = G/G0 and ft the canonical m a p of G onto X. Let /3(e) = x0. Suppose t h a t F is a positive

MEASURE

THEORY

ON

G-SPACES

187

Borel function on Gx X and a is a quasi-invariant cr-finite measure on X. We write (54)

F ~ a

if for each g e G, x-> F(g,x) is a version of da/dct9'^. Theorem 5.30. Let A and A0 be the modular functions of G and G0, respectively. Write (55)

m(h) = A(A)A0(^)-1

{he G0).

Then, for any quasi-invariant a-finite measure a, there exists a strict (G,X,R + )-cocycle r such that r~a. Any such r defines the homomorphism m at x0. Conversely, if r is a strict cocycle defining m at x0, r~afor some quasiinvariant a-finite measure a; r determines a up to a constant multiplying factor. Finally, an invariant o-finite measure on X is necessarily a Borel measure, and there exists one such on X if and only if A(h) = A0(h) for all heG0. Proof. We begin our proof with a simple observation. Let S^G be a subgroup such that G—S is measurable and has measure zero. Then S = G. In fact, if S^G, and g e G—S, gS^G—S so that gS has measure zero. This implies that S is a null set, a contradiction. This said, let us consider a quasi-invariant a-finite measure a on X. Let F be a positive Borel function on G x X such that for almost all g, x->F(g,x) is a version of da/daf*-1* (theorem 5.10). Equation (22) shows that F is a (G,X,B+)-cocycle. Hence, by lemma 5.26 there exists a strict cocycle r such that r = F almost everywhere on GxX. For each g e G, let fg be a version of da/da (9-1) and let S = {g : r(g,x) = fg(x) for almost all x}. Since the fg satisfy (20) and r satisfies (36), it is easily seen that 8 is a subgroup of G. Further, G—S has measure zero. Therefore S = G. In other words, r ~ a . If r' is another strict cocycle such that r' ~ a, then r' = r almost everywhere on GxX. Hence / « / • by lemma 5.25. Thus, R+ being abelian, r' defines the same homomorphism of GQ as r does, at x0. Let ma denote this homomorphism. We claim that ma is independent of a. Suppose that a is another a-finite quasi-invariant measure. Let a be a positive Borel function which is a version of da/da. Then it is quickly verified that (with r ~ a ) (56)

r' :g,x^

a(x)r(g,x)a(g -x)'1

is a strict cocycle such that r ' ~ a \ But then, r'&r, so that ma> = ma. Let m0 denote this homomorphism of G0, determined by all the measures a.

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Suppose now t h a t r is a strict cocycle defining the homomorphism m 0 a t x0. Let ax be a a-finite quasi-invariant measure and rx a strict cocycle such t h a t r1^a1. Since rx also defines m0 at x0, r1^r. Hence there exists a Borel m a p b of X into R + such t h a t (57)

r(g,x) = ^aOr^aOfcter-a;)- 1

for all (g,x) e G x X. If a is the a-finite quasi-invariant measure such t h a t da/da1 = b, then (57) shows t h a t r~a. If a is another a-finite quasiinvariant measure such t h a t r~a also, then a is a constant multiple of a. I n fact, let a be a positive Borel function such t h a t a is a version of da jd<x. Then for each g, a(x) = a(g-x) for almost all x, showing t h a t a is a constant almost everywhere. Suppose now t h a t X admits an invariant a-finite measure, say y. Then the strict cocycle 1 : g, x -> 1 is such t h a t l ~ y . Since 1 defines the trivial homomorphism at trivial. Conversely, let m 0 be trivial and let a be a a-finite quasi-invariant measure. Let r be a strict cocycle defining ra0 a t x0 such t h a t r~a. Since m 0 is trivial, r&l, and so there exists a Borel m a p b of X into R + such t h a t (58)

r(g,x) =

b(g-x)b(x)^

for all (g,x) e Gx X. The equation (58) shows t h a t the cr-finite measure y defined by the equation dyjda — b is invariant. I n other words, X admits a a-finite invariant measure if and only if m 0 is trivial. I t remains to prove t h a t m0 = m. To this end we use the results and notations of lemmas 5.14 and 5.15 and theorem 5.19. Using the argument of lemma 5.29, we construct a Borel m a p d of G into R+, such t h a t (i) d(e) = 1 and d is a locally bounded m a p of G into R +, (ii) there exists a strict cocycle r such t h a t r°(g,g,) = d(gg')d(g,)~1 for all (g,g') e GxG, and (iii) r defines the homomorphism m of G0 at x0. Let y be the Borel measure on G defined by the linear functional (also written y)

y.f^^J{g)d(g)-H^g),

(feCc(0))

Ii being a left H a a r measure on G. We claim t h a t y vanishes on the kernel of the m a p / - ^ / ~. I n fact, suppose / e CC(G) is such t h a t

f f(gh)dp0(h) = 0 J G0

MEASURE

THEORY 1

for all g in G. Multiplying by ud' respect t o /A, we get

ON

189

G-SPACES

where u e CC(G) and integrating with

L (L f^h)d^o(h)y(g)d(g)-^(g) = 0. We interchange t h e order of integration and change g t o gh while integrating with respect t o JX. We then obtain, on using t h e fact t h a t d defines m at; XQ) j

( A O ( A ) - 1 ^u(gh-^)d(g)-^(g)d^g)y^(h)

= 0.

If we now change t h e order of integration once again we obtain (59)

^u~(p(g))d(g)-y(g)dn(g)

= 0.

Since t h e m a p u->u~ maps CC(G) onto CC(X), we can choose u so t h a t u~ is identically 1 on t h e set P[K], where K is a compact subset of G outside of which / is zero. Then (59) leads t o t h e equation (60)

jaf(g)d(g)-1dix(g)

= o.

I n other words, there exists a linear functional y~ on CC(X) such t h a t y ( / ) = y~(/~) for all/eO c (6r). Moreover, we know from lemma 5.15 t h a t given a n y veCc(X), which is real and > 0 , there is a ueCc(G) such t h a t u~—v a n d u>0. This shows t h a t y~ is a nonnegative linear functional. Therefore there exists a Borel measure a on X, such t h a t y~ corresponds t o a. Then, obviously y = a°. Since y is quasi invariant, so is a (corollary 5.17). Fix geG. We shall compute da/da^-v. Let t be a positive Borel version of it on X. Then, by corollary 5.17, t° is a version oidy/dy(a~1\ But, since dy/dfM = d~1i we obtain * V ) = digg'W)-1

= r°(g,g')

for almost all g'. I n other words, for each g e G, x -> r(g,x) is a version of da/dcS9'^. This proves t h a t r~a. B y our definition of m0, m = m0. The above argument shows actually t h a t if r is a locally bounded strict cocycle defining m, there exists a quasi-invariant Borel measure a such t h a t r~a. Suppose now t h a t X admits a n invariant cr-finite measure a. m is then trivial a n d our remark a t t h e beginning of this paragraph shows t h a t there exists a Borel measure y such t h a t 1 ~ y. B u t this implies t h a t a is a constant multiple of y b y corollary 5.21. a is t h u s a Borel measure. The proof of t h e theorem is complete. Corollary 5.31. / / G is unimodular, G/G0 admits an invariant Borel measure if and only if G0 is also unimodular. If G is arbitrary but GQ is compact, then GjG0 admits an invariant Borel measure.

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We end this section with a n example of a unimodular G a n d a nonunimodular G0. Let G=SL(2,C), the group of 2 x 2 matrices g (61)

la g= {

b\ \

(ad-bc

= 1)

of determinant 1. g - > a,b, c, d gives a n injective homeomorphism of G into C4. The group G, equipped with t h e relative topology obtained b y imbedding it as above in C4, is a lcsc group. Let G0 be the subgroup of all elements tatZ, where (62)

ta,z = ^

Z

\

(a,zeC,a^0).

G0 is closed. We claim t h a t G is unimodular and G0 is not, so t h a t X = G/G0 does not admit any CT-finite invariant measure. For the unimodularity of G we shall be more general and prove t h a t G admits no nontrivial continuous homomorphism into R + . Suppose t h a t m is one such. Since m(x) = m(yxy~1) for all x, y in G, it follows t h a t for any matrix x in G with distinct eigenvalues, m(x) = m(y) for all matrices y with t h e same eigenvalues as x. Let G' be t h e set of all matrices x in G with distinct eigenvalues and H the subgroup of diagonal matrices. Then, t h e restriction of m t o H is a continuous homomorphism of H into R + with the property t h a t m(x) = m(y) whenever the diagonal entries of x and y are permutations of one another. I t is easy t o deduce from this t h a t m is identically 1 on ^ . Since for a n y g e G' there is a n a e G such t h a t aga~x e H, m(g) = 1 for all g e G'. Now G' is a dense subset of G and so m is identically 1 on G. I n particular, G is unimodular. GQ is not unimodular. To see this, let (63)

77 = {tat0 : a e C*}.

Then H is a n abelian closed subgroup of G0 and is hence unimodular. If G0 were unimodular, then there would exist a n invariant Borel measure oh X0 = G0IH. N O W in each left #-coset in GQ there exists exactly one element of the form tlt2 (z e C); if we use the complex number z t o parametrize the coset tlfZH, t h e action of G0 on X0 becomes (64)

tCt2:l~>c2Ucz.

I t is obvious t h a t t h e space of complex numbers does n o t admit a n y Borel measure invariant under the maps (64). Thus GQ is not unimodular. I t is of interest t o note in this case t h a t X = GjG0 is actually compact. I n fact, if K denotes the subgroup of unitary matrices in G, it is a n easy consequence of t h e existence of the J o r d a n canonical form of an arbitrary matrix in G, t h a t G = KG0. Hence X is the image of K under the canonical mapping of G. I n particular, X is compact.

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191

6. B O R E L G R O U P S AND T H E W E I L TOPOLOGY Our main concern in this section is to examine how the topology of a lcsc group can be recovered from the measure-theoretic structure. Suppose t h a t G is a lcsc group and t h a t /x is a left H a a r measure. Then it is classical t h a t for a n y Borel set E with /UL(E)>0, the set EE'1 contains a neighborhood of e. This result hints t h a t the topology of G is completely determined by JJL. That this is indeed so was proved by Weil [1] (cf. Halmos [1], pp. 257-289 for a detailed discussion of Weil's results). For our purposes, this result of Weil is somewhat inadequate. The main theorem of this section is a sharpening of Weil's theorem, first proved by Mackey [5]. We shall use this sharpened form of Weil's result in a significant way in Chapter V I I . This sharpening essentially comes out of the following observations. The first point is t h a t starting with only a left invariant measure class on a separable Borel group G one can show t h a t the group possesses a left invariant measure. The second point is t h a t the Weil topology for G is second countable and generates the original Borel structure on G. The third observation now consists in imbedding ( J a s a dense subgroup of a lcsc group G*. The fact t h a t G is standard now implies t h a t G is a Borel set in G*. This leads easily to the identity G = G*. In other words, every standard Borel group admitting a left invariant measure class is actually a lcsc group under its Weil topology. This is the sharpening of Weil's theorem we referred to. I t is due to Mackey [5]. Since the main aim of our analysis is to obtain topological results from measure-theoretic assumptions, it is convenient to start with groups without a n y topology. Throughout this section, G will denote a separable Borel group, i.e., a group G with a Borel structure which converts G into a separable Borel space such t h a t the m a p x, y -> xy ~~1 of G x G into G is Borel. Since the left translations x -> zx are Borel automorphisms, we m a y consider measure classes invariant under these. We call them left invariant. I n an analogous fashion we define right invariant measure classes. A measure class which is both left and right invariant is called bi-invariant. Lemma 5.32. Let Gbea separable Borel group on which there exists a left (or right) invariant measure class. Then G admits a bi-invariant measure class. Proof. Let ^ be a left invariant measure class and A a finite measure in ¥>. We shall prove first t h a t for any Borel set E^G, s -> X(Es) is a Borel function of s. In fact, the map T : s, t -> s, ts is a Borel automorphism of the Borel space G x G and hence T[GxE] is a Borel set in GxG. I t is then obvious t h a t Es = {t: (s,t) e T[G x E]}, and hence the function s -> X(Es) is Borel by Fubini's theorem.

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Define the measure Ax b y (65)

X^E) = f

\(Es)d\(s).

Xx is easily seen to be a measure and A1(G) = \(G)2>0. We claim t h a t the measure class determined by X± is bi-invariant. I t is enough to prove t h a t if X1(E) = 0, then X1(Ez) = X1(zE) = 0 for all ZEG. Suppose \x(E) = 0. Then, as X{Es) > 0, X(Es) = 0 for A-almost all s. Since Xetf and <€ is left invariant X(zEs) = 0 for A-almost all s, showing t h a t X1(zE) = 0. On the other hand, if N is a Borel set such t h a t X(N) = 0 and X(Es) = 0 for SEG-N, then X(z~1N) = 0 and A(#zs)=0 for seG-z^N. This proves t h a t X±(Ez) = 0 also. A similar argument can be given starting with a right invariant measure class. The lemma is proved. Lemma 5.33. Let G be a separable Borel group and let ^ be a bi-invariant measure class. Then & contains left invariant and right invariant a-finite measures. Proof. Let A be a finite measure in # , such t h a t X(G) — l. Define As b y XS(E) = A ^ " 1 ^ ) . Then the measures {As} are all mutually absolutely continuous. Since s->Xs(E) is a Borel function on G, corollary 5.6 is applicable and therefore there exists a Borel function r on GxG such t h a t for A-almost all s, t -> r(s,t) is a version of dX/dX^'^. Since A* and A are mutually absolutely continuous, r(s,t) > 0 for almost all t for A-almost all s. Hence r is positive (A x A)-almost everywhere on GxG. We m a y thus assume t h a t r is positive on GxG. The arguments of theorem 5.10 can now be used to conclude t h a t for (Ax Ax A)-almost all (s,t,u) e GxGxG, r(st,u) =

r(s,tu)r(t,u).

Then there exists a positive Borel function b on G such t h a t (66)

r(s,t) = b(st)/b(t)

for (A x A)-almost all (s,t) e GxG. I n fact, all we have to do is to imitate t h e argument of the first p a r t of the proof of lemma 5.26; the only fact used there is the bi-invariance of the class of null sets involved which is certainly satisfied in our present setup. The measure ^ such t h a t d^dX = b is then cr-finite and belongs to %\ From (66) we conclude t h a t for A-almost all s, d/jildfxis ~1} is equal to 1 almost everywhere. Thus /x = /x(s~1} for A-almost all s. Now the set S of all s E G such t h a t /x = /x (s_1) is a subgroup. Since

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193

this subgroup has a complement of A-measure zero, it must coincide with G (cf. remark in proof of theorem 5.30). This proves t h a t /x is left invariant. A similar argument yields a right invariant measure in the measure class <€. The proof is complete. Let now ffl be a separable Hilbert space and °ll the group of all unitary operators on Jf. We shall give the strong topology for °ll\ i.e., the smallest topology which makes all the maps JJ-^Uf (f e J^) continuous. If { Un} is a sequence of elements in °U, Un^ V e °ll if and only if Unf —> Vf for all / belonging to some set D such t h a t the linear combinations of vectors of D are dense in J^. Lemma 5.34. °tt is a metrizable topological group satisfying axiom of countability and its Borel structure is standard.

the second

Proof. Since 3f is separable, there exists a countable set D dense in Jf7, and the topology for 9/ is clearly the smallest one which makes all the maps U - > Uf (fe D) continuous. As there are only denumerably many maps, it follows from standard topological arguments t h a t the topology for °ll is metrizable and second countable. To prove t h a t °ll is a topological group we must show t h a t the m a p U, V -> UV'1 is continuous. But, as the elements of °ll are unitary, we have, for a n y / e JfJ || Z 7 F - Y - C70 F 0 " Y|| < || F( F0" Y) - F 0 ( F 0 " y ) || + || ?7( F 0 - Y) - t70( F Q - V)||. This inequality proves at once t h a t °ll is a topological group. To prove t h a t the associated Borel structure of ^ is standard, we must exhibit a Borel isomorphism of ^/ with a Borel space which is known to be standard. Let X be the topological product of countably m a n y copies of Jj? and let ^ = {/i,/ 2 5* * •} be a countable orthonormal basis for Jtf*. For each U e %, let £(£/) be the element {Ufl9 Uf2, • • •) of X ; (67)

£:U-+(Ufl9Uf29...).

£ is clearly a homeomorphism of °2l into X. We want to show t h a t the range of £ is a Borel set in X. Clearly, (zl9 z2, • • •) e X lies in the range of £ if and only if {^l3 z 2 , • • •} is an orthonormal basis, i.e., if and only if (zm,zny (68)

= Smn 2

2 KWi)! = 1

(Kronecker delta)

for all i.

m

These are countably m a n y equations and each one of them defines a Borel subset of X. Thus the range of £ is Borel. £ is thus an isomorphism of % with the Borel subspace of X defined by (68). As this is a Borel subset of the standard space X, we can conclude t h a t °ll is standard.

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Let now G be a separable Borel group and let /x be a left invariant (j-finite measure on G. Let Jfu be the Hilbert space of complex valued (equivalence classes of) Borel functions / on G such t h a t ||/|| 2 = £ \f(g)\2d(*(g) < co.

(69)

For a n y g e G, let Lg be the operator of Jfu defined by (70)

(Lgf)(t)=f(g~H)

(teG).

Lg is clearly a unitary operator of Jt?u. We have Le = 1, L

9192 =

L

the identity operator,

9lL92

f

°

r a

U

01> 02 G

G

'

Denote by ^ M the group of all unitary operators on J^u. Lemma 5.35. Let G be a separable Borel group which has a left invariant a-finite measure p. Then the map L(g —> Lg) of G into ^u is a one-one Borel map. If G is standard, its range L[G] is a Borel set in %u and L is a Borel isomorphism of G with L[G]. Proof. I n view of the definition of the Borel structure on ^ , we must, in order to show t h a t L is Borel, prove t h a t for each / e J^, g -> Lgf is a Borel m a p of G into ^ . Since ^ is separable, we can use lemma 5.4 to reduce this t o showing t h a t for each / , / ' e J^, g -> (Lgf,f'y is a Borel function on G. B u t (* denotes complex conjugation)

= f

f(g-H)f'(t)*dp(t),

JG

which is obviously Borel, as g,t-^f(g H)fr(t)* is a Borel function on OxG. We show next t h a t L is one-one. As L is a homomorphism of G into ^u, this reduces t o showing t h a t if g^e, Lgf^f for s o m e / e J f r Suppose t h a t Lgf=f for all f e Jtf^. Since the Borel structure of G is countably generated and /x is a-finite, there exists a sequence {En} of sets of finite /x-measure such t h a t {En} generates the Borel structure of G. Let fn = ^En. Then fn e J^u and the sequence {/n} separates the points of G (since each single point set of G is a Borel set). Let N be a set of ^-measure zero such t h a t for x £ N, (72)

/ . ( r 1 * ) =/„(*)

for all n. The fact t h a t {/n} is a separating sequence now implies t h a t g - 1X = x for all x e G — N. The t r u t h of this for even one xeG — N implies g—e. L is thus one-one. The remaining statements follow from the fact t h a t
MEASURE

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195

We shall now introduce the Weil topology. Let G be a separable Borel group admitting a left invariant a-finite measure /x. Consider for each g e G, the unitary operator Lg in Jf?u. The Weil topology for G is then defined to be the smallest one which makes the maps (73)

g-+LJ

of G into J ^ continuous for all / e J^. I n view of the definition of the strong topology for ^u, it is clear t h a t the Weil topology is t h a t topology for G which makes the m a p g - » Lg of G into %u a homeomorphism. Thus 6r, under the Weil topology, is a topological group satisfying the second axiom of countability. Let E and F be Borel sets of finite measure. We write (74)

P(E,F)

= fi((E-F) u (F-E)).

Then from the relations P(E,F)

= f

\XE(t)~xF(t)\2d^t),

JG P(E,F)

=

P(xE,xF),

it is easy to deduce t h a t # n -> a; in the Weil topology for G if and only if p(xnE,xE) -> 0 for all Borel sets E of finite measure; it is even sufficient to restrict the E9s to a Boolean ring of sets of finite measure which generate the Borel structure of G. Lemma 5.36. If G is a standard Borel group and \i a left invariant a-finite measure on it, the Borel structure generated by the sets which are open in the Weil topology coincides with the original Borel structure. Moreover, if G is a Icsc group, the Weil topology coincides with the original topology. Proof. Let G± be the range of L(g -> Lg). Let &* be the Borel structure for G generated b y the Weil topology and let & be the original Borel structure. Then, L being a homeomorphism of G onto Gl9 L is a Borel isomorphism of (G,&*) with the Borel subspace of ^ defined b y Gx. By lemma 5.35, L is also a Borel isomorphism of (G,&), with the Borel subspace of ^ M defined by Gx. Hence ^ = ^ * . Let us now assume t h a t G is a lcsc group. We want to prove t h a t its topology coincides with the Weil topology. Since both topologies are metrizable, it is obviously sufficient to prove t h a t for a n y sequence {gn} in G, gn-> e in the given topology of G if and only if p(gnE,E) -> 0 for each Borel set E of finite measure. All topological statements, in the argument to follow, refer t o the given topology on G. Let then gn -> e. If {Vs} is a decreasing sequence of compact symmetric (VS=VS~1) neighborhoods of e which form a base for G at e, then for a n y compact set K, it is easy to see

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t h a t p | s {VSK) — K. Fix a Borel set E of finite measure, and let s > 0. Choose a compact K^E so t h a t fji(E — K)<e and choose s so t h a t /x(F s i£ — i f ) < £ . If n0 is chosen so t h a t gne Vs for n>n0, then for all such ?i, P f a n ^ ) < P(gnK,K) + 2P(E,K) < 2fi(VsK-K)

+ 2e

< 4c.

I n the converse direction, suppose ^ n 4> e. By passing to a suitable subsequence we m a y assume t h a t there exists a compact neighborhood V of the identity such t h a t gn $ V for all n. Choose a compact neighborhood W of e such t h a t WW"xc V. Then gnW nW=0 for all w, so t h a t P(gnW, W) = 2/n(W) > 0 . This shows t h a t grn cannot converge to e in the Weil topology. This completes the proof of the lemma. Consider now a standard Borel group G with a nonzero left invariant Borel measure /x. We equip G with the Weil topology. Following standard terminology, we shall say t h a t a subset A c; G is bounded if for each open set £7 in 6r one can find elements x1, • • •, xn e G such t h a t

-4 <= 0 xtU; i=l

I t is obviously enough to require this only for open sets containing e. We shall say t h a t G is locally bounded if some open set containing e is bounded. Notice t h a t open sets in G are Borel (lemma 5.36); if U is open a n d nonempty, there are elements G such t h a t G = \Jn xnU, from which it follows at once t h a t 0 < /JL( U) < oo. Lemma 5.37. G is locally bounded. Every bounded Borel set is of finite measure. Proof. Before we take up the proof of this lemma we make a simple observation. Let E be a Borel set with E = E~X and 0 < fi(E) < oo. Then there is a Borel subset F^E with F = F~* and 0 0 for all x e E and f fd/jLc} for some c>0. This said, we come to the proof of lemma 5.37. Let E be a Borel set such t h a t E = E~1 and 0 < / x ( # ) < o o . P u t , for 0<e<2n(E), N(E : e) = {x : P(xE,E)

< e}.

Since a; -> p(xE,E) is a continuous function on 6r, vanishing a t e, JV(^ : e) is an open set containing e. We claim t h a t 0 < fi(N(E

: e)) < oo.

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197

We need only to prove t h a t fi(N(E : e)) < oo. Now, we have XE(t)XE(t-1X)dfl(x)dH.(t)

jj

= JXE(t)fM(tE)dfJL(t)

GxG

G

= M(^) 2 On the other hand, as E = E~1, jj

Xs^XEit-^d^dfiix)

= jj

=

XEMxEix-tydpWdpix)

fi(xE n E)dfji(x). G

Therefore,

fji(E)2 >

f n(xE n E)dfi(x). N(E:e)

But,

if XGN(E

:e), P(xE,E)

= 2p(E)-2ii(xE

n E) < e

so t h a t /x(^> This proves our claim t h a t 0
(r(E)-el2)p(N(E:e)). : e))
At this stage, we know t h a t sufficiently small open subsets of G are of positive finite measure. Let U be any open set containing 1 with 0 < fji(U)
(n = 1 , 2 , . . . ) -

i= l

Choose an open set L containing e such t h a t LL_1clf n V. Then it is easy to see t h a t the sets x1L, x2L, • • • are disjoint and contained in VV'1. 1 Since VV~ ^U,0 0, we have a contradiction. The last statement is easy. For, let A be a bounded Borel set and W an open set of finite measure. If xx, • • •, xn e G are such t h a t A^(Jf=1 xtW, then fx(A) < nyi{ W) < oo. We shall now use these results to imbed G as a dense subgroup of a lcsc group 6r*. Our argument is based on the observation t h a t G admits a left invariant metric metrizing its Weil topology; in fact, if {En} is a denumerable Boolean ring of Borel sets of finite measure which generate the Borel structure of G, the metric

(75)

8(»,y) = S / ( ! f n / ^ >

2-"

is easily verified to induce the Weil topology; 8 is left invariant.

GEOMETRY

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Lemma 5.38. Let Gbea second countable topological group and let it satisfy the two conditions: (a) (b)

there is a left invariant metric 8 for the topology of G, if {xn} is a 8-Cauchy sequence, then {# n - 1 } contains a 8-Cauchy subsequence.

Let G* be the completion of the metric space (G,8) with G considered as imbedded in 6r*. Then there exists exactly one way to convert G* into a topological group such that G becomes a dense topological subgroup of it. Proof. As 8(xn,xm) = 8(xn~1xm,e), it follows t h a t {xn} is a S-Cauchy sequence if and only if xn~ 1xm -> e as n, m -> GO. Let G* be the completion of G (under 8); we regard G as imbedded in G*. We denote also by 8 the extension of the metric 8 to 6r*, which is a second countable topological space in its metric topology. We shall now show t h a t if {xn} and {yn} are Cauchy sequences in G, so is {xnyn}. Let V be any open neighborhood of e and let us choose an open neighborhood W of e such t h a t W3^ V. Let k be an integer such t h a t yn~1yrn£ W for n,m>k. Choose an open neighborhood W of e such t h a t Wf^W and yk~1W'yk<^W. Finally, select an integer kx>k such t h a t xn~1xm e W for n, m>k±. Then, if n, m>k±, yn

Xn

'%mym

= (yn

Vkf'Vk

\Xn

x

vx)yk'Vk

Urn

eW{yk^W'yk)W <= V. This proves t h a t {xnyn} is Cauchy. From this it follows easily t h a t there is a unique continuous m a p p* of 6r* x 6r* into G* such t h a t p*(x,y)=xy (x,y e G). We define, for x,y e G*, (77)

xy =

p*(x,y).

This product operation is clearly associative and xe = ex = x for all XEG*. If x E G* and {xn} is any sequence in G converging to x, then, for any limit point x' of the sequence {x n _ 1 } we have e. Condition (76b) now implies t h a t x'1 exists and t h a t x^1 converges to x"1. The rest of lemma 5.38 now follows trivially. Corollary 5.39. Let G be a locally bounded second countable topological group whose topology is induced by a left invariant metric 8. Then G satisfies (b) of (76). In this case G* is a Icsc group and for any compact set K of G*, K n G* is a bounded set in G. Proof. Suppose V is a bounded set in G. We claim t h a t any sequence in V has a Cauchy subsequence. To prove this we proceed as follows. Let

MEASURE

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199

{Un} be a sequence of open subsets of G forming a base for the topology of G at e such t h a t U~+1Un + 1^ Un for all n. If {xn} is a sequence in V having infinitely m a n y elements, it follows from the boundedness of V t h a t , for each k, some translate akUk contains infinitely m a n y of the xn. This means t h a t for some infinite set Zk of the integers, xn ~ 1xm e U^1!]^ Uk_x whenever n, m e Zk. By choosing the Z's to decrease with k and using the Cantor diagonal procedure we m a y assure ourselves t h a t {xn} has a subsequence which is Cauchy. We shall now prove t h a t if {xn} is any Cauchy sequence, { # n _ i } has a Cauchy subsequence. I t is enough to exhibit a bounded set containing all but finitely m a n y of the elements £ n - 1 . Let U be a bounded neighborhood of e. Now there is a & such t h a t xn~1xke U for n>k. Hence xn ~1 e Uxk ~x for all n > k; as Uxk ~x is bounded, our assertion is proved. G* can thus be regarded as a second countable topological group and G a dense subgroup. To prove t h a t G* is locally compact, it is enough to exhibit some compact subset of G* with nonnull interior. Let V be any subset of G, bounded and open in G and containing e. Since any sequence in V contains a Cauchy subsequence, V, the closure of V in G*, is compact. Since G is second countable, there is a sequence {xn} in G such t h a t y*, there exists an m such t h a t ym ~ 1yn e V for all n>m. Thus yne ymV for all n>m. Choose an integer k such t h a t ym E xkV. Then yne xkVV for all n>m, showing t h a t y* e xkVV. By the category theorem of Baire, some xkVV contains an interior point. This implies t h a t V V is a compact set with nonempty interior and shows t h a t G* is locally compact. For the last assertion, let K be compact in G* and U an open set in G containing e. Let C7* be an open subset of G* such t h a t U=U* n G. Select an open set F * in G* such t h a t e e V* and 7 * 7 * ^ U*. There exist #i*> #2*5* * * > xk* e £r* such t h a t K^\^Jf=1 xfV*. We choose xi e G

(l
such that xr^eV*.

Then

K^\JtsslxiV*V*^\J?=1ziU*

and so K n ffc(J{c=1 ^£7. We come to our final lemma. Let G be a separable Borel group with a left invariant a-finite measure /x. Lemma 5.40. / / G is standard, then G is already locally compact under its Weil topology. Proof. Under the Weil topology, G is locally bounded and second countable. We also know t h a t the Weil topology is induced by a left invariant metric. Corollary 5.39 is thus applicable and hence we m a y regard G as a dense subgroup of a lcsc group G*. For any Borel set ^ 4 * c £ * 5 A* n G is Borel in G and hence, (78)

p* : A* -> fi{A* n G)

200

GEOMETRY

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is a measure on G*. Now \x is finite for bounded sets and so for any compact K^G*, IJL*(K) = IJL(K n G) ->/*^> uniformly while all the functions /* (g n ) vanish outside a fixed compact set. This implies t h a t

JG*

JG*

and hence

f f*(t)dn*(t) = f f™\t)dn*(t). JG*

JG*

Since this is valid for a l l / * e CC(G*), it is clear t h a t ix* is invariant under left translation by g*. I n other words, /x* is a left H a a r measure. xt* is evidently nonzero. Now G is standard and the identity m a p of G into G* is Borel and oneone. Hence G is a Borel set in G*. B u t then

H*(G*-G) =

fi((G*-G)nG)

= 0. Thus 6r is a subgroup which is also a Borel set with a complement of zero measure in G*. Therefore, the observation we made at the beginning of theorem 5.30 is applicable and enables us to conclude t h a t G = G*. I n other words, G is already locally compact in its Weil topology. Combining lemmas 5.32 through 5.40 we obtain the following theorem. Theorem 5.41. (Mackey [5]). Let G be a standard Borel group. If G admits a left (or right) invariant measure class, then G admits a left or right invariant o-finite measure. Moreover, there exists exactly one topology for G which converts G into a Icsc group and whose corresponding Borel structure is the original one on G. N O T E S ON C H A P T E R V The measure theoretic study of the orbit spaces of locally compact groups has been a very active subject in recent years because of possible applications to representation theory, number theory, statistical mechanics, and so on. For a survey treating m a n y of these aspects see R. Zimmer, Ergodic Theory, group representations, and rigidity, Bull. Amer. Math. S o c , 6 (1982), p p . 383-416.

CHAPTER VI SYSTEMS OF IMPRIMITIVITY

1. D E F I N I T I O N S This chapter discusses the notion of a system of imprimitivity for a group and, in some cases, describes all such systems to within unitary equivalence. These problems were first solved by Frobenius for the case of finite groups. For infinite groups their study is more recent. Their first appearance seems to be in a series of examples constructed by Murray and von Neumann [1], [2] in their theory of rings of operators. Anticipating terminology, we can say t h a t the examples of Murray and von Neumann form a class of ergodic systems of imprimitivity. On the other hand, the first work on transitive systems appeared in 1939. I n his great 1939 paper Wigner underscored their importance for building up the theory of representations of the inhomogeneous Lorentz group and studied, although in quite an implicit fashion, the systems of imprimitivity associated with this group. Also Gel'fand and Naimark [1] based their solution to the problem of describing the unitary representations of the "ax + b" group on an analysis of certain natural systems of imprimitivity. The systematic development of this concept is due to Mackey. I n a series of papers (cf. Mackey [7] for all references) he obtained the main general results of the subject and pointed out the far-reaching scope of the ideas involved therein. The variety of applications of these results is quite impressive. They enable one, among other things, to obtain the equivalence of matrix and wave mechanics; they lead to a concise description and classification of the relativistic wave equations; they make possible a unified approach to the theory of representations of the commutation rules; also, on the purely mathematical side, they lead to the notion of induced representations of locally compact groups which has played an important role in the theory of group representations; and finally, they have very interesting implications in the theory of invariant subspaces and factorization of analytic functions (cf. Helson [1]). We begin with the notion of a representation. Let G be a lcsc group and J$? a Hilbert space (always separable). A (unitary) representation of G in J^ is a homomorphism (1)

U:g-+U„ 201

202

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OF QUANTUM

THEORY

of G into the group °ti of all unitary operators on J f such t h a t for each / e ^f7, the m a p (2)

9-+UJ 7

is continuous from G into Jf . Since °ll is second countable in its strong topology (lemma 5.34), lemmas 5.28 a n d 5.4 show t h a t for the homomorphism g -> Ug to be a representation it is sufficient t h a t (3)

9-*

(f.f'eJT)

are measurable on G for a l l / , f'eJf.A representation U in ^ f is said to be equivalent to a representation £7' in £?' if there exists a unitary isomorphism (4)

W : Jf - > JT

of 2tf onto J f ' such t h a t (5)

Ug' =

WU.W-1

for all g e G. The representation U on J f is said to be irreducible if the only closed linear manifolds of J f which are invariant under all t h e linear operators Ug are 0 and Jf. If G is a locally compact abelian group, it is well known t h a t its irreducible representations are none other t h a n its characters; i.e., continuous homomorphisms of G into the multiplicative group of complex numbers of modulus 1. Two characters are equivalent if and only if they are identical. The theory of characters was developed in its definitive form b y Pontrjagin [1]; this development was used by Weil [1] to obtain the Plancherel formula for locally compact abelian groups. On the other hand, when G is not necessarily abelian, b u t compact, a definitive theory can again be developed. The beginnings of this theory can be traced a t least to the theory of spherical harmonics. B u t in its modern form, the theory is undoubtedly the creation of H e r m a n n Weyl. Using powerful and original (transcendental) methods he obtained the representations of arbitrary compact groups, his version of the Plancherel formula for such groups (the Peter-Weyl theorem), and rounded off the theory with a succinct description of the representations of compact semisimple Lie groups (cf. his papers [3], [4], [5] in Mathematische Zeitschrift and his expositions [1] (to an audience of physicists) and [2]). The Lorentz group is neither compact nor abelian. The study of its representations must therefore be based on the theory of representations of lcsc groups which are neither compact nor abelian. This theory is vast a n d quite recent. A survey of some of this recent work is given in Mackey's address [7]. Suppose t h a t G is a lcsc group. Let X be a standard Borel space. We relate G and X by assuming t h a t X is a Borel 6r-space. Let / be a

SYSTEMS

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IMPRIMITIVITY

203

separable Hilbert space. A system of imprimitivity for G acting in ^ f is a pair (U,P), where P(E - > PE) is a projection valued measure on t h e class of Borel subsets of X (the projections being defined in Jf), and U(g - > Ug) is a representation of G in J f such t h a t the relations (6)

UgPEUg~i

= Pg.E

are satisfied for all g e G and all Borel sets E^X. The pair (U,P) is said to be based on X. The notions of equivalence and irreducibility are defined for systems of imprimitivity in a fashion similar to the corresponding notions for representations. Two systems (U,P) (in J f ) and (U',Pr) (in Jf') based on the same 6r-space X are equivalent if there exists a unitary isomorphism W of £F onto J f ' such t h a t

P/

WPEW'1

=

for all
3tf = © ^ i

and where Ug:(f1,fa,---)-+{U91f1,U,%,---). I n symbols: @{U\Pi).

(U,P) = i

I n order to be able to handle the concepts of irreducibility, equivalence, and so on, it is convenient to transform these geometric concepts into algebraic ones. I t is customary to do this with the help of the notion of the commuting ring. We work with representations, b u t all of our remarks apply also to systems of imprimitivity. Let U be a representation of G in Jf. The commuting ring of U is the set 21 of all bounded operators A in J f such t h a t (8)

UgA

=AUg

for all g e G. Obviously % is an algebra and, if A e % then A* e %. Moreover it is easily seen t h a t 21 is closed under the weak topology. 21 is thus a von Neumann algebra (cf. Section 4, Chapter V I I , Volume I). The

GEOMETRY

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OF QUANTUM

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structure of 2t determines very completely the geometry of the action of U. For example, a closed linear manifold L is invariant under U if and only if the projection PL on L belongs to 21. I n this case, by restricting each Ug to L we obtain a representation of G in L, called the subrepresentation of U defined by L. Thus, every projection in 21 defines a unique subrepresentation of U. This subrepresentation is irreducible if and only if the associated projection, say Q, in 21, is minimal, i.e., if Q'

(9)

=

f{x)dvUtV(x),

where vUtV is, as usual, the complex valued measure on X defined b y

vuJE)

= (PEu,v}

(u,veJf).

vUtU is a finite measure for any u e Jf7 and vu/u(X) = \\u\\2. Moreover,

M/«« 2 = J \m\'dvu.u(*)

{uetf).

The m a p p i n g / -> Af from the algebra of all bounded Borel functions into the algebra of all bounded operators in Jf7 has the properties described below; their verifications are easy: (i) (ii)

(10)

(iii)

/—> Af is an algebra homomorphism, if/ is real, Af is self-adjoint; more generally Af* = Af*,

11,4/11 < sup

\f(x)\.

oceX

Moreover, using standard arguments, we obtain, from (6), (11)

U„AfUa-i 9

w h e r e / ' is the function

= 1

x->f(g~ -x).

A^,

SYSTEMS

OF

IMPRIMITIVITY

205

An important special case arises when X is a locally compact Hausdorff space satisfying the second axiom of countability and the action of G on X is continuous. I n this case, it is perhaps more natural to replace the projection valued measure P by the homomorphism / —> Af (f eCc(X)). We have: Lemma 6.1. Let f -> Af be a homomorphism of the algebra CC(X) into the algebra of all bounded operators on a separable Hilbert space Jf7 satisfying the relations (10) and such that the linear manifolds spanned by the ranges of all the operators Af is dense in 3f. Suppose further that g —> Ug is a unitary representation of G in J f such that (U,A) satisfies (11). Then, there exists exactly one system of imprimitivity (U,P) based on X such that, for each feCc(X), Af is related to P by (9). Proof. For a n y / in CC(X) which is real and > 0 , we observe t h a t Af>0; in fact, / = / i 2 , where f1 is also real and > 0 and Af = Afl2>0. Consequently, the m a p / - ^ (Afu,u) defines a Borel measure vu>u on X. If K is any compact set a n d / i s an element of CC(X) such t h a t 0 < / < 1 and / = 1 on K, the inequality (hi) of (10) shows t h a t

"».«(*) ^ Ml 2 vUtU is thus a finite measure and vutU(X) < \\u\\2. At the same time, if u, v e J^ there exists a unique complex valued measure vUtV such t h a t (12)

= j

fdvu,v

(feCc(X)).

For any Borel set E, the map u, v -> vUiV{E) is *-bilinear and 0 < vUtU(E) < \u\2 for all %£«#. Therefore, there exists a self-adjoint operator PE with 0 < PE < 1 such t h a t (13)

vUtV(E) =

for all u, v E Jf. We want to show t h a t E -> PE is a projection valued measure. Suppose K is a compact set and {fn} is a sequence of real nonnegative functions in CC(X) such t h a t / n j XK pointwise. Then (12) shows t h a t (14)

->

for all u, v e J f ; i.e., Afn converges weakly to PK (=> will denote weak convergence). I f / is any element of GC(X), AfnAf=>PKAf and AfAfn=> AfPK so t h a t PK commutes with all Af. Moreover, if f>0, then ffn \fxK and hence


= J ->

ffndvUtV fdvUtV.

206

GEOMETRY

OF QUANTUM

THEORY

I n other words, if/ e CC(X) is real and > 0, (15)


= f /»„„,„.

If we choose another compact set £ and a sequence {/n'} of real nonnegative elements oiCc(X) such t h a t / n ' j XL> w e obtain, from (15) and t h e convergence Af . => PL, the equation (PLu,PKv}

=

(PKnLu,v>.

Now, for fixed K, E - > (PEu,PKv} and E -> (PEnKu,v} are valued measures which coincide for all compact sets L. Hence incide for all E. For arbitrary b u t fixed E, we now use the coincidence for all K, and conclude t h a t for all Borel sets E and

complex t h e y coresulting F,

(PEu,pFvy = (PEnFu,vy for all u,veJ$f, i.e., PE^F = PEPF' For E = F this shows t h a t PE is a projection. The m a p E - > P £ has now all the properties of a projection valued measure except perhaps t h a t Px m a y not be 1. B u t Px = 1; for, if P x / 1 , there will be a nonzero u e Jt? such t h a t P x w = 0. This would imply t h a t vUtU = 0 and hence t h a t Af*u = 0 for a l l / e O c ( X ) . B u t then u would be orthogonal to the ranges of all the Af, a contradiction. Finally, by a routine transport of structure, the equation UgAfU9-x

= A<„

is seen t o lead t o t h e equation

ugpBua->-

= pg.E.

This proves the lemma. We shall now give some examples of systems of imprimitivity. X a n d G have the usual meanings, i.e., X is a standard Borel (r-space, G is lcsc. Example 1: Let a be a cr-finite invariant measure on X, Jf = J£?2(X,a); Ug and PE are defined by /io\

E

J = XEf> (UJ)(x)=f(g-i-x).

The identities Ugig2=U9iUg2, Ue = l as well as the relations (6) are t h e results of routine calculations. Further, the invariance of a implies t h a t Ug is unitary. For / , / ' e ^ the m a p g, x -> f(g~1-%)f'(x)* is Borel on G x X and hence the m a p

9^

=

jj(g-1-x)f'(x)*da(x)

SYSTEMS

OF

IMPBIMITIVITY

is a Borel map. U is thus a representation and (U,P) imprimitivity.

207

a system of

Example 2: Our second example is a variation on the theme of the first one. Let J f be a separable Hilbert space. We then define, for any a-finite measure a on X, J^ = ^2(X,J^,a), to be the vector space of aequivalence classes of Borel maps / of X into JT such t h a t I/I2 = j

(17)

\f(x)\Ha{x)

< oo;

here | • | denotes the norm in CtT. If we define, for/, / ' G ^ (18)

= j


where <. , . > denotes the inner product in Jf, then 3tf becomes a Hilbert space under < . : . > . If X is a separable Borel space, Mf is separable. We shall now select an arbitrary invariant measure class on X and a G-finite measure a in it. a need not be invariant. For each g e G we select a positive Borel function rg which is a version oidajda^9~1). We then define, for fEje Pjs

f = XE*' Ugf={(rgy)1,2f9>

(19) i.e.,

(20)

(uaf)(x) = W r ^ H ( r ^ ) .

The quasi invariance of a implies t h a t everything in sight is well defined. The identities (20) of Chapter V, which are atisfied by t h e rg, now lead to the conclusion t h a t g -> Ug is a homomorphism of G and t h a t (U,P) satisfies (6). The calculations are straightforward. Moreover, as each Ug is invertible and

\UJV = j W\f9\aMx)

= J r.l/Wl2^'-1^) it follows t h a t each Ug is a unitary operator. I n order to assure ourselves t h a t (U,P) is a system of imprimitivity, it remains only to check t h a t for each fjf'eJJ?, g - » (JJgf : / ' ) is a measurable function on G. Now, by theorem 5.10, there exists a positive Borel function r o n ^ x l such t h a t for almost all g, rg(x) = r(g,x) for a-almost all x. Therefore, for almost allgr, =

jx{r(g,g-^x)Y'\f(g-^x),r(x)yda(x).

GEOMETRY

208

OF QUANTUM

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The right side is a Borel function of g since the integrand is Borel onGxX, so t h a t g - > < Ugf : / ' ) is a measurable function. But, as noted above, this is sufficient to assure us t h a t U is a representation. When X is the phase space of a classical mechanical system, G = R1, and the action of 0 on X is given by the dynamical group of the system, the representation U in example 1 was considered by Koopman [1] (a is t h e Liouville measure). Prompted by this, we shall call the system of example 2 (of which the system in example 1 is a special case) a Koopman system of imprimitivity. I n this example, the fact t h a t {U,P) satisfies (6) depends decisively on the identities satisfied by the rg. I t is natural to suspect t h a t one m a y be able t o construct other systems of imprimitivity, by using more general cocycles. This is substantially the case. We shall develop this idea in the following sections.

2. H I L B E R T SPACES O F VECTOR V A L U E D FUNCTIONS The Hilbert space in which the system of imprimitivity of example 2 of Section 1 acts is a space of J f - v a l u e d functions. This situation is very typical. We devote the present section to a detailed study of certain algebras of operators in such Hilbert spaces. Let J f be a separable Hilbert space. We use the notation | • | to denote the norm in J f a n d <. , . > to denote t h e inner product in Jf] For a n y operator A of JT we write \A\ for its norm. Let 8 be the set of all bounded operators in Jf! The unitary group M of J f is a subset of S. We shall consider S as a Borel space b y giving it the smallest Borel structure which makes all the maps A - > (Au,v > (u, v e J f ) Borel. When we speak of Borel maps from or into S, this is the Borel structure we have in mind even when it is not made explicit. Lemma 6.2. S, equipped with the above Borel structure, becomes a standard Borel space and M is a Borel subset of it. The maps A, B —>• AB and A, B->A±B are Borel from SxS into S, A,u->Au is Borel from SxX* into J f , while A - > A* is a Borel automorphism of S. The induced Borel structure on M coincides with the Borel structure associated with the strong topology on M. Proof. Choose an orthonormal basis {en} in Jf. Let T be the space of all complex matrices (a iy ). We m a y identify T canonically with a product of copies of the complex plane. T thus becomes a standard Borel space. For A e S, we define (21)

ait(A) =

(Ae^y.

SYSTEMS

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IMPRIMITIVITY

209

Obviously, the Borel structure on S is the smallest one which makes all the maps atj(A -> au(A)) Borel. Thus, if t(A) denotes the element {a{j{A)) of T, t(A -> t(A)) is a one-one Borel m a p of 8 onto a set, say S' contained in T, and the Borel structure on 8 is t h a t one for which i i s a Borel isomorphism with the Borel subspace of T defined by 8'. To prove t h a t 8 is standard, it is sufficient to prove t h a t S' is a Borel subset of T. Only the case dim t>T = oc is of interest. Now, if (aXj) is an element of T, it is easily seen to belong to 8' if and only if there is an integer L such t h a t for any integer N, and any N complex numbers cx. . ., cN with rational real and imaginary parts,

2 J2= I a»ct

< L(|Cl|2+...+|cw|a).

Since the conditions involved are only denumerably m a n y and since each condition defines a Borel subset of T, it follows t h a t S' is a Borel set. This proves t h a t 8 is standard. The equations atj(AB)

=

J^alk(A)akj(B), k

a{j{A*) = atJ(A±B) (Au,eky

aji(A)*9

=

aij(A)±aij(B),

= 2


J

tell us immediately t h a t A, u -> Au, A, B -> AB, and A -> A* are Borel maps with appropriate domains and ranges. Since M = {A : A eS, AA* = A*A

= 1},

it follows from these results t h a t M is a Borel set in 8. Finally, the induced Borel structure on M is the smallest one which makes all the maps A —> Au of M into J T (u e J f ) Borel, and is thus clearly the Borel structure associated with the strong topology on M. Corollary 6.3. Suppose Z is a Borel space and z^> A{z), z-> B(z), and z ~^u(z) are Borel mappings of Z into 8, 8, and JfJ respectively. Then z -> A*(z) and z -> A(z)B(z) are Borel maps of Z into 8 and z - > A(z)u(z) is a Borel map of Z into Jf. Consider now a standard Borel space X, a a-finite measure a on X and the Hilbert space je = ^2(X,Jf,a). For each Borel set fcj, (22)

PE'.f-*XEf

(fe#)

GEOMETRY

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is a projection operator, and P(E - > PE) is a projection valued measure. The main concern of this section is with the algebra of bounded operators of J f which commute with all PE. We consider bounded Borel maps b: (23)

b : x -> 6(a), C = sup |6(a)| < oo

of X into S. If / is any Borel function on X with values in J f such t h a t f \f(x)\2da(x)

< oo, then x -> b(x)f(x)

is also Borel and

f |6(a:)/(z)| 2 da(z) < 0 f

|/(*)| 2 tfa(z).

Thus there exists a unique bounded operator, denoted by b~, such t h a t for each / e Jf for a-almost all x, (24)

(&~/)(s) = &(*)/(*).

6~ is an operator in JfJ its norm \\b~ \\
b~PE =

PEb~.

I t is obvious t h a t if b± is another bounded Borel m a p of X into 8 such t h a t b1(x) = b(x) for a-almost all #, b±~ =b~. We regard the collection of bounded Borel maps from X into 8 as an algebra with involution by defining (a1b1 + a2b2)(x) = a1b1(x) + a2b2(x) (26)

(«i,a 2 complex numbers)

(bc)(x) = b(x)c(x), b*(x) = b(x)*,

for all x e X. Lemma 6.4. For any bounded Borel map b of X into 8, b~ is a bounded operator in 34? commuting with P. Conversely, if B is an operator in M* which commutes with P , there exists a bounded Borel map b of X into 8 such that b~ =B. bx~ =b2~ if and only if bx(x) = b2(x) for a-almost all x. Finally, (&i&2)~ =°i~b2~, (&i*)~ = (&i~)*. In particular, b~ is a unitary operator {projection, normal, etc.) if and only if b(x) is unitary (projection, normal, etc.) in CtC for a-almost all x. Proof. We assume t h a t a ( X ) < o o ; the general case is obtained b y an obvious patching up over sets of finite measure which we leave to the reader. For any u e J f we write, for the purpose of this proof, u for the constant function x - > u of X into J f ; u G J f as a(X) < oo. We observe t h a t if B± and B2 are two bounded operators of & which commute with P, and if i ^ u = B2u for all u e Jf, then B1 = B2. I n fact, for

SYSTEMS

OF

IMPRIMITIVITY

211

U

any Borel set E^X, B1(XEU) = B2(XE )From this it follows t h a t Bxf= B2f whenever / is a Borel function on X (with values in J f ) taking only finitely m a n y values. The general case now follows from the fact t h a t such functions are dense in J^. Thus B1 = B2. This done, let B be a bounded operator in 3^ such t h a t B commutes with P . We m a y assume t h a t ||2?|| < 1 . Let {en} be an orthonormal basis for Jfl Let bn be a Borel m a p of X into JT such t h a t bn = Ben. Now | | J 5 | | < 1 ; so, for a n y N complex numbers cx,-'',cN, as B commutes with P , • • +cNeN)\\2

\\BPE{c1e1+'

= ||x s ( ci £e 1 + . •. + cNBeN)\\2 \\xE(ciei+--+cNeN)\\2,

< i.e., WxsiCiBe!*.

• • + cNBeN)\\2

< ( | C l | 2 + • • • + \cN\2)a(E)

for all E, i.e., f \clb1(x)+

• • • +cNbN(x)\2da(x)

< ( | C l | 2 + • • • +\cN\2)a(E)

for all E. For fixed JV and clf • • •, cN, these inequalities (for all E) imply that |c 1 6i(*)+ • • • +cNbN(x)\2

< | C l | 2 + • • • + \cN\2

for a-almost all x. B y varying N and the c's over complex numbers with rational real and imaginary parts, we can assure ourselves t h a t for some a-null set Z we have t h e inequalities

(27)

|CIM*)+---+%M*)|2

< hl2+--- + M 2

valid for all x E X — Z, all integers JV, and all sets (c1? • • •, cN) of jY-tuples of complex numbers with rational real and imaginary parts. From (27) we conclude t h a t for each xeX — Z there exists an operator b(x) in J f with \b(x)\ < 1 such t h a t (28)

br(x) = b(x)er

for all r. We define b(x) = 0 for xe Z. Equation (28) shows a t once t h a t x - > b(x) is a Borel m a p from X into S such t h a t \b(x)\ < 1 for all x and t h a t for the operator b~ (29)

6~e r = Ber

for all r. From (29) and from the observation made at the beginning of the proof, we deduce t h a t B = b~. To prove t h a t b is unique u p to null sets, let &! be such t h a t 6X~ = 6 ~ . Then for each r, b1(x)er = b(x)er for a-almost all x. This shows a t once t h a t b±(x) = b(x) for a-almost all x.

212

GEOMETRY

OF QUANTUM

THEORY

Routine calculations show t h a t (be)~ = 6 ~ c ~, (& + c ) ~ = 6 ~ + c ~ , and (b*)~ =b~*. If b(x) is unitary for almost all x, then it is obvious t h a t 6~ is unitary. Conversely, if&~6~* = &~*6~ = l, then b(x)b(x)* = b(x)*b(x) = l for almost all x, by the essential one-one nature of the map b -^»b~. Thus b(x) is unitary for almost all x. The assertions involving projection and normal operators are proved similarly. The proof of the lemma is complete. Our next lemma deals with families of operators commuting with P . Lemma 6.5. With notation above, let Y be a standard Borel space and (30)

b:y,x-+b(y,x)

a bounded Borel map of YxX into S. For each y e Y, let b(y,-) denote the map x - > b(y,x) of X into 8. Then the map y ->b(y,-)~ is a bounded Borel map of Y into the Borel space of all bounded operators of <#?. Conversely, if y -^ By is a bounded Borel map of Y into the Borel space of all bounded operators in #? such that By commutes with P for all y, and if v is a a-finite measure on Y, there exists a bounded Borel map b of Y x X into S such that (31)

By =

b(y,.)~

for v-almost all y. b is determined uniquely (v x a)-almost everywhere. If each By is unitary, we can arrange matters so that b(y,x) is unitary for all y, x. Proof. We m a y clearly assume t h a t a and v are finite. Let b be a bounded Borel m a p of 7 x 1 into S, say |6(i/,#)|
= j

which is Borel in y as the integrand is Borel in y, x. We also have || By || < C. Conversely, let y -> By be a bounded Borel m a p of Y into the Borel space of all bounded operators of J f and let By commute with P for all y. We m a y assume t h a t | | ^ | | < 1 for all y. For each y e Y we choose a Borel map by of X into S such t h a t \by(x)\ < 1 for all x and By — by~. Let {en} be an orthonormal basis for J f and for any u e J f let us write u for the constant m a p x -^ u on X. Then, for E^X, {ByXEtm : O

=

(by(x)em,en>da(x),

JE

so t h a t for each Borel set y->\

E^X,
is a Borel function of y. Lemma 5.5 is now applicable and hence we can deduce the existence of complex-valued Borel functions rm n on 7 x 1 such t h a t for each m and n, for v-almost all y, (32)


=

rnfTn(y,x)

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for a-almost all x. There is a set N^ Y of v-measure zero such that for each y e Y — N and all m, n, (32) holds for a-almost all x. Since |6y(a;)| < 1, we deduce from (32) and the Fubini theorem that for (v x a)-almost all y, x,

2 Vn.m(y>*)\2 < °°

(33)

n

for each m, and

(34)

2

2

r

».»( ?>*)«•

Z |ci|2+---+|ck|a

for all Jc and all ^-tuples of complex numbers (cl5 • • •, ck) whose real and imaginary parts are rational. Let Z^Y xX be the Borel set of all (y,x) such that (33) and (34) are satisfied for the indicated ranges of m, k, ct. The complement of Z is obviously v x a-null. Let r* m be defined by (35)

r*m(y,x) =

n,m(y>x)

(y>x)G

z

>

[0 (y,x) $ Z. Then the r* m are Borel functions and the inequalities (33) and (34) are satisfied by the r* m for all y, x. Hence for each y, x there exists an operator b(y,x) such that \b(y,x)\ < 1 and (36)

(b(y,x)em,en> =

r*n(y,x).

Equation (36) shows that (y,x) -> b(y,x) is a Borel map of Y x X into S. Now, rnfm = r*fm (^ x a)-almost everywhere on 7 x 1 , so that we easily deduce from (32) that for ^-almost all y, (by(x)em,eny = (b(yfx)em,eny for a-almost all x, i.e., for y-almost all y, By = b(y,-)~. If b' is another bounded Borel map of 7 x 1 into S satisfying (31) for v-almost all y, it follows from lemma 6.4 and Fubini's theorem that b(y,x) = b'(y,x) for (v x a)-almost all y, x. For the last statement, note that if By is unitary for all y, Z = {(y,x) : b(y,x) is unitary} is a Borel set whose complement is of (v x a)-measure zero. All we have to do is to redefine b(y,x) to be 1 for (y,x) $ Z. The proof of the lemma is complete.

3. FROM COCYCLES TO SYSTEMS OF IMPRIMITIVITY We remarked in our example 2 of Section 1 on the connection between the cocycle identities (35) of Chapter V and the identities (6). The main aim of this section is to show that associated with a cocycle taking values

214

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in a unitary group, there is a system of imprimitivity, and t h a t this system is determined u p to equivalence by the cohomology class of the cocycle. This is precisely the reason for our detailed study of cocycles in the previous chapter. We begin with an elementary lemma. Lemma 6.6. Let G be a Icsc group and Jt? a separable Hilbert space. Suppose that L(g -> Lg) is a mapping of G into the set of bounded operators in J$? such that (i) for all f,ffeJ^, g -> (Lgf,f'y is a measurable function, (ii) Lg is unitary for almost all g, and (hi) Lgig2 = LgiL92 for almost all (91^2) tGxG. Then there exists exactly one representation U of G in £? such that Lg = Ug for almost all g. Proof. Let /x be a left Haar measure on G and ££1 the Banach space of /x-summable functions. F o r / ^ / 2 e J£x we introduce their convolution

(/i*/«)w = | / i ( « ) / > r w ) . fi * / 2 is almost everywhere finite and defines an element of J*?1. Define the operator Lf, for f e J^?1, by setting, for u, v e J^, (37)

(Lfu,v}

= j

f(g)(Lgu,v)dn(g).

f->Lf is linear and \\Lf\\ < | | / | | i . A routine use of the Fubini theorem, coupled with (hi) of the lemma, gives us the equation Lf *f = LflLf2. Moreover, the sum of the ranges of the Lf is dense in ^f; for, if v e 3f and

j /(gKL^vyd^g) = 0 for all u e J4? and / e J?1, then, for each u e Jf, (Lgu,v} = 0 for almost all g. Taking a countable dense set of u's, we obtain the equation Lg*v = 0 for almost all g. Hence v = 0. We now apply a known theorem (Loomis [2], p . 128) and obtain a unique homomorphism g -> Ug of G into the group of invertible operators in J f such t h a t || Ug\\ < 1 for each g, g -> Ug is Borel, and for e a c h / e J?1, f f(gKUgu,v>dp(g)

= Lf

(u, v e

jf).

JG

This implies rather easily t h a t Ug = Lg for almost all g. Ug is therefore unitary for almost all g. Now, the set of g, for which Ug is unitary, is a subgroup. Hence Ug is unitary for all g, i.e., U is a representation. The uniqueness of U is obvious. Let X be a standard Borel G-space. We choose an invariant measure class ^ o n l and fix it throughout the rest of this section. Let a be a measure in this class. We introduce a separable Hilbert space J f and

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215

2

denote its unitary group by M. Let J^ = S£ (X,$f,a). By a cocycle we mean a (6r,X,M)-cocycle relative to a, i.e., a Borel map (g,x) - »
(38)

PEf =

XEf,

and for almost all g, (39)

(UJ)(x)

{r3(g-i-x)yl2cp(g,g-i-x)f(g-i-z)

=

for each f e Jf? for almost all x. Moreover, the equivalence class of depends only on <€ and y^, where y0 is the cohomology class of
(U,P)

Proof. Equation (38) defines PE and P(E -> PE) is a projection valued measure in Jf. Let us now define, for each g e G and / e <#?, the Borel function Lgf on X by (40)

(Lgf)(x)

{rg(g-'-x)Y^(9,9-1-x)f(g-1-x).

=

Since cp takes values in M, we have: \(Lgf)(x)\2

=

r9(g-i-x)\f(g-i-x)\*,

from which we conclude t h a t Lgf e J#* and | | i ^ / | | = ||/||. Lg is thus an isometry. I t is actually unitary because its range is the whole of Jf. In fact, if/' e J f and (Lgf : / ' ) = 0 for a l l / 6 Jtf.\ a brief calculation shows t h a t cp(g,x)*-f'(g-x) = 0 for almost all x which gives us the r e l a t i o n / ' = 0 . Next,

x)

for almost all (gi,g2,%) e GxGxX. From this we conclude by a routine calculation t h a t Lgi92 = L9iLg2 for almost all gl9 g2 (for the rg we use identities (20) of Chapter V). We now verify t h a t g->Lg is a measurable map. We choose a positive Borel function r on Gx X such t h a t , for almost all g, r(g,x) = rg(x) for almost all x. Then, for/, f'e^f, <£„/:/'>=

f

{r(g,g-^x)Y'\
JG

for almost all g. Since the integrand on the right is a Borel function on GxX, the integral is a Borel function on G and hence g -> (Lgf : / ' > is a measurable function on G. We are therefore in a position to apply lemma 6.6. Thus there exists a unique representation U of G in J f such t h a t Ug = Lg for almost all g.

216

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We calculate from (40) t h a t for each g e G, LgPE = PgELg for all Borel sets E^X. Therefore, if N is a Borel set of measure zero such t h a t Ug = Lg for g e G — N, then for each g e G — N, (41)

U9PEUa-i

= Pt.s

for all Borel sets E. Now, as U is a representation, the set of all g e G for which (41) holds for all Borel ^ i s a subgroup of G. As the complement of this subgroup has measure zero, we conclude t h a t (41) must be valid for all g e G and all Borel sets E. (U,P) is thus a system of imprimitivity based on X. Its uniqueness is obvious. To show t h a t the equivalence class of (U,P) depends only on the measure class of a and the cohomology class of 0 be a Borel function such t h a t a — dd\da. Let (IP\P') be the system of imprimitivity associated with a and 95. Then W:f->a~ll2f

(42)

is a unitary isomorphism of J?2(X,Jf,a) (39) shows t h a t pE'

onto J*? 2 (X,J^a') and the formula WPEW-1

=

for all E, and for almost all g, Ug' =

WUgW~\

Once again, as the set of all g such t h a t Ug'= WUgW'1 is a subgroup, these equations are valid for all g. Thus (U,P) and (U\Pr) are equivalent. I t remains to examine what happens when we fix a but change 99. Let
b : x -> 6(a)

of X into i f such t h a t 9'(gr,a;) =

b(g-x)(P(gyx)b(x)-1

for almost all (gr,a;) e GxX. For any f e Jf> \f(x)\2

=

\b(x)f(x)\2,

and hence there exists a unitary operator i? of ££2(X,Ctif,<x) such t h a t (cf. Section 2) (44)

B = 6~.

From (44) we obtain (45)

£P£ = P££

SYSTEMS

OF IMPRIMITIVITY 1

for all Borel sets E. Next we compute BUgB' . almost all g e G, (BUgB-if)(x)

=

217

We have, from (39), for

{rg{g-^x)Y'%(x)9(g,g^.x)b(g-^x)-^(g-^x)

=

{rg{g-^x)y'Y(g,g-i.x)f{g^.x)

= (U„'f)(x) for each / e J4? for almost all x. Therefore (46)

Ug' =

BU.B-1

for almost all g. Using our usual arguments, we conclude t h a t (46) is valid for all g. Equations (46) and (45) show t h a t (U,P) and (U',Pf) are equivalent. This completes the proof of the theorem. Corollary 6.8. If

z) = v)

for almost all x, then for each g e G and each f (UJ)(x)

=

{rt{g-1-x)}»!>v{g,g-i-x)f(g-i-x)

for almost all x. I n this case, g -> Lg itself is a homomorphism so t h a t Lg= Ug for all g. Given
4. P R O J E C T I O N V A L U E D M E A S U R E S I n this book we have come across projection valued measures from time to time, but such uses as we have made of them have been quite superficial. The aim of this section is to describe some of the deeper aspects of the theory of projection valued measures. These results are all known and form the Hahn-Hellinger multiplicity theory. We recommend the reader to the accounts of Stone [2], Nakano [1], and Halmos [2]. We shall confine ourselves to a concise description of the main features of this theory. The main problem in this theory is t h a t of determining when two projection valued measures are equivalent. More precisely, let X be a standard Borel space, J f * a separable Hilbert space, and P{ a projection valued measure in MP1 based on X. We say P1 and P 2 are equivalent, P1~P2 in symbols, when there is a unitary isomorphism W of J^1 onto Jf72 such t h a t PE2=WPE1W-1 for all E. The main problem is, given X, to construct canonical forms for the projection valued measures based onl. We begin by choosing a a-finite measure a on X and a separable Hilbert

218

GEOMETRY

OF QUANTUM

space Jf! Let Jtfp = J?2(X,Jjr,a) E -> PE, where (47)

THEORY

and P be the projection valued measure

PEf=XEf

(/e^T).

We write P=P(Jf,a). The first result of the theory is t h a t P(Jf,a)~ P ( J f ' , a ' ) if and only if (i) a and a' define the same measure class, and (ii) dim J f = dim Jf'. If these conditions are satisfied, if J f = Jf', and if a is a positive Borel function which is a version of da /da, the m a p

W:f->a~ll2f is a unitary isomorphism of ^2(X,Jf,a) onto 3?2(X,Cf,a) such t h a t P(Xcc') = WP(Jf,a)W-\ This result motivates us to introduce the following definition. A projection valued measure P (acting in a separable Hilbert space) is said to be homogeneous if it is equivalent to P(Jf^a) for suitable C/f and a. The definition (47) of P(Jf,'<x) shows t h a t for a Borel set E^X, a(E) = 0 if and only if PE = 0. The measure class of a is thus determined by P in a very direct fashion. I t is called the measure class of P. The dimension of J f is said to be the multiplicity of P. Given any measure class ^ on X and any integer n (1
Jf = © JfB>
be the direct sum of the ^ > a n , and for E^X (49)

let PE be defined by

PE : ( / . . A , / * - • •) -+ ( P f ' ' . / . . ^ * ' / i . " ')

")*CT-finitemeasures a and j3 on X are called mutually singular if we can write X = A U B, where A and B are Borel sets with i n 5 = 0 , f}(A ) = a(B) = 0.

SYSTEMS

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219

(when a n = 0 for some n, J^n = 0 ) . Then P is a projection valued measure. We write P = P({J^}, {an}). The main theorem of spectral multiplicity theoryasserts t h a t every projection valued measure on X in a separable Hilbert space is equivalent to P({J^n}, {an}) for a suitable sequence {an}. If {an} and {an'} are two sequences of mutually singular measures on X, the theorem further asserts t h a t P(VQ,

{«„» - P{{JQ,

K'»

if and only if for each n, an and an' are in the same measure class. Given P, we therefore have for each n = oo, 1,2,••• a canonically determined measure class, say ^fn. ^n is called the measure class corresponding to the multiplicity n. For a given n, c€n m a y be 0. P is homogeneous if and only if c€n = 0 for all n except a single value nQ. The construction of the ^ n from P is delicate. We do not need this for our purposes. On the other hand, consider the set of all cr-finite measures a with the property t h a t a(E) = 0 if and only if E is ^ n - n u l l for all n (1
5. FROM SYSTEMS OF IMPRIMITIVITY TO COCYCLES We now examine the problem converse to t h a t studied in Section 3. We resume our concern with a standard Borel Cr-space X and a system of imprimitivity (U,P) based on X, acting in 3f. These will be fixed throughout this section. The main result is t h a t if P is homogeneous, one can construct a cocycle 9? relative to the measure class ^ of P such t h a t (U,P) is equivalent to the system of imprimitivity associated with ^ and (p. We shall use the notations and results described in Section 4 concerning projection valued measures. I n particular, for each n(co, 1, 2, • • •), J^n is a fixed separable Hilbert space of dimension n. We write Mn for the unitary group of J ^ . The norm and inner product in Ctfn are denoted by |. | and <.,.>, while the norm and inner product in 2tf and , # ^ a [a a a-finite measure on X) are denoted by ||. || and < . : . > . Lemma 6.9. With the notation described above, the measure classes ^n corresponding to the various multiplicities n are all invariant under the action of G. In particular, the measure class of P is invariant. Proof. Choose measures an e ^n. Then, P~P({Jfn}, {an}). We must show t h a t each ^n is invariant under G. Let g e G be arbitrary. Let us

220

GEOMETRY

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write QE = Pg.E. Then Q(E->QE) is a projection valued measure in 3tf. The equation UgPBUg~i = Pg.E shows t h a t Q~P. B u t it is obvious t h a t Q ^ P ( { J ^ } , { a / } ) (via the m a p f-^f9) and t h a t the an9 are mutually singular. Hence for each n, an and 9 an define the same measure class. This proves the lemma. We recall t h a t a measure class ? on 1 is said to be ergodic if a n y Borel function / , with the property t h a t f9 =f ^-almost everywhere for each g e G, is constant ^ - a l m o s t everywhere. Lemma 6.10. If the measure class ^ of P is ergodic, then P is homogeneous. If (U,P) is irreducible, <€ is ergodic and so P is homogeneous. Proof. Choose mutually singular finite measures an (n = oo, 1, 2,• • •) such t h a t we then have P~P({3Tn},{an}). Let %\ be the measure class defined by an. Since the am are mutually singular, we can find mutually disjoint Borel sets En (n = cc, 1, 2,- • •) such t h a t X = [J1^n^o0 En and an(X — En) = 0 for all n. Suppose t h a t the measure class ^ of P is ergodic, b u t t h a t for two distinct values of n, ^ n # 0 . Let an be distinct real numbers and l e t / be the Borel function which takes the value an on En. For each n, f is a constant ^ n - a l m o s t everywhere. Since ^ n / 0 implies an(En)y^0, En is not ^ - n u l l for a n y n for which c€n is nonzero. Thus / is not a constant ^-almost everywhere. B u t (by lemma 6.9) as each ^ \ is invariant, f9=f, ^-almost everywhere for each g e G. This is a contradiction. We m u s t therefore have ^ n = 0 for all b u t a single value of n, i.e., P is homogeneous. The second statement would follow if we show t h a t ^ is ergodic whenever (U,P) is irreducible. Suppose now t h a t (U,P) is irreducible b u t ^ is not ergodic. Then there exists a real Borel f u n c t i o n / such t h a t / i s not a constant ^-almost everywhere but, for each geG, fg=f, ^-almost everywhere. Let c be a real number such t h a t , if E0 = {x :f(x) i-e-> either E0 or X — E0 is null, a contradiction. Theorem 6.11. Let (U,P) be a system of imprimitivity acting in £F such that P is homogeneous and of multiplicity n (\
y*^Z{y)

SYSTEMS

OF IMPRIMITIVITY

221

is a one-one correspondence between the set of all cohomology classes of (G,X\Mn)-cocycles relative to %?P and the set of all equivalence classes of systems of imprimitivity of the form (U\Pn,a). Proof. There exists a unitary isomorphism W of J4? onto Jfn>a such t h a t pn,a

WPEW-^

=

for all Borel sets E. We set Ug'=WUgW~\ Then (U,P)~ (U\Pn>a). na Consider now a system (U',P ' ). Let us choose, for each g in G, a positive Borel function rg such t h a t rg is a version of da/da^9'^. For simplicity of notation we write je'=J^nta, P' = Pn>a. Let (V,P') be the Koopman system of imprimitivity, i.e., for each g e G a n d / e <#", (51)

(Vgf)(x)

/(flT 1 -*).

= {rg{g-^x)Y*

Write (52)

Wg=

V

1

^ ' .

Since both (Uf,P') and (V,P') are systems of imprimitivity, it follows t h a t Wg commutes with PE for all E. Further, g -> Wg is a Borel map of G into the unitary group of 3tf". We now use lemmas 6.4 and 6.5. We select, for each g e G, a Borel m a p wg of X onto Mn such t h a t (53)

Wg = wg~,

and a Borel m a p 99' of G x X into Mn, (54)



such t h a t for almost all g, (55) For gr1; gr2 (56)

W„ = e

@>

we

9'{g,-)~.

obtain from (52), Wgig2 = ( J V ^ g i F g 2 ) J F 9 2 .

A quick calculation shows t h a t

Vg^WgiVg2 = K T 1 ' ) - ,

(57)

where, as usual, for x e X, wggf1}(x)

=

wgi{grx).

From (56) and (57) and lemma 6.4, it follows t h a t for each (gvg2)eGx (58)

w9ig2(x)

=

wgi(g2'X)w92(x)

for almost all x. Then (59)


9'(gL>g2-x)x)

G,

222

GEOMETRY

OF QUANTUM

THEORY

for almost all (glyg2,x) e GxGx X. This proves t h a t cp is a (G,X,Mn)cocycle relative to a. Moreover, as Ug' =VgWg, we have, for almost all g, Ug = Vg
{rg(g-1-x)}^cPf(g,g^-x)f(g-^x)

(EV/)(s) =

for / e J f 7 ' . I n other words, (U',Pn'a) is the system of imprimitivity associated with cp' and a. If 9 / is another cocycle satisfying (60), then, for almost all g,
(61)

WU/W-1

= Ug\

for all g, E. There exists (lemma 6.4) a Borel map w, (62) w : x - > w(x) of X into Mn such t h a t W = w~. But then the equation WUg"W~1= Ugf, together with the relations between n(g,g-1 •x)w{g~1 -x)-1

=

cPf(g,g-1-x),

i.e., w{g-x)^p"{g,x)w{x)-1 for almost all (g,x) eGxX. proof of the theorem.

=
This shows t h a t cp'~cp". This completes the

Remark. I t is easy to see t h a t if the cocycle cp' is a coboundary, the corresponding systems of imprimitivity are equivalent to the K o o p m a n systems. Thus, the examples of Section 5 in Chapter V give rise to systems (U,P) which are not equivalent to the Koopman systems.

6. T R A N S I T I V E SYSTEMS I n this section X continues to be a standard Borel (x-space and (U,P) a system of imprimitivity based on X. We shall completely describe all the transitive systems. Our success is due to the detailed knowledge of cocycles on transitive G-spaces which was accumulated in Section 5 of Chapter V. We recall t h a t a measure class ^ is said to be transitive if there exists x0 e X such t h a t X — G-x0 is a ^-null set. I n this case the

SYSTEMS

OF

IMPRIMITIVITY

223

orbit G • x0 is uniquely determined by <% and we shall say t h a t ^ lives on G-x0. A system of imprimitivity (U,P) is said to be transitive if the measure class ^ of P is such; if X ' is the orbit on which ^ lives, we say t h a t the system lives on X'. Note t h a t if (U9P) is transitive, the measure class of P is ergodic, and hence P is homogeneous (lemma 6.10). If xeX and X' = Gx, X' is an invariant Borel set and is hence a standard transitive Borel 6r-space in its own right. If a is a a-finite quasi-invariant measure on X', the measure a : E -> a'(E C\ X') defines a measure class on X which is invariant and which lives on X'. Obviously it is the only measure class living on X' with this property. Let x0 e X and X' = G-x0. Suppose t h a t GXQ = G0 is the stability subgroup at xQ. Let m be a representation of G0 in the Hilbert space J ^ (which we have chosen once for all) of dimension n (la, associated with cp and a. We shall call such systems, obtained by varying a, 9 (but a living on X') systems induced by the representation mofG0. I t is obvious t h a t all such systems are mutually equivalent, homogeneous, transitive, and live on X'. We now prove the basic theorem of this section. Theorem 6.12. Let X be a standard Borel G-space and let (U,P) be a transitive system of imprimitivity based on X such that (i) P is of multiplicity n, and (ii) the measure class of P lives on the orbit X' = G-x0. Then there exists a representation m of the stability group G0 at x0 in 3fn such that (U,P) is equivalent to a system induced by m. The equivalence class of m is determined uniquely by the equivalence class of (U,P) and gives rise to a oneone correspondence {(U,P)} -> {m} of the set of all equivalence classes of transitive systems living on X' onto the set of all equivalence classes of representations of GQ. Moreover, the commuting ring of (U,P) is isomorphic to the commuting ring of m. In particular, (U,P) is irreducible, or a direct sum of irreducible systems, if and only if m is so. Proof. If m± and m 2 are inequivalent representations of G0, theorem 5.27 shows t h a t the (G,X',Mn)-cocycles which correspond to them define distinct cohomology classes. Therefore we can conclude from theorem 6.11 t h a t the systems of imprimitivity induced by m1 and ra2 are inequivalent. By the same token, if m± and m 2 are equivalent, so are the systems t h a t are induced by m± and m 2 . Next, let (U,P) be a transitive system which lives on X'. Let a be a cr-finite measure belonging to the measure class of P. a(X — Xf) = 0

224

GEOMETRY

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THEORY

obviously. Theorem 6.11 now tells us that there exists some n and a (6r,X,ifn)-eoeyele 9 relative to a such that (U,P) is equivalent to the system associated with

', it is clear that (U,P) is equivalent to a system induced by m. The only thing that remains to be proved is the statement about the commuting rings. We assume (as we may) that X = X'. Let 9 be a strict (6r,X,ikfj-cocycle defining the homomorphism m at x0 of G0. We choose a quasi-invariant cr-finite measure a on X and write 3^ = 3%^^. In our usual notation, for each g e G and Borel E^X, p

(63)

Ef=XEf,

(ugf)(x) = K(r W'Wr W(r **)• We use the results and notation of Section 5, Chapter V. We select a Borel map d(g -> d(g)) of G into Mn such that d{gh)—d{g)m{h) for all (g,h)eGxG0, d(e) = l, and g') = 9''Xo))' Suppose that T is an operator in J f which commutes with all PE and all Ug. By lemma 6.4 we can choose a Borel map t, (64)

t:x-^

t(x)

of X into the Borel space Sn of all operators in Jfn such that |£(a;)| < ||T|| for all x and (65)

T = r

If we use the fact that 2T commutes with all Ug, we obtain, after a brief calculation, the identity (66)


x

• x) = t(x)
for each g for almost all x, i.e., for each g e G (67) for almost all xeX.

9>(gr,a?)*(a;) = t{g-x)
ngg'Wigtf)

for almost all (<7,#') EGXG. Substituting the expression for
t°(g) = d(g)rd(g)-i

SYSTEMS

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225

for almost all g. Replacing g by gh (h e G0) and simplifying, we obtain (69)

mihjrmih)-1

= r.

This shows t h a t r is in the commuting ring of m. On the other hand, if we start from a r in the commuting ring of m, the m a p g -> d(g)rd(g~1) is a bounded Borel m a p of X into Sn which is constant on the left 6r 0 -cosets. Hence there is a bounded Borel map t from X into Sn such t h a t t° is given by (68) for all g. Routine calculation leads us from (68) to (67) to (66). At this stage, we know t h a t T = t~ is in the commuting ring of (U,P). The correspondence T ^± r is obviously one-one and is an algebra isomorphism as is easily seen from the formulas (68) and (65) connecting T and T. Moreover, as each d(g) is unitary, (68) also implies t h a t (70)

P(g)* =

d(g)r*d(g)-\

so t h a t T +t r is adjoint-preserving. Since the commuting ring of a representation (or a system of imprimitivity) determines completely the circumstances under which it is irreducible or a direct sum of irreducibles, the proof of the theorem is complete. Corollary 6.13. Let X be a standard transitive Borel G-space, x0 e X, and G0 the stability group at x0. Then every system of imprimitivity based on X is homogeneous and its measure class is the unique invariant measure class on X. There is a canonical one-one correspondence between the equivalence classes of systems based on X and equivalence classes of representations of G0. This correspondence preserves irreducibility as well as the property of being a direct sum of irreducibles. Remark. Theorem 6.12 gives a complete analysis of the transitive homogeneous systems of imprimitivity based on X. I t is natural to ask what happens when we give up the assumption of transitivity. According to lemma 6.10 if the measure class ^ of P is ergodic, then P is homogeneous. Theorem 6.11 then tells us t h a t we have only to study the cohomology classes relative to %\ However, the examples of Section 5 in Chapter V show t h a t the cohomology classes relative to an ergodic invariant measure class are incredibly more complicated t h a n in the transitive case. In view of this it has not been possible to carry out the analysis of ergodic systems of imprimitivity further t h a n t h a t of theorem 6.11. We note here t h a t if there is any ergodic invariant measure class ^ on X, then one can construct irreducible systems of imprimitivity (U,P) based on X such t h a t ^ is the measure class of P . I n fact, let a be a afinite measure in ^ and let cp be any ( ( r ^ l f ^ - c o c y c l e relative to # . Let 3tf = $f\ta and let (U,P) be the system of imprimitivity associated

220

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with op and a. We m a y take Jf± to be the complex number space so t h a t Jf = J?2(X,a). We claim t h a t (U,P) is actually irreducible. I n fact, let T be an operator commuting with (U,P). By our lemma 9.4 there exists a complex valued, bounded Borel function t such t h a t

(Tf)(x) = t(x)f(x) for all / e 3tf. The condition t h a t T commutes with all Ug gives us (67). But, as t(x) is a complex number, it commutes with g-x, is an a-measurable subset of X. We define (71)

a~(E)

=

a(G-E).

Now the fact t h a t D meets each orbit exactly once implies t h a t , for disjoint E, the sets GE are also disjoint. Hence, a~, as defined by (71), is a measure on the Borel space defined by D and a~(D) = 1. We claim t h a t for a n y Borel set E^D, a~(E) is either 0 or 1. If this were not so, we can choose a Borel set E^D such t h a t 0
SYSTEMS

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227

D — En whenever necessary, we may assume that a~(En) = l for all n. Put (72)

E0 = f| En; n=l

a~(E0) = l. Since the sequence {En} separates the points of D, EQ must consist of a single point, say x0. But then a(G-x0) = l, showing that a is transitive. A variation of the argument given just now leads to: Lemma 6.15. Suppose that there exists a sequence {fn} of invariant real valued Borel functions on X which separate the orbits of X, i.e., two points x,y of X lie on the same orbit if and only iffn(x) =fn(y) for all n. Then every ergodic invariant measure class on X is transitive. Proof. Let a be a measure such that a(X) = 1 and let a be quasi invariant and ergodic. Let &' be the smallest Borel structure on X which makes all t h e / n Borel. Every set in £%' is a Borel subset of X which is also invariant. Hence each such set has a-measure either 0 or 1. Let {En} be a sequence of sets which generates &' such that a(En) = 1 for all n. If we put (73)

E0 = H En,

then a(E0) = 1. Now the class of Borel sets which either contain E0 or are disjoint from it is a a-algebra containing all the En. Hence every set in 08' has this property. This implies that E0 is a single orbit; for, if x0 e E0 and X' = G'X0i the fact that t h e / n separate the orbits implies that (74)

X' = C){x :/»(*) =/„(*«>)}, n

and consequently that X'e&'\ X' = E0.

since X' C\Eo^0,

EQc:X', so that

Corollary 6.16. If 0 is compact, then every ergodic invariant measure class on X is transitive. Proof. Using theorem 5.7 it is easy to come down to the case when X is a compact metric space and G acts continuously on X. We shall verify that the condition of lemma 6.15 is satisfied. By the Stone-Weierstrass theorem, C(X) is separable. Let {un} be a countable set dense in C(X), and for u e C{X) let (75)

u(x) =

u(g-x)dp(g),

where /x is the (normalized) Haar measure on G. u is continuous on X. Clearly each u is invariant. We claim that {un} separates the orbits in X.

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Let x, y e X such t h a t un(x) = un(y) for all n. Since {un} is dense in C(X), we deduce from (75) t h a t u(x) — u(y) for all u e C(X). Suppose now x and y were not in the same orbit. Then Gx and G-y are closed disjoint subsets of X, and hence there is a u e C(X) with 0
7. E X A M P L E S A N D R E M A R K S We shall now discuss a number of examples. The simplest example of a transitive space is G itself. If (U,P) is a system based on G, it is transitive and its measure class is the invariant measure class of G. We now apply the results of Section 6. The stability group at e e G is {e} whose only representations are trivial. Any trivial representation is a direct sum of trivial one-dimensional representations. We thus obtain Theorem 6.17. Let G act on itself by left translations. Let 3^ — where /x is a left Haar measure on G, and let P and U be defined by:

^2{^),

(UJ)(x) = f(g-ix). Then (U,P) is an irreducible system of imprimitivity based on G. Any irreducible system based on G is equivalent to this one. An arbitrary system of imprimitivity is a direct sum of irreducible systems. Rotations in the Plane. Let X = C, the complex plane, and let G be the multiplicative group of complex numbers of modulus 1. We put, for z e G and x e C, z-x — zx. X is a Borel 6r-space. For each

d>0,

Ed = {x : \x\ = d} is an orbit and these are all the orbits. Since G is compact, all ergodic measure classes are transitive. For d>0, the stability groups associated with Ed are trivial and hence, corresponding to each d>0, there exists, u p to equivalence, exactly one irreducible system living on Ed. We shall now describe this. Note t h a t Ex has an invariant measure, say a, such t h a t a(E1) = l. If we define the measure ad by f(x)dad(x) J Ed

=

f(dx)da(x), JE±

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229

then ad(Ed) = l and it is the unique invariant measure living on Ed with this property. Let J^d be the Hilbert space if?2(crd). Then, if we define (Ud,Pd) by PB*

(U/f)(x) d

= XEL =

f(z~ix),

d

(U ,P ) is the unique irreducible system living on Ed. When d = 0, Eo = {0} and the stability group is G itself. I n this case, P is trivial and U can be an arbitrary representation of G. If (U,P) is to be irreducible, U will have to be a character of G. The example discussed just now can be generalized. Let us assume t h a t X — Rn and G=SO(n), the group of orientation preserving rotations. The orbits in X are the spheres Sd of radius d > 0 and the origin 0. Let us choose d > 0 and consider the point xd = ( 0 , 0 , . . . , 0 , d ) . Then the stability subgroup xd is canonically isomorphic to SO(n—l). Since G is compact, we know from corollary 6.16 that there are no ergodic systems other t h a n the transitive ones. For each d>0, there are denumerably m a n y inequivalent irreducible transitive systems living on Sd; these correspond one-one to the equivalence classes of irreducible representations of 80(n — 1). However, when we proceed to describe explicitly the irreducible systems living on SXi for example, we run into a difficulty. The theory developed in Section 5 of Chapter V and in the present chapter directs us to start with an irreducible representation, say m, of SO(n — l) and construct a strict (GfSx)-cocycle which defines the representation m at the point (0, • • •, 0, 1). Such cocycles can be constructed, of course, but they depend on the choice of a section for the left coset space SO(n)ISO(n—l). Since canonical sections do not exist, our theory does not yield any simple geometric description of the unique irreducible system of imprimitivity living on S± which corresponds to m. This difficulty, although of no theoretical importance, can be quite irritating when we want to make any calculation. We shall next proceed to discuss a geometric method of writing down systems of imprimitivity which does not rely on the construction of sections. This method is of course not very general, but it covers most of the cases which occur in physics. Special Descriptions. Let I be a standard transitive Borel (r-space, x0 e X. Let G0 be the stability group at x0. We assume t h a t X possesses an invariant a-finite measure, say a. Let m be any representation of G0 in a separable Hilbert space Jfi We assume that there exists a separable Hilbert

230

GEOMETRY

OF QUANTUM

space Jf and a Borel homomorphism operators of C/f' such that

THEORY

m' of G into the group of invertible

(a)

Jf is a closed linear manifold of JT',

(b)

m'(h)u

J

(77)

= m(h)u

J

>

for all h e G0

and

u e Crff.

Let | • | and <(.,. > denote the norm and inner product in Jf'. Note t h a t the operators m'(g) for g e G0 need not be unitary. Let us consider now any point XEX. Since X is transitive, there exists a g e G such t h a t g-x0 = x. We now define the closed linear manifold Jf^c; Jf' by: (78)

Xx =

ro'(0)[Jf].

Equation (78) defines CtiTx without ambiguity. We now introduce an inner product < . , . > x in 3JTX, by choosing g e G such t h a t g • x0 = x and putting (79)

(u,v}x

= (m'ig-^m'ig-^v}

(u9veJTx).

Once again, (79) is well defined. The fact t h a t C/fx and <.>•>* defined and t h a t m is a homomorphism of G implies t h a t (80)

Xg.x

(81)

=

are

we

ll

m\g\Xx\

(m'(g)u,m'(g)v)g.x

=


for all (g,x)eGxX. I n particular, m'(g) is a unitary isomorphism of Jf^ onto J^.^. We now introduce the vector space y of a-equivalence classes of Borel maps f: (82)

f:X-+X",

such t h a t (83)

/(x)eJTx

for all x e X. If / , / ' e y (as usual we make no distinction between functions satisfying (83) and the equivalence classes which they define), we observe t h a t x —> (f(x),f'(x)}x is a Borel function. To prove this, we choose a section c, i.e., a Borel m a p c of X into 6r, (84)

c :x->

c(x)

c(x)-x0

— x

such t h a t c(x0) = e and (85) for all x. Then, (86)

* = <m'(c(s)"*)/ (*),m'(c(z) - i)/'(*)>

and, as the m a p g -> m'(^) is Borel from G into the Borel space of operators on Jf', the Borelian nature of x->{f(x)J'(x))x follows from corollary 6.3.

SYSTEMS

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231

For fe'f", we p u t ll/ll 2 = f Jx

(87)


Let

•** = {/: I/II2 < ^ } -

(88) For (89)

fJ'eJf,

= £xd«(ar)

is clearly finite. < . : . > is an inner product for Jf. We define for g e G and E^X, P

(90)

= XEL = m'(g)f(g-i.x),

EI

(Ugf)(x)

for all / e V. I t is clear t h a t PEf and Ugf e ^ t h a t PE and £7g are linear transformations of y^ and t h a t both J7ff and P £ leave Jf7 invariant; it is also clear t h a t UgPE = PgEUg for all g e (x, and Borel sets E^X. Theorem 6.18. Jf7 is a separable Hilbert space, and (U,P), as defined by (90), is a system of imprimitivity in Jf'. Moreover, the equivalence class of (U.P) is the one induced by the representation m of G0. Proof. We shall use the section c : x -> c(x) satisfying (85) for all x e X. With the help of c we m a p 3tf* on the Hilbert space S£2{X^a). We define for/e^J/by (91)

(Jf)(x)

=

m'(c(x)~i)f(x).

Since x->f(x), g-^m'(g), and x^c{x)~x are Borel maps with appropriate domains and ranges, it follows t h a t Jf is a Borel m a p of X into Jf. Moreover, (86) implies t h a t (92)


=

<(Jf)(x),(Jf)(x)>.

J is also linear and one-one. Therefore we conclude t h a t J is a linear isomorphism of J4? onto ^f2(X,J^,a) which preserves the inner products, i.e.,

= <Jf'.Jf>

(/£•#)•

This proves t h a t J f is a separable Hilbert space and t h a t J is a unitary isomorphism of 34? onto ^2(X,Jf,a). We then p u t Vg (93)

=JUgJ'\

9

QE

9

=

J *EJ

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GEOMETRY

OF QUANTUM

THEORY

(V,Q) is a system of imprimitivity based on X, acting in j£?2(X,Jf^a). Using (90) and (91), an easy computation reveals t h a t f o r / 0 e j£? 2 (X,J^a), (94)

QEf0 = XEU

and (95)

(Vgf0)(x)

=

m'(c(x)-igc(g-i.x))f0(g-i-x).

Now, the element c(g • x) ~ xgc(x) lies in G0 because it sends x0 to . (g. x) = x0. Therefore, if we define (96)


c(g-x)~1

m(c(g-x)-1gc(x)),

then g, x ->

Therefore

(Vgf0)(x)

=


Equations (97) a n d (98) show t h a t (V,Q) is the system associated with m. Since (U,P)~(V,Q)} the proof of the theorem is complete. We shall discuss a number of situations where the conditions (77) connecting G0, G, and m are satisfied for every irreducible representation m of Go. (i) G is compact. I n this case, let m be an irreducible representation of G0. If we decompose the representation of G, induced by m, into irreducible components, and select any irreducible, m! say, which occurs in the decomposition, it follows quite readily from the Frobenius reciprocity theorem (cf. Weil [1]) t h a t the restriction of m! to G0 contains m as a subrepresentation. (ii) G is a connected, simply connected complex semisimple Lie group, GQ compact. Here, G is known to be unimodular so t h a t X admits an invariant Borel measure. From the general theory of the semisimple groups (cf. Helgason [1]), we know t h a t there exists a maximal compact subgroup K containing G0. Let m be an irreducible (finite dimensional) representation of G0 acting in Jf! By (i) above, there exists a finite dimensional Hilbert space J f ' containing J f and an irreducible representation m! of K in J T ' such t h a t m'(h) = m(h) for all h e G0. Now, it is well known (cf. Weyl [2], p. 267) t h a t there exists a homomorphism m" of G into t h e group of invertible operators of X' itself such t h a t m"(k) = m'(k) for ke K, and t h a t m" is even unique if we require t h a t the m a p g -> m"(g) is complex analytic on G. Thus the conditions of theorem 6.18 are satisfied

SYSTEMS

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233

iii this case. In particular, if G0 itself is maximal compact in G, the systems of imprimitivity based on GjGQ can be described in the geometric form of theorem 6.18. We shall do this in Chapter I X , when G = SL(2,C) and G0 = SU(2,€), in connection with the equations for the Dirac electron. We thus have: Theorem 6.19. Let G be a connected, simply connected complex semisimple Lie group, and G0 a maximal compact subgroup of G. Let a be an invariant Borel measure on X = GjG0. Let m be an irreducible representation of G0 in a Hilbert space Jf, and let m'(g -> m'{g)) be the unique complex-analytic homomorphism of G into the group of invertible linear endomorphisms of J>f such that m'(h) = m(h) for all h e G0. Then there exists exactly one map x —> <•>•>* °f X into the set of positive definite inner products on J f x J f such that < . , . }XQ = < . , . } and (u,v)x

=

(m'(g)u,m'(g)v\.x

for all (g,x) e G xX. such that

If J#* is the Hilbert space of Borel maps f of X into Jf

(99)

|!/1| 2 = j

(f{x),f(x))xda(x)

< oo,

and we define P and U by (100) then (U,P) by m.

PsfJ = XXef, J (UJ)(x)

=

m'(g)f(g-i-x),

is a system of imprimitivity

equivalent to the system

induced

Representations in Vector Bundles. The reader might have noticed t h a t the essential point of the proof of theorem 6.18 is contained in the fact t h a t the family of subspaces x -> C/fx transform " covariantly " under the action of G. I t is natural to suspect t h a t a more general formulation of theorem 6.18 can be obtained in the context of such covariant families. This is indeed so, and the natural setup for formulating this is that of a vector bundle, more precisely, a Hilbert space bundle. Let X and B be both standard Borel G-spaces and let X be transitive. We assume t h a t there is a Borel map (101)

TT:B->X

of B onto X with the property t h a t for each g e G, the diagram B-^->

(102)

J X

—

B

234

GEOMETRY

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THEORY

is commutative. If we write, for each g e G, D(g) for the automorphism of B induced by g, and x —> g • x for the automorphism of X induced by g, (102) translates to (103)

*(/%)(&)) = g-TT{b)

for each be B and each g e G. We shall assume also for simplicity that X admits an invariant a-finite measure, say a. We shall say that B is a G-Hilbert space bundle if, for each x e X, (104)

Bx = w-1({*})

is a separable Hilbert space whose natural Borel structure is the one induced on Bx by B and if, for each g e G and x e X, (105)

b -> D(g)b

(b e Bx)

is a unitary isomorphism of Bx onto Bg.x. If we write | - j ^ and <.,. ) a . for the norm and inner product on 2^, (105) means (106)

<6,6'>x =
(b, b' e Bx).

In particular, if Gx is the stability subgroup at x, D(g) leaves Bx invariant for g eGx and g -> D(g) defines a representation of Gx in Bx. We shall now consider sections of the bundle. A section of B is a Borel map/, (107)

f'x^f{x)

of X into 2? such that f(x) e Bx for all x e X. To handle the measurability problems involving these sections, we transform them into Borel maps of X into a single Hilbert space. Choose a point x0 e X and a Borel map c from X into G such that c(#0) = e and C\Xj '

XQ

==

3/

for all x e X. Since the map g,b -> D(#)& of Gx B into 2? is Borel, it follows quite simply that for any section f of B, x-> D(c(x))~1f{x) is a Borel map of X into I?Xo. Conversely, if x ~>f0(x) is a Borel map of X into l?Xo, then x -> Z>(c(#))/0(#) is a section of the bundle B. We denote by ^ the vector space of a-equivalence classes of sections. For any x e X, and any section /, * =


This shows that x-> |/(#)|£ is a Borel function on X. We shall say t h a t / is a square integrable section if

(108)

ll/l2 = J |/(*)|§ia(*) < 00.

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235

Yitff denotes the subset of y consisting of square integrable sections, then £F is a pre-Hilbert space under the inner product (109)

= £


The isomorphism, which associates with each section / , the m a p x —> D(c(x))~1f(x), induces a unitary isomorphism W of Jtf* onto ^2(X,BXQ,a). £F is thus a separable Hilbert space. For any Borel set Es^X we write (HO)

PE!=XEJ

(/6JT).

For each g e G, we define the operator Ug by (111)

(UJ)(x)

= D(g)f(g-i-x)

(fetf).

Computations similar to those of theorem 6.18 then give rise to the following theorem. Theorem 6.20. (U,P) is a system of imprimitivity. / / D° is the representation of the stability subgroup GXQ (of G at x0) in the Hilbert space BXQ, then (U,P) is equivalent to the system induced by D°. As an interesting example of representations in such bundles we mention the following. Let X be a O00 Riemannian manifold, and let each g E G be an isometry of X. We assume t h a t G is a Lie group acting transitively on X in such a fashion t h a t g, x —> g x is a O^-map of G x X into X. Let B be the tangent bundle^ of X. For each g, let D(g) be the differential of the isometry x-^gx. Then all the conditions of the preceding theorem are satisfied. The system of imprimitivity thus acts on the Hilbert space of square integrable vector fields and is equivalent to the system induced by the representation of the stability group G0 at x0 e X defined in the tangent s p a c e | to X at x0. I n Chapter I X , we shall see how the representations associated with the photon can be formulated, in the spirit of theorem 6.20, as acting in the Hilbert space of sections of a certain vector bundle on the light cone, leading thereby in a very simple way to the equations of Maxwell. Ergodic Intransitive Systems. The last example is simply t h a t of a standard Borel 6r-space which has ergodic nontransitive (invariant) measure classes. We take real numbers Al5 A2,- • • which are rationally independent and take X to be a torus having its dimension equal to the number of A's (finite or infinite). Let 6 r = R \ and for teR1 and (112) f Complexified.

* = (£i,£2,"-)

(\Q = 1 for all j )

236

GEOMETRY

OF QUANTUM

THEORY

in X, let (113)

(eUKiU,euH2,--).

tx =

Then, it is classical (Kronecker) that each orbit is dense and that the Haar measure on X is invariant and ergodic but not transitive.

8. SEMIDIRECT PRODUCTS We shall now use the theory of systems of imprimitivity to obtain the description of all irreducible representations of an interesting class of lcsc groups which are neither compact nor abelian. This class of groups includes the inhomogeneous Lorentz group and the group of Euclidean motions as members. For the special case of the Lorentz group, the analysis which we shall give now was carried out by Wigner in his 1939 paper. The general study is due to Mackey and comes out as a simple consequence of the work in Section 6. We begin by introducing the concept of semidirect products. Let A and H be two groups and for each h e H let (114)

th:a-^h[a]

be an automorphism of the group A. We shall assume that h -> th is a homomorphism of H into the group of automorphisms of A so that h[a] — a for all a e A if h = eH, the identity of H, (115)

thlh2 =thlth2

(h^eH).

Equation (115) converts A into an H-space. We shall now make G=

HxA

into a group by defining the multiplication in G by (116)

(h,a)(h',af) = {Kh\ath[a'}).

It is easy to verify that this definition converts G into a group with e = (eH,eA) as its identity (eA being the identity of A). Further, (117)

(M"1 =

(h-^h-^a-1]).

G is called the semidirect product of H and A relative to t and we put (118)

G=

HxtA.

When no confusion can arise as to what t is, we omit it, and write G = Hx'A. But it must be remembered that (i) A and H play very unlike roles in the formation of G, in contrast to the construction of direct products, and (ii) G can be formed only after specifying t, i.e., specifying how / / " a c t s " on A.

SYSTEMS

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237

Suppose now that A and H are lcsc groups and that the map (119)

h,a->h[a]

is continuous from H xA into A. Then, if we equip G with the product topology, it follows quite easily from (116) and (117) that G becomes a topological group. Clearly, G is a lcsc group also. We shall denote this topological group also by HxtA. We shall fix A, H, and t throughout this section. A quick calculation shows that (120)

(h,a)(h,,a,)(h,a)-1 =

{hh'h-1,ah[a']hh'h-1[a-1]).

It follows from (120) that (121)

A-

={(eH,a')}

is a closed normal subgroup of G and that

(122)

(MXetf^'XM - 1 = ( e H } # > _ 1 ) .

In particular, the inner automorphism of G induced by (h,eA) coincides on A with the automorphism (eH,a') -> (eH,h[a']). We put (123)

H~ = {(h,eA)};

then # ~ is a closed subgroup of G. Identifying A with A ~ and H with H~ we find G = AH, (124)

{c} = An

H, 1

h[a] = hah' . The inhomogeneous linear groups are typical examples of semidirect products. In these examples, A is a real (or complex) finite dimensional vector space and a —> h[a] is an invertible linear endomorphism of A. H is thus identified with a subgroup of the group of all invertible (linear) endomorphisms of A, which is usually closed. The Euclidean group and the inhomogeneous Lorentz group fit into this general framework. For the Euclidean group, ^4=R n and H is the group 80(n). For the inhomogeneous Lorentz group, J = R 4 and H is the group of all invertible linear transformations of R4 which preserve the quadratic form coordinates on A). It undoubtedly would be of great interest therefore to develop a theory of the representations of semidirect products. In complete generality this has not been done. However, under certain restrictive hypotheses, we shall construct a theory of representations of semidirect products which goes far enough to yield all physically important results. From now on, through the rest of

238

GEOMETRY

OF QUANTUM

THEORY

the section, we shall assume that A is a closed, abelian, normal subgroup of G, H a closed subgroup of G such that the conditions (124) are satisfied. The first step of our theory is to obtain the representations of A. The irreducible representations are the characters of A, i.e., continuous homomorphisms of A into the group of complex numbers of modulus 1. Let A be the set of all characters-of A. Under pointwise multiplication, A becomes a group also. We shall equip A with the topology of uniform convergence on compacta. Then it can be shown t h a t A becomes a topological group and is in fact a locally compact Abelian group satisfying the second axiom of countability (cf. Pontrjagin [1]). Let us then take a o--finite measure a on A and consider the Hilbert space J#p = J£?2(A,a). For each a e A a n d / e Jf let (Uaf)(x) = x(a)f(x)(x e A), where x(a) is the value of the character x at a. Then a -> Ua is a representation of A. These representations do not exhaust the representations of A. To obtain more general representations we replace a by a projection valued measure P based on A. Let J f be a (separable) Hilbert space and P a projection valued measure based on A acting in 3tf. For each a e A let Ua be the operator in Jf defined by

= jx(a)dVftf.(z)9 where

vu.(E) =
Ua = f

x(a)dP(x).

I t is easy to show t h a t Ua is unitary and t h a t a —> Ua is a representation of A in Jtf. Conversely, it is a known theorem t h a t if U is any representation of A in Jf, there exists a unique projection valued measure P based on A such t h a t (126)

Ua=

f

x(a)dP(x)

for all a e A. P is called the projection valued measure corresponding to U. The formula (126) shows how U can be constructed from P. Thus, for instance, it is quite clear t h a t if T is an operator commuting with all PE, then T commutes with all Ua. Conversely, let T commute with all Ua. Then T commutes with all the spectral projections of all the Ua. Now, if r]a is the continuous m a p x - > x(a) of A into the group M of complex

SYSTEMS

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239

IMPRIMITIVITY a

numbers of modulus 1, it follows from (126) t h a t Q {F -> Pv-i(F)) projection valued measure based on M and

is a

Ua = f zdQa(z). JM

This shows t h a t the Pv-i(F) are the spectral projections of Ua and hence t h a t T commutes with all of them. Since the sets of the form rja~1(F) generate the Borel structure of A, it follows t h a t T commutes with P. The reader who wants to study the (Fourier) analysis of representations of A m a y consult Loomis [2, Chapter V I I ] . Let us now consider the lcsc group G and a representation U of G in a separable Hilbert space Jtf. Our method of analysis of U is essentially the Fourier analysis of the restriction of U to A. We begin this analysis by first constructing the " a d j o i n t " action of H on A. Lemma 6.21. Let h e H. Then, for each x e A there exists one and only one y e A such that (127)

y(a) =

x(h~\a\)

for all aeA. If we write y = h[x], then h, x -> h[x] is continuous H x A into A and A becomes a H-space.

from

Proof. That y is unique and well defined is obvious. The remaining assertions are easy to prove. We omit the proofs. Not every representation of A in 30? can be enlarged to a representation of G in ^f. Our next question is: when can this be done? Lemma 6.22. Let U and V be representations of A and H, respectively, in a separable Hilbert space Jf and let P be the projection valued measure on A corresponding to U. Then, a necessary and sufficient condition that there should exist a representation W of G in ^ whose restrictions to A and H are U and V, respectively, is that (V,P) is a system of imprimitivity for H, based on A. In this case, W is unique. Proof. Let W be a representation of G in J f and let Ua= Wa (aeA), Vh= Wh (h e H). U and V are representations of A and H, respectively. Now hah'1 = h[a] so t h a t (128)

VhUaVh-i

=

UhM

for all (h,a) e H xA. Let P be the projection valued measure on A corresponding to U. Now, a routine calculation based on transport of structure shows t h a t the projection valued measure of the representation a - > VfrUaVh'1 of A is E -> VhPEVh~x, and t h a t corresponding to the

240

GEOMETRY

OF QUANTUM

THEORY

n

representation a -> Uh[a^ is E -> Ph[Ev l view of the uniqueness of the projection valued measures which correspond to representations of A we infer from (128) t h a t (129)

VhPEVh~i

=

Ph[E].

(V,P) is t h u s a system of imprimitivity for H based on A. Conversely, let us start with U, V, and P such t h a t P and V satisfy (129). Then U and V satisfy (128). Define W on G by (130)

Wah = UaVh.

The relations (128) are now enough to secure the fact t h a t W is a homomorphism. Since a^ Ua and h -> V h are Borel, x -> Wx is Borel. W is t h u s a representation of G and its restrictions to H and ^4 are V and £/, respectively. Since G = AH, W is unique. The lemma is proved. The lemma just proved enables us to establish a one-one correspondence between representations of G and systems of imprimitivity of H based on A. Lemma 6.23. A representation W of G is irreducible if and only if the corresponding system of imprimitivity for H based on A is irreducible. Two representations of G are equivalent if and only if the corresponding systems of imprimitivity are equivalent. Proof. Let If be a representation of G in JF and let (V,P) be the corresponding system of imprimitivity, also acting in J^. We use t h e notation of lemma 6.22. If T is an operator in J f , t h e n T commutes with all the Ua (a e A) if and only if T commutes with all PE. From this it follows t h a t T lies in the commuting ring of W if and only if T lies in t h e commuting ring of (V,P). This proves the first assertion. For the second, let Wl be a representation of G in Jf ( , and ( V \ P l ) the corresponding system of imprimitivity (i = l, 2). If T is a unitary isomorphism of J f 1 onto ^f2, it follows from the uniqueness of the correspondence Ul <± Pl t h a t Ua2 = TUa1T-1 for all a e A if and only if P £ 2 = T P / T " 1 for all Borel sets E^A. From this t h e second assertion of the lemma follows quickly. The preceding lemmas and the theory of Sections 5 and 6 reduce t h e problem of describing the irreducible representations of G to the problem of describing the irreducible representations of the various stability subgroups of H with respect to its action on A. I n view of t h e applications t o the study of relativistic equations we proceed to spell out t h e details of the relevant constructions. We choose a point x0 e A and a cr-finite measure a which is quasi invariant on t h e J^-space A and which lives on t h e orbit H[x0] = X. Let HXQ = H0 be t h e stability group a t x0, i.e., (131)

H0 = {h:heH,

h[x0] = x0},

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OF

IMPRIMITIVITY

241

and let m be a n irreducible representation of H0 in a separable Hilbert space Jfl L e t us define Jtf* = JP2(A,Jf,a) a n d let (V,P) be a system of imprimitivity for H based on A, living on the orbit X, and induced b y the representation m. W e p u t , for ah e G (a e A, he H) and / e <#?, (132)

(Wahf)(x)

= x(a)(VJ)(x)

(x e A).

Since PEf=%Ef for all Borel sets E^A, it follows easily t h a t , for t h e representation a - > Ua of A corresponding t o P, (133)

(Uaf)(x)

= x(a)f(x)

(xeA),

so t h a t TF a/l = UaVh. Thus TF is a representation of G. From our work in Sections 5 a n d 6 we know t h a t t h e equivalence class of the system (V>P) depends only on t h e measure class of a a n d t h e equivalence class of m. From lemmas 6.22 a n d 6.23 it follows t h a t W is irreducible. W e shall say t h a t W is associated with x0 a n d m. The following theorem is now an immediate consequence of our work in Sections 5 a n d 6 of t h e present chapter. Theorem 6.24 (Mackey [2] [3]). Let us choose, for each H-orbit (o in A, one point x^ on o>, and an irreducible representation m, of the stability subgroup of H(0 at the point xco. Then the representation ~Wm,l° of G associated with x^ and m is irreducible. Wm,co and Wm',co' are equivalent if and only if the orbits a> and to coincide, and the representations m and m' are equivalent. If the H-orbit structure of A is smooth, then each irreducible representation of G is equivalent to some Wm,co.

N O T E S ON C H A P T E R V I The theory of induced representations is originally due t o Frobenius who studied it for finite groups. I t s generalization t o t h e category of locally compact second countable groups is due t o Mackey. Nowadays Mackey's theory is referred t o as t h e Mackey Machine; it is also known as t h e little group method among physicists. Significant contributions were also made b y I.M. GeFfand a n d M . A . N a i m a r k in their work on representations of complex classical groups (cf. Trudy Mat. Inst. Steklova, 36 (1950), pp. 1-288), a n d b y F . B r u h a t (Bull. Soc. Math. France, 84 (1956), pp. 9 7 205) who developed t h e theory of induced representations for Lie groups with t h e techniques of t h e theory of Schwartz distributions. Especially noteworthy are the various expositions of Mackey in: The Theory of unitary group representations in Physics, Probability, and Number Theory, Benjamin/ Cummings, Reading, Mass., 1978; Harmonic analysis as the exploitation of symmetry—ahistoricalsurvey,Bull. Amer.Math. S o c , 3 (1980),pp.543-698. Unitary representation theory is so vast and currently so active t h a t it is

242

GEOMETRY

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THEORY

not possible to begin to sketch its outlines. For some notable surveys see: I. M. Gel'fand, Proceedings of the International Congress of Mathematicians Stockholm, 1962, pp. 74-85; the articles in Harmonic Analysis on homogeneous spaces, Proceedings of Symposia in Pure Mathematics, Vol. XXVI, Amer. Math. Soc. Providence, R.I., (ed.) C.C.Moore, 1973; and V.S. Varadarajan, Harmonic analysis on real semisimple Lie groups, Proceedings of the International Congress of Mathematicians, Vancouver, 1974, pp. 121-127. The theory for semisimple groups is mainly the creation of Harish-Chandra; for this, see Harish-Chandra, Collected Papers, 4 Vols., Springer-Verlag, New York, 1984; see also Harmonic analysis and representations of semisimple Lie groups, Reidel, Dordrecht, 1980, edited by J. A. Wolf, M. Cahen, and M. de Wilde. Recently, interest has intensified in representation theory of infinite dimensional groups; for a view through the theory of infinite dimensional Lie algebras see the book of Kac and the references therein: V. G. Kac, Infinite dimensional Lie algebras, Birkhauser, Boston, 1983.

CHAPTER VII MULTIPLIERS 1. T H E P R O J E C T I V E G R O U P Earlier in Chapter I I I we showed t h a t in any mathematical description of a quantum mechanical system the requirement of covariance introduces a representation of t h e symmetry group 0 of the system into the group A u t ( ^ ) , where Sf is the set of all states of the system in question. If we assume t h a t the logic of the system is standard, this representation can be replaced by a representation of G into the group of all symmetries of the Hilbert space underlying the logic. If 0° is the subgroup of G consisting of all elements in the same connected component as the identity of G, then it can be shown t h a t when G is a Lie group the symmetry corresponding to each element of G° is a unitary rather t h a n an antiunitary operatorf. However, these unitary operators are not uniquely determined. Each of them can obviously be multiplied by a complex number of modulus 1 (called a phase factor) without changing the induced automorphism of the logic. I t is therefore not immediately obvious t h a t we have a unitary representation of G°, i.e., t h a t the phase factors involved are all removable. I n this chapter we shall examine this question. We remark t h a t it was first studied by Wigner [1]. He proved, among other things t h a t , if G is the inhomogeneous Lorentz group, any representation of G in A u t ( ^ ) can be induced by a unitary representation, at least of the universal covering group of G. This result was the starting point for his classification of relativistic wave equations. Wigner's discussion of this problem, however, was restricted to the Lorentz and rotation groups. I t was Bargmann [1] who first examined these problems systematically. He obtained m a n y general theorems which included Wigner's results as special cases. I n his work on group representations, Mackey was led to some of these f If x e G is of the form y1 for y e G, then the symmetry associated with x is necessarily unitary. Now, if G is a Lie group and G° the connected component of the identity, then for any x of the form exp X (X in the Lie algebra of G), x = y2, where 2/ = exp J X . Since the range of the map X —> exp X contains a neighborhood of the identity, it follows t h a t every element in a certain neighborhood N of the identity of G is mapped into a unitary operator in a given representation. Thus the elements of the group generated by the elements of N are also mapped into unitary operators. But G° coincides with this group. 243

244

GEOMETRY

OF QUANTUM

THEORY

problems from a purely mathematical point of view. I n his paper [6] he formulated the basic concepts of a theory of such phase factors for arbitrary lcsc groups and showed how the subject can be fitted into t h e general framework of the theory of group extensions. This is the point of view taken in this chapter. W i t h every phase factor, or multiplier as we call them, we shall, following Mackey, associate a group extension of the unit circle by the given group and reduce the solution of problems involving t h a t multiplier to problems involving this group extension. The subject then becomes essentially a p a r t of a cohomology theory of certain types of extensions. Our purpose is not to develop a general theory of group extensions. Such a general discussion m a y be found, in various contexts and a t widely differing levels, in the works of Hochschild [1], [2], Calabi [1], Moore [1], Mackey [7], [8], Kleppner [1], and a number of other authors. Our aim is t h e much more modest one of developing in some detail the basic concepts of group extensions, so as to be able to obtain the solutions to the problems arising in connection with the physical space-time symmetry groups. We might mention t h a t H e r m a n n Weyl (cf. [1]), has also emphasized the fact t h a t it is the automorphism of the logic and not the underlying symmetry t h a t is of significance in physics. Some of the facts about projective representations m a y be found in his book. Let J f be a complex separable Hilbert space. We denote by °ll the unitary group of 3f. If we equip °li with the strong topology, we have seen in Chapter V t h a t ^ becomes a metrizable second countable topological group and t h a t its Borel structure is standard. We denote by ^ the set of all elements of °ll which are multiples of the identity operator. 2£ is a closed, normal, subgroup of °U and is central, i.e., each element of 3£ commutes with all the elements of °ll. The group (i)

& =

#/#

is called the 'proactive group of 34?. We write IT for the canonical homomorphism (2)

ir:U-*ir{U)

(UeW)

of °U onto 8P. Theorem 4.27 now tells us t h a t the subgroup of the group of automorphisms of the logic of 3PP, consisting of automorphisms which can be induced by unitary operators, is isomorphic to ^ . We shall equip SP with the quotient topology. Thus, a set A<^0> is defined to be open if and only if TT~1(A) is an open subset of °li. I t is easy t o show t h a t SP becomes a metrizable topological group satisfying the second axiom of countability and t h a t IT is an open continuous m a p of °U onto ^? If we write (3)

T = {z : z e C,

\z\ = 1},

MULTIPLIERS

245

then the kernel of n is precisely the set of operators of the form z\ (z e T), and is hence a compact group. I n all statements involving the topology and Borel structure on &, we shall refer only to the above-described topology of & and its natural Borel structure. Lemma 7.1. The following statements concerning a sequence of elements in % are equivalent: W (ii) (hi)

{Un}n=s012...

\\ -+ \\ for all f,f eX, there exists a sequence {zn} of complex numbers of modulus 1 such thatznUn-> U0in W, 7T(Un)->7r(Uo)in0>.

Proof, (ii) => (iii) obviously. On the other hand, for fixed / , / ' e tf, U -> |<£7/,/'>| is a continuous function on °ll which is constant on each i^T-coset. Hence, by virtue of the fact t h a t 3P has the quotient topology, it defines a continuous function on &. The implication (iii) => (i) follows from the assertion t h a t this function is continuous on ^ We shall complete the proof of the lemma by showing t h a t (i) => (ii). We begin the proof with a simple remark. Suppose U is an operator of je such t h a t | < ^ / J ' > | = | /'> I f o r a l l / , / ' £ . # : Then V = t\ for some t e T. I n fact, for a n y / , Ufe{f}11, so t h a t Uf is a multiple off. Since this is true for each / , U must be a multiple of the identity, U = tl. I t is obvious t h a t \t| = 1, and thus t e T. This said, we proceed to show t h a t (i) => (ii). Replacing Un by UJJQ'1 we m a y assume t h a t U0 = 1. Since || Un \\ < 1, there exists a n operator U with || U|| < 1 and a subsequence {Un]c} such t h a t (Uf,f'} for a l l / j / ' e ^ f (Dunford-Schwartz [1]). From our remark in the previous paragraph, we conclude t h a t U = 11 for some t e T. Since each Un is unitary, UnJ-> tf for each / so t h a t UUk —>- tl. Since we can apply this argument to a n y subsequence of {Un}, it follows t h a t {U^ U2, • • •} is a set with compact closure and t h a t its only limit points are of the form £1 (t e T). Choose a unit vector / and write (for sufficiently large n) zn = ( Unf,f} ~x | < Unf,f} |. We claim t h a t 1 is the only limit point of the sequence {znUn}. If znJJUk -> V, then V = tl for some t e T, and as iznkUnJjy is real and > 0 for all sufficiently large k, t = l. Since {z1U1, z2U2, • • •} has compact closure, we see t h a t znUn -> 1. F o r / , / ' 6 / a n d F G f , we write dftr(ir(V)) = | < F / , f >|. Corollary 7.2. The topology for 0* is the smallest one with respect to which the maps dur are all continuous. If Z is a Borel space and z -> U(z) is a map of Z into %, then z -> TT(U(Z)) is a Borel map of Z into 0> if and only if * -> \ | = |< W > | for a U / , / ' e tf if and only if 7T(V) = TT(W).

GEOMETRY

246

OF QUANTUM

THEORY

^ is hence Hausdorff. Let D be a countable dense subset of Jtf*. Then an easy argument shows t h a t 3^ is also the smallest topology with respect to which the maps dfJ, (f, / ' e D) are continuous. Hence y is metrizable and lemma 7.1 now implies the first statement. The second statement is now clear. Lemma 7.3. Let fe Jf7 be nonzero and let (4)

Qf = {U :Ue<%,

<£//,/> is real and > 0}.

Then Qf is a Borel set in % containing 1 which meets each ^-coset at most once. Moreover, 7r[Qf] is open in & and -n is a homeomorphism of Qf onto Proof. That Qf is a Borel set containing 1 is obvious. If (Ufjy > 0 and t e T, (tUf,fy > 0 if and only if t= 1. Thus Qf meets each iF-coset a t most once. On the other hand, if <£//,/> = 2 ^ 0 , then for V=\t\t~1U, one has < VfJ} = \t| > 0. This proves t h a t (5)

T T - V W / ] ) = {U:UeW,

* 0}.

Equation (5) shows a t once t h a t n[Qf] is open in 0*. I t remains to prove t h a t 7ris a homeomorphism onQ f . Let {Un}n = 0 1 . . . b e a sequence inQ f such t h a t 7T(Un)->7r(U0) in ^ . By lemma 7.1 there exists a sequence {tn} in T such t h a t tnUn^U0 in # . Since \\=-+\\=, it follows t h a t {tn} itself converges to 1. This shows t h a t Un - > U0. As rr is continuous, this proves t h a t IT is a homeomorphism and completes the proof of the lemma. Theorem 7.4. & is a standard Borel group. There exists a Borel map c from & into °U, (6)

c : x - > c(x)

such that (i)

(7)

TT(C(X)) = x

for

all

xe&,

(ii)

c( w (l)) = l,

(iii)

there is an open set containing TT( 1) on which c is continuous.

The range of c is a Borel subset of % which meets each 3£-coset exactly once. Finally, iff is a map of & into some Borel space X, f is Borel if and only if the map IJ -^ f (TT(TJ)) of °ll into X is Borel. In particular, A<^& is Borel if and only if ' rr~1(A) is a Borel set in °ll. Proof. We begin our proof with the construction of c. Let {fl9 • • •, fn, • • •} be a countable dense set of nonzero vectors and let (cf. (4))

(8)

Qn = Qfn-

MULTIPLIERS

247

We now define sets E±, E2,• • •<^°llas follows: (9)

E, = Ql9

and for n > 1,

(10)

En = Qnn " h V : <£%/<> = 0}. t=l

Let 00

(ii)

^ = u tf». n=l

i£ is a Borel subset of ^ . We claim t h a t E meets each iF-coset exactly once. I n fact, let V e % and let m be the smallest of integers i such t h a t /i>#0; it is then obvious t h a t V& nE=V& nEm consists of one element. E is thus a Borel set meeting each iF-coset exactly once. The canonical m a p TT is thus one-one on E and maps E onto 0. Since 0 is a separable Borel space and E is standard, the image of any Borel subset Fs^E under TT is a Borel set in 0. Hence TT is a Borel isomorphism on E. I n particular, 0 is a standard Borel space and the m a p c, which inverts the restriction of TT to E, is a Borel m a p of 0 into ^ such t h a t TT(C(X)) = X for aW.x e 0. We now observe t h a t 2 ^ = d and l e ^ . Hence we use lemma 7.3 to conclude t h a t ^[E-^ is open in 0 and t h a t c is continuous on ir[E{\.

2. M U L T I P L I E R S AND P R O J E C T I V E

REPRESENTATIONS

After these preliminaries involving the projective group, we are in a position to introduce the basic concepts centering around the so-called projective (or ray) representations of a group. We begin with the concept of a multiplier. Let G, K be lcsc groups with K abelian. By a K-multiplier for G we mean a Borel m a p m (12)

m : x, y -> m(x,y)

of G x G into K such t h a t (i)

m(x,yz)m(y,z)

(ii)

m(x,e) = m(e,x) = 1

= m(xy,z)m(x,y) for all

for all

x, y, z e G,

a; e 6r.

Note t h a t we are writing K multiplicatively and t h a t we write 1 for the identity of K. If K = T, the multiplicative group of complex numbers of modulus 1, we omit the prefix K and speak simply of multipliers. If K is additively written, the equations (13) become (i)

m(xy,z) + m(x,y) = m(x,yz) + m(y,z)

(14) (ii)

m(x,e) = m(e,x) = 0

for all

xeG.

for all

x,y,ze

G,

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GEOMETRY

OF QUANTUM

THEORY

1

It follows from (13), on putting y — x' , z = x, that m(x,x~1) = m(x~1,x)

(15) for all x e G.

The set of all i£-multipliers for G is obviously a commutative group under pointwise multiplication. We denote this group by MK'(G). Two K-multipliers m1} m2 for G are said to be similar, mx~m2 in symbols, if there exists a Borel function a on G with values in K, a : # -» a(#) such that

(16)

^ ^ - s ^ " ^

for all x,y EG. Note that a(e) is necessarily equal to 1. If a ^-multiplier m is similar to the multiplier 1 : x, y —> 1, we shall say that m is exact. Thus m is exact if and only if for some Borel map a of G into K, (17)

m(*,*/) =

v U)

a(x)a(y)

for all x, y E G. From (17) it is easily calculated that the exact K-multipliers for G form a subgroup EK{G) of MK\G). We form the quotient group (18)

MK(G) = MK'(G)IEK(G)

and call it the K-multiplier group of G. When i£= T we write M(G) for MT(G) and call it the multiplier group of G. Suppose that m is a multiplier for G. A mapping U:g^Ui

9

of G into the unitary group % of a separable Hilbert space 3f is said to be an m-representation if (i) g-> Ug is Borel, i.e., g -> (Ugf,f'y (19)

is Borel for all

J J

(ii) ff. = l, (hi) ^ = m(x9y)UxUy

for all

x,yEG.

A Borel mapping UgofG into ^ is said to be a projective representation if there exists a multiplier m for # such that U is an m-representation; then m is obviously uniquely determined by U. We shall say that m is the multiplier of U. If £7 is a projective representation of (^ in ^ the map (20)

iroU:g-*7r(Ug),

MULTIPLIERS

249

of G into the projective group ^ is Borel and (iii) of (19) shows t h a t TT o U is a homomorphism. Conversely, we shall prove t h a t every Borel homomorphism of G into 3P arises from a projective representation in this fashion. Theorem 7.5. Let m be any multiplier for G and let U be any m-representation of G acting in £F. Then TT O U is a continuous homomorphism of G into 0*. If V is any projective representation of G in 3f such that TT o U = IT o V, then the multiplier of V is similar to m; and conversely, if m' is a multiplier similar to m, there is an m'-representation V with IT O Y — TT O U. Suppose u

:g->ug

is a Borel homomorphism of G into 0. Then there exists a projective representation U of G in #? such that (21)

u =

TT

o U;

moreover U can be chosen to be continuous in some open subset of G containing 1. Finally, if m is any multiplier for G, there exist m-representations of G. Proof. Let U be a projective representation of G in Jf. Then x-^ir( Ux) is a Borel map of G into 0. Since Uxy is a constant multiple of UxUy, 7r (Uxy) = 7T(Ux)7T(Uy). Thus TT o U is a homomorphism of G into ^ B y lemma 5.28,770 U is continuous. Suppose V is another projective representation of G in Jtf* and let m and m' be the multipliers of U and U', respectively. Then TT O U = TT O U' if and only if for each x e G there is an a(x) 6 T such t h a t Ux' = a(x)Ux. The equation TJX~XTJx' = a(x)l shows t h a t the function a(x -> a(x)) is Borel. If such an a exists, then a direct calculation shows t h a t

(22)

m>(x,y)

=

m(x,y)J^L

for all x, y e G. Thus m~mf. Conversely, suppose m~m''. Then (22) is satisfied for a suitable Borel function a on G with values in T and, by setting Ux =a(x)Ux, we obtain an m'-representation V with TT O U' = TT o

U.

Next, let u : g -> ug be a Borel homomorphism of G into ^ ; u is continuous b y lemma 5.28. We select a Borel m a p c of ^ into ^ satisfying the conditions (i) to (iii) of (7). Let (23)

Ug = c(ug).

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GEOMETRY

OF QUANTUM

THEORY

Since c is Borel and U = c o u, U(g -> Ug) is a Borel map of G into be an open set containing 7r(l) on which c is continuous. Then U is continuous on the open subset u~1(A) of G containing 1. If x, y e G, then it is obvious that there exists an m(x,y) e T such that (24)

Vxy = m(x,y)UxVv.

Since UxyUy~Wx^

= m(x,y)l,

it follows that x, y —> m(x,y) is a Borel map from GxG into T. The equation uxuyz = uxyuz, expressed in terms of the Ug% gives (i) of (13), while the relations uxe = uex = ux lead to (ii) of (13). m is, in other words, a multiplier for G, and U is an m-representation, continuous around e. Finally, let m be an arbitrary multiplier for G. We write J f = j£?2(6r,/xr), where /*r is a right Haar measure on G. For any a;e(? and / e <#?, we define (25)

(Uxf)(g) =

m(gix)-y(gx).

It is readily verified that £7e = l, Uxy = m(x,y)UxUy. The equation || /7 X /|| 2 = ||/1| 2 shows that each Ux is unitary. If/,/' e ^

<£W> = jGMg^)-1f(gx)r(grd^r(g), and as the integrand is a Borel function on G x X, # -> (Uxf,f'} is a Borel function on 6r. £7 is thus an m-representation. The proof of the theorem is complete. Corollary 7.6. Let m be any multiplier for G. Then m is similar to a multiplier which is continuous on some open subset of GxG containing (e,e). Proof. Let U be an m-representation of G, acting in a Hilbert space Jtif. From theorem 7.5 we know that there exists a multiplier m' similar to m and an m'-representation V acting in 3f such that (i) TT O U'—TT O U, (ii) for a suitable open subset N of G containing e, x —> Ux is continuous from N into °ll. Let iV\ be an open set containing e such that N^^N. As tf*y' = m'(x,y)Ux'Uy', and as 6T/ is continuous on N, it is clear that m' is continuous on N-^xN^ Corollary 7.7. Let m be a multiplier for G and U any m-representation of G in J^. Then m is exact if and only if there exists a representation U' of G in 3f such that TT O U' — TT O U. If TJ" is another representation of G in J f such that TT o U" = TT o U, there exists a continuous homomorphism x of G into T such that (26) for all g eG.

Ug" = XgVg'

251

MULTIPLIERS

Proof. The first assertion is an immediate consequence of theorem 7.5. For the second, we note that U'g~1Ug" = xg-l for each g e G, where X9 e T. Then g -> %g ™ Borel and multiplicative. Therefore x is a continuous homomorphism of G into T.

3. MULTIPLIERS AND GROUP EXTENSIONS The obvious problem now facing us is of course that of finding, for given lcsc groups G and K (K abelian), the if-multiplier group MK(G). If, for example, K=T and M(G) is trivial (as it may well be), then we know that every multiplier for G is exact. In this case there is no practical difference between projective and ordinary representations. Our aim is to examine this problem at least for special classes of G and K. We introduce in this section a very useful technique to study the multipliers for a given group, namely the method of group extensions (cf. Mackey [6]). Suppose G and K are two lcsc groups. We shall, as always, assume that K is abelian. By an extension of K by G we mean a triple (H,i,j), where H is a lcsc group, i is an isomorphism of K onto a closed normal subgroup of H, and j is a homomorphism of H onto G with kernel i[K]. We require that i be a homeomorphism and that j be continuous. {H,i,j) is said to be a central extension if i[K] is contained in the center of H. In the language of cohomology theory, (H,i, j) is an extension of K by G if and only if the sequence (27)

0

>K-^->H-^>G—>0

is exact. If (H,i,j) and (H',i',j') are two extensions, then we say that they are equivalent, if there is an isomorphism q of H onto H\ such that, the diagram

(28)

G

K H'

commutes, i.e., for all h e K, (29)

q(i(k)) = i'(k),

and for all h e H, (30)

j'(q(h)) = j(h).

Once again we require that q be a homeomorphism, i.e., q is an isomorphism of the lcsc group H onto the lcsc group H'. We recall that for two lcsc groups A and B, any continuous homomorphism of A into B, which is

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one-one and maps A onto B, is necessarily a homeomorphism (cf. Pontrjagin [1]). We shall now associate with any central extension (H,i,j) of K by G, a K-multiplier for G. Since H is lcsc and i[K] is a closed subgroup, there is a Borel section c based on G, i.e., a Borel m a p c of G into # , (31)

c:G->H

such t h a t c(e) = i(l), the identity of H, and for all x e G, j(c(x))

= x

(cf. theorem 5.11). Therefore, the m a p (32)

x,k->

c(x)i(k)

is a one-one Borel m a p of the standard Borel space Gx K onto the standard Borel space H. Consequently, it is a Borel isomorphism. We denote by d, the inverse of this map, which is a Borel isomorphism of H onto GxK: (33)

d : c(x)i(k) - » x, k.

If x, y e G, c(xy) and c(x)c(y) belong to the same ^[TTj-coset of H, so t h a t (34)

i-1[c{xy)~1c(x)c(y)]

m(x,y) =

is an element of K. Clearly m is a Borel m a p of G x G into i£ and m(x,e) = m(e,x) = 1 for all # e 6r. If we now remember t h a t i[m(x,y)] commutes with all elements of H, we find, for x, y, z e G, i[m(x,yz)m(y,z)]

c(xyz)~1c(x)c(y)c(z)

=

c(xyz)-1c(xy){i[m(x,y)]}c(z)

=

= c(xyz) ~ 1c(xy)c{z)i[m(x,y)] = Thus m is a K-multiplier (35)

i[m(xy.z)m(x,y)].

for G. We note at the same time t h a t

c(x)i(k)c(y)i(k')

=

c(xy)i[m(x,y)kk']

for all x, y e G and k, k! e K. If we use the fact t h a t d is a Borel isomorphism between the Borel spaces H and G x K, then (35) tells us t h a t the definition of a product operation in Gx K given by (36)

(x,k)(y,k')

=

(xy,m(x,y)kk')

converts GxK into a Borel group, and t h a t d provides a Borel group isomorphism of this group with H. The converse to the result described just now is a much more difficult question. Is it possible to associate with each ^-multiplier for G a central extension of K by G\ The answer is affirmative as the following theorem, due to Mackey [6], shows.

MULTIPLIERS

253

Theorem 7.8. Let m be any K-multiplier. If we define for (x,k), (y,k') e GxK, (37) (x,k)(y,k') = (xy,m(xy)kkf), then GxK becomes a standard Borel group under this product, and there exists a unique Icsc topology for GxK such that (a) GxK becomes a Icsc group, denoted by G xmK, under this topology, and (b) the Borel structure of this topology coincides with the product Borel structure on GxK. If (38)

JoM = 9, then (G xmK, i0, j0) is a central extension of K by G. Every central extension of K by G is equivalent to one such. The extensions corresponding to the K-multipliers mx and m2 are equivalent if and only if' m1 and m2 are similar. Proof. We begin with a central extension (H,i,j )ofKhyG and associate with it a i£-multiplier m, using the development contained in the equations (32) through (36). Then d : c(x)i(k) -> (x,k) is an isomorphism of H onto G xmK, where G xmK is considered a group with (37) defining the product operation. G xmK is obviously a standard Borel group and is also Icsc if we equip it with the topology which makes d a homeomorphism. It is obvious that the extensions (H,i,j) and (G xmK, i0,j0) are equivalent. Conversely, let m be a if-multiplier for G. Then under (37), GxK becomes a standard Borel group, as is readily verified with the help of the relations (13). We write G xmK for this group. The relations m(x,e) = m(e,x) = l show that {(e,Jc) : k e K} is a central subgroup of G xmK. Clearly i0(k -> (e,k)) and j0((g,Jc) -> g) are Borel homomorphisms. We shall next prove that if A is a Haar measure on K and //, is a left Haar measure on G, JJL X A is a left invariant measure on G xmK. In fact, if/is a fx x A-summable Borel function on G x K and (g0,k0) e G x K, we have, by Fubini's theorem, j j f((go,h)(9,k))d(fix\)(g,k)

= Jj

GxK

f(g0g,m(g0,g)k0lc)d(fxx\)(g,k)

GxK

=

fa(jKftot0MW))dfito)

= £ ( £ /(flW*)d/*to))«*A(i) = jK ( £ f(g,kW{g))d\(k) = jj f(g,k)d(^xX)(g,k).

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\i x A is thus left invariant. Theorem 5.41 now enables us to conclude t h a t GxmK becomes a lcsc group under its Weil topology. The Borel structure of this topology is by choice the product Borel structure on G x K, and it is the only lcsc topology with this property, as proved in t h a t theorem. The m a p j 0 : g, k - > g is a Borel homomorphism of the lcsc group G xmK onto the lcsc group G and is thus continuous. This proves t h a t i0[K] = kernel (j0) is closed in G xmK. A similar reasoning shows t h a t i0 is continuous also. Therefore (G xmK, i0,j0) is a central extension of K by G. The only thing t h a t remains to be proved is t h a t the extensions GxmiK and G xm2K are equivalent if and only if m1^m2. For a n y i£-multiplier ra, let us write (39) Suppose m1c^m2.

Hm = G

xnK.

Then there exists a Borel map a, a:g-+

a(g)

of G into K such t h a t /

/ = ml{x,y)

m2{x,y)

a(xy) ——

for all x, y e G x G. If we write (40)

q(g,k) = (g,a(9)k),

then it is trivial to check t h a t q is a Borel group isomorphism of Hmi onto Hmk)) = g for all g e G. Let a(g) e K be denned by (41)

q(g,l) =

(gAg))-

a is clearly a Borel m a p of G into K. Then, as (<7,l)(e,&) = (g,k) we find Q(g>k) = (g,a(9))(e,k)

= (gMg)k)-

(in Hm2)

MULTIPLIERS

255

If we now use the fact t h a t q preserves the group structures, we can conclude, after a brief computation, t h a t . m2(x,y)

.

. =

a(xy) mi{x,y)^-^)

for all x, y e Gx G. I n other words, m1~m2. proved.

The theorem is completely

Remark. The essential difficulty of the theorem proved above resides in the fact t h a t when the if-multiplier m is not continuous on GxG, the product topology would no longer serve to make G xmK lcsc. I n fact, if m is continuous on GxG, it is obvious t h a t under the product topology, Hm is a topological group and is even a lcsc group. Since the product topology generates the product Borel structure, it follows from the uniqueness of the Weil topology t h a t the two topologies coincide. Conversely, if the Weil topology for Hm coincides with the product topology, the m a p x - > (x,l) of G into Hm is continuous and hence x, y -> (x,l)(y,l) = (xy,m(x,y)) is also continuous, showing t h a t m is continuous. Note t h a t , in this case, c0 :#—>(#,1) gives a global continuous section for Hmji0[K]9 i.e., c 0 is continuous and Jo(c0(x)) = x for all xeG. This shows t h a t , by using continuous if-multipliers, we can construct only those central extensions (H,i,j) of K by G for which there exists a continuous section c from G into H. There are m a n y examples where such global sections do not exist. For instance, if K is finite but both H and G are compact and connected, no continuous section for Hji[K] can be defined on all of G. An example of this is obtained when we take II to be SU(n,€), the special unitary group in ^-dimensions, and K as the center of H. On the other hand, one can impose certain conditions on K under which any K-multiplier for G will be similar to one which is continuous in some open neighborhood of (e,e) in GxG. Corollary 7.6 tells us t h a t this is so when K=T. More generally, it can be shown t h a t this is the case when K is a compact Lie group, or when K and G are both Lie groups. We shall call a K-multiplier locally continuous if it continues on some open set containing (e,e) in Gx G. The assumption of local continuity for m is of a much less serious nature t h a n the assumption of global continuity. When m is locally continuous, it is almost obvious t h a t one can construct, using local mappings, a fiber space topology for Hm over G which is locally trivial and which coincides with the Weil topology. WTe shall derive this as a corollary of theorem 7.8. The result is stated in corollary 7.10. We need an auxiliary result. Corollary 7.9. Suppose m is locally continuous. Then there exists an open set N around e such that x-> (x,l) is a homeomorphism of N into Hm.

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Proof. We select a symmetric open set Ax containing e with compact closure such that m is continuous on A± xA1 and select A2, A3,- • • such that each An is open, symmetric, contains e, and An + 1An + 1^An for all n. Let {xn} be a sequence in A3, let x0 e A3, and let xn-^ x0. We shall show that (xn,l) -> (#0>1) in the Weil topology of Hm. Write yn = x0~1xn. Then Vn e A2, yn -» e, and (42)

(^ 0 ,l)(i/ n ,l)(e,m(^ 0 ,i/ n )- 1 ) = (s n ,l).

Since the map k -> (e,k) is continuous from i£ into Hm and since m is continuous on A1xA1, (42) shows that (xn,l) will converge to (x0,l) provided (yn9l) -> (e,l). Consider now ^42 x iC =j 0 _1 (^4 2 ). This is an open subset of Hm containing (e,l) and (yn,l) belongs to it for all n. In order to prove that (yn,l) -> (e,l) it is enough to prove that \\L(yntl)f—f\\ —> 0 for all / in J?2(Hm) which vanish outside A2xK. Let D be the set of all functions on GxK, continuous with respect to the product topology and vanishing outside subsets of A2 x K which are compact in the product topology; it is enough to prove that L(Vntl)f-+f for a l l / e D. F i x / e D and let C c A2 and i£x c= i£ be compact sets such t h a t / vanishes outside CxK±. Now,

<W><*,*>=/(^*.^f If # £ (7, yn~1x $C for sufficiently large n and (LVntlf)(x,k) ==f(x,k) = 0 for sufficiently large w. If # e A2, the continuity of m on ^4X x ^ implies that (L(yntl)f){x,k) -+f(x,k) for all keK. Now, |/(x,Jfe)| < B for all (a;,ifc) e G x K and so, \(L(yrifl)f)(x,k)\
(43)

JJ|A,„,i)/-/l 2 ^xA)^0. GxK

We have thus proved that x-> (x,l) is continuous from A3 into Hm. As A3 is compact, it is a homeomorphism. This finishes the proof of the corollary. Corollary 7.10. i / m is continuous around (e,e), there exists an open set N containing e such that the Weil topology on the open set N xK coincides with the product topology. Proof. Let AT be an open set around e such that x -> (#,1) is a homeomorphism of N into Hm. Since (x,k) = (x,l)(e,k) and since x-+ (x,l) and k -» (e,k) are both homeomorphisms, the corollary follows at once.

MULTIPLIERS

257

Remark. Corollary 7.10 shows t h a t when m is locally continuous, Hm, as a fiber space over G, has a locally trivial fibration. For a more intensive study of the topology of Hm we refer the reader to the work of Calabi [1]. The main source of technical complications in our theory is the fact t h a t the topology on Hm in general bears no visible relation to t h a t of G and K. We now describe another corollary to theorem 7.8 which gives an example of the nature of the indirect arguments needed in analyzing Hm. Corollary 7.11. If K and G are both connected (resp. simply connected), so is Hm. If K and G are compact, so is Hm. Both the assertions follow from the fact t h a t i0[K] is a closed subgroup of Hm and G is isomorphic to Hmli0[K], Our first use of these group extensions is a description of the similarity classes of iC-multipliers. Let m be a iC-multiplier for G, and Hm the corresponding group extension. For any Borel m a p a of G into K with a(e) = 1, let ca(x) = (x,a(x))9 ma(x,y)

= m(x,y)

-

I t is easily seen t h a t ma = mfl2 if and only if there is a Borel homomorphism k of G into K such t h a t ax(x) = k(x)a2(x) for all x e G. The m a p ca - > ma establishes a correspondence between multipliers similar to m and Borel sections, and reduces the analysis of the similarity class of m to the analysis of the geometry of Hm as a fiber space over G. Theorem 7.12. ca is continuous (respectively continuous around e) if and only if ma is continuous (respectively continuous around (e,e)). m is exact if and only if there exists a Borel section c which is a homomorphism of G into Hm. Proof. Regarding continuity we shall prove only local assertions; the global ones follow from similar reasoning. Suppose now ca is continuous on an open set N around e. Choose an open set Nx around e such t h a t N^^N. Then the m a p (45)

q:x,y->

ca(xy)ca( y) ~ ^(x)

~x

is continuous from N± x Nx into Hm. A simple calculation yields (46)

q(x,y) =

(e.m^x^y)-1).

Since k - > (e,k) is a homeomorphism, (46) shows t h a t ma is continuous on N1xN1. Conversely, let ma be continuous on N x N for a suitable N. If we form the group Hma, we know from corollary 7.9 t h a t d:x->(xi\)

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GEOMETRY

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THEORY

is a homeomorphism of an open set iVi^iV containing e into Hma. Now, from the relation between m and ma, it follows quickly t h a t p : x, k —> x, Jca(x) ~x is a Borel group isomorphism of Hm on Hma. Hence p is a homeomorphism. Therefore, the m a p p'1 °d is a homeomorphism of Nx into # m . B u t (p'1 o d)(x) = (x,a(x)) = ca(x). Therefore ca is a homeomorphism of N1 into i/ m . Now we come to the question of exactness. By very definition, m is exact if and only if for some a, ma(x,y) = 1 for all x, y e G, i.e.,

<«"

«•*> - S ^ S i

for all x, y e G. A trivial computation shows t h a t (47) is true if and only if x->(x,a(x)) is a homomorphism of (? into Hm. The proof of theorem 7.12 is complete. A trivial reformulation of the second half of this theorem gives the following corollary: Corollary 7.13. m is exact if and only if there exists a subgroup G' of Hm such that G' is a Borel set meeting each i0[K]-coset exactly once, i.e., G' O i0[K] = {(e,l)} and G' -i0[K] = Hm. We shall now make a brief digression by obtaining some more corollaries which m a y serve as illustrations of the use and applicability of the preceding theory. Corollary 7.14. Let G be connected and abelian. Then, for a K-multiplier m to be exact, it is necessary and sufficient that m(x,y) — m(y,x) in some neighborhood of (e,e). Proof. T h a t an exact multiplier is symmetric (when G is abelian) follows at once from (17). Conversely, let N be an open set containing e such t h a t m(x,y) = m(y,x) for all x,y e NxN. Then, the formula for multiplication in Hm shows t h a t (x,k) and (y,k') commute whenever x and y belong to N. Since G and K are connected, so is Hm and hence the elements of the form (x,k) (x e N) generate Hm. This shows t h a t Hm is itself abelian. Since K is isomorphic to Tr x Vs, it follows from the structure of locally compact abelian groups t h a t {e} x K is a direct summand of Hm. Corollary 7.13 now implies t h a t m is exact. Corollary 7.15. There exists a multiplier m' similar to m such that m' is locally bounded as a map ofGxG into K (cf. Section 5 of Chapter V). Proof. Let a be chosen so t h a t ca is a regular Borel section, i.e., x-> (x,a(x)) is a m a p of G into Hm which sends compact sets of G into sets with

MULTIPLIERS

259

compact closure (in Hm). Denning q as in (45), it is obvious t h a t q maps any compact subset of G x G into a subset of Hm which has compact closure. The formula (46) tells us now t h a t ma is locally bounded. We shall end this section with an application to the theory of projective representations. Theorem 7.16. Let m be any multiplier for G and let U(g -> Ug) be an m-representation of G in a (separable) Hilbert space Jf. Then (x,k) - > k~1Ux is a representation of Hm in Jtf*. Conversely, if (xjc) -> V(Xtk) is a representation of Hm such that V^^^k'1! for all k e T, and if we write Ux = V(Xfl), then U(x -> Ux) is an m-representation of G and F(a.tfc) = ^" 1 ?7' x . Proof. The proof needs only minor calculations and is omitted. Corollary 7.17. Let m be any multiplier for G. Then there exists an irreducible m-representation of G. Proof. Let V be a representation of Hm such t h a t F(e>fc) = A;~1l for all keT. Such V exist, by theorem 7.5, because G admits ra-representations. Let us decompose V into a direct integral of irreducible representations V,

(48)

V~ |V
(cf. Mautner [1], Mackey [5]). Then, from the theory of the direct integrals of representations, it is easy to see t h a t for some £ 0 , Ffe°ffc) = ^ ~ 1 l for all k e T (in fact ^-almost all £ will have this property). Write V°= V*o, and let U° be the corresponding m-representation of G. Since F°-r>fc) = ^~ 1 Z7 a , 0 , the irreducibility of V° implies t h a t of U°. Corollary 7.18. / / G is compact and m is a multiplier for G, then there exist finite dimensional m-representations of G. The irreducible m-representations of G are finite dimensional and every m-representation of G is a direct sum of irreducible m-representations. Proof. We observe t h a t Hm is compact (corollary 7.11). The present corollary now follows in exactly the same fashion as the previous one. The proof is even more elementary, since only direct sums, rather t h a n direct integrals, of representations, are involved.

4. M U L T I P L I E R S F O R L I E G R O U P S We shall now make a deeper study of the group extensions Hm under the assumption t h a t K and G are both connected Lie groups. Since K is abelian, it would be a direct product of a real vector group and a torus.

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The first basic result of this section tells us t h a t Hm becomes a Lie group in a natural manner. The way is then opened for the application of results of Lie group theory. We shall use these to show t h a t any Kmultiplier for G is similar to one which is globally analytic on G x G (at least when G is simply connected), and t h a t the elements of the multiplier group are in natural one-one correspondence with equivalence classes of central extensions of the Lie algebra of K by the Lie algebra of G. For general information and terminology involving Lie groups and Lie algebras we refer the reader to the books of Che valley [1], [2], [3] and Jacobson [2]. Throughout this chapter K will denote a connected abelian Lie group and G a connected Lie group. The symbol Tr will denote a torus of dimension r and the symbol Vs will denote a real vector space of dimension s. Both are regarded as Lie groups in the natural sense. Our first aim is to introduce an analytic structure on Hm. The essential step in this is accomplished by the following lemma. Lemma 7.19, Let K be an additive vector group and m a K-multiplier for G. Then there exists a K multiplier m' similar to m such that m! is C°° on GxG. Proof. Note t h a t m satisfies the identities (14). I t is enough to prove this when K is the additive group of real numbers, since the general case will follow by applying the special case to each component. By virtue of corollary 7.15 we m a y assume t h a t m is bounded on each compact subset of GxG. Let /JL be a left H a a r measure and /xr a right H a a r measure forG. Let then K = B1. Choose a (700 function with compact support, s a y / , which is such t h a t (49)

$ f(9)d^(g)

= 1.

Let a be the function x -> a(x), where (50)

a(x) =

m(x,u)f(u)d[x(u). JG

Clearly, a is bounded on each compact set. Let (51)

™>i(x,y) = m(x,y) +

a(xy)-a(x)-a(y).

1

Then m1 is a R -multiplier similar to m and is bounded on compact subsets of GxG. Moreover, m x

i( ,y)

=

{m(x,y) + m(xy,u) — m(y,u) — m(x,u)}f(u)dfjL(u)

=

{m(x,yu) —

m(x,u)}f(u)dfji(u),

MULTIPLIERS

261

using the multiplier identities for m. Thus (52)

m^x.y)

m(x,u)f(y~1u)d^(u)

=

—

JG

m(x,u)f(u)dfji(u). JG

The uniform convergence of the integrals on the right of (52) shows that for each x e G, y —>rn^x^y)is a 0°° function. Define a1 to be the Borel function y-> ax(y), where (53)

a±(y) =

m1(u,y)f1{u)dfxr(u) JG

jfi being a C00 function with compact support and satisfying (54)

j fi(u)drr(u) = 1.

Let (55)

m2(x,y) = m 1 (s,y) + a 1 (a#)-a 1 (a;)--a 1 (y).

Equation (55) shows that m2 is a i?1-multiplier similar to mx and bounded on compact subsets of G x G. Moreover, a brief calculation shows that (56)

m2(x,y) =

m^u^f^ux-^d^u)JG

m1(u,y)f1(u)dfjLr(u). JG

From (56) it follows that m2 is a O00 function of each of its variables when the other is fixed. Finally, define a3 by (57)

a3(x) =

m2{x,u)f(u)dfi(u).

Write (58)

m'(x,y) =

m2(x,y)+a3(xy)-a3(x)-a3(y).

1

Then m' is a R -multiplier similar to m and (59)

m2(x,u)f(y-1u)dyL(u)-

m'(x,y) = JG

m2(y,u)f(u)dp(u). JG

Since x, y -> m2(x,u)f(y~1u) is C™ on GxG for each w, (59) shows that m' is O00 on G x 6r. The lemma is proved. We shall now examine the case when K is not simply connected. We cannot hope for global results but the local version of lemma 7.19 can be proved by the same technique. Lemma 7.20. Let K be a connected abelian Lie group and m0 any Kmultiplier. Then there exists a K-multiplier m 0 ' which is similar to m0 and which is (700 in an open neighborhood of (e,e).

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GEOMETRY

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THEORY

Proof. Since lemma 7.19 takes care of the case K = RS, it is enough to prove the present result when K is a torus and thus when K=T. We m a y (and do) therefore assume t h a t K = T. By corollary 7.6 we m a y assume t h a t m 0 is locally continuous. I t is obvious t h a t we can choose a real bounded Borel function m on G x G such t h a t (60)

m0(x,y)

=

exip{im(x,y))

for all x, y e G, and such t h a t m satisfies the identities characteristic of an R1 -multiplier (cf. (14)) locally, i.e., for some symmetric open set M with compact closure containing e, (61)

m(x,e) = m(e,x) — 0

for all x in M, and (62)

m(xy,z)-\-m(x,y)

= m(x,yz) + m(y,z)

for all x, y, z E M. We now proceed exactly as we did in the previous lemma. We c h o o s e / t o be 0°°, and satisfying (49), b u t with the extra requirement now t h a t / be zero outside a compact subset of M. a is defined as in (50) and m1 as in (51). Since exp(im) = m 0 is a T-multiplier, (51) shows t h a t exp(^'m1) is a T-multiplier similar to m 0 . m1 is bounded o n ^ x ^ and satisfies (61) for x e M and (62) for x, y, z e M. Equation (52) is now valid for all x, y e M, from which we infer t h a t for each x e M, y -> mx(x,y) is C00 in M. We define a1 and m 2 as in (53) and (55), respectively, but after choosing / x such t h a t f1 also vanishes outside M. Once again, (56) is valid for all x, y E M. Proceeding as we did before, we infer finally t h a t m 3 is 0°° on MxM. This shows t h a t exp(im 3 ) = m' is (700 on MxM, and is a Tmultiplier similar to mQ. The lemma is proved. We are now in a position to introduce a differentiable structure on Hm. This will then enable us to treat Hm as a Lie group. From this, t h e introduction of analytic coordinates on Hm will follow in the usual fashion. Theorem 7.21. Let K be a connected abelian Lie group and G a connected Lie group. Let m be any K-multiplier for G. Then there exists a unique analytic structure for Hm (compatible with the Weil topology) which converts Hm into an analytic group. The maps k - > (e,k), of K into Hm, and (g,k) - > g, of Hm onto G, are analytic. Proof. Hm is connected by corollary 7.11. We know t h a t any topological group admits at most one compatible analytic structure which converts it into a Lie group (cf. Che valley [1], pp. 127-129). Thus we are concerned only with existence. Since we m a y pass to K-multipliers similar to m, we m a y assume, after lemma 7.20, t h a t m itself is 0°° around (e,e). F r o m

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corollary 7.10 we know t h a t for some open set M containing e, the Weil topology on M x K coincides with the product topology. We choose M so small t h a t this condition is satisfied and t h a t m is O00 on M x M. We give to M x K the product of the O00 structures on M and K and choose a symmetric open set M± containing e such t h a t MXMX^M. For (x,k) and (y,k') e M1xK,

which shows t h a t the m a p (xjc), (y,k') -> (x,k)(y,k')_1 is (700 on (Mx xK)x (M-^xK). By a standard result in Lie group theory (cf. Che valley [1], pp. 134-135) there exists a O00 structure on Hm converting Hm into a Lie group, and such t h a t the induced C00 structure on MxK coincides with the product (700 structure around (e,l). I n this manner one converts Hm into a connected Lie group. Note, under the assumption t h a t m is O00 around (e,e), t h a t there exist open sets M2^G and K2<^K containing the respective identities, such t h a t M2 x K2 has the product (700 structure. Now, it is known (cf. Pontrjagin [1]) t h a t every C00 Lie group can be given a unique compatible analytic structure. Hm m a y be given this unique analytic structure. The final statement of the theorem follows from the known fact (cf. Chevalley [1], p p . 128-129) t h a t a continuous homomorphism of one analytic group into another is necessarily analytic. The proof of the theorem is complete. We can deduce several corollaries from this theorem. Corollary 7.22. The map z->(x,l) locally C™.

of G into Hm is C00 around e, m being

This follows from the fact t h a t the C°° structure of Hm is, around (e,l), the product structure. Corollary 7.23. Let m be an arbitrary K-multiplier for G. Then there exists a multiplier m! which is similar to m and is O00 {respectively analytic) on all of GxG if and only if there exists a 0°° (respectively analytic) map x->c(x) of G into Hm such that j0(c(x)) = x for all x, i.e., if and only if there exists a global O00 (respectively analytic) section. Proof. The proof is the same as for theorem 7.12; we need only replace " c o n t i n u o u s " by " ( 7 0 0 " and " a n a l y t i c " at the appropriate places. Corollary 7.24. Given any K-multiplier for G, there exists a similar to it which is analytic around (e,e).

multiplier

Proof. This requires the result that, for any analytic group, the coset space determined by a closed subgroup admits a local analytic section

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(cf. Che valley [1], pp. 109-111). Applying this result to the analytic group Hm and the closed subgroup {e} x K —i0\K], we find a Borel section c, c : x -> (x,a(x)) such t h a t for some open set N around e, c is an analytic m a p on N • N. If we define

q(x,y) =

cixyMy)-1^)-1,

then q(x,y) = (e,ma(x,y) ~x) (cf. (46)) and q is an analytic m a p of N x N into Hm. Since k ~> (e,k) is an analytic isomorphism and since {e} x K is a closed analytic submanifold of Hm, we deduce t h a t ma is analytic on N xN. Obviously ma ~ m. This is as far as we can go with elementary methods. To obtain further, and in particular, global results, greater use must be made of Lie group theory. Our aim is to use some of the ideas of Hochschild [1], which in t u r n are based on a very interesting lemma of Malcev [1], to prove t h a t when G is connected and simply connected, there exists a canonical one-one correspondence between the Lie group extensions and the associated Lie algebra extensions. We shall moreover be able to show t h a t analytic multipliers exist in every similarity class and t h a t they are naturally induced by analytic K*-multipliers for G, K* being a covering group of K. We begin with some preliminary remarks. For any analytic homomorphism a of a Lie group A into a Lie group B, we write a for t h e induced homomorphism of the Lie algebras; a(exp J ) = e x p d ( Z ) for X e a, where a is the Lie algebra of A. Let b be the Lie algebra of B a n d let us assume t h a t A and B are both connected and simply connected. Consider now a homomorphism d(X —> d(X)) of the Lie algebra ft into t h e Lie algebra of all endomorphisms of the vector space a such t h a t d(X) is a derivation of a for all X, i.e., d(X)([Y,Z]) = [d(X)(Y),Z] + [Y,d(X)(Z)] for all Y, Z in a. Then there exists a unique analytic homomorphism b —> 8(b) of B into the Lie group of invertible linear transformations of the vector space a such t h a t S(exp X) = exp d(X) for all X eft. Since d(X) is a derivation of a, 8(exp X) is an automorphism of the Lie algebra a. Hence 8(6) is an automorphism of a for all b. The m a p 6, Y -> 8(b) Y is clearly analytic from B x a to a (a being a finite dimensional real vector space, can be considered in an obvious fashion as a real analytic manifold). Since A is simply connected, each 8(b) gives rise to a unique automorphism D(b) of A such t h a t D(b)(exp F) = exp{8(6)(7)} for all Yea. Since b - > 8(b) is a homomorphism, b -> D(b) is a homomorphism of B into the group of all analytic automorphisms of the Lie group A. We t h u s see t h a t A becomes a £-space. Moreover, the equation D(fe)(exp 7 ) = exp{8(6)F} shows t h a t the m a p b,a-^ D(b)a is an analytic m a p of BxA into A. Therefore if we form t h e semidirect product BxDA (cf. Chapter VI),

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B xDA becomes a connected, simply connected Lie group. I t s analytic structure is just the product structure of B x A. As usual, we shall identify A and B with subgroups of B xDA; A will be a closed normal subgroup, and for b e B, aeA, bob'1 will coincide with D(b)a. Moreover, for Yea and Xeh, exp X exp Y exp( — Z ) = exp{exp(c?(Z))7}. This shows that, if we identify the Lie algebra of B xDA with the direct sum b-fa, then for Y e a and X e b, [X,Y] = d(X)Y. Lemma 7.25. Let Q be a Lie algebra over a field of characteristic zero and c^1an ideal in g. Suppose that either dim(a,l'g^ = 1 or g/gx is semisimple. Then gx is a semidirect summand of g, i.e., there exists a Lie subalgebra p such that g = gx + p and gx n p = 0. Proof. If dim(g/g 1 ) = l, then we may take for p any one-dimensional subspace not in g x . I n case g/gx is semisimple, the assertion is a refinement of the well known Levi-Malcev theorem and follows easily from it. Let r be the radical of g. Let 77- be the canonical m a p of g on g/g^ Since 7T[T] is a solvable ideal of g/g1? 77-[r] = 0, so t h a t r c i g ^ Let p' be a semisimple subalgebra of g such t h a t p' n r = 0 and p' + x = aj. Such a p' exists, by virtue of the Levi-Malcev theorem (cf. Chevalley [3], pp. 135-145). As p' n g! is an ideal in p', there exists a subalgebra p (which is even an ideal in p') such t h a t p + (p' n Q1) = p' and p n (p' n gi) = 0 (cf. Chevalley [3], pp. 64-65). p is semisimple and it is trivial to verify t h a t p n Qx=0 and £ + 9i = 9Lemma 7.26 (Malcev [1]). Let H be a connected and simply connected Lie group and K a closed connected normal subgroup of H. Let y be the canonical homomorphism of II onto II\K. Then there exists an analytic map c of IIIK into II such that (63) for all t

y(c(t))

E

= t

H/K.

Proof. We follow Hochschild's proof [1]. Let f) be the Lie algebra of H and ! the Lie subalgebra which corresponds to K. Since K is normal, ! is an ideal in t). Let l) = f)1^l)2^ • • • =5 f),=f be a composition series of f) over !, i.e., each f)i + 1 is an ideal in fy, and is maximal in f)t. We prove lemma 7.26 by induction on r, the length of the composition series. Since f)2 is maximal in t)1? either dim(^ 1 /t) 2 ) = 1 or fy1lt)2 ^ semisimple. Therefore, by lemma 7.25, there exists a subalgebra b of f)x such t h a t fyx = b + f)2 and b n I)2 = 0- For JC e b and 7 e | 2 , write (64)

d(Z)7 =

[X,Y].

Let i7 2 and B be connected simply connected groups corresponding to f)2 and b, respectively. I n view of our earlier remarks, the m a p X -> d(X)

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gives rise to an analytic map b, a -> D(b)a of B x H2 into H2 such that H2 becomes a Z?-space and for each b e B, D(b) is an analytic automorphism of H2. We may therefore form the group B xDH2 and identify its Lie algebra with t}± in such a way that B x {eHJ and {eB} x /7 2 correspond to b and J)2, respectively (eB is the identity of B and eH2 that of H2). Now, i? xDH2 is simply connected and so is H. Hence we may identify H with B xDH2. In other words, we may assume that H2 and B are the analytic subgroups of H corresponding to f)2 and fc, respectively. Then H2 C\ B = {eH} (the identity of H) and H = H2B. Let H — HjK and let y be the canonical homomorphism of H onto IT. Put H2 = y(H2),B — y(B). Now, for each 6 G 5 , Z)(6) leaves K and i/ 2 invariant and so induces an automorphism D(b) o£H2\K. Clearly, 6, a -> jD(6)a is analytic from 2?x (H2jK) into # 2 /iL and so one can form BxD(H2/K). It is obvious that our identification of 11 with B xDH2, gives rise to a natural identification of H with B x^{H2jK). This proves that H2 and 5 are closed in H, that i/ 2 n 5 = {e5) and H = H2B. In particular, y is an isomorphism of B onto B. By the induction hypothesis, there is an analytic map c' (s -> c'(s)) of i/ 2 into # 2 such that y(c'(s)) = s for all s e H2. Define c on H by (65)

c(5y(6)) = c'(s)b

(s eH2,be

B).

Then c satisfies (63). Corollary 7.27. The map (66)

&, t - > c(*)ifc

*«5 aw analytic isomorphism of the manifold K x (H/K) onto the manifold H. Proof. Let £ denote this map. £ is analytic and one-one. For a e H, £ " » = (c(y(a))-*a,y(a)) which is clearly analytic. Corollary 7.28. Let f) be a Lie algebra over the reals, H a connected and simply connected Lie group corresponding to f). Let I be an ideal in f). Then the analytic subgroup K of H which corresponds to I is closed. Moreover both K and H/K are simply connected. Proof. That K is closed is well known (cf. Chevalley [1], pp. 125-127). Corollary 7.27 implies that H is (topologically) homeomorphic to K x H/K, and so the simple connectivity of H implies that of K and H/K. Remark. It is known that if K and H/K are simply connected, so is H (cf. Chevalley [1], pp. 59-60). Thus corollary 7.28, together with this result,

MULTIPLIERS completely solves the problem of simple connectivity for H, K, HjK, when K is closed, connected, and normal.

267

and

Lemma 7.29 (Hochschild [1]). Let G be a connected, simply connected Lie group and (H,i,j) an extension of the connected Lie group K by G. Then there exists an analytic map c of G into H such that (67)

j(c(g)) = g

for all g e G. Proof. Let H* be a connected and simply connected covering group of H and let 8 be the covering homomorphism of H* onto H. Then j o 8 is a homomorphism of H* onto G. Let Kx be the kernel of j o 8 and K^ the component of the identity of Kx. If K1°^K1, H^jK^ would provide a nontrivial covering for G, which is not possible since G is simply connected. Hence K1 = K1°, i.e., K1 itself is connected. From Malcev's lemma we know t h a t there exists an analytic map c* of G into H* such t h a t U ° S)(c%)) = g for all g e G. If we write c = 8 o c*, then c is analytic from G into # and satisfies (67) for all g e G Corollary 7.30. Let G be a connected, simply connected Lie group and K a connected abelian Lie group. If m is a K-multiplier for G, then there exists a K-multiplier m' similar to m which is analytic on GxG. Proof. Let Hm be the extension associated with m. From lemma 10.29 we know t h a t there exists a global analytic section from G into Hm. Then, corollary 7.23 implies t h a t there is a multiplier similar to m which is globally analytic over GxG. Remark. T h a t m' can be chosen to be C00 was proved b y Bargmann [1], We shall use the lemmas of Hochschild and Malcev to obtain a direct relation between central extensions of Lie groups and the corresponding central extensions of Lie algebras. Let !, g be Lie algebras over R with ! abelian. A central extension of ! by g is a triple (Jj,a,/?), where f) is a Lie algebra over R, a is an isomorphism of ! into the center of I), and j8 a homomorphism of f) onto q such t h a t a[f] = kernel (/3), i.e., (68)

[«[!],$] = 0

and (69)

0

> t - ^ f) - £ - • g — > 0

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is an exact sequence. Central extensions (f),a,/5) and (f)'\a,/3') of f by g are said to be equivalent if there exists a (Lie) isomorphism £ of t) onto t)' such t h a t the diagram

(70)

f

_

^

^ _JL_> g

Y

>^

commutes. Suppose if is a connected abelian Lie group, G a connected Lie group, and (H,i,j) a central extension of K by G. Let !, f), and g be the Lie algebras of K, i / , and (7, respectively. Then it is obvious t h a t (fy,i,j) is a central extension of ! by g. We shall say t h a t (f),i,j) is the associated central extension of! by g. A routine transport of structure shows t h a t corresponding to equivalent central extensions of K by G, are associated equivalent central extensions of ! by g. Theorem 7.31. Let G, K be connected Lie groups and g, I the corresponding Lie algebras. Suppose that K is abelian and G is simply connected. For any equivalence class r of central extensions of K by G, let f denote the equivalence class of the associated central extensions of i by g. Then r -> f is a one-one correspondence and its range is the set of all equivalence classes of central extensions of I by g. Proof. We shall first give consideration to the case when K is also simply connected. Let (f),a,/3) be any central extension of f by g. Let H be a connected and simply connected Lie group which corresponds to f). Since K is simply connected, there exists an analytic homomorphism i of K into H such t h a t i = a; similarly there exists an analytic homomorphism j of H onto G such t h a t j = p. The range i[K] being an analytic subgroup of H having the ideal a[f] for its Lie algebra, corollary 7.28 implies t h a t i[K] is closed and simply connected. Hence i is also an isomorphism. On the other hand, i[K] is the connected component containing the identity of the kernel Kx of j . If i[K]^K1, H/i[K] will be a nontrivial covering of G^HjK^ which is impossible as G is simply connected. Therefore i[K] is precisely the kernel of j . I n other words, (H,i,j) is a central extension of K by G and (f),a,j3) is its associated extension. The uniqueness of i and j , once H is specified, implies at once t h a t two extensions of K by G, constructed in this manner, are equivalent if and only if their associated Lie algebra extensions are equivalent. Thus we have proved the lemma in the special case when K is simply connected. Suppose now t h a t K is connected but not simply connected. Let K* be a connected, simply connected Lie group covering K. Let e be the covering homomorphism. Let Z be the kernel of e. I n view of the result for

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the simply connected case, it is enough to set up a natural correspondence between central extensions of K and K* (by G) which respects equivalence. Let (H*,i*,j*) be a central extension of K* by G. Then H* is simply connected. Let H' = H*'/i*Z and let 8 be the canonical homomorphism of H* onto H. Then 8 has discrete kernel. Since kernel (8)^i*[K*], there is a unique homomorphism j of H onto G such t h a t j * = ^ o 8. Since kernel (S) = {* [kernel (e)] there is a unique homomorphism i of K into i / such t h a t 8 o i* = i o £; i is then injective. I t is then obvious t h a t kernel (j) = i[K]. Thus the diagram H*

?

K* -^

\

• H

K

/

Q commutes. Moreover, (H,i,j) is a central extension of K by G. Suppose conversely t h a t (H,i,j) is a central extension of K by G. Let H* be a connected, simply connected Lie group which covers H. Let 8 be the covering homomorphism. Then there exists a unique homomorphism i* of K* into H* such t h a t 8 o i* = i o e. Write j * = j o 8. As G is simply connected, we conclude t h a t kernel (j*) is connected. From this it follows quickly t h a t kernel (j*) = i*[K*]. Hence i*[K*] is simply connected. Since K* is also simply connected, i* is injective. Hence (i/*,^*,j*) is a central extension of K* by G. Moreover, the diagram (71) commutes, and (72)

kernel (8) = i*[Z].

We have thus set up a natural correspondence between central extensions of K and K* by G. Routine diagram chasing now shows t h a t this correspondence respects equivalence. The proof of theorem 7.31 is complete. Let (H*,i*,j*) be a central central extension of K by G. onto H such t h a t (71) is a (H*,i*,j*) covers (through e)

extension of K* by G and let (H,i,j) be a Suppose there is a homomorphism 8 of H* commutative diagram. Then we say t h a t (H,i,j).

Corollary 7.32. Let G be connected and simply connected. Let K, K* and e be as above. Then the map m* -> e o m* maps the set of all analytic (respectively C°°) K*-multipliers onto the set of all analytic (respectively C°°)

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K-multipliers, mx*~ m 2 * if and only if e o m x * ~ e o m2* so that m* —> e o m* induces an isomorphism of MK*(G) onto MK(G). Proof. Let m be an analytic if-multiplier for G. Let Hm be the associated extension. Then Hm has the product analytic structure. Since we can cover the extension Hm by an extension of K* by G, we m a y assume t h a t for some K*-multiplier m ^ , the extension Hm * covers Hm. I n view of corollary 7.30, we m a y assume t h a t rax* is analytic on Gx G so t h a t Hm* has the product analytic structure. Let $ = {((7,1) : g eG}^Hm. Let 8 be t h e covering homomorphism of Hm * onto Hm. Write 8* = 8~1(S). A standard topological argument shows t h a t 8 is a covering m a p of each connected component of 8* onto S. Let S0 be the component of 8* containing the identity (e,l) of Hmi*. T h e n $ 0 is a closed regular analytic submanifold of Hmi. Since 8 is simply connected, 8 is an analytic isomorphism of S0 onto Slm Hence there is an analytic m a p a of G into K* such t h a t for all g e G, (73)

8((gM9))) = (9,1).

Since 8 is a homomorphism, (73) gives (74)

8(xy,m1*(x,y)a(x)a(y))

Now, (xy,a(xy))~1(xy,a(x)a(y)m1*(x.y))

=

8((xy,a(xy)))(e,m(x,y)). is computed to be

so t h a t (74) leads to the equation

8 e,

( ^r mi ^ ,2/) ) = (e'w(a:'2/))'

and consequently,

(75)

Ho^T ^

for all x,y eG. If we write m*(x,y) applied, we have I7g\

/

=

for the element in (75) to which e is

m = e o m*

m* is clearly analytic. This proves the result in the analytic case. The O00 case can be disposed off similarly. The equation e o m* = m implies, on the other hand, t h a t the m a p x, k* -> x, e(k*) is a homomorphism oiHm. on i / m which sets up a covering of t h e extension Hm by Hm*. From theorem 7.31 we now conclude immediately t h a t m^ctm^ if and only if e om1*^.eom2:¥. From this it

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follows at once t h a t m* -> e o m* induces an isomorphism of MK*(G) onto j r *( c(x)) of G into Hm such t h a t j0{c(x)) =x for all xeG (theorem 7.12). Since G and Hm are Lie groups, c is analytic. Hence the range B of c is an analytic subgroup of Hm. If b is the subalgebra of ^ corresponding to B, it is obvious t h a t b n a[t] = 0 and b + a[f] = 1). Conversely, suppose there exists a subalgebra b of I) such t h a t b + a[f] = f) and b n a[f] = 0. Then there exists obviously an isomorphism y of g into f) such t h a t y maps g onto b and /3 o y is the identity. Since G is simply connected, there is a homomorphism c of G into # m such t h a t c — y. Since /S o y is the identity, j 0 o c is locally the identity. Since G is connected, j 0 o c is the identity everywhere. This proves t h a t m is exact. The only problem which is still to be examined is t h a t of classifying the central extensions of ! by g. This can be done very simply as the problem is linear. We now proceed to indicate the standard solution to this problem (cf. Che valley-Eilenberg [1]). Let p be a bilinear map of g x g into ! such t h a t (i) p is skew symmetric, i.e., (77)

p(X,Y)+p(Y,X)

= 0

for all X, F e g , and (ii) p satisfies the identity (78)

p{[X9r\,Z)+p([Y,Z],X)+p([Z,XlY)

= 0

for all X, Y, Z e g. We shall associate with p a central extension of ! by g. Let *)P = 9 x 1 , and let us define, for (X,S) and (X',S') e fjp, (79)

[(X,S),(X',S')]

=

{[X,X'lp(X,X')).

From the identities (77) and (78) it follows t h a t f)p becomes a Lie algebra under the bracket operation (79). Moreover, (79) shows trivially t h a t (X,S) —> X is a homomorphism of t)p onto g, t h a t (0,$) lies in the center of fyp for all Set, and t h a t S -> (0,S) is an isomorphism of I into l) p . Write a0(8) = (0,S), (80) Po(X9S) = X. (f)p,a0iPo) is then a central extension of! by g.

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Lemma 7.34. Every central extension ofI by g is equivalent to an extension I) j, for a suitable bilinear map pof§x§ into I satisfying (77) and (78). \)Pl and \)Vi are equivalent extensions if and only if there exists a linear map q of g into I such that (81) for all

p2(X,Y)-Pl(X,Y)

=

q([X,Y])

I J e g .

Proof. Let (f),a,j8) be a central extension of f by g. By identifying ! with <x[f] and g with a linear manifold supplementary to «[!], we m a y assume t h a t t) = g x f , a = a0> a n ( i P = Po- We have now only to determine the form of [.,.] for pairs of elements of f). Since (X,S)}(X',S') -> [(X,S),(X',S')] is bilinear and skew symmetric and since (X,8) - > X is a homomorphism of 1) onto g, it follows t h a t there exists a skew symmetric bilinear m a p of f) x fy into f, say r, such t h a t (82)

[(X,S),(X',S')]

= ([X,X'],r(X,S;

X',S'))

for all X, X' e g and S,S' et. The fact t h a t (0,#) lies in the center of ^ for all S in ! now implies t h a t (83)

r(0,£; X',S')

= 0

for all I ' e g and $, $ ' e !. The bilinear maps of f) x f) into I which are skew symmetric and satisfy (83) depend only on X and X'. Thus there exists a skew symmetric bilinear map p of g x g into ! such t h a t r(X,S; X',S') = p(X,X') for allJT, X' e g and £, £ ' e t. We have then (84)

[(X,S),(X',S')]

=

([X,X'],p(X,X')).

If we now use the fact t h a t [.,. ] satisfies the Jacobi identities, we see t h a t (78) is satisfied b}^ p. I n other words, the extension t) is equivalent to $p. Suppose now t h a t f)p and t)P2 are two extensions of ! by g corresponding to the bilinear maps p± and p2. I t is easily seen t h a t for any linear m a p q of g into !, the m a p (85)

Lq:(X,S)->(X,q(X)+S)

is a linear m a p of f)Pl into fyP2, sends (0,S) to (0,8), and is such t h a t the diagram *Pt — l J - +

*>»*

(86)

is commutative; and t h a t every linear m a p L of fyPl into fyP2, which sends (0,S) to (0,$), and which is such t h a t (86) is commutative (with L

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instead of Lq), is of the form Lq for suitable q. I t is then clear t h a t Lq is a linear isomorphism of f)Pl onto f)P2. A trivial calculation shows t h a t Lq is a homomorphism of the Lie algebras t)Pl and fyP2 if and only if plt p2 and q are related as in (81). This completes the proof of the lemma. We shall call a skew symmetric bilinear m a p p of g x g into ! closed if it satisfies (78). We shall say t h a t p is exact if there exists a linear m a p q of g into ! such t h a t (87)

3J(Z,r)

= gr([Z,r])

(IJe

9

).

The vector space of all closed skew symmetric bilinear maps of g x g into ! is denoted by W{(Q), and the subspace of exact elements is denoted by @f(g). We write (88)

Wt(q) = W(0)/@f(g).

Corollary 7.35. For apeSDfy'(g), p is exact if and only if {0} x I is a semidirect summand off)v. Proof. If p is defined by (87) for some linear m a p q of g into f, it is immediate t h a t the range h of the m a p X -> (X,q(X)) is a subalgebra of f)p such t h a t a0[f] n b = 0 and a0[f] + b = f)p. Conversely, let b be such a Lie subalgebra. Then for each l e g , let us define the element q(X) e I by the condition t h a t (X,q(X)) eh. q is evidently a linear m a p of g into I, and the condition \(X,q(X)),(Y,q(Y))] = ([X,Y],q([X,Y])) leads to (87). The theory developed so far has established a canonical isomorphism between the groups MK(G) and MK>(G), and also a canonical one-one correspondence between MK*(G) and 9Qfy(g) (G simply connected). Now 5D^i(g) is a vector space. On the other hand, if we write K* additively, it is isomorphic to a vector group, so t h a t MK*(G) also becomes a vector space, with addition as the group operation in it. I t is indeed most natural to expect t h a t the canonical correspondence between the elements of MK*(G) and Wt{o) is actually a linear isomorphism. We shall now prove t h a t this is indeed so. We assume K* = Rn. The i? n -multipliers for G then satisfy (14) and form a vector space over R, and the exact ones form a subspace of it. MRn(G) is thus a vector space. Lemma 7.36. Let G be connected and simply connected. For any analytic En-multiplier m for G, let [m] be the similarity class of m. Let 6[m] be the element of %RR*(Q) which corresponds to the equivalence class of central extensions of Rn by G which are associated with [m]. Then (89) is a linear isomorphism

M-&Im] of MRn(G) onto $0lfl»(g).

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Proof. Let m be an analytic i^-multiplier for G and let Hm be the associated group extension. For (xjc) and (y,k') e Hm ( = GxmRn in our earlier notation), (x,k)(y,kf) = (xy,k + k' + m(x,y)). Now (e,0) is the identity of II m, and Hm has the product analytic structure. We shall now determine the Lie algebra f) of Hm. Following Chevalley [1] we regard elements of f) as left invariant vector fields on Hm\ for each (x,k) e Hm, we identify the tangent space to Hm canonically with Gx x Rn, Gx being the tangent space to G at x. Of course we identify the tangent space to Rn at any of its points with Rn itself in the usual fashion. Let l e g b e the left invariant vector field x -> Xx on G and let u e Rn. We write (X,u) for the left invariant vector field of Hm which assigns to (e,0) the tangent vector (Xe,u). Since left translation by (x,k) is (y,k') —> (xy, Jc + k' + m(x,y)), we see that the tangent vector assigned by (X,u) at (x,k) is given by the expression Xxiu +1 ~r± m(x, exp tX) J Write now for any C00 function / on G x G and Ylt • • •, Yr, Z±, • • •, Zs e \ y,zeG, (90)

f{y; Y^-

Yr: z; Zv .. Zs)

J

= (a r+ v^i- • • d ^ r • .a^s)tl = ...=fr=Ui = ...=Ws=0,

where = Kv^VhYi'

• ex^trYr,zexi^u1Z1-

• -exp^ s Z s ).

Then (X,u) is given by (91)

(X,u) :x,k-+

(Xx, u + m(x : e; X)).

From (91) we find (92) [(X,u),(Y,v)] : x, k-> ([X,Y]X, m(x; X: e; Y)-m(x;

Y : e; X)).

From the identities (14) we obtain, on differentiation, m(x; X :e;Y)

= m{x : e; XY) + m(e; X : e; Y).

This then leads to (93)

\(X,u),(Y,v)]

:x,k->

([X,Y]X, m(x:e;[X,Y])

+ m(e;X : e; Y)-m(e;

Y : e; X))

so that finally, on comparing (93) with (91), we obtain (94)

[(X9u)9(Y,v)] = ([X,Y], m(e; X : e; Y)-m(e;

Y : e; X)).

MULTIPLIERS

275

I n other words, if we write (95)

Bm(X,Y)

= m(e; X : e; Y)-m(e;

Y : e; X)

then the Lie algebra I) of Hm can be identified with g x Rn in a canonical fashion and (96)

[(X,u),(Y,v)]

= ([Z, 7 ] , i U X 7)).

J5m is evidently a skew symmetric bilinear m a p of g x g into Rn. The Jacobi identity applied to (96) leads as before to the fact t h a t Bm is a closed form. We know already t h a t [ra] -> b[m] is a one-one correspondence of MRn(G) onto 9#fl"(g), and (96) shows t h a t 6[m] is the element of 9ftB*(g) defined by Bm. B u t a look at (95) shows t h a t Bm(X,Y) depends linearly on m, for fixed X, F e g . Consequently, [m] -> fe[m] is a linear isomorphism of MRn(G) onto $ft#»(g). This proves the lemma. Collecting all these results together we obtain the following theorem. We write G for the Lie group we have been concerned with. K is a connected abelian Lie group, K* a connected, simply connected covering group of K, and e is a covering homomorphism of K* onto K. Theorem 7.37. Let G be a connected and simply connected Lie group, and let K, K*, and e be defined as above. Then any K-multiplier for G is similar to one which is analytic on GxG. For any analytic (respectively (700) K*-multiplier m*, e o m* is an analytic (respectively C°°) K-multiplier for G, and every analytic (respectively O00) K-multiplier can be represented in this form; m x * and m 2 * are similar if and only if e o rax* and e o m 2 * are similar. If we associate with any equivalence class of central extensions of K by G the equivalence class of associated extensions of I by g, we obtain a one-one correspondence whose range is the set of all equivalence classes of central extensions of I by g. These latter classes are in canonical one-one correspondence with elements of the space Tlt(q). If we write K* additively and treat it as a real vector space, MK*(G) becomes a real vector space and the induced correspondence of MK*(G) with 9ttf(g) is a linear isomorphism. Finally, a K-multiplier G is exact if and only if, in the associated extension ofibyQ, the image of t is a semidirect summand.

5. E X A M P L E S We shall now discuss a number of examples which illuminate various aspects of the theory described in Sections 1 through 4. We restrict ourselves to the case of (ordinary) multipliers and R1-multipliers.

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Covering Groups: There is no direct connection between simple connectedness and exactness. We shall give, further on, examples of connected b u t not simply connected Lie groups admitting only exact multipliers, and of simply connected Lie groups whose multiplier groups are not trivial. There are, however, some elementary relations between multipliers and representations of a group and its covering group t h a t m a y be worthwhile putting down. Let G be a connected Lie group, G* a connected and simply connected covering group of G, and 8 the covering homomorphism. Let x* - » Vx* be an irreducible unitary representation of G* in a Hilbert space Jf. By Schur's lemma, V maps every element in the center of 6* into a scalar multiple of the identity. Since the kernel Z of 8 is contained in the center of G*, it follows at once t h a t V2*x* = x(z*)Vx. f ° r an< z * e ^ a n d #* e G*, X being a character of Z. This implies t h a t for any g e G, all the operators Vg* with 8(g*) = g, define the same element of the projective group of Jf. I n this way, each irreducible representation of G* induces a projective representation of G. The relation between G and G* is also useful in another way. Let m be a multiplier for G and let m* be the multiplier x*, y* -> m(8(x*),8(y*)). It is then easily seen t h a t m -> m* induces a homomorphism, say h, of M(G) into M(G*). Since G* is simply connected, theorem 7.37 can be applied to the study of M(G*). In this way useful global insights into the structure of M(G) may be obtained. However, it must be remembered t h a t h is in general neither injective nor surjective, so t h a t , in this approach to the analysis of M(G) in terms of M(G*), the kernel and the range of h must both be determined. For example, if G is a torus, M(G) is trivial while M(G*) is nontrivial as soon as dim 6 r * > l ; if G is semisimple, M(G*) is trivial, but M(G) will not be (cf. supra). 6r = R5, a vector group in s-dimensions. I n this case we can explicitly describe the multiplier group (cf. also Weyl [1], pp. 273-274). We formulate our conclusions in the form of the following theorem. I t is an easy consequence of the results described in theorem 7.37. Theorem 7.38. For any real skew symmetric bilinear form p on Rs x R 8 , (97)

mp : x, y -> exp

ip(x,y)

is a multiplier for Rs. The mapping which assigns to p, the similarity class of mp, is an isomorphism of the additive group of all skew symmetric bilinear forms on RsxRs onto the multiplier group M(Rs). In particular, all multipliers of R 1 are exact. Proof. I t is trivial to check t h a t mp is a multiplier for any bilinear form p on R*xR* (skew symmetric or not) and t h a t m1>1+3,a = m J , l m 3 , a .

MULTIPLIERS

277

To prove the remainder of the theorem we pass to the covering group R 1 of T and use theorem 7.37. Now, for any bilinear form p,x,y-^p(xiy) is an R 1 -multiplier for Rs and it is exact if and only if it is symmetric (corollary 7.14). I n particular, px^p2 ^ a n d onty # P i ~ P 2 *s symmetric. If Px and p2 are skew symmetric, it is now obvious t h a t px ~p2, if and only ifPi —Pi- If P is a n y bilinear form, we can always write it as px + c, where px is skew symmetric and c is symmetric; then p~p±. These conclusions then carry over to the T-multipliers mv. In order to complete the proof we go back to (95). We write Bp = Bm and prove t h a t the mapping p -> Bp has the set of all skew symmetric bilinear maps of R s x Rs into R 1 for its range (we identify g with R 5 also in the obvious fashion so t h a t exp is the identity map). This is trivial, for, by (95), - iB9(X, Y) = 2p(X, Y)

(X,Ye

R-),

whenever p is a skew symmetric bilinear form on R s x R 8 . The last assertion follows trivially from the fact t h a t 0 is the only skew symmetric bilinear form on R1 x R1. Connected Semisimple Lie Groups: Suppose q is a semisimple Lie algebra over the reals and (fy,a,p) a central extension by it of the abelian Lie algebra !. Since f)/a[f] is semisimple, it follows at once t h a t a[f] is the radical of f). By the Levy-Malcev theorem we conclude t h a t there exists a subalgebra b of ^ such t h a t a[f] + b = I) and «[!] n h = 0. This observation also leads a t once t o the conclusion (cf. theorem 7.37) t h a t if 0 is a connected and simply connected semisimple Lie group, then every multiplier for G is exact. Since the classical groups are semisimple, this result applies to the universal covering groups of these groups. For example, we m a y take GQ to be the connected component of the group of all invertible linear transformations of an s-dimensional vector space leaving invariant a nonsingular symmetric bilinear form. When s>2, G0 is known to be a semisimple Lie group. If G denotes a universal covering group of G0, then all multipliers for G are exact. The examples of greatest interest from the physical point of view arise when 5 = 3 and the bilinear form has a positive definite quadratic form, and when 5 = 4 and the bilinear form has a Minkowskian quadratic form. I n other words, the universal covering groups of the rotation and Lorentz groups (connected) admit only exact multipliers. Inhomogeneous Groups: The symmetry groups of space time cannot be semisimple because they contain the space-time translations as a normal subgroup. Therefore, the results derived in the previous example are inadequate for physical applications and it is necessary to extend their scope. We shall give here one such extension.

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Let G be a connected Lie group and let p(x -> p(x)) be a (not necessarily unitary) representation of G in a finite dimensional real vector space V. The m a p x, v -> p(x)v is analytic from G x V into V and converts F into a 6r-space. The semidirect product (98)

Gp

=

Gx0V

can then be formed. Gp (in the product analytic structure) is a connected Lie group; if G is simply connected, it is a simply connected Lie group. We identify V and G with closed subgroups of Gp. V is normal a n d for x G G, veV, xvx~1 — p(x)v. We shall call Gp the inhomogeneous group associated with G and p. Our purpose now is to prove a theorem from which it follows t h a t under certain conditions the covering group of the inhomogeneous group admits only exact multipliers. We shall call the representation p of G in a real vector space V admissible if no nonzero skew symmetric bilinear form on V x V is invariant under p. The fact t h a t m a n y representations of a group are admissible is contained in the following lemma. Since we work with real vector spaces it has to be formulated in the context of absolutely irreducible representations. A representation n(x -> 7T(X)) of G in a real vector space W is said to be absolutely irreducible if the only endomorphisms of W which commute with all the 7T(X) are scalar multiples of the identity. If G is semisimple, absolute irreducibility implies irreducibility in the usual geometric sense and absolute irreducibility of n is equivalent to the irreducibility of 77 in the complexification of W (G semisimple). The classical Schur's lemma can be formulated for absolutely irreducible representations as follows: if n and IT are absolutely irreducible representations of a semisimple group G, acting in W and W, then the vector space of linear maps L from W into W such t h a t LTT(X) = 7T'(X)L for all x e G is of dimension < 1, the dimension being 1 if and only if n and 7/ are equivalent. Lemma 7.39. Let p be a representation of a connected semisimple Lie group G in V. Suppose that V is a direct sum of p-invariant subspaces V1,- • •, Vs such that for each i, (i) the subrepresentation pt (of p) defined by V{ is absolutely irreducible and (ii) there exists a nonzero symmetric bilinear form on Vt x Vi which is invariant under p{. If the representations px, p2,- • •, ps are mutually inequivalent, then p is admissible. Proof. Let V* be the dual of V and for x e G and teV*, let p*(x)t be t h e element v -> t(p(x~1)v) of V*. Then p*(x)(t -> p*(x)t) is easily verified to be an invertible endomorphism of V*. I t is also easy to verify t h a t p* : x -> p*{x) is a representation of G in V*, the so-called contragredient representation acting in V*. Now any bilinear form B on V x V gives rise

MULTIPLIERS

279

to a linear transformation TB of V into F*, and B is invariant if and only if TB intertwines p and p*, i.e., (99)

TBP(x) = P*(x) TB

for all x e G. Let Fi*<= F * be the annihilator of 2y#i ^ r Then p* leaves each Vt* invariant. From the fact t h a t pt leaves a nonzero bilinear form on V{ x Vt invariant, it follows t h a t there exists a nonzero linear m a p Tu mapping V{ into F f * and F y into 0 for j / i , such t h a t T{ intertwines p and p*. From the standard form of Schur's lemma we conclude, using the mutually inequivalent nature of pl9- • •, ps, t h a t the operators JT 1; • • •, Ts span the vector space of operators which intertwine p and p*. This implies t h a t every operator which intertwines p and p* is a TB for a symmetric bilinear form B on V x V. Hence any bilinear form on F x F left invariant by p is necessarily symmetric. Consequently, p is admissible. Theorem 7.40. Let G be a connected semisimple Lie group and p an admissible representation of G in a real finite dimensional vector space. Then, for the universal covering group Gp* of Gp, every multiplier is exact. In particular, if a(x -> ax) is a representation of Gp into the group of automorphisms of the logic of a {complex separable) Hilbert space ^ there exists a representation U(x* - » Ux*) of G* in J4? such that for each x in G, <xx is induced by Ux* for any x* in Gp* lying above x. If p does not contain the trivial representation as a subrepresentation, U is the only representation of Gp* with this property. Proof. Let us first describe Gp* in a specific manner. connected and simply connected covering group of G and be the covering homomorphism. Then p o e = p* : x* -> representation of G* in V. p* is obviously admissible and group (100)

Gp* =

Let G* be a let e(x* -> x) p(e(x*)) is a we form the

G*xp.V.

I t is clear t h a t Gp* is connected and simply connected and the m a p x*, v -> e(#*), v is the covering map of Gp* on Gp (we identify V and G* with subgroups of 6rp* as usual). If a is the given representation of Gp in the group of automorphisms of the logic of ^ we know (cf. footnote in Section 1) t h a t there is a projective representation V1 of Gp in J f which induces a. We lift V1 to the projective representation V : x*9 v -> V\ix.)tV of Gp*. We now apply theorem 7.5. I n order t h a t a be induced by a representation of Gp* in the manner described in the theorem, it is necessary and sufficient t h a t there is a unitary representation of Gp* which gives rise to the same representation as V in the projective group of Jf; clearly this will happen if and only if the multiplier of V is exact. Also, the representation of Gp* which induces a is unique up to

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multiplication by a continuous homomorphism of Gp* into the group T (corollary 7.7). Thus, in order to prove the assertions of the theorem, it is enough to prove two things: first, that all multipliers for Gp* are exact, and second, that Gp* admits no nontrivial continuous homomorphisms into T under the assumptions made on p. We take up the first point. Let g, fc, and gp be the Lie algebras of G*, V, and Gp*, respectively, t) is an ideal in §p, g a subalgebra in gp, and t> n g = 0, *> + g = 8,.

If p is the representation of g in V that corresponds to the representation p* of 6r*, we shall, by using the canonical isomorphism of t> on V, assume that p and p act also on b. Thus, for Z e t) and l e g , (101)

[X,Z] = p(Z)(Z);

this corresponds to the relation x*vx*~1 = p(e(x*))v (x* e 6?*, v e V). In order to prove that all multipliers for G0* are exact, it is sufficient to prove that iff is any abelian Lie algebra and (fy,p,q) a central extension of ! by gp, then p[l] is a semidirect summand of t). We shall now demonstrate this. We write t>' = g~1(t))

(102) and (103)

fl'

= g-i(8).

q maps g' onto g and the kernel of q is p[l]. Hence we conclude from lemma 7.25 that there exists a subalgebra a such that ar\p[f]=0 and ct+^[f] = g'. q is an isomorphism of a onto g, so that a is semisimple. Since b n g = 0, t>' n a^p[l] so that i>' n ct = 0. In other words, (104)

t ) ' n a = 0,

Now b' is an ideal in I), and so for any X e a, (ad X)(Z) = [X,Z] lies in t)' whenever Z e b ' . Thus Z -> ad Z defines a representation of a in t)'. As p[t] is central, ad X annihilates all the elements of p[t]. Therefore, as a is semisimple, there exists, by Weyl's theorem (cf. Chevalley [3], pp. 70-73) a subspace bx such that f)1 n p[t] = 0, h1 +p[l] = t)r, and h1 is invariant under all ad X, X e a. Let (105)

b = bi + a.

b is a linear manifold, b n p [ ! ] = 0, and &+#[!] = *). O u r claim that ^)[!] is a semidirect summand will be proved if we show that b is a subalgebra

281

MULTIPLIERS

of f). Since [X,7] e ^ whenever X e a and 7 e b 1 ? and since a is a subalgebra, we have only to show that [X, 7] e bx whenever X, 7 are in h±. We shall actually prove that [X,Y] = 0 for X, Y in h±. Now g maps hx onto t), so that the abelian nature of t> implies that, for X, Y e &1? [ X , 7 ] E ^ P ] - We write (106)

#i(X,7) = ^ ( [ X - H )

(X,

YehJ.

Kx is evidently a skew symmetric bilinear map of hx x bi into I. If Z e a and X, 7 e b l5 [X,[7,Z]] + [7,[Z,X]] = 0, since this element is nothing but — [Z,[X,7]], and hence 0 as [X, 7] belongs to the center of f). Therefore,

(107)

x1([z,:n,x)+ir1(r,[z,xi) = o.

Now q is an isomorphism of the Lie algebra a onto g, and the vector space bx onto t>. Therefore there exists a unique skew symmetric bilinear map K of t) x t> into ! such that (108)

i ^ X ' , r ) = K(q(X'),q( Y'))

(X', Y' e bx)

and (109)

K([Z97],X) + K(Y,[Z,X])

= 0

for all X, 7 in fc and Z in g. In other words, (110)

K{p(Z)T,X) + K{7,p(Z)X)

= 0.

Since (110) is valid for all Z in g, we conclude that K is invariant under the representation p. If we choose a basis {^4l3 • • •, An} for I and write

K(X9T) = J ^ ( i ^ M i , then each j£t is an invariant skew symmetric bilinear form on t)xk). Each K% is 0, as p is admissible. This implies that K = 0. Therefore, Kx = 0. This proves that ft = h1 + a is a subalgebra of I). In other words, ^>[f] is a semidirect summand of f). We finally come to the second point, namely, to the proof that under the conditions mentioned in the theorem Gp* admits no one dimensional representation other than the trivial one. If it does, gp = t> + g will admit a nontrivial one-dimensional representation. Therefore there will exist a (complex valued) nonzero linear function A on qp such that A([X,7])=0 for all X, 7 in g^. Since g is semisimple, every element of g is a linear combination of elements of the form [X,7] with X, 7 in g (cf. Chevalley [3], p. 67) so that A = 0 on g. Let A' be the restriction of A toto.Then the equation A([Z,X]) = 0, for X e t) and Z e $, shows that the null space of A' is invariant under p. Since p is semisimple, we can select a subspace complementary to the null space of A' and left invariant by p. This space is

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one-dimensional and so, as g is semisimple, we obtain a nonzero vector fixed by p. This is a contradiction. The proof of the theorem is now complete. This is a basic theorem insofar as relativistic applications are concerned. I t tells us, for example, t h a t any representation of the (connected) inhomogeneous Lorentz group can be obtained from a unitary representation of the covering group. To see this, all we have to do is take V as 224, G as the connected component of the group of Lorentz matrices acting on V, and x -> p(x) as the action of G in V. I t is quite easy to check t h a t p is admissible. I n this special case, theorem 7.40 was obtained by Wigner [1] and, in his work, this theorem was the point of departure for his classification of relativistic wave equations. I n the special case when G is the group of all invertible linear transformations of V leaving a nondegenerate quadratic form on V invariant (dim V>2), theorem 7.40 was obtained b y Bargmann [1]. Compact Groups. Much is known about multipliers for compact groups (cf. Moore [1]). We shall content ourselves with the proof of a simple result. Theorem 7.41. Let G be any compact group satisfying the second axiom of countability. Then its multiplier group is a torsion group, i.e., every element of M(G) is of finite order. If G is a torus, or a connected and simply connected compact Lie group, G admits only exact multipliers. Proof. Let m be a multiplier for finite dimensional m-representation £ - > £ 1/n denote a Borel nth. root of which fixes 1 and whose Tith power being the dimension of V. Then

G. By p(x-> £, i.e., is the

p' : x -> (det

corollary 7.18 there exists a p(x)), acting in V say. Let a Borel m a p of T into itself coordinate function £ - > £, n

p(x))llnp(x)

is a projective representation of G; and p defines the same homomorphism of G into the projective group of V, as p. If m! is the multiplier of p , then m and m' are similar. But det p'{x) = 1 for all x so t h a t for all x, y in G, [m'(x,y)]n = l. Hence (m)n is exact, proving t h a t a n y element of M(G) has finite order. If G is a connected and simply connected compact Lie group, then G is semisimple by a theorem of H. Weyl (cf. Helgason [1]), so t h a t G admits only exact multipliers. Suppose finally t h a t G is a torus, say G=Tr. Let e : v - > e(v) be the covering homomorphism from R r onto Tr. If m is any multiplier for G, ms;v, v' -> m(e(v),e(v'))m a multiplier for R r . Since ms is exact from some integer s > 1, so is mes. B u t the multiplier group of R r is canonically iso-

MULTIPLIERS 1

283

r

morphic to the R -multiplier group of R which is a vector group. Hence mes cannot be exact unless me is itself exact. Therefore, there is a Borel function a from R r to T such th.&tme(v9v')=-a(v+v')/a(y)a(v'). Since e is a local group isomorphism, we can find a Borel function & on Tr, with values in T, and an open set N of Tr containing the identity, such t h a t (111)

m(x9y) =

b(Xy)

b(x)b(y)

for all x, y e N x N. Equation (111) implies t h a t m is exact (cf. corollary 7.14). Groups Which Are Not Semisimple. If the Lie group G is not semisimple, then it m a y admit nonexact multipliers. We shall now sketch the construction of the multiplier group for the inhomogeneous Galilean group. We closely follow Bargmann [1] in this discussion. If (x,t) (x = xl9 x2, x3) denotes the coordinates of space-time points, then an inhomogeneous Galilean transformation is a m a p (112)

r

:x,t->x',t'9

where x' = Wx + tv + u, (U3)

f ,4. t = t + rj, W being a rotation in 3-space (det W= + 1 ) , u, v vectors in 3-space, and iq a real number. The set of all such transformations forms, under composition, the so-called inhomogeneous Galilean group; we have, for the multiplication in this group, (114)

(W,ri,v9u)(W'9ri',v'9u')

= (WW9r] + v' , W + v9Wu' + u + r)'v).

If we associate with r the matrix

(

W

v

0

1

u\ 7)

0 0 1/ then r -> M(r) is a faithful representation. The Galilean group is a semidirect product; the subgroup A of all translations in space-time, (116)

A = {(MAtO}

is normal, the subgroup G0 of transformations (117)

G0 =

{{Wfl,v,0)}

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is a closed subgroup; A and GQ generate the whole inhomogeneous group. In fact, (118)

(l,rj,0,u)(W,0,v,0) = (W,rj,v,u)

and (119)

(W,0,v,0)(l,rj,0,u)( Wfi,vfi) - 1 == (1,77,0,^ + ^ ) ,

which exhibits explicitly the semidirect product nature. We denote the inhomogeneous group by G. The subgroup GQ, unlike the Lorentz group, is not semisimple, and this accounts for some of the more subtle differences between the Galilean theory and Lorentzian theory. Let M and H denote the subgroups of G0 given by (120)

M = {(1,0,^,0)},

(121)

H = {(W ,0,0,0)}.

M is a closed normal subgroup of G0, H is a subgroup with G0 = MH; moreover, (122)

(lf,0,0,0)(l,0,v,0)(lf,0,0,0)- 1 =

{l,0,Wvfi),

so that G0 is once again a semidirect product, with respect to the action of H in M given by (122). Note also that the space motions of the form (W,0,0,u) constitute the subgroup S = UH, where U is the subgroup (of all translations in space) {(1,0,0,^)}. U is a normal subgroup of S and (123)

(PT,0,0,0)(l,0,0,^)(lf,0,0,0)-1 =

(l,0fi,Wu);

therefore once again we have a semidirect product. The first step in determining the multiplier group of G is to examine the central extensions at the Lie algebra level. Theorem 7.37 reduces this to the question of determining the closed skew symmetric bilinear forms on g x g, modulo the exact ones. We shall now do this. As is customary, we denote the subalgebras of g by the German letters which correspond to the Latin letters that denote the analytic subgroups of G. The relations (116) through (123) give considerable insight into the structure of g. a is an ideal in g, g = ct + g0, a n g0 = 0; m is an ideal in g0, g0 = tn + ^), m n f) = 0; g0 and fy are subalgebras of g. Also 3 = u-|-f) is a subalgebra, u is an ideal in §, u n 1) = 0. Since A, M, and U are abelian and since H acts on all of them via the inner automorphisms, H acts naturally on a, m, and u. The differential of this action is the usual adjoint action of f) on a, m, u. Thus, using the standard notation according to which [p,p'] is the linear space spanned by all elements of the form [X, Y] with X ep and Yep', (124)

[f),m] ^ m,

[i),u] c u,

MULTIPLIERS

285

and (125)

[5,tf = 0,

where p is the one-dimensional subalgebra of g which corresponds to the subgroup {(1,^,0,0)} of A. Moreover, from (122) and (123) we see t h a t the representations of H in m and u are equivalent and absolutely irreducible and moreover, leave the Euclidean inner product on these invariant; these are therefore admissible by lemma 7.39. By a direct calculation or by use of the irreducibihty of these representations of f), we find (126)

[%m\ = m,

[$,u] = u.

Moreover an easy computation shows t h a t (127)

|>,m] c u,

[u,m] = 0.

Let B be any closed skew symmetric bilinear form on g x g . Then, for X, Y e a and Z e g 0 , we have, as [X,Y] = 0, (128)

B([Z,X], Y) + B(X9[Z, Y]) = 0.

Equation (128) shows t h a t B is invariant under the representation of G0 in a. Since the action of G0 on A leaves no nonzero bilinear form on Ax A invariant, (128) implies at once t h a t B = 0 on a x a. Note t h a t this has been proved for any closed form B. Next, we observe t h a t g 0 = m + l) satisfies the conditions of theorem 7.40. I n fact, t h e group H is semisimple and its action on M (which is isomorphic to 3-space) as given by (122) is, as observed by us already, admissible. Hence the restriction of B to g 0 x g 0 is exact. Replacing B by a form which differs from B by an exact one, we m a y t h u s assume t h a t (129)

B = 0

on

ax a

and

g0xg0.

We shall study the form B on axq0. Now a + ^ is a Lie algebra and a is an ideal in it. Moreover, the action of H on A, as given b y (119), is admissible by lemma 7.39, since A decomposes into a direct sum of U and the one-dimensional space of time translations. Hence theorem 7.40 applies once again and we conclude t h a t the restriction of B to (ct + f)) x (ct + f)) is exact. Therefore, there exists a linear function q on g, vanishing on m, such t h a t B(X,Y) = q([X,Y]) for all X, Y in ct + I). Since t) is semisimple, [^,^)] = ^ so t h a t (129) implies t h a t q vanishes also on f). If we define B' by (130)

B'(X,Y)

= B(X,Y)-q([X,Y])

(X,

7Ggxg),

then B — B' is exact and (131)

B' = 0

on

ctxct, ctxf),

and

g0 x g0.

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If X G p, Y em and Z efy, the identity B'{X,[Z,Y])

+ B'(Z,[Y,X])

+ B'(Y,[X,Z])

= 0

implies, in view of (131) and the relations [Y,X] e u, [X,Z] = 0 (cf. (127)), that (132)

B'(X,[Z,Y])

= 0.

From (126) and (132) we conclude t h a t (133)

B' = 0

on

pxm.

We are now ready to prove t h a t the set of closed forms, modulo the set of exact forms, is one dimensional. To this end, we must analyze the restriction of B' to it x m. Now, for Z e f), we have, as [u,m] = 0, (134)

B'([Z,X], Y) + B'(X,[Z, Y]) = 0

for all X, Y e it x m. Since H induces equivalent representations on it and rrt, the vector space of bilinear forms o n u x m has a one-dimensional subspace of bilinear forms invariant under the natural action of H. Let C0 be a basis for this space. Let C be defined as the unique symmetric bilinear form on g x g such t h a t C=C0 on u x m and (7 = 0 on all of ax a, Qox9o> ctxf), and pxm. Then (131), (133), and (134) show t h a t B' is a multiple of C. I n other words, the vector space of closed forms modulo t h e subspace of exact forms has dimension < 1. On the other hand, it is easily verified t h a t C is actually closed. If C were exact, and C(X, Y) = q([X,Y]) for all X and Y in g, then the equation [it,m] = 0 would imply t h a t (7 = 0 on it x tn, a contradiction. Thus the space WIRI(Q) has dimension exactly 1. Therefore there exists a nonexact ^ - m u l t i p l i e r m 0 * for the covering group G* of G such t h a t every multiplier for G* is similar to one of the form x*, y* —>• exp[trm 0 *(x*,i/*)] for a suitable real number T. Actually m 0 * can be chosen to induce a nonexact ^ - m u l t i p l i e r mQ for G itself. Bargmann [1] has indicated a simple method for constructing m 0 . Let us write i = (W,rj), ^ = (v,u). Then we can expand m 0 as a series 2n homogeneous of degree n and having coefficients which are functions of £x and f2. Since the £'s are transformed linearly in the group multiplication, it follows t h a t each £i> £2* a n d which are bilinear in Ci and £2 f ° r fixed f l5 £ 2 . If we assume t h a t m0 is of this

287

MULTIPLIERS

form and write down the multiplier identities, we obtain equations which are quite easy to solve. We find (135)

m0(r,r') = <^, W > - < v , W > + T/ ,

where <.,. > denotes the Euclidean inner product on 3-space and (136)

r = (W,rj,v,u),

r' = (W

J>'>')•

It is of course necessary to check that ra0, when lifted to a ^-multiplier m0* for G*, is not exact. But suppose m0* is exact. Then m0*(x*,y*) = m0*(y*,x*) whenever x* and y* commute, and hence m0(x,y) = m0(y,x) if x and y commute and are close to the identity. We contradict this by choosing xs = (l,0,sv,0) and ys = (l,0,0,su), with s near 0; xsys = ysxs but mo(xs>ys)-mo(ys>Xs)= -2s2(u,v}. Theorem 7.42. Let G be the inhomogeneous Galilean group. Then for any real number r (137)

raT

: r, r' -> exp i^ [
is a multiplier for G, where r = (W,r),v,u) and

r' = (W',rj',v',uf).

Every multiplier of G is similar to one of this form. The mT, for distinct r, are not similar. In particular, m% is nonexact for r / 0 .

NOTES ON CHAPTER VII For a closely related treatment of multipliers see K. R. Parthasarathy, Multipliers on locally compact groups, Lecture Notes in Mathematics, No. 93, Springer-Verlag, Berlin, 1969. For a systematic treatment of the cohomology of locally compact second countable groups see the papers of C.C.Moore: Trans. Amer. Math. Soc, 113 (1964), pp. 40-63, 113 (1964), pp. 64-86; 221 (1976), pp. 1-33; 221 (1976), pp. 35-58; see also his review article in Group representations in Mathematics and Physics, Lecture Notes in Physics, No. 6, Springer-Verlag, Berlin, 1970.

CHAPTER VIII KINEMATICS AND DYNAMICS 1. T H E ABSTRACT S C H R O D I N G E R E Q U A T I O N I n this chapter we shall use the main results of Chapters V through V I I to study the dynamical and kinematical properties of quantum mechanical systems. This will lead to the Schrodinger equation, the well known formulas for the position, momentum, and spin observables, and to t h e standard expressions for the Hamiltonians which enter the q u a n t u m theory of the atom. The first problem we study is t h a t of determining the dynamical group of a system, say (S. According to our general principles, we must exhibit the dynamical group as a one-parameter group of convex automorphisms of the convex set of all states of (3. I n this section we obtain the form of the most general dynamical group of a system whose logic is standard. Our analysis leads to the so-called abstract Schrodinger equation of motion. This equation, in its usual form, gives a set of directions for computing the possible values of the energy observable a n d describes the development of the state of the system in time. I n keeping with the spirit of our development, we can say t h a t the Schrodinger equation gives the infinitesimal form of the temporal development of an arbitrary quantum mechanical system. Let us then consider a system <&; the quantum mechanical character of (B is p u t in evidence by the assumption t h a t the logic of <S is standard. We shall actually assume t h a t the logic of (& is t h a t associated with the complex number field. We write Jf for a separable infinite dimensional Hilbert space over C whose logic ££ is isomorphic to t h a t of 3 . We identify j£? with the logic of <&. We write £f for the convex set of all states ofJK According to our earlier discussion in Chapter I I I , the dynamical group of <& is then to be described by a Borel one-parameter group t - > D(t) of convex automorphisms of S^ the Borel structure being such as to make t h e maps t-+(D(t)p)(a) Borel, for all p e S? and a tS£. Now, by Gleason's theorem (cf. Chapter IV), A -+PA is a convex isomorphism of the set of von Neumann operators of 288

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AND

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289

a

trace 1 onto ^ , PA being t h e state a->tr(P A) (aeJ?). Consequently, each D(t) induces an automorphism of the convex set of von Neumann operators of trace 1, so t h a t we can select, for each t, a symmetry Ut ofjj? such t h a t (1)

D(t)pA =

pUtAUt-i

(cf. theorem 4.33). As D(t) = D(t/2)2, it follows t h a t each Ut is a unitary operator. Now, t h e m a p t-+Ut need not have any measurability properties, as t h e choice of Ut, for each ty was made quite arbitrarily. B u t , if u and v are unit vectors of Jtif, and if we write A for t h e projection on t h e space spanned b y u, a n d L for t h e space spanned by v, PUtAUt-l(L)

= K^^>|2-

Consequently, t h e m a p t^\(Utu,v}\2 is Borel for all u, v e J4f. If we therefore write IT for t h e canonical homomorphism of t h e unitary group of c2f onto t h e projective group of ^f, this implies (corollary 7.2) t h a t t h e m a p t->7r(Ut) is a Borel homomorphism of R 1 into the projective group. Consequently it is induced b y some projective representation of R 1 in J f (cf. Chapter V I I ) . Now, according to theorem 7.38, all multipliers of R 1 are exact. Hence there exists a (unitary) representation V (2)

V:t-+Vt

of R 1 in J f , such t h a t for each t, and each von N e u m a n n operator A of trace 1, D(t) is given by (3)

D(t)-PA-*PvtAvr1'

The representation V is determined u p to multiplication b y a character of t h e real line, i.e., if V'(t - > Vt') is another representation satisfying (3), then there is a real number c such t h a t (4)

Vt' = e"°Vt

for all t. To t h e one-parameter group V we can apply Stone's theorem (cf. Riesz-Nagy [1]). According to this theorem, there exists a unique selfadjoint operator H such t h a t (5)

Vt =

ex$(-itH)

for all t; t h e domain <2)H of H is given by (6)

QJH = \v : lim - (Vtv — v) exists L I t->o t J

and, for v e Q)H, (7)

Hv = ilim- (Vtv — v). t~*o t

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GEOMETRY

OF QUANTUM

THEORY

We can thus say, in view of (5) and (3), t h a t H induces the dynamical group t -> D(t). Since the representation V is determined only u p to multiplication b y a character of the real line, D determines H u p to an additive constant, i.e., H and H + c-1 (c real) induce the same dynamical group. Collecting these facts, we have the following theorem: Theorem 8.1. Given any self-adjoint operator H on the complex separable Hilbert space J f of infinite dimension, there exists a unique dynamical group D(t - > D(t)) of Sf such that, for each t, D{t) transforms the state corresponding to the von Neumann operator A into the state corresponding to the von Neumann operator exp( — itH)A exip(itH). H and H + cl (c a real number) induce the same dynamical group. Conversely, any dynamical group D(t —> D(t)) of convex automorphisms of £f is induced in the above fashion by a self-adjoint operator H, and D determines H up to an additive constant. Finally, for any t, D(t) transforms the pure state determined by the unit vector u into the pure state determined by the unit vector exp( — itH)u. Theorem 11.1 is the central result of elementary quantum theory. I t tells us t h a t the dynamical groups of physical systems correspond one-one to one-parameter unitary groups and even to self-adjoint operators, provided we ignore additive constants in the self-adjoint operators. Now, according to our basic principles, the observables of the system are in one-one correspondence with the self-adjoint operators of 3ff. Consequently, we reach the remarkable conclusion t h a t to every dynamical group there can be associated, in a canonical fashion, a class of observables, a n y two of which differ by an additive constant, and t h a t any such class will generate a dynamical group. This conclusion brings us to a very interesting formal analogy with classical mechanics. I n its Hamiltonian form (Chapter I, Volume I), the dynamical group of a classical system is determined also b y a function H on the phase space, i.e., a classical observable, and, since H enters the differential equations in terms of its first-order derivatives, this observable in t u r n is determined by t h e dynamical group only up to an additive constant. Prompted by this analogy, we shall call any self-adjoint operator H, which induces the dynamical group of a system, a Hamiltonian of the system. Given a physical system, the observable which corresponds to a Hamiltonian of the system is undoubtedly a very crucial one, since it describes the dynamics of the system completely. I n physics, this observable is usually identified with the energy of the system. We shall give, further on, some of the reasons t h a t make this identification a very reasonable one. Assuming this identification, we can now recover t h e classical prescription of Schrodinger for the determination of the energy levels of the system. If u is a unit vector of 3tf which describes a pure state of the system under

KINEMATICS

AND

DYNAMICS

291

consideration, and if P(E -> PE) is the projection valued measure giving the spectral decomposition of the operator H, then the probability distribution of energy, when the system is in the state corresponding to u, is the measure E^\\PEu\\2;

(8)

if H has a purely discrete simple spectrum, say {Al5 A 2 ? - } , and if ar u1,u2,-m' e orthonormal eigenvectors of H corresponding to the eigenvalues Al5 A 2 , - - , respectively, then Als A 2 ,--- are the possible values of the energy observable, say En, and, in the given state, (9)

Fr(En = Am) = \\*.

Let us now consider t h e dynamical group induced by the unitary group £ - > e x p ( — itH). I t is well known, and easy to prove in the context of Hilbert spaces (cf. Gel'fand [1]), t h a t there are vectors u in Jf such t h a t t —> exp( — itH)u is an analytic function of t with values in Jtif; and t h a t , moreover, such vectors form a linear manifold which is dense in Jf, and on which H is essentially self-adjoint. If we take such a vector u and write ut = exj)( — itH)u, then we obtain the differential equation (10)

-~

= -iHut,

u0 = u.

The group t -> exp( — itH) can be clearly recovered from (10). The equation (10) is thus the differential equation which describes the motion of the pure states of our system. We shall call it the abstract Schrodinger equation (cf. D i r a c [ l ] , p p . 108-111). From the representation t-+Vt we can now determine how the probability distribution of any observable changes with time. Let £ be any observable and A the self-adjoint operator associated with it. Let E -> PE be the spectral measure of A. If the unit vector uQ represents the state at time t = 0, then the distribution of $ a t time t is the measure qt where (11)

qt(E) = <exip(itH)PE

exV(-itH)u0,u0).

I t is of course quite possible t h a t the distribution of £ a t time t = 0 remains unchanged for all time, no matter what the initial state is. From (11) we infer t h a t the necessary and sufficient condition for this is t h a t Vt~lPEVt

= PE

for all t and all Borel sets E, i.e., (12)

exp( — itH) exp( — isA) = exp( — isA) exp( — itH)

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GEOMETRY

OF QUANTUM

THEORY

for all s a n d t. The equation (12) yields, on differentiation, (13)

[H,A] = 0,

where (13) is interpreted in t h e strong sense, i.e., t h e spectral projections of H a n d A commute. I n other words, t h e observable | must be simultaneously observable with energy. I n analogy with classical mechanics we shall call f an integral of the motion. The reader m a y notice t h a t (13) is also the condition for f t o be an integral of t h e classical motion determined by a classical Hamiltonian H, provided we interpret t h e bracket in (13) as t h e Poisson bracket. Later on, we shall come across further instances of this analogy. In particular, energy is always conserved during motion. We shall conclude this section with a few remarks on t h e "Heisenberg picture." Consider a system whose dynamical development is described b y t h e one-parameter group t - > Vt. The pure states of t h e system t h e n move in time along t h e parametrized curves t - > Vtu. We shall call these curves t h e motions of t h e system or, alternately, t h e dynamical states of the system. Clearly a n y such motion t —> ut is completely determined b y the vector u0. Hence t h e rays of t h e Hilbert space of t h e system can also be regarded as representing t h e dynamical states of t h e system. If u a n d v are two unit vectors which represent t h e dynamical states £ a n d 77, respectively, t h e number |<^,v>| 2 can evidently be interpreted as t h e probability of finding t h e system in t h e dynamical state 77 when a measurement is made on it to find out whether it is in t h e dynamical state £ or not. Thus \(u,v)>\2, which depends only on t h e rays determined b y u a n d v, has a n invariant physical significance. I t is possible t o reformulate t h e general prescriptions of q u a n t u m mechanics in terms of the concept of the dynamical state. We shall indicate how this is done b y computing t h e probability distributions of t h e observables as time changes. Suppose x is a n observable a n d Ax — A its operator. Suppose t h a t t h e system is in t h e dynamical state u0. Then, t h e probability distribution of x a t time t is t h e measure qt

:E-*(PEVtu0,Vtu0},

where E - > PE is t h e spectral measure of A. If we write At = then P* : E - > Vt'1PEVt is t h e spectral measure of At a n d

Vt~xAVt,

qt(E) = < P ^ 0 ^ o > , which describes qt entirely in terms of u0. We see, therefore, t h a t t h e probability distribution of x a t time t can be computed if we replace A by At, b u t evaluate t h e relevant inner products with u0 instead of ut. I n other words, we may regard the change in time as inducing a change in the operators which represent x; the dynamical state remains unchanged.

KINEMATICS

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293

I t must be realized, however, t h a t this is only an alternative language to describe the same state of affairs. In physical literature, this way of looking at things is called the Heisenberg picture and the dynamical states are called the Heisenberg states. Our original way of looking at things is referred to as the Schrddinger picture. If A is the operator which represents the observable x, then at time t, it will be represented by the operator At=Vt~1AVi in the Heisenberg picture. The (formal) time derivative of At at £ = 0 then becomes i[H,A], where we write Vt = exp( — itH): (14)

A =

i[H,A].

We thus obtain the celebrated principle of Heisenberg according to which the time derivative of an observable is i times the commutator of its operator (at time 0) with the energy. Appropriately interpreted, (14) retains its meaning in classical mechanics also. 2. COVARIANCE AND COMMUTATION R U L E S We shall now consider a system such as a particle. A characteristic property of such systems is t h a t one can associate with each of them, a certain 0°° manifold M, which is usually called its configuration space. If we are interested in a classical interpretation of such a system, M would actually be its configuration space. Suppose t h a t © is such a system and t h a t we are interested in a quantum mechanical description of ©. Suppose we still like to treat the configuration of the system as a physical observable. The mathematical way to p u t this in evidence is to assign, for each Borel set E^M, a closed linear manifold S(E) of the Hilbert space Jf of (&; S(E) will then represent the experimental statement t h a t the configuration of © belongs to E. I t is, moreover, natural to assume t h a t the assignment E -> S(E) is a cr-homomorphism of the Borel structure of M onto a Boolean sub -a- algebra of the logic ££ of Jf^ i.e., t h a t the mapping E —> PS(E) = PE, which assigns to E the projection on S(E), is a projection valued measure based on M. Let us now suppose t h a t we have in addition a Lie group G acting differentiably on M and t h a t we want a description of our system which is covariant with respect to G (for example, we may treat a particle in a spherically symmetric potential field in space; M will then be R3 and G will be the rotation group). We then associate, with each element g of G, an automorphism ag of the logic 3? of our system such t h a t a(g -> ag) is a representation of G in Aut(J£f); moreover, in view of the simple meaning of the elements S(E) of ££, we shall require t h a t a and S be related as follows: (15)

ag[S(E)] = S(g[E]).

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GEOMETRY

OF QUANTUM

THEORY

We take the point of view t h a t (15) summarizes the physical assumptions we have made. If G is a connected Lie group, we can select a projective representation U(g —> Ug) of G in Jf7 which induces a. Then (15) can be written as (16)

U„PEUt-1

=

Pgm-

If U is a representation, then (16) expresses the fact t h a t (U,P) is a system of imprimitivity. If U were not a representation but its multiplier is exact, then we m a y change U to a representation so t h a t it still induces a. I n such a case, we m a y therefore say t h a t the existence of a configuration space plus the requirement of covariance leads to a system of imprimitivity, the representation and projection valued measure of which have direct physical significance. The exactness of the multipliers of the projective representations which induce a cannot of course be always taken for granted. B u t in m a n y cases this is the case (cf. theorem 7.40). The equations (16) can be written formally in their infinitesimal forms. Let U be a fixed representation satisfying (16). Let g be the Lie algebra of G. For any l e g and each t, x -> exip(tX) • x is a diffeomorphism of M; and as t varies, we obtain a one-parameter group. Let rx be the vector field of this one-parameter group. For any f eCcc(M) and x e M, we have (by definition), (17)

(rxf)(x)=jt(f(exV(tX)-x))t

= 0.

For any real (7°° f u n c t i o n / on M, let Af be the operator (18)

A, = \

fdP.

JM

Af is possibly unbounded b u t self-adjoint. Then, (16) leads to (19)

U9AfUg->

=

A(fSy

Now, t —> U(exptX) is a representation of E1, and hence there exists a selfadjoint operator Bx such t h a t exp( — itBx) = U{exptX) for all t. Bx is determined uniquely and we have, formally from (19), [A„Ar]

= 0,

[BX,BY]

=

iB[x>Y],

aBx + bBY = B[aX + bYWe shall call the equations (20) the Heisenberg commutation (10), Chapter I).

rules (cf.

KINEMATICS

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295

The observables associated with the operators Af and Bx which enter the relations (20) m a y be given interesting physical interpretations. We consider first the operators Af. We recall t h a t for any set E, 8(E) represents the experimental statement t h a t the configuration of 3 belongs to E. Now, from the formula (18) for Af, we see t h a t the operator Af has the spectral measure J P - > Pf-i(F) (F a Borel set of the line). Hence, if we denote by £ the observable corresponding to Af, the statement t h a t the value of £ lies in F must be represented by the subspace 8(f~1(F)). I t is thus identical with the statement t h a t the configuration of 3> belongs to the set f~1(F). I n other words, if we observe, instead of the configuration x of 3 , the value of/ a t x, we obtain an observable whose associated selfadjoint operator is Af. We shall therefore call these observables configuration observables. Note t h a t the function / represents classically an observable with the same significance. Thus f-> Af is an algebra homomorphism which assigns to each classical configuration observable the "corresponding" quantum observable. For example, if M = Rn a n d / is the kih coordinate function (tl9- • •, tn) -> tk, then Af is the operator which represents the kih configuration coordinate. On the other hand, the vector field rx gives rise to the classical momentum observable corresponding to the symmetries t -> exp(L£) acting on M, and the analogies we have so far developed with classical mechanics lead us to define Bx as the quantum mechanical momentum observable corresponding to the symmetries generated by X. In other words, the covariance expressed by (20) can be formally regarded as describing certain commutation rules between the configuration observables and the momentum observables of the system. We have thus arrived at the real mathematical meaning of the celebrated commutation rules of physics; at the same time our extensive knowledge of systems of imprimitivity enables us to understand why they are so far-reaching.

3. T H E S C H R O D I N G E R

REPRESENTATION

Given M and G, the main problem of course is to determine the pairs (a,S) which satisfy the relations (15). This is, at least in the special case when a is induced by a representation of G, nothing but the problem of determining the systems of imprimitivity for G which are based on M. The analysis of Chapter V I shows t h a t such a determination is completely known when G acts transitively on M. Fortunately, in m a n y physical applications, G is transitive on M, so t h a t these results can be applied. On the other hand, whether or not G acts transitively on M, there always exist the Koopman systems of imprimitivity associated with the invariant measure classes on M. In this section we shall explicitly

296

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describe these latter systems in the form in which they are most familiar in physics. We start with the O00 manifold M and a Borel measure v on it. We write Jf = J?2(M,i>). For any real C00 function / on M we define Af to be the operator of multiplication by / : (21)

A,:f
For any C°° vector field Z on M we define Bz as the operator (22)

Bz:
-iZcp.

Both (21) and (22) are meaningful only when