Quantum Mechanics for Mathematicians

Quantum Mechanics for Mathematicians Leon A.Takhtajan

Graduate Studies in Mathematics Volume 95

American Mathematical Society Providence, Rhode Island

Editorial Board David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann 2000 Mathematics Subject Classification. Primary 81-01.

For additional information and updates on this book, visit

www.ams.org/bookpages/gsm-95

Library of Congress Cataloging-in-Publication Data Takhtadzhian, L. A. (Leon Armenovich) Quantum mechanics for mathematicians / Leon A. Takhtajan.

p. cm. - (Graduate studies in mathematics ; v. 95) Includes bibliographical references and index. ISBN 978-0-8218-4830-8 (alk. paper) 1. Quantum theory. 2. Mathematical physics. I. Title. QC174.12.T343

2008

530.12-dc22

2008013072

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permissionclams . org. © 2008 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America.

The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/

1098 7G 5432 1

131211100908

To my teacher Ludwig Dmitrievich Faddeev with admiration and gratitude.

Contents

Preface

xiii

Part 1. Foundations Chapter 1. Classical Mechanics §1. Lagrangian Mechanics 1.1. Generalized coordinates 1.2. The principle of the least action 1.3. Examples of Lagrangian systems 1.4. Symmetries and Noether's theorem 1.5. One-dimensional motion 1.6. The motion in a central field and the Kepler problem 1.7. Legendre transform §2. Hamiltonian Mechanics 2.1. Hamilton's equations 2.2. The action functional in the phase space 2.3. The action as a function of coordinates 2.4. Classical observables and Poisson bracket 2.5. Canonical transformations and generating functions 2.6. Symplectic manifolds 2.7. Poisson manifolds 2.8. Hamilton's and Liouville's representations §3. Notes and references Chapter 2. Basic Principles of Quantum Mechanics §1. Observables, states, and dynamics 1.1. Mathematical formulation

3

4 4 4 8 15 20 22 26 31 31

33 35 38 39 42 51

56 60

63 65 66

vii

f

V111

1.2. 1.3. §2.

2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. §3.

3.1. 3.2. 3.3. 3.4. 3.5. §4.

uonten tS

Heisenberg's uncertainty relations Dynamics Quantization Heisenberg commutation relations Coordinate and momentum representations Free quantum particle Examples of quantum systems Old quantum mechanics Harmonic oscillator Holomorphic representation and Wick symbols Weyl relations Stone-von Neumann theorem Invariant formulation Weyl quantization The *-product Deformation quantization Notes and references

Chapter 3. Schrodinger Equation §1. General properties 1.1. Self-adjointness 1.2. Characterization of the spectrum 1.3. The virial theorem §2. One-dimensional Schrodinger equation 2.1. Jost functions and transition coefficients 2.2. Eigenfunction expansion 2.3. S-matrix and scattering theory 2.4. Other boundary conditions §3. Angular momentum and SO(3) 3.1. Angular momentum operators 3.2. Representation theory of 50(3) §4. Two-body problem 4.1. Separation of the center of mass 4.2. Three-dimensional scattering theory 4.3. Particle in a central potential §5. Hydrogen atom and S0(4) 5.1. Discrete spectrum 5.2. Continuous spectrum 5.3. Hidden S0(4) symmetry §6. Semi-classical asymptotics - I 6.1. Time-dependent asymptotics

74 75

79

80 86 93 98 162 163 111

118 119 125

128 136 139 146 149 149 159 152 154 155 155

163

179 177 189 189 182 185 185 186

188 193 193

197 199

295

206

iX

Contents

6.2. 6.3. §7.

Time independent asymptotics Bohr-Wilson-Sommerfeld quantization rules Notes and references

Chapter 4.

Spin and Identical Particles

§1.

Spin

1.1. 1.2.

Spin operators Spin and representation theory of SU(2) Charged spin particle in the magnetic field Pauli Hamiltonian Particle in a uniform magnetic field System of identical particles The symmetrization postulate Young diagrams and representation theory of Symnr Schur-Weyl duality and symmetry of the wave functions Notes and references

§2.

2.1. 2.2. §3.

3.1. 3.2. 3.3. §4.

209 212 213

217 217 217 218 221 221 223

224 225 229 232 234

Part 2. Functional Methods and Supersymmetry Chapter 5. Path Integral Formulation of Quantum Mechanics §1. Feynman path integral 1.1. The fundamental solution of the Schrodinger equation 1.2. Feynman path integral in the phase space 1.3. Feynman path integral in the configuration space 1.4. Several degrees of freedom §2. Symbols of the evolution operator and path integrals 2.1. The pq-symbol 2.2. The qp-symbol 2.3. The Weyl symbol 2.4. The Wick symbol §3. Feynman path integral for the harmonic oscillator 3.1. Gaussian integration 3.2. Propagator of the harmonic oscillator 3.3. Mehler identity §4. Gaussian path integrals 4.1. Gaussian path integral for a free particle 4.2. Gaussian path integral for the harmonic oscillator §5. Regularized determinants of differential operators 5.1. Dirichlet boundary conditions 5.2. Periodic boundary conditions 5.3. First order differential operators

239

239 239 242 245 247 249 249 250 251 252

255 255 257 259 260 261 264 268 268 274 278

X

Contents

§6.

6.1. 6.2. §7.

Semi-classical asymptotics - II Using the Feynman path integral Rigorous derivation Notes and references

Chapter 6. Integration in Functional Spaces §l. Gaussian measures 1.1. Finite-dimensional case 1.2. Infinite-dimensional case §2. Wiener measure and Wiener integral 2.1. Definition of the Wiener measure 2.2. Conditional Wiener measure and Feynman-Kac formula 2.3. Relation between Wiener and Feynman integrals §3. Gaussian Wiener integrals 3.1. Dirichlet boundary conditions 3.2. Periodic boundary conditions §4. Notes and references Chapter 7. Fermion Systems Canonical anticommutation relations § 1. 1.1. Motivation 1.2. Clifford algebras §2. Grassmann algebras 2.1. Realization of canonical anticommutation relations 2.2. Differential forms 2.3. Berezin integral §3. Graded linear algebra 3.1. Graded vector spaces and superalgebras 3.2. Examples of superalgebras 3.3. Supertrace and Berezinian §4. Path integrals for anticommuting variables 4.1. Wick and matrix symbols 4.2. Path integral for the evolution operator 4.3. Gaussian path integrals over Grassmann variables §5. Notes and references Chapter 8. Supersymmetry §1. Supermanifolds §2. Equivariant cohomology and localization 2.1. Finite-dimensional case 2.2. Infinite-dimensional case

280 281 282 285

289 289 289 291 293 293 296 299 301 301 302

305

307

307 307 311

314 315 317 319

324 324 326 328 330 330 335 338 340 343 343 345 345 349

Contents

§3.

3.1. 3.2. §4.

4.1. 4.2. 4.3. §5. §6. §7.

Classical mechanics on supermanifolds Functions with anticommuting values Classical systems Supersymmetry Total angular momentum Supersymmetry transformation Supersymmetric particle on a Riemannian manifold Quantum mechanics on supermanifolds Atiyah-Singer index formula Notes and references

xi

354 354 356 359 359 360 362

364 369 370

Bibliography

3 73

Index

383

Preface

This book is based on graduate courses taught by the author over the last fourteen years in the mathematics department of Stony Brook University. The goal of these courses was to introduce second year graduate students with no prior knowledge of physics to the basic concepts and methods of quantum mechanics. For the last 50 years quantum physics has been a driving force behind many dramatic achievements in mathematics, similar to the role played by classical physics in the seventeenth to nineteenth centuries. Classical physics, especially classical mechanics, was an integral part of mathematical education up to the early twentieth century, with lecture courses given by Hilbert and Poincare. Surprisingly, quantum physics, especially quantum mechanics, with its intrinsic beauty and connections with many branches of mathematics, has never been a part of a graduate mathematics curriculum. This course was developed to partially fill this gap and to make quantum mechanics accessible to graduate students and research mathematicians. L.D. Faddeev was the first to develop a course in quantum mechanics for undergraduate students specializing in mathematics. From 1968 to 1973 he regularly lectured in the mathematics department of St. Petersburg State University in St. Petersburg, Russia1, and the author enjoyed the opportunity to take his course. The notes for this book emerged from an attempt to create a similar course for graduate students, which uses more sophisticated mathematics and covers a larger variety of topics, including the Feynman path integral approach to quantum mechanics.

1 At that time in Leningrad, Soviet Union.

xiii

xiv

Preface

There are many excellent physics textbooks on quantum mechanics, starting with the classic texts by P.A.M. Dirac [Dir47J, L.D. Landau and E.M. Lifshitz [LL58], and V.A. Fock [Foc78], to the encyclopedic treatise by A. Messiah [Mes99], the recent popular textbook by J.J. Sakurai [Sak94], and many others. From a mathematics perspective, there are classic monographs by J. von Neumann [vN96] and by H. Weyl [Wey5O], as well as a more recent book by G.W. Mackey [MacO4], which deal with the basic mathematical formalism and logical foundations of the theory. There is also a monumental project [DEF+99], created with the purpose of introducing graduate students and research mathematicians to the realm of quantum fields and strings, both from a mathematics and a physics perspective. Though it contains a very detailed exposition of classical mechanics, classical field theory, and supersymmetry, oriented at the mathematical audience, quantum mechanics is discussed only briefly (with the exception of L.D. Faddeev's elegant introduction to quantum mechanics in [Fad99]). Excellent lecture notes for undergraduate students by L . D Faddeev and O.A. Yakubovskii [FY80] seems to be the only book on quantum mechanics completely accessible to mathematicians2. Recent books by S J Gustafson and I.M. Sigal [GSO3] and by F. Strocchi [StrO5] are also oriented at mathematicians. The latter is a short introductory course, while the former is more an intermediate level monograph on quantum theory rather than a textbook on quantum mechanics. There are also many specialized books on different parts of quantum mechanics, like scattering theory, the Schrodinger operator, C*-algebras and foundations, etc. The present book gives a comprehensive treatment of quantum mechanics from a mathematics perspective and covers such topics as mathematical foundations, quantization, the Schrodinger equation, the Feynman path integral and functional methods, and supersymmetry. It can be used as a one-year graduate course, or as two one-semester courses: the introductory course based on the material in Part 1, and a more advanced course based on Part 2. Part 1 of the book, which consists of Chapters 1-4, can be considered as an expanded version of [FY80]. It uses more advanced mathematics than [FY80], and contains rigorous proofs of all main results, including the celebrated Stone-von Neumann theorem. It should be accessible to a second-year graduate student. As in [FY80], we adopt the approach, which goes back to Dirac and was further developed by Faddeev, that classical mechanics and quantum mechanics are just two different realizations of the fundamental mathematical structure of a physical theory that uses the notions of observables, states, measurement, and the time evolution dynamics. Part 2, which consists of Chapters 5-8, deals with functional methods in quantum .

.

.

2 The English translation will appear in the AMS "The Student Mathematical Library" series.

Preface

xv

mechanics, and goes beyond the material in [FY80]. Exposition there is less detailed and requires certain mathematical sophistication. Though our presentation freely uses all the necessary tools of modern

mathematics, it follows the spirit and tradition of the classical texts and monographs mentioned above. In this sense it can be considered "neoclassical" (as compared with a more abstract approach in [DF99a]). Each chapter in the book concludes with a special Notes and references section, which provides references to the necessary mathematics background and physics sources. A courageous reader can actually learn the relevant mathematics by studying the main text and consulting these references, and with enough sophistication, could "translate" corresponding parts in physics textbooks into the mathematics language. For the physics students, the book presents an opportunity to become familiar with the mathematical foundations and methods of quantum mechanics on a "case by case" basis. It is worth mentioning that development of many mathematics disciplines has been stimulated by quantum mechanics. There are several ways to study the material in this book. A casual reader can study the main text in a cursory manner, and ignore numerous remarks and problems, located at the end of the sections. This would be sufficient to obtain basic minimal knowledge of quantum mechanics. A determined

reader is supposed to fill in the details of the computations in the main text (a pencil and paper are required), which is the only way to master the material, and to attempt to solve the basic problems 3. Finally, a truly devoted reader should try to solve all the problems (probably consulting the corresponding references at the end of each section) and to follow up on the remarks, which may often be linked to other topics not covered in the main text.

The author would like to thank the students in his courses for their comments on the draft of the lecture notes. He is especially grateful to his colleagues Peter Kulish and Lee-Peng Teo for the careful reading of the manuscript. The work on the book was partially supported by the NSF grants DMS-0204628 and DMS-0705263. Any opinions, findings, and conclusions or recommendations expressed in this book are those of the author and do not necessarily reflect the views of the National Science Foundation.

3 We leave it to the reader to decide which problems are basic and which are advanced.

Part 1

Foundations

Chapter 1

Classical Mechanics

We assume that the reader is familiar with the basic notions from the theory of smooth (that is, C°O) manifolds and recall here the standard notation. Unless it is stated explicitly otherwise, all maps are assumed to be smooth and all functions are assumed to be smooth and real-valued. Local coordinates

q= (q',.

qn) on a smooth n-dimensional manifold M at a point q E M

are Cartesian coordinates on cp(U) C I[8n, where (U, cp) is a coordinate chart on M centered at q E U. For f : U - IEgn we denote (f o cp-1)(ql, ... , qn) by 1(q), and we let

of of _ of aq (aql , ... , aqn stand for the gradient of a function f at a point q E Rn with Cartesian coordinates (q1, ... , qn ). We denote by n

A(M) =

Ak(M) k=0

the graded algebra of smooth differential forms on Al with respect to the wedge product, and by d the de Rham differential a graded derivation of ,A' (M) of degree 1 such that df is a differential of a function f E .A° (M) _ C°O(M). Let Vect(M) be the Lie algebra of smooth vector fields on M with the Lie bracket [, ],given by a commutator of vector fields. For X E Vect(M)

we denote by GX and ix, respectively, the Lie derivative along X and the inner product with X. The Lie derivative is a degree 0 derivation of .A(M) which commutes with d and satisfies Lx (f) = X (f) for f E A°(M), and the inner product is a degree -1 derivation of A(M) satisfying ix(f) = 0

1. Classical Mechanics

4

and ix (df ) = X(f) for f

A° (11x1). They satisfy Cartan formulas

Lx =ixod+doiX = (d+i)2, i [X,y] = L o iy - iy o Lx For a smooth mapping of manifolds f : 11x1 ---* N we denote by f* : TM -k .

TN and f* : T * N --- T * 11x1, respectively, the induced mappings on tangent and cotangent bundles. Other notations, including those traditional for classical mechanics, will be introduced in the main text.

1. Lagrangian Mechanics 1.1. Generalized coordinates. Classical mechanics describes systems of finitely many interacting particlesl. A system is called closed if its particles do not interact with the outside material bodies. The position of a system

in space is specified by positions of its particles and defines a point in a smooth, finite-dimensional manifold M, the configuration space of a system. Coordinates on Al are called generalized coordinates of a system, and the dimension n = dim M is called the number of degrees of freedom2.

The state of a system at any instant of time is described by a point q E M and by a tangent vector v E TqM at this point. The basic principle of classical mechanics is the Newton-Laplace determinacy principle which

asserts that a state of a system at a given instant completely determines its motion at all times t (in the future and in the past). The motion is described by the classical trajectory - a path 'y(t) in the configuration space Al. In generalized coordinates -y(t) _ (ql(t), ... , qn(t)), and corresponding derivatives q2 =

q

i

are called generalized velocities. The Newton-Laplace principle is a fundamental experimental fact confirmed by our perception of everyday experiences. It implies that generalized accelerations q = d?q2 dtz are uniquely defined by generalized coordinates q2 and generalized velocities

qZ, so that classical trajectories satisfy a system of second order ordinary differential equations, called equations of motion. In the next section we formulate the most general principle governing the motion of mechanical systems.

1.2. The principle of the least action. A Lagrangian system on a configuration space M is defined by a smooth, real-valued function L on TM x ][8

- the direct product of a tangent bundle TM of M and the time axis3 called the Lagrangian function (or simply, Lagrangian). The motion of a 1A particle is a material body whose dimensions may be neglected in describing its motion. 2 Systems with infinitely many degrees of freedom are described by classical field theory. aft follows from the Newton-Laplace principle that L could depend only on generalized coordinates and velocities, and on time.

1. Lagrangian. Mechanics

5

Lagrangian system (M, L) is described by the principle of the least action in the configuration space (or Hamilton's principle), formulated as follows. Let

P(M)go,to = {'y: [to,tiI -> M, 'Y(to) = 4'0, 'Y(ti) = 4'i}

be the space of smooth parametrized paths in M connecting points qp and ql. The path space P(M) = P(M)QO;to is an infinite-dimensional real Frechet manifold, and the tangent space T.yP(M) to P(M) at ry E P(M) consists of all smooth vector fields along the path ry in M which vanish at the endpoints qo and ql. A smooth path P in P(M), passing through ry E P(M), is called a variation with fixed ends of the path y(t) in M. A variation I' is a family 'y6(t) = I'(t, e) of paths in M given by a smooth map

F: [to,t1} x [-eon 50] - M

such that I'(t, 0) = ry(t) for to < t < ti and I'(to, e) = qo, I'(tl, e) = ql for -Ep < < o. The tangent vector E T.yP(M) E=O

corresponding to a variation 'y( t) is traditionally called an infinitesimal variation. Explicitly,

0) E T(t)M, to c t < tl,

8'y(t) =

where aE is a tangent vector to the interval [-Eu, so] at 0. Finally, a tangential

lift of a path ry [to, ti] -- M is the path ry' [to, ti] - TM defined by ry'(t) _ 'y*(at) E T(t)M, to < t < tl, where at is a tangent vector to [t0,t1J at t. In other words, ry'(t) is the velocity vector of a path 'y(t) at time t.

Definition. The action functional S : P(M) - I[8 of a Lagrangian system (M, L) is defined by

ftl L(/(t),t)dt.

5(7)

= Jto Principle of the Least Action (Hamilton's principle). A path ry E PM describes the motion of a Lagrangian system (M, L) between the position qo e M at time to and the position ql E M at time ti if and only if it is a critical point of the action functional S, =n

for all variations yE(t) of ry(t) with fixed ends.

The critical points of the action functional are called extremals and the principle of the least action states that a Lagrangian system (M, L)

1. Classical Mechanics

6

moves along the extremals4. The extremals are characterized by equations of motion a system of second order differential equations in local coordinates on TM. The equations of motion have the most elegant form for the following choice of local coordinates on TM.

Definition. Let (U, cp) be a coordinate chart on M with local coordinates q = (q', ... , qi). Coordinates n

1

on a chart TU on TM, where v = (v', ... , vn ) are coordinates in the fiber corresponding to the basis dinates.

a

aq

1, ... ,

a

aq n

for Tq M, are called standard coor-

Standard coordinates are Cartesian coordinates on co (TU) C TW ^J Wi x Ian and have the property that for (q, v) E T U and f E C°° (U ), n

o = v afQ

v(f) _ vi q2f 2=1

Let (U, co) and (U', cps) be coordinate charts on M with the transition functions F = (F',. . . , Fn) = cp' o cp-1 : cp(U f1 U') cp'(U fl U'), and let (q, v) and (q', v'), respectively, be the standard coordinates on TU and TU'. We

-

(

i

ln

have q' = F(q) and v' = F* (q)v, where F* (q) _ { aF (q) } a1

is a matrix-

.

valued function on cp(U n U'). Thus "vertical" coordinates v = (v',. , vn) in the fibers of TM - Al transform like components of a tangent vector on M under the change of coordinates on M. The tangential lift 'y'(t) of a path -y(t) in M in standard coordinates on . .

TU is (q(t),(t)) _ (q'(t),...,q(t),'(t),...,q(t)), where the dot stands for the time derivative, so that

L(y (t), t) = L(q(t), q(t), t). Following a centuries long traditions, we will usually denote standard coordinates by

where the dot does not stand for the time derivative. Since we only consider paths in TM that are tangential lifts of paths in M, there will be no confusion 6. 4The principle of the least action does not state that an extremal connecting points qo and q1 is a minimum of S, nor that such an extremal is unique. It also does not state that any two points can he connected by an extremal. 5Used in all texts on classical mechanics and theoretical physics. 6We reserve the notation (q(t), v(t)} for general paths in TM.

1. Lagrangian Mechanics

7

Theorem 1.1. The equations of motion of a Lagrangian system (M, L) in standard coordinates on TM are given by the Euler-Lagrange equations aQ (q(t), 4(t), t)

- dt (aQ (4(t), q(t), t))

= o.

Proof. Suppose first that an extremal ry(t) lies in a coordinate chart U of M. Then a simple computation in standard coordinates, using integration by parts, gives S('7E) ti

d E=0

t

n

tJo i =1 t1

L (q(t, s), q(t, s), t} dt to

EIL

i

qz 8q + qz 5q r7L

dL

to

dt

8qi dt +

Sqi

t

The second sum in the last line vanishes due to the property 8q2(to) = Sqz (t ) = o, i = 1, ... , n. The first sum is zero for arbitrary smooth functions qz on the interval [to, t1] which vanish at the endpoints. This implies that for each term in the sum the integrand is identically zero,

- (Q(t), q(t), t) - dt (aq (q(t), Q(t), t) f = o, i

2 = 1,... ,n.

Since the restriction of an extremal of the action functional S to a coordinate chart on M is again an extremal, each extremal in standard coordinates on TM satisfies Euler-Lagrange equations.

Remark. In calculus of variations, the directional derivative of a functional the Gato derivative S with respect to a tangent vector V E T.yP(M) is defined by =o

where ry is a path in P(M) with a tangent vector V at 'Yo = ry. The result of the above computation (when ry lies in a coordinate chart U C M) can be written as d 8L n 8L

vs -

(1.1)

I1=1 (8qz

-f

o

l

aL

dt aqz)

d aL

aq - dt aq)

(q(t),(t),t)v(t)dt

(q(t),(t),t)v(t)dt.

1. Classical Mechanics

$

n

Here V (t) _

v2 (t)

-Z

is a vector field along the path ry in M. Formula

(1.1) is called the formula for the first variation of the action with fixed ends. The principle of the least action is a statement that 8vS(ry) = 0 for all V E T.-P(M).

Remark. It is also convenient to consider a space P(M) _ {'y [to, tl] -f M} of all smooth parametrized paths in M. The tangent space T.yP(M) to P(M) at ry E P(M) is the space of all smooth vector fields along the path ry in M (no condition at the endpoints). The computation in the proof of :

Theorem 1.1 yields the following formula for the first variation of the action with free ends:

8S= _

tl(aq aL

d aL

- dt aq

ti

v dt + aL v

aq

to

Problem 1.1. Show that the action functional is given by the evaluation of the 1-form Ldt on TM x I over the 1-chain ry on TM x R, S(y) =

fLdt, ry

where y = {(y'(t),t);to C t C t} and Ldt (w,c) = cL(q, v), w E T(q,v)TM, C E Ilk.

Problem 1.2. Let f e C°° (M). Show that Lagrangian systems (M, L) and (M, L+df ) (where df is a fibre-wise linear function on TM) have the same equations of motion.

Problem 1.3. Give examples of Lagrangian systems such that an extremal connecting two given points (i) is not a local minimum; (ii) is not unique; (iii) does not exist.

Problem 1.4. For y an extremal of the action functional S, the second variation of S is defined by a2

DEi

2 E'1=E'2=O

where y 1sis a smooth two-parameter family of paths in M such that the paths y,o and yo,r2 in P(M) at the point yo,o = y E P(M) have tangent vectors V1 and V2, respectively. For a Lagrangian system (M, L) find the second variation of S and verify that for given V1 and V2 it does not depend on the choice of y 1 ,E2 .

1.3. Examples of Lagrangian systems. To describe a mechanical phenomena it is necessary to choose a frame of reference. The properties of the space-time where the motion takes place depend on this choice. The space-time is characterized by the following postulates. Strictly speaking, these postulates are valid only in the non-relativistic limit of special relativity, when the speed of light in the vacuum is assumed to be infinite.

1. Lagrangian Mechanics

9

Newtonian Space-Time. The space is a three-dimensional affine EucliE3 a reference point dean space E3. A choice of the origin o

IR3, where the vector space JR3 carries establishes the isomorphism E3 the Euclidean inner product and has a fixed orientation. The time is onedimensional a time axis IR and the space-time is a direct product E3 x R. An inertial reference frame is a coordinate system with respect to the origin o E E3, initial time to, and an orthonormal basis in IR3. In an inertial frame the space is homogeneous and isotropic and the time is homogeneous. The laws of motion are invariant with respect to the transformations

ro e JR3 and g is an orthogonal linear transformation in IR3. The time in classical mechanics is absolute.

The Galilean group is the group of all mine transformations of E3 x IR which preserve time intervals and which for every t E IR are isometries in E3. Every Galilean transformation is a composition of rotation, space-time translation, and a transformation

rir+vt, tit, where v E IR3. Any two inertial frames are related by a Galilean transformation.

Galileo's Relativity Principle. The laws of motion are invariant with respect to the Galilean group. These postulates impose restrictions on Lagrangians of mechanical sys-

tems. Thus it follows from the first postulate that the Lagrangian L of a closed system does not explicitly depend on time. Physical systems are described by special Lagrangians, in agreement with the experimental facts about the motion of material bodies. Example 1.1 (Free particle). The configuration space for a free particle is M = II83, and it can be deduced from Galileo's relativity principle that the Lagrangian for a free particle is

L = 2mr2.

Here m> 03 is the mass of a particle and r2 = II2 is the length square of the velocity vector r e TrJ3 ^r IR3. Euler-Lagrange equations give Newton's law of inertia,

8 Otherwise the action functional is not bounded from below.

10

1. Classical Mechanics

Example 1.2 (Interacting particles). A closed system of N interacting particles in ]ft3 with masses ml,... , mnr is described by a configuration space M- ][$3N

- R3

X

...X][83 N

with a position vector r = (r1, ... , TN), where ra E j3 is the position vector of the a-th particle, a = 1, ... , N. It is found that the Lagrangian is given by N

L=mares-V(r)=T-V, a=1

where N

T=

2 ma rQ

a=1

is called kinetic energy of a system and V(r) is potential energy. The EulerLagrange equations give Newton's equations

mara = Fa, where

Fa -

aV

is the force on the a-th particle, a = 1, ... , N. Forces of this form are called conservative. It follows from homogeneity of space that potential energy V(r) of a closed system of N interacting particles with conservative forces depends only on relative positions of the particles, which leads to the equation N

>Fa = 0. a=1

In particular, for a closed system of two particles F1 + F2 = 0, which is the equality of action and reaction forces, also called Newton's third law. The potential energy of a closed system with only pair-wise interaction between the particles has the form

V(r) _

E

Vab(ra - rb)

1Ca
It follows from the isotropy of space that V (r) depends only on relative distances between the particles, so that the Lagrangian of a closed system of N particles with pair-wise interaction has the form N

L=

ara a=1

Vab(I?'a - Tb) 1Ca
1. Lagrangian Mechanics

11

If the potential energy V (r) is a homogeneous function of degree p, V7r) _ APV (r), then the average values T and V of kinetic energy and potential energy over a closed trajectory are related by the virial theorem 2T = pV.

(1.3)

Indeed, let r(t) be a periodic trajectory with period T > 0, i.e., r(0) = r(T), r(0) = (r). Using integration by parts, Newton's equations, and Euler's homogeneous function theorem, we get z N

2T =

T N

TJ T N

L

1

r

J o

mararadt

172aTadt = -T J a=1

o

o

a=1

DV rQ ar dt = pV .

a=1

a

Example 1.3 (Universal gravitation). According to Newton's law of gravitation, the potential energy of the gravitational force between two particles with masses ma and mb is

V (ra - rb) _ -G

I

a am1'b

where G is the gravitational constant. The configuration space of N particles with gravitational interaction is

a,b=1,...,N}.

M={(rl,...,rnr) EI[83N

Example 1.4 (Particle in an external potential field). Here M = I[83 and

L=

- V(r,t),

where potential energy can explicitly depend on time. Equations of motion are Newton's equations

av mr=F=-ar.

If V =

is a function only of the distance

the potential field is

called central.

Example 1.5 (Charged particle in electromagnetic field9). Consider a particle of charge e and mass m in 1f83 moving in atime-independent electromagnetic field with scalar and vector potentials co(r) and A(r) (Al(r),Aa(r),

A3(r)). The Lagrangian has the form

9This is a non-relativistic limit of an example in classical electrodynamics.

1. Classical Mechanics

12

where c is the speed of light. The Euler-Lagrange equations give Newton's equations with the Lorentz force,

mr=el E+r x is the cross-product of vectors in 1183, and

E=__! and B =curl A are electric and magnetic10 fields, respectively. Example 1.6 (Small oscillations). Consider a particle of mass m with n degrees of freedom moving in a potential field V(q), and suppose that potential energy U has a minimum at q = 0. Expanding V (q) in Taylor series around 0 and keeping only quadratic terms, one obtains a Lagrangian system which describes small oscillations from equilibrium. Explicitly, L = 2mg2 - Vo(9), where Vo is apositive-definite quadratic form on 1[8n given by n

Vo(q) =2 z,j=1

av 2

ag2aq

Since every quadratic form can be diagonalized by an orthogonal transformation, we can assume from the very beginning that coordinates q = (q',. . , qn) are chosen so that Vo(q) is diagonal and .

n

L = 2m(2 -

(1.4)

i=1

where w1, ... , wn > o. Such coordinates q are called normal coordinates. In normal coordinates Euler-Lagrange equations take the form qz + wi qz = o,

i = 1, ... , n,

and describe n decoupled (i.e., non-interacting) harmonic oscillators with frequencies w1, ... , w,.

Example 1.7 (Free particle on a Riemannian manifold). Let (M, ds2 ) be a Riemannian manifold with the Riemannian metric ds2. In local coordinates

x1,...,xnonM, ds2 = g,,(x)dx'1dx", where following tradition we assume the summation over repeated indices. The Lagrangian of a free particle on M is L(v) = 2 (v,v) = z IIvII2> v E TM, 10Notation B = rot A is also used.

13

1. Lagrangian Mechanics

where (,)stands for the inner product in fibers of TM given by the Riemannian metric. The corresponding functional S(y)

= 2 j otl II'(t)I2dt = 2 Jto

t19wv(x)xµx"dt

is called the action functional in Riemannian geometry. The Euler-Lagrange equations are

ii' = 2 axv x x, axa 1 agLa

g vx

.

.,

and after multiplying by the inverse metric tensor gOv and summation over v they take the form

x + rvv = o,

d = 1, ... , n,

where 1

v

2

Ogva _ Dxa axv + 8x'

are Christofel's symbols. The Euler-Lagrange equations of a free particle moving on a Riemannian manifold are geodesic equations. the metric connection in the be the Levi-Civita connection Let and let t be a covariant derivative with respect to tangent bundle TM Vect (M). Explicitly, the vector field µ

(Vt- (ax"

where

_

(x)

a, r =

For a path ry(t) _ (x(t)) denote by V.y a covariant derivative along ry,

(Vj)'(t)

- d(t) - dt

where

ij_ ijµ(t)aaµ

is a vector field along ry. Formula (1.1) can now be written in an invariant form

ftl

(Vy,y)dt, to

which is known as the formula for the first variation of the action in Riemannian geometry.

Example 1.8 (The rigid body). The configuration space of a rigid body in ][83 with a fixed point is a Lie group G = SO(3) of orientation preserving orthogonal linear transformations in I[83. Every left-invariant Riemannian metric ( , ) on G defines a Lagrangian L : TG -f I[8 by L(v) = (v,v), v E TG. According to the previous example, equations of motion of a rigid body are geodesic equations on G with respect to the Riemannian metric (, ). Let g =X0(3) be the Lie algebra of G. A velocity vector g E T9G defines the angular velocity of the body by 1 2 = (Lg-i )*g E g, where Ly : G -p G are

1. Classical Mechanics

14

left translations on G. In terms of angular velocity, the Lagrangian takes the form

where (, ) is an inner product on g = TeG given by the Riemannian metric The Lie algebra g the Lie algebra of 3 x 3 skew-symmetric matrices has the invariant inner product (u, v) o = - Tr acv (the Killing form), so that ,

).

S2, (A. I, I) for some symmetric linear operator A : g - g which is positive-definite with respect to the Killing form. Such a linear operator A is called the inertia tensor of the body. The principal axes of inertia of the body are orthonormal eigenvectors e1, e2, e3 of A; corresponding eigenvalues

11,12,13 are called the principal moments of inertia. Setting SI =1 e1 + 12e2 +

3e311, we get

L = 2 (I1 2i + I2122 + I3523).

In this parametrization, the Euler-Lagrange equations become Eider's equations I111 = (I2 - I3)12113, 2=(I3-I1)1l113

I2

I31Z3 = (I1 - I2)SZ1[12.

Euler's equations describe the rotation of a free rigid body around a fixed point. In the system of coordinates with axes which are the principal axes of inertia, principal moments of inertia of the body are I1, I2, I3. Problem 1.5. Determine the motion of a charged particle in a constant uniform magnetic field. Show that if the initial velocity v3 = o in the z-axis (taken in the cmyt direction of the field, B = (o, o, B )} , the trajectories are circles of radii r W eB

in a plane perpendicular to the field (the xy-plane), where vt = v + u is the initial velocity in the xy-plane. The centers (x0, y0) of circles are given by cmv2

cm' v1

x° = eB +x, y° = - eB +Y, where (x, y) are points on a circle of radius r.

Problem 1.6. Show that the Euler-Lagrange equations for the Lagrangian L(v) = v I I , v E TM, coincide with the geodesic equations written with respect to a constant multiple of the natural parameter.

Problem 1.7. Prove that for a particle in a potential field, discussed in Example 1.4, the second variation of the action functional, defined in Problem 1.4, is given by

2S =

ti

Jto

J(81r)82rdt,

11 This establishes the Lie algebra isomorphism g by the cross-product.

1R3, where the Lie bracket in I1is given

I. Lagrangian Mechanics

15

'

2

where 6ir, 8Zr E yT PR3, = r(t) is the classical trajectory, J = -m dt 2 I

a_ (t) _

I is the 3 x 3 identity matrix, and

are

2

3

ar doarb

(r(t))J.

.

2

- r2

7

A second-order

u,b=1

linear differential operator 3, acting on vector fields along '-y, is called the Jaco bi operator.

Problem 1.8. Find normal coordinates and frequencies for the Lagrangian system considered in Example 1.6 with Vo (q) = 2 a2

1

(qi+1 _q)2, where qn+1 = q1.

Problem 1.9. Prove that the second variation of the action functional in Riemannian geometry is given by 828 =

Jto

(J(S1'Y), c52'y)dt.

Here 8l'y, 82-y E TPM, 3 = -Vy - R(ry, )'y is the Jacobi operator, and R is a curvature operator a fibre-wise linear mapping R : TM ® TM -* End (TM) of vector bundles, defined by R(, ri) = D[, : TM -f TM, where , ?7 E Vect(M). Problem 1.10. Choosing the principal axes of inertia as a basis in IR3, show / c s _ 113o --iii 11\ that the Lie algebra isomorphism g ^1 l3 is given by g f--122

121

o

(111, 112, 113) E 3.

Problem 1.11. Show that for every symmetric A E End g there exists a symmetric 3x 3 matrix A such that A 11 = All + [1A, and find A for diagonal A. Problem 1.12. Derive Euler's equations for a rigid body. (Hint: Use that L = - 2 Tr ASl2, where 11 = g-1g and 611 = -g-18g Il + and obtain the EulerLagrange equations in the matrix form All + h A = All2 -- Ill A.)

1.4. Symmetries and Noether's theorem. To describe the motion of a mechanical system one needs to solve the corresponding Euler-Lagrange equations a system of second order ordinary differential equations for the generalized coordinates. This could be a very difficult problem. Therefore of particular interest are those functions of generalized coordinates and velocities which remain constant during the motion. Definition. A smooth function I : TM --* I is called the integral of motion (first integral, or conservation law) for a Lagrangian system (M, L) if

dtI

fit)) = 0

for all extremals ry of the action functional.

Definition. The energy of a Lagrangian system (M, L) is a function E on TM x IE8 defined in standard coordinates on TM by n

E(q,, t) = z=1

3L

q

- L(q,

,

t).

1. Classical Mechanics

16

Lemma 1.1. The energy E = cj

L - L is awell-defined function on TM x 9'

I.

Proof. Let (U, co) and (U', cp') be coordinate charts on M with the transition functions F = (F1, ... , Fn) = cp' o co- cp(U r1 U') --} p'(U n U'). Corre-

sponding standard coordinates (q, qr) and (q', q') are related by q' = F(q) and q' = F* (q) q (see Section 1.2). We have dq' = F* (q)dq and dq' = G (q, q} dq + F* (q)dq (for some matrix-valued function G(q, )), so that

dL= =

DL

dq, +

DL

DL

cat

DL

q

---F(q) + ,

dt

DL

., G(q,

q)dq +

dt , F (q)dq + DL cat

DL

DL DL dt. j-dq+--dq+ cat

DL

Thus under a change of coordinates DL

DL

and

.,DL

. =q.DL.,

q

so that E is a well-defined function on TM.

Corollary 1.2. Under a change of local coordinates on M, components of DL

aq (q' q, t)

DL - (aq

'

.

DL \ ' '' aq

transform like components of a 1-form on M.

Proposition 1.1 (Conservation of energy). The energy of a closed system is an integral of motion.

Proof. For an extremal ry set E(t) = E(ry(t)). We have, according to the Euler-Lagrange equations,

dE

d (DL\

dt

dt

-

q + aq q

aq

(d (DL

Since for a closed system

DL. aq q

DL..

)aq )q

at

DL..

DL

aq q

at

8t

= 0, the energy is conserved.

Conservation of energy for a closed mechanical system is a fundamental law of physics which follows from the homogeneity of time. For a general closed system of N interacting particles considered in Example 1.2, N

E= a=1

ma-L=

N

ra + V (r). a=1

17

1. Lagrangian Mechanics

In other words, the total energy E = T + V is a sum of the kinetic energy and the potential energy.

Definition. A Lagrangian L : TM -> II8 is invariant with respect to the diffeomorphism g : M - M if L(g*(v)) = L(v) for all v E TM. The diffeomorphism g is called the symmetry of a closed Lagrangian system (M, L). A Lie group G is the symmetry group of (M, L) (group of continuous symmetries) if there is a left G-action on M such that for every g E G the mapping Continuous symmetries give rise to conservation laws.

Theorem 1.3 (Noether). Suppose that a Lagrangian L : TM - I[8 is invariant under aone-parameter group {g8}j of diffeomorphisms of M. Then the Lagrangian system (M, L) admits an integral of motion I, given in standard coordinates on TM by DL (d9s(q) aL ds n

where X =

a2(q) i=1

a; uq

8=01

Dq

is the vector field on M associated with the flow gs.

The integral of motion I is called the Noether integral. Proof. It follows from Corollary 1.2 that I is awell-defined function on TM. Now differentiating L((gs)*(ry'(t))) = L(-)e(t)) with respect to s at s = 0 and using the Euler-Lagrange equations we get d (Dq 8L 8L da d (DL'\ DL. DL 0 = aq a + aq a = dt aq a + rq dt dt a '

where a(t)= (a1(7(t)),...,a(y(t))). Remark. A vector field X on M is called an infinitesimal symmetry if the corresponding local flow gs of X (defined for each s E IE8 on some US C M) is a symmetry: L o (g8)* = L on U9. Every vector field X on M lifts to a vector field X' on TM, defined by a local flow on TM induced from the corresponding local flow on M. In standard coordinates on TM, n n D aai and X' = az (q) i + a(q) (1.5) X =

'

i=1

aq

i=1

q

q

q

It is easy to verify that X is an infinitesimal symmetry if and only if dL(X') = o on TM, which in standard coordinates has the form (1.6) =1

L a(q)q

n

i=1 yj

. Day q

DL (q), i =

o.

1. Classical Mechanics

18

Remark. Noether's theorem generalizes to time-dependent Lagrangians L TM x IE - R. Namely, on the extended configuration space M1 = M x Ilk. define a time-independent Lagrangian L1 by :

L 1 (q7T,

,) T _ L

4',

q .

T

,T

,

where (q, r) are local coordinates on M1 and (q, r, q, T) are standard coordinates on TM1. The Noether integral I1 for a closed system (Mi, L1) defines an integral of motion I for a system (M, L) by the formula I(q, q, t) = I1(q, t,

4'71).

When the Lagrangian L does not depend on time, L1 is invariant with respect to the one-parameter group of translations r T + s, and the --E. Noether integral 1I = gives I = Noether's theorem can be generalized as follows.

Proposition 1.2. Suppose that for the Lagrangian L : TM - I[8 there exist a vector field X on M and a function K on TM such that for every path ry in M, dL(x')('Y(t))

-

Then n

I=

ai

(q)

aL

;;(q) - K(q,)

i=1

is an integral of motion for the Lagrangian system (M, L).

Proof. Using Euler-Lagrange equations, we have along the extremal 'y, d

DL

rL

q

q

L. _ dK q

Example 1.9 (Conservation of momentum). Let M = V be a vector space, and suppose that a Lagrangian L is invariant with respect to a one-parameter group g3 (q) = q + sv, v e V. According to Noether's theorem,

I=

n vz z=1

qz

is an integral of motion. Now let (M, L) be a closed Lagrangian system of N interacting particles considered in Example 1.2. We have M = V = IR3N, and the Lagrangian L is invariant under simultaneous translation of

1. Lagrangian. Mechanics

19

coordinates ra = (r1, rQ, ra3 ) of all particles by the same vector c E 1R3 . Thus V = (c, ... , c) E R3N and for every c = (c1, c2, c3) E N

I=;1

(13L c -

bra

3 .2+c33L\ .3 =cPi+cP2+cP3 2 okra

2 aL -

-)

1

is an integral of motion. The integrals of motion Pi, P2, P3 define the vector

P= a=

3Ta

(or rather a vector in the dual space to R3), called the momentum of the system. Explicitly,

P=

mara,

so that the total momentum of a closed system is the sum of momenta of individual particles. Conservation of momentum is a fundamental physical law which reflects the homogeneity of space.

Traditionally, p =

q

are called generalized momenta corresponding to

generalized coordinates qz, and Fz = are called generalized forces. In these notations, the Euler-Lagrange equations have the same form

q

p=F as Newton's equations in Cartesian coordinates. Conservation of momentum implies Newton's third law.

Example 1.10 (Conservation of angular momentum). Let M = V be a vector space with Euclidean inner product. Let G = SO(V) be the connected

Lie group of automorphisms of V preserving the inner product, and let g = 50(V) be the Lie algebra of G. Suppose that a Lagrangian L is invariant with respect to the action of aone-parameter subgroup gs (q) = esS q of G on V, where x E g and eX is the exponential map. According to Noether's theorem, n

z= is an integral of motion. Now let (M, L) be a closed Lagrangian system of N interacting particles considered in Example 1.2. We have Al = V = J3N, and the Lagrangian L is invariant under a simultaneous rotation of coordinates ra of all particles by the same orthogonal transformation in 1R3.

20

1. Classical Mechanics

Thus x = (u, ... , u) E X0(3)

X0(3), and for every u E X0(3), N

N

r(

2 aL

1 aL

a_1 \

aL C71'a

is an integral of motion. Let u = u1X1 + 262X2 + 263X3 where Xl = 00 0 ooil (o-iol

) XZ ° \- gg), X3 \ o 0 o J is the basis in X0(3) ll83 corresponding to the rotations about the vectors el, e2, e3 of the standard orthonormal basis in IL83 (see Problem 1.10). We get

I = u1M1 + u2M2 + u3M3,

where M = (Ml, M2, M3) E I[83 (or rather a vector in the dual space to

z0(3)) is given by

M = >Ta X a= ].

The vector M is called the angular momentum of the system. Explicitly, N

M=

7'a X ma ra

7

a=1

so that the total angular momentum of a closed system is the sum of angular momenta of individual particles. Conservation of angular momentum is a fundamental physical law which reflects the isotropy of space.

Problem 1.13. Find how total momentum and total angular momentum transform under the Galilean transformations.

1.5. one-dimensional motion. The motion of systems with one degree

of freedom is called one-dimensional. In terms of a Cartesian coordinate x

on M = R the Lagrangian takes the form

L-2mx2--V(x). The conservation of energy

E=

2 mx2 + V (x)

allows us to solve the equation of motion in a closed form by separation of variables. We have dt

=

m 2

dx

E - V(x)

I. Lagrangian Mechanics

21

The inverse function x(t) is a general solution of Newton's equation dV

mx = dx --,

with two arbitrary constants, the energy E and the constant of integration. Since kinetic energy is non-negative, for a given value of E the actual motion takes place in the region of IIg where V (x) < E. The points where V (x) = E are called turning points. The motion which is confined between two turning points is called finite. The finite motion is periodic -the particle oscillates between the turning points xl and x2 with the period

T (E) = 2m

fdx J1 /E-V(x)

.

If the region V(x) < E is unbounded, then the motion is called infinite and the particle eventually goes to infinity. The regions where V (x) > E are forbidden.

On the phase plane with coordinates (x, y) Newton's equation reduces to the first order system dV

m± =

Trajectories correspond to the phase curves (x(t), y(t)), which lie on the level sets

V (x) = E

2m + of the energy function. The points (sO, 0), where xo is a critical point of the

potential energy V (x), correspond to the equilibrium solutions. The local minima correspond to the stable solutions and local maxima correspond to the unstable solutions. For the values of E which do not correspond to the equilibrium solutions the level sets are smooth curves. These curves are closed if the motion is finite. The simplest non-trivial one-dimensional system, besides the free particle, is the harmonic oscillator with V(x) = 2kx2 (k > 0), considered in Example 1.6. The general solution of the equation of motion is

x(t) = Acos(wt + a), where A is the amplitude, w =

m

is the frequency, and a is the phase

. The energy is of a simple harmonic motion with the period T w mw2A2 and the motion is finite with the same period T for E > o.

Problem 1.14. Show that for V(x) = ---x4 there are phase curves which do not exist for all times. Prove that if V(x) o for all x, then all phase curves exist for all times.

22

1. Classical Mechanics

Problem 1.15. The simple pendulum is a Lagrangian system with M = Si =

R/27rZ and L = 2 92 + cos B. Find the period T of the pendulum as a function of the amplitude of the oscillations.

Problem 1.15. Suppose that the potential energy V(x) is even, V(O) = Q, and V(x) is a one-to-one monotonically increasing function for x o. Prove that the inverse function x(V) and the period T(E) are related by the Abel transform

dx dV T(E) 2J m E dV E-V

-

and x(V) _

o

1

V

T (E)dE V -E1.6.

The motion in a central field and the Kepler problem. The

motion of a system of two interacting particles the two-body problem can also be solved completely. Namely, in this case (see Example 1.2) M = 1R6 and

L=" lrl 2

+ m2r2 2

-

r2I)

Introducing on I[86 new coordinates

r = r 1 - r2 and R _ mlrl +m2r2 m1 + m2

we get

L = 2mR2 + 2µr2 - V where m = ml + m2 is the total mass and µ = mime is the reduced mass ml + m2 of a two-body system. The Lagrangian L depends only on the velocity R of the center of mass and not on its position R. A generalized coordinate with this property is called cyclic. It follows from the Euler-Lagrange equations that generalized momentum corresponding to the cyclic coordinate is conserved. In our case it is a total momentum of the system,

P=91 - mR, 31? so that the center of mass R moves uniformly. Thus in the frame of reference

where R = 0, the two-body problem reduces to the problem of a single particle of mass µ in the external central field V (Jr I) . In spherical coordinates in 1183,

x = r sin i9 cos cp, y = r sin i9 sin cp, z = r cos i9,

where 0 < t9 < ir, 0 < cp < 2ir, its Lagrangian takes the form L = 2µ(r2 + r2t92 + r2 sine i9 cp2)

- V(r).

It follows from the conservation of angular momentum M = µr x r that during motion the position vector r lies in the plane P orthogonal

to Mm n R. Introducing polar coordinates (r,) in the plane P we get X2 =X92 + sine t9 cp2,

so that

L = 2µ(r2 + r2X2) - V (r).

I. Lagrangian Mechanics

23

The coordinate x is cyclic and its generalized momentum µr2X coincides

with Ml if k > 0 and with -IM if X G 0. Denoting this quantity by M, we get the equation

ur2=M,

(1.7)

which is equivalent to Kepler's second 1aw12. Using (1.7) we get for the total energy 2

(1.8)

E = 2µ(r2 -}- r2X2) -I- V (r) = 2µr2 + V (r) + 2µr2 M .

Thus the radial motion reduces to aone-dimensional motion on the half-line r > 0 with the effective potential energy

Veff (r) - V (r) +

M2

where the second term is called the centrifugal energy. As in the previous section, the solution is given by

2J

E - Veff(r)

It follows from (1.7) that the angle x is a monotonic function of t, given by another quadrature dr T M (1.10) x

T2EVeff(T)

y

yielding an equation of the trajectory in polar coordinates.

The set l/11(r) C E is a union of annuli o C rmin C r C rmax C oa, and the motion is finite if o C rmin C r C rmax C oc. Though for a finite motion r(t) oscillates between rrnin and rmax, corresponding trajectories are not necessarily closed. The necessary and sufficient condition for a finite motion to have a closed trajectory is that the angle rmax dr 2tC

rmjn

r2E _ Veii (r)

is commensurable with 27r, i.e., x = 2ir

rn

n

for some m, n E Z. If the angle

x is not commensurable with 2i-, the orbit is everywhere dense in the annulus rmin
lm Vef f (r) = h m V(r) = V C oo, 00

the motion is infinite for E> V

oo

the particle goes to oo with finite velocity

µ(E-V). 12It is the statement that sectorial velocity of a particle in a central field is constant.

1. Classical Mechanics

24

A very important special case is when

V(r)=-.r It describes Newton's gravitational attraction (a > 0) and Coulomb electrostatic interaction (either attractive or repulsive). First consider the case when c> 0 - Kepler's problem. The effective potential energy is

a

M2

Vef.f (r) - - r + 2µr2 and has the global minimum

M2

at ro = aµ . The motion is infinite for E > 0 and is finite for Vo < E 0. The explicit form of trajectories can be determined by an elementary integration in (1.10), which gives

X = cos-1

MM r

+C

rp

.

2µ(E - Vo) Choosing a constant of integration C = 0 and introducing notation

and e=

p = ro

we get the equation of the orbit (trajectory) (1.11)

p = 1-+ e cos X.

r

This is the equation of a conic section with one focus at the origin. Quantity 2p is called the latus rectum of the orbit, and e is called the eccentricity. The choice C = 0 is such that the point with X = 0 is the point nearest to the origin (called the perihelion). When Tlo < E < 0, the eccentricity e < 1 so that the orbit is the ellipse13 with the major and minor semi-axes (1 . 12)

a=

1

p ee =

Correspondingly, rmin = orbit is given by

2

iEl

p 1

+e

,

b

=

rmax =

_=

I MI

1 - e2

p

and the period T of elliptic

1I"

r

.

The last formula is Kepler's third law. When E > o, the eccentricity e> 1 and the motion is infinite the orbit is a hyperbola with the origin as 13The statement that planets have elliptic orbits with a focus at the Sun is Kepler's first law.

I. Lagrangian Mechanics

25

internal focus. When E = 0, the eccentricity e = 1 -the particle starts from rest at oo and the orbit is a parabola. For the repulsive case a < 0 the effective potential energy Ve f f (r) is always positive and decreases monotonically from oo to 0. The motion is always infinite and the trajectories are hyperbolas (parabola if E = 0) p = -1 + e cos

r

with

p=-M2and a/L

2 e=4/1+ 2EM2 µa V

IR the Lagrangian system

Kepler's problem is very special: for every R3 with on (1.13)

has three extra integrals of motion Wl, W2, W3 in addition to the components of the angular momentum M. The corresponding vector W = (W1, W2, W3), called the Laplace-Rurige-Lenz vector, is given by

W=rxM- cxrr

(1.14)

Indeed using equations of motion 1ur = -

lar momentum M = µr x r, we get

ar r3

.

and conservation of the angu-

W=µrx r)r - µr r)r _ ar + a(r 3r)r =0.

Using µ(r x M) r = MZ and the identity (a x b)2 = a2 b2 - (a b)2, we get (1.15)

W2 = rx2 + 2M2 E

where 2

E

2il

a r

is the energy corresponding to the Lagrangian (1.13). The fact that all orbits are conic sections follows from this extra symmetry of Kepler's problem. Problem 1.17. Prove all the statements made in this section.

1. Classical Mechanics

26

Problem 1.1$. Show that if lira Veff (r) = -oo,

r-40

then there are orbits with rmin = o

"fall" of the particle to the center.

Problem 1.19. Prove that all finite trajectories in the central field are closed only when

V (r)= kr2, k > o,

and V (r)= - , a>0. r

Problem 1.20. Find parametric equations for orbits in Kepler's problem.

Problem 1.21. Prove that the Laplace-Runge-Lenz vector W points in the direction of the major axis of the orbit and that of the orbit.

= ae, where e is the eccentricity

Problem 1.22. Using the conservation of the Laplace-Runge-Lenz vector, prove that trajectories in Kepler's problem with E C o are ellipses. (Hint: Evaluate TV r and use the result of the previous problem.)

1.7. Legendre transform. The equations of motion of a Lagrangian system (M, L) in standard coordinates associated with a coordinate chart U C M are the Euler-Lagrange equations. In expanded form, they are given by the following system of second order ordinary differential equations: aL

aqi (4, q) _ dt (aq (q, q))

2L

a2 L 3qz 3qj

In order for this system to be solvable for the highest derivatives for all initial conditions in TAT, the symmetric n x n matrix HL(q,)

a2L

=

(q)l z,7=1

should be invertible on TU.

Definition. A Lagrangian system (M, L) is called non-degenerate if for every coordinate chart U on M the matrix HL(q, q) is invertible on TU.

Remark. Note that the n x n matrix HL is a Hessian of the Lagrangian function L for vertical directions on TM. Under the change of standard coordinates q' = F(q) and q' = F* (q)v (see Section 1.2) it has the transformation law

HL(q,v) = where F* (q)T is the transposed matrix, so that the condition det HL does not depend on the choice of standard coordinates.

0

27

1. Lagrangian Mechanics

For an invariant formulation, consider the 1-form 9L, defined in standard coordinates associated with a coordinate chart U C .Ilk by n

DL

OL =

.i

i=1

dqi =

q

DL

dq .

q

It follows from Corollary 1.2 that 6L is awell-defined 1-form on TM.

Lemma 1.2. A Lagrangian system (M, L) is non-degenerate if and only if the 2-form dBL on TM is non-degenerate.

Proof. In standard coordinates, D2L

dOL -

D2 L dqj n dqi + 3gi3gjdqjn dq i

,

y =1

and it is easy to see, by considering the 2n-form dOL = dBL n

A dOL, n

that the 2--form dOL is non-degenerate if and only if the matrix HL is nondegenerate.

Remark. Using the 1-form 0L, the Noether integral I in Theorem 1.3 can be written as (1.16)

I = zx1(eL),

where X' is a lift to TM of a vector field X on M given by (1.5). It also immediately follows from (1.6) that if X is an infinitesimal symmetry, then

cX'(eL) = o.

(i.17)

Definition. Let (U, cp) be a coordinate chart on M. Coordinates

on the chart T* U ^J Ian x U on the cotangent bundle T* M are called standard coordinates14 if for (p,q) T*U and ,f E C°°(U)

i,

i = 1, ... , n.

Equivalently, standard coordinates on T* U are uniquely characterized by

the condition that p = (pr,.. , pn) are coordinates in the fiber corresponding to the basis dq , ... , dq for TM, dual to the bass 1, ... , n for TqM. .

14 Following tradition, the first n coordinates parametrize the fiber of T * U and the last n coordinates parametrize the base.

I. Classical Mechanics

28

Definition. The 1-form 0 on T*M, defined in standard coordinates by n

0=

z=

pdq

is called Liouville's canonical 1-form. Corollary 1.2 shows that 9 is a well-defined 1-form on T* M. Clearly, the 1-form B also admits an invariant definition

B(u) = p(ir*(u)),

where

26 E T(p9)T*M,

and it : T*M -> M is the canonical projection.

Definition. Afibre-wise mapping TL : TM - T*M is called a Legendre transform associated with the Lagrangian L if eL = TL(e).

In standard coordinates the Legendre transform is given by TL(4, 4) _ (P, 4),

t9L

where p = aq (q, 4)

The mapping 'rL is a local diffeomorphism if and only if the Lagrangian L is non-degenerate.

Definition. Suppose that the Legendre transform TL : TM --* T * M is a diffeomorphism. The Hamiltenian function H : T* M - R, associated with the Lagrangian L : TM - R, is defined by

HoYL=EL=q

-L.

In standard coordinates,

H(p,q)= (p-L(q,t))

,

p- aq aL

where q is a function of p and q defined by the equation p = aq (q, q) through the implicit function theorem. The cotangent bundle T*M is called the phase space of the Lagrangian system (M, L). It turns out that on the phase space the equations of motion take a very simple and symmetric form.

Theorem 1.4. Suppose that the Legendre transform TL : TM -* T*M is a diffeomorphism. Then the Euler-Lagrange equations in standard coordinates

on TM,

-d oL dt t9q

-_

oL 3qi

= o,

2 = 1, ... , n,

I. Lagrangian Mechanics

29

are equivalent to the following system of first order differential equations in standard coordinates on T*M: OH

Pi=- oq

q= ,z

:

i

OH

z=1,...,n.

,

Proof. We have

dH =

OH

d p -}y-

OH

d4

qdP

aq dq

aq dql _ aL

OL

_

\9'd

aq d4

_ r9L

Thus under the Legendre transform, OH and p = q=-8-p

OL d OL oj=jq dt qr

OH

q

Corresponding first order differential equations on T*M are called Hamilton's equations (canonical equations.

Corollary 1.5. The Hamiltonian H is constant on the solutions of Hamilton's equations.

Proof. For H(t) = H(p(t), q(t)) we have dH dt

OH. -q+8q 8P P

OH.

OH 8H

OH OH

Oq 81n

8P aq

-=O.

For the Lagrangian 2

L=m2 -V(r)=T-V,

rElI83,

of a particle of mass m in a potential field V (r), considered in Example 1.4, we have

p= OL =mr. Or Thus the Legendre transform TL : TI3 -+ T*IR3 is a global diffeomorphism, linear on the fibers, and z

H(p, r) _ (pi-L)IT_ = 2 + V(r) = T + V.

I. Classical Mechanics

30

Hamilton's equations

aH p = ap m _ 3H _ a

r =

,

V

are equivalent to Newton's equations with the force F = -

ar

For the Lagrangian system describing small oscillators, considered in Example 1.6, we have p = mq, and using normal coordinates we get n

1

Vo(q) - 2 m P2 + m2

H(7>> 9') _ (p-L(q,))I._m = m2

i=1

Similarly, for the system of N interacting particles, considered in Example ... , pN}, where 1.2, we have p = pa

The Legendre transform linear on the fibers, and

a

mara,

ar a

, . . . ,

N.

is a global diffeomorphism,

TR3N ---+

H(p, r) _ (p-L)IT_m =

2mQa + V(r) = T + v. a=1

In particular, for a closed system with pair-wise interaction, N

pQ

1

H (p, r} =

1

a=1

2m a

+

)

Vab (ra -- rb) .

1

1Ca
In general, consider the Lagrangian n

L=

- V(q), q E Rn,

where A(q) _ {a23 (q)}_1 is a symmetric n x n matrix. We have

aL pi = q

n

=

ij (4)ci',

i

i = 1, ... , n,

j=1

and the Legendre transform is a global diffeomorphism, linear on the fibers, if and only if the matrix A(q) is non-degenerate for all q E R. In this case, n

H(1>>q)= (p9-L(4,4)) _aL = where {az3 (q)}_1 = A-1(q) is the inverse matrix.

2. Hanlltonian Mechanics

31

Problem 1.23 (Second tangent bundle). Let r : TM - M be the canonical projection and let TV (TM) be the vertical tangent bundle of TM along the fibers of rr the kernel of the bundle mapping 7r* : T(TM) --- TM. Prove that there is a natural bundle isomorphism i : TM T(TM).

Problem 1.24 (Invariant definition of the 1-form BL ). Show that BL (v) = dL((i o 7r*)v), where v E T(TM).

Problem 1.25. Give an invariant proof of (1.17). Problem 1.26. Prove that the path 'y(t) in M is a trajectory for the Lagrangian system (M, L) if and only if i-yf (t)(dOL ) -+- dEL (y' (t)} = o,

where 'y'(t) is the velocity vector of the path y' (t) in TM.

Problem 1.27. Show that for a charged particle in an electromagnetic field, considered in Example 1.5, e c

p = mr + A and H(p, r) =

2m

p - ec A + ecp(r).

Problem 1.28. Suppose that for a Lagrangian system (R", L) the Legendre transform TL is a diffeomorphism and let H be the corresponding Hamiltonian. Prove that for fixed q and q the function pq - H(p, q) has a single critical point

at p =

-q

.

Problem 1.29. Give an example of a non-degenerate Lagrangian system (M, L) such that the Legendre transform TL : TM --k T*M is one-to-one but not onto.

2. Hamiltonian Mechanics 2.1. Hamilton's equations. With every function H : T* M -

on the phase space T*M there are associated Hamilton's equations a first-order system of ordinary differential equations, which in the standard coordinates on T*U has the form

p=- 8H oq '

(2.1)

q

_ op

The corresponding vector field XH on T * U, n

(3H1

13H3\

Ha

DH 3

z= gives rise to awell-defined vector field XH on T*M, called the Hamiltonian vector field. Suppose now that the vector field XH on T*M is complete, i.e., its integral curves exist for all times. The corresponding one-parameter group {t}tEa of diffeomorphisms of T*M generated by XH is called the Hamiltonian phase ,flow. It is defined by gt(p, q) _ (p(t), q(t)), where p(t), q(t) is a solution of Hamilton's equations satisfying p(0) = p, q(0) = q.

I. Classical Mechanics

32

Liouville's canonical 1-form 0 on T* M defines a 2-form w = dB. In standard coordinates on T* M it is given by n

w=

dpi A dqz = dp A dq, i=1

and is a non-degenerate 2-form. The form w is called the canonical syrnplectic form on T*M. The symplectic form w defines an isomorphism J : T* (T* M) --- T(T* M) between tangent and cotangent bundles to T* M. For M is T(p?q)T * M ---} every (p, q) E T* M the linear mapping given by 7'(p>q)T'"M.

W(ui,uz) = J-1(u2)(ui), ui,ua E The mapping J induces the isomorphism between the infinite-dimensional vector spaces ,Al(T*M) and Vect(T*M), which is linear over C°O(T*M). If

4 is a 1-form on T*M, then the corresponding vector field J(9) on T*M satisfies

w(X, J(t4)) _ t4(X),

X E Vect(T*M),

and J' (X) _ -iXw. In particular, in standard coordinates,

J(dp) _ q and J(dq) _ -

,

so that XH = J(dH). Theorem 2.1. The Hamiltonian phase flow on T*M preserves the canonical symplectic form. Proof. We need to prove that (gt)*cv = w. Since gt is aone-parameter group of diffeomorphisms, it is sufficient to show that d

= LxHW = 4, (gt) w dt t=o where GXH is the Lie derivative along the vector field XH. Since for every vector field X,

£x(dl) = d(X(f)), we compute GXH (dpi) _ -d I

q

and GXH (dq) = d (ap I

I

so that GxHw =

(GxH (dpi) n dqZ + dpi A £xH (dq2)) i=1 dq2

_

Z-1

\-d \ aq l n

+ dpi n d

\ ap / /

-d(dH) = 0.

2. Hamiltonian Mechanics

33

Corollary 2.2. GXH(B) = d(-H+B(XH)), where B is Liouville's canonical 1-form.

The canonical symplectic form w on T* M defines the volume form wn }

n.

1 _=

n.

A w on T*M, called Liouville's volume form.

wA

n

Corollary 2.3 (Liouville's theorem). The Hamiltonian phase flow on T*M preserves Liouville's volume form. The restriction of the symplectic form w on T* M to the configuration space M is o. Generalizing this property, we get the following notion.

Definition. A submanifold .2' of the phase space T*M is called a Lagrangian submanifold if dim 2' = dim M and w= 0. It follows from Theorem 2.1 that the image of a Lagrangian submanifold under the Hamiltonian phase flow is a Lagrangian submanifold. Problem 2.1. Verify that XH is a well-defined vector field on T*M.

Problem 2.2. Show that if all level sets of the Hamiltonian H are compact submanifolds of T*M, then the Hamiltonian vector field XH is complete.

Problem 2.3. Let 7r : T* M --} M be the canonical projection, and let .2' be a Lagrangian submanifold. Show that if the mapping ?r : 2 -* M is a diffeomorphism, then 2 is a graph of a smooth function on M. Give examples when for some t > 4 the corresponding projection of gt ($) onto M is no longer a diffeomorphism.

2.2. The action functional in the phase space. With every function H on the phase space T* M there is an associated 1-form

B- Hdt=pdq - Hdt on the extended phase space T*M x Ilk, called the Poincare-Cartan form. Let "y: {to, t1] - T* M be a smooth parametrized path in T* M such that = qo and 7r ('Y (t1 )} = q1, where 7r : T* M -* M is the canonical projection. By definition, the lift of a path ry to the extended phase space TM x IE given by a(t) = (y(t),t), and a T*M x Ilk is a path a : [to, t1] path a in T*M x II is called an admissible path if it is a lift of a path yin T* M. The space of admissible paths in T* M x IR is denoted by P (T* M)qo,o .

A variation of an admissible path a is a smooth family of admissible paths aE, where E E [-EO, o] and cro = a, and the corresponding infinitesimal variation is E Tc7P(T*M)go,to =o

(cf. Section 1.2). The principle of the least action in the phase space is the following statement.

1. Classical Mechanics

34

Theorem 2.4 (Poincare). The admissible path Q in T*M x ][8 is an extremal for the action functional

S(Q) =

J

(pdq - Hdt --

tl (pq - H)dt to

if and only if it is a lift of a path y(t) _ (p(t), q(t)) in T*M, where p(t) and q(t) satisfy canonical Hamilton's equations

aH

aH

Proof. As in the proof of Theorem 1.1, for an admissible family aE(t) _ (p(t,E),q(t,E),t) we compute using integration by parts, n

d de

S (a) ) =p

q dpi -- pi q -

___.-

i=1 n

to

i

Sq -

oH api

Spi

dt

Oqu1t

:Pi

0

i=1

Since 5q(to) = Sq(tl) = 0, the path a is critical if and only if 1n(t) and q(t) satisfy canonical Hamilton's equations (2.1).

Remark. For a Lagrangian system (M, L), every path -y(t) _ (q(t)) in the configuration space M connecting points qo and ql defines an admissible path (t) _ (p(t), q(t), t) in the phase space T*M by setting p _ L. If the Legendre transform TL : TM -+ T*M is a diffeomorphism, then S

(P_H)dt=j tl ftl to to

Thus the principle of the least action in a configuration space -Hamilton's principle -follows from the principle of the least action in a phase space. In fact, in this case the two principles are equivalent (see Problem 1.28). From Corollary 1.5 we immediately get the following result.

Corollary 2.5. Solutions of canonical Hamilton's equations lying on the hypersurface H(p, q) = E are extremals of the functional fa Pdq in the class of admissible paths a lying on this hypersurface.

Corollary 2.6 (Maupertuis' principle). The trajectory ry = (q(r)) of a closed Lagrangian system (Al, L) connecting points qp and ql and having energy E is the extremal of the functional

fpdq _

f

/g(T),(T))(T)[aq

2. Hamiltonian Mechanics

35

on the space of all paths in the configuration space M connecting points qo

and ql and parametrized such that H((), q(T)) = E. The functional

so(7) = fpdq is called the abbreviated action15.

Proof. Every path ry = q('r), parametrized such that H ( 4 , q) = E, lifts to an admissible path a = ((r), q (T ), r), a C r C b, lying on the hypersurface q

H(p, q)= E. Problem 2.4 (Jacobi). On a Riemannian manifold (M, ds2) consider a Lagrangian system with L (q, v ) = 2 v M 2 - V (q). Let E> V(q) for all q e M. Show that the trajectories of a closed Lagrangian system (M, L) with total energy E are geodesics for the Riemannian metric ds2 = (E - V (q)}ds2 on M.

2.3. The action as a function of coordinates. Consider anon-degenerate Lagrangian system (M, L) and denote by ry(t; qo, VU) the solution of Euler-Lagrange equations d aL aL _ o

dt aq - aq with the initial conditions 'y(to) = qo E M and 'j'(tp) = vo e T40M. Suppose

that there exist a neighborhood Vo C TvpM of vo and ti > to such that for all v E Vp the extremals ry(t; qp, v), which start at time tp at qp, do not intersect in the extended configuration space M x lib for times to < t < tl. Such extremals are said to form a central field which includes the extremal 7o(t) _ 'y(t; qp, VU). The existence of the central field of extremals is

equivalent to the condition that for every to
Vo

4(t) _ 'Y(t;40,v) E Ut

is a diffeomorphism. Basic theorems in the theory of ordinary differential equations guarantee that for ti sufficiently close to to every extremal ry(t) for to
S(q,t; 90,to) = f L(7 (T))dT, to

15 The accurate formulation of Maupertuis' principle is due to Euler and Lagrange.

1. Classical Mechanics

36

where 'y(r) is the extremal from the central field that connects qo and q. For given qo and to, the classical action is defined for t E (to,t1) and q E U0<<1 U. For a fixed energy E, (2.3)

5(4, t, 40, to) = So(q, t, 40, to) - E(t - to),

where Sp is the abbreviated action from the previous section.

Theorem 2.7. The differential of the classical action S(q, t) with fixed initial point is given by

dS = pdq - Hdt, where p =

(q, q) and H = Pq - L(q, q) are determined by the velocity q of the extremal y(T) at time t.

Proof. Let qE be a path in M passing through q at = 0 with the tangent vector v E TqM ^IEBn, and for e small enough let ryE(T) be the family of extremals from the central field satisfying y(to) = qp and 'y(t) q. For the infinitesimal variation 8y we have &y(to) = 0 and 5'y(t) = v, and for fixed t we get from the formula for variation with the free ends (1.2) that

dS(v) _ This shows that

so that

t

q v.

S4 = p. Setting q(t) = ry(t), we obtain S(q(t),t)= as 4+ as =L,

= L - pq = -H.

Corollary 2.8. The classical action satisfies the following nonlinear partial differential equation (2.4) \q

This equation is called the Hamilton-Jacobi equation. Hamilton's equations (2.1) can be used for solving the Cauchy problem (2.5)

s(q),

s e C°°(M),

for Hamilton-Jacobi equation (2.4) by the method of characteristics. Namely, assume the existence of the Hamiltonian phase flow gt on T*M and consider the Lagrangian submanifold

as(q))

2. Hamiltonian Mechanics

37

a graph of the 1-form ds on M a section of the cotangent bundle i T*M -+ M. The mapping it 2 is one-to-one and for sufficiently small t the restriction of the projection it to the Lagrangian submanifold 2t = gt (2) remains to be one-to-one. In other words, there is t1 > o such that for all o C t C t 1 the mapping lrt = it o gt o (irI2)' : M - M is a diffeomorphism, and the extremals yf , qo, qo) in the extended configuration space M x R, where qo = qo) and (P0, qo) e ', do not intersect. Such extremals ap(po,

are called the characteristics of the Hamilton-Jacobi equation.

Proposition 2.1. For 0 < t
s(qo) +

8(q, t)

0

Here 'y(T) is the characteristic with y(t) = q and with the starting point qo = 'y(O) which is uniquely determined by q.

Proof. As in the proof of Theorem 2.7 we use formula (1.2), where now qo depends on q, and obtain

as as aqo aL aL aqo aq (4) = aqo (qo) aq + aq (q, q) - aqo (40, qo) aq =P, as

since aqo (qo) =

aL

aqo (Qo, 40) Setting q(t) = 'y(t) we get

-8(q(t),t) t so that

as

as

=9q-+-- =L(q,),

as = -H(p,q), -

and S satisfies the Hamilton-Jacobi equation. We can also consider the action S(q, t; qo, to) as a function of both variables q and qo. The analog of Theorem 2.7 is the following statement.

Proposition 2.2. The differential of the classical action as a function of initial and final points is given by

dS = pdq - podgo - H(p7 q)dt + H(po, qo)dto. Problem 2.5. Prove that the solution to the Cauchy problem for the HamiltonJacobi equation is unique.

1. Classical Mechanics

38

2.4. Classical observables and Poisson bracket. Smooth real-valued functions on the phase space T*M are called classical observables. The vector

space C°°(T*M) is an I[8-algebra - an associative algebra over R with a unit given by the constant function 1, and with a multiplication given by the point-wise product of functions. The commutative algebra CO°(T*M) is called the algebra of classical observables. Assuming that the Hamiltonian

phase flow gt exists for all times, the time evolution of every observable f E C°°(T*M) is given by (p,q) E TM. Equivalently, the time evolution is described by the differential equation

ft(p,q) = f(gt(p,q)) = f(p(t),q(t)),

dit

dt

dfs+t ds n

d(.ft o9s) ds

s=0

axaft

(aHaft

Lz-1

apz aqi

aqi ape

- XH(.ft) s=0

-

axaft

axaft

Dp aq

aq

ap'

called Hamilton's equation for classical observables. Setting (2.6)

{f, 9} = X1(9) =

f, 9 E

we can rewrite Hamilton's equation in the concise form

= {H, f},

(2.7)

where it is understood that (2.7) is a differential equation for a family of functions ft on T*M with the initial condition ft(p, q)=o = f(p, q). The properties of the bilinear mapping {, } : C°°(T*M) x C°°(T*M) -> C°°(T*M) are summarized below.

Theorem 2.9. The mapping {

,

} satisfies the following properties.

(i) (Relation with the symplectic form) {f, g} = w(J(df ), J(dg)) = w(X f, Xy). (ii) (Skew-symmetry)

{f,g} = -{g,f}. (iii) (Leibniz rule)

{fg, h} = f{g, h} + g{f, h}. (iv) (Jacobi identity) {f, {g,h}} + {g, {h,

for all f,g,h E C°°(T*M).

2. Hamiltonian Mechanics

39

Proof. Property (i) immediately follows from the definitions of w and J in Section 2.1. Properties (ii)-(iii) are obvious. The Jacobi identity could be verified by a direct computation using (2.6), or by the following elegant argument. Observe that {f, g} is a bilinear form in the first partial derivatives of f and g, and every term in the left-hand side of the Jacobi identity is a linear homogenous function of second partial derivatives of f, g, and h. Now the only terms in the Jacobi identity which could actually contain second partial derivatives of a function h are the following:

{f, {g, h}} + {g, {h, f}} _ (XfXg - X9X f)(IZ). However, this expression does not contain second partial derivatives of h since it is a commutator of two differential operators of the first order which is again a differential operator of the first order! The observable {f, g} is called the canonical Poisson bracket of the observables f and g. The Poisson bracket map {, } : C°O(T*M) xCO°(T*M) -p COO (TM) turns the algebra of classical observables C°°(T*M) into a Lie algebra with a Lie bracket given by the Poisson bracket. It has an important property that the Lie bracket is a bi-derivation with respect to the multiplication in C°O(T*M). The algebra of classical observables CO°(T*M) is an example of the Poisson algebra - a commutative algebra over IIg carrying astructure of a Lie algebra with the property that the Lie bracket is a derivation with respect to the algebra product. In Lagrangian mechanics, a function I on TM is an integral of motion for the Lagrangian system (M, L) if it is constant along the trajectories. In Hamiltonian mechanics, an observable I - a function on the phase space T*M - is called an integral of motion (first integral) for Hamilton's equations (2.1) if it is constant along the Hamiltonian phase flow. According to (2.7), this is equivalent to the condition

{H,I} = 0. It is said that the observables H and I are in involution (Poisson commute).

2.5. Canonical transformations and generating functions. Definition. A diffeomorphism g of the phase space T'*M is called a canonical transformation, if it preserves the canonical symplectic form w on T'*M, i.e., g* (w) = w. By Theorem 2.1, the Hamiltonian phase flow gt is a oneparameter group of canonical transformations.

Proposition 2.3. Canonical transformations preserve Hamilton's equations.

1. Classical Mechanics

40

Proof. From g*(w) =wit follows that the mapping J

T*(T*M) --

T(T*M) satisfies

g*oJog*_J.

(2.8)

Indeed, for all X,Y E Vect(M) we Navels

w(X, Y) = g*(w)(X, Y) = w(g(X), g(Y)) o g, so that for every 1-form i9 on M,

w(X, J(g*(9))) = g*(19)(X) = t9(g(X)) o g = w(g(X), J(9)) o

J(). Using (2.8), we get

which gives

9*(xx) = 9*(J(dH)) = J((9*)-1(dH)) = XK, where K =Hog-1. Thus the canonical transformation if maps trajectories of the Hamiltonian vector field XH into the trajectories of the Hamiltonian vector field XK.

Remark. In classical terms, Proposition 2.3 means that canonical Hamilton's equations

p=

q=

-(p,q)

in new coordinates (P, Q) = g(p, q) continue to have the canonical form

P _ - aQ (P,Q)

,

Q - aP

P,Q)

with the old Hamiltonian function K(P, Q) = H(P, q). Consider now the classical case M = W. For a canonical transformation

(P,Q) = g(p, q) set P = P(P, q) and Q = Q(p,q). Since dP n dQ = dp n dq on T*M

the 1-form pdq - PdQ

the difference between the canonical Liouville 1-form and its pullback by the mapping if - is closed.

From the Poincare lemma it follows that there exists a function F(p, q) on

j2n such that (2.9)

pdq - PdQ = dF(P, 4)

Now assume that at some point (p0, qo) the n x n matrix

DP

(3p) fl --

is non-degenerate. By the inverse function theorem, there exists a neighborhood U of (po, qo) in R2n for which the functions P, q are coordinate functions. The function

S(P,4) = F'(p,Q) + PR x s Since g is a difeomorphism, g* X is a well-defined vector field on Al.

41

2. Hamiltonian Mechanics

is called a generating function of the canonical transformation g in U. It follows from (2.9) that in new coordinates P, q on U,

as

(P, 4)

and Q=(P,q). p

The converse statement below easily follows from the implicit function theorem.

Proposition 2.4. Let S(P, q) be a function in some neighborhood U of a point (Po,qo) E Il82n such that the n x n matrix

a2s

aPaq

I

a2s

_ aPZaq (Po, qo)

(Po, qo)

}n

i

.

is non-degenerate. Then S is a generating function of a local (i. e., defined canonical transformation. in some neighborhood of (Po, qo) in

Suppose there is a canonical transformation (P, Q) = g(p, q) such that H(p, q) = K(P) for some function K. Then in the new coordinates Hamilton's equations take the form

P=0, Q=,

(2.10)

and are trivially integrated:

Q(t) = Q(0) + t( P(0)).

P(t) = P(0),

P

is non-degenerate, the generating function Assuming that the matrix S(P, q) satisfies the differential equation

(P, q), q) =K(P)H

(2.11)

where after the differentiation one should substitute q = q(P, Q), defined by the canonical transformation g-1. The differential equation (2.11) for fixed P, as it follows from (2.3), coincides with the Hamilton-Jacobi equation for

the abbreviated action So = S - Et where E = K(P), H(q

(P,4) ,q) =E.

Theorem 2.10 (Jacobi). Suppose that there is a function S(P, q) which depends on n parameters P = (F1, ... , Pn), satisfies the Hamilton-Jacobi equation (2.11) for some function K(P), and has the property that the n x n matrix

P a2

is non-degenerate. Then Hamilton's equations

4

aH

P=---jqy q= -;; op

1. Classical Mechanics

42

can be solved explicitly, and the functions P(p, q) _ (Pl(p, 4), defined by the equations p =

Proof. Set p =

q

Pn(p, q)), (P, q), are integrals of motion in involution.

as (P, q) and Q = as

,

, q ). By the inverse function the-

orem, g(p, q) _ (P, Q) is a local canonical transformation with the generating function S. It follows from (2.11) that H(p(P, Q), q(P, Q)) = K(P), so that Hamilton's equations take the form (2.10). Since w = dP n dQ, integrals of motion Pl (p, q), ... , Pn(p, q) are in involution. The solution of the Hamilton-Jacobi equation satisfying conditions in Theorem 2.10 is called the complete integral. At first glance it seems that solving the Hamilton-Jacobi equation, which is a nonlinear partial differential equation, is a more difficult problem then solving Hamilton's equations, which is a system of ordinary differential equations. It is quite remarkable that for many problems of classical mechanics one can find the complete integral of the Hamilton-Jacobi equation by the method of separation of variables. By Theorem 2.10, this solves the corresponding Hamilton's equations.

Problem 2.6. Find the generating function for the identity transformation P =

p,Q=q

Problem 2.7. Prove Proposition 2.4. Problem 2.8. Suppose that the canonical transformation g (p, q) = (P, Q) is such that locally (Q, q) can be considered as new coordinates (canonical transformations with this property are called free). Prove that S1 (Q, q) = F(p, q), also called a generating function, satisfies aS1 p=---- and P=-

.

Problem 2.9. Find the complete integral for the case of a particle in R3 moving in a central field.

2.6. Symplectic manifolds. The notion of a symplectic manifold is a generalization of the example of a cotangent bundle T* M.

Definition. Anon-degenerate, closed 2-form w on a manifold Gl is called a symplectic form, and the pair (4', w) is called a symplectic manifold. Since a symplectic form w is non-degenerate, a symplectic manifold 4' is necessarily even-dimensional, dim .' = 2n. The nowhere vanishing 2n-form

wn defines a canonical orientation on 4, and as in the case 4 = T* M, is called Liouville s volume form. We also have the general notion of a n. Lagrangian submanifold.

2. Hamiltonlan Mechanics

43

Definition. A submanifold z° of a symplectic manifold (4', w) is called a Lagrangian submanifold, if dim z° = 2 dim .iGl and the restriction of the symplectic form w to 2 is 0. Symplectic manifolds form a category. A morphism between (4'i, wi) and (4'2, ca2), also called a symplectomorphism, is a mapping f :4'i -f such that cvl = f*(w2). When .iGll =X112 and wl = w2i the notion of a symplectomorphism generalizes the notion of a canonical transformation. The direct product of symplectic manifolds (4', cal) and (4'2, w2) is a symplectic manifold (.4'1 x 2, 7r (w1 ) + 7r2 (w2 )) ,

where i1 and 7r2 are, respectively, projections of 4' x 42 onto the first and second factors in the Cartesian product. Besides cotangent bundles, another important class of symplectic manifolds is given by Kahler manifolds17. Recall that a complex manifold 4 is a Kahler manifold if it carries the Hermitian metric whose imaginary part is a closed (1, 1)-form. In local complex coordinates z = (z',. . , zn ) on .' the Hermitian metric is written as .

n

h=

z)dza ® d. c/9=1

Correspondingly, n 1

g = Re h = 2

_

h( z, z, z )(dz® dz + d® dz)

is the Riemannian metric on .' and n

w = - Im h =

2

h - z, z )dza A d

is the symplectic form on 4 (considered as a 2n-dimensional real manifold). The simplest compact Kahler manifold is CP1 ^J 82 with the symplectic form given by the area 2--form of the Hermitian metric of Gaussian curvature

the round metric on the 2-sphere. In terms of the local coordinate z associated with the stereographic projection CP1 ^C U {oo}, 1

dz A dz

(1+2)2

.

Similarly, the natural symplectic form on the complex projective space (CPn is the symplectic form of the Fubini-Study metric. By pull-back, it defines symplectic forms on complex projective varieties. '7Needless to say, not every symplectic manifold admits a complex structure, not to mention a Kahler structure.

44

1. Classical Mechanics

The simplest non-compact Kahler manifold is the n-dimensional complex vector space Cn with the standard Hermitian metric. In complex coordinates z = (z',.. . , zn ) on Cn it is given by n

h = dz ® dz =

dza ® d2a.

a= In terms of real coordinates (x, y) _ (x',.

.

. ,

xn, y1,

... , yn) on 2n ^J fin,

where z = x -+- iy, the corresponding symplectic form w = - Im h has the canonical form

w=?dzndz=

n

dxan dy a= dx n dy.

a=1

This example naturally leads to the following definition.

Definition. A symplectic vector space is a pair (V, w), where V is a vector space over Ilk and w is a non-degenerate, skew-symmetric bilinear form on V.

It follows from basic linear algebra that every symplectic vector space V has a symplectic basis a basis e1, ... , en, fi,. , fn of V, where 2n = dim V, such that . .

o

and

off,

i, j _ 1, ... , n.

In coordinates (p, q) _ (p1, ... , p, q', ... , qn) corresponding to this basis, and v n

w = dp n dq =

dpi n dqi .

Thus every symplectic vector space is isomorphic to a direct product of the phase planes j 2 with the canonical symplectic form dp n dq. Introducing complex coordinates z = p + iq, we get the isomorphism V ^r Cn, so that every symplectic vector space admits a Kahler structure. It is a basic fact of symplectic geometry that every symplectic manifold is locally isomorphic to a symplectic vector space.

Theorem 2.11 (Darboux' theorem). Let (4', w) be a 2n-dimensional symplectic manifold. For every point x E 4 here is a neighborhood U of x with local coordinates (p, q) = (pi, ... , pn, q, ... , qn) such that on U n

w=dpndq=

dpindgz.

Coordinates p, q are called canonical coordinates (Darboux coordinates).

The proof proceeds by induction on n with the two main steps stated as Problems 2.13 and 2.14.

2. Hamiitonian Mechanics

45

A non-degenerate 2-form w for every x E .mil defines an isomorphism J : Tx J( -> TX.21 by

w(u1,u2) = J'(u2)(ui), U1,U2 E TX4'. Explicitly, for every X E Vect (.mil) and 9 E Al (4') we have

(X)

w(X,

and

J-l(X) _ -Zx(w)

(cf. Section 2.1). In local coordinates x = (xl, ... , x2n) for the coordinate chart (U, cp) on 4', the 2-form w is given by w=2

wia(x) dx A dx', i,j=1

where {wjj (x)}? _, is a non-degenerate, skew-symmetric matrix-valued function on cp(U). Denoting the inverse matrix by we have i = 1, ... , 2n.

Definition. A Hamiltonian system is a pair consisting of a symplectic manifold (4', w), called a phase space, and a smooth real-valued function H on .iGl, called a Hamiltonian. The motion of points on the phase space is described by the vector field

XH = J(dH), called a Hamiltonian vector field.

The trajectories of a Hamiltonian system ((4', w), H) are the integral curves of a Hamiltonian vector field XH on .iGl. In canonical coordinates (p, q) they are described by the canonical Hamilton's equations (2.1),

8H OH p=---, q=-. Suppose now that the Hamiltonian vector field XH on .ill is complete. The Hamiltonian phase flow on 4' associated with a Hamiltonian H is a oneparameter group {t}tER of diffeomorphisms of .mil generated by XH. The following statement generalizes Theorem 2.1. Theorem 2.12. The Hamiltonian phase flow preserves the symplectic form.

Proof. It is sufficient to show that GXHw = 0. Using Cartan's formula

GX=iXod+doix and dw = 0, we get for every X E Vect (.iGl) ,

GXw = (do)().

1. Classical Mechanics

46

Since iX(w)(Y) = cv(X,Y), we have for X = XH and every Y E

that ZXx (w) (Y) = w(J(dH), Y) _ -dH(Y). Thus iXH (w) _ -dH, and the statement follows from d2 = 0.

Corollary 2.13. A vector field X on .iGl is a Hamiltonian vector field if and only if the 1-form iX(w) is exact.

Definition. A vector field X on a symplectic manifold (.ill, w) is called a symplectic vector field if the 1-form ix (w) is closed, which is equivalent to

The commutative algebra C°°(4'), with a multiplication given by the point-wise product of functions, is called the algebra of classical observables.

Assuming that the Hamiltonian phase flow gt exists for all times, the time evolution of every observable f E C°O (4') is given by

ft(x) = f(gt(x)),

x E 4',

and is described by the differential equation

ftx(f)

dt

- Hamilton's equation for classical observables. Hamilton's equations for observables on 4' have the same form as Hamilton's equations on 4' = T*M, considered in Section 2.3. Since

XH(f)

- df(XH) -W(XH, J(df)) =W(XH, Xf),

we have the following.

Definition. A Poisson bracket on the algebra C°°(4') of classical observables on a symplectic manifold (4', w) is a bilinear mapping { C°°(4') x C°°(4") -> C°°(4'), defined by {f,g} = w(X1, Xg), f,g E C°°(4').

}

Now Hamilton's equation takes the concise form (2.12)

understood as a differential equation for a family of functions ft on 4' with the initial condition f. In local coordinates x = (x1, ... , x2n) on 2n

4"

{f,g}(x)=z,7=1

a f(x) ag(x) axz

axe

Theorem 2.14. The Poisson bracket { , } on a symplectic manifold (.ill, w) is skew-symmetric and satisfies Leibniz rule and the Jacobi identity.

2. Hamiltonian Mechanics

47

Proof. The first two properties are obvious. It follows from the definition of a Poisson bracket and the formula [X1,Xg](h) _ (XgX1 - X fXy)(h) _ {g, {f, h}} - {f, {g, h}}

that the Jacobi identity is equivalent to the property (2.13)

[X1,Xg] =

X{f,s}.

Let X and Y be symplectic vector fields. Using Cartan's formulas we get 2[X,Y](CJ) -'CX(2Y(W)) - ZY(LX(W))

= d(iX o iY(W)) + Zxd(zY(w))

= d(w(Y,X)) = ZZ(W),

where Z is a Hamiltonian vector field corresponding to cv(X,Y) E C°O(.ill). Since the 2-form w is non-degenerate, this implies [X, Y] = Z, so that setting X = X f, Y = Xy and using {f, g} = w(X f, X9), we get (2.13). From (2.13) we immediately get the following result. Corollary 2.15. The subspace Ham(.iGl) of Hamiltonian vector fields on .iGl The mapping CO°(4') -> Ham(4'), given is a Lie subalgebra of

by f H X f, is a Lie algebra homomorphism with the kernel consisting of locally constant functions on 4'. As in the case Gl = T*M (see Section 2.4), an observable I - a function on the phase space Gl - is called an integral of motion (first integral) for the Hamiltonian system ((4', w), H) if it is constant along the Hamiltonian phase flow. According to (2.12), this is equivalent to the condition (2.14)

{H,I}=O.

It is said that the observables H and I are in involution (Poisson commute). From the Jacobi identity for the Poisson bracket we get the following result.

Corollary 2.16 (Poisson's theorem). The Poisson bracket of two integrals of motion is an integral of motion.

Proof. If {H, Il} _ {H, I2} = 0, then {H,{Ii,12}} _ {{H,11},12} - {{H,12},I1} = 0. It follows from Poisson's theorem that integrals of motion form a Lie algebra and, by (2.13), corresponding Hamiltonian vector fields form a Lie subalgebra in Vect(.iGl). Since {I, H} = dH(XI) = 0, the vector fields XI are tangent to submanifolds .iGlE _ {x e .iGl : H(x) = E} -the level sets of the Hamiltonian H. This defines a Lie algebra of integrals of motion for the Hamiltonian system ((.4', w), H) at the level set .iGIE.

1. Classical Mechanics

48

Let G be afinite-dimensional Lie group that acts on a connected symplectic manifold (4', cv) by symplectomorphisms. The Lie algebra g of G acts on .mil by vector fields

Xt(.f)(x) = as s=Q X E Vect(.') is a homomorphism of Lie and the linear mapping g algebras, , T] E g [Xt, X ] = X [t,,] , The G-action is called a Hamiltonian action if X are Hamiltonian vector fields, i.e., for every g there is I C°°(4'), defined up to an additive .

constant, such that Xt = X = J(d I

).

It is called a Poisson action if there

is a choice of functions ' such that the linear mapping 1 : g -+ C°°(4') is a homomorphism of Lie algebras, (2.15)

{,I,} =

,ij Eg.

Definition. A Lie group G is a symmetry group of the Hamiltonian system ((4', w), H) if there is a Hamiltonian action of G on .iGl such that

H(g x) = H(x), g E G, x E 4'. Theorem 2.17 (Noether theorem with symmetries). If G is a symmetry group of the Hamiltonian system ((4', cv), H), then the functions , E g, are the integrals of motion. If the action of G is Poisson, the integrals of motion satisfy (2.15).

Proof. By definition of the Hamiltonian action, for every

E g,

0 = X(H) = X(H) = {,H}. Corollary 2.18. Let (M, L) be a Lagrangian system such that the Legendre transform 7L : TM - T*M is a diffeomorphism. Then if a Lie group G is a symmetry of (M, L), then G is a symmetry group of the corresponding Hamiltonian system ((T*M, w), H = EL o TL 1), and the corresponding G-

action on T*M is Poisson. In particular, I _ -Ic o TL 1, where I are Noether integrals of motion for the one-parameter subgroups of G generated

Proof. Let X be the vector field associated with the one-parameter subgroup {es}sea of diffeomorphisms of M, used in Theorem 1.3, and let X' be its lift to TM. We have18 (2.16)

Xt =

-frL)*(X'),

1$The negative sign reflects the difference in definitions of X and X.

2. Hamiltonian Mechanics

49

and it follows from (1.16) that I = i(9) =

B is the canonical

Liouville 1-form on T*M. From Cartan's formula and (1.17) we get

dig = d(iX£ (B)) _ -j(d0) + Gx£ (B) _ -zx£ (W )

so that

J(d) = -J(i(w)) = X, and the G-action is Hamiltonian. Using (1.17) and another Cartan's formula, we obtain

= irx,xi(0) = £x(ix(O)) +

=Xt() Example 2.1. The Lagrangian L = Zmr2 - V(r) for a particle in 1[83 moving in a central field (see Section 1.6) is invariant with respect to the action of the group 50(3) of orthogonal transformations of the Euclidean space I[83. Let ul, u2i u3 be a basis for the Lie algebra 50(3)

corresponding to the rotations with the axes given by the vectors of the standard basis el, e2, e3 for I[83 (see Example 1.10 in Section 1.4). These generators satisfy the commutation relations [ui, uj] _ Eijkuk,

where i, j, k = 1, 2, 3, and ezjk is a totally anti-symmetric tensor, e123 = 1. Corresponding Noether integrals of motion are given by uz = -Mi, where

M1 = (r x P)1 =reps - r3p2, M2 = (r x P)2 = r3pi -rips M3 = (r x p)3 = rlp2 - r2p1

are components of the angular momentum vector M = r x P. (Here it is convenient to lower the indices of the coordinates ri by the Euclidean metric on 1R3.) For the Hamiltonian 2

H=-+V(r) 2 {H,M}=0. According to Theorem 2.17 and Corollary 2.18, Poisson brackets of the components of the angular momentum satisfy

{Mi,Mj} _ -6ijkMk, which is also easy to verify directly using (2.6),

8 f 8g {f,g}(p,r =agar

8 f 8g

ara

I. Classical Mechanics

50

Example 2.2 (Kepler's problem). For every a E I[8 the Lagrangian system on I[83 with

L=2mr2 +--a r

has three extra integrals of motion

the components W1, W2, W3 of the

Laplace-Runge-Lenz vector, given by W_

p

xM-cer

r

m

(see Section 1.6). Using Poisson brackets from the previous example, together with {r2, M3} = -Ejjkrk and {pi, M3} = ---EijkPk, we get by a straightforward computation,

p2

where H -

2m

2H m

and {W,

{W2, M3} = a

- - is the Hamiltonian of

Kepler,s problem.

r

The Hamiltonian system ((4', w), H), dim .iI = 2n, is called completely

integrable if it has n independent integrals of motion Fl = H,. ,F, in involution. The former condition means that dFi (x), ... , dFn (x) E are linearly independent for almost all x E 4. Hamiltonian systems with . .

one degree of freedom such that dH has only finitely many zeros are completely integrable. Complete separation of variables in the Hamilton-Jacobi equation (see Section 2.5) provides other examples of completely integrable Hamiltonian systems. Let ((4', w), H) be a completely integrable Hamiltonian system. Suppose that the level set f = {x e 4 : Fl (x) = fi,. . , F, (x) = fTh} is compact and tangent vectors JdF1,... , JdFT are linearly independent for all x E 4s. Then by the Liouville-Arnold theorem, in a neighborhood of 4' there exist so-called action-angle variables: coordinates I = (Ii,. , In,) E (R>o)'h and cp = (c01, ... , pn) E Tn = (I/27rZ)n such that w = dl Adcp and H = H(11,. . , In). According to Hamilton's equations, .

. .

.

z

o

and

pz

w

Ii

,

z

1,

... , n,

so that action variables are constants, and angle variables change uniformly, Sri (t) = Bpi (o) + wit, i = 1, ... , n. The classical motion is almost-periodic

with the frequencies wi,... , cvn. Problem 2.10. Show that a symplectic manifold (4', w) admits an almost corn-

plex structure: a bundle map j : T4 -+ T.' such that J2 = -id. Problem 2.11. Give an example of a symplectic manifold which admits a complex structure but not a Kahler structure.

2.Hamiltonian Mechanics

51

Problem 2.12 (Coadjont orbits). Let G be a finite-dimensional Lie group, let g be its Lie algebra, and let g* be the dual vector space to g. For U E g* let .' = C7u be the orbit of u under the coadjoint action of G on g. Show that the formula w(u1, u2) = u([xx, x2]),

where u1 = ad*x1 (u), u2 = ad*x2 (u) E (9u, and ad* stands for the coadjoint action of a Lie algebra g on g*, gives rise to a well-defined 2-form on .', which is closed and non-degenerate. (The 2-form w is called the Kirillov-Kostant symplectic form.)

Problem 2.13. Let (4', w) be a symplectic manifold. For x E 4 choose a function q1 on 4' such that q1 (x) = o and dq1 does not vanish at x, and set X = -Xqi. Show that there is a neighborhood U of x e .' and a function p1 on U such that X(q') = 1 on U, and there exist coordinates p1, q1, z1, ... , such that

X= p1

and V _ Xr1 =

aq'

on U

.

Problem 2.14. Continuing Problem 2.13, show that the 2--form w - dp1 A dq1 on U depends only on coordinates z1, ... , and is non-degenerate. z2rt-2

Problem 2.15. Do the computation in Example 2.2 and show that the Lie algebra of the integrals M1, M2, M3, W1, W2, W3 in Kepler's problem at H(p, r) = E is isomorphic to the Lie algebra so(4), if E C o, to the Euclidean Lie algebra e(3), if E = o, and to the Lie algebra so ( 1, 3), if E> o.

Problem 2.16. Find the action-angle variables for a particle with one degree of freedom, when the potential V(x) is a convex function on R satisfying V(x) oo. (hint: Define I = H(p, x) = E.)

pdx, where integration goes over the closed orbit with

Problem 2.17. Show that a Hamiltonian system describing a particle in IR3 moving in a central field is completely integrable, and find the action-angle variables.

Problem 2.18 (Symplectic quotients). For a Poisson action of a Lie group G on a symplectic manifold (4', w), define the moment map P :.' - g* by P(x)( ) =' (x), E g, x E 4', where g is the Lie algebra of G. For every p E g* such that a stabilizer Gp of p acts freely and properly on .gyp = P-1 (p) (such p is called the regular value of the moment map), the quotient M = is called a reduced phase space. Show that MP is a symplectic manifold with the symplectic form uniquely characterized by the condition that its pull-back to 4' coincides with the restriction to p of the symplectic form w.

2.7. Poisson manifolds. The notion of a Poisson manifold generalizes the notion of a symplectic manifold.

Definition. A Poisson manifold is a manifold .mil equipped with a Poisson structure - askew-symmetric bilinear mapping

{, } : C°°(di) x C°°(Ji) which satisfies the Leibniz rule and Jacobi identity.

1. Classical Mechanics

52

Equivalently, .ill is a Poisson manifold if the algebra ,A = C°°(4') of classical observables is a Poisson algebra - a Lie algebra such that the Lie bracket is a bi-derivation with respect to the multiplication in A (a pointwise product of functions). It follows from the derivation property that in local coordinates x = (xl, ... , xN) on 4", the Poisson bracket has the form {.f, 9}(x) _

N 2'

z,j=1

(x)af(x)ag(x) 8xz

axe

(x), called a Poisson tensor, defines a global section r of The 2-tensor the vector bundle T.iGI A T.21 over 4'. The evolution of classical observables on a Poisson manifold is given by Hamilton's equations, which have the same form as (2.12),

= X(f) ={H,f}. The phase flow gt for a complete Hamiltonian vector field XH = {H,

}

defines the evolution operator Ut :.A -> .A by

Ut(.f)(x) = f(gt(x)), .f E A. Theorem 2.19. Suppose that every Hamiltonian vector field on a Poisson manifold (4', { }) is complete. Then for every H E ,A, the corresponding evolution operator Ut is an automorphism of the Poisson algebra ,A, i. e., ,

Ut({f, g}) _ {Ut(f), Ut(g)} for all f,g E A.

(2.17)

Conversely, if askew-symmetric bilinear mapping { , } : C°°(4') x C°°(4') -> C°°(4') is such that XH = {H, } are complete vector fields for all H E A, and corresponding evolution operators Ut satisfy (2.17), then (4', { , }) is a Poisson manifold.

Proof. Let ft = Ut(f ), gt = Ut(g), and19 ht = Ut({ f, g}). By definition, d

dt

If (4', { that

{ft,gt} _ {{H,ft},gt}+{ft,{H,gt}} and dht = {H, ht}. ,

dt }) is a Poisson manifold, then it follows from the Jacobi identity

{{H,ft},gt} + {ft, {H,gt}} _ {H, {ft,gt}}, so that ht and {ft,gt} satisfy the same differential equation (2.12). Since these functions coincide at t = 0, (2.17) follows from the uniqueness theorem for the ordinary differential equations. Conversely, we get the Jacobi identity for the functions f, g, and H by

differentiating (2.17) with respect to t at t = 0. 1 9 Here g, is not the phase flow!

2. Hamiltonian Mechanics

53

Corollary 2.20. A global section r of T.iGI A TAt is a Poisson tensor if and only if

£x17= O for all fEA. Definition. The center of a Poisson algebra ,A is

Z(A)={f EA:{f,g}=0 for all A Poisson manifold (4', { }) is called non-degenerate if the center of a Poisson algebra of classical observables A = C°° (4') consists only of locally constant functions (2(A) = I[8 for connected 4'). Equivalently, a Poisson manifold (4', {, }) is non-degenerate if the Poisson tensor r is non-degenerate everywhere on 4', so that .mil is necessarily an even-dimensional manifold. Anon-degenerate Poisson tensor for every x E .iGl defines an isomorphism J : T4' -f by

ij(ul,ua) = u2(J(ul)), U1, U2 E TxAt. In local coordinates x = (xl, ... , x) for the coordinate chart (U, cp) on .iGl, we have

N

a

j=1

8x

J(dx) =

i = 1,.

. .

,

N.

Poisson manifolds form a category. A morphism between (.21{ , }1) and }2) is a mapping cp : .4'l - 4' 2 of smooth manifolds such that

{focp,gocp}1={f,g}2ocp for all f,gEC°O(./Gl2). A direct product of Poisson manifolds (4", { , }1) and (4'2, { }2) is a Poisson manifold (4', x 4'2i { }) defined by the property that natural projection maps 7rl .iGli x Gl2 -f .iGli and 712 4<1 x Gl2 - 4'2 are Poisson mappings. For f E x Gl2) and (Si, 52) E Gti x 4'2 denote, respectively, by fx) and f) restrictions of f to .ill x {x2} and {x,} x 4'2. x ll2), Then for f,g E {f,g}(xl>x2)

-

{f,g}2(x2).

Non-degenerate Poisson manifolds form a subcategory of the category of Poisson manifolds.

Theorem 2.21. The category of symplectic manifolds is (anti-) isomorphic to the category of non-degenerate Poisson manifolds.

Proof. According to Theorem 2.14, every symplectic manifold carries a Poisson structure. Its non-degeneracy follows from the non-degeneracy of a symplectic form. Conversely, let (4', { , }) be anon-degenerate Poisson manifold. Define the 2-form w on Gl by

w(X,Y) = J-1(Y)(X),

X,Y E Vect(.iGl),

1. Classical Mechanics

54

where the isomorphism J : T*.ill - T.iGI is defined by the Poisson tensor

In local coordinates x _ (x'..... xN) on .ill, (x) dxz n dx3,

w=1
where {ijjj(X)}._1 is the inverse matrix to {if3(x)}_1. The 2-form w is skew-symmetric and non-degenerate. For every f E A let X f = {f, } be the corresponding vector field on .ill. The Jacobi identity for the Poisson bracket { } is equivalent to GXfr = 0 for every f E A, so that ,

Gxfw=O. Since X ,c = Jdf, we have w(X, Jdf) = df (X) for every X E Vect(.ill), so that W(Xf,Xg) = {f,g}. By Cartan's formula, dw(X, Y, Z) = 3 (Lw(Y,Z) - GYw(X, Z) + GZw(X, Y)

-w([X,Y],Z)+w([X,Z],Y) -w([Y,Z],X)), where X, Y, Z E Vect(.iGl). Now setting X = X1, Y = Xy, Z = Xh, we get dW(Xf, Xg, Xh)

-3

lW(Xh, [X1,

`Z'9])

+W(Xf, [Xg, Xh]) +W(X9, [Xh, Xf]))

- g (w(X, X{f,9}) +W(Xf, X{g,h}) +W(X9, X{h,f})) = 3 ({h, {f, g}} + {f, {g, h}} + {g, {h, f}})

The exact 1-forms df, f E .A, generate the vector space of 1-forms .Al (4') as a module over ,A, so that Hamiltonian vector fields X f = Jdf generate the vector space Vect(4') as a module over A. Thus dw = 0 and (.ill, w) is a symplectic manifold associated with the Poisson manifold (.ill, { , }). It follows from the definitions that Poisson mappings of non-degenerate Poisson manifolds correspond to symplectomorphisms of associated symplectic manifolds.

Remark. One can also prove this theorem by a straightforward computation in local coordinates x = (x',.. xN) on 4'. Just observe that the . ,

condition Dx i

Dx

+

Dx

j

= o,

i,7, l = 1, ... , N,

which is a coordinate form of dw = o, follows from the condition N

31 C

z (x)

a

kl

/xl

C7x

J + rl

l

(x)

ar ik /mil k, (x) J+

a /x1 li

C7xi

1) _ 0,

2. Hamiltonian Mechanics

55

which is a coordinate form of the Jacobi identity, by multiplying it three times by the inverse matrix rjzj (X) using

(iP (x)

j

+

= o.

p=I

Remark. Let 4 =

with the Poisson bracket {, } given by the canonical symplectic form w = dp A dq, where (p, q) = (pi,. . . ,p, q', ... , n are coordinate functions on T*IR. The non-degeneracy of the Poisson manifold

(T*Rn, { , }) can be formulated as the property that the only observable f E C°Q (T*IRn ) satisfying

{f,pi}={f,p}=o, {f,ql}=.={f,qn} is f(p, q) = const.

Problem 2.19 (Dual space to a Lie algebra). Let g be a finite-dimensional Lie algebra with a Lie bracket [, ], and let g* be its dual space. For f g E C°° g* define

{f, g}(u) = u ([dl, dg])where

' E g * and T9* * ^1 g. Prove that {, } is a Poisson bracket. (It was introduced by Sophus Lie and is called a linear, or Lie-Poisson bracket.) Show that this bracket is degenerate and determine the center of A = C°° (g*).

Problem 2.20. A Poisson bracket {, } on 4 restricts to a Poisson bracket {, }o on a submanifold K if the inclusion i : Y -} .' is a Poisson mapping. Show that the Lie-Poisson bracket on g* restricts to a non-degenerate Poisson bracket on a

coadjoint orbit, associated with the Kirillov-Kostant symplectic form. Problem 2.21 (Lie-Poisson groups). A finite-dimensional Lie group is called a Lie-Poisson group if it has a structure of a Poisson manifold (G, {, }) such that the group multiplication G x G -k G is a Poisson mapping, where G x G is a direct product of Poisson manifolds. Using a basis l3 , ... , n of left-invariant vector fields on G corresponding to a basis x1, ... , xn of the Lie algebra g, the Poisson bracket {, } can be written as {f1,f2}(g)

z

-

f2

i,j = 1

where the 2-tensor iij (g) defines a mapping r/ G --} A29 by i(g) _

x. Show that the bracket {

}

equips Gwith aLie-Poisson structure if and only if

the following conditions are satisfied: (1) for all g e G, n

Z2k(9) n

- t-i (c(g)i(g) +

where [xi, x j ] = action on A29, i.e.,

+1(9)aliIkZ(9) + + Clprlp2

(9))

c xk; (ii) the mapping i is a group l-cocycle with the adjoint (g1g2) =

r1(g1) + (g2), 9'1, 9'2 E G.

1. Classical Mechanics

56

Problem 2.22. Show that the second condition in the previous problem trivially 'j-1 rzj xi ®xa E for some r = holds when 7] is a coboundary, i(g) = A2 g, and then the first condition is satisfied if and only if the element (r) = [r12, r13 +T23] + [r13, r23] E A3g

rz' x2 ®x ®1, is invariant under the adjoint action of g on A3g. Here r12 = ria 1 ® x ® x j are corresponding ® 1 ® x j , and r23 r13 = =1 elements in the universal enveloping algebra Ug of a Lie algebra g. In particular, G is a Lie--Poisson group if (r) = o, which is called the classical Yang-Baxter equation.

riixi ®x E A2g is such that the matrix {r2 } is non-degenerate, and let {rid } be the inverse matrix. Show that r satisfies the classical Yang-Baxter equation if and only if the map c : A2g --} C, defined by uzxi, y = i 1 vzxi, is a non-degenerate c(x, y) = E=1 rijuiv2, where x = 1 Lie algebra 2-cocycle, i.e., it satisfies

Problem 2.23. Suppose that r =

c(x, [y, z] } + c(z, [x, y]) + c(y, [z, x] ) = 0,

x, y, z E g.

2.8. Hamilton's and Liouville's representations. To complete the formulation of classical mechanics, we need to describe the process of measurement. In physics, by a measurement of a classical system one understands the result of a physical experiment which gives numerical values for classical observables. The experiment consists of creating certain conditions for the system, and it is always assumed that these conditions can be repeated over

and over. The conditions of the experiment define a state of the system if repeating these conditions results in probability distributions for the values of all observables of the system. Mathematically, a state 1u on the algebra A = C°°(4') of classical observables on the phase space V is the assignment

A f I>

E

(IR),

Borel measures on where .9(R) is a set of probability measures on I such that the total measure of IE is 1. For every Borel subset E C R the quantity o C f E} C 1 is a probability that in the state 1a the value of the observable f belongs to E. By definition, the expectation value of an observable f in the state u is given by the Lebesgue-Stieltjes integral

E(f) =

Qo

Adf(a),

f ((-oc, A)) is a distribution function of the measure dµ f. The correspondence f H µ f should satisfy the following natural properties.

where

Eµ (f) I G oo for f E .Ao -the subalgebra of bounded observables. S2. Eµ(1) = 1, where 1 is the unit in A. S 1.

2. Hamiltonian Mechanics

57

S3. For all a, b E I[8 and f, g E .A,

Eµ(a f + bg) = aE1(f) + bEµ(g),

if both Eµ(f) and Eµ(g) exist. S4. If fi = cp o f2 with smooth cp : lib -k I[8, then for every Borel subset EC I[8,

/1f(E) = It follows from property S4 and the definition of the Lebesgue-Stieltjes integral that (2.18)

In particular, E,(f 2) o for all f e A, so that the states define normalized, positive, linear functionals on the subalgebra A0. Assuming that the functional E extends to the space of bounded, piecewise continuous functions on 4, and satisfies (2.18) for measurable functions cp, one can recover the distribution function from the expectation values by the formula

µf(A) = Eµ(e(A- f)), where B(x) is the Heavyside step function,

0(x) =

fi, x>0, 10, x<0.

Indeed, let x be the characteristic function of the interval (-oc, A) C I[8. Using (2.18) and the definition of the Lebesgue-Stieltjes integral we get

E(0(A

-

=

L X(S)df(S) =

((-oo, A))

f(A).

Every probability measure dµ on 4' defines the state on A by assigning20

to every observable f a probability measure µ f = f () on l[8 - a pushforward of the measure dµ on 4' by the mapping f :ill -> I[8. It is defined by

µf(E) _ c(f-1(E)) for every Borel subset E C Ilg, and has the distribution function

= (f'(-oo, A))

=

J

d,

where 4'A(f) _ {x E .ill : f(x) < A}. It follows from the Fxbini theorem that (2.19)

E( f) = f Adµf (A) -

fd.

20There should be no confusion in denoting the state and the measure by j.

1. Classical Mechanics

5$

It turns out that probability measures on Gl are essentially the only examples of states. Namely, for a locally compact topological space .iGl the Riesz-Markov theorem asserts that for every positive, linear functional l on of continuous functions on .mil with compact support, there the space is a unique regular Borel measure dµ on Gl such that

l (f) =

fdt

J

for all

feC(4').

This leads to the following definition of states in classical mechanics.

Definition. The set of states S for a Hamiltonian system with the phase (.mil) of all probability measures on 4'. The space .iGl is the convex set states corresponding to Dirac measures d,c supported at points x E .iGl are called pure states, and the phase space .ill is also called the space of states21. All other states are called mixed states. A process of measurement in classical mechanics is the correspondence

Ax S

(f,i)

-+

(II),

= f*(bt) E

which to every observable f e A and state µ E S assigns a probability measure µ f on I[8 - apush-forward of the measure dµ on .iGl by f. For every Borel subset E C I[8 the quantity 0 < µf(E) 1 is the probability that for a system in the state µ the result of a measurement of the observable

f is in the set E. The expectation value of an observable f in a state µ is given by (2.19).

In physics, pure states are characterized by having the property that a measurement of every observable always gives awell-defined result. Mathematically this can be expressed as follows. Let

a(f) = Eµ ((f -

Eµ(f2)

- Eµ(f)2 > 0

be the variance of the observable f in the state µ. Lemma 2.1. Pure states are the only states in which every observable has zero variance.

Proof. It follows from the Cauchy-Bunyakovski-Schwarz inequality that o(f) = 0 if only if f is constant on the support of a probability measure dµ.

In particular, a mixture of pure states dµx and dµy, x, y E 4', is a mixed state with

so that o(f) > 0 for every observable f such that f(x) L f(y). 21 The space of pure states, to be precise.

2. Hamlltonlan Mechanics

59

For a system consisting of few interacting particles (say, a motion of planets in celestial mechanics) it is possible to measure all coordinates and momenta, so one considers only pure states. Mixed states necessarily appear for macroscopic systems, when it is impossible to measure all coordinates and Remark. As a topological space, the space of states .iGl can be reconstructed from the algebra ,A of classical observables. Namely, suppose for simplicity momenta22.

that .iGl is compact. Then the (C-algebra C = C(4') of complex-valued continuous functions on 4' -the completion of the complexification of the II8-algebra ,A of classical observables with respect to the sup-norm is acommutative (C*-algebra. This means that C(4') is a Banach space with respect to the norm = supxE f(x)I, has a C-algebra structure (associative algebra over (C with a unit) given by the point-wise product of functions such that f ' 9II If ii 19M, and has a complex anti-linear antiinvolution: amap * : C -> C given by the complex conjugation f*(x) = f(x)

and satisfying I' f*_ If P. Then the Gelfand-Naimark theorem asserts that every commutative C*-algebra C is isomorphic to the algebra C(4') of continuous functions on its spectrum the set of maximal ideals of C - a compact topological space with the topology induced by the weak topology on C*, the dual space of C.

We conclude our exposition of classical mechanics by presenting two the time evolution of a Hamiltonian system ((4', {, }), H) with the algebra of observables A = C°°(4') equivalent ways of describing the dynamics

and the set of states S = .9(4'). In addition, we assume that the Hamiltonian phase flow gt exists for all times, and that the phase space .iGl carries a volume form dx invariant under the phase flow23.

Hamilton's Description of Dynamics. States do not depend on time, and time evolution of observables is given by Hamilton's equations of motion,

µ E S,

and

t

= {H, f }, lEA.

The expectation value of an observable f in the state µ at time t is given by

.

E(ft)=f fogtdP=ff(gt(x))p(x)dx, where p(x) _ is the Radon-Nikodim derivative. In particular, the expectation value of f in the pure state dµx corresponding to the point x E .mil is f(gt(x)). Hamilton's picture is commonly used for mechanical systems consisting of few interacting particles. 22Typically, a macroscopic system consists of N 1023 molecules. Macroscopic systems are studied in classical statistical mechanics. 231t is Liouville's volume form when the Poisson structure on .4' is non-degenerate.

I. Classical Mechanics

60

Liouville's Description of Dynamics. The observables do not depend on time

d = 0, fA,

and states dµ(x) = p(x)dx satisfy Liouville's equation dt

-{H,p}, p(x)dx e S.

and Liouville's equation are Here the Radon-Nikodim derivative p(x) _ understood in the distributional sense. The expectation value of an observ-

able f in the state µ at time t is given by Eµr(f) =

ff(x)(t(x))dx.

Liouville's picture, where states are described by the distribution functions p(x) -positive distributions on Gl corresponding to probability measures p(x)dx - is commonly used in statistical mechanics. The equality

E(ft) =Eat (f) for all f e A, µ E S, which follows from the invariance of the volume form dx and the change of variables, expresses the equivalence between Liouville's and Hamilton's descriptions of the dynamics.

3. Notes and references Classical references are the textbooks [Arn89] and [LL76], which are written, respectively, from mathematics and physics perspectives. The elegance of [LL76] is supplemented by the attention to detail in [Go180], another physics classic. A brief overview of Hamiltonian formalism necessary for quantum mechanics can be found in [FY80]. The treatise [AM78] and the encyclopaedia surveys [AG9O], [AKN97] provide a comprehensive exposition of classical mechanics, including the history of

the subject, and the references to classical works and recent contributions. The textbook [Ste83], monographs [DFN84], [DFN85], and lecture notes [Bry95] contain all the necessary material from differential geometry and the theory of Lie groups, as well as references to other sources. In addition, the lectures [God69] also provide an introduction to differential geometry and classical mechanics. In particular, [God69] and [Bry95] discuss the role the second tangent bundle plays in Lagrangian mechanics (see also the monograph [Y173] and [Cra83]). For a brief review of integration theory, including the Riesz-Markov theorem, see [RS8O]; for the proof of the Gelfand-Naimark theorem and more details on C*-algebras, see [StrO5] and references therein.

Our exposition follows the traditional outline in [LL76] and [Arn89], which starts with the Lagrangian formalism and introduces Hamiltonian formalism through

3. Notes and references

61

the Legendre transform. As in [Arn89], we made special emphasis on precise mathematical formulations. Having graduate students and research mathematicians as a main audience, we have the advantage to use freely the calculus of differential forms and vector fields on smooth manifolds. This differs from the presentation in [Arn89], which is oriented at undergraduate students and needs to introduce this material in the main text. Since the goal of this chapter was to present only those basics of classical mechanics which are fundamental for the formulation of quan-

tum mechanics, we have omitted many important topics, including mechanicalgeometrical optics analogy, theory of oscillations, rotation of a rigid body, perturbation theory, etc. The interested reader can find this material in [LL76] and [Arn89] and in the above-mentioned monographs. Completely integrable Hamiltonian systems were also only briefly mentioned at the end of Section 2.6. We refer the reader to [AKN97] and references therein for a comprehensive exposition, and to the monograph [FTO7] for the so-called Lax pair method in the theory of integrable systems, especially for the case of infinitely many degrees of freedom.

In Section 2.7, following [FY80], [DFN84], [DFN85], we discussed Poisson manifolds and Poisson algebras. These notions, usually not emphasized in standard exposition of classical mechanics, are fundamental for understanding the meaning a passage from classical mechanics to quantum mechanics. We of quantization also have included in Sections 1.6 and 2.7 the treatment of the Laplace-Runge-Lenz vector, whose components are extra integrals of motions for the Kepler problem24. Though briefly mentioned in [LL76], the Laplace-Runge-Lenz vector does not actually appear in many textbooks, with the exception of [Go180] and [DFN84]. In Section 2.7, following [FY80], we also included Theorem 2.19, which clarifies the meaning of the Jacobi identity, and presented Hamilton's and Liouville's descriptions of the dynamics.

Most of the problems in this chapter are fairly standard and are taken from various sources, mainly from [Arn89], [LL76], [Bry95], [DFN84], and [DFN85]. Other problems indicate interesting relations with representation theory and symplectic geometry. Thus Problems 2.12 and 2.20 introduce the reader to the orbit to the method of symplectic reduction (see method [K1r04], and Problem 2.18 [Arn89], [Bry95] and references therein}. Problems 2.21- 2.23 introduce the reader to the theory of Lie-Poisson groups (see [Dri86], [Dr187], [STS85], and [Tak9O] for an elementary exposition).

24We will see in Chapter 3 that these extra integrals are responsible for the hidden SO(4) symmetry of the hydrogen atom.

Chapter 2

Basic Principles of Quantum Mechanics

We recall the standard notation and basic facts from the theory of selfadjoint operators on Hilbert spaces. Let 3l° be a separable Hilbert space with an inner product (, ), which is complex-linear with respect to the first argument, and let A be a linear operator in ° with the domain D(A) C - a linear subset of P. An operator A is called closed if its graph F(A) _ {(p, Aco) E ' x ': cp e D(A)} is a closed subspace in 319 x . If the domain of A is denser in ', i.e., D(A) _ , the domain D(A*) of the adjoint operator A* consists of cp E 1° such that there is r/ E 31" with the property that (A/',co)=(b,71) for all bE D(A), and the operator A* is defined by A*cp = An operator A is called symmetric if

(A, cp) _ Acp) for all cc, % E D(A). By definition, the regular set of a closed operator A with a dense domain D(A) is the set p(A) _ {.1 E C I A-AI : D(A) - M' is a bijection with a bounded inverse2},

and for A E p(A), the bounded operator R(A) _ (A - .1I)-1 is called the resolvent of A at A. The regular set p(A) C (C is open and its complement a(A) _ C \ p(A) is the spectrum of A. The subset o(A) of Q(A) consisting of eigenvalues of A of finite multiplicity is called the point spectrum. 1 We consider only linear operators with dense domains. 2 By the closed graph theorem, the last condition is redundant. 63

2. Basic Principles of Quantum Mechanics

64

An operator A is self-adjoint (or Hermitian) if A = A*. Equivalently, A is symmetric and D(A) = D(A*), and for such operators o(A) C Il8. A symmetric operator A is called essentially self-adjoint if its closure A = A** is self-adjoint. For a symmetric operator A the following conditions are equivalent:

(i) A is essentially self-adjoint.

(ii) Ker(A* + iI) = Ker(A* - iI) _ {0}.

(iii) Im(A -}- iI) = Im(A - iI) _ . Asymmetric operator A with D(A) _

is bounded and self-adjoint. An operator A is positive3 if (Ap, cp) > 0 for all Q E D(A), which we denote by A > 0. Positive operators satisfy the Cauchy-Bunyakovski-Schwarz inequality (0.1)

(Aco,)I2 < (Aco, cp)(Ab, b)

for all

cp, c% E D(A).

In particular, (Acp, cp) = 0 implies that Acp = 0. Every bounded positive operator is self-adjoint4. We denote by 2(9°) the cC*-algebra of bounded linear operators on with the operator norm and with the antiinvolution * given by the operator adjoint. Operator A E 2(3) is called compact if it maps bounded sets in 31' into pre-compact sets5. The vector space °(3°) of compact operators on is a closed two-sided ideal6 of the (C*-algebra

An operator A E `'(s') is of trace class if

I= µn(A) <00, n=1

where µn(A) are singular values of A: µn(A) _

An(A) > 0, where An(A)

are eigenvalues of A*A. Equivalently, an operator A E c°() is of trace class if and only if for every orthonormal basis {e}1 for 3° 00

I(Aen,en)I <00. n=1

Since a permutation of an orthonormal basis is again an orthonormal basis, this condition can be replaced by 00

(Ae,e) <00 n=1 3 Non-negative, to be precise. 4This is true only for complex Hilbert spaces. 5The sets with compact closure.

sldeal '('} is the only closed two-sided ideal of f(.91°).

I. Observables, states, and dynamics

65

for every orthonormal basis {e}1 for. The trace of a trace class operator A is defined by

Aen en

'IrA = n=1

and does not depend on the choice of an orthonormal basis {e}1 for A positive operator A E 2(3') is of trace class if

'.

(Aen, en) <00 n=1

for some orthonormal basis {e}1 for 3l°. The space <S°1 of trace class operators in .3'P is a Banach algebra with the norm h='Ir /AA, and is a two-sided ideal (von Neumann-Schatten ideal) in the (C*-algebra 2('). The property

'IrAB = Tr BA for all A E 91i B E is called the cyclic property of the trace. If A E 91, the mapping 1A(B) _

Tr AB, B E 2(z), is a continuous linear functional on 2('), so that 921 C 9(32)*

the dual Banach space to

However, 91

but rather 9i =

and the mapping A H lA is an isomorphism of Banach spaces. Similarly, the mapping B H lB gives an isomorphism An operator A E 97(0) is called a Hilbert-Schmidt operator if AA* E is a Hilbert-Schmidt operator if Y'1. Equivalently, an operator A E

and only if for some orthonormal basis {e}1 for *' 00

n=1

I< oo.

The vector space 92 of Hilbert-Schmidt operators in J't' is a Hilbert space with the inner product (A, B)2 = Tr AB*. The Hilbert-Schmidt space 92 is also atwo-sided ideal in the C*-algebra P(te).

1. observables, states, and dynamics the physical laws at an Quantum mechanics studies the microworld that cannot be adequately described by classical mechanatomic scale ics. The properties of the microworld are so different from our everyday experiences that it is no surprise that its laws seem to contradict the common sense. Thus classical mechanics and classical electrodynamics cannot explain the stability of atoms and molecules. Neither can these theories reconcile different properties of light, its wave-like behavior in interference and diffraction phenomena, and its particle-like behavior in photo-electric emis-sion

and scattering by free photons. The fundamental difference between

2. Basic Principles of Quantum Mechanics

66

the microworld and the perceived world around us is that in the microworld

every experiment results in interaction with the system and thus disturbs its properties, whereas in classical physics it is always assumed that one can neglect the disturbances the measurement brings upon a system. This imposes a limitation on our powers of observation and leads to a conclusion that there exist observables which cannot be measured simultaneously. We will not discuss here these and other basic experimental facts, referring the interested reader to physics textbooks. Nor will we follow the historic path of the theory. Instead, we show how to formulate quantum mechanics using the general notions of states, observables, and time evolution, described in Section 2.8 in Chapter 1. There we have seen that commutativity of the algebra of observables A results in its realization as an algebra of functions on the topological space the space of states and thus brings us to the realm of classical mechanics. Therefore in order to get a realization of observables and states which is different from classical mechanics, we

must assume that the C*-algebra associated with observables is no longer commutative. A fundamental example of a non-commutative C*-algebra is given by the algebra of all bounded operators on a complex Hilbert space, and it turns out that it is this algebra which plays a fundamental role in quantum mechanics! Here we formulate the basic principles of quantum mechanics using precise mathematical language. At this point it should be noted that one can not verify directly the principles lying at the foundation of quantum mechanics. Nevertheless, the validity of quantum mechanics, whenever it is applicable, is continuously being confirmed by numerous experimental facts which perfectly agree with the predictions of the theory7.

1.1. Mathematical formulation. The following axioms constitute the basis of quantum mechanics.

Al. With every quantum system there is associated an infinite-dimensional separable complex Hilbert space in physics terminology called the space of states8. The Hilbert space of a composite quantum system is a tensor product of Hilbert spaces of component systems.

A2. The set of observables d of a quantum system with the Hilbert space consists of all self-adjoint operators on 19. The subset .gyp = d fl 2(f) of bounded observables is a vector space over Il8.

A3. The set of states So of a quantum system with a Hilbert space 3° consists of all positive (and hence self-adjoint) trace class operators M with This refers to non-relativistic phenomena at the atomic scale. 8The space of pure states, to be precise.

1. Observables, states, and dynamics

67

'Ir M = 1. Pure states are projection operators onto one-dimensional subspaces of °. For = 1, the corresponding projection onto C is denoted by Pg,. All other states are called mixed states9. A4. A process of measurement is the correspondence

d x 9 (A,M)- /tA E

(I),

which to every observable A E d and state M E 9 assigns a probability measure /c,q on R. For every Borel subset E C ]l8, the quantity 0 < 1U,q(E) < 1

is the probability that for a quantum system in the state M the result of a measurement of the observable A belongs to E. The expectation value (the mean-value) of the observable A E d in the state M E 9' is

where µA(A) = µA((-oo, ))) is a distribution function for the probability measure µg.

The set of states 9' is a convex set. According to the Hilbert-Schmidt theorem on the canonical decomposition for compact self-adjoint operators,

for every M E 9' there exists an orthonormal set {b7}n 1 in 1'° (finite or infinite, in the latter case N = oo) such that N

(1.1)

M=aP and

N

n= where a7z > o are non-zero eigenvalues of M. Thus every mixed state is a n=1

convex linear combination of pure states. The following result characterizes the pure states.

Lemma 1.1. A state M E 9' is a pure state if and only if it cannot be represented as anon-trivial convex linear combination in 9'.

Proof. Suppose that

P=aM1-}-(1-a)M2i 0
C (P,) _ 0, so that (Mlcp,cp) = 0 for all co E 3j and by (0.1) we get Ml= 0. Since Ml is self-adjoint, it leaves the complementary subspace (CL' invariant, and

from 'Ir Ml = 1 it follows that Ml = P. Therefore, Ml = MZ = P. 91n physics terminology, the operator M is called the density operator.

2. Basic Principles of Quantum Mechanics

68

Explicit construction of the correspondence d x 97 -> (IR) is based on the general spectral theorem of von Neumann, which emphasizes the fundamental role the self-adjoint operators play in quantum mechanics. Definition. Aprojection-valued measure on I[8 is a mapping P 2'() of the Q-algebra (R) of Borel subsets of Il8 into the algebra of bounded operators on .3, satisfying the following properties.

PM1. For every Borel subset E C ll8, P(E) is an orthogonal projection, i.e., P(E) = P(E)2 and P(E) = P(E)*.

PM2. P(O) = 0, P(T1) = I, the identity operator on . PM3. For every disjoint union of Borel subsets, n

00

E = JJE,

P(E)= lim

n=i

P(E) z=1

in the strong topology on

Remark. Similarly, aprojection-valued measure on IIBn is a mapping P (IEBn) satisfying the same properties PM1-PM3.

It follows from PM1-PM3 that P(E1)P(E2) = P(El n E2)

(1.2)

for all El, E2 E

With every projection-valued measure P on IE8 we associate a projectionvalued function P(A) = P((-oo,A)), called the projection-valued resolution of the identity. It is characterized by the following properties. PD 1. P(.\)P(,u)

P(min{A,p}).

PD2.

lim P7) = 0, lim P(A) = I. ->oa

A- +--oo

PD3. µ m P(µ) = P(A). µ
For every cp E the resolution of the identity P(A) defines a distribution function (P(A)cp, co) of the bounded measure on ][8 (probability measure when

Roil = 1). By the polarization identity

(P(A)) =

- (P(A)(-),-)

+ 2(P(A)(cp + ib), cp + i'%) - z(P(A)(cp - ib), co -

z)}

,

1. Observables, states, and dynamics

69

so that (P(A)p, ) corresponds to a complex measure on I[8 - a complex linear combination of measures.

A measurable function f on ][8 is said to be finite almost everywhere (a.e.) with respect to the projection-valued measure P if it is finite a.e. with respect to all measures (Pb, sb), E .3. For separable ° a theorem of von Neumann states that for every projection-valued measure P there exists cp E 9 such that a function f is finite a.e. with respect to P if and only if it is finite a.e. with respect to the measure (Pco, cp).

The next statement is the celebrated general spectral theorem of von Neumann.

Theorem 1.1 (J. von Neumann). For every self-adjoint operator A on the Hilbert space 9 there exists a unique projection-valued resolution of the identity P(A), satisfying the following properties. (i)

D(A)

={

E

:

f

A2d(P(A),

and for every cp E D(A) Acp =

f:AdPA,

defined as a limit of Riemann-Stieltjes sums in the strong topology on . The support of the corresponding projection-valued measure P coincides with the spectrum of the operator A: A E Q(A) if and only if PA((A - e, A + e)) 0 for all E > 0. (ii) For every continuous function f on III, f(A) is a linear operator on

3' with a dense domain

D(f(A))

= l

: f If(A)I2d(P(A),

E

)

o0

defined for cp E D(f (A)) as

f(A)dP(A), = and understood as in part (i). The operator f(A) satisfies f(A)* = f(A)

where f is the complex conjugate function to f, and the operator f(A) is bounded if and only if the function f is bounded on cr(A). For bounded on Q(A) continuous functions f and g,

f f(A)g(A)dP(A), P E .

2. Basic Principles of Quantum Mechanics

70

(iii) For every measurable function f on I[8, finite a. e. with respect to the projection-valued measure P, f(A) is a linear operator on defined as in (ii), where the integral for f(A)cp is now understood in the weak sense: for cp E D(f (A)) and every '/ E

(f(A),

f(A)d(P(A),

=

)

- a Lebesgue-Stieltjes integral with respect to a complex measure.

The correspondence f H f(A) satisfies the same properties as in (ii), where the integrals are understood in the weak sense. (iv) Abounded operator B commutes with A, that is, B(D(A)) C D(A) and AB = BA on D(A), if and only if it commutes with P(A) for all A and, therefore, B commutes with every operator f(A). (v) For every projection-valued resolution of identity P(A) the operator A on defined as in part (1), is self-adjoint.

Using the spectral theorem, the correspondence (A, M) +lated in A4, can be explicitly described as follows.

A, postu-

A5. The probability measure µ,q on IIB, which defines the correspondence d x 92 (I), is given by the Born-von Neumann formula µ,q(E) = Tr P,q(E)M,

(1.3)

EE

(R),

where PA is a projection-valued measure on R associated with the self-adjoint operator A.

Remark. The probability measure IUA on can be considered as a "quantum push-forward" of the state M by the observable A (cf. the discussion in Section 2.8 in Chapter 1). From the Hilbert-Schmidt decomposition (1.1) we get N

N

µa(E) _

_ n=1

N

an = 1,

anll n=1

n,=1

so that indeed 0 < µA(E) < 1. Denote by µA(A) the distribution function of the probability measure µA(A) _ (PA(A)b,'4') for M = P.

Proposition 1.1. Suppose that an observable A E d and a state M E 9 are such that (AIM) < oo and ImM C D(A). Then AM E 91 and (AIM) =TrAM.

In particular, if M = Pand b E D(A), then (AIM) _ (Ab,b)

and

(A2IM) = IIkbII2.

1. observables, states, and dynamics

71

Proof. Let {e}1 be an orthonormal basis for. Since 00

=

,LLA(E) =TPA(E)M

(PA(E)Me, en), n=1

we get for every E E .(I[g), OQ

µa(E) _ µn(E), n=i

where µn are finite measures on IE8 defined by µn(E) _ (PA(E)Me, en). Since

ffdA -

n-1

Jl f din

for every function f integrable with respect to the measure µA, it follows from the spectral theorem that 00

00

(AMen, en) _

00

f Ad(A) = J

00

Ad(A) < oo.

Thus AM E Y1 and (AIM) _ SAM. In particular, when M = P. and (AIM) =

f

(A,

).

Finally, from the spectral theorem and the change of variables formula we get 00

lIAII2

=J

A2d(Pa(A)'b, ) =

f

d(Pa2

o

Corollary 1.2. If (AIM), (A2IM) < oo, then AM E 9'1 and (AIM) _ TrAM.

Proof. Since 00

J

2dNn(\)

C

J

AZdµa(A)

_

(A2IM) c oo,

we get that en E D(AM), and the result follows from the proof of Proposition 1.1.

Remark. It is convenient to approximate an unbounded self-adjoint operator A by bounded operators An = f(A), where fn = X[-n,n] characteristic function of the interval [-n, n]. Assuming that (AIM) exists, we

-a

have

(ABM) _ f°° AdµA(a) = lim

J

n

lim (AIM).

2. Basic Principles of Quantum Mechanics

72

Definition. Self-adjoint operators A and B commute if the corresponding projection-valued measures PA and PB commute, for all E1,EZ E

PA(Ei)PB(Ea) = Ps(Ea)Pa(Ei)

(IR).

The following results, which follow from the spectral theorem, are very useful in applications.

Proposition 1.2. The following statements are equivalent. (i) Self-adjoint operators A and B commute. (ii) For all A, µ E C, Im A, Im µ # 0,

Ra(A)Rµ(B) = Rµ(B)R(A). (iii) For all u, v E I[8, eiuAeivB

=

eivB eiuA .

(iv) For all u E ]R, the operators eiuA and B commute.

Slightly abusing notation1Q, we will often write [A, B] = AB BA = o for commuting self-adjoint operators A and B.

Proposition 1.3. Let A = {A1,. . , A,} be a finite set of self-adjoint, pair.

wise commuting operators on Yt'. There exists a unique projection-valued measure PA on the Borel subsets of W having the following properties.

(i) For every E = E1 x

x En E

(IftT),

PA(E) = PAx (E1) ... (ii) In the strong operator topology,

Ak =

ak d PA,

PAn(En).

k = 1, ... , n,

Rn

where a is the k-th coordinate function on W, ak (x1, ... , x7z) = Xk.

(iii) For every measurable function f on W, finite a. e. with respect to the projection-valued measure PA, f(A1,.. . , An) is a linear operator on Yt' defined by

f(A1,...,A)=

J

fdPA,

where the integral is understood in the weak operator topology. The correspondence f --> f(A1,. . . , An) satisfies the same properties as in part (ii) of the spectral theorem. 1e1n general, for unbounded self-adjoint operators A and B the commutator [A, B] = AB-BA is not necessarily closed, i.e., it could be defined only for tp = o.

73

I. Dbservables, states, and dynamics

The support of the projection-valued measure PA on W1 is called the joint spectrum of the commutative family A = {Ai,. . , A}. .

.

Remark. According to von Neumann's theorem on a generating operator, for every commutative family A of self adjoint operators (not necessarily finite) on a separable Hilbert space 3' there is a generating operator a self-adjoint operator R on ° such that all operators in A are functions of R.

It seems natural that simultaneous measurement of a finite set of observables A = {A1,. . , A,} in the state .I1r1 E 9 should be described by the probability measure /A on Ian given by the following generalization of the .

Born-von Neumann formula:

(1.4) ILA(E) = Tr(PAI (E1) ... PAn (E)M), E = E1 x .. x En E However, formula (1.4) defines a probability measure on Ian if and only if PA1 (E1) ... PAn (En) defines a projection-valued measure on Ian . Since a product of orthogonal projections is an orthogonal projection only when the

projection operators commute, we conclude that the operators A1, ... , An should form a commutative family. This agrees with the requirement that simultaneous measurement of several observables should be independent of the order of the measurements of individual observables. We summarize these arguments as the following axiom. A6. A finite set of observables . A = {A1,. . . , A,- } can be measured simultaneously (simultaneously measured observables) if and only if they form a commutative family. Simultaneous measurement of the commutative family A C d in the state 11x1 E 9 is described by the probability measure /A on R' given by

µA(E) = Tr PA(E)M, E E

(W2),

where PA is the projection-valued measure from Proposition 1.3. Explicitly, PA(E) = PAl (El) ...PAS (En) for E = El x ... x En E r' (Wt). For ev-

ery Borel subset E C I[8n the quantity 0 < µA(E) < 1 is the probability that for a quantum system in the state M the result of the simultaneous measurement of observables A1,.. . , An belongs to E. The axioms A1-A6 are known as Dirac-von Neumann axioms.

Problem 1.1. Prove property (1.2). Problem 1.2. Prove that the state M is a pure state if and only if Tr M2 = 1. Problem 1.3. Prove that the Born-van Neumann formula (1.3) defines a probability measure on IR, i.e., /tA is a a-additive function on

Problem 1.4. Prove all the remaining statements in this section.

2. Basic Principles of Quantum Mechanics

74

1.2. Heisenberg's uncertainty relations. The variance of the observable A in the state M, which measures the mean deviation of A from its expectation value, is defined by

A) _ ((A- (AIM)I)21M) _ (A2IM) - (AIM)2 > p, provided the expectation values (A2IM) and (AIM) exist. It follows from Proposition 1.1 that for M = where b E D(A),

(AIM)I)II2

AM(A) = I

= I-

(Ab,b)2.

Lemma 1.2. For A E d and M E 9' the variance QM (A) = 0 if and only if ImM is an eigenspace for the operator A with the eigenvalue a = (AIM). In particular, if M = P,1,, then b is an eigenvector of A, A = aL'. Proof. It follows from the spectral theorem that QM(A)

_ t:(A - a)2dµA(A),

so that QM (A) = 0 if and only if the probability measure µA is supported at the point a E Ilg, i.e., µ,q({a}) = 1. Since µA({a}) = TrPA({a})M and 'Ir M = 1, we conclude that this is equivalent to Im M being an invariant subspace for PA({a}), and it follows from the spectral theorem that 1m M is an eigenspace for A with the eigenvalue a. Now we formulate generalized Heisenberg's uncertainty relations.

Proposition 1.4 (H. Weyl). Let A, B E d and let M = Pbe the pure state such that

E D(A) n D(B) and A'b,

E D(A) f1 D(B). Then

4 (i[A,B]IM)2.

The same inequality holds for all M E 9', where by definition (i[A, B] M) _ ilmn--goo (2 [`4n ,

Bn]IM)

Proof. Let M = P,1,. Since

[A- (AIM)I,B- (BIM)I] _ [A,B], it is sufficient to prove the inequality (A211v[)(B211vI) ? 4 (We

have for all a E ][S,

I+

a2(Bb, Bb) - ia(Ab, Ab) + (Ab, Ab) = a2(B2b, ) + c(i[A, B]b, b) + (A2b, b) 0,

so that necessarily

b)(B2b, ) > (i[A,BJ,b)2.

1. observables, states, and dynamics

75

The same argument works for the mixed states. Since

= n->oo lim QM(An)a1vt(Bn) (see the remark in the previous section), it is sufficient to prove the inequality

for bounded A and B. Then using the cyclic property of the trace we have for all a E 1I8,

0 <'Ir((A + iaB)M(A + iaB)*) = Tr((A + iaB)M(A - iaB)) =a2 Tr BMB + is TrBMA - is TAMB + Tr AMA 'Ir B2 M + a T(i [A, B] M) -}- Tr A2 M,

so that 4 Tr A2 M TrB2 M > 'Ir(i [A, B] M) 2.

LII

Heisenberg's uncertainty relations provide a quantitative expression of the fact that even in a pure state non-commuting observables cannot be measured simultaneously. This shows a fundamental difference between the process of measurement in classical mechanics and in quantum mechanics.

1.3. Dynamics. The set d of quantum observables does not form an algebra with respect to an operator product". Nevertheless, a real vector space o of bounded observables has a Lie algebra structure with the Lie bracket

i[A, B] = i(AB - BA),

A, B E moo.

Remark. In fact, the (C*-algebra 2`a°(') of bounded operators on f has a structure of a complex Lie algebra with the Lie bracket given by a commutator [A, B] = AB - BA. It satisfies the Leibniz rule [AB, Cl = A[B, C] + [A, C]B,

so that the Lie bracket is a derivation of the C*-algebra In analogy with classical mechanics, we postulate that the time evolution of a quantum system with the space of states is completely determined by a special observable H E d, called a Hamiltonian operator (Hamiltonian for brevity). As in classical mechanics, the Lie algebra structure on .cj leads to corresponding quantum equations of motion. Specifically, the analog of Hamilton's picture in classical mechanics (see Section 2.8 in Chapter 1) is the Heisenberg picture in quantum mechanics, where the states do not depend on time dt

11The product of two non-commuting self-adjoint operators is not self-adjoint.

2. Basic Principles of Quantum Mechanics

76

and bounded observables satisfy the Heisenberg equation of motion

t

(1.5)

= {H,

A E moo,

where

{, }n=[,]

(1.6)

is the quantum bracket -the

dependent Lie bracket on . The positive number h, called the Planck constant, is one of the fundamental constants in

physics12.

The Heisenberg equation (1.5) is well defined when H E .moo. Indeed, let U(t) be a strongly continuous one-parameter group of unitary operators associated with a bounded self-adjoint operator H,

t E I.

U(t) =

(1.7)

It satisfies the differential equation

ind

HU(t) = U(t)H, dtt) = so that the solution A(t) of the Heisenberg equation of motion with the initial condition A(0) = A E co is given by (1.8)

A(t) = U(t)-lAU(t).

(1.9)

In general, a strongly one-parameter group of unitary operators (1.7), associated with aself-adjoint operator H, satisfies differential equation (1.8) only on D(H) in a strong sense, that is, applied to cp E D(H). The quantum dynamics is defined by the same formula (1.9), and in this sense all quantum observables satisfy the Heisenberg equation of motion (1.5). The evolution

operator Ut d -> d is defined by U(A) = A(t) = U(t)-lAU(t), and is an automorphism of the Lie algebra cp of bounded observables. This is a quantum analog of the statement that the evolution operator in classical mechanics is an automorphism of the Poisson algebra of classical observables (see Theorem 2.19 in Section 2.7 of Chapter 1). By Stone's theorem, every strongly-continuous one-parameter group of unitary operators13 U(t) is of the form (1.7), where

D(H) _ {cp E

: li o

I cp exists}

and Hcp = ih o

I

J

The domain D(H) of the self-adjoint operator H, called the infinitesimal generator of U(t), is an invariant linear subspace for all operators U(t). 1 2The Planck constant has a physical dimension of the action (energy x time). Its value = 1.054 x 10-27 erg x sec, which is determined from the experiment, manifests that quantum mechanics is a microscopic theory. 13According to a theorem of von Neumann, on a separable Hilbert space every weakly measurable one-parameter group of unitary operators is strongly continuous.

77

1. Observables, states, and dynamics

We summarize the presented arguments as the following axiom.

A7 (Heisenberg's Picture). The dynamics of a quantum system is described by the strongly continuous one-parameter group U(t) of unitary op-

erators. Quantum states do not depend on time, 9 M H-> M(t) = M E 9, and time dependence of quantum observables is given by the evolution operator Ut,

d A --> A(t) = U1(A) = U(t)-lAU(t) E .sad. Infinitesimally, the evolution of quantum observables is described by the Heisenberg equation of motion (1.5), where the Hamiltonian operator H is the infinitesimal generator of U(t). The analog of Liouville's picture in classical mechanics (see Section 2.8

in Chapter 1) is Schrodinger's picture in quantum mechanics, defined as follows.

A8 (Schrodinger's Picture). The dynamics of a quantum system is described by the strongly continuous one-parameter group U(t) of unitary operators. Quantum observables do not depend on time, d A H-> A(t) _ A E d, and time dependence of states is given by the inverse of the evolution operator Ut 1 = U_t,

M(t) = U_(M) = U(t)MU(t)-1 E 9. M Infinitesimally, the evolution of quantum states is described by the Schrodinger equation of motion (1.10)

(1.11)

dot

= -{H, M}h, M E ,

where the Hamiltonian operator H is the infinitesimal generator of U(t). Proposition 1.5. Heisenberg and Schrodinger descriptions of dynamics are equivalent.

Proof. Let µA(t) and (/t)A be, respectively, probability measures on 1I8 associated with (A(t), M) E dx <S° and (A, M(t)) E dx <S° according to A3-A4,

where A(t) = U(A) and M(t) = U_t(M). We need to show that µA(t) _ (pt)A It follows from the spectral theorem that PA(t) = U(t)-1PAU(t), so that using the Born-von Neumann formula (1.3) and the cyclic property of the trace, we get for E E (R), Tr(U(t)-1pa(E)U(t)M)

='IrPA(t)(E)M =

=Tft(PA(E)U(t)MU(t)-i)

_ Tr PA(E)M(t) _ (1ut)A(E).

Corollary 1.3. (A(t)IM) _ (AIM(t)). In analogy with classical mechanics (see Section 1.4 of Chapter 1), we have the following definition.

2. Basic Principles of Quantum Mechanics

78

Definition. An observable A E d is a quantum integral of motion (or a constant of motion) for a quantum system with the Hamiltonian H if in Heisenberg's picture dA(t) dt

It follows from Proposition 1.2 that A E d is an integral of motion if and only if it commutes with the Hamiltonian H, so that, in agreement with (1.5),

{H,A}n= 0. This is a quantum analog of the Poisson commutativity property, given by formula (2.14) in Section 2.6 of Chapter 1. It follows from (1.11) that the time evolution of a pure state M = P, is given by M(t) = where /(t) = Since D(H) is invariant under U(t), the vector e/>(t) = U(t)% satisfies the time-dependent Schrodinger equation (1.12)

iii

= Hc/>

with the initial condition b(0) = zb.

Definition. A state M E 9 is called stationary for a quantum system with Hamiltonian H if in Schrodinger's picture dM(t) _ 0 dt

The state M is stationary if and only if [M, U(t)] = 0 for all t, and by Proposition 1.2 this is equivalent to

{H,M}n= 0, in agreement with (1.11). The following simple result is fundamental.

Lemma 1.3. The pure state M = P, is stationary if and only if i,b is an eigenvector for H,

and in this case G(t) = e

t

Proof. It follows from U(t)P, = that b is a common eigenvector for unitary operators U(t) for all t, U(t),L5 = c(t)I = 1. Since U(t) is a strongly continuous one-parameter group of unitary operators, the continuous function c(t) _ (U(t),b, L) satisfies the equation c(tl + t2) = c(ti)c(t2) for all t1, t2 E I[8, so that c(t) = e- * t for some

theorem b E D(H) and Hb _ ),.

E I. Thus by Stone's

2. Quantization

79

In physics terminology, the eigenvectors of H are called bound states. The corresponding eigenvalues are called energy levels and are usually denoted by

E. The eigenvalue equation H = E is called the stationary Schrodinger equation.

Problem 1.5. Show that if an observable A is such that for every state M the expectation value (AM(t)) does not depend on t, then A is a quantum integral of motion. (This is the definition of integrals of motion in the Schrodinger picture.)

Problem 1.6. Show that a solution of the initial value problem for the time-dependent Schrodinger equation (1.12) is given by Y

(t) =

e- tAdP1ra

where P7) is the resolution of identity for the Hamiltonian H.

Problem 1.7. Let D be a linear subspace of

f= J

f (s)U(s)t/ ds,

fE

, consisting of Gdrdirtg vectors (R),

where 9(R) is the Schwartz space of rapidly decreasing functions on R. Prove that D is dense in and is invariant for U(t) and for the Hamiltonian H. (Hint: Show

that U(t)1 1 ft E D, where ft (s) = f (s - t), and deduce H1 =

2. Quantization To study the quantum system one needs to describe its Hilbert space of states 4' and the Hamiltonian H - a self-adjoint operator in 4" which defines the evolution of a system. When the quantum system has a classical analog, the procedure of constructing the corresponding Hilbert space 4" and the Hamiltonian H is called quantization.

Definition. Quantization of a classical system ((4', { }), H is a one-to-one mapping Qn ,A -f d from

the set of classical observables .A = C°°(4') to the set d of quantum the set of self-adjoint operators on a Hilbert space . The observables map Qh depends on the parameter h> 0, and its restriction to the subspace of bounded classical observables .,4o is a linear mapping to the subspace moo of bounded quantum observables, which satisfies the properties Ain

2Q1(Rn(.fi)Rn(.fa) + Qn(f2)Qn(fl)) _ flf2

and

i o Q1({Qn(.fi),Qn,(.fa)}n) _ {fl,f2} for all 11,12 E A0. 14 Notation HC is used to distinguish the Hamiltonian function in classical mechanics from the Hamiltonian operator H in quantum mechanics.

2. Basic Principles of Quantum Mechanics

80

The latter property is the celebrated correspondence principle of Niels the Hamiltonian operator for a Bohr. In particular, H F-4 Qn(H) = H quantum system.

Remark. In physics literature the correspondence principle is often stated in the form

Quantum mechanics is different from classical mechanics, so that the correspondence f H Qn(f) cannot be an isomorphism between the Lie algebras of bounded classical and quantum observables with respect to classical and quantum brackets. It becomes an isomorphism only in the limit h - 0 when, according to the correspondence principle, quantum mechanics turns into classical mechanics. Since quantum mechanics provides a more accurate and refined description than classical mechanics, quantization of a classical system may not be unique.

Definition. Two quantizations Qand Qof a given classical system are said to be equivalent if there exists a linear mapping ((4', { }), o W and l' o l4 = A such that Qh2 = Gll .,4 ,

id.

For many "real world" quantum systems the systems describing actual the corresponding Hamiltonian H does not depend physical phenomena on a choice of equivalent quantization, and is uniquely determined by the classical Hamiltonian function H.

2.1. Heisenberg commutation relations. The simplest classical system with one degree of freedom is described by the phase space I[82 with coordinates p, q and the Poisson bracket { }, associated with the canonical symplectic form w = dp n dq. In particular, the Poisson bracket between

classical observables p and q -momentum and coordinate of a particle has the following simple form:

{p,q} = 1.

(2.1)

It is another postulate of quantum mechanics that under quantization classical observables p and q correspond to quantum observables P and Q self-adjoint operators on a Hilbert space , satisfying the following properties.

CR1. There is a dense linear subset D C 3' such that P : D - D and CR2. For all

L' E D,

(PQ

-

-i8.

2. Quantization

81

CR3. Every bounded operator on 1f which commutes with P and Q is a multiple of the identity operator I.

Property CR2 is called the Heisenberg commutation relation for one degree of freedom. In terms of the quantum bracket (1.6) it takes the form

{P,Q}n= I, which is exactly the same as the Poisson bracket (2.1). Property CR3 (2.2)

is a quantum analog of the classical property that the Poisson manifold }) is non-degenerate: every function which Poisson commutes with p and q is a constant (see the last remark in Section 2.7 of Chapter 1). The operators P and Q are called, respectively, the momentum operator and the coordinate operator. The correspondence p H-> P, q H-> Q with P and Q satisfying CR1-CR3 is the cornerstone for the quantization of classical systems. The validity of (2.2), as well as of quantum mechanics as a whole, is confirmed by the agreement of the theory with numerous experiments. (T2, {

,

Remark. It is tempting to extend the correspondence p H P, q --> Q to all observables by defining the mapping f(p, q) H f(P, Q). However, this approach to quantization is rather naive: operators P and Q satisfy (2.2) and do not commute, so that one needs to understand how f(P, Q) - a "function of non-commuting variables" - is actually defined. We will address this problem of the ordering of non-commuting operators P and Q in Section 3.3.

It follows from Heisenberg's uncertainty relations (see Proposition 1.4),

that for any pure state M = P with / e D, aM(P)UM(Q)

2

This is a fundamental result saying that it is impossible to measure the coordinate and the momentum of a quantum particle simultaneously: the more accurate the measurement of one quantity is, the less accurate the value of the other is. It is often said that a quantum particle has no observed path, so that "quantum motion" differs dramatically from the motion in classical mechanics. It is now straightforward to consider a classical system with n degrees of ... , pn } freedom, described by the phase space R2n with coordinates p =

and q = (q1,. . . , qn), and the Poisson bracket {

,

}, associated with the

canonical symplectic form w = dp n dq. The Poisson brackets between clashave momenta and coordinates of a particle sical observables p and q the form (2.3)

{pk,pl}=O,

{qk,ql}=o,

{p,ql}=,

k, l = 1, ... , n.

2. Basic Principles of Quantum Mechanics

82

Corresponding momenta and coordinate operators P = (Pi,.. . ,Pte) and Q = (Q1,. Qn) are self-adjoint operators that have a common invariant dense linear subset D C , and on D satisfy the following commutation relations: (2.4)

{Pk,Pj}n=O,

{Qk,Ql}a=o,

{Pk,Q1}n=8L1, k, l = 1, ... , n. These relations are called Heisenberg commutation relations for n degrees of freedom. The analog of CR3 is the property that every bounded operator on

19 which commutes with all operators P and Q is a multiple of the identity operator I. The fundamental algebraic structure associated with Heisenberg commutation relations is the so-called Ileisenberg algebra. Definition. The Heisenberg algebra Fjwith n degrees of freedom is a Lie algebra with the generators e 1, ... , en, fi, ... , f n, c and the relations [ek,c] = o, [fk,c] =0, [eC,fi] =8c, (2.5) c, k, l = 1, ... , n. The invariant definition is the following. Let (V, w) be a 2n-dimensional symplectic vector space considered as an abelian Lie algebra, and let g be a one-dimensional central extension of V by a Lie algebra 2-cocycle given by the bilinear form w. This means that there is an exact sequence of vector spaces (2.6)

and the Lie bracket in g is defined by {x,y] = W (x,y)c,

where , y are the images in V of elements x, g e g, and c is the image of 1 under the embedding R c-* g, called the central element of g. A choice of a symplectic basis e1,. . . , en, Ii, ... , f n for V (see Section 2.6 of Chapter 1)

establishes the isomorphism g ^Ij n , and relations (2.5) are obtained from the Lie bracket (2.7). By Ado's theorem, the Heisenberg algebra Cjn is isomorphic to a Lie subalgebra of a matrix algebra over IE. Explicitly, it is realized as a nilpotent subalgebra of the Lie algebra 91n+2 of to + 2) x (n + 2) matrices with the elements n

(nkf+vek)+c =

a

0

ul

u2

0

0

0

0

vi

0

0

0

0

v2

\0

0

0

0

0

...

k=1

...

2. Quantization

83

Remark. The faithful representation 1n - gCn+2, given by (2.8), is clearly reducible: the subspace V = {x = (Xi, ... , xn+2 } E Ian+2 : xn+2 = 0} is an invariant subspace for I with the central element c acting by zero. However, this representation is not decomposable: the vector space cannot be written as a direct sum of V and a one-dimensional invariant subspace for This explains why the central element c is not represented by a diagonal matrix with the first n + 1 zeros, but rather has a special form given by 2.8)

Analytically, Heisenberg commutation relations (2.5) correspond to an irreducible unitary representation of the Heisenberg Lie algebra Ctn. Recall that a unitary representation p of Cin the Hilbert space is the linear mapping p :Can -p id the space of skew-Hermitian operators in such that all self-adjoint operators ip(x), x E Can, have a common invariant dense linear subset D C and satisfy

-

y

P([x,

P E D.

Formally applying Schur's lemma we say that the representation p is irreducible if every bounded operator which commutes with all operators ip(x) is a multiple of the identity operator I. Then Heisenberg commutation relations (2.5) define an irreducible unitary representation p of the Heisenberg Lie algebra Can in the Hilbert space ° by setting (2.9)

P(.fk) _ -iPk,

P(ek) _ -iQk,

k = 1, ... ,n,

P(c) _ -ihl.

Since the operators Pk and Qk are necessarily unbounded (see Problem 2.1), the condition Pk Pz cp = PZ Pk cp for all co E D

does not necessarily imply (see Problem 2.2) that self adjoint operators Pk and PZ commute in the sense of the definition in Section 1.1. To avoid such "pathological" representations, we will assume that p is an integrable representation, i.e., it can be integrated (in a precise sense specified below) to an irreducible unitary representation of the Heisenberg group Hn a connected, simply-connected Lie group with the Lie algebra Ctn. Explicitly, the Heisenberg group is a unipotent subgroup of the Lie algebra SL (n + 2, I) with the elements

g=

1

u1

0

1

u2

...

0

...

D

a v

1

...

0

v2

9

?

0

0

0

0

0

0

un

2. Basic Principles of Quantum Mechanics

84

The exponential map exp : (jn --} Hn is onto, and the Heisenberg group Hn is generated by two n-parameter abelian subgroups n

exp uX = exp

n

uk fk

,

exp vY = exp

u, v E fin,

vkek

and a one-parameter center exp ac, which satisfy the relations n

(2.10)

exp uX exp vY = exp(-uvc) exp vY exp uX,

uv =

2Lkvk.

k=0

Indeed, it follows from (2.5) that [uX, vY] = ----uvc is a central element, so that using the Baker-Campbell-Hausdorff formula we obtain

exp uX exp vY = exp(- uvc) exp(uX + vY), exp vY exp uX = exp ( 2 uvc) exp(uX + vY).

In the matrix realization, the exponential map is given by the matrix expo-

nential and we get euX = I + uX, e' = I + vY, and e°`° = I + ac, where I is the (n + 2) x (n + 2) identity matrix. Let R be an irreducible unitary representation of the Heisenberg group Hn in the Hilbert space a strongly continuous group homomorphism

R : Hn - ( (), where ( (3) is the group of unitary operators in

'. By Schur's lemma, R (e) °) = e z '°` I, A E IlL. Suppose now that A = h, and define two strongly continuous n--parameter abelian groups of unitary operators

U(u) = R(exp uX ), V (v) = R(exp vY),

u, v E IEBn.

Then it follows from (2.10) that unitary operators U(u) and V(v) satisfy commutation relations

U(u)V(v) =

(2.11)

(v)U(u),

called Weyl relations. Let P = (F1,.. , Pn) and Q = (Q1, ... , Q') be, .

respectively, infinitesimal generators of the subgroups U(u) and V (v), given by the Stone theorem, Pk _ Z aU(k)

au

u=0

(v) and Q k = i aV avk

k = 1, ... , n. v=0

Taking the second partial derivatives of Weyl relations (2.11) at the origin u = v = 0 and using the solution of Problem 1.7 in the previous section, we easily obtain the following result.

Lemma 2.1. Let R : Hn -f £W() be an irreducible unitary representation and let P = of the Heisenberg group Hn in 1° such that R(eap) = , Pn) and Q = (Q1, ... , Qfl) be, respectively, infinitesimal generators (F1, of the strongly continuous n-parameter abelian subgroups U(u) and V(v).

2. ouantization

85

Then formulas (2.9) define an irreducible unitary representation p of the Heisenberg algebra Can in

'.

The representation p in Lemma 2.1 is called the differential of a represen-

tation R, and is denoted by dR. The irreducible unitary representation p of n is called integrable if p = dR for some irreducible unitary representation R of Hn .

Remark. Not every irreducible unitary representation of the Heisenberg algebra is integrable, so that Weyl relations cannot be obtained from the Heisenberg commutation relations. However, the following heuristic argument (which ignores the subtleties of dealing with unbounded operators) is commonly used in physics textbooks. Consider the case of one degree of freedom and start with

{P,Q}n= I. Since the quantum bracket satisfies the Leibniz rule we have, for a "suitable"

function f ,

{f(P), Q}h = f'(P). In particular, choosing f(P) = e-iuP = U(u), we obtain U(u)Q - QU(u) = fituU(u)

or

U(u)QU(u)-1 = Q -I- hul.

This implies, for a "suitable" function g,

U(u)g(Q) = g(Q + hul)U(u), and setting g(Q) = e-ZVQ = V(v), we get the Weyl relation. We will prove in Section 3.1 that all integrable irreducible unitary representations of the Heisenberg algebra Can with the same action of the central element c are unitarily equivalent. This justifies the following mathematical formulation of the Heisenberg commutation relations for n degrees of freedom.

A9 (Heisenberg's Commutation Relations). Momenta and coordinate operators P = (F1, ... , Pn) and Q = (Q1,.. ,Qom) for a quantum particle with n degrees of freedom are defined by formulas (2.9), where p is an .

integrable irreducible unitary representation of the Heisenberg algebra Can

with the property p(c) _ -ihI. Problem 2.1. Prove that there are no bounded operators on the Hilbert space .3° satisfying [A, B] = I.

Problem 2.2. Give an example of self-adjoint operators A and B which have a common invariant dense linear subset D C ° such that ABSp = BASp for all cp e D, but eiA and e2B do not commute.

2. Basic Principles of Quantum Mechanics

86

Problem 2.3. Prove Lemma 2.1. Glint: As in Problem 1.7, let D be the linear set of Garding vectors

f f (u, v)U(u)V(v)dnudnv, f E

(R2),

E

,

2r

where 9'(R) ) is the Schwartz space of rapidly decreasing functions on 1R2.)}

2.2. Coordinate and momentum representations. We start with the case of one degree of freedom and consider two natural realizations of the Heisenberg commutation relation. They are defined by the property that one of the self adjoint operators P and Q is "diagonal" (i.e., is a multiplication by a function operator in the corresponding Hilbert space). In the coordinate representation, ° = L2(IR, dq} is the L2-space on the configuration space IR with the coordinate q, which is a Lagrangian subspace of defined by the equation p = o. Set

D(Q) = SPEC :

g2

(q)I2dq < oo

and for Sp E D(Q) define the operator Q as a "multiplication by q operator", (Qco)(q)

q E T,

=

justifying the name coordinate representation. The coordinate operator Q is obviously self-adjoint and its projection-valued measure is given by (2.12)

(P(E)p)(q) = XE(q)co(q),

where xE is the characteristic function of a Borel subset E C I. Therefore

supp P =R and Q(Q) :R. Recall that aself-adjoint operator A has an absolutely continuous spectrum if for every 4' E °, IRbM = 1, the probability measure v1,(E) = (PA(E)L', sb),

EE

(R),

is absolutely continuous with respect to the Lebesgue measure on R. Lemma 2.2. The coordinate operator Q has an absolutely continuous spectrum IR, and every bounded operator B which commutes with Q is a function of Q, B = f(Q) with f E L°O(]E8).

Proof. It follows from (2.12) that v(E) = fE I/(q)I2dq, which proves the first statement. Now a bounded operator B on ' commutes with Q if and only if BP(E) = P(E)B for all E E (R), and using (2.12) we get (2.13)

XEB((P)

Choosing in (2.13) E = El and cp = xE2, where El and E2 have finite Lebesgue measure, we obtain B(XEl ' XE2) = B(XE1nE2) = XE1B(XE2)

- XE28()CE1),

2. Quantization

87

so that denoting fE = B(XE) we get supp fE C E, and IE1IE1flE2 - fE2JE1nE2

for all El, E2 E %(R) with finite Lebesgue measure. Thus there exists a measurable function f on IE8 such that lIE = fEIE for every E E (R) with finite Lebesgue measure. The linear subspace spanned by all XE E L2(R) is dense in L2 (I[8) and the operator B is continuous, so that we get

(Bc)(q) = f(q)(q) for all

cp E L2(l[8).

Since B is a bounded operator, f e L°° (T1) and

I IBII = I If I I oo

Remark. By the Schwartz kernel theorem, the operator B can be represented by an integral operator with a distributional kernel K(q, q'). Then the commutativity BQ = QB implies that, in the distributional sense,

(q-q')K(q,q') _ 0, so that K is "proportional" to the Dirac delta-function, i.e., K(q, q') = f(q)S(q

-

with some f E L°°(II8). This argument is usually given in the physics textbooks.

Remark. The operator Q has no eigenvectors

the eigenvalue equation

has no solutions in LZ(IEg). However, in the distributional sense, this equation

for every A E I[8 has a unique (up to a constant factor) solution cpa (q) _ S(q - A), and these "generalized eigenfunctions", also called eigenfunctions of the continuous spectrum, combine to a Schwarz kernel of the identity operator I on L2(IE8). This reflects the fact that operator Q is diagonal in the coordinate representation. Remark. Normalization of the eigenfunctions of the continuous spectrum cpa (q) can be also determined by the condition that for every A E IE8 the function a

IA(4) =

f Pu(4)dµ, a ao

satisfies (2.14)

o o O a+o - a 2 = 1

Here Ao E I[8 is fixed and does not enter (2.14). Indeed, in our case _ -the characteristic function of the interval (A0, A) -so that I I a+o -

aII2 = O.

2. Basic Principles of Quantum Mechanics

88

For a pure state M =

Ikb= 1, the corresponding probability mea-

sure µQ on II8 is given by

µQ(E) = u(E) - fE

IE E

Physically, this is interpreted that in the state P,,L, with the "wave function" b(q), the probability of finding a quantum particle between q and q + dq is I'i4'(q)j2dq. In other words, the modulus square of a wave function is the probability distribution for the coordinate of a quantum particle. The corresponding momentum operator P is given by a differential operator

hd

P

i dq the Sobolev space of absolutely continuous func-

with D(P) = W"2(Il.) tions f on I[8 such that f and its derivative f' (defined a.e.) are in L2(]f8). The operator P is self-adjoint and it is straightforward to verify that on D = C(IEg), the space of smooth functions on ]E8 with compact support,

QP-PQ=ihl. Remark. The operator P on ° has no eigenvectors

the eigenvalue

equation

Pc=pp, pEI, has a solution cp(q) = const x

which does not belong to L2(I[8). We will see later that the family of normalized eigenfunctions of the continuous spectrum 1

combines to a Schwartz kernel of the inverse h-dependent Fourier transform

operator, which diagonalizes the momentum operator P. In the distributional sense,

f

= 8(p - p').

Remark. As for the case of the coordinate operator, the normalization of the eigenfunctions of the continuous spectrum c,op(q) of the momentum operator can be determined from the condition (2.14). Indeed,

p(9) =

fPce*kqdk o

=

, (e*q _ J

Zq

so that

p-fO(q) - p(q) =

(p+20)e sin

2h

.

2. Quantization

89

I-

Using the elementary integral, we obtain 2

/'00

J

S1g2 q dq

=

2irc2fii,

so that c = Proposition 2.1. The coordinate representation defines an irreducible, unitary, integrable representation of the Heisenberg algebra.

Proof. To show that the coordinate representation is integrable, let U(u) _ and V (v) = e-ivQ be the corresponding one-parameter groups of unitary operators. Clearly, (V(v)co)(q) = e-ZVgcp(q) and it easily follows from the Stone theorem (or by the definition of a derivative) that (U(u)p)(q) _ cp(q - hu), so that unitary operators U(u) and V (v) satisfy the Weyl relation (2.11). Such a realization of the Weyl relation is called the Schrodinger representation.

To prove that the coordinate representation is irreducible, let B be a bounded operator commuting with P and Q. By Lemma 2.2, T = f(Q) for some f e L°O(I[g). Now commutativity between T and P implies that

TU(u) = U(u)T for all

u E I[8,

which is equivalent to f(q - fitu) = f(q) for all q, u E IEB, so that f = const LI a.e. on R.

To summarize, the coordinate representation is characterized by the property that the coordinate operator Q is a multiplication by q operator and the momentum operator P is a differentiation operator,

Q=q and P=-hd i dq Similarly, momentum representation is defined by the property that the

momentum operator P is a multiplication by p operator. Namely let ' L2 (If, dp) be the Hilbert L2-space on the "momentum space" R with the coordinate p, which is a Lagrangian subspace of R2 defined by the equation o. The coordinate and momentum operators are given by q Q=ih

d dp

and P=p,

and satisfy the Heisenberg commutation relation. As the coordinate representation, the momentum representation is an irreducible, unitary, integrable representation of the Heisenberg algebra. In the momentum representation, the modulus square of the wave function (p) of a pure state M = P,

2. Basic Principles of Quantum Mechanics

90

= 1, is the probability distribution for the momentum of the quantum particle, i.e., the probability that a quantum particle has momentum between pand p -F- dp is (p)2dp. Let L2 (l1) --> L( II) be the h-dependent Fourier transform operator, defined by

-

---

00

1

e *pq Sp(q)dq.

Here the integral is understood as the limit cp = limn_

in the strong

topology on L2 (1l), where 1

= By Plancherel's theorem, and

-n

e-

pq c (q)dq.

is a unitary operator on

Q=Q', P =

`nP

,

so that coordinate and momentum representations are unitarily equivalent. In particular, since the operator P is obviously self adjoint, this immediately shows that the operator P is self-adjoint. For n degrees of freedom, the coordinate representation is defined by setting ° = L2 (W , where dnq = dq- . dg7z is the Lebesgue measure on W1, and n Q=q=(q,...,q), 1

no

(no

no

Here Rn is the configuration space with coordinates q - a Lagrangian subspace of ][82n defined by the equations p = 0. The coordinate and momenta

operators are self-adjoint and satisfy Heisenberg commutation relations. Projection-valued measures for the operators Qk are given by where E E (It) and k : Il8n --> ][8 is a canonical projection onto the k-th component, k = 1, ... , n. Correspondingly, the projection-valued measure P for the commutative family Q = (Q1, ... , Qfl) (see Proposition 1.3) is defined on the Borel subsets E C ]f8n by

The family Q has absolutely continuous joint spectrum W. Coordinate operators Q1,. . . , Qfl form a complete system of commuting observables. This means, by definition, that none of these operators is a function of the other operators, and that every bounded operator commuting with Q1, ... , Qfl is a function of Q1, ... , QTh, i.e., is a multiplication by f(q)

2. Quantization

91

operator for some f E L°°(lt8n). The proof repeats verbatim the proof of Lemma 2.2. For a pure state M = IkL= 1, the modulus square of the wave function is the density of a joint distribution function µQ for the commutative family Q, i.e., the probability of finding a quantum particle in a Borel subset E C I(8n is given by

IiR(E) =

f

The coordinate representation defines an irreducible, unitary, integrable representation of the Heisenberg algebra Ctn. Indeed, n-parameter groups of unitary operators U(u) = and V (v) = e-ivQ are given by

(U(u))(q) _ P(4 - hu),

(V(v))(q) =

and satisfy Weyl relations (2.11). The same argument as in the proof of Proposition 2.1 shows that this representation of the Heisenberg group Hn, called the Schrodinger representation for n degrees of freedom, is irreducible. In the momentum representation, = LZ(IEBn, dnp), where dnp _ dpl dpn is the Lebesgue measure on ll8n, and

p

= (Pi,...,Pn).

Here IIBn is the momentum space with coordinates p - a Lagrangian subspace of I[82n defined by the equations q = 0. The coordinate and momentum representations are unitarily equivalent

by the Fourier transform. As in the case n = 1, the Fourier transform L2(W) -> L2(llgn) is a unitary operator defined by

f

(p) _ lira

(2h)_f12fe_P(q)dnq,

N-*oo

4 CN

where the limit is understood in the strong topology on L2 (Wn ). As in the case n = 1, we have

k_

Qk=

k=1,...,n.

In particular, since operators Pl, ... , Pn are obviously self-adjoint, this immediately shows that P1,. . , Pn are also self-adjoint. .

Remark. Following Dirac, physicists denote a vector 4' E by a ket vector a vector cp E r in the dual space to 3' ^M° is a complex anti-

linear isomorphism) by a bra vector (co, and their inner product by (b). In standard mathematics notation, and

(Ap)=(IAIb),

2. Basic Principles of Quantum Mechanics

92

where A is a linear operator. Fom a physics point of view, Dirac's notation is intuitive and convenient for working with coordinate and momentum representations. Denoting by Iq) = 8(q - q') and Ip) = (2ith)_f/2e*pq the set of generalized common eigenfunctions for the operators Q and P, respectively, we formally get

Q= 44'), Pp) =pjp), where operators Q act on q', and

(q) = f 8(q - q')(q')dq' = n (pI) _ (2h)_f12f e(4)dn9' _ n

as well as (qq') = 8(q - q'), (pp') = p - p'). Though in our exposition we are not using Dirac's notation, these formulas would help the interested reader "translate" the notation used in physics textbooks to standard mathematics notation. Remark, we will show in Section 3.2 that every Lagrangian subspace of the symplectic vector space Ij2n with the canonical symplectic form w = dp A dq gives rise to an integrable, unitary, irreducible representation of the Heisenberg algebra This is the simplest example of the real polarization, which for a given symplectic manifold (4', w) is defined as an integrable distriF)n.

bution {2}xE.

of Lagrangian subspaces 2 of tangent spaces T4'. The

notion of a polarization plays a fundamental role in geometric quantization:

it allows us to construct (under certain conditions) the Hilbert space of states Yt' associated with the classical phase space (4', w). In the linear case .' = IR2n every Lagrangian subspace 2 in gives rise to a real polarization by using the identification TXIl27z ^J R2n. In particular, for the

coordinate representation, .2' is given by the equation q = 0, and for the momentum representation by the equation p = 0. The corresponding Hilbert .3° space consists of functions on IR2n which are constant along the fibers of the polarization. Problem 2.4. Give an example of a non-integrable representation of the Heisenberg algebra.

Problem 2.5. Prove that there exists Sp E .3' = L2 (lll, dq) such that the vectors P(E)o, E E }R}, where P is a projection-valued measure for the coordinate operator Q, are dense in i°.

Problem 2.6. Find the generating operator for the commutative family Q = (Q1,..

Q"). Does it have a physical interpretation?

Problem 2.7. Find the projection-valued measure for the commutative family P = (P1, ... ,P,) in the coordinate representation.

93

2. Quantization

2.3. Free quantum particle. A free classical particle with one degree of freedom is described by the phase space 1E82 with coordinates p, q and the Poisson bracket (2.1), and by the Hamiltonian function z

(2.15)

The Hamiltonian operator of a free quantum particle with one degree of freedom is

P2

and in coordinate representation is given by h2 d2 H0

2m dq2

= L2(118, dq) with D(Ho) = W2,2(II8) the Sobolev space of functions in L2(1[8), whose generalized first and second

It is a self-adjoint operator on derivatives are in L2(IE8).

The operator Ho is positive with absolutely continuous spectrum [0, oo) of multiplicity two. Indeed, let fjo = L2(ll8>0, C2; dam) be the Hilbert space of C2-valued measurable functions W on the semi-line IIB>p = (0, oo), which are square-integrable with respect to the measure do(A) = Za dA,

o = {w(A) _

() 11112 = f (Ii(A)12 +

oo}

.

It follows from the unitarity of the Fourier transform that the operator llo : L2(R, dq) -

((v1) Io, where I and Io are, respectively, identity operators in )' and fjo. The operator G1lo establishes the isomorp, and since in the momentum representation Ho is a phism L2 (I, dq) multiplication by 3p2 operator, the operator GlloHoP41 is a multiplication

is unitary, lo °llo = I and

by A operator in ho.

Remark. The Hamiltonian operator Ho has no eigenvectors -the eigenvalue equation has no solutions in L2 (I[8) . However, for every A = 2m k2 > 0 this differential

equation has two linear independent bounded solutions

In the distributional sense, these eigenfunctions of the continuous spectrum combine to a Schwartz kernel of the unitary operator Gllo, which establishes

94

2. Basic Principles of Quantum Mechanics

the isomorphism between 3' = L2 (I[8, dq) and the Hilbert space fjp, where Ho acts as a multiplication by A operator. The normalization of the eigenfunctions of the continuous spectrum is also determined by the condition (2.14):

lim

-

1

where iY(4)

()2 = 1

lim

1 ()

- Y(+)

(-)

-

=0

t4(4)dp

_ .Ika

The Cauchy problem for the Schrodinger equation for a free particle,

inddtt)

(2.16)

(O)

_

is easily solved by the Fourier transform. Indeed, in the momentum representation it takes the form

.h(P,t) =(p,t), at

(p,O)=(p),

2m

so that (p,t) = e _

(p ) .

In the coordinate representation, the solution of (2.16) is given by 1

(2.17)

J°°

b(q, t) = 2?r L

-00

e p' (p, t)dp =

1

°a

J-oo

arp e*xCp,q,t}tl dp,

where p2

X(p,4,t)

2m

+p4t

Formula (2.17) describes the motion of a quantum particle, and admits the following physical interpretation. Let initial condition in (2.16) be such that its Fourier transform b _ ,Fpb) is a smooth function supported in a neighborhood Uo of po E lI8 \ {0}, 0

f

Uo, and

(p)I2dp =1.

Such states are called "wave packets" .Then for every compact subset E we have

(q,t)2dq=0.

(2.18)

Itl Since

C I[8

E

f

(q,t)2dq =1

for all t, it follows from (2.18) that the particle leaves every compact subset of IEg as t -* oo and the quantum motion is infinite. To prove (2.18), observe

that the function (p,q,t) -the "phase" in integral representation (2.17)

2. Quantization

95

- has the property that aP I > C > 0 for all p E Uo, q E E and large enough ti. Integrating by parts we get 1

(q, t) =

fo

e*t)t(p)dp (p) ax(pagat)

\e

dpi

p

so that uniformly on E,

(q,t) = O(1t1') as

ti - °

By repeated integration by parts, we obtain that for every n E ICY, uniformly on E,

(q,t) = O(1t1Th), so that 4'(q,t) = O(iti-°°). To describe the motion of a free quantum particle in unbounded regions, we use the stationary phase method. In its simplest form it is stated as follows.

The Method of Stationary Phase. Let f,g E C°O(Il8), where f is realvalued and g has compact support, and suppose that f has a single nondegenerate critical point xp, i.e., f'(xo) = 0 and f"(xo) O°

iN f e

_ g(x}dx

-00

0. Then

1

a

(Ni27T

fxo)l)

eiNf (moo

}+ 4 sgn f (xo }g(x

o) +

1

as N ---} oo.

Applying the stationary phase method to the integral representation (2.17) (and setting N = t), we find that the critical point of X(p, q, t) is po =

with x" (pp ) _ -

0, and

t

t a domain as t -* oo, the wave function (q, t) is supported on m Uo where the probability of finding a particle is asymptotically different from zero. At large t the points in this domain move with constant velocities V = m, p E U0. In this sense, the classical relation p = my remains valid in the quantum picture. Moreover, the asymptotic wave function 'bo satisfies

o(q,t)idq

2

=

= 1,

2. Basic Principles of Quantum Mechanics

96

and, therefore, describes the asymptotic probability distribution. Similarly, setting N = -It , we can describe the behavior of the wave function b (q, t)

ast -> -oo. Remark. We have 1imk1 0 ',b(t) = o in the weak topology on. Indeed, for every cp

we get by Parseval's identity for the Fourier integrals,

((t),

OQ

)=

z

t

a (p)e 2m dp,

and the integral goes to zero as ti -f oa by the Riemann-Lebesgue lemma. A free classical particle with ri degrees of freedom is described by the phase space R2n with coordinates p = (p1, ... , pn) and q = (q',. . , qn), the Poisson bracket (2.3), and the Hamiltonian function .

12

p2

2

2m 2m The Hamiltonian operator of a free quantum particle with n degrees of freedom is P2 1 (P+ a ... + pna H0

2m

2m

and in the coordinate representation is h2

H0=, 2m where

(O2

_ q/- \ ql/ +...+ \

2

is the Laplace operator15 in the Cartesian coordinates on l[8n. The Hamil= L2(IIBn, dnq) with D(Ho) _ tonian Ho is a self-adjoint operator on the Sobolev space on W. In the momentum representation, W2'2(1I8n) H0

2m

- a multiplication by a function operator on

= LZ(lEBn, dnp).

The operator Ho is positive with absolutely continuous spectrum [0, oo) of infinite multiplicity. Namely, let Sn-1 = {n E I[8n : n2 = 1} be the (n-1)dimensional unit sphere in IEBn, let do be the measure on 8"-1 induced by the Lebesgue measure on I[8n, and let C_{f:

Sn-i

C: lI

151t is the negative of the Laplace-Beltrami operator of the standard Euclidean metric on

2. Quantization

97

I

Let fj pn = L2 (I[8>o, Cj; dQn) be the Hilbert space of Cj-valued measurable

the measure dQn(A) _ (2mA), $)(n)

0

with respect to

(0, oo), square-integrable on

functionsls 'I' on

_ {w

: I1g>o --

,

II1l2 =

f

d(A) < oo

the corresponding Hilbert space for one degree When n = 1, X01}o n} , of freedom. The operator o : L2(Ian, dnq) -j o

= (a),

2m rz},

(A)(n) =

(n) . In the mois unitary and establishes the isomorphism L2 (IIi , dnq) ^J mentum representation Ho is a multiplication by 2mp2 operator, so that the operator oHoo ' is a multiplication by A operator in 5i0 .

Remark. As in the case n = I, the Hamiltonian operator Ho has no eigenthe eigenvalue equation vectors

Hob=A has no solutions in L2 (Wi). However, for every A > 0 this differential equation has infinitely many linearly independent bounded solutions rz

q} = (2irh)1 e

2ma 7zq

parametrized by the unit sphere S'. . These solutions do not belong to L2 (W ), but in the distributional sense they combine to a Schwartz kernel of

the unitary operator 4o, which establishes the isomorphism between ° = L2(W, d7zq) and the Hilbert space 0 where Ho acts as a multiplication by A operator.

As in the case n = 1, the Schrodinger equation for free particle,

n(dtt)

Ho(t),

(0)

_

is solved by the Fourier transform

(q,t) _

(2h)_2fn

For a wave packet, an initial condition b such that its Fourier transform ,/') is a smooth function supported on a neighborhood Uo of Po E z% _ ]I8n \ {0} such that 0

Uo and IRn 1

1 s That is, for every f E Ij the function ( f , 'I') is measurable on IE

o.

2. Basic Principles of Quantum Mechanics

98

the quantum particle leaves every compact subset of Ian and the motion is infinite. Asymptotically as t i --} oc, the wave function iJ'(q, t) is different from o only when q = t, p E U0. Problem 2.8. Find the asymptotic wave function for a free quantum particle with n degrees of freedom.

2.4. Examples of quantum systems. Here we describe quantum systems that correspond to the classical Lagrangian systems introduced in Section 1.3 of Chapter 1. In Hamiltonian formulation, the phase space of these systems, except for the last example, is a symplectic vector space R2n with canonical coordinates p, q and symplectic form w = dp A dq.

Example 2.1 (Newtonian particle). According to Section 1.7 in Chapter 1, a classical particle in RT moving in a potential field V(q) is described by the Hamiltonian function 2

H (7>> 4) =

2m + V (q) Assume that the Hamiltonian operator for the quantum system is given by

P 2

H =

+ V,

with some operator V, so that coordinate and momenta operators satisfy Heisenberg equations of motion (2.19)

P = {H, P}n, Q = {H, Q}n.

To determine V, we require that the classical relation q ` p between the rn

velocity and the momentum of a particle is preserved under the quantization, i.e.,

mSince

Q_P

{P2, Q}n = 2P, it follows from (2.19) that this condition is equivalent

to

k = 1, ... , n.

[V, QkI = 0,

It follows from Section 2.2 that V is a function of commuting operators Qi,. . , Qn, and the natural choice17 is V = V (Q). Thus the Hamiltonian operator of a Newtonian particle is .

z

H=-+V(Q), 17Confirmed by the agreement of the theory with the experiments.

2. Quantization

99

in agreement with H = HC(P, Q)18. In coordinate representation the Hamiltonian is the Schrodinger operator (2.20)

H = - 2m

+ V (q)

with the real-valued potential V (q).

Remark. The sum of two unbounded, self adjoint operators is not necessarily self-adjoint, and one needs to describe admissible potentials V(q) for which H is a self-adjoint operator on L2(Wl, dnq). If the potential V(q) is a real-valued, locally integrable function on IRA, then the differential operator (2.20) defines a symmetric operator H with the domain C (Wn ) twice continuously differentiable functions on Ian with compact support. Potentials for which the symmetric operator .H has no self-adjoint extensions are obviously non-physical. It may also happen that H has several self-adjoint extensions19. These extensions are specified by some boundary conditions at infinity and there are no physical principles distinguishing between them. The only physical case is when the symmetric operator H admits a unique self-adjoint extension, that is, when H is essentially self-adjoint. In Chapter 3 we present necessary conditions for the essential self-adjointness. Here we only mention the von Neumann criterion that if A is a closed operator and D(A) -= °, then H = A* A is a positive self adjoint operator.

Example 2.2 (Interacting quantum particles). In Lagrangian formalism, a closed classical system of N interacting particles on was described in Example 1.2 in Section 1.3 of Chapter 1. In Hamiltonian formalism, it is described by the canonical coordinates r = (ri, ... , TN), the canonical momenta p = pN}, ra, pa E and by the Hamiltonian function N

(2.21)

H p, r =

2

pa -E- V r , 2ma a=1

where ma is the mass of the a-th particle, a = 1, ... , N (see Section 1.7 in Chapter 1). The corresponding Hamiltonian operator H in the coordinate representation has the form N

h2

(2.22)

a=i

a

In particular, when

V (r)=

V (ra - rb), 1
181n the special case H( p, q) = 1(p) + g(q) the problem of the ordering of non-commuting operators P and Q does not arise. 1 s This is the case when the defect indices of H are equal and are non-zero.

2. Basic Principles of Quantum Mechanics

100

the Schrodinger operator (2.22) describes the N-body problem in quantum mechanics. The fundamental quantum system is the complex atom, formed by a nucleus of charge Ne and mass M, and by N electrons of charge -e and mass m. Denoting by R E Il83 the position of the nucleus, and by rl, ... , TN the positions of the electrons and assuming that the interaction is given by the Coulomb attraction, we get for the Hamiltonian function (2.21)

H(P,p,R,r)=

PZ 2M

N Pa a _1

2m

N

Net

a_1

IRTai

e2

IabI

where P is the canonical momentum of the nucleus. The corresponding Schrodinger operator H in the coordinate representation has the form20 N

N

Ne + H=-----L-2m-Da- 1R-TaI 2

21l/1

2

a_1

2

a_1

2

e

1
ra - rbi

In the simplest case of the hydrogen atom, when N = 1 and the nucleus consists of a single proton21, the Hamiltonian is

2

2

e2

H rp- rel' where rp is the position of the proton and re is the position of the electron. As the first approximation, the proton can be considered as infinitely heavy, so that the hydrogen atom is described by an electron in an attractive Coulomb field where now r = re - rp. The corresponding Hamiltonian operator takes the form

mlWe

2 H=---L--. e2

(2.23)

will solve the Schrodinger equation with this Hamiltonian H and determine its energy levels in Section 5.1 of Chapter 3.

Example 2.3 (Charged particle in an electromagnetic field). A classical particle of charge e and mass m moving in the time-independent electromagnetic field with scalar and vector potentials (r) and A(r), r E 1[83, is described by the Hamiltonian function

H(p,r) =

z

(- A)+e(r)

(see Problem 1.27 in Section 1.7 of Chapter 1). The corresponding classical velocity vector v = {H, r} is given by

20Ignoring the fact that electron has spin, see Chapter 4. 21In the case of hydrogen-1 or protium; it includes one or more neutrons for deuterium, tritium, and other isotopes.

2. Quantization

101

and its components v = (Vi, v2i v3) have non-vanishing Poisson brackets:

{V1,V2} _ --4-B3,

{v3,vl} _ -m2CB2,

{v2,v3} =

where B = (B1, B2i B3) are components of the magnetic field B = curl A. The Hamiltonian operator of a quantum particle is

H=

(2.24)

2m

(P A)

2

+ ecp(r)

-the Schrodinger operator of a charged particle in an electromagnetic field. The corresponding quantum velocity vector V = {H, Q}n is given by the same formula as in the classical case,

V=Pand its components V = (V1, V2i V3) have non-vanishing quantum brackets: {V1, V2}h _ -m2cB3i

{V2, V3}h _ -m2CB1i

{V3, Vl} _ -m2cB2.

Thus in the presence of a magnetic field the three components of a quantum velocity operator no longer commute and cannot be measured simultaneously.

Example 2.4 (Free quantum particle on a Riemannian manifold). The phase space of a classical particle of mass m = 1 moving on a Riemannian manifold (M, g) is the cotangent bundle T*M, and the corresponding Hamiltonian function is given by

x) = where g(x) is the inverse of the metric tensor g,1,,(x), and

(p,x) =

1

n\

are standard coordinates on T*M (see Section 1.7 of Chapter 1). The Hilbert = L2(M, dµ), where dµ = g(x)dnx, space of the quantum system is g(x) = det(g,(x)), is the measure associated with the density of the Riemannian metric (Riemannian volume form if M is oriented). When canonical coordinates (p, x) are only locally defined on T*M, it is not possible to construct corresponding operators P and Q. Still, one can always define the Hamiltonian operator by 2

(2.25)

H=

2 Og,

where Dy =

1

aµ

(gv) µa ax

is the Laplace-Beltrami operator of the Riemannian metric g on M. Note that in this case there is a non-trivial problem of the ordering of noncommuting operators in the quantization of H (p, x), which arises if in a

2. Basic Principles of Quantum Mechanics

102

coordinate chart on M one replaces canonical coordinates p and x by P and Q. The formula 2

L9

-gµv'x1axaaxv +

shows that local expressions defined by

HC(P,Q) = 2 combine to awell-defined self-adjoint operator H on f given by (2.25). Thus even when M = I[8n and canonical coordinates (P, x) are globally (2.26)

defined on T*I[8n, the correct formula for HC(P, Q) the one which extends to general Riemannian manifolds - is given by (2.26), where the second term

represents the "quantum correction" to the naive expression g'(Q)PµP

2.5. Old quantum mechanics. The formulation of quantum mechanics presented here goes back to 1925-1927, and is due to Heisenberg, Schrodinger, Born, Jordan, and Dirac. It replaced the old quantum theory, proposed in 1913 by Bohr, which was based on Rutherford's planetary model of the atom. In the old theory, the energy levels of a one-dimensional quantum system correspond to the closed orbits of the associated classical Hamiltonian system which satisfy the Bohr- Wilson-Sommerfeld quantization rule (BWS rule)

pdq = 2irh(n+ 2), where n is a non-negative integer, and integration goes over the closed orbit in the phase space R2. Bohr-Wilson--Sommerfeld quantization rules also apply to completely integrable Hamiltonian systems with several degrees of freedom (see Section 2.6 of Chapter 1). Namely, let F1 = FN be N independent integrals of motion in involution. The BWS quantization rules are

7dC = 2irh(n7 + 4 lnd y), 'Y

where integration goes over all 1--cycles y in the Lagrangian submanifold A = {(p,q) E R2N He (p, q) = E, F2 (p7 q) = E2, ... , FN (p, q) = EN } and ind y e Z is the so-called Maslov index of a cycle ry in A. In the onedimensional case the closed orbit is topologically a circle and its Maslov index is 2. It will be shown in Section 6.3 of Chapter 3 that, in general, BohrWilson-Sommerfeld quantization rules only give asymptotics of the energy levels as h - o. However, for integrable systems with extra symmetry, such as the harmonic oscillator and the Kepler problem, the BWS quantization rules determine the energy levels exactly. We will show this in the next section for the harmonic oscillator, and in Section 5.1 of Chapter 3 for the Kepler problem.

2. Quantization

103

2.6. Harmonic oscillator. The simplest classical system with one degree of freedom, besides the free particle, is the harmonic oscillator. It is described by the phase space R2 with the canonical coordinates p, q, and the Hamiltonian function (2.27)

Hc(p,4)

p2

mw2g2

- 2m +

2

(see Sections 1.5 and 1.7 in Chapter 1). Hamilton's equations

p= {H,p} _ -mw2q,

q = {Hc,q} = m

with the initial conditions pp, qo are readily solved, (2.28)

p(t) = pp cos wt - mwgp sin wt,

(2.29)

q(t) = qp coswt + 1 pp sinwt, mw

and describe the harmonic motion. As in Section 2.6 of Chapter 1, it is convenient to introduce complex coordinates on the phase space on IEg2 ^(C, (2.30)

z=

(wq + ip),

1

2W

z=

(wq - ip). 2cv

We have

{z,}_-, m

(2.31)

Hc(z, z)-mcvlzj2,

so that Hamilton's equations decouple,

z - HG, z} = --iwz,

z=

z} = iwz,

and are trivially solved,

z(t) = e-iwtzo,

(2.32)

z

eiwto.

Here

zo =

(wqo + apo),

zo -

2w

(wqo - Zpo)

For the quantum system, the corresponding Hamiltonian operator is

H

P2 mw2Q2 2m + 2 '

and in the coordinate representation f = L2 (I, dq) it is a Schrodinger operator with a quadratic potential, h2 d2 mw2g2

H=-----+ 2m dq2

2

The quantum harmonic oscillator is the simplest non-trivial quantum system, besides the free particle, whose Schrodinger equation can be solved explicitly. It appears in all problems involving quantized oscillations, namely in molecular and crystalline vibrations. The exact solution of the harmonic

2. Basic Principles of Quantum Mechanics

104

oscillator, described below, has remarkable22 algebraic and analytic properties. Temporarily set m = 1 and consider the operators (2.33)

a=

2w

(wQ - iP),

which are quantum analogs of complex coordinates (2.30). The operators a and a* are defined on W"2(R) f1 W1>2(R), where W12(IE8) _ and it is easy to show that cc* is the adjoint operator to a and a** = a, so that a is a closed operator. From the Heisenberg commutation relation (2.2) we get the canonical commutation relation [a, a'"] = I

(2.34) on W2'2(ll8) fl W22(i). Indeed,

as = PZ + wZQ2

iw

[P'

Q]-

PZ + w2Q2

1

2wfii

+ 2I'

P2 + W ZQ2

1

GW /G

G

and

P2 + w2Q2

ZW

LW IZ

GW 2

1r

so that (2.34) holds on W2'2(IR) n W2'2(]E8), where and

i22(I) _ ' (W2z(ll8)),

H = wh (a*a + ZI) = cvh (aa* - 2I). In particular, it follows from the von Neumann criterion that the Hamiltonian operator H is self-adjoint. The operators a, a* and N = a*a satisfy the commutation relations

[N, a] _ -a, [N, a*] = a*, [a, a*] = I. These commutation relations correspond to the irreducible unitary repre(2.35)

sentation of afour-dimensional solvable Lie algebra Cj associated with the introduced in Section 2.1. Namely, C) is a Lie Heisenberg algebra Cj = algebra with the generators e, f, h, and c, where e, f, c satisfy the relations of the Heisenberg algebra Cj, and

[h,e] =-f,

[h,f] =wee,

[h,c] =0.

The irreducible integrable representation p of the Heisenberg algebra F) (see (2.9) in Section 2.1) extends to a unitary representation of the Lie algebra 22The algebraic structure of the exact solution of the harmonic oscillator plays a fundamental role in quantum electrodynamics and in quantum field theory in general.

105

2. Quantization

F) by setting p(h)

= -iwN =

P2

aQa

Remark. In invariant terms,. the Lie algebra F) is aone-dimensional right extension of the Heisenberg algebra F),

0---)I[8--0. It is defined by the F)-valued Lie algebra 1-cocycle r e Z' ([j, F)) - a derivation of F), given on generators by

r(e) = f,

r(f) _ -wee, r(c) = 0.

Explicitly, if h E F) is such that h = 1 E I[8 under the projection C) -> ][8, then identifying elements in F) with their images under the embedding F) -- F), we have

[x+Qh,y+/3h] _ {x,y] - ar(y) + f3r(y),

x, y E

F).

It is due to this Lie-algebraic structure of commutation relations (2.35) that Heisenberg equations of motion for the harmonic oscillator can be solved exactly. Namely, we have a = {H, a}n _ -iwa, a* _ {H, a*}n = iwa*,

so that

a(t) = e-iwtao, a* (t) = eiwtaO. Comparing with (2.32) we see that solutions of classical and quantum equations of motion for the harmonic oscillator have the same form! Next, using commutation relations (2.35) and positivity of the operator N, we will solve the eigenvalue problem for the Hamiltonian H of the harmonic oscillator explicitly by finding its energy levels and corresponding eigenvectors. We will prove that the eigenvectors form a complete system

of vectors in 3, so that the spectrum of the Hamiltonian H is the point spectrum. This is a quantum mechanical analog of the fact that classical motion of the harmonic oscillator is always finite.

The algebraic part of the exact solution is the following fundamental result.

Proposition 2.2. Suppose that there exists a non-zero ri = 1, 2, ... , such that Then the following statements hold.

(i) There exists i/o E 3l°, IkL'oII = 1, such that 2 iw%o

E D (an)nD ( (a* )n ),

2. Basic Principles of Quantum Mechanics

106

(ii) The vectors

n=

(a*)n}o

E°, n = O,1, 2, ... ,

are orthonormal eigenvectors for H with the eigenvalues itw(n+ 2),

= hw(n + (iii) Restriction of the operator H to the Hilbert space o - a closed subspace of spanned by the orthonormal set is essentially self-adjoint.

Proof. Rewriting commutation relations (2.35) as

Na = a(N - I) and Na* = a*(N + I), and putting A _ hw(µ + 2 ), we get for all n > 0,

Nanz/ _ (,u - n)an and N(a*)n _ (µ + n)(a*)nb.

(2.36)

Since N > 0 on D(N), it follows from the first equation in (2.36) that there ano,,I' exists nno > _ 0 such that ano 0 but ano+l_ 0. Setting w E y we get

an0

ado = 0 and

(2.37)

0.

Since H = huv(N + 2I), this proves part (i). To prove part (ii), we use commutation relations (2.38)

[a, (a*)n] = n(a*)n-i

which follow from (2.34) and the Leibniz rule. Using (2.37)-(2.38), we get (2.39)

(x 2bn = n + 1 zbn+l ,

abn =

so that y,o

(a*nin) - v,a

- IiII2 = ... - IIoII2 = 1.

From the second equation in (2.36) it follows that Non =non, so n are normalized eigenvectors of H with the eigenvalues hw(n+ 2). The eigenvectors L'n are orthogonal since the corresponding eigenvalues are distinct and the operator H is symmetric. Finally, part (iii) immediately follows from the fact that, according to part (ii), the subspaces Im (H f iI) are dense in

, which is the criterion of essential self-adjointness. Remark. Since the coordinate representation of the Heisenberg commutation relations is irreducible, it is tempting to conclude, using Proposition 2.2, that _ °. Namely, it follows from the construction that the linear span of vectors n - a dense subspace of - is invariant for the operators Pand Q. However, this does not immediately imply that the projection

2. Quantization

107

operator IIo onto the subspace 313 commutes with self-adjoint operators P and Q in the sense of the definition in Section 1.1. Using the coordinate representation, we can immediately show the existence of the vector 'c/>o in Proposition 2.2, and prove that _ . Indeed, equation ado = 0 becomes a first order linear differential equation

Chd4 +w)

o

= 0,

so that X0(9) _

and

f°°edq=

1.

The vector '4'o is called the ground state for the harmonic oscillator. Correspondingly, the eigenfunctions

n(4')_

d

1

1

n!

c,vq - fit-

2wfii

dq

n

0

are of the form Pn(q)e X42, where Pn(q) are polynomials of degree n. The

following result guarantees that the functions {fin}form an orthonormal basis in L2(Il8, dq).

Lemma 2.3. The functions qne-q2, n = 0, 1, 2, ... , are complete in L2(I[8, dq).

Proof. Let f e L2 (I[8, dq) is such that

ffqqne_2dq=o, n=0,1,2..... The integral Do

F(z)

= f00

f(q)e2dq

is absolutely convergent for all z E C and, therefore, defines an entire function. We have F(n)(0) = in

f(q)2dq = 0,

n = 0,1,2,...

so that F(z) = 0 for all z E C. This implies the function g(q) = f(q)e2 E is the "ordinary" (h = 1) Fourier L1(JR)nL2(I[8) satisfies (g) = 0, where transform. Thus we conclude that g = 0.

2. Basic Principles of Quantum Mechanics

108

The polynomials .Pn are expressed through classical Hermite-Tchebyschef

polynomials H, defined by Hn (q) = (_1)ne

q2 do _ q2 e , dqn

n = 0,1,2,...

Namely, using the identity e2

do dq

dn- '

d

e_q2

e2dqn_'e _q 2]

q -- d

q

_

_ (-1)n

(q-

d

n _ 9?

we obtain

=4

1

W

e

-"92

W

Cj)

Hn (

We summarize the obtained results as follows.

Theorem 2.1. The Hamiltonian h2 d2

mw 2 q H=--------+ 2 2m dq

2

of the quantum harmonic oscillator with one degree of freedom is a selfadjoint operator on = L2 (R, dq) with the domain D(H) = W2'2(R) n W22(1[8). The operator H has pure point spectrum

H = Inn, n=0,1,2,..., with the eigenvalues an = licv(n+ 2). Corresponding eigenfunctions n form an orthonormal basis for )1° and are given by (2.40)

n(4) - mw c/__re-mwq2 Hn 1

mw 1

where Hn(q) are classical Hermite-Tchebyscheff polynomials.

Proof. Consider the operator H defined on the Schwartz space .9(I[8) of rapidly decreasing functions. Since the operator H is symmetric and has a complete system of eigenvectors in 9(IE8), the subspaces Im(Hfil) are dense

°, so that H is essentially self-adjoint. The proof that its self-adjoint closure (which we continue to denote by H) has the domain W2'2(Il8) fl in

W22(Ilg) is left to the reader.

Remark. The energy levels of the harmonic oscillator can be also obtained from the Bohr-Wilson-Sommerfeld quantization rule. Indeed, the corresponding classical orbit with the energy E is an ellipse in the phase plane

2. Quantization

109

given by the equations q(t) = Acos(wt+a), p(t) _ -mwAsin(wt+a), where E = Zmw2A2 (see Section 1.5 in Chapter 1). Thus

pdq = J J dp n dq = 7rmwA2 =

W

E,

and the BWS quantization rule gives the energy levels E= fizcv(n+ 2)

Remark. Since the energy levels of the Hamiltonian H are equidistant by hv, the quantum harmonic oscillator describes the system of identical "quanta" with the energy fluv. The ground state 10) _ L'o, in Dirac's notation,

is the vacuum state with no quanta present and with the vacuum energy 21w, and the states In) _ n consist of n quanta with the energy huv(n+ 2). According to (2.39), the operator a* adds one quantum to the state In) and is called a creation operator, and the operator a destroys one quantum in the state 1) and is called an annihilation operator. An example of the harmonic oscillator illustrates the dramatic difference between the motion in quantum mechanics and in classical mechanics. The

classical motion in the potential field V(q) = Zmw2g2 is finite: a particle with energy E moves in the region wq) G m , whereas there is always anon-zero probability of finding a quantum particle outside the classical region. Thus for the ground state energy E = 2 this probability is

p(q)j2dq =

0.1572992070. Ji

The classical harmonic oscillator with n degrees of freedom is described by the phase space IR2n with the canonical coordinates p, q, and the Hamiltonian function fl mw22 qp2

-

H (p, q) _ 2m +

,

2

where w i , ... , wn > o (see Sections 1.3 and 1.7 in Chapter 1).

The corresponding Hamiltonian operator is

p2 H=-+ 2m

' mw Q j=1

2

and in the coordinate representation )' = L2 (R, dnq) is a Schrodinger operator with quadratic potential,

H = --

2

2m

+

n mw2q .

-1

2

2. Basic Principles of Quantum Mechanics

110

The Hamiltonian H is a self-adjoint operator with D(H) = W2'2(W) n W2'2(ll8n) and a pure point spectrum. The corresponding eigenfunctions

-

L'ki (ql)

...

where k = (k1,. . , kn) and L'k (q3) are eigenfunctions (2.40) with w = .

form an orthonormal basis for L2(IEBn, dnq). The corresponding energy levels are given by

The spectrum of H is simple if and only if hw1, ... , hwn are linearly independent over Z. The highest degeneracy case is w1 = = Wn = W, when the multiplicity of the eigenvalue

+

Ak =

)

j=1

is the partition function p( kk)) the number of representations of the integer k I = k1 + -}- kn as a sum of nnon-negative integers. Setting m = 1 and introducing the operators23 (2.41) aj =

1

2W Z

1

(wQ+iP), aj =

(wQ-iPj), j = 1,...,n,

2W L

we get canonical commutation relations for creation and annihilation operators for ra degrees of freedom, (2.42)

[aj,aj] = o,

{a,afl = o,

[a,afl =

j, l = 1, ... , n,

which generalize relation (2.34) for the one degree of freedom. The operators a j , a and Nj = j = 1, ... , n, satisfy commutation relations (2.43)

[Ni, al] = -Sjlai, In particular, the operator

[Ni, of ] = 8jiai ,

j,1 = 1, ... , n.

n

j=1

satisfies

j=1

[N,a]=-a, [N,a]=a, j=1,...,n,

and H= Commutation relations (2.42) and (2.43) correspond to the irreducible unitary representation of the solvable Lie algebra 1 n with 3n + 1 generators e j , fj, h j , and c, where e3, fj, c satisfy the relations in the Heisenberg algebra and

[h,e1] =

[h,f1] = ojjwI ei,

23Here, using the standard Euclidean metric on

[h,c] = o,

j, l = 1, ... , n.

we lowered the indices for Qj .

2. Quantization

111

Problem 2.9. Show that (HIM)

for every M E .9, where H is the

2

Hamiltonian of the harmonic oscillator with one degree of freedom.

Problem 2.10. Let q(t) = A cos(wt + a) be the classical trajectory of the harmonic oscillator with m = 1 and the energy E = 2 w2A2, and leta be the probability measure on IR supported at the point q(t). Show that the convex linear combination of the measures , o C a C 27r, is the probability measure on I with the distribution function (q) = ` - 42 } where 9is step Y q} the Heav side A2-q2' function.

Problem 2.11. Show that when n - oo and /t -* o such that hw (n -+- 2 ) = 2 w2 A2 remains fixed, the envelope of the distribution function kl'7 (ci)12 on the interval q C A coincides with the classical distribution function ,a(q) from the previous problem. (Hint: Prove the integral representation e-q2 Hn(q) =

27z+1

2

e-' yn cos(2gy - 2nir)dy, o

and derive the asymptotic formula

=

1

A

2

w

cos

sin_ 1 q

A2

whenh- oandh(n+ )= 2wA2,

- - ql + qA2 2

1

2

q

I
Problem 2.12. Complete the proof of Theorem 2.1.

Problem 2.13 (The N-representation theorem). Let ' E .9(R). Show that the L2-convergent expansion zb = where cn = (, n}, converges in .9(R). (Hint: Use Nia = non . }

Problem 2.14. Show that the operators E23 = a a, i, j = 1, ... , n, satisfy the commutation relations of the Lie algebra sl(n, C).

2.7. Holomorphic representation and Wick symbols. Let QZ

=

C=

IIoII2

-

Icnl2 <00 nn==0 o

be the Hilbert £2-space. The choice of an orthonormal basis {fin}°O_o for LZ(IEB, dq), given by the eigenfunctions (2.40) of the Schrodinger operator for the harmonic oscillator, establishes the Hilbert space isomorphism L2(][8, dq) £2 00

=

L 2 ( Ilk, dq )

cn , --} 0= {c}) o E £2, n=o

where

=

n)

=

L

(q)(q)dq,

2. Basic Principles of Quantum Mechanics

112

since the functions n are real-valued. Using (2.39) we get *

OD

OQ

00

= >cna *'n = \/n+ 1 cn n+ 1 =

b e D(a*),

cn - l n

n=1

n=o

n=o and

= >1 00

00

00

a _ >cnabn _ n=0

E

n=0

n=1

so that in Q2 creation and annihilation operators a* and a are represented by the following semi-infinite matrices:

a=

0

/i

0

0

0

0

0

\/

0

0

0

9

9

9

V 9

a* = ...

As a result,

0

0

0

0

/i

0

0

0

0

'

0

0

0

0

0

0000

0100

0020

N=a*a=

so that the Hamiltonian of the harmonic oscillator is represented by a diagonal matrix, H =qua (N + 2) = diag{ 2 iw, 2 fiiw, 2 fiuv, ... }.

This representation of the Heisenberg commutation relations is called the representation by occupation numbers, and has the property that in this representation the Hamiltonian H of the harmonic operator is diagonal. Another representation where H is diagonal is constructed as follows. be the space of entire functions f(z) with the inner product Let (2.44)

(f, 9) _

f f(z)g(z)e_2d2z,

where d2 z = 2 dz n dz is the Lebesgue measure on C ^l[82. It is easy to check

that

is a Hilbert space with the orthonormal basis ,f (z)

zn

i, n.

n=0,1,2,...

The correspondence Q2

C=

I> f(z) -

Cnfn(z) E n=o

113

2. Quantization

establishes the Hilbert space isomorphism £2 ^. The realization of a Hilbert space 4° as the Hilbert space of entire functions is called a polo-morphic representation. In the holomorphic representation,

a*=z,

a=

d

dz

d +-2 and H=hw z dz 1

,

,

and it is very easy to show that a* is the adjoint operator to a. The mapping 00

00

nfnlz E

Cn4'n H .f z _ n=0

n=0

establishes the isomorphism between the coordinate and holomorphic representations. It follows from the formula for the generating function for Hermite-Tchebyscheff polynomials,

n= n=0

is an integral operator

that the corresponding unitary operator U : 4°

U(z) = J

U(z, q)c%(q)dq

with the kernel (2.45)

(q)f(z) _

U(z,9') _ n=o

Another useful realization is a representation in the Hilbert space °1J of anti-holomorphic functions f(2) on (C with the inner product

(f,g) _ _J f (z)9(z)e-

zl2d2z,

given by

a =z'

a

d

dz

Remark. Holomorphic and anti--holomorphic representations correspond to different choices of a complex polarization of the symplectic manifold R2.

By definition, a complex polarization of a symplectic manifold (4', w) is an integrable distribution on . of complex Lagrangian subspaces of the C. As the notion of real polarization, complexified vector spaces Tx the notion of complex polarization plays a fundamental role in geometric quantization. In particular, holomorphic representation corresponds to the ®JR C ^C2 given by the equation complex Lagrangian subspace in z = o, where z and z are complex coordinates on C2. (Note that here z is not a complex conjugate to z!) The equation z = o defines a complex Lagrangian subspace which corresponds to the anti-holomorphic representation.

114

2. Basic Principles of Quantum Mechanics

The anti-holomorphic representation is used to introduce the so-called Wick symbols of the operators. Namely, let A be an operator in which is a polynomial with constant coefficients in creation and annihilation operators a* and a. Using the commutation relation (2.34), we can move all operators

a* to the left of the operators a, and represent A in the Wick normal form as follows: (2.46)

Alma*)lam

A= 1,m

By definition, the Wick symbol A(z, z) of the operator A is A(z, z) _ `Almzlzm (2.47) ,m

It is a restriction of a polynomial A(v, z) in variables v and z to v -= z. In order to define Wick symbols of bounded operators in , we consider the family of coherent states (or Poisson vectors) I , v e C, defined by

zEC. They satisfy the properties

acv =

(2.48)

v E C.

Indeed, the first property is trivial, whereas the "reproducing property" immediately follows from the formula

v1z -

(2.49)

DD

f(v)f(),

n=0

where fn(z) = f(z), n = 0, 1, 2, ... , is the orthonormal basis for . We also have

(f,g) - f(f,v)(g,v)e_2d2v.

(2.50)

Now for the operator A in the Wick normal form (2.46) we get, using the first property in (2.48),

A zy v} =

v) = l,m

A(v, z)(

Airn(am Z,m

,

Therefore,

A(v, z)since by the reproducing property,

=

),

(v) = e1z.

,

a)

2. Quantization

115

Definition. The Wick symbol A(z, z) of a bounded operator A in the is a restriction to v = z of an entire function A(v, z) in Hilbert space variables v and z, defined by A(v, z)

Theorem 2.2. Wick symbols of the bounded operators on

have the fol-

lowing properties.

(i) If A(z, z) is the Wick symbol of an operator A, then for the Wick symbol of the operator A* we have A* (z, z) = A(2, z) and A*(v,z)

A(2,t).

(ii) For f E 11, (Af)(2) _

f A(z, v)f (v)ed2v.

(iii) Areal-analytic function A(z, z) is a Wick symbol of a bounded operator A in 1J if and only if it is a restriction to v = z of an entire function A(v, z) in variables v and z with the property that for every f e 1J the integral in part (ii) is absolutely convergent and defines

a function in . (iv) If A1(2, z) and AZ(z, z) are the Wick symbols of operators Al and A2, then the Wick symbol of the operator A = AlA2 is given by

A(z, z) = 1 f Al (z, v)AZ(v, z)ed2v. Proof. We have A*(v, z) =

I) = e-vz(I, Acv) = e-vz(AT, Tz)

which proves (i). To prove (ii), we use the reproducing property to get

(Af)(2) = (Af,) = (f,A*z) =

I

f()(A*z)()e_2d2v.

Using the reproducing property once again we have

(A*(Jz)(ii) = (A*4,Iv) = A* (t , z)(I

,

I ) -= e' A(2, v),

which proves (ii). Property (iii) follows from the definition and the uniform

boundness principle, which is needed to show that the operator A on ,

2. Basic Principles of Quantum Mechanics

116

defined by the integral in (ii), is bounded. We leave the standard details to the reader. To prove (iv), by using (2.50) and (i) we get e2

A(z, z) = 1

--

(AiA2z, z) = e-1z12 (A2z, Az)

(Az, )(A2, v)e

1

r f Al (z, v)A2 (v, z)e

22)d2v

--(v-z}(v-z} d2v.

Remark. Properties (i) and (iv) remain valid for the polynomials in a* and

a -operators of the form (2.46). A matrix symbol A(, z) of a bounded operator A in the Hilbert space is a restriction to v = z of an entire function A(v, z) in variables v and z, defined by the following absolutely convergent series:

A(v,z)=

(2.51)

(AJmJn)fn(V)fm(Z). m,n=0

The matrix and Wick symbols are related as follows.

Lemma 2.4. For a bounded operator A in the Hilbert space , A(v, z) = eVZA(v, z).

Proof. Using (2.49) we get 1

1n(v)fm(z) f (AJm)(u)fn(u)e2d2u

1f

evzA(v, z)

Changing the order of summation and integration is justified by the absolute convergence.

Corollary 2.3. If Al(z, z) and A2(z, z) are matrix symbols of operators Al and A2, then the matrix symbol of the operator A = AlA2 is given by

A(z, z) _ if Ai(,v)A2(,z)e2d2v. Proof. The proof immediately follows from part (iv) of Theorem 2.2 and Lemma 2.4.

It is straightforward to generalize these constructions to n degrees of freedom. The Hilbert spacers defining the holomorphic representation is

2. Quantization

117

the space of entire functions f(z) of n complex variables z = (Zi,...

zn)

with the inner product fcm

where 1z12 = z1 +-

f(z)g(z)e_I2d2Thz < oo,

d2 zn is the Lebesgue measure

-+ zn and den z -- d2 z1

on C1 ^IR2n. The functions

fmfz) =

zm1...znn 1

m1, ... , mn = 0,1, b.

,

,

where m -- (ml, ... , m) is a mufti- index, form an orthonormal basis for Corresponding creation and annihilation operators are given by a

* aj=zj,

j-= 1,...,n.

The Hilbert spacers of anti-holomorphic functions f() on Cn is defined by the inner product

f

(f,9) _

(2.52)

f(z)9(z)e-2d2nz

< oo,

and the creation and annihilation operators are given by * a3= z,

aj _

a Ozj

j = 1, ... , n.

,

= &7z, where vz = v1 z1 + The coherent states are satisfy the reproducing property

+ vnzn, and

f(i)=(f,4,), fEd, v E ctn. The Wick symbol A(z, z) of a bounded operator A on n is defined as a restriction to v = z of an entire function A(v, z) of 2n variables v = (Vi,.

,

vrt}

and z(Zi,.

.

. ,

z n , given iven by b

A(v, z) = e-vz (AIz,

j).

We have

(Af)() - n7f f A(z, v).f n

(v)e-v(v-z)d2nv,

.f E n,

and the Wick symbol A(z, z) of the operator A = AlA2 is given by

A(z z) = 1

J

Al(z v)AZ(v

z)e-(v-z)(v-z)d2n,U

where Al (z, z) and A2(z, z) are the Wick symbols of the operators Al and A2.

118

2. Basic Principles of Quantum Mechanics

The matrix symbol A(z, z) of a bounded operator A on !n is defined as a restriction to v = z of an entire function A(v, z) of 2n variables v = (v1,. . , vn ) and z = (Zi,. ,z7), given by the absolutely convergent series .

. .

DQ

A(v,z)=

(lc,m=0

where k = (kl, ... , kn), m = (mi,. , mn) are multi-indices, and Jm() _ fm(Z). The matrix and Wick symbols of a bounded operator A are related . .

by

A(v,z) = evzA(v, z)

.

Remark. The anti-holomorphic representation is very useful in quantum mechanics and especially in quantum field theory, where it is called holomorphic representation (with respect to the variables z). Slightly abusing terminology, in Part 2 we will also call it holomorphic representation. Problem 2.15. Find an explicit formula for the unitary operator establishing the Hilbert space isomorphism ^L2 (ll , d' q). Problem 2.16. Prove that for the bounded operator A the functions A(v, z) and A (v, z) are entire functions of 2n variables.

Problem 2.17. Let A be a trace class operator onn with the Wick symbol A (. , z). Prove that

'IAA = 1

Trn

c

A z z e-fzi2d2nz.

Problem 2.18. Show that the Wick symbol A(z, z) of the product A = Ai ... Al is given by A(z, z) =

i 1

ft Az (z, zi_1 ) ... Al (z1, z) exp

zk (zk_1 - zk)d2z1 ... d2zi_1,

where z0 = z = z, zl = z, and Ak (z, z) are the Wick symbols of the operators Ak.

3. Weyl relations Let R be an irreducible unitary representation of the Heisenberg group Hn in Hilbert space °. It follows from Schur's lemma that R(ea0) = for some

It E IfB, where I is the identity operator on . If It = 0, the n-parameter abelian groups of unitary operators U(u) = R(e) and V(v) = R(e) commute, so using Schur's lemma once again we conclude that there exist p,q E IEBn such that U(u) = e-i'PI and V(v) = e-Z"4I. Therefore, in this case the irreducible representation R is one-dimensional and is parametrized by a vector (p, q) E I[82n. When It 0, unitary operators U(u) and V(v) satisfy the Weyl relations (3.1)

U(u)V(v) = eZ`'vV(v)U(u),

3. Weyl relations

119

which admit the Schrodinger representation, introduced in Section 2.2. It turns out that every unitary irreducible representation of the Heisenberg group Hn is unitarily equivalent either to aone-dimensional representation with parameters (P, q) E II82n, or to the Schrodinger representation with some h 0. The physically meaningful case corresponds to Ii > 0, and

representation with -h is given by the operators U-1(u) = U(-u) and V(v).

3.1. Stone-von Neumann theorem. Here we prove the following fundamental result.

Theorem 3.1 (Stone-von Neumann theorem). Every irreducible unitary representation of the Weyl relations for n degrees of freedom,

U(u)V(v) = evV(v)U(u), is unitarily equivalent to the Schrodinger representation.

Proof. It is sufficient to consider n = 1, the general case n > 1 is treated similarly. Set

S(u,v)=e

ihtLV 2

U(u)V(v).

The unitary operator S(u, v) satisfies

S(u, v)* = S(-u, -v),

(3.2)

and it follows from the Weyl relation that (3.3)

u1v2-u2v1)S(ui

S(ul, vi)s(uz, v2) = e 2

Define a linear map W : L' (1R2) ->

W(f) =

+ u2, vi + v2).

called the Weyl transform, by

2 2 f(u,v)S(u,v)dudv.

Here the integral is understood in the weak sense: for every '1i b2 E

- 2ir

2

f(u,v)(S(u,v)1,2)dudv.

The integral is absolutely convergent for all 1i L'2 E bounded operator W (f) satisfying

IIW(f)I C The Weyl transform has the following properties. WT1. For f E Ll(][82),

W(f)* = W(f*) where

',

f*(uv) = f(-u,-v).

and defines a

2. Basic Principles of Quantum Mechanics

120

WT2. For f E L1(182), S(ui, vi)W (.f)S(u2,v2) = W (J), where

f(u, v) = e

u1 - u2, V - v1 - V2).

J

WT3. Ker W = {o}. WT4. For 11,12 E L1 W(f1)W(f2) = W (f 1 *n f2), where

(fi *hf2) (u,v) -

27r

e2

L2

u - u', V - v'} f2 (u', v')du'de'.

The first two properties follow from the definition and (3.2) and (3.3). and To prove WT3, suppose that W (f) = 0. Then for every all u', v' E I[8 we have

0=

(W(f)S(n',v')bi,S(u',v')b2) 1 271

f

v)(u, v)(S(u,

2)dudv.

2

Therefore f(u,v)(S(u,v)1,''2) = 0 a.e. on ]I82, so that f = 0. To prove WT4, we compute (T'V(f1)T'V(f2)L'1,

(T'V(f2)L'1, T,V(fi)*L,2)

I52)

=

1

271

=

L

f2(u2,v2)(S(u2,v2)i,W(fi)*2)du2dv2 2

12ir J 1

r22 1

_e2

2

f2(u2,v2)(W(fi)S(u2,v2)i,2)du2dv2

2f

/

2

fi(ui - 4d2i 111 - 'UZ)f2lu2,v2)

(fiWe

(u1v2-u2v1)(,S(ui,

R2

have fl *12 E L1 (R2), and it follows from the associativity of the operator product and property WT3 that the linear mapping L1(I[82) x Ll(Il82)

-

L1(I182)

defines a new associative product on L1(1[82) Ii *h (12 *n .f3) _ (fi *h .f2) *h .fs

for all

fig 12,13 E L1(I[82).

When fii = 0, the product *h becomes the usual convolution product on L' (R2).

121

3. Weyl relations

the closed subspace of 3° For every non-zero L' E 3° denote by is invarispanned by the vectors S(u, v) for all u, v E I[8. The subspace ant for all operators U(u) and V(v). Since the representation of the Weyl relation is irreducible, There is a vector c%o E 31 for which all inner products of the generating can be explicitly computed. Namely, let of vectors S(u,

_.

fo(u,v) _ and set

Wo = W(fo) Then W0 = Wo and W0S u, v Wo = e_ 4 rut+v2) Wo.

In particular, W = Wo, so that Wo is an orthogonal projection. Indeed, using WT4, we have Wos(u, v)Wo = W(fo *h Jo) where

fo(u', v') = e

'-u'v) fo(u' -

u, v' - v).

By "completing the square" we obtain o u', v') _ (f a *nf)(

where e_

I(u',v')

= Shifting contours of integration to Im u" = -v - v', Im v" = u + u' and substituting = u" -- u --- u' + iv + iv', j = v" - v - v' - in - in', we get

I(u, v )e 'ddr'=1. 271

Now let °o = Im Wo For every 4b 1, b2 E

(S(n1, v1)'b1, s(u2,

a2

by property WT3. a non-zero closed subspace of b1, W0 L'2 = ' b2, and we have Wo (S(ui, vi)Wo?,bi, S(u2, v2)Wo'b2}

= (W0S(-n2, -v2)S(ui, vi)Woi, x[12} =e

u1v2lU2v1 )

(W0S(ui - u2, v1 - v2)Wo,bi, /'2) 4{(u1 u2)2+(v1_v2)2}(W0b1,

= e 2 tu1v2-u2v1) = e 2 1ulv2Tu2v1)

/'2) 4

?)2}2}

(cj1,

'`f`2).

122

2. Basic Principles of Quantum Mechanics

This implies that the subspace is one-dimensional. Indeed, for every such that (b1,2) = 0, we have that the corresponding sub-

_

spaces and 2 are orthogonal. Since at least one of the vectors %1i b2 is 0. Let

for every non-zero c%,

_ (C '%o,kI-'o = 1, and set

= S(a,/3)bo,

a,8 E Ift. The closure of the linear span of the vectors 'a,p for all a, Q E IE8 is °. We have

(bc3,

=e

!1

and

Next we consider the Schrodinger representation of the Weyl relation in L2(Tft, d4):

(U(u))(q) _ P(q - fitu), (S(u,

hu).

eZa

For the corresponding operator Wo we have

W=

(o)(q)

e - 4 (u2+v2) e

J It

,

00

2 uv-ivq

e - 4 {u2+(u- q}2}

7r

Sp (q

- hu)dudv

o q - hu }du

P00

1

e

so that Wo is a projection onto the one-dimensional subspace of L2 (R, dq) q2, IkooIP = 1. Let spanned by the Gaussian exponential po (q)

_-e

= S(ae, 13)coo

Since the Schrodinger representation is irreducible, the closed subspace spanned

by the functions ofor all a, C3 E IR is the whole Hilbert space L2(IR, dq). (This also follows from the completeness of Hermite-Tchebyscheff functions, since pas q) _ We have (Pc,j3, toy,) = e

and S(u,

=

3. Weyl relations

123

For i/ - i J c/

° define

E

n

E L2(R, dq).

= i=1

Since inner products between the vectors bacoincide with the inner prodwe have ucts between the vectors so that ill is awell-defined unitary operator between the subspaces spanned Thus ill extends to the unitary by the vectors {/,}a,ER and operator GIL

:

- L2 (I[8, dq) and G1l S(a, /3) = S(a, ,3)W

for all a, 3E Itg.

Lemma 2.1 in Section 2.1 is now an easy corollary of the Stone-von Neumann theorem.

Corollary 3.2. The self-adjoint generators P = (Pi,. ,P) and Q = . .

(Q1,. Qfl) of the n-parameter abelian groups of unitary operators U(u) and V(v) satisfy Heisenberg commutation relations.

Proof. It follows from the Stone theorem that in the Schrodinger representation the generators P and Q are momenta and coordinate operators.

Remark. For n degrees of freedom e- E t )e_line--Zt

S(u, v) = eT 2

and formally using the Baker-Campbell-Hausdorff formula, we get

S(u, v) =

(3.4)

Thus the Weyl transform

W(f)=

f(uv)e_i(UP+VQ)dnudnv

(27r) n

can be considered as a non-commutative Fourier transform valued generalization of the "ordinary" Fourier transform

J(p,q)

- (2) n JIl 1

n

an operator-

f (u, v)ev4}dnu dnv.

Remark. The Stone-von Neumann theorem is a very strong result. In particular, it implies that creation and annihilation operators for n degrees of freedom,

a= ----(Q+iP),

a* =

2. Basic Principles of Quantum Mechanics

124

where P and Q are the corresponding generators24 satisfying Heisenberg commutation relations, are unitarily equivalent to the corresponding operators in the Schrodinger representation. Thus there always exists a ground , annihilated by the operators a = (al, ... , an } . state a vector 'L'o E The corresponding statement no longer holds for quantum systems with infinitely many degrees of freedom, described by quantum field theory, where one needs to postulate the existence of the ground state (the "physical vacuum" ). = L2 (IRn, dnq) The Weyl transform in the Schrodinger representation can be described explicitly as a bounded operator with an integral kernel.

Namely,

(8(u, v)i)(q) = e uv-ivq (q -._. hu) ,

(3.5)

and for 'L'1, b2 E L2 (W , dnq ) and f

L' (IR2n ) l L( R) ) we have

I f(u,v)(S(u,v)i,2)dThudv f e 2 uv-,f iv4 I dnu (2ir)'

f

dnq},

1

(W(f)b1,2) = (21r) 1 n

E L2

2n

f

j

e

=1 \

II8

where

K(9,4'') =

(3.6)

1

(27th)n

f f(,v)e'dv.

The interchange of the order of integrations is justified by the Ftitbini theorem. Thus W (f) is an integral operator with the integral kernel K(q, q'),

(W(f))(q)

f K(9,9 )(9')dTy9,

= By the Plancherel theorem,

iE L2(, d"4).

f so that W (f) is a Hilbert-Schmidt operator on . The linear subspace 2T

J 27

L' (R2) rl L2(R2) is dense in L2(Ilg2n) and the Weyl transform extends to the isometry W : L2(R2) -* 92, where 922 is the Hilbert space of HilbertSchmidt operators on = L2(IIBn, dnq). Since every Hilbert-Schmidt operator on 3° is a bounded operator with the integral kernel in L2(]I82n), the mapping W is onto. Thus we have proved the following result. 24Here it is convenient to lower the indices by the Euclidean metric on Rn and use Q = (Q1 , .. y Qn).

3. Weyl relations

12 5

Lemma 3.1. The Weyl transform W dunes the isomorphism L2(I2)

9.

^r

Problem 3.1. Let P = (P1,.. ., P) and Q = (Qi,. ,Q) ) be self-adjoint op.

.

erators25 on° satisfying Heisenberg commutation relations (2.4) on some dense

domain D in ', and such that the symmetric operator H = >1(P + Q ), defined on D, is essentially self-adjoint. Also suppose that every bounded operator which commutes with P and Q is a multiple of the identity operator in f. Prove

that P and Q are unitarily equivalent to momenta and coordinate operators in the Schrodinger representation. (Hint: By the Stone--von Neumann theorem, it is sufficient to show that P and Q define an integrable representation of the Heisenberg algebra. One can also prove this result directly by using commutation relations between the operators H, a*, a and the arguments in Section 2.6. This would give another proof of the Stone-von Neumann theorem.)

Problem 3.2. Prove formula (3.4).

3.2. Invariant formulation. Here we present acoordinate-free description of the Schrodinger representation. Let (V, w) be afinite-dimensional symplectic vector space, dim V = 2n. Recall that (see Section 2.1) the Heisenberg algebra g = g(V) is cone-dimensional central extension of the abelian Lie algebra V by the skew-symmetric bilinear form cv. As a vector space,

9=VERc, with the Lie bracket

{u-+-cc,v+/1c] =w(u,v)c,

u,v E V, c,/9 E R.

Every Lagrangian subspace £ of V defines an abelian subalgebra £ EB Ire of g. A Lagrangian subspace .e' is complementary to £ in V if £ EB .e' = V. A

choice of the complementary Lagrangian subspace £' to £ establishes the isomorphism gl (e EB Rc)

.e'.

The Heisenberg group G = G(V) is a connected and simply-connected Lie group with the Lie algebra g. By the exponential map, G D V EB IRe as a manifold, with the group law exp(vi + a1 e) exp(v2 + a2c) = exp(vl + v2 + (a1 + a2 + 2w(v1, v2)}C),

where v1, v2 E V, a1, a2 E IR. A choice of the symplectic basis for V establishes the isomorphism G(V ) ^r Hwith the matrix Heisenberg group defined in Section 2.1. The volume form d2nv A dc, where d2nv e A2nV* is a volume form on V and V * is the dual vector space to V, defines a bi-invariant Haar measure on G. Every Lagrangian subspace £ defines an 25 See previous footnote.

126

2. Basic Principles of Quantum Mechanics

abelian subgroup L = exp(f 11c) of G, and a choice of the complementary Lagrangian subspace £' establishes the isomorphism

G/L

(3.7)

The isomorphism A2nV * ^Anf* A ATht * gives rise to the volume form dv' on 1?' and defines a measure dg on the homogeneous space G/L. The measure dg is invariant under the left G-action and does not depend on the choice of

For a given h E R, the function x : L -+ C, x (exp (v + ac)) = ez

v E V, c e III,

defines a one-dimensional unitary character of L, 11, 12 E L.

x(1112) = x(i1)x(12),

Definition. The Schrodinger representation S of the Heisenberg group C(V), associated with a Lagrangian subspace £ of V, is a representation of G induced by the one-dimensional representation x of the corresponding abelian Lie group L,

S = IndG x. By definition of the induced representation, the Hilbert space of the representation S consists of measurable functions f : G -- C satisfying the property f(9l) = (l)'f(g), 9 E G, l E L, and such that

IfI2 = fIL I f(g)2dg < oo. Corresponding unitary operators SQ(g), g E G, are defined by the left translations, (Se(9)f)(9') =

f E , g E G.

In particular,

(S(expcc)f)(if) = f(exp(-cc)) = f(gexp(-cw)) = so that SP(exp ac) = where I is the identity operator on . For every Lagrangian subspace £ of V, the representation SP is unitarily equivalent to the Schrodinger representation. This can be explicitly shown as follows. By (3.7), a choice of a complementary Lagrangian subspace £' gives rise to the unique decomposition if = exp v" exp(v + ac), and for f E we get

f(s) = f(expv'exp(v + ac)) = e-Z f (exp v'),

v E Q, v' E Q', a E R.

3. Weyl relations

12 7

Thus every f E is completely determined by its restriction to exp £' and the mapping --} L2 (F]', dnv'}, defined by :

91 (f ) (v') = f(exp iv'),

v' E

is the isomorphism of Hilbert spaces. For the corresponding representation

S =in L2(e', dnv') we have

(S(expv)i/)(v') = ezW(v,v'(v'), v E £, v' E (S(expu')?/')(v') = /(v' - Fin'), u', v' E £'. Let e1,... en, f1, ... , fn be the symplectic basis for V such that ,

and

£'=Rf1EF...EIRf,,

and let (p, q) = (P1, ... , p, q1, ... , of ) be the corresponding coordinates in V (see Section 2.6 in Chapter 1). Then L2(', dnv') ^L2 (Rn, dq) and S (exp uX ) = U(u)

and S (exp vY) = V(v),

where uX = >lukfk, vY = k-1 vke/c, and U(u) = e iu1, V(v) = e_it Q

are, respectively, ri-parameter groups of unitary operators corresponding to momenta and coordinate operators .P and Q. The mapping *h the new associative product on L' (R2) introduced in the last section can also be described in invariant terms. Let B = G/Fh

be the quotient group, where Fh = {exp(c) E G n E Z} is a discrete central subgroup in G, and let db = d2nv A d* cx be the left-invariant Haar measure on Bh, where d* cu is the normalized Haar measure on the circle R/ 7L. The Banach space L1 (Bh, db) has an algebra structure with respect to the convolution:

*B Pa)(b) =

fs

Pi Pa E L1(Bh,db).

The correspondence Ll (V, d2nv)

f(v) --> cp(exp(v + ac)) = e-Z f (v) E Ll

defines the inclusion map L' (V, d2v) y Ll L' (V, d2nv) is a subalgebra of for 11,12 E Ll(V d2nv),

db)

db), and the image of

db) under the convolution. We have

(cpl *Bh cp2)(expv) = f cpl(exp(u + ac))cp2(exp(-u - ac) expv)d2nvd*a

=J V

=

JV

fl(u)f2(v - u)e 2

(f=l*hf2)(v),

vEV

d2nv

2. Basic Principles of Quantum Mechanics

128

Problem 3.3. Prove that the Weyl transform is the restriction to L1 (V, d2v) of the mapping cp H

L1(Bn, db)

E

JBfi

Problem 3.4. Let £1, £2, £3 be Lagrangian subspaces of V. Define the triple Maslov index T(1?1, £2, £3) as a signature of the quadratic form Q on £1 £2 £3, defined by Q(xl, x2, x3) = w`x1, x2)+ w(x2, x3)+ w(x3, x1). For Lagrangian subspaces £1, £2 of V ,j1 be the unitary operator establishing ^32 and intertwining representations Sj1 and the Hilbert space isomorphism S4?2,

2,1S1(g) =

Prove that

ins.

g e G.

=

Il , where Il is the identity operator

Problem 3.5. Let Sp ( V ) be the symplectic group of (V, w)

the subgroup of GL ( V ), preserving the symplectic form w. The group Sp ( V ) acts as a group of automorphisms of the Heisenberg group G by the formula h exp(v + ac) = exp ( h v + ac), h e Sp (V ), v e V. Let R be an irreducible unitary representation of G in the Hilbert space '. Show that there exist unitary operators U(h), defined up to multiplication by complex numbers of modulus 1, such that

U(h)R(g)U(h)1 = R(h g),

h e Sp(V), g E C,

and that U defines a projective unitary representation of Sp ( V ) in °, called the Shale- Weil representation. Find a realization of the Hilbert space° such that the 2--cocycle of the Shale-Weil representation is explicitly computed in terms of the Maslov index.

3.3. Weyl quantization. The Weyl transform, introduced in Section 3.1, defines a quantization of classical systems associated with the phase space with coordinates p = (P1,. . , p7z ), q = (q',. , qn) , and the Poisson bracket { , } associated with the canonical symplectic form w --= dp A dq. . .

.

Let ?(R2) = Wo where W is the Weyl transform and is the inverse Fourier transform, be the linear mapping of the Schwartz space 9'(R2n) of complex-valued

functions of rapid decay on 1I82n into the Banach space of bounded operators 2(a) on = L2(I[8n, dnq). It is given explicitly by the integral

= ()m fa2n f(u, v)S(u, v)dmu dtmv, understood as a limit of Riemann sums in the uniform topology on where

f(u, v) _ 7'(f )(u, v) _ (27r) l

JR2T

f

3. Weyl relations

129

and S(u, v) = e- 2 "'vU(u)V (v). It follows from (3.6) that I(f) is an integral operator: for every L' E L2(Rn dnq)

((f))(q) = f K(q, q')(q')dq', where

K(q,q') =

1

j(qjq, v)e

&1

1')dnv

Since the Schwartz space is self-dual with respect to the Fourier transform, K E 9(][8n x ][8n) C L2(IIgn' x II8"''), so that (f) is a Hilbert-Schmidt operator.

Using the property S(u, v)* = S(-u, -v) we get

= (j) so that classical observables -real-valued functions on II82n, correspond to quantum observables -self-adjoint operators in '. It follows from the

property WT3 in Section 3.1 that the mapping J' is injective; its image Im

and the inverse mapping I-1 are explicitly described as follows.

Proposition 3.1. The subspace Im 4 C 2`L°(a) consists of operators B E 9'1 such that corresponding functions g(u, v) _ din Tr(B S(u, v)-1) belong to the Schwartz class 9(ll82n); in this case W(g) = B. The inverse mapping -1 = o W-1 is given by the Weyl's inversion formula

J(u,v) = hft((f)S(u,v)1),

,f E

92"'1. 1

Proof. First consider the case n = 1. Let p2 + Qa

H=

2

be the Hamiltonian of the harmonic oscillator with m = 1 and w = 1. According to Theorem 2.1, the operator H has a complete orthonormal system of real-valued eigenfunctions bn(q), given by Hermite-Tchebyscheff functions, with the eigenvalues fii(n + 2 ), so that the inverse operator H-1 E

92. The operator H'I(f) is an integral operator with the kernel 2 (ll2 jQ + g2)K(q, q'), which is a Schwartz class function on I[82, so that H'1(f) E 92. The operator

'I(f)=H'H(f)

is a product of Hilbert-Schmidt operators and, therefore, is of trace class. = L2(I[8), we get Using the orthonormal basis {5n(q)}o for (3.8)

r

r

Cmnn q}m q'}

K(q, q') = m,n=O

130

2. Basic Principles of Quantum Mechanics

where emn, _

gg2

K(4',

and the series (3.8) converges in L2(I[82). Since K E x(1[82), it follows from

the N-representation theorem (see Problem 2.13) that this series is also convergent in the 97(I[82) topology. It is this fact which allows us to set q' = q in (3.8) and to get the expansion 00

cmn' n r

K(q, q) =

(q),

m,n=4

which converges in 9(I[8). Thus we obtain

T(f) =

00

n=o

00

((f),) _

_,.,.

n=O

- J-

K(4,4)d9,

where the interchange of orders of summation and integration is legitimate. The general case n > 1 is similar. We consider the operator

H- p2+Qz 2

and use the fact that operators HI(f) and H-n are Hilbert-Schmidt. To prove the formula

f

= K(q,q)dn9, we expand the kernel K(q, q') using the orthonormal basis {(q)}0 for L2(I[8n), where

-

bki (qi) ... jkn (q), k _ (kl,... From the explicit form of the kernel K we get Lk(q)

(3.9)

(f) _

(21h)n J J

f(o,

v)e-ZV9dnv dn(j

,

ICn).

= h-n f (o> 0),

which gives the inversion formula for u = v = 0. To get the inversion formula for all u, v E IEgn, it is sufficient to apply (3.9) to the function where

u-u ) _ f('u + v + and to use S(u, v)-1 = S(u, v)* = S(-u, -v) and the property WT2 in f,(u/

,

v,

Section 3.1,

W(f)8(-u, -v) = W(f,). To complete the proof, we need to show that Im F consists of all B E 911 with the property that the function g(u, v) _ din Tr(B S(u, v)-1) belongs to the Schwartz class. By the inversion formula,

g(u, v) = h' T(I(f) S(u, v)'),

3. Weyl relations

131

where f =fi(g), so it is sufficient to prove that if

Tr(BS(u,v)1) = 0 for all u, v E IEBn, then B = 0. Indeed, multiplying by f(u, v) and integrating, we get

TrBW(f)* =0 for all f E 9(Il82n). Since Schwartz class functions are dense in L2(I[82n), by Lemma 3.1 we get Ti- BB* = 0 and, therefore, B = 0.

Corollary 3.3.

= (2h)

2n

f(p,q)dpdq.

Remark. By the Schwartz kernel theorem, the operator S(u, v) is an integral operator with the distributional kernel e

uv-zvq

q - q' _ hu

so that

S(u,v) = Thus, as is customary in physics texts,

h-n fn

h-n,f (0, 0)

Let Ao = 5I2(I[82n, Il8) C .A be the subalgebra of classical observables on ][82n of rapid decay, and let d f1 be the space of bounded quantum observables.

Proposition 3.2. The mapping Ap

f --> (f) E J?p is a quantization,

i. e., it satisfies A

((f1)(f2) + (f2)(f1)) = flf2

and the correspondence principle

lim -1 ({(fi),

n,-o

(f2)}h) _ {fig f2},

fig 12 E .Ao,

where

{(fi),

(f2)}h =

[(f1), (f)]

and {fl, f2} =

afl aft _ afl aft

a aq

are, respectively, the quantum bracket and the Poisson bracket.

aq aP

2. Basic Principles of Quantum Mechanics

132

Proof. In terms of the product *introduced in Section 3.1 we have

I1(I(fi)F(f2)) =

(Ji

and it follows from the property WT4 that (271)2ff

(3.10)

2n

f 2nfi(ui,vi).f2(u2,v2)

e 2 (v,lv2-u2

Using the expansion e

2

(u1v2-u2v1)

- 1 + 2 ulv2 - u2v1) + 0(h2(uiv2 - u2v1)2)

as h - 0, and the basic properties of the Fourier transform,

9(uf(u, v)) =

9)

and

(vJ(u, v)) = zaq(p, 4),

we get from (3.10) that as h --* 0, q)

_ (flf2)(p, q) - 2 {.fl, f2}(p, q) + 0(h2).

Using skew-symmetry of the Poisson bracket completes the proof.

The quantization associated with the mapping f' = Wo 1 is called the Weyl quantization. The correspondence f F-> (f) can be easily extended to the vector space L1(R2) the image of L ' under the Fourier transform, which is a subspace of C (IRL2n ). More generally, for f E the space of tempered distributions on the corresponding kernel 4'±q' K(q, q') = (2irh) n an f(p,)e*_')dhlp, considered as a tempered distribution on IRn x R, is a Schwartz kernel of the linear operator jfl)F (f) : 9'(I) -* In particular, the constant function f = 1 corresponds to the identity operator Iwith K(q, q') = S(q - q'). In terms of the kernel K(q, q') the Weyl 1

(3.11)

inversion formula takes the form

f(p,q) = f K(4-v,4 +

(3.12)

2v)e*dnv.

The distribution f(p, q) defined by (3.12) is called the Weyl symbol of the operator in L2(][8n, dnq) with the Schwartz kernel K(q, q'). Examples below describe classes of distributions f such that the operators (f) are essentially self-adjoint unbounded operators on the domain 97(W')

C

I'2(I8'',dn4).

133

3. Weyl relations

Example 3.1. Let f = f(q) E LP(In) for some 1 < p < oo, or let f be a - oo. In the distributional sense,

polynomially bounded function as

f(u, v) = (27r)f/26(u)f(v) so that

K(4,9') =

(h/)n f

2v(4+4')dnv

1

= 8(4 - Q')f (4) = f(q)8(q - q'). Thus the operator (f) is a multiplication by f(q) operator on L2(IEBn). In particular, coordinates q in classical mechanics correspond to coordinate

operators Q in quantum mechanics. Similarly, if f = f(p), then (f) _ 1(P). In particular, momenta p in classical mechanics correspond to the momenta operators P in quantum mechanics.

Example 3.2. Let 2

H= 2 + V (q) be the Hamiltonian function in classical mechanics. Then H = 4(H) is the corresponding Hamiltonian operator in quantum mechanics, P2 H 2m +V(Q) In Chapter 3 we give necessary conditions for the operator H to be essentially self-adjoint.

Example 3.3. Here we find f E 9'(IE82n)' such that

f) =PEG - a pure state Pg,, where t' E L2(IL8n), IkbII = 1. The projection integral operator with the kernel 1 n (2h)

is an

and we get from (3.11),

-

Lfl

Introducing q+ = 2 (q + q'), q- = 2 (q - q')> we obtain v) = hTh or

f(u, v)

f

-h f

(q_ + q+)(q+ - q_)edq, 2hu)eZVgdnq

n

Assuming that b does not depend on fii, we get in accordance with Corollary 3.3,

P(u,v)

1

- lim t 0 (27r

f(u,v)

11

(2ir)n Jn I q) I

2 e 2vy d"'1J.

2. Basic Principles of Quantum Mechanics

134

Thus in the classical limit fii - 0 the pure state P. in quantum mechanics becomes a mixed state in classical mechanics, given by a probability measure dµ = p(p, q)dnp dnq on ][82n with the density

p(p,q) It describes a classical particle at rest (P = 0) with the distribution of coordinates given by the probability measure on IIBn. When b(q) _ eh1'o4cp(q), where cp(q) does not depend on h, the corresponding density is

p(p, q) = (p - po)I(q)I2. Remark. The Weyl quantization can be considered as a way of defining a function f(P, Q) of non-commuting operators P = (P1, ... , Pn) and Q = (Q1 ... , Qfl) by setting l

f(P,Q) =

In particular, if f(p, q) = g(p) + h(q), then f(P, Q) = g(P) + h(Q).

For f(p, q) = Pq = plgl +

-}- pngn we get, using (3.11),

f(P,Q)

PQ+QP 2

This shows that the Weyl quantization symmetrizes products of the noncommuting factors P and Q. In general, let f be a polynomial function,

f(p,q)=

(3.13)

kkIj3N where for the mufti-indices cx = (cvi. ... , a?)

=p and J = al + ... following formula (3.14)

1

... pn

+

4' _

and 3 = (i3. (ql)/91

.

. ,

...

-f- Qn Using (3.11), we get the

_ ,C31 -f-

I(f)=

Here Sym(PaQ4) is a symmetric product, defined by t

(3.15)

(uP -I- vQ)k

where uP + vQ = u1P1 +

.._.}_ unPn + v1Q' + ... + vnQT and

3. Weyl relations

13 5

Remark. In addition to the Weyl quantization I, consider also the

and 2 : 9(R22)

pings I1 : 9(IE82n) ->

l(f)-

map-

defined by

(2iir)f2f(u,v)e8(u,v)dudv

and

2(,f) - (2ir)n

2n

j(u,v)eS(u,v)dnudnv,

where f = -1(f) is the inverse Fourier transform. Though 1 and c12 no longer map Ao into the real vector space o of bounded quantum observables, they satisfy all the properties in Proposition 3.2. It follows from (3.5)

that for f E 9'(R2n) the operators

K

}(q,q)

(f) and 2(f) are integral operators

with the integral kernels (2irh)'

an

fp,q'np

and

K2(9',4) _

1

(2

/

h)n J

f(p,q)e*P(')np,

respectively. As in the case of the Weyl quantization, these formulas extend the mappings f --> (f) and f F-k 2(f) of the space I[g2n) of tempered distributions on 1[82n. In particular, if f(p, q) is a polynomial function (3.13), then (3.16)

PaQ

I'l (f) _

and (3.17)

2 (,f) _

Therefore the mapping f H-+

(f) is called the 1nq-quantization, and the mapping f ti I2 (f) -the qp-quantization. Corresponding inversion formulas are (3.18)

f(p,q)=f n K i(q - v>

4)edv

and (3.19)

f(p,q)=

f

K2(4,q' -

The distribution f(p, q) defined by (3.18) is called the pq-symbol of an operator with the Schwartz kernel Kl (q, q'), and the distribution f(p, q) defined by (3.19) the qp-symbol of an operator with the Schwartz kernel K2 (q, q'). The qp-symbols are commonly used in the theory of pseudodifferential operators. It follows from (3.18) and (3.19) that if f(p, q) is a pq-symbol of the (f)*. operator '(f), then f(p,q) is a qp-symbol of the adjoint operator

2. Basic Principles of Quantum Mechanics

136

Problem 3.6. Prove formula (3.14). Problem 3.7. Prove formulas (3.16)-(3.17). Problem 3.8. Prove that Weyl, pq, and qp-quantizations are equivalent. (Hint: Find the relations between Weyl, pq, and qp-symbols of a given operator.)

: 9(II82n) ied in the previous section, defines a new bilinear operation y2(II82Th) X .9(R2n) -a (][82n) on 92(]2n) by the formula

3.4. The *-product. The Weyl quantization

(3'), stud-

fl *h f2 = -lf/fl\Fff2 This operation is called the *-product26. According to (3.10), (3.20)

(.f1

f2)(1>> q) = (271) 2n

2' zf1(ul, vl)f2(u2, v2)

ez2 (u1v2-v,2v1)-2(ui+u2)p-z(vi+v2)9dnu1 d1'u2dnv1dnv2

The *-product on 92(Il82n) has the following properties.

1. Associativity:

fl*h(f2*hf3) - (fl*hf2)*nf3. 2. Semi-classical limit:

(1' *nf2)(1>> 4) _ (fif)(p,q) - Z {fi,f}(p,q) + 0(h2) as h-40. 3. Property of the unit: I *7 1 = 1 *h .f,

where 1 is a function which identically equals 1 on R2. 4. The cyclic trace property:

T(fl fix f2) - r (f2 *ji fl), where the (C-linear map T : 9(][82n) - (C is defined by 1

(271h)

/

f(p,q)dThpdq.

JIR2n

Property 1 follows from the corresponding property for the product * (see Section 3.1), property 2 follows from Proposition 3.2, and properties 3 and 4 directly follow from the definition (3.20). The complex vector space 9(I[82n) (C 1 with the bilinear operation *is an associative algebra over (C with unit 1 and the cyclic trace r, satisfying the correspondence principle, lim tz 0 (fl

*h 12 - 12 *.fl) - {f',

26Also called Moyal product in physics.

.fa}.

3. Weyl relations

137

Consider the tensor product of Hilbert spaces L2 (I2) ® L2

L2 (JR2n X ][$2n)

(]l2n)

and define the unitary operator Ul on L2(ll82n) ® L2(II) by

Ul_e -2 where

a

a

11

a

J

a

a

n

k

k=1

It follows from the theory of Fourier transform that for fl, 12 E

12)) (p1,ql,p2,q2) _ (2)2n

1I82n),

fl(ul,vl)f2(u2,v2).

L2m llg2m

.e

Similarly, defining the unitary operator U2 on L2 (I2) ) 0 L( R2) ) by U2

e

-2

pl

1

,

we get

(U2(f1 ® f2)} (pl,ql,p2,q2) =

1

(2)2n JR2fl JjR2n

fl(ul,vl)f2(u2,v2).

.e

Finally, define the unitary operator Un on L2(R2) 0 L(R2) by Un - U1 U2-'

8p®8q-Dq®8p) =e- 2iha 8

9

8

l

,

9

the point-wise product of and denote by m : 9 (IR2n ) ® 9 functions, (m(fi 0 f2))(p, q) = fi(p, q)f 2 (p, q). Then the *-product can be written in the following concise form:

fl *t 12 = (m o Un) (f1 012)

(3.21)

In analogy with the Poisson bracket {

,

} on J272 associated with the

canonical symplectic form w = dp A dq,

fi, fil- 3fi (912 tJlJZJ3p3q y

(911 312

aq ap

we introduce the notation a2

_

a2

5glap2 Then it follows from the theory of Fourier transform that formula (3.21) for the *-product can be rewritten as (9p

(3.22)

aq

(fl*nf2)(p,q) _

ap

aq

(e

2{

'

aplag2

}fi(pi,4i)f2(p2,42))

P1=P2=P 9i =92=9

138

2. Basic Principles of Quantum Mechanics

Thus we have shown that on 9(IE8212) the *-product can be equivalently defined by (3.20), or by (3.21) and (3.22). The latter representation has the advantage that the formal expansion of the exponential e- 2 { } into the power series gives the asymptotic expansion of the *-product as h - 0. Namely, define the bidifferential operators Bk

byBk =mo{

}k

9'(IEg2n")

®

9(j2n1J -

l

1

fork? land J3 =m.

Lemma 3.2. For every fl, f2 E y7(2n) and l E ICY there is C> 0 such that for all p, q E 82n

`j (

k

4) - 2kk Bk(.fl, f2)(p, 4') < Ch1+1

(fl

k=O

as

h __ 0.

Succinctly, 00

(3.23)

(fi *f2)(7>> q) _

Bk(.f1,

f2)(p, 4) +

0(h°O)

k=O

Proof. Expanding the exponential function e 2 u1 -u2v1 } into the power series and repeating the proof of Proposition 3.2 gives the result. Finally, we get another integral representation for the *-product. Applying the Fourier inversion formula to the integral over dnu1dnv1 in (3.20), we get (fi *,, f2)(1>> 4') _

JJR2n fi(- 2v2, q + 2u2)f2(u2, v2)

(2ir)n

z

2

1

(2)2n L2rz I2n f1 - v2, q + 2u2}f2(p2, q2). e -2u2p---iv2 q+iu2p2 +iv2 Q2 do p2 do g2 dnu2 dnv2 ,

and changing variables p1 = p -- 2 v2, q1 = q + u2, we obtain 1

1

(f1 ' f21 (p, ql = .E

pi 4-pql

1

(7rh)2n JR2n

R2n

fi(pi,qi)f2(p2,q2)

p2 -9'2p1 +pq2 -p2 q} drz

p

do

'

do

p 2 do q2.

Let be a Euclidean triangle (a 2-simplex) in the phase space vertices (p, q ), (p1, q1 ), and (p2, q).. It is easy to see that

p1q - pq1 + 4'1P2 - 42P1 + pq2 - p2q = 2

w,

with the

3. Wey] relations

139

which is twice the symplectic area of the sum of oriented areas of the projections of onto two-dimensional planes (Pi, q'),.. . , (pa, qn ). Thus we have the final formula (3.24)

1

(fi *f2)(p, 9) _ (h)2n 2n 2n fi(pi, .e

do

Wdo ldo ldo

p

1

ql)f2(p2, q2)

q p2 q2 which is a composition formula for the Weyl symbols.

Remark. It is instructive to compare formulas (3.23) and (3.24). The latter formula represents the *-product on 9 (1R2n ) as an absolutely convergent integral, and is equivalent to the Weyl quantization. The former formula is an asymptotic expansion of the *-product as h -+ o, and does not capture all properties of the Weyl quantization. In general, the power series in (3.23) diverges; for polynomial functions this series becomes a finite sum and gives a formula for the *-product of polynomials. Problem 3.9 (Composition formula for pq-symbols). Let f1 (p, q) and f2 (p, q) be, respectively, the pq-symbols of the operators 11(f ) and I1(f2). Show that the pq-symbol of the operator II(f1)1(f2) is given by

f (p,4) _

(/t)n

I

f1(p, q1)f2(p1, q}e

nqx

(Hint: Use the formula for (m o U?) (fi ® f2).)

Problem 3.10 (Composition formula for qp-symbols) . Let Ii

q} and f 2 (p, q)

be, respectively, the qp-symbols of the operators 12(fl) and 2(f2). Show that the qp-symbol of the operator 2(f1)I2(f2) is given by f (p, q) =

1

,f

f1 p1 q}f2 a

_ q1}e

1R2n

(Hint: Use the formula for (m o U2) ) (f'

®f2

).

}

Problem 3.11. Using (3.24) prove that the *-product is associative.

Problem 3.12. For classical observable f (p, q) define the *-exponential (the

analog of the evolution operator) by 00

exp* f

=) n=o

f * f * ... * f. n

Compute exp (-itH ), where HC (p, q) is the Hamiltonian function (2.27) of the harmonic oscillator.

3.5. Deformation quantization. Here we consider the quantization procedure from a formal algebraic point of view as the deformation theory of associative algebras. Let ,A be a C-algebra (or an associative algebra with unit over a field k of characteristic zero) with a bilinear multiplication map

140

2. Basic Principles of Quantum Mechanics

mo : A ® .A - A, which we will abbreviate as a b = mo(a, b). Denote by (C[[t}] the ring of formal power series in t with coefficients in C, C[[t]]

- {antm : an E (C nn==0 o

and let A be the (C[[t]]-algebra of formal power series in t with coefficients in A. The multiplication in At is a (C[[t]]-bilinear extension of the multiplication in ,A, which we continue to denote by mo. The algebra ,At is 7G-graded, n=0

where An = tE,.4, so that Am Am, C Ate,+n,.

Definition. A formal deformation of a (C-algebra ,A with a multiplication mo is an associative algebra At over the ring C[[t]] with a (C[[t]]-bilinear multiplication map mt : At ®q[t]] At - At such that mt(a, b) = a b +

tn1"12n,(a, b)

n=1

for all a, b e A, where mn : A

A - ,A are bilinear mappings.

It follows from (C[[t]]-bilinearity of mt that the associativity condition is equivalent to (3.25)

mt(mt(a, b), c) = mt(a, mt(b, c))

for all a, b, c E A.

Definition. Two formal deformations mt and mt of a (C-algebra A are equivalent if there is a C[[t]]-linear mapping Ft : At -+ At such that

(i) for every a e A,

Ft(a) =a+t7(a), n=1

where fn : ,A - A are linear mappings; (ii) for all a, b E A,

Ft(iht(a, b)) = mt(Ft(a), F(b)). The bilinear maps mn satisfy infinitely many relations which are obtained by expanding (3.25) into formal power series in t. The first two of them, arising from comparing the coefficients at t and t2, are a ml (b, c) - ml (a b, c) + ml (a, b c) - ml (a, b) c = 0,

3. Weyl relations

141

and

a m2 (b, c) - m2 (a b, c) -F- m2 (a, b c) - m2 (a, b) c

= mi(mi(a, b), c) - mi(a, mi(b, c)). In general

a mn(b, c) - mn(a b, c) -I- mn(a, b c) - mn(a, b) c n-1

- m(a, m_(b, c))).

_(a, b), c)

j=1

The main tool for understanding these equations and studying the deformation theory of associative algebras is the Hochschild cohomology. Namely, let M be an ,A-bimodule, i.e., a left and right module for the C-algebra A.

Definition. The Hochschild cochain complex C' (,A, M) of a C-algebra .A with coefficients in .A-bimodule M is defined by the cochains Cn (.A, M) = Homc (.A®n, M),

i.e., n-linear maps f(ai,. do : Cn (A, M) ---} Cn+

. .

,

an) on A with values in M, and the differential

M}

y

(dnf)(ai, a2, ... , an+i) = ai f(a2,..

.

,

n

(-1)f(ai, ... , aj__1,aj

+

aj+l? aj+2 ... , an+1}

j=1

+ (_1)Ti+if(ai, ... , an) an+i. We have d2 = o, i.e., do+1 o do = o, and the cohomology H' (A, M) of the complex (C(A,M),d), H n (,A, M) = ker do / Im do _ 1,

is called the Hochschild cohomology of the algebra A with coefficients in the .A-bimodule M. In the deformation theory of associative algebras we have the simplest

non-trivial case M- .A with the left and right ,A-actions given by the multiplication map. The associativity equation (3.25) can be written as (d2mi)(a, b, c) = o,

(d2m)(a, b, c) =

(m(m_(a, b), c) --- m j (a, mn_ j (b, c)} }

,

a, b, c E A.

j=1

It is quite remarkable that the Hochschild cochain complex C' (,A, ,A) carries an additional structure of a graded Lie algebra, which plays a fundamental

142

2. Basic Principles of Quantum Mechanics

role in studying the associativity equation (3.25). Namely, for f E C"" (,A, A) and g e Cn (,A, A) let

(f 0 g)(a,...

,

m- I

-1)3,f (ai, ... , aj glaj+I ? .. , aj+n} aj+n+I,

. .

,

Qrn+n_ I ),

j =O

and define (-1)('n-1)(n-1)9 of. [.f, 9]G =fog The linear mapping [, ]c: x Cn(,A,A) -* (Cm+n-1(A,.A) satisfies

[f, 9jG = _(_1)(m-1)(n-1)[9 fIG, and is called the Gerstenhaber Considering the restriction of a multiplication mt to A® A as formal power series in t with coefficients in C (A, .A) and using the Gerstenhaber bracket, we can rewrite (3.25) in the succinct form: bracket27.

(3.26)

[mt,mnt]G = 0.

Let ,A be a commutative (C-algebra, and let At be a formal deformation of A. Define the bilinear map {, } : A ® ,A -; A by (3.27)

{a,b} = ml (a, b) - ml (b, a),

a, b E A.

Lemma 3.3. A formal deformation At of a commutative (C-algebra A equips A with a Poisson algebra structure with the bracket { }. ,

Proof. Consider the equation dµl (a, b, c) = 0. Subtracting from it the equation with a and c interchanged, and using the commutativity of A, we obtain Interchanging b and c, we get

{a.c,b} -{a,b.c} =a.{c,b}-b. {a,c}, and interchanging further a and c, we obtain {a. c,b} - {c,a b} = c. {a,b} - b {c,a}. Adding the first and third equations and subtracting the second equation gives {a'b,c} =a.{a,c}-+-b.{a,c},

so that the skew-symmetric cochain { } E C2 (,A, A) satisfies the Leibniz rule. To prove the Jacobi identity, observe that ,

(3.28)

{a, b} = mt(a'

b)

t mt(b' a)

mod t.At.

27Together with the cup product of cochains, the Gerstenhaber bracket equips C' (A, ,A) with the structure of a Gerstenhaber algebra.

3. Weyl relations

143

Using associativity condition (3.25) (associativity modulo t2,At is sufficient) we get {{a, b}, c} + {{c, a}, b} + {{b, c}, a}

_

((mt(mt(a, b) - mt(b, a)), c)

-mt(c, mt(a, b) - mt(b, a)) -f- mt(mt(c, a) - mt(a, c), b) -mt(b, mt(c, a) - mt(a, c)) + mt(mt(b, c) - mt(c, b), a) -mt(a, mt(b, c) - mt(c, b))) mod t,At = 0. This result motivates the following definition.

Definition. A deformation quantization of a Poisson algebra (A, {, }) with the commutative product mo : A ®c A -* A is a formal deformation At of an algebra A such that the multiplication map mt satisfies (3.28). By Lemma 3.3, every formal deformation ,.4t of a commutative algebra A is a deformation quantization of the Poisson algebra (A, { , }) with the Poisson bracket given by (3.27). According to Lemma 3.2, a deformation quantization of the Poisson algebra (9(R2'), { , }), where the Poisson bracket is associated with the canonical symplectic form w = d1n n dq, is the algebra 9'(I[82n)[[t]], where t = -ifii and It is considered as a formal parameter, and the multiplication map is given by the *-product. The following statement is a formal algebraic analog of the representation (3.22) for the *-product.

Theorem 3.4 (Universal deformation). Let A be a commutative (C-algebra with the multiplication map mo, and let cpl and c02 be two commuting derivations of A, i. e., linear maps cpl, cp2 : 4 -> A satisfying Leibniz rule and

(

(P2 = (p2 0 (p1.

Then

{a,b} = (pi (a) ' cp2 (b) - (p2 (a) (pi (b),

a, b E A,

defines a Poisson bracket on A, and the formula mt = mp o etc,

4) _ (pl ®

called the universal deformation formula, gives a deformation quantization of the Poisson algebra (A,{ , }).

Proof. The Jacobi identity for the bracket { } follows from the commutativity of the derivations cpl and cp2. To show that At is a deformation quantization of the Poisson algebra (,A, { , }), we need to verify the associativity condition (3.25) for the map mt. Let O ,,4 - A ®c ,A be the ,

coproduct map,

0(a)=ai1+1®a, aEA.

2. Basic Principles of Quantum Mechanics

144

It extends to a C-linear mapping from Homc(,A, A) to Homc(,A®2, A®2), which we continue to denote by O, and 0(cp) = 'p® id + id ®cp,

cp E Homy (A, .A).

To prove (3.25), we observe that the Leibniz rule and the binomial formula yield the following identity for a derivation cp of ,A: etc° o m0 = mp o et0(cp} .

Using commutativity of cpl and cP2i we obtain

mt(a, mt(b, c)) = mo(ett(a ®

c))))

_ (moo (id®mo))(et(id®0)(etid®(a ® b® c)) b ® c)), _ (moo (id where (id

cpl ® 0(cP2) E ,A®3. Similarly,

0 b 0 c)),

mt(mt(a, b), c) _ (mo a (mo 0

where (L®id)() = 0(cp1)®cp2 E .A®3. Since mao(id®mo) = moo(mo®id), the associativity of the multiplication mt is equivalent to

id = (id®z)() + id ®,

(O ® id)

which is obviously satisfied since [ = cpi 0 cp2.

In general, 2n pair-wise commuting derivations cpi on a commutative algebra A define a Poisson bracket on ,,4, n

('p(a)

{a, b} _ i=1

'

'p(b) - 'p(a)

'

and deformation quantization of the corresponding Poisson algebra (A, {, } ) is given by the universal deformation formula

A deformation quantization of a Poisson manifold (4', { }) (see Section 2.7 in Chapter 1) is, by definition, a deformation quantization of the corresponding Poisson algebra of classical observables (C°°(4'), { , }) with the property that the linear maps mn, n > 1, are bidifferential operators28. The formal power series expansion (3.22) of the *-product is a deformation ,

quantization of the Poisson manifold (1t82n, {, }), where the Poisson bracket corresponding to the canonical symplectic form w = dp n dq. 28 This property ensures that the unit in Poisson algebra is preserved under the deformation.

3. Weyl relations

145

Problem 3.13. Show that C' (,A, ,A) is a graded Lie algebra with respect to the Gerstenhaber bracket, i.e., that [ , ]c satisfies the graded Jacobi identity (_1)(m_1)(P_1)[f, [g,

h]] + (-1)(1)(P-')[h, [1'9]] + (_l)(m_l)(n_l)[g, [hfj] -0

for all f E Cm (,A, .A), g E CT (.A, .A), h E Cp (A, .A).

Problem 3.14. Verify that df = [f,mo]G, where f E Cn (A, ,A) and do is the Hochschild differential.

Problem 3.15. Show that deformation quantizations associated with Weyl, pq, and qp-quantizations are equivalent. (In fact, all deformation quantizations of the canonical Poisson manifold (lR2', { , }) are equivalent.}

Problem 3.16. Let g be a finite-dimensional Lie algebra with a basis x1, ... , xn, and let g* be its dual space equipped with the Lie-Poisson bracket {, } (see Problem 2.20 in Section 2.7 of Chapter 1). For u E g* and x = 1ixz, y = z 1 r7xz E g let euG

lagetxetc-ac-y}

=

aaQlu, tlrr, c r/9

where = ()1 ...

(i)3i ... (rr)', be the

formal group law

the "Baker- C ampb ell- Hausdorff series" . Show that the product

mt(fl, 12) =

t)Dafl(u)D f2(u),

f1, f2 E C°°(g*),

where for a multi-index CE = (cEi, ... , c) and f E C°° (g* ), D' f = a deformation quantization of the Poisson manifold (g*, { , }).

Problem 3.17. Let (G, {, }) be a Lie-Poisson group, where (g) = --r+Adr 1g r and non-degenerate r satisfies the classical Yang-Baxter equation (see Problems 2.21 - 2.23 in Section 2.7 of Chapter 1). Using Baker-Campbell-Hausdorff series for the one-dimensional central extension of g by the 2-cocycle c the Lie algebra g (see Problem 2.23 in Section 2.7 of Chapter 1), show that there exists an element F E (Ug ® Ug ) [[t]] of the form F = 1 ---- tr + 0(t2), satisfying

(z®id)(F)(F®1) _ (id®)(F)(1®F), where

is the standard coproduct in Ug.

Problem 3.18. Let FL and FR be the images of F and F-1 under the identifications of the universal enveloping algebra Ug with the algebras of left and right-invariant differential operators on G (see Problem 2.21 in Section 2.7 of Chapter 1), and let" = FL O FR. Show that the product mt = mo o` gives a deformation quantization of the Poisson-Lie group (G, { , } ) such that

o mt = (mt ® mt ) o , is the standard coproduct on functions on G, ( f )(g1, g) = f (gig), ), f E C°° (G). The Hopf algebra (C°°(C),mt, , S), where S is the standard antipode on is called the quantum group corresponding to the Poisson-Lie G, S( f )(g) = f group G, and is denoted by Gq, q = et.

where

146

2. Basic Principles of Quantum Mechanics

Problem 3.19. Let R = a(F-1)F E (Ug ® Ug)[[t]], where c is a permutation an involution of Ug ® Ug, defined by a(a ® b) = b ® a, a, b E Ug. Show that R = 1 - tr + 0(t2) and satisfies the quantum Yang-Baxter equation R12 R13 R23 =231312 (see Problem 2.22 in Section 2.7 in Chapter 1 for notation).

4. Notes and references Dirac's classical monograph [Dir47] is the fundamental text. Other classical ref-

erences are physics textbooks [and [LL58]. The monograph [Sak94] is a popular text for graduate courses in physics departments. Another useful physics reference is the encyclopedic treatise [Mes99], which discusses at length the origin of quantum theory and the development of its mathematical formalism. An elementary textbook [PW35] provides an introduction to the old quantum theory, including Bohr-Wilson-Sommerfeld quantization rules, and applications of quantum mechanics to chemistry. We refer the interested reader to these sources for physical formulation and the origin of quantum mechanics, including a discussion of basic experimental facts: the double-slit experiment and particle-wave duality. Quantum mechanics does not account for single measurement outcomes in a deterministic way, and the quantum process of measurement admits different interpretations. The accepted one is the so-called Copenhagen interpretation, in which a measurement causes an instantaneous "collapse" of the wave function describing the quantum system.

Monographs [Rud87, BS87, AG93], as well as [RS8O, RS75], contain all the necessary material from the theory of operators in Hilbert spaces; the latter treatise also covers the theory of distributions, Fourier transform, and self-adjointness. The book [Kir76] introduces the reader to various methods in the representation theory,

and the monograph [BR86}, written for theoretical physicists, contains all the necessary information on the representation theory of Lie groups and Lie algebras.

A succinct introduction to the method of stationary phase and other asymptotic methods is given in the classical text [Erd56] and in [01v97J; see also Appendix B in [BW97]. Properties of Hermite-Tchebyscheff polynomials29 can be found in the classical monograph [Sze75], as well as in many other reference books on special functions and orthogonal polynomials.

Our exposition in Section 1.1 follows a traditional approach to the mathematical foundation of quantum mechanics based on Dirac-von Neumann axioms. These axioms go back to von Neumann's classical monograph [vN96], with more detailed discussion in [MacO4]; see also [BS91], notes to Chapter VIII of [RS8O], and references therein. Another mathematical description of quantum mechanics is based 29Following V.A. Fock [Foc78], in Section 2.6 we used the more appropriate name HermiteTchebyscheff polynomials instead of the customary Hermite polynomials (see [Ger5U] for the relevant historic arguments).

4. Notes and references

147

on C*-algebras and the Gelfand-Naimark-Segal reconstruction theorem, and can be found in [StrO5] and references therein.

Our exposition in Section 2 follows the outline in [FY80]; see [Ber74] for the general mathematical formulation of the quantization problem and the recent survey [AEO5]. Having in mind a more advanced audience than for [FY80], in Sections 2.1 and 2.2 we included a complete mathematical treatment of Heisenberg commutation relations and coordinate and momentum representations. In Sections 2.2 and 2.3 we use the same normalization of the eigenfunctions of continuous spectrum as in [Foc78]; it corresponds to the orthogonality relation in Section 2.2 of Chapter 3. In our exposition in Section 2.6, which is otherwise standard, we include a proof of the completeness of the eigenfunctions (q), which is usually only mentioned in the physics textbooks. Holomorphic representation in quantum mechanics was introduced by V.A. Fock in 1932 [Foc32], where the scalar product was given in terms of the orthonormal basis of monomials fm (z). In terms in of the integral (2.52) this scalar product was defined by V. Bargmann [Bar6l]; holomorphic representation is also called Fock-Bargmann representation. Discussion of Wick symbols of operators in Section 2.7 essentially follows [BS91]; more details can be found in the original paper [Ber7lb]. The method of geometric quantization is only briefly mentioned in remarks in Sections 2.2 and 2.7; the interested reader

is referred to monographs [GS77, Woo92], lecture notes [SW76, BW97] and the survey [KirOO].

Our proof of the celebrated Stone-von Neumann theorem in Section 3.1 essen-

tially follows the original paper of von Neumann [vN3l]. The main instrument was introduced by Weyl in the classical of the proof the Weyl transform monograph [Wey5O] (see [RosO4] for the history and generalizations of the Stonevon Neumann theorem). Invariant formulation of the Stone-van Neumann theorem, discussed in Section 3.2, as well as the relation with the triple Maslov index, methaplectic group, Shale-Well representation, and applications to number theory, can be found in [LV8O]. We refer to [BS91] for the extended discussion of pq and qp-quantizations as their relations with the Weyl quantization, as well as for more details on the *-product, introduced in Section 3.4. The beautiful formula (3.24) for belongs to Berezin [Ber7la]. the *-product composition of the Weyl symbols

The notion of the deformation quantization was introduced in [BFF+ 78a, BFF+78b], where the solution of Problem 3.12 can be found. The fundamental theorem that every Poisson manifold admits a deformation quantization was proved by Kontsevich in [KonO3]. More on Hochschild cohomology and the deformation theory of associative algebras can be found in the survey [VorO5] and references therein. The deformation theory point of view on the evolution of physical theories, like a passage from classical mechanics to quantum mechanics emphasized in Faddeev's lectures [FY80], can be found in [F1a82] and [Fad98].

148

2. Basic Principles of Quantum Mechanics

Most of the problems in this chapter are rather standard and are mainly taken from the references [RS8O, BS91]. Other require more sophistication and are aimed at introducing the reader to a new topic. Thus the asymptotic expansion in Problem 2.11, which can be found in monograph [Sze75], is an example of

semi-classical asymptotics, and Problem 3.1, taken from {gives a criterion when the irreducible unitary representation of the Heisenberg algebra is integrable.

Problems 3.17-3.19 introduce the reader to the theory of quantum groups (see [Jim85, Dr186, Dri87, RTF89] ) and are taken from [Dri83] (see also {The

fundamental result that every Lie-Poisson group admits a deformation quantization described in Problem 3.18 has been proved in {EK96] (see also [EnrOl]).

Chapter 3

Schrodinger Equation

1. General properties Here we describe the general properties of the Schrodinger operator for a quantum particle in W moving in a potential field. To simplify the notation, in this chapter we set h = 1 and m = , and use x = (Xi, ... , x?) for Cartesian coordinates on Iftf . Recall (see Section 2.4 in Chapter 2) that the corresponding Schrodinger operator is given by the formal differential expression

H= -O + V(im), where V (x) is areal-valued, measurable function on W. Denote by (02 Ho-_o-_ ax

2

+...+ax82

2

nJ

the Schrodinger operator of a free quantum particle of mass m = on Ian, also called the kinetic energy operator, and by V the multiplication by V(x) operator, also called the potential energy operator, so that (1.1)

H=H0-FV.

According to Section 2.3 of Chapter 2, the operator Ho is self-adjoint and

unbounded on P = L2(1[8), with the domain D(Ho) = W2'2(ll8n) and absolutely continuous spectrum [0, oo), and the operator V is self-adjoint and is bounded if and only if V (x) E L°O(ll8n), when IVII _ VIk The sum Hp+ V is not necessarily self-adjoint, and the first major mathematical problem of quantum mechanics is to characterize potentials V (x) for which the formal

differential expression (1.1) uniquely defines aself-adjoint operator H on . We start by presenting some useful criteria for the self-adjointness. 149

3. Schrodi roger Equation

15 0

1.1. Self-adjointness. The main result here is the Kato-Rellich theorem on perturbation of self-adjoint operators.

Definition. Let A and B be densely defined operators in . The operator B is smaller than A in the sense of Kato if D(A) C D(B) and there exist a, b ER with a <1 such that for all c/> E D(A), (1.2) lBbll allAcbll +blI. Equivalently, the operator B is smaller than A in the sense of Kato, if there exist a < 1 such that for all L' E D(A), (1.3)

Theorem 1.1 (Kato-Rellich). If A is a self-adjoint operator with domain D(A) and B is a symmetric operator smaller than A in the sense of Kato, then H = A + B with D(H) = D(A) is a self-adjoint operator. Proof. Recall that the operator H is self-adjoint if and only if Im(H+AI) _ Im(H - Al) _ for some A E i IfB, and hence for all A E C \ Q(H). Since A = A*, for every A E i Il8 we have RA _ (A - AI)-1 E 2`c°(,3) and ImRa = D(A). Now

H - Al

(I + BRA)(A - Al), and in order to prove that Im(H - Al) _ 3' it is sufficient to show that for Al Ilarge enough IIBRaII < 1, since then by Neumann series I -}- BRA is an invertible bounded operator and Im(I + BRa) _ . Indeed, for all cc E we have the inequalities IIRacpII < jjiIcpIl

and

IARAsoII C

which follow from the equation l= llAll2 + AI2ll'll2 by setting cp = (A - AI)'b. Now using (1.2) with b =Rasp, we get IBRAcOII
the multiplication by V(x) operator.

Theorem 1.2. Let V = Vl + 172i where Vl(x) E L2(1183) and VZ(x) E L00(I[83). Then H = Ho+V is a self-adjoint operator and D(H) =

W2,2(R3).

Proof. It is sufficient to show that V is smaller than Ho in the sense of Kato on C( R3) C W2'2(R3). Denoting by II ' Iloo the norm in L°°(I[83), for

cc E C( R3) we have llVccll

II Viii llcciio + llV2ll

iiccll

I. General properties

151

Let h(p) =1n2. Denoting by I ' 1 the norm in Ll(]l83) and using the Fourier transform and the Cauchy-Bunyakovski-Schwarz inequality, we get

(2)3/2II = sup f xER3 I

I(h + 1)-lII I+ 1)cp

Now replace c(p) by cp,.(p) =

IkI1i, kII

1ii 3

- r3/2jI, and IhcrI = (2)3/2I

c(I I=

r > o. Since we get l

=

OQ

Iko Ii=

<

r

r such that a =

r-1'2(2ir)-3/2CIIVijj

<1

completes the proof.

Corollary 1.3. The Schrodinger operator of a complex atom, considered in Example 2.2 in Section 2.4 of Chapter 2, is essentially self-adjoint on cl o (R3(N+1)1 1

Proof. We consider only the special case of the Hamiltonian of the hydrogen atom, e2 H=-L---, r=IxI. r

Writing V = x1V + (1 - xl)V = Vi + V2i where X1 is the characteristic function of the unit ball Bl = {x E I[83 : x < 1}, we have Vl(x) E L2(1E83) and 1/2(x) E L°°(IE83).

Another useful criterion applies to real-valued potentials V (x) E L( W1), the space of locally bounded a.e. functions on W. In this case the operator H, defined by the formal differential expression (1.1), is symmetric on Cp (]I8n) and we have the following result.

Theorem 1.4. If V(x) E L (IEBn) is bounded from below, V(x) > C a.e. on Rn, then the Schrodinger operator H = Hp+V is essentially self-adjoint on In fact, a much more general statement holds.

Theorem 1.5 (Sears). Suppose that the potential V(x) e L(W) for all x E IIBn satisfies the condition

V(x) > -Q(jx), where Q(r) is an increasing continuous positive function on [0, oo) such that dr

Q(r) = oo.

JO

3. Schrodinger Equation

15 2

Then the Schrodinger operator H = H0 + V is essentially self adjoint on Problem 1.1. Prove the general case of Corollary 1.3 (Hint: Derive the estimate (1.3) for each term in the corresponding potential energy operator.)

Problem 1.2 (Kato's inequality). Let 5 E L1 (W ) be such that

b, defined

in the distributional sense, is represented by a function in L oc (lien ). Prove that in the distributional sense,

1u . Re uuu , where it i s assumed that

if u(x) = 0. (The distribution T E 7(R n is non-negative if T( gyp)

u) =

0

u(x) 0 for all

non-negative co E $ (lRn ). }

Problem 1.3. Prove Theorem 1.4 using Kato's inequality. (Hint: Show that if E L2 (lR TL ) in the distributional sense satisfies (-z + V(x) + C -+- 1) = 0, then

=D.} Problem 1.4. Prove that one-dimensional Schrodinger operators with unbounded below potentials V(x) = x and V(x) = _x2 are essentially self-adjoint on Co (l ).

1.2. Characterization of the spectrum. The second major mathematical problem in quantum mechanics is to describe the spectral properties of the Schrodinger operator H. Here we present some general results characterizing the spectrum of H. The first basic result is the following.

Theorem 1.6. Suppose that V(x) e L (IPJ) satisfies lim V (x)= oo. Then the operator H has a pure point spectrum: there exists an orthonormal consisting of eigenfunctions of H with eigenvalues basis {n}nEN for a C A2 C .
7Z f' 7L 7

and lim An = oo. n--too

Recall that the essential spectrum Qess(A) of a self-adjoint operator A consists of all non-isolated points of Q(A) and of eigenvalues of infinite multiplicity. The following result gives a sufficient condition for the essential spectrum of the Schrodinger operator to fill [0, oo).

Theorem 1.7. Suppose that V = Vl + VZ, where Vl(x) E Lq(IIBn), 2q > n for n > 4 and q > 2 for n < 41, and V2(x) E L°°(W) satisfies

lim V2(x)=0. IaI-oo

Then Qess(H) _ [0, oo), so that a(H) f1 (-oc, 0) consists of isolated eigenvalues of H of finite multiplicity. 'In the special case n = 4 one has q> 2.

153

1. General properties

The next result gives a sufficient condition for the Schrodinger operator with decaying potential to have only negative eigenvalues.

Theorem 1.8 (Kato). Suppose that V(x) E L°O(Il8n) and

lim IxIV(x) = 0. Then the Schrodinger operator H = Ho + V has no positive eigenvalues.

Recall that the absolutely continuous spectrum and the singular spectrum of a self-adjoint operator A on 3° are defined, respectively, by sac (A) s are closed subspaces } , where a (A a ) and a(A) = a : A ac and

for A defined as follows. Let PA be the projection-valued measure for the self-adjoint operator A and let v = (PA'b, b) be the finite Borel measure consists of o and all o. Then on Ilk corresponding to E ' E 31° such that the measure v' is absolutely continuous with respect to such that consists of o and all b e the Lebesgue measure on III, and is continuous singular with respect to the Lebesgue measure the measure v on III.

Theorem 1.9. Suppose that the potential V(x) E L°°(R) for some e > 0 satisfies

V(x)

O(IxI')

as

fx - oc.

Then the Schrodinger operator H = Hp -}- V has no singular spectrum and Qac(H) _ [0,oo). Moreover, Q(H) f1 (-oo, 0) consists of eigenvalues of H of finite multiplicity with the only possible accumulation point at 0. Finally, for the physically important case n = 3 there is a useful estimate for the number of eigenvalues of the Schrodinger operator.

Theorem 1.10 (Birman-Schwinger). Suppose that V(x) E L00(R3) and (y) V (x)V a3 a3 I x - y

3

3

Then for the total number N of eigenvalues of the Schrodinger operator

H = Ho + V, counted with multiplicities, we have

N<

dsxdsy.

I

1 2 Ilgs

ygs

12

Problem 1.5. Prove all results stated in this section. (Hint: See the list of references to this chapter.)

154

3. Schrodinger Equation

1.3. The virial theorem. Let H = Ho -}- V be the Schrodinger operator whose potential is a homogeneous function on I[8n of degree p, i.e., V (ax)

_

aPV(x). The virial theorem in quantum mechanics is the relation between the expectation values of the kinetic and potential energy operators Ho and V in the stationary state.

Theorem 1.11 (The virial theorem). Let /'

= 1, be the eigenfunction of the Schrodinger operator with the homogeneous potential of degree p, and let To = (Hod, /) and Va = (Vb, b) be the corresponding expectation values of kinetic and potential energy operators. Then 2T0 = pVo

Proof. It follows from the Schrodinger equation

-+V(x)b=A that

T0 + V0 = -(z, ) + (Vb,

) = A.

For every a> 0 the function ba(x) _ /(ax) satisfies -02ba -}- CLP+2V(x)4'a =

so that by differentiating this equation with respect to a at a = 1 and using Euler's homogenous function theorem, we get (1.4)

-0 + V(x) _ A + (2A - (p +

Since Li E D(H) and H is self-adjoint,

((H- AI)) _ (,(H- AI)) _ 0, and we obtain from (1.4) that 2A _ (p+2)Vo, which completes the proof.

O

Remark. Since energy levels of quantum systems are related to the closed orbits of corresponding classical systems, Theorem 1.11 is the quantum analog of the classical virial theorem (see Example 1.2 in Section 1.3 of Chapter 1

Problem 1.6 (The Raleigh-Ritz principle). Prove that A is an eigenvalue of a self-adjoint operator H with the eigenfunction b if and only if

is a critical point

of the functional F (cp) = ((H - I) w, cp) on D(H). Problem 1.7. Derive the virial theorem from the Raleigh-Ritz principle.

2. One-dimensional Schrodinger equation

155

2. One-dimensional Schrodinger equation The Schrodinger operator on the real line has the form ger operator

the one-dimensional Schrodin-

2

H

+ V(x),

where the real-valued potential V(x) E LoC(I[8), the space of locally integrable functions on R. The corresponding eigenvalue problem

Hb=Ab reduces to the second order ordinary differential equation

-oo <x < oo,

-y" -{- V (x)y =key,

(2.1)

where it is convenient to set A = k2. In this section, we will study in detail the one-dimensional Schrodinger equation (2.1) for the case when

f(i + x)IV(x)jdx < oc.

(2.2)

Condition (2.2) is a mathematical formulation of the physical statement that

the potential V(x) decays as - oo. It allows us to compare solutions y(x,k) of the Schrodinger equation (2.1) with the solutions of the Schrodinger equation for the free quantum particle, which corresponds to the case V = 0 in (2.1). All results in this section hold for the real-valued potential V(x) E L (R) satisfying (2.2); to simplify the presentation we will additionally assume that the potential V (x) in (2.2) is continuous on I.

2.1. Jost functions and transition coefficients. Let 00

Q(x) = J

00

and Ql(x) = f Q(s)ds.

x Since V E L' (R), we have the Fubini theorem we get

Q(x) = 0, and using condition (2.2) and

00

al fix)=J

J

I V fit) dtds =J

S

J

t

V fit) dsdt =J

(t-x)IV(t)Idt
so that a E L1(x, oo) for all x E ll8 and lim00 Ql (x) = 0. Similarly, the functions

Q(x) = J x -00

and

Ql(x) = J

&(s)ds

-00

satisfy lim_00 mi(x) = Ql (x) = 0. Condition (2.2) ensures that for real k the differential equation (2.1) has solutions fi(s, k) and f2(x, k),

3. Schrodinger Equation

156

uniquely defined by the following asymptotics: eikx + orll

fi(x, k) = f2(x, k)

=

J

+ o(l)

e_

as

x - oc,

as

x -- ---- oc .

They are called Jost solutions and play a fundamental role in the theory of aone-dimensional Schrodinger equation. Theorem 2.1. For real k the differential equation (2.1) has solutions fi(x, k) and f2(x, k) satisfying the following properties.

lIc) lIc)

(i) Estimates for k E ]I8: - 11

(i(X) - 1(x +

- 11

(&1(x)

-j

-

(ii) Asymptotics as lxi - oo: lim e-' x-00

fi (x, Ic)

lim

1,

lim ei f 2 (x, Ic) =

x-#00

e-z' fi (x7 lc) = 2k,

lim ei' fl (x, Ic) _ ----2k

1,

x-+ -00

(iii) Analyticity: Ii (x, k) and f2(x, Ic) admit analytic continuation to the upper half-plane Im k > 0, and are continuous functions on Im k > 0, uniformly in x on compact subsets of ll8. (iv) The conjugation property:

fi(x,Ic) = fi(x, -k),

f2(x,k) = f2(x, -k), Im k > 0. (v) Estimates in part (i) hold for Im k > 0. For k 0, e- Im kx

1

ICI

fi(x, Ic) -

eIm kx ICI

e-ikxl

l

1

(vi) Asymptotics as I l --> oo, Im k > 0: e_'fi(x,Ic)

=

1+O(1Ic1'),

ef2(x,Ic) =

1+O(1Ic1').

Proof. It follows from the method of variation of parameters that differential equation (2.1) with the boundary condition limxy00 e-2kx fl(x, k) = 1 is equivalent to the integral equation (2.3)

sin k(x - t)

fi(x, Ic) = eikx Jx

V (t)fi (t, k)dt.

2. One-dimensional Schrodinger equation

Setting cp(x, k) =

157

fl(x, k), we get

(x,k)=1- f

00 1 - e-2ik(x-t) V (t)cp(t, k)dt.

2ik

For Im k > 0 the integral equation (2.4) is of Volterra type and can be solved by the method of successive approximations. Namely, we look for the solution in the form

(x,k) _

(2.5)

n=0

where cpp(x, k) = 1 and 00 1

+1(x, k) = -

=

-

V(t)Spn (t, k)dt

2zk

k)ds dt.

Ix( It ko(x,k)I

n!

Indeed, this estimate is true for n = o, and using the Fubini theorem and the induction hypothesis we get

Ik) I <

\

t

1

_ nl f \f V (t) Ivl (t)ndsJ f dt

If 00

n,

<-n!JI

(f°° V (t) a(t)ndt) ds

1

x

a(s)ai(s)Thds

=

ai(x)Th+

(n+1)!

Thus

By the Weierstrass M-test the function cp(x, k), defined by convergent series (2.5), is analytic for Im k > 0. It is also continuous up to Im k = 0 uniformly in x on compact subsets of ll8.

158

3. Schrodinger Equation

The first estimate in part (i) now follows from (2.4) and (2.7). Namely, using IV(t)I _ -Q'(t) and integration by parts, we obtain

I(x,k) - i

00

c

e al(x)

1

J f+ 1

(t X+I

1

-(t -

-e

IV(t)Ilw(t,k)Idt

2zk

IL

x+ikl

+J

i

I Q(t)dt +

ICI

J +1

V(t)Idt

ICI

= ea1(ai(x) - a(x+

)).

For k = 0 this estimate should be understood as cp(x, 0) - 1 < o1(x)e°1(x). The function fl (x, k) = k) satisfies the integral equation (2.3), and (2.7) allows us to differentiate under the integral sign in (2.3). This proves that fi(x, k) satisfies the differential equation (2.1) and the first estimate in part (i). In particular, e-Zk fl (x, k) = 1 for Im ft > 0. Differentiating (2.3) and using cos k(x-t)I < 1 for t > x and Imft > 0, we obtain e-In'k(t-x)1

(2.8)

e-

fl(x, k) - ik

Vt

Qxe

t,

,

x

so that e-Zkx fi (x, k) = ik for Im k > 0. The conjugation property follows from the uniqueness of the solution f' (x, k) satisfying the estimate in part (i). To prove the uniqueness, denote by (x, k) the homogeneous solution of the integral equation (2.4) with the property that a(x) = (t,k) G oo for all x E R. Consider the inequality IL

Idt, \

00

(ft

which follows from the homogeneous form of equation (2.4). Repeating the

proof of the estimate (2.6), we get (x,k) (x)_ and passing to n. the limit n -> oo gives X(x, k) = 0. To prove the first estimate in part (v), consider the inequality (2.9)

(x, k) - 1 <

which follows from (2.4) for k

-

u'(t)(t, k) - 1dt

0. Iterating (2.9) we get

f(x,k) - ij

2. One-dimensional Sehrodinger equation

159

The asymptotic in part (vi) follows from the estimate in part (v), which completes the proof for the solution Ii (x, k). The existence and analytic properties of the solution f2(x, k) are proved similarly by considering the integral equation

f2(x, k) =

(2.10)

e-zkx +

/'

Corollary 2.2. For Im k > 0 and k lim e-ikx( fl (x, k) -ixfl(x, k)) = 0,

Sin k(x - t)V(t),f2(t,

k)dt.

0,

lim

k) +ixf2(x, k)) = 0,

where the dot stands for the partial derivative with respect to k.

Proof. Differentiating (2.4) with respect to k we obtain the following integral equation for cp(x, k) = e-Zk'(fl(x, k) - ixfl(x, k)):

e_2(t) (x,k)=g(x,k)_f °O 1 -2ik V (t)(t,k)dt, where g(x,k)

=

1 Ic

J

(t -

xx)e-2ik(-t)V

(t)cp(t, k)dt +

(1 - (x, k)).

It follows the estimate in part (i) of Theorem 2.1 and (2.7) that g(x, k) and repeating the proof of the estimate (2.7), we obtain that (x,k)J < l(x)e2a1. Since 0, this proves the statement for fi (x, k). The corresponding result for 12 (x, k) is proved similarly.

For real k 0 the pairs fi(x,k), fi(x,-k) = fi(x,k) and f2(x,k), f2(x, -k) = f2(x, k) are fundamental solutions of the differential equation (2.1). Indeed, the Wronskian W(yl,y2) = yiy2 - ylyof two solutions of (2.1) does not depend on x, and we get from part (ii) that

W(fi(x,k),fi(x, -Ic)) = lim W(fi(x,Ic),fi(x, -Ic)) = 2ik, W (f2(x,Ic),f2(x, -k)) = lim W(f2(x,k),.f2(x, -k)) _ -2ik. -00

Therefore for such k we have

f2(x,k) = a(Ic)fi(x, -Ic) + b(Ic).fi(x, Ic), where the transition coefficients a(k) and b(k) are given by (2.11)

(2.12) a(k) = 2IcW fi (x,

.fz(x> k)), b(Ic) -

and satisfy a(k) = a(-k), b(k) = b(-k). Similarly we find that (2.13)

fi(x, k) = a(k)f2(x, -k) - b(-k)f2(x,Ic)

3. Schrodinger Equation

166

Back substituting (2.13) for fi(x, k) and fi(x, -k) into (2.11), for real k we find the so-called normalization condition,

0

a(k)2 =1 + b(k)2.

(2.14)

It follows from parts (iii) and (vi) of Theorem 2.1 that coefficient a(k) admits

analytic continuation to Im k > 0 and

a(k) = 1+

(2.15)

as

jkj -+ oo

Moreover, the function ka(k) is continuous in Im k > 0. Normalization condition (2.14) implies that all zeros of a(k) are in the upper half-plane Im k > 0. Let kp be such a zero, a(kp) = 0. It follows from the first equation in (2.12) that W(fl(x, ko), f2(x, ko)) = 0, so that the Jost solutions fi (x, ko), 12 (x, ko) are linearly dependent,

fi(x,ko) = cof2(x,/co),

0. It follows from part (i) of Theorem 2.1 that for Im k > 0 the solution fi (x, k) is exponentially decaying as x -j oo, and the solution f2(x, k) is exponentially decaying as x -k -oo. Therefore fi( , ko) E L2(R) is an eigenfunction of the Schrodinger operator H with the eigenvalue Ao = ko Since H is symmetric, its eigenvalues are real: for some co

AoMfi(' ,k0)112 = f (_f'(x,ko) + V(x).fi(x,ko))fi(x> ko) dx

f fi(x, ko)(-.fi (x, ko) +V(x).fi(x, ko)) dx = oll.fi(' , as is easily seen by integrating by parts twice; due to part (i) of Theorem 2.1 and estimate (2.8) the boundary terms vanish as Ix --> oo. Thus ko = ixo is

pure imaginary with xo > 0 so that Ap = -xo < 0, and the corresponding eigenfunction fi(x,ixo) is real-valued.

Proposition 2.1. The function a(k) in the upper half-plane Im k > 0 has only finitely many pure imaginary simple zeros ki = ixl, and

a(i1) _ -iclIIf2(' ,zxl)II2,

l = 1,...,n,

where the dot stands for the derivative and fi(x,ixi) = cl f2(x, ixl). The function 1/a(k) is bounded in some neighborhood of k = 0 in Im lv > 0.

Proof. It follows from (2.14) and (2.15) that k = 0 is the only possible accumulation point of zeros of a(k) in Im k > 0. First suppose that functions Ii (x, k) and 12 (x, k) are linearly independent for k = 0. Then the continuous in Im k > 0 function ka(k) satisfies limkyo ka(k) = W (fl (x, 0), f2(x, 0)) 0. Therefore a(k) 0 in some neighborhood of 0, and a(k) has only finitely many zeros in Im k > 0.

2. One-dimensional Schrodinger equation

161

The case when fi(x, 0) = cf2(x, 0), c 0, is more subtle. Suppose that there is a converging to 0 subsequence kn = ixn of zeros of a(k) in Im k > 0. It follows from part (i) of Theorem 2.1 that there exists A > d such that

for all yc > Owe have fi(x,i)> 2e-x for x > A and f2(x,i)> 2e' for x
If a

00

fi(x, Z1-tn)fl(x, O)dx, A

1 2

(x,i).f2(x,0)dx ?

00

4x

Using integration by parts, we get, as before,

n J-00

OG

fi(x,

O)dx =

J-00

(fl (x,

V (x)fl(x, 2YLn))fl(x, O)dx

= Lf1,m)(f',0) - V(x)fi(x,O))dx = 0. On other hand, we have oo

(2.16)

0= I f1(x,i)f1(x,O)dx+ I AA fi(x,i-cn)fi(x,O)dx A

A

-I- C. Cn

J

f(x,

O)dx.

Since fi (x, k) is continuous on Im k > 0, uniformly in x on the compact subsets of Il8, we have fA

fA

lim

J

fi(x,in)fi(x,O)dx=

Using

lim c cn = c 7L--too lim

fl-00

f1(x, i n)

t

,

'fi

n

J

fi(x,O)2dx> O. A

f'(x,O) f2(x,O) -

C2 >0,

we obtain from (2.16) that for large enough n,

0>

4Xn

This is clearly a contradiction, so a(k) has only finitely many zeros. The proof of local boundness of 1/a(k) in this case is left to the reader. Next, consider the differential equation (2.1) together with the equation obtained from it by differentiating with respect to k: _y" + V(x)y _ (2.17) -y/' + V (x)y =key + 2ky. Set y = fi(x, k) in equation (2.17), y = f2(x, k) in equation (2.18), and multiply (2.17) by f2(x, k) and (2.18) by fi(x, k). Subtracting the resulting equations, we obtain (2.18)

w(.fi(x, k), fa(x, k))' = 2k fi(x, k).f2(x, Ic).

3. Schrodinger Equation

162

Similarly, setting y = 12 (x, k) in (2.17), y = fi (x, k) in (2.18), cross multiplying and subtracting, we obtain

-W(fi(x, k), f2(x, k))' = 2kfi(x, k)f2(x, k). Therefore

W (fl(x, k), f2(x, k))

(2.19)

x

-A

- W(.fi(x,k),f2(x, k))

(2.20)

ffi(x,k)f2(x,k)dx,

A

-A fA

=2k / fi(x,k)f2(x,k)dx.

Now suppose that a(ko) = 0. We have

W(fi(x, k), f2(X, k)) = 2ika(k),

so that differentiating with respect to k and setting k = ko gives W (fi(x> Ico), f2(x, ko)) + W (fi(x, ko), 12(x, ko)) = 2ikoa(ko).

Since fi(x,ko) = co f2 (x, ko), it follows from Theorem 2.1 and Corollary 2.2

that boundary terms in (2.19)-(2.20) vanish as A - oo, and we obtain, using that f(x, ko) is real-valued:

a(ko) _ -i

J fix,k0f2(x,kodx =

The eigenvalues of the Schrodinger operator H are simple. Indeed, the

function fl (x, ix) is a solution of the differential equation (2.1) for ) _ -x2 < 0, which decays exponentially as x - oo. Since the Wronskian of any two solutions of (2.1) is constant, the second solution of (2.1), linearly independent with fi (x, ix), is exponentially increasing as x - oo. Thus

every exponentially decaying as x -> oo solution of (2.1) for ) _ -yc2 is a constant multiple of fl(x,iyc). In particular, this proves that the point spectrum of H is simple. This result also follows from the simplicity of zeros of the function a(k) and the eigenfunction expansion theorem, proved in the next section.

Problem 2.1 (The oscillation theorem). Let a1 = -x? c

c an = xn c

o be the eigenvalues of the one-dimensional Schrodinger operator H. Prove that corresponding eigenfunctions f1 (x, ixl) have exactly Z -1 simple zeros, t = 1, ... , n.

Problem 2.2. Prove that for Im k > o the Jost solution fx (x, k) admits the representation f 1 (x, k) = eikx +

Ki (x, t)eiktdt,

where O0

K(x, x) = 2

V(t)dt x

and K1 (x, t) C 2 2 t

x2 t }

2. One-dimensional Schradinger equation

163

Moreover, the kernel K1 (x, t) is differentiable and x

x, t) + 4V }I c

with the similar inequality for (x, t). Correspondingly, for Im k C o the Jost solution f2 (x, k) admits the integral representation x

f2 (x, k) = e-z x + f K2 (x, t)e-z tdt, o0

where the kernel K2 (x, t) satisfies similar estimates. (Hint: Show that (2.3) is equivalent to the integral equation oa oe t+(s-x) Kl (x, t) V(s)ds + 2 K1 (s, u)duds V(s) 2 t

x

x

t-(s-x)

with the condition K1 (x, t) - = o for t C x, which is solved by the method of successive approximations.)

Problem 2.3. Show that transition coefficients a(k) and b(k) have the representations a(k) = 1 -

1

O°

2ik -00

V (x)dx -

O°

1

2ik

a

A(t)ettdt,

1

b(k) = ---

2ik

jO°

where A(t) E L1 (o, oo) and B(t) E L1 (-oo, oo).

Problem 2.4. Show that for Im k > o the transition coefficient a(k) satisfies the dispersion relation

f

1

°° log(l + Ib(p)2) dp) n k - ix

p-k

jlk+zxi.

2.2. Eigenfunction expansion. Here we explicitly construct the resolvent kernel for the one-dimensional Schrodinger operator H, and show that it is self-adjoint with the domain consisting of functions % E L2(IEg), which are twice differentiable on I[8, and such that -%" -}- V(x)i4' E L2(I[8). Using the method of complex integration we derive the eigenfunction expansion theorem for the operator H, which generalizes the corresponding result for

the operator Ho of a free quantum particle, considered in Section 2.3 of Chapter 2. For A E (C \ [0, oo) let

fi(x, k)f2(y, k) (2.21)

y) =

{IfiYkv2xk 2ika(k)

if x > J>

if x

where the branch of the function k _ on C \ [0, oo) is defined by the condition that Im k > 0. For fixed x and y the function Ra (x, y) is meromorphic on (C \ [0, oo) with simple poles at A1 _ -x?, 1 = 1, ... , n. For fixed

3. Schrodinger Equation

164

Ai the function Ra (x, y) = RA (y, x) is continuous in x and y, and it a follows from part (v) of Theorem 2.1 and (2.15) that

Ra(x,y) C C e

(2.22)

ICI

C > 0.

Al defines a bounded integral operator RA in

The kernel Ra(x, y) for A L2 (1[8) by the formula

(R)(x) = fR(xYdY. Indeed, it follows from (2.22) that for

k12 IIRII2 < p00

=

C2 I

fo

J-OCJ-00

2

E LZ(II8),

f

e-Im-yl

(y)I dy f 2dx

Iyl+'00

e-

IRAI<

2C IkIImk

Lemma 2.1. The operator H is self-adjoint and RA = (H - Al)' -for A E (C \ {[o, oo) U {A,,.

. .

,.fin} }, where I is the identity operator in L2(l[8).

Proof. Let g E LZ(Ilg). As in the variation of parameters method, it follows from (2.21) that for A E C \ {[0, oo) U {A,,. . . , An}} the function y =Rag E L2 (I) is twice differentiable a.e. on IEl and satisfies the differential equation

-y" + V(x)y = Ay + g(x). Thus (H - AI)RA = I and, in particular, Im(H ± iI) = L2(l[8), so that H is self-adjoint.

Let fjo be the Hilbert space of (C2-valued functions (A) _ (') on [0, oo) with the norm 00

=j

(I,(A)

+

where dQ(A) = -dA and /3> 0 for A > 0 (see Section 2.3 of Chapter 2, where one should put fii = 1 and m = 2) For A > 0 set .

(2.23)

u( x,) ) = ()fi(X), j = 1, 2,

2. One-dimensional Schrodinger equation

165

and for L' E C( R) define the C2-valued function l/' on [0, oo) with com-

ponents (WL)1 and (i'L')2 by (2.24)

()(A) = 1 f

(x)u(x,

j = 1, 2.

)dx,

By Proposition 2.1 the function °ll b is bounded at A = 0, and for large A using differential equation (2.1) and integration by parts we get

-

12rt I (-n (x) t V (x)(x))ui (5,

)dx,

which shows that Gll% E

Let P be the orthogonal projection on the subspace of 1' = L2(II8) l = 1, ... , n. The functions spanned by the eigenfunctions L'j(x) = fi(x, zL'1 are real-valued and orthogonal, so that for L' E 3' 72

(P)(x) =

Theorem 2.3. The operator W extends to a partial isometry operator 3

11 * P1= I -- P and

* = Io,

and establishes the isomorphism (I- P).3 ^J Sf o. Here I and Io are, respectively, identity operators in ° and 5io. For the spectrum of the corresponding Schrodinger operator H we have o(H) = {-x,.. . -x} U [D, oa}, and ,

where APP = P3f' and a = (I - P)Yt' are, respectively, invariant subspaces associated with pure point and absolutely continuous spectra of H. is a multiplication by A operator in jo, so that the The operator absolutely continuous spectrum [6, oo} of H has multiplicity two, and the operator ' "diagonalizes" the restriction of H to the subspace

Proof. To proof the relation Gll*W = I - P the so-called completeness relation - it is sufficient to establish the following classical eigenfunction expansion formula for b e Co (I[8):

(x) = (2.25)

1 f (ui(xk)J

b(y)u2(y,

b(y)ui(y, k)dy + u2(x7k)

+ IIII l

f

(y)j(y)dy,

3. Schrodinger Equation

166

where all integrals are absolutely convergent, and we put /5 = k, do(A) _ dk. Indeed, assuming that (2.25) holds, we get for , cp E C( R)

()=J b(x)c(x)dx=__J c(x)(J (ui(xk)j + u2(x,k)

f

(y)ui(y,k)dy

(Y)u2(Yk)dY)dk)dx+ rb

(b,'co)o+L 1I2,)1,c°), and the interchange of integrals over where (, )o is the inner product in x and k is justified by the Fubini theorem. Thus

(Qi'Wo)o =

(2.26)

for /', cp E Cp (I[8), and we conclude that Gll extends to a bounded operator from ° and fjo satisfying (2.26) for all , cp E .

To prove the eigenfunction expansion formula (2.25), for A E C \ [0, oo) set g(x, A) _ (Rah) (x). The function g(x, A) for fixed x is meromorphic in A E (C \ [0, oo) with simple poles at A _ Al, l = 1, ... , n. It follows from (2.21) and Proposition 2.1 that

Resa=9(x,A)

- a(i f(x, zxl) f f2(y,ix1)/'(y)dy= LCj

Since HRa =RAH = I + ARa, we have

-g"(x, A) + V(x)9(x, A)

_

(x) + A9(x, A),

so that

g(x,A) = -(x) + where cp =

E Co(I[8). From (2.22) we get

(R'p)(x)

as

A -+ 00,

so that (2.27)

9(x,A)_ -(x) +

as jIt

follows from Proposition 2.1 and (2.21) that (2.28)

9(x, A) =

as A-0.

For 0 < e < 1 < N let C = Ce,nr be the contour consisting of the arg 6 < ir, oriented following parts: (i) the arc C of a circle A = eeie, e clockwise; (ii) the arc Cnr of a circle A = Neie, nr < arg B < r, oriented anti-clockwise, where Nsin enr = e sin s; (iii) the segments Z of the straight

2. One-dimensional Schrodinger equation

167

lines Im A = fe Sin E which connect the boundaries of the arcs. Choose 6 and N such that all poles Al of g(x, A) are inside C, and consider the integral

I

27ri

fg(xA)dA.

On one hand, by the Cauchy residue theorem n

I=

Res1g(x, A) =

(b,b1)

-

1Li 119

=1

On the other hand, it follows from (2.27) and (2.28) that lim

27fi

cT,

g(x,A)dA=-(x)

1

lim --}o 2iri

and

g(x, a)da = o.

Thus we obtain n

1

I

i

ll2

lim lim

,

1

e-+0 N-+oo

-

f°°

(2.29)

p

1 -

I

J

2iri \ z+

g(x , A)dA-

R0(x, y) - RA_o(x,

J[(xA)dA) z_

/

where

Rafio(x,y)

- limRaf2E(x,y)

To compute the difference Ra+io(x, y) -Ra_2o(x, y), observe that on the cut A > 0 we have A + i0 = k > 0 and A - i0 = -k <0. It follows from (2.21) that for x > y Ra+2o (x, y) - Ra-zo (x, y)

- - 2ik (fl(x,

(y, k)

_k))

+ fi(x,

and using the equations fi(x,k)

=

f2(Y, -k)

=

- ___

a(_k)f2

+

b(-k)f2( k),

(k)f1(Y k)

which follow from (2.11) and (2.13), we obtain Ra+zo(x, y) -

2kIa(k)12

=

(fi(x, k).fi(y, k) +.f2(x, k).f2(y, k))

(u(x, k)ul(y, k) + u2(x,

k)),

where A = k2. Substituting this into (2.29) and using the symmetry Ra (x, y) _ Ra(y, x) of the resolvent kernel, we get the eigenfunction expansion (2.25).

Let Qlac be the restriction of the operator Il to the subspace

(I -

_

It follows from the completeness relation that the operator llaC

3. Schrod inger Equation

168

is an isometry, so that Im ( = Im

is a closed subspace of 5o . Thus to verify the orthogonality relation lC * = Io, it is sufficient to show that ac

Im ( = 5o. Using integration by parts we easily get that a self-adjoint operator acHI'ac 1 is a multiplication by A operator in Im ( with the domain

'D(H). Moreover,

(R = R(o} ', p E C' [o, oo}, where R(°) is the resolvent of the multiplication by A operator in 5o, so that Now it follows from Lemma 2.2 in Im W is an invariant subspace for Section 2.2 of Chapter 2 that there are Borel subsets E1, P22 C [o, oc} such

that

ImGll = {

- (xEi1

-

X1 0 .

E

If, say, Ei = [0, oo) \El has positive Lebesgue measure, then for A = k2 E Ei we have

ul(x,

J

=0 for all

c% E Co(]I8),

so that u(x, k) = 0 for all x E IE8 - a contradiction. Therefore, ImGl( =

Let 1o

° -p 5o be the corresponding unitary operator for the

Schrodinger operator Ho of a free particle, constructed in Section 2.3 of -f Chapter 2 (with h = 1 and m = 2 ), and set U = Corollary 2.4. The restriction of H to the absolutely continuous subspace ac is unitarily equivalent to Ho, = UH0U-1. HI

Remark. In physics literature, completeness and orthogonality relations, understood in the distributional sense, are usually written as n 1

2

k)ui(y, k) + u2(x, k)u2(y, k))dk +

l

8(x - y)

and k)uk(x,p)dx =

p),

where i, k = 1, 2 and k, p> 0. When transition coefficients a(k) and b(k) are differentiable for k 02, the orthogonality relation can be derived from the identity (2.30)

dW

(k2

-

This is the case when V(x) satisfies f °(i + x 2 )I V ( x )dx C oo.

2. One-dimensional Schrodinger equation

169

which immediately follows from the differential equation (2.1). Namely, con-

sider the case j = k = 1 and integrate (2.30) over [-N, N]. Using (2.13) and asymptotics of Jost solutions, we get N

N

INui(x,k)ui(x,p)dxei(k-T')N

i a(k)a(p)

(k2-p2)a(k)a(p)W(fi(x,k),fi(x,p))

-

a(k)a(p)e-i(k-p)N

k-p

-N

+ b(k)b(p)eZ(k-p)N

p))e2(k+P)N 1 +o(1) as N -oo. k+p

+

l

Since k, p > 0, by the Riemann-Lebesgue lemma the distributional limit as N ---* o0 of the terms in the third line is zero. Since a(k) and b(k) are assumed differentiable, in the second line we can replace a(p) and b(p) by a(k) and b(k), respectively, since the difference goes to zero as N -* oo by the Riemann-Lebesgue lemma. Finally, using (2.14) we obtain `v

= 2N oo 1v1oo,f ui(x,k)ui(x,p)dx N

sink - p)N k-p

=28(k-p),

where the last equality is the distributional form of the orthogonality relation for the Fourier transform.

Remark. The eigenfunctions of the continuous spectrum u3 (x, A) also satisfy the normalization condition (2.14) in Section 2.2 of Chapter 2: lim

1 (U+-- UUj,ft -- U=

1=1 2

where 2

Uk(x)-

Jk0

2/d - Jko

f3(x,p) (

p)

dpi

j=1,2.

Remark. The qualitative structure of the spectrum of the Schrodinger opx) _ A of erator H is determined by the structure of the level sets x) = p2 + V(x). Namely, condition the classical Hamiltonian function V (x) = 0 implies that the level sets for A > 0 are non-compact and the classical motion is unbounded in both directions, and these values of A fill the absolutely continuous spectrum [0, oo) of H of multiplicity two. For A <0 the level sets are compact and classical motion is periodic. According to BWS quantization rules (see Section 2.5 in Chapter 2), the energy levels correspond to the closed orbits, and condition (2.2) - eigenvalues of H guarantees that H has only finitely many eigenvalues.

170

3. Schrodinger Equation

Problem 2.5. Find the energy levels for the potential

V(x)=-

cosh2 xx

,

Vo>0.

In particular, show that when Vo = 2x2 there is only one eigenvalue E =

Problem 2.6. Show that the operator Ra --- Ra°} , where

(Ho - )I)-' and

) E C \ [0, oo}, is of the trace class, and

Tr R - RCo} - a

}

d to da

ga().

(Hint: Use Wronskian identities (2.19)-(2.20) from the previous section.}

2.3. S-matrix and scattering theory. It was shown in the previous section that for potentials V(x) satisfying (2.2) the operators H = Ho + V and Ho have the same absolutely continuous spectrum. Among many partial isometry operators in which establish the unitary equivalence between and Ho, there are two operators W± of fundamental physical signifiH1 cance. They are defined as (2.31)

Wj. = t-±oo lim e1Hte-ztHo,

where the limit is understood in the strong operator topology, and are called wave operators (or Moller operators). In general, the wave operators exist

for a Schrodinger operator H = Ho + V on ][8n with the potential V(q) decaying sufficiently fast as -* oo. In this section, we will show that for potentials V(x) satisfying (2.2) the limits (2.31) exist. The wave operators satisfy the partial isometry relation (2.32)

Indeed, strong limits of unitary operators preserve the inner products, so that W o) = (, cp) for all L', E Convergence An -k A as n -k oc in the strong operator topology does not necessarily imply that An --} A*, so that we cannot conclude that W± W = I. Thus in general the wave operators are not unitary. In fact, it can be shown that Im W =

= (I - P)3 or, equivalently, I -- P,

(2.33)

where P is the orthogonal projection on the subspace app. Though the proof of (2.33) is rather non-trivial, it is easy to show that Im W C (I - P) '. Indeed, let b E ' be a bound state an eigenvector for H,

H = \b. Then for every cp

we have

(Wco, b) = lim ez \t t--}±oo

(e-itH0 P,

/:) = 0

2. One-dimensional Schrodinger equation

171

by the Riemann-Lebesgue lemma, as we have seen in Section 2.3 of Chapter 2.

Assuming that wave operators exist and satisfy (2.33), it is easy to prove

that

f(H)W = W+ f (Ho)

(2.34)

for any measurable function f on R. Indeed, according to the spectral theorem, it is sufficient to prove this property for the functions f(\) = e2T) for all r E R. In this case, (2.34) immediately follows from the identity eiT.Hei(t-T)He-i(t-T} = by passing to the limits t -* foo. Using (2.34) and (2.32) we conclude Ho,

so that restrictions of the wave operators to the subspace

establish the

unitary equivalence between Hand Ho. Remark. The physical meaning of the wave operators is the following. The one-parameter group U(t) = e-itH of unitary operators describes the evolution of the quantum particle moving in ashort-ranged potential field. For

large ti the particle with positive energy moves far away from the center and as -j oo its evolution is described by the one-parameter group Uo(t) = e-itHo which corresponds to the free motion. Mathematically this is expressed by the fact that for every cp_ E there exists a vector L' E 3° such that , - e-itHp - I I- 0> t 1i I

and such a vector is given by /' = W_cp_. Similarly, for every cp+ E MP the vector / = W+cp+ satisfies

lim ie_itx -

t->oo

0.

As a physical interpretation, orthogonality of Im W to the subspace °PP is explained by the fact that for all times t bound states are localized near the potential center, whereas the free quantum particle goes to infinity as

Iti-oo. The wave operators are used to describe the scattering of the quantum particle by the potential center. Namely, given cp_ E there is = W_cp_ E 31° such that the solution fi(t) = U(t) of the non-stationary Schrodinger equation with Hamiltonian H and initial condition b(0) _ as t - -oo approaches the solution cp_(t) = Uo(t)cp_ of the non-stationary Schrodinger equation with free Hamiltonian Ho and initial condition cp_ (0) _ cp_ . Since Im W_ = Im W+, there is cp+ E such that L' = W+cp+ and

3. Sehrodznger Equation

172

the solution /(t) of the non-stationary Schrodinger equation with Hamiltonian Hand initial condition b(0) _ as t -> oo approaches the solution cp+(t) = Up(t)p+ of the non-stationary Schrodinger equation with free Hamiltonian Ho and initial condition cp+(0) = cp+. The passage from the solution cp_ (t) of the free Schrodinger equation describing the motion of quantum particle at t = -oo to the solution cp+(t) describing its motion at t = oo is the result of scattering of the quantum particle by a potential center. All information about the scattering is contained in the scattering operator S (also called S-matrix in physics), which relates initial and final conditions cp_ and cp+:

From (2.32) and b = W_cp_ = W+cp+ we immediately get that

S = W+W_, and from (2.32)-(2.34) we obtain that the scattering operator S is a unitary operator in and commutes with the free Hamiltonian Ho: (2.35)

S*S = SS* = I and SHo = HoS. This is the outline of anon-stationary approach to scattering theory. In the stationary approach to scattering theory, the wave operators Wj. and the scattering operator S are constructed as integral operators in the coordinate representation with integral kernels expressed through special solutions of the stationary Schrodinger equation. Here we present this construction for the one-dimensional Schrodinger operator considered in the previous section.

Namely, let ul(x, k) and u2 (x, k) be solutions of the one-dimensional Schrodinger equation (2.1) given by (2.23), where /) = k > 0. It follows from Theorem 2.1 and (2.11), (2.13) that these solutions have the following asymptotics:

(k)e-}- 0(1)

(2.36)

ul(x, k) =

(2.37)

u( x, k) = e2kx + s21(k)e-ikx + 0(1)

as

x -> oo,

as

x -> -oo,

as

x -+ oo,

as

x -> -oo,

and s12(k)e' + o(l)

(2.38)

u2(x,k) = e-ikx +

(2.39)

ua(x, k) - S22(k)e-ikx + 0(1)

where (2.40)

s11(k) = s22(k) -

ak

,

s12(k) =

a

s21(k) _

-

Solutions ul (x, k) and u2(x, k) are called scattering solutions, the function sll(k) is called the complex transmission coefficient, and functions s12(k) and

2. One-dimensional Schrodinger equation

173

s21(k), respectively, are called right and left complex reflection coefficients.

Correspondingly, T = sll(k)12 is called the transmission coefficient and -the refection coefficient. The 2 x 2 matrix R= Csll(k)

s12(k)l

521ck

is called the scattering matrix. It follows from (2.14) that the scattering matrix is unitary, S*(k)S(k) = I, where I is the 2 x 2 identity matrix, and satisfies S(k) = S(-k). In particular, unitarity of the scattering matrix implies that T + R = 1, which is called conservation of probability. The physical interpretation of the solutions ul(x, k) and u2(x, k) is the following. For simplicity, suppose that the potential V (x) is smooth and vanishes for xI > A, so that asymptotics (2.36) and (2.38) become equalities for x > A, and asymptotics (2.37) and (2.39) become equalities for x < -A. In this case elements szj (/c) of the scattering matrix are smooth functions fork # 0. As in Section 2.3 of Chapter 2, let cpl(k) and c02(k) be the wave packets: smooth functions on (0, oo) supported in some neighborhood Uo of ko > 0. The functions, called scattering waves, Oa

b(x,t)

=

cps (k)uj (x, k)e-

2

tdk,

= 1, 2,

are solutions of the time-dependent Schrodinger equation i

t

= Hb.

In the region x > A the solution '/1(x, t) can be simplified as follows:

bi(x,t)

f

00

1(X, t) _ f

when x > A,

(k)eZkx-Zk2tdk

Pi (k)su

(k)e-2)dk, when x < -A.

1(k) (ei-2t + S21

Using the method of stationary phase (see Section 2.3 in Chapter 2) we get

as t -> -oo,

1(x,t) =

when x > A,

i(x,t) = llt

i

when x <-A,

()e 4t +Za +O(1t1-1),

and as t -f 00,

h(x,t) = i(x,t) =

1

1

t

)sU

x --

2t

ix2 _ iir

e 4t

4+

4 x+01 x() 21 2

2t

s

e 4t

2t

_

D(t

),

when x > A,

when x C -A.

3. Schrodinger Equation

174

Assuming that the neighborhood Uo is "sufficiently small", we see as in Section 2.3 of Chapter 2 that as t - -oo the solution ',b1 (x, t) represents a plane wave with amplitude koi(ko) moving from x = -oo to the right, toward the potential center, with velocity v = 2ko. When t --> oo, the solution 1(x, t) is a superposition of two plane waves: the transmitted wave with times the original amplitude, located to the right of the amplitude I potential center and moving toward oo with velocity v, and the reflected wave with amplitude factor located to the left of the potential center and moving toward -oo with velocity -v. Corresponding transmission and reflection coefficients are T= ii (ko)J2 and R= IAnalogously,

we have as t - -oo,

72 (-) e at

2(x,t) = ltl

+O(1t1-1),

when x<-A,

e/>2(x,t) =

andast - oo,

2(x,t) _ '2(X, t) _

when x > A,

1

1

7

x ()s12 ()ex 4t - 4 i+O(t-1), x

when x > A,

x

P2 (-) s22 (-) e at - 4 -}- O (t-1), when x < -A.

As t - -oo, the solution '2(X, t) is a plane wave with amplitude cp2(ko) moving from x = oo to the left, toward the potential center, with velocity -v. When t - oo, the solution b2(x, t) is a superposition of two plane waves: the transmitted wave with the same amplitude factor located to the left of the potential center and moving toward -oo with velocity -v, and the reflected wave with amplitude factor s12(ko)I, located to the right of the potential center and moving toward oo with velocity v. Formulas (2.40) show that transmission and reflection coefficients for the solution /2(x, t) are the same as for the solution %1(x, t), and are independent of the direction of travel. This is the so-called reciprocity property of the transmission coefficient.

The solutions ul(x, k) and u2(x, k) of the stationary Schrodinger equation have the property that corresponding solutions 4'1(x, t) and b2 (x, t) of the time-dependent Schrodinger equation become the plane waves as t -* -oo. In accordance with this interpretation, we denote them, respectively, by u( x, k) and u( x, k). Solutions of the stationary Schrodinger equation,

ul+)(x k)

= u2(x, k) = u2(x, -k) and u2+)(x k) = ui(x, k) = ui(x, -k), admit a similar interpretation: they correspond to the solutions of the timedependent Schrodinger equation which become the plane waves as t -3 oo.

2. One-dimensional Schrodinger equation

175

Introducing U+(x, k) =

u2

k) (x, k)

and

(u(x, )

U_ (x, k) =

u2- (x, k)

(4+),

we get by a straightforward computation using (2.11) and (2.13) that (2.41)

U+(x,k) = S(k)U_(x,k).

The following result establishes equivalence between stationary and nonstationary approaches to scattering theory.

Theorem 2.5. For the one-dimensional Schrodinger operator H = Ho + V with potential V (x) satisfying condition (2.2) the wave operators exist and are given explicitly by TIT VV±

Here l4

/0

1(0.

--> fjo are the integral operators given by

1f

=

(x)Ut(x,

)dx,

and o :

-+ 5o establishes the unitary equivalence between the operator H0 in 3f and the multiplication by ) operator in Sao, 1

f(x) e-ifx e2 °°

dx.

The scattering operator S in 3f and the multiplication by the scattering matrix S(y5) operator in 5o are unitarily equivalent:

s = o s(V5)P/o. Proof. The formula for the scattering operator immediately follows from the definition of the operators Gllt, formula (2.41), and orthogonality relation ill Il* = Io (see Theorem 2.3). To prove that W± _ it is sufficient

_

to show that for all

P2

E fJo, where cpl(k) and p2(k) are smooth

functions on (0, oo) with compact support,

lim l _ e-itHoIo )J1II - 0 Indeed, setting X(t) _ (e-ztx94* _ e-itxocYio )I we have r

1

1

J0

°OtJ ((u1 (x, l£) -eikx

+(u2f)(x k) - e-ikx)P2(k))e-Zk2tdk

Ias 3. Schrodinger Equation

176

and it is not difficult to estimate the integral (2.42)

x (t)112 =

f

t - foo. Namely, it follows from the Riemann-Lebesgue lemma that for any fixed A > 0 the contribution to (2.42) from the interval [-A, A] goes to 0 as t -* boo. In integrals over -oo < x < -A and A < x < oo we replace ui (x, k) by their asymptotics (2.36)-(2.39). Using estimates in Theorem 2.1, part (v), it is easy to show that for sufficiently large A the difference can be made arbitrarily small uniformly in t. To finish the proof it remains to show that for every continuous function c,o(k) on (0, oo) with compact support the integrals

J( t)

=

2

rf

(k)e_2tdk

dx

JA

a nd

J (t) =

1

rA

2 I J o0

f

2

00

dx

vanish as t -> foo. Consider, for instance, the integral Ji+)(t). Given e > 0, there exists a smooth function i(k) on (0, oo) with compact support [a, Q] such that

ko-?7IL2o <. We have by the Plancherel theorem, 2

J(t) = 2 Ja f(((k)

_ikx-ik tdk dx

1

f

2

00

DO

dx + 00

f

dx 00

((k) -

f

2

00

(k)e_th2tdk

2

2

00 r1(k)e-7-t2tdk

dx.

To estimate the remaining integral we use integration by parts to obtain 00

00

i(x + 2kt) and since

x+2kt

00

c for x,t > 0, we have - x+2at 00

00

dx c CZ A

0

dk,

e

dx

A (x+2t)2

_

C2

A + tat

dx

2. One-dimensional Schrodinger equation

177

2

Choosing t >

2C , we obtain J( t) <E. Other integrals are treated simi-

larly.

Corollary 2.6. The wave operators satisfy orthogonality and completeness relations

I - P.

I and

Proof. The result of Theorem 2.3, proved for the operator ( = W , obviously holds for the operator +. Thus relations (2.32) and (2.33) follow from the corresponding orthogonality and completeness relations for the operators Qi'+.

Remark. The orthogonality relation (2.32) is trivial once the existence of the wave operators is established. Thus Theorem 2.5 gives another proof of the orthogonality relation in Theorem 2.3. Problem 2.7 (The Cook's criterion). In abstract scattering theory, prove that the wave operators W exist if for all cp E

L

IIVe_ttH0,oMdt < oo.

Problem 2.8. Find the scattering matrix 8(k) for the potential V(x) from Problem 2.5 and show that when Vo = 2x2, the scattering matrix is diagonal.

Problem 2.9 (Quantum tunneling). Find the scattering matrix 8(k) for the rectangular potential barrier: V(x) = o for x <0 and x> 2a, and V(x) = Vo > o for o C x C 2a. Show that when E varies from o to Vo, T increases from D to the penetration of a potential barrier by a quantum particle. (1 + V0a2 )-1

2.4. Other boundary conditions. Here we consider two examples of the Schrodinger operator 2

H= - x2 + V (x) with the potential V (x) having different asymptotics as x - boo.

Example 2.1. Suppose that the potential V (x) satisfies

j (1+xftV(x)-c2Idx 0. The Schrodinger operator H has negative discrete spectrum, G )n < 0, and abconsisting of finitely many simple eigenvalues )1 < solutely continuous spectrum [0, oo), which is simple for 0 < A < c2 and is of multiplicity two for A > c2. This qualitative structure of the spectrum is determined by the structure of the level sets H0(p, x) _ A of the classix) = p2 + V (x): the eigenvalues could only cal Hamiltonian function appear for compact level sets, when the classical motion is periodic, while the values of A with non-compact level sets, where the classical motion is

3. Schrodinger Equation

178

infinite, belong to the absolutely continuous spectrum. The absolutely con-

tinuous spectrum has multiplicity one or two depending on whether the corresponding classical motion is unbounded in one or in both directions, i.e., when 0c2. These results can be proved using methods in Section 2.1. Namely, set A = k2 and define the function k1 =2 --- c2 by the condition that Im k1 o for Im k o. In particular, sgn kl = sgn k for k real, Iki > c. The differential equation

-y" + V(x)y = k2y

(2.43)

for real k has two linear independent solutions fi(x, k), fi(x, -k) = fi(x, k), where fl (x, k) has the asymptotics

fi(x,k)=e+o(1)

as

x -> oo.

For real k, Ik > c, equation (2.43) also has two linear independent solutions f2(x, Ic), f2(x, -Ic) = f2(x, -k), where f2(x, k) has the asymptotics

f2(x,k)=e_1x+o(1)

as

x

-oo.

(Actually, these solutions satisfy estimates similar to that in Theorem 2.1.) As in Section 2.1, for real k, II > c, we have f2(x, Ic) =

a(k)fi(x, -Ic) + b(Ic)fi(x, Ic),

where

(2.44) a(k) =

1 W (fi(x,k),f2(x,k)), b(k) = 1

2ik W (.f2(x, k)> fi(x, -Ic)),

2iIc

and a(-k) = a(k), b(-k) = b(k). However, W(f2(x,k), f2(x,-k)) _ -2ik1i so that for real Ic, Icj > c, we have

f'(x, Ic) =

a(k)f2(x, -Ic) -

b(-k)f2(x, Ic).

From here we obtain the normalization condition

- Jb(k)12 =

,

k > c.

For fixed x solutions fi(x, k) and f2(x, k) can be analytically continued to the upper half-plane Imk > 0. For -c < k < c solution f2(x, k) is realvalued and satisfies f2(x, Ic) =

a(Ic)fi(x, -Ic) + a(k)fi(x, -Ic),

where a(k) is still given by the same formula (2.44). The function a(k) does

not vanish for real k and admits meromorphic continuation to the upper half-plane Im k > 0 where it has finitely many pure imaginary simple zeros

ixn which correspond to the eigenvalues A1 = K,

,

An - -xn'

Z. One-dimensional Schrd dinger equation

179

The absolutely continuous spectrum of the Schrodinger operator H fills [0, oo). For 0 < A = k2 < c2 the spectrum is simple and

u(x,A) =

0< k
f2(x,k), a(k)

are the corresponding normalized eigenfunctions of the continuous spectrum. For A = k2 > c2 the spectrum has multiplicity two and

ul(x,A) = aril f2(x k) l

u2lx,kl =

1

rkl \

filx

k> c,

1

are the corresponding normalized eigenfunctions of the continuous spectrum. Denoting by L'1(x) the normalized eigenfunctions of H corresponding to the eigenvalues A1, we get the eigenfunction expansion theorem: for c% E L2(I[8), n (x)

fC

=

C1t(x) +

+ f(Ci(k)ui(x,

where C1 _ (,

C(k)u(x, k)dk

k) + C2(k)u2(x, k))dk,

l = 1, ... , n, and

foe

10O

(x)n(x,k)dx,

C( k)= _

J

j=1,2.

Example 2.2. Suppose that the potential V (x) grows as x -* -oo and decays as x --> oo:

f(1+xI)V(x)Idx
for all

a,

and there exists xo E I[8 such that the spectrum of the Sturm-Liouville problem

_y" + V (x)y = Ay,

-oo <x < xo,

and

y'(xp) = 0

is bounded below and discrete. The latter condition is a quantitative formulation of the property oo. Then for every real k there exists a solution f(x, Ic) of the differential equation (2.43) with the asymptotics f(x, k) = eZkx + 0(1)

as

x -> oo,

and there is a function s(k) such that u(x, Ic)

= f(x,

-Ic) + s(Ic)f(x, Ic)

3. Schrodinger Equation

180

is square integrable over (-cc, a) for any real a. The Schrodinger operator H has simple absolutely continuous spectrum [0, cc) and discrete spectrum consisting of finitely many negative eigenvalues. Functions u(x, k) are normalized eigenfunctions of the continuous spectrum and the corresponding eigenfunction expansion theorem has the following form: for every L' E L2 (R), n

(x) _

Cast (x) + f c(k)u(x,k)dk, l=1

w here C1 = (sb,l

L = 1, ... , n, and

Problem 2.10. Give an explicit form of the eigenfunction expansion theorem for the potential V(x) = ex.

Problem 2.11. Find energy levels for the Morse potential V(x) = e-2

- 2e-°e v,

a >0. Problem 2.12. Give an explicit form of the eigenfunction expansion theorem for the potential V(x) = Fx, describing the motion of a quantum particle in a homogeneous field; according to Problem 1.4, the corresponding Schrodinger operator is self-adjoint. (Hint: Solve the Schrodinger equation explicitly in the momentum representation, and express normalized eigenfunctions of the continuous spectrum in the coordinate representation in terms of Airy-Fock functions, defined in Section 6.2.)

Problem 2.13. Give an explicit form of the eigenfunction expansion theorem for the potential V(x) -- - 2 kx2, where k > o (according to Problem 1.4, the corresponding Schrodinger operator is self-adjoint) .

3. Angular momentum and SO(3) 3.1. Angular momentum operators. In Chapter 1 (see Example 1.10 in Section 1.4) for a classical particle in I[83 we introduced the angular mo-

mentum vector M = x x p with components M 1 = x2P3 - X3p2,

Mc = x3p1 - x1p3,

1VJc3 = lp2 ---- x2731

According to Example 2.1 in Section 2.6 of Chapter 1, they have the following Poisson brackets with respect to the canonical Poisson structure on T* R3 :

(3.1) {JV[cl,IV[c2} = - M 3,

{M2,M3} - -Mcl, {M3,M1} =

---

c2

The square of the angular momentum M = M + M 2 + M 3 satisfies

{M, M1} = {M, M2} = {M, M3} =0.

3. Angular momentum and S0(3)

181

If the Hamiltonian function 2

H(p,x) = f +V(x) is invariant under rotations, V(x) = V(jxI), then components of the angular momentum are the integrals of motion

{H,M1} _ {H,M2} _ {H,M3} = 0 and {H,M} = 0. This can also be verified directly using Poisson brackets (3.2)

and {Mj,xk} _ -Ejklxl,

{MCj,pk} _ -Ejklpl

z,j,k = 1,2,3,

where ejkl is a totally anti-symmetric tensor, e123 = 1. Correspondingly, in quantum mechanics the components of the angular

momentum operator M = Q x P are defined by

Ma = QsPi - QiPs, Ms = Q21's - QsP2, where Q = (QQ2, Q3) and P = (P1, P2i P3) are, respectively, coordinate and momentum operators. Since operators Qi and Pk commute for Ml = Q2P3 - Q3P2,

k, there is no ordering problem when defining quantum angular momentum operators. It follows from Heisenberg commutation relations that their quantum brackets are the same as the corresponding Poisson brackets (3.1): i

(3.3)

{Mi,M2} _ -M3i

{M2,A'13}h _ -Ml,

{M3,M1}a _ -M2.

Equivalently, (3.4)

[M2, Mss = ihMl,

[Min Mad = ihMs,

{Ms, Mid = ihM2.

The operator of the square of the total angular momentum MZ = Mi + M2 + M3 satisfies [M2, Ml] _ [M2, M2] _ [M2, M3] = 0.

Correspondingly, for the Hamiltonian operator 2

H=-+V(Q) with spherically symmetric potential V (x) = V(Ix) operators Ml, M2i M3 and, therefore, M2, are quantum integrals of motion: [H, Ml] _ [H, M2] _ [H, M3] = 0 and

[H, M2] = 0.

This can be verified directly by using quantum brackets (3.5)

{M3,Pk}h _ -ejkiPi,

{Mj,Rk}fii = -Ejklnwl,

j,k,l = 1, 2, 3,

which are the same as Poisson brackets (3.2), and follow from Heisenberg commutation relations.

3. Schrodinger Equation

182

In the coordinate representation

= L2(ll83, d3x) the operators of angular momentum are given by the following first order self-adjoint differential operators:

a - x2 Nr1= i x3 axe C

(x1

al

3x3j '

MZ - Zh

a - x3 ax

Nr3 = ihx2

a - xi axl ax2

3

C

,

al 1

They have the property that

M1b=M2b=M3b=0 for any spherically symmetric smooth function z/(x) _ 'L'(IxI). In other words, angular momentum operators act only on the angle coordinates in I[83. Namely, let

x = r sin t9cos cp,

where o <9 C ir, o C

C 27r, be the spherical coordinates in 1R3. Explicit

computation gives

a Mi = ifii (sin a + got 9 cos M2 = -iii 3 = --- i h

Thus 2

cos cp

i

.

2/1 a (sin 9 1

al

-cot 9 sin cp

-

so that the operator -

x3 = r cos t9,

x2 = r sin 9 sin gyp,

1 2M

-f-

sine 9 acp2

M2 is the spherical part f the Laplace operator

in I, 1

r2 ar

r 8r

- i2r 1

2M2

.

3.2. Representation theory of SO(3). Quantum operators of angular momentum are related with the representation theory of the rotation group 80(3) the group of 3 x 3 orthogonal matrices with determinant 1. 80(3) is a compact Lie group, isomorphic to the real projective space II81P3 as a smooth manifold. There is a Lie group isomorphism 80(3) ^SU(2)/{fI}, where SU(2) is a Lie group of 2 x 2 unitary matrices with determinant 1.

3. Angular momentum and SO(3)

183

The Lie algebra so(3) of 50(3) is athree-dimensional Lie algebra of 3 x 3 skew-symmetric matrices, with the basis

/00 1

/00 0

x1=(o o-1 0

1

,

x2=(

0

0

-1

0 o), xg = 0

/0-i 0

0

1

0

0

0

0

0

The matrices Xl, X2i X3 correspond, respectively, to the one-parameter sub-

groups of 50(3) consisting of rotations about coordinate axes in I. They satisfy commutation relations [X1, X2] = X3, [X2, X3] = Xl, [X3, Xl] = X2,

which are similar to (3.4). To establish the connection between quantum angular momentum operators and representation theory of S0(3), consider the regular representation R of 50(3) in ° = L2(]I83, d3x), defined by

(R(g)b)(x) _ L'(9-lx)>

9 E 50(3), b E .

Lemma 3.1. We have R(euhX1+u2X2+u3X3)

` e-*

Proof. Put uX = u1X1 + u2X2 + u3X3 and uM = u1M1 + u2M2 + u3M3. It follows from the Stone theorem and direct computation that R(e uX ) =

zh

u=o

j=1,2,3,

where self adjoint operators M1, M2, M3 are given by (3.6)-(3.8). Next, for fixed u1, u2, u3, consider the following one-parameter group U(t) = R(etu) of unitary operators in f. By the Stone theorem, U(t) . = where

A=i

I

dt t-o

U(t) = 1 uM.

Remark. We have M3 = ihp(X3), where p = dR is the corresponding regular representation of the Lie algebra xo(3)

in .

All irreducible unitary representations Rl of the Lie group S0(3) are finite-dimensional and are parametrized by non-negative integers l > 0. The corresponding irreducible representation pl = dRl of the Lie algebra zo (3) in 21 + 1-dimensional complex vector space U can be explicitly described as follows. Introduce Hermitian operators T3 = ipl (X ), j = 1, 2, 3, which satisfy commutation relations [Ti, T2] = iT3,

[7's, Ti] = ZT2,

[7'2, 7'3] = iTl.

3. Schrodinger Equation

1 84

The vector space V1 has an orthonormai basis {ejm}1 such that (3.10)

(Ti - iT2)ei,,,t = -

(l + m)(l - m + 1)elm-li

(3.11)

(Ti + iT2)el,,,,, _ -

(l - m)(l + m + 1)el,,,,+1,

(3.12)

T32jm = mElm.

In particular, (Ti +iT2)elj = 0, so that V is the highest weight module. The representation pl is irreducible, and we have by Schur's lemma

T2 = l(l + l)T, where Il is the identity operator in V. This can also be verified directly by using (3.10)-(3.12).

Remark. When l is ahalf-integer, i.e., l E 2 + 7G>0, and m = -l, -l + 1, ... , l -1,1, formulas (3.10)-(3.12) still define an irreducible highest weight representation pl of the Lie algebra 50(3) of dimension 21 + 1, and every irreducible n-dimensional representation of so (3) is isomorphic to the representation pl with l = n21. For half-integer 1, representations pi are not integrable: they give rise to two-valued representations of SO(3), the socalled spinor representations. However, considered as representations of the Lie algebra zu(2) s0(3), pl correspond to the irreducible unitary representations of the Lie group SU(2). It follows from the representation theory of SU(2) that for 1, l' E 27G>o, Z+z'

(3.13)

Vj®Vc a

Il-z'I

the so-called Clebsch-Gordan decomposition. In physics, it corresponds to the addition of angular momenta.

The regular representation R of SO(3) in ducible. We have (3.14)

= L2(II83, d3x) is not irre-

L2(SZ, dn) ® L2(lI8>o, r2dr),

where do is the measure on S2 induced by the Lebesgue measure on Il83. The group S0(3) acts by rotations in the first factor of the tensor product (3.14), whereas in the second factor it acts as the identity operator. Thus the problem of decomposing the regular representation R reduces to finding S0(3) invariant subspaces of the Hilbert space L2(S2, dn). The result is the orthogonal sum decomposition (3.15)

L2(S2, dn) _

1J1,

l=o

4. Two-body problem

185

where l V. Under this isomorphism, the orthonormal basis Yim in which corresponds to the basis elm in V1 is given by the normalized spherical functions 1

2?

m=--l,... ,l,

(COSi9), ezmPm l

where Pr(x) are normalized associated Legendre polynomials, 2 _m 1 x 2 dxl-m (x2 - 1)1, x < 1. 2l!

2l-}-1 I

m

(l - m)!

2

The self-adjoint operator M2 in the Hilbert space L2(S2, dn) has pure point spectrum consisting of eigenvalues h21(l + 1) of multiplicities 21 + 1, l = 0, 1, 2, ... , and the decomposition (3.15) gives its eigenfunction expansion: for every /' E L2(S2, dn), 00

f2ir fir

l

ClmYjm ,

where

Clr,,, = J 0

l=Q m=-l

J0

cp)Ym(, cp) sin 9dt9dcp.

Problem 3.1. Prove all results in this section. (Hint: See the list of references to this chapter.)

4. Two-body problem 4.1. Separation of the center of mass. Consider the Schrodinger operator for the two-body problem (see Example 2.2 in Section 2.4 in Chapter 2)

H = -i h2

112

2m1

2m2

+ V(xi - x2).

Introducing

X-

m1x1 + m2x2 m1 + m2

and x = x1 - x2,

the coordinate of the center of mass and the relative coordinate, we get

H= --x112

12

+ V (x),

mlm2

where M = ml -{- m2 is the total mass and µ = is the reduced ml -f- m2 mass. The Hamiltonian operator H can be diagonalized by the method of separation of variables. Namely, consider the following decomposition of the

two-body Hilbert space ° = L2(IE86) into the tensor product of Hilbert spaces: (4.1)

= L2(]I83, d3X) ® L2(II83, d3x).

3. Schrodinger Equation

186

The operator

H=-Z

(4.2)

acts as the identity operator in the second factor of (4.1), and the operator 2

H=

(4.3)

+ V(x)

acts as the identity operator in the first factor of (4.1). For the solutions of the eigenvalue problem

Hb= E

(4.4)

in the product form

b(X,x) = the variables are separated:

and H(x) =

HXW(X) = E1W(X)

E = El + E2. Since

(j"kx 3

7i(X1 -

k2 = 2ME1,

e

27r

the quantum two-body problem reduces to the problem of a quantum particle moving in the potential field, and is described by the Hamiltonian operator (4.3) in the Hilbert space L2(1I83, d3x).

4.2. Three-dimensional scattering theory. Here we outline the scattering theory for the Schrodinger operator H = -O + V (x) (where we put h = 1 and µ = 2) with rapidly decreasing potential. Specifically, assume that the bounded, real-valued function V (x) on Ifg3 satisfies (4.5)

V(x) =

as

lxi - o0

for some e > 0. Then for every k e I[83 the Schrodinger equation (4.6)

-b(x) + V(x)b(x) = k2(x), k =

has two solutions u()(x, k) satisfying the following asymptotics as xi - oo: fikr (4.7) (x k) = e2kx + f (k,w,n) e r

where k = kw, x = rn. Asymptotics (4.7) are called Sommerfeld's radiation conditions. To prove the existence of solutions u(±) (x, k), one should consider the following integral equations: (4.8)

u(t)(x k) = eikx +

f

G(x - y,k)V(y)u(y, k)d3y, 3

4. Two-body problem

187

where

1 e+ikr 47r

r

Integral equations (4.8) are equivalent to the Schrodinger equation (4.6) with Sommerfeld's radiation conditions (4.7), and are called Lippman-Schwinger equations. Their analysis uses Fredholm theory and Kato's Theorem 1.8. Solutions u(+) (x, k) are called stationary scattering waves. They are analogous to the solutions u(x, k) for the one-dimensional case, where the vector w e S2 replaces the index j = 1, 2. The absolutely continuous spectrum of H fills [o, oc} and has a uniform infinite multiplicity, parametrized by the two-dimensional sphere Solutions u(+)(x, k) are normalized eigen-functions of the continuous spectrum. In general, the operator H has finitely many negative eigenvalues al C o of finite multiplicities ml, 1 = 1, ... , n. In terms of the operators --given by

=

(2)

a

f3

(x)u(x,k)d3x,

completeness and orthogonality relations take the form

h'±a' =I,

l? 4j

where P is the orthogonal projection on the invariant subspace for the operator H associated with the pure point spectrum. Now let

W= where 3o =

- is the inverse Fourier transform. As in Section 2.3, one

can prove the following result.

Theorem 4.1. The operators W are wave operators for the Schrodinger operator H. The corresponding scattering operator S = W+W_ has the form where S is the integral operator,

_ (k,) + 27r ,2 f(k,w,w(k,w')dw', s and f(k,w,oi) = f()(k,w,w'). The function f(k, w, w') is called the scattering amplitude.

Problem 4.1. Prove that if suP

J

xEfl3 ]3

V (y) j

d3 y < 1 kr-yI

then the Lippman-Schwinger integral equations can be solved by Neumann series and for fixed x and w solutions u(±) (x, k), k = kw, admit analytic continuation to

188

3. Schrodinger Equation

the upper half-plane Im k > o. Show also that in this case the Schrodinger operator H has no eigenvalues.

Problem 4.2 (Born's approximation). Show that u(+)(x, k) = ei ' + o(1) as k --+ aa, and deduce from this that

f

n w) +

1

47r

f

x d3x = o(1)

as

k

o0 s

3

uniformly on n, w e S2.

Problem 4.3. Prove unitarily of the operator S' using Schrodinger equation (4.6), radiation conditions (4.7), and the Green's formula.

Problem 4.4 (The optical theorem). Show that f (k, n, w) Zdn

L2

Im f (k, w, w).

(Here the left-hand side is the total cross-section in direction w at the energy

E-k2)

Problem 4.5. Let ) = -x <0 be the eigenvalues of H with multiplicities ml, l = 1, ... , n. Prove that for fixed x and w solutions u(+) (x, k) admit a meromorphic

continuation to the upper half-plane Im k > o with poles of orders ml at ix1,

l = 1,...,n.

Problem 4.6. Prove that the wave operators W± exist using the non-stationary approach. (Hint: Show that when V(x) satisfies (4.5), Cook's criterion, formulated in Problem 2.7, is applicable.)

4.3. Particle in a central potential. The eigenvalue problem for the Schrodinger operator 2

H= --/X+V(x)

µ simplifies, when H commutes with the 50(3)-action in

° = L2(1[83, d3x):

[H,T(g)J=0 for all g E SO(3). This reduces to the conditions (4.9)

[H,M]=0, i=1,2,3,

and is equivalent to the property that the potential V is spherically symmetric, V (x) = V (r), r = In particular [H, Mss _ [H, M2] = 0,

where M2 = Mi + M2 + M, and operators M3 and M2 are commuting quantum integrals of motion for the Hamiltonian H. As follows from results in Section 3.2, one can look for the solutions of the eigenvalue problem

H'% = Eb

4. Two-body problem

189

satisfying

M3 = m and M2b_ h21(l + 1)b, m = -l, ... ,1. Using (3.9), we get

H=--Ir -1++V(r), so that in accordance with the decompositions (3.14) and (3.15), we look for the solutions in the form b(x) = Rc(r)Ym(n), x = rn, where Y,,,, are normalized spherical functions. Separating the variables, one gets the following ordinary differential equation for the function Rl(r): h2

h21ulr2

d

2µr2 dr (r2 ddrl) +

1)

Rl + V (r)Rl = ERI.

Introducing

f1(r) = rRl(r), we obtain the so-called radial Schrodinger equation (4.10)

-2µ

dr2

+22µr2

+V(r)ft = Eft

Since for the continuous potential V(x) the solution b(x) is also continuous, equation (4.10) should be supplemented by the boundary condition fl(0) _ o.

The radial Schrodinger equation looks similar to the Schrodinger equation for aone-dimensional particle, if one introduces the so-called effective potential Veff'(r) = V(r) +

h21 (l -f 1)

2µr2

'

where the second term is called the centrifugal energy. However, since fi is defined only for r > 0, equation (4.10) is equivalent to equation (2.1) with the potential satisfying V (x) = oo for x < 0, which describes the infinite potential barrier at x = 0. Since the radial Schrodinger operators 2

2 (

2

Hi=----FV(r)+ 2µ dr2

)

2µr2

are obtained from the three-dimensional Schrodinger operator z

H= -+V(r) µ by separation of variables, the operators Hl with boundary condition fi(O) _ 0 are self-adjoint in L2(0, oo) whenever H is aself-adjoint operator in L2(1t83).

3. Schrddinger Equation

190

In particular, it follows from Theorem 1.9 in Section 1.2 that when the bounded potential V satisfies

V(r) = O(r) as

r -f 00 for some e > 0, then H1 are self-adjoint operators with simple absolutely continuous spectrum filling [0, oo), and negative eigenvalues with possible accumulation point at 0. If V(r)

= O(r2) as

r

- oc,

then the operators Hl have only finitely many negative eigenvalues, and for l large enough have no eigenvalues at all. The same conclusion holds if (4.11)

< o.

J

Such potentials V (r) are called short-range potentials. Decaying at infinity potentials which do not satisfy (4.11) are called long-range potentials.

For ashort-range potential V (r) the differential equation (4.10) for

E

0 has two linearly independent solutions fi(r) satisfying the following asymptotics as r -> oo: (4.12)

where x =

f(r) = e(1 + 0(1)), -2µE is pure imaginary, Im x > 0 for E > 0, and x > 0

for E < 0. When r -k 0, the most singular term in the radial Schrodinger equation is given by the centrifugal energy. Since the elementary differential equation

.f -

l(l

r

21).f

has two linearly independent solutions r-l and rl+l, the solution f1(r) satisfying the boundary condition fj(0) = 0 has the asymptotics (4.13)

fl(r) = Cr' l+-}- 0(1) as r --* 0,

and is uniquely determined (up to a constant). Since

.fl(r) = clf+(r) + c2.fl (r), for some constants Cl and C2 depending on E, the differential equation (4.10) has no square-integrable solutions for E> 0. The corresponding solution fl(r) is bounded on [0, oo), and is an eigenfunction of the continuous spectrum. This agrees with the description in Section 1.6 of Chapter 1, since

for E > 0 the classical particle in the central potential Veff(r) goes to infinity with finite velocity. For E < 0 the equation Cl (E) = 0 determines the eigenvalues of Hi, which are simple. This also agrees with the classical picture, since classical motion is finite for E < 0. For a short range potential the equation Cl (E) = 0 has only finitely many solutions. When

4. Two-body problem

191

Vee(r) > 0 for r > 0

the case of repulsive potential

the operator Hl

has no eigenvalues.

Remark. When V (r) = 0, the radial Schrodinger operator Hl has only simple absolutely continuous spectrum filling [0, oo). The substitution

fj(r)=J(),

k = Ix =

and

0,

reduces differential equation (4.10) to the Bessel equation

dJ

Zd2J

2

-v)J=O 2

of the half-integer order v = l + 2. The corresponding solution regular at 0 the Bessel function of the first kind is given explicitly by z

+( 1 d

2

l sin e

and (4.14)

as

It can be shown that fEl(T)

krl

/kr - V fii J1+2

satisfy the normalization condition (2.14) in Section 2.2 of Chapter 2: lim

(4.15)

where k =

1

f0o /

J

I

JI

2µE and dQ(E) =

k+O

2

fEl(r)dU(E)) dr = 1, 2E dE = dk. The corresponding eigen-

function expansion theorem is the special case of the classical Fourier-Bessel

transform for integer 1, which generalizes the Fourier sine transform: for every f E L2(0,oo),

f(r) = f cc(E)fEl(r)do(E),

cc(E)

0

=

f

f(r)fEZ(r)dr.

0

In general, for every l = 0, 1, ... , denote by fEl(r) the solution of the radial Schrodinger equation (4.10) with the following asymptotics: (4.16)

sin(

fEl(r)= V

-

l

+o(1) as r - oo.

The function Sl(k), k = 2µE, is called the phase shift. It follows from (4.14) that for the free case Si(k) = 0. The eigenfunctions of the continuous spectrum fEj(r) satisfy the same normalization condition (4.15) as the

3. Schrodinger Equation

192

eigenfunctions for the free case. The function S1(k) = e2ib1(") plays the role

of the scattering matrix for the radial Schrodinger operator Hi. It admits a meromorphic continuation to the upper half-plane Im k > 0 with simple poles k = iyckl = i -2µEkl, where Ekl are the eigenvalues of Hl. Let Eol < E21 < < Ep,1-11 < 0 be the eigenvalues of Hl, and let fai(r), j = 0, ... , Nl - 1, be the corresponding normalized eigenfunctions. By the oscillation theorem, the eigenfunctions f3l(r) have j simple zeros in (0, oo). The functions

jlm(x) =

fi(r) I m(fl), m = -l, .

. . ,

1,

are normalized eigenfunctions of the Schrodinger operator H with eigenvalues E1, , and the functions

= -1,... Elm(X) = IEI(T)Ylm(fl)m r are normalized eigenfunctions of the continuous spectrum. The corresponding eigenfunction expansion theorem for the Schrodinger operator H with spherically symmetric potential states that for every /' E LZ(IlS3, d3x), 00

(x)

l

oo

00

- 1=0 m=-l f Clm(E

l

N1-1

c71m4'jlm(x),

l=0 m=-l j=0

where

J3

C7lm -

(x)Y'jlm(x)Cl3x.

3

Remark. In physics, it is traditional to call the parameter j in the eigenfunction aim (x) the radial quantum number and denote it by nr. The parameter 1 is called the azimuthal quantum number, and the parameter m the magnetic quantum number. This terminology originated from the old quantum theory, where to each value Ekl there corresponds a classical orbit. The parameter n n,- + 1 + 1 is called the principal quantum number, so that n,, = n -1--1 is always the number of zeros of the corresponding radial eigenfunction f3l(r).

Remark. In general, the eigenvalues of a Schrodinger operator H with spherically symmetric potential have multiplicity 21 + 1. For special potentials, due to the extra symmetry of a problem, there may be "accidental degeneracy" with respect to the azimuthal quantum number 1. This is the case for the Schrodinger operator of the hydrogen atom, considered in the next section. Problem 4.7. Prove all results stated in this section. (Hint: See the list of references to this chapter.)

5. Hydrogen atom and SO(4)

193

Problem 4.8. Find the energy levels of a particle with angular momentum l = 0 in the centrally symmetric potential well V(r) _ -Vo C 0 when 0 C r C a and

V(r) =0when r>a. Problem 4.9. Find the spectrum of the Schrodinger operator with the potential V(r) - ar-2 + br2, a, b> 0. Problem 4.10. Prove that the scattering wave u(x, k) = u(+)(x, k) in the central potential is given by

u(x, k) _ r 1/ 2

(2l + l=0

where x k = kr cos t9 and, as in Section 4.2, It = 1 and

=2.

Problem 4.11. Using the result of the previous problem, show that 00 1

f(k,n,w) _ .f(/c,cost9) = 2k (21 + 1)(SL- l)Pj(cos?v) 1=0

the partial wave decomposition of the scattering amplitude lowing formula for the total cross-section

and get the fol-

°O

c7t0t (k) =

2

>(2l + 1) sine 81(k). 1=o

5. Hydrogen atom and SO(4) The hydrogen atom is described by the long-range potential

V(r)=- ar - the Coulomb potential, where a > 03. The corresponding eigenvalue problem for the Schrodinger operator with Coulomb potential is called the Coulomb problem. Here we present its exact solution using the so-called Coulomb units It = 1, = 1, and a = 14 .

5.1. Discrete spectrum. To determine the discrete spectrum, it is convenient to put 2E _ -yc2 <0, so that the eigenvalue equation (4.10) becomes (5.1)

ii+

C2

-r

_ l(l + 1) r2

- x J1 .fi = 0. 2

Asymptotics (4.12)-(4.13) for the short-range potentials suggest that for the long-range Coulomb potential one can look for asquare-integrable solution in the following form:

ft(r) =

rl+ie-"-A(r)

3For the hydrogen atom cx = e2, where e is the electron charge. The case a = Zee, where Z = 2, 3, ... , corresponds to the hydrogen ions. 4When a = e2, Coulomb units coincide with atomic units.

194

3. Schrodinger Equation

Substituting it into (5.1), we get the equation (5.2)

A'± (2(l ±1)

- 2x)Ai

-{-

2yc(i+ 1))11i

(2_

-0

which can be solved by the power series

Ai(r) _

(5.3)

akrk. k=D

Substituting these power series into (5.2) yields the following recurrence relation for the coefficients ak:

2x(k±l+l)-2

k = 1, 2, ... ,

ak+i - (k+ 1) (k + 2l +2) ak,

where ao L 0. The power series (5.3) converges for all r > 0 by the ratio test. When x > 0 is such that ak 0 for all k, we have lim

(k±1)ak+1 = 2x CLk

Thus for every e > 0 there is N such that fork > N all ak are of the same sign, say ak > 0, and 2x --ak _ k+1

ak+1

.

Then for k > N we get

ak > C

(2x - )k

with some constant C> 0 (depending on ), and since for fixed N the sum of the first N terms in (5.3) grows like rN as r --> oo, for r large enough we obtains Ai (r) > Ce(2x-e}r _ C1rN > C2e(2-e}r.

This proves that for such values of

the function Ii (r) is not square-

integrable on (0, oo ). However, for the special values 1

the power series (5.3) terminates: Al (r) becomes a polynomial Akl (r) of order

k and fkl(r) =

E L2(0, oo). Setting n = k+l+l, we get an

explicit formula for the eigenvalues 1

En

2n2'

n = 1, 2, ... .

5It can be shown that 111(r) = Ce2 T (1 + o(1)) as r -* oo.

5. Hydrogen atom and SO(4)

195

For fixed n the integer l varies from 0 to n - 1, and for each l there are

fr)

21+ 1 orthogonal eigenfunctions Y,,,, (n) with the eigenvalue E. Thus r the overall multiplicity of the eigenvalue En is n-1

(21 + 1) = c=o

The eigenfunctions fkl(r) of the radial Schrodinger equation (5.1) corresponding to the eigenvalue En can be expressed in terms of classical Laguerre polynomials. Namely, for yc = 1 the substitution x = 2r reduces the n n differential equation (5.2) to

xQ"+(p+ I -x)Q'+kQ =0, where p = 21+1 and Q(x) = Al (r). This is the differential equation satisfied by the associated Laguerre polynomials Q(x), defined bys k

Q(x) _

e-xxk+p

which are polynomials of order k with leading coefficient (-1)', having k zeros in (0, oo). For every p the associated Laguerre polynomials {Q(x)}0 are orthogonal polynomials on (0, oo) with respect to the measure e-xxpdx, and have the property 00

L

e-xxp+iQk(x)2dx = k!(k + p)!(2k +p + 1).

The orthonormal eigenfunctions of the Schrodinger operator H for the hydrogen atom corresponding to the eigenvalue En have the form (5.4)

/)k1m()/c,(1), (-'-)

/

L

where n = k + l + 1, l = o, ... , n - 1, and m = - 1, ... , l

nl

l

e .

Remark. Returning to the physical units a = e2, where e is the electron charge, we get for the energy levels of the hydrogen atom

e4 En L?"Lr

2/t' 2

where ,u is the reduced mass of the electron and the nucleus. In particular, the

ground state energy is E1 = -13.6 eV; its absolute value is the ionization energy the energy required to remove the electron from the hydrogen atom. Using Bohr's formula for the frequencies of the spectral lines hWmn = En -- Em, 6 Polynomials L(x) = (_1)m

' Q"

(x) are also being used.

3. Schrodinger Equation

196

we get for the hydrogen atom µe4 wrr,,n

1

1

m2

m C n.

When n = 1 and m = 2, 3, ... , we get classical Lyman series, n = 2 and m = 3, 4, ... give Balmer series, and n = 3 and m -= 4, 5, ... give Paschen series. These series of spectral lines were discovered experimentally long before the formulation of quantum mechanics. Remark. The energy levels of the hydrogen atom can also be determined from Bohr-Wilson-Sommerfeld quantization rules. Namely, in spherical coordinates (r, 9, cp) in Ills the Lagrangian of Kepler's problem takes the form (see Section 1.6 in Chapter 1)

a r

L = 2µ(r2 + r2t92 -}- r2 sin2 z9 cp2) + -,

and the corresponding generalized momenta are

Pr = ,ii',

p =µr2 sine z9 cP

The BWS quantization conditions (5.5)

ycrdr=2irn(k+),

(5.6) (5.7)

where integration goes over the closed orbit of the Kepler problem with the energy E G 0, exactly determine the energy levels En, n = k + l + 1. Indeed, we have p, = Mc3i where M _ (Mcl, is the classical angular momentum. It is constant along the orbit, so that (5.7) determines the eigenvalues of operator M3 (see Section 3.2). To evaluate fpd9, we use polar coordinates (r, x) in the orbit plane P (see Section 1.6 in Chapter 1). Since X2 = t92+sin2 9cp2, along the orbit we get pxdX = where px - µrZX = IMI. Condition (5.6) now follows from (5.8)

pXdx =

2),

which is the quantization rule for the square of total angular momentum7. Finally, to evaluate f prdr, we use equations (1.7) and (1.11) in Section 1.6 71t gives the quantized values h2 (Z + ) 2 , which for large I agree well with the eigenvalues h21(1 + 1) of the operator M2.

5. Hydrogen atom and 50(4)

197

of Chapter 1 and get

/

M

dr -

prdr = µrdr =

l l

2

2

dx = pX

2

o

(1 + el

X)ZdX,

where 0 < e < 1 is the eccentricity of the orbit. We have 127r

e2 sine x

pr dr =

X

p(1+ecosX)2_2

pX

(

1 - e2

)

as can be shown by using the substitution z = eiX and the Cauchy residue theorem8. It follows from equation (1.12) in Section 1.6 of Chapter 1 that

a

1-e2= MC! /[ET and we obtain from (5.5) and (5.8), µa2

El _ -En =

,

n=k+1+1.

2n2h2

5.2. Continuous spectrum. Set 2E = k2, where k > o, and consider the equation C2

.fig + r

(5.9)

_ 1(1+1)

+k2J1

r2

,fi =0.

Substitution rl+ie-ikrFl (r)

f1(r) = reduces (5.9) to (5.10)

Fi' + (2(l ±1)

r

- 2ik) Fl +

(2

2ik(l -{- 1))

r

Fi-o.

The solution of equation (5.10) satisfying F1(0) = 1 can be explicitly written as

F, (rl - F (1 -4- 7 -L- ia 91 -!- 9 9 krl Z

>

>

'

)=

1 is

where F(a, y, z) is a confluent hypergeometric function, defined by the absolutely convergent series Oa

F(c,7,z)

=

r(cx -I- n)r(y )zn

T(a)F( + n) n!

for all a and y

o, --1, -2..... The confluent hypergeometric function is an entire function of the variable z and satisfies the differential equation

zF"+(y-z)F'-aF=o. $ This integral was evaluated by Sommerfeld in 1916.

3. Schrodinger Equation

198

For a = -k and -y = p + 1, where k, p = 0, 1, 2, ... , the confluent hypergeometric function reduces to associated Laguerre polynomials p

(

P).

F(-k,p+1,x),

considered in the previous section. For Re ry > Re a > 0 the function F(a, ry, z) admits the integral representation i

- t)-a-leztdt f obtained by the Laplace method. The confluent hypergeometric function (5.11)

F'(a,'Y, z) =

F(a)F(7- a)

to-1(1

satisfies the functional equation (5.12)

F(a, ry, z) = ezF(ry - a, ry, -z)

and has the following asymptotics as z - oo: (5.13)

(i + o(z-1))

F(a> 7, z) =

+O(z-1)).

T(a)

ri+le-ikrF(l It follows from (5.12) that f1(r) = + 1 + iA, 21 + 2, 2ikr) is real-valued and it follows from (5.13) that the function

(5.14)

IE1(r) =

(2k)1'

/(2l + 1)!

e2 P(l +1-

has the following asymptotics as r -k oo: (5.15)

fEl(r)=

sin (kr + Alog 2kr - l2

-I-

Si) ,

where (5.16)

Sl(k) = argl'(l + 1 - iA),

A

=

ZE' is the phase shift. The partial S-matrix for the Coulomb problem is = F(l + 1 - iA) = 81(k)

P(l+l+iA)It

admits meromorphic continuation to the upper half-plane Im k > 0 and has simple poles at k = ix31, which correspond to the eigenvalues Eki of the radial Schrodinger operator H1. The radial eigenfunctions of the continuous spectrum (5.14) satisfy normalization condition (4.15).

Remark. It is instructive to compare asymptotics (5.15) for the Coulomb potential, which is long-range, with the corresponding formula (4.16) for the general short-range potential. The long range nature of the Coulomb interaction manifests itself by the extra logarithmic term A log 2kr in (5.15).

5. Hydrogen atom and SO(4)

lgg

The eigenfunction expansion theorem for the Schrodinger operator H of the hydrogen atom has the same form as in Section 4.3: the eigenfunctions kl,,,,(x) are given by (5.4) where k = 0, 1, ... , Nl = oo, and the eigenfunctions of the continuous spectrum are given by

fE1(r)() 1=0,1,..., m = -l, ... ,1.

E

5.3. Hidden 50(4) symmetry. As we have seen in Section 1.6 of Chapter 1, classical system with the Hamiltonian function He (p, x ) =

p2 2m

- -cx , r

in addition to the angular momentum MG, has three extra integrals of motion given by the Laplace-Runge-Lenz vector

W='-m x M - cxxr c

According to Example 2.2 in Section 2.6 of Chapter 1, the integrals M and W for the Kepler problem have the Poisson brackets {Mc3,Mck} = -EjklTk[cl,

{Wc3,Mck} = -E klWcl,

{Wcj, Wck} = 2HcEjk1Mcl,

where j, Ic,1 = 1, 2, 3 and E123 = 1.

The quantum Kepler problem is the Coulomb problem. In the coordinate representation f = L 2 (R3 , d3 x ), its Hamiltonian is

P2

cx

r where r = x and we put m = 1. The quantum Laplace-Runge-Lenz vector 2

the Laplace-Runge-Lenz operator W 2

is defined by

(PxM MXP), r

or in components Wj _ -Ejkl `PkM1 + MIPk) 2

r

,

j = 1, 2, 3.

Here Qz are multiplication by xi operators, and it is always understood that there is a summation over repeated indices 1, 2, 3. Using commutation relations (3.5), we also have (5.17)

W=PxM-

r

- z P=M - xP_

r +ihP,

where the term ihP plays the role of a "quantum correction". The following result reveals the hidden symmetry of the Coulomb problem.

3. Schrodinger Equation

200

Proposition 5.1. The Schrodinger operator H of the hydrogen atom has quantum integrals of motion M and W, {H,M]

i = 1,2,3,

{H,W] = D,

satisfying W M= M W= 0 and a2 + 2HM2 + 2h2H.

W2

(5.1s)

Moreover, self-adjoint operators M and W have the following commutation relations: [Wj, Mk] =

[Mi, Mk] =

[Wj, Wk] _

Proof. We know that [H,M] = 0. To prove that W3 are quantum integrals of motion, we first compute [P2,

9i]. r

It follows from Heisenberg's

commutation relations that Q3 ,

(5.19) [

r1J

r

so that using Leibniz rule and relations r2 = xi + x2 + x3, [Ml, r] = 0, and Q3Pk - QkP3 - EjklMl, we obtain f P2,

Q'J =

Mj

L

Now using the first equation in (5.17) and (5.19), we get r

{H,Wj]

2

=L_

r

jklPkMl -

aQ - ifiiP

Q3 Ml

r

+

r Ml +

Q3

0. q

It is easy to prove the relation W M = W1M1 + W2M2 + W3M3 = 0. Indeed, it follows from the definition of M and commutativity of Pk and Qj for k l that

and using the first equation in (5.17) and commutation relations for the components of the angular momentum, we immediately get W M = 0. To prove that M W = 0, one should use the second equation in (5.17).

5. Hydrogen atom and 50(4)

201

Verification of (5.18) is more involved. We have

aQ r +

=I

WZ =

(EmnPmMn

= Ejkl(EjmnMlPkPmMn - alPk + ae

r

- ihMlPkPj) -- aEjmn

+ i a h9jPj -+ i j rnn Pj Pm Mn - Z a hF

r

r2

r aQ

r

q

PmMn

+

= a I + n2P2 + EjklEjmnMlPkPmMn - xEjklq M+rPkMl - ZCkhLPG,

9J r

.

Since operators M3 and 13 commute, the identity (a x b)2 = a2 b2 - (a. b)2 is still applicable and we get

jklEjmnMlPkPrn,Mn. _ (MxP).(PXM) = M2P2 - (M P)2 = M2P2. Using (5.19) we easily obtain (summation over repeated indices)

[,9]r

/

r

+1

r3)

r

Since [M2, r] = 0 it follows from M = Q X P that

/

1 M2

_ 2M z

PkMll = M21 + (MtPk Q r r r r + r / Putting everything together, we get (5.18). It is also not difficult to establish commutation relations between M3 and Wk. Using (3.5) and properties of ejkl, we get Ejkl

[M,Wk]

r', EkmnPmMn ---

- ihPk

r

= 2h(EjmpEkmnPpMn + EjnpEkmnPmMp) - Z tEjkl

= ih(PjMk - MkPj) -

jki

r (arl + ihPl) = i1jklWi r

+ ihPl

Finally, to establish commutation relations between components of W, we use the representations (5.20)

= PZQ - (P Q)P - aQ W = Q P2 - P(Q P) - aQ r r

.

The first formula in (5.20) follows from the first formula in (5.17) by using EjklPkMl = EjklElmnPkRmPI =

ElkjPkRkPj)

= Pk(PkQj - i3Qk) - QJP2 - 13(P Q) - 2ih13,

3. Schrod inger Equation

202

and relation P Q = Q P - 3ihI. The second formula in (5.20) follows from the first by using Heisenberg's commutation relations. Now we have jkl wk Wi] = 2ejmnWmWn

(P2Qm

=

- (P. Q)Pm -

Qrn) (QP2 - Pn(Q ' P) -

Qnl

= 2M (-P2 (Q P) + (P ' R)P2 + a (4l P) - (P' Q) a J

_ -2ihM (P2

- 2a _ r

where we have used [M, P. Q] = 0 and

= [p. Q, iJ r

This proves that r

[Wk,Wi] _

Let 33 = PH(-oo, 0)J, where PH is the projection-valued measure for the Schrodinger operator H. Since self-adjoint operators M and W commute with H, the subspace is an invariant subspace for these operators. The Schrodinger operator H is non-negative on

so that on this subspace

the operator (-2H)-1/2 is well defined. Now on o we set

j(±) = (M ± (-2H)"2W). It follows from Proposition 5.1 that self-adjoint operators the commutation relations (5.21)

J(±) J±)[ ,

7

k

]

on 3t3 satisfy

[jc+)' 4-)] =p

=

and (5.22)

= (j(-))2

= --1

(h21+

2H ) Equations (5.21) are commutation relations for the generators of the Lie algebra z0(4), which correspond to the Lie isomorphism 50(4) ^X0(3) EBX0(3), and exhibit the hidden SO(4) symmetry of the Coulomb problem! Together with (5.22), they allow us to find the energy levels pure algebraically. Namely, the eigenvalues of the operators (J(+))2 and (J(-))2 are, respectively, h211(ll + 1) and h212 (l2 + 1), and it follows from (5.22) that ll = l2 = l so that the corresponding eigenvalue of H is 4

2h2n2'

m=21+1.

Assuming that (5.23)

vi® v1, iEE Zoo

5. Hydrogen atom and SO(4)

203

where summation goes over all non-negative integer and half-integer values of 1, we get from the Clebsch-Gordan decomposition (3.13) that the multiplicity of the eigenvalue En is 21

--

dim V1®

(21+1)2

(2j+1)

j=0

To prove the orthogonal sum decomposition (5.23), one should consider the Schrodinger equation for the hydrogen atom in the momentum representation. Using spherical coordinates in I[83 and the elementary integral /'O° sin r

dr=-, 2 71

r

J

a

it is easy to show that for every b E 9'(ll83) 1

(5.24)

27r2fii

where L, _

/;(q) dsq

Rs 7 -

=

flb(x)e-d3x

1

(v1)3 as r

'

(/) is the h-dependent Fourier transforms. Let

be the

eigenfunction of H with the the eigenvalue E < 0. It follows from (5.24) that in the momentum representation the corresponding Schrodinger equation the eigenvalue equation Ei4' -takes the form

p) _ A

(5.25)

()3q

f

J3 P - 4' 2

where po = /-2pE and A = . Next, consider the homogeneous coorpo h dinates p in JR3 as coordinates of the stereograPhic projection of the unit po

sphere S3 in IR'. Namely, denoting by n the unit vector from the origin to the north pole of S3, we get for the point u e S3 corresponding to p E po

u=

p2 - po

2 p 2+ po

rz +

2po

p2+ po2

7.

For the volume form on S3 we have d52 =

(2p)3 d3(p2+p)3"

L

Using

(p2 +p)(q2 +p)

- v2 f

(2Po)2 s Since

x is the variable in the coordinate representation, here q is another variable in the

momentum representation.

3. Schradrngcr Equation

204

where v e S3 corresponds to q E I[83, and introducing po

(p2+p)2

W(u) _

1

(2Po)2

po

w e can rewrite equation (5.25) as

3

(5.27)

1t(v)

A2

'Y(u) _

(5.26)

2

2 (u)I2dc2. = J 3

dSty.

2

P 2pop0

(p) 2d3p =

Indeed, it follows from the Schrodinger equation that

gg2µ1 pgs

1f

(2I (E - V(r))I(x)2d3x,

7

and by the virial theorem (see Section 1.3) we get

If

2

d3x =

-

1

FG

1

2

pg3

f Vfr)I(x)2d3x,

where V (r) _ - a , so that

r

f3V(r)I(x)2d3x=2Ef

3

(x)2d3x=2Ef

3

(p)2d3p.

This shows that the eigenvalue problem for the Schrodinger equation is equivalent to the eigenvalue problem for the integral equation (5.26) in L2(S3, dS2). The latter is a classical problem in the theory of spherical haru E ][84, is monics. Namely, the function G(u) _ where u = the fundamental solution for the Laplace operator O in IE84, a2

a2

a2

a2

° - aui + au2 + au3 +

au4,

and equation (5.26) is the equation satisfied by the spherical harmonics on S3: the eigenfunctions of the spherical part 00 of the Laplace operator on R4, defined by

u 8u + u2 u3a 8u (3a)

o.

It is well known from the representation theory of the Lie group SO(4) that spherical harmonics are restrictions to S3 of homogeneous harmonic polynomials in I[84 of degree it - 1, it e N. Correspondingly, equation (5.26) is obtained as a limit lul -k 1 of the Green's formula for the homogeneous harmonic function of degree it - 1 in the unit ball in II84, by using the

6. Semi-classical asymp t o ti cs -1

205

property of a double layer potential. This gives A = n, which once again establishes the exact formula for the energy levels of the hydrogen atom. The isomorphism the decomposition

L2(S3, dS2) follows from (5.27), and (5.23)

from

L2(S3, dSfl LE 27G>o

of the regular representation of SO(4) into the direct sum of irreducible components. One can also obtain the explicit form (5.4) of the eigenfunctions

by using the representation theory of 50(4). We leave all these details to the interested reader. Remark. For E > 0 one should consider the subspace = PH (0, oa)3 and define j(±) = (Al ± (2H) '/2W). Instead of (5.21), these operators satisfy commutation relations of the Lie algebra 50(3,1) of the Lorentz group 50(3, 1). The problem of finding eigenfunctions of the continuous spectrum for the Coulomb problem can be solved by using harmonic analysis on the three-dimensional Lobachevsky space. Problem 5.1. Show that components of angular momentum M are generators of so (4) that correspond to the infinitesimal rotations in the subspace of j 4 with coordinates (o, p), and components of the Laplace-Runge-Lenz vector W correspond to the infinitesimal rotations in the (pops), (Po,P2), and (P0P3) planes in JR4.

6. Semi-classical asymptotics - I Here we describe the relation between classical and quantum mechanics by considering the behavior of the wave function b(q, t) -the solution of the time dependent Schrodinger equationlo

ih at

(6.1)

2

= - mod + v(q)

as h - 0. The substitution _S(q,t;h)

(q,t)

(6.2)

reduces (6.1) to the following non-linear partial differential equation: (6.3)

9S

1

2

i/I

8 + 2m (aq) + - 2m OS. It is remarkable that (6.3) differs from the Hamilton-Jacobi equation (2.4) for V(q)

2

the Hamiltonian function q) = 2m + V (q), considered in Section 2.3 of Chapter 1, only by the term in the right-hand side which is proportional to fit. Thus as fii -* 0, equations of motion in quantum mechanics turn into classical equations of motion. 10 Here it is convenient to denote Cartesian coordinates on R' by q = (q , ... , q)..

3. Schrodinger Equation

206

For the stationary state

(q,t) _ (q)e_*Et the substitution (6.2) be-

comes (6.4)

The corresponding non-linear partial differential equation 1

2m \aq J

2m

OQ

differs from the corresponding Hamilton-Jacobi equation for the abbreviated action, considered in Section 2.5 of Chapter 1, by the term proportional to h in the right-hand side. In this section we consider semi-classical asymptotics: asymptotics of

the partial differential equations (6.3) and (6.5) as fii - 0. They describe the precise relation between quantum mechanics and classical mechanics, and provide a quantitative form of the correspondence principle, discussed in Section 2 of Chapter 2. In particular, using semi-classical asymptotics for the stationary Schrodinger equation, we derive Bohr-Wilson-Sommerfeld quantization rules, postulated in Section 2.5 of Chapter 2.

6.1. Time-dependent asymptotics. Here we consider the problem of finding short-wave asymptotics - asymptotics as h -* 0 of the solution L'n(q, t) of the Cauchy problem for the Schrodinger equation (6.1) in one dimension, a

a

at - -2m aq2 + with the initial condition

= (q)e*8. It is assumed that the real-valued functions cp(q) and s(q) are smooth, s(q), cp(q) E C°O(]I8), and that the "amplitude" cp(q) has compact support. The substitution (6.2) is c/>n(q, t) = and differential equation (6.3) takes the form

as

fas12 (q) at + 2m aq J + V 1

ih aZs 2m 8q2

To determine the asymptotic behavior of S(q, t,11) as h - 0, we assume that as h -* 0 00

S(q,t,h) _ n=0

t

6. Semi-classical asymptatics - I

207

and substitute this expansion into (6.6). Comparing terms with the same powers of fii we obtain that So (q, t) satisfies the initial value problem 2

(6.7)

--+-----)

+V(q)=0,

and (6.8)

So(q,t)=0

whereas Si (q, t) satisfies the differential equation (6.9)

aSl + 1 aso asl at m aq 8q

1

82S0

2m aq2 '

the so-called transport equation, and (6.10)

Si (9,t) t=o = cP(4)

The functions Sn(q,t) for n> 1 satisfy non-homogeneous differential equations similar to (6.9). The initial value problem (6.7)-(6.8) is the Cauchy problem for the Hamilton-Jacobi equation with the Hamiltonian function 2

H (p,q) = 2m + V q) considered in Section 2.3 of Chapter 1. According to Proposition 2.1 in Section 2.3 of Chapter 1, the solution of (6.7)-(6.8) is given by the method of characteristics, (6.11)

So(q,t) = s(qo) +

f

L(ry

Here L(q, q) = 2mg2 - V (q) is the Lagrangian function, and y(r) is the characteristic -the classical trajectory which starts at qp at time T = 0 with the momentum pp = (qo), and ends at q at time T = t, where qo is uniquely determined by q. (We are assuming that the Hamiltonian phase flow gt satisfies the assumptions made in Section 2.3 of Chapter 1.) It follows from Theorem 2.7 in Section 2.3 of Chapter 1 that along the characteristic, aq0

= m dt (t),

so that (6.12)

(at + aq) s1(7(t),t) = dts1(7(t),t)

Now we can solve the Cauchy problem (6.9)-(6.10) for the transport equation explicitly. Consider the flow art IR - I[8, defined in Section 2.3 of

3. Schrodinger Equation

208

Chapter 1, and denotell by ry(Q, q; T) the characteristic connecting points q

at r = 0 and Q = 7rt(q) at r = t (under our assumptions the flow art is a diffeomorphism and the mapping q F--> Q is one to one). Differentiating the equation

Q Q,t)= m -t (Q,9; t) with respect to q we obtain 8Q 82 So aQ2

d (DQ\

Q t aq = m agat (Qq; t)- m dt a2ry

8q

so that (6.9) can be rewritten as d

aQ S,(Q,t) Id = -log, q

and using (6.10) we obtain

S,(Q,t) =

(q)

aQ

Therefore, (6.13)

h(Q, t) _

(q)

(q)2

es(Q>q;t)+3(e))(1

+ O(h)),

where S(Q, q; t) is the classical action along the characteristic that starts at

q at time r=0 and ends at Q at time T = t. The rigorous proof that (6.13) is an asymptotic expansion as h -> 0 uses the assumptions made in Section 2.3 of Chapter 1, and is left to the interested reader. Here we just mention that asymptotics (6.13) is consistent with the conservation of probability: for any Borel subset E C IEB,

L It)I2dQ = fE

(q)2dq +

t

as h -+ 0, where Et = irt(E). Remark. When assumptions in Section 2.3 of Chapter 1 are not satisfied, the situation becomes more complicated. Namely, in this case there may be several characteristics ryj (T) which end at Q at r = t having q3 as their corresponding initial points. In this case,

(q)2 e 1

(Q, t) _

(qj) aQ

22N'7 1

+ o(h)),

where jc E 7L is the Morse index of the characteristic rye. It is defined as the number of focal points of the phase curve (q(r), p(r)) with initial data 11 There should not be any confusion with the quantum coordinate operator Q.

6. Semi-classical asymptotics --- I

qj and pj _

209

q (q) with respect to the configuration space IEB. It is a special

case of a more general Maslov index.

The case of n degrees of freedom is considered similarly. Under the assumptions in Section 2.3 of Chapter 1, the solution n(q, t) of the Schrodinger equation (6.1) with the initial condition bn(9,t) It=o

cP(q)e *S(4)

=

where real-valued functions s(q) and cp(q) are smooth and cp(q) is compactly supported, has the following asymptotics as h -> 0: (6.14)

Grt(4,t) _ P(9) det

((q)) r

0(h)).

6.2. Time independent asymptotics. Here we consider semi-classical asymptotics for the one-dimensional Schrodinger equation

h2 d - 2m dx + V

(6.15)

Ezb.

This asymptotic method is also known as the WKB method (after G. Wentzel, H. Kramers, and L. Brillouin). The substitution (6.4) is bn(x) = and equation (6.5) takes the form 1

2m

(da'\2 + V (x) _ E + iii d2Q dx 2m dx2 J

As in the previous section, we use the expansion 00

(x,h) _

L(-Zh)nQn(x), n=0

and for the first two terms obtain (6.16)

ZmQO2

= E - V (x)

and

i=

- 2 vo

Let p(x) = fim(E - V(x)) be the classical momentum of a particle moving in a potential V (x) with the energy E. The solution of the first equation in (6.16) is given by

Qo = ±J p(x)dx, and from the second equation in (6.16) we get Ql = - 2 log p.

The wave function in the WKB approximation has the form (6.17)

bn(x)

-

/__51(C1e

+ c2e-* f

0(h)),

210

3. Schrodinger Equation

where p(x) is real in the classical region V (x) <E, and is pure imaginary in the classically forbidden region V (x) > E (see Section 1.5 of Chapter 1). Note that the asymptotic (6.17) is valid only if Q'2 K hQ-", which in the first approximation gives

d( h lx k\p(x)

Introducing g the classical force F =

<< 1 .

-d d ,this condition can be written as

(6.18)

m 3F
p(x) is small. In particular, it is not applicable near the turning points where E = V (x) and, therefore, p(x) = 0. Let x = a be a turning point. If Fo = F(a) 0, using the approximation E - V (x) Fp(x - a) near x = a, one can replace (6.15) by

= Fo(x-a). This differential equation can be explicitly solved by the Laplace method. Its bounded solution is (x)

where

=

(2mFo\

=

2

)

(a

-

is the Airy-Fock function, defined by the improper integral

=

V"

cos (t3 + t) Cit. 0

The large asymptotics of the Airy-Fock function, obtained by the steepest descent method, are the following: (6.19)

)=

(6.20)

)-

(1

sin

+as

4l (1+0(II 2))

as

-oo.

Using asymptotics (6.19)-(6.20) one can obtain connection formulas relating WKB approximations for classical and forbidden regions. The rigorous justification of this approach is technically rather involved and we will not

present it here. Instead we highlight the simplest case when there is only one turning point x = a with Fp > 0, so that x
ei.fa

6. Semi-classical asymptotics - I

211

whereas in the classical region x C a it is oscillating, WKB

(x) =

(Cle

fa p(s)ds + C2e- * f p(s)ds

=

B

sin

p(s)ds+a).

As follows from (6.18), the WKB approximation near the turning point

I x - a >>

3 x = a remains valid whenever , which is equivalent 2 mFo to the condition f » 1. On the other hand, when Ix -- a CC 1 the wave function has the asymptotic

n(x) = C()(1 + 0(h)}. To determine the unknown phase a and to find the relation between coeffi-

cients A, B, and C, we consider the domain 2 mFo

CC x - aj CC 1 and compare the WKB approximation for the wave function with the large asymptotics of the Airy-Fork function. Namely, for x C a we have >> 1 and

-/a 3m

-

Fo - x )3

J

a

Ip(s)Ids,

- 6 2m Fo

4

Comparing (6.19) with the WKB wave function for x < a we obtain that 2A = 6 2mhF0C. For x > a we have
2 3

2 =

2 2mFo (x 3h

1

- a) 2

f p(s)ds' x

4

/ifi

a

6 2mhFo

.

Comparing (6.20) with the WKB wave function for x < a we obtain that

a = 4 and B = 6 2mhFpC, so that B = 2A. Thus for this example the WKB wave function is

e f Ip(s)Ids

A

l

J

2A

sin( fax p(s)ds + 1

when x
r

4)

when x > a.

The case when there is only one turning point x = b with F0 C o, so that x > b is a classically forbidden region, reduces to the previous example by reversing the orientation x F-k -x. As the result, we obtain 2B

6.22j

pox}

sin 1 f x p(s)ds +

$

e

-' fb p(s}Ids

iptx} f

4 )

when x C b,

when x > b.

Problem 6.1 (Penetration through a potential barrier). Consider a potential a potential V(x) such that {x e Ilk : V(x) > E} =

barrier for a given energy E (a, b). Show that

TWKB = e* fa Ip(x)Idx

Verify directly that transmission coefficient for the potential in Problem 2.9 in the semi-classical approximation is given by this formula.

3. Schrodinger Equation

212

Problem 6.2 (Above barrier reflection). Suppose that a potential V(x) admits analytic continuation into the upper half-plane, and let E be the energy such that E > V(x) for all real x. Suppose that there is only one complex xo such that V (xo) = E. Show that in the semi-classical approximation WKH -= e_

Im fa °

where p(x) = /2m(E - V (x )} and a E I (the choice of a does not change the imaginary part of the integral in the exponent). Verify directly that the reflection coefficient for the potential in Problem 2.9 in the semi-classical approximation is given by this formula.

Problem 6.3. Find transmission and reflection coefficients for the parabolic barand verify directly that in a potential V(x) = -- 2kx2, where k > o the semi-classical approximation they satisfy, for E C o and E > o, respectively, formulas in Problems 6.1 and 6.2. rier

6.3. Bohr-Wilson-Sommerfeld quantization rules. The WKB method allows us to determine energy levels in the semi-classical approximation. Consider, for simplicity, the finite motion of aone-dimensional particle in a potential well: in a potential V (x) at the energy E such that there are two turning points a and b. The classical region is a < x < b, and the motion is periodic with the period

T=2Jbdx=2mJbdx= a

x

a

2m

P

dx Jb

a

./E-V(x)

The regions x b are forbidden (see Section 1.5 of Chapter 1). It follows from (6.21)-(6.22) that the WKB wave function exponentially decays in the forbidden regions x < a and x > b. Using (6.21) we see that in the classical region the WKB wave function has the form wKB(x) _

Asin

J p(s)ds + 4

,

while using (6.22) we obtain 'wKB(x) _

1b

Bx sin 2B

sin

p(er )

(1fbp(s)ds + 2 -

(

fX

+

These two expressions define the same function in a < x < b if and only if there is a positive integer n such that b

(6.23)

p(x)dx + 2 = (n+1)7r,

1

a

7. Notes and references

213

in which case A = (-1)'B. Equation (6.23), written in the form

f

b

2m(E - V(x)) dx = 7rh(fl +

,

determines the semi-classical energy levels. It follows from (6.21)-(5.22) that the integer n is equal to the number of zeros of the WKB wave function. In accordance with the oscillation theorem, the case n = o corresponds to the

ground state, case n = 1 to the next energy level state, etc. Denoting, as in Chapters 1 and 2, coordinate x by q, we can rewrite (6.23) as pdq = 2irfii(n + 2 ),

(6.24)

where integration goes over the closed classical orbit in the phase plane II82 with canonical coordinates p, q. Condition (6.24) is the famous BohrWilson-Sommerfeld quantization rule for the case of one degree of freedom

(see Section 2.5 of Chapter 2). We emphasize that, in general, the BWS quantization rule is applicable only for large n and gives the semi-classical asymptotic of energy levels E. However, as was shown in Section 2.6 of Chapter 2, the energy levels of the harmonic oscillator obtained by the BWS quantization rule are exact. The BWS rules are also exact for the energy levels of the hydrogen atom (see Section 5.1). This semi-classical analysis can be generalized for quantum systems with n degrees of freedom which correspond to completely integrable classical Hamiltonian systems (see Section 2.6 of Chapter 1). Bohr-WilsonSommerfeld quantization rules take the form (6.25)

'

pdq =

4 indry).

Here integration goes over all 1-cycles -y in the Lagrangian submanifold and A _ {(p,q) E 2n Hc(p,4) = E,F2(1>> 9) = E2i... , F(7>> 4) = indry is the Maslov index of a cycle ry in A. For the action-angle variables (I, cp) the integration in (6.25) goes over the basic 1-cycles on the n-torus Tn, and we obtain quantization conditions Ii = (n2 + 2)h, i = 1, ... , n.

7. Notes and references The classical text [LL58] is the source of many basic facts on the Schrodinger equation, written from the physics perspective. The textbook [Foc78J, another classic, shows meticulous attention to detail. There are numerous mathematics papers and monographs devoted to the different aspects of the Schrodinger equation, and here we mention only the sources being used. For the self-adjointness criteria in Section

1.1 we refer the reader to the encyclopaedia survey [and to the treatise

214

3. Schrodinger Equation

[RS75], and references therein; in particular, in [RS 75] one can find the proof that the Schrodinger operator for a complex atom is essentially self adjoint. For the proof of Sears theorem, see [BS91]. Theorems 1.6 and 1.7 (the latter in the special case V1 o) in Section 1.2 are proved in [B S91] and [RSS94]; see [RS 78] for the proof of Theorem 1.7 in the general case, and for other generalizations. The proof of the Kato theorem can be found in [RS78]; see also [RSS94]. For the proof of the Birman-Schwinger bound, see [RS78]. The monograph [HS96], besides the introduction to spectral theory, contains the proofs of many results in Sections 1.1-1.2; see also the monograph [CFKSO8] for these and other results. Section 2 is based on the fundamental paper [Fad64], classical surveys [Fad59, Fad74], and the monograph [Mar86], devoted to the inverse scattering problems. For more information and details on the material in Section 2.1 and Problems 2.2, 2.3, 2.4, see [Fad64] and [Mar86]. The complex integration of the resolvent kernel is a powerful method for proving the eigenfunction expansion theorem, especially in the presence of the absolutely continuous spectrum, and in Section 2.2 we followed the elegant approach from [Fad59]; Problem 2.6 is taken from [Fad74]. Section 2.3 is based on [Fad59, Fad64, Fad741; see also the monograph [NewO2] for a comprehensive exposition of the scattering theory from the physics perspective, and the recent book [Yaf92] for its abstract mathematical formulation. We refer the reader to [Fad74] and references therein for more details on the material in Section 2.4.

Section 3 is fairly standard; from a physics point of view, its material can be found in almost any textbook on quantum mechanics; see [LL58, Foc78] for a clear exposition. It was H. Weyl who introduced Lie groups into quantum mechanics [Wey5O], and nowadays representation theory of Lie groups is a part of the theoretical physics curriculum; see, e.g., [BR86]. There are also many mathematics textbooks and monographs on representation theory of compact Lie groups; see, e.g., [K1r76, FH91], as well as [Vi168]. Section 4 contains the standard material presented in any textbook on quantum mechanics; see [LL58, Foc78, Mes99] for the physics presentation. Our goal here was precise mathematical formulation of the basic facts; for the proofs of all results in this section (and of the problems as well), and for their generalizations, we refer the reader to the treatise [RS79].

The solution of the eigenvalue problem for the hydrogen atom, given by E. Schrodinger in 1926, was the major triumph of quantum mechanics. Before that, the energy levels of the hydrogen atom were obtained by N. Bohr in 1913 using the Bohr

model (later replaced by the Bohr-Sommerfeld model and the BWS quantization rules), and in 1926 by W. Pauli by using the quantum Laplace-Runge-Lenz vector. Sections 5.1 and 5.2 are standard and our exposition follows [LL58, Foc78]. We started Section 5.3 by presenting Pauli's approach (see [BR86] for more details on representation theory). The hidden So (4) symmetry of the hydrogen atom was discovered by V.A. Fock, and the end of Section 5.3 is based on [Foc78] (see

7. Notes and references

215

[B166a, BISFb] for more details and for the treatment of the absolutely continuous spectrum). There exists an extensive literature on semi-classical asymptotics and the WKB

method, which we just barely touched on in Section G. We refer to [LL58] and [Dav76] for a physics discussion and to [GS77, BW97] for a mathematical introduction. Detailed exposition of the one-dimensional WKB method can be found in [01v97] a reference book on special functions used in this chapter. We refer to

the monographs [and [for the detailed exposition of the much more complicated mufti-dimensional WKB method.

Chapter 4

Spin and Identical Particles

1. Spin 1.1. Spin operators. So far we have been tacitly assuming that the Hilbert d3x). Based on this assumption, -- L2 in Section 5.1 of Chapter 3 we have found the energy levels of the hydrogen atom. In particular, in the ground state the third component M3 of the quantum angular momentum operator M has eigenvalue o. However, the famous Stern-Gerlach experiment has shown that the electron in the hydrogen atom also has a "magnetic angular momentum", whose third component in the ground state may take two values which differ by a sign. Therefore in addition to the "mechanical angular momentum" operator .M with components M1, M2, M3, the electron also has the "internal angular momentum" operator S with components Si, 52, 53, called spin. The spin describes internal degrees of freedom of the electron and is independent of its position ® C2, in the space'. The Hilbert space of states of the electron is 3s = i.e., it has twice as many states than a spinless particle. The Hilbert space consists of two-component vector-valued functions on space for a quantum particle is

i;pwhere

=1

3

2 fix}

Esc

I3 I

217

218

4. Spin and Identical Particles

To every observable A in 3f there corresponds an observable A ® I2 in given by the 2 x 2 block-diagonal matrix ( o A observables of the form I ® S, where I is the identity operator in° and S is a self-adjoint operator in C2, describe the inner degrees of freedom and commute with all observables A ® I2. The complete set of the observables in °S consists of the operators Qi ® I2, Q2 0 I2, Q3 0 I2 and I® Si, I® 82,10 S3, where Q are coordinate operators, and S3 are spin operators traceless self-adjoint operators in C2, satisfying the same commutation relations as the quantum operators of the angular momentum, ,

),

.

[8k, 82] = ihS3,

[S2, 83] = illS1,

[S3, Si] = ilS2.

In terms of the standard basis el = (o) , e2 = (°) of C2, S = 2 Q j 1, 2, 3, where Q are the so-called Pauli matrices

ci1=10 1o)' ci2=.0

i

1

o)'

0

-i)'

which are commonly used in physics. It is also convenient to represent W E as a function 'b(x, Q) of two variables, where x e I[83 and Q takes two

values 2 and -2, by setting b(x, 2) _ jJl(x) and b(x, -2) _ c%2(x). Every function b(x, Q) is a finite linear combination of the functions (x)X(Q)

-

the decomposable elements of the tensor product L2(I[83) ® C2. Here we are identifying (C2 with the complex vector space of functions x : {-2, 2} -* (C by assigning to the standard basis el, e2 the functions Xl, X2, defined by

Xl (2) = 1, Xl (- 2) = 0 and X2() = 0, x2 (- 2) = 1. This identification is also commonly used in physics. In this notation the operator S3 becomes a multiplication by fiia operator,

S3b(x,ci) = hcib(x,ci), and

Si(x, ci) =

hjcij(x,

-ci),

S25(x,

ci) =

-ihcib(x, -a).

Here and in what follows we will always assume that spin operators act only

on the variable Q, and will often write S instead of I ® S. The operator S2 = Si + SZ + S3 is called the square of the total spin operator, and 52 = h2s(s + 1)I2, where s = Qf = 2 is a total spin. In physics terminology,

the electron has spin 2.

1.2. Spin and representation theory of SU(2). Mathematical inter-

pretation of spin is provided by the representation theory of the Lie algebra 511(2), discussed in Section 3.2 of Chapter 3. Namely, let i A2=---ciA3 =---c 3 2a

1. Spin

219

be the standard basis of the Lie algebra su(2) - 2 x 2 traceless skewHermitian matrices satisfying commutation relations [A2, A3] = A1,

[A1, A2] -= A3,

[A3, Al ] = A2,

and let pi2 be the two-dimensional (fundamental) representation of su(2) in Vi C2. The spin operators are given by Si = iItp

1, 2, 3,

and p1 is called the representation of spin 2 . In general, representation ps in the 2s + 1-dimensional complex vector space Vs is called the representation

of spin s e 2 Z>0. The action of operators T = ips (A ) in Vs is described by formulas (3.10)-(3.12) in Section 3.2 of Chapter 3 (where 1 is replaced by s). There is another explicit realization of the representation ps in the vector spaces of polynomials 1(z) of degree not greater than 2s. Namely, the mapping (s + m). (s

m} .

establishes the isomorphism between the vector space Vs with the inner product determined by the orthonormal basis esm, and the vector spaces with the inner product

(l.l)

/

1f,g)

(2s + 1)! 7r

f

f (z)g(z)

d

Tl - iT2 = z,

f g E S.

J

in S are given by

Corresponding operators T3 = i ps (1.2)

2

1 + z212s+2 d z

Tl -I- iT2 =

z2ddz

- 2sz,

d

T3 = zz - s.

It is easy to verify that Tl, T2i T3 are Hermitian operators with respect to the inner product (1.1). Representation ps of the Lie algebra 5u(2) is integrable, ps = dRs, where RS is an irreducible unitary representation of the Lie group SU(2), defined by -f- a)2s f

a)

g= I

I E SU(2), f E mss.

Remark. The spin 2 representation R in (C2 can also be explicitly described as

R2 (g)z =

/3J

(z2)

,

z = Iz2 )C2,

and is called the fundamental representation.

4. Spin and Identical Particles

220

Remark. Representation Rs of SU(2) can be lifted to a family of representations of the general unitary group U(2) acting in the same vector space mss. Namely, every g E U(2) can be written as g =ago, where a2 = det g and is determined up to a sign, and go E SU(2). Then for every integer l with the property that 2s + l is even, the formula R1,3(9') = a1Rs(9o)

defines an irreducible unitary representation of U(2) in S. The condition that 2 + s is an integer ensures that Rl,s does not depend on the choice of a sign in the definition of a. The representation R1,3 is the highest weight representation of U(2) with the dominant weights A1 = 2 + s > A2 = 2 - s, A1 + A2 = 1. We will denote the corresponding U(2)-module as Va,l, where A _ (A1, A2) is a dominant weight with A1 + A2 = 1. As an SU(2)-module, VS, where s = (A1 A2)

-

Returning to the basic principles of quantum mechanics, formulated in Section 1 of Chapter 2, we now extend the first postulate Al to the case of particles with spin.

AlO (Particles with Spin). The Hilbert space of states for a quantum particle of a spin s E {0,

,

1, 2 , ... } is

5 = L2(3) ® s where Vs is a 2s + 1-dimensional complex vector space of the irreducible representation R5 of the Lie group SU ( 2 ) of spin s. The spin operators S = (Si, S2 , S3 ) are given by Si = ihps (A ),

j = 1, 2, 3,

where A1, A2, A3 are standard generators of the Lie algebra 5u(2) and ps = dRs is the corresponding representation of Eu(2) in V3. Particles with integer spin are called bosons, and particles with half-integer spin are called fermions.

As in the case of spin 2, it will be convenient to represent 'I' E 3S as a function l(x, a) of two variables, where x E R3 and the variable a takes 2s+1 values from the set {-s, -s+1,... , s-1, s}. For a given a, the function %(x, a) is the k-th component, k = s - Q + 1, of the 2s + 1-dimensional vector Y(x). Every function b(x, a) is a finite linear combination of the functions 'cb(x)X(Q) the decomposable elements of the tensor product L2(lE3J 1 ® C2s+1 where we are identifying (C2S+1 with the complex vector space of functions x {-s, -s + 1, ... , s - 1, s} -+ (C by assigning to the elements ek of the standard basis el, ... , e23 +1 the functions xk, defined by Xk(s + 1 - k) = 1 and being 0 otherwise.

2. Charged spin particle in the magnetic field

221

The complete set of observables for a quantum particle of spin s consists of position operators Qi ® Its+1, Q2 ® I2s+1, Q3 ® I2s+1, where I28+1 is

the identity operator in VS, and spin operators I ® S1, I ® S2, I ® 83. The total angular momentum operators J = (J1, J2, .13) in s are self adj oint operators in given by

J=M®I28+1+I®S. They satisfy the same commutation relations as the spin operators, [J1, J2] = ihJ3,

[J2, J3] = ihJ1,

ihJ2.

[J3, J1]

As angular momentum operators M (see Lemma 3.1 in Section 3.2 of Chapter 3), the total angular momentum operators J are associated with the unitary representation of the Lie group SU(2) in the Hilbert space .9t = L2 (R3, d3x) ® V. Namely, let R be the unitary representation of SU(2) in = L2 (R3 , d3 x ) corresponding to the ad j oint representation of SU (2 ) in

5u(2) ^1 3,

(R(g))(x) _ b(Adg-1 x),

g E SU(2),

c% E

where x o` = x171 + X2U2 + X3cT3.

Lemma 1.1. We have (R ®Rs )(ea1Ai+a2A2+a3A3)J = e-* `Q1J1+a2J2+a3J3} ,

where a = (a1, a2, a3) E Il. Problem 1.1. Derive formulas (1.2) and show that operators T1, T2, T3 are Her-mitian with respect to the inner product (1.1).

Problem 1.2. Prove Lemma 1.1.

2. Charged spin particle in the magnetic field 2.1. Pauli Hamiltonian. The classical particle of charge e moving in the electromagnetic field with the vector potential A(x) and the scalar potential cp(x) is described by the Hamiltonian function

x) -

e

z

(P- A) + e
2m (see Problem 1.27 in Section 1.7 of Chapter 1). As we have seen in Example 2.3 in Section 2.4 of Chapter 2, the corresponding Hamiltonian operator of the spinless particle is given by (2.1)

H° =

2

2m

\P cAl + e(x).

However, quantum particles with spin interact with the magnetic field, so that the Hamiltonian H°, which is an operator in the Hilbert space _

4. Spin and Identical Particles

222

L2(II83, d3x), is not adequate when s 0. The correct prescription is to include the spin degrees of freedom and consider the Hamiltonian operator

in the Hilbert space 3S = 3' ® V. For the case of an electron -particle with spin 2

the corresponding Hamiltonian is given by

(2.2)

H

2m (PA

PA = P - A.

Q)2 + e(x),

Using the basic facts that Pauli matrices anti-commute and satisfy Qi = cr3 = I2 and 0102 = 203,

0203 = 201,

0301

we readily obtain

(PA. )2

-

(1

e

e

2

cAll + \P2 +i [Pi - Ai, P2 - A2] s + i

e

2

cA2l + \P3 [P2

e

-

2

cA3l

P3 - A3] Ql

e

+2 Pg - A3, P1 - CA1] U2

(P - eA) c

2

- ec (Blal + B20'2 + B3U3).

Here B = (B1, B2i B3) is the magnetic field, B =curl A, and in the last line we have used Heisenberg commutation relations (cf. a similar computation in Example 2.3 in Section 2.4 of Chapter 2). The Hamiltonian (2.2) takes the form

11-

(2.31

he

' 2mc' and is known as the Panli Hamiltonian; the quantity 1u is the total magnetic moment of an electron. It is customary to put ,u = -SLOB, where

= 0.927 x 102o erg me gauss is the so-called Bohr magneton. The corresponding time-dependent Schrodinger equation ,aB _ 2

2h

at = xb

in this context is called the Pauli wave equation.

Remark. The Pauli wave equation is obtained from the relativistic Dirac equation for the electron in the limit c -+ oo by neglecting the terms of order

The corresponding Hamiltonian for a quantum particle of spin s and charge e has a similar form ji_ Shs

2. Charged spin particle in the magnetic field

223

where S = (S1, S2, 53) are spin operators of spin s, quantity u is the total magnetic moment of a particle, and the operator jc is its "intrinsic" magnetic moment.

2.2. Particle in a uniform magnetic field. Here we consider the problem of finding the energy levels of a charged spin 2 particle moving in a constant uniform magnetic field. Choosing the z-axis along the direction of

the field, B = (O,O,B), we get Al(e) _ -By, A2(x) = A3(x) = 0, and cp(x) = 0. The Pauli Hamiltonian takes the form

FI -

(2.4)

(Pi + By

2m

2

c

L

1

+ PZ + P3 - µBQ3. J

The eigenvalue equation Hb(x,a)= E(x,a)

in S reduces to two (for each value of a = f 2) eigenvalue equations a

1

By) + P2 + P3 2m (Pi + c J in

- µBQ = E

'. Separating the variables

(x) = we obtain 2

(2.5)

2

- m X + m2 B y -

2

) Jo)2X = (E + µBQ 2m

X

where

yo = - eB and

I mB .

Equation (2.5) coincides with the eigenvalue problem for the one-dimensional harmonic oscillator with the frequency cvB, solved in Section 2.6 of Chapter 2. Thus we obtain 2

(2.6)

E _ (n + 2 )has + 2m - µBQ,

n=0,1,...; o= f 1

.

The first term in this formula gives discrete energy levels, called Landau levels, for the motion in the plane perpendicular to the magnetic field. The spectrum of Pauli Hamiltonian (2.4) is absolutely continuous and fills [(hWB - µB), oo) with variable infinite multiplicity. It can be visualized as a countable union of branches [(n + )hWB - µBQ, oo), each with a multiplicity parametrized by ]I8 x {f1} (all kl E ][8 and fk3 for fixed k3

224

4. Spin and Identical Particles

correspond to the same E). Corresponding normalized eigenfunctions of the continuous spectrum are

x' k 1, k3} =

1

4

7flwB

1

1x+k3z)_--2

e

2irh

(y-yo)2

In ( V

Fl

y

a

where - ao
E- (n++a)pBB+--. 2m In this case there is an extra degeneration of the spectrum: energy levels with n and a = and those with n + 1 and a = - 2 coincide. Problem 2.1. Derive the eigenfunction expansion theorem for the Hamiltonian (2.4).

Problem 2.2. Show that the operators c

X0=Q1+

eB

P2

o=- eBc Pi,

and

which correspond to the coordinates of a center of a circle (see Problem 1.5 in Section 1.3 of Chapter 1), are quantum integrals of motion, i.e., commute with the Hamiltonian H, but {Xo,Yo}h =

C

jI.

3. System of identical particles Consider the quantum system of N particles of spins si,... , 8N. According to the basic principles of quantum mechanics, the Hilbert space of states of the system is given by N=

s1

®... ®

SN =

L2 3 }

®... ® L2 (R3 ) ®V81 ® ... ® VSO, . N

In terms of the variables = (x2, az), where xi e 1R3 and a e {-s, -sj + 1, ... , sz - 1, sz}, the total wave function of the system can be written as ... , EN). The total wave function is a finite linear combination of the product functions X151, ... , N} = where W (x 1, ... , xN )

L2 (R3N ) is a coordinate part, and x (a1, ... , aN ) is

a spin part of the total wave function. The spin part, in accordance with the tensor product structure V81 0 0 VSN, is a finite linear combination of the products Xk1(a1) ... xkN (UN) of functions Xk, defined in Section 1.

3. System of identical particles

225

The Hamiltonian of the system of N-particles (without the spin interaction) in the coordinate representation has the form N 2 N HN = Vik (xi - xk), V (xi)+ i=1

2Lz

i+

1Ci
i=1

where the first term is the operator of kinetic energy of the system of N particles, the second term describes the interaction of the particles with the external field, and the last term describes their pair-wise interaction. Note that though the Hamiltonian HN acts only on the coordinate part ... , xN} of the total wave function, the physical properties of the system do depend on the spins of the particles.

3.1. The symmetrization postulate. For the system of N identical particles of mass m and spin s the corresponding Hilbert space is

N=

(3.1)

The irreducible representation RS of the Lie group SU(2) in the vector space vs naturally defines a representation RS®N in V®N, ® UN) = R3(9)vi ® ... 0 R3(9)v1v,

9 E SU(2).

The corresponding representation pr of the Lie algebra su(2) is given by N

®viv) _

vi ® ... ® p3(x) vk 0 ... ® UN,

x E 5u(2).

=1 The total spin operators S = (S1, 52i S3) of the system of N identical particles are given by N

(3.2)

Si -

S(k),

j = 1, 2, 3.

k=1

I2s+1 ® ® ihp8(A3) ® 0 I2S+ are the spin operators of the k-th particle; they act non-trivially only in the k-th factor of the tensor

Here

product V®N. Representations R?N and p3N are reducible and contain all represen-

tations V with spins l 2 Z>a such that sN - l are non-negative integers. Indeed, the square S2 of the total spin operator has the form S'2

= S? + s2 + S3 = S_ S+ + S3 ( S3 + I )

,

where S = Si ± iS2 and I here is the identity operator in V®N. It is easy to see from (3.2) that the eigenvalues of the operator S3 are -hsN, h(-sN + 1), ... , h(sN - 1), hsN. Now by Schur's lemma, restriction of S'2 to the irreducible representation V is h21(l + 1) times the identity operator in V. Since V contains a highest weight vector which is annihilated by the

226

4. Spin and Identical Particles

operator S+ and is an eigenvector of the operator 83 with eigenvalue hi, we conclude that V®N contains all irreducible representations with spins 1 E 2 Zoo such that sN - i are non-negative integers. The problem of determining the multiplicities with which representations V appear in the decomposition V®N into the direct sum of irreducible representations will be discussed below. It plays a fundamental role in classifying the energy levels of the system of N identical particles of spin s. There is also a natural action of the symmetric group SymN on N elements on YN, given by it E SymN.

The corresponding Hamiltonian in this case has the form

HN = -

(3.3)

2m

N

i+

U

i=1

Vext (Xi ) +nt i=1

1Ci
-

}

and commutes with the SymN-action: [HN, .P] = o

for all

71 E SymN-.

Thus the operator HN can be restricted to the SymN-invariant subspaces

in particular to the subspace}.lvof totally symmetric functions

of

1, ...

P=

and to the subspace

for all

7r E SymN,

NA} of totally anti-symmetric functions

P.= (-i))i

1, ... ,N , }

for all n E SymN,

where E(7r) is the parity of a permutation 71. In classical mechanics, it is possible to label the identical particles (say by the integers 1, 2, ... , N), and to trace the trajectory of each labeled particle separately. It follows from the Heisenberg uncertainty relations that in

quantum mechanics it is impossible to trace the time evolution of labeled particles separately. In other words, in quantum mechanics identical particles

are truly "indistinguishable", so that the Hilbert space (3.1) contains "too many states". It is another postulate of quantum mechanics which specifies "correct" Hilbert space of states for the system of identical particles.

All (The Symmetrization Postulate). The Hilbert space of states for the system of N identical particles of spin s is the totally symmetric subspace N of the N-fold tensor product (L2(R3) ® Vs ®N for the case of bosons

(particles with integer spin), and is the totally anti-symmetric subspace (A}

N

for the case of fermions (particles with half integer spin).

3. System of identical particles

227

Remark. The symmetrization postulate implies the Pauli exclusion principle, which states that no two fermions can occupy the same state. The Pauli exclusion principle is fundamental for atomic and molecular physics, and for the whole body of chemistry. For the case of identical particles of spin 0 it follows from the symmetriza-

tion postulate that the corresponding Hilbert space is the subspace of all totally symmetric functions W(xl, ... , xN) in L2(1[83") and in order to find the energy levels of the system, one needs to solve the Schrodinger equation

HN =END

(3.4)

only in the totally symmetric subspace of N. Solutions of the eigenvalue problem (3.4) that do not belong to this subspace have no physical interpretation. For the spin 2 particles the situation is different: the symmetry properties of the coordinate part of the total wave function depend on the symmetry properties of the corresponding spin part. Consider first the simple example of a system of two electrons. In this case we have the direct sum decomposition of Hilbert spaces, S)

A)

which immediately follows from the representation

(I(,,2) -

= Using the identity W(xl,x2)x(a1,Q2)

- 'I'(x2,x1)x(a2,cr1)

= 2 I'(x1,x2)+ W(x2,xl))\X(al,Q2) - X(a2,Ql)) +21I'(x1, x2) we can represent

- W(x2, xl))lx(al, cT2) + x(a2, a1)),

as a direct sum of Hilbert spaces A} 2

where

S0 = (L2(R3)®L2(R3))®A2 C2 consists of finite linear combinations of the product functions 'I1,

)_

W(xi,x2)X(ai,a2), Where W(xl, x2) E L2(R3) ® L2(IEg3) is symmetric and X(ai, a2) E n2 (C2 is anti-symmetric, and

1 = (L2(R3) ®

Sym2 (C2

4. Spin and Identical Particles

228

consists of finite linear combinations of the product functions where W (x i, X2) is anti-symmetric and x(ai, cr2) is symmetric. The vector space 112 C2 is one-

dimensional and is spanned by the function xoo, which corresponds to the vector 12 (e1®e2-e2®e1) E C2 O C2,

and the vector space Sym2C2 is three-dimensional and is spanned by the functions iii, xio, Xi-i, which correspond to the vectors ei ® ei,

(ei ® e2 + e2 ® ei ), e2 ® e2 E C2 ® C2.

Comparing the decomposition C2 ® C2 = 112 C2

Sym2 C2

with the Clebsch-Gordan decomposition

V1 ®V1 = V e Vi , 2 2

discussed in Section 3.2 of Chapter 3, we see that2 n2 cC2 = Vo and Sym2 (CZ =

Vl. In physics terminology, Vo is the spin singlet space, and Vl is the spin triplet space. We summarize the obtained results in the following statement.

Proposition 3.1. For a system of two identical particles of spin 2 the total wave function of spin 0 has the form W(xl, x2)Xoo(cri, 0-2), where 'I'(xl, x2)

is symmetric and satisfies the Schrodinger equation (3.4). The total wave function of spin 1 has the form W(xl, x2)xi-,,,,(Q1, a2), where m = -1,0, 1, and 'Y(xl, x2) is anti-symmetric and satisfies the Schrodinger equation (3.4). Remark. The coordinate part 'I'(xl, X2) of the total wave function has the usual probabilistic interpretation. Thus for a simple example of a system of two electrons with V;nt(x) = 0, the coordinate part of the total wave function has the form W(xl, x2) =

where b1 (x) and b2 (x) are two one-electron coordinate wave functions, the

plus sign refers to the spin singlet state, and the minus sign refers to the spin triplet state. The corresponding probability density is 2lbl(xl)l2lL'2(x2)2 + kb1(x2)2112(x1)12

f2 where the last term is called the exchange energy. It follows from this formula that when two electrons are in a spin triplet state, there is a zero probability of finding them at the same point x E ][83. However, when electrons are in

a spin singlet state, there is a non-zero probability of finding them at the same point in space, which is due to the exchange energy. This can also be verified directly by checking the action of total spin operators.

3.

System of identical particles

229

Consider now the system of N particles of spin 2. In this case the total wave function cI1i ... , N) E (A ) satisfies

HN =END and S2 = s(s + 1)tI,

(3.5)

and describes the bound state with total spin s, where 2 N - s is a nonnegative integer. It seems natural to assume that if one finds a coordinate wave function W(xl, ... , xnr) satisfying (3.4), and a spin wave function QN) which is the eigenfunction of S2, then the total wave function

Ii,.

,

N) can be obtained by the anti-symmetrization:

(-1)W(x(1),...,x(N))X(a(1),...,aN)).

1,... , N) _ 7rESymnr

However, it may happen that for a coordinate wave function W(xl,... xN) satisfying (3.4), the corresponding total wave function ... , nr) is identically zero! The requirement that the total wave function ... , nr) is not identically zero determines admissible symmetry properties of the coordinate wave function I'(xl,... xnr), which are described by the irreducible representations of the symmetric group Sym. ,

,

3.2. Young diagrams and representation theory of SymN. The number of irreducible representations of the symmetric group SymN is the num-

ber of its conjugacy classes, which in turn is the number of partitions ... A = (A1, ... , an ) of N : N = ) +. . + an , where A1 A 1. De.

a

note by Par(N) the set of all partitions of N. To each partition A e Par(N) there is an irreducible representation GA of SymN, constructed as follows. Let YA be the Young diagram associated with A E Par(N) a collection of N boxes arranged into n left-justified rows, with the first row containing al boxes, the second row containing a2 boxes, etc. Thus the diagram

corresponds to the partition 10 = 5 + 3 + 2. A Young tableau A associated with the Young diagram Ya is an assignment of N integers 1, 2, ... , N to the N boxes such that each box gets a different integer; denote by TA the set of all Young tableaux associated with the partition A E Par(N). A canonical Young tableau AA E TA is obtained by numbering the boxes consecutively along the rows from left to right. For a Young tableau A E TA, say the canonical one, define two subgroups of SymN: Row(A) _ {ir E Symn, : ir preserves the rows of A}, Col(A) _ {ir E SymN : it preserves the columns of A}.

4. Spin and Identical Particles

230

Now let 2t = CSymN be the group algebra of SymN a complex vector space with the basis {e1,.}EsymN and the multiplication map defined by Denote by R the regular representation of SymN in Qt, e7,-1 defined by

R(ir)x = e- x, x E 2t. With a given Young tableau A one can associate two elements in the group algebra 2t: the row symmetrizer

sir,

RA

7rERow(A)

and the column ant i-symmetrizer CA =

:i:

(-i)ir.

7rECo1(A)

Define the Young symmetrizer by HA -= CARA E 2t,

and consider the left Ga = 2t HA, and the representation Ta : SymN - End GA, where SymN acts by left multiplication. The following result is fundamental. Theorem 3.1. Each TA is an irreducible representation of SymN, and representations TA and T, are not isomorphic if A fit. The dimension of the representation T), is given by the Frobenius formula dA =

N

li = ai + n - i, i = 1, ... , n,

(lz - l ),

,

1<3

and IIA = daIIA. Every irreducible representation of Symnr is isomorphic to the representation TA for some partition A E Par(N).

Remark. It follows from the representation theory of finite groups that

R=

dAGA

End GA as algebras),

(2t =

AEPar(N)

AEPar(N)

so that

4.

N!= AEPar(N)

There is another expression for dimensions dA as a product over the boxes of the Young diagram YA, given by the so-called hook-length formula. 3Another choice of a Young tableau A E Ta gives a SymN-module equivalent to G.

3. System of identical particles

231

Remark. One can explicitly construct the bases in the subspaces GA by the following procedure. Let A be the standard Young tableau for a partition

A E Par(N): a Young tableau with the condition that in each row of YA, the numbers in the boxes are strictly increasing from left to right, and in each column strictly increasing from top to bottom. Let it E SymN be the unique permutation such that A = 7r(AA) and put eA = Then the vectors eA, where A runs through all standard Young tableaux for the partition A, form a basis of GA. In other words, the basis elements are obtained by successive symmetrization on the rows of Ya with a subsequent anti-symmetrization on the columns.

The partition A = (N) corresponds to the Young diagram with only one row and produces the trivial representation of Sym. The partition A = (1, ... ,1 ) corresponds to the transposed Young diagram a diagram with only one column

and produces another one--dimensional rep-

resentation, the alternating representation it (-1)} In general, let A' = (Ai, ... , A) be the partition conjugated to the partition A, which is .

defined by transposing the Young diagram Ya by interchanging the rows and columns (A is the number of terms in the partition A which are greater than

or equal to i). Then (3.6)

TAB = TA O T(1,...,1)

Other partitions correspond to the symmetries of mixed type.

The next result will play a crucial role in determining the symmetry properties of coordinate wave functions.

Proposition 3.2. The tensor product TA ® T, contains the trivial representation of Sym if and only if a = A, in which case it has multiplicity one. In terms of a basis {e}1 in GA such that the representation TA is given by orthogonal da x da matrices tA(7r)zj, da

to r} jzej,

TA

i = 1, ... , da,

j=1

the one-dimensional subspace of the trivial representation is spanned by the vector e ® ei E GA ® GA. Correspondingly, the tensor product TA ® T contains the alternating representation of Sym if and only if ,u = A', in which case it has multiplicity one. In terms of a basis {e}1 in GA' such that the representation TA' is given by the matrices ta' (7r) jj = (_1)6()tA(1r)jj, the one-dimensional subspace of the alternating representation is spanned by the vector

ei ® e E GA ® GA'.

Problem 3.1. Prove all the statements in this section. (Hint: See the list of references to this chapter).

4. Spin and Identical Particles

232

3.3. Schur-Weyl duality and symmetry of the wave functions. In addition to the diagonal action of the Lie group SU(2) in the tensor product (C2)®"', discussed in Section 3.1, there is also a left action of a symmetric group Symn, by permuting factors4: 7r E SymN.

7'(7r)(vl ® ... ® UN) - v_1(1) ® ... ®

For every partition A E Par(N) denote by IIa the Young symmetrizer corresponding to the canonical Young tableau Aa, and by S),C2 the image of the operator T(Ha) in ((C2)®N. The subspace S),C2 is zero-dimensional when the Young diagram Ya has more than two rows, and is an irreducible U(2)-module, called the Weyl module, when the Young diagram Y), has one or two rows. For the partition A _ (N) the Weyl module S),C2 = SymN (C2 and is isomorphic to the highest weight module Va,N with the dominant weight (N, 0), defined in Section 1.2. For the partition A _ (A1, A2) the Weyl module S),C2 is isomorphic to the highest weight module VAN with the dominant weight A. Remark. This construction is a special case of the general construction given by H. Weyl, which applies to the symmetric group action in the tensor product V®N, where V is afinite-dimensional vector space. The Weyl modules S),V associated with the Young diagram Ya are zero-dimensional when the number of rows is greater than n =dim V, and are isomorphic to the highest weight modules of GL(V) with the dominant weight a when the number of rows is less than or equal to n. The association V H SaV is called the Schur functor. The following result, the Schur-Weyl duality, establishes explicit pairing

between irreducible representations of U(2) and SymN that appear in the decomposition of the representation of SU(2) x Symin ((CZ)®N into the irreducible components. Theorem 3.2 (Schur-Weyl duality). There is a, direct sum decomposition into irreducible components of the representation RN x T: z

(C2)®N aEPar(N,2)

where Par(N, 2) is the set of partitions of N with number of terms less than or equal to 2.

Setting A1 = 2 + s, A2 = 2 - s (cf. Section 1.2), from the Frobenius formula we obtain (3.7)

d),= dim Ga = 12 N s

I - (Z

N

- 1)

41t moves the vector in the i-th place of the tensor product to the it (i)-th place.

3. System of identical particles

233

Now return to the main problem of finding total wave functions satisfying (3.5). It follows from the Schur-Weyl duality that there are (2s +1)da linear independent spin wave functions X,,,,j (di,. , ow) of spin s satisfying (3.8)

S2)Cm3= 3(s +1)Xmj

and

S3xmj = 17LXmj,

where m = -s, -s + 1, ... , s - 1, s and j = 1, ... , da. Here da corresponds to the partition A _ (2 + s, 2 - s) (or A _ (N) for s = 2) and is given by (3.7). The spin s representation ps of SU(2) acts on the index m of the spin wave function x,,,,,j(Ql, acts on Or), and the representation TA of the index j. The total wave function has the form ,

dA

(3.9)

m(1,

xN)Xm.7(Q1,

..., UN),

j=1

where m = -s, ... , s. It follows from Proposition 3.2 that the total wave function fir,, is totally anti-symmetric if and only if the coordinate wave functions W (x1,. . . , xN) transform according to the conjugated representation Tai of SymN, (3.10)

(PI/)(x1,.

. . ,

xN)

-

da

tA' (7r)jil'j (xl, ...,XN),

i = 1,...,da.

j=1

We summarize these results as follows.

Theorem 3.3. Coordinate wave functions 1Y (x1,. , xN) associated with the total wave function of spin s of a system of N identical particles of spin Z are those solutions of the N-particle Schrodinger equation (3.4) that transform according to the representation Tai of the symmetric group Sym, where A' is a conjugated partition to A _ (2 + s, 2 - s). The spin wave functions xr,,j(Ql, ,UN) satisfy (3.8), and corresponding total wave funcEN) are given by (3.9). In general, for a given energy level tions E there are 2s + 1 linearly independent total wave functions of spin s, . .

0 < s < 2 , and 2 -s is an integer. Remark. For the highest value of total spin s = 2 the coordinate wave function 'Y(xl, ... , xN) is totally anti-symmetric. Since (N)' _ (1,. .,1) .

Par(N, 2) when N > 2, coordinate wave function for the case N> 2 is never totally symmetric. Using explicit description of the Young symmetrizer in the previous section, we obtain the following result.

Corollary 3.4 (V.A. Pock). The coordinate wave function tY(xl, , xnr) associated with the total wave function of spin s is characterized by the following symmetry properties.

234

4. Spin and Identical Particles

(i) W(xl, ... , xnr) is anti-symmetric with respect to the group of arguments xl,... , xk, k = 2 -{- s. (ii) 1Jr(xl, ... , xN) is anti-symmetric with respect to the group of arguments xk+l s xN (iii) 1I1(x1, ... , xN) satisfies

,

... , XN} =

W(x1,... , Xi_l, x +l, Xi+1, ... ,

X, xk+2, ... , xN).

i=1

The case of N identical particles of arbitrary spin s is treated similarly. The symmetry properties of coordinate wave functions can be obtained by using Schur-Weyl duality for the action of U(l) x Symin (Cl, where l = 2s + 1. It has the form (Cl)®N

=

EF?

Sail ® Ga,

AEPar(N,l)

where Par(N,1) is the set of partitions of N with number of terms less than or equal to 1, and SC1 is the Weyl module corresponding to the highest weight representation of U(l) with the dominant weight a. Problem 3.2. Prove Corollary 3.4.

Problem 3.3 (one-electron approximation). Suppose that V t (x) = 0, and let (x),. , (x), k 2 , be the one-electron wave functions: eigenfunctions . .

of the one-electron Schrodinger operator H1 (see (3.4)). Show that the coordinate wave function I1(xk+1) ...

...

k(xk)

where stands for the matrix determinant, satisfy the symmetry properties in Corollary 3.4.

4. Notes and references The notion of electron spin is fundamental for the agreement of quantum mechanics with experiments. See [Mes99] for a thorough discussion of the principal evidence supporting the hypothesis of electron spin (Zeeman effect, Stern-Gerlach exper-

iment). Classic texts [LL58, Foc78] also introduce the concept of spin from a physics perspective. Representation theory of the Lie group SU(2) the simplest compact non-abelian Lie group is discussed in detail in [Vi168]; for the description of all irreducible representations of U(n) see [FH91]. Our discussion of the Pauli Hamiltonian in Section 2.1 follows [Foc78]; the case of a particle in a uniform magnetic field in Section 2.2 was first considered by L.D. Landau in 1930 (see

[Problem 2.2 shows that in the presence of a magnetic field the operators

4. Notes and references

235

No and Y0, associated with coordinates of classical trajectories of a charged particle, do not commute and cannot be measured simultaneously.

The Pauli wave equation in Section 2.1 is non-relativistic, i.e., it does not take into account the postulates of special relativity. The correct quantum description of the electron in an external electromagnetic field was proposed by Dirac and is given by the Dirac equation; the Pauli wave equation is obtained as its non-relativistic limit [Foc78]. The Dirac equation is a single particle equation which describes an electron and a particle with charge -e a positron. In order to describe quantum processes of creation and annihilation of particles, one needs to pass from quantum mechanics to quantum field theory, which deals with systems with infinitely many degrees of freedom, and we refer to [1Z80] for an introduction.

The symmetrization postulate, formulated in Section 3.1, is another fundamental principle of quantum mechanics, and we refer to [Mes99] for its physical motivation. The N-particle Schrodinger equation describes all the properties of atoms. However, it is almost impossible to solve this equation analytically (for the potentials in atomic physics), and even numerically when N > 2; the case of the helium atom (N = 3) illustrates the difficulties of the problem. For this purpose various approximate methods like the one-electron approximation, the Hartree-Fork method and its modifications, have been developed (see [LL58, Foc78] for the details and [FY80] for a clear introduction). A correct description of the properties of atoms and an explanation of D.I. Mendeleev's periodic table of elements, based on these methods, has been a major triumph of quantum mechanics. For a detailed account we refer the interested reader to [LL58, Foc78] and [FY80]. The N-particle Schrodinger equation, in principle, also describes5 all properties of molecules studied in chemistry. However, this "reductionism" does not work due to the complexity of the quantum N-body problem, and approximate methods like Hartree-Fock approximation play a central role in quantum chemistry. The vector space C2 ® C2 describes the spin degrees of freedom of two elec-

trons, and the spin states which are not represented as a tensor product u ® v are called entangled. Entangled states are used to describe the Einstein-PodolskiRozen paradox, a quantum experiment, which apparently violates the principle of locality (see its description in [Be187, Sak94] ). The fact that quantum mechanics is correct and allows for an accurate description of the microworld is beyond any doubt, and is confirmed by numerous physical experiments. The EinsteinPodolski-Rosen paradox is only a "paradox" because classical intuition does not always correspond to the physical reality. Moreover, it follows from the Bell inequalities [Be187] that any attempt to turn quantum mechanics into a deterministic theory by introducing the so-called hidden variables is fallacious; we refer to

the web site http://en.wikipedia.org/wiki/EPRparadox for further references. The vector space C2 ®

.

® (C2, describing the spin degrees of freedom of several

51n this sense, quantum mechanics is a "theory of everything" for chemistry.

236

4. Spin and Identical Particles

identical particles of spin 2 , also plays a fundamental role in quantum computation theory. This is a rapidly developing field, and we refer the interested reader to the monograph [KSVO2] for the mathematical introduction, and to the web site http://en.wikipedia.org/wiki./Quantum_computer for updates on the specialized literature. Another rapidly developing and exciting new application of quantum mechanics is quantum cryptography, reviewed at http://en.wikipedia.org/wiki /Quantum cryptography.

Section 3.2 is a crash course on the representation theory of the symmetric group SymN a beautiful piece of classical mathematics which has many interesting applications in modern combinatorics and representation theory. Our goal here was to give a concise and clear presentation of the basic necessary facts, and

we refer to the classic text {Wey5O] and to [for detailed accounts of representation theory, and to [GW98} for the Schur-Weyl duality. In Section 3.3 we emphasize the role of Schur-Weyl duality in studying the symmetry properties of coordinate wave functions. Physics textbooks like [LL58], [Dav76] and [Sak94] carefully treat the basic examples, but are somewhat vague in describing the form

of the total wave function of N particles. The monograph [is very precise in this regard, but in his presentation V.A. Fock is trying to reduce the usage of group theory to a minimum. Our goal in Section 3.3 was to fill this gap and to give a clear description of the symmetry properties of coordinate and spin parts of the total wave function by using the language of representation theory. For the case of N particles of spin 2 this problem was solved by V.A. Fock in 1940; see [Foc78], as well as in yet another classic [Wig59]. For more applications of group theory to quantum mechanics we refer to the latter text, and to [Wey5O].

Part 2

Functional Methods and Supersymmetry

Chapter 5

Path Integral Formulation of Quantum Mechanics 1. Feynman path integral Here we present Feynman's path integral approach to quantum mechanics, which expresses the propagator of a quantum system with the Hamiltonian H the kernel of the evolution operator U(t) = eas a "sum over trajectories" of the corresponding classical system.

1.1. The fundamental solution of the Schrodinger equation. Recall (see Section 1.3 of Chapter 2) that in terms of the evolution operator U(t) _ e-*tH, the solution b(t) of the initial value problem for the time-dependent Schrodinger equation (1.1)

(1.2)

(t) =Hi(t), b(t)j0 =

iii

2of

is given by fi(t) = U(t)%. For the Hamiltonian operator 2

H=-+V(Q)

a quantum particle in IEBn moving in the potential field V (q), the initial value problem (1.1)-(1.2) in coordinate representation becomes the following 239

5. Path Integral Formulation of Quantum Mechanics

240

Cauchy problem: 2

ih-

(1.3)

= --2m

at (q,t)_0 =

(1.4)

+ V(q),

where D is the Laplace operator on I[8n. Under rather general conditions

on the potential V(q) (i.e., when V E L( II') is bounded from below) the Cauchy problem (1.3)-(1.4) has a fundamental solution: a function K(q, q', t) which satisfies partial differential equation (1.3) with respect to the variable q in the distributional sense (i.e., K(q, q', t) is a weak solution of the Schrodinger equation), and the initial condition

K(q,q',t)_0 = 8(q - q').

(1.5)

The solution of the Cauchy problem (1.3)-(1.4) can be formally written as

(q',t)

(1.6)

=f

K(q',q,t)(q)dq,

where the integral is understood in the distributional sense. For 2b E L2(R) formula (1.6) should be understood as (q', t)

f

K(4/, q,

l.i.m. J

n K(q , q, t)W(Q)dnq,

where l.i.m. stands for the limit in the L2-norm. In general, q, t) is only a weak solution of the Schrodinger equation (1.3), and it is a regular solution when L' belongs to the Garding domain Do (see Problem 1.7 in Section 1.3 of Chapter 2). The fundamental solution K(q, q', t) is the distributional kernel (in the sense of the Schwartz kernel theorem) of the evolution operator U(t). Using the group property U(t + t') = U(t)U(t'), we can rewrite (1.6) in the form (1.7)

(q', t') =

f

K(q', t'; q, t) (q,

q, t) 2 has a K (q', t' t) physical meaning of conditional probability distribution of finding a quantum particle at a point q' E Ian at time t' provided that it was at the point

gEIFJ at time t. Remark. In physics terminology, the distributional kernel K(q', t'; q, t) of the evolution operator U(t' - t) is called the complex probability amplitude, or simply the amplitude or propagator. In Dirac's notation, K(q', t'; q, t) = (q', t' I q, t).

Finding the propagator of a given quantum system is a fundamental problem of quantum mechanics. When the spectral decomposition of the Hamiltonian operator H is known, the propagator can be obtained in

1. Feynman path integral

241

a closed form. Namely, suppose for simplicity that H has a pure point spectrum, i.e., there is an orthonormal basis of the Hilbert space 31' = L2 (fin, dnq) consisting of the eigenfunctions {'/)(q)}0 of H with the eigenvalues E. We have

ea=

enn(q),

(q) = =a

fan (q)(q)dq,

so that

e Enenn q

(U(t)b) (q)

n=o and

where the series and integrals converge in the L2-sense. If the change of orders of summation and integration was justified, we could write (1.$)

T = t' - t.

e-

K(q', t, q, t) = n =0

The series (1.8) converges in the distributional sense, and gives a representation of the propagator in terms of the spectral decomposition of H. A similar representation of the propagator exists when the Hamiltonian H has absolutely continuous spectrum. Consider the simplest case of a free quantum particle with the Hamiltonian operator

p2 H0=-. 2m The corresponding Schrodinger equation can be solved by the Fourier method (see Section 2.3 in Chapter 2), and we obtain 2

= l.i.m.(27rh)2

fn

(p, t)dnp

= l.i.m.R K(q', t'; q, t)`(q, t)dnq, n

where (1.9)

'(q', t', q,t) =

1

e

(2irh)

p(4'-q)---

T }dnp

n

Using the classical Fresnel integral formula eiac

x

a e iri sgn 4

a

T = t' - t.

242

5. Path Integral Formulation of Quantum Mechanics

and completing the square in (1.9), we obtain the following expression for the propagator of a free quantum particle: 7' imT (q_q')2 m (1.11) K(4'',t'7 qy t) = e

(2ihT

'

where i 2= e- T and T> o. Remark. Formally replacing the actual "physical" time t in the Schrddinger equation 2

2m

D 4'

for a free quantum particle of mass m by the Euclidean time (or pure imaginary time) -it, we obtain the heat (diffusion) equation 1 8u

u

D 8t

on W' with the diffusion coefficient D = 2 It is quite remarkable that m in agreement with this formal procedure the propagator (1.11) for a free quantum particle is obtained from the heat kernel on IRn by the analytic continuation T F -p -iT.

Problem 1.1. Show that R

lim

R--- o0

0

sin x2 dx = lim

by integrating the function e _

R-> o0

z2

0

1

cos x2 dx = 2-

2

over an appropriate "pizza-sliced" contour.

Problem 1.2. Show directly that

b(q',t) =

i tH ) = l.i.m. e -)(q

n

2

(_ 2712Lt )

fn

e

Problem 1.3. Give an expression for the propagator when H is a Schrddinger operator with rapidly decreasing potential, considered in Section 2 in Chapter 3.

1.2. Feynman path integral in the phase space. For a general Hamiltonian operator H = H0 + V, where V = V ( Q ), there is no simple formula for the propagator K (q', t'; q, t) like (1.11). This is because operators H0 and V do not commute, so that e- tH e_ * tH0 e- *tV. It was Feynman's fundamental discovery that there is another representation of the propagator for a quantum system in terms of the corresponding classical system. We start by describing Feynman's approach for the case of a quantum particle with one degree of freedom. It is based on the so-called Lie--Kato-Trotter product formula, which allows us to express the exponential ei(A+$) of two non-commuting self adjoint operators in terms of the individual exponentials e' and eiB.

1. Feynrnan path integral

243

Theorem 1.1 (Lie-Kato-Trotter product formula). Let A and B be selfadjoint operators on 9° such that A+B is essentially self-adjoint on D(A)f1

D(B). Then foribE°,

eb = lim n->oo

Proof. We consider only the special case when A and B are bounded operators, which was already proved by Sophus Lie. Set C, = e2(A)/n and Dn = etA/neiB/n. We have the telescopic sum

Cn - Dn = 'n - Cn-1 Dn +'n-1Dn -

C'n-2Dn

...nDn- -Dn

n--1

n -- D n }Dn k=o

and since by the Taylor formula Cn -

_ for some constant c> 0,

we obtain

C- D

n

< This proves the result with convergence in the uniform topology.

We will always assume that the Hamiltonian H = Ho + V is essentially self-adjoint on D(H0) f1 D(V), so that the Lie-Kato-Trotter formula is applicable. (According to Theorem 1.2 in Section 1.1 of Chapter 3, for the case n = 3 we can assume that V = V1 + V2, where V1 E L2(I3) and

the Lie-Kato-Trotter formula to A = - Ho and B = - V, we obtain that in the strong operator topology V2 E L00 (1R3 ). } Applying

(1.12)

e_iTH

i0t

= n-oo lim (e_H0 e_

i0t V)n,

t = -, n

where T = t' --- t. In the coordinate representation the operator e_! a multiplication by e_ *

"(

is

mot operator, and the distributional kernel of the

operator e- P H0 is given by (1.9), where T is replaced by t. Thus the z* xot distributional kernel K(q', q; t) of the operator e is given by H0 e (1.13)

K(q

, q,

t) .L /

27rh -oo

dp.

e

Since the kernel of a product of two operators is a composition of corresponding individual kernels, for the distributional kernel Kn (q', t'; q, t) of the operator (e_PH0 e- x" ) n we obtain the following integral representation: (1.14)

Kn(q', t'; qy t)_

... Ian-

n-1

=4

n-1

K(qk+1, qk; t)dqk,

_

5. Path Integral Formulation of Quantum Mechanics

244

where qo = q, qn = q'. Now replace each factor K (qk+1 , qk; t) in (1.14) by its integral representation (1.13), where the corresponding variable of integration is denoted by pk, k = o, ... , n -i. Changing the order of integrations in the resulting (2n ---1 )-fold integral and using (1.12), we obtain the following remarkable representation of the propagator of a quantum particle as a limit of multiple integrals when the number of integrations goes to infinity: (1.15)

K(q', t'; q, t) = lim -fin (q', t1; q, t) n---* oo

= lim 1. f exp { k= (pk(qk+1 - 4'k) 2n-1

4k)Ot)

.

11

.

k=1

d2 7rhk

2

q) = 2 + V (q) is the classical Hamiltonian function, and qp =

Here q,qn=q'.

Heuristically, formula (1.15) admits the following interpretation. To every point (pa, pi,... , pn_1, q1, ... , q,_1) E JR2n-1 assign a piece-wise linear path a in the extended phase space JR2 x R of the classical particle,

defined by the following time slicing procedure. Let tk = t + kOt and a(T) = (pfr),q(r),r), where p(r) = pk, q(r) = q+ (T -- tk

qk+1

qk

,

tk+1 - tk

and r E [tk, tk+11 , k = o, ... , n ---- 1. Then for the Riemann integrable potentials V(q) we have n-1

(p(q+i - 4'k) - HC(pk> qk)Ot) =

(1.16)

S(Q) -I- 0(1),

as

n ---- oo,

k=0

where s(ue) _

f

(pdq -

f

t'

t

-

is the action functional of a classical system with the Hamiltonian function q) (see Section 2.2 in Chapter 1). This suggests to interpret (1.15) as a kind of "integral" over the space P(JR2)9;t of all paths a(T) _ (p(r), q(T),T) in the extended phase space ]I82 x I[8 such thatl q(t) = q and q(t') = q', which was used in the formulation of the principle of the least action in the phase space (see Section 2.2 in Chapter 1). Thus we put (1.17)

K(q', t'; q, t) =

I

e* s(a} pI'q,

1 Note that there is no condition on the values of p(r) at the endpoints.

I Feynman path integral

245

.

where the "measure" ?1p1q on P(JR2)

q(1.18 '

is given by 1

)

dpdq, i1p1q = lim dpo 11 n---}oo 2irh 2irh k=1

Representation (1.17)-(1.18) is the famous Feynman path integral in the phase space for the propagator of a quantum particle. It expresses the propagator K(q', t'; q, t) as the "weighted sum" over all possible "histories" of the classical particle, the paths a E P (R2 )q't', where each path a has a complexweight exp{S(a)}. On one hand, this representation clearly shows the fundamental difference between classical mechanics and quantum mechanics. Thus in classical mechanics, the particle moves along classical trajectories, which are the critical points (extrernals) of the action functional S(a),

whereas in quantum mechanics all possible paths contribute to the complex probability amplitude K (q', t'; q, t). On the other hand, representation (1.17)-(1.18) clearly points at the relation between quantum mechanics and classical mechanics in the semi-classical limit h - 0. Namely, as h - 0 the weights exp{ S(a)} become rapidly oscillating and their contributions to (1.17) will mutually cancel, unless the action is almost constant. The latter happens near the critical points, which give the major contribution to (1.17). This is how classical trajectories of a particle emerge from its quantum description, which will be discussed in detail in Section 6.1.

Remark. The Feynman path integral in the phase space in not an integral in the sense of abstract integration theory, since the formal expression p/Iq does not define a measure2 on the path space P(R2) '. Besides, the functional exp{S(a)} has absolute value 1 and cannot be integrable with respect to any measure d,u with the property /i( P (R2 )q't') = oc. The rigorous mathematical meaning of (1.17)-(1.18) is the original formula (1.15), which expresses the propagator as a limit of multiple integrals as the number of integrations goes to infinity. This explains the apparent "paradox" that quantum mechanics is defined entirely in classical terms by (1.17). It is not so, since there is no "intrinsic" definition of (1.17) in classical terms, besides the time slicing procedure (1.15), which also requires the special choice of the approximation (1.16) of S(u) by the Riemann sums.

1.3. Feynman path integral in the configuration space. Using the Fresnel integral (1.10) for the integration over p in (1.13), we obtain the fol-

lowing formula for the distributional kernel of the operator e- P D e V '2 V 2irihOt 21n abstract measure theory the measure is non-negative and countably additive.

5. Path Integral Formulation of Quantum Mechanics

246

Repeating the time slicing procedure from the previous section, instead of (1.15) we now get n

m

K(q', t'; q, t) = lim

(1.19)

2(t)2

n--oo

n-1 X

f... f exp Ilgn-1

fi

(

2

k=0

4k+1 - 4k

Ot

2

n-1 - V (9k)

)

Ot fi dQk J k=1

Assuming that qj = q(t), k = 0, ... , n, for some smooth path 7(T) = q(r) in

where q'o = q and qn = q', we obtain as n - oo, qk+ 1 - Qk Ot C

l - V(k)) 2

t = 8(7) + o(1),

where

f

L(y

Jt

L(q(T), q(T))dT,

L(q, q) = 2mg2 - V(q),

is the action functional of a classical particle of mass m moving in the potential field V(q) (see Section 1.3 in Chapter 1). This suggests to interpret

the limit of multiple integrals (1.19) as the Feynman path integral in the configuration space (1.20)

K(q', t'; q, t) =

esq.

e

Here P(R) ' is the space of smooth parametrized paths y in the configura tion space R connecting points q and q', and the "measure" 1q on is given by (1.21)

q = lim

n n---1

dq

(2t)

k=1

As the Feynman path integral in the phase space, the Feynman path integral in the configuration space is not actually an integral in the sense of integration theory, and the correct mathematical meaning of (1.20)-(1.21) is given by (1.19). Formula (1.20) expresses the propagator K(q', t'; q, t) as the sum over all histories in the configuration space of classical particle -paths t , by assigning to each path ry a complex weight exp{ ry E *S(ry)}.

I. Feynman path integral

247

Remark. Convergence in (1.19), as well as in (1.15), is understood in the distributional sense. In particular, for every & E L2(I[g), m 2 (e_*Tb)(q)

= I1Ynoo (2t)

n1 m qk+l Z{>t

2Ot

Rn

-

V (qk)

Ot k=1

where all integrals are understood as f -

n

2

dqk

= limRiqR' and all limits

in the L2-sense.

Remark. It is said in physics textbooks that the Feynman path integral in the configuration space is obtained from the Feynman path integral in the

P(R)'

phase space by evaluating the Fresnel integral over gyp, (1.22)

Je

f: (r q2-v(q)}drgq =

e ft

Note that the symbol !?Iq has two different meanings: in the left-hand side of (1.22) it is defined by (1.21), whereas in the right-hand side it is defined as a part of (1.18).

1.4. Several degrees of freedom. Let P2 H=-+V(Q) 2m

be the Hamiltonian operator of a quantum particle in Ian moving in the potential field V (q). As in the case of one degree of freedom, by using the Lie-Kato-Trotter product formula, the propagator K(q', t'; q, t) is expressed as a limit of multiple integrals when the number of integrations goes to infinity, (1.23)

2Xp

lim

K(q', t'; q, t) 2

(pk(qk+1 - qk)

Hc (pky qk),Ot}

k=0

N -1 (27Th) n

k-1

(2irh) n

"

Here H( p, q) = p2 + V(q) is the classical Hamiltonian function, and qo = 2m q, qN = q'. This representation is symbolized by the Feynman path integral

in the phase space 4 = T*In, (1.24)

K(q', t'; q, t) =

J

t

e

f(-c(q}}dTpga

5. Path Integral Formulation of Quantum Mechanics

24$

where lam

N -1

dpo n

H (2ir) n=1

d

do qk (27rh)n

and P(4') 4space ' is the space of all admissible paths a in the extended phase

4 x I connecting points (q, t) and (q', t') (see Section 2.2 in Chapter

1

Equivalently, the propagator K(q', t'; q, t) can be written as N-oa 2ir

.

x

772

K(q', t', q, t) = lim

(1.25)

N--1

exp k=O

m (qk+i -- qk

t

2

nN

hjti \ - V qk )) t

2

N--- i

Hdngk

J k=1

This formula is symbolized by the Feynman path integral in the configuration space, (1.26)

e * f t L(4,4)dTgq,

K q', t'; q, t = P(M)4,t

where L(q, q) = 2 rngr - V(q) is the corresponding Lagrangian, rtN Ni m

q = lim

N-}oc

(2t)

dT qk ,

and P(M)' qspace is the space of smooth parametrized paths in the configuration M connecting points q and q'. The precise mathematical meaning of formula (1.26) is the same as of (1.20).

Remark. Formally, definition (1.24) of the Feynman path integral in the phase space can be extended to the case (4',w,H), where 4 = T* M and w = dB, the canonical Liouville 1-form on 4. Namely, one can obtain a distributional kernel K (q', t'; q, t) by using the same time slicing as in (1.24)

and replacing the 1-form pdq by 0, and the volume form dnp,dnq by an appropriate multiple of the Liouville volume form on 4. Similarly, representation (1.26) can be formally extended to the case of general Lagrangian system (M, L), where the configuration space Al is a Riemannian manifold, by using the same time slicing and replacing dnq by the Riemannian volume form on M. However, in general it is not clear for which unitary operators these Feynman integrals are actual distributional kernels. Moreover, even in the case M = Rn with the standard Euclidean metric, representations (1.26) and (1.24), where H is a Legendre transform of L, do not necessarily agree if L is not of the form "kinetic energy minus potential energy". However,

2. Symbols of the evolution operator and path integrals

249

V(x) (cf. Example 1.7 in Section 1.3 of Chapter 1), then with appropriately defined time slicing, Feynman path integrals (1.26) and (1.24) agree, and are equal to the propagator for the Hamiltonian operator

when L =

H= g+V 2

Here g is the Laplace operator of the Riemannian metric on M, introduced in Example 2.4 in Section 2.4 of Chapter 2.

Remark. Formula (1.24) could serve as a heuristic tool which enables us, in some cases, to quantize the classical Hamiltonian system (4', w, He ). In general this is a rather non-trivial problem, especially when the phase space .' is a compact manifold.

2. Symbols of the evolution operator and path integrals In Sections 2.7 and 3.3 of Chapter 2 we introduced Wick, pq-, qp-, and Weyl symbols of operators. For the evolution operator U(T) = e- T these symbols can also be represented by Feynman path integrals. Here we consider only the case of one degree of freedom, since generalization to several degrees of freedom is straightforward.

2.1. The pq-symbol. Let Fl (p, q, T) be the pq-symbol of the evolution operator U(T). Using (1.15) and formula (3.18) in Section 3.3 of Chapter 2 - the relation between the pq-symbol of an operator and its distributional

kernel - we get o F1 p, q, T =

(2.1)

exp

= lim

-i

n-i

k0

2rz-2

K(q _ v, T; q, O)e*dv n-1

(pk(qk+1 - qk ) -c p qk ) t} j

k1

dpi-1dq , 27th

where pn--1 = p and qo = qn = q. Formula (2.1) can also be obtained from the composition formula for pq-symbols (see Problem 3.9 in Section 3.4 of Chapter 2). Indeed, it is sufficient to observe that the pq-symbol of zat ° zoty Hc(paq), and to represent the is f(, q) = e! the operator e")n as the multiple integral pq-symbol of the operator (eH0 e-

... f f(p, 4'n-l).f (pn-2, 4n-2) ....f (p2, Q2)f (poi q 2 n-2 x exP

(p k=0

- pk)(9k - 4'k+1)

n-1 dpk-ld4k 2ith '

fi

J k=1

250

5. Path Integral Formulation of Quantum Mechanics

where qo = q. In the same spirit as (1.15), formula (2.1) can be written as the following Feynman path integral for the pq-symbol: (2.2)

Fi (p, 4, 7') =

f

where

12(p,q) (]l82) _ {Q :

[0, T] -+ I[82 : Q(0) = Q(T) _ (p, q) E II82} is the

s(a) = f pdq -

space of parametrized loops in the phase space ]I82 which start and end at the point (p, q).

Remark. The mathematical meaning of (2.2) is the original formula (2.1) with a special choice of the approximation of S(Q) by Riemann sums.

2.2. The qp-symbol. Let F2 (p, q, T) be the qp-symbol of the evolution operator U(T). In this case instead of formula (1.12), convenient for pqsymbols, one should consider the equivalent representation

e- * TH = lim (e_PVe_PH0)n.

(2.3)

n-+oo

The distributional kernel K(q', q; t) of the operator e-

aT A.t

zF

V e-

H0 is given

by

(2.4)

1

k(q', qa ' t) =

00

ii:

dp,

e

and as in Section 1.2 we obtain n-1 (2.5)

K(q', t'; q, t) = lim

[.111 k(qkl, q ; t) IRn-1

- X11

... f

n--1

dq k=1

k=O

i n-1

exp{-fii

(pk(qk+i - 9'k) -

4'k+1)Ot)J

dpo

k=4

IIg2n-i

n

n-1

k=1

k

2irfii

Using formula (3.19) in Section 3.3 of Chapter 2, we get from (2.5), 00

(2.6)

F2(p, q, T) =

K(q, T; q + v, 0)e pvdv

n-1

= n-oo lim

[... f exp

?

n-1 (pk(qk+i - qk ) _ HC (pk, qk+i

k=D

dpi- dq

k=1

where po = p and qo = qn = q. Formula (2.6) can also be obtained from the composition formula for qp-symbols (see Problem 3.10 in Section 3.4 of Chapter 2) by observing that the qp-symbol of the operator e- x K tV e7 H° xot is again e- He (p'q) . In the same spirit as (1.15), formula (2.6) can be written

it

2. Symbols of the evolution operator and path integrals

251

as the following Feynman path integral for the qp-symbol: (2.7)

f

Fa (p, 4',7') _

e

c (p,Q) (a2)

This expression looks exactly like (2.2), so that the formal path integral representation does not distinguish between pq- and qp-symbols! Of course, the mathematical meaning of (2.7) is the original formula (2.6), which requires a special choice of the approximation of 8(u) by Riemann sums, which is different from the one used for the pq-symbol. Problem 2.1. Deduce formula (2.6) from the composition formula for qp-symbols.

2.3. The Weyl symbol. Let F (p, q; T) be the Weyl symbol of the evolution operator U(T). It follows from the Weyl inversion formula formula (3.12) in Section 3.3 of Chapter 2 that F(p, q, T) _

fK(q_

2

v, T; q +2 v, o)e*dv.

However, for general potentials V(q) one cannot perform integration over v by using either (1.15) or (2.5), and we choose another approach. Namely, denote by U(t) an operator3 with the Weyl symbol e- Hc(p,q)Lt, and suppose

that

U(T) = lim U(Lt)n.

(2.8)

n-oo

Using the composition formula for the Weyl symbols, formula (3.24) in Section 3.3 of Chapter 2, we can express the Weyl symbol as

f (2.9)

-(q k

F(p, q, T) = lim

-

k)\/ek+l

-

k))

J

- k-0

f

J

n-2

exp{? (2

c

Otl} 4/c

((Pk

- k)(r/k+l - r/k)

k-1

where o = p, r1o = q and n_1 = pn-1, 7Jn-1 = q?-i. As for formula (1.15), formula (2.9) can be written as the following Feynman path integral for the Weyl symbol: (2.10)

F(p, q,T) e

fo

3There is no easy way to express it in terms of e- ° and e-

pc'q zoo .

7

252

5. Path Integral Formulation of Quantum Mechanics

where

pq r = lim

n-1 ,

irh

k=i

and "integration" goes over the space of real-valued functions p(t), q(t), fi(t),

and i1(t) on {0,T] satisfying (0) = p, q(T)

q and (T) = p(T), rj(T) _

Remark. Expression (2.10) looks very different from (2.2) and (2.7). However, the variables (t) and i(t) appear only linearly in the exponential in (2.10), and the integration over them can be performed explicitly. Indeed, using integration by parts (and ignoring subtleties with the boundary terms) we can transform the integral over to

Je which equals the product of delta-functions q(t)) over [O, T] and allows us to replace 2r)(t) by q(t) in (2.10). Thus we obtain the formula F (p, q, T)}

=

e* fo

j

p

q7

which looks exactly like (2.2) and (2.7)! This heuristic argument shows that the Feynman path integral, understood naively, "erases" the difference between various symbols of the evolution operator. Of course, one should always specify particular finite-dimensional approximation of S(cr) in order to get correct results. Problem 2.2. Find the integral kernel of the operator U(L1t).

Problem 2.3. Derive formula (2.9).

2.4. The Wick symbol. Instead of the variables z and z, used in Section 2.7 of Chapter 2, it is convenient to use variables a = z and a = z, which introduce explicit h-dependence in the calculus of Wick symbols. Let H(a, a) be the Wick symbol of a Hamiltonian operator H. Here we derive a formula for the Wick symbol U (a, a; T) of the evolution operator U(T) = TH. Denote by U(Lt) an operator with the Wick symbol e* TI.

7

and as in the previous section, suppose that (2.11)

U(T) = n->oo U(ot)n.

Using the formula for the composition of Wick symbols (see Theorem 2.2 and

Problem 2.18 in Section 2.7 of Chapter 2), we get the following expression

2. Symbols of the evolution operator and path integrals

253

for the Wick symbol Un(a, a; T) of the operator U(Lt)n: Un(a, a; T) =

(2.12)

f... cn-i

fexp

{(a(an_i - a) - iH(a, an-1)Ot

r

n-1 d2a

(ak-i-ak) - iH(czk ak-i)Ot)/ J 11

k=1

where ap = a. Setting an = a, we can rewrite the sum in (2.12) as n

ak) + a(an - a) - iH(ak, ak-1)Ot), k=1

and obtain U (a, a; T ) = lim Un (a, a; T ) = iim n-oa n-oo

(2.13)

.. . cn-1

1

exp

r

n-1 d2a

n

{((ak(ak_i - ak) + a(an - a) - iH(ak, ak-1)Ot)/ J 11 k=1

k=1

As in the previous examples, we can represent formula (2.13) by the following Feynman path integral for the Wick symbol: (2.14)

U(a, a; T) _

J

e

f' (iakH(a,a))dt+ a(a(T}-a}

fa(T)=a a(O)=a

where

n-I d2aj

1a da =n#oo11 lim TT_. irh k=1

Here integration goes over all complex-valued functions a(t) and a(t) satisfying boundary conditions a(0) = a and a(T) = a, and such that a(t) is complex-conjugated to a(t) for 0
Remark. It should be emphasized that in (2.14) the values a(0) and a(T) are not complex-conjugated to the fixed boundary values a(0) = a and a(T) = a, but rather are "variables of integration". This should be compared with formula (1.17), where the boundary values for the function q(T) are fixed, and the boundary values for the function p(T) are free. The difference

is that at t = 0 we fix the boundary value of the function a(t), while at t = T we fix the boundary value of another function a(t).

5. Path Integral Formulation of Quantum Mechanics

254

Remark. The Wick symbol U(a, a; -iT) of the operator e-hTH = U(-iT) can also be represented by U (a, aa; --iT) = lim Un (a, a; -iT) = lim n- oa

cm-'

. ' n-= 2 a

r1

k=1

(2.15)

- k=1

I

nh

e- * f" (aa+H(a,a))dt+ *a(a(T)_a)a

a(T)=a a(D)=a

which is the so-called Euclidean path integral the Feynman path integral with respect to the Euclidean time, which will be discussed in Section 2.3 of

Chapter 6. In particular, if the Hamiltonian H has a pure point spectrum and e-TH is of trace class, then using the relation between the trace and Wick symbols (see Problem 2.17 in Section 2.7 of Chapter 2), we obtain from (2.15)

'Ire- hTx =

(2.16)

I

e- K fa (aa+H(a,a))dtaa 7

a(0)=a(T)

la(0)=a(T)}

where now integration goes over all complex-conjugate functions a(t) and a(t) satisfying periodic boundary conditions a(0) = a(T) and a(0) = a(T), and n

Ia1a = lim e n---goo

- an an

d a, nr h

Indeed, it follows from the identity n-1 (2.17) ak(ak-1 - ak) + a(a_1 - a) k=1

1 n-

1

{a(a_i - ak) + (ak+1 - ak)ak} + 2 {a(a_i - a) + (al - a)a}

2 k=1

that when a and a are variables of integration, one can put an = a and ap = a, which together with an = a and ao = a imply periodic boundary conditions. Thus n-1

n,

ak) + a(an-1 - a) = 2 k=1

_1

{ak(ak_1 - ak) + (ak+1 -

3. Feynman path integral for the harmonic oscillator

which in the limit n -p oo becomes fT 2

J0

255

fT

(-aa + aa)dt = -

J0

aadt

due to the periodic boundary conditions.

In the holomorphic representation, the matrix symbol K(a, a; T) of the evolution operator plays the role of a propagator K(q', T; q, o) in the coordinate representation. Using the relation between matrix and Wick symbols (see Lemma 2.4 in Section 2.7 of Chapter 2), we obtain e fo (zaa-H(a,a}}dt+ a

K (a, a; T =

(2.18)

a(O)=a J {a(T)=a)

Problem 2.4. Find the relation between the Wick symbol of the evolution operator and the propagator. (Hint: Use the analog of formula (2.45) in Section 2.7 of Chapter 2.)

3. Feynman path integral for the harmonic oscillator As a first impression, one may think that Feynman path integrals are not very practical. Indeed, they are defined as limits of multiple integrals when the number of integrations goes to infinity, and it seems difficult to evaluate them. In fact, this is not so: Feynman path integrals turned out to be very useful for various computations, and for several important cases they can be evaluated exactly. Here we consider the basic example4, the Feynman path integral for the harmonic oscillator.

3.1. Gaussian integration. Evaluation of Feynman path integrals simplifies when corresponding finite-dimensional integrals can be computed exactly for every n (or for large enough n). This is the case for the important class of Gaussian integrals.

Lemma 3.1 (Gaussian integration). Let A be apositive-definite, real symmetric n x nmatrix. We have e-r

(3.1)

where (

,

A

)

(

,4}n dq

JdetA

e2

A ,

) stands for the standard Euclidian inner product in

4The simplest case of a free particle provides a trivial example, since the Lie-Kato-Trotter product formula reduces to e" = (e n A )n , and formula (1.14) gives the same answer as (1.11) for every n.

5. Path Integral Formulation of Quantum Mechanics

256

Proof. Completing the square we obtain

- 2 (A9, 4) + (p> q) _ - 2 (A(q - A-1p), (4 - A-1p)) + 2 (A1p,p), so by the change of variables q = x + A-1P the integral reduces to the °dnx, which is evaluated by diagoe- 2 standard Gaussian integral nalizing the matrix A and using the one-dimensional Gaussian integral

Formula (3.1) is one of the fundamental mathematical facts which is used in many disciplines, from probability theory to number theory. For applications to quantum mechanics, one needs a version of Lemma 3.1 when the decaying exponential factor is replaced by the oscillating exponential factor, as in (1.10).

Corollary 3.1. Let A be a real, non-degenerate symmetric n x n matrix. Then n

(3.2)

fn

dnq

e

e

,

Vl det Al

where the integral is understood in the distributional sense as lima and v is the number of negative eigenvalues of A.

iqI
Proof. Formula (3.2) follows from (3.1) by analytic continuation. It also can be proved directly by completing the square, diagonalizing the matrix A, and using the F'resnel integral formula (1.10).

There is also a complex version of Gaussian integration, given by the following analog of Lemma 3.1.

Lemma 3.2 (Gaussian integration in complex domain). Let C be a complex n x n matrix such that its Hermitian part 2 (C + C*) is positive-definite. We have

fe_'9zd2mz= det C n

(3.3)

n

) stands for the standard Hermitian inner product in C, a, b E C, and d2nz = d2z1 ... d2zn. where (

,

Problem 3.1. Prove Lemma 3.2.

3. Feynman path integral for the harmonic oscillator

257

3.2. Propagator of the harmonic oscillator. The classical harmonic oscillator with one degree of freedom is described by the Lagrangian function

L(q, q) = 2m(g2 - w2g2). The corresponding Hamiltonian operator for a quantum harmonic oscillator is given by P2

H 2m+

mw2Q2 2

The following result is an exact computation of the propagator K(q', t'; q, t) for the harmonic oscillator, using the Feynman path integral in configuration space.

Proposition 3.1. The Feynman path integral for the harmonic oscillator is explicitly evaluated as follows:

1 .P

ft

e

q

V

mw e 2 s riwT sin wT

{(q2+q 2) coswT-2qq'}

'

)q,t

where for

w

(v + 1),

= TU
w

--e-47t _ 7rxu

-

2

v E N, we have

mw 1/ 2irh sin wT

When T - Tv, the right-hand side converges in the distributional sense to e- 9v 8(q - q') for even v, and to e- 9 c(q -+- q') for odd v.

Proof. In this case the (n - 1)-fold integral in (1.19) is Gaussian and can be computed exactly using (3.2). Namely, we have n-1

((qk+1 - qk)Z - e24'k) _ (An-i4, q) - 2(p, 4) + q2 + q' 2 k=O

= wOt, q = (qi,. . qn-1), p = (q, o, ... , o, q') are vectors in and An_1 is the following three--diagonal (n - 1) x (n - 1) matrix: where

. ,

An_1 =

2-e2 -1 0 -1 2-e2 -1 -1 2-E2 0 0

0

0

0

0

0

0

0

0

0

0

0

2-s2 -1 -1 2-e2

5. Path Integral Formulation of Quantum Mechanics

258

It follows from (3.2) that n-1

2

2irihOt

= e-

(3.4)

Lni-e

dqk k=1

m

7riv

n-1

aqk+1--qk)2-E2q}

k^ e

det A_1

that {q2+q _ jjn

1

' V 2iri vn_ 1 is the number of negative eigenvalues of the matrix A_1. It

where v is easy to find det An_1 and (A_1p, p). Namely, put an = det An. Expanding det An with respect to the last row, we obtain the three-term recurrence relation an+1 =

X3.5}

f'-.

-

2\

n n = o, 1, b, ... ,

- an-1 ,

with the initial conditions a_ 1 = o and ao = 1. Recurrence relation (3.5) has two linearly independent solutions zn and z-n, where 2 = z -+ z-1, and a solution an satisfying these initial conditions is given by

-

zn+1

an

z_n_1 - z_1

z_ From here it is easy to obtain that the eigenvalues of the matrix An are given by Ak =

z+z- 1

-2cos

'irk

n+1

,

k = 1,...,n.

Since e = Wn we have z = ei9 where 0 _ E + D (n _ 2 ) and sin n0

_

sin

(1+O(n -1 )} as n -} oo, sine and for n large enough the matrix An_ 1 has exactly v negative eigenvalues whenever T
1.=

which are given by

B11=BnT 1 n 1=

CLn_2

sin(n - 1)0

an_ 1

sin n0

'

B1n-1 T Bn-11

1

sin n0

an

sine

.

Thus we get

/2-(A q2+q

(2 sin n 11p, p) = sin nB

0

cos

(2n 2 1)0 (q2 + q 2)

_ 2 sin 8

qq/

.

Using these formulas and passing to the limit n -> oo in (3.4), we obtain the expression for the propagator for the case T T. The limit T T is evaluated by using the following standard formula in the theory of distributions: llm

t-+o

1

e

`7rt

z(x--y)2 2t

in

= e 4 c (x - y) .

3. Feynman path integral for the harmonic oscillator

259

Remark. It follows from Proposition 3.1 that in the limit w - 0 the propagator of the harmonic oscillator turns into the propagator (1.11) of a free particle.

Remark. For even v, special values Tv are integer multiples of the period w of the harmonic oscillator (see Section 1.5 in Chapter 1), so when t' - t = Ti,.,

the extremal connecting q at time t and q' at time t' exists if and only if q' = q. Correspondingly, when t' - t = T, for odd v, the extremal connecting q and q' exists if and only if q' = -q. For the general case t' - t 74 T, the extremal connecting q at time t and q' at time t' exists for all q and q'. The integer v is the Morse index of the trajectory q(T) the number of negative eigenvalues of the corresponding Jacobi operator J, J=-m

d2

dT

t
-mw2,

with Dirichlet boundary conditions (see Problem 1.7 in Section 1.3 in Chapter 1). Problem 3.2. Compute Weyl, pq, and qp-symbols of the evolution operator for the harmonic oscillator by using: (a) formula (3.2) and path integral representations from Section 2; (b) formula (3.G) and formulas (3.12), (3.18) and (3.19) in Section 3.3 of Chapter 2.

Problem 3.3. Show that the matrix symbol of the evolution operator for the

T - 2 wT} by using: (a) series (1.8) harmonic oscillator is K(a, a; T) = in holomorphic representation; (b) Lemma 3.2 and path integral representation from Section 2; (c) formula (3.6) and result of Problem 2.4. exp{aae_

3.3. Mehler identity. It is instructive to compare the closed expression (3mw

q, t} =

.G}

hsin wT

e 2h Size wT

{(q2+q 2) cos wT-2qq'} l

for the propagator of the harmonic oscillator with the series (1.8). Putting

x=

mw 2

q,y=

rvw

q' and using the explicit formula for the normalized

/L

eigenfunctions

=

1

mw 2

1

2nn

e

9 Hn(x)

corresponding to the eigenvalues En = hi(n+ 2), where Hn(q) are classical Hermite-TchebyschefF polynomials (see Section 2.6 in Chapter 2), we obtain the series

K(q',t';q,t) =

Hn(x)Hn(y),

5. Path Integral Formulation of Quantum Mechanics

260

which converges in the distributional sense. Setting z = e_ T and comparing with (3.6), we get the formula 00

(3.7)

zn n

-H(x)H(y) =

1

1- z

2 exp

2xyz - (x2 + y ) z2 ,

1- z2

where z

±1, and the square root in the right-hand side is understood as in Proposition 3.1. When z C 1, formula (3.7) is the classical Mehler identity from the theory of Hermite-Tchebyschefl polynomials. Thus we obtained a distributional form of the Mehler identity for = 1 by computing the propagator of the harmonic oscillator in two different ways. Formula (3.6) shows that the propagator K(q', t'; q, t) is a smooth funcj

tion of q, q' and T = t' - t whenever T

Tv, and it is singular at T = T.

Corresponding eigenvalues of the evolution operator U(TU ) are so that when v is even we have U (T) ) = e 9v I, and therefore

K (q , t + T ; q, t) _

2},

2 ' -q

iriU e

in perfect agreement with Proposition 3.1. For odd v using the series (1.8) we obtain Oa

= e+

(-1)P, n=o

where Pn are projection operators on the eigenspaces

H(-q) = (-1)'H(q), in this case we have

4'n

of H. Since

_'irw e

2 q+q'}

which again agrees with Proposition 3.1.

4. Gaussian path integrals It was already mentioned in Section 1 that in the semi-classical limit h -> 0 the leading contribution to the propagator K(q, t'; q, t) is given by the classical trajectory qcl(T). This suggests to represent the paths ry = q(T) E P(I[8)9;t as q(T) = q01(T) +y(T), where y(T) the quantum fluctuation part - satisfies Dirichlet boundary conditions y(t) = y(t') = 0. It follows from the principle of the least action (see Section 1.2 in Chapter 1) that (4.1)

S(qt + y) = S+

f (m- V"(ql(T))y2)dT + higher order terms in y,

where Sc1 = is the classical action. Similarly, for the case of several degrees of freedom, (4.2)

S(qt + y) = S1 + 2

f

,7(y)y dT + higher order terms in y,

4. Gaussian path integrals

261

where J is the corresponding Jacobi operator (see Problem 1.7 in Section 1.3 of Chapter 1). It is remarkable that the Gaussian path integral f t'3(1 dT iy

e

(4.3)

Jy(t')=O y(t)=0

over the fluctuating part can be evaluated explicitly in terms of the regularized determinant of the second order differential operator J. Here we do this calculation for the case of the free particle and harmonic oscillator, and give another interpretation of formulas (1.11) and (3.6) for the propagators. The Gaussian path integral (4.3) also plays a fundamental role in the semi-classical asymptotics which we discuss in Section 6.1. In general, formula (4.2), with higher order terms in y, and Gaussian integration over y, constitute a basis for the perturbative expansion of the propagator.

4.1. Gaussian path integral for a free particle. The propagator of a free quantum particle is (4.4)

K(q', t'; q7 t) =

'm

e

27ri hT

im(q-q1 2

I

2hT

im e2

t' 2 ft q dT q

and the corresponding classical trajectory is

qi()=q+(r-t) q-q T

Tt -t.

Using the decomposition q(T) = gc1(T) + y(T), where y(t) = y(t') = 0, we obtain

S(q) = where

= -21

f mdT = Sl - S(y), rn

,2

dT =

rm(q - q')2 2T

t

Assuming that 1q = 1y under the "change of variable" q = qci -+- y, we can rewrite the Feynman path integral for a free particle as K(q', t'; q, t)= e

Sc1

J

e

ft

5 y(t')=O

y(t)=4

Remarkably, the classical contribution e K Sc1 = e

zm2hT ')2

exactly reproduces the exponential factor in the propagator for a free particle. The integral over the fluctuating part the Gaussian path integral for a free particle does

not depend on q and q' and, as we know, coincides with the prefactor in (4.4).

5. Path Integral Formulation of Quantum Mechanics

262

A more conceptual way to interpret this result is the following. Let d A=-D2' D= dT'

be the second order differential operator on the interval [t, t'] with Dirich-

let boundary conditions y(t) = y(t') = 0. The operator A is self-adjoint on L2(t, t'). For any real-valued, absolutely continuous function y(T) satisfying Dirichlet boundary conditions and y, y E L2(t, t'), we have by using integration by parts (Ay, y)

-

y dr =

2dr.

t

t

The "integrand" in the fluctuation factor

I

e2imftt y dT

3

fy(t')=O y(t)=O

is the exponent of the quadratic form of the operator A, and in accordance with the finite-dimensional formula (3.1) it is natural to expect that this Gaussian path integral is proportional to (det A)- 1 of course, the problem here is to understand what we mean by a determinant of a differential operator. Clearly, it should be defined by some regularization of the divergent infinite product [J1 an, where ) are non-zero eigenvalues of A. The most natural and useful regularization is given by the so-called operator zeta-function. Namely, let A be a non-negative self-adjoint operator

in the Hilbert space ° with pure point spectrum o C a C a2 C ..., such that for some ae > o the operator (A + I) -a is of trace class. Then the zetafunction (A(s) of the operator A is defined for Re s > a by the following absolutely convergent series:

(A(s)=

1 1t

If BA(s) admits a meromorphic continuation to a larger domain containing the point s = 0 and is regular at s = 0, then we define a regularized determinant of A by (4.5)

det'A = exp {

-

(0)}.

Here the prime on the symbol det indicates that zero eigenvalues are excluded from the definition of an operator zeta-function. In the special case when 0 is not the eigenvalue of A, it is customary to denote the regularized

4. Gaussian path integrals

263

determinant of A by det A. We will also write

det'A = ]J' fin, A >0

where the prime indicates that the infinite product is regularized by the operator zeta-function. We have CAA (s) = CA (s) (s) for c> o, so that det'cA = c' o} det'A, which shows that CA(0) plays the role of "regularized scaling dimension" of

the Hilbert space 9° (with respect to the operator A). When dime = n C oo and A > o, then A (o)= n and ( (o)= log a1 + + lag an, and we recover the usual definition of det A.

This basic outline works for the general case of elliptic operators on a compact manifold M. In quantum mechanics, only determinants of differential operators on one-dimensional5 manifolds M = [t, t'] or M = Si appear. Their systematic study will be presented in Section 5, while here we consider the simplest case of the second derivative operator A - = -.D2 associated urn 2 with a free particle. The corresponding eigenvalues of A are an = (-f-) , n=12... and we have for the operator zeta-function ,

T S

CA(s) =

r

where C(s) is the Riemann zeta-function. Using classical formulas C(0) _

-2

and C'(0) _ - 2log 2ir, we obtain (4.6)

and ((0) _ - log T -log

1

(A(O) _ - 2

-log 2T.

Thus for the operator A = -D2 on the interval [t, t'] with Dirichlet boundary conditions we have (4.7)

det A = 2T.

The formula

e9 ftt' ydr ti

1

2

y

m 27rihT

y(t)=0

agrees with our interpretation that the Gaussian path integral is proportional to (det A). The coefficient of proportionality cmah _ is determined by comparison with the actual propagator for a free particle. Remark. In Section 3.1 of Chapter G we will prove that for Gaussian Wiener integrals the corresponding constant is . 5 Higher-dimensional manifolds are used in quantum field theory.

5. Path Integral Formulation of Quantum Mechanics

264

Problem 4.1. Show that the propagator of a particle in a constant uniform field f is given by the following formula: 2

m

/,

+

27.3

fi 2m

,

e

.

(Hint: Use Lagrangian function L = 2 mq2 + fq.)

4.2. Gaussian path integral for the harmonic oscillator. It is remarkable that the same interpretation (with the same constant cm,n) holds for the harmonic oscillator. Namely, according to Proposition 3.1 we have e tm f t' (q2-w2g2)dTcjYq

K(q', t'; q, t) =

mw eXp{ imw 2 +q /2 )coswT-2qq) ((q 2h sin wT sin wT

Now solving classical equations of motion with the boundary conditions q(t) = q and q(t') = q', we readily compute that for T Tv,

Sit = 2 J

W 29)dT - 2 s n wT ((q2 + q

t

2)

cos wT - 24q ),

so that e *Sd' is exactly the exponential factor in the propagator. For the the Gaussian path integral for the harmonic oscillator

fluctuating factor

- we have

1

e2

m

=

f t' ( 2-w2y2)dT

7r2h det Aw '

y(t)=o

-

where Aw = -D2 w2 is the second order differential operator on the interval [t, t'] with Dirichlet boundary conditions. Thus to justify, in this case, the interpretation of the fluctuating factor in terms of the regularized 2 sinwT when T T. determinant, we need to show that det Aw = w

At a heuristic level, this can be done by the following beautiful computation, which goes back to Euler. Namely, the eigenvalues of Aw are 7rn 2 w2 (o is not an eigenvalue for Aw since T L Tv ), and an w } = (--) T

-

we have

AW det A0

O°

n=1

A(w) n an (o)

-

°° n=1

wT 2 2det

1

22) -

sin wT wT

Since det Ao = 2T, we get the result! For the rigorous derivation, it is more where w > o, inconvenient to consider the positive-definite operator

stead of the operator A.

4. Gaussian path integrals

265

Lemma 4.1. Let Aiw = -D2 + w2 be the second order differential operator on the interval {t, t'] with Dirichlet boundary conditions. Then 2 sink wT

det AzW =

w

Proof. Denoting Ci(s) = CA(s), we have for Re s > 2 , 00

1

e

(s) =____Jo

C*

1

2F(s)

Jo

_w2

n2 n2

D°

e

rx

O° _w2x t9 e

s

T

xs

T2

dx x

dx x

1

2w 2s'

eis the Jacobi theta series. Using the Jacobi

where 9(x) = inversion formula

x

we get the following representation: f°° -wZx 1 T

222I'(s) JO

+

s_ ! x8

rx)

e

TI'(s - Z)

+

1

(T' T

_ -2

x ao

00

Jo

e

2w23

(4.8)

dx n2T2

>ii:

n=1

2 r() wWs Zs-i

23 +

nT s 2

r(s)

xS2 - 1 dx x

1

00

+

e

I

n=1

w

K 1(2wnT),

where

K8(x)

u

e o

u

x > o,

is Macdonald's K-function (modified Bessel function of the second kind). Since K8(x) = D (e _x ) as x ---} oc, uniformly in s E C on compact subsets, it follows from representation (4.$) and the Weierstrass M-test that Caw (s) admits meromorphic continuation to the entire s-plane with simple poles sT (s) = 1, we obtain C(O) = ---- 2 . Using at s E -- 2 -+- Z>o. Since classical formulas e_x

2x

2

2

and P(2) _

..

Ki(x) = K_1(x) =

(4.9)

we obtain from (4.8), 00

ds

(o) = logw

wT T n--1

1

e-2nWT

n

= log w - wT - log(1 -- e-2wT ),

266

5. Path Integral Formulation of Quantum Mechanics

so that (O)1 _ 2 sink wT

det AzW = exp

ds

J

0

w

Corollary 4.1. Let B = -D2 +iwD be the second order differential operator on the interval [t, t'] with Dirichlet boundary conditions. Then det $2W =

det A.

Proof. The substitution y(T) = eZwT f ('r) transforms the eigenvalue problem BZWy = ay to the eigenvalue problem AW f = Af.

Let I be the identity operator in L2(t, t'). In addition to the regularized determinant det'A of the differential operator A = -D2, it is possible to define the so-called characteristic determinant of A - an entire function det(A - Al) on the complex A-plane, whose only simple zeros are the eigenvalues An = (ten) 2 T Indeed, for Re(A1 - A) > 0 consider the zeta-function Oa

(4.10)

00 1

(A-\I(5) _

e--s

_ a} s n=1 (fi n

n=1

'

where we are using the principal branch of the logarithm. As before, CAAl(s)

is absolutely convergent for Res > 2, admits a meromorphic continuation to the complex s-plane, and is regular at s = 0. Thus for Re(A1 - A) > 0 we define 00

det(A - Al) _ fl'(An - A) = eXP { n=1

- alas AI (°)},

which is a holomorphic function of A. Now suppose that det(A - Al) is already defined for Re(Anr - A) > 0 and is holomorphic. To extend it to the domain Re(AN+1

- A) > 0, we set

N det(A - Al)

_

00

fl(Ak - A) fi' (An -

A),

n=N+1

k=1

where the regularized product is defined by the "truncated" zeta-function 00

CA-al (s) -

/fi

n=N+1 `n

As

as Oa

(4.11)

fi' (An - A) - exp { n=N+1

a (N+l)

as

4. Gausslan path integrals

267

It requires to prove that A I (s) admits a meromorphic continuation to the complex s-plane and is regular at s = 0. This is done by using the representation

I(s) __ 2rrs1 f Ji t

c(AN+1)

1

eN+l (x)xs d

eN+l (x)xs x + 2r(s)

,

w here

_) x-1.

(x) -

9N+1(x)

n= 1

Since Re (An -- A) > o for all n > N, the first integral in this formula is absolutely convergent for all s E C and represents a holomorphic function. Using the Jacobi inversion formula and expanding e', e-F'1x, ... , into power series in x, we conclude, just as before, that the second integral admits a meromorphic continuation to the s-plane with simple poles at s E - 2 + Zoo. Since for M > N 00

M

fl' (A - A)

00

II (A - A) fi' (A - A),

n=N+1

k=N+1

n=M+1

det(A - Al) is well defined, and is an entire function of A with simple zeros

at A = A. To obtain a closed (and simple) formula for det(A - Al), observe that for Re(A1 - A) > 0 it follows from (4.10) that (s)

°°

=

1

(A - A)s+

This series is absolutely convergent for Re s> -2, so that for Re(A1-A) > 0 we have 00

(4.12)

d

to det A - AI

a2 - _aaAA_AI(0)=AA.

By analytic continuation, formula (4.12) is valid for all A 2 = ) ,we have

n(T

00 1

T

A-A n

1

A. Since

1 2\/cotvT-

so that

det(A - AI) _

sin V5T

and comparison between formula det(A + w2I) = det Aiw and Lemma 4.1 gives c = 2. Thus we have proved the following result.

5. Path Integral Formulation of Quantum Mechanics

268

Lemma 4.2. The characteristic determinant det(A - Al) of the operator A = -D2 on the interval [t, t'] with Dirichlet boundary conditions is well defined, and is an entire function of A. It is given explicitly by

det(A - Al) = det A [J

(l -

A

n-1

)_

2sinv'T '/

Problem 4.2. Prove the Jacobi inversion formula. (Hint: Use the Poisson summation formula

00

00

n=-00

n=-00

f(2irn),

where f E 9 (IR) and f is the Fourier transform of f . }

Problem 4.3. Prove formula (4.9).

Problem 4.4. Let LW

m

x2+

.2+z2 + eB x'

±

---

be the Lagrangian of a classical particle moving in the constant uniform magnetic field B = (0,0, B ). Show that the propagator of the corresponding quantum particle (see Example 2.3 in Section 2.4 of Chapter 2) is given by m , wT im (z -- z'}2 2 K(r , t , r, t)-exp + wcatwT[(x - x ) 2 27 riT sin wT 2h T +(y y')2] + 2w(xy' -

-

(Hint: Use Corollary 4.1. )

5. Regularized determinants of differential operators Here we study the characteristic determinant det(A - Al) of the SturmLiouville operator

A=-D2+u(x), D-, on the interval [0, T] with Dirichlet or periodic boundary conditions, and its generalizations to the matrix case.

5.1. Dirichlet boundary conditions. Suppose that u(x) E Cl ([O, T], ][8). The operator A is self-adjoint on L2(0, T) with domain D(A) _ {y(x) E W2°2(O,T) : y(0) = y(T) = 0},

where W2'2(0, T) is the Sobolev space. It has a pure point spectrum with ,accumulating to oo. Moreover, simple eigenvalues A1
as n - oo,

22 (5.1)

1

2

7,2 + c - O(n- ),

where

c=

/'T

J

0

u(x)dx.

5. Regularized determinants of differential operators

269

Consider first the case A > o. Putting 00

9A(t) = n=1

we have for Re s > 2 , 00

(5.2)

J

(A(s) =

4A(t)ts

dt

A(t)ts+L 0A(t)ts -. t I'(s) t

P(s)

Since OA(t) = O(e-alt) as t -> oo, the first integral in (5.3) converges absolutely for all s E C and represents an entire function. Using asymptotics (5.1) and the Jacobi inversion formula we get as t - 0, a 1 A(t) = (t9 (1+O(t))= where

T a_=7=

(5.4)

1 ao=--

and t9A(t) = O(/). Thus for the second integral in (5.3) we have r(S)

J l A(t)ts

t - (S

a2)1

a0

r(s) + SF(s) + r(s) Jol A(t)ts

t'

so that it admits a meromorphic continuation to the half-plane Re s> and is regular at s = o. Therefore we can define

1

2

00

det'A = fi' An = exp { - ds (0)}. n=1

Now repeating verbatim the arguments in the proof of Lemma 4.2, we see

that for Re(AN - A) > 0 the truncated zeta-function '1'(s) admits a meromorphic continuation to Re s> - 2 , and is regular at s = 0. Defining the regularized product 1IETh=N+l(An -A) by the same formula (4.11), we see

that

N

00

det(A - Al) _ [J(Ak - A) fi' (fin - A) k=1

n=N+1

is an entire function of A with simple zeros at A. To remove the assumption

A>0, replace A by A = A + (a - A1)I > 0, where a > 0. Thens det(A - Al) = det(A - (A + A1

- a)I),

6The definition of det'A does not depend on the choice of a > 0.

270

5. Path Integral Formulation of Quantum Mechanics

and det' A =

det A

if 0 is not an eigenvalue of A, if 0 is an eigenvalue of A.

a-1 det(A + Al)

Let yl (x, A) be the solution of the second order differential equation

-y" + u(x)y = Ay

(5.5)

on [0, T], satisfying initial conditions (5.6)

y(0,A)=1.

yi(O,A)=0,

It is known that for every 0 < x < T, the solution yl (x, a) is an entire function of A of order 2 and as A -* oo, (5.7)

yi(x,A) =

si

x

(IA_1eRex).

The entire function d(A) = yl (T, A) has simple zeros at the eigenvalues .fin of the operator A, and has the following Hadamard product representation: (5.8)

d(A)=cA5 fi (i_). 12

Here c is aconstant, S = 1 if 0 is an eigenvalue of A, and b = 0 otherwise.

Theorem 5.1. The characteristic determinant det(A - Al) is given by the simple formula

det(A - Al)

2d(A).

Moreover,

det(A-AI) det'A

(1 - - A1 jo n

fin)

'

which fixes the constant in (5.8) as c = Z (-1)det'A. Proof. Since both functions are entire, it is sufficient to prove the equality det(A - Al) = 2d(A) for Re(A1 - A) > 0. In this case, using 00

(A-i(s) - F(s)

T' e-(A-aI)tts dt

J

for Re s > 2 , and differentiating under the integral sign we get 3A-j(s) =

r1s

/'00

J

- ttsdt, 'I4e- (Aa7)

5. Regularized determinants of differential operators

271

which is now absolutely convergent for Re s > - 2 Differentiating with respect to s at s = o, we obtain .

a2

f Tre)cdt

AI)-1

0

It follows from (5.1) that the operator Ra = (A - Al)-' -the resolvent of A - is of trace class. Thus all our manipulations are justified and we arrive at the following very useful formula:

log det(A - AI) _

(5.9)

which generalizes the familiar property of finite-dimensional determinants. To compute the trace in (5.9), we use the representation of Ra for A as an integral operator with the continuous kernel RA(x, ) (cf. formula (2.21) in Section 2.2 of Chapter 3). Namely, let y2(x, A) be another solution of (5.5) with boundary conditions y2 (T, A) = 0 and y(T, A) = 1, so that

W(yi, ya)(A) _ yi(x, A)y2(x, A) -yi(x, A)ya(x,A) _ -d(A).

Using the method of variation of parameters, for the solution of the inhomogeneous equation

-y"+u(x)y=Ay+f(x), A7A, satisfying Dirichlet boundary conditions we get

f Ra(x,e)f T

y(x) = where

_yl(x, A)y2(, A)

Ra(x,e) _

(5.10)

-

d(A) A)y2(x,A) d(A)

if x

if x > .

Since Ra is a trace class operator on L2 (0, T) with the integral kernel Ra (x, ), which is a continuous function on [0, T] x [0, T], its operator trace equals the "matrix trace", 'IRa

= JTR(xx)dx

1

T

We evaluate the last integral by the same computation used in the proof of Proposition 2.1 in Section 2.1 of Chapter 3. Namely, put (x, A) = (x, A), Ua and consider the following pair of equations:

a

-+ u(x)?J, = a 1 + y1, -y'2 + u(x)y2 = Aye.

5. Path Integral Formulation of Quantum Mechanics

272

Multiplying the first equation by y2 (x, A), the second equation by yl (x, A) and subtracting, we obtain y1 y2 = y1 y - y

y2 = - W `y1 , y2)',

so that T

f

yi(x,A)y2(x,A)dx

= - W(i,y2).

This formula is valid for any two solutions of the differential equation (5.5). Using boundary conditions for the solutions yl and y2, we finally get T (5..12)

yi(x, A)y2(x, A)dx

L

=

i1(T, A).

Thus we have proved that for Re(A1 - A) > 0,

TrRa =

(5.13)

d 1°g d(A)

which implies that

det(A - Al) = C d(A)

(5.14)

for all A E cC and some constant C. It follows from (5.7) that

d(-µ) _

(5.15)

eVT

(1

as

-+-

µ -> +oo.

Thus in order to determine the constant C in (5.14), it is sufficient to compute the asymptotics of det(A + µI) as µ -i -}-oo We have CA+I(s)

- I(s)

A(t)e J1

t + rrsl / 1 A(t)e -µt is t

lis dt

The first integral is an entire function of s whose derivative at s = o exponentially decays as u - boo. For the second integral we have

rls\

is

dt

_ rlsl 11A(t)e_tts dt l

J

1

+

J 1

a_ 1

+) e-µt isdtt

.

as t - 0, the first integral is absolutely convergent as µ - +oo. For the for Res > -2 and its derivative at s = 0 is Since 9A(t) =

5. Regularized determinants of differential operators

273

remaining integral we have 1

(°)

1 az Jo

t s

= a-2

t t

e

P(s)

(

i I 2) - µ

\I'(s

+a

J

-t s-2 dt e t t

dt\

(F(s) It is elementary to show that the s-derivative of this integral at s = 0 has asymptotics finally obtain

2 /i- ao log jC + O(e-µ/2) as µ -> +oo. Using (5.4) we

det(A+I)

evT (i+o(-)) = _7

as j - + oc, and comparison with (5.15) gives C = 2.

Remark. When zero is not an eigenvalue of A, its inverse A' is a trace class operator and

det(A - AI) det A

- detF(I -

where detF is the Fredholm determinant.

Remark. In Section 6.1 we will use Theorem 5.1 for evaluating the fluctuating factor in the semi-classical asymptotics of the propagator, and in for evaluating Gaussian Wiener integrals. Section 3.1 of Chapter 6 A similar result holds for the matrix-valued Sturm-Liouville operator

with Dirichlet boundary conditions. Namely, let U(x) _ {u(x)}1 be a Cl-function on [0, T] which takes values in real, symmetric n x n matrices, and consider

A = -D2I + U(x), where In is the n x n identity matrix. The differential operator A with Dirichlet boundary conditions is self-adjoint on the Hilbert space L2([0, T], C) of C'-valued functions, and has a pure point spectrum accumulating to oc. Its

regularized determinant det'A and characteristic determinant det (A - Al), where I is the identity operator in L2([0, T], Ca), are defined as in the n = 1 case. Let Y (x, A) be the solution of the differential equation

-Y" + U(x)Y = AY satisfying initial conditions

Y(0, A) =0,

Y'(O,A) =Ins

and put D(A) = det Y(T, A). The entire function D(A) has properties similar to that of d(A), and the following analog of Theorem 5.1 holds.

274

5. Path Integral Formulation of Quantum Mechanics

Proposition 5.1. The characteristic determinant det(A - Al) is given by the formula

det(A - Al) = 2nDO. Moreover,

det(A - AI) _ s (-A) det'A

fi(i__),n

n

where S is the multiplicity of the eigenvalue A = 0.

Problem 5.1 (Gelfand-Levitan trace identity). Prove that

(A-A) - c} = 2T

ai.(alrla - u(0)+u(T)

1

n1

where An°} = (lrri)2 and c =

4

'

f' u(x)dx.

5.2. Periodic boundary conditions. As in the previous section, we assume that u(x) E C1 ([0, T], Ift) . The Sturm-Liouviiie operator A = _D2 -+ u(x) with periodic boundary conditions is seif-adjoint on L2(0, T) with the domain D(A) _ {y(x) E W2'2(0, T) : y(0) = y(T) and y'(0) = y '(T)}. It has a pure point spectrum with the eigenvalues

.gyp < 1 G 2 < ... < 2n-1

2n < .. .

accumulating to oo. Moreover, as n ---> oo, (5.16)

+ c + 0(n-2),

A2n-1 =

A2n =

7,2

+ c + 0(n-2),

where c is the same as in (5.1). Replacing, if necessary, A by A - (A0 + a)I with a > 0, we can always assume that Ao > 0, and define e-.ant

A(t) =

t>0.

n=0

Using asymptotics (5.16), we get that as t

A(t) = et

l

- 0,

(1+0(t)) =

+ 0(),

where a_ = 2T as in (5.4), but ao = 0. This allows us to define the regularized determinant det'A and the characteristic determinant det(A Al) exactly as in the previous section. Since ao = 0, we now get as µ -* +oo (5.17)

det(A+µI) = eVT (1 +

5. Regularized determinants of deferential operators

275

(see the end of the proof of Theorem 5.1). Here we denote7 by yl (x, A) and y2 (x, A) solutions of the Sturm-Liouville

equation (5.5) satisfying initial conditions yl (0, A) = 1, yi (0, A) = 0 and y2 (0, A) = 0, y2 (0, A) = 1. Solutions yl and y2 are linearly independent for all A and the matrix

Y(x

-

(yi(x,A) A)

y2(x, A)/

satisfies the initial condition Y(0, A) = 12, where I2 is the 2 x 2 identity matrix, and has the property det Y(x, A) = 1. For fixed x the matrix Y(x, A) is an entire matrix-valued function of A having the following asymptotics as

cos /5x

(5.18)

sin V'x

cos V'Xx

By definition, the monodromy matrix of the periodic Sturm--Liouville problem is the matrix T(A) = Y(T, A). The monodromy matrix satisfies det T(A) = 1 and is an entire matrix-valued function. The following result is the analog of Theorem 5.1 for the periodic boundary conditions.

Theorem 5.2. One has det(A - Al) _ -det2(7'(A) - Ia) _ y1(7', A) + y2(7', A) - 2,

l

where det2 is the determinant of a 2 x 2 matrix. Moreover,

det(A - Al) det'A

_ (-A)8fio

(1 -

where {fin}o are the eigenvalues of A, and 0 < S < 2 is a multiplicity of the eigenvalue A = 0.

Proof. The proof follows closely the proof of Theorem 5.1, and we will assume that A > o. First, in exact analogy with (5.9) we obtain that for A

d log det(A - Al) _ - T where Ra = (A - AI)-1. To get a closed expression for the integral kernel R(x, ) of the operator Ra, we use the same variation of parameters method as in the previous section, but now for periodic boundary conditions. As a There should he no confusion with the notation in the previous section.

5. Path Integral Formulation of Quantum Mechanics

276

result, we obtain that the symmetric, continuous kernel Ra(x, ) is given for x < by the formula R(x,

) = - (yi(x, A), y2(x, A)) (T(A) - I2)1T(A) =

2

{(T(A)

- 12)-'T(A)Z(x,e;A)},

where 1r2 in the last formula is the matrix trace and Z(

A)- ( y1(x,A)y2(,A)

-

-yi(x,A)yi(,A)

y2(x,A)y2(,A) -y2(x,A)yi(e,A)

As in the proof of Theorem 5.1, we need to compute fZ(x,x;A)dx. It readily follows from formula (5.11) and the definition of the monodromy

matrix that Therefore

f

TZ(x,x;A)dx

da logdet(A-AI) ='I42 ((T(A)

= T'(A)T(A).

- I2)-1T(A) I

= d logdet2(T(A)-I2),

and det(A - Al) = Cdet(T(A) - I2). To determine the constant C we set A _ -µ - +oo and compare asymptotics (5.17) with the asymptotics

det2(T(-µ) - IZ) = 2 -T(-µ) = 2 - 2 cosh T(1 + O(µ-2 ))

= _eT(1 + which follows from (5.18). Thus C = -1. Remark. In Section 3.2 of Chapter 6 we will use Theorem 5.2 for calculating Gaussian Wiener integrals over the loop spaces.

In the special case u(x) = 0, we have sin V5x yl (x, A) = cos V5x and y2 (x, A) =

so that (5.19)

det(-D2 - Al) = 2(cos

T - 1) _ -4

sine

VT

.

2

Setting A _ -w2 <0, for the operator AzW = -D2 + w2 we get (5.20)

det Aiw = 4 sinh2 2 ,

and also (5.21)

det Ap - - lim

det(A - Al) A

= lim

w-#p

det'2AiW

w

= '2.

5. Regularized determinants of differential operators

277

Remark. Using that the spectrum of AiW consists of double eigenvalues an w } = n = 1, 2, ... , and of the simple eigenvalue w2, we can T derive formulas (5.20)-(5.21) directly, as was done in Section 4.2 for Dirichlet

boundary conditions. There is also an analog of the heuristic computation in Section 4.2: det Aia

2

2

det'Ao - W 11 An(O)2 -

W

ZTZ 2 4 2 WT 11 (1 + 4W2n2) = 7,2 sink 2 .

A similar result holds for the general second order differential operator A = -D2 + v(x)D + u(x) on the interval [0, T] with periodic boundary conditions. Namely, repeating the proof of Theorem 5.2, we have the following

Theorem 5.3. For the differential operator A = -D2 + v(x)D -}- u(x) one has

det A - AI = -e^ 2 fo

xdet2 T A -12 = e- a fo v(x)dam (yi(T,A) + i T A -1- efo v(}dam

where solutions yl,2 (x, a) and the monodromy matrix T(a) are defined by the same formulas as for the case v(x) = 0. In particular, the following result will be used in Section 2.2 of Chapter 8.

Corollary 5.4. det'(-D2 + wD) = 2W sinh 2 . Proof. The proof is an elementary computation, using Theorem 5.3, an explicit form of the solutions y1,2(x, A), and the formula

det'(-D2 + wD) _ - o det(-D2

L

Remark. Since the operator -D2 + wD on [0, T] with periodic bound-

(T)

(T)

2

ary conditions has simple eigenvalues An (a) _ ,n= + iw -oo, ... , oo, we can also repeat the heuristic computation in Section 4.2:

det'(-D2+ wD) _ nH1 (1 + det'(-D2)

w2T2

2

sinh

2T. cv

A similar result holds for the matrix-valued Sturm-Liouville operator

with periodic boundary conditions. Namely, let U(x) _ {u(x)}1 be a C1-function on [0, T] which takes values in real, symmetric n x n matrices, and consider

A = -D2In + U(x).

5. Path Integral Formulation of Quantum Mechanics

278

The differential operator A with periodic boundary conditions is self-adjoint on the Hilbert space L2 ( [o, T], C71) of CT-valued functions, and has a pure

point spectrum accumulating to oc. Its regularized determinant det'A and characteristic determinant det (A --- Al), where I is the identity operator in L2 ( [o, T], C71), are defined as in the n = 1 case. Let Yi (x, a) and Y2 (x, a) be

the solutions of the differential equation

-Y" + U(x)Y = aY satisfying, respectively, the initial conditions

= In , Y1 (o, a) = o and Y2 (o, a) = o, Y2 (o, a) - = In. The monodromy matrix T(A) is defined as the following 2n x 2n block Y1 (o, a)

matrix:

(Yi(T,A) Y' (T, a)

Y2(T, A) Y2 (T, a)and

is a matrix-valued entire function. The analog of Theorem 5.2 is the following statement.

Proposition 5.2. The characteristic determinant is given by the formula det(A - Al) = (-1)71det271(T(A) - I2n), where det271 is the determinant of a 2n x 2n matrix, and

det (A - Al) det A

-

A o

(i

_)' A

an

where 8 is the multiplicity of the eigenvalue A = o.

Problem 5.2, Prove Theorem 5.3. Problem 5.3. Derive Corollary 5.4.

5.3. First order differential operators. Here we continue to assume that u(x) E C1 ([0, T], I[8), and consider the first order differential operator

A = D + u(x) on the interval [0, T] with periodic boundary conditions y(0) = y(T). The equation y -+- u(x)y = Ay

has an explicit solution y(x) = Ce-fo u(r)dr, which is periodic if and only if A = an, where .T

An = uo + 2Tn, n e 7G,

and uo = T

u(x)dx. 0

Thus the spectrum of the operator A coincides with the spectrum of the operator Ao = D +110.

5. Regularized determinants of differential operators

279

Proposition 5.3. For no > o,

- e-u°T,

det(D + u(x)} = 1

and det'D = T for uo = o.

Proof. The zeta-function of the operator A with no > o is given by the series n=-0o an

where ) s = e-5 log with the principal branch of the logarithm. This series is absolutely convergent for Re s> 1. Introducing the Hurwitz zeta-function 00

1

n=o

I n _L

Ui

where Re a > 0 and Re s> 1, we can rewrite cA (s) as s

BA(S) _

Z,

(e

X23 ((5, a) + e( (s, a)) + u ,

a =1-

o

2

i.

It is well known that the Hurwitz zeta-function admits a meromorphic continuation to the whole s-plane with single simple pole at s = 1 with residue 1, and

((0,a) = 2 - a'

as

(0,a) = log T(a) - 2 log 2.

Using the classical formula

F(1+z)I'(l-z)=

7rz

sin 7rz'

we obtain

ds (0) = log

I- log uoT +

u0T

=-log(e 2 -e 2 )+--, 2 so that det(D -{- u(x)) = 1 - e-u0T. Finally, CD(s) = and we get

det D -- lim uo4

det' (D

--- + n)

uo

uo S)

= T.

Remark. For uo <0 one should use the branch of the logarithm with the cut along the positive semi-axis, and the above arguments give det(D + u(x)} = 1

- eu°T.

5. Path Integral Formulation of Quantum Mechanics

280

Remark. One can also consider the operator A = D + u(x) on the interval [O, T] with anti-periodic boundary conditions y(0) _ -y(T). The corresponding eigenvalues are

iri(2n+ 1) , n E Z, T and the passage from periodic to anti-periodic boundary conditions amounts to replacing no by up -+- T . It follows from Proposition 5.3 that for up > o,

det(D+u(x))= l+e_u0T. Remark. Proposition 5.3 is very useful for calculating Gaussian path integrals in the holomorphic representation, discussed in Section 2.4. As an example, consider the harmonic oscillator with the Wick symbol H (a, a) = w (da + 2h). Formula (2.16) expresses the trace Tr e- Tx as a path integral in the holomorphic representation. On the other hand, using the explicit form of the eigenvalues En = wh(n + ), we immediately get oo

e

e

-1 nT =

n=o

1-e

= 2 sink wT' '

Comparing with (2.16), we obtain that (5.22)

I

e

- fo

1

det ( D + w )

fa(o)=a(T) la(0)=a(T)

which should be considered as a special analog of the finite-dimensional Gaussian integration in the complex domain -formula (3.3). Problem 5.4. Give a direct proof of formula (5.22).

6. Semi-classical asymptotics - II Here we consider the semi-classical asymptotics

the asymptotics of the

propagators K (q', t'; q, t) as /t - o. We compare the heuristic method, based on the Feynman path integral representation (1.26), with the rigorous

analysis, based on the short-wave asymptotics, derived in Section 6.1 of Chapter 3. 8 Here the dependence on the Planck constant h is introduced explicitly.

281

6. Semi-classical asymptotlcs W- II

6.1. Using the Feynman path integral. Start with the Lagrangian L(q, q) = Zmg2 - V (q) for a classical particle with one degree of freedom. The propagator Kn(q', t'; q, t) is given by the Feynman path integral (1.20), and we will formally apply the stationary phase method to investigate its

behavior as h -> 0. As in Section 4, we assume that there is a unique classical trajectory qcl(T) connecting points q and q' at times t and t', set q(T) = qcl(T) + y(T), and consider the expansion (4.1), i.e., /'t'

5(q) = Sci + 2m J (y2 - u(T)y2)dT + 0(y3), t

and

where u(T) = /'t'

Soi= J

t

According to the stationary phase method (see Section 2.3 in Chapter 2), the leading contribution to the Feynman integral (1.20) as h - 0 comes from a critical point of the action functional -the classical trajectory qcl(T). Thus we obtain as fii - 0,

K

(6.1)

'

t',

--

t r e SGT

[m

e

ft (Y -u (r) y2 }dr

y

fy(t')=O y(t)=0

fi51q ,

4', t) } .

det A eXP5i l Here A is the corresponding Jacobi operator - a second order differential

operator -DZ - u(T) on the interval [t, t'] with Dirichlet boundary conditions. (We are assuming that the potential V (q) is sufficiently smooth so that

u(T) E Cl([t,t']).) Formula (6.1) is a remarkably simple expression which shows a deep relation between semi-classical asymptotics of a quantum mechanical propagator and classical motion.

the regularized determinant of the Remark. When u(T) = differential operator A = -D2 - u(T) on the interval [t, t'] with Dirichlet boundary conditions can be expressed entirely in terms of the classical trajectory qcl(T). Namely, differentiating Newton's equation (6.2)

mj

-V'(q),

satisfies the differwith respect to r, we find that the function y(T) = 0, the function9 ential equation Ay = 0. When y(t) = Y1(T) 9Here we assume that y(t)

0, so that V'(q)

y(T)

0.

5. Path Integral Formulation of Quantum Mechanics

282

satisfies initial condition (5.6) (where the interval [0, T] is replaced by the interval [t, t']). According to Theorem 5.1, we have in this case,

det A = 2y1(t') _ -2m

(6.3)

V'(q) In order to find the solution y( r) of the differential equation Ay = 0 for the case y(t) L 0, observe that the Wronskian yly - yyi of its two solutions is constant on [t, t']. Using (5.6) we get

and solving this differential equation we obtain y1(T)

- y(T)y(t)Jf

dS T

c

y (s)

Thus we get the formula t'

(6.4)

det A =

2(T)'

which expresses the fluctuating factor in the semi-classical asymptotics of the propagator in terms of the classical motion. Similarly, for the case of n degrees of freedom, when

L=

- V(q),

and V (q) E C3 (IIBn, I), we obtain (6.5)

K

' ' (q,t,q,t)

m

/detA

as h -* D, where A = -D2 -- U(r) and z

n

2

(c1(T))} aq

aqiaq3

6.2. Rigorous derivation. Let K(q, q', t) be the fundamental solution of the Schrodinger equation the solution of the Cauchy problem (1.3) and (1.5). Since Kn(q', t'; q, t) = Kn(q', q, T), where T = t'-t, we need to find the asymptotics of a fundamental solution K( q, q', T) as h - 0. The solution of this problem can be divided into two parts.

1. Find the short-wave asymptotics - asymptotics as It - 0 of the solution T) of the Cauchy problem for the Schrodinger equation ah

at - - 2m aq2 + v

with the initial condition

6. Semi-classical asymptotlcs -- II

283

where s(q), (q) E C°°(]I8, ]18), and the amplitude support. 2. Using the representation a (q

f

- qo) -

(q) has compact

00

where cp has compact support and satisfies o(go) 1, express Kn(Q, qo, T) as an integral over d of solutions of the Schrodinger

equation with initial amplitude co(q) and initial phase s(q, ) _ (q - qo). Use asymptotics from part 1 to evaluate the resulting integral by the stationary phase method as h

- 0.

The problem in the first part was solved in Section 6.1 in Chapter 3, so we proceed to the second part. For fixed , let /'p(q, t; ) be the solution of the Schrodinger equation with the initial amplitude (q) and the initial phase s(q) _ (q - qo). By the superposition principle, (6.6)

f

90,7') -

7', )d

Let y(t; q, ) be the classical trajectory -the solution of Newton's equation (6.2) with the initial conditions

(O;q,) = 4 and

(6.7)

(O;q,)

m As in Section 2.3 of Chapter 1, here we assume that the mapping q H Q = ry(T; q, ) is a diffeomorphism, and denote by q = Q) the corresponding inverse function, (6.8)

Q), ) = Q.

'Y(T;

The asymptotics of 'bh(Q, T; c') as h -> 0 is given by formula (6.13) in Section

6.1 of Chapter 3, where p(q) _

q = , so that the characteristic ending

at Q has the initial momentum e and the initial coordinate q = q(, Q). Substituting this expression into (6.6), we obtain 1

°°

Kh(R,vo,T) x exp {

l

znrt

2

-00

(s(Q,q(,Q);T) + (q(,Q)

-

o))}

To apply the stationary phase method to this integral, we need to find the critical points of the function S(Q, q(, Q); T) + (q(c, Q) - qo) Using the formula as

(Q,q;T)

= -p = -,

5. Path Integral Formulation of Quantum Mechanics

284

which follows from Proposition 2.2 in Section 2.3 of Chapter 1, we obtain

(S(Q,q(,Q);T) +(q(,Q) - qo))

_ - a Q) + q(,Q) - 4'0 +

Q) = q(, Q) - 40,

so that a single critical point o is determined from the equation qo. The prefactor in the stationary phase method is given by

Q) _

1

-2

(qo)

which can be simplified using cp(go) = 1 and the equation 8ry ey aq

aaqa

which follows from (6.8). Thus the final expression for the semi-classical asymptotics is (6.9)

Kn(Q,qo;T)

1

It is remarkable that formula (6.9), after the identification Q = q' and qo = q, coincides with formula (6.1)! Indeed, they have exactly the same exponential factors, and equality of the corresponding prefactors follows from the following result.

Lemma 6.1. Let ry(t; q, ) be the classical trajectory with the initial conditions (6.7). Then (6.10)

a (T;q,) - 2m det A,

where A is the differential operator -D2 - u(t) on the interval [0, T] with Dirichlet boundary conditions, and u(t) -

V"(-y(t, q, )) m

Proof. Differentiating equation (6.2) with respect to , we obtain that y(t) _ (t;q,) satisfies the differential equation

mu = -V"(7(t,q,))y, so that Ay = 0. Differentiating initial conditions (6.7) with respect to , we get 1

y(O)=O and y = m, so that by Theorem 5.1, det A = 2my(T).

7. Notes and references

285

Remark. One can get another remarkable formula for the prefactor in representation (6.9). Namely, differentiating (Q, q; T) _ - with respect to Q we get

a

a2s

agaQ - - aQ' where Q = -y(T; q, ). Thus (6.9) can be rewritten entirely in terms of the classical action

i

2

1

S(Q,go;T

agaQ (Q, qo; T)

.

Remark. When assumptions in Section 2.3 of Chapter I are not satisfied, there are several characteristics connecting points qo and Q, and the situation becomes more complicated. In this case we have Kn(Q, qo, T)

(T;qo,j)2

e

J

where j is the initial momentum, and µj is the Morse index of the characteristic ry(t; qp, ) (see Section 6.1 in Chapter 3). The case of n degrees of freedom is considered similarly. Using shortwave asymptotics (6.14) in Section 6.1 of Chapter 3, we obtain as h - 0, (6.12)

Kn(Q,90, T)

where Q = ry(t, qo,

so).

a

(2irih)- 2 det I aQ 9'0 o))

- qa

9Q _ OgDQ _._ (6.12) can be rewritten as

Here det

028 ago9Q

e S(Q,go;T)

Using the equation

___ -

K(Q,qo;T)

2

(2irih)1 det

J 328

n

l 3gz3QaQ qz 7

a,j=1

2 (30Q(Q,qo;T))

x

S(Q,go;T

is known as the van Vieck determinant.

Problem 6.1. Justify all computations in this section.

7. Notes and references The path integral approach to quantum mechanics was developed by Feynman in his 1942 Princeton thesis, and was published in [Fey48]. In addition to the classic

text [the best introduction to Feynman path integrals in configuration space, written from a physics perspective

we also refer the reader to a modern

286

5. Path Integral Formulation of Quantum Mechanics

textbook [DRO1]. The Feynman path integral in the phase space was introduced by Feynman [Fey5l] in 1951 and by Tobocman [Tob56] in 1955. Nowadays it is a very useful method in quantum field theory (see, e.g., Faddeev's Les Houches lectures [Fad76] and the monograph [FS91]). Our derivation of the Feynman path integral in Section 1 is standard and follows monograph [RS75], which also contains a complete proof of the Kato-Lie-Trotter formula. As we mentioned in Section 1, this approach establishes only the L2-convergence of finite-dimensional approximations like (1.15) and (1.23) to the propagator. We refer the reader to the paper [Fuj8O] for the proof of the convergence in other topologies on functional spaces, and to the monograph [AHK76] for the rigorous definition of Feynman path integrals as infinite-dimensional Fresnel integrals.

In Section 2 we follow the outline in [Ber7la] and [Fad 76, F S 91 ] , and rigorously derive formulas (2.1) and (2.6) for the pq and qp-symbols of the evolution operator by using explicit relations between the symbols and the propagator. However, corresponding formulas (2.9) and (2.13) for the Weyl and Wick symbols of the evolution operator were derived only heuristically. As was already emphasized in [Ber7la}, in this case one needs to justify formulas (2.8) and (2.11). This nontrivial problem was only recently solved in [Dyn98] for a large class of symbols, and we refer to this paper for further details and references. However, following [Fad76, FS91], we carefully treat boundary conditions for the Wick symbol, deriving the correct expression (2.14) (as opposed to the formula in [Ber7la]). Material in Section 3 is standard, and in our exposition we made a special emphasis on the details of computation related to the Morse index. The formula for the propagator of the harmonic oscillator in Proposition 3.1 is called the Feynman-Souriau formula in [DROI]. The relation between formula (3.G) for the propagator of the harmonic oscillator and the Mehler identity for Hermite-Tchebyscheff polynomials, was stated in [FH65]. For the answer to Problem 3.2 we refer the reader to [Ber7la] (after correcting the typos), and for the elegant heuristic solution of Problem 3.3 to [FS91].

Besides having a conceptual meaning, the path integral formalism is a very convenient computational tool, since it allows us to use, albeit at a heuristic level, such methods of finite-dimensional integration as change of variables, integration by parts, and stationary phase approximation. There is a vast literature devoted to the applications of Feynman path integration in quantum physics. We only mention a perturbative expansion of the propagator using Feynman diagrams [FH65], which is nowadays the main computational method in quantum mechanics and quantum field theory (see also lectures [Kaz99] for the rigorous treatment of a finitedimensional example). For more applications see [DROI], and references therein.

The idea of computing Gaussian path integrals by evaluating separately the classical contribution and the fluctuation factor goes back to [FH65], and Problems 4.1 and 4.4 are taken from this source. In Section 4 we emphasized the role of the

7. Notes and references

287

zeta--function regularized determinants of differential operators, by considering the

simplest example of the operator A = -D2. We refer the reader to the textbook [Apo76] for the basic properties of the Riemann zeta-function. Our proof of Lemma 4.1 can be considered as a simplified one-dimensional analog of the derivation of the first Kronecker limit formula given in [Lan87]. For the case of Laplace operators on

compact Riemannian manifolds, the definition (4.5) of a regularized determinant det'A (under the name analytic torsion) was given in [RS7I], and the operator zeta-function CA (s) was introduced in [MP49]. A similar notion of a perturbation determinant goes back to M.G. Krein [Kre62]; according to Problem 2.6 in Section 2.2 of Chapter 3, the transition coefficient a (fi) is the perturbation determinant of H - XI, where H is a one-dimensional Schrodinger operator. We also note that regularized determinants of differential operators have been extensively used in quantum field theory. The corresponding definition was given by V.A. Fock in 1937 and J. Schwinger in 1951, and is similar to (4.5). It is currently known [1Z80] as the Fock-Schwinger proper time method.

In general, the proof of meromorphic continuation of the zeta-function CA (s) of an elliptic operator A, and its regularity at s = o, uses the theory of complex powers developed in [See67] (or the short-time asymptotics t --+ o of Tr e-At the trace of a heat kernel of A when A is non-negative [G1195]). In Section 5 we are using an elementary approach to the short-time asymptotics of the heat kernel, which is based on the large n asymptotics of the eigenvalues of the corresponding boundary value Sturm-Liouville problems. The monograph [LS91] contains all the facts used in Sections 5.1 and 5.2. The coefficients a_ . and a in formula (5.4), and in their analogs for the periodic case, are called Seeley coefficients. Our proof of the

key relation (5.13) and its analog for the periodic case uses Wronskian identities (5.11), and goes back to the papers [Fad57, BF6o] on the trace identities for onedimensional Schrodinger operators (see also Problem 2.6 in Section 2.2 of Chapter

3). The theorem that "the operator trace is equal to the matrix trace", used in Sections 5.1 and 5.2, as well as the properties of the Fredholm determinant, can be found in the classic monograph [GK69J. Problem 5.1 the Gelfand-Levitan trace identity is taken from the paper [GL53] (see also [Dik58] for its generalization). Properties of the Hurwitz zeta-function, used in Section 5.3, are proved in [For

the general approach to characteristic determinants of the n-th order differential operators on the interval with matrix coefficients and Dirichlet or periodic boundary

conditions, we refer to [BFK91, BFK95]. Our exposition in Section 6 follows [GS77}, with simplifications arising from considering the case of one degree of freedom. Lemma 6.1 establishes the equivalence between the heuristic approach in Section 6.1 using path integrals, and the rigorous

approach in Section 6.2 using short-wave asymptotics. We refer to [GS77] and [MF81] for the details. Also, see [Foc78] for the relation between the van Vleck determinant and canonical transformation in classical mechanics, and [DRO1] for the examples.

Chapter 6

Integration in Functional Spaces

the inIn the previous chapter we studied the propagator K(q', t; q, o) by using its repretegral kernel of the evolution operator U(t) = e sentation by the Feynman path integral. Here we replace the physical time t by the Euclidean time -it, and study the integral kernel of the semigroup

e' for t > o by using its representation by the Wiener integral.

1. Gaussian measures Here we consider the simplest example of Gaussian measures, which are naturally defined for finite-dimensional and infinite-dimensional vector spaces, and are used in many areas of analysis and probability theory. The basic reis the main tool of the Wick theorem sult of the Gaussian integration the perturbation expansion in quantum mechanics and quantum field theory, conveniently expressed through Feynman diagrams.

1.1. Finite-dimensional case. Let A be a positive-definite, real symmetric n x n matrix. A basic formula of Gaussian integration is

Jan

e

(A44)dn q

=

VdetA

(see Lemma 3.1 in Section 3.1 of Chapter 5). The corresponding Gaussian measure associated with the matrix A is a probability measure ,cA on IEBn, 289

6. Integration in Functional Spaces

290

defined by (1.2)

dICA (q)

d q.

e

=

The measure /1A is a mean-zero probability measure on Ian with the covariance C = A-1. When A = In the n x rz identity matrix the corresponding measure is denoted by c, .

As follows from Lemma 3.1 in Section 3.1 of Chapter 5,

f e' dltA(q) =

(1.3)

r n

and by analytic continuation, (1.4)

I

LTh

d A q} = lim

R-+D°

f

ei(P

)

e

2

I

The function (271) 2 e- (G,7)) is a Fourier transform of the measure A .

Theorem 1.1 (Wick theorem). d In (v1,q).(vN,q)A(q)

_

N

0,

is odd,

vz N 2s even, 1vily vi2 . . . where the sum goes over all possible pairings (il, i2), ... , (iij, iN) all partitions in pairs of the set {1, 2, ... , N}. y

7

Proof. Applying to (1.3) the directional derivative along the vector v E I[872

- the differential operator 72

av - v P

- k=1IVk apk

(differentiation under the integral sign is clearly legitimate), we obtain (1.5)

f (v,q)e()d(q) _

(Gv>p)e,

Setting here p = 0 we get

f (v,q)dA(q)=0, n while applying to (1.5) another av and setting p = 0 afterwards, we obtain

L

(v,q)(v',q)d,UA(q)

_ (Gv,v').

To get the general result, we differentiate (1.3) N times along the vectors v1, ... , vN and set p = 0.

1. Gaussian measures

291

Problem 1.1. Let vi, w E IRn be such that (Av, w) = o, i, j = 1, ... , N, and let F and H be bounded measurable functions on IRN . Show that the functions f(q) = F((vi, q), ... , (vs, q)} and h(q) = H((wl, q), ... , (wa, q)} satisfy

ff

(q)

f

1.2. Infinite-dimensional case. Let Y = R°° be the Cartesian product of countably many copies of R equipped with the Tychonofr topology, and let OQ

2(R)=x={xj}iEY:JIxjj

2

}

i=1

be the real Hilbert space with the scalar product (x,y) = >iXjyj. In particular, the Hilbert space 31° contains all elements of finite support: the elements x E 1" such that xi = o for sufficiently large i. The Gaussian measure p on Y is defined by the direct product of the Gaussian measures p1,

p=p00=p1 x1-1 x ... x p1 x ...

.

be the set of x E) for some n,

More precisely, the measure p is defined as follows. Let

cylindrical subsets of 1": C E ' if C = pn 1(E1 x

where pn : 1" -* In is the projection on the Cartesian product of the first n factors, and E1, ... , En are Borel subsets of R. Then we set p(C) = p1 (E1 )... p1 (En), and extend p to the whole cr-algebra generated by

by using the Kolmogo-

rofF extension theorem. In particular, if F(x) = f(xi,. bounded measurable function on R, then

. .

,

x1), where f is a

Fdp = f fdp.

(1.7)

In

Equivalently, the Gaussian measure p is characterized by the following property.

Lemma 1.1. The measure µ is a unique probability measure on 1" such that for all v E "Y" with finite support,

L Proof. The proof immediately follows from (1.4), since measures µn are uniquely determined by their Fourier transforms. Remark. The Gaussian measure p can be heuristically represented by 00

dp = (27r)00e2 fldx. i=1

6. Integration in Functional Spaces

292

Here the "divergent to o" product (2ir)_00e- 2 II

compensates the "diver-

II

gent to oc" product flO° dxi. Now for a = {a2} E 1" let

3a= xEl":

oQ

oQ

i=1

}

The following result is a version of Kolmogoroff's celebrated 0-1 law in probability theory.

Proposition 1.1. Jo

2,fCX,

1

ifaEYt.

In particular, µ(') = 0. Proof. Let La be the characteristic function of the seta C V, n

xa(x) = lira lim exp E-+o fl -+00

{

2

2

i

i=1

Twice applying the dominated convergence theorem, we get

a} =

I

xadp = lira lim

s- O n-oo

exp

-

]fin

n

= lim lim

E--+0 fl-00

i=1

and the product fl1(1 + 2 a? ) is convergent if and only if a E. Remark. For v E that

let vin} = (v1, ... , vn, o, o, ... ). It follows from

L so that the sequence of functions Fn (x ) = (v(), x ), x E 1", is a C auchy sequence in L2 ('V , d a), and it converges in L2 to the function F (x). Abusing

notation, we write F(x) = (v,x) E L2(1",d). Thus though a(9') = o, Lemma 1.1 and, consequently, Wick's theorem, hold for v E

.

Problem 1.2. Prove that there is no probability measure i on° such that = un(E) for every cylindrical subset C = p'(E) of ', where p7--} :

Rn is the natural projection, and E is a Borel subset of Ian . However, show that there exists a finitely-additive, non-negative function v on cylindrical subsets of satisfying this property.

2. Wiener measure and Wiener integral

293

Problem 1.3. Show that the formula n

({x C

: x1 C 11,...,xn C In}) =

ftl(kIk), k=1

where klk = (kcxk, k/3k) for Ik = (ak, 13k) C R, defines a probability measure v on

V such that ii(') = 1.

2. Wiener measure and Wiener integral 2.1. Definition of the Wiener measure. Here we define the probability measure on the space Sf = C([o, oc}, Ian; o) of parametrized continuous paths

in Ian starting at the origin, called the Wiener measure. It is related to the Brownian motion: the diffusion process in IE J with diffusion coefficient D > o, which is described by the probability density P(q1,

(2.1)

q; t) = (4Dt)e

that a particle with initial certainty of being at point q C Ian is diffused in time t to point q' C Ian . It will be convenient to compactify Ian by adding a point at infinity, Ian = Ian U {oc} c S. Let OCt
be the Cartesian product of copies of Ian parametrized by IEbo. Equipped with the Tychonoff topology, S2 is a compact topological space the space of C tm } all parametrized paths in IR 7 For every partition tm = {o C ti C and every F C C (W2 x x Wi ), define o C C() ) by .

m

Y) = F'('Y(ti), ... ,'Y(tm,))

for all

y E S2.

We denote by Cfin(S2) the subspace of C(S2) spanned by the functions cp for all partitions tm and for all continuous functions F. Define a linear functional l on C(SZ) by the following formula: (2.2) Rn

n F(q1,...

,

qm)P(qm, qm_1; tm -- tm_1)..

... P(q1, o; t1)dng1 ... dngm. It follows from the semi-group property fRn

P(q', g1; t' -- t1 )P(q1, q; t1

- t)dngl = P(g', g7 t' - t}

.

6. Integration in Functional Spaces

294

(also called Kolmogoroff's equation in probability theory) that the functional l is well defined. The functional l is positive: l(cp) > 0 for cp > 0, satisfies l(1) = 1, and Il(o)I

k°lloc = Sup ryE

)I

The subspace Cfin(S2) separates points in S2 and 1 E Cfin(S2), so that by the Stone-Weierstrass theorem Cfin(SZ) is dense in C(52). Now the functional l has a unique extension to a continuous positive linear functional on C(SC) with norm 1, and by the Riesz-Markoff theorem, there exists a unique regular Borel measure on Sl with µ(SZ) = 1 such that

fdw. The measure µis called the Wiener measure. The integral over the Wiener measure is called the Wiener integral.

Remark. The Riesz-Markoff theorem provides a natural way of defining measures in various problems of functional analysis. In general, it ensures the existence of a Baire measure a measure defined on the cr-algebra of Baire sets. However; for compact spaces a Baire measure has a unique extension to a regular Borel measure a measure defined on the cr-algebra generated by all open subsets. A Borel measure ji is regular, if for every Borel set E C Sl,

(E)

Iinfii(U), E C U, U is open, sup µ(K), K C E, K is compact and Borel.

The space Il is "so large" that its cr-algebras of Baire and Borel sets are different.

Proposition 2.1. The Wiener measurew is supported on continuous paths starting at the origin, i. e., 1. Replacing the probability density P ( q1, o; t1) by P (q1, qo; ti) in definition (2.2), for a fixed qo E 1W1 we obtain a Wiener measure 1u, which is supported on the space (4 = C([o, oc}, fin; qo ) of continuous paths in 1W'

that start at qo. Remark. The support of the Wiener measure ,U can be characterized more precisely as follows. For o C a C 1 let ft, be the subspace of Q of Holder continuous paths of order a: SZa =

'y E SZ : tgu

o

It

(t)-(t')

< oo

2. Wiener measure and Wiener integral

295

Then

11, ifOCaC, o,

C a C 1.

if

Remark. It seems natural to define Wiener measure by the following construction. Set for simplicity n = 1, and for every partition tm = {o C t1 C C tm } and intervals (ai, /3i ), ... , (am, /3m), define the measure of a cylindrical set by the formula (2.3)

(Ci)=J

I31

al

/3m

...Jam

-tm_] )... P(q1, o; t1)dq1 ... dqm.

P(qm, qm-1; tm

By the Kolmogoroff extension theorem and the semi-group property, p ex-

tends to a measure on the 7-algebra generated by the cylindrical subsets of 1, which we continue to denote by p. However, the set 'it' of continuous paths starting at o turns out to be non-measurable! Specifically, one can show that

p) = o

and p* 1, where p* (E) and p* (E) denote, respectively, the inner and outer measures of the subset E C Q. To remedy this situation, one should from the beginning define cylindrical sets Ct as consisting of continuous paths only, and define p(Ct) by the same formula as above. Then the measure p extends to a ualgebra generated by cylindrical subsets of , and coincides with the Wiener measure P w .

Remark. The same formula (2.3) can be used to define the Wiener measure on the space C([0, in, Ift; o) of continuous functions on the interval [o, in] which

vanish at t = o; in this case tm is a partition of [o, in]. This is the classical definition of the Wiener measure given by Wiener himself. The following result will be used for representing the integral kernel of the one-parameter semigroup em for t > o by the Wiener integral. Proposition 2.2. Let the real-valued function V E C(W) be bounded below. Then for every t o the function .2 defined by :

-

F l y) = e_ fo

,

is integrable with respect to the Wiener measure, and N

L t dpw = ' pan ... f exp -

V (4k)Ot } P(qlv, 4N-1;t) .. . k=1

... P rq1> t }do Q1 ... do O; ` >

t = Nt

.

6. Integration in Functional Spaces

296

Proof. For y E N

t

f

V `/1'(T

)"

dT -

Nurn

k=1

where tk = kOt. Since by definition every function k 1V('y(tk))Ot is measurable with respect to the Wiener measure on °, the function .Ft is measurable as a point-wise limit of a sequence of measurable functions. The

function ,Ft is bounded and, therefore, is integrable on `t' with respect to the Wiener measure. Finally, by the dominated convergence theorem,

L

dlimo f exP

N k=1

and the result follows from (2.2).

Remark. Note the limit in Proposition 2.2 exists because the function .Ft is integrable, and not the other way around. This is similar to an elementary calculus argument that the limit lim

n-oo

2 (1++...+-1ogn

1

1

is convergent. Here [x] stands for the largest integer not greater than x. Problem 2.1. Prove all statements about the support of Wiener measure.

Problem 2.2. Construct the Wiener measure by defining it on the cylindrical subsets of

by formula (2.3).

Problem 2.3. Prove that the Wiener measure p. on C([o, 7r],

o) for D = 4

actually coincides with the measure 11 defined in Problem 1.3. (Hint: Show that the mapping 'y(t) -* y(t} -- y(0) establishes the isomorphism between the space of O([0, it], R) of functions orthogonal to 1 and C([0, it], R; 0), and use the Fourier sine coefficients for the embedding O([0, yr], IR) c 1".)

2.2. Conditional Wiener measure and Feynman-Kac formula. Let

q4' - y E

§?l : 'y(t) = q, y (t')

- q'

t
be the space of all parametrized paths in IRTh which start at q E Ian at time

t and end at q' E Ian at t', and let q,q be the corresponding subspace of continuous paths. The conditional Wiener measure Cq,q on

is defined

2. Wiener measure and Wiener integral

297

similarly. We replace a positive linear functional 1 on C(1) by a positive linear functional 1q,q' on C (I q,q ), which for o E Cfin (SZq,q' ) is defined by lq,q'

_R

aan

n

.F(q1,... , qm)P(q', qm; t' - tm).. . t)d'zg1

... P(q1, q; t1 --

... dfgm,

where t C t1 C ... C t,n C t' and co ('y ) = P ('Y (t 1 ), ... , 'y (tm )} . Then lq,q'

the conditional Wiener measure As in the case of the Wiener measure lq?q' is supported on continuous paths and /iq,q' ` q,q' )

= P(q', q; t' - t).

Let

H=.gyp+V=

P2 2m

+V(Q)

be the Schrodinger operator on L2 (R, dnq) with continuous, real-valued and bounded below potential V (q). Denote by Ln(q', t'; q, t), t' > t, the heat kernel the integral kernel of the diffusion operator e- H. Here is the main result of this section. Theorem 2.1 (Feynman-Kac formula).

Ln(q',t'; q,t) = 1

f

e-k f

,(y),

4>4

where /1q,q' is the conditional Wiener measure.

Proof. We have by the Lie-Kato-Trotter product formula

lim (e_H0 e_

V)N,

N-*oo

where T = t' - t. Let LAN} (q', t'; q, t) be the integral kernel of the operator (e7"°e"h )N. Computing it as in Section 1.3 of Chapter 5, and using the definition of the conditional Wiener measure we obtain N exp --1 v(7(tk)) t d q,q' L( q', t'; q, t) =

fk=1

('Y).

Evaluating the limit N -+ oc by the dominated convergence theorem completes the proof.

298

6. Integration in Functional Spaces

Using the conditional Wiener measure it is easy to define a measure p , on the space

fi i':(t)='7(t')}

£={yE

t
of free parametrized loops in Ian. Namely, the space G is a disjoint union over q e Wz of the spaces flq,q of the parametrized ioops based at q e R. By definition, a function cp : G -* C is called integrable, if its restrictions P 1q q are measurable with respect to p q,q for all q E I', and if the function fcq,q

(p dpq,q :

---} C is Lebesgue integrable on W'. Then

f

p

cp

f

=

cp

diiq,q d`nq.

Suppose that the Schrodinger operator H = H0 + V with continuous, real-valued, and bounded below potential V(q) has a pure point spectrum, and that e_' is of trace class 1. The following result is a useful corollary of the Feynman-Kac formula.

Corollary 2.2.

e- fo

Tr e4H =

V (y(t)}dtd loop (y)

G

Proof. The heat kernel L(q', T; q, o) the integral kernel of the trace class operator e4H is a continuous function of q and q', so that by the theorem2 that "the operator trace equals the matrix trace", used in Sections 5.1-5.2 of Chapter 5, we get Tr e

TH

=

L (q, T; q, 4)dnq.

The result now follows from the Feynman-Kac formula and equation (2.4).

Remark. The Wiener measure on the loop space G = {'y E C([O, T], Il8) : 'y(O) = y(T)}

in terms of the Fourier coefficients 00

2irint

cne T

ry(t) = n=-oo

can be expressed as follows: oQ

(2.5)

diw p =

c

rn

I-I () (21rn)2e_f(2)2k2d2cn.

n=1

This is the case when V(q) - oa as q -r oc "fast enough", i.e., V(q) 2 See Section 7 of Chapter 5 for the reference.

2. Wiener measure and Wiener integral

299

In particular, (2.6)

f

f(7)d0P(7) =

m

V'2hT L oa (L0

.f

, yo

p

o

dx.

Here £ is the space of loops yo with CO = o, and £ = JR x Lo according to the decomposition y = x + 70, where 0f' yo (t)dt = o so that x = co (y). Problem 2.4. Deduce from the Feynman-Kac formula that for every L E L2 (Wi),

(e_*T0+V))(q) =

q

(7(t))e- f V( (t)}dtd q(y).

Problem 2.5. Prove formula (2.5) and deduce (2.6) from it. (Hint: Use Problem 2.3.)

2.3. Relation between Wiener and Feynman integrals. It is very instructive to compare Feynman and Wiener integrals. Heuristically, the conditional Wiener measure can be written as

tivq,

1g T e- , f q2 dT

(2.7)

where

N N- I

In4 -- lim

(2.$)

m

2

Of course, neither the "measure" c1,q exists: the infinite product (2.$) is divergent, nor the trajectory y = q(r) is, in general, differentiable: the integral ft g2dr is divergent. However, due to the presence of a negative sign in the exponential, we have the indeterminate form "infinity over infinity", and the resulting expression (2.7) can be given a precise meaning as the Wiener

measure. The corresponding "measure" for the Feynman path integral is obtained by replacing h by iii in (2.7)--(2.8), and the exponential factor no longer compensates for the divergence of q, since it is a complex number of modulus 1 for a differentiable trajectory, and has no meaning for a non-differentiable one. However, as we have seen in Sections 1.3-4.4 of Chapter 5, the Feynman

path integral allows us to represent the propagator of a quantum particle in a profound way, which shows a deep relation between quantum and classical mechanics. Namely,

Kn(q', t'; q, t) =

e* S(Y)q, I f4(t')=4' 4(t)=4

i

where S(y) = jtt L(q, )dr is the action functional evaluated on a trajectory 'y = q(T), and L(q, q) = 2mg2 - V (q) is the corresponding Lagrangian

6. Integration in Tnctional Spaces

300

function. One also can formally rewrite the Wiener integral for the integral kernel Ln(q', t'; q, t) the Feynman-Kac formula in a similar form, Je- ft

Ln q', t'; q, t = Jq(t')=q' q(t)=q

where E(q, qr) = mqr + V(q) is the energy function. However, in this representation we no longer see the presence of the corresponding action functional, and connection with the classical mechanics is lost. The precise relation between Wiener and Feynman integrals for a quantum particle is the following. As in the case of the Feynman-Kac formula, suppose that the real-valued potential V(q) E C(W) is bounded below. Using the Lie--Kato-Trotter formula, it is easy to show that the heat kernel Ln(q', t'; q, t), defined for h> 0, admits an analytic continuation into the half-plane Re h> 0. Then

K (q', t'; q, t) = lim

(2.9)

t'; q, t).

Remark. Let f be a smooth bounded function on R such that f'(x) = _x2

) as lxi -* oc. The Gaussian integral f_00f(x)e dx is absolutely convergent, whereas the integral f0000 f(x)e2dx is only conditionally cono (I x I

1

vergent. The formula (00

-00

lim f(x)e2dx = e._..}0+

f 00

f(x)e()x2dx

_00

interprets it as the limit - 0 of an integral with respect to the complexvalued Gaussian measure dx. It is tempting to extend this interpretation to Wiener integrals, and for a complex diffusion coefficient D with Re D > 0 to define a complex--valued Wiener measure by the same formula (2.2). However, a theorem of Cameron states that the linear functional 1, defined by an analog of (2.2) for ReD > 0, is no longer bounded on Cfin (S ), so that this approach does not work. Problem 2.6. Let l be the functional on

defined by (2.2) where P(q', q, t)

is given by (2.1) with ReD > 0.

(i) Prove that when Re D = 0, sup

{ll(o)l: 11c01100 =1, p('y) = F('r(t))} _ 00.

t>O,FEC(I(8)

(ii) Prove that when Re D > 0, Sup t,,, , F E C (1fF" )

{Il(co)1: Roll00 =1, p(y) =

F('y(ti),...,'v(tm))}

= (\fiiii)

3. Gaussian Wiener integrals

301

3. Gaussian Wiener integrals In Section 4 of Chapter 5 we evaluated Gaussian Feynman integrals in terms of the regularized determinants of corresponding differential operators. Here we consider the same problem for Gaussian Wiener integrals.

3.1. Dirichlet boundary conditions. Let d

A = -DZ + u(t),

D = fit,

where u e Cl ([0, T] ), be the Sturm-Liouville operator on the interval [0, T] with Dirichlet boundary conditions. The following result is fundamental.

Theorem 3.1. Suppose that u(t) > 0. Then

I

e

fo u(t)y2(t)dtd

2

oo

m

(y)

7rh det A

o,o

Proof. Using Proposition 2.2 and finite-dimensional Gaussian integration formula (1.1), we get m e-2 fo u(t)y2(t)dtdao,o(y)

n

= lim

fn-+oo

.. .

(_m

Jan -1

n-1

m

((yk+1 k

-p

n-1

- yk)2 -f-

dyk

m

= hm

n- c V 27rhOt det An_i

(cf. the proof of Proposition 3.1 in Section 3.2 of Chapter 5). Here yo = yn =

0, tk = kit> t =

n

>

and

An _ i =

al

-1

0

0

0

-1

a2

-1

0

0

0

-1

a3

0

0

0 0

0 0

0 0

... ...

an_2

-1

-1 an.._ 1

where ak = 2 + U(tk) (Ot)2, k = 1,... ,n - 1. Let yin) be t times the principal minor of order /c -- 1 of the matrix An_ 1 corresponding to its upper--left corner, so that initial conditions 2

det An_i. The sequence yin) satisfies the

20t + O((Ot)3),

3

3L]t + O((L1t)3),

302

6. Integration in Functional Spaces

and expanding the order k determinant yin} with respect to the last row, we obtain the recurrence relation 2y (n}

= u tk t 2

n}

k=3

... n - L

Since

lim u (tk ) = u(t)

k,n---}oo

lira tk = t,

when

k,n- -goo

it follows from the method of finite-differences for solving initial value problem for ordinary differential equations that lim y(n) = k,n-oo

where y(t) is the solution of the differential equation

_y" + u(t)y = o, with the initial conditions (n)

y(O) = lim

yen}

n-- oo

= o,

y'(o) = lim y3 n--;oo

- y (n)

t

2

= 1.

Using Theorem 5.1 in Section 5.1 of Chapter 5, we finally obtain

lim t det A_1 = lim n ynfin} n--+oo

n-ao

_ T = 12 det'A.

Remark. The same computation also works for Feynman path integrals, and gives a rigorous proof of the formula e im

fo (y2-u(t)y2)dt = y

fy(T)=O

m

\/irihdetA'

y(0)-0

where A = -D2 - u(t) is apositive-definite operator (see Section 6.1 of Chapter 5). In particular, the coefficient c,,,,h from Section 4 of Chapter 5 is indeed equal to

3.2. Periodic boundary conditions. Let A = -D2 + u(t), where u E Cl ([0, T] ), be the Sturm-Liouville operator on the interval [0, T] with periodic boundary conditions. Here we prove the following analog of Theorem 3.1.

Theorem 3.2. Suppose that u(t) > 0. Then

f

e

-

f o u(t)Y2(t)dtd1llooP(y) =

1 7

det A where G is the space of free loops in I[8 parametrized by the interval {0,T].

3. Gaussian Wiener integrals

303

Proof. As in the proof of Theorem 3.1, we have

f

m ii

((Yk+i

n

m

e- f'u(t)y2(t)dtd1ooP(y) = urn

(2nt)

2

- Yk)2 -f 26(tk)(Ot)2yk)

[J dyk

1 .

det An

Here yo = yn and An is the following n x n matrix:

fao

-1

0

...

0

-1 al -1 0 -1 a2

A=

0

\-i

_1\

0

0

0

0

0

0

CLn_2

0

0

-1

1

an-1

where ak = 2 + u(tk) (Ot)2, k = 0,1,... ,n - 1. We compute det An by the following elegant argument. First, note that the real A is an eigenvalue of A if only if the difference equation (3.1)

-(Yk+i -I- Yk-i - 2iJk) + u(tk)(Ot)2yk = Ayk, k = 0,... ,n -1,

with initial conditions y_1 and yo has a "periodic solution" - a solution {Yk}i satisfying yn-1 = y-1 and yn = yo. For given A denote by v(A) and v2(A) solutions of (3.1) with the corresponding initial conditions v(A) 1, volt (A) = 0 and v(A) =0, v2( A) = 1, and put Tn(A)

(V$

(A)

V2(A) v72( A)

It is easy to show that the discrete analog of the Wronskian

(A)vk2 (A)

-

does not depend on k, so that det Tn(A) = 1. Since every solution yk of the initial value problem for (3.1) is a linear combination of the solutions v'( A) and v2( A), we have 2) vkl) (A)vk_i(A)

(Yn;i)

= Tn(A)

From here we conclude that A is an eigenvalue of the matrix An if and only if det(T(A) - I2) = 0, and the multiplicity of A is the multiplicity of a root of this algebraic equation. Since

=

O(an-l)

vn2)(A)

_ (-A)n +

O(An-i)

as

A -- oo,

6. Integration in Functional Spaces

304

we obtain

det(An - .1In) _ - det(Tn(A) - I2) = v$'1(A) + v(A) -2. It remains to compute

det(An, - AIn). Denote by and correspondingly, two solutions of the difference equation (3.1) with the initial conditions y( A) =yolk (A) = 1 and y(A) = 0, y( A) = Lit. We have

vk2 (A)=yk2)

and

yk2) (A)

so that (2)

det(An -

(3.2)

1)

AIR) = y(n

1(A) +

a} -

n

-2.

x

Now it follows from the method of finite differences that

lim y'(A) = y1 (t, a),

(3.3)

lim y(2) (A) = y2 (t, a

k,n oa

k,n

o0

as limk,n 0 tk - = t, where y1,2 (t, a) are two solutions of the differential equation

-y" + u(t)y = Ay with the initial conditions yl (0, A) = 1, yl (0, A) = 0 and y2 (0, A) = 0, y2(0, A) = 1. Using (3.2)-(3.3) and Theorem 5.2 in Section 5.2 of Chapter 5, we finally obtain lim det(An, - AIn) = yi(T, A) + y2(T, A) - 2 = det(A - Al).

n-->oo

Example 3.1. Corollary 2.2 and Theorem 3.2 can be used to compute Tr e- TH for the harmonic oscillator H =

1(p2 + rrz2w2Q2). We have

2m 1

Tre-TH =

det AiW

where AiW = -D2 + w2, and by formula (5.20) in Section 5.2 of Chapter 5, 1

(3.4)

2 sink wT 2

Of course, since the eigenvalues are En = hw(m + 2 ), we can get the same result by using geometric series

Tr e *TH =

e *TEn

= eT

O°

n-o

e---wTn

=

1

2 sink 2

Similar results hold for the general second order differential operator

A = -D2 + v(t)D + u(t) on the interval [0, T] with periodic boundary conditions.

4. Notes and references

305

Theorem 3.3. Suppose that det A > 0, where A = -DZ + v(t)D + u(t). Then

L

fo (v(t)y(t)y(t)+u(t)y2(t))dtdi loop

e-

(y)

=

1

det A

Example 3.2. Let A = -DZ + wD. It follows from Theorem 3.3 and (2.6) that for E > 0 1

/det(A + El) m 2irhT

00

f e-

f

2dx

oo

e1* = T f fo

e

-

fo

loop (y) fo (wy(t)y(t)+Ey2 (t) )dtdw

e- fo

wp

o

(yo ) fo (w'yo ( t) yo ( t } +eyv ( t} } dtd,loop w

where we have used the decomposition y(t) = x + yo (t), 0f' yo (t)dt = 0. Here Go is the subset of the free loop space G which consists of loops with zero constant term in the Fourier series expansion. Using corollary 5.4 in Section 5.2 of Chapter 5, we obtain Tft T e-fo2wyo (t)yo (t)dtdjcwloop(yo) = lim /det(A + El) Go det'A wT

(3.5)

- (sinh

i

2

2

We will use this result in Section 2.2 of Chapter 8. Problem 3.1. Derive formula (3.4) using (2.5).

Problem 3.2. Prove Theorem 3.3.

4. Notes and references There is a vast literature on Wiener integration theory and Brownian motion, and here we present, in a succinct form, only the very basic facts. Material in Section 1 is standard, and our exposition follows the exercise section in [Rab95]. For necessary facts from probability theory, including Kolmogoroff's extension theorem and an introduction to the stochastic processes, we refer to the classic treatise

[Loe77, Loe78] and recent text [KhoO7]. In particular, the statement () = o in Proposition 1.1 follows from the strong law of large numbers.

Th elegant construction of the Wiener measure in Section 2 belongs to E. Nelson [Ne164], and our exposition, including the proof of the Feynman-Kac formula, follows [RS75]. We refer the reader to Ito-McKean's classic monograph [1M74] for the construction of the Wiener measure from the probability theory point of view; along this way Problems 2.1 and 2.2 get solutions. The classic book [Kac59]

306

6. Integration in Functional Spaces

and lectures [Kac8O] by M. Kac are another excellent source of information on Wiener integration and its applications in different areas of mathematics. The relation between Wiener and Feynman path integrals in Section 2.3 belongs to E.

Nelson [Problem 2.6 is taken from R.H. Cameron's paper [Cam63] (see also [RS75]); this result shows that there is no complex-valued analog of Wiener measure associated with the complex diffusion coefficient with Re D > o (as opposed to the statement made in [GY56]). Theorems proved in Section 3 serve as a rigorous foundation for our discussion

of the Gaussian Feynman path integrals in Section 4 of Chapter 5. Our proof of Theorem 3.1 in Section 3.1, which uses the method of finite differences, follows the

outline in [GY56] which attributed it to [Mon52J. The proof of Theorem 3.2 in Section 3.2 seems to be new.

The Euclidean quantum mechanics, obtained by replacing the physical time t

by the Euclidean (or imaginary) time -it, is ultimately related to the theory of stochastic processes. It can be formulated by a set of axioms of the one-dimensional Euclidean quantum field theory, and we refer the interested reader to [StrO5] for the detailed discussion.

Chapter 7

Fermion Systems

1. Canonical anticommutation relations 1.1. Motivation. In Sections 2.6 and 2.7 of Chapter 2 we have shown that the Hilbert space ^L (III, dq) of a one-dimensional quantum particle can

be described in terms of the creation and annihilation operators. Namely, the operators1

a=

1

(Q - iP) and a =

1

(Q + iP)

satisfy the canonical commutation relation {a, a*] = I

on W2'2(IR) n W22(II8), and the vectors

k= where

o(q) = (rh) r 4 e-

1 q2

ki

o,

= 0,1, 2, .. .

7

E ° satisfies aL'0 = 0, form an orthonormal

basis for i°. The corresponding operator N = a*a is self-adjoint and has an integer spectrum,

Nk = kk, k = 0,1,2 , ... . Similarly, for several degrees of freedom ° ^L2 (W , do q ), the creation and annihilation operators are given by

(1.1) a _

1

(Q - iPk) and ak =

1

(Qk + iPk) , k = 1, ... , n,

1 Here in comparison with Section 2.6 of Chapter 2 we put w = 1. 307

7.Fermion Systems

308

and satisfy canonical commutation relations (1.2)

[ak, at] = [a%, at] = o

and

Ic,1 = 1, ... , n.

[ak, at] = 6ia',

The ground state, the vector bo (q) = (irh)1 e2 E

,

has the property

k = 1, ... , n,

ak?/)o = 0,

and the vectors

tat}k ...

... ,n = D, 1, 2,

'yo,

r

1c4 .... 'in .

form an orthonormal basis for The operator N = adjoint and has an integer spectrum: N ybk1= (I1 +... + .

.

is self-

s

and the Hilbert space 1C decomposes into the direct sum of invariant subspaces 00

(1.3) k=0

- the eigenspaces for the operator N. However spin operators, introduced in Chapter 4, satisfy algebraic relations of different type. Namely, consider the operators o- _ _

(Sl f iS2), where Si and S2 are spin operators of a quantum particle of spin 2 (see Section 1.1 of Chapter 4). Using the explicit representation of spin operators by Pauli matrices, we get

-i o)' (0 O\

+= (0

1

o

The operators o are nilpotent, Q = 0, and satisfy the anticommutation relation

cT+ d_ + '-+ = 12, where 12 is the identity operator in C2. Introducing the notion of an anticommutator of two operators, [A,B]+ = AB + B A, we see that the operators a = o-_ and a* = o+ satisfy canonical anticommutation relations

_ [a*, a*]+ = 0 and [a, a*]+ = I2. The vector eo = (°) has the property aeo = 0, and together with the vector [a, a]+

a*eo = (o) they form an orthonormal basis of C2. The matrix

N = a*a =

(%\1

(c73 + 12)

=o

Ol

o)

1. Canonical anticommutation relations

309

has eigenvectors eo and a*eo with eigenvalues 0 and 1. Thus in complete analogy with the previous discussion, we say that a and a* are fermion creation and annihilation operators for the case of one degree of freedom. The Hilbert space of a Fermi particle is = C2, and the vector eo is the ground state. It is straightforward to generalize this construction to the case of several degrees of freedom. Namely, canonical anticommutation relations have the form

= 8k1', k,1 = 1, ... , n, where I is the identity operator, and creation operators aj are adjoint (1.4)

[ak, al] + = {a, ai ]+ = 0 and

[ak, ai ] +

to annihilation operators aJ in the fermion Hilbert space J. Canonical anticommutation relations (1.4) can be realized in the Hilbert space F = (C)® n = C2as follows: (1.5) k---1

(1.6)

k-1

k = 1, ... , n. The ground state, the vector bo = eo O (1.7)

akbo = 0,

- ® eo E

F, satisfies

k = 1, ... , n,

and the vectors

= (*)ki ... (a)ko, k1, ... , n = 0,1, form an orthonormal basis forF The operator (1.8)

.

Ti

N=> a aj k=1

is self adjoint and has an integer spectrum: T-

+ ... +

k)/'k,

and the Hilbert space 3' decomposes into the direct sum of invariant sub-

(1.9)F

spaces

k=O

the eigensp aces of N.

Remark. The fermion Hilbert space 3F is isomorphic to the spin part of the Hilbert space of n quantum particles of spin 2 , discussed in Section 3.1 of Chapter 4. The corresponding fermion creation and annihilation operators can also be used for describing spin degrees of freedom. The fundamental importance of the fermion and boson Hilbert spaces -= (C2)®Ti and

310

7. Fermion Systems

B = L2(I[8n, dnq) -manifests itself in quantum field theory, which formally corresponds to the case n = oo, and describes quantum systems with infinitely many degrees of freedom. Lemma 1.1. Realization (1.5)-(1.6) of canonical anticommutation relations in the fermion Hilbert space M°F is irreducible: every operator in which commutes with all creation and annihilation operators a and ak, is a multiple of the identity operator. Proof. It is sufficient to show that if {0}

V CF is an invariant subspace for all operators ak and a, then V =F . Indeed, every non-zero E V can be written in the form k,,

a

and let c1,...,kn bk1,...,k be any of its non-zero components with maximal

degree k1 + we obtain

+ k. Using canonical anticommutation relations and (1.7),

an ... a1

= c0,

c = clU1,...,k7z

0,

so that L)o E V. Applying creation operators to ,Li0, we get v = .

Remark. The realization of canonical anticommutation relations in the fermion Hilbert space °F is analogous to the representation by the occupation numbers of canonical commutation relations, discussed in Section 2.7

of Chapter 2, and is called representation by the occupation numbers for fermions. It should be emphasized that the algebraic structure of the former relations allows their realization in a finite-dimensional Hilbert space, while the algebraic structure of the latter relations warrants the infinitedimensional Hilbert space. Analogously to (1.1), coordinate and momentum operators for fermions are defined by (1.10)

Qk =

(ak+a),

k

k -- ak},

k = 1, ... , n.

As follows from (1.4), they satisfy the following anticommutation relations:

(1.11) [Qk,Ql]+=[Pk,Pj]+=hSklI and [Pk,Q1J+=0,

k, l = 1, ... , n

- a fermion analog of Heisenberg commutations relations, introduced in Section 2.1 of Chapter 2. The following result is a fermion analog of the Stone-von Neumann theorem from Section 3.1 of Chapter 2. Theorem 1.1. Every irreducible finite-dimensional representation of canonical anticommutation relations is unitarily equivalent to the representation by occupation numbers in the fermion Hilbert space F.

1. Canonical anticommutation relations

311

Proof. Let V be the Hilbert space which realizes the irreducible representation of canonical anticommutation relations (1.4). First of all, there is Po E V, IkooII = 1, such that aico = ... = ancoo = o. Indeed, choose any non-zero co E V; ifcPa174 o, replace it by the vector a o, which obviously satisfies al (alo1 ) = o. If a2 (aicp) 74 o, replace it by a2al p,

which is annihilated by ai and a2, etc. In finitely many steps we arrive at a non-zero vector cp annihilated by the operators al, ... , an, and coo = c / I I Now consider the subspace Vo of V, spanned by the vectors I I

(a*)k1

... (a*)coo, ki, ... ,n = 0,1.

It follows from (1.4) that Vo is an invariant subspace for all operators

a and

a, so that Vo = V. Since operators a and aj are adjoint with respect to the inner product in V, it is easy to see, again using canonical anticommutation relations (1.4), that vectors cok1,...,k form an orthonormal basis for V. The mapping V ........ kn /'k1..... n E 35' establishes the Hilbert space

isomorphism V ^J f'. Problem 1.1. Express the spin operators

of the k-th particle in the system of n spin 2 particles (see Section 3.1 of Chapter 4)in terms of fermion creation and

annihilation operators in )t''.

1.2. Clifford algebras. We have seen in Section 2.1 of Chapter 2 that the Heisenberg Lie algebra is the fundamental mathematical structure associated with canonical commutation relations. Similarly, the fundamental mathematical structure associated with the canonical anticommutation relations is Clifford algebra.

Let V be a finite-dimensional vector space over the field ft of characteristic zero, and let Q : V ---} ft be a symmetric non-degenerate quadratic form

on V, i.e., Q(v) =

(v, v), v E V, where c : V ®k V -} ft is a symmet-

ric non-degenerate bilinear form. The pair (V, Q) is called quadratic vector space.

Definition. A Clifford algebra C(V, Q) = C(V) associated with a quadratic vector space (V, Q) is a k-algebra generated by the vector space V with relations

v2=Q(v)'l, vEV. Equivalently, Clifford algebra is defined as a quotient algebra

C(V) = T(V)/J, where J is atwo-sided ideal in the tensor algebra T(V) of V, generated by the elements u ® v + v U v) 1 for all u, v E V, and 1 is the unit

7. Fermion Systems

312

in T(V). In terms of a basis {e}1 of V, the Clifford algebra C(V) is a k-algebra with the generators e1, ... , e,, satisfying the relations [e,ej]+ = eiej + ejei = 2I(ei, ej) . 1, i, j = 1, ... , n. When k = C (or any algebraically closed field of characteristic zero), there always exists an orthonormal basis for V a basis {e}L1 such that c (ei, ek ) = 8ik. In this case for every dimension n there is one (up to an isomorphism)

Clifford algebra Cn with generators el,... , en and relations i, k - 1, ... , n.

Remark. If k = I, there exist non--negative integers p + q = n and an isomorphism V ^Iftn such that This classifies Clifford algebras over R.

Definition. A left module S for a Clifford algebra C(V) is a finite-dimensional

k-vector space S with the linear map p : C(V) ® S - S such that p(ab ® s) = p(a ® p(b ® s)}

foralla,bEC(V) andseS. The fermion Hilbert space F, introduced in the previous section, is an irreducible C2n--module. Indeed, it follows from canonical anticommutation relations (1.4) that self-adjoint operators (1.12)

'Y2k-1 = a/ + a

(1.13)

rY2k = --i(ak

-at),

k = 1, ... , n,

satisfy the relations v+

(1.14)

281

I,

v = 1, ... , 2n,

where I is the identity operator in °F. We define the action of the Clifford algebra C2n on 31fF by setting

p(l)=I and

p(e1) - Y1y p = 1, ... , 2n, and extending it to a C-algebra homomorphism p : C2n -k End ( 3F ). Rela-

tions (1.14) show that the map p admits such an extension.

Proposition 1.1. The homomorphism p : C2n --} End(') is a C-algebra isomorphism.

Proof. It follows from Lemma 1.1 that the representation p is irreducible: every operator in °F which commutes with all elements of the C-algebra p(C2n) is a multiple of the identity operator. Then by Wedderburn's theorem ') , the map p is p(C2n ) = End (3 ) , and since dim C2n = 22n = dim End ( an isomorphism.

1. Canonical anticommutation relations

313

Remark. The structure of a Clifford algebra with an odd number of generators is different. Thus the mapping p(ek) = Qk, where 0k, k = 1, 2, 3, are Pauli matrices (see Section 1 of Chapter 4), defines an irreducible representation of C3 in 3' = C2. However, in this case C3 End(C2) ® C [e] , where E = iele2e3 and satisfies E2 = 1.

where N = a a3. We define the chirality operator by r = r2 I. Moreover, we have Since the operator N has an integral spectrum, [F,y]+=O, p = 1, ... , 2n. (1.15) ---

Indeed, as follows from (1.4),

Nab = a( N + I)

and

_a*eN

and

Nab = a( N - I),

so that

a eN-1} = -a e7riN.

Thus r anticommutes with all a, a, and hence with all 'y1. Since F2 = I, the operators

Pt=2(z±r) are orthogonal projection operators and we have a decomposition into the subspaces of positive and negative chirality spinors. It follows from (1.15) that AISO Since

a

7 F } =,

=1,...,2n.

_ I - 2aj a , = -i y2 --....1y2j, we have

r = (-i)y1 ... y2n. Remark. When n = 2, 4 x 4 matrices `yl, 72, 73,74 are celebrated Dirac gamma matrices (for the Euclidean metric on R4), and F = 'y5. Problem 1.2. Show that the definition of a Clifford algebra C(V) is compatible with the field change: if k C K is a field extension and VK = K ®k Vf, then C(VK) = K ®k C(Vk).

C C- = C(V) be the natural filtration s C r. of a Clifford algebra C(V), where Cr is spanned by the elements v1 ...

Problem 1.3. Let C. 1 = C° C C1 C Let

n

Grgr/v` _ Grk/G.k-1 l

J

k=1

be the associated graded algebra. Show that the skew-symmetrizer map

v n ... A v ,' r1

(-1) .

QESymr

vT

7. Fermion Systems

314

establishes a Z-graded algebra isomorphism A' (V) exterior algebra of V.

Cgr(V), where A' (V) is the

Problem 1.4. Formulate and prove the analog of Proposition 1.1 for Clifford algebras with an odd number of generators.

2. Grassmann algebras Grassmann algebras algebras with anticommuting generators are necessary for the semi-classical description of fermions. The corresponding mathematical definition is the following.

Definition. A Grassmann algebra with n generators is a C-algebra Grn with the generators O1, ... , Bn satisfying the relations

00+00=O, i,j = 1,...,n. In particular, these relations imply that generators of a Grassmann algebra are nilpotent: 0l = = 0n = o. Equivalently, a quotient of a free C-algebra C (81, ... , 0?), generated by 01, ... , 0n, by the two-sided ideal J generated by the elements O Oj + 9jB , i, j = 1, ... , n.

Remark. It follows from (1.11) that in the semi-classical limit h - * o fermion operators Pk and Qk, k = 1, ... , 2n, satisfy the defining relations of Grassmann algebra Gr2n .

Comparison with the polynomial algebra a quotient of a free C-algebra C (xl , ... , xn ) by the two-sided ideal gener-

ated by the elements xx -x3xz, i, j = 1, ... , n shows that the Grassmann algebra Grn can also be considered as a polynomial algebra in anticommuting variables U1, ... , O. In what follows we will always use Roman letters for commuting variables and Greek letters for anticommuting variables, so

that Grn = C[01, ... , 0n].

Needless to say, the polynomial algebra C [xl, ... , xn] is isomorphic to the symmetric algebra of the vector space spanned by x1,... , xn, and the Grassmann algebra C[91,. . . , 9n] is isomorphic to the exterior algebra AV of the vector space V = C81 CBn with the basis B1, ... , On. The Grassmann algebra Grn is a complex vector space of dimension 2n and is Z-graded: it admits a decomposition n

(2.1)

GTn =

GTn k=0

2. Grassmann alge bras

315

into homogeneous components Grn of degree k and dimension (), k = o, ... , n, where Gr = Namely, denote by the degree of homogeneous elements in the Grassmann algebra, a = k for a E Grn Then multiplication in Grn satisfies Grn Grn C Grn+l, where Grn+ o if k + 1 > n, and is graded-commutative: .

a3 = (-1)9a

(2.2)

for homogenous elements a, 3 E Grn. The elements of the Grassmann algebra Grn of even degree are called even elements, and those of odd degree odd elements.

2.1. Realization of canonical anticommutation relations. The Grassmann algebra provides us with the explicit representation of canonical anticommutation relations by multiplication and differentiation operators, which is analogous to the holomorphic representation for canonical commutation relations (see Section 2.7 of Chapter 2). Namely, let

z-

:Grn--Grn

be left partial differentiation operators, defined on homogeneous monomials

i

... i

by

-

-1 } Z-1 iiz

Y

via ...

1=1 Y

where Oi denotes the omission of the factor 9il . The differentiation operators

are of degree -1 and satisfy the graded Leibniz rule, 3

(a3) =

ova

_

as

Ian

Bi

a8i

Remark. One can also introduce right partial differentiation operators by

(-1)8 . Big ...

-

...

z=1

which satisfy the following graded Leibniz rule: a ,C3 }

aei

- a (si) + (-

1 ) CI 1

(

4)

s.

To distinguish between the left and right partial derivatives of f e Grn, we a a will denote them, respectively, by f

and f-.

7. Fermion Systems

316

As a complex vector space, Grassmann algebra Grn carries a standard inner product defined by the property that homogeneous monomials Bit ...

for all 1

(9zl

(2.3)

...

... 9ji ) = 8kz 8i1j1 ...

By checking on homogeneous monomials, it is elementary to verify that

f,9 E Grn

(aif,9) = (f,gig), ^

n

where 9i are left-multiplication by 9i operators in Grn, so that 9i = 3. It is also easy to verify that the operators Bi and 19i satisfy the anticommutation relations [O,O]+

= [ioi]+ =0

and

+ = 8ijI,

[Bi,

i, j = 1, ... , n,

where I is an identity operator in Grn. Thus we have the following result.

Proposition 2.1. The assignment

F,...,k7z

... Bn' E Grn

establishes an isomorphism Grn between the fermion Hilbert space of n identical particles, and the vector space of the Grassmann algebra with n generators. It preserves decompositions (1.9) and (2.1) and has the property that ai

9i

and ai

a ,

i = 1, ... , n.

Using Proposition 2.1, it is also very easy to verify that the representation of canonical anticommutation relations in the fermion Hilbert space ' is irreducible. Indeed, suppose that B E End (Grn ) commutes with all h operators Bi and l?,. Then

a(B(1))=B(a(1))=0, i=1,...,n. The only solution of the equations al f = = 8n f = 0 is f =c 1, so that B(1) = c 1. Since B commutes with all creation operators 921 we obtain

B=cI.

Remark. We will show in Section 2.3 that by using the notion of Berezin integral, the inner product (2.3) in Grn can be written in a form (2.11), which is similar to the definition of the inner product in the holomorphic representation, given by formula (2.52) in Section 2.7 of Chapter 2.

2. Grassm ann. alge bras

317

2.2. Differential forms. An algebra of differential forms in anticommuting variables B1, ... , On is a C-algebra 1

with odd generators B1, ... , Bn and even generators dot,... , dBn satisfying relations

Bi dB = d8 Big

i, j = 1, ... , n.

Equivalently, Sin is a symmetric tensor product over C of Grassmann algebra Grn and polynomial algebra C [d01, ... , dBn] , and every w E SZn can be uniquely written as 00

(2.4)

w = >fk(del)kl ... (dBn,), fkEGr, =o

where k = (k1,.. ,kn ) is a multi-index and fk = o for all k, except finitely many. By definition, the degree Iw, of a homogeneous component w, = E SZn is fkl, the degree of fk E Grn. (dBn) fk (d01)" .

Remark. It is instructive to compare the algebra SZn with the algebra of polynomial differential forms on Ctn. On the one hand, Stn is an infinitedimensional algebra in commuting variables dO1,... , dBn with coefficients in a finite-dimensional algebra Grassmann algebra C[01,. . . , On]. On the other hand, the algebra of differential forms in commuting variables x1, ... , xn is a finite--dimensional algebra in anticommuting variables dx1, ... , dxn with coefficients in an infinite-dimensional polynomial algebra C [x1, ... , xn] .

The analog of the exterior (de Rham) differential on the algebra 11 is the mapping d : tin --f Stn, defined by OQ

dw -

n

afk doi (dot )... (dO),

fk E Grn,

k=O i

where w is given by (2.4). It can also be written as d = z 1 dOz3i, where it is understood that 3i (dO3) = o for all i, j = 1, ... , n.

Lemma 2.1. The exterior differential d on 1I satisfies the graded Leibniz rule,

d(wlw2) = dwl w2 + (-1)"widw2, for homogeneous wl, and is nilpotent, d2 = 0. Proof. The graded Leibniz rule for d follows from the corresponding property of partial differentiation operators r,. The property d2 = o follows from the commutativity of "differentials" dOi and anticommutativity of the partial derivatives 3. The next result is an analog of the Poincare lemma for differential forms in anticommuting variables.

7. Fermion Systems

318

Lemma 2.2. Suppose that w E Qn is closed, dw = o. Then w is exact: there is 7] E Stn such that w = dry.

Definition. A 2-form w E Stn, n

w E Grn,

w=

(2.5)

2,3=1

is called a symplectic form on a Grassmann algebra if it is closed, dw = o,

and is non-degenerate: the n x n symmetric matrix {}=i is invertible in Grn.

In particular, the 2-form w with constant coefficients wi3 E C is always closed, and w is symplectic if and only if the matrix is non-degenerate. Since every quadratic form over C can be written as a sum of squares, we can always assume that generators 9, ... , 9n are chosen such that the symplectic form w with constant coefficients has a canonical form n dOdOz.

z-1

With every symplectic form (2.5) there is an associated Poisson bracket on Grassmann algebra, defined by

(f)wii(g),

{f,g}=

(2.7)

f,gEGr,

z,j=1

is the inverse matrix to {w'3 }j=1' and we are using notation for the left and right partial differentiation operators, introduced in the previous section. The following result an analog of Theorem 2.9 in Section 2.4 of Chapter 1 is fundamental for formulating Hamiltonian mewhere {wig

chanics for the systems with anticommuting variables, which we will discuss

in Chapter 8.

Proposition. 2.2. Suppose that all coefficients wZ of a closed symplectic form w are even. Then the Poisson bracket map

{, }

Grn x Grn --k Grn

satisfies the following properties.

(i) (Graded skew-symmetry)

{f,g} _ -(-1)Ifllsl{g, f}. (ii) (Graded Leibniz rule)

{fg, h} = f{g, h} + (-i)g{f, h}.

2. Grassmann algebras

319

(iii) (Graded Jacobi identity)

{f, {g, h}} + (-1){g, {h, f}} + (_1)1lzI(1fHID{h, {f, g}} _ 0 for all f,g,h E Grn. Proof. Part (i) follows from (2.2), the property

(_)IfIfaei, aezf and the condition that coefficients ware even. The graded Leibniz rule follows from the corresponding property of right partial differentiation operators. The graded Jacobi identity for the Poisson bracket {f,g}=

which corresponds to the canonical symplectic form (2.6), can be verified by a direct computation. The proof of a general case is left to the reader. Problem 2.1. Complete the proof of Lemma 2.1. Problem 2.2. Prove Lemma 2.2. Problem 2.3. Complete the proof of Proposition 2.2.

2.3. Berezin integral. There is a principal difference between differential forms in commuting and in anticommuting variables. The former can be differentiated and integrated, and the differential and integral are related by the Stokes' formula. However, the latter can only be differentiated. Still, there is an analog of the integration over anticommuting variables. Definition. The integral on a Grassmann algebra Grn with an ordered set of generators Oi, ... , E (Berezin integral) is a linear functional B : Grn --- C, defined by

B(f) =

f12...n,

where

fil...ik6i ... 0ik E Grn. k-o 1 CiI <... Cik cn

It is traditional to write the Berezin integral in the form

B(f) =

f .f(e)del ...den,

where 8 = (Or,. , 6n), as if f = f(Oi,. . , B) was actually a "function of anticommuting variables". It follows from the definition of partial differentiation operators that .

. .

J

f (6)d81

... dBn = Bn ... ael .f

7. Fermion Systems

320

which implies

ff(o)dol...de=o. aez This leads to the following integration by parts formula for the Berezin integral:

f(O)

(ia) (O)g(O) dO1 ..den

g (O)dOi . . . d0n --

.

for homogeneous f, g E Grn .

Remark. The Berezin integral is not an integral in the sense of integration theory. It is defined as a linear functional on a Grassmann algebra Grn and it depends on the ordering of the generators e1, ... , en of Grn, which is symbolized by de1... den. For each a E Symn, f(6)dOi .. .

den = (-i)) f(6)dO(l) ...

where s(a) is the parity of a permutation a. Remark. Using the embeddings Grk C Grn for k C n, physicists usually define the Berezin integral as a "repeated integral" starting from the following "one-dimensional integrals" : d0z = o,

i - 1, ... , n.

ezdez = 1,

Lemma 2.3 (Change of variables for Berezin integral). Let e1, ... , en and e1, ... , en be two sets of generators of the Grassmann algebra Grn, related by

ei = >-1 azjej, where the n x n matrix A = {a}1 is non-degenerate. Then 1

f(O)d0

.

.

.

f(O)dB1

d0 n =

.

.

.d6n,

det A

n

where 1(6) = f(O) = f ( 1

Proof. By multi-linear algebra, f12n = f12Th det A.

Remark. According to the lemma, the "density" det ... de7z has the transformation law (2.8)

de1

... dBn =

1

n

det A

d91

... de n

under the change of variables e - = a 1 ai j Oj . This differs from the usual change of variables formula for the Lebesgue integral

(2.9) f f(xi,..

.

,

xn)dxl

... dxn = det A J n

f(y...

,2Jn)dyi ...dyne

2. Grassmann algebras

321

1 az j y3. Of course, the or dx 1 ... dxn = det A l dy1 ... dy7z, where xi = > Berezin integral is rather a multiple derivative than an integral with respect to a measure, which explains this profound difference.

Let A -= {aid }j=1 be an n x n skew-symmetric matrix. For even n = 2m its Pfaian Pf (A) is defined by l

l

(-1) mm .2 Q E symn

n 1}a(n)a

(1)Q(2)

where E(u) is the parity of a permutation a. By definition, Pf (A) = o when n is odd.

Proposition 2.3 (Gaussian integration for anticommuting variables). Let A = {a}_1 be an n x nskew-symmetric matrix. Then (1)

J

exp { 2

dBn = Pf (A). i,j=1

(ii) For any non-degenerate n x n matrix C Pf (CACt) = Pf (A) det C.

Pf(A)2 = det A.

Proof. Part (i) obviously holds for odd n since the integrand is an even element of Grn. By the definition of the Pfaffian, we get for n = 2m, n

1

m aij OiOj

mm .2

= Pf (A) e1 ... n a

i,j =1

and expanding the exponent into power series, we obtain

J

exp

2

aBiB

dBl ... dBn = Pf(A)

J

Bi

... 6 dBl ... dBn = Pf(A).

Part (ii) follows from part (i) and Lemma 2.3. Part (iii) is a classical result, which can be proved by the Berezin integral as follows. Suppose first that A is real-valued. There exists an orthogonal matrix C with determinant 1 such that 0

f-A1 CAC_ 1

=

A1

0

0

0

0

0

322

7. Fermion Systems

is ablock-diagonal matrix. Using part (ii) with this matrix C, we obtain Pf (A) =

dB2m = Al

... a m

so that Pf (A)2 = det A. This relation holds for complex-valued A, since both sides are polynomials in variables aij, 1 C i <j C n, which coincide for real

a3. For a Grassmann algebra Gr2n = C {Oi, ... , Bn, Ul, ... , 0n] with 2n generators denote by f d6dO = f d l dOl ... dOnd0n the corresponding Berezin integral,

f(O,)dOda

yen aen

gel gel

Lemma 2.4. For any n x n matrix A = {aij? j,l, exp

dude = det A. i,j=1

Proof. It follows from the definition of a matrix determinant that Ti

n 1

= det A 0101... Bnon.

ni

Definition. An involution on a Grassmann algebra Grn over C is a complex anti-linear mapping Grn f f* Grn satisfying (f*)* = f and (fg)* = g* f * for all f,g E Grn .

The Grassmann algebra CC[01, ... , On, U1, ... , Bn] has a natural involution defined on generators by (Oi)* = U1, (0iY * = 91, ... , (0) * = Bn, (On) * = On.

In particular, for

k=O

E

JZ1...Zk i1

... ei E Grn C Gr2n

1Ci1C...CzkCn

we have

f(9)*

fzl...ik 92

=

,

... ix E Gr2n.

k=o 1Ci1 C...Cik Cn

The next lemma expresses the inner product on the Grassmann algebra Grn, introduced in Section 2.1, in terms of the Berezin integral.

2. Grassmann algebras

323

Lemma 2.5. The standard inner product (2.3) on the Grassmann algebra Grn = C[01,. .

.

,

Bn] is given by the following Berezin integral over the

Grassmann algebra Gr2n = (C[Bl,

,

Bn, 9i,.

,

(fl,f2) = ffi(O)f2(O)e00dOd,

(2.11) where

Proof. Put f'(9) =

... B and f2(6) = 8j1 ...

9.

It is clear that the

integral (2.11) is o, unless /c = 1 and i1 = j 1, ... , i = 3k , in which case we have

ez, Dil ... Dik! = foil O

.

.

. °zk9zk ...

... O O ... O dOdO O Jn dOd8 = 1.

The following result was already stated in Section 2.1. Lemma 2.5 allows us to prove it in a way that is reminiscent of a holomorphic representation (see Section 2.7 of Chapter 2).

Corollary 2.1. The operators 3 and O, i = 1, ... , n, are adjoint with respect to the inner product on Grn .

Proof. Using Lemma 2.5, formulas aZ f (B) = 0, aZe-ee = 6ie-ee, and the integration by parts formula, we obtain (az.f1,.f2) =

fafi(e)f2(o)e°dode _(_1)ifhi+1s2i /.fi(e).f2(e)

eie-eed9d9

J _(_l)+is2i ffi(e)oif2(o)e_eeded6 _ (fi,BZfz),

since the last integral and (fl, Bi f2) are both 0 unless fil -}- f2 is odd. Problem 2.4. Evaluate the Berezin integral n

exp

2

n

aOi8 +

rIkO

k=1

where rh, ... , i7n are Grassmann variables.

Problem 2.5. Prove formula (2.10).

d01

... din

7. Fermion Systems

324

Problem 2.6. Prove that (e)e-eedede

.f

=

the constant term of expansion of f(9) into the sum of monomials in C{01,. .. , On].

3. Graded linear algebra 3.1. Graded vector spaces and superalgebras. The notions of graded vector spaces and superalgebras, introduced in this section, allow us to consider commuting and anticommuting variables on the same footing. Definition. A 7L/27L graded vector space (graded vector space for brevity, or super vector space) over (C is a vector space W with a decomposition

W=W°EDWl into even and odd subspaces. The elements in W° U W1 \ {0} are called W° U Wl \ {0} -* {0, 1} such homogeneous and the parity is a map that We reserve the notation V for ordinary (even) vector spaces, denoting graded vector spaces by W. If W is finite-dimensional, we define graded dimension as a pair (dim W°, dim W1), usually denoted by no Ii, where n = dim W2, i --= = 0,1. When W° = C and W 1 = C, the corresponding A fermion Hilbert space graded vector space W is denoted by is a graded vector space with the even and odd subspaces given by decomposition (1.9):

o k even

The graded dimension of °F is 2n-1 2n-1.

Direct sums and tensor products of graded vector spaces are defined in the same way as for ordinary vector spaces. For the direct sums the homogeneous subspaces are defined by

(WiW2)k=WeW, k=0,1, and for the tensor products they are defined as W1

W2 k=0,1. W® W,

W2 }' =

,

i+j=k

The difference between ordinary and graded vector spaces becomes transparent in the definition of the corresponding tensor categories. Namely, the associativity morphism cwt W2 C'V3

: W1 ® (W2 ® W3 )- (W1 O W2 )® W3

3. Graded linear algebra

325

for graded vector spaces is defined by the same formula

C123 (W, ® (w2 ® w3)) = (w, 0 W2) 0 w3 as in the case of ordinary vector spaces, whereas the commutativity morphism Qwl w2 W, 0 WZ

WZ 0 Wl

is defined for the homogeneous elements by UW1W2(wi ® w2) _

2U1.

The tensor algebra T (W) of a graded vector space W is defined using the associativity morphism. However the exterior algebra A' W and symmetric algebra Sym(W) of W are defined as quotient algebras of T (W) by using the commutativity morphism. Namely,

Sym(W) = T(W)/I, where I is the two-sided ideal in T(W) generated by wl 0 w2 - Q(w2 ® wl), W1, W2 E W, and

A(W) = T(W)/J, where J is the two-sided ideal in T(W) generated by wl 0 w2 + 0(w2 ® WI), wl, w2 E W. Here a = Qw,yp is the commutativity morphism.

Definition. Let W = W° W 1 be a graded vector space. Aparity-reversed vector space IIW is a graded vector space with (IIW)° = W1 and (11W)' = W°.

It immediately follows from the definitions that for even vector space V (3.1)

sym(nv) =

sym(v).

and

Definition. A superalgebra over (C is a graded vector space A = A° E3 Al with a (C-algebra structure such that 1 E A° and

A° A° C A°,

A°

.

A' c A',

Al A' c A°.

A superalgebra A is a commutative superalgebra if

a . b - (_1)HbIb.a for homogeneous elements a, b E A.

An even superalgebra A is just an ordinary (C-algebra.

Definition. A left (super) module for a superalgebra A is a graded vector space M with the linear map

such that a m I _ (aI + ImD mod 2 for homogeneous a E A and m E M,

for alla,bEAandmEM.

7. Fermion Systems

326

Problem 3.1. Let a E Symn . Show that the isomorphism W1 ® ... ® Wn ^Wy_ 1 (1) ® ... ® We-1 (n),

induced by the commutativity morphism of graded vector spaces, is well defined does not depend on the representation of a as a product of transpositions in Symn .

Problem 3.2. Show that for a graded vector space W the algebras Sym(W) and A' (W) are superalgebras.

3.2. Examples of superalgebras. Example 3.1 (Tensor algebra). The tensor algebra 00

T(V)=

V®'k,

V° = c 1,

=o of an even vector space V is a superalgebra. The multiplication is given by the tensor product, and even and odd subspaces by V®k T(V)o = V®k T(V)1 = k odd

k even

Example 3.2 (Symmetric algebra). The symmetric algebra Sym(V) of an even vector space V is a commutative algebra. A choice of a basis x1,.

. .

, xn

of V establishes the isomorphism Sym(V) ^C[xl,... xn] - a polynomial algebra in commuting variables xl,... , xn. ,

Example 3.3 (Exterior algebra). The exterior algebra A' V of an even vector space V is a commutative superalgebra, with multiplication given by the wedge product; the even and odd subspaces are images of the corresponding subspaces of T (V) under the surjective mapping

T(V) - AV = T(V)/J, where J is the two-sided ideal in T(V), generated by the elements u®v+v®u,

u, v E V. According to (3.1), A(V) = Sym(IIV). The choice of a basis 81i ... , Bn of the odd vector space IIV establishes the isomorphism cc[el,

... ,en].

Here C[01,. . . , On] is the Grassmann algebra - a polynomial algebra in anticommuting variables 81i ... , 6n.

Example 3.4 (Algebra of differential forms). Let M be an n-dimensional manifold. The graded algebra A' (M) of smooth differential forms on M is a commutative superalgebra. Example 3.5 (Clifford algebra). The Clifford algebra C(V) of a quadratic vector space (V, Q) is a superalgebra, with the multiplication and grading descending from the tensor algebra T(V) under the surjective mapping

T(V) - AV = T(V)/J,

3. Graded linear algebra

327

where now J is the two-sided ideal in T(V), generated by the elements u®v+v®uv) 1, u, v E V. The natural map V y C(V) is injective, and V is identified with its image in C(V). The elements of V are odd in C(V). The fermion Hilbert space F is a left supermodule for C2n, and the C-algebra isomorphism p : C2n --> End(3F) (see Proposition 1.1) is an isomorphism of superalgebras.

Example 3.6 (Graded matrix algebra). Let W be a graded vector space. The vector space End(W) of all endomorphisms of W is a graded vector space: the even subspace consists of all endomorphisms which preserve the grading in W, and the odd subspace consists of those which reverse the grading. The vector space End(W) is a superalgebra with the product given

by the composition of endomorphisms, and W is a module for End(W). When W = the superalgebra End(W) is usually denoted by Its elements can be conveniently represented by 2 x 2 block matrices A_-

A11

A12

A 21

A 22

'

where A11, A12, A21, and A22 are, respectively, matrices of orders p x p, p x q,

q x p, and q x q. The even and odd elements of Mat (p q) are, respectively, block-diagonal and anti-diagonal matrices (' A2 2) and (A12 Ai 2 ) .

Example 3.7 (Lie superalgebra). A graded vector space g is called a Lie superalgebra (or, super Lie algebra) if it carries a Lie superbracket mapping [ , ] : g 0 g - g satisfying the following properties.

a linear

(i) (Super skew-symmetry) [x,y]

(-1)Ix11yl [y, x]

for homogeneous x, y E g.

(ii) (Super Jacobi identity) [x, [y, z]] + (_1)I(II) [y, [z, x]] + (_l) Izl(Fxl+IvI) [z, [x, y]] =0 for homogeneous x, y, z E g.

According to Proposition 2.2, a Grassmann algebra Grn with a Poisson bracket (2.7) is a Lie superalgebra. Corresponding to classical simple Lie algebras there are associated Lie superalgebras. Problem 3.3. Verify all the statements in this section. Problem 3.4. Show that a superalgebra A carries a Lie superalgebra structure with a Lie superbracket defined by

[a, b] = ab - (-1)ba for homogeneous a, b E A.

7. Fermion Systems

328

3.3. Supertrace and Berezinian. Let A = A°

A' be a commutative

superalgebra. We have the following general notion of a graded matrix algebra.

Definition. A graded matrix algebra with coefficients in A is a superalgebra

Mat(pq) of 2 x 2block-matrices

A= (A,,

Ai21

Azi A22

where All, A12, A21, and A22 are, respectively, matrices of orders p x p, p x q, q x p, and q x q with elements in A. The element A E Mat(pq) is even if corresponding matrices All and A22 consist of even elements of ,,4, and matrices A12 and A21 consist of odd elements of A. The element

A E Mat(pq) is odd if matrices All and A22 consist of odd elements of ,A, and A12 and A21 consist of even elements of A. The graded vector is a superalgebra with the product given by the matrix space multiplication.

The algebra Mat(pq) in Example 3.6 corresponds to the case A = C; another interesting example is when A is a Grassmann algebra. It is quite remarkable that such basic notions of linear algebra as trace and determinant admit non-trivial generalizations for graded matrix algebras.

Definition. A supertrace on a graded matrix algebra is a linear mapping Mat(pq) -+ A, defined by

All - (_l)IA

Trs A =

Aaa,

A = (Ali Aial A2i Aaa)

E

Mat,q(plq),

where 'Ir is the ordinary matrix trace.

Proposition 3.1 (The cyclic property of the supertrace). 'IBS AB = (_l)I4HBI ftS BA,

A, B E Mat,q(plq)

Proof. We check only the case when both 2 x 2block-matrices A and B Other cases are treated similarly and are are even elements of left to the reader. We have, using the cyclic property of the trace, 'IBS AB = Pr(AiiBii + A12B2i) - `I'r'(A2iBia + A22B22)

=

B21Ai2) - Tr(-BiaAzi + B22A22) _ Ts BA.

The superalgebra End() corresponds to the case ,A = C and is isoThe supertace on End(') is given by

morphic to (3.2)

Trs A = Tr All - Tr A22 = PrAT,

A E End(),

329

3. Graded linear algebra

where I' is the chirality operator (see Section 1.2). It has the following 2 x 2 block-matrix form: F

I

o

-o -I)'

where I stands for the identity operator inF and °, . It is a non-trivial problem to define a superdeterminant

a natural

analog of the determinant for graded matrix algebras, which is multiplicative and generalizes the rule det eA = eTr A. The corresponding notion, which is defined only for invertible A E Mat, (p q), was introduced by F.A. Berezin. It is now commonly called Berezinian and is denoted by Ber(A) . Consider first the case of even diagonal A = (An 1 A22)' The relation

defines the Berezinian by

Ber(A) = det All det A1. Thus in this case the even q x q matrix A22 is necessarily invertible, i.e., det A22 is an invertible element of the commutative algebra A°, so that the inverse matrix A22 exists. Now consider the general even A E Mat,A(pq), and suppose that A22 is invertible. It is easy to verify the following analog of the Gauss decomposition: (A11 A2i

Aia A22)

- (0

0l

AiaA(Aii - AiaAAzi Iq

)\

0

I

0

A22/ \A2a Aai Iq

'

where Ip and Iq are, respectively p x p and q x q identity matrices. This justifies the following definition.

Definition. Let A E

be even and such that the corresponding

even q x q matrix A22 is invertible. Then the Berezinian (superdeterminant) of A is given by

Ber(A) = det(All

- A12A22 A21) det A22

.

Assuming that the p x p matrix All is invertible, we get the decomposition (A11 Ail Aai A22)

( I0(All

- \A21Aii Ie) \ 0

0

A22 - Aai`411 Ail)

IAii Ail

(0

Iq

)'

which suggests that also Ber(A) = det All det(A22

- A21Ali A12)-1

It is a fundamental fact that for even invertible A E formulas for the Berezinian coincide.

these two

330

7. Fermion Systems

Theorem 3.1. Let A = (Aii Aial

Aai A22

be an even element of

Then

(i) A is invertible if and only if the matrices All and A22 are invertible. (ii) For invertible A, Ber(A) is an invertible element of ,A° satisfying

Ber(A) = det(All -A12A22 A2i) det A22 = det All det(A22 -A21Aii A12)-1

and

Ber(A-1) = Ber(A)-1 (iii) If even A, B E

are invertible, then

Ber(AB) = Ber(A)Ber(B). Problem 3.5. Complete the proof of Proposition 3.1.

Problem 3.6. Prove Theorem 3.1.

4. Path integrals for anticommuting variables 4.1. Wick and matrix symbols. Here we describe the calculus of Wick and matrix symbols of operators in the fermion Hilbert space F, which is analogous to the treatment of Wick and matrix symbols in the holomorphic representation in Section 2.7 of Chapter 2. As well as in the bosonic case,

it is a tradition to work in anti-holomorphic representation by usingF = C[01,.. . , O,] as the fermion Hilbert space with the annihilation and creation operators

ak-Bk and

(4.1)

a,

k

1,

... , n.

The inner product (2.11) takes the form (fl,f2)

= ffi()f2()e°°ddO,

and the monomials 1Cil<...
parametrized by subsets I = {ii,. basis in

'.

. .

,

ik} c {1,.

.

. ,

n}, form an orthonormal

4. Path integrals for anticommuting variables

331

Definition. A matrix symbol of an operator A : A(6, 6)

is an element

... , Bn, B1, ... , 9n], defined by

C[01,

A(, 6) =

(Afj, fi)fi()fj() I.J

(Afj,fj)1

... 9jl.

Here summation goes over all subsets I = {i1, ... , ij } and J = { j 1, ... , ji} of the set {1,. . n}, and as in Section 2.3, we denote by 1(6) a natural involution on the Grassmann algebra C[e1, ... , On, 01, ... , On], .

,

fj(6) = Bjl ... 9j1 for A:

fj() =

...

According to Proposition 1.1, C2n ^End(3 ), so that every operator F -f can be uniquely represented in a Wick normal form as

follows:

A=

a a1 ... aj1.

KIJ I, J

F ---} 3' is an element

Definition. A Wick symbol of an operator A : A(6, 6) E

,Vin, 01,... ,On], defined by

KIJ

A(6,6) =

Bj1

... O

.

I,J

Remark. The definition of matrix and Wick symbols in the fermion case repeats verbatim the corresponding definition for the bosonic case in Section 2.7 of Chapter 2. Note, however, that in the fermion case the product Bj1 ...Bbl in the definition of the matrix symbol has reverse ordering, as is required by the inner product (4.2) in (C[91i ... , On]. To matrix and Wick symbols A(8, 8) and A(8, 8) of an operator A one canonically associates elements A(6, a), A(9, a) and A(a, B), A(a, B) in the larger Grassmann algebra

C[a,(,V,V] = C[a1,...,an,ai,...,C n,Oi,...,BnO1,...,Bn], by replacing, correspondingly, 9z by ai and 9 by a . The incomplete Berezin integral f dada on C[a, ix, 6,6] is defined by

I fdada =

0Cxn cian

...

-

o

f ocx1 3c1 _

f e C [a, a, 6, 6]

,

and has the property (4.3)

J

h(8, )g(a, a)dada = h(8, 8)

J

g(a, a)dada.

7. Fermion Systems

332

We will also use the incomplete Berezin integral f d9d9, defined by

-

JVCL4 =

-

,1

7

feC[c,0,6].

As follows from the proof of Corollary 2.1, (ak.fi).fze-eed9d6

= -(-1)Ifl1+lf21 ffl(akf2)e°ddo,

where operators a and ak are given by (4.1 ), and Ii, 12 E C [a, a, 0, 8]. The next result shows that the matrix symbol of an operator A in .3F, which is just a 2n x 2n matrix, can be considered as an integral kernel in anticommuting variables!

Lemma 4.1. Let A(8,0) be the matrix symbol of an operator A in Then for every 1(0) E

(A!) (8) =

(4.5)

A(6, )f(a)edQda.

J

Proof. It is sufficient to verify (4.5) for f = 1K, where K = {kl, ... , k} {1,. , n}. Using (4.3) and Lemma 2.5, we get

C

. .

f A(,

a)fx(a)e-««dada

= I,J

(Afj, fi)fi() f (AfK, fI)fI() = (AfK)().

= I

Next, we introduce Grassmann analogs of the coherent states (see Section 2.7 of Chapter 2). Put

= eea =

fj()fj(a) and a(6) = e-e« - y f(a)f).

As in the bosonic case, elements (4.6)

,

II-

ak a = c k a and ak4 a =

C [a, a, 0,0] satisfy k = 1, ... , n.

It is very easy to express the matrix symbol of an operator in terms of the coherent states.

Lemma 4.2. Let A(a, a) be the matrix symbol of an operator A in F. Then

4. Path integrals for an tlcomm u ting Vary a bles

333

Proof. The proof is a straightforward computation (cf. the proof of Lemma 2.4 in Section 2.7 of Chapter 2):

(Afj,fi)fi(i)fj() (Afjfj(), fi(a)fi) =

= I,J

ia).

E

The next result is an exact analog of Lemma 2.4 in Section 2.7 of Chapter 2.

Lemma 4.3. The matrix and Wick symbols of an operator A in tF are related by

A(, a) =

a).

Moreover, A(ca, 8) = eaBA(a, 8) and A(8, a) = eeixA(9 ,a). Proof. As follows from the definition of coherent states, (an

an\

acx

Now representing the operator A in a Wick normal form and using properties (4.4) and (4.6), we obtain

(A, L) =

KI J (a I,J

--

... az aj1 ... ajl a y L

>(_1)+kKiJ(aji

. ail a, aik ... Qz1 }

I,J I,J

(_1)ktKijcj1

=

-

x.71 [xii ... [xik `a, -a}

I, J

= A(a, a)(,

) a= eaaA(cx, a).

The same computation, after replacing ca by

, or Ia bye, proves the

remaining two formulas.

Thus we have shown that 2n x 2n matrices -operators in the fermion

Hilbert space F -can be considered as integral operators in anticommuting variables with the integral kernels given by matrix or Wick symbols. The next result is an exact analog of Theorem 2.2 in Section 2.7 of Chapter 2, and establishes the calculus of symbols for operators in

Theorem 4.1. Let Al and AZ be operators in 3j' with matrix symbols A1(6,6) and A2 (8, B) and Wick symbols A1(6,6) and A2 (8, 8) following formulas hold.

.

Then the

7..Fermion Systems

334

(i) The matrix and Wick symbols of the operator A = AlA2 are given by

A(9, 8) = J Al(8, a)AZ(a, 9)e-aadada,

A(8, 8) = fA1(, a)A2(a, (ii) The trace and supertrace of an operator A in F are given by 'Ir A =

f A(,6)edd6 =

fA(,o)e_29Oddo,

A = f A(9, 9)e-eed6d9 = f A(B, 9)ded6. Proof. Part (i) for matrix symbols is proved by the following straightforward computation:

J

-

Al(B, a)A2(a, e)e-aadada

(Aifj, fi) (Azfz,, 1x) f I,J K,L

(AlfJ,fI)(A2fL,fJ)fI()fL() I,J,L

_

(A2fz,,fJ)(fJ,Aifr)fz(e)fL(e) r,J,L

_

(AafL,Ai.fr).fr(e).fL(e) _ > (AiAz.fL,.fr).fr(e)fL(e) I,L

I,L

A(e, e). The corresponding formula for the Wick symbols now follows from Lemma 4.3. The proof of part (ii) is also straightforward. We have

J

A(8, 9)e-eeded8 =

>(Afj, Ii)

r.

f

fl(e)f,(e)e-eedede

(Af1,f1) _ 'IrA. Similarly,

(Afj, fi)f fj()fj()e°°d6de

f ale, eke-eBd9d9 =

(-1)1(Af1,f1) = Tr8A,

= r

4. Path integrals for anticommuting variables

335

where Il denotes the cardinality of the subset I C {1, ... , n}. Corresponding formulas for the Wick symbol follow from Lemma 4.3. O Problem 4.1. Verify directly all results in this section for the simplest case of one degree of freedom, when

°F = C2.

Problem 4.2. Show that the Wick symbol of the product A = Ai ...A1 is given by

A(g, e) _

f... fAt (e, at-i) ... Ai (ai, e) eXP {

t-i ak (ak-i - aft) + 6(a1_1 - 8) k=1

daldal ...day-ldal-1 where a0 = g and Ak (0, 0) are the Wick symbols of the operators Ak.

Problem 4.3. Prove that the Wick symbol ['(0,9) of the chirality operator F is e--2ee.

4.2. Path integral for the evolution operator. Let H be a Hamiltonian of a system of n fermions an operator in 3 with the Wick symbol H(8, 9). Here we express the Wick symbol U(6, 9; T) of the evolution operator U(T) = e--iTH by using the path integral over Grassmann variables. Our exposition will be parallel to that in Section 2.4 of Chapter 5, with obvious simplification due to the fact that fermion Hilbert space °F is finitedimensional. Namely, the following elementary result replaces assumption (2.11 ), made in Section 2.4 of Chapter 5.

Lemma 4.4. Let U Xt} be the operator with the Wick symbol e_iH(O,e)°t Then

U(T) =N-lim U(t)',

where

Ot = N.

-Proof. The Wick symbol of the operator R(Ot) = I - iHOt - U(Ot) is a polynomial in Lt with Grassmann algebra coefficients, which starts with the term (zt)2. It is easy to see that I< c(t)2 for some c> 0, and

U(T) _ 1

(I - iHOt)N = iv' oo`U(Ot) + R(Ot))N =

U(Lt)N.

Using the formula for the composition of Wick symbols (see Theorem 4.1 and Problem 4.2), we can represent the Wick symbol UN (B, 9; T) of the operator U(Ot)N as an (N -- 1)-fold Berezin integral. Namely, consider the anticommuting variables rxk = {c,. .. , a}, ak = {,.. , rx }, k = 1, ... , N -1 generators of the Grassmann algebra with involution and .

J

7. Fermzon Systems

336

denote ikc k =

M,c4, etc. Then

/

N

UN(, e; T) - 1 ... f exP {

'

(aN -9)

(ak(ak_1 - ak) k=1

N-1

-iH([Yk, ak-1)Ot)}

fj dakdak,

k=1

where ao = B, and we put aN = 8. It follows from Lemma 4.4 that (4.7)

rr(e, e; T) = N

UN (e, e; T) = N f. . .

f N-1

N

ak(ak-1 - ak) -f- (aN -6) - 2H(CYk, ak-1)Ot)}

exP {

dakdak.

k=1

k=1

This formula looks exactly the same as the corresponding formula (2.13) for the Wick symbol of the evolution operator in Section 2.4 of Chapter 5! Accordingly, we interpret the limit N -> oo as the following Feynman path integral for Grassmann variables (or Grassmann path integral): (4.8)

U(6,6;T)-

1

6i

f0(itxa-H(a,cx)}dt+9(o(T)-8)

for &

fa(T)=

Here the "integration" goes over all functions a(t), a(t) with anticommuting values2 on the interval [0, T], satisfying boundary conditions a(0) = 8, a (T) = 8, and N-1

1G1a°1la = fi da(t)da(t) = lim fJ dakdak. 0
N-+oo

k=1

As in Section 2.4 of Chapter 5, for 0
Remark. It should be emphasized that the only rigorous meaning of the Grassmann path integral (4.8) is the limit of multiple Berezin integrals in (4.7). However, as we have seen already in Chapter 5, it is very useful to pretend that the path integral has an independent definition, and formally work with it as if it was an actual integral. 2 For every 0 C t
4. Path integrals for anticommuting variables

337

Using Theorem 4.1, it is easy to express the supertrace of the evolution

operator U(T) - an operator in afinite-dimensional Hilbert space F as a Grassmann path integral. We have

'fts e-iTx = N

f UN(6, B; T)d6d9 =Eli

N

f..N-1J exp {(a(T) -6)

- ak) - iH(ak, ak-1)Ot)} k=1

J

ei f

1

-

dakdakd9d9

a

k=1

a Grassmann path integral with periodic boundary conditions. Here periodic boundary conditions emerge from ao = 8 and EN = 8 in the same way as in Section 2.4 of Chapter 5. Namely, since B and 8 are now variables of integration, the identity (2.17) in Section 2.4 of Chapter 5, which is valid

in the case of anticommuting variables as well, ensures that aN =Band ao = 8. Denoting by A the corresponding "Grassmann loop space" the space of all functions with anticommuting values a(t) and a(t) conjugated with respect to the Grassmann algebra involution and satisfying periodic boundary conditions a(0) = a(T) and a(0) = a(T) - we can rewrite the previous formula as T[

ei f

e_iTH

91L.

A

Replacing the physical time t by the Euclidean time -it and T by -iT, we get the Grassmann integral representation for the Wick symbol of the

operator U(-iT) = e-Tx U(6, e; -ZT) _

I

e- fo

7

and for the supertrace, (4.9)

Tr e_TH =

e- f T (aa+H(a,a)}dt A

Problem 4.4. Express the matrix symbol of the evolution operator as the Grassmann path integral.

Problem 4.5. Show that Tr e-TH =

1 a(o)=-a(T)

a(0}=-a(T}

e- f

338

7. Fermion Systems

the Grassmann path integral with anti-periodic boundary conditions.

4.3. Gaussian path integrals over Grassmann variables. For simplicity, here we consider only the case n = 1. As in Section 5.3 of Chapter 5, for u(t) E Cl ([0, T], I[g) we put fT

uo = T

u(t)dt and D =

J0

t,

and consider on the interval [0, T] the first-order differential operator D + u(t) with periodic boundary conditions. The following result evaluates the simplest Gaussian path integral for Grassmann variables.

Theorem 4.2. We have e- fo (acs+u(t)aae)dta

= det(D + u t

=1-e

-u°T.

IA

Proof. Using Lemma 2.4, we get

£

= lim

N-*oo

-j e

e- fp N

k)(tk)akk_1t) [I dakdak

'=' (k(k_1

k=1

= lira det AN . N--} oo

Here ao = aN, ao = aN, tk = kOt, and AN is the following N x N matrix: 0

0

0

bN

bl

1

0

0

0

0

b2

1

0

0

0

0

0

1

0

0

0

0

bN_1

1

1

AN

:

where bk = -1 + u(tk)Ot. We have N

N

det AN = 1 - (_1)Nflbk = 1 - fl(i - u(tk)Ot), k=1

k=1

so that

lim det AN = 1 - e_ fo u(t)dt = 1 - e-u0T. Using Proposition 5.3 in Section 5.3 of Chapter 5 completes the proof.

CI

Example 4.1 (The fermion harmonic oscillator). The fermion analog of the harmonic oscillator is the Hamiltonian H = 2 w (a*a -- aa* ) = w (a* a -- 2 I) = w (N - 2},

4. Path integrals for anticommuting variables

339

where a* and a are creation and annihilation operators in the one fermion Hilbert space .F - C2 (see Section 1.1). The Wick symbol H is H (o, a) 2 }. Now using (4.9) and Theorem 4.2, we obtain Tr s e-TH

ez n

e- fo

=

e- fo (aa+waa)dt a s= e a 1_ e-WT = 2

sinh2

.

Of course, the same result can be obtained directly since e-TH is just a 2 x 2 matrix with eigenvalues e and e1 which correspond respectively,

to the eigenspacesF andF TrS e-TH

.

Thus

- e = 2 sinh cyT 2.

=e 2

Remark. We have the following analog of formula (2.6) of Chapter 6 for path integrals over Grassmann variables: (4.10)

IA

f(a, a) cu =

fD f(/9, B, /9, 9)

where

d9d9,

i9

T

a(t)=6+8(t)

and

f

a(t)dt = o,

and Ao is the space of Grassmann loops ,fi(t), j3(t) with zero constant term.

Example 4.2. Using formula (4.10) and Theorem 4.2, we get

det(D + w) = j e-.fo dt

e -wT l9B d 9d IA0 e fo

=wT

e

fo

dt

-

so that

e- f '

dt

1 - e -wT det ( D + w) = lim w-0 wT w- O wT

= hm

Equivalently,

g- fo

(4.11)

1

det,D,

a(t))

-T where det'D = T (see Proposition 5.3 in Section 5.3 of Chapter 5). Introc1J/(j1,13 -

ducing

Bi(t) _

(/9(t) +a(t))>

BZ(t) _

(/9(t)

-

that B (t) = O( t), j = 1, 2, we can rewrite (4.11) as

e.f(B1ei+B2e2)dtBB IA0

-

det'D - -

1.

7. Fermion Systems

340

Since Pf'(D) =

det'D =

we have

e-2 fo

(4.12)

J_Pf'(D)

= 1.

Here the domain of integration Ao(I[8) consists of all functions B(t) with values in the Grassmann algebra over ][8, which have zero constant term and satisfy periodic boundary conditions 9(0) = B(T). We will use formulas (4.10) and (4.12) in Section 2.2 of Chapter 8.

Problem 4.6. Show that

I

e- f

1 + e-u0T

the regularized determinant of the operator D -+- u(t) on [0, T] with anti-periodic boundary conditions.

Problem 4.7. For the fermion harmonic oscillator verify that Tr e-TH = 2 cosh 2 directly, and also by using results of Problems 4.5 and 4.6. Problem 4.8. Prove formula (4.10). (Hint: Follow the proof of Theorem 4.2, and 1 /3 = 0.)

use the decomposition oe = 0+ /3k, where 13o = 13N and

Problem 4.9. Give a direct proof of formula (4.12).

5. Notes and references Canonical anticommutation relations, discussed in Section 1, were introduced by P. Jordan and E. Wigner in 1928 [JW28]. We refer the reader to F.A. Berezin's classical monograph [Ber66] for a comprehensive mathematical treatment of canonical commutation and anticommutation relations. Though [Ber66] is mainly devoted to quantum systems with infinitely many degrees of freedom, which are studied by quantum field theory, it also discusses the simpler case of finitely many degrees of freedom. A self-contained mathematical introduction to Clifford algebras, their representations, and other topics can be found in [VarO4I and references therein. The fundamental idea that systems with Grassmann variables appear as a semi--classical

limit of fermions was formulated by I.L. Martin in 1959 [Mar59b, Mar59a]. Independently, F.A. Berezin in [Bern l] introduced Grassmann variables for rigorous mathematical description of the second quantization for fermion systems by using generating functionals for vectors and operators. Material in Section 2 differenbelongs to F.A. Berezin, tial and integral calculus on the Grassmann algebra graded linand our exposition follows [Ber66] and [Ber87]. The superalgebra ear algebra which we very briefly describe in Section 3, was also discovered and developed by F.A. Berezin [Ber87]. The references [Man97], [Fre99], and [DM99] provide the reader with a more abstract mathematical treatment, while lecture notes [VarO4] supplement the category theory approach of [DM99] with

5. Notes and references

341

motivation from physics. We refer to [Ber6l, Ber66] for the general discussion of the path integral over Grassmann variables, and of the matrix and Wick symbols of operators in the fermion Hilbert space. In Section 4 we follow the elegant exposition

in [as in Section 2.4 of Chapter 5, we carefully treat boundary conditions for Grassmann path integrals for the Wick symbols (as opposed to the formulas [Ber7la]).

Chapter 8

Supersymmetry

1. Supermanifolds The coordinate vector space V = W has a natural structure of a smooth manifold, which to every open subset U C W1 assigns a commutative IItalgebra C°° (U) of all smooth functions on U. The assignment U F-* C°°(U)

for all open U C Ian defines a sheaf of commutative III-algebras (commutative

rings) on a topological space W, and turns it into a ringed space. Every smooth n-dimensional manifold M is a ringed space a topological space with a sheaf of commutative rings, which is locally isomorphic to the ringed space IR'z. It is easy to see that this definition is equivalent to the standard one, defined by gluing coordinate charts. The notion of a supermanifold generalizes this idea by using local models associated with graded vector spaces.

Namely, let W = be the coordinate graded vector space of dimension p q over III. The following definition formalizes the intuitive idea that odd coordinates on W are anticommuting.

Definition. A supermanifold

is a topological space W with a sheaf of commutative III-superalgebras (supercommutative rings) over III, called the structure sheaf, defined by the assignment

for all open U c Rp, where C°° (U)[9' , ... , 9] is a Grassmann algebra with generators 9', ... , 9q over the commutative ring C°° (U ). 343

8. Supersymmetry

344

Remark. Elements of C°O(U)[Bl, ... , 04] have the form

f=

fze7,

BI = Bil

... Bz

and

fi E C'°°(U)

I

for I = {il, ... , ik} C {l, ... , q}, and are called functions on a supermanifold ][gPI9 over U. Thus is a coordinate space with even coordinates x = (x',. , xp) and odd coordinates 8 = (B1, ... , Bq), and we will write f E . .

as f(x,8). is a pair (X, OX) - a topoDefinition. A supermanifold of dimension logical space X with a sheaf OX of supercommutative rings over I[8, called the structure sheaf, which is locally isomorphic to Supermanifolds form a category: a morphism between supermanifolds (X, OX) and (Y, COY) is a continuous map cp : X --> Y, together with a sheaf map cp* : COY - OX over cp, a collection of homomorphisms of supercommutative rings over I[8,

cpv : OY(V) -Ox(cp-1(V)), V C Y open, which commute with restriction maps of the sheaves. To every vector bundle E of rank q over an ordinary p-dimensional mandefined ifold Mthere is an associated supermanifold HE of dimension by reversing parity in the fibers of E. Namely, for every open U C M let C°°(U, E) be the space of all smooth sections of E over U. It is a free C°°(U)module generated by q sections Bl, ... , Bq. Consider them as generators of

the Grassmann algebra (this explains using Greek letters), and define the structure sheaf of IIE by the assignment U F-f C°°(U)[01, ... , 99]. It can be shown that every supermanifold is isomorphic (non-canonically) to HE for some vector bundle E over M. Remark. Supermanifolds have many more morphisms than vector bundles, since one is allowed to mix even and odd variables. Thus for the supermanifold R"2 with even coordinate x and odd coordinates 01, 02 the mapping

,(x)=x+0'02,

(0i)0i, i=1,2,

is an isomorphism of 1l2, but it is not induced by an isomorphism of a trivial rank 2 vector bundle over I. Using these definitions, one can develop differential geometry of supermanifolds and the corresponding theory of integration, which is quite similar to that for ordinary manifolds (see Section 7 for the references). Here we will only consider the simplest example of supermanifolds HTM, where M is an ordinary manifold. It is quite remarkable that the basic notions of differential geometry on a manifold M can be formulated in terms of the supermanifold HTM.

2. Equivariant cohomology and localization

345

Namely, let A' (M) be a commutative superalgebra of smooth differential

forms on the n-dimensional manifold M, and let C°°(IITM) be a commutative superalgebra of global sections of the structure sheaf of IITM -the superalgebra of functions on a supermanifold IITM. The isomorphism

A(M)

C°° (IITM)

allows us to interpret differential forms on M as functions on IITM. Indeed, to every cep E A(M), given in local coordinates on U C M by n ... n dxi2,

wp =

we assign wp (x, 9) =

(x)Oil

...

O

E C°° (U)[Ol , ... , Bn] .

1Ci'<...
It trivially follows from the definition of differential form and of IITM that under a change of coordinates (x) transform like coefficients of differential forms, and 8', ... , Bn transform like components of tangent vectors, so that wp (x, 9) is a well-defined function on the supermanifold IITM. Correspondingly, the de Rham differential d gives rise to an odd vector field 8 on IITM, n

=

1

B

ax

,

with the property that 82 = o. Finally, the supermanifold IITM carries a canonical volume form dxd B = dx ' ... dxn dB 1 ... d8', which is well defined due to the opposite change of variables formulas for the ordinary and Berezin's integrals (see Section 2.3 of Chapter 7). The integration of a top differential form over M reduces to the integration of a corresponding function over IITM with respect to the canonical volume form,

f Mwn=

f

HTM

Remark. Sometimes in the mathematics literature, in order to emphasize the opposite change of variables formulas for the ordinary and Berezin's integrals, the volume form on the supermanifold IITM is denoted by dxd9 ' . Problem 1.1. Prove all statements in this section.

2. Equivariant cohomology and localization 2.1. Finite-dimensional case. Let M be a compact orientable n--dimensional manifold with the circle action an action of abelian group U(1) =

346

S. Supersymmetry

Sl -and let V E Vect(M) be the vector field corresponding to this action, V (f) (x) -

_ dt to

f(eit.x),

x E M.

I

Consider the linear operator

where iv is the inner product operator with V. Using the identification ,,4'(M) ^C°O(IITM), in local coordinates x = (xl, ... , xn) on U C M we can represent D by (2.1)

D=

(o-_aaµ µ=1

- v` ae I '

v

where V = µ=1

ax'

It follows from the Carton formula (see Chapter 1), or directly from (2.1),

that D2 = --LV,

where Lv is a Lie derivative. Thus D is a differential on the subcomplex .f4' (M)Sl of S1-invariant differential forms on M. The cohomology of this complex is an equivariant cohomology in the Cartan formulation. Since d and iv have, respectively, degrees 1 and -1, equivariantly closed forms on M necessarily have several components. Namely, the equation Da = o for a= ap E A(M) is equivalent to the following system of equations:

dap = ivap+2, p = o, ... , n - 2,

and

dan_1 = o.

Let MV be a zero locus of the vector field V the set of fixed points of the circle action. We have the following fundamental property of equivariantly closed differential forms. Lemma 2.1. An equivariantly closed differential form an a compact manifold M is equivariantly enact on M \ Mv.

Proof. Let a E .A' (M) be such that Da = 0. We want to find a differential form A on M \ My such that a = DA on M \ MV. Suppose there is a form on M \ My, with components of odd degrees, satisfying D = 1. Setting A

DA

To construct such a form , choose an 51-invariant Riemannian metric g on M, and denote by ,6 E Ai (M) the 1-form on M dual to the vector field V with respect to the Riemannian metric, /3 = (V, ). Since the metric g is Sl-invariant, GVg = 0 and, therefore, Gv/i = 0. We have D,3 = K + SZ,

where K = -11V112

and SZ = d,6.

2. Equl Variant cohomology and localization

347

Since K E AB(M) does not vanish on M \ My,

_ /3(D/3)-1 = K (1 + K-152)-1 =

l= L2J

i-O

is awell-defined form on M \ My satisfying D = 1.

Corollary 2.1. The top component of an equivariantly closed form on M is exact on M \ M. Remark. In local coordinates x = (xi, ... , xn) on M the Riemannian metric ghas the forms ds2 = and equation Gug = 0 the condition that V is a Killing vector field with respect to the metric g can be written as

(2.2)

gVv' +

= 0.

Here Oµ is a covariant derivative with respect to the vector field aaµ . For the 1-form ,6 = gv"dxµ we also have

DQ = -gvµv" + wdxµ n dxv, where w, = gpv' By (2.2), the n x n matrix w(x) _ {w(x)},...., is skew-symmetric. For (2.3)

even n = 21, orientation on M allows us to define Pf(w(x)) as the product A1 ... )l of above-diagonal elements in the canonical form2 of a skewsymmetric matrix w(x) (see Section 2.3 of Chapter 7). Correspondingly, Pf(w)(x)(detg(x))-2 does not depend on a choice of local coordinates, and defines a function on M. For every x E M there is also a linear mapping L : TAM - TIM, defined by LAW = (VwV)(x), where Vyy is a covariant derivative with respect to W E TIM. Since det L det g(x) = Pf (cv(x))2, we define (2.4)

detLx = Pf(w(x)) /det g(x)

For x E MV the mapping L does not depend on the choice of an S'-invariant

metric g, and is given by the matrix

µ,v=1.

axv

According to Lemma 2.1, the integral fM a of an equivariantly closed differential form a localizes on the zero locus Mv. A quantitative result is given by the following statement.

Proposition 2.1. Suppose that Da = 0 on M. Then for every Sl -invariant ,3 E Al (M) the integral fM aetD does not depend on t. We are assuming the summation over repeated indices. 2 Obtained by using orthogonal transformation with determinant 1.

S. Supersymmetry

34$

Proof. Denote this integral by Z(t). We have dZ dt

J

_ fM

=0, f where the first integral is zero by Stokes' theorem, and the second integral M

M

is zero by the conditions Da = 0 and D2,Q = 0.

This result is usually used to evaluate Z(0) = fM a by computing the asymptotics of Z(t) as t --> oo. For a suitably chosen 1-form Q the leading contribution is given by the integral over a tubular neighborhood of the zero locus of the component K = (D/3)o E .A°(M), which can be computed in the closed form as t -* oo. The simplest case of a localization phenomenon is when the circle action on M has only isolated fixed points, i.e., when My

is a finite set and L is non-degenerate for x E M. In particular, in this case n is even.

Theorem 2.2 (Berline-Vergne localization theorem). Let M be a compact oriented 21-dimensional manifold with a circle action which has only isolated fixed points. Then for every equivariantly closed form a E

IM

ao(x)

a=(27r)1 xEMv

det L

Proof. Let ,3 be the 1-form introduced in the proof of Lemma 2.1, so that in local coordinates S1 = d/3 = w, dx11 n dxv. Using Proposition 2.1 and the

identification A(M) ^C°° (IITM), we get

IL

a = lim

t-*oo HTM

(x}vv

a (x,

dxdO.

We have in the distributional sense, t

(2.5)

lim

t--*oo

7t

det

(x)vy (x}

x

=Vx

the main formula of the method of steepest descent (cf. the method of stationary phase in Section 2.3 of Chapter 2) and (2.6)

Tim 2t c->

_L

1

etwµv

S(e) Pf(w(x)) - the main formula of Gaussian integration over Grassmann variables (see

Proposition 2.3 in Section 2.3 of Chapter 7). Since ap(x) = a(x, 0), we obtain the result by using formulas (2.5)-(2.6), (2.4), and

S(V (x)) _ L xi EMv

det Lxi

Problem 2.1. Prove formulas (2.2)-(2.4).

8(x-x).

2. Equivariant cohonioiogy and localization

349

Problem 2.2. Prove formulas (2.5)-(2.G) and complete the proof of Theorem 2.2.

2.2. Infinite--dimensional case. An important class of infinite-dimensional manifolds is given by the loop spaces. Let M be a compact orientable n-dimensional manifold and let

£(M) where Sl = IE8/7G, be its free loop space. The loop space G(M) is an infinitedimensional Frechet manifold, and the tangent space T.yG(M) to G(M) at a point ry E G(M) is

T-L(M) = F(Sl,y*(TM)) - the vector space of smooth vector fields V = {v(t) E T,y(t)M, 0 < t G 1} along the loop y in M. The loop space G(M) has a canonical circle action given by the action of Sl on itself by rotations. The vector field corresponding to this action is the velocity vector field 'y(t) along the loop ry in M. The fixed points ry of the action are constant loops, so that G(M)y = M. The vector field ry on G(M) - a generator of the circle action on G(M) is analogous to the vector field V on M, considered in the previous section. As in the finite-dimensional case, denote by D = d - i.y the equivariant differential on the complex ,,4'(G(M))sl of equivariant differential forms on GM. Using the identification A(G(M))CO°(IITG(M)), we have the following infinite-dimensional analog of representation (2.1): (2.7)

F DF - L1 (o(t)88)

(t))) dt,

F E C°°(IITG M

Here we are using functional derivatives (as in calculus of variations), local

coordinates x = (x',.. , xn) on M, and associated standard coordinates .

and IIT,y(t)M, so that -'y(t) _ (±'(t),.. . xn(t)) E T,y(t)M and B(t) _ (6'(t),. , 9n(t)) E HT(t)M. As in the finite-dimensional case, in

,

. .

(2.8)

D2=_f

8x/2 (t) + eµ(t)

(t))

dt =

To define the infinite-dimensional analog of a 1-form ,3, we choose a Riemannian metric g vdx1 dx'1 on M and consider an S1-invariant Riemannian metric on £(M), obtained by the integration of g along a loop3, 1

(Vl, V2)ry = J (vi(t),v2(t))dt,

Vi, V2 E T,yG(M).

0

3Here and in what follows v1 (t), v2 (t)} always stands for the inner product in T ) M given by the Riemannian metric g.

S. Supersymmetry

350

The 1-form Q on G(M) is dual to the vector field 'v with respect to the Riemannian metric on G(M), and is given by the following function on IITG(M): 1

(2.9)

,

1

B) = 0f ((t), e(t))dt = f0

9uv('Y(t))xµ(t)B"(t)dt.

Lemma 2.2. The 1-form ,Q on G(M) is Sl invariant, and the function -28 on IITG(M) is given by 8(7,9) =

(2.10)

2 f (I(t)II2 + (O(t),V(t)9(t)))dt,

where V.y is a covariant derivative along ry. The first term in -28 is an analog of the function K in the finite-dimensional case, and the second term is an analog of the 2-form SZ.

Proof. The 1-form 41 is 81-invariant by definition. We have D,Q = d,E1 - i.y/3,

where the second term, afunction -i.y/3 on IITG(M), clearly gives the first term in (2.10). Now using that in the distributional sense S

t) = SPµS(t - S)

and bPµdtS(t

- S) _ -&S (t -

we compute the term d,3 as follows: d41(7, e) =

-f

0

1

f

0

f 1 f 1 eP(s) SxP(s) s 0

o

1

S)\

BPISJ

8) - b9'

\

fl

I Bv(t)dsdt

(x(t))xµ(t)eP(t)ev(t))dt.

axP

It follows from the definition of Christoffel's symbols (see Example 1.7 in Section 1.3 of Chapter 1) and anticommutativity of 9µ that

xBPB" -

(2.11)

and we obtain d41(7, e) =

f 1 gµv(x(t)) (O(t) + 0

=1 (V(t)9(t),6(t))dt.

2. Equivariant co.homoiogy and localization

351

As in Chapter 6, one can develop the integration theory on G(M) with respect to the Wiener measure associated with a Riemannian metric on M by using the corresponding heat kernel4. One can also integrate differential forms over G(M), provided that G(M) is orientable. As in the finitedimensional case, the latter is defined by the condition that the structure

group of an G(SO(n))-bundle G(FM) -p G(M), where FM - M is the frame bundle, reduces to the connected component of the identity. It follows from the transgression homomorphism

HZ(M,Za) -' FI1(G(M),Z2) that the image of the second Stiefel-Whitney class of M is the obstruction to orientability of G(M). In particular, if M is a spin manifold, then G(M) is orientable, and if M is simply connected, then this is also a necessary condition.

In what follows we will not attempt to develop the integration theory on G(M), and will formulate and "prove" the infinite-dimensional version of the general Berline-Vergne localization theorem at a heuristic level only. The key observation, by Lemma 2.2, is that the differential form e-S on G(M) is equivariantly closed, therefore the functional integral f(M) e-S localizes reduces to the finite-dimensional integral over G(M),y = M, the zero locus of the vector field y.

To give the quantitative formulation of this result, recall the notion of the A-genus. Namely, the A-genus of a Riemannian manifold (M, g) is a differential form A(M), defined as follows. Let R E ,AZ(M, zo(TM)) be the Riemannian curvature of M, and let e be a local orthonormal frame of TM. Put

R= (Rep,

E A2 (M),

and let

j(R)

det

sinh R/2

R/2 ) be an even degree differential form on M. It does not depend on the choice of a local orthonormal frame of TM, and the A-genus of the manifold M is defined by5 1

A(M) = j(R)

1

= det

(sinhR/2R/2)

2 E A(M).

It is a closed differential form, whose cohomology class does not depend on the choice of a Riemannian metric on M. 4In Chapter 6 we considered the case of M = R' with the standard Euclidean metric. This is the geometer's definition of the A-genus; in topology one replaces R by (2iri)R.

S. Supersymmetry

352

Remark. In local coordinates on M, 1

Rev

R/Iv

n dxu ,

where R/Ivpa is the Riemann curvature tensor.

Theorem 2.3. Let M be a compact orientable manifold such that the free loop space £(M) is orientable. Then Jc(Nt)

e_s

_ (2i) -2 L

A(M).

Proof. The proof is at a heuristic level of rigor. The integral

[G(M) es = IIITG(M) localizes at the zero locus G(M).y = M, so it is sufficient to integrate over a small tubular neighborhood of IITM in IITG(M). For this aim, consider Riemann normal coordinates of the normal bundle N(IITM) to IITM at and a point (xO, Bp) E IITxoM, given by y(t) _ (y1(t),.. . yn(t)) E fi(t) _ (ij1(t),. , rf1(t)) E IITXoM, satisfying ,

. .

J y(t)dt = 0

and

J r(t)dt = 0,

µ = 1, ... , n.

0

0

Coordinates (Ey(t), E(t)), where real e is sufficiently small, correspond to a point (expxo(ey(t)), P(6 t)(Bo+'q(t))), in the normal bundle N(IITM), where expxo is the exponential map, and PEE t) is a parallel transport in IITM from as s changes from 0 to E. xo along the path Since we are integrating over an arbitrarily small tubular neighborhood of IITM in IITG(M), it is sufficient to expand S(ry, B) up to terms of second order in local coordinates (y(t), ra(t)). Since in Riemann normal coordinates around xo on M

+o(Ilxll2) and Rµvpv(xo) -

s,v(x) =

raUQ

axe

_ ar v ax

we readily obtain that at (X0, Bo) E S(ry, B) = Sp(xp, Bp; y, rJ) + higher order terms in yµ(t), r7µ(t), where So(xo,eon y,q)

i

(2.12)

=

2 f ((t)yµ(t) - Rµv(xo)yµ(t)yv(t) 1

Rµv(xO)

- 2Rµvpo(x0)epep

2. Equivariant cohomology and localization

353

Now using property (2.6) in Section 2.2 of Chapter 6 (where m = fii = 1 and T = 1), formula (4.10) in Section 4.3 of Chapter 7, and performing Gaussian integration, we get

f

e_S = (27r)_

Ill dxodOo HTM

(M)

N(HTM)

ft (xo,

=(27r)

Oo)D}-

dxodOo,

HTM

where we have also used formula (4.12) in Section 4.3 of Chapter 7. Since R= is a skew-symmetric matrix with even matrix elements, there is an orthogonal matrix C of determinant 1 such that {R}_1

0

I-ri

rl

0

0l

0

0

0

O

rm

CRC-1 =

\0

O

O 0

..

It is easy to see that second order matrix differential operators

-rI D and

-D2I2 +

- D2I2 + I

91 D,

Or

where 12 is the 2 x 2 identity matrix, have the same spectrum. Using formula

(3.5) in Section 3.2 of Chapter 6 with w = firk, we get

- R,µvD) _ fi det' (_D212

det'(-D2

+ \O rr

k=1

0 o -rk/ D/

m

_ fi det'(-D2 - irkD)det'(-D2 + irkD) k=1

IT (sin rk/212 = rk/2

k-1

det(/2) R/2 n R11

= j(iR). Therefore, using the identification C°°(IITM)

f

e-S (M)

= (2)- 2 f

we finally obtain

j (iR(x, B) )- 2 dxd9

ITM

=(2iYfA(M). M

Remark. Using formulas (2.6) in Section 2.2 of Chapter 6 and (4.10) in Section 4.3 of Chapter 7, we see that the same result holds for the space

354

8. Supersymmetry

£j(M) of loops on M parametrized by the interval I = [o, T],

f

= (2i) 2

TCI (M)

A(i). M

Problem 2.3. Derive formula (2.8). Problem 2.4. Derive formulas (2.11) and (2.12). Problem 2.5. Justify the arguments in the "proof" of Theorem 2.3. (Hint: See the references in Section 7.)

3. Classical mechanics on supermanifolds As was mentioned in Chapter 4, spin is a pure quantum notion which has no classical analog, and the same applies to fermion systems, considered in Chapter 7. However, one can formally consider particles with anticommuting coordinates and formulate classical mechanics on supermanifolds. Though

such systems have no physical interpretations, their formal quantization yields fermion systems. This allows us to interpret particles with odd degrees of freedom as semi-classical limit of fermions.

3.1. Functions with anticommuting values. Any smooth map f : M --> N of smooth manifolds M and N gives rise to a Frechet algebra homomorphism f* C°°(N) -p where f*(co) = cp o f for cp E C°O(N). Conversely, any Eechet algebra homomorphism F : C°°(N) - C°°(M) is of this form for some smooth map

f :MAN.

By definition, a map (morphism) between supermanifolds X and Y is a homomorphism of superalgebras

F C°°(Y) where C°° (X) and C°° (Y) are commutative superalgebras of global sections

of corresponding structure sheaves OX and COY (see Section 1). We will denote by Map(X, Y) the space of all maps between supermanifolds X and Y. The simplest cases are the following.

1. X =

-odd one-dimensional supermanifold, and Y = M

an ordinary (even) manifold.

-

2. X = IEg (or Si and I = [to, ti}) -even one-dimensional manifold, and Y = I[8OIn -odd n-dimensional coordinate vector space. 6 Classical mechanics, which describes physical phenomena at the macroscopic level, necessarily uses commuting coordinates.

3. Classical mechanics on superman folds

355

In the first case every homomorphism of superalgebras F : C°°(M) - IEg[B] has the form F(p) = A(cp) + B(cc)B, cp E C°°(M), where A : C°°(M) - Il8 is an 1[8-algebra homomorphism, and B : C°°(M) -> C°°(M) is a derivation of the commutative algebra C°°(M). By the GelfandNaimark theorem, there exists x E M such that A(c) = cp(x), and B(cp) dcp(v) where v E TIM. Thus every map f : -- M is given by a point (x,v) E TM. However, in the second case we have a homomorphism of superalgebras

F : R[O',.

. . ,

9fl]

and since the commutative algebra COO (I[8) contains no nilpotents,

F(6') = ... = F(O) = 0. This result is clearly unsatisfactory, since every map ][8 is trivial: k (t) = 0, k = 1, ... , n. To remedy this situation, one should consider the analog of Grothendieck's functor of points from algebraic geometry. Namely, introduce an auxiliary Grassmann algebra W = IlS[rjl, ... , and consider the product? W x 1[8 with the superalgebra of functions CO° (I[8) [i1 The "W-points" of Map(Il8, ][8°In) are defined by homomorphisms of commu-

tative superalgebras

F : R[O',

- C°°(IR)[7)',. In this case it is customary to write (3.1)

Bk(t)

. . .

,

,

= F(Bk) = a( t)1 + ... + a(t),

k = 1, ... , n, where real-valued ai (t), ... , a(t) E C°° (I[8). It is convenient to think of O' (t) as smooth functions of variable t with anticommuting values, and define (3.2)

Bk(t)

= ai

-I- ...+

where the dot stands for the derivative. To emphasize that Grassmann variables B (t) are "real-valued", we will always assume that W is a set of fixed points in a Grassmann algebra with involution over (C, so that

ek(t)

k = 1, ... , n.

In the same way one defines W-points of Map(Sl, IE8°ft) and Map(I, IlB9In),

Remark. This definition of the space of maps Map(I, Il8°In) (and also of the spaces Map(I[8, I[8°In) and Map(Sl, I[8°In)) is motivated by the definition

used by physicists, who introduce an infinite-dimensional auxiliary Grassmann algebra W, so that for every t the values 6k (t) are "independent real Grassmann variables". Geometrically this corresponds to the family of spaces parametrized by W.

8. Supersymmetry

356

Remark. One can also define functions on the space Map(I, Il8) -the functionals of functions with anticommuting values Bk(t). Thus for the simplest example of the quadratic functional tl

s(e) = 2

e(t)e(t)dt, Ito

where we put n = 1 and Bl(t) = 8(t), using (3.1)-(3.2) we obtain S(B)

- ?2 J tl (ak(t)al(t) -

E

.

o

Here the factor i =

ensures that ir1kr/ are real Grassmann elements, i.e., In general, the space of "functions" on Map(I, I[8°) is A'(Map(I, IIBn)*) ® W, where Map(I, I[8n)* is the (topological) dual to the vector space Map(I, ][8n), and W is a Grassmann algebra with infinitely many generators.

Vector fields with anticommuting values are naturally associated with the path space of a smooth manifold M. Namely, let

P1(M) = {y: I - M}, where I = [to, tl], be the space of smooth parametrized paths in M. Consider the supermanifold IITM (see Section 1), and for every y E Pr(M) let ry* (HTM) be the pullback bundle over I. By definition, a vector field with anticommuting values 8(t) is a section of ry*(IITM) over I. In local coordinates x = (xl, ... , xn), 6(t)

=

01L(t)

E HT(t)M.

Thus

IITP1(M) = {(y, 9) ; 'y E Pi(M), 0 E r(I, ry*(HTM)}}. Problem 3.1. Show that S E A' (Map (.I, R)*) ® W, where S is the quadratic functional given by (3.3).

3.2. Classical systems. Here we consider the basic examples of classical systems on supermanifolds.

Example 3.1 (Free classical particle of spin 2 ). The configuration space is the supermanifold HTIR3 the tangent bundle to R3 with reversed parity of the fibers, with even and odd coordinates x = (x', x2, x3 ) and 6 = (0', 2, 0). The action functional S : IITP1 (IE3 ) --- W, where W is .

3. Classical mechanics on supermanifolds

357

some auxiliary Grassmann algebra, is defined by ti

s(x(t), e(t)) =

j

L(x(t), x(t), e(t))dt

to

1

_-

(3.4)

2

ftl

Jto

(me2 + i66)dt,

(x(t), 8(t)) E IITPI(R3).

Here

L(x(t), (t), 8(t)) = 2 (mx2(t) + i6(t)9(t)) is the Lagrangian function, which is real-valued since iO(t)O(t)

-iO(t)0(t) = i9(t)O(t).

As in Section 1.2 of Chapter 1, we readily derive classical equations of motion as the following Euler-Lagrange equations:

x(t) = 0 and e(t) = 0. Canonically conjugated momenta8 are defined by pk

=

= mxand k =--L=--O,kk=1,2,3, aek

2

and the Hamiltonian function H is given by the Legendre transformation,

H-=pay+Oir-L=

p2

2m

In agreement with classical equations of motion 0(t) = o, the Hamiltonian H does not depend on the odd variables. The phase space is a supermanifold with real coordinates p, x, 8 and the symplectic form

w = dp n dx - d9d9. In accordance with Section 2.2 of Chapter 7, Poisson brackets between the odd variables are given by (3.5)

{O',O1}_-i8,

k, l = 1, 2, 3.

Poisson brackets satisfy the following involution property: (3.6)

{fl,f2} _ (_1)f11f2I{j1,j2},

11,12 E C[B1, 92, B3].

We will see in Section 5 that quantization of this system describes the free quantum particle of spin 2. sHere the indices are raised by the standard Euclidean metric on

I.

8. Supersymmetry

358

Example 3.2 (Classical spin 2 particle in constant magnetic field). This system is described by the configuration space with real coordinates x, B and Lagrangian function L = 2 (mx2(t) + i9(t)e(t) - i(B x 8)8). The corresponding phase space and symplectic form are the same as in the previous example, and the Hamiltonian function is 2

H= !_+ 2 (Bx6)0.

(s.7)

We will see in Section 5 that quantization of the Hamiltonian (3.7) yields the Pauli Hamiltonian.

Example 3.3 (Free particle on If). Generalizing Example 3.1, consider the configuration space HTII the tangent bundle to W2 with reversed parity of the fibers, with even and odd real coordinates x = (x1, ... , xn) and B = (9', ... , 9n ). The Lagrangian function (3.8)

L(x(t),(t), B(t)} =

x

z8}

yields the same Euler-Lagrange equations as in Example 3.1,

i=o and 6=o. The corresponding phase space is a supermanifold 1 2m1m with real coordinates p, x, 0 and the symplectic form z

w = dp n dx - d6d0, and the Hamiltonian function is 2

H = 2m

.

Example 3.4 (Fee particle on IITM). Let (M, g) be a Riemannian manifold with Riemannian metric g,,dxPdx". Similar to the construction in Section 2.2, we consider the phase space IITM with the Lagrangian function

L= 2 (IIII2+i(0,V6)),

(3.9)

where V is a covariant derivative along the path x(t) (which is also customarily denoted by D/Dt), and B(t) E IITXM. The corresponding action functional is defined by 1

J2

(III

S=2 to

+ z(e,

4. Supersymmetry

359

or in local coordinates x = (x',.

. .

, xn )

on M,

tl

I 9 (x (t)) (+

dt

v

o

tl

ft g,L(x(t)) (L±v + BµBv + o

We will see in the next section that this system describes a supersymmetric particle on a Riemannian manifold. Problem 3.2. Develop Lagrangian and Hamiltonian formalisms for classical mechanics on supermanifolds.

Problem 3.3. Prove formula (3.6).

Problem 3.4. Show that Euler-Lagrange equations for the system with La-

grangian (3.9) are

`

Dx' 9a, Dt =

1

2

and

D9'

Dt

.= 9

equations for a spinning particle in a gravitational field.

Problem 3.5. Describe the phase space and find the Hamiltonian function for Example 3.4.

4. Supersymmetry The Lagrangian system considered in Example 3.3 in the previous section possesses remarkable symmetries.

4.1. Total angular momentum. First of all, the Lagrangian (3.8) is obvi-

ously invariant with respect to the action of the orthogonal group G = SO(n) (4.1) L(g x, g B) = L(x> 6), 9 E G. The corresponding conserved quantity -the Noether charge J E g* -the dual space to the Lie algebra g =so(n), can be obtained as follows (cf. the proof of Theorem 1.3 in Section 1.4 of Chapter 1). Consider an infinitesimal change of even and odd coordinates

x(t) H x(t) = x(t) + ax(t), e(t) F-+ B(t) - e(t) + ye(t), and using 080 = -80 0, compute SL = L( -}- bx, 8 + 88) - L(x, 8) up to the second order terms in Sx and 88, aL = mx 8x + 2 (b9 8 + B Se)

_ -mx Sx +

(x Sx) + 2 (86O - B 89) +

_ -mx Sx + m -(x Sx) + i89 B + i dt

2 dt

(e se)

2 dt .

080)

8. Supersymmetry

360

Thus on the solutions of Euler-Lagrange equations x = 0, 8 = 0 we have SL = d dt

mxSx + z 8 58). 2

Now using (4.1) with g = e, where u = {u},=1 is a skew--symmetric n x n matrix, we see that SL -- = 0 for Sx = uv x' and S8 = euv ' . Choosing the standard basis in the space of skew-symmetric n x n matrices, we obtain

that components

x"x) - i8Ev

(4.2)

are integrals of motion:

J =0

dt on solutions of the Euler-Lagrange equations. In particular, for n = 3 we get

J1 := P3 = M1 M2

- i0203 - 28381,

J2 := J31 J12 = M3 -- i8162, J3 where M1, .112, M3 are components of the angular momentum M of a particle in l3 (see Section 1.4 in Chapter 1). We will see in the next section that

after quantization the vector J - = (J1, J2, J3 ) becomes the total angular momentum operator of a quantum particle of spin 2 in R3. Remark. In fact, the Lagrangian (3.8) is invariant under the action of G x G on

so that both the angular momentum m(x ? - x h ) in W and

the "Grassmann angular momentum"

in R01

are conserved.

4.2. Supersymmetry transformation. It is quite remarkable that in addition to the symmetries discussed in the previous section, the Lagrangian (3.8) is also invariant under special transformations on Wh1n that mix even and odd coordinates. Namely, let ('y(t), 8(t)) E IIT.P1(W ), where 8(t) E IITy P1 (Wn ) is a section over I = [to, t1] of the pull-back by 7 of the tangent bundle of TRn with reverse parity of the fibers,

8(t) = O(t)

E HT(j)M.

Consider the following infinitesimal change of coordinates which mixes even and odd variables: (4.3)

x(t) H x(t) + S6x(t) and 8(t) H 8(t) -+--

where (4.4)

S6x (t) = iE8 (t) E T (t) Iln

and 5E8(t) = -m cic (t) E

and E is an odd real element. Then for SQL = L(x + SEX, 8 + 888)- L(x, 8)

Ian,

4. Supersymmetry

36 1

we obtain

sEL = mxsEx+ 2(SEee+ebEe

= mx9- --(E:x9+9Ex) = 22 (x e9 + e9 x) 22Ed

=

t(9x).

Thus for periodic boundary conditions the action functional tl

S('y,9)

=j

L('y(t),O(t))dt

co

is invariant under the change of coordinates (4.3), B) = 0 for all periodic (y,O) E IITPI(ll8n).

The infinitesimal transformation (4.3) is called the supersymmetry transformation.

Remark. We emphasize that invariance of the action under the supersymmetry transformations holds for all (8(t)) with periodic boundary conditions, and not only on equations of motion! Introducing the quantity

Q = im9x = i6 = iBµpµ, called the generator of supersymmetry (or the supercharge), we can summarize the above computation as (4.5)

SEL

2

Another remarkable property is that the Lagrangian L can be recovered from the supercharge Q. Namely, the simple computation

5Q =

x + 9 8Ei) = m(-m2 - i9 0)

gives (4.6)

-2miEL =SEQ.

Using (3.5) we obtain (4.7)

{Q,Q}

-ip2 = -2miH,

which shows that the Hamiltonian H can also be recovered from the supercharge Q.

362

8. Supersymmetry

Remark. Geometrically, supersymmetry transformation is just an equivariant differential on the space GI(IIBn) of free loops on li8n parametrized by the interval I = [t0, tl], considered in Section 2.2 (for the interval [0, 1]). Namely,

set m = 1 and consider the Wick rotation t H -it to the Euclidean time, so that the sypersymmetry transformation (4.3)-(4.4) becomes

8Ex(t) =fee(t),

(4.8)

8E9(t) _ -iex(t).

Now we immediately see that formulas (4.8) can be written as

ieDx(t) and 8E9(t) = iED6(t), where D is the equivariant differential (2.7), tl

D = f (e(t)(t) o

which satisfies D2 = -G.y. The corresponding Euclidean supercharge Q coincides with the function Q on IITG(IIBn), given by formula (2.9), so that tl

R=

J

Bµ(t)xµ(t)dt, o

where we are using the standard Euclidean metric on ll8n. It follows from Lemma 2.2 that p ti

DQ = -

(i(t)i(t) + e(t)e(t))dt = -2s(x(t), e(t)), to

where now S stands for the Euclidean action. The invariance of the action S under the supersymmetry transformation is the property that S is equivariantly closed, DS = 0.

Problem 4.1. Verify that d

= 22E1E2-,

dt where e1, E2 are odd variables, and deduce from it formula (4.7).

4.3. Supersymmetric particle on a Riemannian manifold. The classical system with Lagrangian (3.9), considered in Example 3.4 in Section 3.2, describes a supersymmetric particle on a Riemannian manifold (M, g). Namely, for (y, B) E IITPI(M) define the supersymmetry transformation x(t) H x(t) + 8Ex(t) and B(t) 8(t) + 8E9(t), in local coordinantes x(t) and B(t), by the same formula (4.4),

5x(t) = ie9(t) E IITy(t)M, where e is an odd real element. (4.9)

8E9(t) _ -m(t) E IIT(t)1V1,

Lemma 4.1. Supersymmetry transformation (4.9) does not depend on the choice of local coordinates.

4. Supersyrnmetry

363

Proof. Let x = (1,.

. .

,

xn) be another local coordinates on M, xµ

f(x), µ = 1, ... , n. Along the path ry = x(t), 3fb

i(t) ax (x(t))xv(t) where

and 8(t) = µ

= -(x(t)O(t).

OIL(t)

We have F

ax (x(t))SexL(t) =

(x(t))Eev(t)

= 2e(t),

and 3fIL

SEeµ(t)

32f/1

- ax

aavaxa

v

= -m---(x(t))th"(t) +

_ -me(t),

32p

ax ax (x(t))r(t)9"(t)

since Bµ(t) anticommute.

As in the previous section, we define the supercharge Q by

Q(t) = im((t), 0(t)). The main result of this section is the following statement.

Proposition 4.1. Under the supersymmetry transformation (4.9), (4.10)

SEQ = -2miEL,

where (4.11)

L= 2 Iis

the Lagrangian of a free particle on IITM. Also (4.12)

2 dt .

SEL

Proof. The derivation of (4.10) is a straightforward computation using formula (2.11). Formula (4.12) is proved by another computation, similar to the one used to prove (4.5) in the previous section.

Corollary 4.1. The action functional .ti

S('y,9)

= Jo

L(y(t),0(t))dt

is invariant under the supersymmetry transformations, 8ES(ry, B) = 0

for all periodic (y, B) E IITPI(M).

$. Supersymmetry

364

considered in SecRemark. As for the example of a free particle on tion 4.2, the Euclidean version of the sypersymmetry transformation (4.9) corresponds to the equivariant differential (2.7) on the space GI(M) of free loops on M parametrized by the interval I = [to, ti]. The corresponding Euclidean supercharge Q (for m = 1) coincides with the function ,3 on IITGI(M) given by formula (2.9), and Proposition 4.1 reduces to the statements that DQ = -28 and DS = 0, where S is the Euclidean action, proved

in Lemma 2.2.

Problem 4.2. Prove all the formulas in this section.

Problem 4.3 (Superfield formalism). For d ry, B) E HTM define the superfield by X(t) = x(t) + O(t), where j is an auxiliary Grassmann variable, and let - . Show that D=

-

cat

f9(X(t))k(t)V(X)(t)dtdr =

t

- fo (+ (0, O))dt

twice the Euclidean action of a supersymmetric particle on a Riemannian manifold M.

5. Quantum mechanics on supermanifolds Here we describe quantum systems which correspond to classical systems in Section 3.2. According to the correspondence principle (see Section 2 of Chapter 2), for quantization of even coordinates we replace Poisson brackets {, } by i[, ], where i = /i1i and [, ] is the commutator9. For quantization of odd coordinates, in accordance with Section 1 of Chapter 7, we replace corresponding Poisson brackets by i[, ]+, where [ ]+ is the anticommutator. ,

Example 5.1 (Quantum particle of spin Z). The corresponding phase space with even coordinates p _ (p1,P2, p3) and x = is a supermanifold (x1, x2, x3), and odd coordinatesl0 e = (el, 02i B3), with canonical Poisson brackets

O,} = iS,, ,a, v = 1, 2, 3. Q3), which correQuantum operators P = (P,, P2, P3) and Q = (Q,, {p,,

S,v

and

spond to canonical coordinates p and x, satisfy Heisenberg commutation relations (see Section 2.1 of Chapter 2), while operators ® - = (Ei, e2, e3), which correspond to the anticommuting coordinates 9, satisfy the following anticommutation relations: Cr

y

U + = 5/2v1,

it, v = 1, 2, 3.

9 Here we put It = 1. '° Here it is convenient to lower all indices by the standard Euclidean metric on IRE.

5. Quantum mechanics on supermanifolds

365

Since operators \/ Oµ define a representation of a Clifford algebra C3, the only irreducible realization of (5.1) is given by

e=7=o, ,u=1,2,3,

(5.2)

where oµ are Pauli matrices (see Section 1.1 of Chapter 4). Thus the Hilbert space of the system is '1f = L2(I[83) ® cC2 -the Hilbert space of a quantum particle of spin 2 and the Hamiltonian operator is

p2 H=-. Using (5.2) and the multiplication table of Pauli matrices, we obtain the following form for quantum Noether integrals of motion (4.2):

1=M+S, where M is the angular momentum operator (see Section 3.1 of Chapter 3), and S = 2o. Thus J is the total angular momentum operator of a quantum spin 2 particle (see Section 1.2 of Chapter 4).

Example 5.2 (Quantum spin 2 particle in constant magnetic field). The Hilbert space is the same as in the previous example, while the Hamiltonian operator which corresponds to (3.7) takes the form

B H=+(Bxe)e=-2m 2m 2

2

2

2

Thus operator H is the Pauli Hamiltonian with total magnetic moment µ = 2 (see Section 2.1 of Chapter 4).

Example 5.3 (Supersymmetric quantum particle on IE). The phase space is a supermanifold IlS2nIn with even coordinates p = (P1,. , pn) and x _ (x1,. . , xn), and odd coordinates 8 = (Or, ... , Bn), with canonical Poisson brackets

{p, xU} = 5

{O, O} = i8,,, i, v = 1, ... , n. Quantum operators P = (P1, ... , P1) and Q = (Q,. . , Q7), which correspond to canonical coordinates p and x, satisfy Heisenberg commutation relations, and operators o = (Ei,. . , E), which correspond to the antiand

[p,

.

.

commuting coordinates 9, satisfy the following anticommutation relations: (5.3)

v = 1, ... , n.

The operators \ Q' define a representation of a Clifford algebra C. When n is even, the only irreducible realization of (5.3) is 1

S. Supersymmetry

366

are operators in F = (C2)®d, d = 2, given by formulas (1.12)(1.13) in Section 1.2 of Chapter 7. For odd n, operators 'y act in (C2)®d, where d = [2 ] (see Problem 1.4 in Section 1.2 of Chapter 7). In both cases, the Hilbert space of the system is = L2(W) ® F = L2(R) ® the Hamiltonian operator is given by where ry1

H=

PZ 2m

,

and quantum Noether integrals of motion are

= QP -- QvP - ie,LoU . Quantization of the supercharge gives the operator (5.4)

i

Q = 2O1,Pµ =

a

-v 1

'YµPµ

where

1

as

is the Dirac operator on W'. The Dirac operator is skew-adjoint in

When n is even, decomposition 3' =

into the subspaces of positive and negative chirality spinors (see Section 1.2 of Chapter 7) gives decomposition

_ and using that

,

it)

where f = L2(R) ®

(see Section 1.2 of Chapter 7), we can represent the Dirac operator in the following 2 x 2block-matrix form: y,

(0 Here the operator have

:

-

0 \$+ is called the chiral Dirac operator. We

[Q,QI+=2Q2=2=_2rnH,

so that 2mH is the Dirac Laplacian -. In matrix form, 2mH =

0

Example 5.4 (Supersymmetric quantum particle on Riemannian manifold). We have seen in Example 2.4 in Section 2.4 of Chapter 2 that though for a free quantum particle on a Riemannian manifold (M, g) it is not pos-

sible to construct operators P and Q which correspond to standard local coordinates (p, x) on T*M, the Hamiltonian operator H is well defined as the Laplace-Beltrami operator of the Riemannian metric. It is remarkable

5. Quantum mechanics on supermanifolds

367

that consistent quantization of a supersymmetric particle requires M to be a spin manifold, and for even n = dim M the corresponding supercharge operator Q coincides with the Dirac operator! Namely, let Spin(n) be the spin group: connected, simply connected Lie group which is a double cover of SO(n). The oriented Riemannian manifold (M, g) of dimension n is said to have a spin structure (and is called a spin manifold) if the bundle SO(M) of oriented orthonormal frames over M principal 50(n)-bundle can be extended to the principal Spin(n)-bundle Spin(M). This is equivalent to the condition that for some open covering

M - UEA Ua transition functions t: Ua fl Ua - SO(n) of a tangent bundle TM can be lifted to the transition functions Tai : Ua n Uj -* Spin(n); t= p(TaQ), where p : Spin(n) -* SO(n) is the canonical projection. The manifold M is a spin manifold if and only if its second Stief£el-Whitney class w2 E H2 (M, 7G2) vanishes; in this case different spin structures are parametrized by Hl (M, 7G2) ^Hom(7r1(M), 7L2), where 7rl (M) is the fundamental group of M. For even n = 2d, the irreducible representation p of the Clifford algebra Cn in 3' ^C2d (see Section 1.2 of Chapter 7) defines unitary representation R of the spin group Spin(n) in which commutes with the parity operator F. By definition, the spinor bundle S on an evendimensional spin manifold M is a Hermitian vector bundle associated with the principal Spin(n)-bundle Spin(M) by the unitary representation R. In

other words, S is a complex vector bundle with the transition functions R(T) Ua fl U- U(2d), where U(2d) is the group of unitary 2d x 2d matrices. Decomposition of vector spaces = 3F defines the decomposition

s=s+es_ of the spinor bundle s into the bundles S+ and S__ of positive and negative chirality spinors. The Dirac operator $ : C°° (M, 5) - C°° (M, s) of the even-dimensional spin manifold M is defined as follows. Let s be the connection on the spinor bundle s induced by the Levi-Civita connection in the tangent bundle TM. Then in the coordinate chart U C M with local coordinates x = (x1, ... , xn) we have (5.5)

where Oµ is the covariant derivative with respect to the vector field aaµ over U, and ryµ(x) are endomorphisms of the spinor bundle S over U satisfying (5.6)

['(x),y1'(x)} = 2g'(x)I,

where I is the identity endomorphism. It is easy to show that there is an open covering M = UG.EA Ua such that y1(x) exist over each Ua, and that local

8. Supersymmetry

368

expressions (5.5) give rise to a globally defined operator $ : C°°(M, S) C°O(M,S). Equivalently, if E C°°(M,S) is given by _ {E}aA, a then on Ua n Ua - F, where a =

=

(5.7)

where $a is given by (5.5) for U = Ua. We also have

: C°°(M,S±) - C°°(M,8). The Hermitian metric fin the spinor bundle S and the Riemannian metric g on M allow us to define the Hilbert space global sections of S,

of square integrable

e r(M,S): II2 II(x)IId(x)
where $+ :

+ --f

is the chiral Dirac operator on a spin manifold M.

Remark. The Dirac operator from the previous example is the Dirac operator on the spin manifold Rn with the standard Euclidean metric; the corresponding spinor bundle is a trivial Hermitian vector bundle W x

.

The space of states of a quantum supersymmetric particle on a spin manifold (M, g) is the Hilbert space M of square-integrable sections of the spinor bundle S over M. The corresponding supercharge operator Q and Hamiltonian operator H are given by the same formulas as in the previous example,

Q=

$ and

[Q, Q]+ = 2Q2 =

$2

= -2mH.

The operator

2mH =

p

is the Dirac Laplacian on a spin manifold M. Problem 5.1. Identify the spin group Spin(n) as a subgroup in the group of invertible elements of the Clifford algebra

Problem 5.2. Prove relations (5.7).

G. Atiyah-Singer index formula

369

6. Atiyah-Singer index formula Let M be an even-dimensional, oriented, compact spin manifold with the Riemannian metric g. Operators $J+ and +$+ are, respectively, selfadjoint on Hilbert spaces and and have pure point spectra consist,

ing of non-negative eigenvalues of finite multiplicity with the only accumulation point at oc. In particular, complex vector spaces ker -$+ = ker $+ and

ker +$+ = ker $+ are finite-dimensional, and we define the index ind+ of the chiral Dirac operator $+ by

ind + = dim ker+ -- dim ker. We have the following basic result.

Theorem 6.1 (McKean-Singer). For every T> o, = Trs eT2 = Tr, e_Tc++ - Ti:

ind

On the other hand, we have seen in Example 5.4 in the last section that -c = 2H, where H is the Hamiltonian operator of free supersymmetric particle of mass m 1 on the spin manifold M. Thus for every T> o, 2

ind + = T s e-2TH.

In Section 2.2 of Chapter 6 and in Sections 4.2 and 4.3 of Chapter 7 we developed a formalism for expressing traces and supertraces of the evolution

operator in Euclidean time by path integrals. Using these results, we can, at a physical level of rigor, represent the supertrace Try e-2TH by the path integral

f

(6.1)

HT1(M)

Here 1

SE(y, B) = 2

f

2T

(0

is the Euclidean action of a supersymmetric particle on a Riemannian manifold M, obtained from the Lagrangian function (4.11) by replacing time

t by the Euclidean time -it, and GI (M) is the space of free loops on M parametrized by the interval [0, 2T]. Whenil 2T = 1, the integral in (6.1) coincides with integral f(i) e-S, considered in Section 2.2. Using Theorem 2.3, we obtain ind

_ 'IBS e-H = (2i) 2 J A(M), M

which is the celebrated Atiyah-Singer formula for the index of the Dirac operator on a spin manifold! "According to a remark in Section 2.2, the same result holds for every T > 0.

370

8. Supersymmetry

Remark. Integrating over Grassmann variables in (6.1), we obtain (6.2)

Trs e_H = c(M)

Pf (Vy)d,

where d,au, is the Wiener measure on the loop space £ (M) associated with the Riemannian metric g on M. It can be shown that when M is a spin manifold, the Pfaffzan Pf (V.y) of a covariant derivative operator along 'y E £ (M) is a well-defined function on £(M). Thus formula (6.2), as opposed to the local computation in the derivation of Theorem 2.3, captures global properties of the manifold M. Problem 6.1. Prove the McKean--Singer theorem. Problem 6.2. "Derive" formula (6.1). (Hint: See the references in the next section.)

7. Notes and references The goal of Section 1, besides giving a definition of a supermanifold, was to introduce the isomorphism .A' (M) ^C°° (IITM), emphasized by E. Witten [Wit82a, Wit82b], and commonly used in the physics literature. For a systematic introduction to supermanifolds, we refer to the classic texts [Kos77, Ber87], as well as to

the modern sources [Man97, DM99, Varo4] and references therein. A detailed exposition of equivariant cohomology and localization in the finite-dimensional case can be found in the monograph [BGVO4}; our proof of the Berline-Vergne localization theorem in Section 2.1 follows [SzaOO]. Our presentation of the infinitedimensional case, based on [BT95, Szaoo], is at a physical level of rigor. Theorem 2.3 was originally formulated by E. Witten in his famous path integral derivation of the Atiyah-Singer formula for the index of the Dirac operator. Witten's original approach was lucidly presented by Atiyah [At185]. Our exposition follows [Wit99b],

with special attention to constant factors; see also [AG83, A1v95], as well as [BT95, Szaoo]. For a detailed explanation of the functor of points and a discussion of classical mechanics on supermanifolds, see [Fre99]; our examples of classical systems are taken from [Supersymmetry introduced in Sections 4.2.4.3 is called N = 2 supersymmetry and is obtained as a reduction of N = 1 Supersymmetry; see lectures [DF99b, Fre99, Wit99a, Wit99b] and [CFKSO8] for more details and references. In particular, [Fre99, Wit99a] describe useful superfield formalism for producing supersymmetric Lagrangians, sketched in Problem 4.3 in Section 4.3. Material in Section 5 is based on [A1v95]; we refer the reader to the monograph [BGVO4] for the invariant definition of Dirac operators on spin manifolds and related topics. It should be emphasized that our derivation of the Atiyah-Singer formula in Section 6 is purely heuristic. Rigorous justification of this approach using Ito-Malliavin

7. Notes and references

371

stochastic calculus was made by J.-M. Bismut [Bis84a, Bis84b, Bis85]; it turns out that formula (6.2) in Section 6 requires an extra factor: the exponential of the integral along the path of a multiple of a scalar curvature. When T - o, this rigorous formula for the index coincides with the heuristic expression (6.2). Still, the integration of a differential form of "top degree" over the loop space in [Bis85} remains formal, and is a challenging open problem. Finally, we refer the interested reader to [ASW9O] for a path integral derivation of the H. Weyl character formula, and to [Wit99b] for the treatment of the Dirac operator on the loop space.

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Index

*-product, 136 action classical, 35, 208, 260 functional, 5 abbreviated, 35 algebra of classical observables, 38, 46 angular momentum conservation of, 20 internal, 217 operator, 181, 217 vector, 20, 180 angular velocity, 13 anti-holomorphic representation, 113 asymptotics semi-classical, 206, 280 short-wave, 206

Berezin integral, 319 incomplete, 331 Berezinian, 329 Born-von Neumann formula, 70 bosons, 220 canonical anticommutation relations, 308 commutation relations, 104, 110, 307 coordinates, 44 transformation, 39 free, 42 generating function of, 41 chirality operator, 313 classical trajectory, 4 Clebsch-Gordan decomposition, 184 Clifford algebra, 311

completeness relation, 165, 187 complex probability amplitude, 240 configuration space, 4 extended, 18 conservation law, 15 coordinate operator, 81, 86 for fermions, 310 representation, 86 coordinates even, 344 normal, 12 odd, 344 correspondence principle, 80 Coulomb potential, 193 problem, 193

Darboux' theorem, 44 deformation quantization, 143 degrees of freedom, 4 internal, 217 determinant characteristic, 266 Fredholm, 273 regularized, 262 van Vleck, 285 Dirac bra and ket vectors, 91 gamma matrices, 313 Laplacian, 366, 368 operator chiral, 366, 368

on R, 366 on a spin manifold, 367 Dirac-von Neumann axioms, 73

384

distribution function, 67 joint, 91 energy, 15 centrifugal, 23 conservation of, 16 effective potential, 23 kinetic, 10

potential, 10 total, 17 equations geodesic, 13 of motion, 4, 6, 7 of a rigid body, 13

quantum, 75 Euclidean path integral, 254 Euler's equations, 14 Euler-Lagrange equations, 7, 28 evolution operator in classical mechanics, 52 in quantum mechanics, 76 expectation value of classical observables, 58 of quantum observables, 67 extremals, 5 central field of, 35

Index

on Grassmann algebra, 321 measure finite-dimensional, 289 infinite-dimensional, 291 path integral, 261 for a free particle, 261 for Grassmann variables, 338 for the harmonic oscillator, 264 Wiener integral for Dirichlet boundary conditions, 301 for periodic boundary conditions, 302 generalized accelerations, 4 coordinates, 4 cyclic, 22 forces, 19 momenta, 19 velocities, 4 graded dimension, 324 matrix algebra, 327, 328 vector space, 324 Grassmann algebra, 314 inner product, 316, 322 with involution, 322 path integral, 336

fermions, 220

Feynman path integral for Grassmann variables, 336 for the pq-symbol, 250 for the qp-symbol, 251 for the harmonic oscillator, 257 for the Weyl symbol, 251 for the Wick symbol, 253 in the configuration space, 246, 248 in the phase space, 245, 247 Feynman-Kac formula, 297 first variation with fixed ends, 8 with free ends, 8 force, 10 conservative, 10 Fourier transform, 91 frame of reference, 8

inertial, 9 Fresnel integral, 241, 245, 247 fundamental solution, 240 Galilean group, 9 transformation, 9 Galileo's relativity principle, 9 Gato derivative, 7 Gaussian integral, 255 in complex domain, 256

Hamilton's canonical equations, 29 equations for classical observables, 38, 46 picture, 59, 75 principle, 5 Hamilton-Jacobi equation, 36, 205, 207 Hamiltonian action, 48 function, 28, 45 operator, 75 of N particles, 99 of a complex atom, 100 of a Newtonian particle, 98 phase flow, 31, 45 system, 45 vector field, 31, 45 harmonic oscillator classical, 12, 21 quantum, 103, 109 fermion, 338 Heisenberg algebra, 82, 311 commutation relations, 81, 82 equation of motion, 76 group, 83 picture, 75 uncertainty relations, 74 Hilbert space, 63

Index

boson, 309 fermion, 309 holomorphic representation, 113 index

of a chiral Dirac operator, 369 inertia principal axes of, 14 principal moments of, 14 tensor, 14 integral first, 15 Noether, 17 of motion, 15, 39 quantum, 78 Jacobi inversion formula, 265 operator, 15, 259, 261, 281 theta series, 265 Jost solutions, 156

Kepler's first law, 24 problem, 24 second law, 23 third law, 24 kinetic energy operator, 149 Lagrangian, 4 function, 4 submanifold, 33, 43 subspace, 86, 89 system, 4 closed, 9 non-degenerate, 26 Laplace-Runge-Lenz operator, 199 vector, 25, 50, 199 Legendre transform, 28 Levi-Civita connection, 13 Liouville's canonical 1-form, 28 equation, 60 picture, 60, 77 theorem, 33 volume form, 33 Lippman-Schwinger equation, 187 localization theorem Berline-Vergne, 348 for the free loop space; 352 Lorentz force, 12 mass, 9 reduced, 22 total, 22

385

Maupertuis' principle, 34 measurement in classical mechanics, 56 in quantum mechanics, 67 momentum, 19 conservation of, 19 operator, 81, 88 for fermions, 310 representation, 89 monodromy matrix, 275, 278 Morse index, 208, 259 motion finite, 21 infinite, 21

Newton's equations, 10 law of gravitation, 11 law of inertia, 9 third law, 10 Newton-Laplace principle, 4 Noether theorem with symmetries, 48 Noether's theorem, 17 observables classical, 38 quantum, 66 complete system of, 90 simultaneously measured, 73

operator adjoint, 63 annihilation, 109, 307 fermion, 309 closed, 63 compact, 64 creation, 109, 307 fermion, 349 differential first-order, 278 essentially self-adjoint, 64 Hilbert-Schmidt, 65 matrix-valued Sturm-Liouville

with Dirichlet boundary conditions, 273

with periodic boundary conditions, 277 of trace class, 64 self-adjoint, 64 Sturm-Liouville, 268 with Dirichlet boundary conditions, 268

with periodic boundary conditions, 274 symmetric, 63, 64 operator symbol pq, 135 qp, 135 matrix, 116

386

for fermions, 331 Weyl, 132 Wick, 114 for fermions, 331

orthogonality relation, 168, 187

particles), 4 charged, in electromagnetic field, 11 free, 9 on Riemannian manifold, 12 in a potential field, 11 interacting, 10 quantum free, 93 Pauli exclusion principle, 227 Hamiltonian, 222, 365 matrices, 218 wave equation, 222 Pfaffian, 321 phase shift, 191 phase space, 28, 45 extended, 33 Planck constant, 76 Poincare-Cartan form, 33 Poisson action, 48 algebra, 39 bracket, 46, 52 canonical, 39 on Grassmann algebra, 318 manifold, 51 non-degenerate, 53 structure, 51 tensor, 52 theorem, 47 potential effective, 189 long-range, 190 repulsive, 191 short-range, 190 potential energy operator, 149 potential field, 11 central, 11 principle of the least action in the configuration space, 5 in the phase space, 33 projection-valued measure, 68, 72 resolution of the identity, 68 propagator, 240, 245 of a free quantum particle, 242 propogator for the harmonic oscillator, 257

quantization, 79

Index

pq, 135 qp, 135 rules of Bohr-Wilson-Sommerfeld, 102, 213

Weyl, 132

quantum bracket, 76 quantum number azimuthal, 192 magnetic, 192 principal, 192 radial, 192 reflection coefficient, 173 regular set, 63 regularized product, 263, 266 representation by occupation numbers, 112 for fermions, 310 resolvent operator, 63 rigid body, 13 ringed space, 343

scattering amplitude, 187 matrix, 173 operator, 172, 187 solutions, 172 theory non-stationary, 172 stationary, 172 Schrodinger equation of motion, 77 radial, 189 stationary, 79 time-dependent, 78 operator, 99, 149 of a charged particle, 101 of a complex atom, 100, 151 of a harmonic oscillator, 103 of a hydrogen atom, 100, 151 one-dimensional, 155 radial, 189 picture, 77 representation, 89 for n degrees of freedom, 91 Schur-Weyl duality, 232 Sommerfeld's radiation conditions, 186

space of states classical, 58 quantum, 66 spectral theorem, 68 spectrum, 63 absolutely continuous, 86, 93, 96, 153 essential, 152 joint, 73 point, 63, 105 singular, 153

Index

spin, 217 manifold, 367 operators, 218 singlet space, 228 structure, 367 total, 218 triplet space, 228 standard coordinates on T* M, 27

on TM, 6 state, 4 bound, 79 ground, 107, 308, 309 stationary, 78 states coherent, 114 for fermions, 332 in classical mechanics, 58 mixed, 58 pure, 58 in quantum mechanics, 66 mixed, 67 pure, 66 superalgebra, 325 commutative, 325 Lie, 327 supercharge, 361 supermanifold, 344 functions on, 344 supersymmetry generator, 361 supersymmetry transformation, 361 supertrace, 328 symmetrization postulate, 226 symmetry, 17 group, 17, 48 infinitesimal, 17 symplectic form, 42 canonical, 32 on Grassmann algebra, 318 manifold, 42 vector field, 46 system closed, 4 quantum, 66 composite, 66 systems on supermanifolds classical, 356 quantum, 364 time Euclidean, 242 physical, 9, 242 slicing, 244 trace of operator, 65

387

transition coefficients, 159 transmission coefficient, 173 transport equation, 207 turning point, 21, 210 variance, 74 variation infinitesimal, 5 with fixed ends, 5 virial theorem in classical mechanics, 11 in quantum mechanics, 154 wave

operators, 170 plane, 174 scattering, 173 stationary, 187 wave function, 88 total, 224 coordinate part of, 224 spin part of, 224 Weyl

inversion formula, 129 module, 232 quantization, 132 relations, 84, 118 transform, 119, 124 Wick normal form, 114 for fermions, 331 theorem, 290 Wiener integral, 294 measure conditional, 296 on C([0, oa}, W ; 0), 294 on C([0, oo}, W ; qo), 294 on the free loop space, 298 WKB method, 209 wave function, 210 Young diagram, 229 symmetrizer, 230

tableau, 229 canonical, 229 zeta-function Hurwitz, 279 operator, 262 Riemann, 263

Titles in This Series 95 Leon A. Takhtajan, Quantum mechanics for mathematicians, 2008 94 James E. Humphreys, Representations of semisimple Lie algebras in the BGG category 0, 2008 93 Peter W. Michor, Topics in differential geometry, 2008 92 I. Martin Isaacs, Finite group theory, 2008 91 Louis Halle Rowen, Graduate algebra: Noncommutative view, 2008 90 Larry J. Gerstein, Basic quadratic forms, 2008 89 Anthony Bonato, A course on the web graph, 2008 88 Nathanial P. Brown and Narutaka Ozawa, C*-algebras and finite-dimensional approximations, 2008

87 Srikanth B. Iyengar, Graham J. Leuschke, Anton Leykin, Claudia Miller, Ezra Miller, Anurag K. Singh, and Uli Walther, Twenty-four hours of local cohomology, 2007

86 Yulij Ilyashenko and Sergei Yakovenko, Lectures on analytic differential equations, 2007

85 John M. Alongi and Gail S. Nelson, Recurrence and topology, 2007 84 Charalambos D. Aliprantis and Rabee Tourky, Cones and duality, 2007 83 Wolfgang Ebeling, Functions of several complex variables and their singularities (translated by Philip G. Spain), 2007

82 Serge Alinhac and Patrick Gilrard, Pseudo-differential operators and the Nash-Moser theorem (translated by Stephen S. Wilson), 2007

81 V. V. Prasolov, Elements of homology theory, 2007 80 Davar Khoshnevisan, Probability, 2007 79 William Stein, Modular forms, a computational approach (with an appendix by Paul E. Gunnells), 2007

78 Harry Dym, Linear algebra in action, 2007 77 Bennett Chow, Peng Lu, and Lei Ni, Hamilton's Ricci flow, 2006 76 Michael E. Taylor, Measure theory and integration, 2006 75 Peter D. Miller, Applied asymptotic analysis, 2006 74 V. V. Prasolov, Elements of combinatorial and differential topology, 2006 73 Louis Halle Rowen, Graduate algebra: Commutative view, 2006 72 R. J. Williams, Introduction the the mathematics of finance, 2006 71 S. P. Novikov and I. A. Taimanov, Modern geometric structures and fields, 2006 70 Sean Dineen, Probability theory in finance, 2005 69 Sebastian Montiel and Antonio Ros, Curves and surfaces, 2005 68 Luis Caffarelli and Sandro Salsa, A geometric approach to free boundary problems, 2005

67 T.Y. Lam, Introduction to quadratic forms over fields, 2004

66 Yuli Eidelman, Vitali Milman, and Antonis Tsolomitis, Functional analysis, An introduction, 2004

S. Ramanan, Global calculus, 2004 64 A. A. Kirillov, Lectures on the orbit method, 2004 63 Steven Dale Cutkosky, Resolution of singularities, 2004 62 T. W. Korner, A companion to analysis: A second first and first second course in 65

analysis, 2004

61 Thomas A. Ivey and J. M. Landsberg, Cartan for beginners: Differential geometry via moving frames and exterior differential systems, 2003

60 Alberto Candel and Lawrence Conlon, Foliations II, 2003

TITLES IN THIS SERIES

59 Steven H. Weintraub, Representation theory of finite groups: algebra and arithmetic, 2003

58 Cedric Villani, Topics in optimal transportation, 2003 57 Robert Plato, Concise numerical mathematics, 2003 56 E. B. Vinberg, A course in algebra, 2003 55 C. Herbert Clemens, A scrapbook of complex curve theory, second edition, 2003 54 Alexander Barvinok, A course in convexity, 2002 53 Henryk Iwaniec, Spectral methods of automorphic forms, 2002 52 Ilka Agricola and Thomas Friedrich, Global analysis: Differential forms in analysis, geometry and physics, 2002

51 Y. A. Abramovich and C. D. Aliprantis, Problems in operator theory, 2002 50 Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, 2002 49 John R. Harper, Secondary cohomology operations, 2002 48 Y. Eliashberg and N. Mishachev, Introduction to the h-principle, 2002 47 A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi, Classical and quantum computation, 2002

46 Joseph L. Taylor, Several complex variables with connections to algebraic geometry and Lie groups, 2002

45 Inder K. Rana, An introduction to measure and integration, second edition, 2002 44 Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, 2002 43 N. V. Krylov, Introduction to the theory of random processes, 2002 42 Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, 2002 41 Georgi V. Smirnov, Introduction to the theory of differential inclusions, 2002 40 Robert E. Greene and Steven G. Krantz, Function theory of one complex variable, third edition, 2006

39 Larry C. Grove, Classical groups and geometric algebra, 2002 38 Elton P. Hsu, Stochastic analysis on manifolds, 2002 37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, 2001

36 Martin Schechter, Principles of functional analysis, second edition, 2002 35 James F. Davis and Paul Kirk, Lecture notes in algebraic topology, 2001 34 Sigurdur Heigason, Differential geometry, Lie groups, and symmetric spaces, 2001

33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G. Bartle, A modern theory of integration, 2001 31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods of financial mathematics, 2001

30 J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, 2001 29 Javier Duoandikoetxea, Fourier analysis, 2001 28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000 27 Thierry Aubin, A course in differential geometry, 2001 26 Rolf Berndt, An introduction to symplectic geometry, 2001 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberta Candel and Lawrence Conlon, Foliations I, 2000

For a complete list of titles in this series, visit the

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