Projective Loop Quantum Gravity I. State Space
PProjective Loop Quantum GravityI. State Space
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
Instead of formulating the state space of a quantum field theory over one big Hilbert space, it has been proposedby Kijowski [14] to describe quantum states as projective families of density matrices over a collection of smaller,simpler Hilbert spaces. Beside the physical motivations for this approach, it could help designing a quantum statespace holding the states we need. In [24] the description of a theory of Abelian connections within this frameworkwas developed, an important insight being to use building blocks labeled by combinations of edges and surfaces. Thepresent work generalizes this construction to an arbitrary gauge group G (in particular, G is neither assumed to beAbelian nor compact). This involves refining the definition of the label set, as well as deriving explicit formulas torelate the Hilbert spaces attached to different labels.If the gauge group happens to be compact, we also have at our disposal the well-established Ashtekar-LewandowskiHilbert space, which is defined as an inductive limit using building blocks labeled by edges only. We then show thatthe quantum state space presented here can be thought as a natural extension of the space of density matrices over thisHilbert space. In addition, it is manifest from the classical counterparts of both formalisms that the projective approachallows for a more balanced treatment of the holonomy and flux variables, so it might pave the way for the developmentof more satisfactory coherent states. Contents a r X i v : . [ g r- q c ] N ov Introduction
The context of the present work is a formalism, originally introduced by Jerzy Kijowski [14], inwhich the state space of a quantum field theory is presented in the form of a projective limit: thekey idea is, instead of describing quantum states as density matrices over a (very) large Hilbertspace, to describe them as families of density matrices over a collection of ‘small’ Hilbert spaces.The labels indexing these Hilbert spaces are to be thought as selecting finitely many degrees offreedom out of the considered infinite dimensional theory. Whenever the degrees of freedomretained by some label η are also covered by a finer label η � , the small Hilbert space H η associatedto η should be identified as a tensor product factor in H η � : this allows to formulate, in terms ofthe corresponding partial traces, the projective consistency conditions that the families of densitymatrices representing our quantum states need to fulfill. This formalism has been further developedby Andrzej Okołów [23, 24, 25], achieving in particular its application to models involving real-valued connections (in [24, section 5] and [25]). Our goal here is to expand this line of researchto the construction of suitable projective quantum state spaces for theories of connections havinga possibly non-Abelian gauge group. We will rely strongly on a previous series of articles [17, 18],in which the projective approach has been investigated at a fairly general level.Our motivation for adopting this approach lies in its potential to deliver bigger state spaces [18,subsection 2.2], as it sidesteps the need to specialize into a single representation of the algebra ofobservables. This may help to ensure that the quantum theory will actually contain the particularstates we are looking for. More specifically, we will be interested in applications to Loop QuantumGravity (LQG, [1, 31]), and to the construction of semi-classical and related states in this context.There seems indeed to exist serious obstructions [30, 16, 10] to find such states within the Ashtekar-Lewandowski Hilbert space used in LQG [2], arising from the intrinsic asymmetry in the role playedby the configuration and momentum variables (ie. the holonomies and fluxes, see eg. the discussionin [5]). This asymmetry can be traced back to the fact that the formalism is build on a vacuum whichis an eigenstate of the flux observables (thus having maximal uncertainties in the holonomies). Thestates are then obtained as discrete excitations around this vacuum. The trouble is that, no matterhow many discrete excitations are piled up on top of the vacuum, this will never be sufficient tomask this initial bias.Note that the situation here differs crucially from similar constructions routinely used when doingquantum field theory on a flat Minkowski background: the vacuum state there is a coherent statewith respect to a certain set of canonically conjugate variables (as provided by mode decomposition),so it does not favor half of the variables to the detriment of the others. This is however not anoption for LQG, because the Ashtekar-Lewandowski vacuum is the only diffeomorphism invariantstate at our disposal [21]: using a genuinely non-diffeomorphism-invariant state as vacuum would(in much the same way as just argued) lead to a quantum theory breaking the diffeomorphisminvariance, which would be many orders of magnitude more disastrous than the above concerns.The expectation that the projective approach could allow for a more balanced treatment of theconfiguration and momentum variables is supported by the respective classical precursors of bothformalisms. While the inductive limit construction underlying the Ashtekar-Lewandowski Hilbertspace can be seen as emerging from a projective limit of configuration spaces [22, 31], we emphasizedin [18, section 3] that projective quantum state spaces are naturally obtained as the quantization ofa projective limit of phase spaces.Following the general procedure laid out in [18, subsection 3.1], we will, once equipped with an ppropriate label set (theorem 2.16), proceed to set up a factorizing system of configuration spaces [17,def. 2.15] (theorem 3.7): this is indeed a fairly generic way of writing down a projective limit of phasespaces, provided that a family of real polarizations can be chosen consistently across all partialphase spaces. We can then quantize each of these partial theories in the corresponding positionrepresentation and the consistency of the polarizations ensures that the resulting small Hilbertspaces can eventually be arranged into a projective system of quantum state spaces. Equivalently,the tensor product factorizations needed for the quantum projective system can be read out directlyfrom the classical factorizing system [18, prop. 3.3].One might be worried that relying on the position representation to perform the quantization ofthe individual partial theories would reintroduce an unwanted singularization of the configurationvariables. This is however not the case, because they only play a special role as far as thequantization of the finite dimensional small phase spaces is concerned: this is therefore comparableto the choice of a representation in quantum mechanics (in opposition to quantum field theory),which is known to be rather innocuous [32]. Indeed, the small phase spaces we will be workingwith will be cotangent bundles on Lie groups, whose position representation (namely, the L spaceon the considered group) can be shown to be unitarily equivalent to a suitably defined momentumrepresentation [9, 5]. Although we will thus be using the same building blocks as in the usualapproach, the critical difference comes from the alternative way of gluing them together to composethe state space of the full theory: design choices in this regard are precisely the most likely to haveirreversible consequences on the final quantum theory.On the other hand, working in the position representation is convenient to study the relationbetween these two approaches. Any factorizing system of configuration spaces yields, by forgettingabout the more precise information it provides regarding the links between the partial theories,an associated projective system of configuration spaces. On the quantum side, we then havean embedding, mapping the states over the corresponding inductive limit Hilbert space into thequantum projective state space [18, prop. 3.5]. We will make use of this device to understand howthe state space we are proposing here extends the established one (theorem 3.20, props. 3.21 and3.22).As a side remark, for the construction we will be presenting here, the gauge group does notneed to be compact, nor do we have to impose any particular restriction beyond the assumptionof a finite-dimensional Lie group (should it be of any use, even countable discrete groups fall intothis category, as the -dimensional case). By contrast, the Ashtekar-Lewandowski vacuum (andtherefore the Hilbert space built on it) only exists if the gauge group is compact (note that thejust highlighted results concerning the relation between the projective and inductive approaches ofcourse only apply when the latter can at all be defined). The possibility of setting up a quantumstate space in the case of a non-compact gauge group, although not in the focus of our interest,might find application in the treatment of the complex Ashtekar variables, requiring SL (2 , C ) asgauge group (see eg. the discussion in [23] and references therein). In the following Σ will denote a finite-dimensional, analytic manifold [20], � its dimension ( � � G a finite-dimensional Lie group and g its Lie algebra. Label set
The construction routinely employed in LQG relies on an inductive limit of ‘small’ Hilbert spaces,or, equivalently, on a projective limit of finite dimensional configuration spaces, with buildingblocks labeled by graphs . Each graph corresponds to a selection of position variables, namely theholonomies along its edges.In principle, we could associate to any such graph a corresponding phase space, as the cotangentbundle on its configuration space. However, if we now consider a big graph γ � and a subgraph γ of γ � , there is no preferred way of defining a projection from the phase space M γ � thus associated to γ � into the phase space M γ associated to γ . In order to define unambiguously such a projection, wewould indeed have to specify how the impulsion variables described by M γ should be transportedto M γ � (having in mind that a downward projection between the phase spaces is dual to an upwardinjection between the algebras of observables). As pointed out in [17, prop. 2.10], a projectionbetween the phase spaces encapsulates, at least locally, the same information as a factorization of M γ � into a Cartesian product of M γ times a ‘complementary’ phase space M γ � →γ . In addition, ifthe projection we are considering is compatible with the splitting of the phase spaces into positionand momentum variables, this factorization should go down to a factorization C γ � ≈ C γ � →γ × C γ ofthe underlying configuration spaces.In other words, if we specify not only which configuration variables are to be retained by γ ,but also which momentum variables, we have a preferred choice of complementary configurationvariables: they are simply characterized by their vanishing Poisson brackets with the retainedmomentum variables. By contrast, if we are only provided with a projection between configurationspaces, we cannot single out a choice of complementary variables within C γ � that would span C γ � →γ . These considerations suggest that the desired projective structure should rely on labels thatare made not only of edges but also of surfaces, whose role will be, for each label η , to select whichfluxes are to be the momentum variables associated to η . Besides, such mixed labels clearly soundspromising in view of giving the holonomies and fluxes a more symmetric status.The need to include surfaces in the labels was already recognized by Okołów in [24, 25]. The labelset he was using is however not immediately applicable to the non-Abelian case, which requires,as we will see, to impose more restrictive conditions on the relative disposition of the edges andsurfaces. The reason why complications emerge in the non-Abelian case is the following. Asmentioned above, a projection from the phase spaces associated to a finer label η � into the oneassociated to a coarser label η is dual to an embedding of the algebra of observables selected by η into the algebra of observables selected by η � . Moreover, this embedding is linear and preservesthe Poisson brackets [17, prop. 2.2], ie. it is an injective algebra morphism. But this requires thatthe vector space generated by the observables associated to η , within the algebra of η � , should beclosed under Poisson brackets, and that the algebra structure thus induced by η � should match theone seen from η .This is a rather harmless requirement in the Abelian case, for there the flux operators commutewith each other and the Poisson bracket of a flux variable with an holonomy variable is just aconstant (possibly depending on the intersection of the corresponding edge and surface), so theset of observables associated to a collection of edges and surfaces will be automatically closed underPoisson brackets. The aim of the present section will therefore be to determine which collections f edges and surfaces are admissible when the gauge group is arbitrary, and to check that the labelset they are forming, although much reduced, is still directed.Actually there do exist possibilities to write the state space of a theory of connection in projectiveform while using labels made of edges only: two such models have been for example proposed in[23] and, at the classical level, in [29]. In subsection 3.2, we will discuss in more details how anon ambiguous choice of complementary variables is achieved in these proposals without explicitlyreferring to the momentum variables in the definition of the labels, and why the thus obtainedprojective structures would altogether not fit our purpose. To fix our notations and definitions, we begin by writing down what we mean precisely by edgesand surfaces, and we recall a few elementary properties, that we will use again and again in thefollowing [31, section II.6].As a technical side note, the class of edges we are considering here is pretty restrictive (namely,they are fully analytic edges embedded in a single analytic path), in contrast to the class of semi-analytic edges commonly used in LQG [31, section IV.20]. This is purely for convenience, and asfar as the construction of the quantum state space is concerned, it has absolutely no incidence(for this restricted class of edges is a cofinal part of the more usual one, so that in particular thecorresponding inductive limits of Hilbert spaces are identical, see subsection 3.2 and in particularthe proof of theorem 3.20 for more details).
Definition 2.1
An analytic, encharted edge in Σ is an analytic diffeomorphism ˘ � : U → V , where U is an open neighborhood of [0 , × { } �− in R � , and V is an open subset of Σ. We call ˘ L edges the set of all encharted edges, and for ˘ � ∈ ˘ L edges we define its starting point � (˘ � ) := ˘ � (0 , � (˘ � ) := ˘ � (1 ,
0) and its range � (˘ � ) := ˘ � �[0 , × { } �− �.We say that ˘ � , ˘ � ∈ ˘ L edges are equivalent, and we write ˘ � ∼ ˘ � , iff: � (˘ � ) = � (˘ � ) & � (˘ � ) = � (˘ � ) .This defines an equivalence relation on ˘ L edges . Its set of equivalence classes will be denoted by L edges . An element � ∈ L edges is called an edge, and we can define its starting point � ( � ), itsending point � ( � ) and its range � ( � ), since these are the same for any representative of � . Proposition 2.2
Let � ∈ L edges and let � � = � � be two distinct points in � ( � ). Then, there exists aunique edge � [ �,� � ] ∈ L edges such that: � � � [ �,� � ] � ⊂ � ( � ) , � � � [ �,� � ] � = � & � � � [ �,� � ] � = � � . (2.2. )We denote by � − the reversed edge � − := � [ � ( � ) ,� ( � )] . We also define a strict, total order on thepoints of � ( � ) by: ∀� ∈ � ( � ) , � ( � ) < ( � ) � ⇔ � ( � ) � = � ∀�, � � ∈ � ( � ) \ {� ( � ) } , � < ( � ) � � ⇔ � � � [ � ( � ) ,� ] � � � � � [ � ( � ) ,� � ] � (2.2. )For any � � = � ∈ � ( � ) and any � � = � ∈ � � � [ � ,� ] �, we have: � � [ � ,� ] � [ � ,� ] = � [ � ,� ] , so in particular � � − � − = � , � � − � [ � ,� ] = � [ � ,� ] and � � [ � ,� ] � − = � [ � ,� ] ; � � � [ � ,� ] � = {� ∈ � ( � ) | � � ( � ) � � ( � ) � } if � < ( � ) � {� ∈ � ( � ) | � � ( � ) � � ( � ) � } if � > ( � ) � ; � < ( � [ � ,� ] ) � ⇔ � � < ( � ) � if � < ( � ) � � > ( � ) � if � > ( � ) � . Proof
Existence and uniqueness.
Let � � = � � ∈ � ( � ) and let ˘ � : U → V be a representative of � .Let � � = � � ∈ [0 ,
1] such that ˘ � ( �,
0) = � and ˘ � ( � � ,
0) = � � . The map φ , defined by: φ : U → R � ≈ R × R �− ( τ, � ) �→ ( � + τ ( � � − � ) , � ) ,is an analytic diffeomorphism onto its image W := φ �U� , with φ (0 ,
0) = ( �, φ (1 ,
0) = ( � � , φ �[0 , × { } �− � = [ �, � � ] × { } �− . Next, U is an open neighborhood of [0 , × { } �− in R � , thus also of [ �, � � ] × { } �− . Hence, U := φ − �U� is an open neighborhood of [0 , × { } �− in R � . Defining W � = φ �U � and V := ˘ � �W � � , ˘ � := ˘ �| W � →V ◦ φ | U →W � is an encharted edge,with � (˘ � ) = � , � (˘ � ) = � � and � (˘ � ) = ˘ � �[ �, � � ] × { } �− � ⊂ � (˘ � ).Let ˘ � : U → V be an encharted edge such that � (˘ � ) = � , � (˘ � ) = � � and � (˘ � ) ⊂ � (˘ � ). Since V is an open neighborhood of � (˘ � ) in Σ, W := (˘ � ) − �V � is an open neighborhood of [0 , × { } �− in R � and φ = �˘ � − � ◦ � ˘ � | W � : W → U is an analytic diffeomorphism onto its image. Moreover,we have φ (0 ,
0) = ( �, φ (1 ,
0) = ( � � ,
0) and φ �[0 , × { } �− � ⊂ [0 , × { } �− . So, by theintermediate value theorem, φ �[0 , × { } �− � = [ �, � � ] ×{ } �− , and therefore � (˘ � ) = � (˘ � ). Sincewe also have � (˘ � ) = � = � (˘ � ), ˘ � and ˘ � are two representative of the same edge � [ �,� � ] .Prop. 2.2. then follows immediately from eq. (2.2. ). Order on � ( � ) . That eq. (2.2. ) unambiguously defines is a strict order on � ( � ) (ie. an irreflexive andtransitive relation) can be checked directly. Moreover, if ˘ � is a representative of � and �, � � ∈ [0 , � ( �,
0) = � and ˘ � ( � � ,
0) = � � , we have, from the previous point: � < ( � ) � � ⇔ � < � � .In particular, < ( � ) is therefore a total order.Let � � = � ∈ � ( � ) and � � = � ∈ � � � [ � ,� ] �. Using the explicit expression above for arepresentative of � [ � ,� ] , there exist � � = � ∈ [0 ,
1] and � � = � ∈ [ � , � ] such that ∀� � , ˘ � ( � � ,
0) = � � and we have: � � [ � ,� ] � = ˘ � �[ � , � ] × { } �− � and � < ( � [ � ,� ] ) � ⇔ � � − � � − � < � − � � − � � ,which yields props. 2.2. and 2.2. . � Proposition 2.3
We say that � , � � � , � � ∈ L edges are composable iff there exist an edge � ∈ L edges and points � , � , � � � , � � in � ( � ) such that: � ( � ) = � < ( � ) � < ( � ) � � � < ( � ) � �− < ( � ) � � = � ( � ) & ∀� ∈ { , � � � , �} , � � = � [ � �− ,� � ] . (2.3. )Then � is uniquely determined by � , � � � , � � and we write � = � � ◦ � � � ◦ � . Moreover, the followingproperties holds: � − � , � � � , � − are composable and � − = � − ◦ � � � ◦ � − � ; ∀� � � ∈ { , � � � , �} , � � , � � � , � � are composable and � [ � ( � � ) ,� ( � � )] = � � ◦ � � � ◦ � � ; if, for all � ∈ { , � � � , �} , there exist composable edges � �, , � � � , � �,� � ∈ L edges such that � � = � �,� � ◦ � � � ◦ � �, , then � , , � � � , � ,� , � � � , � � � , � �, , � � � , � �,� � are composable and: � = � �,� � ◦ � � � ◦ � �, ◦ � � � ◦ � � � ◦ � ,� ◦ � � � ◦ � , . Proof
Let � , � � � , � � ∈ L edges and let � ∈ L edges and � , � , � � � , � � ∈ � ( � ) be as in eq. (2.3. ). Let˘ � be a representative of � and � , � , � � � , � � ∈ [0 ,
1] such that ˘ � ( � � ,
0) = � � for all � � � . Then, usingauxiliary results from the proof of prop. 2.2, eq. (2.3. ) can be rewritten as:0 = � < � < � � � < � �− < � � = 1and ∀� ∈ { , � � � , �} , ˘ � �[ � �− , � � ] × { } �− � = � ( � � ) & ˘ � ( � �− ,
0) = � ( � � ) .Thus, � ( � ) = � � � =1 � ( � � ) and � ( � ) = � ( � ), and therefore � is uniquely determined by � , � � � , � � .Then, props. 2.3. to 2.3. can be checked using props. 2.2. to 2.2. . � Definition 2.4
A graph is a finite set of edges γ ⊂ L edges such that: ∀� � = � � ∈ γ, � ( � ) ∩ � ( � � ) ⊂ {� ( � ) , � ( � ) } ∩ {� ( � � ) , � ( � � ) } .We denote the set of graphs by L graphs and we equip it with the preorder (reflexive and transitiverelation): ∀γ, γ � ∈ L graphs ,γ � γ � ⇔ � ∀� ∈ γ, ∃ � , � � � , � � ∈ γ � , ∃ � , � � � , � � ∈ {± } � � = � � � � ◦ � � � ◦ � � � . (2.4. )(The transitivity of � follows from props. 2.3. and 2.3. .) As a warming up for the more difficult proof of directedness that we will carry out in subsec-tion 2.2 (where we will be dealing with labels that are made of edges and surfaces), we recall herewhy the set of analytic graphs L graphs is directed [22, 31]. Note that it is only in lemma 2.6 (and inits analogue for the intersection of an edge with a surface, viz. lemma 2.10) that the analyticity ac-tually plays a role. Hence, any class of edges (and surfaces) that could provide such an intersectionproperty would do as well for the whole construction [4]. roposition 2.5 Let � γ be a finite set of edges. Then, there exists γ ∈ L graphs such that ∀� ∈ � γ, {�} � γ .In particular, L graphs , � is a directed preordered set. Lemma 2.6
Let �, � � ∈ L edges such that: ∀� ∈ � ( � ) \ {� ( � ) } , ∃ � � ∈ � ( � ) � � ( � ) < ( � ) � � < ( � ) � & � � ∈ � ( � � ) . (2.6. )Then, there exists � ∈ � ( � ) \ {� ( � ) } such that � � � [ � ( � ) ,� ] � ⊂ � ( � � ) . Proof
Let ˘ � : U → V , resp. ˘ � � : U � → V � , be a representative of � , resp. � � . Eq. (2.6. ) can berewritten: ∀� ∈ ]0 , , ∃ � � ∈ ]0 , � [ � ˘ � ( � � , ∈ � ( � � ) . (2.6. ) U being an open neighborhood of [0 , × { } �− in R � there exists � ∈ ]0 ,
1] such that ] −�, � [ ×{ } �− ⊂ U . Now, the map � � �→ ˘ � ( � � ,
0) is continuous from ] −�, � [ into Σ and � ( � � ) is compact, so � ( � ) = ˘ � (0 , ∈ � ( � � ) ⊂ V � . Hence, since V � is an open subset of Σ, there exists � � ∈ ]0 , � ] such that˘ � �] − � � , � � [ × { } �− � ⊂ V � .Thus, we can define a map ψ :] − � � , � � [ → R �− by: ψ : ] − � � , � � [ → R �− � �→ � ◦ � ˘ � � � − ◦ ˘ � ( �,
0) ,where � : R � ≈ R × R �− → R �− is the projection map on the second Cartesian factor. ψ is analytic as a composition of analytic maps and, from eq. (2.6. ), 0 is an accumulation point of ψ − � � , hence ψ ≡ ψ � :] − � � , � � [ → R by: ψ � : ] − � � , � � [ → R � �→ � ◦ � ˘ � � � − ◦ ˘ � ( �,
0) ,where � : R � ≈ R × R �− → R is the projection map on the first Cartesian factor. ψ � is acontinuous, injective map (combining ψ ≡ � and ˘ � � ), ψ � (0) ∈ [0 , ) there exists � �� ∈ ]0 , � � [ such that ψ � ( � �� ) ∈ [0 , ψ � � [0 , � �� ] � ⊂ [0 , � := ˘ � (0 , � �� ) ∈ � ( � ) \ {� ( � ) } , we have � � � [ � ( � ) ,� ] � ⊂ � ( � � ). � Proof of prop. 2.5
Intersection of 2 edges.
Let � , � ∈ L edges . We define: C ( � , � ) := {� ∈ L edges | � ( � ) ⊂ � ( � ) ∩ � ( � ) } ,and: � ( � , � ) := � � ∈ � ( � ) ∩ � ( � ) �� ∀� ∈ C ( � , � ) , � ∈ � ( � ) ⇒ � ∈ {� ( � ) , � ( � ) } � .Then, for any � ∈ � ( � ) \ {� ( � ) } we have, by applying lemma 2.6 to � = � , [ �,� ( � )] , � � = � andusing prop. 2.2: � � < ( � ) � � � ∀� �� ∈ � ( � ) , � � < ( � ) � �� < ( � ) � ⇒ � �� /∈ � ( � )� or � � � , [ � � ,� ] � ⊂ � ( � ) ,and therefore: ∃ � � < ( � ) � � ∀� �� ∈ � ( � ) , � � < ( � ) � �� < ( � ) � ⇒ � �� /∈ � ( � , � ) .Similarly, for any � ∈ � ( � ) \ {� ( � ) } , applying lemma 2.6 to � = � , [ �,� ( � )] , � � = � yields: ∃ � � > ( � ) � � ∀� �� ∈ � ( � ) , � < ( � ) � �� < ( � ) � � ⇒ � �� /∈ � ( � , � ) .Hence, choosing a representative of � and using the explicit form of < ( � ) from the proof ofprop. 2.2, we can, for any � ∈ � ( � ), construct an open neighborhood V � of � in � ( � ) such that � ( � , � ) ∩ V � ⊂ {�} . Since � ( � ) is compact, we thus have that � ( � , � ) is finite.Let � ∈ � ( � ) ∩ � ( � ) \ � ( � , � ). Using prop. 2.2. together with the definition of � ( � , � ), thereexists � � ∈ � ( � ) such that: � � < ( � ) � and � � � , [ � � ,� ] � ⊂ � ( � ) .Thus, using again the explicit form of < ( � ) in terms of some representative of � , we can define � inf ∈ � ( � ) by: � inf = inf < ( � � � � ∈ � ( � ) �� � � < ( � ) � & � � � , [ � � ,� ] � ⊂ � ( � )� ,and � inf ∈ � ( � ) (for � ( � ) ∩ � ( � ) is closed in � ( � )). Moreover, for any � �� ∈ � ( � ) with � inf < ( � ) � �� < ( � ) � , there exists � � ∈ � ( � ) such that: � � < ( � ) � �� < ( � ) � and � � � , [ � � ,� ] � ⊂ � ( � ) ,therefore � �� ∈ � ( � ). Hence, � � � , [ � inf ,� ] � ⊂ � ( � ) . On the other hand, if there exists � ∈ C ( � , � )with � inf ∈ � ( � ), then � = � , [ � ( � ) ,� ( � )] , thus there exists � �� ∈ {� ( � ) , � ( � ) } such that: � �� � ( � ) � inf < ( � ) � and � � � , [ � �� ,� ] � ⊂ � ( � ) ,so � inf = � �� . Therefore, � inf ∈ � ( � , � ). Similarly, we can construct � sup ∈ � ( � , � ) such that � < ( � ) � sup and � � � , [ �,� sup ] � ⊂ � ( � ) . To summarize, we have proved: ∀� ∈ � ( � ) ∩ � ( � ) \ � ( � , � ) ,∃ � inf , � sup ∈ � ( � , � ) � � inf < ( � ) � < ( � ) � sup & � � � , [ � inf ,� sup ] � ⊂ � ( � ) . (2.6. ) Intersection of 3 edges.
Let � , � , � ∈ L edges and consider � ∈ � ( � ) ∩ � ( � ) ∩ � ( � ) with � /∈� ( � , � ) ∪ � ( � , � ). Then, for � = 1 ,
2, there exist � �� , � ��� ∈ � ( � ) such that: � �� < ( � ) � < ( � ) � ��� and � � � , [ � �� ,� ��� ] � ⊂ � ( � � ) .Hence, defining � � := max < ( � � � � , � � � and � �� := min < ( � � � �� , � �� �, we have: � � < ( � ) � < ( � ) � �� and � , [ � � ,� �� ] ∈ C ( � , � ) .Thus, � /∈ � ( � , � ). In other words, we get: � ( � , � ) ∩ � ( � ) ⊂ � ( � , � ) ∪ � ( � , � ) . Directedness of L graphs . Let � γ be a finite subset of L edges , and define: (� γ ) := � � ,� ∈ � γ � ( � , � ) .Let � ∈ � γ . From the previous point, we have: � ( � , � γ ) := � (� γ ) ∩ � ( � ) = � � ∈ � γ � ( � , � ) ,and since all � ( � , � ) are finite, so is � ( � , � γ ). Moreover, {� ( � ) , � ( � ) } = � ( � , � ) ⊂ � ( � , � γ ), sothere exist � � � � � , � � � , � � � � ∈ � ( � ) such that: � ( � ) = � � < ( � ) � � < ( � ) � � � < ( � ) � � � � = � ( � ) and � ( � , � γ ) = � � � , � � � , � � � � � .Hence, � = � , � � ◦ � � � ◦ � , , where � ,� := � , [ � � �− ,� � � ] .Let � , � ∈ � γ and � ∈ { , � � � , � � } . Suppose that there exists � ∈ � ( � ,� ) \ {� � �− , � � � } such that � ∈ � ( � ) . By definition of � ,� , we then have � ∈ � ( � ) ∩ � ( � ) \ � ( � , � ), so from eq. (2.6. ), ∃ k � � − , ∃ k � � � � � � � , [ � � k ,� � k� ] � ⊂ � ( � ) .Thus, we have in particular � ( � ,� ) ⊂ � ( � ). Moreover, � ( � ,� ) ∩ � ( � , � γ ) = � ( � ,� ) ∩ � (� γ ) = � ( � ,� ) ∩� ( � , � γ ) = {� � �− , � � � } . Hence, there exist � ∈ { , � � � , � � } such that:� � � �− = � � �− & � � � = � � � � or � � � �− = � � � & � � � = � � �− � .In other words, we have proved: ∀� , � ∈ � γ, ∀� ∈ { , � � � , � � } , � � ( � ,� ) ∩ � ( � ) ⊂ {� � �− , � � � } � or � ∃ � ∈ { , � � � , � � } , ∃ � = ± � ,� = � � ,� � .Moreover, we have from prop. 2.2. : ∀� ∈ � γ, ∀� � = � ∈ { , � � � , � � } , � ( � ,� ) ∩ � ( � ,� ) ⊂ {� � �− , � � � } ,so we get: ∀� , � ∈ � γ, ∀� ∈ { , � � � , � � } , ∀� ∈ { , � � � , � � } , � � ( � ,� ) ∩ � ( � ,� ) ⊂ {� � �− , � � � } � or � ∃ � = ± � ,� = � � ,� � . (2.6. )Finally, we define the finite subset γ � := � � ,� �� � ∈ � γ, � ∈ { , � � � , � � } � ⊂ L edges and we canconstruct γ ⊂ γ � , such that: ∀� ∈ γ � , ∃ ! � = ± � � ∈ γ .From eq. (2.6. ), we then have: ∀�, � � ∈ γ, � � ( � ) ∩ � ( � � ) ⊂ {� ( � ) , � ( � ) } � or � � = � � � .Therefore γ ∈ L graphs and, by construction, ∀� ∈ � γ, {� } � γ .In particular, for any γ, γ � ∈ L graphs , there exists γ �� ∈ L graphs , such that ∀� ∈ γ ∪ γ � , {�} � γ �� ,hence γ, γ � � γ �� . � e now come to the surfaces. Our notion of surfaces is quite limiting here, since we restrictourselves to fully analytic, ‘round’ surfaces. Like our class of edges, this is mostly a matterof convenience, and it would be relatively harmless to relax our definition (eg. we could cut anarbitrary compact piece out of an analytic plane, instead of only considering disk-shaped surfaces).Anyway, the flux operators will not be labeled directly by surfaces, but rather by finite intersectionsand differences thereof (in a sense that will be made precise in prop. 3.3), and those run through aconsiderably larger class of geometrical objects. Definition 2.7
An analytic, encharted surface in Σ is an analytic diffeomorphism ˘ S : U → V , where U is an open neighborhood of { } × B ( �− in R � ( B ( �− being the closed unit ball of R �− ), and V is an open subset of Σ. We call ˘ L surfcs the set of all encharted surfaces, and for ˘ S ∈ ˘ L surfcs wedefine its range � � ˘ S � := ˘ S � { } × B ( �− �.We say that ˘ S : U → V , ˘ S : U → V ∈ ˘ L surfcs are equivalent, and we write ˘ S ∼ ˘ S , iff: � � ˘ S � = � � ˘ S � & ˘ S � U � ∩ � R + × R �− �� = ˘ S � U � ∩ � R + × R �− �� ,where U � (resp. U � ) is an open neighborhood of { } × B ( �− in U (resp. in U ), and R + is the setof non-negative reals. This defines an equivalence relation on ˘ L surfcs . Its set of equivalence classeswill be denoted by L surfcs . An element S ∈ L surfcs is called a surface, and we can define its range � ( S ), since it is the same for any representative of S . To determine the symplectic structure (or, equivalently, to specify, on the quantum side, howthe flux operators act in the position representation), one has to discuss the relative positioning ofedges with respect to surfaces. It will make the construction in subsection 3.1 appreciably simplerto consider ‘one-sided’ fluxes, that only interact with the edges reaching the surface from one side.Also, we will impose that all edges having a non-trivial interaction with a given surface should start from that surface (reorienting them if need be), so that flux operators always act at the beginningof edges. Thus, we classify the edges adapted to a surface as being above, below or indifferent toit (instead of the slightly different classification as outside/inside/up/down [31, section II.6.4]).Since the surfaces we are considering are closed, one might be worried that an edge touchingsome surface precisely on its rand would have an unclear positioning: this is however not thecase, for our surfaces have been defined in def. 2.7 as being embedded within an open analyticplane, which extends beyond the rand and allows to distinguish between above and below in aneighborhood of the surface. In particular, an edge intersecting the rand of a surface can only beindifferent to that surface if it runs along the same analytic plane.Examples of well-positioned edges are shown in fig. 2.1. Note that all figures in the present articlewill be drawn in the case � = 2 (where both edges and surfaces are one-dimensional), as this issufficient to illustrate most aspects of the construction (we will comment on subtleties arising inthe physically more relevant case � = 3 when appropriate).Given some surface, it is well-known that any edge can be subdivided into parts adapted to thatsurface [31, section II.6.4]. As announced earlier, this is the second place where the requirementfor analyticity plays a critical role: it ensures that an edge cannot cross the plane of the surfacemore than finitely many times (fig. 2.2). Thus, we can cut this edge at each intersection point, and,splitting again each section in two parts, we can reorient these parts so that they start from the e represent edges by arrows(going from � ( � ) to � ( � ) )and surfaces by double lines. Figure 2.1 – Examples of edges above (on the left, assuming the surface is oriented upward) andindifferent to a surface (on the right)surface.
Proposition 2.8
Let � ∈ L edges and S ∈ L surfcs . We say that: � is indifferent to S , and we write � � S , if there exist a representative ˘ S : U → V of S and � , � � � , � � ∈ L edges such that: � = � � ◦ � � � ◦ � & ∀� ∈ { , � � � , �} , � ( � � ) ∩ � ( S ) = ∅ or � ( � � ) ⊂ ˘ S �U � � ,where U � := U ∩ � { } × R �− �; � is above S , and we write � ↑ S , if there exist a representative ˘ S : U → V of S and � , � ∈ L edges such that: � = � ◦ � , � � S, � ( � ) ∩ � ( S ) = {� ( � ) } & � ( � ) \ {� ( � ) } ⊂ ˘ S �U + \ U � � ,where U + := U ∩ � R + × R �− �; � is below S , and we write � ↓ S , if there exist a representative ˘ S : U → V of S and � , � ∈ L edges such that: � = � ◦ � , � � S, � ( � ) ∩ � ( S ) = {� ( � ) } & � ( � ) \ {� ( � ) } ⊂ ˘ S �U − \ U � � ,where U − := U ∩ � R − × R �− � and R − is the set of non-positive reals.We have the following properties: these 3 cases are mutually disjoint; if � � S , then � − � S ; if � , � ∈ L edges are such that � = � ◦ � , then, for any � ∈ � � , ↑, ↓ � : � � S ⇔ � � S & � � S . he points in � ( �, S ) are markedby crosses. Figure 2.2 – Adapting an edge to a given surface
Proof
Assertion 2.8. follows from the definition of the equivalence relation in def. 2.7, assertions2.8. and 2.8. from the properties of subedges (prop. 2.2) and edge compositions (prop. 2.3). � Proposition 2.9
For any � ∈ L edges and any S ∈ L surfcs , there exist � , � � � , � � ∈ L edges and � , � � � , � � ∈ {± } such that: � = � � � � ◦ � � � ◦ � � & ∀� ∈ { , � � � , �} , � � � � S with � � ∈ � � , ↑, ↓ � . Lemma 2.10
Let � ∈ L edges and S ∈ L surfcs . Then, there exists � ∈ � ( � ) \ {� ( � ) } such that � [ � ( � ) ,� ] � S with � ∈ � � , ↑, ↓ � . Proof If � ( � ) /∈ � ( S ), then there exists an open neighborhood of � ( � ) in � ( � ) that does not intersects � ( S ), for � ( S ) is compact. Hence, choosing some representative of ˘ � and using the explicit expressionfor the range of a subedge (from the proof of prop. 2.2), there exists � ∈ � ( � ) \ {� ( � ) } such that � � � [ � ( � ) ,� ] � ∩ � ( S ) = ∅ , so � [ � ( � ) ,� ] � S .We now assume � ( � ) ∈ � ( S ) and we pick out representatives ˘ � : U → V of � and ˘ S : U � → V � of S . U is an open neighborhood of [0 , × { } �− in R � , V � an open neighborhood of � ( S )in Σ, and ˘ � (0 , ∈ � ( S ). Hence, as in the proof of lemma 2.6, there exists � ∈ ]0 ,
1] such that˘ � �] −�, � [ × { } �− � ⊂ V � and we can define an analytic map ψ :] − �, � [ → R by: ψ : ] − �, � [ → R � �→ � ◦ � ˘ S � − ◦ ˘ � ( �,
0) ,where � : R � ≈ R × R �− → R is the projection map on the first Cartesian factor.If there exists � � ∈ ]0 , � [ such that ∀� ∈ ]0 , � � [ , ψ ( � ) > < � := ˘ � ( � �� , � �� ∈ ]0 , � � [, and we have � [ � ( � ) ,� ] ↑ S (resp. ↓ S ). Hence, there only remains to consider: ∀� � ∈ ]0 , � [ , ∃ � , � ∈ ]0 , � � [ � ψ ( � ) � & ψ ( � ) � ∀� � ∈ ]0 , � [ , ∃ � ∈ ]0 , � � [ � ψ ( � ) = 0 .Therefore, 0 is an accumulation point of ψ − � � , so ψ ≡
0. Defining � = ˘ � ( � � ,
0) for some � � ∈ ]0 , � [,we thus get � [ � ( � ) ,� ] � S . � roof of prop. 2.9 Transversal crossings � ( �, S ) . Let � ∈ L edges and S ∈ L surfcs . We define: � ( �, S ) := � � ∈ � ( � ) ��� ∃ � � ∈ � ( � ) \ {�} � � [ �,� � ] ↑ S or � [ �,� � ] ↓ S � . (2.10. )Let � ∈ � ( � ) \ {� ( � ) } . Applying lemma 2.10 to � [ �,� ( � )] and S , there exists � � < ( � ) � such that � [ �,� � ] � � S with � � ∈ � � , ↑, ↓ � . Let � ∈ � ( � ) such that � � < ( � ) � < ( � ) � and assume that thereexists � � ∈ � ( � ) \ {�} such that � [ �,� � ] � � S with � � ∈ � � , ↑, ↓ � . Since � ∈ � ( � [ �,� � ] ) \ {�, � � } ,there exists � �� ∈ � � ( � [ �,� � ] ) \ {�, � � } � ∩ � � ( � [ �,� � ] ) \ {�, � � } �. From prop. 2.8. , � [ �,� �� ] � � S . On theother hand, we have either � [ �,� � ] = � [ � �� ,� � ] ◦ � [ �,� �� ] ◦ � [ �,� ] (if � �� < ( � ) � ) or � [ �,� � ] = � [ �,� � ] ◦ � [ � �� ,� ] ◦ � [ �,� �� ] (if � �� > ( � ) � ), so using twice prop. 2.8. (together with prop. 2.8. ), we get � [ �,� �� ] � S . Therefore,prop. 2.8. yields � � = � .Thus, for any � ∈ � ( � ) \ {� ( � ) } , there exists � � < ( � ) � such that: ∀� ∈ � ( � ) , � � < ( � ) � < ( � ) � ⇒ � /∈ � ( �, S ) .Similarly, for any � ∈ � ( � ) \ {� ( � ) } , there exists � � > ( � ) � such that: ∀� ∈ � ( � ) , � < ( � ) � < ( � ) � � ⇒ � /∈ � ( �, S ) .As in the proof of prop. 2.5, this ensures that � ( �, S ) is finite. Subedges with no transversal crossing are indifferent.
Let � � = � � ∈ � ( � ) such that � ( � � ) ∩ � ( �, S ) = ∅ , with � � := � [ �,� � ] . Applying lemma 2.10 to � � and recalling the definition of � ( �, S ), there exists � �� ∈ � ( � � ) \ {�} such that � [ �,� �� ] � S . This allows to define � sup ∈ � ( � � ) \ {�} by: � sup := sup < ( �� ) � � �� ∈ � ( � � ) \ {�} �� � [ �,� �� ] � S � .Applying lemma 2.10 to � [ � sup ,� ] , there exists � �� such that: � � ( � � ) � �� < ( � � ) � sup and � [ � sup ,� �� ] � S .On the other hand, by definition of � sup , there exists � ��� such that: � �� < ( � � ) � ��� < ( � � ) � sup and � [ �,� ��� ] � S .Combining props. 2.8. and 2.8. , we thus get � [ �,� sup ] � S .Let � �� ∈ � ( � � ) \ {�, � � } such that � [ �,� �� ] � S . Applying lemma 2.10 to � [ � �� ,� � ] , there exists � ��� suchthat: � �� < ( � � ) � ��� � ( � � ) � � and � [ � �� ,� ��� ] � S ,hence � [ �,� ��� ] � S (using again prop. 2.8. ), and therefore � �� < ( � � ) � sup . So we have � sup = � � .Together with the previous point, this implies: ∀� � = � � ∈ � ( � ) , � � � [ �,� � ] � ∩ � ( �, S ) = ∅ ⇒ � [ �,� � ] � S . (2.10. ) Well-positioned subedges.
Let � � = � � ∈ � ( � ) such that � � � [ �,� � ] � ∩ � ( �, S ) ⊂ {�} . Applyinglemma 2.10 to � [ �,� � ] , there exists � �� ∈ � � � [ �,� � ] � \ {�} such that � [ �,� �� ] � S with � ∈ � � , ↑, ↓ � . If �� = � � , then � [ �,� � ] � S . Otherwise, we have � � � [ � �� ,� � ] � ∩� ( �, S ) = ∅ , so from eq. (2.10. ), � [ � �� ,� � ] � S ,and therefore � [ �,� � ] � S (using � [ �,� � ] = � [ � �� ,� � ] ◦ � [ �,� �� ] together with prop. 2.8. ). Thus, we have proved: ∀� � = � � ∈ � ( � ) , � � � [ �,� � ] � ∩ � ( �, S ) ⊂ {�} ⇒ � [ �,� � ] � S with � ∈ � � , ↑, ↓ � . (2.10. ) Decomposition of � adapted to S.
Since � ( �, S ) is finite there exists � � κ ∈ { , } and � , � , � � � , � � ∈ � ( � ) such that: � ( � ) = � < ( � ) � < ( � ) � � � < ( � ) � �− < ( � ) � � = � ( � ) ,and: � ( �, S ) = {� k + κ | k ∈ N , k + κ � �} . (2.10. )For � ∈ { , � � � , �} , we define: � � = �+1 if � + κ is odd − � + κ is even ,and � � = � � � [ � �− ,� � ] . From eqs. (2.10. ) and (2.10. ), there exists � � ∈ � � , ↑, ↓ � such that � � � � S .Moreover, we have � = � � � � ◦ � � � ◦ � � ◦ � � . � As argued at the beginning of the present section, a satisfactory projective limit of phase spacesfor conjugate holonomy and flux variables requires labels containing not only edges but also sur-faces. The difficulty is that we cannot prevent the surfaces in a label to intersect wildly, for thiswould void the hopes for directedness: if a surface S belongs to some label, and a surface S belongs to some other label, there has to be, in a directed label set, a label containing both S and S at the same time. On the other hand, the set of variables described by a label should be closedunder Poisson brackets: as already stressed, the algebra of observables associated to some label η will be mounted by pullback into the algebra of any finer label η � in a way that preserves thePoisson brackets [17, prop. 2.2], so the brackets between two variables should be correct all the wayfrom the very first label in which these two variables appear. Since we know that fluxes associatedto intersecting surfaces do not commute (at least as soon as the gauge group G is non-Abelian),the additional variables arising as their Poisson brackets should therefore be included as soon asthese surfaces are considered.Thus, whenever the surfaces in a label η intersect, the flux variables supported on their inter-section are naturally among the observables selected by η . Accordingly, the momentum variablesassigned to this label are not attached to the individual surfaces in η , but rather to so called ‘faces’,which enumerate all possible non trivial ways of positioning an edge with respect to these surfaces.It might be that different collections of surfaces actually result in the same set of momentum vari-ables, which motivates the equivalence relation introduced in def. 2.12. Also, the ordering of thecorresponding equivalence classes is prescribed by the comparison of their associated algebras ofmomentum variables (as will become clear in prop. 3.8). Proposition 2.11
Let � λ be a finite set of surfaces. For � : � λ → � � , ↑, ↓ � , S �→ � S , we define: � (� λ ) := � � ∈ L edges ��� ∀S ∈ � λ, � � S S � .In particular (abusing notations by writing � for the constant map S �→ � ), we have the set of alledges that are indifferent to every surface in � λ : F � (� λ ) = � � ∈ L edges ��� ∀S ∈ � λ, � � S � .The set of faces in � λ is defined as: F (� λ ) := � F � (� λ ) ��� � : � λ → � � , ↑, ↓ � � F � (� λ ) � = ∅ & � �≡ � � ,(where � �≡ � stands for {S | � S � = � } � = ∅ ). In addition, we define: F any (� λ ) := � F ∈ F (� λ ) F ,and, for F , F � ⊂ L edges : F � ◦ F := {� ◦ � | � ∈ F , � ∈ F � , and � , � are composable } .We have the following properties: the elements of F (� λ ) are disjoints; for any F ∈ F (� λ ) ∪ � F � (� λ )� , F � (� λ ) ◦ F = F ; F � (� λ ) = � � ∈ L edges ��� ∀� � = � � ∈ � ( � ) , � [ �,� � ] /∈ F any (� λ )� ; for any � ∈ L edges there exist � , � � � , � � ∈ L edges and � , � � � , � � ∈ {± } such that: � = � � � � ◦ � � � ◦ � � and ∀� ∈ { , � � � , �} , � � ∈ F any (� λ ) ∪ F � (� λ ) . Proof
Assertions 2.11. and 2.11. follow from prop. 2.8. and 2.8. respectively. Assertion 2.11. . From Prop. 2.8, we have: F � (� λ ) ⊂ � � ∈ L edges ��� ∀� � = � � ∈ � ( � ) , � [ �,� � ] /∈ F any (� λ )� .We now want to prove the reverse inclusion.Let � ∈ L edges such that for any � � = � � ∈ � ( � ), � [ �,� � ] /∈ F any (� λ ). Let S � ∈ � λ and � � = � � ∈ � ( � )such that � [ �,� � ] � � S � with � � ∈ � � , ↑, ↓ � . Choose an ordering of the finitely many remainingsurfaces � λ \ S = {S , � � � , S � } , and define � � and � � for � ∈ { , � � � , �} such that: � � ∈ � � � [ �,� �− ] � \ {�} and � [ �,� � ] � � S � ,by applying inductively lemma 2.10 to � [ �,� �− ] and S � . From prop. 2.8. , � [ �,� � ] ∈ F � (� λ ) where � : � λ → � � , ↑, ↓ � is defined by ∀� ∈ { , � � � , �} , � S � := � � . Since � [ �,� � ] /∈ F any (� λ ), �≡ � , therefore � � = � .Hence, � ( �, S � ) = ∅ (where � ( �, S � ) has been defined in eq. (2.10. ) ). So, using eq. (2.10. ) with = � ( � ) and � � = � ( � ), � � S � . As this holds for any S � ∈ � λ , � ∈ F � (� λ ). Assertion 2.11. . Let � ∈ L edges . We define: � ( �, � λ ) := � S∈ � λ � ( �, S ) . � ( �, � λ ) is finite, for � λ is finite and each � ( �, S ) is finite. Moreover, eq. (2.10. ) becomes: ∀� � = � � ∈ � ( � ) , � � � [ �,� � ] � ∩ � ( �, � λ ) ⊂ {�} ⇒ � [ �,� � ] ∈ F any (� λ ) ∪ F � (� λ ) .Thus we can form a decomposition of � adapted to � λ exactly like in the last step of the proof ofprop. 2.9. � Definition 2.12
We define on the set of finite subsets of L surfcs an equivalence relation by:� λ ∼ � λ � ⇔ F (� λ ) = F (� λ � ) .Its set of equivalence classes will be denoted by L profls . An element λ ∈ L profls is called a profile,and we can define its set of faces F ( λ ) and the corresponding set of indifferent edges F � ( λ ), sincethese are the same for any representative of λ (thanks to prop. 2.11. ). Proposition 2.13
We equip L profls with the binary relation: ∀λ, λ � ∈ L profls ,λ � λ � ⇔ � ∀F ∈ F ( λ ) , ∃ F , � � � , F � ∈ F ( λ � ) � F = F � ( λ ) ◦ � � � =1 F � � .For any two finite sets of surfaces � λ, � λ � , we have:� � λ � profl � � � λ ∪ � λ � � profl ,where [ · ] profl denotes the equivalence class in L profls .In particular, L profls , � is a directed preordered set. Proof
Let � λ, � λ � be two finite sets of surfaces and F ∈ F (� λ ). There exists � : � λ → � � , ↑, ↓ �, with ��≡ � , such that F = F � (� λ ). We define: F (� λ, � λ � , � ) := � F � � (� λ ∪ � λ � ) ��� F � � (� λ ∪ � λ � ) ∈ F (� λ ∪ � λ � ) & � � | � λ = � � .Let � ∈ F ( F � = ∅ by definition of F (� λ )). By applying inductively lemma 2.10 to the surfacesin � λ � and using prop. 2.8. (as in the proof of prop. 2.11. ), there exists � ∈ � ( � ) \ {� ( � ) } and anextension � � : � λ ∪ � λ � → � � , ↑, ↓ � of � such that � [ � ( � ) ,� ] ∈ F � � (� λ ∪ � λ � ) . Let � � ∈ � � � [ � ( � ) ,� ] � \ {� ( � ) , �} .From prop. 2.8. , we have � [ � ( � ) ,� � ] ∈ F � � (� λ ∪ � λ � ) and � [ � � ,� ( � )] ∈ F � (� λ ). Therefore, � ∈ F � (� λ ) ◦ F � � (� λ ∪ � λ � ).In particular, F � � (� λ ∪ � λ � ) � = ∅ and, since ��≡ � , we also have � � �≡ � . Thus, F � � (� λ ∪ � λ � ) ∈ F (� λ, � λ � , � ).So F (� λ, � λ � , � ) � = ∅ and there exists F , � � � , F � ∈ F (� λ ∪ � λ � ) such that: F (� λ, � λ � , � ) = {F , � � � , F � } . nd we just proved that F ⊂ F � (� λ ) ◦ � � � =1 F � .Now, let � , � be composable edges such that � ∈ F � for some � ∈ { , � � � , �} and � ∈ F � (� λ ).By definition of F (� λ, � λ � , � ), � ∈ F , hence, by 2.11. , � ◦ � ∈ F . Therefore, F � (� λ ) ◦ � � � =1 F � ⊂ F .So, we have � � λ � profl � � � λ ∪ � λ � � profl .To prove that L profls , � is a directed preordered set, only the transitivity of � remains to bechecked. Let λ, λ � , λ �� ∈ L profls with λ � λ � and λ � � λ �� . Using the definition of � on L profls togetherwith prop. 2.11. , we have: F � ( λ �� ) ⊂ F � ( λ � ) ⊂ F � ( λ ) .Then, for any F ∈ F ( λ �� ), we can use prop. 2.11. to write: F � ( λ ) ◦ F � ( λ �� ) ◦ F = F � ( λ ) ◦ F = F � ( λ ) ◦ F � ( λ ) ◦ F ,so we get: F � ( λ ) ◦ F � ( λ � ) ◦ F = F � ( λ ) ◦ F , (2.13. )and therefore λ � λ �� . � Finally, we are ready to describe what our labels should be, keeping in mind that each label ismeant to be associated with a small, finite dimensional phase space, on which the observables itselects can be represented. As underlined many times, this phase space should be big enough fortheir Poisson algebra to be correctly reproduced. Yet it should not be too big either, otherwisethe projection between the phase spaces corresponding to two labels η � η � would not be uniquelycharacterized by the sole prescription of how its pullback should mount the observables from η to η � .These considerations reveal that edges and faces should comes in conjugate pairs. In particular, ifthe label contains intersecting surfaces, it should also contain edges that will probe the intersectionfrom every side, so that the additional momentum variables promoted above are supplied withsuitable conjugate configuration variables (fig. 2.3). Definition 2.14
We define the label set L by: L := �( γ, λ ) ∈ L graphs × L profls �� ∃ χ : γ → F ( λ ) bijective � ∀� ∈ γ, � ∈ χ ( � )�.For η = ( γ, λ ) we define its underlying graph γ ( η ) := γ and profile λ ( η ) := λ , its set of faces F ( η ) := F ( λ ) and its set of indifferent edges F � ( η ) := F � ( λ ), as well as the unique bijective map χ η : γ ( η ) → F ( η ) such that ∀� ∈ γ ( η ) , � ∈ χ η ( � ) (uniqueness follows from the fact that the faces in F ( η ) are disjoints, see prop. 2.11. ).We equip L with the product preorder, defined by: ∀η, η � ∈ L , η � η � ⇔ � γ ( η ) � γ ( η � ) & λ ( η ) � λ ( η � )� . igure 2.3 – Examples of valid labels It is of critical importance for the intended construction of a projective state space that the labelset L should be directed (eg. the pivotal ‘three-spaces consistency’ condition gets truly useful incombination with the directedness of the label set). Since both L graphs and L profls are directed ontheir own, so is L graphs × L profls , thus it is sufficient to show that L is a cofinal part of L graphs × L profls .In other words, given some arbitrary graph γ and profile λ , we want to construct a finer graph γ � and a finer profile λ � that are adapted to each other in the sense of def. 2.14.For this, we will proceed in successive steps. First we will subdivide the edges of γ to adaptthem to λ in the sense of prop. 2.9. Next we will add a bunch of small surfaces to ensure there isnever more than one edge belonging to a given face, and we will add a few small edges to populatethe faces that does not yet contain one edge. Finally we will add a few more small surfaces sothat every edge has its fellow face. Note that the order of these steps is important, for we haveto ensure that what has been achieved at a given point will be preserved by the subsequent steps.Also, we should take care that the whole procedure only requires a finite sequence of operations:graphs have been defined as finite sets of edges, while profiles arise from finite sets of surfaces,thus adding infinitely many edges or surfaces, or subdividing an edge into infinitely many parts,would not lead to a valid label. Definition 2.15
For any γ ∈ L graphs and any λ ∈ L profls , we define: M (1)( γ,λ ) := � χ : γ → F ( λ ) ∪ � F � ( λ )� �� ∀� ∈ γ, � ∈ χ ( � )� ; M (2)( γ,λ ) := � χ ∈ M (1)( γ,λ ) ��� ∀F ∈ F ( λ ) , ∀�, � � ∈ χ − �F � , � = � � � ; M (3)( γ,λ ) := � χ ∈ M (2)( γ,λ ) ��� ∀F ∈ F ( λ ) , χ − �F � � = ∅ � ; M (4)( γ,λ ) := � χ ∈ M (3)( γ,λ ) ��� χ − � F � ( λ )� = ∅ � = � χ : γ → F ( λ ) �� χ bijective & ∀� ∈ γ, � ∈χ ( � )� . igure 2.4 – Adding surfaces to separate the edges into different faces Theorem 2.16 L , � is a directed preordered set. Lemma 2.17
Let γ ∈ L graphs and λ ∈ L profls . Then, there exists γ � ∈ L graphs , such that γ � γ � and M (1)( γ � ,λ ) � = ∅ . Proof
This follows from prop. 2.11. and the definition of � on L graphs (def. 2.4). � The next step is to deal with the faces that contain more than one edge of the graph. The keyidea here is to add some small surfaces with respect to which these edges have distinct positionings:thus, they will not belong to the same face any more. If they have different starting points, wecan simply add, for each of them, a small surface going through its starting point, with respect towitch it is, say, above (as we do for the edge on the right in fig. 2.4). If many edges start from thesame point of the face, we will add, for each � among these, a small surface arranged so that � isindifferent to it, while all other edges starting from that point are either above or below it: this isanother way of ensuring that two different edges will have a distinct positioning with respect to atleast one of the added surfaces (see the two edges on the left in fig. 2.4).Given an edge � and a bunch of other edges � � � � starting from the same point, we therefore wantto construct a surface that contains an initial subedge of � and intersect each � � transversally atthe common starting point � ( � ) = � ( � � ) : in two dimensions, where "surfaces" are one-dimensional,we can cut such a surface out of the very analytic curve that provides � ; in higher dimension, thefamily of surfaces that contains � is parametrized by continuous parameters, so we just need topick out one that passes in between the finitely many edges starting from � ( � ) . However, if theanalytic extension of � beyond � ( � ) is among the edges � � � � , it will automatically be in the analyticplane of any surface containing an initial subedge of � , and thus indifferent to that surface, asillustrated on fig. 2.5. Note that it may seems at first a very unlucky special case, that preciselythe analytic extension of � also belongs to the graph, but we should keep in mind that this verysituation is produced in great numbers when we subdivides the edges of γ to adapt them to thegiven λ (fig. 2.2). To deal with this case we need to also include an additional small surface withrespect to which � is of the "above" type, because its analytic continuation will then be of the"below" type: this will ensure that these two edges do not belong to the same face at the end. Toavoid laborious case distinctions in the proof, we add surfaces quite liberally, and, for any initialface F containing more than one edge, and any edge � belonging to F , we will systematically add igure 2.5 – Only a transversal surface can separate an edge from its analytic continuationFigure 2.6 – Dealing with an edge that ends precisely where surfaces will have to be addedtwo small surfaces, one along � and the other transverse to � . Proceeding this way we probablyend up with much more faces that would have been strictly necessary to separate the edges intodistinct faces: we generate a lot of new faces that do not contain any edge at all. On the otherhand, we will have to deal with such faces in a latter step anyway, so it does not cost us more toadd more of those in the present step.It is by contrast crucial that we preserve what has been achieved in the previous step, namelythat all edges of the graph are adapted to the new profile. When adding a surface for an edge � ,we can make it as small as we want around � ( � ) without prejudice to the requirements above. Inparticular, we can ensure that it does not intersect any edge of the graph that does not go through � ( � ) . Moreover, for any edge � � that start at � ( � ) , the analyticity of edges and surfaces will ensurethat, provided the surface is chosen small enough around � ( � ) , � � will either be above or below it orindifferent to it. The only edges � � that might be problematic here are therefore those that contains � ( � ) but not as starting point. Since � and � � are edges of a graph, � ( � ) should then be � ( � � ) , andwe deal preventively with this potential source of difficulties by subdividing � � and reorienting thesecond part so that it now starts from � ( � ) (see fig. 2.6). Note that, by doing so, we admittedlyincrease the number of edges, but not the number of those for which we will then need to addsmall surfaces, for the first part of � � takes the place of � � in χ ( � � ) , while its second part is in F � ( λ ) . Thus this does not cause infinite recursion, and the total number of surfaces added duringthe present step is finite, as it should. Lemma 2.18
Let γ ∈ L graphs and λ ∈ L profls such that M (1)( γ,λ ) � = ∅ . Then, there exists γ � ∈ L graphs and λ � ∈ L profls , such that γ � γ � , λ � λ � and M (2)( γ � ,λ � ) � = ∅ . Proof
Construction of γ � . Let χ ∈ M (1)( γ,λ ) . We define: γ (1 ,χ ) := � � ∈ γ �� ∃ � � � = � � χ ( � ) = χ ( � � ) ∈ F ( λ )� ,and: B � γ (1 ,χ ) � := {� ( � ) | � ∈ γ (1 ,χ ) } . et � ∈ γ . Since γ is a graph, � ( � ) ∩ B � γ (1 ,χ ) � ⊂ {� ( � ) , � ( � ) } . If � ( � ) /∈ � ( � ) ∩ B � γ (1 ,χ ) �,we define γ �� := {�} . Otherwise, we choose some point � ∈ � ( � ) \ {� ( � ) , � ( � ) } and we define γ �� := � � [ � ( � ) ,� ] , � [ � ( � ) ,� ] � . γ � := � �∈γ γ �� is a graph such that γ � γ � and, from prop. 2.8, there exists χ � ∈ M (1)( γ � ,λ ) , satisfying B � γ � (1 ,χ � ) � = B � γ (1 ,χ ) � . Moreover, we now have for any � ∈ γ � , � ( � ) ∩ B � γ � (1 ,χ � ) � ⊂ {� ( � ) } . Construction of λ � . Let � ∈ γ � (1 ,χ � ) and define: γ � ( � ) := {� � ∈ γ � | � ( � ) ∈ � ( � � ) } = {� � ∈ γ � | � ( � ) = � ( � � ) } .We choose a representative ˘ � : U → V of � and a real � > B ( � ) � ⊂ U (where B ( � ) � isthe closed ball of radius � and center 0 in R � ). We define:˘ S ˘ �,� : U � → V� �→ ˘ � ( � � ) with U � := � � ∈ R � �� � � ∈ U � .Since { } × B ( �− ⊂ B ( � ) ⊂ U � , ˘ S ˘ �,� is an analytic, encharted surface in Σ. We denote by S ˘ �,� thecorresponding surface (def. 2.7). We also define, for any θ ∈ S ( �− (with S ( �− the unit sphere in R �− ): R θ : R × R �− → R × R �− ( �, � ) �→ ( −θ��, � + ( � − θ�� ) θ ) . R θ is an analytic diffeomorphism R � → R � and: R θ � { } × B ( �− � ⊂ R θ � B ( � ) � = B ( � ) ⊂ U � .Hence, ˘ S ˘ �,�,θ defined by ˘ S ˘ �,�,θ := ˘ S ˘ �,� ◦ R θ : R − θ �U � � → V is an analytic, encharted surface in Σ.We denote by S ˘ �,�,θ the corresponding surface.Let � � ∈ γ � ( � ) \ {�} . We define: K ( �,� � ) = � θ ∈ S ( �− �� ∃ � ∈ � ( � � ) \ {� ( � � ) } � � � [ � ( � � ) ,� ] � S ˘ �,�,θ � .Since � ( � � ) = � ( � ) = ˘ � (0) ∈ � � S ˘ �,�,θ � for any θ ∈ S ( �− , we have from prop. 2.8 (together with thedefinition of the equivalence relation in def. 2.7): ∀θ ∈ K ( �,� � ) , ∃ � � ∈ � ( � � ) \ {� ( � � ) } � � � � � [ � ( � � ) ,� � ] � ⊂ ˘ � � U ∩ R θ � { } × R �− �� ,where we have used:˘ S ˘ �,�,θ � R − θ �U � � ∩ � { } × R �− �� = ˘ � � U ∩ R θ � { } × R �− �� .Suppose now that there exist � − θ , � � � , θ �− ∈ S ( �− , linearly independent in R �− ,such that ∀� � � − , θ � ∈ K ( �,� � ) . Then, there exists � �� ∈ � ( � � ) \ {� ( � � ) } such that: � � � � [ � ( � � ) ,� �� ] � ⊂ ˘ � � U ∩ �− � � =1 R θ � � { } × R �− �� = ˘ � � U ∩ � R × { } �� .Since � ( � � ) = ˘ � (0) and � ( � ) ∩ � ( � � ) ⊂ {� ( � ) , � ( � ) } (for � � = � � and both belong to the graph γ � ), we ave, by the intermediate value theorem, � � � � [ � ( � � ) ,� �� ] � = ˘ � � [ −α, × { }� for some α >
0. Thus, � � [ � ( � � ) ,� �� ] ↓ S ˘ �,� .Accordingly, we define: γ � ( �, := {�} ∪ � � � ∈ γ � ( � ) ��� ∃ � �� ∈ � ( � � ) \ {� ( � � ) } � � � [ � ( � � ) ,� �� ] ↓ S ˘ �,� � ,and γ � ( �, := γ � ( � ) \ γ � ( �, . Then, for any � � ∈ γ � ( �, , K ( �,� � ) has measure zero in S ( �− (for example withrespect to the standard measure on S ( �− if � �
3, and with respect to the counting measure if � = 2). Hence, there exists θ ∈ S ( �− such that ∀� � ∈ γ � ( �, , θ /∈ K ( �,� � ) .Now, from lemma 2.10 and prop. 2.8. , there exist, for any � � ∈ γ � ( � ) , � � � ∈ � ( � � ) \ {� ( � � ) } and � � � ,�, , � � � ,�, ∈ � � , ↑, ↓ � such that: � � [ � ( � � ) ,� �� ] � � � ,�, S ˘ �,� and � � [ � ( � � ) ,� �� ] � � � ,�, S ˘ �,�,θ .By construction, we have: ∀� � ∈ γ � ( �, \ {�} , � � � ,�, = ↓ and ∀� � ∈ γ � ( �, , � � � ,�, � = � .On the other hand, we can check from the definition of S ˘ �,� and S ˘ �,�,θ that: � �,�, = ↑ and � �,�, = � .Thus, we get ∀� � ∈ γ � ( � ) \ {�} , � � � � ,�, , � � � ,�, � � = � � �,�, , � �,�, � .Next, using def. 2.7 and prop. 2.8, there exists � �� � ∈ � � � � [ � ( � � ) ,� �� ] � \ {� ( � � ) , � � � } such that: � � � � [ � ( � � ) ,� ��� ] � \ {� ( � � ) } ⊂ ˘ S ˘ �,� � U � ∩ D � ��,�, ∩ R θ � D � ��,�, �� , (2.18. )where D � := { } × R �− , D ↑ := ( R + \ { } ) × R �− and D ↓ := ( R − \ { } ) × R �− . Since:� � � ∈γ � \γ � ( � ) � ( � � ) ∪ � � � ∈γ � ( � ) � � � � [ � ��� ,� ( � � )] �is a compact set of Σ that does not contain � ( � ), there exists an open neighborhood W of � ( � ) in V such that: ∀� � ∈ γ � \ γ � ( � ) , � ( � � ) ∩ W = ∅ and ∀� � ∈ γ � ( � ) , � � � � [ � ��� ,� ( � � )] � ∩ W = ∅ .˘ � − �W � is an open neighborhood of 0 in R � , hence there exists � � ∈ ]0 , � ] such that B ( � ) � � ⊂ ˘ � − �W � .We define: S �, := S ˘ �,� � and S �, := S ˘ �,� � ,θ .For k ∈ { , } , we have � � S �,k � ⊂ W , therefore: ∀� � ∈ γ � \ γ � ( � ) , � � � S �,k and ∀� � ∈ γ � ( � ) , � � [ � ��� ,� ( � � )] � S �,k .In addition, we get from eq. (2.18. ): ∀� � ∈ γ � ( � ) , � � [ � ( � � ) ,� ��� ] � � � ,�,k S ˘ �,k , igure 2.7 – Adding a small edge to populate a face that was empty thus, using prop. 2.8. : ∀� � ∈ γ � ( � ) , � � � � � ,�,k S ˘ �,k .To summarize, we have proven that, for any � ∈ γ � (1 ,χ � ) there exist a finite set of surfaces � λ �� := {S �, , S �, } and a map χ �� : γ � → F �� λ �� � ∪ F � �� λ �� � such that: ∀� � ∈ γ � , � � ∈ χ �� ( � � ) & ∀� � ∈ γ � \ {�} , χ �� ( � � ) � = χ �� ( � ) .Finally, we choose � λ ⊂ L surfcs such that λ = � � λ � profl and we define a profile λ � by: λ � := � λ ∪ � �∈γ � (1 ,χ� ) � λ � profl .Then, λ � λ � (prop. 2.13) and the map χ �� : γ � → F ( λ � ) ∪ � F � ( λ � )� , given by: ∀� � ∈ γ � , χ �� ( � � ) = χ � ( � � ) ∩ � �∈γ � (1 ,χ� ) χ �� ( � � ),belongs to M (2)( γ � ,λ � ) . � We will now turn to populating faces that do not contain any edge yet. The basic idea is to picksome edge in the concerned face and to add it to our graph (remember that faces have been definedas non empty set of edges in prop. 2.11). Since the edge we add is chosen in a face of λ , the resultof the first step is preserved, and since we add a single edge per face, and only for those faces thatdo not already contain an edge of the graph, there is no problem with the second step either. Theonly precaution required here is therefore to make sure that the added edges, together with thealready present ones, form a graph, ie. intersects only at their extremities. Now, if we have pickedsome edge � in a face F , any initial subedge of edge of � will also be in F , and will do just aswell for our purpose. This way, if � intersects another edge � � (either an edge of the graph, or oneof the to-be-added edges), and if this intersection takes place away from � ( � ) , we can simply make � shorter to avoid it (fig. 2.7 shows an edge being added to a face that were initially empty: wemake the added edge small enough so that it stays away from any preexisting edges). Moreover, itcannot be that � has a common initial subedge with another edge � � for this would mean that � � belongs to F , in contradiction with the preserved validity of step two (eg. the fact that we do notget more that one edge per face, as underlined above). igure 2.8 – An edge that runs along a surface (or an intersection of surfaces) may need to besubdivided when populating the corresponding facesThe only kind of intersection that cannot be fixed by shortening � is therefore the situation inwhich � ∩ � � = {� ( � ) } . As illustrated in fig. 2.8, it might not be possible to prevent this by a carefulchoice of � , because � ( � ) has to belong to every surface involved in the face F (this is indeeda necessary condition for � to belongs to F ): it might force � ( � ) to be in the interior of someedge � � . Not that this concern is not an artifact of the dimension two: even when the surfaceshave dimension � − > , there are faces that arises at the intersection of surfaces, and thisintersection could be one-dimensional, with an edge running along it. Thus, the present step mightrequire, beside the addition of new edges, also the subdivision of preexisting edges of the graph.Such a subdivision does not, however, threaten the two previous steps, nor does it lead to infiniterecursion in the present step, for the first part of the subdivided edge � � will belong to χ ( � � ) , andevery subsequent parts will be of the F � ( λ ) type. Lemma 2.19
Let γ ∈ L graphs and λ ∈ L profls such that M (2)( γ,λ ) � = ∅ . Then, there exists γ � ∈ L graphs ,such that γ � γ � and M (3)( γ � ,λ ) � = ∅ . Proof
Auxiliary result: Intersection of edges belonging to different faces.
Let � ∈ L edges and F ∈ F ( λ ) such that � ∈ F . Suppose that there exists � � = � ∈ � ( � ) and F � ∈ F ( λ ) ∪ � F � ( λ )� suchthat the subedge � [ �,� ] ∈ F � . Then, from prop. 2.8. , there exists � � ∈ � � � [ �,� ] � \ {�, �} such that � [ �,� � ] ∈ F � . Since � � /∈ {� ( � ) , � ( � ) } , we are in one of the following situations: � = � [ � � ,� ( � )] ◦ � [ �,� � ] if � ( � ) = � < ( � ) � � < ( � ) � ( � ) � = � [ � � ,� ( � )] ◦ � [ �,� � ] ◦ � [ � ( � ) ,� ] if � ( � ) < ( � ) � < ( � ) � � < ( � ) � ( � ) � = � � [ �,� � ] � − ◦ � [ � ( � ) ,� � ] if � ( � ) < ( � ) � � < ( � ) � = � ( � ) � = � [ �,� ( � )] ◦ � � [ �,� � ] � − ◦ � [ � ( � ) ,� � ] if � ( � ) < ( � ) � � < ( � ) � < ( � ) � ( � ) .Hence, from props. 2.8. and 2.8. , we have either F � = F (if � = � ( � ) ) or F � = F � ( λ ) (otherwise).Let � , � ∈ L edges and F , F ∈ F ( λ ) such that � ∈ F , � ∈ F and: ∀� ∈ � ( � ) \ {� ( � ) } , ∃ � � ∈ � ( � ) � � ( � ) < ( � ) � � < ( � ) � & � � ∈ � ( � ) .Then, from lemma 2.6, there exists � ∈ � ( � ) \ {� ( � ) } such that � � � , [ � ( � ) ,� ] � ⊂ � ( � ), hence � , [ � ( � ) ,� ] = � , [ � ( � ) ,� ] . Since � , [ � ( � ) ,� ] ∈ F � = F � ( λ ) (by definition of F ( λ ) ), the previous argumentapplied to the subedge � , [ � ( � ) ,� ] of � , together with prop. 2.11. , implies that F = F .Thus, we have proven that, for any � , � ∈ L edges and any F � = F ∈ F ( λ ) such that � ∈ F and � ∈ F , there exists � ∈ � ( � ) \ {� ( � ) } such that � � � , [ � ( � ) ,� ] � ∩ � ( � ) ⊂ {� ( � ) , �} . Hence,there exists � � ∈ � � � , [ � ( � ) ,� ] � \ {� ( � ) } such that � � � , [ � ( � ) ,� � ] � ∩ � ( � ) ⊂ {� ( � ) } . onstruction of γ � . Let χ ∈ M (2)( γ,λ ) and define: F (2 ,χ ) ( λ ) := � F ∈ F ( λ ) �� χ − �F � = ∅ � .For any F ∈ F (2 ,χ ) ( λ ) , we choose an edge � F ∈ F , and we define: γ (2 ,χ ) := {� F | F ∈ F (2 ,χ ) ( λ ) } .Let � ∈ γ (2 ,χ ) and F ∈ F (2 ,χ ) ( λ ) such that � ∈ F . For any � � ∈ γ , � ∈ � F with � F = χ (� � ) ∈ F ( λ ) \ F (2 ,χ ) ( λ ). And for any � � ∈ γ (2 ,χ ) \ {�} , there exists � F ∈ F (2 ,χ ) ( λ ) \ {F } such that � � ∈ � F . Hence,for any � � ∈ γ ∪ γ (2 ,χ ) \ {�} , there exists � �, � � ∈ � ( � ) \ {� ( � ) } such that � � � [ � ( � ) ,� �, � � ] � ∩ � (� � ) ⊂ {� ( � ) } .Since γ ∪ γ (2 ,χ ) \ {�} is a finite set, there exists � � such that: ∀ � � ∈ γ ∪ γ (2 ,χ ) \ {�} , � � � [ � ( � ) ,� � ] � ∩ � (� � ) ⊂ {� ( � ) } .We define γ �� := � � [ � ( � ) ,� � ] � and χ �� : γ �� → F ( λ ) ∪ � F � ( λ )� , � [ � ( � ) ,� � ] �→ F . Then, for any � � ∈ γ �� , wehave � ( � � ) ⊂ � ( � ) , � � ∈ χ �� ( � � ) and: ∀ � � ∈ γ ∪ γ (2 ,χ ) \ {�} , � ( � � ) ∩ � (� � ) ⊂ {� ( � � ) , � ( � � ) } .Let � ∈ γ . The set {� (� � ) | � � ∈ γ (2 ,χ ) & � (� � ) ∈ � ( � ) } is finite, hence there exist � � � � , � � � , � � � ∈ L edges such that � = � � � ◦ � � � ◦ � and: {� (� � ) | � � ∈ γ (2 ,χ ) & � (� � ) ∈ � ( � ) } ⊂ � � � � =1 {� ( � � ) , � ( � � ) } .We define γ �� := {� , � � � , � � � } . We also define the map χ �� : γ �� → F ( λ ) ∪ � F � ( λ )� by: χ �� ( � ) := χ ( � ) & ∀k ∈ { , � � � , � � } , χ �� ( � k ) := F � ( λ ) .Then, for any � � ∈ γ �� , we have � ( � � ) ⊂ � ( � ) , � � ∈ χ �� ( � � ) (combining � ∈ χ ( � ) with props. 2.8. and2.3. ) and: ∀ � � ∈ γ \ {�} , � ( � � ) ∩ � (� � ) ⊂ � ( � � ) ∩ {� ( � ) , � ( � ) } ⊂ {� ( � � ) , � ( � � ) } ,where we have used that γ ∈ L graphs . For any � � ∈ γ (2 ,χ ) , we have by definition of γ �� and γ � � � : ∀� � ∈ γ �� , ∀ � � � ∈ γ � � � , � ( � � ) ∩ � (� � � ) ⊂ � ( � � ) ∩ � � � � =1 {� ( � � ) , � ( � � ) } ⊂ {� ( � � ) , � ( � � ) } .Additionaly, we have from props. 2.3 and 2.2: ∀� � � = � �� ∈ γ �� , � ( � � ) ∩ � ( � �� ) ⊂ {� ( � � ) , � ( � � ) } .Finally, we construct a finite set of edges γ � as γ � := � �∈γ∪γ (2 ,χ ) γ �� , and we define a map χ � : γ � → F ( λ ) ∪ � F � ( λ )� by: ∀� ∈ γ ∪ γ (2 ,χ ) , ∀� � ∈ γ �� , χ � ( � � ) := χ �� ( � � )( χ � is well-defined since the γ �� for different � are disjoints by construction). By putting togetherwhat we have proven above, we get: ∀� � � = � � � ∈ γ � , � ( � � ) ∩ � (� � � ) ⊂ {� ( � � ) , � ( � � ) } , igure 2.9 – Adding a small surface through the middle of a still unpaired edgeFigure 2.10 – Accidental extra faces need to be populated thus γ � ∈ L graphs , and, by definition of γ �� for � ∈ γ , γ � γ � . We also have ∀� � ∈ γ � , � � ∈ χ � ( � � )and: ∀F ∈ χ �γ� , χ �− �F � = � � �� � ∈ χ − �F � � & ∀F ∈ F (2 ,χ ) ( λ ) , χ �− �F � = γ �� F .Hence, using F ( λ ) = F (2 ,χ ) ( λ ) � � χ �γ� \ � F � ( λ )�� and χ ∈ M (2)( γ,λ ) , we obtain that χ � ∈ M (3)( γ,λ ) . � Finally, we want to consider those edges that do not yet belong to any face. If � is such an edge,we will let a small surface cut it through the middle, subdivide � accordingly, and reorient theparts so that they start from the added surface (fig. 2.9): thus one part will be above the surface,and the other below. Since this surface goes through an interior point � � of � , we can ensure,by making it small enough, that it does not cross any other edges of the graph (by definition of agraph, � � can not belong to any other edge). Thus, the results of the first two steps are preserved.However, if � lies inside some preexisting surface, the achievement of the third step might haveto be restored at this point: beside the two faces now populated by the two parts of � , there might beadditional new faces corresponding to intersections of the added small surface with the preexistingones. If this occurs, we will need to add a few edges to populate those faces (fig. 2.10), but we canmake sure that all added edges, together with the two parts of � , intersects at most at their startingpoints. Also, by making the added edges shorter if required, we can prevent them to intersect anyother edge of the graph. Thus, there is no need for further subdivision of the edges (in contrast tothe situation depicted in fig. 2.8 that could arise in the previous step), and we have achieved thegoal announced at the beginning of the present subsection. Lemma 2.20
Let γ ∈ L graphs and λ ∈ L profls such that M (3)( γ,λ ) � = ∅ . Then, there exists γ � ∈ L graphs and λ � ∈ L profls , such that γ � γ � , λ � λ � and M (4)( γ � ,λ � ) � = ∅ . Proof
Let � λ ⊂ L surfcs such that λ = �� λ � profl and χ ∈ M (3)( γ,λ ) . We define: γ (3 ,χ ) := � � ∈ γ �� χ ( � ) = F � ( λ )� & γ � := γ \ γ (3 ,χ ) . hus, χ �γ � � ⊂ F ( λ ) and, by definition of M (3)( γ,λ ) , χ := χ| γ � → F ( λ ) is bijective.For each � ∈ γ (3 ,χ ) , we choose a representative ˘ � : U � → V � of � and define � � := ˘ � (0 � ,
0) . Since,for any � ∈ γ (3 ,χ ) , � � � ∈γ\{�} � ( � � ) is a compact set that does not contain � � (for � � ∈ � ( � ) \{� ( � ) , � ( � ) } and γ ∈ L graphs ), there exists an open neighborhood W � of � � in V � such that: ∀� � ∈ γ \ {�} , � ( � � ) ∩ W � = ∅ .Next, {� � | � ∈ γ (3 ,χ ) } is finite and Σ is Hausdorff, hence there exists a family � W �� � �∈γ (3 ,χ ) of disjoint open subsets of Σ such that, for any � ∈ γ (3 ,χ ) , W �� is an open neighborhood of � � in W � . Thus, wehave: ∀� ∈ γ (3 ,χ ) , W �� ∩ � � � ∈γ\{�} � ( � � ) ∪ � � � ∈γ (3 ,χ ) \{�} W �� � = ∅ . (2.20. )Let � ∈ γ (3 ,χ ) . There exists � > { � } × B ( �− � ⊂ ˘ � − �W �� � and we define:˘ S � : U �� → V � �, � �→ ˘ � ( � + 0 � , � � ) with U �� := { ( �, � ) | ( � + 0 � , � � ) ∈ U � } .Since { } × B ( �− ⊂ U �� , ˘ S � is an analytic, encharted surface in Σ. We denote by S � thecorresponding surface. We can check from the definition of ˘ S � that � ( S � ) ⊂ W �� and: � ↑ := � [ � � ,� ( � )] ∈ F �,↑ & � ↓ := � [ � � ,� ( � )] ∈ F �,↓ ,where F �,↑ := � � � ∈ F � ( λ ) ��� � � ↑ S � � and F �,↓ := � � � ∈ F � ( λ ) ��� � � ↓ S � � (using � ∈ χ ( � ) = F � ( λ )together with props. 2.8. and 2.8. ). Defining γ ��, := {� ↑ , � ↓ } , we moreover have: ∀� � ∈ γ ��, , � ( � � ) ∩ {� ( � ) , � ( � ) } ⊂ {� ( � � ) } , (2.20. )and ∀� � � = � � � ∈ γ ��, , � ( � � ) ∩ � (� � � ) ⊂ {� ( � � ) } = {� (� � � ) } . (2.20. )Next, we define: F ( �,↑ ) ( λ ) := � F ∈ F ( λ ) ��� ∃ � � ∈ F � � � ↑ S � � & F ( �,↓ ) ( λ ) := � F ∈ F ( λ ) ��� ∃ � � ∈ F � � � ↓ S � � .For each F ∈ F ( �,↑ ) ( λ ) (resp. F ∈ F ( �,↓ ) ( λ ) ), we define: F �,F,↑ := � � � ∈ F ��� � � ↑ S � � (resp. F �,F,↓ := � � � ∈ F ��� � � ↓ S � � ),and we choose an edge � �,F,↑ ∈ F �,F,↑ (resp. � �,F,↓ ∈ F �,F,↓ ). We also define: γ ��, := {� �,F,↑ | F ∈ F ( �,↑ ) ( λ ) } ∪ {� �,F,↓ | F ∈ F ( �,↓ ) ( λ ) } .The sets {F �,↑ , F �,↓ } , {F �,F,↑ | F ∈ F ( �,↑ ) ( λ ) } and {F �,F,↓ | F ∈ F ( �,↓ ) ( λ ) } are disjoints subsetsof F �� λ ∪ {S � } � . Hence, defining: F ( � ) �� λ ∪ {S � } � := {F �,↑ , F �,↓ } ∪ {F �,F,↑ | F ∈ F ( �,↑ ) ( λ ) } ∪ {F �,F,↓ | F ∈ F ( �,↓ ) ( λ ) } , ( � ) �� λ ∪ {S � } � ⊂ F �� λ ∪ {S � } � and there exists a bijective map χ �, : γ ��, ∪γ ��, → F ( � ) �� λ ∪ {S � } �such that, for any � � ∈ γ ��, ∪ γ ��, , � � ∈ χ �, ( � � ) .Let � � ∈ γ ��, . There exists F � := χ �, ( � � ) ∈ F �� λ ∪ {S � } � such that � � ∈ F � . Moreover, for any� � � ∈ γ ��, ∪ γ ��, \ {� � } , there exists � F � := χ �, (� � � ) ∈ F �� λ ∪ {S � } � \ {F � } such that � � � ∈ � F � . Thus,using the auxiliary result from the proof of lemma 2.19, there exists, for any � � � ∈ γ ��, ∪ γ ��, \ {� � } ,a point � � � , � � � ∈ � ( � � ) \ {� ( � � ) } such that � � � � [ � ( � � ) ,� ��, � �� ] � ∩ � (� � � ) ⊂ {� ( � � ) } . Hence, there exists � � � ∈ � ( � � ) \ {� ( � � ) } such that: ∀ � � � ∈ γ ��, ∪ γ ��, \ {� � } , � � � � [ � ( � � ) ,� �� ] � ∩ � (� � � ) ⊂ {� ( � � ) } .Since W �� is an open subset of Σ containing � ( � � ) (for � ( � � ) ∈ � ( S � ) ⊂ W �� ), there exists � �� � ∈� � � � [ � ( � � ) ,� �� ] � \ {� ( � � ) } such that � � � � [ � ( � � ) ,� ��� ] � ⊂ W �� . Also, we have � � � � [ � ( � � ) ,� ��� ] � ⊂ � ( � � ) and, fromprop. 2.8. , � � [ � ( � � ) ,� ��� ] ∈ F � .Defining γ ��, := � � � [ � ( � � ) ,� ��� ] ��� � � ∈ γ ��, � and γ �� := γ ��, ∪ γ ��, , there exists therefore a bijective map χ � : γ �� → F ( � ) �� λ ∪ {S � } � such that, for any � � ∈ γ �� , � � ∈ χ � ( � � ) .In addition, we have: ∀� � ∈ γ ��, , � ( � � ) ⊂ � ( � ) & ∀� � ∈ γ ��, , � ( � � ) ⊂ W �� , (2.20. )and ∀� � ∈ γ ��, , ∀ � � � ∈ γ �� \ {� � } , � ( � � ) ∩ � (� � � ) ⊂ {� ( � � ) } . (2.20. )But, since for any � � ∈ γ ��, , � ( � � ) ∈ � ( S � ) , and for any � � � ∈ γ ��, , � (� � � ) ∩ � ( S � ) ⊂ {� (� � � ) } , eq. (2.20. )together with eq. (2.20. ) implies: ∀� � � = � � � ∈ γ �� , � ( � � ) ∩ � (� � � ) ⊂ {� ( � � ) } . (2.20. )Now, we define: γ � := γ � ∪ � �∈γ (3 ,χ ) � γ �� � & � λ � := � λ ∪ � �∈γ (3 ,χ ) {S � } .The fact that γ is a graph, together with eqs. (2.20. ), (2.20. ), (2.20. ) and (2.20. ), ensures that γ � is again a graph. Moreover, by definition of γ ��, for � ∈ γ (3 ,χ ) , γ � γ � , and, from prop. 2.13, λ � λ � where λ � := �� λ � � profl . Next, we define: F (0) (� λ � ) := � F ∩ F � (� λ � \ � λ ) ��� F ∈ F (� λ )� ,and, for any � ∈ γ (3 ,χ ) , F ( � ) (� λ � ) := � F ∩ F � �� λ � \ (� λ ∪ {S � } )� ��� F ∈ F ( � ) (� λ ∪ {S � } )� .Since, for any � ∈ γ (3 ,χ ) , � ( S � ) ⊂ W �� , we have:� � � ∈ L edges ��� ∃ � � ∈ � ↑, ↓ � � � � � � S � � ⊂ {� � ∈ L edges | � ( � � ) ∈ � ( S � ) } {� � ∈ L edges | � ( � � ) ∈ W �� } . (2.20. )Therefore, for any � � = � � ∈ γ (3 ,χ ) , we get, using W �� ∩ W � � � = ∅ :� � � ∈ L edges ��� ∃ � � , � � � ∈ � ↑, ↓ � � � � � � S � & � � � � � S � � � = ∅ .This, together with the definition of F ( � ) (� λ ∪ {S � } ) for � ∈ γ (3 ,χ ) , implies: F (� λ � ) ⊂ � �∈{ }�γ (3 ,χ ) F ( � ) (� λ � ) .In addition, we can define for any � ∈ { } � γ (3 ,χ ) a bijective map χ �� : γ �� → F ( � ) (� λ � ) by: ∀� � ∈ γ � , χ � ( � � ) := χ ( � � ) ∩ F � (� λ � \ � λ ) & ∀� � ∈ γ �� , χ �� ( � � ) := χ � ( � � ) ∩ F � �� λ � \ (� λ ∪ {S � } )� .For any � ∈ { } � γ (3 ,χ ) , χ �� satisfies: ∀� � ∈ γ �� , � � ∈ χ �� ( � � ) ,as follows from the corresponding property of χ � together with eqs. (2.20. ), (2.20. ) and (2.20. ). Inparticular, we thus have: ∀� ∈ { } � γ (3 ,χ ) , ∀F ∈ F ( � ) (� λ � ) , F � = ∅ ,so, we obtain: F (� λ � ) = � �∈{ }�γ (3 ,χ ) F ( � ) (� λ � ) .Since the domains of the bijective maps χ �� for � ∈ { } � γ (3 ,χ ) are disjoints, as well as their images,they can be combined into a well-defined bijective map χ � : γ � → F ( λ � ) . Hence, M (4)( γ � ,λ � ) � = ∅ . � Proof of theorem 2.16
Let ( γ, λ ) , ( γ � , λ � ) ∈ L . Since L graphs , � and L profls , � are directed sets(props. 2.5 and 2.13), there exists ( γ , λ ) ∈ L graphs × L profils such that γ, γ � � γ and λ, λ � � λ .By chaining lemmas 2.17 to 2.20 and using the transitivity of � on L graphs and L profls , there exists( γ �� , λ �� ) ∈ L graphs × L profils such that γ � γ �� , λ � λ �� and M (4)( γ �� ,λ �� ) � = ∅ . Thus, ( γ �� , λ �� ) ∈ L and( γ, λ ) , ( γ � , λ � ) � ( γ �� , λ �� ) . � The labels introduced in the previous section are meant to identify corresponding small algebrasof observables, and the ordering has been chosen such that, whenever η � η � , there is a naturalinjection of the algebra labeled by η into the one labeled by η � . By carefully adjusting under whichconditions a collection of edges and surfaces can be turned into a label, we have ensured thatthese algebras of observables can be represented on small phase spaces M η and M η � respectively,and that the identification between observables on M η and M η � unambiguously prescribes a suitable rojection from M η � into M η .Moreover, this projection is compatible with the symplectic structures [17, def. 2.1], and is actuallyof the form that were considered in [18, theorem 3.2] (as will be shown in prop. 3.8). Thus, weexpect from this theorem that the obtained projective system of symplectic manifolds goes down toa factorizing system on the underlying configuration spaces [17, def. 2.15]. Indeed, we will provethat it is the case by giving the explicit expressions for the factorization maps (with the addedbenefit that no further restriction need to be imposed on the finite-dimensional Lie group G , while[18, theorem 3.2] have been derived in the case of simply-connected groups). Once we have sucha factorizing system of configuration spaces, it can be straightforwardly quantized into a projectivequantum state space (along the lines of [24] and [18, subsection 3.1]).For setting up the quantum state space in subsection 3.1, the group G will neither be required tobe Abelian, nor compact. However, if G actually happens to be compact, a different constructionis also possible: in this case, we have at our disposal a family of normalizable measures on theconfiguration spaces, and this family of measure is compatible with the factorizations, so thatinstead of assembling the small Hilbert spaces (obtained by quantization of the small phase spacesusing a position polarization) into a projective structure, we can assemble them into an inductivelimit Hilbert space. As exposed in [18, prop. 3.5], there is then a natural injection mapping thedensity matrices on the inductive limit Hilbert space into the projective state space. Interestingly,this inductive limit can in our case be identified with the Ashtekar-Lewandowski Hilbert space[22]. This may at first seem surprising, since the former is made of building blocks labeled by edgesand surfaces, while the latter use labels which are just graphs. The trick here is that the injectionsdefining this inductive limit in fact do not depend on the disposition of the surfaces in the labels:the injection that mount the Hilbert space associated to a label η into the one associated to a finerlabel η � turns out to only depend on the underlying graphs γ ( η ) and γ ( η � ) , so that the inductivesystem labeled by elements of L actually collapses into an inductive system simply labeled by graphs.This is in fact the very observation that was spelled out at the beginning of section 2: projectionsbetween configuration spaces (which are the ingredients of an inductive limit construction) are lessspecific than factorizations, and it was precisely in order to distinguish between different possiblefactorizations corresponding to the same projection map that we had to introduce surfaces in thelabels. We start by attaching to each label η a configuration space C η , as well as a momentum space P η (later on, we will rely on left translations to identify the cotangent bundle T ∗ ( G ) with G × g ∗ , andthus T ∗ ( C η ) with C η × P η ): C η is nothing but the configuration space routinely associated in LQGto the graph γ ( η ) , while momentum variables are assigned to the faces of the label, as announcedin subsection 2.1.Also, we introduce a few notations to discuss how the holonomy along an edge, or the fluxthrough a face, can be related to the variables in C η and P η (provided η is fine enough to describe thedesired observable). Since we want to deduce the correct projective structure from the interpretationof the labels in terms of observables, it is particularly important that the relation between theseobservables and the variables in the small phase spaces should be unambiguous: this will ensure hat the factorization maps are well-defined, and will be essential for proving the so-called three-spaces consistency [17, eq. (2.11. ) and fig. 2.2] (combined with the directedness of L , this consistencycondition indeed expresses the concern for an univocal meaning of the variables attached to a label).For a clean labeling of the flux observables we formulate in prop. 3.3 a notion of ‘free-standing’faces. In prop. 2.11 the different faces corresponding to a certain collection of surfaces {S} havebeen defined as particular sets of edges, and each such face F can only contain edges that areadapted to every surface in {S} . In particular, an edge can be prevented from belonging to F simply because it crosses transversally some surface of the collection, even this surface is in realityunrelated to the face F . Thus, a given flux operator is described in different profiles by different setof edges, and its characterization by a more intrinsic set is only obtained after compensating thiseffect. Definition 3.1
For any η ∈ L , we define its associated configuration space: C η := {� : γ ( η ) → G} ≈ G γ ( η ) ,where γ ( η ) < ∞ denotes the number of edges in γ ( η ). Since G is a finite-dimensional Lie group, C η is a finite-dimensional smooth manifold.Similarly, we define the corresponding momentum space: P η := {P : F ( η ) → g ∗ } ≈ ( g ∗ ) F ( η ) ,which is a finite-dimensional real vector space. Proposition 3.2
Let � ∈ L edges . We define: L graphs /� := {γ ∈ L graphs | {�} � γ} .For any γ ∈ L graphs /� , there exists a unique map � γ→� : { , � � � , � γ→� } → γ (with � γ→� �
1) suchthat: � = � γ→� ( � γ→� ) � γ→� ( � γ→� ) ◦ � � � ◦ � γ→� (1) � γ→� (1) , (3.2. )where, for any k ∈ { , � � � , � γ→� } : � γ→� ( k ) := �+1 if � � � γ→� ( k )� < ( � ) � � � γ→� ( k )� − � � � γ→� ( k )� > ( � ) � � � γ→� ( k )� . (3.2. )Moreover, � γ→� then induces a bijection { , � � � , � γ→� } → H γ→� , where: H γ→� := {� � ∈ γ | � ( � � ) ⊂ � ( � ) } . (3.2. ) Proof
Let γ ∈ L graphs /� . Since {�} � γ , there exist � γ→� � � γ→� : { , � � � , � γ→� } → γ such that: � = � γ→� ( � γ→� ) � γ→� ( � γ→� ) ◦ � � � ◦ � γ→� (1) � γ→� (1) ,where, for any k ∈ { , � � � , � γ→� } , � γ→� ( k ) is defined as in eq. (3.2. ). By definition of the compositionof edges (prop. 2.3), � γ→� is injective and for any k ∈ { , � � � , � γ→� } , � γ→� ( k ) ∈ H γ→� .Next, let � � ∈ H γ→� , and let � ∈ � ( � � ) \ {� ( � � ) , � ( � � ) } . Since � ∈ � ( � ), there exits k ∈{ , � � � , � γ→� } such that � ∈ � γ→� ( k ), so, γ being a graph, � � = � γ→� ( k ). Thus, � γ→� induces bijection { , � � � , � γ→� } → H γ→� . In particular, � γ→� = H γ→� .Finally, defining the map µ : H γ→� → � ( � ) by: ∀� � ∈ H γ→� , µ ( � � ) := � � ( � � ) if � ( � � ) < ( � ) � ( � � ) � ( � � ) if � ( � � ) > ( � ) � ( � � ) ,prop. 2.3 implies that µ ◦ � γ→� is strictly increasing (using < on { , � � � , � γ→� } and < ( � ) on � ( � )),hence the uniqueness. � Proposition 3.3
For any
F ⊂ L edges , we define: F ⊥ := � � ∈ L edges �� ∀� � = � � ∈ � ( � ) , � [ �,� � ] /∈ F � ,and we will denote by L faces the set: L faces := � λ∈ L profls � F ⊥ ◦ F �� F ∈ F ( λ )� .In addition, we define for any F ∈ L faces : L profls /F := � λ � ∈ L profls �� ∃ λ � λ � , ∃ F ∈ F ( λ ) � F = F ⊥ ◦ F � .Then, for any F ∈ L faces and any λ � ∈ L profls /F , we have: F = F ⊥ ◦ � F � ∈H λ�→F F � , (3.3. )where H λ � →F := � F � ∈ F ( λ � ) �� F � ⊂ F � . Proof
Let λ ∈ L profls , F ∈ F ( λ ), F := F ⊥ ◦ F and λ � ∈ L profls such that λ � � λ . From props. 2.8. and 2.8. , we have: F ⊥ = F ⊥ ◦ F ⊥ & F ⊥ = � F ⊥ ◦ F � ⊥ = F ⊥ .Hence, for any F � ∈ H λ � →F : F ⊥ ◦ F � ⊂ F ⊥ ◦ F = F ⊥ ◦ F ⊥ ◦ F = F ,therefore F ⊥ ◦ � F � ∈H λ�→F F � ⊂ F .Now, by definition of the preorder � on L profls (prop. 2.13), there exist F � , � � � , F �� ∈ F ( λ � ) ( � � F = F � ( λ ) ◦ � � � =1 F �� .For any � ∈ { , � � � , �} , F � ( λ ) ◦ F �� ⊂ F implies F �� ⊂ F ⊂ F (using again props. 2.8. and 2.8. ),so F �� ∈ H λ � →F . And from prop. 2.11. , F � ( λ ) ⊂ F ⊥ , therefore: F ⊂ F ⊥ ◦ � � � =1 F �� ⊂ F ⊥ ◦ � F � ∈H λ�→F F � . � η � η Figure 3.1 – Classification of the edges in η � � η : a tag ( κ ) denotes an edge belonging to H ( κ ) η � →η (prop. 3.4)Props. 3.2 and 3.3 make it possible to unambiguously attach a physical interpretation to thevariables in C η and P η , and are therefore at the root of the relation between the variables assignedto a label η and the ones assigned to a finer label η � . Although we will directly give an analyticexpression for the factorization map φ η � →η : C η � → C η � →η × C η and we will not explicitly make useof [18, theorem 3.2], the proof we gave for this result provides the right hints regarding why such afactorization map does exist, and how it should be defined so that it leads to the desired projectionbetween the phase spaces.The key idea is that the momentum variables assigned to the label η can be mounted into thephase space associated to η � , and therefore correspond to certain vector fields on C η � . Each orbitunder the finite transformations generated by these vector fields can then be naturally identifiedwith C η (for the relation between the configuration variables in C η and C η � yields a projection from C η � into C η which intertwines the action of these transformations). The complementary space C η � →η can thus be taken as the corresponding quotient space, which itself can be identified with thepreimage of some point in C η (eg. the function mapping every edge in γ ( η ) to the identity element in G ) under the projection C η � → C η . Note that this is simply the non-linear version of the proceduredescribed in [24, section 3.4].This prescription can equivalently be expressed as the realization that the space C η � →η , togetherwith the projection from C η � into C η � →η , can be completely specified by identifying a maximal set ofvariables in C η � which are not acted upon by the fluxes retained in the label η . On the other hand,the projection from C η � into C η is obtained by writing down the edges in η as compositions of edgesin η � . Thus, the first step toward the determination of the factorization map is to state preciselyhow the edges and faces of the label η lie within η � . For this, we will classify the edges of η � intovarious categories depending on whether they belong to some face and/or are part of some edge of η . Note that no edge in γ ( η � ) can be a subedge of two different edges in γ ( η ) , nor can it belong to η � η η � → η Figure 3.2 – Complementary variables related to edges of type ‘0’ or ‘2’, edges of type ‘1’ do notdemand any extra complementary variabletwo different faces in F ( η ) , nor can a subedge of an edge � ∈ γ ( η ) belongs to any face in F ( η ) but χ ( � ) . So we are left with the 4 options listed in prop. 3.4 and depicted in fig. 3.1.Clearly, the group variables corresponding to edges of η � of type ‘0’ or ‘2’ (those that do not belongto any face among F ( η ) ) qualify as complementary variables (they are not acted upon by the fluxesretained in η ). Also, knowing the holonomy along some edge of η , as well as the holonomies alongall edges of η � that compose it except the first one, the holonomy along this first part (which is oftype ‘1’) can be reconstructed. Hence, the group variables corresponding to edges of type ‘1’ can besafely droped when extracting C η � →η (fig. 3.2).Dealing with the group variables corresponding to edges of type ‘3’ is slightly more subtle. Suchan edge � � ∈ γ ( η � ) belongs to some face F ∈ F ( η ) without being the initial part of the conjugateedge � ∈ γ ( η ) . To build a group variable invariant under the flux corresponding to F , we willcompose the holonomy along � − (which ends in F ) followed by the holonomy along � � (whichstarts in F ). In this way the action of the flux through F cancel out, while the variable in C η � thatcorresponds to the considered edge � � can still be reconstructed from the variables in C η � →η and C η (fig. 3.3). Note that we could equally well take the composition of the holonomy along � − followedby the holonomy along � � , with � the first part of � (in its decomposition into edges of η � ). Thesetwo alternatives only differ by a function of the group variables attached to the remaining parts of � , which already belongs to C η � →η (these remaining parts being edges of type ‘2’ as outlined above),so it is nothing but a change of coordinates on C η � →η , and therefore of no consequences for theconstruction. Proposition 3.4
Let η � η � ∈ L . We define: H (0) η � →η := {� � ∈ γ ( η � ) | ∀� ∈ γ ( η ) , � ( � � ) �⊂ � ( � ) & � � /∈ χ η ( � ) } ;and, for any � ∈ γ ( η ) : H (1) η � →η,� := {� � ∈ γ ( η � ) | � ( � � ) ⊂ � ( � ) & � � ∈ χ η ( � ) } ; η � η η � → η Figure 3.3 – Complementary variables related to edges of type ‘3’ H (2) η � →η,� := {� � ∈ γ ( η � ) | � ( � � ) ⊂ � ( � ) & � � /∈ χ η ( � ) } ; H (3) η � →η,� := {� � ∈ γ ( η � ) | � ( � � ) �⊂ � ( � ) & � � ∈ χ η ( � ) } ;Then,� H (0) η � →η � ∪ � �∈γ ( η ) � H (1) η � →η,� , H (2) η � →η,� , H (3) η � →η,� �is a partition of γ ( η � ) .Additionally, we define: ∀κ ∈ { , , } , H ( κ ) η � →η := � �∈γ ( η ) H ( κ ) η � →η,� . Proof
For any � ∈ γ ( η ), � H (1) η � →η,� , H (2) η � →η,� , H (3) η � →η,� � is a partition of: H (4) η � →η,� := H (1) η � →η,� ∪ H (2) η � →η,� ∪ H (3) η � →η,� = {� � ∈ γ ( η � ) | � ( � � ) ⊂ � ( � ) or � � ∈ χ η ( � ) } .Since we have: H (0) η � →η = � � � ∈ γ ( η � ) ��� ∀� ∈ γ ( η ) , � � /∈ H (4) η � →η,� � ,there only remains to prove that the H (4) η � →η,� for � ∈ γ ( η ) are mutually disjoint.Let � ∈ γ ( η ) and let � � ∈ γ ( η � ) such that � ( � � ) ⊂ � ( � ) . There exists � ∈ � ( � � ) \ {� ( � � ) , � ( � � ) } . Fromprop. 2.2, we get � /∈ {� ( � ) , � ( � ) } , thus ∀ � � ∈ γ ( η ) \ {�} , � /∈ � (� � ) , for γ ( η ) is a graph. Therefore, ∀ � � ∈ γ ( η ) \ {�} , � ( � � ) �⊂ � (� � ) . Moreover, � ∈ χ η ( � ) and for any � � ∈ γ ( η ) \ {�} , χ η (� � ) � = χ η ( � ) , sousing the auxiliary result at the beginning of the proof of lemma 2.19, � � /∈ χ η (� � ) . Hence, we haveproved: ∀� � = � � ∈ γ ( η ) , ∀� � ∈ γ ( η � ) , � � ( � � ) ⊂ � ( � ) ⇒ � � /∈ H (4) η � →η, � � � . ow let � ∈ γ ( η ) and let � � ∈ γ ( η � ) such that � � ∈ χ η ( � ) . Then, from the previous point, we get ∀ � � ∈ γ ( η ) \ {�} , � ( � � ) �⊂ � (� � ) . And, since the elements of F ( η ) are disjoints and χ η is bijective, wealso have ∀ � � ∈ γ ( η ) \ {�} , � ( � � ) /∈ χ η (� � ) . Therefore, we obtain the desired result: ∀� � = � � ∈ γ ( η ) , H (4) η � →η,� ∩ H (4) η � →η, � � = ∅ . � Proposition 3.5
Let η � η � ∈ L and � ∈ γ ( η ). We have {�} � γ ( η � ), and making use of prop. 3.2,we define: � η � →η,� := � γ ( η � ) →� , � η � →η,� := � γ ( η � ) →� & � η � →η,� := � γ ( η � ) →� .We then have: � η � →η,� (1) = +1 , H (1) η � →η,� = {� η � →η,� (1) } & H (2) η � →η,� = {� η � →η,� ( k ) | k > } , (3.5. )therefore � η � →η,� = H (2) η � →η,� + 1 and � η � →η,� induces a bijection { , � � � , � η � →η,� } → H (1) η � →η,� ∪ H (2) η � →η,� .Also, for any F ∈ F ( η ), we have, using the notations of prop. 3.3 with F = F ⊥ ◦ F , λ ( η � ) ∈ L profls /F and: H λ ( η � ) →F = H (1 , η � →η,F := � F � ∈ F ( η � ) �� χ − η � ( F � ) ∈ F � = � χ η � ( � � ) ��� � � ∈ H (1) η � →η,χ − η ( F ) ∪ H (3) η � →η,χ − η ( F ) � .(3.5. ) Proof
Let � ∈ γ ( η ). From eq. (3.2. ) � η � →η,� induces a bijection into its image: H γ ( η � ) →� = H (1) η � →η,� ∪ H (2) η � →η,� ,and from eq. (3.2. ) together with props. 2.8. and 2.8. : � η � →η,� (1) � η�→η,� (1) ∈ χ η ( � ) & ∀k > , � η � →η,� ( k ) /∈ χ η ( � ).Then, writing � η � →η,� (1) � η�→η,� (1) = � [ �,� � ] , there exists � �� ∈ � � � [ �,� � ] � \ {�, � � } ⊂ � ( � ) \ {� ( � ) , � ( � ) } suchthat � [ �,� �� ] ∈ χ η ( � ). This can only hold if � = � ( � ), ie. � η � →η,� (1) = +1. Thus, we get: � η � →η,� (1) ∈ H (1) η � →η,� & ∀k > , � η � →η,� ( k ) ∈ H (2) η � →η,� .Let F ∈ F ( η ) and F = F ⊥ ◦ F . For any F � ∈ H λ ( η � ) →F , we have χ − η � ( F � ) ∈ F � ⊂ F , hencethere exist � ∈ F and � ∈ F ⊥ such that χ − η � ( F � ) = � ◦ � . But from prop. 2.8. together with F � ( η � ) ⊂ F � ( η ) (as in the proof of prop. 2.13), we have � ∈ F � ( η � ) ⊂ F � ( η ), so χ − η � ( F � ) ∈ F . Thus,we get H λ ( η � ) →F ⊂ H (1 , η � →η,F . Reciprocally, let F � ∈ F ( η � ) such that χ − η � ( F � ) ∈ F ⊂ F . Then, fromeq. (3.3. ), there exist � ∈ F �� , for some F �� ∈ H λ ( η � ) →F , and � ∈ F ⊥ , such that χ − η � ( F � ) = � ◦ � .And since χ − η � ( F � ) ∈ F � , we also have � ∈ F � , therefore F � = F �� ∈ H λ ( η � ) →F (for the elements of F ( η � ) are disjoint from prop. 2.11. ). This proves H (1 , η � →η,F ⊂ H λ ( η � ) →F , hence H λ ( η � ) →F = H (1 , η � →η,F . � Proposition 3.6
Let η � η � ∈ L and define: C η � →η := � � (0) : H (0) η � →η → G � × � � (2) : H (2) η � →η → G � × � � (3) : H (3) η � →η → G � . ike C η , C η � →η is a finite-dimensional smooth manifold.To any � η � ∈ C η � we associate maps � η : γ ( η ) → G , � (0) η � →η : H (0) η � →η → G , � (2) η � →η : H (2) η � →η → G , � (3) η � →η : H (3) η � →η → G by: ∀� � ∈ H (0) η � →η , � (0) η � →η ( � � ) := � η � ( � � ) ; ∀� ∈ γ ( η ) , � η ( � ) := � � η�→η,� � k =2 [ � η � ◦ � η � →η,� ( k )] � η�→η,� ( k ) � � [ � η � ◦ � η � →η,� (1)]with the convention that products of group elements are ordered from right to left: ∀� , � � � , � � ∈ G, � � k =1 � k := � � � � � � � � ; ∀� � ∈ H (2) η � →η , � (2) η � →η ( � � ) := � η � ( � � ) ; ∀� ∈ γ ( η ) , ∀� � ∈ H (3) η � →η,� , � (3) η � →η ( � � ) := � η � ( � � ) � � � η ( � )� − (with � η ( � ) from 3.6. ).Then, the map φ η � →η : � η � �→ � (0) η � →η , � (2) η � →η , � (3) η � →η ; � η is a diffeomorphism C η � → C η � →η × C η . Proof φ η � →η is smooth for G is a Lie group. Next, for any � η ∈ C η and any � � (0) η � →η , � (2) η � →η , � (3) η � →η � ∈ C η � →η we define a map � η � by: ∀� � ∈ H (0) η � →η , � η � ( � � ) := � (0) η � →η ( � � ) ; ∀� ∈ γ ( η ) , ∀� � ∈ H (1) η � →η,� , � η � ( � � ) := � k = � η�→η,� � � (2) η � →η ◦ � η � →η,� ( k )� −� η�→η,� ( k ) � � η ( � ) ; ∀� � ∈ H (2) η � →η , � η � ( � � ) := � (2) η � →η ( � � ) ; ∀� ∈ γ ( η ) , ∀� � ∈ H (3) η � →η,� , � η � ( � � ) := � (3) η � →η ( � � ) � � η ( � ) .Having a partition of γ ( η � ) (prop. 3.4) ensures that � η � is well-defined and, again because G is aLie group, the map � φ η � →η : � (0) η � →η , � (2) η � →η , � (3) η � →η ; � η �→ � η � is a smooth map C η � →η × C η → C η � . We cancheck that � φ η � →η ◦ φ η � →η = id C η� and φ η � →η ◦ � φ η � →η = id C η�→η × C η , thus φ η � →η is a diffeomorphism. � In order for the previously defined factorization maps to provide a valid factorizing system, theyshould fulfill the three-spaces consistency condition [17, eq. (2.11. ) and fig. 2.2]: given three labels η � η � � η �� , the variables discarded when going down in one step from η �� to η should be the sameas the ones discarded when going, in two successive steps, first from η �� to η � and then from η � to η .In other words, we need a map φ η �� →η � →η identifying C η �� →η with C η �� →η � × C η � →η , in agreement withthe factorization maps φ η �� →η , φ η �� →η � and φ η � →η . This will ensure that there is no ambiguity as tohow the variables associated to the label η are to be extracted from the phase space correspondingto η �� .To ascertain that this is indeed the case, we have to distinguish the different ways, for an edge � �� of η �� , to be positioned with respect to the edges and faces of η � and η : this yields 13 inequivalent ossibilities, as depicted in fig. 3.4. Only one of them is compatible with � �� being of type ‘1’ fortransition η �� → η , while, in the 12 others case, the group variable attached to � �� contributes tothe variables of C η �� →η , in terms of which φ η �� →η � →η then needs to be appropriately specified (see thepoints 3.7. to 3.7. of the proof below).Note that we could have made use of [17, prop. 2.17] to obtain the three-spaces consistency ofthe factorization maps from a similar condition formulated at the level of the projections betweenthe small phase spaces [17, def. 2.3 and fig. 2.1]: that these projections fulfills such a condition canindeed be read out from their expression (that will be given in prop. 3.8). Yet, using this resultwould require G to be connected: a restriction that appears quite artificial when the factorizationmap φ η � →η in prop. 3.6 has been expressed solely in terms of the group operations (multiplicationand inverse). Preferably, the three-spaces consistency can be obtained in full generality directlyat the level of the factorization maps: once the correct explicit expression for φ η �� →η � →η has beendeduced from the one for φ η � →η , it is a straightforward (albeit rather fastidious) check that [17,eq. (2.11. )] holds. Theorem 3.7
Let η � η � � η �� ∈ L . There exists a diffeomorphism φ η �� →η � →η : C η �� →η → C η �� →η � × C η � →η such that:� φ η �� →η � →η × id C η � ◦ φ η �� →η = �id C η��→η� × φ η � →η � ◦ φ η �� →η � (aka. [17, eq. (2.11. )] ).This provides a factorizing system of smooth manifolds ( L , C , φ ) × [17, def. 2.15]. Proof
Let η � η � � η �� ∈ L . Let � ∈ γ ( η ) and, for any k ∈ { , � � � , � η � →η,� + 1 } , define: � ( k ) η �� →η � →η,� := k− � � =1 � η �� →η � ,� η�→η,� ( � ) .Using prop. 2.3 together with the uniqueness of � η �� →η,� (prop. 3.2), we then get � η �� →η,� = � ( � η�→η,� +1) η �� →η � →η,� and, for any k ∈ { , � � � , � η � →η,� } and any � ∈ � � ( k ) η �� →η � →η,� + 1 , � � � , � ( k +1) η �� →η � →η,� � : � η �� →η,� ( � ) = � η �� →η � ,� η�→η,� ( k ) � � − � ( k ) η �� →η � →η,� � if k = 1 or � η � →η,� ( k ) = +1 � η �� →η � ,� η�→η,� ( k ) � � ( k +1) η �� →η � →η,� + 1 − � � otherwise , � η �� →η,� ( � ) = � η �� →η � ,� η�→η,� ( k ) � � − � ( k ) η �� →η � →η,� � if k = 1 or � η � →η,� ( k ) = +1 −� η �� →η � ,� η�→η,� ( k ) � � ( k +1) η �� →η � →η,� + 1 − � � otherwise .Next, for any F ∈ F ( η ), we have: H (1 , η �� →η,F = � F � ∈H (1 , η�→η,F H (1 , η �� →η � ,F � (the proof is similar to the one for the second part of prop. 3.5).In particular, using eqs. (3.5. ) and (3.5. ), we then get, for any � ∈ γ ( η ): ,
2) (3 ,
3) (3 ,
1) (2 , , , ,
2) (1 , , , ,
0) (3 , η �� η � η η �� → η � η � → ηη �� → η Figure 3.4 – Three-spaces consistency: in the illustration of η �� , a tag ( κ � ,κ ) denotes an edge belongingto H ( κ � ) η �� →η � ,� for some � in H ( κ ) η � →η and a tag (0) denotes an edge belonging to H (0) η �� →η � (for betterreadability we have tagged only one edge of each type) . H (0) η �� →η = H (0) η �� →η � ∪ � � � ∈H (0) η�→η H (4) η �� →η � ,� � ∪ � � � ∈H (2) η�→η H (3) η �� →η � ,� � ∪ � � � ∈H (3) η�→η H (2) η �� →η � ,� � ; H (1) η �� →η,� = � � � ∈H (1) η�→η,� H (1) η �� →η � ,� � ; H (2) η �� →η,� = � � � ∈H (1) η�→η,� H (2) η �� →η � ,� � ∪ � � � ∈H (2) η�→η,� � H (1) η �� →η � ,� � ∪ H (2) η �� →η � ,� � � ; H (3) η �� →η,� = � � � ∈H (1) η�→η,� H (3) η �� →η � ,� � ∪ � � � ∈H (3) η�→η,� � H (1) η �� →η � ,� � ∪ H (3) η �� →η � ,� � � ;Now, for any � � (0) η �� →η , � (2) η �� →η , � (3) η �� →η � ∈ C η �� →η , we define: ∀� � ∈ H (0) η � →η , � (0) η � →η ( � � ) := � � η��→η�,�� � k =2 � � (0) η �� →η ◦ � η �� →η � ,� � ( k )� � η��→η�,�� ( k ) � � � � (0) η �� →η ◦ � η �� →η � ,� � (1)�(well-defined since H (1) η �� →η � ,� � ∪ H (2) η �� →η � ,� � ⊂ H (0) η �� →η ); ∀� � ∈ H (2) η � →η , � (2) η � →η ( � � ) := � � η��→η�,�� � k =2 � � (2) η �� →η ◦ � η �� →η � ,� � ( k )� � η��→η�,�� ( k ) � � � � (2) η �� →η ◦ � η �� →η � ,� � (1)�(well-defined since H (1) η �� →η � ,� � ∪ H (2) η �� →η � ,� � ⊂ H (2) η �� →η ); ∀� � ∈ H (3) η � →η , � (3) η � →η ( � � ) := � � η��→η�,�� � k =2 � � (0) η �� →η ◦ � η �� →η � ,� � ( k )� � η��→η�,�� ( k ) � � � � (3) η �� →η ◦ � η �� →η � ,� � (1)�(well-defined since H (1) η �� →η � ,� � ⊂ H (3) η �� →η and H (2) η �� →η � ,� � ∪ H (0) η �� →η � ,� � ⊂ H (2) η �� →η ); ∀� �� ∈ H (0) η �� →η � , � (0) η �� →η � ( � �� ) := � (0) η �� →η ( � �� )(well-defined since H (0) η �� →η � ⊂ H (0) η �� →η ); ∀� �� ∈ H (2) η �� →η � , � (2) η �� →η � ( � �� ) := � � (0) η �� →η ( � �� ) if � �� ∈ H (0) η �� →η � (2) η �� →η ( � �� ) if � �� ∈ H (2) η �� →η (well-defined since H (2) η �� →η � ⊂ H (0) η �� →η ∪ H (2) η �� →η ); ∀� � ∈ γ ( η � ) , ∀� �� ∈ H (3) η �� →η � ,� � ,� (3) η �� →η � ( � �� ) := � (0) η �� →η ( � �� ) � � � (0) η � →η ( � � )� − if � � ∈ H (0) η �� →η � (0) η �� →η ( � �� ) � � � (2) η � →η ( � � )� − if � � ∈ H (2) η �� →η � (3) η �� →η ( � �� ) � � � η�→η,� k =2 � � (2) η � →η ◦ � η � →η,� ( k )� � η�→η,� ( k ) if � � ∈ H (1) η �� →η,� (with � ∈ γ ( η ) ) � (3) η �� →η ( � �� ) � � � (3) η � →η ( � � )� − if � � ∈ H (3) η �� →η (using � (0) η � →η , � (2) η � →η and � (3) η � →η from 3.7. to 3.7. ; well-defined since H (3) η �� →η � ,� � ⊂ H (0) η �� →η in the firsttwo cases, and H (3) η �� →η � ,� � ⊂ H (3) η �� →η in the last two cases). hen, the map φ η �� →η � →η : � (0) η �� →η , � (2) η �� →η , � (3) η �� →η �→ � (0) η �� →η � , � (2) η �� →η � , � (3) η �� →η � ; � (0) η � →η , � (2) η � →η , � (3) η � →η is smooth C η �� →η → C η �� →η � × C η � →η .Let � η �� ∈ C η �� , and define: � � (0) η �� →η � , � (2) η �� →η � , � (3) η �� →η � ; � η � � := φ η �� →η � � � η �� � ∈ C η �� →η � × C η � ; � � (0) η � →η , � (2) η � →η , � (3) η � →η ; � η � := φ η � →η � � η � � ∈ C η � →η × C η ; � � (0) η �� →η , � (2) η �� →η , � (3) η �� →η ; � η � := φ η �� →η � � η �� � ∈ C η �� →η × C η ; � � (0) η �� →η � , � (2) η �� →η � , � (3) η �� →η � ; � (0) η � →η , � (2) η � →η , � (3) η � →η � := φ η �� →η � →η � � � (0) η �� →η , � (2) η �� →η , � (3) η �� →η � .Using the definitions of φ η � →η (from prop. 3.6) and φ η �� →η � →η � , we can check that:� � (0) η �� →η � , � (2) η �� →η � , � (3) η �� →η � ; � (0) η � →η , � (2) η � →η , � (3) η � →η ; � η � == � � (0) η �� →η � , � (2) η �� →η � , � (3) η �� →η � ; � (0) η � →η , � (2) η � →η , � (3) η � →η ; � η � .In other words, [17, eq. (2.11. )] is fulfilled. But this also ensures that φ η �� →η � →η is a diffeomorphismfor φ η �� →η � , φ η � →η , and φ η �� →η are diffeomorphisms (prop. 3.6). Together with the directedness of L proved in theorem 2.16, this yields the desired factorizing system [17, def. 2.15]. � Finally, we wrap up the classical side of the construction by writing down the symplectic manifold M η attached to a label η , the projection from M η � into M η (as prescribed by the factorization φ η � →η ,for η � η � ), and the expression of the holonomy and flux observables over M η . These explicitformulas validate a posteriori the intuitive arguments that were repeatedly asserted above (onaccount of the underlying physical interpretation of the labels), the aim of all previous definitionsbeing indeed to eventually arrive at props. 3.8 and 3.9 below. Note that, as announced at thebeginning of the present section, the projections M η � → M η are of the form that were consideredin [18, theorem 3.2] (making C η into a Lie group via pointwise multiplication and inverse). Proposition 3.8
To any η ∈ L we associate the symplectic manifold M η := T ∗ ( C η ) (with itscanonical symplectic structure as a cotangent bundle), which we identify with C η × P η (def. 3.1) via: L η : M η → C η × P η �, � �→ �, � F �→ � ◦ � T � � �→ R ( F ) η,� � ��� ,the map R ( F ) η,� � ∈ C η being defined for F ∈ F ( η ), � ∈ G , and � ∈ C η by: ∀� ∈ γ ( η ) , R ( F ) η,� � ( � ) := � � ( � ) � � if χ η ( � ) = F� ( � ) else .For any η � η � ∈ L , we define π η � →η : M η � → M η as π η � →η = � � η � →η ◦ � φ η � →η , where � � η � →η : T ∗ ( C η � →η × C η ) ≈ T ∗ ( C η � →η ) × T ∗ ( C η ) → T ∗ ( C η ) is the projection on the second Cartesian factor and� φ η � →η : T ∗ ( C η � ) → T ∗ ( C η � →η × C η ) is the cotangent lift of φ η � →η . Then, ( L , M , π ) ↓ is a projectivesystem of phase spaces [17, props. 2.16 and 2.13]. oreover, we have: L η ◦ π η � →η ◦ L − η � : C η � × P η � → C η × P η � η � , P η � �→ � η , P η ,where � η and P η are given in terms of � η � and P η � by: ∀� ∈ γ ( η ) , � η ( � ) = � � η�→η,� � k =2 [ � η � ◦ � η � →η,� ( k )] � η�→η,� ( k ) � � [ � η � ◦ � η � →η,� (1)] , (3.8. )and ∀F ∈ F ( η ) , P η ( F ) = � F � ∈H (1 , η�→η,F P η � ( F � ) . (3.8. ) Proof
For any η , L η is a diffeomorphism M η → C η × P η by definition of χ η (def. 2.14).Let ( � η � , � η � ) ∈ M η � and ( � η � , P η � ) = L η � ( � η � , � η � ) . We have:� φ η � →η � � η � ; � η � � = � � (0) η � →η , � (2) η � →η , � (3) η � →η , � η ; � η � ◦ � T φ η�→η ( � η� ) φ − η � →η �� ,with � (0) η � →η , � (2) η � →η , � (3) η � →η , � η constructed from � η � as in prop. 3.6 (in particular, � η is given byeq. (3.8. )). Thus: π η � →η � � η � ; � η � � = � � η ; � η � ◦ � T � η φ − η � →η � � (0) η � →η , � (2) η � →η , � (3) η � →η , · ��� ,and finally L η ◦ π η � →η � � η � ; � η � � = � � η ; P η � , where P η is given in terms of � η � , � η � by: ∀F ∈ F ( η ) , P η ( F ) = � η � ◦ � T � �→ φ − η � →η � � (0) η � →η , � (2) η � →η , � (3) η � →η , R ( F ) η,� � η �� .Now, for any F ∈ F ( η ) , � ∈ G and � � ∈ γ ( η � ) , we get, using the explicit expression for φ − η � →η from the proof of prop. 3.6: φ − η � →η � � (0) η � →η , � (2) η � →η , � (3) η � →η , R ( F ) η,� � η � ( � � ) = � � η � ( � � ) � � if χ η � ( � � ) ∈ H (1 , η � →η,F � η � ( � � ) else .Therefore, ∀F ∈ F ( η ) , P η ( F ) = � F � ∈H (1 , η�→η,F P η � ( F � ) . � Proposition 3.9
Let � ∈ L edges and let η ∈ L such that γ ( η ) ∈ L graphs /� (prop. 3.2). For any δ ∈ C ∞ ( G, R ) (with C ∞ ( G, R ) the set of smooth functions G → R ), we define h ( �,δ ) η ∈ C ∞ ( M η , R )by: ∀ ( �, � ) ∈ M η , h ( �,δ ) η ( �, � ) := δ � � γ ( η ) →� � k =1 [ � ◦ � γ ( η ) →� ( k )] � γ ( η ) →� ( k ) � .Then, for any η, η � ∈ L such that γ ( η ) , γ ( η � ) ∈ L graphs /� , we have: ∀δ ∈ C ∞ ( G, R ) , h ( �,δ ) η ∼ h ( �,δ ) η � ,with the equivalence relation of [17, eq. (2.4. )]. Moreover {η ∈ L | γ ( η ) ∈ L graphs /� } � = ∅ , thus we an associate to any δ ∈ C ∞ ( G, R ) a well-defined observable h ( �,δ ) ∈ O ↓ ( L , M ,π ) [17, def. 2.4].Let F ∈ L faces and let η ∈ L such that λ ( η ) ∈ L profls /F (prop. 3.3). For any � ∈ g , we defineP ( F,� ) η ∈ C ∞ ( M η , R ) by: ∀ ( �, � ) ∈ M η , P ( F,� ) η ( �, � ) := � F � ∈H λ ( η ) →F � ◦ � T � � �→ R ( F � ) η,� � �� ( � ) .Then, for any η, η � ∈ L such that λ ( η ) , λ ( η � ) ∈ L profls /F , we have: ∀� ∈ g , P ( F,� ) η ∼ P ( F,� ) η � .And since � η ∈ L �� λ ( η ) ∈ L profls /F � � = ∅ , we can associate to any � ∈ g a well-defined observableP ( F,� ) ∈ O ↓ ( L , M ,π ) . Proof
Let � ∈ L edges . By chaining lemma 2.17 to 2.20, there exists ( γ, λ ) ∈ L such that {�} � γ (and[ ∅ ] profl � λ where [ · ] profl denotes the equivalence class in L profls ), hence {η ∈ L | γ ( η ) ∈ L graphs /� } � = ∅ . Let η such that γ ( η ) ∈ L graphs /� and let η � � η . Then, γ ( η � ) ∈ L graphs /� . Moreover, we can express � γ ( η � ) →� in terms of � γ ( η ) →� and of the � η � →η,� � for � � in the image of � γ ( η ) →� (like we did above in theproof of theorem 3.7). Together with eq. (3.8. ), we readily check that: ∀δ ∈ C ∞ ( G, R ) , h ( �,δ ) η ◦ π η � →η = h ( �,δ ) η � .Finally, for any η, η � such that γ ( η ) , γ ( η � ) ∈ L graphs /� , there exists η �� � η, η � , hence: ∀δ ∈ C ∞ ( G, R ) , h ( �,δ ) η ◦ π η �� →η = h ( �,δ ) η �� = h ( �,δ ) η � ◦ π η �� →η � ,ie. for any δ ∈ C ∞ ( G, R ), h ( �,δ ) η ∼ h ( �,δ ) η � .Let λ ∈ L profls , F ∈ F ( λ ) and F = F ⊥ ◦ F . Again, there exists ( γ � , λ � ) ∈ L such that λ � λ � (and ∅ � γ � ), hence � η ∈ L �� λ ( η ) ∈ L profls /F � � = ∅ . Let η such that λ ( η ) ∈ L profls /F and let η � � η .Then, λ ( η � ) ∈ L profls /F and we have (through a reasoning similar to the proof for the second part ofprop. 3.5): H λ ( η � ) →F = � F � ∈H λ ( η ) →F H (1 , η � →η,F � . (3.9. )Combining this with eq. (3.8. ), we get: ∀� ∈ g , P ( F,� ) η ◦ π η � →η = P ( F,� ) η � .Like above, this ensures that for any η, η � such that λ ( η ) , λ ( η � ) ∈ L profls /F and for any � ∈ g ,P ( F,� ) η ∼ P ( F,� ) η � . � Quantizing the classical formalism set up above is then a straightforward application of theprescriptions for quantization in the position representation that were detailed in [18, props. 3.3and 3.4]. roposition 3.10 Let µ be a right-invariant Haar measure on G . For η ∈ L , we can, using thenatural identification C η ≈ G γ ( η ) , define a measure µ η ≈ µ γ ( η ) on C η (actually the identification C η ≈ G γ ( η ) is not unique, since it depends on an ordering of the edges in γ ( η ); yet the measure µ η is independent of this choice).Then, there exists a family of smooth measures µ η � →η on each C η � →η for η � η � ∈ L , such that( L , ( C , µ ) , φ ) × is a factorizing system of measured manifolds [18, def. 3.1]. Defining: ∀η ∈ L , H η := L ( C η , �µ η ) ; ∀η � η � ∈ L , H η � →η := L ( C η � →η , �µ η � →η ) and:Φ η � →η : H η � → H η � →η ⊗ H η ≈ L ( C η � →η × C η , �µ η � →η × �µ η ) ψ �→ ψ ◦ φ − η � →η ;there exists a family of Hilbert spaces isomorphisms �Φ η �� →η � →η � η � η � � η �� such that ( L , H , Φ) ⊗ is aprojective system of quantum state spaces [18, def. 2.1]. Proof
The right-invariant Haar measure µ comes from a smooth, right-invariant volume form on G .Hence, for any η there exists a smooth, right-invariant volume form ω η on C η such that µ η is themeasure associated to ω η ( ω η is not unique, as there is a freedom in the orientation of C η ; however,it is sufficient here to just pick an ω η for each C η ). In particular, µ η is therefore a smooth measureon C η . Defining: R η,� : C η → C η � �→ ( � �→ � ( � ) � � ( � )) ,the right-invariance of ω η can be written as: ∀� ∈ C η , R ∗η,� ω η = ω η . (3.10. )Let η � η � ∈ L , � η := dim C η , � η � := dim C η � and � η � →η := � η � − � η . We choose a basis � , � � � , � � η in T ( C η ) (where ∈ C η is the map � �→ ∈ G ) and we define a smooth volume form ω η � →η on C η � →η by: ∀� ∈ C η � →η , ∀� , � � � , � � η�→η ∈ T � ( C η � →η ) ,ω η � →η,� ( � , � � � , � � η�→η ) := � φ − ,∗η � →η ω η � � ( �, ) �( � , , � � � , ( � � η�→η , , (0 , � ) , � � � , (0 , � � η )� ω η, ( � , � � � , � � η ) .Now, for any � ∈ C η , we define � � ∈ C η � by: ∀� ∈ γ ( η ) , ∀� � ∈ H (1) η � →η,� ∪ H (3) η � →η,� , � � ( � � ) = � ( � ) ;and ∀� � ∈ H (0) η � →η ∪ H (2) η � →η , � � ( � � ) = .Using the explicit expression for φ − η � →η from the proof of prop. 3.6, we can check that: ∀� ∈ C η , R η � , � � ◦ φ − η � →η = φ − η � →η ◦ �id C η�→η × R η,� � . pplying eq. (3.10. ), we thus get, for any � ∈ C η : ω η, ( � , � � � , � � η ) = ω η,� ( U ,� , � � � , U � η ,� )and ∀� ∈ C η � →η , ∀� , � � � , � � η�→η ∈ T � ( C η � →η ) , � φ − ,∗η � →η ω η � � ( �, ) �( � , , � � � , ( � � η�→η , , (0 , � ) , � � � , (0 , � � η )�= � φ − ,∗η � →η ω η � � ( �,� ) �( � , , � � � , ( � � η�→η , , (0 , U ,� ) , � � � , (0 , U � η ,� )� ,where ∀� ∈ { , � � � , � η } , U �,� := [ T R η,� ] ( � � ) ∈ T � ( C η ) . And since U ,� , � � � , U � η ,� is a basis of T � ( C η ), we get: φ − ,∗η � →η ω η � = ω η � →η × ω η .Therefore, defining, for any η � η � , µ η � →η to be the smooth measure on C η � →η associated to thevolume form ω η � →η , we have: ∀η � η � ∈ L , φ − ,∗η � →η µ η � = µ η � →η × µ η ,and, using the 3-spaces consistency condition [17, eq. (2.11. )] in the factorizing system ( L , C , φ ) × ,this also implies: ∀η � η � � η �� ∈ L , φ − ,∗η �� →η � →η µ η �� →η = µ η �� →η � × µ η � →η .Thus, ( L , ( C , µ ) , φ ) × is a factorizing system of measured manifolds, from which a projective systemof quantum state spaces ( L , H , Φ) ⊗ can be constructed as described in [18, prop. 3.3]. � Proposition 3.11
Let � ∈ L edges and let η ∈ L such that γ ( η ) ∈ L graphs /� . For any δ ∈ C ∞ ( G, R ),h ( �,δ ) η fulfills the quantization condition [18, eq. (3.4. )] and can be quantized as a densely defined,essentially self-adjoint operator �h ( �,δ ) η on H η (with dense domain D η ), given by: ∀ψ ∈ D η , ∀� ∈ C η , � �h ( �,δ ) η ψ �( � ) := δ � � γ ( η ) →� � k =1 [ � ◦ � γ ( η ) →� ( k )] � γ ( η ) →� ( k ) � ψ ( � ) . (3.11. )Moreover, we have for any η, η � ∈ L such that γ ( η ) , γ ( η � ) ∈ L graphs /� : ∀δ ∈ C ∞ ( G, R ) , �h ( �,δ ) η ∼ �h ( �,δ ) η � ,with the equivalence relation of [18, eq. (2.3. )], thus we can associate to any � ∈ L edges and any δ ∈ C ∞ ( G, R ) a well-defined observable �h ( �,δ ) ∈ O ⊗ ( L , H , Φ) [18, prop. 2.5].Let F ∈ L faces and let η ∈ L such that λ ( η ) ∈ L profls /F . For any � ∈ g , P ( F,� ) η fulfills thequantization condition and can be quantized as a densely defined, essentially self-adjoint operator � P ( F,� ) η on H η , given by: ∀ψ ∈ D η , ∀� ∈ C η , � � P ( F,� ) η ψ �( � ) := � � T � �→ ψ � R ( F ) η,� � �� ( � ) , (3.11. ) he map R ( F ) η,� � ∈ C η being defined for � ∈ G and � ∈ C η by: ∀� ∈ γ ( η ) , R ( F ) η,� � ( � ) := � � ( � ) � � if � ∈ F� ( � ) else .Moreover, we have for any η, η � ∈ L such that λ ( η ) , λ ( η � ) ∈ L profls /F : ∀� ∈ g , � P ( F,� ) η ∼ � P ( F,� ) η � ,thus we can associate to any F ∈ L faces and any � ∈ g a well-defined observable � P ( F,� ) ∈ O ⊗ ( L , H , Φ) . Proof
Hamiltonian vector field projected on C η . Let η ∈ L . Using left-translated exponentialcoordinates around a point � ∈ C η (cf. the proof of [18, theorem 3.2]), we can show that thesymplectic structure Ω η on M η is given by: ∀� ∈ C η , ∀P ∈ P η , ∀�, � � ∈ T � ( C η ) , ∀�, � � ∈ P η , � L − ,∗η Ω η � �,P �( �, � ) , ( � � , � � )� == � �∈γ ( η ) � � ( χ η ( � )) ◦ � ( � ) η,� − ( � ) − � ( χ η ( � )) ◦ � ( � ) η,� − ( � � ) + P ( χ η ( � )) �� � ( � ) η,� − ( � ) , � ( � ) η,� − ( � � )� g � ,where L η : M η → C η × P η was defined in prop. 3.8 and for any � ∈ γ ( η ): ∀� ∈ C η , � ( � ) η,� − := � T � � �→ � ( � ) − � � ( � )� : T � C η → g .Let r ∈ C ∞ ( C η , R ). For any ( �, P ) ∈ C η × P η and any � ∈ P η , we have:� � r ◦ L − η � ( �,P ) (0 , � ) == Ω η,L − η ( �,P ) � X r ,L − η ( �,P ) , � T ( �,P ) L − η � (0 , � )�= � L − ,∗η Ω η � �,P �� T L − η ( �,P ) L η � � X r ,L − η ( �,P ) � , (0 , � )�= � �∈γ ( η ) � � χ η ( � )� ◦ � ( � ) η,� − �� T L − η ( �,P ) γ η � � X r ,L − η ( �,P ) �� , (3.11. )where X r is the Hamiltonian vector field of r and γ η : M η → C η is the bundle projection in M η = T ∗ ( C η ).Now, for any �, � ∈ C η and any � ∈ γ ( η ), we have:� � �→ R ( χ η ( � ) ) η,� � � ◦ � � � �→ � ( � ) − � � � ( � )� ( � ) = R ( χ η ( � ) ) η,� ( � ) − � � ( � ) � : � � �→ � � ( � � ) if � � = �� ( � � ) else .Therefore, we get for any � ∈ C η :� �∈γ ( η ) � T � �→ R ( χ η ( � ) ) η,� � � ◦ � ( � ) η,� − = id T � ( C η ) .For any � ∈ C η and any υ ∈ T ∗� ( C η ), we define � ( � ) υ ∈ P η by: F ∈ F ( η ) , � ( � ) υ ( F ) := υ ◦ � T � �→ R ( F ) η,� � � ,so that eq. (3.11. ) becomes: ∀ ( �, P ) ∈ C η × P η , υ ◦ � T L − η ( �,P ) γ η � � X r ,L − η ( �,P ) � = � � r ◦ L − η � ( �,P ) (0 , � ( � ) υ ) .Thus, r fulfills the quantization condition [18, eq. (3.4. )] if and only if there exists a smooth vectorfield X r on C η such that: ∀ ( �, P ) ∈ C η × P η , ∀υ ∈ T ∗� ( C η ) , � � r ◦ L − η � ( �,P ) (0 , � ( � ) υ ) = υ � X r ,� � .If this is the case, we can then define �r as a densely defined operator on H η by: ∀ψ ∈ D η , ∀� ∈ C η , ��r ψ �( � ) := �r( �,
0) + � µ η X r ( � )� ψ ( � ) + � � � r ◦ L − η � ( �,P ) (0 , � ( � ) �ψ � ) ,where div µ η X r ∈ C ∞ ( C η ) is defined by L X r µ η = �div µ η X r � µ η [18, def. A.12]. Taking D η as the setof smooth, compactly supported functions on C η , Stokes’ Theorem ensures that �r is a symmetricoperator. Holonomies.
Let � ∈ L edges and let η ∈ L such that γ ( η ) ∈ L graphs /� . For any δ ∈ C ∞ ( G, R ), wehave: ∀ ( �, P ) ∈ C η × P η , h ( �,δ ) η ◦ L − η ( �, P ) = δ � � γ ( η ) →� � k =1 [ � ◦ � γ ( η ) →� ( k )] � γ ( η ) →� ( k ) � ,hence h ( �,δ ) η fulfills the quantization condition with X h ( �,δ ) η := 0. This yields the expression ineq. (3.11. ) for �h ( �,δ ) η . Next, since δ is real-valued, we get, using bump functions: ψ ∈ Ker � �h ( �,δ ) η † ± � � ⇔ ∀φ ∈ D η , � C η �µ η h ( �,δ ) η φ ∗ ψ = ∓� � C η �µ η φ ∗ ψ⇔ h ( �,δ ) η ψ = ∓� ψ a.e. in � C η , µ η � ⇔ ψ = 0 a.e. in � C η , µ η � ,so the symmetric operator �h ( �,δ ) η is essentially self-adjoint [26, theorem VIII.3]. Finally, for any η, η � ∈ L such that γ ( η ) , γ ( η � ) ∈ L graphs /� , we have h ( �,δ ) η ∼ h ( �,δ ) η � (prop. 3.9), so [18, prop. 3.4] ensuresthat �h ( �,δ ) η ∼ �h ( �,δ ) η � , and therefore that an observable �h ( �,δ ) can be consistently defined on S ⊗ ( H , L , Φ) . Fluxes.
Let
F ∈ L faces and let η ∈ L such that λ ( η ) ∈ L profls /F . For any � ∈ g , we have: ∀ ( �, P ) ∈ C η × P η , P ( F,� ) η ◦ L − η ( �, P ) = � F � ∈H λ ( η ) →F P ( F � )( � ) ,hence P ( F,� ) η fulfills the quantization condition with: ∀� ∈ C η , X P ( F,� ) η ,� := � F � ∈H λ ( η ) →F � T � �→ R ( F � ) η,� � � ( � ) . ow, through a reasoning similar to the proof for the second part of prop. 3.5, we have:� � ∈ γ ( η ) �� � ∈ F � = � � ∈ γ ( η ) �� ∃ F � ∈ H λ ( η ) →F � χ η ( � ) = F � � , (3.11. )therefore: ∀� ∈ C η , X P ( F,� ) η ,� = � T � �→ R ( F ) η,� � � ( � ) ,which yields the expression in eq. (3.11. ) for � P ( F,� ) η since the flow τ �→ R ( F ) η,� τ� of X P ( F,� ) η preservesthe right-invariant measure µ η on C η . Moreover, by the dominated convergence theorem, we have: ψ ∈ Ker � � P ( F,� ) η † ± � � ⇔ ∀φ ∈ D η , −� � C η �µ η ( � ) � T � �→ φ ∗ � R ( F ) η,� � �� ( � ) ψ ( � ) = ∓� � C η �µ η ( � ) φ ∗ ( � ) ψ ( � ) ⇔ ∀φ ∈ D η , ��τ ���� τ =0 � C η �µ η ( � ) φ ∗ � R ( F ) η,� τ� � � ψ ( � ) = ± � C η �µ η ( � ) φ ∗ ( � ) ψ ( � ) ⇔ ∀φ ∈ D η , ∀τ ∈ R , � C η �µ η ( � ) φ ∗ � R ( F ) η,� τ� � � ψ ( � ) = � ±τ � C η �µ η ( � ) φ ∗ ( � ) ψ ( � ) ,thus, using again bump functions, together with the invariance of the measure under right transla-tions: ψ ∈ Ker � � P ( F,� ) η † ± � � ⇔ ∀τ ∈ R , ψ ◦ � � �→ R ( F ) η,� −τ� � � = � ±τ ψ a.e. in � C η , µ η � ⇔ ∀τ ∈ R , �ψ� η = � ±τ �ψ� η ⇔ ψ = 0 a.e. in � C η , µ η � ,hence � P ( F,� ) η is essentially self-adjoint. Finally, like above, prop. 3.9 together with [18, prop. 3.4]ensures that the � P ( F,� ) η , defined for each η ∈ L such that λ ( η ) ∈ L profls /F , can be consistentlyassembled into an observable � P ( F,� ) on S ⊗ ( H , L , Φ) . � The projective state space set up in the previous subsection regrettably cannot be displayed asthe rendering [17, def. 2.6] of a continuous classical theory of connections: if we were to defineprojection maps from the infinite dimensional phase space of such a theory [31, section IV.33]into the various small phase spaces M η , these maps would not be smooth, and in fact they wouldnot even be surjective. Indeed, the holonomy, resp. flux, variables are obtained by smearing theconnection, resp. its conjugate ‘electric field’, along singular geometrical objects (respectively - and � − -dimensional, in contrast to a smearing by a smooth function on the � -dimensional manifold Σ ). Moreover, these non-smooth flux variables from the continuous theory actually combine bothfaces of a given surface: the ‘one-sided’ flux variables we introduced above have no equivalent interms of appropriate smearings of the electric field, while the fluxes attached to submanifolds ofdimension less than � − (arising at the intersection of surfaces) should vanish according to thecontinuous theory. On the other hand the inclusion of these additional, seemingly non-physicalobservables is forced upon us by the regularization of the Poisson brackets [31, section II.6.4]: inother words, they are the price for trying to somewhat restore a notion of compatibility of theprojection map with the symplectic structure (similar to what was expressed by [17, def. 2.1] in thecase of smooth projections). We refer to the LQG literature for further discussion of these issues(see eg. [29, section 6]).Instead of exploring the relation between the formalism just constructed and the continuousclassical theory of interest, we will take the Ashtekar-Lewandowski Hilbert space H �� [2, 22] as astarting point, and investigate in which sense the quantum projective state space from prop. 3.10can be understood as a reasonable extension of the space of states defined over H �� . This analysishas to to be carried out in the case of a compact group G , since this is a prerequisite for H �� to exist: from the perspective we are adopting here, the compactness ensures that the measures µ η � →η will be normalizable, which in turn allows to pick out a natural ‘reference state’ ζ η � →η in each H η � →η , and thus to identify H η with the vector subspace � ζ η � →η � ⊗ H η in H η � ≈ H η � →η ⊗ H η . Thisprovides an inductive system of Hilbert spaces, whose limit will reproduce H �� .As stressed at the beginning of subsection 2.1, the limit of an inductive system is not affectedif one restricts its label set to some cofinal part. This is the reason why we have so far onlyconsidered graphs with fully analytic edges. Still, the use of graphs with semianalytic edges [31,sections II.6 and IV.20] is favored in LQG, for, although they yields the same Hilbert space, theypresent it in a form more convenient for writing the action of semianalytic diffeomorphisms (which,unlike fully analytic ones, can be local). Therefore, we briefly sketch below how to switch back tothe semianalytic class. In this subsection the gauge group G is assumed to be compact , and the measure µ (introducedin prop. 3.10) is taken to be the normalized Haar measure on G . Definition 3.12
Let k ∈ { , , � � � , ∞} . We define the set L ( k )edges of ( k )-edges like in def. 2.1,by requiring encharted ( k )-edges to be C k -diffeomorphisms instead of analytic ones. In analogyto props. 2.2 and 2.3, we define the range � ( � ), the beginning and ending points � ( � ) and � ( � ),( k )-subedges � [ �,� � ] (for � � = � � ∈ � ( � ) ), the reversed ( k )-edge � − , and the order < ( � ) on the rangeof a ( k )-edge � , as well as the composition of ( k )-composable ( k )-edges. These have the sameproperties as in the analytic case, since the proofs of props. 2.2 and 2.3 did not made use of theanalyticity.An analytic encharted edge is also an encharted ( k )-edge, and two analytic encharted edge areequivalent in ˘ L edges iff they are equivalent in ˘ L ( k )edges . Thus, we have a natural injection of L edges into L ( k )edges . In the following, we will always identify L edges with the image of this injection andwrite L edges ⊂ L ( k )edges . roposition 3.13 We define ��-graphs as finite sets of (1)-edges γ ⊂ L (1)edges such that: ∀� ∈ γ, ∃ � , � � � , � � ∈ L edges ⊂ L (1)edges , (1)-composable � � = � � ◦ � � � ◦ � ; ∀� � = � � ∈ γ, � ( � ) ∩ � ( � � ) ⊂ {� ( � ) , � ( � ) } ∩ {� ( � � ) , � ( � � ) } .We denote the set of ��-graphs by L �� and we equip it with the preorder � defined as in def. 2.4.Then, L graphs , � is a cofinal subset of L �� , � , so in particular L �� , � is a directed preorderedset. Proof
By construction L graphs ⊂ L �� , and the order � between two elements of L graphs coincideswith their order as elements of L �� (indeed, if an analytic edge is the composition of (1)-composableanalytic edges, then, by definition, these edges are composable in L edges ).Next, let γ ∈ L �� and for any � ∈ γ , choose γ � = {� , � � � , � � } ⊂ L edges such that � = � � ◦� � �◦� .We have: ∀� ∈ γ, ∀ � � � = � � � ∈ γ � , � (� � ) ∩ � (� � � ) ⊂ {� (� � ) , � (� � ) } ∩ {� (� � � ) , � (� � � ) } ,and ∀� � = � � ∈ γ, ∀ � � ∈ γ � , ∀ � � � ∈ γ � � ,� (� � ) ∩ � (� � � ) ⊂ � (� � ) ∩ � (� � � ) ∩ � ( � ) ∩ � ( � � ) ⊂ � (� � ) ∩ {� ( � ) , � ( � ) } ∩ � (� � � ) ∩ {� ( � � ) , � ( � � ) }⊂ {� (� � ) , � (� � ) } ∩ {� (� � � ) , � (� � � ) } ,by definition of the composition of edges. Therefore γ � := � �∈γ γ � ∈ L edges , and we have γ � γ � . � We recall here the classical construction underlying the Ashtekar-Lewandowski Hilbert space,namely the composition of a projective limit of configuration spaces. The projection maps involvedhere match exactly the ones induced by the projections between phase spaces considered in prop. 3.8(modulo the straightforward extension to semianalytic graphs). However, the momentum variablesdo not come into play in this context, since we are not setting up an actual projective limit ofsymplectic manifolds, so we can directly use graphs as labels, without having to decorate themwith faces.
Proposition 3.14
Let γ � γ � ∈ L �� . Then, for any � ∈ γ , there exists a unique map � γ � →γ,� : { , � � � , � γ � →γ,� } → γ � (with � γ � →γ,� �
1) such that: � = � γ � →γ,� ( � γ � →γ,� ) � γ�→γ,� ( � γ�→γ,� ) ◦ � � � ◦ � γ � →γ,� (1) � γ�→γ,� (1) ,where, for any k ∈ { , � � � , � γ � →γ,� } , � γ � →γ,� ( k ) is defined from � γ � →γ,� as in eq. (3.2. ). Proof
The proof works exactly as in the analytic case (prop. 3.2). � Proposition 3.15
For any γ ∈ L �� we define the finite-dimensional smooth configuration space C γ := {� : γ → G} (like in def. 3.1). And for any γ � γ � ∈ L �� , we define the map π γ � →γ : C γ � → C γ by: � � ∈ C γ � , ∀� ∈ γ, π γ � →γ ( � � )( � ) := � � γ�→γ,� � k =1 [ � � ◦ � γ � →γ,� ( k )] � γ�→γ,� ( k ) � . π γ � →γ is smooth, for G is a Lie group, and it is moreover surjective.In addition, for any γ � γ � � γ �� ∈ L �� , we have: π γ � →γ ◦ π γ �� →γ � = π γ �� →γ . (3.15. ) Proof
Let γ � γ � and let � ∈ C γ . For any � � = � � ∈ γ , we have: ∀ � � ∈ � γ � →γ,� �{ , � � � , � γ � →γ,� }� , ∀ � � � ∈ � γ � →γ,� � �{ , � � � , � γ � →γ,� � }� ,� (� � ) ∩ � (� � � ) ⊂ {� (� � ) , � (� � ) } ∩ {� (� � � ) , � (� � � ) } ,as in the proof of prop. 3.13, hence � γ � →γ,� �{ , � � � , � γ � →γ,� }� ∩ � γ � →γ,� � �{ , � � � , � γ � →γ,� � }� = ∅ .Moreover, � γ � →γ,� is injective (by definition of the composition of edges), hence the map � � ∈ C γ � given by: ∀� ∈ γ, � � ◦ � γ � →γ,� (1) = � ( � ) � γ�→γ,� (1) , ∀� � ∈ γ � � � � /∈ {� γ � →γ,� (1) |� ∈ γ} , � � ( � � ) = ,is well-defined and is such that π γ � →γ ( � � ) = � . Thus, π γ � →γ is surjective.Finally, eq. (3.15. ) follows from the uniqueness of � γ �� →γ,� as in the proof of theorem 3.7. � As mentioned earlier, there are many projections between the phase spaces T ∗ ( C γ � ) and T ∗ ( C γ ) (with γ � γ � ) that can reproduce a given projection between the configuration spaces C γ � and C γ , or,equivalently, many ways of choosing in C γ � a set of variables complementary to the ones comingfrom C γ . In the proof of prop. 3.16, we choose, for each pair of graphs γ � γ � , a factorizationof C γ � as C γ � −γ × C γ consistent with the projection π γ � →γ defined in prop. 3.15. The injective maps τ γ � ←γ between the Hilbert spaces H γ that serve as building blocks for H �� can then be understoodas arising from the corresponding factorization H γ � ≈ H γ � −γ ⊗ H γ , via the selection of a ‘referencestate’ ζ γ � −γ in H γ � −γ . Because this reference state is taken as the constant function ζ γ � −γ ≡ on C γ � −γ , the thus obtained identification of H γ with a vector subspace of H γ � in the end does notdepend on which particular factorization of C γ � has been used, but solely on the projection π γ � →γ from C γ � into C γ . Also, as announced earlier, the need for a compact gauge group G manifestsitself in this approach as a condition for ζ γ � −γ to be a normalizable element of H γ � −γ (otherwise themap τ γ � ←γ would not be well-defined).Of course, it is not really necessary for assembling the inductive limit H �� to ever introducesuch factorization maps (a more standard path being to directly check that π γ � →γ,∗ µ γ � = µ γ ). Still,it is worth looking closer at the particular family of factorizations elected below. The projectionsbetween phase spaces to which they give rise turn out to be precisely the ones that were consideredin [29, def. 3.8] and although they do not fulfill the three-spaces consistency needed for a projectivesystem (expressed in [17, def. 2.3], or, at the level of the factorization maps, in [17, eq. (2.11. )]), thiscan be quickly fixed: by somewhat tightening the ordering among graphs (requiring, in addition toeq. (2.4. ), that the first part of an edge � ∈ γ , in its decomposition into edges of γ � , should beoriented like � itself, ie. that � = +1 ), we can, without voiding the directedness of L �� , rescue this ivotal consistency condition.So why did we go through the intricacy of carefully setting up a label set with edges andsurfaces if an apparently valid projective system could readily have been built over L �� ? If weexamine carefully the structure of observables that would arise from such a projective structure(and in particular, if we compare it to the one obtained in the previous subsection), we realize thatits momentum variables are fluxes carried by single edge germs (defining the germ of an edge � asthe equivalence class consisting of all edges sharing an initial subedge with � ).This sheds light on how a projective limit of phase spaces can at all be constructed using labelsthat seems to only know about configuration variables. What makes it possible, is the availability ofa preferred pairing of conjugate variables, binding each configuration variable to its own particularmomentum variable (eg. the holonomy along an edge with the flux carried by its germ). This pairingis such that whenever an edge � belongs to some graph γ , its companion flux does not act on anyother edge of γ (indeed, edges in a graph cannot share a common subedge). In this way, any graph,as it selects specific holonomy variables, selects at the same time their conjugate flux variables (andthe slight sharpening of the ordering described above ensures that if γ � � γ the fluxes thus attachedto γ are also in γ � ).A similar mechanism underlies the projective quantum state space built in [23]. In this work,holonomies along analytic loops were used as a complete set of independent configuration variables,which again provides an a priori pairing of conjugate variables, since the selected configurationvariables can be thought as coordinates and mapped to their dual differential operator.In both cases, the resulting factorizing systems leads to a theory whose basic momentum variableshave no equivalent in the continuum classical theory. The trouble is then that fluxes associated tonon-degenerate surfaces (aka. ( � − -dimensional ones) cannot be represented on the correspondingquantum projective state space. If we however insist on using regular holonomies and fluxes asour primary variables (so as to implement an algebra of observables that separates the points in thecontinuum classical theory), then there is no way of choosing beforehand a canonical pairing thatwould, as described above, automatically provides a suitable set of canonically conjugate variablesfor any arbitrary graph γ . Proposition 3.16
For any γ ∈ L �� , we define the measure µ γ ≈ µ γ on C γ ≈ G γ (as in prop. 3.10)and the Hilbert space H γ := L ( C γ , �µ γ ). Next, for any γ � γ � ∈ L �� , we define the map τ γ � ←γ by: τ γ � ←γ : H γ → H γ � ψ �→ ψ ◦ π γ � →γ . τ γ � ←γ is an isometry and, for any γ � γ � � γ �� ∈ L �� : τ γ �� ←γ � ◦ τ γ � ←γ = τ γ �� ←γ . (3.16. )We define the Ashtekar-Lewandowski Hilbert space H �� as the (norm completion of ) the inductivelimit of the system �� H γ � γ∈ L �� , � τ γ � ←γ � γ � γ � � . For γ ∈ L �� , we will denote by τ �� ←γ the naturalisometric injection of H γ in H �� . Proof
Let γ � γ � ∈ L �� . We define: γ � − γ := γ � \ {� γ � →γ,� (1) | � ∈ γ} , s well as the finite dimensional smooth manifold C γ � −γ := �� � : ( γ � − γ ) → G � . The smooth map φ γ � −γ given by: φ γ � −γ : C γ � → C γ � −γ × C γ � � �→ � � | γ � −γ , π γ � →γ ( � � ) ,is a diffeomorphism (as can be checked by expressing its inverse like in the proof of prop. 3.6).Moreover, defining for any � ∈ C γ the maps T ( γ ) γ � ,� : C γ � → C γ � and R γ,� : C γ → C γ by: ∀� � ∈ C γ � , ∀� � ∈ γ � , � T ( γ ) γ � ,� ( � � )� ( � � ) = � � ( � � ) � � ( � ) if ∃� ∈ γ / � � = � γ � →γ,� (1) & � γ � →γ,� = +1 � ( � ) − � � � ( � � ) if ∃� ∈ γ / � � = � γ � →γ,� (1) & � γ � →γ,� = − � � ( � � ) else ,and ∀� ∈ C γ , ∀� ∈ γ, [ R γ,� ( � )] ( � ) = � ( � ) � � ( � ) ,we have φ γ � −γ ◦ T ( γ ) γ � ,� = �id C γ�−γ × R γ,� � ◦ φ γ � −γ . Let ω γ , resp. ω γ � , be a right invariant volumeform on C γ , resp. C γ � , such that µ γ , resp. µ γ � , is the corresponding measure. Because the Haarmeasure on a compact group is left-invariant as well as right-invariant, there exists a smooth map ε : C γ → { +1 , − } such that, for any � ∈ C γ , T ( γ ) ,∗γ � ,� ω γ � = ε ( � ) ω γ � . Then, we can, like in the proofof prop. 3.10, construct a smooth volume form ω γ � −γ on C γ � −γ such that � φ − ,∗γ � −γ ω γ � � = ω γ � −γ × � εω γ �.Therefore, there exists a smooth measure µ γ � −γ on C γ � −γ such that φ γ � −γ,∗ µ γ � = µ γ � −γ × µ γ . And from µ γ � C γ � = 1 = µ γ � � C γ � �, we get µ γ � −γ � C γ � −γ � = 1.Thus, for any measurable function ζ : C γ → R + , we have, by Fubini’s theorem:� C γ �µ γ ( � ) ζ ( � ) = � C γ� �µ γ � ( � � ) ζ ◦ π γ � −γ ( � � ) ,so that τ γ � ←γ is well-defined as a map H γ → H γ � and is an isometry. Finally, eq. (3.16. ) followsfrom eq. (3.15. ). Note.
Denoting by γ �� the ��-graph γ � − γ ⊂ γ � , we have µ γ � −γ = µ γ �� . Indeed, for any measurablefunction ζ : C γ � −γ → R + :� C γ�−γ �µ γ � −γ ( � ) ζ ( � ) = � C γ�−γ × C γ �µ γ � −γ ( � ) �µ γ ( � ) ζ ( � ) (for µ γ ( C γ ) = 1)= � C γ� �µ γ � ( � � ) ζ � � � | γ � �= � C γ�� × C γ� �µ γ �� ( � �� ) �µ γ � ( � � ) ζ ( � �� )(with γ � := γ � \ γ �� ∈ L �� ; γ � = γ �� ∪ γ � implies � C γ � , �µ γ � � ≈ � C γ �� , �µ γ �� � × � C γ � , �µ γ � � )= � C γ�� �µ γ �� ( � �� ) ζ ( � �� ) (for µ γ � ( C γ � ) = 1). � inally, we also recall the definition of the holonomy and flux operators on H �� [3, 1, 31]. Indeed,if we want to investigate the relation between the Ashtekar-Lewandowski construction and the justdeveloped projective formalism, it is not enough to produce a map σ between the state spaces: weshould also check that σ is dual to the map α that transports the observables according to theirphysical interpretation. Actually, the second part of prop. 3.19 shows that the map σ is uniquely specified as soon as we require it to intertwine the implementation of holonomies and exponentiatedfluxes on both sides. Proposition 3.17
Let � ∈ L edges and define: L �� /� := {γ ∈ L �� | {�} � γ} .For any γ ∈ L �� /� and any δ ∈ C ∞ ( G, R ), we have on H γ a densely defined, essentially self-adjointoperator �h ( �,δ ) γ on H γ (with dense domain D γ ) given by: ∀ψ ∈ D γ , ∀� ∈ C γ , � �h ( �,δ ) γ ψ � ( � ) := δ � � γ→{�},� � k =1 � � ◦ � γ→{�},� ( k )� � γ→{�},� ( k ) � ψ ( � ) .Moreover, for any γ � � γ , we have γ � ∈ L �� /� and: ∀ψ ∈ D γ , τ γ � ←γ ( ψ ) ∈ D γ � & �h ( �,δ ) γ � ◦ τ γ � ←γ ( ψ ) = τ γ � ←γ ◦ �h ( �,δ ) γ ( ψ ) .Thus, the family � �h ( �,δ ) γ � γ∈ L �� /� provides a densely defined, essentially self-adjoint operator �h ( �,δ )�� on H �� . Proof
Let γ ∈ L �� /� and δ ∈ C ∞ ( G, R ). Taking D γ = C ∞ ( C γ , C ) ⊂ H γ (this matches thecompactly supported smooth functions used in prop. 3.11 since C γ is compact), �h ( �,δ ) γ is well-definedand essentially self-adjoint (actually, it is a bounded operator). Moreover, we have: ∀� ∈ C γ , δ � � γ→{�},� � k =1 � � ◦ � γ→{�},� ( k )� � γ→{�},� ( k ) � = δ �� π γ→{�} ( � )� ( � )� . (3.17. )Now let γ � � γ . By transitivity of � on L �� , γ � ∈ L �� /� . For any ψ ∈ D γ , τ γ � ←γ ( ψ ) ∈ D γ � (for π γ � →γ is smooth) and, using eq. (3.15. ): ∀� � ∈ C γ � , � �h ( �,δ ) γ � ◦ τ γ � ←γ ( ψ )� ( � � ) = δ �� π γ � →{�} ( � � )� ( � )� [ ψ ◦ π γ � →γ ] ( � � ) = � τ γ � ←γ ◦ �h ( �,δ ) γ ( ψ )� ( � � ) . L �� /� is a cofinal part of L �� (for L �� is directed), and this allows us to construct a symmetricoperator �h ( �,δ )�� on the vector subspace D �� ⊂ H �� , defined as the inductive limit of vector spaces� D γ � γ∈ L �� , � τ γ � ←γ | D γ → D γ� � γ � γ � (without any completion). D �� is dense in H �� and �h ( �,δ )�� is bounded,hence essentially self-adjoint. � Proposition 3.18
Let
F ∈ L faces (prop. 3.3) and define: L �� /F := � γ ∈ L �� �� ∀� ∈ γ, ∀� � = � � ∈ � ( � ) , � � [ �,� � ] ∈ F ⇒ � ∈ {� ( � ) , � ( � ) } �� . et γ ∈ L �� /F and define, for any � ∈ C γ and any � ∈ G , the map T ( F ) γ,� � ∈ C γ by: ∀� ∈ γ, T ( F ) γ,� � ( � ) := � ( � ) � � if � ( �, F ) = {� ( � ) }� − � � ( � ) if � ( �, F ) = {� ( � ) }� − � � ( � ) � � if � ( �, F ) = {� ( � ) , � ( � ) }� ( � ) if � ( �, F ) = ∅ ,where � ( �, F ) := � � ∈ � ( � ) �� ∃� � � = � � � [ �,� � ] ∈ F � ⊂ {� ( � ) , � ( � ) } . Then, for any � ∈ g , we have adensely defined, essentially self-adjoint operator � P ( F,� ) γ on H γ (with dense domain D γ ) given by: ∀ψ ∈ D γ , ∀� ∈ C γ , � � P ( F,� ) γ ψ � ( � ) := � � T � �→ ψ � T ( F ) γ,� � �� ( � ) ,and, for any � ∈ G , we have a unitary operator �T ( F,� ) γ on H γ given by: ∀ψ ∈ H γ , ∀� ∈ C γ , � �T ( F,� ) γ ψ � ( � ) := ψ � T ( F ) γ,� � � .Moreover, L �� /F is a cofinal part of L �� and for any γ � γ � ∈ L �� /F : ∀ψ ∈ D γ , τ γ � ←γ ( ψ ) ∈ D γ � & � P ( F,� ) γ � ◦ τ γ � ←γ ( ψ ) = τ γ � ←γ ◦ � P ( F,� ) γ ( ψ ) ,and ∀ψ ∈ H γ , �T ( F,� ) γ � ◦ τ γ � ←γ ( ψ ) = τ γ � ←γ ◦ �T ( F,� ) γ ( ψ ) .Thus, the family � � P ( F,� ) γ � γ∈ L �� /F provides a densely defined, essentially self-adjoint operator � P ( F,� )�� on H �� , while the family � �T ( F,� ) γ � γ∈ L �� /F provides a unitary operator �T ( F,� )�� . Proof
Let γ ∈ L �� /F and � ∈ g . Taking as dense domain D γ = C ∞ ( C γ , C ) ⊂ H γ , � P ( F,� ) γ iswell-defined, and, using the invariance of the measure µ γ under left- and right-translations, we cancheck that it is a symmetric operator with Ker� � P ( F,� ) γ † ± � � = { } (like in the proof of prop. 3.11),therefore it is an essentially self-adjoint operator. The left- and right-invariance of the measurealso ensures that, for any � ∈ G , T ( F ) γ,�,∗ µ γ = µ γ , so �T ( F,� ) γ is a well-defined unitary operator on H γ .Now, let γ ∈ L �� and let λ ∈ L profls such that there exists F ∈ F ( λ ) with F = F ⊥ ◦ F . Since L graphs is cofinal in L �� (prop. 3.13), there exists γ � ∈ L graphs with γ � � γ , and by subsection 2.2,there exists ( γ �� , λ �� ) ∈ L with γ �� � γ � and λ �� � λ . Next, let � �� ∈ γ �� and suppose that thereexists � � = � � ∈ � ( � �� ) such that � �� [ �,� � ] ∈ F . Then, using F = F ⊥ ◦ F together with λ �� � λ , thereexists � �� ∈ � � � �� [ �,� � ] � \ {�, � � } ⊂ � ( � �� ) \ {� ( � �� ) , � ( � �� ) } such that � �� [ �,� �� ] ∈ F �� for some F �� ∈ F ( λ �� ).But since � �� ∈ χ ( γ �� ,λ �� ) ( � �� ), this can only be the case if � = � ( � ) (props. 2.8. and 2.8. ). Therefore γ �� ∈ L �� /F . Thus, L �� /F is cofinal in L �� .Let � ∈ L (1)edges such that � ( �, F ) ⊂ {� ( � ) , � ( � ) } and let � , � � � , � � ∈ L (1)edges , � , � � � , � � ∈ {± } uch that � = � � � � ◦ � � � ◦ � � . Then, using: ∀ � � ∈ F , ∀� ∈ � (� � ) \ {� ( � ) } , � � [ � ( � ) ,� ] ∈ F ,we have: � � � � � � � ∈ � ( � � , F ) ⇔ � � = 1 & � ( � ) ∈ � ( �, F )� , � � � � � � � ∈ � ( � � , F ) ⇔ � � = � & � ( � ) ∈ � ( �, F )� .Thus, for any γ � γ � ∈ L �� /F , we can check that: ∀� � ∈ C γ � , ∀� ∈ G, π γ � →γ � T ( F ) γ � ,� � � � = T ( F ) γ,� � π γ � →γ ( � � )� . (3.18. )Hence, for any � ∈ G , the unitary operators �T ( F,� ) γ defined on each H γ for γ ∈ L �� /F can beassembled into a unitary operator �T ( F,� )�� on H �� , while, for any � ∈ g, a densely defined, symmetricoperator � P ( F,� )�� can be constructed on the dense subspace D �� ⊂ H �� , defined as the inductive limitof vector spaces � D γ � γ∈ L �� , � τ γ � ←γ | D γ → D γ� � γ � γ � .Finally, let ψ ∈ Ker� � P ( F,� )�� † ± � � and define, for γ ∈ L �� /F , ψ γ ∈ H γ such that τ �� ←γ � ψ γ � is theorthogonal projection of ψ on the closed vector subspace τ �� ←γ � H γ � . Then, ψ γ ∈ Ker� � P ( F,� ) γ † ±� � = { } , so: ψ ∈ � γ∈ L �� /F τ �� ←γ � H γ � ⊥ = � γ∈ L �� τ �� ←γ � H γ � ⊥ = { } .Hence, � P ( F,� )�� is essentially self-adjoint. � Proposition 3.19
Let
F ∈ L faces and � ∈ G . For any η ∈ L such that λ ( η ) ∈ L profls /F , we define aunitary operator �T ( F,� ) η on H η by: ∀ψ ∈ H η , ∀� ∈ C η , � �T ( F,� ) η ψ �( � ) := ψ � R ( F ) η,� � � ,where R ( F ) η,� : C η → C η was defined in prop. 3.11. Then, for any η, η � ∈ L such that λ ( η ) , λ ( η � ) ∈ L profls /F :�T ( F,� ) η ∼ �T ( F,� ) η � ,so this provides an element �T ( F,� ) ∈ A ⊗ ( L , H , Φ) [18, def. 2.3].Let η ∈ L and denote by I η the algebra of bounded operators on H η generated by:� �h ( �,δ ) η ��� δ ∈ C ∞ ( G, R ) & � ∈ γ ( η )� ∪ � � T ( F ⊥ ◦F,� ) η ���� � ∈ G & F ∈ F ( η )� . f ρ, ρ � are (self-adjoint) positive semi-definite, traceclass operators on H η such that: ∀A ∈ I η , Tr H η ρ A = Tr H η ρ � A ,then ρ = ρ � . Proof
Definition of � T ( F,� ) . For any η ∈ L such that λ ( η ) ∈ L profils /F , we have R ( F ) η,�,∗ µ η = µ η (for µ isinvariant under right-translations) hence �T ( F,� ) η is a well-defined and unitary operator on H η .Next, for any η such that λ ( η ) ∈ L profls /F and any η � � η , we have λ ( η � ) ∈ L profls /F . Moreover,using eqs. (3.11. ), (3.9. ) and (3.5. ), we can check that:�id C η�→η × R ( F ) η,� � ◦ φ η � →η = φ η � →η ◦ R ( F ) η � ,� .Thus, we get Φ η � →η ◦ �T ( F,� ) η � = �id H η�→η × �T ( F,� ) η � ◦ Φ η � →η . Therefore, the directedness of L ensuresthat, for any η, η � ∈ L such that λ ( η ) , λ ( η � ) ∈ L profls /F , �T ( F,� ) η ∼ �T ( F,� ) η � (with the equivalence relationdefined in [18, eq. (2.3. )]), so we can define an element �T ( F,� ) ∈ A ⊗ ( L , H , Φ) . Definition of � h (Δ) η and � T ( � ) η . Let η ∈ L and Δ : γ ( η ) → C ∞ ( G, R ). Then we can define an element�h (Δ) η ∈ I η by:�h (Δ) η := � �∈γ ( η ) � h ( �, Δ( � ) ) η .The right-hand side does not depend on the ordering of the product and we have: ∀ψ ∈ H η , ∀� ∈ C η , � �h (Δ) η ψ � ( � ) = � �∈γ ( η ) [Δ( � ) ◦ � ] ( � ) ψ ( � ) .Next, for any F ∈ F ( η ) and any � ∈ G , we have λ ( η ) ∈ L profls /F ⊥ ◦F and, with the help of prop. 2.8: ∀� ∈ C η , ∀� ∈ γ ( η ) , � R ( F ⊥ ◦F ) η,� � � ( � ) := � � ( � ) � � if χ η ( � ) = F� ( � ) else .Hence, for any � ∈ C η we can define a unitary operator �T ( � ) η ∈ I η by:�T ( � ) η := � F∈ F ( η ) � T ( F ⊥ ◦F,�◦χ − η ( F )) η ,and we have (the product ordering being again irrelevant): ∀ψ ∈ H η , ∀� ∈ C η , � �T ( � ) η ψ � ( � ) = ψ ( ��� ) ,where, for any �, � � ∈ C η , � � � � denotes their pointwise multiplication as maps γ ( η ) → G . Characterization of ρ ∈ S η by its evaluations over I η . Let ρ, ρ � be non-negative traceclass operators n H η and φ, φ � ∈ H η . Let � > � � := � , as well as: � := min � � � H η ρ , � � H η ρ � , > C -vector space generated by C ∞ ( G, R ) is dense in L ( G, �µ ) ( C ∞ ( G, R ) ⊂ L ( G, �µ ) for G iscompact), hence the C -vector space generated by {⊗ �∈γ ( η ) Δ( � ) | Δ : γ ( η ) → C ∞ ( G, R ) } is dense in� �∈γ ( η ) L ( G, �µ ) ≈ H η (using the isometric identification provided by Fubini’s theorem). Thus, thereexist a finite family of maps �Δ � : γ ( η ) → C ∞ ( G, R )� � � � L and a finite family of complex numbers( µ � ) � � � L such that:����� φ − L � � =1 µ � [ ⊗ γ ( η ) Δ � ]����� H η < � �φ � � H η ,where, for any Δ : γ ( η ) → C ∞ ( G, R ), the vector ⊗ γ ( η ) Δ ∈ H η is defined by: ∀� ∈ C η , [ ⊗ γ ( η ) Δ] ( � ) := � �∈γ ( η ) [Δ( � ) ◦ � ] ( � ) .Similarly, there exist �Δ �� � : γ ( η ) → C ∞ ( G, R )� � � � � L � and � µ �� � � � � � � L � (0 � L, L � < ∞ ) such that:����� φ � − L � � � � =1 µ �� � [ ⊗ γ ( η ) Δ �� � ]����� H η < � �φ� H η .Thus, we have:����� �φ � | ρ | φ� − L � � =1 L � � � � =1 µ �� � ∗ µ � �⊗ γ ( η ) Δ �� � | ρ | ⊗ γ ( η ) Δ � � ����� < � �φ� H η Tr H η ρ �φ� H η + �φ � � H η � Tr H η ρ �φ � � H η + � Tr H η ρ �1 + �φ� H η � �1 + �φ � � H η � � � � , (3.19. )and similarly for ρ � . We define: � := � � L� =1 � L � � � =1 � |µ � | |µ �� � | � �∈γ ( η ) � Δ � ( � ) � ∞ � Δ �� � ( � ) � ∞ � > δ ∈ C ∞ ( G, R ), �δ� ∞ := sup �∈G |δ ( � ) | ( < ∞ for G is compact). Note that, for anyΔ : γ ( η ) → C ∞ ( G, R ), we have the bound:��� �h (Δ) η ��� A η � � �∈γ ( η ) � Δ( � ) � ∞ ,where � · � A η denotes the operator norm on the algebra A η of bounded operators over H η [18,def. 2.3], as well as: �⊗ γ ( η ) Δ � H η � � �∈γ ( η ) � Δ( � ) � ∞ using the fact that µ η is normalized).Now, from the spectral theorem, together with the non-negative and traceclass conditions, thereexist an orthonormal family �� ψ k � � k � � K in H η (with 0 � � K � ∞ ) and a family of strictly positivereals ( � k ) � k � � K such that: ρ = � K � k =1 � k ��� � ψ k �� � ψ k ��� & � K � k =1 � k =: Tr H η ρ < ∞ .Let K < ∞ such that ���Tr H η ρ − � Kk =1 � k ��� < � k � K , there exists ψ k ∈C ∞ ( C η , C ) ⊂ H η such that:���� ψ k − ψ k ��� H η < min �1 , � H η ρ + 1 � .Thus, we have:����� L � � =1 L � � � � =1 µ �� � ∗ µ � �⊗ γ ( η ) Δ �� � | ρ | ⊗ γ ( η ) Δ � − K � k =1 L � � =1 L � � � � =1 � k µ �� � ∗ µ � �⊗ γ ( η ) Δ �� � | ψ k � �ψ k | ⊗ γ ( η ) Δ � ����� < L � � =1 L � � � � =1 |µ �� � | |µ � | �⊗ γ ( η ) Δ �� � � H η �⊗ γ ( η ) Δ � H η � � K � k =1 � k � H η ρ + 1 � � � � , (3.19. )and, for any � ∈ C η :����� L � � =1 L � � � � =1 µ �� � ∗ µ � Tr H η � ρ �h (Δ � ) η �T ( � ) η �h (Δ ��� ) η � − K � k =1 L � � =1 L � � � � =1 � k µ �� � ∗ µ � � ψ k ���� �h (Δ � ) η �T ( � ) η �h (Δ ��� ) η ���� ψ k ������ < L � � =1 L � � � � =1 |µ �� � | |µ � | �����h (Δ � ) η �T ( � ) η �h (Δ ��� ) η ���� A η � � K � k =1 � k � H η ρ + 1 � � � � . (3.19. )Similarly, there exist a finite family � ψ �k � � k � K � of functions in C ∞ ( C η , C ) (with 0 � K � < ∞ ) anda finite family � � �k � � k � K � of strictly positive reals such that the equivalents of eqs. (3.19. ) and(3.19. ) are fulfilled for ρ � . We define: � := � � K � k =1 L � � =1 L � � � � =1 � k |µ � | |µ �� � | �ψ k � ∞ �⊗ γ ( η ) Δ � � ∞ > ζ ∈ C ∞ ( C η , C ), �ζ� ∞ := sup �∈ C η |ζ ( � ) | . Similarly, we define � � using � ψ �k � � k � K � and� � �k � � k � K � .Let � � � L � and k � K . The function ζ � � ,k , defined on C η × C η by: ∀� ∈ C η , ∀� ∈ C η , ζ � � ,k ( �, � ) := [ ⊗ γ ( η ) Δ �� � ] ( ��� ) ψ k ( ��� ) ,is smooth by construction. Hence, for any � ∈ C η , there exists an open, measurable neighborhood V ( � ) k,� � of in C η and an open, measurable neighborhood W ( � ) k,� � of � in C η such that: � � ∈ W ( � ) k,� � , ∀� ∈ V ( � ) k,� � , |ζ � � ,k ( � � , � ) − ζ � � ,k ( �, ) | < � ∀� � ∈ W ( � ) k,� � , ∀� ∈ V ( � ) k,� � , |ζ � � ,k ( � � , � ) − ζ � � ,k ( � � , ) | < � .Now, since C η is compact, there exists a finite subset H k,� � of C η such that C η ⊂ � �∈H k,�� W ( � ) k,� � , so,defining V k,� � := � �∈H k,�� V ( � ) k,� � , V k,� � is an open, measurable neighborhood of in C η and we have: ∀� � ∈ C η , ∀� ∈ V k,� � , |ζ � � ,k ( � � , � ) − ζ � � ,k ( � � , ) | < � .Next, defining V := � Kk =1 � L � � � =1 V k,� � , we get, for any measurable subset � V ⊂ V , any k � K and any � � � L � : ∀� ∈ C η , ������ � V �µ η ( � ) [ ⊗ γ ( η ) Δ �� � ] ( ��� ) ψ k ( ��� )� − � µ ( � V ) [ ⊗ γ ( η ) Δ �� � ] ( � ) ψ k ( � )����� < µ ( � V ) � . (3.19. )Similarly, we have an open, measurable neighborhood V � of in C η such that, for any measurablesubset � V ⊂ V � , any k � � K � and any � � � L � , the equivalent of eq. (3.19. ) is fulfilled for ψ �k � (withrespect to � � instead of � ). We define V � := V ∩ V � .For any � ∈ C η , � � V � is an open, measurable neighborhood of � in C η , hence, from the compacityof C η , there exist � , � � � , � M in C η (1 � M < ∞ ) such that C η ⊂ � M� =1 � � � V � . For � � M wedefine: V � := ( � � � V � ) ∩ � C η \ �− � � =1 � � � V � � .Thus, ( V � ) � � � M is a partition of C η into finitely many measurable parts. Moreover, we have, forany � � M : � − � � V � ⊂ V � ,so that, for any k � K and any � � � L � , eq. (3.19. ) yields: ∀� ∈ C η , ����� �⊗ γ ( η ) Δ �� � | ψ k � − M � � =1 µ ( V � ) ��T ( � � ) η �h (Δ ��� ) η ψ k �( � )����� == ������� C η �µ η ( � ) [ ⊗ γ ( η ) Δ �� � ] ( ��� ) ψ k ( ��� )� − M � � =1 � µ ( V � ) [ ⊗ γ ( η ) Δ �� � ] ( ��� � ) ψ k ( ��� � )������ � M � � =1 ������ � − � � V � �µ η ( � ) [ ⊗ γ ( η ) Δ �� � ] ( ��� � � � ) ψ k ( ��� � � � )� − � µ ( � − � � V � ) [ ⊗ γ ( η ) Δ �� � ] ( ��� � ) ψ k ( ��� � )����� < � .Therefore, for any k � K , any � � L and any � � � L � , we get:����� �ψ k | ⊗ γ ( η ) Δ � � �⊗ γ ( η ) Δ �� � | ψ k � − M � � =1 µ ( V � ) � ψ k ����h (Δ � ) η �T ( � � ) η �h (Δ ��� ) η ��� ψ k ������ ������ C η �µ ( � ) ψ ∗k ( � ) [ ⊗ γ ( η ) Δ � ] ( � ) � �⊗ γ ( η ) Δ �� � | ψ k � − M � � =1 µ ( V � ) ��T ( � � ) η �h (Δ ��� ) η ψ k �( � )������ < � �ψ k � ∞ �⊗ γ ( η ) Δ � � ∞ .Now, using eqs. (3.19. ) to (3.19. ) together with the definition of � , this implies:����� �φ � | ρ | φ� − L � � =1 L � � � � =1 M � � =1 µ �� � ∗ µ � µ ( V � ) Tr H η � ρ �h (Δ � ) η �T ( � � ) η �h (Δ ��� ) η ������ < � � ,and the same holds for ρ � .Finally, if ρ, ρ � are such that: ∀A ∈ I η , Tr H η ρ A = Tr H η ρ � A ,we thus have: ∀φ, φ � ∈ H η , ∀� > , |�φ � | ρ | φ� − �φ � | ρ � | φ�| < � ,and therefore ρ = ρ � . � We can now formulate the relation between the inductive construction just reviewed and theprojective construction from subsection 3.1, by displaying how an arbitrary state over H �� can beunambiguously identified with a projective family of density matrices over the Hilbert spaces H η .Note that, as stressed above, we are not merely stating that there exists some injective map σ between the state spaces (which would only be an assertion about the respective cardinalities, withvery little physical content): we also make sure that the mapping of the states considered hereintertwines the evaluation of observables in agreement with their physical interpretation.By using [18, prop. 3.5] (which is itself a straightforward application of the more general result in[18, theorem 2.9]), we first obtain a map from the state space of an inductive limit of Hilbert spacesbuilt over the label set L . As previously announced, the insensitivity of the injections with respectto the faces in each label then allows to collapse this inductive system into a simpler one, builtover a subset of L graphs (namely those analytic graphs that are the underlying graph of some label).Finally, since this set of graphs is cofinal in L �� (as follows from L being cofinal in L graphs × L profls and L graphs in L �� ), the corresponding inductive limit can be identified with H �� . Theorem 3.20
There exist maps σ : S �� → S ⊗ ( L , H , Φ) and α : A ⊗ ( L , H , Φ) → A �� (where S �� is the spaceof self-adjoint, positive semi-definite, traceclass operators over H �� , A �� is the space of boundedoperators on H �� , and S ⊗ ( L , H , Φ) and A ⊗ ( L , H , Φ) were defined respectively in [18, def. 2.2 and prop. 2.4] )such that: α is a C ∗ -algebra morphism; for any � ∈ L edges and any δ ∈ C ∞ ( G, R ), α � �h ( �,δ ) � = �h ( �,δ )�� , while for any F ∈ L faces and any � ∈ G , α � �T ( F,� ) � = �T ( F,� )�� ; for any ρ ∈ S �� and any A ∈ A ⊗ ( L , H , Φ) , Tr H �� ( ρ α ( A )) = Tr ( σ ( ρ ) A ) ; . σ is an injective map; σ � S �� � = �� ρ η � η∈ L ����� sup η∈ L inf η � � η Tr H η� ρ η � Θ η � |η = 1� , where S �� is the space of density matricesover H �� and the bounded operator Θ η � |η is defined on H η � by: ∀ψ ∈ H η � , ∀� ∈ C η � →η , ∀� ∈ C η , �Θ η � |η ψ � ◦ φ − η � →η ( �, � ) := � C η�→η �µ η � →η (� � ) ψ ◦ φ − η � →η (� �, � ) . Proof
Auxiliary inductive limit of Hilbert spaces.
For any η � η � ∈ L , we define the map τ η � ←η : H η → H η � by: ∀ψ ∈ H η , ∀� η � ∈ C η � , � τ η � ←η ( ψ )�( � η � ) := ψ � � �→ � � η�→η,� � k =2 [ � η � ◦ � η � →η,� ( k )] � η�→η,� ( k ) � � [ � η � ◦ � η � →η,� (1)]� .As was shown in [18, prop. 3.5], τ η � ←η is well-defined, and � L , � H η � η∈ L , � τ η � ←η � η � η � � is an inductivesystem of Hilbert spaces whose limit we denote by H ��� . For any η ∈ L , we call τ ��� ←η thenatural injection of H η into H ��� . Also by [18, prop. 3.5], there exist maps � σ : S ��� → S ⊗ ( L , H , Φ) and� α : A ⊗ ( L , H , Φ) → A ��� (where S ��� , resp. A ��� , denote the space of non-negative traceclass operators,resp. of bounded operators, over H ��� ) satisfying: ∀ρ ∈ S ��� , ∀A ∈ A ⊗ ( L , H , Φ) , Tr H ��� � ρ � α ( A )� = Tr�� σ ( ρ ) A � .Moreover, � σ is injective and:� σ � S ��� � = �� ρ η � η∈ L ����� sup η∈ L inf η � � η � C η�→η × C η�→η � (2) µ η � →η ( �, � � ) � C η �µ η ( � ) ρ η � � φ − η � →η ( �, � ) , φ − η � →η ( � � , � )� = 1� ,where S ��� is the space of density matrices over H ��� and, for any density matrix ρ η over H η , ρ η ( · , · )denotes the integral kernel of ρ η .To further specify � α , we now fetch its explicit definition from the proof of [18, theorem 2.9].First, we can define, for any η ∈ L , an Hilbert space H ��� →η , and an Hilbert space isomorphismΦ ��� →η : H ��� → H ��� →η ⊗ H η . H ��� →η is given as an inductive limit and we have, for any κ � η ∈ L ,a natural isometric injection τ ��� ←κ→η : H κ→η → H ��� →η satisfying:( τ ��� ←κ→η ⊗ id H η ) ◦ Φ κ→η = Φ ��� →η ◦ τ ��� ←κ .Then, � α is the C ∗ -algebra morphism A ⊗ ( L , H , Φ) → A ��� such that, for any η ∈ H η and any boundedoperator A η on H η :� α �[ A η ] ∼ � = Φ − →η ◦ �id H ��� →η ⊗ A η � ◦ Φ ��� →η ,where [ A η ] ∼ denotes the equivalence class of A η in A ⊗ ( L , H , Φ) [18, eq. (2.3. )]. In particular, for any κ � η , we have:� α �[ A η ] ∼ � ◦ τ ��� ←κ = τ ��� ←κ ◦ Φ − κ→η ◦ �id H κ→η ⊗ A η � ◦ Φ κ→η . dentification of H ��� with H �� . For any η ∈ L , we define Γ η→γ ( η ) : H η → H γ ( η ) as the identity mapon H η = L ( C η , �µ η ) = L ( C γ ( η ) , �µ γ ( η ) ) = H γ ( η ) . Then, for any η � η � ∈ L , we have γ ( η ) � γ ( η � )and (from props. 3.2 and 3.14):Γ η � →γ ( η � ) ◦ τ η � ←η = τ γ ( η � ) ←γ ( η ) ◦ Γ η→γ ( η ) .Thus, there exists an isometric injection Γ ��� → �� : H ��� → H �� satisfying: ∀η ∈ L , Γ ��� → �� ◦ τ ��� ←η = τ �� ←γ ( η ) ◦ Γ η→γ ( η ) .Moreover, � γ ∈ L �� �� ∃η ∈ L � γ ( η ) = γ � is cofinal in L �� (from subsection 2.2 and prop. 3.13),hence:� η∈ L Γ ��� → �� ◦ τ ��� ←η � H η � = � γ∈ L �� τ �� ←γ � H γ � ,is dense in H �� , and therefore Γ ��� → �� is an Hilbert space isomorphism.Now, we define, for any ρ ∈ S �� : σ ( ρ ) := � σ �Γ − → �� ◦ ρ ◦ Γ ��� → �� � ,and for any A ∈ A ⊗ ( L , H , Φ) : α ( A ) := Γ ��� → �� ◦ � α ( A ) ◦ Γ − → �� .The points 3.20. , 3.20. and 3.20. follow from the corresponding properties of � σ and � α . Moreover,we have, for any η ∈ L and any bounded operator A η on H η : α �[ A η ] ∼ � ◦ τ �� ←γ ( η ) = τ �� ←γ ( η ) ◦ Γ η→γ ( η ) ◦ A η ◦ Γ − η→γ ( η ) . (3.20. )Let � ∈ L edges , δ ∈ C ∞ ( G, R ) and η ∈ L such that γ ( η ) ∈ L graphs /� . Then, γ ( η ) ∈ L �� /� and,using again props. 3.2 and 3.14, we can check that:Γ η→γ ( η ) ◦ �h ( �,δ ) η ◦ Γ − η→γ ( η ) = �h ( �,δ ) γ ( η ) ,hence: α � �h ( �,δ ) � ◦ τ �� ←γ ( η ) = τ �� ←γ ( η ) ◦ �h ( �,δ ) γ ( η ) = �h ( �,δ )�� ◦ τ �� ←γ ( η ) .Now, α is a C ∗ -algebra isomorphism, so in particular an isometry, and � γ ( η ) �� η ∈ L � γ ( η ) ∈ L graphs /� �is cofinal in L �� so that:� η∈ L / γ ( η ) ∈ L graphs /� τ �� ←γ ( η ) � H γ ( η ) � = � γ∈ L �� τ �� ←γ � H γ � is dense in H �� . Therefore, α � �h ( �,δ ) � = �h ( �,δ )�� .Next, let F ∈ L faces , � ∈ G and η ∈ L such that λ ( η ) ∈ L profls /F . From eq. (3.3. ) and prop. 2.8,we get: ∀� ∈ γ ( η ) , ∀� � = � � ∈ � ( � ) , � � [ �,� � ] ∈ F ⇔ � � = � ( � ) & χ η ( � ) ∈ H λ ( η ) →F �� .Hence, γ ( η ) ∈ L �� /F and, using eq. (3.11. ) : � ∈ C η = C γ ( η ) , T ( F ) γ ( η ) ,� � = R ( F ) η,� � .Thus, we get:Γ η→γ ( η ) ◦ �T ( F,� ) η ◦ Γ − η→γ ( η ) = �T ( F,� ) γ ( η ) .Like above, this ensures that α � �T ( F,� ) � = �T ( F,� )�� , for � γ ( η ) �� η ∈ L � λ ( η ) ∈ L profls /F � is cofinal in L �� .Finally, for any η � η � ∈ L , we define ζ η � →η : C η � →η → C by: ∀� ∈ C η � →η , ζ η � →η ( � ) = 1 .Observe that �ζ η � →η � H η�→η = µ η � →η � C η � →η � = µ η � � C η � � / µ η � C η � = 1. Hence, the operator Θ η � |η iswell-defined and bounded on H η � , for it is given by:Θ η � |η = Φ − η � →η ◦ � |ζ η � →η � � ζ η � →η | ⊗ id H η � ◦ Φ η � →η . (3.20. )And, since for any non-negative traceclass operator ρ η � on H η � we have:� C η�→η × C η�→η � (2) µ η � →η ( �, � � ) � C η × C η � (2) µ η ( �, � � ) δ µ η ( �, � � ) ρ η � � φ − η � →η ( �, � ) , φ − η � →η ( � � , � � )� == � ψ∈ ��� η �Φ − η � →η ( ζ η � →η ⊗ ψ ) �� ρ η � �� Φ − η � →η ( ζ η � →η ⊗ ψ )� ,(with ��� η some orthonormal basis of H η ), we get: σ � S �� � = � σ � S ��� � = �� ρ η � η∈ L ����� sup η∈ L inf η � � η Tr H η� ρ η � Θ η � |η = 1� . � The injective map σ allows to identify the space S �� of states over H �� with a certain subset in thespace S ⊗ ( L , H , Φ) of all projective states. The results below suggests that S ⊗ ( L , H , Φ) can even be thoughtas a closure of S �� . Indeed, S ⊗ ( L , H , Φ) is complete, in the sense that a net of projective states admitsa limit as soon as it converges over each H η , and σ � S �� � is dense in the same sense, namely therestrictions of its elements over each H η fill a dense subset of the associated space S η of densitymatrices (in fact, they fill all S η ). This is somewhat reminiscent of the Fell’s theorem [8], but, whilethe Fell’s theorem tells us that S �� is dense in the space of all states [11, part III, def. 2.2.8] over the C ∗ -algebra A ⊗ ( L , H , Φ) [18, prop. 2.4], with respect to the weakest topology that makes the evaluationmaps ρ �→ Tr ρ A continuous, we show here that S �� is dense in the (presumably smaller) set of allprojective states with respect to a much stronger topology. Proposition 3.21
For any η ∈ L , we define on S ⊗ ( L , H , Φ) [18, def. 2.2] the semimetric (aka. possiblydegenerate metric, see [7, section IX.10]) � ( η ) by: ∀ � ρ η � � η � ∈ L , � ρ �η � � η � ∈ L ∈ S ⊗ ( L , H , Φ) , � ( η ) �� ρ η � � η � ∈ L , � ρ �η � � η � ∈ L � := �� ρ η − ρ �η �� ,where � · � denotes the trace norm on the space of traceclass operators over H η (see [18, lemma 2.10] nd/or [27]).The family of semimetrics � � ( η ) � η∈ L equips S ⊗ ( L , H , Φ) with the structure of a complete uniform space[7, sections IX.11 and XIV.9], and σ � S �� � is dense in the induced topology. Proof
For any η , the space S η of density matrices over H η equipped with the metric � ( η ) η definedby: ∀ρ η , ρ η � ∈ S η , � ( η ) η [ ρ η , ρ η � ] := �� ρ η − ρ �η �� ,is complete (for the traceclass operators over H η form, with respect to the trace norm, a Banachspace [27] in which S η is a closed subset). Hence, � η∈ L S η has a structure of complete and Hausdorffuniform space as a Cartesian product of complete metric spaces [7, theorem XIV.9.4].Let η � η � ∈ L . For any self-adjoint traceclass operator δ on H η � , we have, writing δ = δ (+) −δ ( − ) with δ (+) and ( −δ ( − ) ) respectively the positive and negative parts of δ : � Tr η � →η δ� � ��Tr η � →η δ (+) �� + ��Tr η � →η δ ( − ) �� = Tr H η Tr η � →η δ (+) + Tr H η Tr η � →η δ ( − ) = Tr H η� δ (+) + Tr H η� δ ( − ) = �δ� .Thus, Tr η � →η is a continuous map between the metric spaces � S η � , � ( η � ) η � � and � S η , � ( η ) η �, so its graphis closed in their Cartesian product. The projective limit S ⊗ ( L , H , Φ) is therefore a closed subset of� η∈ L S η , hence inherits a structure of complete and Hausdorff uniform space, which is preciselythe one induced by the family of semimetrics � � ( η ) � η∈ L .Let ρ = � ρ η � η∈ L ∈ S ⊗ ( L , H , Φ) . For any η ∈ L , we define ρ ( η ) ∈ S �� by: ρ ( η ) := τ �� ←γ ( η ) ◦ Γ η→γ ( η ) ◦ ρ η ◦ Γ − η→γ ( η ) ◦ τ +�� ←γ ( η ) .From theorem 3.20. and eq. (3.20. ), we have: ∀A η ∈ A η , Tr H η �� σ ( ρ ( η ) )� η A η � = Tr H �� � ρ ( η ) α �[ A η ] ∼ �� = Tr H η � ρ η A η � ,hence � σ � ρ ( η ) �� η = ρ η . Thus, the net � σ � ρ ( κ ) �� κ∈ L converges to ρ with respect to the family ofsemimetrics � � ( η ) � η∈ L (like in the proof of [17, prop. 2.7]). Therefore, σ � S �� � is dense in S ⊗ ( L , H , Φ) forthe corresponding topology. � Finally, we want to check that the projective quantum state space S ⊗ ( , H , Φ) is not a mere rewritingof the space of density matrices over H �� , ie. that σ , while being injective, is not surjective, andyields a strict embedding of S �� in S ⊗ ( , H , Φ) . To exhibit a projective state that cannot be realized as adensity matrix on H �� , we will, in line with the previous result, consider a sequence of states over H �� , which, although it does not converge in S �� , does converge in S ⊗ ( L , H , Φ) .To this intend, we cut some analytic edge � into infinitely many pieces (with an accumulationpoint at one extremity of the edge) and we denote by ψ ( � ) the state in H �� that assigns to the � � � � � ∞ � � � � � � ∞ � ∞ � � � � � � ∞ Figure 3.5 – A few labels having 3 as the smallest possible choice for � η (the dashed line indicatesthe base edge � )first � pieces a given, non-trivial, sample wavefunction χ . It can then be shown that the sequence � σ ��� ψ ( � ) � � ψ ( � ) ���� η (as a sequence in � , with η held fixed) is eventually constant: indeed theevaluations on ψ ( � ) of the observables covered by the label η freeze as soon as � exceeds a certain( η -dependent) threshold � η (fig. 3.5), and we have proved in prop. 3.19 that these evaluationscompletely specify a state over H η .Next, we need to confirm that the thus constructed projective state is not in the image of σ .Let us look closer at the characterization of this image given in theorem 3.20. . The orthogonalprojection Θ η � |η selects in H η � those states that do not depend on the complementary variables η � → η . Letting the upper label η � become infinitely fine corresponds, in H �� , to the orthogonalprojection on the fixed-graph subspace H γ ( η ) . Thus, if we now let the lower label η get finer andfiner, we will recover the state we started from, provided it was a state on H �� to begin with. Bycontrast, for a state that is not realizable on H �� , chopping those parts of the state that depart fromthe Ashtekar-Lewandowski vacuum over the η � → η degrees of freedom and taking first the netlimit on η � , we loose a significant part of the state, no matter how fine η : such states are not justexcitations around the �� -vacuum, but differ from it all the way down to infinitely fine labels.Now, for any label η and any integer M , there exists a set of labels η � � η , which is cofinalin L and such that, for each η � in this set, the holonomies along M distinct pieces of � can bediscovered among the complementary variables η � → η . Since the state we are considering hereattributes to each such piece a distribution χ distinct from the uniform one, its agreement withthe �� -vacuum, as far as the η � → η degrees of freedom are concerned, can thus be bounded by anexponentially decreasing function of M . Taking first the limit on finer and finer η � , we can then let M goes to infinity, so that the overlap, over the degrees of freedom beyond η , between this stateand the �� -vacuum is actually zero. If it would be a state in S �� its projection on any finite-graphsubspace H γ ( η ) should therefore vanish, in contradiction with the state having unit trace. roposition 3.22 The map σ is not surjective. Proof
A family of states in H �� . Let � ∈ L edges and let ˘ � : U → V be a representative of � . Wedefine � � = � ( � ), � ∞ = � ( � ) and, for any � � � � = ˘ � � �� + 1 ,
0� .For any � �
1, we define: γ ( � ) := � � [ � k− , � k ] �� 1 � k � � � ∈ L �� .Next, we choose χ ∈ L ( G, �µ ) such that �χ� = 1 and � G �µ ( � ) |χ ( � ) | < χ := 1 /√� V where V is the indicator function of a measurable region of G with0 < � := µ ( V ) < � �
1, we define ψ ( � ) γ ( � ) ∈ H γ ( � ) by: ∀� ∈ C γ ( � ) , ψ ( � ) γ ( � ) ( � ) := � � k =1 [ χ ◦ � ] � � [ � k− , � k ] � ,and ψ ( � ) := τ �� ←γ ( � ) � ψ ( � ) γ ( � ) � ∈ H �� . We have �� ψ ( � ) �� H �� = ��� ψ ( � ) γ ( � ) ��� H γ ( � ) = 1. Evaluation of the η-observables is ( � ) -eventually-constant. Let η ∈ L . From prop. 2.11. , thereexist � ∈ � ( � ) \ {� ( � ) } , � = ± F ∈ F ( η ) ∪ � F � ( η )� such that � � [ �,� ( � )] ∈ F . Let � � ∈� � � [ �,� ( � )] � \ {�, � ( � ) } . Then, � � < ( � ) � ( � ) and, from prop. 2.8: � [ � ( � ) ,� � ] ∈ F � ( η ) or � [ � ( � ) ,� � ] ∈ F ∈ F ( η ) .Moreover, for any � � ∈ γ ( η ), applying lemma 2.6 to � − and � � yields: ∃� ∈ � ( � ) \ {� ( � ) } � � � � � [ � ( � ) ,� ] � ⊂ � ( � � ) or � � � [ � ( � ) ,� ] � ∩ � ( � � ) ⊂ {� ( � ) } � .Thus, there exists � � < ( � ) � ( � ) such that: ∀� � ∈ γ ( η ) , � � � � [ � ( � ) ,� � ] � ⊂ � ( � � ) or � � � [ � ( � ) ,� � ] � ∩ � ( � � ) ⊂ {� ( � ) } � .Therefore, there exists � η � � � � [ � ( � ) ,� �η ] � ∩ � � � ∈γ ( η ) � ( � � ) ⊂ {� ( � ) } or ∃ � � ∈ γ ( η ) � � � � [ � ( � ) ,� �η ] � ⊂ � ( � � ) ,and: � [ � ( � ) ,� �η ] ∈ F � ( η ) or ∃ F ∈ F ( η ) � � [ � ( � ) ,� �η ] ∈ F .Let � > � � � η . We have: � � � [ � ∞ ,� � ] � ∩ � � � ∈γ ( η ) � ( � � ) ⊂ {� ∞ } or ∃ � � ∈ γ ( η ) � � � � [ � ∞ ,� � ] � ⊂ � ( � � ) , (3.22. )and: � [ � ∞ ,� � ] ∈ F � ( η ) or ∃ F ∈ F ( η ) � � [ � ∞ ,� � ] ∈ F . (3.22. )Let γ := γ ( � ) ∪ � � [ � ∞ ,� � ] � ∈ L �� . For any � � ∈ γ ( η ), L �� /� � is cofinal in L �� and, for any F ∈ F ( η ), �� /F ⊥ ◦F is cofinal in L �� , hence there exists γ � ∈ L �� , with γ � γ � , such that: ∀� � ∈ γ ( η ) , γ � ∈ L �� /� � & ∀F ∈ F ( η ) , γ � ∈ L �� /F ⊥ ◦F .Next, let � γ := γ ( � ) ∪ � � [ � ∞ ,� � ] � ∈ L �� . We have � γ � γ for: γ ( � ) ⊂ γ ( � ) & � [ � ∞ ,� � ] = � − � � ,� � +1 ] ◦ � � � ◦ � − � �− ,� � ] ◦ � [ � ∞ ,� � ] .Thus, we can define φ γ � − ( � ) − ( � ) := � γ � − ( � ) − ( � ) ◦ �id C γ�−γ × φ γ− � γ � ◦ φ γ � −γ , where φ γ −γ : C γ → C γ −γ × C γ has been defined for any γ � γ ∈ L �� in the proof of prop. 3.16 and � γ � − ( � ) − ( � ) is givenby: � γ � − ( � ) − ( � ) : C γ � −γ × C γ− � γ × C � γ → C γ � − ( � ) − ( � ) × C ( � ) − ( � ) � � , �, � �→ ( � � , � ) , � ,with C γ � − ( � ) − ( � ) := C γ � −γ × C � γ and C ( � ) − ( � ) := C γ− � γ . Using the definition of φ γ −γ for γ � γ togetherwith eq. (3.15. ), we get: ∀� � ∈ C γ � , φ γ � − ( � ) − ( � ) ( � � ) = � � � | γ � −γ , π γ � → � γ ( � � )� , π γ � →γ ( � � ) | γ ( � ) − ( � ) , (3.22. )where γ ( � ) − ( � ) := γ − � γ = � � [ � k− , � k ] �� � + 1 � k � � � .We define H γ � − ( � ) − ( � ) := L � C γ � −γ , �µ γ � −γ � ⊗ L � C � γ , �µ � γ � and H ( � ) − ( � ) := L � C γ− � γ , �µ γ− � γ � , sothat φ γ � − ( � ) − ( � ) provides a unitary map Φ γ � − ( � ) − ( � ) :Φ γ � − ( � ) − ( � ) : H γ � → H γ � − ( � ) − ( � ) ⊗ H ( � ) − ( � ) ψ �→ ψ ◦ φ − γ � − ( � ) − ( � ) .Since γ ( � ) , γ ( � ) � γ � , we have: ψ ( � ) = τ �� ←γ � ◦ τ γ � ←γ ( � ) � ψ ( � ) γ ( � ) � & ψ ( � ) = τ �� ←γ � ◦ τ γ � ←γ ( � ) � ψ ( � ) γ ( � ) � ,and eq. (3.22. ) yields, for any ( � � , � ) ∈ C γ � − ( � ) − ( � ) and any � ∈ C ( � ) − ( � ) :Φ γ � − ( � ) − ( � ) ◦ τ γ � ←γ ( � ) � ψ ( � ) γ ( � ) � (( � � , � ) , � ) = ψ ( � ) γ ( � ) ◦ π γ � →γ ( � ) ◦ φ − γ � − ( � ) − ( � ) (( � � , � ) , � )= ψ ( � ) γ ( � ) ◦ π � γ→γ ( � ) ( � ) = ψ ( � ) γ ( � ) � �| γ ( � ) �= � � k =1 [ χ ◦ � ] � � [ � k− , � k ] � ,as well as:Φ γ � − ( � ) − ( � ) ◦ τ γ � ←γ ( � ) � ψ ( � ) γ ( � ) � (( � � , � ) , � ) = � � k =1 � χ ◦ � π γ � →γ ( � ) ◦ φ − γ � − ( � ) − ( � ) (( � � , � ) , � )�� � � [ � k− , � k ] �= � � k =1 [ χ ◦ � ] � � [ � k− , � k ] � � � k = � +1 [ χ ◦ � ] � � [ � k− , � k ] � .Thus there exist ψ γ � − ( � ) − ( � ) ∈ H γ � − ( � ) − ( � ) and ζ ( � ) − ( � ) , χ ( � ) − ( � ) ∈ H ( � ) − ( � ) such that: ψ ( � ) = τ �� ←γ � ◦ Φ − γ � − ( � ) − ( � ) � ψ γ � − ( � ) − ( � ) ⊗ ζ ( � ) − ( � ) � , ψ ( � ) = τ �� ←γ � ◦ Φ − γ � − ( � ) − ( � ) � ψ γ � − ( � ) − ( � ) ⊗ χ ( � ) − ( � ) � . (3.22. )In addition, �ζ ( � ) − ( � ) � H ( � ) − ( � ) = �χ ( � ) − ( � ) � H ( � ) − ( � ) , for �� ψ ( � ) �� H �� = 1 = �� ψ ( � ) �� H �� .Now, let γ �� := � � �� ∈ γ � �� � ( � �� ) ⊂ � � � [ � � , � � ] �� ∪ � � [ � ∞ , � � ] � ∈ L �� . We have � γ � γ �� � γ � , hence: ∀� � ∈ C γ � , φ γ �� − � γ ◦ π γ � →γ �� ( � � ) = π γ � → ( γ �� − � γ ) ( � � ) , π γ � → � γ ( � � ) ,and, since ( γ �� − � γ ) ⊂ ( γ � − γ ), we get, for any ( � � , � ) ∈ C γ � − ( � ) − ( � ) and any � ∈ C ( � ) − ( � ) : π γ � →γ �� ◦ φ − γ � − ( � ) − ( � ) (( � � , � ) , � ) = φ − γ �� − � γ � � � | γ �� − � γ , � � . (3.22. )Next, for any � � ∈ γ ( η ), we define: γ � ( � � ) = {� �� ∈ γ � | � ( � �� ) ⊂ � ( � � ) & � ( � �� ) �⊂ � ( � ) } ∪ {� �� ∈ γ �� | � ( � �� ) ⊂ � ( � � ) } ∈ L �� . γ � ( � � ) � γ � and, from eq. (3.22. ), γ � ( � � ) ∈ L �� /� � , so π γ � →{� � } = π γ � ( �� ) →{� � } ◦ π γ � →γ � ( �� ) . Moreover, since γ � ( � � ) ⊂ ( γ � − γ ) ∪ γ �� , we have, for any ( � � , � ) ∈ C γ � − ( � ) − ( � ) and any � ∈ C ( � ) − ( � ) : ∀� �� ∈ γ � ( � � ) , � π γ � →γ � ( �� ) ◦ φ − γ � − ( � ) − ( � ) (( � � , � ) , � )� ( � �� ) == � π γ � → ( γ � −γ ) ◦ φ − γ � − ( � ) − ( � ) (( � � , � ) , � )� ( � �� ) = � � ( � �� ) if � �� ∈ ( γ � − γ )� π γ � →γ �� ◦ φ − γ � − ( � ) − ( � ) (( � � , � ) , � )� ( � �� ) = � φ − γ �� − � γ � � � | γ �� − � γ , � �� ( � �� ) if � �� ∈ γ �� .Thus, π γ � →γ � ( �� ) ◦φ − γ � − ( � ) − ( � ) (( � � , � ) , � ) does not depend on � . Therefore, using eq. (3.17. ), there exists, forany δ ∈ C ∞ ( G, R ), an operator A ∈ A γ � − ( � ) − ( � ) (with A γ � − ( � ) − ( � ) the algebra of bounded operatorson H γ � − ( � ) − ( � ) ) such that: � h ( � � ,δ ) γ � = Φ − γ � − ( � ) − ( � ) ◦ � A ⊗ id H ( � ) − ( � ) � ◦ Φ γ � − ( � ) − ( � ) .Let F ∈ F ( η ). Since � γ � γ �� � γ � , eq. (3.22. ) implies: ∀� � ∈ C γ � , φ γ � − ( � ) − ( � ) ( � � ) = � π γ � → ( γ � −γ ) ( � � ) , π γ �� → � γ ◦ π γ � →γ �� ( � � )� , π γ � →γ ( � ) − ( � ) ( � � ) .But since γ � ∈ L �� /F ⊥ ◦F and � [ � ∞ ,� � ] ∈ F � with F � ∈ F ( η ) ∪ � F � ( η )�, we have ( γ � − γ ) , γ �� & γ ( � ) − ( � ) ∈ L �� /F ⊥ ◦F and, moreover: ∀� � ∈ γ ( � ) − ( � ) , � ( � � , F ⊥ ◦ F ) = ∅ ,so that ∀� ∈ G, ∀� ∈ C ( � ) − ( � ) , T ( F ⊥ ◦F ) γ ( � ) − ( � ) ,� � = � . Thus, using eq. (3.18. ), we get, for any � ∈ G andany � � ∈ C γ � : φ γ � − ( � ) − ( � ) � T ( F ⊥ ◦F ) γ � ,� � � � = � T ( F ⊥ ◦F )( γ � −γ ) ,� ◦ π γ � → ( γ � −γ ) ( � � ) , π γ �� → � γ ◦ T ( F ⊥ ◦F ) γ �� ,� ◦ π γ � →γ �� ( � � )� , π γ � →γ ( � ) − ( � ) ( � � ) .Hence, eq. (3.22. ) yields, for any ( � � , � ) ∈ C γ � − ( � ) − ( � ) and any � ∈ C ( � ) − ( � ) : T ( F ⊥ ◦F ) γ � ,� � φ − γ � − ( � ) − ( � ) �( � � , � ) , � �� = φ − γ � − ( � ) − ( � ) � T ( F ⊥ ◦F )( γ � − ( � ) − ( � )) ,� ( � � , � ) , � � ,where T ( F ⊥ ◦F )( γ � − ( � ) − ( � )) ,� ( � � , � ) := � T ( F ⊥ ◦F )( γ � −γ ) ,� � � , π γ �� → � γ ◦ T ( F ⊥ ◦F ) γ �� ,� ◦ φ − γ �� → � γ � � � | γ �� − � γ , � �� . herefore, there exists, for any � ∈ G , an operator A ∈ A γ � − ( � ) − ( � ) such that: � T ( F ⊥ ◦F,� ) γ � = Φ − γ � − ( � ) − ( � ) ◦ � A ⊗ id H ( � ) − ( � ) � ◦ Φ γ � − ( � ) − ( � ) .So we have proved that for any A �� in:� � h ( � � ,δ )�� �� δ ∈ C ∞ ( G, R ) & � � ∈ γ ( η ) � ∪ � � T ( F ⊥ ◦F,� )�� �� � ∈ G & F ∈ F ( η ) � ,there exists A ∈ A γ � − ( � ) − ( � ) such that: A �� ◦ τ �� ←γ � = τ �� ←γ � ◦ Φ − γ � − ( � ) − ( � ) ◦ � A ⊗ id H ( � ) − ( � ) � ◦ Φ γ � − ( � ) − ( � ) .Theorem 3.20. and 3.20. then implies that for any A η ∈ I η , there exists A ∈ A γ � − ( � ) − ( � ) such that: α �[ A η ] ∼ � ◦ τ �� ←γ � = τ �� ←γ � ◦ Φ − γ � − ( � ) − ( � ) ◦ � A ⊗ id H ( � ) − ( � ) � ◦ Φ γ � − ( � ) − ( � ) ,hence, using the expression for ψ ( � ) and ψ ( � ) from eq. (3.22. ) : ∀A η ∈ I η , � ψ ( � ) , α �[ A η ] ∼ � ψ ( � ) � H �� = � ψ ( � ) , α �[ A η ] ∼ � ψ ( � ) � H �� . Constructing a projective state from the ψ ( � ) . For each η ∈ L , we choose � η as above, and wedefine: ρ η := � σ � �� ψ ( � η ) � � ψ ( � η ) �� �� η .Let η � η � ∈ L and let � � � η , � η � . From the previous point, together with theorem 3.20. , wehave, for any A η ∈ I η :Tr H η ρ η A η = � ψ ( � η ) �� α �[ A η ] ∼ � ψ ( � η ) � H �� = � ψ ( � ) �� α �[ A η ] ∼ � ψ ( � ) � H �� = Tr H η � σ � �� ψ ( � ) � � ψ ( � ) �� �� η A η .Hence, the second part of prop. 3.19 implies: ρ η = � σ � �� ψ ( � ) � � ψ ( � ) �� �� η ,and similarly for ρ η � . But σ � | ψ ( � ) �� ψ ( � ) | � ∈ S ⊗ ( L , H , Φ) [18, def. 2.2], so we get:Tr η � →η ρ η � = Tr η � →η � σ � �� ψ ( � ) � � ψ ( � ) �� �� η � = � σ � �� ψ ( � ) � � ψ ( � ) �� �� η = ρ η ,and therefore ρ := � ρ η � η∈ L ∈ S ⊗ ( L , H , Φ) . ρ is not in the image of σ . Let η ∈ L and � = � η �
1. Let M � odd integer and � = � + M . By definition of � η we have, for any � such that 0 � � � M − � � +2 � +1 /∈ � � � ∈γ � � ( � � ) ,where γ � := γ ( η ) \ � � � ∈ γ ( η ) �� � ( � [ � ∞ ,� � ] ) ⊂ � ( � � )� . If there exists � � ∈ γ ( η ) such that � ( � [ � ∞ ,� � ] ) ⊂ � � � � � ∞ � � � � � � � � � ∞ � ∞ � � � � � � � � � ∞ � � � Figure 3.6 – Construction of the auxiliary label � η with M = 3 for the labels of fig. 3.5 (recalled inlight gray) � ( � � ) (note that there can be as much one such � � for γ ( η ) is a graph), we define γ � ∈ L graph by: γ � = � � � [ � ( � � ) ,� ] , � � [ �,� ( � � )] � if � ( � � ) < ( � � ) � < ( � � ) � < ( � � ) � ( � � )� � � [ � ( � � ) ,� ] � if � ( � � ) < ( � � ) � < ( � � ) � = � ( � � )� � � [ �,� ( � � )] � if � ( � � ) = � < ( � � ) � < ( � � ) � ( � � ) ∅ if � ( � � ) = � < ( � � ) � = � ( � � ) with {�, �} = {� ∞ , � � } ,otherwise we define γ � = ∅ . By construction, we have γ \γ � � γ � ∪ � � [ � ∞ ,� � ] � and, for any � such that0 � � � M − � � +2 � +1 /∈ � � �� ∈γ � � ( � �� ) . Since � � � ∈γ � ∪γ � � ( � � ) is compact, ˘ � − � V \ � � � ∈γ � ∪γ � � ( � � )� ⊂U is an open neighborhood of � � + 2 � + 1 � + 2 � + 2 ���� 0 � � � M − × { } �− in R � so there exists R > � in { , � � � , ( M − / } :� � + 2 � + 1 � + 2 � + 2 � × B ( �− R ⊂ U & ˘ � �� � + 2 � + 1 � + 2 � + 2 � × B ( �− R � ∩ � � � ∈γ � ∪γ � � ( � � ) = ∅ (where B ( �− R is the closed ball of radius R and center 0 in R �− ). Thus, this allows us to constructa label � η ∈ L such that (fig. 3.6): γ (� η ) = � γ := � � [ � � ,� �− ] , � [ � � ,� ∞ ] � ∪ � � � 1. Since Φ η �� → ( � ) − ( � ) , Φ η �� →η � and Φ η � →η are Hilbert space isomorphism,Φ η �� → ( � ) − ( � ) ◦ Φ − η �� →η � ◦ �id H η��→η� ⊗ Φ − η � →η � thus induces a unitary (injective) map from:Vect� ψ η �� →η � ⊗ ζ η � →η ⊗ ψ η ��� ψ η �� →η � ∈ H η �� →η � , ψ η ∈ H η �(where · denotes the completion) into H η �� → ( � ) − ( � ) ⊗ {ζ ( � ) − ( � ) } . Therefore, we get, using thecharacterization of Θ η � |η from the proof of theorem 3.20 (eq. (3.20. ) ) together with [18, def. 2.3]:Tr H η� ρ η � Θ η � |η == Tr H η� � σ ���� ψ ( � � ) � � ψ ( � � ) ����� η � Θ η � |η = Tr H η�� � σ ���� ψ ( � � ) � � ψ ( � � ) ����� η �� Φ − η �� →η � �id H η��→η� ⊗ Θ η � |η � Φ η �� →η � = Tr H η�� � σ ���� ψ ( � � ) � � ψ ( � � ) ����� η �� Φ − η �� →η � ◦ �id H η��→η� ⊗ Φ − η � →η ��id H η��→η� ⊗ |ζ η � →η � � ζ η � →η | ⊗ id H η � �id H η��→η� ⊗ Φ η � →η � ◦ Φ η �� →η � Tr H η�� � σ ���� ψ ( � � ) � � ψ ( � � ) ����� η �� Φ − η �� → ( � ) − ( � ) �id H η��→ ( � ) − ( � ) ⊗ |ζ ( � ) − ( � ) � � ζ ( � ) − ( � ) | � Φ η �� → ( � ) − ( � ) .This implies, using theorem 3.20. and eq. (3.20. ) :Tr H η� ρ η � Θ η � |η � �Φ η �� → ( � ) − ( � ) Γ − η �� →γ ( η �� ) τ γ ( η �� ) ←γ ( �� ) ψ ( � � ) γ ( �� ) ��� �id H η��→ ( � ) − ( � ) ⊗ |ζ ( � ) − ( � ) � � ζ ( � ) − ( � ) | � ������Φ η �� → ( � ) − ( � ) Γ − η �� →γ ( η �� ) τ γ ( η �� ) ←γ ( �� ) ψ ( � � ) γ ( �� ) � H η��→ ( � ) − ( � ) ⊗ H ( � ) − ( � ) . (3.22. )Let � �� ∈ C η �� and ��� � � (0) , � � � (2) , � � � (3) � , � � ; � � � := φ η �� → ( � ) − ( � ) ( � �� ) . Let � γ (0) := H (0) η �� → � η and � γ (2) := H (2) η �� → � η .Since � γ (0) , � γ (2) ⊂ γ ( η �� ) ∈ L graphs , we have � γ (0) , � γ (2) ∈ L graphs ⊂ L �� and � γ (0) , � γ (2) � γ ( η �� ). Moreover,the expression for φ η �� → � η (prop. 3.6) together with the uniqueness part of prop. 3.14 yields:� � � (0) = � �� | � γ (0) = π γ ( η �� ) → � γ (0) ( � �� ) , � � � (2) = � �� | � γ (2) = π γ ( η �� ) → � γ (2) ( � �� ) , & π − γ � →γ ( � ) − ( � ) (� � ) , � � = φ � γ− { � [ �∞,�� ] } ◦ π γ ( η �� ) → � γ ( � �� ) .Using eq. (3.15. ) and the expression for φ γ −γ from the proof of prop. 3.16, the last relation abovebecomes:� � = π γ ( η �� ) →γ ( � ) − ( � ) ( � �� ) & � � = π γ ( η �� ) → { � [ �∞,�� ] } ( � �� ) .Next, since � [ � � ,� � ] ∈ F � (� η ) and � � �∈ � γ � (� � ) = � ( � [ � � ,� ∞ ] ), we have, for any � � ∈ γ ( � ) and any � ∈ �1 , � � � , � γ ( η �� ) →γ ( � ) ,� � � : � γ ( η �� ) →γ ( � ) ,� � ( � ) ∈ F � (� η ) & ∀ � � ∈ � γ, � � � γ ( η �� ) →γ ( � ) ,� � ( � )� �⊂ � (� � ) .Hence � γ ( η �� ) →γ ( � ) ,� � ( � ) ∈ H (0) η �� → � η , so that γ ( � ) � � γ (0) . Similarly, if � � � � + 1, � [ � � ,� ∞ ] ∈ � γ implies� � [ � � +1 ,� ∞ ] � � � γ (2) (we have � � [ � � +1 ,� ∞ ] � � γ ( η �� ) for � � [ � � ,� � +1 ] � � γ ( � � ) � γ ( η �� ) and � � [ � � ,� ∞ ] � � � γ � γ ( η �� ) ), as well as γ ( � � ) − ( � +1) � � γ (2) (if � � happens to be bigger than � + 2). Then, using repeatedlyeq. (3.15. ), we get: ∀k ∈ { , � � � , �} , π γ ( η �� ) →γ ( �� ) ( � �� ) � � [ � k− ,� k ] � = π � γ (0) →γ ( � ) (� � � (0) ) � � [ � k− ,� k ] � , ∀k ∈ {� + 1 , � � � , �} , π γ ( η �� ) →γ ( �� ) ( � �� ) � � [ � k− ,� k ] � = � � � � [ � k− ,� k ] �(note that � � � � as underlined earlier), ∀k ∈ {� + 2 , � � � , � � } , π γ ( η �� ) →γ ( �� ) ( � �� ) � � [ � k− ,� k ] � = π � γ (2) →γ ( �� ) − ( � +1) (� � � (2) ) � � [ � k− ,� k ] �(of course, this only applies if � � � � + 2).If � � > � we also need the evaluation of π γ ( η �� ) →γ ( �� ) ( � �� ) on � [ � � ,� � +1 ] . For this, we notice that� � [ � ∞ ,� � ] � � γ [ �,� +1] � γ ( η �� ), where γ [ �,� +1] := � � [ � � ,� � ] , � [ � � ,� � +1 ] , � [ � � +1 ,� ∞ ] �. Using the explicitexpression for π γ [ �,� +1] → { � [ �∞,�� ] } together with � � [ � � ,� � ] � � γ ( � ) − ( � ) and � � [ � � +1 ,� ∞ ] � � � γ (2) (and againrepeatedly applying eq. (3.15. )), we get:� � ( � [ � ∞ ,� � ] ) = π γ ( η �� ) → { � [ �∞,�� ] } ( � �� )( � [ � ∞ ,� � ] )= π γ ( η �� ) →γ [ �,� +1] ( � �� )( � [ � � ,� � ] ) − � π γ ( η �� ) →γ [ �,� +1] ( � �� )( � [ � � ,� � +1 ] ) − � π γ ( η �� ) →γ [ �,� +1] ( � �� )( � [ � � +1 ,� ∞ ] ) − π γ ( � ) − ( � ) → { � [ ��,�� ] } (� � )( � [ � � ,� � ] ) − � π γ ( η �� ) →γ ( �� ) ( � �� )( � [ � � ,� � +1 ] ) − � π � γ (2) → { � [ �� +1 ,�∞ ] } (� � � (2) )( � [ � � +1 ,� ∞ ] ) − = � � � k = � +1 � � ( � [ � k− ,� k ] )� − � π γ ( η �� ) →γ ( �� ) ( � �� )( � [ � � ,� � +1 ] ) − � π � γ (2) → { � [ �� +1 ,�∞ ] } (� � � (2) )( � [ � � +1 ,� ∞ ] ) − ,so that: π γ ( η �� ) →γ ( �� ) ( � �� )( � [ � � ,� � +1 ] ) = � � � k = � +1 � � ( � [ � k− ,� k ] ) � � � ( � [ � ∞ ,� � ] ) � π � γ (2) → { � [ �� +1 ,�∞ ] } (� � � (2) )( � [ � � +1 ,� ∞ ] )� − .We want to use the thus obtained relations between π γ ( η �� ) →γ ( �� ) and φ η �� → ( � ) − ( � ) in order to refor-mulate eq. (3.22. ). We first consider the case � � = � . Then, we have:Φ η �� → ( � ) − ( � ) Γ − η �� →γ ( η �� ) τ γ ( η �� ) ←γ ( �� ) ψ ( � � ) γ ( �� ) = ψ η �� → ( � ) − ( � ) ⊗ χ ( � ) − ( � ) ,where: ∀ �� � � (0) , � � � (2) , � � � (3) � , � � ∈ C η �� → ( � ) − ( � ) ,ψ η �� → ( � ) − ( � ) �� � � (0) , � � � (2) , � � � (3) ; � � � := � � k =1 � χ ◦ π � γ (0) →γ ( � ) (� � � (0) )� ( � [ � k− ,� k ] ) & ∀ � � ∈ C ( � ) − ( � ) , χ ( � ) − ( � ) �� � � := � � k = � +1 � χ ◦ � � �( � [ � k− ,� k ] ) .Thus, eq. (3.22. ) becomes:Tr H η� ρ η � Θ η � |η � �ψ η �� → ( � ) − ( � ) � H η��→ ( � ) − ( � ) ��� �ζ ( � ) − ( � ) | χ ( � ) − ( � ) � H ( � ) − ( � ) ��� .And, since ��� ψ ( � � ) γ ( �� ) ��� H γ ( �� ) = 1, we get:Tr H η� ρ η � Θ η � |η � ��� �ζ ( � ) − ( � ) | χ ( � ) − ( � ) � H ( � ) − ( � ) ��� �χ ( � ) − ( � ) � H ( � ) − ( � ) = ��� G �µ ( � ) χ ( � )�� M �χ� M � �� G �µ ( � ) |χ ( � ) | � M .We now consider the case � � > � . Here, we get: ∀ �� � � , � � � ∈ C η �� → ( � ) − ( � ) , ∀ � � ∈ C ( � ) − ( � ) , Φ η �� → ( � ) − ( � ) Γ − η �� →γ ( η �� ) τ γ ( η �� ) ←γ ( �� ) ψ ( � � ) γ ( �� ) �� � � , � � ; � � � == ψ η �� → � η �� � � � χ ( � ) − ( � ) �� � � χ � β η �� → � η �� � � � � � � ( � [ � ∞ ,� � ] ) − � β ( � ) − ( � ) �� � �� ,where: ∀ �� � � (0) , � � � (2) , � � � (3) � ∈ C η �� → � η ,ψ η �� → � η �� � � (0) , � � � (2) , � � � (3) �:= � � � k =1 � χ ◦ π � γ (0) →γ ( � ) (� � � (0) )� ( � [ � k− ,� k ] )� � � � � k = � +2 � χ ◦ π � γ (2) →γ ( �� ) − ( � +1) (� � � (2) )� ( � [ � k− ,� k ] )� , η �� → � η �� � � (0) , � � � (2) , � � � (3) � := � π � γ (2) → { � [ �� +1 ,�∞ ] } (� � � (2) )( � [ � � +1 ,� ∞ ] )� − , & ∀ � � ∈ C ( � ) − ( � ) , β ( � ) − ( � ) (� � ) := � +1 � k = � � � ( � [ � k− ,� k ] ) − .Thus, eq. (3.22. ) now reads:Tr H η� ρ η � Θ η � |η � � C η��→ � η �µ η �� → � η (� � � ) ��� ψ η �� → � η �� � � ���� � C ( � ) − ( � ) × C ( � ) − ( � ) �µ (2)( � ) − ( � ) (� �, � � � ) χ ∗ ( � ) − ( � ) �� � � χ ( � ) − ( � ) �� � � �� G �µ ( � ) χ ∗ � β η �� → � η �� � � � � � − � β ( � ) − ( � ) �� � �� χ � β η �� → � η �� � � � � � − � β ( � ) − ( � ) �� � � �� ,while the normalization condition ��� ψ ( � � ) γ ( �� ) ��� H γ ( �� ) = 1 yields:1 = � C η��→ � η �µ η �� → � η (� � � ) ��� ψ η �� → � η �� � � ���� � C ( � ) − ( � ) �µ ( � ) − ( � ) (� � ) ��� χ ( � ) − ( � ) �� � ���� � G �µ ( � ) |χ ( � ) | (the measure µ being invariant under the transformation � �→ � � � − � � for any � , � ∈ G )= � C η��→ � η �µ η �� → � η (� � � ) ��� ψ η �� → � η �� � � ���� (since �χ� = 1).Moreover, for any � , � , � � ∈ G , the Cauchy-Schwarz inequality ensures that:����� G �µ ( � ) χ ∗ � � � � − � � � χ � � � � − � � � ����� � �χ� = 1 ,so we again get:Tr H η� ρ η � Θ η � |η � � C η��→ � η �µ η �� → � η (� � � ) ��� ψ η �� → � η �� � � ���� � C ( � ) − ( � ) × C ( � ) − ( � ) �µ (2)( � ) − ( � ) (� �, � � � ) ��� χ ∗ ( � ) − ( � ) �� � ���� ��� χ ( � ) − ( � ) �� � � ����= �� G �µ ( � ) |χ ( � ) | � M .Since χ has been chosen so that � G �µ ( � ) |χ ( � ) | < 1, there exists, for any � > 0, an odd integer M � G �µ ( � ) |χ ( � ) | � M < � . Thus, there exists, for any η ∈ L and any � > η � � η such that:Tr H η� ρ η � Θ η � |η < � .Hence, for any η ∈ L , inf η � � η Tr H η� ρ η � Θ η � |η = 0, and, therefore, sup η∈ L inf η � � η Tr H η� ρ η � Θ η � |η = 0. On theother hand theorem 3.20. implies: σ � S �� � = � ρ � = � ρ �η � η∈ L ����� sup η∈ L inf η � � η Tr H η� ρ �η � Θ η � |η = Tr ρ � � ,and we have Tr ρ = Tr H η ρ η = �� ψ ( � η ) �� H �� = 1 (for some η ∈ L ), so ρ /∈ σ � S �� �. � Outlook So we were able to show that the construction developed in [24] can be generalized from thelinear case to an arbitrary gauge group G . The key ingredient is still the same, namely the use oflabels defined as collections of edges and surfaces. However, the somewhat involved algebra builtby the holonomies and fluxes in the case of a non-Abelian group requires to be more restrictive asto which such collections qualify as labels. The factorization maps, which are the central objects ofthe formalism, can then be expressed explicitly in terms of the group operations, so that no furtherrestrictions on the Lie group G are needed.In the case of a compact gauge group, which will be the most relevant for us (since we areinterested in the application to LQG, with G = SU (2) ), we have a clear picture of how thisalternative state space relates to the well-established Ashtekar-Lewandowski one. Recall that ourmotivation for this construction was to extend the latter, to try and cure the difficulties arising inthe search for good semi-classical states. The observation of prop. 3.22, confirming that we indeedhave gained new states in the process, is in this respect particularly important.The state we used to prove this result can even be seen as a first step toward the design ofsatisfactory semi-classical states. Indeed, if we take as ‘pattern’ χ a coherent state (eg. a Hallstate [12], which is the generalization over a compact Lie group of a Gaussian state), we obtaina projective state yielding a narrow distribution for infinitely many holonomies (namely the onesalong the infinitely many pieces of the base edge � ), while such a state could not exist over H �� .Still, this would not yet be a state suitable for the study of the semi-classical limit, where wewould need states presenting narrow distributions for a full set of holonomies and fluxes. Thereremain in fact further obstructions to this endeavor, the understanding and overcoming of whichwill be the topic of forthcoming work.Another issue that will have to be addressed thoroughly before the projective formalism canprovide a serious alternative to the successful inductive one, is how to solve at least the Gaussand diffeomorphism constraints [1, 3, 31]. In [17, section 3] we proposed a strategy to deal withconstraints in the projective context, with the help of a suitably defined regularization scheme (thisproposal was developed at the classical level, ie. in the setting of a projective limit of symplecticmanifolds, however we also displayed on an example in [19, subsection 3.2] how a similar approachcould be implemented at the quantum level).Note that while the Gauss constraints are well-adapted to the inductive structure underlyingthe Ashtekar-Lewandowski Hilbert space (ie. they leave the fixed-graph subspaces H γ invariant,which allows for their straightforward resolution in H �� ) they are not adapted (in the sense of[17, subsection 3.1]) to the projective structure we have introduced: while gauge transformationspreserve the algebra of holonomies attached to a graph, they do not preserve the algebra of fluxesattached to a profile. In fact, fluxes do not at all transform nicely under gauge transformations. Apopular method to circumvent this difficulty is to use, instead of the standard fluxes, appropriately‘anchored’ ones [29, def. 3.5]: by choosing, for each face, a supporting system of paths, we canparallel transport the electric field at each point of the face back to a common root, thus forming anobservable with better transformations properties. Yet, a complete solution based on this device willrequire some more work, because again one needs to ensure, at the same time, that the labels we areusing can be properly associated to an algebra of observables (ie. that the observables assigned to agiven label form a subset closed under Poisson brackets), and that they build a directed preordered et (with a preorder that respects the relations between the associated algebras of observables).Similarly the techniques developed for the resolution of the diffeomorphism constraints in H �� [3] cannot be directly imported into the projective formalism because they critically rely on havingstates made up of discrete excitations. To make progress on this issue, we will have to understandhow the input needed for a resolution along the lines of [17, section 3] can be set up in a backgroundindependent manner. In a more speculative line, it might also be possible to combine the knownregularization scheme for the Hamiltonian constraint [28] with the strategy from [17, section 3] inorder to arrive at a constructive description of a space of states solving the quantum dynamics ofgravity. Acknowledgements This work has been financially supported by the Université François Rabelais, Tours, France,and by the Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany (via the Bavarian EqualOpportunities Sponsorship – Förderung von Frauen in Forschung und Lehre (FFL) – PromotingEqual Opportunities for Women in Research and Teaching).This research project has been supported by funds to Emerging Field Project “Quantum Geometry”from the FAU Erlangen-Nuernberg within its Emerging Fields Initiative. A References [1] Abhay Ashtekar. Lectures on Non-perturbative Canonical Gravity . Number 6 in AdvancedSeries in Astrophysics and Cosmology. World Scientific, 1991. 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