Projective Limits of State Spaces II. Quantum Formalism
PProjective Limits of State SpacesII. Quantum Formalism
Suzanne Lanéry � and Thomas Thiemann Institute for Quantum Gravity, Friedrich-Alexander University Erlangen-Nürnberg, Germany Mathematics and Theoretical Physics Laboratory, François-Rabelais University of Tours, France
November 11, 2014
Abstract
In this series of papers, we investigate the projective framework initiated by Jerzy Kijowski [13] and AndrzejOkołów [19, 20], which describes the states of a quantum theory as projective families of density matrices.After discussing the formalism at the classical level in a first paper [15], the present second paper is devotedto the quantum theory. In particular, we inspect in detail how such quantum projective state spaces relateto inductive limit Hilbert spaces and to infinite tensor product constructions. Regarding the quantization ofclassical projective structures into quantum ones, we extend the results by Okołów [20], that were set upin the context of linear configuration spaces, to configuration spaces given by simply-connected Lie groups,and to holomorphic quantization of complex phase spaces.
Contents
A.1 Prequantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43A.2 Holomorphic representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.3 Position representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
B References 54 a r X i v : . [ g r- q c ] N ov Introduction
While finite dimensional symplectic manifolds are comparatively easy to quantize, allowing theformulation of rather systematic procedures (such as geometric quantization [27], of which a briefaccount is given in appendix A), quantizing infinite dimensional ones (aka. field theories) is sub-stantially more involved: typically, we have to rely on some additional insight, telling us how tobreak them down into a stack of finite dimensional truncations and how to afterwards reassemblethe pieces into a consistent quantum theory. This motivates an approach to quantum field theoryintroduced in [13, 20], where each such truncation get represented on a ‘small’ Hilbert space, beforesewing these partial quantum theories together into a projective structure. In section 2, we willreview this formalism, before inspecting how the quantum state spaces thus obtained compare tothose provided by other quantization methods, that are also assembled from ‘small’ Hilbert spaces,yet sewed in a different manner.As stressed in [15, section 1], there is a correspondence between projective limits of symplecticmanifolds on the classical side and projective limits of state spaces on the quantum side. Accord-ingly, one can try to formulate a quantization program to turn a classical projective system into aquantum one. In [20] Andrzej Okołów established such a quantization prescription: by identifyingappropriate assumptions, he was able to set up what we would call, in the terminology of [15,def. 2.15], a factorizing system of linear configuration spaces, which could then be quantized in aprojective form. He subsequently used this construction to obtain the kinematical state space of acertain theory of quantum gravity [19]. In subsection 3.1, we will extend this result to configurationspaces given as simply-connected Lie groups. This is meant as a preparation for a correspondingtreatment of loop quantum gravity, but could probably have applications to other gauge field theo-ries as well. Additionally, holomorphic quantization will be discussed in subsection 3.2, followingthe lines of geometric quantization (note that the quantum projective structure used in [16, subsec-tion 3.2] could be seen as arising from such an holomorphic quantization, although we will arriveat it from a different perspective).Note that the heuristic picture that was presented at the beginning of [15, section 2] as a justifi-cation for the projective formalism becomes a bit more involved when we go over to the quantumtheory. In particular, we had justified the directedness of the label set L by arguing that, givenany two experiments, involving observables included respectively in the labels η and η � , we shouldbe able to describe the simultaneous realization of both experiments, hence the need for a label η �� � η� η � . However, this argument obviously does not hold any more in the quantum theory, wherecomplementarity forbids the simultaneous measurement of non-commuting observables. As a wayout, we could simply decide to restrict the elementary observables (the ones that are accountedfor in L ) to a set of mutually compatible observables, but that would force us to hard-code in thetheory which observables we intend to actually measure (and to drop those that could a priori bemeasured but will not). This in turn creates severe difficulties, because we need to prescribe howto select in advance the set of those truly measured observables without spoiling predictivity (seeeg. the concerns raised in [4]).In the following, we bypass this discussion by assuming that we get the kinematical quantumstate space via the quantization of a nice classical projective limit, so that it will not be a problem,for any finite set of kinematical observables to represent them on a ‘simple’ Hilbert space, whetherthese kinematical observables can be simultaneously measured or not. On the other hand it is tobe expected that the algebra generated by a finite set of dynamical quantum observables will not be asily represented: already on the classical side we had underlined that a finite set of dynamicalclassical observables may generate an intricate Poisson-algebra (recall the comment before [15,def. 3.21]). Like in the classical formalism we must therefore expect that the exact dynamicalobservables will have to be approximated by approached ones, which in particular build a tractablealgebra. The crucial point, that goes back to Jerzy Kijowski [13], is that quantum states will be realizedas projective families of density matrices , and not as families of vector states. This is actuallya repercussion of the specific viewpoint of this formalism, namely that labels in L stand for aselection of observables (and not eg. for a selection of states). Indeed, in order to project a statefrom a more detailed partial quantum theory, represented on an Hilbert space H η � , to a coarserone, with Hilbert space H η , we need a map that will retain from a state only the features neededto compute expectation values of the observables on H η : this is what the partial trace on a tensorproduct factor accomplishes but it can only be defined as a map between density matrices (thepartial trace of a pure state can be a mixed state and conversely).While this has been previously rather seen as a weakness of the construction (see the discussionin [20, section 6.2]), we argue that such a generalized framework will be, in practice, indistinguish-able from a more traditional theory on a Hilbert space: given a particular experiment, we should beable to select an η such that everything takes place within H η . Moreover it seems advantageous tostart with a very large kinematical state space, thus avoiding the inherent arbitrariness of restrictingit a priori to some particular subspace, and to wait until there is a real need for such a restriction,together with clear requirements on how to perform it (in the light of [15, section 3], this could bebecause we are forced to consider only the states for which the regularization scheme, needed toimplement the dynamics, converges).In order to clarify the claim made above, that quantum state spaces described as projective limitstend to be ‘bigger’, we will examine precisely how they compare to those built on inductive limits ofHilbert spaces (theorem 2.9) or on infinite tensor products (theorem 2.11): these are constructionsthat incorporate ingredients somewhat similar to the projective approach, but differ notably byseeking a global Hilbert space for the quantum theory. In this subsection, we present the projective approach to quantum field theory [13, 20], formulatingit in a form as close as possible to the classical formalism set up in [15, section 2]. Indeed, theprojective systems of quantum state spaces described here can be seen as the direct equivalents η �� H η �� →η � ⊗ H η � H η �� →η � ⊗ H η � →η ⊗ H η H η �� →η ⊗ H η φ η �� →η � φ η �� →η φ η � →η φ η �� →η � →η Figure 2.1 – Three-spaces consistency for projective systems of quantum state spacesof the factorizing systems we had on the classical side [15, subsection 2.3], replacing Cartesianproducts of classical phase spaces by tensor products of Hilbert spaces, in accordance to the basicprinciples of quantum mechanics (in particular, the three-spaces consistency from [15, fig. 2.1] isstraightforwardly transformed into its quantum version illustrated in fig. 2.1).While we had convinced ourselves that the factorizing systems are quite generic among theprojective systems of classical phase spaces (see the argument laid in [15, prop. 2.10]), their quantumcounterparts seem to be even more broadly applicable (as we will show in the case of loop quantumcosmology in forthcoming work, there are systems that are classically not of the factorizing type,yet do admit a quantum formulation within the framework presented below).
Definition 2.1
A projective system of quantum state spaces is a quintuple:� L � � H η � η∈ L � � H η � →η � η � η � � � φ η � →η � η � η � � � φ η �� →η � →η � η � η � � η �� �where: L is a preordered, directed set (we denote the pre-order by � , its inverse by � ); � H η � η∈ L is a family of Hilbert spaces indexed by L ; � H η � →η � η � η � is a family of Hilbert spaces indexed by {η� η � ∈ L | η � η � } , such that dim( H η→η ) =1 for all η ∈ L ; � φ η � →η � η � η � is a family of isomorphisms of Hilbert spaces φ η � →η : H η � → H η � →η ⊗ H η indexedby {η� η � ∈ L | η � η � } such that φ η→η is trivial (by isomorphism of Hilbert spaces we mean a(bijective) unitary map between Hilbert spaces); � φ η �� →η � →η � η � η � � η �� is a family of isomorphisms of Hilbert spaces φ η �� →η � →η : H η �� →η → H η �� →η � ⊗ H η � →η indexed by {η� η � � η �� ∈ L | η � η � � η �� } such that φ η �� →η � →η is trivial whenever two labelsamong η� η � � η �� are equal and: η � η � � η �� ∈ L � ( φ η �� →η � →η ⊗ id H η ) ◦ φ η �� →η = (id H η��→η� ⊗ φ η � →η ) ◦ φ η �� →η � . (2.1. )Whenever possible, we will use the shortened notation ( L � H � φ ) ⊗ instead of � L � � H η � η∈ L � � H η � →η � η � η � � � φ η � →η � η � η � � � φ η �� →η � →η � η � η � � η �� �. Definition 2.2
Let ( L � H � φ ) ⊗ be a projective system of quantum state spaces. For η ∈ L , we define S η the space of (self-adjoint) positive semi-definite, traceclass operators on H η and S η the space ofdensity matrices: S η = � ρ η ∈ S η �� Tr H η ρ η = 1� .For η � η � ∈ L , we define:Tr η � →η : S η � → S η ρ η � �→ Tr H η�→η ( φ η � →η ◦ ρ η � ◦ φ − η � →η ) .From eq. (2.1. ), we have: ∀η � η � � η �� ∈ L � Tr η � →η ◦ Tr η �� →η � = Tr η �� →η .Hence, � L � � S η � η∈ L � �Tr η � →η � η � η � � forms a projective system and we denote its projective limitby S ⊗ ( L � H �φ ) . The maps Tr η � →η being linear under conical combinations (ie. under addition andmultiplication by positive reals), S ⊗ ( L � H �φ ) forms a cone (ie. we can equip it with a notion of additionand positive multiplication).Now, for all η � η � ∈ L � Tr η � →η � S η � � = S η , hence � L � � S η � η∈ L � �Tr η � →η � η � η � � also forms aprojective system. Accordingly, a state on ( L � H � φ ) ⊗ is a family � ρ η � η∈ L such that ∀η ∈ L , ρ η isa density matrix over H η and ∀η � η � � Tr η � →η ρ η � = ρ η . We denote the space of states by S ⊗ ( L � H �φ ) . The projective structure conferred to the space of states comes with a natural inductive structurefor the observables [20, section 6.2]. Thus, we can define a C ∗ -algebra A ⊗ ( L � H �φ ) of bounded opera-tors, as the inductive limit of the algebras A η that live over each H η . Then, the states defined abovecan be seen as states on A ⊗ ( L � H �φ ) in the sense of [10, part III, def. 2.2.8]. Looking at states over a C ∗ -algebra is in a sense more fundamental than looking at density matrices over a specific repre-sentation of this algebra. Indeed, any such representation can be split into cyclic components, andeach cyclic component arises from a state over the algebra via the GNS construction. Reciprocally,any state ρ over the algebra defines a corresponding GNS representation, and the states that can berepresented as density matrices over this representation form the folium of ρ (see also prop. 2.8 onthis point). The irreducible representations are precisely the ones that arise from pure states (statesthat cannot be written as a non trivial convex superposition of other states) [10, part III, theorem2.2.17], and Fell’s theorem [6] tells us that, whenever a state yields a faithful GNS representation,its folium will be, in a definite sense, dense in the set of all states over the algebra.Note, however, that the framework presented here differs substantially from the one of AlgebraicQuantum Field Theory [10, part III]. In both cases one is looking at states over an inductive limit ∗ -algebra. But here the building blocks of our algebra of observables will be in practice very small algebras: each A η , instead of being meant to include all the operators needed to interpretany arbitrary experiment taking place in some given region of spacetime, should be thought as onlycontaining the operators needed for the description of finitely many experiments. More deeply, thepurpose of giving the algebra of observables an inductive limit structure is in our case not so muchto encode additional physically essential information (eg. the localization of the operators andassociated causal structure) but rather to arrive at a description of the space of states as concreteand as convenient as possible (by building it from small representations that are well under controland suitable for calculation). Of course, we could combine both aspects, by decorating the projectivestructure with this extra information: for example, we could map a region in spacetime to the setof all η that can be seen as contained in this region. Definition 2.3
We consider the same objects as in def. 2.2. For η ∈ L , we denote by A η thealgebra of bounded operators on H η and, for η � η � ∈ L , we define: ι η � ←η : A η → A η � A η �→ φ − η � →η ◦ �id H η�→η ⊗ A η � ◦ φ η � →η .By definition ι η � ←η is injective and, from eq. (2.1. ), we have: ∀η � η � � η �� � ι η �� ←η � ◦ ι η � ←η = ι η �� ←η . (2.3. )Accordingly, an operator over S ⊗ ( L � H �φ ) is an equivalence class in � η∈ L A η for the equivalence relationdefined by: ∀η� η � ∈ L � ∀A η ∈ A η � ∀A η � ∈ A η � �A η ∼ A η � ⇔ � ∃η �� ∈ L / η � η �� � η � � η �� & ι η �� ←η � A η � = ι η �� ←η � � A η � �� (2.3. )The space of operators over S ⊗ ( L � H �φ ) will be denoted by A ⊗ ( L � H �φ ) . For A = [ A η ] ∼ ∈ A ⊗ ( L � H �φ ) and ρ = ( ρ η ) η∈ L ∈ S ⊗ ( L � H �φ ) , we can define Tr ρA := Tr H η ρ η A η . The definition of the equivalence relationensures that this is well-defined. Proposition 2.4
For η � η � ∈ L , the map ι η � ←η is an injective C ∗ -algebra morphism (ie. an injective,isometric ∗ -algebra morphism). Hence, A ⊗ ( L � H �φ ) can be equipped with a normed ∗ -algebra structureas an inductive limit of C ∗ -algebras. And denoting by A ⊗ ( L � H �φ ) the completion of A ⊗ ( L � H �φ ) withrespect to the operator norm, A ⊗ ( L � H �φ ) is a C ∗ -algebra.Then, for all ρ ∈ S ⊗ ( L � H �φ ) , Tr ( ρ · ) can be extended by continuity as a state over A ⊗ ( L � H �φ ) . Proof
That ι η � ←η is a C ∗ -algebra morphism for any η � η � is ensured by φ η � →η being a Hilbertspace isomorphism and by the properties of the tensor product of operators.Next, let ρ ∈ S ⊗ ( L � H �φ ) , and let [ A η ] ∼ � [ B η � ] ∼ ∈ A ⊗ ( L � H �φ ) . For any η �� � η � � η and any �� � ∈ C , wehave: Tr ρ ( � A + � B ) = Tr H η�� ρ η �� � � ι η �� ←η ( A η ) + � ι η �� ←η � ( B η � )� = � Tr ( ρ A ) + � Tr ( ρ B ) ; . Tr ( ρ A ) = Tr H η ( ρ η A η ) � �A η � = �A� ; Tr ( ρ ) = Tr H η ( ρ η id H η ) = 1 ; Tr ( ρ A + A ) = Tr H η ( ρ η A + η A η ) � � Proposition 2.5
For η ∈ L , we denote by O η the algebra of densely defined (possibly unbounded)normal operators on H η . An observable over a projective limit of quantum state spaces S ⊗ ( L � H �φ ) isdefined as an equivalence class in � η∈ L O η in analogy to def. 2.3.The space of observables over S ⊗ ( L � H �φ ) will be denoted by O ⊗ ( L � H �φ ) . For O = [ O η ] ∼ ∈ O ⊗ ( L � H �φ ) , wecan define the spectrum ς ( O ) of O as the spectrum ς ( O η ) of any representative O η of O and, for W a measurable subset of ς ( O ), we can define I W ( O ) as the equivalence class [ I W ( O η )] ∼ ∈ A ⊗ ( L � H �φ ) ofthe spectral projector I W ( O η ).Hence, for a state ρ = ( ρ η ) η∈ L ∈ S ⊗ ( L � H �φ ) , we can define the probability of measuring O in W as ρ [ O ∈ W ] := Tr ρ I W ( O ). Proof ς ( O η ) being independent of the choice of a representative O η comes from: ∀η � � η� ς � φ − η � →η ◦ �id H η�→η ⊗ O η � ◦ φ η � →η � = ς � O η � ,where we used [21, theorem VIII.33] together with the fact that φ η � →η is an Hilbert space isomorphism.That O η ∼ O η � implies I W ( O η ) ∼ I W ( O η � ) comes from: ∀η � � η� I W � φ − η � →η ◦ �id H η�→η ⊗ O η � ◦ φ η � →η � = φ − η � →η ◦ �id H η�→η ⊗ I W ( O η )� ◦ φ η � →η . � In order to investigate the relations between the spaces of quantum states assembled this way andmore standard constructions, we will use the same tool as we used to make the connection betweenclassical projective structures and infinite dimensional symplectic manifolds, namely extensionsand restrictions of the label set. As in [15, subsection 2.2], the strategy will be to extend the labelset by adding to it a greatest element (associated to the ‘big’ Hilbert space to which we want tomake contact), before restricting ourselves to this greatest element alone (thus ending with a trivialprojective system that can be identified with the target Hilbert space). A bit unintuitively, the non-trivial switch of state space occurs during the first step: this is because a greatest element forms bydefinition a cofinal part, which makes the second step innocuous.
Proposition 2.6
Let ( L � H � φ ) ⊗ be a projective system of quantum state spaces and let L � be adirected subset of L . We define the map: : S ⊗ ( L � H �φ ) → S ⊗ ( L � � H �φ ) � ρ η � η∈ L �→ � ρ η � η∈ L � . σ is conically linear (ie. compatible with addition and multiplication by positive scalars).Moreover, we have a map α : A ⊗ ( L � � H �φ ) → A ⊗ ( L � H �φ ) such that: ∀ρ ∈ S ⊗ ( L � H �φ ) � ∀A ∈ A ⊗ ( L � � H �φ ) � Tr ( ρ α ( A )) = Tr ( σ ( ρ ) A ) , (2.6. )and α is a C ∗ -algebra morphism.If L � is cofinal in L , we have in addition that σ and α are bijective maps. Proof
The proof works in the same way as in the classical case [15, prop. 2.5]. The conical linearityof σ and morphism property of α comes from their definition (with α being defined in analogy to theclassical case). Eq. (2.6. ) can be first checked for S ⊗ ( L � H �φ ) and A ( L � � H �φ ) and expanded by continuityand conical linearity. � Proposition 2.7
In particular, if L admits a greatest element Λ, there exist bijective maps σ : S ⊗ ( L � H �φ ) → S Λ ( S Λ being the space of (self-adjoint) positive semi-definite, traceclass operators over H Λ ) and α : A Λ → A ⊗ ( L � H �φ ) such that ∀ρ ∈ S ⊗ ( L � H �φ ) � ∀A ∈ A Λ � Tr ( ρ α ( A )) = Tr Λ ( σ ( ρ ) A ). Proof
This is an application of prop. 2.6 for the cofinal part L � = { Λ } of L , using the obviousidentification of S ⊗ ( { Λ }� H �φ ) with S Λ and A ⊗ ( { Λ }� H �φ ) with A Λ . � If we choose a particular state in a projective limit of quantum state spaces, we can use itas a vacuum to construct a corresponding GNS representation of the inductive limit algebra ofobservables [20, section 6.2], and this representation will naturally inherit a structure of inductivelimit Hilbert space (like the Hilbert space used as starting point in LQG, see [1]). We will specializein the case where the vacuum state we are using projects as a pure state on every H η . Whetherthere exists such a pure projective state of course depends on the specific projective structure underconsideration, however we will be interested in situations where the natural vacuum turns out tobe of this type (notably in prop. 3.5 and [16, prop. 3.17]). In this case, the inductive limit Hilbertspace is obtained from a collection of ‘reference states’ ζ η � →η ∈ H η � →η , that allows us to see thetensor product factor H η as a vector subspace in H η � . Any density matrix over such an inductivelimit H ζ can then be unambiguously mapped to a state in the projective limit, but the conversetypically does not hold, and in fact, we can formulate a handy condition to check if a state on theprojective structure has its counterpart as a density matrix on H ζ . Proposition 2.8
Let ( L � H � φ ) ⊗ be a projective system of quantum state spaces and let ρ =� ρ η � η∈ L ∈ S ⊗ ( L � H �φ ) . For all η ∈ L , ρ η is a state over A η and we denote by H GNS ρ η � ( · ) GNS the GNSrepresentation constructed from this state [10, section III.2.2].Then, for all η � η � ∈ L , there exists an injective linear map τ η � ←η : H GNS ρ η → H GNS ρ η� . τ η � ←η isisometric onto its image and satisfies: ∀A η ∈ A η � τ η � ←η ◦ A GNS η = � ι η � ←η ( A η )� GNS ◦ τ η � ←η . (2.8. ) oreover, for all η � η � � η �� ∈ L , we have τ η �� ←η � ◦ τ η � ←η = τ η �� ←η , hence we can define an Hilbertspace H GNS ρ as (the completion of ) the inductive limit of � L � � H GNS ρ η � η∈ L � � τ η � ←η � η � η � � and H GNS ρ can be identified with the GNS representation of A ⊗ ( L � H �φ ) arising from the state ρ (prop. 2.4).If in addition there exists, for all η ∈ L , a vector ζ η ∈ H η such that ρ η = |ζ η � � ζ η | , then H GNS ρ η ≈ H η and for all η � η � there exists a vector ζ η � →η ∈ H η � →η such that the map τ η � ←η is givenby: ∀ψ ∈ H η � τ η � ←η ( ψ ) = φ − η � →η � ζ η � →η ⊗ ψ � . (2.8. ) Proof
Explicit description of H GNS ρ η . Let η ∈ L . By applying the spectral theorem to the (self-adjoint) positive semi-definite, traceclass, normalized operator ρ η , there exist N η ∈ N � {∞} , anorthonormal family � ζ ( k ) η � k � N η in H η and a family of strictly positive reals � � ( k ) η � k � N η such that: ρ η = � k � N η � ( k ) η �� ζ ( k ) η � � ζ ( k ) η �� ,with � k � N η � ( k ) η = 1.We define the map Ψ ρ η by:Ψ ρ η : A η → H + η ⊗ H η A �→ � k � N η � � ( k ) η � ζ ( k ) η ��� ⊗ ��� A ζ ( k ) η � ,where H + η is the topological dual of H η equipped with its natural Hilbert space structure. The mapΨ ρ η is well-defined, for we have:� k � N η ����� � ( k ) η � ζ ( k ) η �� ⊗ �� A ζ ( k ) η ����� = � k � N η � ( k ) η ���� A ζ ( k ) η ��� � �A� . (2.8. )Moreover, it has following properties: Ψ ρ η is C -linear; ∀A� B ∈ A η � Ψ ρ η ( A B ) = �id H + η ⊗ A � Ψ ρ η ( B ) ; ∀A� B ∈ A η � �Ψ ρ η ( A ) � Ψ ρ η ( B )� H + η ⊗ H η == � k�k � � N η � � ( k ) η � � ( k � ) η � ζ ( k � ) η ��� ζ ( k ) η � ⊗ � A ζ ( k ) η ��� B ζ ( k � ) η �= � k � N η � ( k ) η ⊗ � ζ ( k ) η �� A + B ζ ( k ) η � = Tr H η ρ η A + B ; Ψ ρ η � A η � = Vect �� � ( k ) η � ζ ( k ) η ��� ⊗ |ψ� | k � N η � ψ ∈ H η � ‘ ⊂ ’: by definition of Ψ ρ η ; ‘ ⊃ ’: by considering operators of the form �� ψ � � ζ ( k ) η �� )= Vect �� � ( k ) η � ζ ( k ) η ��� | k � N η � ⊗ H η =: K ρ η ⊗ H η .Therefore, we can identify H GNS ρ η with K ρ η ⊗ H η and we have: ∀A ∈ A η � A GNS = id K ρη ⊗ A . Definition of the injections τ η � ←η . Let η � η � ∈ L . We define a C -linear map: τ η � ←η : Vect �� � ( k ) η � ζ ( k ) η �� | k � N η � ⊗ H η → Vect �� � ( k � ) η � � ζ ( k � ) η � ��� | k � � N η � � ⊗ H η � by: ∀k � N η � ∀ψ ∈ H η � τ η � ←η �� � ( k ) η � ζ ( k ) η �� ⊗ |ψ� � == � k � � N η� �� � φ − η � →η � � ⊗ ζ ( k ) η � ��� ζ ( k � ) η � � � � ( k � ) η � � ζ ( k � ) η � ��� ⊗ �� φ − η � →η ( � ⊗ ψ )�= � � � √ρ η � ◦ φ − η � →η � � ⊗ ζ ( k ) η ��� ⊗ �� φ − η � →η ( � ⊗ ψ )� ,where ( � ) is an orthonormal basis of H η � →η and √ρ η � is defined via spectral resolution. We have: ∀k � N η � ∀ψ ∈ H η � ���� τ η � ←η �� � ( k ) η � ζ ( k ) η �� ⊗ |ψ� ����� == � ��� � � √ρ η � ◦ φ − η � →η � � ⊗ ζ ( k ) η � �� √ρ η � ◦ φ − η � →η � � � ⊗ ζ ( k ) η �� � φ − η � →η � � � ⊗ ψ � �� φ − η � →η ( � ⊗ ψ )�= � � � φ − η � →η � � ⊗ ζ ( k ) η � �� ρ η � ◦ φ − η � →η � � ⊗ ζ ( k ) η �� �ψ� = � ζ ( k ) η �� �Tr η � →η ρ η � � ζ ( k ) η � �ψ� = � ( k ) η �ψ� = ����� � ( k ) η � ζ ( k ) η �� ⊗ |ψ� ���� .Therefore τ η � ←η is well-defined and can be extended as an injection H GNS ρ η → H GNS ρ η� , which isisometric onto its image. For A η ∈ A η we can check directly that eq. (2.8. ) is satisfied, using A GNS η = id K ρη ⊗ A η .Next, we have:� k � N η Tr H η� ρ η � ι η � ←η ��� ζ ( k ) η � � ζ ( k ) η ��� = Tr H η ρ η = 1 � k � N η �k � � N η� �� � ( k � ) η � ���� ζ ( k � ) η � ��� φ − η � →η � � ⊗ ζ ( k ) η ����� ,but since all � ( k � ) η � are strictly positive and � k � � N η� � ( k � ) η � = 1, this implies: ∀k � � N η � � � k � N η �� ���� ζ ( k � ) η � ��� φ − η � →η � � ⊗ ζ ( k ) η ����� = 1 = ��� ζ ( k � ) η � ��� ,and therefore: ∀k � � N η � � ζ ( k � ) η � = � k � N η �� � φ − η � →η � � ⊗ ζ ( k ) η � ��� ζ ( k � ) η � � �� φ − η � →η � � ⊗ ζ ( k ) η �� . (2.8. )With this we can now prove: ∀A η ∈ A η � τ η � ←η ◦ Ψ ρ η ( A η ) == � k � N η �k � � N η� �� � φ − η � →η � � ⊗ ζ ( k ) η � ��� ζ ( k � ) η � � � � ( k � ) η � � ζ ( k � ) η � ��� ⊗ �� φ − η � →η � � ⊗ A η ζ ( k ) η ��= � k � � N η� � � ( k � ) η � � ζ ( k � ) η � ��� ⊗ ��� ι η � ←η � A η � ζ ( k � ) η � �= Ψ ρ η� ◦ ι η � ←η � A η � . (2.8. )Thus, for η � η � � η �� ∈ L , τ η �� ←η � ◦ τ η � ←η = τ η �� ←η follows from eq. (2.3. ) together with point2.8. above. Inductive limit Hilbert space as GNS representation of A ⊗ ( L � H �φ ) . Let H GNS ρ be (the completion of )the inductive limit of � L � � H GNS ρ η � η∈ L � � τ η � ←η � η � η � � . Eq. (2.8. ) ensures that we can consistentlyassemble the family of maps �Ψ ρ η � η∈ L into a map Ψ ρ : A ⊗ ( L � H �φ ) → H GNS ρ , and, by eq. (2.8. ), wecan extend this map to A ⊗ ( L � H �φ ) . Now, the properties of the individual Ψ ρ η ensure that Ψ ρ has thefollowing properties: Ψ ρ is C -linear; ∀η� η � ∈ L � ∀ [ A η ] ∼ � [ B η � ] ∼ ∈ A ⊗ ( L � H �φ ) � ∀η �� � η� η � � Ψ ρ �[ A η �� B η �� ] ∼ � = � A GNS η �� Ψ ρ η�� ( B η �� )� ∼ ; ∀A� B ∈ A ⊗ ( L � H �φ ) � � Ψ ρ ( A ) � Ψ ρ ( B ) � H GNS ρ = Tr ρ A + B ; Ψ ρ � A ⊗ ( L � H �φ ) � = H GNS ρ .Therefore, we can identify H GNS ρ with the GNS representation of A ⊗ ( L � H �φ ) arising from the state ρ . Note.
This result could be proved at a more abstract level, by directly using eq. (2.8. ) to defineτ η � ←η . Here we gave the explicit expressions as an added bonus. ure projective state. We now assume that for all η ∈ L N η = 0 and we define ∀η ∈ L � ζ η := ζ (0) η .Thus ∀η ∈ L � K ρ η ≈ C and therefore H GNS ρ η ≈ H η . Then, for η � η � , eq. (2.8. ) becomes: ζ η � = � � BON of H η�→η � φ − η � →η � � ⊗ ζ η � �� ζ η � � �� φ − η � →η � � ⊗ ζ η �� ,hence, defining ζ η � →η := � � BON of H η�→η � φ − η � →η � � ⊗ ζ η � �� ζ η � � |�� , we get ζ η � = φ − η � →η � ζ η � →η ⊗ ζ η � .Inserting into the definition of τ η � ←η and applying the identification K ρ η ≈ C provides eq. (2.8. ). � Theorem 2.9
Let ( L � H � φ ) ⊗ be a projective system of quantum state spaces and suppose thereexists a family of vectors � ζ η � →η � η � η � such that: ∀η � η � � ζ η � →η ∈ H η � →η & �ζ η � →η � = 1; ∀η � η � � η �� � φ η �� →η � →η ( ζ η �� →η ) = ζ η �� →η � ⊗ ζ η � →η .We define an Hilbert space H ζ as (the completion of ) the inductive limit of � L � � H η � η∈ L � � τ η � ←η � η � η � �,where the injective maps τ η � ←η are defined as: ∀η � η � ∈ L � τ η � ←η : H η → H η � ψ �→ φ − η � →η � ζ η � →η ⊗ ψ � .Then, there exist maps σ : S ζ → S ⊗ ( L � H �φ ) and α : A ⊗ ( L � H �φ ) → A ζ ( S ζ being the space of (self-adjoint)positive semi-definite, traceclass operators over H ζ and A ζ the algebra of bounded operators on H ζ ) such that: ∀ρ ∈ S ζ � ∀A ∈ A ⊗ ( L � H �φ ) � Tr H ζ ( ρ α ( A )) = Tr ( σ ( ρ ) A ); σ is injective; σ � S ζ � = �� ρ η � η∈ L ����� sup η∈ L � inf η � � η Tr H η� � ρ η � Θ η � |η �� = 1� ,where S ζ is the space of density matrices over H ζ and:Θ η � |η := φ − η � →η ◦ � |ζ η � →η ��ζ η � →η | ⊗ id H η � ◦ φ η � →η . We will, in the proof below, rely heavily on the so-called trace norm, which, for a positivetraceclass operator is just its trace. The reason why this is the appropriate norm for our purpose istwofold. First, it plays nicely with the partial traces, since the trace norm of a partial trace of ρ isalways bounded by the trace norm of ρ itself (it is obviously equal in the case of a positive ρ , andthe bound follows by decomposing a general ρ into positive and negative parts, or by invoking thenext point). Second, it supports the physical interpretation of quantum states, revolving around theevaluation of observable expectation values, since the trace norm of ρ is precisely the norm of thecontinuous linear functional A �→ Tr ρ A defined on the algebra of bounded operators (this can beproven using the polar decomposition of ρ [21, theorem VI.10] ). An additional advantage is thatthe traceclass operators form a Banach space with respect to this norm [22]. emma 2.10 Let H be an Hilbert space. For any traceclass operator ρ on H we define its tracenorm �ρ� (aka. Schatten-norm with � = 1 [22]) by: �ρ� := Tr � ρ + ρ .Let ( J α ) α be a family of closed vector subspaces of H , forming a directed preordered set underinclusion, and such that: H = � α J α .Define Θ α to be the orthogonal projection on J α .The following statements hold: for any (self-adjoint) positive semi-definite, traceclass operator ρ on H , the net (Θ α ρ Θ α ) α converges in trace norm to ρ ; if ( ρ α ) α is a net of (self-adjoint) positive semi-definite, traceclass operators on H such that: ∀α� α � / J α ⊂ J α � � ρ α = Θ α ρ α � Θ α ,and if sup α Tr ρ α = � < ∞ , then there exists a (self-adjoint) positive semi-definite, traceclassoperator ρ on H such that ρ α = Θ α ρ Θ α and Tr ρ = � . Proof
The trace norm is well-defined, since for any traceclass operator ρ on H , ρ + ρ is a self-adjointpositive semi-definite operator on H , so its square-root can be defined by spectral resolution, andthis square-root is traceclass (by definition of ρ being traceclass). Statement 2.10. . Let ρ be a (self-adjoint) positive semi-definite, traceclass operator on H . Thereexist real numbers � k � k ∈ N ) and vectors ψ k in H with �ψ k � = 1 such that: ρ = � k � k |ψ k � � ψ k | & � k � k = Tr ρ � α : �ρ − Θ α ρ Θ α � � � k � k �� |ψ k �� ψ k | − | Θ α ψ k �� Θ α ψ k | �� .Let ε > N ∈ N such that:� k>N � k � ε H is the completion of the union of the J α (which are directed with respect to inclusionsubordinate to the labels α ), there exists, for every k � N , an α k such that �ψ k − Θ α k ψ k � � ε .And since the family ( J α ) α is directed under inclusion, there exists α such that � k � N J α k ⊂ J α . Let α � such that J α � ⊃ J α . Then, we have: ∀k � N� �ψ k − Θ α � ψ k � � ε k � N , the non-zero eigenvalues of |ψ k � � ψ k | − | Θ α � ψ k � � Θ α � ψ k | are: ± = µ ± µ �1 − µ µ := �ψ k − Θ α � ψ k � ,each one with multiplicity 1. So from µ � ε µ � �ψ k � = 1), we have:�� |ψ k � � ψ k | − | Θ α � ψ k � � Θ α � ψ k | �� = |λ + | + |λ − | � µ � ε �ρ − Θ α � ρ Θ α � � � �� k � N � k ε ε � ε . Statement 2.10. . Since the family ( J α ) α is directed and each J α is a vector subspace of H , J := � α J α is a vector subspace of H and, by hypothesis, J is dense in H .For any ψ� ψ � ∈ J , we define: ρ ψ�ψ � := �ψ� ρ α ψ � � for α such that ψ� ψ � ∈ J α . ρ ψ�ψ � is well-defined, since there exists α ψ , resp. α ψ � , such that ψ ∈ J α ψ , resp. ψ ∈ J α ψ , hence thereexists α such that ψ� ψ � ∈ J α ; and if α � is an other index such that ψ� ψ � ∈ J α � , then there exists α �� with J α � J α � ⊂ J α �� , so we have: �ψ� ρ α ψ � � = � Θ α ψ� ρ α �� Θ α ψ � � = �ψ� ρ α �� ψ � � = � Θ α � ψ� ρ α �� Θ α � ψ � � = �ψ� ρ α � ψ � � .Moreover, ( ψ� ψ � ) �→ ρ ψ�ψ � is a positive semi-definite, sesquilinear form on J and: ∀ψ� ψ � ∈ J � |ρ ψ�ψ � | � � �ψ� �ψ � � ,hence, there exists a positive semi-definite, self-adjoint, bounded operator ρ on H , such that: ∀ψ� ψ � ∈ J � ρ ψ�ψ � := �ψ� ρ ψ � � .So, for any α and any ψ� ψ � ∈ H , we have: �ψ� Θ α ρ Θ α ψ � � = � Θ α ψ� ρ Θ α ψ � � = � Θ α ψ� ρ α Θ α ψ � � = �ψ� ρ α ψ � � ,therefore ρ α = Θ α ρ Θ α .Now, suppose ρ would not be traceclass. Then, there would exist a finite orthonormal family ψ k , k ∈ { � � � � � N} such that: N � k =1 �ψ k � ρ ψ k � > � + 1 ,Next, like in the proof of statement 2.10. , we can find α satisfying: ∀k � N� ∀α � / J α � ⊃ J α � �ψ k − Θ α � ψ k � � N (2 � + 1) .Hence, using �ρ� � � (where � · � is the operator norm): N � k =1 �ψ k � ρ ψ k � � N � k =1 � � Θ α � ψ k � ρ Θ α � ψ k � + 2 � + 1 N (2 � + 1) � Tr ρ α � + 1 � � + 1 ,which would then be contradictory.Lastly, ρ being a (self-adjoint) positive semi-definite, traceclass operator on H , the first statementimplies that the net ( ρ α ) α converges in trace norm to ρ , hence lim α Tr ρ α = Tr ρ . So Tr ρ � � and, forany ε >
0, there exists α ε such that: ∀α � / J α � ⊃ J α ε � Tr ρ α � � Tr ρ + ε .Therefore, for any α , choosing α � such that J α ∪ J α ε ⊂ J α � , we have:Tr ρ α � Tr ρ α � � Tr ρ + ε .Thus, � � Tr ρ , hence Tr ρ = � . � Proof of theorem 2.9
Existence of σ and α satisfying 2.9. . The inductive limit defining H ζ isconsistent since for all η � η � � η �� ∈ L and for all ψ ∈ H η : τ η �� ←η � ◦ τ η � ←η ( ψ ) = φ − η �� →η � ◦ �id H η��→η� ⊗ φ η � →η � − � ζ η �� →η � ⊗ � ζ η � →η ⊗ ψ ��= φ − η �� →η ◦ � φ η �� →η � →η ⊗ id H η � − �� ζ η �� →η � ⊗ ζ η � →η � ⊗ ψ �= φ − η �� →η � ζ η �� →η ⊗ ψ � = τ η �� ←η ( ψ ) .Additionally, for all η ∈ L , we call τ ζ←η the injective map H η → H ζ .Let η ∈ L . We define an Hilbert space H ζ→η as (the completion of ) the inductive limit of� {κ ∈ L | κ � η} � � H κ→η � κ � η � � τ κ � ←κ→η � κ � � κ � η �, where the injective maps τ κ � ←κ→η are definedas: ∀κ � � κ � η� τ κ � ←κ→η : H κ→η → H κ � →η ψ �→ φ − κ � →κ→η ( ζ κ � →κ ⊗ ψ ) .We can prove that ∀κ �� � κ � � κ � η� τ κ �� ←κ � →η ◦ τ κ � ←κ→η = τ κ �� ←κ→η in a way similar to above,using:�id H κ��→κ� ⊗ φ κ � →κ→η � ◦ φ κ �� →κ � →η = � φ κ �� →κ � →κ ⊗ id H κ→η � ◦ φ κ �� →κ→η , (2.9. )which can be proved by acting on both sides with � · ⊗ id H η � ◦ φ κ �� →η and using repeatedly eq. (2.1. ).Additionally, for all κ � η , we call τ ζ←κ→η the injective map H κ→η → H ζ→η .Then, we can combine the isomorphisms φ κ→η : H κ → H κ→η ⊗ H η defined for κ � η into anisomorphism φ ζ→η : H ζ → H ζ→η ⊗ H η , for we have, for all κ � � κ � η : φ κ � →η ◦ τ κ � ←κ = � τ κ � ←κ→η ⊗ id H η � ◦ φ κ→η ,as can be shown using eq. (2.1. ).Similarly, we can combine the isomorphisms φ κ→η � →η : H κ→η → H κ→η � ⊗ H η � →η defined for κ � η � � η into an isomorphism φ ζ→η � →η : H ζ→η → H ζ→η � ⊗ H η � →η , for we have, for all κ � � κ � η � � η : κ � →η � →η ◦ τ κ � ←κ→η = � τ κ � ←κ→η � ⊗ id H η�→η � ◦ φ κ→η � →η ,as can be shown using eq. (2.9. ).Moreover, we have (again from eq. (2.1. ) ):� φ ζ→η � →η ⊗ id H η � ◦ φ ζ→η = �id H ζ→η� ⊗ φ η � →η � ◦ φ ζ→η � .Now, if we define L ζ := L � {ζ} and extend the preorder on L to L ζ by requiring ∀η ∈ L � η ≺ ζ , we can therefore assemble these objects into a projective system of quantum state spaces� L ζ � H � φ � ⊗ .Using prop. 2.6, we then have maps σ � : S ⊗ ( L �{ζ}� H �φ ) → S ⊗ ( L � H �φ ) and α � : A ⊗ ( L � H �φ ) → A ⊗ ( L �{ζ}� H �φ ) ,and, using prop. 2.7, we have maps σ − ζ : S ζ → S ⊗ ( L �{ζ}� H �φ ) and α − ζ : A ⊗ ( L �{ζ}� H �φ ) → A ζ . Hence, wedefine: σ := σ � ◦ σ − ζ & α := α − ζ ◦ α � . Properties of σ (2.9. and 2.9. ). For all η � η � , we define:Θ η � |η := φ − η � →η ◦ � |ζ η � →η ��ζ η � →η | ⊗ id H η � ◦ φ η � →η = τ η � ←η τ + η � ←η ,which is the orthogonal projection on the image of τ η � ←η in H η � , and, for all η ∈ L , Θ ζ|η , which isthe orthogonal projection on the image of τ ζ←η in H ζ and satisfy:Θ ζ|η ◦ τ ζ←η = τ ζ←η & ∀η � � η� Θ ζ|η ◦ τ ζ←η � = τ ζ←η � ◦ Θ η � |η .We start by deriving a useful identity, for η � η � ∈ L and A a self-adjoint, traceclass operatoron H η � : τ + η � ←η A τ η � ←η = Tr H η�→η �� |ζ η � →η ��ζ η � →η | ⊗ id H η � � φ η � →η A φ − η � →η � � |ζ η � →η ��ζ η � →η | ⊗ id H η � �= Tr η � →η �Θ η � |η A Θ η � |η � . (2.9. )Let ρ ζ ∈ S ζ . For κ ∈ L , we define � ρ κ := Θ ζ|κ ρ ζ Θ ζ|κ , and δ κ := ρ ζ − � ρ κ . � ρ κ is a positivesemi-definite, traceclass operator, and, from lemma 2.10. , δ κ converges in trace norm to 0.Moreover, for any κ ∈ L , δ κ is self-adjoint, so we can write δ κ = δ + κ − δ −κ where δ ±κ arethe positive and negative parts of δ κ (defined by spectral resolution), hence � σ ( δ ±κ )� η are positivesemi-definite, self-adjoint operators on H η , and from the conical linearity of σ : ∀κ ∈ L � σ ( ρ ζ ) = σ (� ρ κ ) + σ ( δ + κ ) − σ ( δ −κ ) =: σ (� ρ κ ) + σ ( δ κ ) ,hence: ∀κ ∈ L � ∀η ∈ L � � σ ( ρ ζ )� η = ( σ (� ρ κ )) η + ( σ ( δ κ )) η .Additionally, we have:Tr H η � σ ( δ ±κ )� η = Tr � σ ( δ ±κ ) � = Tr H ζ � δ ±κ α ( )� = Tr H ζ � δ ±κ id H ζ � = Tr H ζ δ ±κ ,where ∈ A ⊗ ( L � H �φ ) is the equivalence class of id H η . So, we get: κ ∈ L � ���( σ ( δ κ )) η ��� � ���� σ ( δ + κ )� η ��� + ���� σ ( δ −κ )� η ��� = Tr H ζ δ + κ + Tr H ζ δ −κ = �δ κ � ,therefore the net �( σ (� ρ κ )) η � κ∈ L converges in trace norm to � σ ( ρ ζ )� η .Now, for η � � κ ∈ L , we have:( σ (� ρ κ )) η � = Tr ζ→η � � ρ κ = Tr ζ→η � Θ ζ|η � � ρ κ Θ ζ|η � = τ + ζ←η � � ρ κ τ ζ←η � = τ η � ←κ τ + ζ←κ ρ ζ τ ζ←κ τ + η � ←κ = φ − η � →κ � |ζ η � →κ ��ζ η � →κ | ⊗ � τ + ζ←κ ρ ζ τ ζ←κ � � φ η � →κ , (2.9. )and, for η � � κ � � κ ∈ L :Θ η � |κ ( σ (� ρ κ � )) η � Θ η � |κ = τ η � ←κ τ + κ � ←κ τ + ζ←κ � ρ ζ τ ζ←κ � τ κ � ←κ τ + η � ←κ = ( σ (� ρ κ )) η � .Hence, for all η� κ ∈ L , and for all η � ∈ L such that η � � η and η � � κ , we have:Tr η � →η Θ η � |κ � σ (� ρ η � )� η � Θ η � |κ = ( σ (� ρ κ )) η . (2.9. )On the other hand:����Tr η � →η Θ η � |κ � σ ( ρ ζ )� η � Θ η � |κ � − ( σ (� ρ κ )) η ��� = ����Tr η � →η Θ η � |κ � σ ( ρ ζ )� η � Θ η � |κ � − �Tr η � →η Θ η � |κ � σ (� ρ η � )� η � Θ η � |κ ���� � ���Θ η � |κ � σ ( ρ ζ ) − σ (� ρ η � )� η � Θ η � |κ ��� = ���Θ η � |κ � σ ( δ η � )� η � Θ η � |κ ��� � �δ η � � ,as can be shown by decomposing the self-adjoint operator δ η � in positive and negative parts.Therefore, we have:lim η � � κ�η Tr η � →η �Θ η � |κ � σ ( ρ ζ )� η � Θ η � |κ � = ( σ (� ρ κ )) η , (2.9. )where the limit is taken in the trace norm. And we can now take the net limit on κ :lim κ∈ L lim η � � κ�η Tr η � →η �Θ η � |κ � σ ( ρ ζ )� η � Θ η � |κ � = � σ ( ρ ζ )� η . (2.9. )Now, for ρ ζ � = ρ �ζ ∈ S ζ , there should exist κ ∈ L such that:� τ + ζ←κ ◦ ρ ζ ◦ τ ζ←κ � � = � τ + ζ←κ ◦ ρ �ζ ◦ τ ζ←κ � ,which, from eq. (2.9. ), implies: ∀η � κ� ( σ (� ρ κ )) η � = � σ (� ρ �κ )� η , ut, using eq. (2.9. ), ( σ (� ρ κ )) η can be computed from σ ( ρ ζ ), hence σ ( ρ ζ ) � = σ ( ρ �ζ ). Therefore, σ | S ζ isinjective, so, from the conical linearity of σ , σ is injective.Then, for ρ ζ ∈ S ζ , we have (from eq. (2.9. )):lim η∈ L lim η � � η Tr H η� � � σ ( ρ ζ )� η � Θ η � |η � = Tr H ζ ρ ζ = 1 ,and, since the net �Tr H η� � � σ ( ρ ζ )� η � Θ η � |η � � η � � η is decreasing, while the net:�lim η � � η Tr H η� � � σ ( ρ ζ )� η � Θ η � |η � � η∈ L = �Tr H ζ �� ρ η � � η∈ L ,is increasing, the limits are given respectively by the infimum and by the supremum, so: σ ( ρ ζ ) ∈ �� ρ η � η∈ L ∈ S ⊗ ( L � H �φ ) ����� sup η∈ L � inf η � � η Tr H η� � ρ η � Θ η � |η �� = 1� .To prove that this condition indeed characterizes σ � S ζ � , we now consider � ρ η � η∈ L ∈ S ⊗ ( L � H �φ ) suchthat: sup η∈ L � inf η � � η Tr H η� � ρ η � Θ η � |η �� = 1 .Let η ∈ L . We have 0 � inf η � � η Tr H η� � ρ η � Θ η � |η � = µ η �
1. We consider the net �ˇ ρ η � |η � η � � η , whereˇ ρ η � |η is a positive semi-definite, traceclass operator on H η defined by:ˇ ρ η � |η := Tr η � →η �Θ η � |η ρ η � Θ η � |η � .For η �� � η � � η ∈ L , we have:ˇ ρ η � |η − ˇ ρ η �� |η = Tr η �� →η � � ι η �� ←η � (Θ η � |η ) − Θ η �� |η � ρ η �� � ι η �� ←η � (Θ η � |η ) − Θ η �� |η �� ,and ι η �� ←η � (Θ η � |η ) − Θ η �� |η = φ − η �� →η � ◦ ��id H η��→η� − |ζ η �� →η � ��ζ η �� →η � | � ⊗ Θ η � |η � ◦ φ η �� →η � ,hence ˇ ρ η � |η − ˇ ρ η �� |η is also a positive semi-definite, traceclass operator on H η . Its trace is Tr H η� � ρ η � Θ η � |η � − Tr H η�� � ρ η �� Θ η �� |η � = Tr H η ˇ ρ η � |η − Tr H η ˇ ρ η �� |η , therefore �Tr H η ˇ ρ η � |η � η � � η is decreasing and converges to µ η .Thus, �ˇ ρ η � |η � η � � η is a Cauchy net and, since the traceclass operators form a Banach space withrespect to the trace norm [22], it converges in trace norm to a positive semi-definite, traceclassoperator ˇ ρ η on H η , with Tr H η ˇ ρ η = µ η .Moreover, for κ � κ � ∈ L , we have: τ + κ � ←κ ˇ ρ κ � τ κ � ←κ = lim η � κ � τ + κ � ←κ ˇ ρ η|κ � τ κ � ←κ = lim η � κ � ˇ ρ η|κ = ˇ ρ κ (using eq. (2.9. ) ).Hence, since sup κ∈ L Tr H κ ˇ ρ κ = 1, there exists, from lemma 2.10. , an operator ˇ ρ ζ ∈ S ζ satisfying: κ ∈ L � τ + ζ←κ ˇ ρ ζ τ ζ←κ = ˇ ρ κ .Therefore, we have: ∀η� κ ∈ L � ∀η � � η� κ� � σ � ��ˇ ρ ζ � κ � � η = Tr η � →η Θ η � |κ ˇ ρ η � Θ η � |κ (using eqs. (2.9. ) and (2.9. ) ),hence, applying for η � = κ � η ∈ L :� σ �ˇ ρ ζ � � η = lim κ � η Tr κ→η ˇ ρ κ = lim κ � η lim κ � � κ Tr κ→η ˇ ρ κ � |κ On the other hand, we can show as above that for any κ � κ � , ρ κ − ˇ ρ κ � |κ is a positive semi-definite,traceclass operator on H κ , with trace smaller than 1 − µ κ . Thus, ρ = σ �ˇ ρ ζ �. � Finally, we also consider the case of infinite tensor products [25, 24]. Given a family of Hilbertspaces ( J λ ) λ∈ F , we can build its infinite tensor product (ITP) H F , which will in general be a non-separable Hilbert space. On the other hand, we can also build a projective system of quantum statespaces where the ‘small’ Hilbert spaces are given by tensor products of finitely many J λ . In thiscase, we still can map density matrices on H F to states in the projective limit, but this mappingwill no longer be injective, because we can define considerably more observables over the infinitetensor product, and we can use them to distinguish between states that are indistinguishable if wesolely use the algebra of observables defined over the projective system. However, if we believe thatthese latter observables (which can be sensible only to correlations between finitely many J λ ) are theonly experimentally measurable ones, additional distinctions between states might be objectionable.Interestingly, the ITP H F , while being a really huge Hilbert space, still fails (except in absolutelydegenerate cases) to reproduce the full state space of the projective system: this can be traced backto the fact that the latter allows to model states that are patently more ‘statistical’ than any staterealizable on H F . Also, grouping the tensor product factors J λ into finite tensor products before performing the ITP construction generically gives rise to inequivalent Hilbert spaces (ie. ITP’s arenot associative, see [25, section 4.2] ), while such a grouping does not affect the projective statespace (as a consequence of prop. 2.6). Theorem 2.11
Let ( J λ ) λ∈ F be a family of Hilbert spaces and define: L := { Λ ⊂ F | < ∞} equipped with the preorder ⊂ ; ∀ Λ ∈ L � H Λ := � λ∈ Λ J λ ; ∀ Λ ⊂ Λ � ∈ L � H Λ � → Λ := H Λ � \ Λ with φ Λ � → Λ the natural identification H Λ � → H Λ � \ Λ ⊗ H Λ .Then, we can complete these elements into a projective system of quantum state spaces ( L � H � φ ) ⊗ .Let H F be infinite tensor product of ( J λ ) λ∈ F . There exist maps σ : S F → S ⊗ ( L � H �φ ) and α : A ⊗ ( L � H �φ ) → A F such that: ∀ρ ∈ S F � ∀A ∈ A ⊗ ( L � H �φ ) � Tr H F ( ρ α ( A )) = Tr ( σ ( ρ ) A ); if {λ ∈ F | dim J λ > } is infinite, σ is neither injective nor surjective. roof Clearly, ( L � ⊂ ) is a directed set and, defining, for any Λ ⊂ Λ � ⊂ Λ �� ∈ L , φ Λ �� → Λ � → Λ as thenatural identification H Λ �� \ Λ → H Λ �� \ Λ � ⊗ H Λ � \ Λ , we obtain a projective system of quantum statespaces ( L � H � φ ) ⊗ . Existance of σ and α satisfying 2.11. . The ITP H F arising from ( J λ ) λ∈ F can be written as [25,chapter 4]: H F = � [ � ] H [ � ] ,where the [ � ] are equivalence classes in �( � λ ) λ∈ F ∈ ( J λ ) λ∈ F ����� � λ∈ F �� �� λ � J λ − � λ ) λ∈ F � ( � λ ) λ∈ F ⇔ � λ∈ F �� �� λ � � λ � J λ − H [ � ] is (the completion of ) the inductive limit of � L � �� λ∈ Λ J λ � Λ ∈ L � � τ � Λ � ← Λ � Λ ⊂ Λ � �,the inductive maps τ � Λ � ← Λ being defined as: ∀ Λ ⊂ Λ � � τ � Λ � ← Λ : � λ∈ Λ J λ → � λ∈ Λ � J λ ψ �→ �� λ∈ Λ � \ Λ � λ � ⊗ ψ ,for � some representative of [ � ] such that ∀λ ∈ F � �� λ � J λ = 1.Now, we can identify H [ � ] with the Hilbert space H ζ � constructed as in theorem 2.9 for the family� ζ � Λ � → Λ � Λ ⊂ Λ � given by: ∀ Λ � ζ � Λ → Λ = 1 & ∀ Λ � Λ � � ζ � Λ � → Λ := � λ∈ Λ � \ Λ � λ .Hence, as in the proof of theorem 2.9, we can construct for all Λ ∈ L an Hilbert space H [ � ] → Λ and an Hilbert space isomorphism φ [ � ] → Λ : H [ � ] → H [ � ] → Λ ⊗ H Λ , and for all Λ ⊂ Λ � , we can constructan Hilbert space isomorphism φ [ � ] → Λ � → Λ : H [ � ] → Λ → H [ � ] → Λ � ⊗ H Λ � → Λ , satisfying:� φ [ � ] → Λ � → Λ ⊗ id H Λ � ◦ φ [ � ] → Λ = �id H [ � ] → Λ � ⊗ φ Λ � → Λ � ◦ φ [ � ] → Λ � .Now, we define: ∀ Λ ∈ L � H F → Λ = � [ � ] H [ � ] → Λ , φ F → Λ = � [ � ] φ [ � ] → Λ & ∀ Λ ⊂ Λ � � φ F → Λ � → Λ = � [ � ] φ [ � ] → Λ � → Λ .Note that H F → Λ can also be identified as the ITP of ( J λ ) λ∈ F \ Λ .Defining L := L ∪ { F } and extending the preorder on L to L by ∀ Λ ∈ L � Λ ⊂ F , we thus havea projective system of quantum state spaces � L � H � φ � ⊗ .As in the proof of theorem 2.9, we can then define σ and α by first using prop. 2.7 to go from H F to � L � H � φ � ⊗ and then using prop. 2.6 to go from � L � H � φ � ⊗ to ( L � H � φ ) ⊗ . Properties of σ (2.11. ). Let ρ F ∈ S F . For Λ ∈ L , we have: σ ( ρ F )� Λ = Tr F → Λ ρ F = Tr H F → Λ � φ F → Λ ◦ ρ F ◦ φ − F → Λ �= � [ � ] Tr H [ � ] → Λ � φ [ � ] → Λ ◦ Π [ � ] ◦ ρ F ◦ Π +[ � ] ◦ φ − � ] → Λ � ,where Π [ � ] : H F → H [ � ] is the orthogonal projection on H [ � ] .Thus, σ does not see the correlations between different H [ � ] that might be contained in ρ F , andtherefore σ cannot be injective if there exist more than one equivalence class (as will be the case if {λ ∈ F | dim J λ > } is infinite, see below). Indeed, if ψ� ψ � are two normalized states in H F with ψ ∈ H [ � ] , ψ � ∈ H [ � ] and [ � ] � = [ � ], then: σ � 12 |ψ��ψ| + 12 |ψ � ��ψ � | � = σ ����� ψ + ψ � √ ψ + ψ � √ ρ = ( ρ Λ ) Λ ∈ L ∈ S ⊗ ( L � H �φ ) and suppose there exists ρ F ∈ S F such that ρ = σ ( ρ F ). If Tr H F ρ F = 0, then ρ F = 0, hence ρ = 0. Therefore, if ρ � = 0, we have Tr H F ρ F >
0, andtherefore there should exist at least one [ � ] such that Tr H [ � ] �Π [ � ] ρ F Π +[ � ] � >
0. Using theorem 2.9,we then have:sup Λ ∈ L � inf Λ � ⊃ Λ Tr H Λ � � ρ Λ � Θ � Λ � | Λ � � = Tr H [ � ] �Π [ � ] ρ F Π +[ � ] � > � Λ � | Λ = φ − � → Λ ◦ ��� ζ � Λ � → Λ �� ζ � Λ � → Λ �� ⊗ id H Λ � ◦ φ Λ � → Λ .Now, we suppose that we have a infinite part Γ ⊂ F such that ∀γ ∈ Γ � dim J γ >
1, and, for all γ ∈
Γ, we choose � γ and � γ two normalized vectors in J γ that are orthogonal with each other. Wedefine: ∀ Λ ∈ L / Λ ⊂ Γ � ∀ � ε γ � γ∈ Λ ∈ { � } Λ � � ( ε )Λ := � γ∈ Λ � ε γ γ ∈ H Λ .Then, we choose some ( � λ ) λ∈ F \ Γ , with ∀λ ∈ F \ Γ � � λ ∈ J λ and �� λ � J λ = 1. We define, for anyΛ ∈ L : ∀ � ε γ � γ∈ Λ ∩ Γ ∈ { � } Λ ∩ Γ � � ( ε )Λ := �� λ∈ Λ \ Γ � λ � ⊗ � ( ε γ )Λ ∩ Γ and ρ Λ := 12 ∩ Γ) � ( ε ) ��� � ( ε )Λ �� � ( ε )Λ ��� .We can check that ( ρ Λ ) Λ ∈ L ∈ S ⊗ ( L � H �φ ) and we have, for any � and any Λ ⊂ Λ � :Tr H Λ � � ρ Λ � Θ � Λ � | Λ � = � γ∈ (Λ � \ Λ) ∩ Γ Tr J γ � ��� � γ �� � γ �� + �� � γ �� � γ ��� |� γ ��� γ | � � λ∈ (Λ � \ Λ) \ Γ Tr J λ � |� λ ��� λ | |� λ ��� λ | �= � γ∈ (Λ � \ Λ) ∩ Γ 12 ���� � γ � � γ ��� + ��� � γ � � γ ��� � � λ∈ (Λ � \ Λ) \ Γ Tr J λ � |�� λ � � λ �| � .Thus, we get for any � and any Λ ⊂ Λ � the bound: r H Λ � � ρ Λ � Θ � Λ � | Λ � � ∩ Γ) � ∩ Γ) .Hence, ∀ Λ ∈ L � inf Λ � ⊃ Λ Tr H Λ � � ρ Λ � Θ � Λ � | Λ � = 0 and sup Λ ∈ L � inf Λ � ⊃ Λ Tr H Λ � � ρ Λ � Θ � Λ � | Λ � � = 0 ,so ( ρ Λ ) Λ ∈ L /∈ Im σ , and therefore σ is not surjective. � The motivation of [15, section 2] was to pave the way for a better understanding of how aquantum projective structure as described in the previous section can be constructed starting froma classical field theory. The procedure we have in mind here, is, given an infinite dimensionalsymplectic manifold, to first build its rendering by a system of finite dimensional manifolds (thepartial theories, that encapsulate insights from a careful analysis of how measurements are doneexperimentally), and then quantize this projective system (with the aim of getting a quantum theoryassembled from ‘small’ Hilbert spaces, on which calculations should be workable).In this section, we will consider two basic, yet fairly generic cases, namely position and holo-morphic representations, assuming that we have a factorizing system on the classical side (see [15,subsection 2.3]). In both cases, the key prerequisite is that the polarizations, that endow eachsymplectic manifold M η with the additional structure needed for quantization (the choice of con-figuration variables or the complex structure, respectively), should be compatible, in an appropriatesense, with the projections defining the projective system. In this section, all manifolds are assumed to be smooth finite dimensional manifolds and all mapsbetween them are assumed to be smooth.
The starting point for position quantization will be a projective limit of classical phase spacesarising from a factorizing system of configuration spaces as described in [15, prop. 2.16]. Then,there is only one additional ingredient required, namely we need to find a family of measures onthese configuration spaces that are intertwined by the factorization maps. With this, constructingthe projective system of quantum state spaces is a straightforward generalization of [20, subsections3.4.3 to 3.4.5], since an L -space over a Cartesian product of measure spaces has a natural tensorproduct factorization.Surely, given an arbitrary projective system of phase spaces, it will in general not be possible torewrite it as arising from a factorizing system of configuration spaces. However, we consider intheorem 3.2 an important case where we indeed get such a factorizing form automatically, namely hen each individual phase space can be identified with the cotangent bundle on a simply-connectedLie group (assuming some appropriate compatibility conditions between the projections and theseidentifications). The idea is that the group structure, together with the favorable topology, fillsexactly the gap between the local result from [15, prop. 2.10] and the global factorization we wantto have. Also, using Haar measures, we can easily build a family of measures for this factorizingsystem.Note that this result in particular covers the situation considered in [20] (looking at R � asan additive Lie group), while answering the question raised in this reference, as to whether theconstruction can be generalized to non-Abelian gauge theories. To make the relation clearer betweenthe objects in [20] and the ones we are using here, let us look in more detail at the assumptionsof theorem 3.2. That each ‘small’ phase space M η is a cotangent bundle on a simply-connectedLie group, equipped with its canonical symplectic structure, is a weaker version of assumptions2, 3b and 4 in [20]. The most crucial assumption is that we start from a projective system ofphase spaces: on the one hand, the compatibility of the projections with the symplectic structuresprovides the seeds of the desired factorizations, on the other hand its three-spaces consistencycondition will turn into the corresponding condition for the quantum projective system (eq. (2.1. )).This is ensured in [20] by assumptions 3a and 6. Finally, the condition 3.2. , correspondingto the rest of assumption 6 in [20], ensures the compatibility of the projection maps both withthe configuration polarizations (so that the factorization of the phase spaces will descend to afactorization of the configuration spaces) and with the group structure (otherwise we would not beable to really make use of this structure). Note that, thanks to the compatibility of the projectionswith the symplectic structures, simply assuming that the map, besides acting independently on theposition and momentum variables, is linear in the momentum variables is sufficient to ensure afull compatibility with the group structure, as expressed by eqs. (3.2. ) and (3.2. ). Definition 3.1
A factorizing system of measured manifolds is a factorizing system of smooth, finitedimensional, manifolds ( L � C � φ ) × [15, def. 2.15] such that: for all η ∈ L , C η is equipped with a smooth measure µ η (def. A.12); for all η ≺ η � ∈ L , C η � →η is equipped with a smooth measure µ η � →η (on C η→η we will use thecounting measure); for all η ≺ η � ∈ L , φ η � →η is volume-preserving, and for all η ≺ η � ≺ η �� ∈ L , φ η �� →η � →η isvolume-preserving; in other words, we require: ∀η ≺ η � � φ − �∗η � →η µ η � = µ η � →η × µ η ,and ∀η ≺ η � ≺ η �� � φ − �∗η �� →η � →η µ η �� →η = µ η �� →η � × µ η � →η . Theorem 3.2
Let ( L � M � π ) ↓ be a projective system of phase spaces such that: ∀η ∈ L � M η = T ∗ ( C η ) where C η is a simply-connected Lie group; by relying on left translations,we thus have an identification L η : M η → C η × Lie ∗ ( C η ) ; ∀η � η � � π η � →η = L − η ◦ ( ρ η � →η × λ η � →η ) ◦ L η � where ρ η � →η is a map C η � → C η and λ η � →η is a linear map Lie ∗ ( C η � ) → Lie ∗ ( C η ) .Then, there exists a factorizing system of measured manifolds ( L � ( C � µ ) � φ ) × such that ( L � M � π ) ↓ rises from ( L � C � φ ) × (in the sense of [15, prop. 2.16]). Proof
Conditions on ρ η � →η and λ η � →η . Let η ∈ L and � ∈ C η . There exists an open neighborhood U of 0 in Lie( C η ) such that the map:Ψ � : U → C η X �→ � � exp( X ) ,is a diffeomorphism onto its image, hence it provides a local coordinate system around � in C η . Wecan lift it to a local trivialization of the cotangent bundle M η = T ∗ ( C η ) :�Ψ � : U ×
Lie ∗ ( C η ) → M η X � � �→ � � exp( X ) � � ◦ [ T X Ψ � ] − .Using [15, eq. (2.16. )], we then get, for all � ∈ Lie ∗ ( C η ) and for all ( �� � ) � ( � � � � � ) ∈ Lie( C η ) × Lie ∗ ( C η ):Ω M η � �Ψ � (0 �� ) �� T (0 �� ) �Ψ � � ( �� � ) � � T (0 �� ) �Ψ � � ( � � � � � )� = � � ( � ) − � ( � � ) .Next, we have: L η ◦ �Ψ � : U ×
Lie ∗ ( C η ) → C η × Lie ∗ ( C η ) X � � �→ � � exp( X ) � � ◦ [ T X Ψ � ] − ◦ [ T ( � � exp( X ) � · )] = � ◦ ad X id Lie( C η ) −� − ad X ,where we have used:[ T exp( X ) (exp( −X ) � · )] ◦ [ T X exp] = id Lie( C η ) − � − ad X ad X ,as follows from the Baker–Campbell–Hausdorff formula.Therefore, for all �� � ∈ C η × Lie ∗ ( C η ) and for all ( �� � ) � ( � � � � � ) ∈ T � ( C η ) × Lie ∗ ( C η ), the symplecticform on M η is given by:� L − �∗η Ω M η � ( ��� ) �( �� � ) � ( � � � � � )� = � � ◦ L − η�� ( � ) − � ◦ L − η�� ( � � ) + � �� L − η�� ( � ) � L − η�� ( � � )� Lie( C η ) � ,where L η�� := [ T ( � � · )] : Lie( C η ) → T � ( C η ). This allows us to express the map · : T ∗ ( M η ) → T ( M η )(defined from the symplectic structure as in [15, def. 2.1]) and we have for any �� k ∈ T ∗� ( C η ) × Lie( C η ) :( �� k ) ◦ � T L − η ( ��� ) L η � = � T ( ��� ) L − η � � L η�� ( k ) � � �[ k� · ] Lie( C η ) � − � ◦ L η�� � .Let η � η � . We can now formulate the conditions on ρ η � →η and λ η � →η for π η � →η to be compatiblewith the symplectic structures as:[ T � � ρ η � →η ] ◦ L η � �� � ◦ λ ∗η � →η = L η�ρ η�→η ( � � ) , (3.2. )and � λ ∗η � →η ( · ) � λ ∗η � →η ( · )� Lie( C η� ) = λ ∗η � →η �[ · � · ] Lie( C η ) � . (3.2. ) C η as a Lie subgroup of C η � . π η � →η being surjective, so is λ η � →η , thus λ ∗η � →η : Lie( C η ) → Lie( C η � )is injective, and, from eq. (3.2. ), it is a Lie algebra morphism. Therefore, λ ∗η � →η � Lie( C η ) � is aLie subalgebra of Lie( C η � ) so there exists a unique connected Lie subgroup � C η of C η � such that �� C η � = λ ∗η � →η � Lie( C η ) � [26, theorem 3.19]. � C η is an immersed submanifold in C η � and its tangentspace at � � ∈ � C η is given by: T � � �� C η � = L η � �� � ◦ λ ∗η � →η � Lie( C η ) � .Let � � � � � ∈ C η � and define: κ � � �� � : C η � → C η � � �→ ρ η � →η ( � � � � � ) � ρ η � →η ( � � � � � ) − .For any k ∈ Lie( C η ), we have:� T κ � � �� � � ◦ λ ∗η � →η ( k ) == � T ρ η�→η ( � � ) � · � ρ η � →η ( � � ) − �� ◦ � T � � ρ η � →η � ◦ L η � �� � ◦ λ ∗η � →η ( k ) ++ � T ρ η�→η ( � � ) � ρ η � →η ( � � ) � ( · ) − �� ◦ � T � � ρ η � →η � ◦ L η � �� � ◦ λ ∗η � →η ( k )= � T ρ η�→η ( � � ) � · � ρ η � →η ( � � ) − �� ◦ L η�ρ η�→η ( � � ) ( k ) + � T ρ η�→η ( � � ) � ρ η � →η ( � � ) � ( · ) − �� ◦ L η�ρ η�→η ( � � ) ( k )(using eq. (3.2. ) )= � T � � �→ ρ η � →η ( � � ) � � � � − � ρ η � →η ( � � ) − �� ( k ) = 0 .With this, we get, for any � � ∈ � C η :� T � � κ � � �� � � � T � � �� C η �� = � T � � κ � � �� � � ◦ L η � �� � ◦ λ ∗η � →η �Lie � C η ��= � T κ � � �� � � � � �� � � ◦ λ ∗η � →η �Lie � C η �� = { } ,thus, � C η being connected by definition, κ � � �� � is constant on � C η for any � � � � � ∈ C η � . In particular,applying with � � = gives: ∀� �� ∈ C η � � ∀� � ∈ � C η � � ρ η � →η ( � �� � � � ) = � ρ η � →η ( � �� ) � � ρ η � →η ( � � ) , (3.2. )where ∀� � ∈ C η � � � ρ η � →η ( � � ) := ρ η � →η ( ) − � ρ η � →η ( � � ) .Therefore, � ρ η � →η | � C η → C η is a smooth group homomorphism, and, moreover, its derivative at is aLie algebra isomorphism, for we have using eq. (3.2. ):[ T � ρ η � →η ] ◦ λ ∗η � →η = L − η�ρ η�→η ( ) ◦ [ T ρ η � →η ] ◦ λ ∗η � →η = id Lie( C η ) .Hence, � ρ η � →η | � C η → C η is a Lie group isomorphism, for � C η is connected and C η is simply-connected [26,prop. 3.26]. We will denote by Λ η � ←η : C η → � C η its inverse. Factorizing system.
We define C η � →η := � ρ − η � →η � � . � ρ η � →η has surjective derivative at each point, so C η � →η is a smooth manifold as level set of a constant rank map [17, theorem 5.22]. Next, we definethe map φ η � →η by: η � →η : C η � → C η � →η × C η � � �→ � � � �Λ η � ←η ◦ � ρ η � →η ( � � )� − � ρ η � →η ( � � ) . φ η � →η is well-defined for, using eq. (3.2. ), we have for all � � ∈ C η � :� ρ η � →η � � � � Λ η � ←η �� ρ η � →η ( � � ) − �� = � ρ η � →η ( � � ) � � ρ η � →η ( � � ) − = .To prove that φ η � →η is a bijective map, we define a map � φ η � →η by:� φ η � →η : C η � →η × C η → C η � �� � �→ � � σ − � Λ η � ←η ( � ) ,where σ := Λ η � ←η � ρ η � →η ( )� . Using again eq. (3.2. ), we can check that φ η � →η ◦ � φ η � →η = id C η�→η × C η and � φ η � →η ◦ φ η � →η = id C η� . Since both φ η � →η and � φ η � →η are smooth, φ η � →η is a diffeomorphism.From eq. (3.2. ), we have: ∀� ∈ C η � [ T � Λ η � ←η ] = L η � � Λ η�←η ( � ) ◦ λ ∗η � →η ◦ L − η�� .Thus, for any �� � ∈ C η � →η × C η , the derivative of φ − η � →η satisfies: ∀� ∈ T � ( C η ) � � T ��� φ − η � →η � (0 � � ) = L η � � φ − η�→η ( ��� ) ◦ λ ∗η � →η ◦ L − η�� ( � ) . (3.2. )So we get, for any � � � � � ∈ T ∗ ( C η � ): π η � →η ( � � � � � ) = � ρ η � →η ( � � ) � � � ◦ L η � �� � ◦ λ ∗η � →η ◦ L − η�ρ η�→η ( � � ) �= � � η � →η ◦ φ η � →η ( � � ) � � � ◦ [ T � � φ η � →η ] − (0 � · )� ,where � η � →η : C η � →η × C η → C η is the projection on the second Cartesian factor. We can now use [15,prop. 2.17] to build from these objects a factorizing system ( L � C � φ ) × that gives rise to ( L � M � π ) ↓ . Volume forms.
For any η ∈ L , we choose a non-zero � η -form � ω η on Lie( C η ) (with � η := dim C η )and we define a right-invariant volume form ω η on C η by: ∀� ∈ C η � ∀� � � � � � � � η ∈ T � ( C η ) � ω η ( � � � � � � � � η ) := � ω η � R − η�� � � � � � � R − η�� � � η � ,where R η�� := T ( · � � ) . We call µ η the smooth measure arising from the volume form ω η .Let η � η � , � ∈ C η � →η and w � � � � � w � η�→η ∈ T � ( C η � →η ) = Ker � T � ρ η � →η � (with � η � →η :=dim C η � →η = � η � − � η ). The map: α : Lie( C η ) � η → R � � � � � � � � η �→ � ω η � � R − η � �� ( w ) � � � � � R − η � �� ( w � η�→η ) � A η � ←η�� ( � ) � � � � � A η � ←η�� ( � � η )� ,where A η � ←η�� := Ad � ◦ λ ∗η � →η ◦ Ad − ρ η�→η ( ) , is a � η -form on Lie( C η ), so there exists ω η � →η�� ( w � � � � � w � η�→η ) ∈ R such that: α ( � � � � � � � � η ) = ω η � →η�� ( w � � � � � w � η�→η ) � ω η ( � � � � � � � � η ) .Now, using the expression for φ − η � →η given above, we have, for any �� � ∈ C η � →η × C η : w ∈ T � ( C η � →η ) � R − η � �φ − η�→η ( ��� ) ◦ � T ��� φ − η � →η � ( w�
0) = R − η � �� ( w ) ,and, from eq. (3.2. ), we also have: ∀� ∈ T � ( C η ) � R − η � �φ − η�→η ( ��� ) ◦ � T ��� φ − η � →η � (0 � � ) = A η � ←η�� ◦ R − η�� ( � ) ,where we have used that λ ∗η � →η = T Λ η � ←η with Λ η � ←η a group homomorphism. With this, we cancheck that φ − �∗η � →η ω η � = ω η � →η ∧ ω η . In particular, this implies that ω η � →η is a smooth volume form on C η � →η . Thus, defining µ η � →η to be the corresponding smooth measure, we get φ − �∗η � →η µ η � = µ η � →η × µ η .Finally, for any η � η � � η �� , φ η �� →η � , φ η � →η and φ η �� →η are volume-preserving, hence so is φ η �� →η � →η × id C η (using [15, eq. (2.11. )] ) and therefore φ η �� →η � →η itself. � Proposition 3.3
Let ( L � C � φ ) × be a factorizing system of measured manifolds. We define: for η ∈ L , H η := L ( C η � �µ η ); for η ≺ η � ∈ L , H η � →η := L ( C η � →η � �µ η � →η ), and:Φ η � →η : H η � → H η � →η ⊗ H η ψ �→ ψ ◦ φ − η � →η ,with the natural identification of L ( C η � →η � �µ η � →η ) ⊗L ( C η � �µ η ) with L ( C η � →η × C η � �µ η � →η ×�µ η ).Then, we can complete these objects to build a projective system of quantum state spaces( L � H � Φ) ⊗ . Proof
We define: for η ≺ η � ≺ η �� ∈ L ,Φ η �� →η � →η : H η �� →η → H η �� →η � ⊗ H η � →η ψ �→ ψ ◦ φ − η �� →η � →η ,with the natural identification of L ( C η �� →η � � �µ η �� →η � ) ⊗ L ( C η � →η � �µ η � →η ) with L ( C η �� →η � × C η � →η ��µ η �� →η � × �µ η � →η ); for η ∈ L , H η→η := C and we define Φ η→η to be the natural isomorphic identification between H η and C ⊗ H η ; for η � η � ∈ L , we define Φ η � →η→η (resp. Φ η � →η � →η ) to be the natural isomorphic identificationbetween H η � →η and H η � →η ⊗ C (resp. C ⊗ H η � →η ).That Φ η � →η for η ≺ η � defines an Hilbert space isomorphism comes from the volume-preservingproperty of φ η � →η and from the fact that L ( C � �µ ) ⊗ L ( C � � �µ � ) can be unitarily identified with L ( C × C � � �µ × �µ � ) (thanks to Fubini’s theorem). Similarly, Φ η �� →η � →η for η ≺ η � ≺ η �� is an Hilbertspace isomorphism.We now just need to check the three-spaces consistency condition eq. (2.1. ). We consider η ≺ η � ≺ η �� (since the condition is trivially satisfied whenever two labels are equal): ∀ψ ∈ H η � �Φ η �� →η � →η ⊗ id H η � ◦ Φ η �� →η ( ψ ) == � ψ ◦ φ − η �� →η � ◦ � φ − η �� →η � →η ⊗ id C η � � ψ ◦ φ − η �� →η � � ◦ �id C η��→η� ⊗ φ − η � →η � (using [15, eq. (2.11. )] )= �id H η��→η� ⊗ Φ η � →η � ◦ Φ η �� →η � ( ψ ) . � To argue that the quantum projective system composed above actually provides a quantizationof the classical one (as specified by the factorizing system of configuration spaces we startedfrom), we need to say how classical observables on the latter are turned to quantum observableson the former. For this, we import the prescriptions of geometric quantization (summarized inappendix A.3, especially in prop. A.14, and rewritten here more explicitly using the benefit ofworking in a phase space given as a cotangent bundle). Thus, for each η , we can formulate thequantization condition (the choice of preferred configuration variables is tied to a selection of whichobservables can be directly quantized) as well as the definition of the quantized observables. Thekey statement is that the compatibility conditions imposed on the family of measures is sufficientto ensure that these prescriptions, supplied separately for each η , will fit readily into a coherentpicture. Proposition 3.4
We consider the same objects as in prop. 3.3. Let ( L � M � � φ ) × be the factorizingsystem of phase spaces constructed from ( L � C � φ ) × as in [15, prop. 2.16] and let � = [ � η ] ∼ ∈ O × ( L � M � � φ ) [15, prop. 2.13].If there exists a representative � η of � such that: ∃ X � η ∈ T ∞ ( C η ) / ∀ ( �� � ) ∈ M η � [ T ( ��� ) γ η ] � X � η � ( ��� ) � = X � η �� , (3.4. )where T ∞ ( C η ) is the space of smooth vector fields on C η , and γ η : M η = T ∗ ( C η ) → C η is the bundleprojection, then this condition is satisfied by all representatives of � . Accordingly, we define: O × ( L � C �φ ) := � � ∈ O × ( L � M � � φ ) ��� ∃ � η ∈ � satisfying eq. (3.4. ) � .For � η ∈ � ∈ O × ( L � C �φ ) , we can define � � µ η η as a densely defined operator on H η (with dense domain D η ⊂ H η ) by: ∀ψ ∈ D η � ∀� ∈ C η � � � µ η η ( ψ )( � ) := ψ ( � ) � η ( ��
0) + � [ �ψ ] � � X � η �� � + � ψ ( � ) �div µ η X � η � ( � ) ,where div µ η X � η is defined by L X �η µ η = �div µ η X � η � µ η (def. A.12).Then, the application:� · µ : O × ( L � C �φ ) → O ⊗ ( L � H � Φ) [ � η ] ∼ → �� � µ η η � ∼ ,is well-defined ( O ⊗ ( L � H � Φ) has been defined in prop. 2.5). Proof
Quantization condition.
For η � η � ∈ L , we define π η � →η : C η � → C η , λ η � →η : C η � → C η � →η ,and � π η � →η : M η � → M η , � λ η � →η : M η � → M η � →η , such that: � � ∈ C η � � φ η � →η ( � � ) = � λ η � →η ( � � ) � π η � →η ( � � )� & ∀� � � � � ∈ M η � � � φ η � →η ( � � � � � ) = �� λ η � →η ( � � � � � ) � � π η � →η ( � � � � � )� .From [15, prop. 2.16], we then have: ∀η � η � ∈ L � γ η ◦ � π η � →η = π η � →η ◦ γ η � & γ η � →η ◦ � λ η � →η = λ η � →η ◦ γ η � .Let � η ∈ C ∞ ( M η � R ) satisfying eq. (3.4. ) and let η � � η . Using the previous identity, we have: ∀� � � � � ∈ M η � � [ T � � π η � →η ] ◦ [ T � � �� � γ η � ] � X � η ◦ � π η�→η � ( � � �� � ) � == � T � π η�→η ( � � �� � ) γ η � ◦ [ T � � �� � � π η � →η ] ��� π ∗η � →η �� η � � � �� � �= � T � π η�→η ( � � �� � ) γ η � �[ �� η ] � π η�→η ( � � �� � ) � (using [15, eq. (2.1. )] )= � T � π η�→η ( � � �� � ) γ η � � X � η � � π η�→η ( � � �� � ) � = X � η �π η�→η ( � � ) ,and: ∀� � � � � ∈ M η � � [ T � � λ η � →η ] ◦ [ T � � �� � γ η � ] � X � η ◦ � π η�→η � ( � � �� � ) � == � T � λ η�→η ( � � �� � ) γ η � →η � ◦ � T � � �� � � λ η � →η � ��� π ∗η � →η �� η � � � �� � �= 0 ,since �� π ∗η � →η �� η � � � �� � ∈ �Ker [ T � � �� � � π η � →η ]� ⊥ = Ker � T � � �� � � λ η � →η � .Therefore, [ T � � �� � γ η � ] � X � η ◦ � π η�→η � ( � � �� � ) � only depends on � � . If we now define: ∀� � ∈ M η � � X � η ◦ � π η�→η �� � := [ T � � � γ η � ] � X � η ◦ � π η�→η � ( � � � � ,we have X � η ◦ � π η�→η ∈ T ∞ ( C η � ) and: ∀� � � � � ∈ M η � � [ T � � �� � γ η � ] � X � η ◦ � π η�→η � ( � � �� � ) � = X � η ◦ � π η�→η �� � .Thus, � η ◦ � π η � →η also fulfills eq. (3.4. ) and we moreover have: ∀� � ∈ C η � � [ T � � φ η � →η ] � X � η ◦ � π η�→η �� � � = �0 � X � η �π η�→η ( � � ) � . (3.4. )On the other hand, for � η ∈ C ∞ ( M η � R ), if there exists η � � η such that � η ◦ � π η � →η satisfyeq. (3.4. ), then, in the same way as above: ∀� � � � � ∈ M η � � � T � π η�→η ( � � �� � ) γ η � � X � η � � π η�→η ( � � �� � ) � = [ T � � π η � →η ] � X � η ◦ � π η�→η �� � � .Therefore, since � π η � →η is surjective, � η also satisfy eq. (3.4. ) , and again we have eq. (3.4. ) . uantized observable. Let � η ∈ � ∈ O × ( L � C �φ ) . We start by deriving an identity for div µ η X � η :�div µ η� X � η ◦ � π η�→η � µ η � = L X �η ◦ � πη�→η µ η � = φ ∗η � →η � L φ − �∗η�→η � X �η ◦ � πη�→η � φ − �∗η � →η µ η � �= φ ∗η � →η � L φ − �∗η�→η � X �η ◦ � πη�→η � µ η � →η × µ η � (using def. 3.1. )= φ ∗η � →η � L ( � X �η ) µ η � →η × µ η � (using eq. (3.4. ) )= φ ∗η � →η � µ η � →η × � L X �η µ η ��= ��div µ η X � η � ◦ π η � →η � µ η � .Hence, it follows:div µ η� X � η ◦ � π η�→η = �div µ η X � η � ◦ π η � →η . (3.4. )Now, let ψ ∈ Φ − η � →η � H η � →η ⊗ D η � (where D η is the domain of � � µ η η and the ⊗ is to be understoodas a tensor product of vector spaces, that is without any completion in contrast to a tensor productof Hilbert spaces). Then, we have: ∀�� � ∈ C η � →η × C η � �id H η�→η ⊗ � � µ η η � ◦ Φ η � →η ( ψ )( �� � ) == ψ ◦ φ − η � →η ( �� � ) � η ( �� � � ∂ � ψ ◦ φ − η � →η � ( ��� ) � X � η �� �+ � ψ ◦ φ − η � →η ( �� � ) �div µ η X � η � ( � )= ψ ◦ φ − η � →η ( �� � ) � η ◦ � π η � →η ( φ − η � →η ( �� � ) �
0) + � [ �ψ ] φ − η�→η ( ��� ) � X � η ◦ � π η�→η �φ − η�→η ( ��� ) � ++ � ψ ◦ φ − η � →η ( �� � ) �div µ η� X � η ◦ � π η�→η � ◦ φ − η � →η ( �� � ) (using eqs. (3.4. ) and (3.4. ) )= �Φ η � →η ◦ � � η ◦ � π η � →ηµ η� � ( ψ )( �� � ) .Therefore, ∀� η � � η � ∈ � ∈ O × ( L � C �φ ) � � � µ η η ∼ � � µ η� η � . � We close this subsection with an application of theorem 2.9: under the additional hypothesis thatthe measures are normalized to unity, we can construct an inductive limit of Hilbert spaces fromthe H η , whose space of states is naturally embedded in the one of the projective structure developedabove. As long as all C η have finite volume (hence in particular if they are compact), it is alwayspossible to consistently normalize the measures to unity. Note however that, depending on theprojective structure under consideration, it may not be possible to equip all C η with normalizablemeasures fulfilling the factorization requirement def. 3.1. (see eg. the models considered in [18, 19] , n particular the discussion in [18, section 1.1] ). Proposition 3.5
We consider the same objects as in prop. 3.3, and we now additionally assume: ∀η ∈ L � µ η ( C η ) = 1.Then, we can also construct an Hilbert space H ⊕ as (the completion of ) the inductive limit of� L � � H η � η∈ L � � τ η � ←η � η � η � �, where the injective maps τ η � ←η are defined as: ∀η � η � ∈ L � τ η � ←η : H η → H η � ψ �→ ψ ◦ � η � →η ◦ φ η � →η , with � η � →η : C η � →η × C η → C η ( �� � ) �→ � .There exist maps σ : S ⊕ → S ⊗ ( L � H � Φ) and α : A ⊗ ( L � H � Φ) → A ⊕ ( S ⊕ being the space of (self-adjoint)positive semi-definite, traceclass operators over H ⊕ and A ⊕ the algebra of bounded operators on H ⊕ ) such that: ∀ρ ∈ S ⊕ � ∀A ∈ A ⊗ ( L � H � Φ) � Tr H ⊕ ( ρ α ( A )) = Tr ( σ ( ρ ) A ); σ is injective; σ � S ⊕ � = �� ρ η � η∈ L ����� sup η∈ L inf η � � η � C η�→η × C η�→η ���� � � C η �� ρ η � � φ − η � →η ( �� � ); φ − η � →η ( � � � � )� = 1�where S ⊕ is the space of density matrices over H ⊕ and ρ η ( · ; · ) is the integral kernel of ρ η . Proof
This is an application of theorem 2.9, where for η � η � ∈ L , we define: ζ η � →η ≡ ∈ H η � →η .We have ∀η � η � � �ζ η � →η � = 1, since µ η � →η ( C η � →η ) = µ η � ( C η � ) / µ η ( C η ) = 1, and ∀η � η � � η �� � Φ η �� →η � →η ( ζ η �� →η � ) ≡ ≡ ζ η �� →η � ⊗ ζ η � →η . � Finally, note that, as far as the construction of the quantum projective state space and observablesthereof is concerned, we can actually dispense from having a factorizing system of measures.Indeed, if we just have families � µ η � η∈ L and � µ η � →η � η � η � of smooth measures, which do not satisfythe compatibility conditions from def. 3.1. , we can rely on the canonical identification introducedin prop. A.15 to relate the position representation built on the measure µ η � with the one built onthe measure φ ∗η � →η ( µ η � →η × µ η ) . Provided this conversion is incorporated in the definition of thequantum projective structure, one can check that the three-spaces consistency condition still holds.In contrast, the consistency of the measures is essential for the inductive construction of prop. 3.5,where it ensures the compatibility of the reference states ζ η � →η . We now turn to the holomorphic representation. In order to get the scalar product right, we annot spare, when doing holomorphic quantization, a formulation using a prequantum bundle B η (see [27, section 8] and appendix A, def. A.2) constructed over M η (this differs from the previoussubsection, for when dealing with configuration representation, the relevant part of the bundlestructure is flat and, as a result, the prequantum bundle only needs to be taken into accountif a unified context for describing various representations is demanded). Therefore, we begin byexamining how to arrange prequantum bundles built on the M η into a form of factorizing structuresuitable for quantization. More precisely, we are looking for a way to connect the B η bundles thatwill provide the required tensor product factorizations of the corresponding L -spaces of bundlecross-sections (the prequantum Hilbert spaces, see [27, section 8] and def. A.3).To address this question, we go back to the basics underlying the tensor product decomposition ofthe L -space of complex-valued functions over a Cartesian product A × B : there, the tensor productof a function on A with a function on B is obtained as their pointwise product. Accordingly, whatwe need in the hermitian line bundle case is an operation to make the ‘product’ of a point in thebundle above A with a point in the one above B , and we want this operation to be valued in thebundle we happen to have above A × B . Definition 3.6
Let M , N � and N be three smooth, finite dimensional, manifolds, and let φ : M → N � × N be a diffeomorphism. Let B M , B N � and B N be hermitian line bundles (def. A.1), respectivelywith base M , N � and N . We call a smooth map ζ : B N � × B N → B M a factorization of B M compatible with φ iff: φ ◦ Π B M ◦ ζ = Π B N � × Π B N , where Π B M , Π B N � and Π B N are the bundles projections of B M , B N � and B N respectively; ∀� � ∈ B N � � ∀� ∈ B N � |ζ ( � � � � ) | = |� � | |�| ; ∀� � ∈ B N � � ∀� ∈ B N � ∀λ � � λ ∈ C � ζ ( λ � � � � λ� ) = λ � λ ζ ( � � � � ) . Proposition 3.7
We consider the same objects as in def. 3.6. Moreover, we assume that N � and N are equipped with smooth measures µ N � and µ N , and we equip M with the smooth measure µ M := φ ∗ ( µ N � × µ N ). Then, there exists a unique Hilbert space isomorphism:Φ ζ : L ( M → B M � �µ M ) → L � N � → B N � � �µ N � � ⊗ L ( N → B N � �µ N ) ,such that: ∀� � ∈ L � N � → B N � � �µ N � � � ∀� ∈ L ( N → B N � �µ N ) � Φ ζ �� ζ ( � � � � )� = � � ⊗ � ,where ∀� � � � ∈ N � × N � � ζ ( � � � � ) ◦ φ − ( � � � � ) := ζ � � � ( � � ) � � ( � )�. Proof
We define H M := L ( M → B M � �µ M ), and similarly H N � and H N .We first want to prove that Vect �� ζ ( � � � � ) ��� � � ∈ H N � & � ∈ H N � is dense in H M . It is well-defined as a vector subspace of H M for ∀� � ∈ H M � ∀� ∈ H N � � ζ ( � � � � ) is a cross-section of B M (from def. 3.6. ) and ���� ζ ( � � � � )��� M = �� � � N � ��� N (from def. 3.6. and Fubini’s theorem), hence� ζ ( � � � � ) ∈ H M .The cross-sections with compact support are dense in H M , and, by partition of the unity, they re linear combinations of cross-sections with compact support of the form W := φ − �V � × V � ,where V � is a trivialization patch for B N � and V a trivialization patch for B N . Given a non-zerocross-section � , resp. � � , of B N � | V � , resp. B N | V , we define a non-zero cross-section � �� of B M | W by: ∀� � � � ∈ V � × V � � �� ◦ φ ( � � � � ) := ζ � � � ( � � ) � � ( � )� .Thus, using the trivialization defined by � � , resp. � , � �� , to identify the vector subspace of H N � ,resp. H N , H M , of cross-sections with compact support in V � , resp. V , W , with L � V � � �µ N � | V � �,resp. L � V � �µ N | V �, L � W � �µ N | W �, the restriction � ζ W of � ζ to these vector subspaces is given by:� ζ W ( � � � � � � � ) = � � � ⊗ � � ◦ � φ| W →V � ×V � � �� (from def. 3.6. ).Hence, its image is dense in L � W � �µ N | W � = L � V � � �µ N � | V � � ⊗ L � W � �µ N | W � (which equalityfollows from def. 3.6. and Fubini’s theorem).Now, the application:� ζ : H N � × H N → H M ( � � � � ) �→ � ζ ( � � � � )is a bilinear map (from def. 3.6. ) and satisfies (from def. 3.6. and Fubini’s theorem): ∀� � � � � ∈ H N � � ∀�� � ∈ H N � �� ζ ( � � � � ) � � ζ ( � � � � )� M = �� � � � � � N � ��� �� N .Hence, there exists a unique Hilbert space isomorphism Φ − ζ : H N � ⊗ H N → Vect Im� ζ = H M ,such that ∀� � ∈ H N � � ∀� ∈ H N � Φ − ζ ( � � ⊗ � ) = � ζ ( � � � � ). � With this, we can now present the announced factorizing structure for prequantum bundles. Asusual, we need to require an appropriate ‘three-spaces consistency’ that will support the correspond-ing consistency of the projective limits we are ultimately interested in (fig. 3.1 looks slightly differentfrom what we had for factorizing system of phase spaces in [15, fig. 2.2], because we are forcedto define the maps ζ in the direction opposite to our standard convention for factorizing maps).Note that we also have a compatibility condition involving the connection of the prequantum bun-dles, that will come into play when (pre-)quantizing observables and expressing the holomorphiccondition. Definition 3.8
Let ( L � M � φ ) × be a factorizing system of finite dimensional phase spaces [15,def. 2.12]. A factorizing system of prequantum bundles for ( L � M � φ ) × is a quadruple:�� B η � ∇ η � η∈ L � � B η � →η � ∇ η � →η � η � η � � � ζ η � →η � η � η � � � ζ η �� →η � →η � η � η � � η �� �such that: ∀η ∈ L , � B η � ∇ η � is a prequantum bundle for M η (def. A.2); ∀η � η � ∈ L , � B η � →η � ∇ η � →η � is a prequantum bundle for M η � →η (except for the case η = η � : M η→η has only one element and B η→η = C ); ∀η � η � ∈ L , ζ η � →η : B η � →η × B η → B η � is a factorization of B η � compatible with φ η � →η : M η � → η �� B η �� →η � × B η � B η �� →η � × B η � →η × B η B η �� →η × B η ζ η �� →η � ζ η �� →η ζ η � →η ζ η �� →η � →η Figure 3.1 – Three-spaces consistency for factorizing systems of prequantum bundles M η � →η × M η ; ∀η � η � ∈ L , ∀� ∈ B η � →η � ∀� ∈ B η � � T ��� ζ η � →η � �Hor � ( B η � →η � ∇ η � →η ) × Hor � ( B η � ∇ η )� = Hor ζ η�→η ( ��� ) ( B η � � ∇ η � ) , (3.8. )where Hor � ( B η � ∇ η ) is defined as the ∇ η -horizontal subspace of T � ( B η ) for � ∈ B η , andHor � ( B η � →η � ∇ η � →η ) is defined similarly for � ∈ B η � →η ; ∀η � η � � η �� ∈ L , ζ η �� →η � →η : B η �� →η � × B η � →η → B η �� →η is a smooth map such that: ζ η �� →η ◦ ( ζ η �� →η � →η × id B η ) = ζ η �� →η � ◦ (id B η��→η� × ζ η � →η ) . (3.8. ) Def. 3.8 seems to require a lot, so it is reassuring that, at least in the topologically trivial case,we can construct such a structure for any factorizing system of phase spaces satisfying nothingbut the quantization rule [27, section 8.3], which is anyhow mandatory to ensure the existence ofprequantum bundles for the M η . Theorem 3.9
Let ( L � M � φ ) × be a factorizing system of finite dimensional phase spaces such that: ∀η ∈ L � M η is simply-connected; ∀η ∈ L , ∀ S a closed oriented 2-surface in M η , � S Ω η ∈ π Z , where Ω η is the symplecticstructure of M η .Then there exists a factorizing system of prequantum bundles for ( L � M � φ ) × . Proof
For η � η � ∈ L , we have that M η � →η is simply-connected, otherwise M η � ≈ M η � →η × M η would not be simply-connected.Besides, for any oriented 2-surface S η � in M η � , we have: S η� Ω η � = � φ η�→η ◦ S η� φ − �∗η � →η Ω η � = � φ η�→η ◦ S η� Ω η � →η × Ω η = � S η�→η Ω η � →η + � S η Ω η , (3.9. )where S η � →η , resp. S η , is the projection on M η � →η , resp. M η , of φ η � →η ◦ S η � (which is an oriented2-surface in M η � →η × M η ). In particular, if S η � →η is a closed oriented 2-surface in M η � →η , applyingeq. (3.9. ) to S η � := φ − η � →η ◦ � S η � →η × {� η } �, where � η is any point in M η , gives:� S η�→η Ω η � →η = � S η� Ω η � ∈ π Z .Let � � �η � η∈ L ∈ S × ( L � M �φ ) and let η ∈ L . The construction in [27, section 8.3] tells us that, thanksto the conditions 3.9. and 3.9. , we can construct a prequantum bundle � B η � ∇ η � for M η in such away that B η can be identified with the equivalence classes in:�� � η � � η � γ η � �� � η ∈ M η � � η ∈ C � γ η is a piecewise smooth path from � �η to � η � ,for the equivalence relation:�� � η � � η � γ η � � � � �η � � �η � γ �η � � ⇔ � η = � �η � �η = � η exp � −� � Σ η ( γ η �γ �η ) Ω η � ,where Σ η ( γ η � γ �η ) is any oriented 2-surface in M η such that ∂ Σ η ( γ η � γ �η ) = γ �η− � γ η . Moreover, the ∇ η -parallel transport along some path γ �η in M η is then given by: P ∇ η γ �η �� γ �η (0) � � η � γ η � � � = � γ �η (1) � � η � γ �η � γ η � � .Since we proved above that, for all η � η � , M η � →η also fulfills these conditions, we can makethe same construction to obtain a prequantum bundle � B η � →η � ∇ η � →η �, using as origin the point � �η � →η ∈ M η � →η , defined by φ η � →η ( � �η � ) = ( � �η � →η � � �η ) .Now, for η � η � ∈ L , we define ζ η � →η : B η � →η × B η → B η � by: ζ η � →η �[ � η � →η � � η � →η � γ η � →η ] � � [ � η � � η � γ η ] � � := � φ − η � →η ( � η � →η � � η ) � � η � →η � η � φ − η � →η ( γ η � →η � γ η )� � .This is a well-defined map, for we have, using eq. (3.9. ):exp � −� � Σ η� � φ − η�→η ( γ η�→η � γ η ) � φ − η�→η ( γ �η�→η � γ �η )� Ω η � � == exp � −� � Σ η�→η � γ η�→η � γ �η�→η � Ω η � →η � exp � −� � Σ η ( γ η � γ �η ) Ω η � .Moreover, we can check that it fulfills defs. 3.6. to 3.6. .Let � η � ∈ M η � and ( � η � →η � � η ) = φ η � →η ( � η � ). Let γ η and γ η � →η be piecewise smooth paths from � �η to � η , and � �η � →η to � η � →η , respectively. We can choose local coordinates around � η � →η in M η � →η nd around � η in M η . Hence, we have diffeomorphisms ψ η � →η : [ − � � η�→η → U η � →η , resp. ψ η :[ − � � η → U η , where � η � →η := dim M η � →η , resp. � η := dim M η , and U η � →η , resp. U η , is an openneighborhood of � η � →η in M η � →η , resp. of � η in M η . This provides us local trivializations of thebundles B η � , B η � →η and B η , by: ∀� η � →η ∈ [ − � � η�→η � ∀� η � →η ∈ C � Ψ η � →η ( � η � →η � � η � →η ) = � ψ η � →η ( � η � →η ) � � η � →η � χ � η�→η � γ η � →η � � , ∀� η ∈ [ − � � η � ∀� η ∈ C � Ψ η ( � η � � η ) = � ψ η ( � η ) � � η � χ � η � γ η � � , ∀� η � →η � � η ∈ [ − � � η�→η × [ − � � η � ∀� η � ∈ C � Ψ η � ( � η � →η � � η � � η � ) = � φ − η � →η � ψ η � →η ( � η � →η ) � ψ η ( � η )� � � η � � φ − η � →η � χ � η�→η � γ η � →η � χ � η � γ η �� � ,where χ � η�→η : τ �→ ψ η � →η ( τ � η � →η ) and χ � η : τ �→ ψ η ( τ � η ) .And we have: ∀� η � →η � � η ∈ [ − � � η�→η × [ − � � η � ∀� η � →η � � η ∈ C � Ψ − η � ◦ ζ �Ψ η � →η ( � η � →η � � η � →η ) � Ψ η ( � η � � η )� = ( � η � →η � � η � � η � →η � η ) ,Therefore, ζ η � →η is smooth.Then, for γ �η � →η a path in M η � →η , and γ �η a path in M η , we have: P ∇ η� φ − η�→η ( γ �η�→η � γ �η ) ◦ ζ η � →η �� γ �η � →η (0) � � η � →η � γ η � →η � � � � γ �η (0) � � η � γ η � � � == � φ − η � →η � γ �η � →η (1) � γ �η (1)� � � η � →η � η � φ − η � →η � γ �η � →η � γ η � →η � γ �η � γ η �� � = ζ η � →η � P ∇ η�→η γ �η�→η �� γ �η � →η (0) � � η � →η � γ η � →η � � � � P ∇ η γ �η �� γ �η (0) � � η � γ η � � � � ,hence P ∇ η� φ − η�→η ( γ �η�→η � γ �η ) ◦ ζ η � →η = ζ η � →η ◦ � P ∇ η�→η γ �η�→η � P ∇ η γ �η � . Therefore, eq. (3.8. ) is fulfilled.Lastly, for η � η � � η �� ∈ L , we can in a similar way define a map ζ η �� →η � →η : B η �� →η � × B η � →η andwe have: ζ η �� →η ◦ ( ζ η �� →η � →η × id B η ) �[ � η �� →η � � � η �� →η � � γ η �� →η � ] � � [ � η � →η � � η � →η � γ η � →η ] � � [ � η � � η � γ η ] � � == � φ − η �� →η � φ − η �� →η � →η ( � η �� →η � � � η � →η ) � � η � � � η �� →η � � η � →η � η � φ − η �� →η � φ − η �� →η � →η ( γ η �� →η � � γ η � →η ) � γ η �� � = � φ − η �� →η � � � η �� →η � � φ − η � →η ( � η � →η � � η )� � � η �� →η � � η � →η � η � φ − η �� →η � � γ η �� →η � � φ − η � →η ( γ η � →η � γ η )�� � (using [15, eq. (2.11. )] )= ζ η �� →η � ◦ �id B η��→η� × ζ η � →η � �[ � η �� →η � � � η �� →η � � γ η �� →η � ] � � [ � η � →η � � η � →η � γ η � →η ] � � [ � η � � η � γ η ] � � , herefore eq. (3.8. ) holds. � The last ingredient we need in order to perform prequantization are measures on the M η and M η � →η , and they should be compatible, like we asked when setting up the configuration repre-sentation. But this is in fact something we can get automatically and in a very straightforwardway from the structure ( L � M � φ ) × , since a symplectic form gives us a natural volume form andthe compatibility of the symplectic forms is enough to ensure the compatibility of their associatedvolume form. Proposition 3.10
Let ( L � M � φ ) × be a factorizing system of finite dimensional phase spaces. Wedefine: for η ∈ L , the volume form ω η := 1( � η / η ∧ � � � ∧ Ω η = 1( � η / ∧� η / η on M η (where � η =dim M η ) and the corresponding smooth measure µ η on M η ; for η ≺ η � ∈ L , the volume form ω η � →η := 1( � η � →η / η � →η ∧ � � � ∧ Ω η � →η = 1( � η � →η / ∧� η�→η / η � →η on M η � →η (where � η � →η = dim M η � →η ) and the corresponding smooth measure µ η � →η on M η � →η .Then, this equips ( L � M � φ ) × with a structure of factorizing system of measured manifolds (def. 3.1). Proof
That ω η , resp. ω η � →η , is a nowhere-vanishing top-dimensional form on M η , resp. M η � →η , canbe checked in local Darboux coordinates.What is left to prove is the compatibility of these definitions of the volume forms with the maps φ η � →η and φ η �� →η � →η (def. 3.1. ). For η ≺ η � , we have: φ − �∗η � →η ω η � = 1( � η � / φ − �∗η � →η Ω η � � ∧� η� / = 1( � η � / η � →η × Ω η � ∧� η� / (since φ η � →η is a symplectomorphism)= 1( � η � →η / � η / η � →η � ∧� η�→η / ∧ �Ω η � ∧� η / = ω η � →η ∧ ω η ,hence φ − �∗η � →η µ η � = µ η � →η × µ η , and similarly, for η ≺ η � ≺ η �� : φ − �∗η �� →η � →η µ η �� →η = µ η �� →η � × µ η � →η . � On the grounds of the preliminaries developed so far, the prequantization of a factorizing systemof prequantum bundles is actually very similar to what we did for the position quantization, and,again, the link connecting the classical structure and the (pre-)quantum one is demonstrated byexposing the correspondence between observables. roposition 3.11 Let ( L � M � φ ) × be a factorizing system of finite dimensional phase spaces,equipped with a structure of factorizing system of measured manifolds according to prop. 3.10, andlet �� B η � ∇ η � η∈ L � � B η � →η � ∇ η � →η � η � η � � � ζ η � →η � η � η � � � ζ η �� →η � →η � η � η � � η �� � be a factorizing system ofprequantum bundles for ( L � M � φ ) × .We define: for η ∈ L , H preQ η := L ( M η → B η � �µ η ); for η ≺ η � ∈ L , H preQ η � →η := L ( M η � →η → B η � →η � �µ η � →η ), and Φ preQ η � →η := Φ ζ η�→η : H preQ η � → H preQ η � →η ⊗ H preQ η .Then, we can complete these elements into a projective system of quantum state spaces ( L � H preQ � Φ preQ ) ⊗ . Proof
The proof works like the one for prop. 3.3, using prop. 3.7 and: ∀η ≺ η � ≺ η �� � ∀� �� ∈ H preQ η �� →η � � ∀� � ∈ H preQ η � →η � ∀� ∈ H preQ η � Φ − ζ η��→η� ◦ �id H preQ η��→η� ⊗ Φ − ζ η�→η � ( � �� ⊗ � � ⊗ � ) = � ζ η �� →η � � � �� � � ζ η � →η ( � � � � )�= � ζ η �� →η �� ζ η �� →η � →η ( � �� � � � ) � � � (from eq. (3.8. ) and [15, eq. (2.11. )] )= Φ − ζ η��→η ◦ �Φ − ζ η��→η�→η ⊗ id H preQ η � � � �� ⊗ � � ⊗ � � . � Proposition 3.12
We consider the same objects as in prop. 3.11. For � η ∈ C ∞ ( M η � R ), we definethe prequantization � � η of � η as a densely defined operator on H preQ η (def. A.3).Let � η ∈ C ∞ ( M η � R ) and � η � ∈ C ∞ ( M η � � R ), such that � η ∼ � η � (the equivalence relation isdefined in [15, eq. (2.4. )], where we use π η � →η from [15, eq. (2.13. )] ). Then, � � η ∼ � � η � (with theequivalence relation defined in eq. (2.3. ) ).Hence, a classical observable � = � � η � ∼ ∈ O × ( L � M �φ ) [15, prop. 2.13] defines a prequantum observ-able � � := �� � η � ∼ ∈ O ⊗ ( L � H preQ � Φ preQ ) (prop. 2.5). Proof
Let � η ∈ C ∞ ( M η � R ) and let η � � η . Let � � ∈ Φ preQ �− η � →η � H preQ η � →η ⊗ D preQ η � (where D preQ η ⊂ H preQ η is the dense domain of � � η and the ⊗ is to be understood as a tensor product of vector spaces). Wedefine:Φ preQ η � →η ( � � ) =: � α � α ⊗ � α , with ∀α� � α ∈ D preQ η .Then, we have: ∀�� � ∈ M η � →η × M η � �Φ preQ �− η � →η ◦ �id H preQ η�→η ⊗ � � η � ◦ Φ preQ η � →η ( � � )� ◦ φ − η � →η ( �� � ) = � α ζ η � →η � � α ( � ) � � � η ( � ) � α ( � ) + �∇ η� X �η � α ( � )��= � α � η ◦ π η � →η ◦ φ − η � →η ( �� � ) �� ζ η � →η ( � α � � α )� ◦ φ − η � →η ( �� � ) ++ � � α � ∇ η � � T φ − η�→η (0 �X �η ) � ζ η � →η ( � α � � α )� ◦ φ − η � →η ( �� � ) (using eq. (3.8. ) and def. 3.6. )= �� � η ◦ π η � →η � � � + � ∇ η � � X �η ◦πη�→η � � � ◦ φ − η � →η ( �� � )= � � � η ◦ π η � →η ( � � )� ◦ φ − η � →η ( �� � ) .Therefore, we have ∀� η ∈ C ∞ ( M η � R ) � ∀η � � η� � � η ∼ � � η ◦ π η � →η . Hence, ∀� η ∈ C ∞ ( M η � R ) � ∀� η � ∈ C ∞ ( M η � � R ) � � � η ∼ � η � ⇔ � � η ∼ � � η � �. � Finally, we obtain the advertised holomorphic representation for a choice of Kähler structure onthe symplectic manifolds M η . Requiring the factorizing maps to be holomorphic is enough to ensurethat the holomorphic subspaces of the prequantum Hilbert spaces H η set up above will correctlydecompose over the already arranged tensor product factorizations, as can be shown by proving thecorresponding factorizing properties of the orthogonal projections on these (closed) vector subspaces. Definition 3.13
A factorizing system of Kähler manifolds is a factorizing system of phase spaces( L � M � φ ) × [15, def. 2.12] such that: for all η ∈ L , M η is equipped with a complex structure J η such that � M η � Ω η � J η � is a Kählermanifold (def. A.5); for all η ≺ η � ∈ L , M η � →η is equipped with a complex structure J η � →η such that � M η � →η � Ω η � →η � J η � →η �is a Kähler manifold; for all η ≺ η � ∈ L , φ η � →η is holomorphic, and for all η ≺ η � ≺ η �� ∈ L , φ η �� →η � →η is holomorphic. Proposition 3.14
We consider the same objects as in prop. 3.11, but we now moreover assume that( L � M � φ ) × is a factorizing system of Kähler manifolds. We define: for η ∈ L , H Holo η := H preQ η ∩ Holo � M η → B η � (prop. A.6); for η ≺ η � ∈ L , H Holo η � →η := H preQ η � →η ∩ Holo � M η � →η → B η � →η � and for η ∈ L , H Holo η→η := H preQ η→η = C .Then, for all η � η � , Φ preQ η � →η � H Holo η � � = H Holo η � →η ⊗ H Holo η and for all η � η � � η �� , Φ preQ η �� →η � →η � H Holo η �� →η � = H Holo η �� →η � ⊗ H Holo η � →η . Hence, defining: for η � η � ∈ L , Φ Holo η � →η := Φ preQ η � →η ��� H Holo η� → H Holo η�→η ⊗ H Holo η ; and for η � η � � η �� ∈ L , Φ Holo η �� →η � →η := Φ preQ η �� →η � →η ��� H Holo η��→η → H Holo η��→η� ⊗ H Holo η�→η ; L � H Holo � Φ Holo ) ⊗ is a projective system of quantum state spaces. Proof
First, for every η ∈ L , we can define a complex structure on B η in the following way: for � ∈ B η , with � = Π B η ( � ), we have T � � B η � = Hor � � B η � ∇ η � ⊕ T � �Π − B η ( � )�, where Hor � � B η � ∇ η �can be identified T � ( M η ), and thus equipped with the lift of the complex structure J η�� , while on T � �Π − B η ( � )�, the multiplication by � provide a natural complex structure. With this, a cross-sectionof B η is holomorphic if and only if it is holomorphic as a map M η → B η .Similarily, for every η � η � ∈ L , we have a complex structure on B η � →η . Eq. (3.8. ), togetherwith defs. 3.6. , 3.6. and the holomorphicity of φ η � →η , ensures that ζ η � →η is holomorphic as a map B η � →η × B η → B η � .For η ∈ L , Φ preQ η→η is a trivial identification, hence the desired result holds. Thus, we consider η ≺ η � ∈ L . We first want to prove that Φ preQ η � →η ◦ Π Holo η � ◦ Φ preQ �− η � →η = Π Holo η � →η ⊗ Π Holo η , where Π Holo η � : H preQ η � → H Holo η � is the orthogonal projection on the closed vector subspace H Holo η � in H preQ η � , and Π Holo η ,Π Holo η � →η are defined analogously.Let � ∈ H preQ η � →η , � ∈ H preQ η and define � = Π Holo η � →η � and � = Π Holo η � . By definition of Φ preQ η � →η , we have: ∀�� � ∈ M η � →η × M η � Φ preQ �− η � →η � � ⊗ � � ◦ φ − η � →η ( �� � ) = ζ η � →η ( � ( � ) � � ( � )) .But, as a composition of holomorphic maps, ( �� � ) �→ ζ η � →η ( � ( � ) � � ( � )) is holomorphic. Hence, φ η � →η being holomorphic, Φ preQ �− η � →η � � ⊗ � � ∈ H Holo η � .Let � � ∈ H Holo η � . Using the volume-preserving property of φ η � →η , we compute:� � � � Π Holo η � ◦ Φ preQ �− η � →η ( � ⊗ � )� H η� = � � � � � ζ η � →η ( �� � )� H η� = � M η�→η �µ η � →η ( � ) � M η �µ η ( � ) � � � ◦ φ − η � →η ( �� � ) � ζ η � →η ( � ( � ) � � ( � ))� B η� .For �� � ∈ M η � →η × M η we define � � ( � ) ∈ B �η such that: ∀� � ∈ B �η � � � � ◦ φ − η � →η ( �� � ) � ζ η � →η ( � ( � ) � � � )� B η� = �� � ( � ) � � � � B η , � � ( � ) is well-defined, since the left-hand side is a C -linear function of � � (from def. 3.6. ).Since � � ∈ H Holo η � , the map � �→ � � ◦ φ − η � →η ( �� � ) is holomorphic, and for any (local) anti-holomorphiccross-section � � of B η , the map: � �→ � ζ η � →η ( � ( � ) � � � ) � � � ◦ φ − η � →η ( �� � )� B η� ,is holomorphic (for the connection is a U (1)-connection, so the parallel transport preserve the scalarproduct). Therefore, the cross-section � � is holomorphic. Moreover, using def. 3.6. : ∀� ∈ M η � |� � ( � ) | � �� � � ◦ φ − η � →η ( �� � )�� |� ( � ) | ,thus, by Fubini theorem, for almost every � in � M η � →η � �µ η � →η �, � � ∈ H Holo η .Hence, for almost every � ∈ M η � →η : M η �µ η ( � ) � � � ◦ φ − η � →η ( �� � ) � ζ η � →η ( � ( � ) � � ( � ))� B η� = � M η �µ η ( � ) �� � ( � ) � � ( � ) � B η = �� � � �� H η = � � � � Π Holo η � � H η = � M η �µ η ( � ) � � � ◦ φ − η � →η ( �� � ) � ζ η � →η ( � ( � ) � � ( � ))� B η� .And we can prove in a similar way that, for almost every � ∈ M η � →η :� M η�→η �µ η � →η ( � ) � � � ◦ φ − η � →η ( �� � ) � ζ η � →η ( � ( � ) � � ( � ))� B η� == � M η�→η �µ η � →η ( � ) � � � ◦ φ − η � →η ( �� � ) � ζ η � →η ( � ( � ) � � ( � ))� B η� .Therefore, we arrive at:� � � � Π Holo η � ◦ Φ preQ �− η � →η ( � ⊗ � )� H η� = � � � � Φ preQ �− η � →η ( � ⊗ � )� H η� .Since this holds for all � � ∈ H Holo η � and we have already proved that Φ preQ �− η � →η � � ⊗ � � ∈ H Holo η � , we have:Π Holo η � ◦ Φ preQ �− η � →η ( � ⊗ � ) = Φ preQ �− η � →η ( � ⊗ � ) ,which gives us the announced result:Φ preQ η � →η ◦ Π Holo η � ◦ Φ preQ �− η � →η = Π Holo η � →η ⊗ Π Holo η .Hence, Φ preQ η � →η � H Holo η � � = Φ preQ η � →η ◦ Π Holo η � ◦ Φ preQ �− η � →η � H preQ η � →η ⊗ H preQ η � = Π Holo η � →η ⊗ Π Holo η � H preQ η � →η ⊗ H preQ η � = H Holo η � →η ⊗ H Holo η . And the relation involving Φ preQ η �� →η � →η can be proved in a similar way. � Note that in an holomorphic representation, the evaluation of the holomorphic wavefunction ata given point in phase space is a bounded linear form (via an argument similar to the proof ofprop. A.6), hence is dual to a vector in the Hilbert space: this defines the coherent state centeredaround this classical point. Now, if we choose an element in the projective family of symplecticmanifolds we started from, ie. a projective family of points � � η � η∈ L , we can form a projective familyof quantum states, by considering, in each H Holo η , the coherent state centered around � η . This familyof states will moreover be of the form considered in theorem 2.9, so we can apply this result to geta corresponding inductive limit Hilbert space and characterize its range in the quantum projectivestate space (for example, the Fock representation we will consider in [16, prop. 3.17] can be obtainedin this manner). hile we have discussed extensively how the classical structures presented in [15, section 2]can be converted into their quantum analogues, we have not yet formalized how to infer from thestrategy exposed in [15, section 3] a program for dealing with constraints at the quantum level.Nevertheless, an example will be considered in [16, subsection 3.2], suggesting how such a programwould look like.Our main motivation for studying this projective approach to quantum field theory being itsapplication to quantum gravity, we will in forthcoming work construct projective quantum statespaces closely related to the Hilbert spaces currently used in Loop Quantum Gravity (LQG) andLoop Quantum Cosmology (LQC). An important goal here is to obtain states in the full theorythat are almost symmetric both in configuration and momentum variables, and that we couldidentify with the states of the reduced theory. This problem is closely related to the search forgood coherent states and this is where extending the space of states can make a difference. Also,symmetry reducing a theory is mathematically the same as imposing second class constraintsand will thus involve the construction of an appropriate regularization strategy for their quantumimplementation.Other interesting directions for further work include the development of quantization prescrip-tions going beyond the framework laid out in section 3. The ultimate goal would be to havea general procedure, building upon geometric quantization, to quantize any projective system ofclassical phase spaces, as soon as we give us a consistent family of polarizations thereon.In particular, it should be possible to relax the requirement of having a global factorization.Recall that in our discussion of the local factorization result [15, prop. 2.10], we had identified twodifferent kinds of obstructions that could prevent it from holding globally. In both cases, we cansketch a route for proceeding to quantization nevertheless. The first kind of obstruction is realizedwhen M η � cannot be written as a Cartesian product, but at least can be seen as an open subsetof a bigger manifold � M η � := M η � →η × M η . This suggests to deal with this situation by a slightgeneralization of def. 2.1, allowing H η � to be a closed vector subspace in a bigger Hilbert space � H η � := H η � →η ⊗ H η . Then, the density matrices over H η � could be seen as density matrices over � H η � ,with support restricted to H η � . Thus, it would still be possible define a map Tr η � →η : S η � → S η byfirst embedding S η � in � S η � and then tracing over H η � →η . While such a map could no longer be seenas a partial trace over a tensor factor in H η � →η , it should still retain the properties that we reallyneed for the formalism to make sense (in particular, appropriate compatibility with the evaluationof expectation values). The other obstacle for a global factorization is illustrated by taking M η � asa covering space of M η : in this case we still have the option of writing M η � � M η � →η × M η with adiscrete space M η � →η , but we have to accept that the identification will not be everywhere smooth:there will be cuts, and the disposition of these cuts will, when going over to the quantum theory,be imprinted in the precise interpretation of the observables (we will provide an example of thisprocedure when investigating LQC). Acknowledgements
This work has been financially supported by the Université François Rabelais, Tours, France.This research project has been supported by funds to Emerging Field Project “Quantum Geometry”from the FAU Erlangen-Nuernberg within its Emerging Fields Initiative. Appendix: Geometric quantization
The aim of this appendix is to import a few definitions and properties from geometric quan-tization, that are needed in particular for section 3. We try here to give a short self-containedintroduction, leading rapidly to the definition of the holomorphic representation on a Kähler mani-fold [27, sections 8.4 & 9.2], and of the position representation arising from a choice of configurationvariables on a symplectic manifold [27, sections 4.5 & 9.3]. Accordingly, we skip advanced aspects,including underlying insights and technical subtleties.
In this appendix all manifolds are assumed to be smooth, finite dimensional manifolds and allmaps between them are assumed to be smooth.
A.1 Prequantization
Definition A.1
An hermitian line bundle is a vector bundle ( B � Π B � M ) associated to a U (1)-principal bundle on a smooth manifold M via the standard action of U (1) on C . Since the C -linearstructure and the Hermitian product � · � · � : �� � � �→ � ∗ � � on C is preserved under the action of U (1),each fiber of B can be equipped with a natural hermitian structure.Any connection in the U (1)-principal bundle defines a covariant derivative ∇ on B , and we canassociate to its curvature a Lie( U (1)) ≈ R -valued 2-form D∇ on M , such that for any cross-section � of B and any vector fields X � Y on M :[ ∇ X � ∇ Y ] ( � ) = ∇ [ X�Y ] ( � ) + � D∇ ( X � Y ) � . (A.1. ) Definition A.2
Let M be a symplectic manifold (with symplectic structure Ω). A prequantumbundle ( B � ∇ ) on M is an hermitian line bundle B , with base M , equipped with a connection, withcorresponding covariant derivative ∇ and such that: D∇ = − Ω .On M , we define the symplectic volume form ω := 1 � ! Ω ∧ � � � ∧ Ω = 1 � ! Ω ∧� (where � := dim M / µ ω . Definition A.3
Considering the same objects as in def. A.2, we define the prequantum Hilbert space H preQ as the space L ( M → B � �µ ω ) of (equivalence classes up to almost-everywhere equality of )cross-sections of B whose norm, defined using the hermitian structure on B , is square-integrablewith respect to µ ω .For � ∈ C ∞ ( M � C ), we define the prequantization � � of � as a (densely defined) operator on H preQ by: ∀� ∈ D � ⊂ H preQ � � � � := � � + � ∇ X � � ,where ∇ X � := ∇ X Re( � ) + � ∇ X Im( � ) . roposition A.4 Let � � � ∈ C ∞ ( M � C ). Then:�� � � � � � = � � {� � �} .and: ∀�� � � ∈ D � � �� � ∗ ( � ) � � � � = � �� � � ( � � )� . Proof
Let � ∈ D ��� (defining the common domain D ��� such that both � � � � � and � � � � � are well-defined; since D ��� contains at least the compactly supported smooth cross-sections, it is dense in H preQ ). Using eq. (A.1. ) we have:�� � � � � � ( � ) = − � ∇ X � � ∇ X � � ( � ) + � � X � ( � ) � − � � X � ( � ) � = −∇ [ X � � X � ] ( � ) + � Ω � X � � X � � � + � Ω � X � � X � � � − � Ω � X � � X � � � = −∇ X {���} ( � ) + � {� � �} � = � � {� � �} ( � ) .Let �� � � ∈ D � . We have for all X vector field on M : ∀� ∈ M � � X ��� � � � ( � ) = �∇ X ( � )( � ) � � � ( � ) � + �� ( � ) � ∇ X ( � � )( � ) � , (A.4. )for ∇ comes from a U (1)-connection. Hence, we get:�� � ∗ ( � ) � � � � = � �µ ω ( � ) � � ∇ X ∗� ( � )( � ) � � � ( � )� +� �µ ω ( � ) �� ∗ � ( � ) � � � ( � ) � = − � �µ ω ( � ) � � X � ��� � � � ( � ) +� �µ ω ( � ) �� ( � ) � � ∇ X � ( � � )( � ) � +� �µ ω ( � ) �� ( � ) � � � � ( � ) � = � � ��� � � � ( L X � ω ) +� �µ ω ( � ) � � ( � ) � � � ( � � )( � )�(using Stokes theorem [17, theorem 10.23]; M is assumed to be without boundary, or D � isrequired to ensure suitable fall-off conditions)= � �µ ω ( � ) � � ( � ) � � � ( � � )( � )� = � �� � � ( � � )�(for X � generates symplectomorphisms, thus preserving the symplectic volume form ω ). � The prequantization of M leads to a faithful representation of the full Poisson-algebra C ∞ ( M � C ) .However, this representation is typically much too big (as is to be expected from the Groenewold-Van-Hove theorem [9] and generalizations thereof [8]), so the next step will be to implementadditional prescriptions yielding a physically admissible Hilbert space (at the cost of restrictingwhich observables can be quantized). .2 Holomorphic representation To discuss holomorphic quantization we need to equip M with an almost complex structure J (def. A.5. ), which is required to be integrable (def. A.5. , ensuring the existence of local holomorphiccoordinates, thus making J into a complex structure for M ) and compatible with the symplecticstructure Ω (def. A.5. ). An additional positivity requirement (def. A.5. ) allows to define from Ω and J a Riemannian metric on M (the so-called Kähler metric) and makes M into a Kählermanifold [14, section IX.4]. Definition A.5
A Kähler manifold ( M � Ω � J ) is a symplectic manifold ( M � Ω) equipped with asmooth field J satisfying: ∀� ∈ M � J � is an endomorphism of T � ( M ) such that J � = − id T � ( M ) ; ∀X � Y ∈ T ∞ ( M ) � [ JX � JY ] − J [ X � JY ] − J [ JX � Y ] − [ X � Y ] = 0 (where T ∞ ( M ) is the space ofsmooth vector fields on M ); ∀� ∈ M � ∀�� w ∈ T � ( M ) � Ω � ( J � �� J � w ) = Ω( �� w ) ; ∀� ∈ M � ∀� � = 0 ∈ T � ( M ) � Ω � ( �� J � � ) > Proposition A.6
Let M be a Kähler manifold and ( B � ∇ ) be a prequantum bundle on M . We definethe holomorphic quantization H Holo of M to be H Holo := H preQ ∩ Holo( M → B ), where Holo( M → B )is the space of holomorphic cross-sections of B :Holo( M → B ) := {� ∈ C ∞ ( M → B ) | ∀� ∈ M � ∀� ∈ T � ( M ) � ∇ J� � = � ∇ � �} . H Holo is a closed vector subspace of H preQ , hence is itself an Hilbert space. Proof
Let ( � α ) α∈A be a net in H Holo that converges (for the norm � · � preQ ) to � ∈ H preQ .Let � ∈ M . There exist a neighborhood U of � in M , holomorphic coordinates � � � � � � � � (2 � := dim M ) on U and a real valued function K ( �� � ∗ ) on U such that [27, section 5.4]: ∀� � ∈ U� Ω � � = � ∂ K∂� � ∂� ��∗ ( � � ) �� � ∧ �� ��∗ ,and from A.5. ∂ K∂� � ∂� ��∗ has to be a positive definite hermitian matrix at every point in U . Thesymplectic measure is then given over U by: µ ω := β µ ( � ) C = 2 � det� ∂ K∂� ∂� ∗ � µ ( � ) C (where µ ( � ) C is the standard measure on C � ).We choose � ∈ Π − B ��� (the fiber of B above � ), with |�| = � −K ( � ) / , and we define the cross-section � of B | U by: � ( � ) = � & ∀� � ∈ U� ∀� ∈ T � � ( M ) � ∇ � � ( � � ) = − � ∂K∂� � ( � � ) � �� � � � � ( � )� � ( � � ) .We can check using eq. (A.1. ) that this characterizes a well-defined cross-section of B | U and wehave moreover: � � ∈ U� ∀� ∈ T � � ( M ) � ∇ J� � ( � � ) = � ∇ � � ( � � ) ,and [ � ��� �� ] � � ( � ) = −�K � � ( � ) ��� �� ( � � ) ,so � is an holomorphic cross-section of B | U and ∀� � ∈ U� |� ( � � ) | = � −K ( � � ) / .Next, for all α ∈ A we can define � α as the holomorphic function � α : U → C such that ∀� � ∈ U� � α ( � � ) = � α ( � � ) � ( � � ). Similarly, we define � : U → C such that ∀� � ∈ U� � ( � � ) = � ( � � ) � ( � � ).Let ε > U be a closed ball (with respect to the coordinates � � � � � � � � ) of center � andradius � > U ⊂ U and ∀� � ∈ U � β ( � � ) � −K ( � � ) > ε . Let U be the closed ball of center � and radius �/
2. For all � � ∈ U we call U � � the closed ball of center � � and radius �/
2. Hence, U � � ⊂ U . For � an holomorphic function on U → C , we have: ∀� � ∈ U � � ( � � ) = 8 � � � � � � k =1 � �/ � k �� k � � ( � � )= 4 � π � � � � � � k =1 � �/ �� k � π � k �θ k � � � � � + � � � �θ � � � � � � � � �θ � �� ,hence: ∀� � ∈ U � |� ( � � ) | � � 4 π � � � �� U �� �µ ( � ) C ( � ) |� ( � ) | � � � 4 π � � � π � � � � � U �� �µ ( � ) C ( � ) |� ( � ) | (by convexity of � �→ � ) � π � � U �µ ( � ) C ( � ) |� ( � ) | .Therefore, for all α� α � ∈ A : ∀� � ∈ U � |� α ( � � ) − � α � ( � � ) | � π � � U �µ ( � ) C ( � ) |� α ( � ) − � α � ( � ) | � ε π � � U �µ ω ( � ) |� α ( � ) − � α � ( � ) | |�| � ε π � �� α − � α � � preQ ,hence the net � � α | U � α∈A converges uniformly to a function � � : U → C . Cauchy’s integral formulaimplies that � � is holomorphic on the interior of U . On the other hand, the net � � α | U � α∈A convergesin L -norm to � | U (for ∀α ∈ A� �� � α | U − � | U �� � √ε �� α − �� preQ ), hence � � = � | U ( µ ω -almost-everywhere). Therefore, � ∈ H Holo . � Since we restrict the quantum Hilbert space to H Holo , we should also restrict the admissible bservables to be the ones that stabilize H Holo (note that we do not discuss here whether theintersection of H Holo with the dense domain of such an observable will also be dense in H Holo ; thisis a non-trivial question, for the usual tools based on bump functions are not available in theholomorphic class).
Proposition A.7
We consider the same objects as in prop. A.6. We define: O Holo � C := {� ∈ C ∞ ( M � C ) | ∀Y ∈ T ∞ ( M ) � ∃ Z ∈ T ∞ ( M ) / [ Y + � JY � X � ] = Z + � JZ } ,where T ∞ ( M ) is the space of smooth vector fields on M .Then, for all � ∈ O Holo � C � � � stabilizes H Holo . Proof
Let � ∈ O Holo � C and � ∈ D � ∩ H Holo . Let � ∈ M and � ∈ T � ( M ). Then, there exists Y ∈ T ∞ ( M )such that Y � = � (we can construct such a vector field using local smooth coordinates around � and an appropriate bump function). Since � ∈ O Holo � C , there also exists Z ∈ T ∞ ( M ) such that[ Y + � JY � X � ] = Z + � JZ . Hence: ∇ J�� � � � � = � ∇ JY � � � � ( � )= � � ( ∇ JY � )� ( � ) + �Ω ( X � � JY ) � � ( � ) + � ( ∇ JY ∇ X � � ) ( � )= � � ( ∇ JY � )� ( � ) − � � ∇ [ X � � J Y ] � � ( � ) + � ( ∇ X � ∇ JY � ) ( � ) (using eq. (A.1. ))= � � ( ∇ JY � )� ( � ) + ( ∇ Z + � JZ � ) ( � ) + � ∇ [ X � � Y ] � � ( � ) + � ( ∇ X � ∇ JY � ) ( � )= � � � ( ∇ Y � )� ( � ) + � ∇ [ X � � Y ] � � ( � ) − ( ∇ X � ∇ Y � ) ( � ) (using � ∈ H Holo )= � ∇ Y �� � � � ( � ) = � ∇ �� � � � � ,therefore � � � ∈ H Holo . � Proposition A.8
We consider the same objects as in prop. A.7. Let � be a nowhere-vanishingcross-section of B such that � ∈ H Holo and let µ � be the measure on M defined by µ � = ��� �� µ ω .Then, the map:Φ � : L ( M � �µ � ) ∩ Holo( M ) → H Holo ψ �→ ψ � ,is an Hilbert space isomorphism.If � ∈ O Holo � C and ψ ∈ Φ − � � D � � , we have:� � � ψ := �Φ − � � � Φ � � ψ = � ψ + � ( � X � ψ ) + � X �� ψ , (A.8. )where X �� is defined by 2 ∇ X � � = X �� � . Proof
Let � ∈ H Holo . Since � is a nowhere-vanishing holomorphic cross-section there exists aunique smooth function ψ : M → C such that � = ψ � . Moreover, for all � ∈ M and all � ∈ T � ( M ): ( � � ψ ) � = � ∇ � � − � ψ ∇ � � = ∇ J� � − ψ ∇ J� � = ( � J� ψ ) � ,hence ψ ∈ Holo( M ). Moreover: ��� Holo = � M �µ ω ( � ) �ψ ( � ) � ( � ) � ψ ( � ) � ( � ) � = � M �µ � ( � ) ψ ∗ ( � ) ψ ( � ) = �ψ� �µ � .Therefore, ψ ∈ L ( M � �µ � ). But since we have Φ � ( ψ ) = � and ��� Holo = �ψ� �µ � , Φ � is an Hilbertspace isomorphism.Eq. (A.8. ) can be checked from the definition of �( · ) (def. A.3). � A.3 Position representation
We now turn to the position representation. We describe a choice of configuration variables asa map γ from the phase space into the configuration space. The typical example occurs when M isgiven as a cotangent bundle (with its canonical symplectic structure) in which case γ is simply theprojection on the base manifold. Definition A.9
Let M be a symplectic manifold. A configuration space for M is a manifold C anda surjective map γ : M → C such that: ∀� ∈ M � Im( T � γ ) = T γ ( � ) ( C ); ∀� ∈ M � Ker( T � γ ) = �Ker( T � γ )� ⊥ := {� ∈ T � ( M ) | ∀w ∈ Ker( T � γ ) � Ω M �� ( �� w ) = 0 } . ∀� ∈ C � γ − �{�}� is connected. Definition A.10
Let M be a symplectic manifold, ( C � γ ) be a configuration space for M and ( B � ∇ )be a prequantum bundle on M . A configuration quantum bundle on C is an hermitian line bundle B C , with base C , and a smooth map Γ : B → B C such that: ∀� ∈ B � Π B C ◦ Γ( � ) = γ ◦ Π B ( � ) (where Π B and Π B C are the bundle projections); ∀� ∈ B � ∀λ ∈ C � Γ( λ � ) = λ Γ( � ) ; ∀� ∈ B � | Γ( � ) | = |�| ; ∀� ∈ B � Ker T � Γ ⊂ Hor � ( B � ∇ ) (where Hor � ( B � ∇ ) is defined as the ∇ -horizontal subspaceof T � ( B ) ). Proposition A.11
Let M be a symplectic manifold, ( C � γ ) be a configuration space for M and( B � ∇ ) be a prequantum bundle on M . If, for all � ∈ C , γ − �{�}� is simply-connected, then thereexists a configuration quantum bundle ( B C � Γ) on C . Proof
Definition of B C . Let � ∈ C and let � ∈ γ − ��� . Since the derivative of γ is surjective atany point in M (def. A.9. ), we have, by the rank theorem [17, theorem 5.13], T � � γ − ��� � = Ker T � γ .So, using def. A.9. , T � � γ − ��� � = T � � γ − ��� � ⊥ , hence Ω � | T � ( γ − ��� ) = 0. Therefore, if we call� B � � ∇ � � the restriction of ( B � ∇ ) over γ − ��� , the connection ∇ � is flat. herefore, � �→ Hor � ( B � ∇ ) ∩ Ker [ T � ( γ ◦ Π B )] is a smooth involutive tangent distribution on B ,so by the global Frobenius theorem [17, theorem 14.13], it defines a foliation of B . Moreover, ifΛ is a leaf of this foliation, there exists � ∈ C such that Π B � Λ � = γ − ��� and Π B | Λ →γ − ��� is adiffeomorphism. Indeed, the leaf Λ being connected by definition, γ ◦ Π B is constant on Λ, so thereexists � ∈ C such that Π B � Λ � ⊂ γ − ��� , ie. Λ ⊂ B � , and, γ − ��� being simply-connected, Λ isjust a global horizontal cross-section of � B � � ∇ � � [14, corollary II.9.2].We define B C as the set of all leaves and Γ : B → B C as the quotient map. Since γ ◦ Π B isconstant on a leaf, we can define a map Π B C on B C such that γ ◦ Π B = Π B C ◦ Γ. Moreover, forany leaf Λ and any λ ∈ C , λ� Λ is also a leaf, therefore, we can define an action of C on B C suchthat ∀� ∈ B � ∀λ ∈ C � Γ( λ � ) = λ Γ( � ). And since ∇ is a U (1) connection, the norm | · | on B isconstant on each leaf, so we also can define a norm on B C such that ∀� ∈ B � | Γ( � ) | = |�| . Local description of the quotient.
Let � ∈ M and let � = γ ( � ). Let U be an open neighborhoodof � in M and φ : U × C → B a local trivialization of the bundle B . Since T γ is surjective atany point in M , there exist, by the rank theorem, open neighborhoods V of � in C , W of 0 in R � ( � := dim M / C ), and U of � in U , and a diffeomorphism φ : V × W → U such that γ ◦ φ is the first projection map V × W → V .Next, by definition of a foliation, there exist a neighborhood T of φ ( ��
0) in φ �U × C � , neigh-borhoods Q and R of 0 in R � +2 and R � respectively and a diffeomorphism ψ : Q × R → T such that for any � ∈ B C there exists a (possibly empty) countable subset Q � ⊂ Q with:Γ − ��� ∩ T = ψ �Q � × R � . (A.11. )Let V , W and S be neighborhoods of � in V , 0 in W and 0 in C respectively, such that φ �φ �V × W � × S � ⊂ T , and define: ψ : V × S → Q �� λ �→ π Q ◦ ψ − ◦ φ ( φ ( �� � λ ) ,where π Q : Q × R → Q is the first projection map. Since we have:[ T (0 � ψ ] �{ } × T ( R ) � = Hor φ ( �� ( B � ∇ ) ∩ Ker [ T φ ( �� ( γ ◦ Π B )]and [ T ( �� φ ] �� T �� �φ � � T � ( V ) × { } � × C � ∩ Hor φ ( �� ( B � ∇ ) ∩ Ker [ T φ ( �� ( γ ◦ Π B )] = { } ,� T �� ψ � is surjective, thus invertible, so by the inverse function theorem [17, theorem 5.11], wecan narrow W and S so that there exists a neighborhood Q of 0 in Q with ψ inducing adiffeomorphism V × S → Q .Now, we define T := ψ �Q × R � ∩ φ �φ �V × W � × C � and: φ : T → V × S × W � �→ ψ − ◦ π Q ◦ ψ − ( � ) � π W ◦ φ − ◦ Π B ( � ) .Precomposing φ by φ ◦ ( φ × id C ), we can check that [ T φ ( �� φ ] is injective, thus invertible, sousing again the inverse function theorem, there exist neighborhoods T , V , W and S of φ ( ��
0) in B , � in C , 0 in R � and 0 in C respectively, and a diffeomorphism φ : V × W × S → T satisfying,for all �� λ ∈ V × S : � ∈ B C / Γ − ��� ∩ Im φ = φ �{�} × W × {λ}� & Π B C ( � ) = � (A.11. )(using eq. (A.11. ) and the injectivity of the restriction of Π B to the leaf Γ − ��� , together with φ �{�} × W × {λ}� = ψ �{ψ ( �� λ ) } × R � ∩ Im φ ) and for all �� w� λ ∈ V × W × S : ∀µ ∈ C / µλ ∈ S� φ ( �� w� µλ ) = µ φ ( �� w� λ ) (A.11. )(we can first check this for w = 0, and then use the previous point, since for all � ∈ B C and all µ ∈ C , Γ − �µ��� = µ � Γ − ��� ).Finally, using eq. (A.11. ), we can extend S to be all C while still satisfying eqs. (A.11. ) and(A.11. ). Compatibility of the local descriptions.
Let � � � ∈ M such that γ ( � ) = γ ( � ) =: � . There existsa path κ : [0 � → γ − ��� such that γ (0) = � and γ (1) = � (simple connectedness implies pathconnectedness).Using the preceding point, for all � ∈ [0 �
1] there exist open neighborhoods V � of � in C , W � of 0in R � and U � of κ ( � ) in M , and a diffeomorphism φ � : V � × W � × C → Π − B �U � � satisfying eq. (A.11. )and eq. (A.11. ). For any � ∈ [0 � π � the projection map π � : V � × W � × C → V � × C andwe define the smooth map Γ � := π � ◦ φ − � : Π − B �U � � → V � × C .Next, there exist � � � � � � � N− (1 � N < ∞ ) such that ( U � � ) � � � N is an open cover of κ � [0 � � (where we set � = 0 and � N = 1), and ∀� � N − � κ − �U � � ∩ U � � +1 � � = ∅ .We define V = � � � � N− γ �U � � ∩ U � � +1 � . γ is an open map (for T γ is surjective at any point),therefore V is an open subset of C , and for any � � N −
1, there exists � ∈ [0 �
1] such that κ ( � ) ∈ U � � ∩ U � � +1 , hence � = γ ◦ κ ( � ) ∈ U � � ∩ U � � +1 . Thus, V is an open neighborhood of � in C .Let � � N −
1. The maps Γ � � | Π − B � U �� ∩U �� +1 ∩γ − �V � � →V × C and Γ � � +1 | Π − B � U �� ∩U �� +1 ∩γ − �V � � →V × C are smooth,surjective, their derivatives are surjective at each point and they are constant on each other levelsets (using eq. (A.11. ) ), therefore the rank theorem implies [17, prop. 5.21] the existence of adiffeomorphism Φ � : V × C → V × C such that: ∀� ∈ Π − B � U � � ∩ U � � +1 ∩ γ − �V � � � Γ � � ( � ) = Φ � ◦ Γ � � +1 ( � ) .Thus, eq. (A.11. ) leads to: ∀� ∈ V � ∀λ ∈ C � ∃� ∈ B C / Γ − ��� ∩ Im φ � � = φ � � �{�} × W � � × {λ}� & Γ − ��� ∩ Im φ � � +1 = φ � � +1 ◦ �Φ − � �{�} × W � � +1 × {λ}� ,where �Φ � is defined naturally from Φ � as a map �Φ � : V × W � � +1 × C → V × W � � +1 × C .Defining Φ := Φ ◦ � � � ◦ Φ N− : V × C → V × C and �Φ : V × W � N × C → V × W � N × C , wehave: ∀� ∈ V � ∀λ ∈ C � ∀w ∈ W � ∀w ∈ W � Γ ◦ φ ( �� w � λ ) = Γ ◦ φ ◦ �Φ − ( �� w � λ ) .This way we have proved that for any � � � ∈ M such that γ ( � ) = γ ( � ) =: � , there exist pen neighborhoods V of � in C , W and W of 0 in R � , � U of � in M and � U of � in M ,diffeomorphisms � φ : V × W × C → Π − B � � U � and � φ : V × W × C → Π − B � � U �, and an injectivemap ψ : V × C → B C , such that: ∀� ∈ V � ∀λ ∈ C � ∀w ∈ W / � ψ ( �� λ ) = Γ ◦ � φ / ( �� w� λ�∀� ∈ V � ∀λ ∈ C � Π B C ◦ ψ ( �� λ ) = � and ∀�� w ∈ V × W / � ∀λ ∈ C � � φ / ( �� w� λ · ) = λ � φ / ( �� w� · ) . Topological, differentiable and bundle structures on B C . We equip B C with the final topologyinduced by Γ (so that U ⊂ B C is open iff Γ − �U� is open in B ). The previous point, together with γ ◦ Π B = Π B C ◦ Γ, ensures that Γ is an open map for this topology (because the preimage of theimage of an open subset of B is an open subset of B ), and that we can use the local descriptions ofthe quotient to define a bundle structure on B C , with respect to which Γ will be a smooth surjectivemap with surjective derivative at each point. We can check that this structure is then compatiblewith the projection Π B C and the action of C on B C defined above. � Since the cross-sections of B that are ∇ -horizontal over the level sets of γ are typically non-normalizable, we need to introduce a measure on C . In general, there is however no preferredchoice for this measure, hence we will associate to any smooth measure [7, section 11.4] on C acorresponding Hilbert space and we will restore the independence with respect to the choice ofmeasure by providing identifications between these different Hilbert spaces. Definition A.12
Let C be a smooth manifold. A smooth measure µ on C is a Borel measure on C such that, for any smooth coordinate chart φ : U → R � ( � := dim C ) on an open subset U of C ,there exists a smooth, nowhere vanishing, strictly positive function α φ : U → R satisfying: µ| U = α φ � φ ∗ µ ( � ) R � ,where µ ( � ) R is the Lebesgue measure on R � .In particular, the measure µ ω associated to a nowhere vanishing volume form ω on C is a smoothmeasure on C .For any smooth vector field X on C we define its divergence with respect to µ as the smoothfunction div µ X on C satisfying: L X µ = (div µ X ) µ .Finally, for any two smooth measures µ� µ � on C there exists a unique strictly positive smoothfunction α on C such that µ � = α µ . Definition A.13
We consider the same objects as in def. A.10. Let µ be a smooth measure on C . The position representation with measure µ is the Hilbert space H µ Pos := L ( C → B C � �µ ) ofcross-sections of B C with square-integrable norm with respect to µ . s underlined above, trimming the prequantum representation down to a physically pertinent sizecomes at the price of restricting the algebra of observables that can be quantized. In the positionrepresentation, this quantization condition requires that the Hamiltonian flow of an admissibleobservable should send level sets of γ onto level sets of γ . In the case of a cotangent bundle,the quantizable functions are therefore the ones that depend at most linearly on the momentumvariables. Proposition A.14
We consider the same objects as in def. A.13. We define: O Pos := � � ∈ C ∞ ( M � R ) �� ∃ X � ∈ T ∞ ( C ) � ∀� ∈ M � T � γ � X ��� � = X ��γ ( � ) � ,where T ∞ ( C ) is the space of smooth vector fields on C , and: O Pos � C := {� ∈ C ∞ ( M � C ) | Re � � Im � ∈ O Pos } .Then, for � ∈ O Pos , we can define the quantization � � µ of � as a densely defined operator on H µ Pos by: ∀� ∈ D µ� � ∀� ∈ C � �� � µ � � ( � ) := � ( � ) � ( � ) + � Γ � ∇ X ��� � � ( � )� + � µ X � � ( � ) � ( � ) ,where � is any point in γ − ��� , � � is the cross-section of B such that Γ ◦ � � = � ◦ γ , anddiv µ X � ∈ C ∞ ( C � R ) is such that L X � µ = �div µ X � � µ . For � ∈ O Pos � C , we define � � µ := � Re( � ) µ + �Im( � ) µ .Moreover, we have for all � � � ∈ O Pos � C : {� � �} M ∈ O Pos � C , �� � µ � � � µ � = � � {� � �} µ ,and ∀�� � � ∈ D µ� � � � � � � � µ ( � )� = �� � ∗ µ ( � � ) � � � . Proof
Let � ∈ O Pos and � ∈ D µ� . The cross-section � � such that Γ ◦ � � = � ◦ γ is well-defined, sincefor any � ∈ C , � ∈ γ − ��� and w ∈ Π − B C ��� , there is a unique � ∈ Π − B ��� such that Γ( � ) = w (this follows from def. A.10. ). We now want to prove that � ( � ) � ( � ) + � Γ � ∇ X ��� � � ( � )� does not dependon the choice of � in γ − ��� .Let V be any smooth vector field on M such that ∀� ∈ M � V � ∈ Ker T � γ . We have ∇ V � � = 0(since ∀� ∈ M � � V � � ( � ) ∈ Ker T � ( � ) Γ ⊂ Hor � ( � ) ( B � ∇ ) using def. A.10. ), therefore: ∀� ∈ M � ∇ V � ∇ X ��� � � ( � ) = � ∇ V � � ∇ X ��� � � � ( � )= ∇ [ V � � X ��� ] � � ( � ) − � Ω M �� � V � � X ��� � � � ( � ) ,and T γ � [
V � X � ] � = 0 (using ∀� ∈ M � V � ∈ Ker T � γ and T � γ � X ��� � = X ��γ ( � ) ), hence ∀� ∈ M � ∇ V � ∇ X ��� � � ( � ) = � ( � V � � ) � � ( � ). Therefore: ∀� ∈ M � ∇ V � � � ( � ) � � ( � ) + � ∇ X ��� � � ( � )� = 0 . (A.14. )Let � ∈ B and let V ��� be the ∇ -horizontal lift on B of the vector field V on M . Using T γ ( V ) = 0 together with def. A.10. , we have [ T Γ( � ) Π B C ] ◦ [ T � Γ] ( V ��� � ) = 0, so there exists � ∈ C such that T � Γ ( V ��� � ) = [ T ( · Γ( � ))] ( � ) = T � Γ ◦ [ T ( · � )] ( � ) (where we used def. A.10. to get the econd equality). Thus, using def. A.10. , V ��� � − [ T ( · � )] ( � ) ∈ Hor � ( B � ∇ ), therefore � = 0, and T � Γ ( V ��� � ) = 0.Hence, eq. (A.14. ) becomes:� T � � �→ Γ � � ( � ) � � ( � ) + � ∇ X ��� � � ( � )��� ( V ) = 0 ,so Γ [ � � � + � ∇ X � � � ] = � ( � ◦ γ ) + � Γ ( ∇ X � � � ) is constant on the level sets of γ . This ensures that � � µ � is well-defined as a cross-section of B C .Let � � � ∈ O Pos (since � · µ is C -linear, and [ · � · ] , { · � · } are C -bilinear, it is enough to consider R -valued functions to prove the commutator relations). Using the characterization of O Pos , we have: ∀� ∈ M � T � γ � X {�� �}�� � = T � γ �� X � � X � � � � = � X � � X � � γ ( � ) ,hence {� � �} ∈ O Pos with X {�� �} = � X � � X � �.Let � ∈ D µ��� (where, as in the proof of prop. A.4, the common domain D µ��� , such that both � � µ � � µ � and � � µ � � µ � are well-defined, is dense in H µ Pos ). Like in the proof of prop. A.4, we have:� � + � ∇ X � � � + � ∇ X � � � � = � {� � �} � � − ∇ X {���} � � .On the other hand, we can rewrite the definition of � � µ as: � �� � µ � � = � � + � ∇ X � + � µ X � � ◦ γ � � � (A.14. )thus: � ��� � µ � � � µ � � � = � � {� � �} − ∇ X {���} + 12 � � X � �div µ X � � ◦ γ − � X � �div µ X � � ◦ γ �� � � .Next, we have: � X � �div µ X � � ◦ γ − � X � �div µ X � � ◦ γ = � � X � �div µ X � � − � X � �div µ X � �� ◦ γ ,and L X � � L X � µ � = � � X � �div µ X � �� µ + �div µ X � � �div µ X � � µ , therefore, using X {�� �} = � X � � X � �: � X � �div µ X � � ◦ γ − � X � �div µ X � � ◦ γ = − �div µ X {�� �} � ◦ γ .Hence, using eq. (A.14. ) for � {� � �} µ :�� � µ � � � µ � � = � � {� � �} µ � .Lastly, let � ∈ O Pos � C and �� � � ∈ D µ� . Using def. A.10. , we have ∀� ∈ M � � � � � ( � ) � � � ( � ) � = �� � ◦ γ ( � ) � � ◦ γ ( � ) � and combining eq. (A.14. ) with eq. (A.4. ) : ∀� ∈ M � �� � � � �� � µ � � ( � ) = � �div µ X � � ◦ γ ( � ) � � � � � � �� ( � ) + � � X ��� � � � � � � �� ( � ) + � �� � ∗µ � � � � � � ( � ) ,therefore: ∀� ∈ C � � � � � � � µ � � ( � ) = � �div µ X � � ( � ) �� � � �� ( � ) + � � X ��� �� � � �� ( � ) + � � � ∗µ � � � � � ( � ) . ow, using Stokes theorem [2, theorem 7.7] ( C is assumed to be without boundary, or D µ� isrequired to ensure suitable fall-off conditions) and the definition of X � , we have:� C �µ ( � ) �div µ X � � ( � ) �� � � �� ( � ) + � X ��� �� � � �� ( � ) = � C L X � �� � � �� �µ � = 0 ,thus � � � � � � µ � � = � � � ∗µ � � � � � . � Proposition A.15
We consider the same objects as in def. A.10. Let µ and µ � be two smoothmeasures on C and let H µ Pos and H µ � Pos be the corresponding position representations. Then thereexists a Hilbert space isomorphism Φ µ→µ � : H µ Pos → H µ � Pos such that: ∀� ∈ O Pos � C � � � µ � = Φ µ→µ � ◦ � � µ ◦ Φ − µ→µ � . (A.15. )Moreover, we can define these maps in such a way that for any three smooth measures µ , µ � ,and µ �� , on C , Φ µ→µ �� = Φ µ � →µ �� ◦ Φ µ→µ � . Thus, this family of maps provides a position representation H Pos , that can be consistently identified with H µ Pos for any µ . Proof
Let µ and µ � be two smooth measures on C . Then, there exists a unique α ∈ C ∞ ( C � R ∗ + ) suchthat µ � = α µ (def. A.12). We define Φ µ→µ � by:Φ µ→µ � : H µ Pos → H µ � Pos � �→ √α � .The factor 1 √α ensures that Φ µ→µ � is a unitary map and we can check that for any three positivevolume forms µ , µ � , and µ �� , Φ µ→µ �� = Φ µ � →µ �� ◦ Φ µ→µ � . In particular, Φ µ→µ � is then invertible, hence itis a Hilbert space isomorphism.Lastly, eq. (A.15. ) follows from: ∀� ∈ O Pos � √α � � X � √α � + �div µ � X � � = �div µ X � � . � B References [1] Abhay Ashtekar and Chris Isham. Representations of the Holonomy Algebras of Gravity andnon-Abelian Gauge Theories.
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