Geometry of quantum theory 2nd edition

In his charac- teristic way, he discovered that the set of experimental statements of keep-a qukeep-antum mechkeep-anickeep-al system formed keep-a projective geometry—the projective ge

Geometry of Quantum Theory

Second Edition

V.S Varadarajan

Geometry of Quantum Theory

Second Edition

Springer

Mathematics Subject Classification (2000): 81-01

Library of Congress Control Number: 2006937106

ISBN-10: 0-387-96124-0 (hardcover)

ISBN-13: 978-0-387-96124-8 (hardcover)

ISBN-10: 0-387-49385-9 (softcover) e-ISBN-10: 0-387-49386-7

ISBN-13: 978-387-49385-5 (softcover) e-ISBN-13: 978-0-387-49386-2

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© 2007,1985,1970,1968 Springer Science+Business Media, LLC

The first edition of this book was published in two volumes: Volume I in 1968 by D Van Nostrand Company, Inc., New York; and Volume II in 1970 by Van Nostrand Reinhold Company, New York All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media LLC, 233 Spring Street, New York,

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9 8 7 6 5 4 3 2 1

TO MY PARENTS

PREFACE TO VOLUME I

OF THE FIRST EDITION

The present work is the first volume of a substantially enlarged version of the mimeographed notes of a course of lectures first given

by me in the Indian Statistical Institute, Calcutta, India, during 1964-65 When it was suggested that these lectures be developed into

a book, I readily agreed and took the opportunity to extend the scope

of the material covered

No background in physics is in principle necessary for ing the essential ideas in this work However, a high degree of mathematical maturity is certainly indispensable It is safe to say that I aim at an audience composed of professional mathematicians, advanced graduate students, and, hopefully, the rapidly increasing group of mathematical physicists who are attracted to fundamental mathematical questions

understand-Over the years, the mathematics of quantum theory has become more abstract and, consequently, simpler Hilbert spaces have been used from the very beginning and, after Weyl and Wigner, group representations have come in conclusively Recent discoveries seem to indicate that the role of group representations is destined for further expansion, not to speak of the impact of the theory of several complex variables and function-space analysis But all of this pertains to the world of interacting subatomic particles; the more modest view of the microscopic world presented in this book requires somewhat less The reader with a knowledge of abstract integration, Hilbert space theory, and topological groups will find the going easy

Part of the work which went into the writing of this book was supported by the National Science Foundation Grant No GP-5224 I have profited greatly from conversations with many friends and colleagues at various institutions To all of them, especially to R Arens, R J Blattner, R Ranga Rao, K R Parthasarathy, and S R S Varadhan, my sincere thanks I want to record my deep thanks to

my colleague Don Babbitt who read through the manuscript carefully, discovered many mistakes, and was responsible for significant im- provement of the manuscript My apologies are due to all those whose work has been ignored or, possibly, incorrectly (and/or insufficiently) discussed Finally, I want to acknowledge that this book might never

vii

viii PREFACE TO VOLUME I OF THE 1ST EDITION

have made its way into print but for my wife She typed the entire manuscript, encouraged me when my enthusiasm went down, and made me understand some of the meaning of our ancient words,

To her my deep gratitude

* Bhagavadgita, 2:U7a

PREFACE TO THE SECOND EDITION

I t was about four years ago t h a t Springer-Verlag suggested t h a t a revised edition in a single volume of m y two-volume work m a y be worthwhile

I agreed enthusiastically b u t t h e project was delayed for m a n y reasons, one

of t h e most i m p o r t a n t of which was t h a t I did not have a t t h a t time a n y clear idea as t o how t h e revision was to be carried out Eventually I decided

t o leave intact most of t h e original material, b u t make the current edition a little more up-to-date b y adding, in t h e form of notes t o t h e individual chapters, some recent references a n d occasional brief discussions of topics

n o t t r e a t e d in t h e original t e x t The only substantive change from t h e earlier work is in t h e t r e a t m e n t of projective geometry; Chapters I I through V of

t h e original Volume I have been condensed a n d streamlined into a single Chapter I I I wish t o express m y deep gratitude t o Donald B a b b i t t for his generous advice t h a t helped me in organizing this revision, a n d t o Springer-Verlag for their patience and understanding t h a t went beyond

w h a t one has a right t o expect from a publisher

I suppose an a u t h o r ' s feelings are always mixed when one of his books t h a t

is comparatively old is brought out once again The progress of Science in our time is so explosive t h a t a discovery is hardly m a d e before it becomes obsolete; and yet, precisely because of this, it is essential t o keep in sight t h e origins of things t h a t are t a k e n for granted, if only to lend some perspective

t o w h a t we are trying t o achieve All I can say is t h a t there are times when one should look back as well as forward, a n d t h a t t h e ancient lines, p a r t of which are quoted above still capture the spirit of m y thoughts

Pacific Palisades, V S VARADARAJAN Dec 22,1984

* Bhagavadgita, 2:47a

ix

TABLE OF CONTENTS

Introduction x v

CHAPTER I

Boolean Algebras on a Classical Phase Space 1

1 The Classical Phase Space 1

2 The Logic of a Classical System 6

3 Boolean Algebras 8

4 Functions 12 Notes on Chapter I 17

CHAPTER I I

Projective Geometries 18

1 Complemented Modular Lattices 18

2 Isomorphisms of Projective Geometries Semilinear Transformations 20

3 Dualities and Polarities 22

4 Orthocomplementations and Hilbert Space Structures 26

5 Coordinates in Projective and Generalized Geometries 28

6 Functions of Several Observables 62

7 The Center of a Logic 63

8 Automorphisms 67

Notes on Chapter I I I 70

CHAPTER IV

Logics Associated with Hilbert Spaces 72

1 The Lattice of Subspaces of a Banach Space 72

2 The Standard Logics: Observables and States 80

xii TABLE OF CONTENTS

3 The Standard Logics: Symmetries 104

4 Logics Associated with von Neumann Algebras 112

5 Isomorphism and Imbedding Theorems 114

Notes on Chapter IV 122

CHAPTER V

Measure Theory on C?-Spaces 148

1 Borel Spaces and Borel Maps 148

2 Locally Compact Groups Haar Measure 156

3 G-Spaces 158

4 Transitive ^-Spaces 164

5 Cocycles and Cohomology 174

6 Borel Groups and the Weil Topology 191

Notes on Chapter V 200

CHAPTER VI

Systems of Imprimitivity 201

1 Definitions 201

2 Hilbert Spaces of Vector Valued Functions 208

3 From Cocycles to Systems of Imprimitivity 213

4 Projection Valued Measures 217

5 From Systems of Imprimitivity to Cocycles 219

1 The Projective Group 243

2 Multipliers and Projective Representations 247

3 Multipliers and Group Extensions 251

4 Multipliers for Lie Groups 259

5 Examples 275 Notes on Chapter VII 287

CHAPTER VIII

Kinematics and Dynamics 288

1 The Abstract Schrodinger Equation 288

2 Co variance and Commutation Rules 293

3 The Schrodinger Representation 295

4 Affine Configuration Spaces 300

5 Euclidean Systems: Spin 303

6 Particles 312 Notes on Chapter VIII 315

TABLE OF CONTENTS xiii

CHAPTER IX

Relativisitic Free Particles 322

1 Relativistic Invariance 322

2 The Lorentz Group 330

3 The Representations of the Inhomogeneous Lorentz Group 343

4 Clifford Algebras 348

5 Representations in Vector Bundles and Wave Equations 356

6 Invariance Under the Inversions 372

INTRODUCTION

As laid down by Dirac in his great classic [1], the principle of superposition of states is the fundamental concept on which the quan- tum theory of atomic systems is to be erected Dirac's development

of quantum mechanics on an axiomatic basis is undoubtedly in ing with the greatest traditions of the physical sciences The scope and power of this principle can be recognized at once if one recalls that it survived virtually unmodified throughout the subsequent transi- tion to a relativistic view of the atomic world It must be pointed out, however, that the precise mathematical nature of the superposition principle was only implicit in the discussions of Dirac; we are in- debted to John von Neumann for explicit formulation In his charac- teristic way, he discovered that the set of experimental statements of

keep-a qukeep-antum mechkeep-anickeep-al system formed keep-a projective geometry—the projective geometry of subspaces of a complex, separable, infinite dimensional Hilbert space With this as a point of departure, he carried out a mathematical analysis of the axiomatic foundations of quantum mechanics which must certainly rank among his greatest achievements [1] [3] [4] [5] [6]

Once the geometric point of view is accepted, impressive quences follow The automorphisms of the geometry describe the dynamical and kinematical structure of quantum mechanical systems,

conse-thus leading to the linear character of quantum mechanics The

covariance of the physical laws under appropriate space-time groups consequently expresses itself in the form of projective unitary repre- sentations of these groups The economy of thought as well as the unification of method that this point of view brings forth is truly immense; the Schrodinger equation, for example, is obtained from a representation of the time-translation group, the Dirac equation from

a representation of the inhomogeneous Lorentz group This ment is the work of many mathematicians and physicists However, insofar as the mathematical theory is concerned, no contribution is more outstanding than that of Eugene P Wigner Beginning with his famous article on time inversion and throughout his great papers

develop-on relativistic invariance [1] [3] [4] [5] [6], we find a beautiful and coherent approach to the mathematical description of the quantum mechanical world which achieves nothing less than the fusion of group theory and quantum mechanics, and moreover does this without

XV

me-my indebtedness to Professor Mackey's lectures and to the books and papers of von Neumann and Wigner is immense and carries through this entire work

There exist today many expositions of the basic principles of quantum mechanics At the most sophisticated mathematical level, there are the books of von Neumann [1], Hermann Weyl [1] and Mackey [1] But, insofar as I am aware, there is no account of the technical features of the geometry and group theory of quantum me- chanical systems that is both reasonably self-contained and comprehen- sive enough to be able to include Lorentz invariance Moreover, recent re-examinations of the fundamental ideas by numerous mathematicians have produced insights that have substantially added to our under- standing of quantum foundations From among these I want to single out for special mention Gleason's proof that quantum mechanical states are represented by the so-called density matrices, Mackey's extensive work on systems of imprimitivity and group representations, and Barg- mann's work on the cohomology of Lie groups, particularly of the physically interesting groups and their extensions All of this has made possible a conceptually unified and technically cogent development of the theory of quantum mechanical systems from a completely geometric point of view The present work is an attempt to present such an approach

Our approach may be described by means of a brief outline of the contents of the three parts that make up this work The first part begins by introducing the viewpoint of von Neumann according to which every physical system has in its background a certain orthocomplemented lattice whose elements may be identified with the experimentally verifiable prop- ositions about the system For classical systems this lattice (called the logic of the system) is a Boolean a-algebra while for quantum systems it is highly nondistributive This points to the relevance of the theory of complemented lattices to the axiomatic foundations of quantum mechanics

In the presence of modularity and finiteness of rank, these lattices pose into a direct sum of irreducible ones, called geometries A typical example of a geometry is the lattice of subspaces of a finite dimensional vector space over a division ring The theory of these vector geometries is taken up in Chapter II The isomorphisms of such a geometry are induced in

decom-a ndecom-aturdecom-al fdecom-ashion by semilinedecom-ar trdecom-ansformdecom-ations Orthocomplementdecom-ations are induced by definite semi-bilinear forms which are symmetric with

INTRODUCTION

respect to suitable involutive anti-automorphisms of the basic division ring

If the division ring is the reals, complexes or quaternions, this leads to the Hilbert space structures In this chapter, we also examine the relation between axiomatic geometry and analytic geometry along classical lines with suitable modifications in order to handle the infinite dimensional case also The main result of this chapter is the theorem which asserts that an abstractly given generalized geometry (i.e., one whose dimension need not be finite) of rank >4 is isomorphic to the lattice of all finite dimensional subspaces of a vector space over a division ring The division ring is an invariant of the lattice

The second part analyzes the structure of the logics of quantum cal systems In Chapter I I I , we introduce the notion of an abstract logic ( = orthocomplemented weakly modular a-lattice) and the observables and states associated with it It is possible that certain observables need not be simultaneously observable It is proved that for a given family of observ- ables to be simultaneously measurable, it is necessary and sufficient that the observables of the family be classically related, i.e., that there exists a Boolean sub a-algebra of the logic in question to which all the mem- bers of the given family are associated Given an observable and a state, it is shown how to compute the probability distribution of the observable in that state In Chapter IV, we take up the problem of singling out the logic of all subspaces of a Hilbert space by a set of neat axioms Using the results of Chapter II, it is proved that the standard logics are precisely the projective ones The analysis of the notions of an observable and a state carried out in Chapter I I I now leads to the correspondence between observables and self-adjoint operators, and between the pure states and the rays of the underlying Hilbert space The automorphisms of the standard logics are shown to be induced by the unitary and antiunitary operators With this the von Neumann program of a deductive description

mechani-of the principles mechani-of quantum mechanics is completed The remarkable fact that there is a Hilbert space whose self-adjoint operators represent the observables and whose rays describe the (pure) states is thus finally established to be a consequence of the projective nature of the underlying logic

The third and final part of the work deals with specialized questions The main problem is that of a covariant description of a quantum mechanical system, the covariance being with respect to suitable symmetry groups of the system The theory of such systems leads to sophisticated problems of harmonic analysis on locally compact groups Chapters V, VI, and VII are devoted to these purely mathematical questions The results obtained are then applied to yield the basic physical results in Chapters VIII and IX

In Chapter VIII, the Schrodinger equation is obtained and the relations between the Heisenberg and Schrodinger formulations of quantum mechanics are analyzed The usual expressions for the position, momentum, and energy observables of a quantum mechanical particle are shown to be inevitable consequences of the basic axioms and the requirement of covariance In addition, a classification of single particle systems is obtained

INTRODUCTION

in terms of the spin of the particle The spin of a particle, which is so

charac-teristic of quantum mechanics, is a manifestation of the geometry of the

configuration space of the particle

The final chapter discusses the description of free particles from the relativistic viewpoint The results of Chapters V, VI and VII are used to obtain a classification of these particles in terms of their mass and spin With each particle it is possible to associate a vector bundle whose square integrable sections constitute the Hilbert space of the particle These abstract results lead to the standard transformation formulae for the (one particle) states under the elements of the relativity group By taking Fourier transforms, it is possible to associate with each particle a definite wave equation In particular, the Dirac equation of the free electron is obtained in this manner The same methods lead to the localization in space, for a given time instant, of the particles of nonzero rest mass The chapter ends with an analysis of Galilean relativity It is shown that the free particles which are governed by Galilei's principle of relativity are none

other than the Schrodinger particles of positive mass and arbitrary spin

With this the program of obtaining a geometric view of the tum mechanical world is completed It is my belief that no other ap- proach leads so clearly and smoothly to the fundamental results It may be hoped that such methods may also lead to a successful de- scription of the world of interacting particles and their fields The realization of such hopes seems to be a matter for the future

quan-V S VARADARAJAN

CHAPTER I BOOLEAN ALGEBRAS ON A CLASSICAL

PHASE SPACE

1 THE CLASSICAL PHASE SPACE

We begin with a brief account of the usual description of a classical

mechanical system with a finite number of degrees of freedom Associated with such a system there is an integer n, and an open set M of the n- dimensional space Rn of n-tuples (xu x2, • • •, %n) of real numbers, n is called the number of degrees of freedom of the system The points of M represent the possible configurations of the system A state of the system

at any instant of time is specified completely by giving a 2n-tuple (#i> #2> * • • > xn> Pi> * *' > Pn) s u c n that (xl9 • • •, xn) represents the configura- tion and (ply • •, pn) the momentum vector, of the system at that instant

of time The possible states of the system are thus represented by the

points of the open set M x Rn of R2n The law of evolution of the system is specified by a smooth function H on M x Rn, called the Hamiltonian of the system If (x^t), • • •, xn(t), Pi(t), - - -, pn(t)) represents the state of the system at time t, then the functions xt(-)t Pii-), i = l, 2, • • •, n, satisfy

the well known differential equations:

For most of the systems which arise in practice these equations have

unique solutions for all t in the sense that given any real number t0, and a point (x^, x2°, • • •, xn°, p±°, • • •, pn°) of M x J?71, there exists a

unique differentiate map t -> {x^t), • • •, xn(t), px{t), • • •, pn(t)) of R1 into

Mx Rn such that x{( •) and p{( •) satisfy the equations (1) with the initial conditions

(2) xt(tQ) = a?,0, pt(*0) = ft0, » = 1, 2, • • •, n

If we denote by 5 an arbitrary point of M x Rn, it then follows in the standard fashion that for any t there exists a mapping D(t)(s -> D(£)s)

1

2 GEOMETRY OF QUANTUM THEORY

of M x Rn into itself with the property that if s is the state of the system

at time t09 D(t)s is the state of the system at time t+t0 The mations D(t) are one-one, map M x Rn onto itself and satisfy the equations:

transfor-D(0) = J (the identity mapping),

(3) D{-t) = D(t)~\

D(tx+t2) = D(tx)D{t2)

If, in addition, H is an indefinitely differentiate function, then the D(t) are also indefinitely differentiate and the correspondence t —> D(t) defines a one-parameter differentiate transformation group of M x Rn so

that the map t, s -> D(t)s of R1xMxRn into Mx Rn is indefinitely

differentiate The set MxRn of all the possible states of the system is

called the phase space of the system

In the formulation described above, the physical quantities or the

observables of the system are described by real valued functions on M x Rn

For example, if the system is that of a single particle of mass m which

moves under some potential field, then n = 3, M = R3, and the Hamiltonian

H is given by

(4) H(xl9x29x39pl9p29p3) = ^ (Pi2 +P22 +P32) + V(xl9x2ix3)

The function s -> (p±2 +p22 +^32)/2m is the kinetic energy of the particle

and the function s —> V(xl9x2,x3) is the potential energy of the particle The function s -+ pt (i = 1, 2, 3) represents the ^-component of the

momentum of the particle In the general case, if/ is a function on M x Rn

which describes an observable, then/(s) gives the value of that observable

when the system is in the state s

This formulation of the basic ideas relating to the mechanics of a classical system can be generalized significantly (Mackey [1], Sternberg [1]) Briefly, this generalization consists in replacing the assumption that

M is an open subset of Rn by the more general one that M is an abstract

C00 manifold of dimension n The set of all possible configurations of the system is now M9 and for any x e M, the momenta of the system at this configuration are the elements of the vector space Mx*9 which is the dual

of the tangent vector space Mx of If at a; The phase space of the system

is then the set of all possible pairs (x,p)9 where x e M and p G MX* This set, say S9 comes equipped with a natural differentiate structure under which it is a C00 manifold of dimension 2n9 the so-called cotangent bundle

of M The manifold 8 admits further a canonical 2-form which is where nonsingular and this gives rise to a natural isomorphism J of the

every-module of all O00 vector fields on S onto the module of all 1-forms (both

being considered as modules over the ring of (700 functions on S) The

BOOLEAN ALGEBRAS ON A CLASSICAL PHASE SPACE 3

dynamical development of the system is then specified by a C00 function

H on S, the Hamiltonian of the system If t -> s(t) is a curve representing

a possible evolution of the system, then we have the differential equations:

(5) *$ = [J-i(dH)](e{t))

Here ds{t)jdt is the tangent vector to 8 at the point s(t) along the curve

t -> s{t) and J~1(dH) is the vector field on 8 corresponding to the 1-form dH; the right side of the equation (5) being the value of this vector field

at the point s(t) of 8 In the special case when M is an open set in Rn and

#i> #2> * * *> xn a r e ^n e global affine coordinates on M, 8 is canonically identified with M x Rn and, under this identification, J goes over into the

map which transforms the vector field

2At(dldxi)+2Bl(dldPl)

into the 1-form

n n

i=l i=l

The equation (5) then goes over to (1) (cf Chevalley [1], Helgason [1] for

a discussion of the general theory of differentiable manifolds)

In this general setup, the dynamical development of the system is given

by the integral curves of the vector field J~1(dH) It is necessary to assume that the integral curves are defined for all values of the time parameter t One can then use the standard theory of vector fields to deduce the existence of a diffeomorphism D(t) of 8 for each t such that the correspondence t -> D(t) satisfies the conditions (3), and the map

t, s —> D(t)s of R1 x 8 into 8 is C°° If the system is at the state s at time

f0, then its state at time t+t0 is D(t)s The physical observables of the system are then represented by real valued functions on S A special class

of Hamiltonian functions, analogous to (4), may be defined in this general framework Let # - > < , >x be a C00 Riemannian metric on M, < , yx

being a positive definite inner product on Mx x Mx For each x e M, we then have a natural isomorphism p -> p* of Mx* onto Mx such that

p(u) = (u,p*}x for all p e Mx* and for all u e Mx The analogue of (4) is then the Hamiltonian H given by

where V is a C°° function on if The function x,p-> (p*,p*}x then represents the kinetic energy of the system in question

It may be pointed out that one can introduce the concept of the

momenta of the system in this setup Let

4 GEOMETRY OF QUANTUM THEORY

be a one-parameter group of symmetries of t h e configuration space M, i.e., y(t - > y t ) is a one-parameter group of C00 diffeomorphisms of i f onto

itself such t h a t t h e m a p t, x - > yt(#) oi R x xM into i f is O00 The

infinitesi-mal generator of y is a (700 vector field, say Xy, on M; for a n y # e i f a n d

a n y real valued C00 function / defined around x,

WKX) = {a/(^*))} t o

-Xy defines, in a n a t u r a l fashion, a 00 0 function ytxy on 8 I n fact, if a; e M

a n d # e M x *,

fiy(x t p) = 2>(Xy(z))

(here X y (x) denotes t h e t a n g e n t vector t o M a t x which is t h e value of

X Y a t x) The observable corresponding t o t h e function /zy is called the

momentum of t h e system corresponding to the one-parameter group of symmetries y If M = R n , if x lf • • •, x n are t h e global affine coordinates on

^ = x t cos £ + a;y sin t, y 5 = —x { sin t + #y cos £,

t h e n t h e observable corresponding t o juy.* is called t h e angular momentum with respect to a rotation in the i-j plane A straightforward calculation shows t h a t in the case when M = R n , S = B n x R n , and x 1 , -,x n >p 1 ,' i p n

are global coordinates on 8 ((x l9 • • • , # „ , p l9 • • •, p n ) depicts 2?= iPi(dXi) x ) 9

fi Y c(x,p) = 0^ +- - -+c n p n ,

and

Suppose now t h a t M is a general C00 manifold and 8 its cotangent bundle

I f / a n d g are two 0°° functions on $, t h e n we can form J~ 1 (df) 9 which is a

O00 vector field on 8, and apply it t o g t o get another (700 function on #,

denoted by [f 9 g]:

(8) [/,</] = (J-Hdf))g

[f 9 g] is called the Poisson Bracket o f / with g If we use local coordinates

x l9 • • •, x n on i f and t h e induced coordinates #!, • • •, x n , p l9 • • •, p n on 8 (so t h a t (x l9 - • •, x n9 p l9 • - ',p n ) represents ^iP^dXi)), t h e n J goes over

BOOLEAN ALGEBRAS ON A CLASSICAL PHASE SPACE 5

into the map which (locally) sends ^iA^djdx^+^B^djdp^ into

— 2t B% dxt + 2 i A{ dpi, and [f,g] becomes

The m a p / , gr —> [f,g] is bilinear, skew symmetric, and satisfies the identity

as is easily verified from (9) If X is any O00 vector field on M and /xx is

the <7°° function on 8 denned by

PxfaP) = P(x(x))>

then one can verify, using (9), that

Pax + bY = fl/zx+6/xy (a, b constants),

where [-X,F] is the Lie bracket of the vector fields X and Y If / is any

C00 function on M a n d /0 is the lifted function on 8, i.e.,

f°(x,p) =/(*),

then we may use (9) once again to check that

for any C°° vector field X on M

In many problems, there is a Lie group 6r which acts on M and provides the natural symmetries of the problem For g e G we write a; -» g-x for the symmetry associated with g and assume that g,x->g-x is C00 from

GxM into AT In such problems, one restricts oneself to the momenta specified by the one-parameter groups of M If g is the Lie algebra of G (cf Chevalley [1]) and if we associate for l e g , the vector field on M denoted by X also and defined by

(X/)(x) = ( | / ( e x p t X * ) )t_o, then we obtain the relations

MEX.Y] — [MXJ/^YL

(10)

[/xX)/°] = (X/)°

between the configuration observables / ° and the momentum observables

fjux These relations are usually referred to as commutation rules

6 GEOMETRY OF QUANTUM THEORY

2 T H E LOGIC O F A CLASSICAL SYSTEM

We shall now examine t h e algebraic aspects of a general classical system

I n view of t h e discussion carried out just now, it is clear t h a t for a n y

classical system (B there is associated a space 8 called t h e phase space of

© The states of t h e system are in one-one correspondence with t h e points

of 8 The notion of a state is so formulated t h a t if one knows t h e s t a t e

of t h e system a t a n instant of time t 0 , a n d also t h e dynamical law of

evolution of t h e system, t h e n one can determine t h e state of t h e system

a t time t + t 0 The observables or physical quantities which are of interest

t o t h e observer are t h e n represented b y real valued functions on S If / is

t h e function corresponding t o a particular observable, its value f(s) a t t h e point s of S is interpreted as t h e value of t h e physical q u a n t i t y when t h e system is in t h e state s If s is t h e state of t h e system a t time t 0 , we can write D(t)s for t h e state of t h e system a t time t + t 0 We t h u s have a trans- formation D(t) of 8 into itself For each t, D(t) is invertible a n d m a p s 8 onto itself The correspondence t -> D(t) satisfies t h e equations (3)

t - > D(t) is then a one-parameter group of transformations of 8 I t is called

t h e dynamical group of t h e system <3

These concepts m a k e sense in every classical system I n t h e case of a n y such system t h e most general statement which can be m a d e about it is one which asserts t h a t t h e value of a certain observable lies in a real

n u m b e r set E If t h e observable is represented b y t h e function / on 8,

t h e n such a s t a t e m e n t is equivalent t o t h e statement t h a t t h e state of t h e

system lies in t h e setf~ 1 (E) of the space 8 I n other words, t h e physically

meaningful statements t h a t can be m a d e about t h e system are in

corre-spondence with certain subsets of 8 The inclusion relations for subsets

naturally correspond t o implications of statements I n mathematical terms, this means t h a t a t t h e background of t h e classical system there is a

Boolean algebra of subsets of t h e space S, t h e elements of which represent

t h e statements about t h e physical system I t is n a t u r a l t o call this Boolean

algebra t h e logic of t h e system

Suppose now t h a t @ is a system which does n o t follow t h e laws of classical mechanics Then one cannot associate with it a phase space in general I t is nevertheless meaningful t o consider t h e totality of experi-mentally verifiable statements which m a y be made about t h e system

This collection, which m a y be called t h e logic of @, comes equipped with

t h e relations of implication and negation which convert it into a plemented partially ordered set For a classical system this partially ordered set is a Boolean algebra Clearly, it is possible t o conceive of

com-mechanical systems whose logics are not Boolean algebras We take the point of view that quantum mechanical systems are those whose logics form some sort of projective geometries and which are consequently nondistributive

BOOLEAN ALGEBRAS ON A CLASSICAL PHASE SPACE 7 lattices W i t h such a point of view it is possible t o understand t h e role

played by simultaneously observable quantities, the uncertainty relations, and t h e complementarity principles These phenomena, which are so peculiar t o q u a n t u m systems, will then be seen t o be consequences of t h e

nondistributive n a t u r e of t h e logic in t h e background of the system (&

I t might seem a bit surprising t h a t the basic assumption on a q u a n t u m system is t h a t its logic is not a distributive lattice I t would be n a t u r a l to argue t h a t statements about a physical system should obey the same rules as t h e rules of ordinary set theory The well known critiques of von

N e u m a n n a n d Heisenberg address this question (von N e u m a n n [1], Birkhoff-von N e u m a n n [1], Heisenberg [1]) The point is t h a t only

experimentally verifiable statements are to be regarded as members of the

logic of t h e system Consequently, as it happens in m a n y questions in

atomic physics, it m a y be impossible t o verify experimentally statements

which involve t h e values of two physical quantities of t h e system—for example, measurements of the position and m o m e n t u m of an electron One can verify statements about one of t h e m b u t not, in general, those which involve both of t h e m W h a t t h e basic assumptions imply is t h a t

t h e statements regarding position or m o m e n t u m form two Boolean algebras of t h e logic b u t t h a t there is in general no Boolean algebra which contains both of these Boolean subalgebras

sub-Before beginning an analysis of t h e logic of general q u a n t u m mechanical systems it would be helpful t o recast at least some of the features of the formulation given in section 1 in terms of t h e logic of the classical system

I n the first place it is n a t u r a l to strengthen t h e hypothesis and assume

t h a t the logic of a given classical system © is a Boolean cr-algebra, say j£?,

of subsets of 8, t h e phase space of <S Suppose now, t h a t an observable

associated with t h e system is represented by t h e real valued function / on

8 The statements concerning the observable are t h e n those which assert

t h a t its value lies in an arbitrary Borel set E of the real line and these are represented by t h e subsets f~ x (E) of 8 The observable can t h u s be repre-

sented, without a n y loss of physical content, equally b y t h e m a p

E -^f~ 1 (E) of t h e class of Borel subsets of the real line into ££ The range

of this mapping is a sub-a-algebra, say 3P f Suppose g is a real valued Borel

function on t h e real line Then, the observable represented by the function

g of (s ->g(f(s))) can also be represented by the m a p E ->f~ 1 (g" 1 (E)) from which we conclude t h a t J?g of is contained in Jjf f

I n order to formulate t h e general features of a classical mechanical

system in terms of its logic J?, it is therefore necessary to determine to

w h a t extent an abstract a-algebra ££ can be regarded as a a-algebra of subsets of some space 8; further to determine the class of mappings from

t h e a-algebra of Borel sets of t h e real line into S£ which correspond to real valued functions on 8; and to clarify the concept of functional

8 GEOMETRY OF QUANTUM THEORY

dependence in this general context W e shall now proceed t o a discussion

of these questions

3 B O O L E A N A L G E B R A S

Let j£? be a n o n e m p t y set ££ is said t o be 'partially ordered if there is a relation < between some pairs of elements of j£? such t h a t (i) a < a for all

a in S£\ (ii) a < b and b < a imply a = b; (hi) a < b and b < c imply a<c If ££

is partially ordered, there is at most one element called t h e null or zero element and denoted by 0, such t h a t 0 < a for all a in <£ Similarly there

is at most one element called the unit element and denoted by 1, such t h a t

a < 1 for all a in ££\ More generally, for any nonempty subset F of j£? there exists at most one element c of ££ such t h a t (i) a<c for all a e F; (ii) if d

is a n y element of ££ such t h a t a<d for all ae F, t h e n c<d W e shall write \J aeF a for c whenever it exists If F is a finite set, say F={a lf • • •, an},

it is customary t o write \/?= I ai o r ai v a 2 V • • • V a n instead of \/ aeF a

I n a n analogous fashion, for any subset F of ££ there exists a t most one element c such t h a t (i) c < a for all a e F; (ii) if d is a n y element of J£? such

t h a t d<a for all a e F, t h e n d<c; we denote it by /\ aeF a whenever it exists If F is a finite set, say F = {a lf • • •, a n }, we often write /\"=:i a t or

a 1 Aa 2 A'-'Aa n instead of /\ aG F a - The partially ordered set ££ is called

a lattice if

(i) 0 and 1 exist in j£? and 0 ^ 1 ,

(ii) \] a and f\ a exist for all finite subsets F of «£?

aeF aeF

Suppose t h a t j£? is a lattice Given a n y element a of j£?, an element a' of j£? is said to be a complement of a if a A a' = 0 and a v a' = 1 a is t h e n a complement of a' j£? is said to be complemented if, given a n y element,

there exists at least one complement of it I t is obvious t h a t 0 and 1 have

t h e unique complements 1 and 0, respectively A lattice ££ is said t o be distributive if for a n y three elements a, 6, c of j£?, t h e identities

a A (b v c) = (a A b) v (a A c),

a V (b A c) = (a V b) A (a V c) are satisfied A complemented distributive lattice is called a Boolean algebra A Boolean a-algebra J? is & Boolean algebra in which /\ aeF a a n d

N/aeF a exist for every countable subset F of j £ \

E v e r y element in a Boolean algebra has a unique complement Suppose

in fact t h a t j£? is a Boolean algebra and t h a t a is an element with t w o

complements a ± a n d a2 Then, one has

a 1 = a x A (a v a2) = (ai A a) v (#i A a2) = «i A a2 < a2;

BOOLEAN ALGEBRAS ON A CLASSICAL PHASE SPACE 9 similarly, a 2 <a ± , so t h a t a 1 = a 2 The unique complement of a is denoted

by a' Using t h e s t a n d a r d manipulations of set theory it is easy t o show

t h a t (Aae F aY= Vae F a' a n d (\Z a eF^Y = A

ae p o/ for a n y finite subset F

of J*f If j£? is a Boolean cx-algebra, t h e n t h e same identities remain valid

even when F is countably infinite If ££ is a n y Boolean algebra a n d a, b are elements in it with a<b, c = a' Ab is t h e unique element of S£ such t h a t

a A c = 0 a n d a Vc — b\ c is called the complement of a in b Since c = b A a', c<a' (cf Birkhoff [1] a n d Sikorski [1] for t h e general theory of Boolean

algebras a n d a-algebras)

A homomorphism of a Boolean algebra ££ 1 into a Boolean algebra S£ 2 is

a m a p h of S£ x into j£?2 such t h a t (i) &(0) = 0, A(l) = l ; (ii) h{a') — h(a)' for

all a in jSfx; (hi) h(avb) = h(a)v h(b), h(a Ab) = h(a) Ah(b) for all a, b i n ^

If h is a homomorphism a n d a < b, then ft(a) < h(b) An isomorphism of ^

onto j£?2 is a homomorphism h of J ^ onto jSf2 such t h a t A(a) = 0 if and only

if a = 0; in this case h is also one-one

The class of all subsets of a n y set is a Boolean algebra under set

inclusion a n d set complementation However, obviously this is n o t t h e most general Boolean algebra since infinite unions and intersections exist

in it Suppose now t h a t X is a topological space The class of subsets of X

which are b o t h open a n d closed (open-closed) is obviously a Boolean algebra A well known theorem of Stone [1] asserts t h a t every Boolean algebra is isomorphic t o one such a n d t h a t , if we require t h e topological space t o be compact Hausdorff as well as totally disconnected, it is essentially uniquely determined by t h e Boolean algebra W e recall t h a t a

compact space is said t o be totally disconnected if every open subset of it

can be written as a union of open-closed subsets We shall call a compact

Hausdorff totally disconnected space a Stone space

L e t j£? be a Boolean algebra A subset Ji of ££ is called a dual ideal if

t h e following properties are satisfied:

(i) 0 ^ ^ ,

(ii) if a e Jt a n d a < b, then b e J£,

(iii) if a, b e Jt, t h e n a A b e Jt

Jt is said t o be maximal if it is properly contained in no other dual ideal

The naturalness of t h e notion of maximal dual ideals can be seen in t h e

following way Let X be a Stone space and 3? = ££(X) t h e Boolean algebra

of all open-closed subsets of X Then, for a n y x e X, t h e collection Jf(x),

where

Jt(x) = {A : A e £>, x e A},

is easily seen t o be a maximal dual ideal; it is also easy t o check t h a t t h e

correspondence x - > Jt{x) is one-one if we notice t h a t X is Hausdorff

The concept of maximal dual ideals is central in t h e proof of Stone's theorem

10 GEOMETRY OF QUANTUM THEORY

Suppose that j£? is an arbitrary Boolean algebra Using Zorn's lemma

one can show easily that maximal dual ideals of -£? exist Let X — X(S£)

be the set of all maximal dual ideals of ££\ For any a G ^ w e define Xa by

Xa = {Jt : Jt e X, a e Jt),

X0= 0, the null set, and X± = X We shall say that a subset A^X is open if A is the union of sets of the form Xa This definition defines the structure of a topology on X called the Stone topology We now have: Theorem 1.1 (Stone [1]) Let ££ be a Boolean algebra and let X = X(JP)

be the space of all maximal dual ideals of ££\ Then, equipped with the Stone topology, X becomes a Stone space The map a -> Xa is then an isomorphism

of ££ with the Boolean algebra of all open-closed subsets of X X is determined

by ££, among the class of Stone spaces, up to a homeomorphism More generally, let X and Y be Stone spaces and let J?(X) and <¥{Y) be their respective Boolean algebras of open-closed subsets If u is any isomorphism

of &{Y) onto S£(X), there exists a homeomorphism h of X onto Y such that (12) u(A) = h-\A) {A e &(Y))\

moreover, h is uniquely determined by (12)

This theorem is very well known and we do not give its proof The reader may consult the books of Birkhoff [1], Sikorski [1], and the paper

of Stone [1] for the proof

Corollary 1.2 Let X be a Stone space and let <£ = J£(X) be the Boolean algebra of open-closed subsets of X If t -> Dt(— oo<t < oo) is any one- parameter group of automorphisms of j£f, there exists a unique one-parameter group t -> ht of homeomorphisms of X onto itself such that for all t and AeJ?,Dt(A) = ht-HA)

Proof Theorem 1.1 ensures the existence and uniqueness of each ht If

tl9 t2 are real, then htl+t2 and htl o hi2 induce the same automorphism

Dtl +t2 of Se, so that htl +t2 =hh o hh

The theorem of Stone shows that there is essentially no distinction between an abstract Boolean algebra and a Boolean algebra of sets If one deals with Boolean cr-algebras, the situation becomes somewhat less straightforward We shall now describe the modifications necessary when one replaces Boolean algebras by Boolean cr-algebras

If o^! and o£?2 a r e Boolean cr-algebras, and h a map of J ^ into j£?2> ^ is

called a a-homomorphism if (i) A(0) = 0, h(l) = l; (ii) h(a') = h(a)' for all

a e ££x; and (hi) if F is any subset of ££x which is finite or countably

infinite, h(\JaeF a) = \JaeF h(a) and h{/\aeF a) = /\aeF h(a) Suppose S£x

BOOLEAN ALGEBRAS ON A CLASSICAL PHASE SPACE 11

a n d j£?2 are t w o Boolean cr-algebras a n d h a cr-homomorphism of ££ x onto

££ 2 The set t A r = {a : a e ££ x , h(a) = 0} is a subset of Sf f1 with t h e properties:

(i) 0 e /T, 1 ^ JV\ (ii) if a e JV a n d b < a, t h e n b e Jf\ (iii) if F is a countable subset of JV, \/ a€F a e JV* ^ is called t h e kernel of & Suppose conversely

££ is a Boolean cr-algebra a n d i/T a subset of ££ with properties (i) t o (iii) listed above W e shall say t h a t elements a a n d b of 3? are equivalent, a~b, if a A b' a n d 6 A a' are in c/T I t is easily verified t h a t ~ is a n equiv- alence relation L e t J ^ be t h e set of all equivalence classes, a n d for a n y a

in «£?, let a denote t h e unique equivalence class containing a W e define a<b whenever there are elements a in a, a n d 6 in 5 such t h a t a<b I t is

t h e n easily shown t h a t ££ is a Boolean cr-algebra whose zero a n d unit elements are, respectively, 0 a n d 1, a n d t h a t t h e m a p a->a is a a- homomorphism of 3? onto ££ with kernel JV* We write JS? = ££\Jf

Theorem 1.3 (Loomis [1]) Let 3? be a Boolean a-algebra Then there

exists a set X, a a-algebra Sf of subsets of X, and a a-homomorphism h of

Sf onto &

Proof L e t X be a Stone space such t h a t t h e lattice <JS?' = J?(X) of

open-closed subsets of X is isomorphic t o t h e Boolean algebra j£? L e t Sf denote t h e smallest cr-algebra of subsets of X containing ££' W e denote

by U a n d n t h e operations of set union a n d set intersection for subsets of

X, and b y V a n d A t h e lattice-theoretic operations in J£ a n d S£'

If A l9 A 2 , - • - is a n y sequence of sets in j£?', t h e n \/ n A n = A exists in

££' since ££' is isomorphic t o ££ a n d 3? is a cr-algebra Since A is t h e smallest element of ££' containing all t h e A n , it follows t h a t t h e set A — {J n A n

cannot contain a n y element of ££' as a subset The sets in 3" form a base for t h e topology of X a n d hence we conclude t h a t A — [J n A n cannot contain a n y nonnull open set Since ( Jn An is open, this shows t h a t

A — Un^ 4ni s a closed nondense set

Consider now t h e class yx of all sets A e Sf with t h e property t h a t for some B in J*?', (A — B) U (B — A) is of t h e first category If B x a n d B 2

are elements of j£?' such t h a t (A — B t ) U (B i — A) is of t h e first category (i = l , 2), t h e n it will follow t h a t (B x -B 2 ) U (B 2 -B ± ) is of t h e first

category, which is n o t possible (by t h e category theorem of Baire) unless

B X = B 2 Thus, for a n y A in S^ 1 there exists a unique B = h x (A) in <£' such t h a t (A-B)U(B-A) is of t h e first category Clearly <£''c^,

and for 4 e-Sf", ^ ( ^ ) = ^ 1

We claim t h a t ^ is a cr-algebra Since

(A-B) u ( J 3 - 4 ) = ( ^ ' - 5 ' ) u ( £ ' - 4 ' ) (primes denoting complementation in X), we see t h a t for a n y A in , 9 ^ ,

12 GEOMETRY OF QUANTUM THEORY

A' is in Sf-L a n d h^A^ — h^A)' Suppose A l9 A 2i • • • is a n y sequence in

y x Write J8n = *1( 4n)> A = \Jn A n , B=\/ n B ni B 0 = {J n B n B y w h a t we said above, B — B 0 is a closed nondense set Moreover, as B 0 ^B, we have

(A-B) U (B-A) £ { ( i l - B o ) U ( £0- 4 ) } U ( B - B0)

£ U U n -B n ) U ( B » - i in) } U ( B - B0)

-n

As all members of t h e right side are of t h e first category, this proves t h a t

A e £f x a n d h 1 (A) = \/ n h 1 (A n ) I n a similar fashion we can show t h a t

p | n A n lies in Sf x a n d ^ ( f l n A n ) = /\ n

h^An)-The conclusions of t h e preceding p a r a g r a p h show t h a t £P l9 is a Boolean

cr-algebra <^<Sf Since ^ contains JSP', S? 1 = Sf Moreover, we see a t t h e same time t h a t h x is a cr-homomorphism of Sf onto JS?' If we write h = koh ±

where k is an isomorphism of JSf" onto <£?, t h e n & is a or-homomorphism

of y onto i f

Remark Let £f be t h e cr-algebra of Borel sets on t h e unit interval [0,1],

JV t h e class of Borel sets of Lebesgue measure 0, and S£ — SPjJf Then 3?

is a Boolean cr-algebra W e can obviously define Lebesgue measure A as a countably additive function A on jSf; A is strictly positive in t h e sense t h a t

for a n y a^O of j£f, A(a) is positive F r o m this it follows t h a t a n y family

of mutually disjoint elements of ££ is countable On t h e other hand, since

£f is countably generated, so is ££ However, any cr-algebra of subsets of some space X which is countably generated can be proved to have atoms,

t h a t is, minimal elements Since j£? does not have atoms, j£? cannot be isomorphic t o a n y cr-algebra of sets

4 F U N C T I O N S

W e now t a k e u p t h e second question raised in section 2, namely, t h e

problem of describing t h e calculus of functions on a set X entirely in

t e r m s of t h e Boolean a-algebra of subsets of X with respect t o which all

these functions are measurable The results are summarized in theorems 1.4 a n d 1.6 of this section

Let X be a n y set of points x and Sf a Boolean cr-algebra of subsets of X

A f u n c t i o n / f r o m X into a complete separable metric space Y is said t o be

^ - m e a s u r a b l e iff~ 1 {E) e ¥ for all Borel sets E^ Y I f / i s ^ - m e a s u r a b l e ,

t h e mapping E ->f~ x (E) is a cr-homomorphism of t h e cr-algebra 3S(Y) of Borel subsets of Y into £P Suppose now £f is an abstract Boolean cr- algebra We shall t h e n define a Y-valued observable associated with S£ t o be

a n y cr-homomorphism of 38 (Y) into ££ If Y = B 1 , t h e real line, we call these observables real valued a n d refer t o t h e m simply as observables

F r o m our definition of cr-homomorphisms we see t h a t a m a p u(E - > u(E))

BOOLEAN ALGEBRAS ON A CLASSICAL PHASE SPACE 13

of «^( Y) into ££ is a Y-valued observable associated with 3? if and only if (i) u(0) = O, u(Y) = l; (ii) u(Y-E) = u(E)f for all E in ^ ( 7 ) ; (iii) if

Elt E2, • • • is any sequence of Borel sets in Y, u([Jn En) = \/n u(En) and u(C)nEn) = /\nu(En)

Theorem 1.4 Let X be a set, £f a Boolean a-algebra of subsets of X and

h a a-homomorphism of 6? onto a Boolean a-algebra ££\ Suppose further that u(E -> u(E)) is any (real valued) observable associated with <£? Then there exists an 6f-measurable real valued function f defined on X such that (13) u(E) = h{f-\E))

for all Borel sets E^R1 f is essentially unique in the sense that if g is any

^-measurable real valued function defined on X such that u(E) = h(g~1(E)) for all Borel sets E^R1, the set {x : x e X, f{x)^g(x)} belongs to the kernel

ofh

Proof We begin with a simple observation Suppose A and B are two

subsets of X in £f such that A^B, and c any element of j£f such that h(A)<c<h(B) Then we can select a set C in £f such that A ^C^B and h(C) = c In fact, since h maps Zf onto »£?, there exists Cx in ^ such that

hiCJ^c If we define C = (C1 n B) U A, then A ^C^B while

1 <i, j<n We shall construct An + 1 as follows Let (il9 i2, • • •, in) be the permutation of (1, 2, • • •, n) such that ri± <ri2 < • • • <rin Then, there exists a unique k such that ^ik<^n + i<rik+1 (w e define rio = — oo and

rin+i — +oo), and by the observation made in the preceding paragraph,

we can select An + 1 in £f such that Ailc^An + 1^Aik+i (we define Aio = 0 ,

^ =Jf) The collection {Alf A2, • • •, An + 1} then has the same perties relative to rl9 r2, • • •, rn + 1 as {Alt • • •, An} had relative to

pro-ri> ri-> ' ' •> ?V It thus follows by induction that there exists a sequence

^4^ A2- • - of sets in ^ with the properties (a) and (b) As

A(fV,) = A «W) = «(A *>/) = 0,

we may, by replacing Ak by Ak — f\jAi if necessary, assume that

14 GEOMETRY OF QUANTUM THEORY

p | ; Aj = 0 Further h{[Jj A5) = N/y u(D,) = u(V, Dt) = 1 so that h(N) = 0, where N = X — (Jj Aj We now define a function/ as follows:

so that h(f~1(E)) = u(E) whenever E = Dk for some & Since the class of

all E for which this equation is valid is a Boolean a-algebra, we conclude that h(f~1(E)) = u(E) for all Borel sets E

It remains to examine the uniqueness Let g be any real valued measurable function on X such that h(g~1(E)) = u(E) for all Borel sets E Then, if we write Dk for R1 - Dkt

Lemma 1.5 Let X be a set, Sf a o-algebra of subsets of X and f an

Im-measurable making of X into Rn Suppose £f~ ={f~1(F) : F e &(Rn)} Then to any £f ~-measurable real function c on X there corresponds a real valued Borel function c~ on Rn such that c(x) = c~(f(x)) for all x e X

Proof Since c is £f~ -measurable, there exists a sequence cn (?i = 1, 2, • • •)

of ^"-measurable functions such that (i) each cn takes only finitely many

values; (ii) cn(x) -> c(x) for all x e X For any n, let Anl, An2, • • •, Ank be

disjoint subsets of X whose union is X such that cn is a constant, say ani,

on Ani, the ani being distinct for i = 1, 2, • • •, kn Since cn is ^ ~ -measurable,

Anie Sf~\ so there exists a Borel set Bni of Rn such that ^4ni=/_1(jBni)

BOOLEAN ALGEBRAS ON A CLASSICAL PHASE SPACE 15

(* = 1, 2, • • •, kn) Replacing Bni by Bni — \Jj<iBnj if necessary, we may

assume that the Bni are disjoint Let us define the function cn~ on Rn as follows:

Clearly c~ is Borel Since cn(x) = cn~(f(x)) and lim cn(x) exists and is equal

to c(x) for all a: in X, c~(f(x)) = c(x) for all x in JT

Let = ^ b e a Boolean a-algebra ££x<^££ is said to be a sub-a-algebra if (i) 0, 1 e jSfij (ii) if a e S?^ then a' e ££^\ (iii) if a1? a2, • • • are in ££x, then

\/n an a nd An an a r e m «^i- A sub-d-algebra J^x is said to be separable if there exists a countable subset D of j£? such that j£?x is the smallest sub-a-

algebra of <£ containing D

Theorem 1.6 (i) Let ££ be a Boolean a-algebra and u(E -> u(E)) an

observable associated with ££\ Then the range J?u = {u(E) : E e ^(R1)} of u

is a separable Boolean sub-a-algebra of ££ Conversely, if ££x is a separable Boolean sub-a-algebra of ££, there exists an observable u associated with ££ such that S£x is the range of u

(ii) Let ut (i = l, 2, • • •, n) be observables associated with <Sf, and S^{

(i = ly 2, • • •, n) their respective ranges Suppose j£?0 is the smallest algebra of ££ containing all the J^ Then there exists a unique o-homomorphism

sub-a-u of &(Rn) (the a-algebra of Borel subsets of the n-dimensional space Rn) onto J£Q such that for any Borel set E of R1, ui(E) = u(pi~1(E)), where p{ is the projection (t1,t2,- • •, tn) -» tt of Rn onto R1 If y is any real valued Borel function on Rn, the map E->u(<p~1(E))(E e ^(R1)) is an observable associated with & whose range is contained in J?0 Conversely, if v(E -> v(E))

is any observable associated with j£? such that the range of v is contained in

j£?0, there exists a real valued Borel function 9 on Rn such that v(E) — u(y ~ 1(E)) for all E

Proof If u is an observable with range ££u, S£u is obviously the smallest

sub -a- algebra of ££ containing all the u(E), where E is any open interval

of R1 with rational end points This shows that ££u is separable

Suppose conversely that ££x<=,££ is a separable sub-a-algebra of ££

By theorem 1.3 there exists a set X, a a-algebra £f of subsets of X, and a (T-homomorphism h of Sf onto J^ Let {^4n:% = l , 2 , - - - } b e a countable

family of sets of Sf such that ££^ *s the smallest sub a-algebra of ££

16

containing all the h(An) We denote by £P0 the smallest a-algebra of subsets

of X containing all the An The function

c : a - > (XA^*), XA 2 {*)> •" •, XAS*)> "')

(where XA denotes the function which is 1 on A, and 0 on X — A) is measurable from X into the compact metric space Y which is the product

im-of countably many copies im-of the 2-point space consisting im-of 0 and 1

Moreover, it is obvious that each An is of the form c~1(F) for some Borel set J ^ c 7 , and hence £f0 = {c-1(F) : F Borel in 7} Now, by a classical theorem (Kuratowski [1]), there exists a Borel isomorphism d of Y onto

R1, so that the function cx : x -> d(c(x)) is an ^-measurable real valued function and 6^0 = {c1~1(E) : E Borel in R1} If we now define, for any Borel set E of R1, u(E) by the equation

u(E) = McrHE)),

then u is an observable associated with ££ whose range is ££x This proves (i)

We now come to the proof of (ii) Suppose ult u2, • • •, un are observables

associated with j£?, having ranges Sfl9 • • •, j£?n, respectively Each j£?t is separable and hence j£?0, the smallest sub-a-algebra of ££ containing all the ££{, is also separable Let X, £f\ and h have the same significance

as in the proof of (i) By theorem 1.4, there exists a real valued

im-measurable function ft on X such that ui(E) = h(fi~1(E)) for all Borel subsets E of R1 Let / be the map x-> (fx(x), • • -,/n(#)) of X into Rn Then / is ^-measurable The map u : F ->h(J~1(F)) is then a a- homomorphism of &(Rn) into <£ such that ui(E) = u{pi-1(E)) for all

E etffliR1) Since &(Rn) is the smallest cr-algebra of subsets of Rn

con-taining all the sets pi~1(E)) it is clear that the range of u is j£?0 The

uniqueness of u is obvious

For any real Borel function <p on Rn, E -^u{q>~1(E)) is an observable

associated with j£? whose range is obviously contained in j£?0 Suppose

now that v is an observable associated with 3? whose range «5?vcjg?0

If we use the notations of the previous paragraph, and define £f~ by

&>- ={ / - i (Jp ) :Fe@{Rn)}, then h maps £f~ onto J5?0 Applying theorem 1.4 to £f~ and v9 we infer the existence of a real valued y ~ -measurable Borel function c o n l

such that h(c~1(E)) = v(E) for all Borel sets E of R1 By lemma 1.5, since

c is S?~ -measurable, there exists a real valued Borel function <p on Rn

such that c(x) — q>(?{x)) for all x e X If now 2£ is any Borel set on the line,

BOOLEAN ALGEBRAS ON A CLASSICAL PHASE SPACE 17

Remark The uniqueness of u, guaranteed by (ii) of theorem 1.6 shows

that u is independent of the constructs X, <9*, and h, used in its tion Consequently, for any real Borel function cp on Rn, the map

construc-E -> u((p~1(E)) is uniquely determined by ul9 u2, • • •, un and cp It is natural to denote this observable associated with ££ by <p(ux, u2i • • •, un)

NOTES ON CHAPTER I

1 For a beautiful modern introduction to classical mechanics, see the

book of V.I.Arnold, Mathematical Methods of Classical Mechanics,

Springer-Verlag, New York, 1978

CHAPTER II PROJECTIVE GEOMETRIES

1 COMPLEMENTED MODULAR LATTICES

The point of departure of our discussion of quantum phenomenology is the observation that the partially ordered set of experimentally verifiable state- ments associated with an atomic system cannot be expected to possess the distributivity properties characteristic of the Boolean algebras associated with classical systems The simplest and most interesting of the mathe- matical structures that model such systems are the projective geometries, namely, the lattices of subspaces of vector spaces More precisely, let D be

a division ring, and let V be a vector space of finite dimension n ^ 2 over D;

we shall always suppose our vector spaces to be left vector spaces (cf Jacobson [1]) Then j£?=j£?(F,D), the lattice of linear subspaces of V partially ordered by inclusion, is called the projective geometry of V This

chapter will begin with a brief review of the basic properties of projective geometries Although these are too simple to serve as models for realistic quantum systems, they already possess many of the fundamental features

of the more complex systems

The first example of an abstract projective geometry that anyone comes across is the projective plane, described axiomatically in terms of its points, lines, and their incidence properties To define higher dimensional pro- jective spaces one replaces incidence by partial order and uses a dimension function to characterize the hierarchies of points, lines, planes, and so on; the properties of incidence that are characteristic of a projective geometry are then summarized by certain natural properties of the dimension function

We recall that a lattice 3? is complemented if for any aeJ? there is a

b e S£ such that a Kb = 0 and a yb = 1 A complemented lattice JS? is modular if (1) a,2>,CGJ^, c < a => a A (bye) = (a A 6) V c

Boolean algebras are modular; for any division ring D and any finite

dimensional vector space V over D, ^(V,D) is a modular lattice which is

not a Boolean algebra if dim(F) > 2 We next introduce the notion of a

lattice of finite rank A chain in a lattice is a sequence (a^i^i^n of elements at

of the lattice such that ax < a2 < < an, with ax ^ 0 and a{ / a , for i <j; n is called the length of the chain, and the rank of the lattice is the supremum of

18

PROJECTIVE GEOMETRIES 19

t h e lengths of all possible chains An element of a lattice is called a point if

it is nonzero and minimal If

i' 2' * * *' n are points, t h e y are called pendent if x { <£ V j V t ^ for * = 1,2, , n The lattice ^(VJ)) has r a n k dim(V),

inde-its points are t h e one-dimensional subspaces, and independence is just t h e

usual linear independence For a n y lattice 3? a n d a n y a / 0 in iff, let us write J?[0,a] for t h e set of all elements b of ££ with 0<b<a\ under t h e partial ordering inherited from J?, ^[0,a] is also a lattice If j£? is of finite rank, so is J^[0,a]; and we refer to its rank as the dimension, dim(a), of a

W e use t h e convention t h a t dim(0) = 0

If J^f is a complemented modular lattice of finite r a n k n, it is n o t difficult

t o show t h a t t h e function a->dim(a) has t h e following properties:

(i) dim(0) = 0, dim(a) is an integer ^ 0 for all a;

(ii) if a < 6, dim (a) ^ dim(6), with equality only for a = b;

(2) (hi) dim(ayb) + dim(aAb) — dim(a) -f dim(&) for all a, b;

(iv) if s = dim(a) > 0, t h e n s is t h e m a x i m u m n u m b e r of independent

points contained in a; and a family of independent points

contained in a has sum equal t o a if and only if it has s elements

I n particular dim(l) = n and dim(#) = 1 for a n y point x The converse is also

t r u e ; if ££ is a complemented lattice of finite r a n k admitting a function d:J?->Jl satisfying (i)-(iii) of (2) and such t h a t d(x) = l for all points x, then ££ is modular a n d d(a) = d i m ( a ) for all a

The lattice J£?(F,D) is modular, dim being the usual linear space dimension A t t h e other extreme is the Boolean algebra of all subsets of a finite set, dim being t h e cardinality To single out the geometries we need

t h e notion of irreducibility An element of a complemented modular lattice

j£P is called central if it has a unique complement in ££\ which is written as a'; the reason for this terminology is the easily proved fact t h a t a e j£? is central

if and only if for all x e J*?

(3) x = {x/\a)\f{x/\a')

The set of all central elements is called the center oi<£; it is a Boolean algebra, which coincides with 3? only when J$? itself is a Boolean algebra & is called irreducible if 0 and 1 are its only central elements A geometry is an irreducible

complemented modular lattice of finite rank The lattices

JS?(F,D) ( d i m ( F ) ^ 2 ) are geometries

E v e r y complemented modular lattice of finite r a n k m a y be viewed in a canonical manner as a direct sum of geometries over its centre More

precisely, let ££ be a complemented modular lattice of finite r a n k and let

c v ,c k be t h e minimal (nonzero) elements of t h e center of 3? Let

J^,- = JSf[0,cJ a n d let & = ££ X x x Se h

20 GEOMETRY OF QUANTUM THEORY

where the partial order in ££ is denned by saying that

(xl9 ,xk) < (yl9 ,yk)

if and only if xt < y{ for 1 ^ i^ k Then each J^- is a geometry, and the map that takes x e££ to

(xAcv xAc2, , xAck)eJ?

is an isomorphism of ££ with J*?, the inverse map taking (x1, ,xk) to

xx\/ \fxk For arbitrary geometries S^i9 ££ denned as above is a

comple-mented modular lattice whose center has (0, 0,1,0, ,0) (1 in the j t h

place) as its minimal elements (l^j^k)

The usual geometrical terminology with which we are familiar may be introduced as follows Let JS? be a complemented modular lattice of finite

rank If aeS£ and dim(a) = l, a is a 'point', if dim(a) = 2, a is a line; if dim(a) = 3, a is a plane If a = b vc where b and c are distinct points, a is called the line joining these points, and 6, c are said to be on a; if d is a point < a,

6, c, d are said to be collinear It can be proved that ££ is a geometry if and

only if every line has at least three points on it

The fundamental theorem of classical geometry is that every geometry of rank ^ 4 is isomorphic to some j£?( F,D) wher the division ring D is deter- mined uniquely up to isomorphism; and that if the rank is 3, this is true if

(and only if) the plane in question is Desarguesian We shall discuss this

question a little later At this time we shall take a closer look at the

geometries S£ (F,D)

2 ISOMORPHISMS OF PROJECTIVE GEOMETRIES

SEMILINEAR TRANSFORMATIONS

The first problem that arises naturally is to obtain a description of all

possible isomorphisms between j£?(F,D) and J?(V',D') We suppose the

vector spaces to be always finite dimensional and that the scalars act from the left To formulate the answer we need the concept of semilinear trans- formations Let D, D' be two division rings which are isomorphic and let

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

PROJECTIVE GEOMETRIES 21

If L is a <7-linear isomorphism of V with V, then for any linear subspace

M <= 7, its image X[lf] under L is a linear subspace of V, and Jf ^ ^ [ i l f ]

is an isomorphism of ^ ( F , D ) with j£?(F',D'), denoted by £z, We can now formulate the main result

Theorem 2.1 There exists an isomorphism between if?(F,D) and ^ ( F ' j D ' )

if and only if D and D' are isomorphic and dim( V) = dim( 7') / / £ ^s aw morphism of J*?(F,D) wi£A J5?(F',D'), there is an isomorphism a(D ^ D ' ) awd

i'so-a i'so-a-linei'so-ar isomorphism L(V^> V) such thi'so-at | = £L If ^-(D^D') is i'so-another isomorphism and L'( V ^ V) a r-linear isomorphism, then £L = £L' if and only

if for some d e D, d ^ 0, we have

For the proof we refer to Baer [1], If D = D' it makes sense to call an

isomorphism £(j£?(F,D) ^ J^(F,D')) linear if there is a linear isomorphism L(V^> V) such that £ = £L Theorem 2.1 leads to

Theorem 2.2 In order that every isomorphism ofJ?(V,D) with j£?(F',D) be linear it is necessary and sufficient that every automorphism of D be inner

The division rings of greatest importance in physics are R, the field of real numbers, C, the field of complex numbers, and H, the division ring of

"Hamiltonian" quaternions

(a) R The identity map of R is its only automorphism, i.e., R is rigid So

every isomorphism between real projective geometries is linear and the linear map is determined up to a multiplicative constant

(b) C One knows from the theory of algebraic fields that there are infinitely many automorphisms (cf Bourbaki [1], pp 114-115) However,

the identity and complex conjugation (c^c*) are the only analytically

well-behaved ones (e.g., measurable, bounded, etc.) Since C is commutative, any isomorphism between two complex geometries determines uniquely the automorphism of C associated with it

(c) H, the quaternions Let

1 = (o i ) ' j l =( o -if' J 2 =( - i o)' h==[i o)'

where i is the usual square root of — 1 in the complex number field We

define H to be the R-linear span of 1 and ja (1 ^ a < 3 ) ; since ja 2 = —1

(1 ^ a ^ 3 ) and j«j&= — j&ja=jc whenever (abc) is an even permutation of

(123), H is an (associative) algebra It is easy to verify that it is the algebra generated by ja with the above relations Let

|q| = +det(q)*= +(?o2 + <Zi2 + <Z22 + <?32)i

where q = qQl + qxjx + q2j2 + <lzh (Qo> Qa e R) • Then | • | is a norm on H which is multiplicative, i.e., |qq'| = |q||q'|, and q is invertible in H if and only if

22 GEOMETRY OF QUANTUM THEORY

q ^ 0 Thus H is a division ring For q e H , say q = ^01 + 2i<a<3#<ja> i t s

conjugate q* is the quaternion g^ —2i<a<s 9«Ja; t n e m aP •I""*1!* i s a n

involutive anti-automorphism of H A quaternion of norm 1 is called a unit quaternion A quaternion is a unit quaternion if and only if its matrix is

so that the unit quaternions constitute the group SU(2,C) of 2 x 2 unitary matrices of determinant 1 Two quaternions are said to be in the same class

if they can be transformed into each other by an inner automorphism; for

q, q' to be in the same class it is necessary and sufficient that

3 DUALITIES AND POLARITIES

We shall now examine the anti-automorphisms and mentations of the projective geometries j£?(F,D) It will turn out that orthocomplementations essentially arise only from Hilbert space structures,

orthocomple-at least when D is one of R, C, and H

We begin with the concept of the dual of a division ring Let D be a division ring We define D° to be the ring with the same elements as D, with the same rule for addition, but reversing the order of multiplication D° is a

division ring, said to be dual to D Isomorphisms of J)1 with D2° may be

viewed as anti-isomorphisms of T>1 with D2 If V is a vector space over D, its dual, defined as the space of D-linear maps of V into D, becomes a vector

space over D° if we define scalar multiplication by writing

( « ' / ) W = / W a (aeD°,veV9feV*);

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

(8) dim( Jf) + dim( Jf °) = dim( V)

and so M = M00 The map M->M° is obviously an inclusion-reversing section of J£?( F,D) with «J^( F*,D°) We call it a duality, and use the same term

bi-to refer bi-to any inclusion-reversing bijection of one geometry onbi-to another

To determine the most general duality we need the concept of a

non-singular semibilinear form Let 6 be an anti-automorphism of D, F a vector space over D A 0-bilinear form is a map < , ) : x, y ~> (x, y) of F x F into D

with the following properties:

(9) (i) {x1 + x<L,y) = {x1,y) + (xz,y) , m

For any anti-automorphism 0 of D, the map

(»!, ,rrn), (yls ,yn)->Xiyie + — +Xnyn0

is a nonsingular 0-bilinear form on Dn x Dn

Theorem 2.3 Let V be a vector space of dimension n over D; 6, an

anti-automorphism of D; and (.,.), a 6-bilinear form on Vx F Then ( , ) is

nonsingular if and only if it satisfies either of (101) or (lOr) For any linear

subspace M of V let

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

Then, if(.,.)is nonsingular, M->M' is a duality of\££(V ,D) If n^3, every duality of j£?(F,D) arises in this manner, for suitable 6 and (.,.) The pair

6', (.,.)' induces the same duality as 6,(.,.) if and only if there is a nonzero deD such that for all x, y e V, c e D,

In particular, if D is commutative, 6' = 6

Proof For y eV let tyeV* be defined by ty(u) = (u,y) (ueV) Then

t (y->ty) is a 0-linear map of F into F* Clearly (lOr) is equivalent to the

24 GEOMETRY OF QUANTUM THEORY

s t a t e m e n t t h a t t is injective while (101) is equivalent t o t h e s t a t e m e n t t h a t (0) is t h e annihilator in V of the range of t; i.e., t h a t t is surjective The first

s t a t e m e n t is now clear Suppose now t h a t ( , ) is nonsingular Then t is

a 0-linear isomorphism a n d so, as M' = t~ 1 (M°) we see t h a t M->M' is a duality of &(F,D) Conversely, if f is a duality of J^(F,D), v :M-> (^(M)) 0

is a n isomorphism of J*?(F,D) onto ^ ( F * , D ° ) This shows already t h a t

(13) dim(M) + dim(i(M)) = n

Moreover, theorem 2.1 gives t h e existence of an anti-automorphism 6 of D and a 0-linear isomorphism t of V with V* such t h a t M° = t[£(M)] for all M in«S?(F,D) If we set {x,y) = (ty)(x) (x,ye V), it is immediate t h a t ( , ) is a nonsingular 0-bilinear form and (j(M) = M f for all M If 6' and ( , ) ' is another pair and t y '{x) = (x,y)' (x, y e V), 0', (.,.}' also give rise to £ if and only if t and t' induce t h e same isomorphism of j£?(F,D) with «^f(F*,D°)

Theorem 2.1 now leads t o t h e required result This completes t h e proof of theorem 2.3

A duality £ of j£?(F,D) which is involutive is known as a polarity, this means i f = |(£(Jf)) for all M A polarity is called isotropic if ilf <= | ( J f ) for all one-dimensional M

Theorem 2.4 Let V be a vector space of dimension n^3 over D Then

J?(V,D) admits an isotropic polarity if and only if D is commutative and

d i m ( F ) is even In this case, if£is an isotropic polarity and 2N = dim(V), we can find a nonsingular skew-symmetric bilinear form (.,.) such that i-(M) = M' for all M Moreover, we can find a basis {x 1 ,y 1 , ,x N ,y N } for V such that (x i ,x j ) = (y i ,y j ) = 0, (x i ,y j )=-(y j ,x i ) = S ij (l^i,j^N) In particular, if £ and £' are two isotropic polarities of3?{V',D), there is a linear automorphism a ofSf( F,D) such that £' = af cr1

Proof Let f be an isotropic polarity of j£?(F,D) B y theorem 2.3 there

exists a n anti-automorphism 6 of D a n d a nonsingular 0-bilinear form

< , ) giving rise t o f Since f is isotropic (x,x) = 0 for all xe V Replacing

x b y x+y we see t h a t < , ) is skew symmetric If now c e D,

<z,i/> c^ = -c(y,x) = c(x,y);

choosing x, y such t h a t (x,y) = 1 we see t h a t 0 is t h e identity This means

t h a t D is commutative a n d < , > is bilinear The remaining statements are standard Theorem 2.4 is proved

W e now consider nonisotropic polarities A 0-bilinear form ( , } is called

symmetric if (x,y) 0 = (y,x) for all x,yeV Notice t h a t this property is relative to 6 We observe t h a t if ( , > is n o t identically zero, 6 is necessarily involutive; t h e n t h e set of values of (.,.) is all of D, and

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

PROJECTIVE GEOMETRIES 25

Lemma 2.5 Let dim( V) ^ 2, ( , ) a nonsingular d-bilinear form such that

(x,x) = (x,x)d^O for some xeV Suppose (.,.) induces a polarity Then this polarity is nonisotropic, 9 is involutive, and (.,.) is symmetric

VrooLIfu,veV9(u9v) = OoB'V^ (D-u)'o(D-u) g (D-v)'o(v,u) = 0

Let c = (x,x) Then x $(D -x)' and dim((D •#)') = % — 1, so that

F = (D-z)0(D-a;)'

If u,ve(D'x)', ((u,v)x — u,x + cv) = 0, so that {x-\-cv,(u,v)x — u) = 0, proving that {u,v)e — (v,u) Now (.,.) is not identically zero on

(D.z)'x(D.z)',

and so this already proves that 9 is involutive The symmetry of {.,.) on

V xV now follows from direct computation Lemma 2.5 is proved

Theorem 2.6 Let 9 be an involutive anti-automorphism of D, V a vector

space of dimension n over D, and (.,.) a nonsingular symmetric 6-bilinear form on V xV Then the duality corresponding to ( , ) is a nonisotropic polarity unless D is commutative and of characteristic 2 Conversely, let n^3 and let £bea nonisotropic polarity o/J§?(F,D) Then there exists an involutive anti-automorphism 9 of D and a nonsingular symmetric 9-bilinear form ( , ) on V x V such that £ is induced by {.,.) If 9' and {.,.)' is another pair,

£ is also induced by them if and only if there is a d^O in D such that for all x,yeV,ceD

(x,y)f = (x,y)d, c6' = d~xced

In this case de — d

Proof Given 9 and ( , ) , the corresponding duality is a polarity If this

were isotropic we can argue as in theorem 2.4 to conclude that D is

com-mutative, 9 = identity, and < , ) is skew symmetric As ( , ) is symmetric

it follows that D has characteristic 2 Conversely, let n^3 and let | be a nonisotropic polarity of J>?(F,D) Then £ corresponds to a pair 90, ( , ) Choose x e V such that (x,x)0 = c0^0 Writing

If (x,y) = 1, then (y,x) — 1 and so {{x,yy)e' — {y,x)' = d while

{{x,yyf' = dd' = d-1ded, also So we have dd = d