Fortschritte der Physik / Progress of Physics: Volume 34, Number 4 [Reprint 2022 ed.] 9783112613689, 9783112613672

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FORTSCHRITTE DER Volume 34-1986 Number 4 PHYSIK PROGRESS OF PHYSICS Board of Editors

F. Kaschluhn A. Lösche R. Rompe

Editor-in-Chief F. Kaschluhn

Advisory Board

A. M. Baldin, Dubna J . Fischer, Prague G. Höhler, Karlsruhe K. Lanius, Berlin J . Lopuszanski, Wroclaw A. Salam, Trieste D. V. Shirkov, Dubna A. N. Tavkhelidze, Moscow I. Todorov, Sofia J . Zinn-Justin, Saclay CONTENTS: A. Amann Observables in TF*-Algebraic Quantum Mechanics E. Sezgin The Spectrum of D = 11 Supergravity via Harmonic Expansions on Si X S1 N. S. Craigie andTheories J . Stern Effective Gauge of Composite W, Z and Massless Fermions

AKADEMIE-VERLAG • BERLIN ISSN 0015-8208

Fortschr. Phys., Berlin 34 (1986) 4, 167-293

167-215 217-259 261-293

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FORTSCHRITTE DER PHYSIK V O L U M E 34

1986

NUMBER 4

Fortschr. Phys. 34 (1986) 4, 1 6 7 - 2 1 5

Observables in W*-Algebraic

Quantum Mechanics

ANTON AMANN

Laboratory of Physical Chemistry, ETH-Zentrum, CH - 8092 - Ziirioh, Switzerland

Summary The problem of the conceptual and formal characterization of observables in quantum mechanics is of central importance. The very name 'observable' rests on the view that one deals with observable, i.e. measurable quantities. This view gave way to a more subtle analysis, involving a distinction between observables and their measurement. Observables are those quantities which characterize a system intrinsically. They are not 'measurable in a naive sense. Often their value is accessible but only indirectly and through extended theoretical reasoning. A careful examination of classical mechanics shows that even there say the position of a particle cannot be determined. Measurements never tell if a particle's site has rational or irrational coordinates, they give merely an interval for it. In quantum mechanics the matter is still more intricate. Independently of the difficulties associated with measurements, one can define observables with regard to the kinematical group of a quantum system (e.g. the Galilei group), namely as 'suitably' transforming operators under an action of this group. For example, operators for position and momentum can be characterized by their transformation properties under Galilei symmetries in both classical and quantum mechanics without appealing to the correspondence principle. Outside this particular situation one gets into difficulties, since the word 'suitably' has then no specified meaning. In the present work these difficulties are eliminated: Every group G acts in a natural way (by translation from the left) on the complex- or real-valued group functions. This action is called Ad A. An operator then is said to transform 'suitably' under a representation a of G if it transforms just as a function / : G -> C under Ad A. In this way observables can be defined with respect to arbitrary (locally compact, separable) kinematical groups. The traditional observables can be shown to fit into this scheme. The general frame of this work is given by W*-algebraic quantum mechanics. Therein a physical system is described by a kinematical group G, a TF*-algebra Jtl and a representation tx of G into the symmetries (automorphisms) of JPl. JT*-algebras have a relatively simple mathematical structure and admit a quantum-logical interpretation. Systems of classical mechanics as well as of quantum mechanics can be dicussed in the W*-formalism. An important part in W*-algebraic quantum mechanics is taken by elementary systems. They are characterized by the ergodicity of the automorphic representation a of the kinematical group. For elementary systems, which are they only ones to be discussed here, the existence of observables in the above group-theoretical sense is equivalent to the integrability of the representation a. Integrable ergodic Mr*-systems are open to structural investigation. The basic W*-algebra of such a system is isomorphic to the tensor-product of a commutative TF*-algebra (a classical part) and a TT*-algebra with trivial center (a quantum-mechanical part). A complete structure- and classification-theory is developed for the ergodic integrable W™-systems of an abelian group and for the ergodic integrable type I JF*-systems of an arbitrary group. 1

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A . AMANN,

Observables

Integrable ergodic Galilei systems (i.e. elementary Galilei systems with observables in the above sense) always admit a time operator. On the contrary, such an operator does not arise within the usual description of classical and quantum-mechanical systems. This deficiency is removed in either case by the introduction of a time observable Í1, which commutes with all other operators.

Contents 0. 1. II. 11.1. 11.2. 11.3. III. 111.1. 111.2. 111.3. 111.4. 111.5. IV. '

0.

Introduction Wr*-algebraic quantum mechanics What are observables? Observables in Wr*-systems of abelian groups Observables in W*-systems of arbitrary groups Integrable W*-systems Structural theory of ergodic integrable JF*-systems A survey The Stinespring construction Classical and quantum mechanical W*-systems Integrable ergodic W*-systems of abelian groups Integrable ergodic W*-systems of type I Galilei systems and the time operator

•

168 171 177 177 179 182 190 190 191 193 194 199 202

Appendix

205

References

214

Introduction

I n quantum chemistry and physics it is a matter of routine to compute expectation values of physical quantities (observables). Nevertheless no entirely satisfactory answer has been found to the problem of characterizing such quantities conceptually. In practice one appeals to Bohr's concept of a correspondence between classical mechanics and quantum mechanics. Therewith certain aspects of molecular systems can be described correctly, but the problem of observables is in no way solved. The correspondence principle does not provide more than a recipe for the solution of numerical questions. Although such a recipe may be useful for practical purposes, it is no substitute for a consistent theory. Despite all the mathematical refinements of quantization procedures the conceptual difficulties of the problem have not been clarified. As long ago as 1927, during the early period of quantum mechanics, observables were investigated from a different point of view. At that time H. Weyl introduced irreducible ray representations of symmetry groups (on a Hilbert space) to characterize quantum systems. In his conception observables correspond to real functions on the group; by use of a particular translation rule they are represented as self-adjoint Hilbert space operators. This procedure abandons the correspondence principle and permits the derivation of spin-operators, i.e. observables which do not arise in classical mechanics. The most important idea taken from Weyl's work was t h a t of using a symmetry group. Reshaped into what is called a kinematical group, it incorporates the idealizations and abstractions of a theorie. As P R I M A S ( [ 1 ] , [ 2 ] ) observed, phenomena and observable facts are only possible by abstractions from 'unimportant' details. What is important and what is irrelevant depends on the respective point of view. Every point of view emphasizes certain properties and neglects others, and thus results in a kind of caricature. The best example of a kinematical group is the Galilei group. The Galilei group is responsible for the abstractions which are concomitant with the introduction of the Euclidian space. I t restricts the possible movements of a "free particle" in three-di-

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mensional space (cf. LEVY-LEBLOND [3], WIGNER [4]) and to this extent determines its kinematics. In a modern approach, however, kinematical groups have gained a far broader significance. They are not at all restricted to describing abstractions in time and space: Chemical kinetics (with the scaling group) and thermodynamics (the respective kinematical group is not yet known) in their respective simplest versions are theories without reference to space and thus without space-time idealizations. Every element of a kinematical group corresponds to an admissible change between 'equivalent' points of view (for the Galilei group this is a change of coordinate system) and is described in the formalism by a symmetry of the relevant structure; in quantum mechanics such a symmetry is a unitary (or antiunitary) mapping of the underlying Hilbert space onto itself. Since the multiplication of group elements corresponds to the product of the respective symmetries, one gets a representation of the kinematical group by symmetries, in particular a ray representation by (anti-)unitary operators. R a y representations arise because states in quantum mechanics are described by Hilbert space rays &nd not by Hilbert space vectors. Let us return to Weyl's idea: He wanted to relate observables to the kinematical group, to introduce 'group observables'. Slightly modified in the details, the essence of this proposal has been preserved: An operator on a Hilbert space is called an observable (cf. (LEVY-LEBLOND [5], [6]) if this operator transforms 'suitably' under a group representation. The best example to illustrate this concept of observables is given by the Galilei group and the (vector-) operators for position and momentum, Q and P. Under rotations Q and P behave like vectors in three-dimensional space, under space translations and pure Galilean transformations (cf. LEVY-LEBLOND [5] and chapter I) they remain invariant up to a multiple of the unit operator. I t turns out (cf. LEVY-LEBLOND [5]) that irreducible (ray) representations of the Galilei group can even be classified according to the existence of position and momentum operators. Those representations admitting operators Q and P are uniquely determined by their mass and spin. On the other hand there are irreducible true representations of the Galilei group 'with mass 0' (s. INONU, WIGNER [7]; VOISIN [8]; LEVY-LEBLOND [5]) which do not admit observables in this group-theoretical sense. The simple transformational properties of the position- and momentum operators have been of vital importance for the development of a group-theoretical treatment of the problem of observables and for the renaissance of the Galilei group itself. A formal scheme has been developed — the imprimitivity systems — which allows the investigation of ray representations with simply transforming operators (such as Q and P). The mathematical aspects have been developed by MACKEY ([9]), whereas the physical aspects are largely covered by the work of Jauch and Piron (cf. JAUCH [10], PIRON [11]). As an underlying structure, Jauch and Piron do not primarily use a Hilbert space, but rather the set of its closed subspaces. Those subspaces (or their respective projections) form a lattice, i.e. a partially ordered set, which contains the infimum and supremumof any two of its elements. The lattice of subspaces of a Hilbert space contains atoms (the one-dimensional subspaces) and is the starting point of what is called quantum logics. There one has complete and orthomodular lattices (propositional systems) as fundamental ingredient for the description of physical systems. Quantum mechanical as well as classical systems fit into this scheme (the latter correspond to Boolean lattices). The kinematical group is no longer represented by unitary operators on a Hilbert space, but by symmetries of the lattice (i.e. mappings which preserve its structure). Appropriately modified, imprimitivity systems can be formulated in arbitrary lattices. Thus, within quantum logics, a deeper understanding of the correspondence principle is possible: The similarity between classical and quantum mechanical systems of a certain kinematical group rests on the existence of lattice elements with the same transformational properties. 1*

170

A. AMANN, Observables

Besides quantum logics different ways have been found to include classical and quant u m mechanical systems in a common structural setting. Here a central position is taken by algebras. Elements of an algebra can be added together or multiplied with each other (or with scalars). The simplest algebra is formed by the complex numbers C. Contrary to arbitrary algebras C is commutative, i.e. the order of multiplication does not play a role. "Observables" of a classical mechanical system (the functions on its phase space) as well as of a quantum mechanical system form an algebra. A common example in quantum mechanics is the algebra of 2 X 2-matrices for the description of the spin of an electron. Algebras can be handled more easily than lattices of quantum logics, but they do not have the same axiomatic foundation. However, within a certain class of algebras, namely TF*-algebras, the advantages of quantum logics can be combined with those of the algebraic theory. The projections of a JF*-algebra (i.e. elements p of the algebra with p 2 = p = p*) form a complete orthomodular but in general non-atomic lattice. This lattice can be interpreted as the proposition lattice of a physical system in the sense of quantum logics (cf. P R I M A S [ 1 2 ] , [ 2 ] ) . I t is Boolean if and only if the PT*-algebra is commutative, i.e. classical systems correspond to commutative JF*-algebras. The symmetries of the projection lattice can be extended to (Jordan-)automorphisms of the algebra. Thus in the PF*-formalism a physical system with the kinematical group G is described by a PF*-algebra and a (Jordan-)automorphic representation of G. I t is within this scheme t h a t we will again take up the problem of observables. The simple transformational properties of position and momentum operators have been mentioned above. Unfortunately, observables of this type are exceptional. I n more difficult cases — such as angular momentum and spin — one often uses Lie structures defined on the algebras of classical and quantum mechanics by the Poisson bracket and the anticommutator, respectively. Since kinematical groups are usually Lie groups, it seems natural to seek a representation of the corresponding Lie algebra and to introduce observables in this way. If possible, representations of the Lie group and of the Lie algebra are used. The connection between those two sorts of representations is given by infinitesimal generators of the one-parameter subgroups of the Lie group. This programme can be put into effect for the Lie relation [Q, P ] = 1 as well as for the rotation group with the Lie relations [£;,£,] = e ¿ , A ,

i,j,k

6(1,2,3}

(e is the antisymmetric tensor) (cf. F A L K [13], [14]). I n other cases, however, one soon runs into difficulties. For example, the 'mass 0' irreducible representations of the Galilei group on a Hilbert space give rise to a (Lie-)representation of the Lie algebra of the Galilei group, but here one cannot identify observables. However, infinitesimal generators of (represented) one-parameter subgroups of a group appear to be observables provided the respective systems has observables in the sense of suitable transforming operators. Thus, the Lie structure may be helpful, but it is certainly no solution to the problem of observables. As far as the group properties alone are concerned, i.e. neglecting the Lie structure, the following observation is useful: Every group G acts in a natural way on the complex-valued group functions by (Ad%)(f))(S)=f(g-iS),

g,s e G,

(/ : G > C is a complex-valued function on G).

Fortschr. Phys. 84 (1986) 4

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For the special case of the Galilei group the transformational properties of Q and P can be realized by simple (real-valued) functions on the group (see chapter I). I t is not difficult to extend this procedure to more complicated situations1). We shall therefore call an operator (or a set of operators transforming among themselves) observable(s), if there are functions on the respective kinematical group which transform in the same way. Thus the proposal of H. Weyl — namely that observables correspond to functions on the group — can be combined with the idea that observables have to be characterized by their transformational properties. This paper aims to give a well-defined formal meaning to the concept of observables as 'suitably' transforming operators (chapter I I ) and to investigate and classify the structure of systems with observables (chapter I I I ) . In an introductory chapter the TF*-formalism and its connection with classical and quantum-mechanical theories is presented. The final chapter contains a discussion of Galilei systems and time operators in them.

I.

TF*-algebraic Quantum Mechanics Preliminary remarks

Different ways have been taken to widen the original quantum mechanical formalism and to embed quantum mechanics and classical theories into a broader structural setting. They can be summarized under the key-words — quantum logics — algebraic quantum mechanics — "convex state approach". All three have in common the abandonment of von Neumann's irreducibility postulate. This postulate — characteristic of pioneer quantum mechanics2) —denies the existence of classical observables (superselection rules), i.e. of physical quantities which commute with all other operators of the respective system and which have a sharp value (unchanged by measurements) in every admissible state of the system. Examples of classical observables are charge and mass of particles, chirality and nuclear frame of molecules, and temperature and chemical potential of substances. In quantum logics (cf. BIBKHOIT, VON NEUMANN [ 1 5 ] ; JAUCH [ 1 0 ] ) the underlying structure is given by an orthomodular lattice, in algebraic quantum mechanics (SEGAL [ 1 6 ] ; HAAG, KASTLER [ 1 7 ] ; PRIMAS [ 2 ] ) by a Jordan-, 0 * - or JF*-algebra and in the "convex state approach" (cf. L U D W I G [ 1 8 ] ; D AVIES, L E W I S [ 1 9 ] ) by a convex space. None of those structures will be given precedence merely on grounds of fundamental considerations. After all, there exist procedures for translating from one 'language' into another. Situations which can only be described in one of them are rather artificial and without importance for physics and chemistry. Another point of view is decisive: A physical system is not fully determined by a given lattice, an algebra or a convex space. An orthomodular lattice, for example, only informs us about the logic of a system, e.g. if this logic allows classical properties, if it is Boolean or not. Additional structure is very often obtained group-theoretically by considering a representation of the system's kinematical group. I t is then especially attractive to proceed the other way round, i.e. to determine and classify all possible realisations of a fixed group. 1)

That spin operators f i t into this scheme and can be uniquely characterized by their transformational properties will be shown in a forthcoming paper. 2 ) Here this term refers to the quantum theory of 1930 (cf. PRIMAS [2]: Sect. 1.3).

172

A. AMANN, Observables

Up to now it is impossible to study group representations on lattices or convex spaces. In contrast to this, representation theory of groups on algebras, especially TF*-algebras, has been taken up successfully by the mathematicians (cf. OLESEN, PEDERSEN, TAKESAK I [ 2 0 ] ; ALBEVERIO, H O E G H - K R O H N [ 2 1 ] ; H O E G H - K R O H N , L A N D S T A D , STORMER [ 2 2 ] ;

DESCHREYE [23]). This is why we choose W*-algebras as fundamental structure. In the following it will be shown how to embed classical mechanics and pioneer quantum mechanics into the JF*-formalism. A comprehensive review of the use of W*algebras in chemistry and physics and its connection to quantum logics and the "convex state approach" is given by PRIMAS ( [ 2 ] ) . The mathematical aspects are treated in the monographs by DIXMIER ( [ 2 4 ] ) , S A K A I ( [ 2 5 ] ) and TAKESAKI ( [ 2 6 ] ) . JF*-systems In TF*-algebraic quantum mechanics a physical system with the kinematical group G is described by — a TF*-algebra d i and — a mapping « : ( ? — > Aut ,M of 0 into the automorphism group Aut JH of the •-algebra Jl with the properties: ( i ) txtl o rxgz =

>xgm,

(a is a r e p r e s e n t a t i o n )

gu g2eG

(ii) For every operator x € JH, the function {G 9 g > ocg(x)} is continuous with respect to the c-weak topology a(