aa r X i v : . [ qu a n t - ph ] F e b Manifold Topology, Observables and GaugeGroup
G. Morchio and F. StrocchiDipartimento di Fisica, Università di Pisa, Pisa, Italy
Abstract
The relation between manifold topology, observables and gauge groupis clarified on the basis of the classification of the representations of thealgebra of observables associated to positions and displacements on themanifold. The guiding, physically motivated, principles are i) locality,i.e. the generating role of the algebras localized in small, topologicaltrivial, regions, ii) diffeomorphism covariance, which guarantees the in-trinsic character of the analysis, iii) the exclusion of additional local de-grees of freedom with respect to the Schroedinger representation. Thelocally normal representations of the resulting observable algebra areclassified by unitary representations of the fundamental group of themanifold, which actually generate an observable, “topological”, subalge-bra. The result is confronted with the standard approach based on theintroduction of the universal covering ˜ M of M and on the decompositionof L ( ˜ M ) according to the spectrum of the fundamental group, whichplays the role of a gauge group. It is shown that in this way one obtainsall the representations of the observables iff the fundamental group isamenable. The implications on the observability of the PermutationGroup in Particle Statistics are discussed. Math. Sub. Class.: 81Q70, 81R15, 81R10Key words: Quantum topological effects, gauge groups, identical particles1
For the analysis of quantum mechanical systems with a non-trivial manifold M as configuration space, the rôle of the topology of M has been investigatedby many authors [1], [2], [3] [4], [5], [6], [7]. The involved fundamental issuesinclude the control of the representations of the Diffeomorphisms Group of themanifold [8], the identification of the observable algebras and the classificationof their representations. Similar problems arise for the formulation of LocalQuantum (Field) Theory on space-time manifolds [9].Historically, the main approaches to Quantum Mechanics (QM) on mani-folds, the Functional Integral formulation [10, 4] and Geometric Quantization(GQ) of symplectic manifolds [11, 12, 13, 14], produced substantially equiv-alent results. Apart from the possible presence of an “internal” space, suchmethods lead in fact to Schroedinger QM in L ( ˜ M ) , ˜ M the universal coveringspace of M , with π ( M ) playing the rôle of a gauge group. The centre of therepresentation of the gauge group in L ( ˜ M ) is observable and reduces L ( ˜ M ) to inequivalent Quantum Mechanical “sectors”, with a multiplicity given bythe action of the gauge group.The result can be seen as a generalization of Dirac treatment of IdenticalParticles [15], which identifies the observable algebra through the invarianceunder the Permutation Group, which therefore plays the role of a gauge group,and classifies its representations by those of the gauge group. In three spacedimensions, the relation with Dirac treatment of identical particles is indeedvery close, since the Permutation Group coincides with the fundamental groupof the configuration manifold of identical particles [16].In view of their implications on basic issues, however, the above treat-ments leave open substantial questions. In fact, they rely on “quantization”prescriptions, associating a quantum system to a “corresponding” classical one,without an independent examination the physical basis and interpretation ofthe adopted approach. The situation is therefore quite different with respectto the case of QM in R d , where the approach based on Weyl algebras providesa clear and unique result, through the Von Neumann uniqueness theorem.In particular, the question arises whether an observable role is associatedto the entire group π ( M ) , or only to its center, as indicated by Dirac analysis.To this purpose, clearly, the identification of the QM observables on the basisof physical principles becomes the decisive issue, also because no comparablephenomena appear for classical particles.The need of a description in terms of observables and states has been em-phasized by Landsman [6, 16]. His approach is in fact based on the constructionof an observable algebra, which is obtained by “Deformation Quantization” on ˜ M , taking a quotient with respect to the action (on ˜ M ) of π ( M ) , whichtherefore plays the role of a gauge group. In Landsman’s analysis, the samegroup also appears as a unitary group in the observable algebra, and in sucha role it provides the classification of its representations. Thus, on one sideLandsman’s treatment follows the approach and confirms the conclusions ofthe previous literature: the fundamental group of the manifold plays the roleof a gauge group , which identifies the observable algebra starting from ˜ M ; onthe other, Landsman’s classification of the irreducible representations of theobservable algebra does not involve the gauge group, but rather an isomorphic observable group. Clearly, two questions arise:First, since ˜ M is taken as a starting point in all the above approches, onemay ask whether the above results depends on such a choice, which does notseem to have a direct physical interpretation. In any case, as argued above, adirect foundation on physical principles is lacking.Second, if ˜ M needs not to appear in the formulation, the classificationof the representations in terms of π ( M ) as a gauge group is in question.Moreover, even if the different formulations can be compared, the classificationin terms of π ( M ) as a gauge group and as an observable group are not a priorirelated and to which extent they may coincide is an open problem.An answer to the first question has been given in ref. [7], which avoids theintroduction of ˜ M and directly identifies the observable algebra as generatedby coordinates and momentum variables associated to vector fields on M . Theconstruction only employs the manifold M , the identification of the observ-ables does not make any reference to ˜ M and no action of π ( M ) as a gaugegroup appears. The irreducible representations of the observable algebra arethen classified by a representations of π ( M ) , which appears within each rep-resentation space . By irreducibility of the observables in each sector, such arepresentation of π ( M ) is automatically given by strong limits of observables.The purpose of this note is twofold: first, to construct the observable al-gebra on the basis of clear physical principles, holding independently of thetopology of the manifold. The identification of the observable algebra will beperformed purely in terms of a collection of local algebras, localized in “small”,topologically trivial, regions. This also allows for a purely local analysis ofthe degrees of freedom which may appear, as a consequence of diffeomorphisminvariance, in addition to Schroedinger QM. The local and global effects aretherefore separated completely, and the role of the topology precisely appearswhen the local (essentially Schroedinger) descriptions are glued together.The second aim is to confront the resulting representations, classified byan observable π ( M ) , with those obtained by the Dirac gauge group method. Outline of the strategy and main results
The Observable Algebra for the description of a quantum particle on a d -dimensional manifold M is constructed according to the following physicallymotivated principles and with the following results:1. The local structure a) Locality : The observable algebra is generated by a collection of local algebrasassociated to (topologically trivial) “small” regions O , technically defined asregions homeomorphic to a subdisk of an open disk (homeomorphism to anopen disk would not be enough to exclude, e.g., the entire space R d ).b) Diffeomorphism Covariance : The identification of the observables is exclu-sively linked to the manifold structure. Locally, it amounts to independenceof coordinates, as well as of a metric, special transformation groups etc.; glob-ally, it provides the intrinsic geometric links between the observable algebraslocalized in different regions, just as Poincaré covariance does in relativisticquantum field theory.c)
Positions and "local Movements" on the manifold : The local algebras aregenerated by position and “local trasportation” variables, the latter identifiedwith the unitary groups G ( O ) describing displacements along (all, localized)vector fields with support in O .The local algebras are therefore assumed to be generated by C functions α ( x ) with support in O , as position variables, and by unitaries U ( g ) represent-ing the diffeomorphisms groups G ( O ) . They are identified with the CrossedProducts Π( O ) ≡ C ( O ) × G ( O ) , defined by the relation U ( g ) α ( x ) U ( g ) ∗ = α ( g − x ) , representing the action of diffeomorphisms g ∈ G ( O ) on functionson O .The algebras Π( O ) generate Π( M ) ≡ C ( M ) × G L ( M ) , as global observ-able algebra, with G L ( M ) the group generated by all the G ( O ) (see Section2.3). No independent global variables are therefore introduced and the topol-ogy of M only affects the result of algebraic operations on local observables.2. The Local One (Spinless) Particle Interpretation
As a consequence of diffeomorphism invariance, the local algebras admit manyinequivalent representations (also describing many particle systems); hence,for a one-particle description, a selection is needed to avoid, at the local level,the presence of additional degrees of freedom, with respect to those appearingin Schroedinger QM. Spin and internal degrees of freedom, excluded by theabove requirement, may be added by an explicit extension of the observablealgebra.The representation of the local algebras can be taken in spaces of the form H ( O ) = L ( O , dµ ) × K ( O ) , (see Sect.2 for the exact characterization), with the association of unitary“cocycles” V g ( x ) in K ( O ) , g ∈ G ( O ) , x ∈ O , to local displacements.Up to unitary equivalence, such cocycles are characterized by those repre-senting the subgroup G ( O , x ) ⊂ G ( O ) which leaves a point x ∈ O stable; theunitary class of their representations is independent from the point and theregion.Such cocycles, describing modifications of local displacements by operationswhich leave the point invariant, have the physical interpretation of describing“internal degrees of freedom”. Their triviality , which excludes such degrees offreedom, is required by a
Local One Particle Interpretation and turns out tobe equivalent to the Schroedinger representation of the local algebras Π( O ) ,up to a multiplicity ( Locally Schroedinger condition ).The exclusion of internal degrees of freedom is actually a general questionfor representations of observable algebra which include momenta describing”movements” besides those associated to translations. In fact, it appears al-ready in the case of one spinless quantum particle in R d with momenta indexedby the generators of the euclidean group, where non-trivial cocycles are associ-ated to rotations and describe spin; more generally, the same problem appearsfor the representation of the stability group of a point in the case of the al-gebra constructed from a group G and associated to the manifold G/H , H asubgroup of G [18].As a result, on the basis of the above principles, we obtain a completeidentification of the local observable algebras and of their representations; they simply describe Schroedinger QM, up to a multiplicity, in diffeomorphismcovariant variables .The control of QM of a particle on a manifold is then reduced to theclassification of the representations of the global algebra Π( M ) ≡ C ( M ) ×G L ( M ) which reduce to a multiple of the Schroedinger representation whenrestricted to Π( O ) , for all O .Equivalently, in the spirit of Local Quantum Theory [19], the local observ-able algebras can be identified with the weak closures A ( O ) of Π( O ) in (any)LS representations of Π( M ) . The global observable algebra for one particle on M is then identified with the algebra A ( M ) generated by them in the sum ofsuch representations. LS representations of Π( M ) coincide with locally normalrepresentations of A ( M ) .3. Classification of Locally Schroedinger (LS) representations
By the above results, the classification of the LS representations of Π( M ) isreduced to the analysis of the cocycles V g n ...g ( x ) associated to products oflocalized diffeomorphisms g i ∈ G ( O i ) . They are shown to depend only on thetopological equivalence class of the path from x to g n . . . g x given by g n . . . g ,and to define a unitary representation of the path grupoid, which is classified,up to unitary equivalence, by its restriction to closed paths with (any) fixedbase point, i.e., by a representation of π ( M ) .All unitary representations of π ( M ) are shown to define admissible co-cycles, so that the correspondence between LS representations of Π( M ) andunitary representations of π ( M ) is one to one. If M is simply connected, the analogue of Von Neumann uniqueness theorem holds: the (locally normal)representation of A ( M ) is unique (and Schroedinger), up to a multiplicity.The above representations of π ( M ) , multiplied by projections over smallregions O , are shown to describe observables corresponding to the transport ofthe particle, starting in the region O , along the corresponding loops. Moreover,for compact M , A ( M ) contains a subalgebra isomorphic to the group algebraof π ( M ) . The classification of the representations is therefore always givenby an observable representation of π ( M ) .4. The gauge and observable realizations of π ( M ) A comparison with the Dirac gauge group approach [15] is provided by theanalysis of the Schroedinger representation of A ( M ) in L ( ˜ M ) . Identifying ˜ M with a space of pairs x ∈ M , γ ∈ π ( M ) , one has L ( ˜ M ) ∼ L ( M ) × l ( π ( M )) . The usual “gauge” action of π ( M ) in ˜ M is given by ψ ( x, γ ) → ψ ( x, γ ◦ δ ) , i.e., by its right regular representation in l ( π ( M )) . It clearly commuteswith the position observables and also with the unitaries U ( g ) , which act asin the Schroedinger representation in L ( M ) , combined with the left regularrepresentation of π ( M )) in l ( π ( M )) (see eq. 3.7).It follows that, in the representation of A ( M ) in L ( ˜ M ) , two commutingunitary representations of π ( M )) appear, respectively as a gauge group and as an observable group. Being given by (a multiple of) the left and rightregular representations, the two representations are unitarily equivalent; theygenerate Von Neumann algebras which are the commutant one of the other in l ( π ( M )) , so that their centres coincide and give rise to the same reduction of L ( ˜ M ) . Therefore, the classifications of the representations of A ( M ) in L ( ˜ M ) given by the observable and by the gauge group coincide .As a result, only the completeness of the Dirac approach is in question, i.e.whether all the (locally normal) irreducible representations of A ( M ) appearin the reduction of L ( ˜ M ) . Equivalently, whether all the irreducible repre-sentations of π ( M ) appear in the (possibly integral) reduction of its regularrepresentation. The answer to this questios is known and rather simple: thishappens if and only if the group is amenable .If π ( M ) is not amenable, the role of its regular representation and of L ( ˜ M ) is completely different, since even the identity representation of π ( M ) ,corresponding to the ordinary Schroedinger representation of the observables,is not present in the reduction. Proposition 3.3 below shows that in this caseall the representations of A ( M ) can still by obtained in suitable Hilbert spacesof functions on ˜ M (with non L scalar products), but the role of the gaugegroup is in general lost.5. Implications for Identical Particles
Particularly significant is the case of N identical particles, which can be de-scribed as the quantum system associated to the N -particle manifold M S ,defined by identifying configurations obtained by permuting the particles andexcluding the set ∆ of coincident points [17].In the case of N particles in Euclidean space of dimension greater thantwo, the fundamental group of M S is the permutation group S N and ˜ M S maybe identified with R Nd \ ∆ . From the above results it follows thati) since S N is finite (and therefore amenable), the Dirac representation ofthe observable algebra in L ( R Nd ) ( ∆ being here irrelevant) contains all itsirreducible representations, classified by the representations of S N in L ( R Nd ) as a gauge group.ii) an observable unitary representations of the Permutation Group is presentin L ( R Nd ) , unitarily equivalent to the gauge representation. In each reductionspace, the gauge and observable representations are complex conjugates.The observable representation acts (see Sect. 3) by physical operations whichshift the position of each particle (as a gauge invariant variable, independentof particle labels), along paths which interchange the position of the particles.Apart from the abelian case, N = 2 , the (gauge invariant) operators imple-menting such actions have nothing to do with gauge trasformations, which infact act on the particle labels and are not observable.Clearly, the presence of the observable representation of S N allows for adirect physical (and topological) reinterpretation of the Dirac classification, interms of unitaries describing permutations as physical operations. In the standard case of a quantum mechanical system with R d as configura-tion space, the observables are usually generated by the Cartesian coordinatesin R d and the associated momenta; Schroedinger Quantum Mechanics thenarises as the unique (regular) representation of the Weyl algebra, generatedby the exponentials of such variables. On the other hand, when the configura-tions are described by a C ∞ connected ( d -dimensional) manifold M no globalcoordinate system is available and, even locally, there are no distinguishedcoordinate systems, nor intrinsic finite dimensional transformation groups.Following the general philosophy of Local Quantum Theory [19], we startby considering “local” open regions O , topologically trivial and topologicallylocalizable , in the sense that they are proper subsets of larger similar regions.In the following, M will indicate any C ∞ connected manifold. Definition 2.1
An open region
O ⊂ M is called small if there is a diffeo-morphism of a neighbourhood O ′ of O taking O ′ to an open ball, | x | < , x ∈ R d , and O to the open ball | x | < / . Hereafter, O will always denote a small region. Clearly, each point of M has a small open neighbourhood O and a denumerable set of such regionscovers M .Our guiding principle for the construction of the local observable algebrasis to recognize as fundamental the covariance under diffeomorphisms, linkingthe observables to the intrinsic geometry of M . For each O we introduce:i) as “ position observables ” the C ∞ (complex valued) functions α ( x ) , x ∈ M ,with compact support in O , generating, with the Sup norm and together witha common identity , a diffeomorphism invariant C ∗ algebra C ( O ) ; then, C ( O ) is isomorphic to the algebra generated by the continuous functions on O , vanishing at its boundary, and by the constant functions.ii) as “ local generalized momenta ”, the vector fields v ∈ L ( O ) , L ( O ) the Liealgebra of C ∞ vector fields with compact support in O ; by compactness oftheir support, they integrate to one parameter groups g λv , λ ∈ R , generatingdiffeomorphism groups G ( O ) , with a common identity e and G ( O ) ⊂ G ( O ) for O ⊂ O . In physical terms, G ( O ) describes displacements localized in O ,acting on (any) configuration of the system.The vector space of the pairs { α, g } , α ∈ C ( O ) , g ∈ G ( O ) , with the operations { α , g } { α , g } = { α α g , g g } , α g ( x ) ≡ α ( g − x ) { α, g } ∗ = { ¯ α, g − } , defines the crossed product C ∗ algebra Π( O ) ≡ C ( O ) × G ( O ) .In the construction of the crossed product, G ( O ) is taken as a topologicalgroup with the discrete topology and Π( O ) is generated by the finite sums P i α i g i , α i ∈ C ( O ) , g i ∈ G ( O ) , with norm P i Sup x | α i ( x ) | , see, e.g., [20]. Inthe following, for simplicity, we adopt the notations: U ( g ) ≡ { , g } , α ( x ) ≡{ α, e } ; then the basic crossed product algebraic relations read U ( g ) − = U ( g ) ∗ , U ( g ) α ( x ) U ( g ) ∗ = α ( g − x ) , Supp α ⊂ O , g ∈ G ( O ) . (2.1)The local “position” algebras C ( O ) satisfy isotony, C ( O ) ⊂ C ( O ) for O ⊂ O and generate, in the Sup norm on M ( being identified with the function on M ), the C ∗ algebra C ( M ) of continuous functions on M , if M is compact,and on its one-point compactification ˙ M = M ∪ { x ∞ } (the Gelfand spectrumof C ( M ) ) otherwise. In the latter case, the diffeomorphisms of G ( O ) extendto diffeomorphisms of ˙ M , with gx ∞ = x ∞ ; α ( x ∞ ) ≡ for α ( x ) in C ∞ ( O ) , forall O .The local groups G ( O ) coincide with the connected component of the iden-tity of the diffeomorphism group of O (see [8]). They obviously satisfy isotonyand generate a group G L ( M ) , uniquely associated to M , defined by the formalproducts of their elements modulo the group relations holding in each region O . Its elements are therefore strings g . . . g n , modulo the equivalence relationdefined by any sequence of replacements of a substring of elements localizedin some O by another string with the same product in G ( O ) . G L ( M ) actson M by diffeomorphisms (depending in fact only on the equivalence class of g . . . g n ), which will be denoted by the same symbol. G L ( M ) formalizes, inthe construction of the observable algebra, the principle that finite sequencesof local operations still define physical operations and are only constrained bythe validity of all local relations .We therefore associate to M the crossed product C ∗ algebra Π( M ) ≡ C ( M ) ×G L ( M ) , defined as above, with eqs.(2.1) satisfied for all α ∈ C ( M ) and g ∈ G L ( M ) . Π( M ) is generated by the local algebras Π( O ) and is invariantunder Diff ( M ) , the entire diffeomorphism group of M , since both C ( M ) and G L ( M ) are invariant.As we shall see, diffeomorphism invariance makes such algebras, alreadyfor regions O , much richer than, e.g., Weyl algebras in R d ; in fact, they admitmany inequivalent representations, with different physical interpretation. Thishappens because additional independent “momentum” variables appear, as aconsequence of diffeomorphism invariance. In order to classify the physically relevant representations of Π( M ) , we startby characterizing, under physically motivated conditions, the representationsof its local subalgebras Π( O ) , which generate it.First, in order to ensure the existence of local momenta, we consider repre-sentations of Π( M ) in separable Hilbert spaces, satisfying, for each O , strongcontinuity in λ of the local groups U ( g λv ) , v ∈ L ( O ) .Then, the representations of Π( M ) , as well as of Π( O ) , are characterizedby the following Proposition, which applies to a generic manifold; for economyof notation, it is stated for Π( M ) . Proposition 2.2
A representation π of Π( M ) in a separable Hilbert space,with U ( g λv ) strongly continuous in λ ( regular representation ) is unitarilyequivalent to one in H π = L ( M , dµ ) × K ⊕ K ∞ , (2.2) K and K ∞ separable Hilbert spaces, dµ equivalent to the Lebesgue measure on M (in any system of local coordinates).Identifying the elements of H π with L functions ψ ( x ) , x ∈ ˙ M , taking valuesin K for x ∈ M and in K ∞ for x = x ∞ , the action of the representatives ofthe elements of Π( M ) is given by: π ( α ) ψ ( x ) = α ( x ) ψ ( x ) , π ( U ( g )) = C g V g , (2.3) C g ψ ( x ) ≡ ψ ( g − x ) J g ( x ) / , V g ψ ( x ) = V g ( x ) ψ ( x ) , (2.4) J g ( x ) ≡ [ dµ ( g − x ) /dµ ( x )] , V g ( x ) a family of unitary operators, in K for x ∈M and in K ∞ for x = x ∞ , weakly measurable in x , satisfying C − h V g ( x ) C h = V g ( hx ) , (2.5) V g ( hx ) V h ( x ) = V gh ( x ) , (2.6) a.e. in x ∈ M .Two (regular) representations π , π of Π( M ) in L ( M , dµ ) × K i ⊕ K ∞ ,i , i =1 , , are unitarily equivalent iff there exists a weakly measurable family of iso-metric operators S ( x ) from K to K for x ∈ M , and from K ∞ , to K ∞ , for x = x ∞ , such that S ( gx ) V (1) g ( x ) S ( x ) − = V (2) g ( x ) . (2.7)0The proof is essentially the same as in [7], Lemma 3, only strong continuityof U ( g λv ) for v ∈ L ( O ) being required; the irreducibility condition is replacedby separability of H π . The compactification of M arises as the Gelfand spec-trum of C ( M ) and gives rise in general to a representation of Π( M ) , in K ∞ ,which assignes the null value to all functions with compact support and reducesto a unitary representation of G L ( M ) by V g ( x ∞ ) .Thus, up to unitary equivalence, the regular representations of the localalgebras Π( O ) are given in L ( O , dµ ) × K ( O ) ⊕ K ∞ ( O ) by eqs.(2.3),(2.4), with α ∈ C ( O ) , g ∈ G ( O ) . Given a regular representation of Π( M ) , by Proposition2.2, the corresponding representation spaces are K ( O ) = K for all O and K ∞ ( O ) = L ( M \ O , dµ ) × K ⊕ K ∞ . By Proposition 2.2, the classification of the representations of Π( M ) re-duces to that of the unitary operators V g ( x ) . For V g ( x ) ≡ and K onedimensional, the representation in L ( M , dµ ) defines the Schroedinger rep-resentation π S (Π( M )) .Eq.(2.6) becomes a group cocycle relation if the excluded zero measuresubset can be taken independent of g and h . In fact, in this case, the operators V g ( x ) are well defined as maps from g ∈ G L ( M ) to the group U of unitaries in K depending on x ∈ M \ A , A of zero measure; eq.(2.6) is then the cocyclerelation associated to the map g → ϕ ( g ) , Aut ( U ) ∋ ϕ ( g ) : V ( x ) → V ( gx ) . Definition 2.3
A representation π of Π( M ) will be called cocycle-regular if it is unitarily equivalent to a regular representation where eqs.(2.6) define agroup cocycle relation, for almost all x in M . In a cocycle-regular representation of Π( M ) the operators V g ( x ) provide arepresentation of the stability groups G ( O , x ) ≡ { g ∈ G ( O ) : gx = x } , x ∈ O in K , for almost all x ∈ M .For different O ⊂ M , x ∈ O , the groups G ( O ) are isomorphic, and thesame applies to G ( O , x ) . In fact, it is enough to consider O , O disjoint, x i ∈ O i ; then, there exists a region O ⊃ ( O ∪ O ) and a diffeomorphism g ∈ G ( O ) transforming O in O and x in x , constructed, e.g., by contracting O i to sufficiently small balls around x i , interpolated by local translations in acylinder around a path from x to x .The analysis of local cocycles allows for a characterization of the represen-tations of the local algebras.1 Proposition 2.4
A cocycle-regular representation π of Π( O ) , by Proposition2.2 in L ( O , dµ ) × K ( O ) ⊕ K ∞ ( O ) , defines unitary representations R ( O , x ) of G ( O , x ) in K ( O ) , for almost all x ∈ O , all unitarily equivalent and a unitaryrepresentation R ∞ of G ( O ) in K ∞ ( O ) . The corresponding unitary equivalenceclasses determine the cocycles V g ( x ) , and therefore π (Π( O )) , up to unitaryequivalence.Conversely, any pair of unitary representations of G ( O , x ) and G ( O ) , respec-tively R, R ∞ , in spaces K, K ∞ (strongly continous in the parameters of one-dimensional subgroups) determines a cocycle-regular representation π R,R ∞ of Π( O ) , in L ( O , dµ ) × K ⊕ K ∞ .Proof. By definition of small regions, there exist O ′ ⊃ O ∼ { y ∈ R d , | y | < } and a subgroup T of G ( O ′ ) acting in O as translations, τ ( a ) , sending y to y + a ,for y, y + a ∈ O . Given x ∈ O , for all x in O there is a (unique) translation, τ ( x − x ) sending x to x . For all g ∈ G ( O ) , two applications of eq. (2.6) give V − τ ( gx − x ) ( x ) V g ( x ) V τ ( x − x ) ( x ) = V τ ( x − gx ) g τ ( x − x ) ( x ) , (2.8)a.e. in x, x . The operators in the r.h.s. are indexed by elements of G ( O , x ) and, by eq. (2.6), give a unitary representation of it. By the equivalence cri-terium, eq. (2.7), taking S ( x ) = V − τ ( x − x ) ( x ) , the representation of Π( O ) isunitarily equivalent to that given by V T,x g ( x ) ≡ R x ( τ ( x − gx ) g τ ( x − x o )) , (2.9)with R x ( h ) ≡ V h ( x ) (2.10)a unitary representation of G ( O , x ) in K ( O ) . Clearly, V T,x h ( x ) = R x ( h ) .For x = x ∞ , by the cocycle equation and the invariance of x ∞ , R ∞ ( g ) ≡ V g ( x ∞ ) (2.11)gives a unitary representation of G ( O ) in K ∞ ( O ) . For different x ∈ O thegroups G ( O , x ) are isomorphic and their representations R x are unitarilyequivalent.Conversely, for given x , T , by eqs. (2.9)(2.11), any pair of (strongly continu-ous) representations R, R ∞ of G ( O , x ) , G ( O ) in separable Hilbert spaces K , K ∞ define operators V T,x g ( x ) satisfying eqs. (2.6) and, by Proposition 2.2, acocycle-regular representation π R,R ∞ of Π( O ) . By eqs.(2.8),(2.9), the repre-sentations of Π( O ) given by R , R ∞ , are unitarily equivalent for different T, x .2Examples of non-trivial R are given by representations ρ n of Π( O ) in L ( O × O × . . . × O , dµ ( x ) . . . dµ ( x n )) ≡ L ( O n ) defined by ρ n ( α ( x )) ψ ( x , . . . , x n ) = α ( x ) ψ ( x , . . . , x n ) ,ρ n ( U ( g )) ψ ( x , . . . .x n ) , = ψ ( g − x , . . . , g − x n ) Π i J g ( x i ) / . Here, K = L ( O n − ) and G ( O , x ) is represented there by the unitary changeof variables Π ni =2 C g ( x i ) . In these examples, K ∞ = 0 . The representations of Π( O ) of the above examples describe states of n par-ticles in O , the position of the first particle being described by α ( x ) and theother particles by the variables U ( g ) . Thus, already for local algebras and in-dependently of the topology of M , additional conditions are needed in order toselect the representations of Π( M ) with a local one-particle interpretation.Given a cocycle-regular representation π of Π( M ) , the unitary equivalenceclass of the representations R ( O , x ) , R ∞ ( O ) do not depend on O , by theargument before Proposition 2.4. A non-trivial R ( O , x ) describes an additionalaction of U ( g ) , g ∈ G ( O ) , besides the change of variables C g , i.e., additionallocalized degrees of freedom with respect to the Schroedinger positions andmomenta in O . Similarly, a non trivial R ∞ ( O ) describes, in K ∞ ( O ) , an actionof G ( O ) on states localized outside O and must be excluded if π ( G ( O )) has toact locally.The same conclusion is obtained considering that the regions O are diffeo-morphic to R d , so that the representations of Π( O ) should be compared (theycannot be distinguished in a diffeomorphism invariant way) with those arisingin a diffeomorphism invariant formulation of QM of a particle in R d .In fact, the condition R ( R d , x ) = R ∞ ( R d ) = I identifies the Schroedingerrepresentation π S (Π( R d )) , up to unitary equivalence, multiplicity and the ad-dition of a trivial representation (“states at infinity”); R ( O , x ) = R ∞ ( O ) = I identifies the Schroedinger representation π S (Π( O )) in L ( O , dµ ) , apart fromunitary equivalence, multiplicity and the addition of a trivial representation,given by π ( α ( x )) = 0 , π ( U ( g )) = 1 , always allowed for the description of stateslocalized in M \ O .Thus, we are led to consider representations of Π( M ) with R ( O , x ) = I and R ∞ ( O ) = I , for all O ⊂ M . Since R ∞ ( O ) = I also implies R ∞ = I ,i.e., a trivial representation of Π( M ) in K ∞ , we set in the following K ∞ =0 and therefore, by Proposition 2.2, consider representations of Π( M ) in L ( M , dµ ) × K .3By the above analysis, such conditions fix the representations of the localalgebras : by Propositions 2.2, 2.4, each of them is unitarily equivalent, in L ( M , dµ ) × K , to a representation with V g ( x ) = I for all g ∈ G ( O ) , i.e.,to the restriction of π S (Π( M )) to Π( O ) , hereafter called the Schroedingerrepresentation of Π( O ) in L ( M , dµ ) , with a multiplicity given by K . Definition 2.5
A representation π of Π( M ) , with K ∞ = 0 , is LocallySchroedinger (LS) if it is cocycle-regular and, for all
O ⊂ M , R ( O , x ) , R ∞ ( O ) are the identity; equivalently, if the representations of Π( O ) are uni-tarily equivalent to their Schroedinger representation in L ( M , dµ ) , apart froma (common, at most denumerable) multiplicity. The LS condition is a non-trivial restriction already at the algebraic level,since π S (Π( O )) is not a faithful representation of Π( O ) . In fact, e.g., for U ( g ) , α ∈ Π( O ) , if gx = x , ∀ x ∈ Supp α ( x ) , π S (( U ( g ) − α = 0) whereas, in Π( O ) , || ( U ( g ) − α || = 2 Sup | α ( x ) | . Remark.
The LS condition is also equivalent to the local validity of the Lie-Rinehart relations of the generators of Diff ( M ) as a module on C ∞ ( M ) [7],allowing to express them in terms of d independent momenta: T P i α i v i = X i ( α i T v i + T v i α i ) , ∀ α i ∈ C ∞ ( O ) , ∀ v i ∈ L ( O ) . (2.12)Eqs.(2.12) hold in fact in the Schroedinger representation and the conversefollows from the proof of Theorem 3.5 in [7].By the LS condition, for any region O , one may take, as the observablealgebra in O , the Von Neumann closure A ( O ) of π S (Π( O )) . As operator al-gebras in the space of the sum, ρ (Π( M )) , of the LS representations of Π( M ) ,the Von Neumann algebras A ( O ) generate a C ∗ algebra A ( M ) , which can betaken as the (one-particle) observable algebra associated to the entire manifold. A ( M ) contains ρ (Π( M )) as a weakly dense subalgebra, so that representa-tions of A ( M ) define representations of Π( M ) and unitarily equivalent LSrepresentations of Π( M ) define unitarily equivalent representations of A ( M ) .Therefore, the LS representations of Π( M ) coincide with the locally normal ([19], p.131) representations of A ( M ) .By irreducibility of the Schroedinger representation of Π( O ) in L ( O , dµ ) ,the algebras A ( O ) are isomorphic to the algebra B ( H ) of all bounded operatorsin a separable Hilbert space. However, the Schroedinger condition is purelylocal and the immersion of the local algebras in A ( M ) gives rise to a nontrivial “bundle” structure, associated to the topology of M , which plays theessential role in the classification of the representations of A ( M ) .4 By Proposition 2.2 and Definition 2.5, a LS representation of Π( M ) is givenby a collection of representations of the local algebras Π( O ) , O ⊂ M , allin L ( M , dµ ) × K and determined by unitary local intertwiners with theSchroedinger representation. The associated cocycles are locally trivial andour aim is to characterize the cocycles V g ( x ) , g ∈ G L ( M ) , (eq.(2.3)) associatedto products of localized diffeomorphisms; by Proposition 2.2, they identify therepresentation of Π( M ) . The analysis will consist of the following steps.a) Local Schroedinger intertwiners
The following Proposition characterizes the local cocycles for a LS repre-sentation of Π( M ) . Proposition 2.6
A regular representation π of Π( M ) , with K ∞ = 0 , is LS ifffor (one and therefore) all O , there exist (unique) weakly measurable unitaryoperators in K , W O ( y, x ) , x, y ∈ O such that, ∀ g ∈ G ( O ) , V g ( x ) = W O ( gx, x ) , ∀ x ∈ O ; V g ( x ) = 1 , ∀ x / ∈ O . (2.13) They satisfy W O ( y, x ) W O ( x, z ) = W O ( y, z ) , W O ( y, x ) = W O ( x, y ) − , (2.14) W O ( y, x ) = W O ( y, x ) , ∀ x, y ∈ O , O ⊂ O . (2.15) Proof.
By eq. (2.7) the quasi-equivalence of the representation of Π( O ) to itsSchroedinger representation in L ( M , dµ ) × K implies the existence of weaklymeasurable unitary operators W O ( x ) in K , such that, ∀ g ∈ G ( O ) , V g ( x ) = W O ( gx ) W O ( x ) − , ∀ x ∈ O , V g ( x ) = 1 , ∀ x / ∈ O . (2.16)Then, W O ( y, x ) ≡ W O ( y ) W O ( x ) − (2.17)is well defined for all x, y ∈ O and weakly measurable, since W O ( x ) is measur-able in x . Eqs.(2.13), (2.14), (2.15) immediately follow from the definitions.Conversely, eqs.(2.13),(2.14) imply eqs.(2.16), with W O ( x ) ≡ W O ( x, x ) (forany choice of x ∈ O , apart from a zero measure subset). Therefore, therepresentation of Π( O ) , in L ( M , dµ ) × K , is unitarily equivalent to theSchroedinger representation in the same space, with multiplicity given by K .5b) Homotopy classes of products of local intertwiners
As a next step, on the basis of eqs. (2.13) (2.14) (2.15), we show that,for g = g n . . . g , g i ∈ G ( O i ) , V g ( x ) is given by a string of local factors W O i ( x i +1 , x i ) , x i +1 = g i x i ; the resulting operators will be shown to be in-variant under small displacements of the intermediate points x i and thereforeindexed by the homotopy class of the corresponding path.In fact, given g ∈ G L ( M ) , g = h N . . . h , by eq.(2.6) V g ( x ) is representedby a product of terms of the form V h i ( y i ) . Dropping all the factors with h i y i = y i (equal to the identity by eq.(2.13)), we are left with a subset g . . . g n , g i ∈ G ( O i ) , such that gx = g n . . . g x , x i , x i − ∈ O i x i ≡ g i x i − = x i − , x ≡ x , x n ≡ gx. Then, by eq.(2.13), one has V g ( x ) = V g n ( x n − ) . . . V g ( x ) = W O n ( gx, x n − ) W O n − ( x n − , x n − ) . . . W O ( x , x ) . (2.18) Lemma 2.7
Given x, y ∈ M , a regular path γ ( y, x ) starting at x and endingat y , a partition of γ , γ = γ n ( y, x n − ) ◦ . . . ◦ γ ( x , x ) , open sets O . . . O n with O i ⊃ γ i and unitary operators W O ( y, x ) in K satisfying eqs.(2.14),(2.15), theoperators W ( y, x, γ ) ≡ W O n ( y, x n − ) W O n − ( x n − , x n − ) . . . W O ( x , x ) , (2.19) only depend on x, y and on the homotopy class [ γ ] of γ ( y, x ) . They satisfy thecomposition law W ( y, x, [ γ ]) W ( x, z, [ δ ]) = W ( y, z, [ γ ] ◦ [ δ ]) . (2.20) For almost all x , the topological operators W ( x, x, [ γ ]) provide a unitaryrepresentation R x ([ γ ]) of π ( M ) , all belonging to the same equivalence class [ R ] .Two systems of (weakly measurable) unitary operators, W , W ( y, x, [ γ ]) , sat-isfying eq.(2.20), are related by (weakly measurable) unitaries S ( z ) , W ( y, x, [ γ ]) = S ( y ) W ( y, x, [ γ ]) S ( x ) − (2.21) iff, for some (and then for all) x ∈ M , the corresponding systems for closedpaths are unitarily equivalent: R x ([ γ ]) = S ( x ) R x ([ γ ]) S − ( x ) , (2.22) equivalently, iff the corresponding equivalence classes [ R i ] , i = 1 , , coincide. Proof.
Given two choices { γ i , O i } and { γ ′ i , O ′ i } , a “combined” partition of γ isdefined by the sequence { y k } obtained by ordering the points x i , x ′ j ; each piece γ ′′ i ( y i , y i − ) of the so obtained partition is contained in some O ′′ i ⊂ O k ∩ O ′ k ′ forsome pair k, k ′ . Therefore, by eq.(2.15) the operators W ( y, x, γ ) associated to { γ ′′ i , O ′′ i } coincide with both the operators constructed from γ i and γ ′ i . Sincethe sets O i can be kept fixed for a small deformation of γ , the result onlydepends on the homotopy class of γ . Independence from the partition of γ ◦ δ immediately implies eq.(2.20).By eqs.(2.20), for all x, y , fixed a path δ from y to x , [ γ ] → [ δ − ◦ γ ◦ δ ] is a bijection between (the equivalence classes of) the closed paths with basepoint x and those with base point y , and W ( y, y, [ δ − ◦ γ ◦ δ ]) = W ( x, y, [ δ ]) − W ( x, x, [ γ ]) W ( x, y, [ δ ]) . (2.23)Then, eq.(2.21) implies eq.(2.22) yielding the unitary equivalence of the rep-resentations R x of π ( M ) for different x ∈ M ; conversly, it is easy to checkthat, given x and S ( x ) satisfying eq.(2.22), S ′ ( y ) ≡ W ( y, x, γ ) S ( x ) W ( y, x, γ ) − (2.24)satisfies eq.(2.21), for any choice of curves γ starting at x and ending at y andit is measurable in x if the curves depend continuously on x in an open set M with complement of zero measure, as in the proof of Proposition 2.4.c) Classification by the fundamental group
By the above results, in a LS representation π of Π( M ) , the operators π ( U ( g )) representing a product of localized diffeomorphisms, g = g n . . . g , aregiven by eqs.(2.3), with V g ( x ) = W ( gx, x, [ γ g ]) , (2.25)where W ( gx, x, [ γ g ]) is defined by eqs.(2.18),(2.19) and γ g is the integral curve,from x to gx , defined by g n . . . g .By eq.(2.7) two LS representations π i of Π( M ) are unitarily equivalentiff the corresponding W operators are related by eq.(2.21), and therefore, byLemma 2.7, iff the associated equivalence classes [ R π i ] of representations of π ( M ) coincide. Hence, the LS representations of Π( M ) are classified by thecorresponding unitary representations of π ( M ) . The following Lemma showsthat all the unitary representation of π ( M ) appear in the classification.7 Lemma 2.8
Any unitary representation R of π ( M ) in a (separable) Hilbertspace K defines a LS representation π R of Π( M ) in L ( M , dµ ) × K , with thesame equivalence class, [ R π R ] = [ R ] Proof.
By Proposition 2.2, given R we have to construct weakly measurableunitary operators V g ( x ) satisfying eq.(2.6).To this purpose, we fix a point x ∈ M and associate to each x ∈ M apath δ ( x, x ) from x to x . As in the proof of Proposition 2.4, such pathscan be taken continuous in x , in the C topology of paths, for all x in a set M ⊂ M , with a complement of zero measure. Then, to each γ ( y, x ) from x to y , ∀ x, y ∈ M , we associate the closed path β ( γ ( y, x )) ≡ δ ( x , y ) ◦ γ ( y, x ) ◦ δ ( x, x ) , (2.26)with δ ( x , y ) ≡ δ ( y, x ) − . Clearly, the equivalence class [ β ] only depends on [ γ ] . Composing two paths, γ ( y, x ) , γ ( x, z ) , gives rise to the composition of thecorresponding images in π ( M ) : β ( γ ( y, x ) ◦ γ ( x, z )) = δ ( x , y ) ◦ γ ( y, x ) ◦ δ ( x, x ) ◦ δ ( x , x ) ◦ γ ( x, z ) ◦ δ ( z, x ) == β ( γ ( y, x )) ◦ β ( γ ( x, z )) . (2.27)Given a unitary representation R of π ( M ) in K , we then define, for all g ∈ G L ( M ) , x ∈ M (omitting for simplicity the equivalence class notation forthe paths) V g ( x ) = W ( gx, x, γ g ) ≡ R ( β ( γ g ( gx, x ))) , (2.28)with, as above, γ g ( gx, x ) the integral curve associated to x and g . For gx = x = x , eq.(2.28), identifies R ([ γ g ]) with the topological factors W ( x , x , γ g ) representing π ( M ) with base point x .By the above continuity property of paths, V g ( x ) is locally constant in x and gx in M ; it is therefore strongly continuous in the parameters of one-dimensionalsubgrous of G L ( M ) and defines a cocycle: by eqs.(2.28),(2.27), ∀ x ∈ M V g ( hx ) V h ( x ) = R ( β ( γ g ( ghx, hx ))) R ( β ( γ h ( hx, x ))) == R ( β ( γ gh ( ghx, x ))) = V gh ( x ) . We have therefore a unitary representation of Π( M ) in L ( M , dµ ) × K , which isregular and cocycle-regular, with R as the associated representation of π ( M ) .For the proof of the LS property, eqs.(2.13),(2.16), consider x, z ∈ O , g ∈ G ( O ) ;since [ β ] only depends on [ γ ] , using eq.(2.27), ∀ γ ( gx, z ) , γ ( z, x ) ⊂ O ,V g ( x ) = R ( β ([ γ g ( gx, x )])) = R ( β ([ γ ( gx, z )])) R ( β ([ γ ( z, x )])) , R ( β ([ γ ( y, z )])) only depend on y , forfixed z .Eq.(2.28) also provides a direct relation, in π R , between the topologicaloperators associated to closed paths with different base points: by using theexistence, for any closed γ , of a path γ g with the same base point x and [ γ g ] = [ γ ] , one has W ( x, x, γ ) = R ( β ([ γ ])) = W ( x , x , β ( γ )) . (2.29)In conclusion, we have: Theorem 2.9
The Locally Schroedinger representations π of Π( M ) , equiv-alently, the locally normal representations of the observable algebra A ( M ) ,are given by eqs.(2.3),(2.25). Up to unitary equivalence, they are classifiedby the associated unitary equivalence class of representations of π ( M ) , [ R ] .Any unitary representation R of π ( M ) (in a separable space) define a LSrepresentation π R of Π( M ) , equivalently, a locally normal representations of A ( M ) , with [ R ] the associated representation of π ( M ) . Corollary 2.10
The commutant and the centre of π R ( A ( M )) ′′ , in the repre-sentation space L ( M , dµ ) × K , are given by the commutant R ′ and the centre R ′ ∩ R ′′ of the representation R of π ( M ) in K : π R ( A ( M )) ′ = I × R ′ , (2.30) π R ( A ( M )) ′ ∩ π R ( A ( M )) ′′ = I × ( R ′ ∩ R ′′ ) . (2.31) Corollary 2.11 (Analogue of Von Neumann uniqueness theorem) If M issimply connected, the locally normal representation of A ( M ) is unique, upto multiplicity and coincides, up to unitary equivalence, with the Schroedingerrepresentation in L ( M , dµ ) .Proof of Corollary 2.10 . A (bounded) operator A in L ( M , dµ ) × K commutingwith all the multiplication operator π R ( α ( x )) is a multiplication operator A ( x ) in K . By eq.(2.26) with x = x , for all y ∈ M , β ( δ ( y, x )) is the identity.Taking a diffeomorphism g , sending x into y , such that [ γ g ( y, x )] = [ δ ( y, x )] ,the vanishing of the commutator of A ( x ) with all the operators π R ( U ( g )) implies, by eq.(2.28), that A ( x ) is constant, a.e. in x , i.e., A = I × A K .Since A commutes with π R ( U ( g )) , A K commutes with V g ( x ) and therefore, byeq.(2.28), it commutes with the representation R of π ( M ) in K . Conversely,9by eq.(2.28), operators I × B , with B ∈ R ′ , commute with π R ( A ( M )) . Thisimmediately implies π R ( A ( M )) ′′ = B ( L ( M , dµ )) × R ′′ and eq.(2.31) follows. Theorem 2.9 and Corollary 2.10 classify the representations of A ( M )) in termsof weak limits of local observables, given by the centre of the associated repre-sentation R of π ( M ) . They leave open the question of whether the operatorsof R , and those of R ′ ∩ R ′′ , already belong to the “quasi local” observablealgebra A ( M ) . Theorem 2.12
For all x ∈ M there exists O ∋ x such that, for all R , π R ( A ( M )) contains the topological operators W ( x, x, γ ) P O , eq.(2.19), with P O the projection on L ( O , dµ ) × K ; actually they belong to (the representationof ) the algebra generated by a finite number of algebras A ( O i ) , O i containedin a neighbourhood of γ .Furthermore, if M is compacti) A ( M ) contains a “topological” C ∗ -subalgebra T ( M ) , isomorphic to the groupalgebra of π ( M ) (generated in norm by the sum of its unitary representa-tions). For all R , π R ( T ( M )) is generated by the operators W ( x , x , γ ) rep-resenting π ( M ) in K and the representations of T ( M ) classify the (locallynormal) representations of A ( M ) .ii) if π ( M ) is finite, T ( M ) is a Von Neumann algebra and its centre Z classifies the (locally normal) representations of A ( M ) ;iii) if π ( M ) abelian, the spectrum of T ( M ) labels the factorial (locally nor-mal) representations of A ( M ) .Proof. The topological factors W ( x, x, [ γ ]) can be represented in terms oflocalized observables as follows. Given g = g n . . . g , g i localized in O i , with gx = x for some x ∈ O , by eqs. (2.25),(2.3) one has, ∀ y ∈ O W ( gy, y, [ γ g ]) = V g ( y ) = C − g π ( U ( g n ) . . . U ( g )) , (2.32)with γ g ( gy, y ) ⊂ ∪ i O i ; moreover, for all y ∈ O ⊂ O , such that g ( O ) ⊂ O ,by eqs. (2.19),(2.14), one has W ( gy, y, [ γ g ]) = W O ( gy, x ) W ( x, x, [ γ g ( x, x )]) W O ( x, y ) . (2.33)0In a representation π R , one can choose O such that, by eq.(2.28), the operators W O in eq.(2.33) are the identity, and therefore W ( x, x, [ γ g ]) P O = C − g π R ( U ( g n ) . . . U ( g )) P O , (2.34)with P O the projection on L ( O , dµ ) × K . π R ( U ( g n ) . . . U ( g )) P O maps L ( O , dµ ) × K into L ( g ( O ) , dµ ) × K . A projector P g ( O ) can therefore be inserted in the r.h.s. of eq. (2.34) and C − g P g ( O ) , asan operator in L ( O , dµ ) , belongs to the (weakly closed) algebra A ( O ) , byirreducibility of the Schroedinger representation of A ( O ) in L ( O , dµ ) .By the argument before eq.(2.29), eq.(2.34) applies to all [ γ ] ∈ π ( M ) , withalmost any x as base point, and therefore gives the unitary representationsof π ( M ) , characterizing the representations of A ( M ) directly in terms ofobservables localized along the paths γ .i) Since γ → β ( γ ) has an inverse γ ( β ) = δ ( x, x ) β δ ( x , x ) , by eq.(2.29) one has R ( β ) = W ( x, x, γ ( β )) (2.35)with β any closed path with base point x and x almost any point in M .Since, as above, there is a γ g in the equivalence class of γ ( β ) , eqs.(2.35),(2.34)imply R ( β ) P O = W ( x, x, [ γ g ]) P O (2.36)for all closed paths β with base point x , almost all x ∈ M , γ g ∈ [ γ ( β )] .If M is compact, it may be covered by a finity family {O α } and therefore bydisjoint measurable sets O α ⊂ O α ; multiplying eq.(2.36) on the right by P O α and summing over α , one gets R ( β ) = X α W ( x α , x α , [ γ g α ]) P O α ≡ π R ( A ( { x α , g α } )) (2.37)with A ( { x α , g α } ) ∈ A ( M ) , by eq.(2.34).Since this holds for any R and all LS representations of Π( M ) are unitarilyequivalent to a π R , the observable algebra A ( M ) , which has been defined inthe sum of the LS representations of Π( M ) , contains a C ∗ -subalgebra T ( M ) isomorphic to the group algebra of π ( M ) . By eq.(2.29), the generators of T ( M ) are represented in π R by W ( x , x , [ β ]) . By Theorem 2.9, the represen-tations of T ( M ) classify the (locally normal) representations of A ( M ) .ii) If π ( M ) is finite, its representations are finite dimensional, apart frommultiplicities, and therefore norm and weak closures coincide.1iii) If π ( M ) is abelian, the Von Neumann algebra generated by the repre-sentations of T ( M ) is the algebra of Borel functions (with the weak topologydefined by Borel measures) on its spectrum, which therefore indexes the fac-torial representations.If the diffeomorphism ˜ g defined by g = g n . . . g belongs to the connectedcomponent of the identity of the diffeomorphism group of O , then C g = π R ( U (˜ g )) and the representation of the topological factors, eq.(2.34), onlyinvolves the local algebras Π( O i ) . If M is not orientable, this is not in generalthe case, and C g only belongs to the weakly closed algebra A ( O ) [21].The classification in terms of a locally Schroedinger description and uni-tary representations of π ( M ) was also obtained in [7], for the crossed productalgebra C ( M ) × Diff ( M ) c , Diff ( M ) c the universal covering group of Diff ( M ) ,under a “Lie-Reinhardt” condition, which, for the local algebras, is equivalentto the LS property. However, the starting point is different in the two ap-proaches: C ( M ) × Diff ( M ) c includes from the beginning the operations oftransport on closed paths as independent variables.On the contrary, in the present approach, the observable algebra is gen-erated by the local algebras A ( O ) , associated to topologically trivial regions,homeomorphic to R d , and therefore assumed to be represented as in ordinarySchroedinger QM, apart from unitary equivalence and multiplicity. Topologi-cal effects arise from the collection of local Schroedinger Quantum Mechanicaldescriptions . They appear in products of local observables and are character-ized by the representation of operations of physical trasport along non-trivialclosed paths, constructed as a sequence of strictly localized operations.The result for C ( M ) × Diff ( M ) c actually follows from the present analysisthrough the extension of eq.(2.25) to the representatives U ( g λv ) of all the one-parameter subgroups of Diff ( M ) c , which follows, as in [7], Proposition 4.4.,from the identification of the corresponding generators. π ( M ) In his treatment of N identical particles, Dirac proposed a solution of twoproblems: the explicit identification of the observables and the classificationof their representations. In terms of operator algebras, his strategy was:i) to consider the algebra of N distinguishable particles, which may be takenas the Weyl algebra for N degrees of freedom A W (3 N ) , or as the algebra2 B ( L ( R N )) of all bounded operators in L ( R N ) , and to define the observablealgebra A S ( N ) of N identical particles as the subalgebra invariant under thepermutation group S N , which thus plays the role of a gauge group.ii) to obtain representations of A S ( N ) by the decomposition of the unique (bythe Von Neumann theorem) representation of A W ( N ) , or of the unique nor-mal (irreducible) representation of B ( L ( R N )) . Both are defined in L ( R N ) and the decomposition is given by their commutant, generated by the gaugerepresentation of S N in L ( R N ) .Since the configuration manifold M S of N identical particles in three di-mensions (introduced above) has S N as fundamental group and R N as uni-versal covering manifold, Dirac strategy may be translated as a recepee forQuantum Mechanics on a manifold M , consisting in the following steps:i) the configuration manifold M is replaced by its universal cover, ˜ M ;ii) the observable algebra A ( M ) is identified as the subalgebra of B ( L ( ˜ M , d ˜ µ )) ( d ˜ µ locally in the equivalence class of the Lebesgue measure) invariant underthe action of π ( M ) in L ( ˜ M ) ;iii) the representations of A ( M ) are obtained by the decomposition of L ( ˜ M ) according to the representations of π ( M ) .Such an approach had a substantial influence on the treatments of QMon manifolds; it provides a natural framework for the introduction of ˜ M , alsosuggested by the Aharonov-Bohm phenomenon and naturally associated to theFunctional Integral approach, and for the associated role of π ( M ) as a gaugegroup. The question then arises about the physical justification of the choicesunderlying the Dirac strategy and about the completeness of the classificationof the representations of the observable algebra obtained in that way. A relatedquestion is whether, in that approach, the role of the fundamental group isonly that of gauge transformations.The aim of this section is to discuss the relation between Dirac strategyand the treatment of QM on manifolds discussed in Section 2, which only relieson physically motivated principles. The essential results are the following:A. If π ( M ) is amenable (i.e., it admits an invariant mean, which is alwaysthe case for finite and abelian groups), then:i) a representations of π ( M ) by observable operators is always present in L ( ˜ M ) , unitarily equivalent to its representation as a gauge group;ii) the gauge and the observable topological classification of the representationsof A ( M ) in L ( ˜ M ) are the same. The observable and gauge representations of π ( M ) associated to irreducible representions of A ( M ) coincide, apart froma complex conjugation;iii) the irreducible (locally Schroedinger) representations of A ( M ) are all con-3tained in the reduction of L ( ˜ M ) .B. If π ( M ) is not amenable, results i) and ii) still hold, but the reduction of L ( ˜ M ) does not contain all the irreducible (locally Schroedinger) representa-tions of A ( M ) , and in fact even its ordinary Schroedinger representation is not obtained in the reduction.In the derivation of the above result, the central role is played by thefollowing facts:i) the gauge representation of π ( M ) in L ( ˜ M ) is a multiple of its right regularrepresentation;ii) consequently, corresponding left regular representations can be constructedand are observable;iii) completeness of the regular representation, i.e. the presence of all theirreducible representations in its (possibly integral) decomposition is equivalent [22] to amenability of π ( M ) .Non amenable groups may appear for manifolds describing relevant physi-cal situations, e.g. in the case of a plane with n holes, n > , where the funda-mental group is freely generated by n elements and therefore non amenable.[23]For non amenable groups, even if the role of L ( ˜ M ) and of the gauge groupare lost, a modification of Dirac strategy still applies: all the irreducible LSrepresentations of A ( M ) can still be obtained by the action of coordinatesand vector fields of M on Hilbert spaces of wavefunctions on ˜ M , with suitable(non L ) scalar products . π To prove the above results, we have to confront the “extended Schroedinger”representation of A ( M ) in L ( ˜ M ) arising in the Dirac strategy with the clas-sification given by Theorem 2.9.a) The standard representation of ˜ M A standard representation of ˜ M , together with the gauge action of π ( M )) on it, is obtained as follows:Denoting, as before, by γ ( y, x ) the continuous paths in M from x to y ,given x ∈ M , ˜ M can be identified as the space of pairs ˜ M = { ξ ≡ ( x, [ γ ]); x ∈ M , γ = γ ( x, x ) } , (3.1)4with manifold structure given by the basis of open sets ˜ O ( x, [ γ ]) ≡ { ( y, [ γ y ]); y ∈ O x ⊂ M , [ γ y ] = [ γ ( y, x ) ◦ γ ( x, x )] , γ ( y, x ) ⊂ O x } , (3.2)indexed by and homeomorphic to the small neighbourhoods O x of x . Theresulting space covers M and is simply connected. π ( M )) acts on ˜ M by its right action on the paths γ ( x, x ) , r ( η ) : ( x, [ γ ]) ( x, γ ◦ η − ) , (3.3) η ∈ π ( M ) with base point x , and clearly M = ˜ M /r ( π ( M )) . (3.4) r ( π ( M )) has therefore the role of a gauge group, associated to the redundantdescription given by ˜ M . No identification of M with a subset of ˜ M is assumedat this point, nor is it implied by eqs.(3.3)(3.4).b) The standard representation of A ( M ) in L ( ˜ M ) A standard representation of A ( M ) in L ( ˜ M ) is provided by multiplicationof pairs ( x, [ γ ]) by functions of x and by the action of localized diffeomorphismsof M on ˜ M . In fact, for all O , diffeomorphisms of M localized in O , g λv , definediffeomorphisms of ˜ M , by g λv ( x, [ γ ]) ≡ ( g λv x, [ γ g ( gx, x ) ◦ γ ]) , with γ g ( gx, x ) the integral curve of v , as above (the homotopy class of thecurve in the r.h.s. only depending on [ γ ] ).On ˜ M we consider measures d ˜ µ , locally in the class of the Lebesgue mea-sure on the disks to which the ˜ O x are homeomorphic. A ( M ) is then repre-sented in L ( ˜ M , d ˜ µ ) by its “extended Schroedinger” representation ˜ π S , withaction given by equations of the form of eqs.(2.3), with K one-dimensional and V g = 1 ˜ π S ( α ) ψ ( x, [ γ ]) = α ( x ) ψ ( x, [ γ ]) (3.5) ˜ π S ( U ( g λv )) ψ ( x, [ γ ]) = ψ ( g − x, [ γ g ( g − x, x ) ◦ γ ]) J g ( x, [ γ ]) / . (3.6)As above, J g ( ξ ) ≡ [ d ˜ µ ( g − ξ ) /d ˜ µ ( ξ )] , ξ ≡ ( x, [ γ ]) .Eqs.(3.5),(3.6) reproduce the representation adopted in the Dirac approach;actually, if A ( ˜ M ) is identified with the Dirac “enlarged algebra”, one of theDirac steps, i.e. the choice of its Schroedinger representation, is forced by theuniqueness result for simply connected manifolds, Corollary 2.11.5The (Dirac) gauge representation of π ( M ) is given by its right action in ˜ M ,which defines a unitary representation ψ ( x, [ γ ]) ψ ( r ( η − )( x, [ γ ])) J η ( x, [ γ ]) / in L ( ˜ M , d ˜ µ ) , commuting with ˜ π S ( A ( M )) by eqs.(3.5),(3.6), with J η a Jaco-bian factor as in eq.(3.6).If π ( M ) is finite, since localized diffeomorphisms of M act in finite union ofsmall regions in ˜ M , ˜ π S ( A ( M )) is contained in π S ( A ( ˜ M )) and coincides withits subalgebra invariant under the gauge representation of π ( M ) .c) Equivalence of ˜ π S ( A ( M )) to a representations in L ( M ) × l ( π ( M )) In order to confront ˜ π S ( A ( M )) with the classification in Theorem 2.9, itis convenient to convert it to the form given by Proposition 2.2. Introducing δ ( x, x ) and β ( γ ) as in the proof of Lemma 2.8, eq. (2.26), β ( γ ( x, x )) ≡ δ ( x , x ) ◦ γ ( x, x ) , [ β ] depends only on [ γ ( x, x )] and belongs to π ( M ) , with x as base point.The correspondence is invertible and ˜ M can therefore be represented as ˜ M = { ˜ x = ( x, [ β ]); x ∈ M , [ β ] ∈ π ( M ) } = M × π ( M ) , (3.7)In this representation, ˜ M consists of copies of M , indexed by π ( M ) andpermuted by its action, eq.(3.3). Clearly, also the left action of π ( M ) , l ( η ) : ( x, [ β ]) ( x, [ η ◦ β ]) , (3.8)acts by permutation of the copies of M in ˜ M and commutes with the rightaction. Contrary to the right action, the left action depends on the aboveconstruction, indexed by the family of paths δ ( x, x ) , equivalently, by thecorresponding family of embeddings of M in ˜ M .Actually, as it is clear by a transformation to the first representation of ˜ M ,eq.(3.1), the above left action of π ( M ) only depends upon the family of iso-morphisms between the fundamental groups with different base points x ∈ M given by the paths δ ( x, x ) . It is immediate to verify that such isomorphismsare unique (i.e. δ ( x, x ) independent) iff π ( M ) is abelian ; in this case, theright and left actions coincide, up to an inversion.Since, apart from a set of zero measure, all points of ˜ M have a neighbour-hood with [ β ] fixed, the measure on ˜ M can be chosen, in the same measureclass, as d ˜ µ ( x, [ β ]) = dµ ( x ) , independent of [ β ] . This provides the unitaryequivalence L ( ˜ M , d ˜ µ ) ∼ L ( M , dµ ) × l ( π ( M )) . (3.9)6In this space, the representations ˜ π S ( A ( M ) takes the form, for simplicitywithout change in notation, ˜ π S ( α ) ψ ( x, [ β ]) = α ( x ) ψ ( x, [ β ]) (3.10) ˜ π S ( U ( g λv )) ψ ( x, [ β ]) = ψ ( g − x, [ θ − ( g, x ) ◦ β ]) J g ( x ) / . (3.11)with θ − ( g, x ) ≡ δ ( x , g − x ) ◦ γ g ( g − x, x ) ◦ δ ( x, x ) ∈ π ( M ) . (3.12)d) The left and right regular representation of π ( M ) in L ( ˜ M ) Eq.(3.11) gives the action of U ( g ) in terms of the left regular represen-tation R l of π ( M ) in l ( π ( M )) , ( I × R l ( θ )) ψ ( x, [ β ]) = ψ ( l ( θ − )( x, [ β ])) (3.13)On the other hand, the right action of π ( M ) in ˜ M defines the right reg-ular representation R r of π ( M ) in l ( π ( M )) and a unitary representation I × R r ( η ) in L ( ˜ M , dµ ) , ( I × R r ( η )) ψ ( x, [ β ]) = ψ ( r ( η − )( x, [ β ])) , (3.14)which, by eqs. (3.10), (3.11), commutes with ˜ π S ( A ( M )) .We recall that the left and right regular representations R l , R r of a discretegroup G in l ( G ) (with basis e g , g ∈ G ) are unitarily equivalent and generateisomorphic Von Neumann algebras N l , N r , which are the commutant one ofthe other, N ′ l = N r , N ′ r = N l . Therefore, their centres coincide and give thecentral decomposition of both R l and R r in l ( G ) . Moreover, for any centralprojection P , the left and right representations in the corresponding space P l ( G ) have P e , the identity in G , as a cyclic vector and are defined bycomplex conjugate matrix elements ( P e , R l ( g ) P e ) = ( P e , R r ( g ) P e ) . Keeping the same notation for G = π ( M ) , it follows from Theorem 2.9and eqs.(3.11),(3.13) that N l ∩ N ′ l is the centre of the Von Neumann closureof the observable algebra, ˜ π S ( A ( M )) ′′ . Since, by the irreducibility of theSchroedinger representation, the algebra generated by ˜ π S ( α ) and ˜ π S ( U ( g λv )) ,with θ ( g, x ) = 1 a.e. in x , is weakly dense in L ( M ) ), ˜ π S ( A ( M )) ′ = N r . We have therefore:7
Theorem 3.1
In the representation ˜ π S ( A ( M )) in L ( ˜ M , d ˜ µ ) ∼ L ( M , dµ ) × l ( π ( M )) the observable factors classifying the representations of A ( M ) (Theorem 2.9)are given (a.e.) by the left regular representation of π ( M ) in l ( π ( M )) ,eq.(3.13). π ( M ) also acts as a gauge group by its right regular representation, eq.(3.14),which generates the commutant of ˜ π S ( A ( M )) .The observable and gauge representation of π ( M ) are unitarily equivalent.The centres of the Von Neumann algebras generated by them coincide and givethe same reduction of ˜ π S ( A ( M )) . e) Completeness of the Dirac approach
If (and only if) π ( M ) is amenable, its regular representation contains [22]all its irreducible representations (in the weak, integral decomposition, senseif it is infinite). In its reduction, as recalled above, central projectors givesrise to left (observable) and right (gauge) representations R and ¯ R , defined bycomplex conjugate matrix elements on a cyclic vector. This gives Theorem 3.2 If π ( M ) is amenable, the central decomposition of ˜ π S ( A ( M )) in L ( ˜ M , d ˜ µ ) is given by L ( ˜ M ) = X i L ( M ) × K oi × K gi , (3.15) the sum ranging over all the irreducible representations R i of π ( M ) . π ( M ) acts as an observable group in K oi , with representation R i , and as a gauge groupin K gi , with a complex conjugate representation ¯ R i . The sum is replaced by anintegral if π ( M ) is infinite, with all the irreducible representations appearingin the support of the corresponding measure. If π ( M ) is not amenable, Eqs.(3.10),(3.11) still allow for the constructionof all the irreducible LS representations of A ( M ) on suitable, non L Hilbertspaces of wavefunctions on ˜ M .In fact, given an irreducible unitary representation R of π ( M ) in a (au-tomatically separable) Hilbert space K and a non-zero vector v ∈ K (cyclicby irreducibility), a scalar product on the vector space V of finite linear com-binations P g λ g e g , e g ∈ π ( M ) is defined by ( e g , e h ) R ≡ ( R ( g ) v, R ( h ) v ) K (3.16)8The left representation ρ l of π ( M ) in V , ρ l ( g ) e g ≡ e g ◦ h , preserves the abovescalar product since ( ρ l ( g ) e h , ρ l ( g ) e m ) R = ( R ( g ◦ h ) v, R ( g ◦ m ) v ) K = ( R ( h ) v, R ( m ) v ) K = ( e h , e m ) R . It therefore extends to the Hilbert completion V R of V , where it is unitarilyequivalent to R , the two representations being given by the same expectationson the cyclic vectors e , v .Then, on the completion H R of the space of complex functions ψ ( x, [ β ]) ,on ˜ M ∼ M × π ( M ) , with finite support in the second argument and scalarproduct ( ψ ( x, [ β ]) , χ ( x, [ β ])) ≡ Z ( ψ ( x, [ β ]) , χ ( x, [ β ])) R dµ ( x ) (3.17)eqs.(3.10), (3.11), (3.12) define a representation of A ( M ) having R as associ-ated representation of π ( M ) . By Theorem 2.9, we have therefore: Proposition 3.3
All the irreducible, locally normal, representations of A ( M ) are unitarily equivalent to representations, defined by eqs. (3.10), (3.11), (3.12),on Hilbert spaces of functions on ˜ M , with scalar product defined by eq.(3.17). If the corresponding representation of π ( M ) is finite dimensional, the scalarproduct can be chosen so that the gauge group is represented in the resultingspaces. In fact, in this case, eq.(3.16) can be substituted by ( e g , e h ) R ≡ T r K ( R ( e g ) − R ( e h )) . (3.18)The right action of π ( M ) , ρ r ( e g ) e h ≡ e h ◦ g − , then defines a unitary representation of π ( M ) in ¯ V R , and therefore in H R ,commuting with the representation of A ( M ) and complex conjugate to R .9 References [1] Goldin, G.A.: Diffeomorphism Groups and Quantum Configurations, in