classical and quantal systems of imprimitivity

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Classical and quantal systems of imprimitivity N. Giovannini Citation: Journal of Mathematical Physics 22, 2389 (1981); doi: 10.1063/1.524821 View online: http://dx.doi.org/10.1063/1.524821 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/22/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Square-integrable imprimitivity systems J. Math. Phys. 41, 4833 (2000); 10.1063/1.533382 Frames from imprimitivity systems J. Math. Phys. 40, 5184 (1999); 10.1063/1.533024 Nontransititive imprimitivity systems for the Galilei group J. Math. Phys. 31, 1859 (1990); 10.1063/1.528683 Classical and quantal pseudoergodic regions of the Henon–Heiles system J. Chem. Phys. 88, 5688 (1988); 10.1063/1.454529 Markov systems of imprimitivity J. Math. Phys. 18, 1832 (1977); 10.1063/1.523497 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 138.251.14.35 On: Thu, 18 Dec 2014 16:02:44

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Page 1: Classical and quantal systems of imprimitivity

Classical and quantal systems of imprimitivityN. Giovannini Citation: Journal of Mathematical Physics 22, 2389 (1981); doi: 10.1063/1.524821 View online: http://dx.doi.org/10.1063/1.524821 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/22/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Square-integrable imprimitivity systems J. Math. Phys. 41, 4833 (2000); 10.1063/1.533382 Frames from imprimitivity systems J. Math. Phys. 40, 5184 (1999); 10.1063/1.533024 Nontransititive imprimitivity systems for the Galilei group J. Math. Phys. 31, 1859 (1990); 10.1063/1.528683 Classical and quantal pseudoergodic regions of the Henon–Heiles system J. Chem. Phys. 88, 5688 (1988); 10.1063/1.454529 Markov systems of imprimitivity J. Math. Phys. 18, 1832 (1977); 10.1063/1.523497

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Classical and quantal systems of imprimitivity N. Giovannini Departement de Physique Thiorique, Universite de Gene've, 1211 Gene've 4, Switzerland

(Received 23 April 1981; accepted for publication 5 June 1981)

We discuss from a group-theoretical point of view, a simple framework in which the classical and the quantal state spaces appear in a unified way. The framework is characterized by the use of (possibly continuous) direct unions of Hilbert spaces. Symmetries are correspondingly represented by families of (anti-) unitary operators. We consider here the associated notions of projective representations as well as the corresponding observables. These observables, classical or quantal, are defined in terms of a slight generalization of the notion of a system of imprimitivity of Mackey.

PACS numbers: 02.20. + b, 03.20. + i, 03.65.Bz

INTRODUCTION

There exists a simple way for describing the classical and the quantal state spaces of an elementary physical sys­tem in a single mathematical framework. It is to consider (topological) direct unions of Hilbert spaces. Classical phys­ics, where a state is usually given by a point in the phase space, corresponds then to the extreme case where each Hil­bert space is of dimension one and the union is precisely taken over this phase space. Usual quantum physics, on the other hand, corresponds to the other extreme case, where the union is trivial and the space is thus a single Hilbert space. More generally and in the intermediate cases, such a frame­work is able to describe situations where parts of a system have a quantal behavior, whereas other aspects are of the classical type. Think for example of superselection sectors, or of a spin-spin interaction in a crystal (with fixed positions and momenta) or of a measurement procedure of a quantal system with a classical device.

On the other hand, the fact that we can compare differ­ent models (classical with quantal and, as we shall see, rela­tivistic with nonrelativistic in particular) in a unified math­ematicallanguage is of course of great importance for the understanding of the results and of the difficulties encoun­tered in each separate context.

In a previous work, I we have shown that, conversely, one can derive the classical and quantal state space, for a spin less particle, in a unified way, when considering these direct unions as representation spaces for the corresponding kinematical symmetry groups. More precisely we consid­ered those representations for which there exist sufficiently faithful operators for each observable which corresponds to the system. This construction is based on the following brief­ly sketched points of view.

The state of a physical system is specified by the possi­ble outcomes of the measurements that we can choose to perform on it. To these measurements correspond the obser­vables, and to the outcomes correspond (subsets ot) the spec­tra of these observables. Let us consider the space r of all spectra of the observables that correspond to some given system. A typical example of such a space r is the phase space, with the observables position and momentum. On this

space r, the usual physical equivalence principles corre­spond to a natural action which forms the defining represen­tation of a group. We have called the corresponding groups kinematical in order to emphasize that they are independent by definition of the dynamics and of the interactions, as the latter neither changes the representation nor the interpreta­tion of the observables. It is only in a second step, when the state spaces have been specified, that it will be possible to consider the corresponding dynamics. This distinction is es­sential if we want to build up theories able to describe more than only free particles.

One of the main results of Ref. 1 was that both in the relativistic and nonrelativistic contexts there are two and only two solutions for the corresponding state spaces: the classical and quantal ones. The framework provides thus also by the way a new approach to quantization.

In order to generalize these models, in particular if we want to consider particles with an arbitrary spin or for the discussion of the nonrelativistic limit of the relativistic case we, however, need to consider a more general framework where arbitrary projective representations on such direct union spaces are considered. A first step in this direction was the solution of the corresponding co homological problem (families of phase factor systems). This solution has been giv­en in Ref. 2.

In the present paper we continue to analyze the math­ematical basis of our approach, and we give the general solu­tion for the state spaces and for the associated observables, the former in terms of group representations and the latter in terms of a generalization of the notion of systems of imprimi­tivity of Mackey.3 We have also taken advantage of the pre­sent generalization for rewriting in a more direct algebraic way some of our previous results. I

The paper will be organized as follows. In the first part we briefly recall some axiomatic justifications of our ap­proach and then discuss the structure of the state spaces in terms of carrier spaces of group representations. In the sec­ond part we give the general solution of the representation problem and in part three we discuss the associated observ­able representations. Finally in part four we illustrate our results at the hand of a simple representative example. The main applications of the theory will be discussed in a sepa­rate paper. 4

2389 J. Math. Phys. 22(11). November 1981 0022-2488/81/112389-08$01.00 @ 1981 American Institute of Physics 2389

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1. K SPACES AND PROJECTIVE K REPRESENTATIONS

A K-space is defined as being a direct union over some (Borel) index set S of a family of (isomorphic) Hilbert spaces (that will be assumed here to be complex and separable)

K = V cW's. s55

(1.1)

The elements of K (respectively, the pure states) are thus given by some index soES and a corresponding vector (re­spectively, a ray) in cW'so'

The choice of( 1.1) as a possible state space for a physical system can be justified by a pragmatic argument (see the Introduction), but also from an axiomatic point of view, which is interesting to briefly mention here (for details we refer to Ref. 5). On K there exists, namely, a natural so-called lattice of propositions denoted by ..?(K) and defined as fol­lows. A projection in K is a family of projectors (Ps I in the corresponding Hilbert spaces; in the set of all projections one can define a lattice structure using the lattice structure which is based on the operations of unions and intersections of closed subs paces in each single Hilbert space. More pre­cisely the inclusion, compatability, complementary, and in­tersection in 2'(K) are defined in turn by

(i) {P, I < {Qs I iff Ps < Qs '<is iff PsQs = Ps '<is,

(ii) (Psl+-+(Qsl iffPs+-+Qs '<is iff [P" Qs] =0 '<is,

(iii) (Ps )' = (P; I = {I w, - P,},

(iv) {Ps }/\ {Qs I = (PJ\ Qs I = {sn-~~(psQsr}. (1.2)

The above algebraic structure allows application of the notion of morphism in a straightforward way and it follows from the representation theorem of Piron5 that (up to small technical restrictions) any lattice of propositions corre­sponding to an elementary physical system is isomorphic to such a 2' (K). In other words, the space (1.1) is then indeed the most general state space for the corresponding system, and the projections (1.2) then correspond to the properties of the system. We refer to Ref. 5 for a more precise review of these aspects.

We can now define the notion of symmetry in this framework.

Definition 1.1: A symmetry in K is an automorphism of the corresponding lattice of propositions, i. e., an invertible map of Y(K) onto itself that preserves its algebraic structure.

The essential result that we shall need in the sequel is the following generalization of the theorem of Wigner.

Theorem 1.2 (Ref.6): Every symmetry of a proposition system defined by a family (cW's,sES I is given by a permuta­tion 1T of the index set S and a family of unitary or antiunitary operators

UrrjS): dY's-cW'rrjs)' (1.3)

Moreover, each Us is unique up to a phase.

It follows directly from this theorem that for each pair of elements g., g2 in a group G of symmetries we have the following generalized projective representation relations:

U,( g.)U1T, '(s)( g2) = w,( g.,g2)U,(g.g2)'

2390 J. Math. Phys., Vol. 22, No. 11, November 1981

where w s ( g.,gz)EU(I) is a phase that may depend ong., gz, ands.

As a consequence of the above theorem we are thus led to the following.

Definition 1.3 (Ref. 2): Aprojective K-representation ofa group G in a space K is a set of automorphisms U (g):K-K that satisfies the following four conditions:

(i) U(e) = lK' e the unit of G, (1.4)

(ii) U(g.)U(gz)=J1(g.,gz)U(g.g2) (1.5)

with J1 a family (ws I of c-numbers oflength 1, J1:G X G_(U(I))S, its action on an element lfIofKbeing giv­en by (J1 (g.,g2)1fI) s = ws( g.,g2j(1fI Is, (iii) U (g) can be decomposed as

U( g) = L (g).~O( g), (1.6)

whereL(g) = (Ls(g)J isa family of(anti-) unitary operators acting in each dY's' i. e., L,( g)EU&(cW',) and ~O( g) is a permu­tation operator acting (only) on the variable s:

(1.7)

We denote gs, by a slight abuse of notation, as the image of s under the permutation induced by g on S.

(iv) U ( g) is continuous in g on S X cW' (with for cW' the strong topology).

It follows from the above definition that L, as a map­ping on S X G, satisfies the following equations:

(i) L,(g.)Lg , ,,(gz)=w,(g.,g2)Ls (g.,gz),

(ii) L,(e) = 1 w,' '<isES, ( 1.8)

i. e., L satisfies (projective) normalized G-S-~(dY')-cocycle equations,7 and the problem can thus be tackled using cocy­cle techniques.

Another important consequence of the above definition is that Sis then a G-space in the sense ofMackeyK: symmetry products being defined by composition we indeed can define a natural antihomomorphic map p from G to the automor­phisms of S, with p( g)s = g-·s and this map p satisfies, as required,

(P( g.)p( g2))(S) = p( g.)(tu( g2)S)),

p( g.)p( gz) = p( gzg.),

p(e)s = s, '<isES. ( 1.9)

In general S thus splits under the action of G into dis­joint orbits but in the particular case where the action of Gis transitive, S can be identified with a quotient space S=G I H, with H = Stab So, the (closed) stabilizer of an (arbitrary) point soES.

Definition 1.4: A projective K-representation is called irreducible if and only if

(i) the action of G on S is transitive,

(ii) the setL,,, (H), soES arbitrary, H = Stabso is an irreducible projective representation of H in ,JY,,,.

Obviously this definition does not depend on the choice of So and it follows by restriction from Definition 1.3 that L,,, (H) is always a unitary lantiunitary representation of H in (~s()·

N. Giovannini 2390

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Defining the commutant as the set of commuting fam­ilies lAs J , A s EsP (£'s ), it is easy to see that a representation is irreducible if and only if the commutant is trivial (the proof is as in Ref. 1).

Similarly one can define equivalent representations as representations intertwined by an automorphism of K, i. e.,

Definition 1.5: Two projective K-representations U and U' of G are called equivalent if and only if there exists a (Borel) isomorphism r:S-.S' in the corresponding base spaces and a family of unitary operators V = ! Vs ' sES I, r{:£'s-'£'r(sl such that 'VgEG

U'(g) = V.U(g).V-I. (1.10)

The first problem is the one of the possible families of phase factors which may occur in (1.3). Analogous to the usual case it follows from the associativity that they are not arbitrary but satisfy the following generalized comultiplier equations:

where 7]( g) denotes the complex conjugation in the case where U ( g) is a family of antiunitary operators. In Ref. 2 we have investigated in detail the solutions, called K-comulti­pliers, of these equations and we have proven the following:

Theorem 1.6: Let U be an irreducible (unitary) projec­tive K -representation. Then the K -multiplier n is equivalent to a K-muItiplier n' given by

U);(gl,g2)=U\,(v(s,gd, v(gils,g2))' (1.12)

with soES arbitrary, H = Stabso, and vis, g) is the (unique) element of H defined by

vIs, g) = k (s)·g·(k (g-IS))-I, (1.13)

where k (s) is, for each s, the (fixed) representative of the unique coset class in G I H satisfying the equation k (s) -I.SO = s. Conversely each equivalence class [U) JEN 2(H, U (1)) gives rise, via the same formula (1.12) with U)su = U), to one and only one equivalence class of solutions en] of(1.11).

For more details on this cohomological problem we re­fer to Ref. 2. In words, the above theorem says that n is, up to equivalence, uniquely defined by its double restriction from S X G to So X Stabso, for some arbitrary soES, and that conversely each solution of this double restriction gives rise to a solution n. Moreover this correspondence preserves the equivalences.

Finally let us briefly show how the so-called lifting pro­cedure') generalizes, i. e., how it is possible to construct a group G fl in such a way that the projective K-representa­tions of G with K-multiplier n can be lifted to ordinary (vec­tor) K-representations of this larger group G n. We therefore consider G as given, K = VsdY's' and U some irreducible projective K-representation of G with n arbitrary but fixed. The group G {) is defined as follows. It has elements

(CP, g), CPE[ U(l)]S, gEG, (1.14)

i. e., cP is a family of phases, cP = 1477, l. sES, 477sEU(l). The product in G fl is given by

(CPI' gtl(CPz, g2) = (CPl'; (gdCP2·n (gl' g2)' glg2), (1.15)

2391 J. Math. Phys., Vol. 22, No.11, November 1981

where the product in (U (1))5 is defined term by term and the action; is given by

;(g)CP = U(g)-<1>.U(g)-1 (1.16)

or, equivalently, as can easily be computed, and in compo­nents, by

(1.17)

with 7]( g) as in (1.11). Using the K-comultiplier equations (1.11) it is easy to

see that this product is associative. Together with a unit ele­ment (lK,e) with e the unit ofG, G n forms a group and the projective K-representation U defines an ordinary represen­tation V of G [} with

def

V(CP, g) = cp·U( g). (1.18)

Conversely, given any (vector) K-representation Vof G n

which satisfies

V(CP,e) = CP, (1.19)

one can construct a projective K-representation U of G with

U( g) = V(lK' g). (1.20) Obviously, G {) appears as an extension with the normal sub­group of phases in general not in the center, even if all L, ( g) in (1.6) are unitaryCthis is in opposition to the usual case)

0-.[ U (I)P-.G [}-.G~l, (1.21)

the extension being characterized by the factor system nand the mapping; of G in the automorphisms of (U (l))S

Similarly to the usual case, 10 if the mapping S-.CP, which can be considered as a cross section in the bundle of base S and fiber U (1), is supposed to be continuous for all cP and if G is continuous then G fl is continuous. Analogously if these cross sections are measurable and if G is a measurable group then G {) is measurable, too.

2. PROJECTIVE KREPRESENTATIONS

In this section we shall, in a first part, construct explicit examples of solutions for Eqs. (1.3)-( 1.7), i. e., explicit exam­ples of projective K-representations. In a second part we shall then show that this construction is exhaustive, i. e., we shall give the general solution of this problem.

Thus let G and S be given. As we are interested in ele­mentary systems, we may suppose without loss of generality (see Definition 1.4) that the action of G on S is transitive, i. e., S=G I H, H = Stabso, soES. Let £ denote this action.

Furthermore, let D be some projective unitary repre­sentation of H, with multiplier U), in some carrier space dY'(D). Then letKbe the direct union OYer S of copies of this space.

K = V, (,/r(D )),. (2.1)

On this space K we define the following operators 'VgEG:

U(g) = L (g).£(g), (2.2)

where L (g) = I L,( g)) and

L, (g) = D (v(s, g)), (2.3)

with vis, g) as given in (1.13). Let us just remark here that these formulas are formally the same as in the nonprojective

N. Giovannini 2391

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case, I this in contradiction to the usual inducing procedure for projective representations in Hilbert spaces where addi­tional phase factors need to be inserted.9

It follows straightforwardly from (2.3) that

L,( gl)Lg , 's( gz) = UJ(v(s, gl),v( gl- IS, gz))D (v(s, glgz)), (2.4)

where we have used the G-S-H-cocycle property7 of vIs, g),

vIs, gtlv( glls, g2) = vIs, gjg2)' (2.5)

Comparing (2.4) with (1.8), using (2.3) again, and applying the last part of Theorem 1.6, it is now easy to see that (2.2) and (2.3) defines a projective K-representation in the space (2.1) and with K-multiplier fl given by

(fl (gj, g2)), = UJ(v(s, gl),V(gl- IS, g2))' (2.6)

The above construction is of course nothing else than a gen­eralization of the inducing procedure of Mackey . 8 The corre­sponding representation will therefore be called aK-induced representation. The important result is now the following converse.

Theorem 2.1: Let Ube any irreducible projective K­representation with K -multiplier fl, then U is equivalent to a K-induced representation.

Proof It follows from the irreducibility that we may assume without loss of generality that S~G / H. Let us choose a set of fixed right coset representatives k (S)EG, sES. Then trivially with (1.13) eachgcan be written as

g = k (s) -I.vis, g).k (g-IS), 'VsES

and correspondingly the unitary operators in the families L = ! L, ( g) J, defined by the given representation U, can be written

Ls( g) = L,(k (S)-I.V(S, g).k (g-IS)). (2.7) We may expand this expression, using (1.8) (i) iteratively, and we find

L,( g) = UJ,- I(k (S)-I ,vIs, g)k (g-IS))'W':;- I (v(s, g),k (g-IS))

'UJs" (k (s),k (s)-lL ,-:: I(k (s)).Lso (v(s, g)).L,o (k (g-IS)).

(2.8)

Using Theorem 1.6 we may also assume without loss of gen­erality thatfl is already in the form (1.12) and normalized (cf. Ref. 2). Using now (1.13) and (l.l2) withg l = k (S)-I one easily finds that, 'VgzEG

UJ,(k (S)-I, gz) = UJ s" (k (so), g2) = 1,

where we have used the normalization of fl. The other phases in (2.8) can be seen to be trivial by similar straightfor­ward calculations so that (2.8) reduces to

L,( g) = (Ls,,(k (S)))-I Ls,,(v(s, g))Ls,,(k (g-IS)). (2.9)

Then using (1.10), with S' = S and which then reads, in components,

(VU'(g)V-I), = V,U,(g),vg '"

one recognizes that, with V = I Vs I and def

V, = L,,, {k (s)),

(2.10)

(2.11)

the operators (2.9) can be brought back to the form (2.3), as

2392 J. Math. Phys., Vol. 22, No. 11, November 1981

the equivalent G-S-~ (.JY')-cocycle then reads

L ;( g) = L,.(v(s, g)) (2.12)

and this achieves the proof of the theorem. This theorem thus implies that the problem of classify­

ing all irreducible projective K-representations of a group G is reduced to the problem of the usual projective representa­tions in Hilbert spaces of some closed subgroups H of G. In the next section we shall see which subgroups are relevant for our purposes.

3. SUPERSYSTEMS OF IMPRIMITIVITY

As in Ref. 1, and for a given physical system with a given kinematical symmetry group G, not any K-space car­rying a representation of G will be considered as a possible state space for the system. We shall in addition require that, in K, the observables are faithfully represented, in a sense that we now want to make precise.

As mentioned in the Introduction, the state of the sys­tem is characterized by the possible outputs of the various measurements that we have chosen to eventually perform on it. We thus have a priori a given set of observables !A j. Let r A denote the set of possible values for the observable A, i. e., the spectrum of A. We now demand that, for each (Borel) subset..1 of r A there exists an operator P ~ in K which selects the states having in actuality the property of having, for A, a value within..1. It follows from the representation (1.2) of the properties in K, that P~ should be a projection in K, i. e., a family of projectors in the corresponding family of Hilbert spaces

P~ = {(P~)J, (P~),E.o/(£'.). (3.1)

Furthermore, it should satisfy the following measure theoretical properties:

(i) P: = OK' P~A = lK'

(ii) P~,n4, = P~,Pt,

(iii) P1e.Jrll, = .2l~, for ..1 i l..1 j if ijj (i. e., disjoint)

lET

and I is countable. (3.2)

In (3.2) the sum and the products are meant as sum, respec­tively, products of the individual projectors in the families. On the other hand it is also by its action on r = ! FA I that the group G is defined. We thus correspondingly require these mappings to transform covariantly, i. e.,

(3.3)

In the particular case where S is a singleton [hence K is a (single) Hilbert space], the conditions (3.2) and (3.3) are noth­ing else than the conditions of systems of imprimivity of Mackey? In analogy with this terminology we shall call in general such mappings supersystems oJimprimitivity (s.s.oj.) as in Ref. 1.

Example: Let ..1E,q(J (S), the Borel sets of S, and X Il the corresponding characteristic function. Then

(3.4)

obviously satisfies (3.1) and (3.2). Moreover, using (2.2) one

N. Giovannini 2392

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directly finds

U(g)X.d U(g)-l = 5'(g)·X.d·5'(g)-l

=Xg.d' (3.5)

hence (3.4) also satisfies (3.3) and thus defines a supersystem of imprimitivity.

We thus see with this simple example that S always corresponds to one (or more) observable. On the other hand, we have seen that it can also be associated with an homogen­eousspace G IH ofG. This is nothing but a particular case of a more general relationship between observables and homo­geneous spaces that we shall also detail in the sequel. .

Now let K and Ube given. Because we want to consider elementary systems we may assume without loss of general­ity that U is irreducible. Let then P be some s.s.oj. based on the Borel subsets of some G-space T. We may also assume that the corresponding action is transitive (or else P would define more than one observable). Hence T can be identified with some quotient space G / H l' with H 1 = Stabto' to some (arbitrary) element of T.

On the other hand, if P defines an observable, T corre­sponds to some subset of r. The relationship between both is easy to recognize: HI ~Hy, where r is some fixed value ofthe observable corresponding to P and Hy is its stabilizer.

We thus have, V FEffJ (T), projections P Fin K satisfying (3.2) and (3.3); hence, in each Hilbert space cW's we have

(il P~ = O;r" P '[ = 1*,','

(ii) p~,rJ', = P~IP~', (iiij(PU,F,)s = IP/i, F;jfj for ii=j,

i

as well as the covariance condition

(U(g)-lPFU(g))s =P~ 'F.

(3.6)

(3.7)

These last equations can be rewritten, using the decomposi­tion of U in (1.6), as

Ls(g)-IP~LAg)=P~:~, (3.8)

where we have for simplicity omitted the imbedding maps of cW's in JY'gS induced by 5' ( g).

Consider now the double restriction of (3.8) to some elements soES and for the group elements hEll ~ G with H = Stabso' We obtain

(3.9)

As (3.6) is true for each s, hence for So, we th us have, together with (3.9), that this restriction, i. e., the set! P ~,FEffJ(T) 1 is, in cW's", a usual system of imprimitivity of Mackey for H on T. In order to apply the theorem ofimprimitivity we. howev­er, need one more property, namely the transitivity of the action of the group. But this property is in general lost in the above restriction. In order to get out of this difficulty we have to analyze in some more detail the action of H on T.

Under the action of H, the space Tsplits in general into disjoints orbits! TI-" /-tEM, with M some index set j. It is easy to verify that this set M can be identified with the set of H:H I

double cosets: each t can indeed be written as I (t ).to, to fixed, I (t) some coset representative ofG IH I • hence t and t' are in the same orbit if I (t ).to = h·1 (t ')-to for some h IElll' i. e., iff

2393 J. Math, Phys .• Vol. 22, No. 11, November 1981

I (t) = h·1 (t ').hl'

i. e., iff I (t) and I (t ') are in the same H:HJ double coset. Then let tl-' ET

Ji, some It. and H t = Stab/l" then one

easily recognizes that TI' ~H /HrJ/t· Proposilion 3.1: The coset space H ~ I HrJ/ ~ can be

identified with a (closed) subset St of S, i. e., one can write

H~ = u,,(H~rJ/).k'(s). (3.10) .SES I

where k 'Is) are some coset representatives of G / H. Proof: Let s be arbitrary, tl' fixed in TI" and H ~

= StablJi' If k (s)ll'tTI-' then no heR can bring i~ back in r;, because the orbits are disjoint. Hence the equatIOn h·k (s)t = t has no solution h. If, however, k (s)tl' ETI-' then I' I' , k (S)I = ht for some h, hence (h - k (s))Ellli. We may cho~eh - t'k (s) as new coset representative k '(s) and obvious­ly (H~rJ/).k '(s)<;;;;'H~. Denoting by S~ alIsEG /H, with this property we thus have

u (H~rJ/)k '(s)~Hli. S€S1~

Con versel y, each h I Ell It can be written as h·k (s) and as h·k (s)tl' = tl-' one hassES~ hence finally the result asserted in (3.10).

Then let G denote the stabilizer of the whole orbit TI" '" . Obviously G -:JH and we have the followmg. Propositio~ 3.2: The coset space G

I/ H can be identified

with a (closed) subset SI' of S, i. e.,

GI-' = us~Hk (s), (3. I 1)

whereS~dSI" VI'EM. Proof: gEG I' implies h.gEG 1'. Conversely if k (s)"G I-' for

some s then there exists at least one tE TI-' such that k (s)tiTI' . But then h·k (S)II' ir;" 'tf heR because the orbits are disjoint, hence there can be no element of GI' in this coset. The condi­tionsESl'meansh·k (s)EGI', 'tfhEll, henceh·k (s)tl-'ETI" There thus exists an element h 'eR such that h·k (s)·tl, = h '.tl' and (h r 'h·k (s)Ell~; hence sESt·

We may now come back to (3.8), the restriction at some So, for H = Stabso of the arbitrary but given s.s.oj. P. In order to recover the transitivity property we restrict it fur­ther to the Borel subsets of T", :

Ps" :F" ---...9 (,7/"s..J, FI' E,W(TI') ~ ffJ(T). (3.12)

This restricted mapping still satisfies the covariance condi­tion (3.9) and two of the measure theoretical conditions (3.6) (ii) and (3.6) (iii). It does, however, not necessarily satisfy the first one, as for FI' = Tp. the image of(3.12) is not necessarily the identity.

Proposition 3.3: Let So be arbitrary but fixed and suppose F is some H-invariant Borel subset of T, then

P~, =A·l w ,. with A =0 or A = 1.

Proof: It follows from (3.9) and the H-invariance that P ~ commutes with Lsn (h ). 'tf heR. But as U is irreducible it foHows from Definition 1.4 (ii) and the lemma of Schur that P ~ = AI:.!',,' That A = 0 or 1 follows from the fact that P~, is a projector [(3.6) (ii)].

N. Giovannini 2393

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Page 7: Classical and quantal systems of imprimitivity

In particular for the orbits Til- one has thus

P;:=AIl-IK", AIl-E{O,I}. (3.13)

Proposition 3.4: There exist one and only one 11-0eM such that (for So fixed) All-o = 1.

Proof Suppose this equation holds for at least two (dis­joint) orbits Til- and TIl-" Then from Proposition 3.3, as

T" uT,,' is H-invariant and from (3.6) (iii) we find that P :,~UT~ = (A" + A,,' )1$ . As A" + Au' is either ° or 1 Au and A , " ~ r"

cannot be both equal to 1. On the other hand, if all AI' = 0, then P =0 and it cannot thus satisfy the s.s.oj. conditions. Hence there is exactly one orbit T"o such that A"o = 1.

Conversely, for a given orbit T"o there may be, howev­

er, more seS for which P ;". = 1 K,'

Proposition 3.5: The set Sil-o for which P ;'" = l.,y, is the same as the set SIl- in Proposition 3.2, for 11- = 11-0'

Proof As follows from (2.3) and from Theorem 2.1, we may assume without loss of generality that Ls (k (s)) = 1£. so that, with (3.8), P; = p~,(s)F, VseS, FeP4(T). Using now" Proposition 3.5 we thus have

P; = lw, iff k(s)F= T"o' (3.14)

hence, withF= TILo ' (3.14) reads

P T", = 1.. iff k (s)T = T S )¥s Jl-o i1-0'

i. e., k (s)eG"o as asserted. It now follows from (3.13) and Proposition 3.4 that Ps

restricted, as in (3.12) on the orbit T"o determined by this 0

Proposition 3.4, does satisfy all usual axioms of an s.oj. for H = Stabso. As the action is now transitive Tl'o S!!H f H' where, as is easy to compute,

H' =H~°nH.

Collecting all these results we now have proven the following.

(3.15)

Theorem 3.6: Let P be a (transitive) s.s.oj. based on TS!!G fH I, for the irreducible projective K-representation U of G, then P is given, up to equivalence, by

pF = pkls)F = p1kls)F)nT"" .'\ "'0 So '

(3.16)

where 7;", is the unique H-orbit in Tfor which P ~" = 1 w,.'

Moreover the mapping (3.16) defines a usual system of impri­mitivity on ,JY',o and based on H f HnH ~o, with H ~o = StabtILo ' some tll-oeTI,o'

The importance of the above theorem lies in the fact that P is thus completely determined by its restriction, at so, on H add on some orbit T"o' and that this restriction forms a usual system ofimprimitivity.

The main point is now to reverse the procedure and to show that each ordinary system of imprimitivity on dYso gives rise to a s.s.oj. on K. Let us therefore suppose that we have given soeSS!!G IH, H = Stabso and toeTS!!G fHI, HI = Stabto' and TlLo = H·toS!!H fHnH l. Suppose further­more that D is an irreducible representation of H in some Hilbert space cW'(D), which is induced from HnHl, i. e., via Mackey's theorem ofimprimitivity, that there exists an ordi­nary s.o.i. Q based on TILo '

Q: /!lJ (7;Lo )-.9' pY(D )) (3.17)

2394 J. Math. Phys., Vol. 22, No. 11, November 1981

with 9 (dY(D )) the set of all projectors in dY(D ). One can now consider the K -space

K = V (J¥'(D)1s SEG/H

(3.18)

and, on K, consider (the irreducible) representation U de­fined in (2.2) and (2.3),

Us ( g) = D (v(s, g))'5s( g). (3.19)

In this framework we now define the following mapping:

P:P4(T)_9(K) (3.20)

with def

P; = Q(k(s)FnTILJ (3.21)

Theorem 3.7: The mapping constructed in (3,21) is a supersystem of imprimitivity for U in (3.19) on the space K given by (3.18).

Proof We have to verify in turn the conditions (3.6) and (3.7) using (3.21) and the fact that Q satisfies the usual s.o,i. properties on T"o'

(i)P; = Q(k(s)TnT"J = Q(T"J = L,y. Vs andP~ = Q(0nT"o) = Q(0) = 0,

(ii) p;.nF, = Q ([k (S)Flnk (s)F2]nTILo ) = Q ((k (s)Flnn

T"o )n(k (S)F2nT"o)) = p;'. P;' The product of two families of projectors being the family of the product, this verifies the condition (3.6) (ii).

(iii) Suppose F l lF2' then k (s)Fdk (S)F2' Vs and thus (k (s)FlnT".,lHk (S)F2nT",J We thus obtain that p~.UF, = Q ((k (s)F1uk (s)F2)nT"o) = Q ((k (S)FlnTll-o)u

uQ (k (S)F2nT"o) = Q ((k (S)F1 )nT"o) + Q (k (s)F2nT"o) = P;' + P~', hence (3.6) (iii) is also verified, the sum of two

families being the family of the individual sums. (iv) It still remains to verify (3,7): Using the expression

(3.19) for U and the fact that Q is an ordinary s.o.i. for D, we find

(U(g)PFU(g)-I)s =D(v(s,g))P: 'sD-I(v(s,g))

= D(v(s, g))Q(k (g-ls)Fn7;'o)D -1(V(S, g))

= Q (v(s, g)( k (g-ls)FnTll-o ])

= Q «(v(s, g)k (g-l s)F)n7;Lo)' (3.22)

where we have used that, as vIs, g)EH, it leaves Tll-o invariant. Now using the definition (1.13) of the cocycle v, i. e., v(s, g) = k (s)g(k (g-IS))-1 and inserting in (3.22) we find

(U(g)PFU(g)-I)s = Q(k(s)gFnT"J

= p~F, VseS

and this achieves the proof of the theorem. Example: As each representation of H is trivially in­

duced from itself, we have on dY(D ) with H = HI a usual s.oj. on the I-point set So with

Q(so) = l:.rIDI' Q(0) = O:.rID)·

Using the definition (3.21) one correspondingly finds, as s.s.oj. on K, VEEYJ(G fH),

E {I if soEk (s)E P s = Q (s)Enso) = ° otherwise '

N. Giovannini 2394

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Page 8: Classical and quantal systems of imprimitivity

hence

(3.23)

and we recover in this way the example given in (3.4). It is useful to remark at this point that the subgroup HI

is uniquely defined by the corresponding observable only up to conjugation. Different choices corresponding to different H:H I double coset elements can give in principle different (even possibly inequivalent) s.s.oj. The choice of HI is, how­ever, restricted by the following consideration. S always de­termines itself as above one or more observables so that His the kinematical symmetry group corresponding to all the remaining observables. The subgroup HI is then, for each of these observables, uniquely determined up to conjugation by some hEll. The H:H I double coset element is then fixed.

Reversing the above procedure we thus find all aIlowa­ble K spaces if we proceed as foIlows. In a first step, for each subset of observables ! B J ~ ! A I with corresponding kine­matical symmetry subgroup H, H kG, we have to find those projective unitary representations which admit usual sys­tems ofimprimitivity for each observable in !B I (and no other). For this, we can use the theorem of imprimitivity of Mackey and thus ask whether these representations are equivalent to representations induced by the corresponding subgroups or not. In a second step we construct U, the space K, and the s.s.oj. as in (3.19), (3.18), and (3.21), and we find in this way all remaining observables.

The observables found in the first step are the usual quantal observables as they correspond to spectral measures in separable Hilbert spaces. The observables found in step 2 then play the role of superselection parameter or, in other words, are of the classical type, as they commute with all other ones and have a purely discrete point spectrum.

We shaIl see in the next section and in Ref. 4 how the above procedure applies in concrete physical examples and show then that the (quite mathematical) above framework is indeed the physicaIly relevant one for our purposes.

4. EXAMPLE

In this last section we shaIl thus consider a simple repre­sentative physical example of application of the method found in the last section. As previously mentioned the main applications, i. e., the discussion of the classical and quantal, relativistic and nonrelativistic states spaces for elementary particles, will be treated in a separate paper.4

Consider therefore the two-dimensional phase space n with coordinates p and q. This space corresponds then to the space r discussed in Sec. 3, as it contains the possible values of the observables position and momentum. Let G be the group of translations in p and q. This group is determined by the physical postulates that there exists no absolute zero for the position nor for the momentum. We thus have G~JR2 with elements (w,a) and action

def

(w,a)( p,q) = (p - w,q - a).

Let us now determine the possible state spaces for this system.

(4.1)

Suppose first that both p and q are classical in the sense

2195 J. Math. Phys., Vol. 22, No. 11, November 1981

of Sec. 3. Then H is trivial and /Jrs is one-dimensional 'rIs = (p,q)Ef1. We thus have

(4.2)

and a state consists of a point (p,q) and a ray in the corre­sponding complex plane. This framework can thus be identi­fied with the usual framework of classical mechanics in one space dimension. The observables are straightforwardly ob­tained as in (3.23) as given by

PLlp./'(p,q) = XLlP(P)f(P,q),

PLlJ'(p,q) = XLlq(q)f(P,q), (4.3)

with tJ.p and tJ.q Borel sets on the real line andf( p,q) any state. The observables are thus given, as is usual, by the pro­jections on the corresponding subsets 0; the phase space.

Suppose then that either p or q is quantal, say q. Then H ~ JR and K = V~ (/Jrp ) with /Jrp carrying an irreducible projective representation of JR which admits an observable position. But, as is easy to see, irreducibility implies that %p is one-dimensional whereas, by the Mackey imprimitivity theorem, Yrp admits an observable position if and only if it is isomorphic to y2(JR). Hence there can be no solution in this case.

Suppose finaIly that both p and q are quantal. Then H = G~JR2. The vector irreducible representations are one­dimensional and, as above, they cannot thus admit observa­bles for the position nor for the momentum. There is, howev­er, one single family of projective rerr'!sentations in

K = ,y2(JR)

with elementsf(x), and which is given by

U(a)f(x) =f(x - a),

U(w)f(x) = exp(iAxW)f(x), AElR.

(4.4)

(4.5)

The observables p and q are given by the foIlowing spectral measures:

PLlqf(x) = XLlq(x)f(x),

PLlPf(k) = XLJP(k )J(k), (4.6)

withJ(k) the usual Fourier transform off Equivalently, these observables are given by the operators

if(x) = xf(x),

N(x) = - iA - lax fix) (4.7)

so that, with of course A = fz- I, we recover the usual frame­work of quantum physics in one space dimension.

We thus see by this simple example that our formalism provides a very direct method which allows us to derive the usual classical and quantal state spaces in a simple unified way. Moreover these two solutions are seen to be the only ones. We have not only built in this way a common language for both frameworks but we have done it in a way which is independent of the dynamics and which is only based on the realization of the properties of a physical system, in accor­dance with the points of view exposed in the Introduction.

'N. Giovannini and C. Pi ron, Helv. Phys. Acta 52,518 (1979). 2N. Giovannini, Lett. Math. Phys. 5,161 (1981), 'G. W. Mackey, Proc. Math. Acad. Sci. USA 35,537 (1949).

N. Giovannini 2395

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'N. Giovannini, J. Math. Phys. 22, 2397 (1981). 'C. Piron, Foundations of Quantum Physics (Benjamin, Reading, Mass., 1976).

6C. Piron, in Fundamenti di Meccanica Quantistica, International School of Physics Enrico Fermi, Yol. IL, edited by B. d'Espagnat (Academic, New York, 1971), p. 274.

7y. S. Yaradarajan, Geometry of Quantum Theory, Yol. II (Yan Nostrand

2396 J. Math. Phys., Vol. 22, No. 11, November 1981

Reinhold, New York, 1970), Chaps. 8 and 9. "G. W. Mackey, The Theory of Unitary Group Representations (University of Chicago, Chicago, 1976).

9G. W. Mackey, Acta Math. 99, 265 (1958). 10K. R. Parthasarathy, Multipliers on Locally Compact Groups, Springer

Lect. Notes Math. 93 (Springer, Berlin, 1969).

N. Giovannini 2396

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