Projective Limits of State Spaces I. Classical Formalism
PProjective Limits of State SpacesI. Classical Formalism
Suzanne Lanéry , and Thomas Thiemann Institute for Quantum Gravity, Friedrich-Alexander University Erlangen-Nürnberg, Germany Mathematics and Theoretical Physics Laboratory, François-Rabelais University of Tours, France
November 11, 2014
Abstract
In this series of papers, we investigate the projective framework initiated by Jerzy Kijowski [6] and AndrzejOkołów [14, 15], which describes the states of a quantum (field) theory as projective families of densitymatrices. The present first paper aims at clarifying the classical structures that underlies this formalism,namely projective limits of symplectic manifolds. In particular, this allows us to discuss accurately the issueshindering an easy implementation of the dynamics in this context, and to formulate a strategy for overcomingthem.
Contents a r X i v : . [ g r- q c ] N ov Introduction
An important step toward the quantization of a classical theory is the choice of a home for thekinematical quantum states: typically, we look for an Hilbert space supporting a representation ofan algebra of selected kinematical observables. As long as we only deal with finitely many degreesof freedom, a comprehensive survey of the available options might still be within reach. But theextent and implications of this initial choice tend to get dramatically more involved in the case offield theory, where the huge algebra of kinematical observables can give rise to an intricate forestof representations. Unfortunately, it is hard to concisely formalize which requirements the electedrepresentation should satisfy. In the worst case, we are left with the ‘trial and error’ method: picksome representation with attractive properties and check whether the next steps of the quantizationprogram work well on it or not.These next steps can fail for various reasons, one of them being that we committed ourselves to aspace of kinematical states that, at a closer look, does not support the states we are really interestedin. In particular, the space of physical quantum states, solving the dynamical constraints of thetheory, should be rich enough. Refined Algebraic Quantization [4] is a way to look for physicalstates out of the representation we initially choose as the space of kinematical states; however,there are unfavorable situations, where we do not really know how to construct the additionalinput it requires.Also, our space of states should contain the ‘coherent states’ needed to explore the semi-classicallimit of the theory: we would like to associate to any point in the classical phase space a cor-responding quantum state, suitably peaked around that point (see [8, 3] for a discussion of thisproblem in the case of Loop Quantum Gravity, together with possible ways to circumvent it).These issues motivate the search for alternative ways of building the space of kinematical states.Here, we will focus on a formalism first introduced by Jerzy Kijowski in the late ‘70s [6] andfurther developed by Andrzej Okołów recently [14, 15]. The idea is to work in a setting that is moregeneral than Hilbert spaces, and allows us to rely more heavily on the physical interpretation ofthe kinematical observables, namely how they are measured in practice. This tends to give statespaces that are bigger, but nevertheless technically easier to handle. In particular, we thus startwith better chances to find the particular states we are looking for.In the present work, we try to develop this formalism at a fairly general level, going beyond theextensive studies that have been carried out in special cases so far. To this intent, we will start bya detailed exposition of projective limits of symplectic manifolds, that build the natural classicalcounterpart of the quantum state spaces we want to discuss. An important observation is that suchprojective limits admit, at least locally, a preferred factorized description (prop. 2.10). Therefore,we will look more closely at those projective systems where the factorization holds globally: notonly they are often more convenient, they also reflect the core properties of the structures weare considering, so they are well-suited to get a first hold of complex questions. This will be inparticular comfortable when turning to the quantum formalism in [9], but we will always try tosketch some ideas on how to strengthen those of our results that make explicit use of such a globalfactorization.Specific difficulties arise when trying to deal with constraints in this approach to (quantum) fieldtheory. In section 3, we will take advantage of having at our disposal a classical precursor ofthe formalism to analyze this question without having to deal at the same time with the inherent ubtleties of the quantum dynamics. We will outline a suitable strategy, with the aim of doingjustice both to the deep physical meaning of the issues at hand and to their practical significancefor computations. This strategy will be applied to two simple toy models in [10]. Unless otherwise stated, all symplectic manifolds will be smooth manifolds with smooth symplecticstructures, and all maps between them will be smooth. Where infinite dimensional manifolds areconsidered, these are Banach-modeled smooth manifolds, and symplectic structures on them arealways strong symplectic structures [2, chap. VII].
The aim of this section is to describe the classical structures that, while underlying the con-structions considered in previous works [6, 14, 15], have not been explicitly analyzed so far. Thediscussion of the physical interpretation will follow closely the one that has been given in thesereferences.The idea of the projective framework is to assemble a complicated classical theory (typicallya field theory) from a collection of easier, smaller, truncated classical theories, by appropriatelysewing them together. The motivation for this is twofold.From the physical point of view, even when considering a theory with an infinite number ofdegrees of freedom, any given realistic experiment will involve only a finite number of observables,since measuring an infinite number of observables would require infinite time as well as infinitememory space (in fact, this means that any experiment can only measure a finite number of boolean observables, but we will not be that radical here, and will satisfy us with small truncated theoriesthat are described by finite dimensional phase spaces). We will therefore think of the small partialtheories as spanned by a finite number of elementary degrees of freedom. By "elementary", wemean those that can be measured in one experimental step, hence the justification for the choiceof a collection of truncations should ultimately come from a careful analysis of what concreteexperiments actually measure.From a technical point of view, the smaller and easier theories are meant to be a convenientarena to develop systematic ways of calculating physical predictions. Indeed, a theoretical modelwill then be optimally useful if it comes with finite algorithms prescribing how to compute, at agiven precision, the outcome of any arbitrary experiment.Note however that the intuitive understanding just sketched has some weak points. One of themis that, even if we are considering only finitely many observables, it might occur that the Poisson-algebra they are generating cannot live on a finite dimensional symplectic manifold. Anotherproblem is related to the formulation of deterministic predictions while considering only finitelymany degrees of freedom out of a field theory. Our viewpoint here is that these problems shouldnot be relevant for the kinematical observables (these are supposed to build an easy algebra, andthe question of writing down predictions does not belong to the kinematical level). Therefore, wepostpone this discussion to section 3, where we will refine the present framework to take intoaccount the dynamics. .1 Projective systems of classical phase spaces Having a collection of partial theories is not enough, we need to say how to connect them togetherin a consistent way (ie. we do not want our physical predictions to depend on the particular partialtheory in which we computed them). To look at this question, we consider two partial theories M and N , where M is a more detailed description of the physical system at hand, in the sense thatall degrees of freedom that are retained by N are also retained by M . The link between them hasthen two dual aspects. On the one side, we want to associate, with any state in M , a state in N , by forgetting the details we presently do not need. On the other side, we want to identify theobservables that can be defined on N with a subalgebra of the ones that can be defined on M .Given a specific experiment, any partial theory big enough to describe that experiment (ie hostingat least all the observables involved in it) should lead to the same predictions. In other words, thetwo identifications mentioned above (downward identification of the states and upward identifica-tion of the observables) should intertwine the evaluation of an observable on a state.These considerations lead to the following formulation of how some degrees of freedom, spanninga symplectic manifold N , can be seen as being extracted out of a bigger symplectic manifold M :what we need is a projection π : M → N , and we will mount observables on N to observableson M by taking their pullback. We impose a compatibility condition between the projection π and the symplectic structures of M and N to ensure that the Poisson bracket computed betweentwo observables in N is identified with the one computed between the corresponding observablesmounted in M . Definition 2.1
A smooth, surjective map π : M → N between two smooth (possibly infinitedimensional) symplectic manifolds M , Ω M and N , Ω N is said to be compatible with the symplecticstructures iff: ∀� ∈ M , ∀υ ∈ T �π ( � ) ( N ) , υ = T � π ( π ∗ υ ) (2.1. )where T �π ( � ) ( N ) is the topological dual of T π ( � ) ( N ), T � π is the differential of π at � , and υ (resp. π ∗ υ )is the unique vector in T π ( � ) ( N ) (resp. T � ( M )) such that υ = Ω N ,π ( � ) ( υ, · ) (resp. π ∗ υ = Ω M ,� ( π ∗ υ, · )). Proposition 2.2 If π : M → N satisfies def. 2.1 and f , g : N → R are smooth maps on N , then {f , g} N ◦ π = {f ◦ π, g ◦ π} M where {·, ·} N (resp. {·, ·} M ) denotes the Poisson brackets on N (resp. M ). Proof
Eq. (2.1. ) is equivalent to: ∀� ∈ M , ∀µ, υ ∈ T ∗π ( � ) ( N ) , µ ( υ ) = π ∗ µ ( π ∗ υ ).Using the definition of the Poisson brackets, we therefore have: ∀� ∈ M , {f , g} N ◦ π ( � ) = �g π ( � ) � �f π ( � ) � = � ( g ◦ π ) � � � ( f ◦ π ) � � = {f ◦ π, g ◦ π} M ( � ). � Next, the collection of partial theories, together with the projections between them, can be ar-ranged into a structure of projective limit. Such a construction has been considered for example in η �� M η � M η π η �� →η � π η �� →η π η � →η Figure 2.1 – Three-spaces consistency for projective systems of phase spaces[18].That the label set L indexing the partial theories should be directed is manifest if we go backto the interpretation of these small theories as the arenas to describe specific experiments: if wewant to describe an elaborate experimental protocol, combining two sub-experiments, that can bedescribed respectively in M η and M η � , we need a symplectic manifold M η �� , containing the degreesof freedom in M η as well as the ones in M η �� , in order to model the full experiment. And thethree-spaces consistency condition (fig. 2.1) ensures that the connection between a bigger partialtheory M η �� and a smaller one M η is unambiguous, namely that it coincides with the identificationwe get if we perform the truncation in two successive steps, going first from M η �� to an intermediary M η � and then from M η � to M η .With this structure for the state space, the observables naturally build an inductive limit, which isconsistent with the discussion above regarding the mounting of observables and indeed correspondsto the standard construction when looking at functions on a projective limit. Definition 2.3
A projective system of phase spaces is a triple � L , � M η � η∈ L , � π η � →η � η � η � � where: L is a preordered, directed set (we denote the pre-order, ie. a reflexive and transitive binaryrelation, by � , its inverse by � ); � M η � η∈ L is a family of symplectic manifolds indexed by L ; � π η � →η � η � η � is a family of surjective maps π η � →η : M η � → M η indexed by {η, η � ∈ L | η � η � } such that π η � →η is compatible with the symplectic structures, π η→η = id M η and ∀η, η � , η �� ∈ L , η � η � � η �� ⇒ π η �� →η = π η � →η ◦ π η �� →η � .Whenever possible, we will use the shortened notation ( L , M , π ) ↓ instead of � L , � M η � η∈ L , � π η � →η � η � η � �.The projective limit of ( L , M , π ) ↓ , denoted by S ↓ ( L , M ,π ) , is the space: S ↓ ( L , M ,π ) := ( � η ) η∈ L ∈ � η∈ L M η ������ ∀η � η � , π η � →η ( � η � ) = � η .On S ↓ ( L , M ,π ) we put the initial topology with respect to the family of projections � π η � η∈ L where: π η : S ↓ ( L , M ,π ) → M η ( � η � ) η � ∈ L �→ � η =: [( � η � ) η � ∈ L ] η . efinition 2.4 An observable over a projective limit of phase spaces S ↓ ( L , M ,π ) is an equivalence classin � η∈ L C ∞ ( M η , R ) for the equivalence relation defined by: ∀η, η � ∈ L , ∀f η ∈ C ∞ ( M η , R ) , ∀f η � ∈ C ∞ ( M η � , R ) ,f η ∼ f η � ⇔ � ∃η �� ∈ L / η � η �� , η � � η �� & f η ◦ π η �� →η = f η � ◦ π η �� →η � � (2.4. )The space of observables over S ↓ ( L , M ,π ) will be denoted by O ↓ ( L , M ,π ) . The definition of the equiv-alence relation ensures that the evaluation f ( � ) = f η ( � η ) of an element of f = [ f η ] ∼ of O ↓ ( L , M ,π ) on apoint � = ( � η ) η∈ L in S ↓ ( L , M ,π ) is well-defined. From prop. 2.2 the Poisson bracket of two elementsof O ↓ ( L , M ,π ) is well-defined as an element of O ↓ ( L , M ,π ) ( ∀η � � η, f η ◦ π η � →η ∈ [ f η ] ∼ , hence, L beingdirected, we can find a common label to compute the Poisson bracket). A question that occurs frequently when working with the structure introduced above, is to askwhat happens if we restrict ourselves to a directed subset L � of the label set L . It is immediate thata state � � η � η∈ L in the projective structure based on L defines a state � � η � η∈ L � in the one based on L � , simply by throwing away all the � η for η ∈ L \ L � . But this map from S ↓ ( L , M ,π ) into S ↓ ( L � , M ,π ) willin general neither be injective nor surjective.The injectivity might fail because the structure based on L � retains less observables than thestructure based on L , and states that can, thanks to these additional observables, be distinguishedin the latter may be indistinguishable in the former. That also the surjectivity might fails is moresubtle: it can occur if L has a label η that is above an infinite number of labels in L � . Then, givena state � � η � η∈ L � in S ↓ ( L � , M ,π ) , it may indeed not be possible to find an � η that will project correctly onall the � η � for η � ∈ L with η � � η .In the particular case of L � being cofinal in L , we can however completely identify the twoprojective structure, since we can reconstruct any thrown away � η for η ∈ L \ L � by projectingdown from some η � ∈ L � above η . Proposition 2.5
Let ( L , M , π ) ↓ be a projective system of phase spaces and let L � be a directedsubset of L . We define the map: σ : S ↓ ( L , M ,π ) → S ↓ ( L � , M ,π ) � � η � η∈ L �→ � � η � η∈ L � .Then, we have a map α : O ↓ ( L � , M ,π ) → O ↓ ( L , M ,π ) such that: ∀� ∈ S ↓ ( L , M ,π ) , ∀f ∈ O ↓ ( L � , M ,π ) , α ( f )( � ) = f ( σ ( � )) , (2.5. ) nd: ∀f , g ∈ O ↓ ( L � , M ,π ) , {α ( f ) , α ( g ) } = α � {f , g} � . (2.5. )If L � is cofinal in L , we have in addition that σ and α are bijective maps. Proof
It’s an immediate check that σ is indeed valued in S ↓ ( L � , M ,π ) .Then, for f = [ f η ] ∼, L � ∈ O ↓ ( L � , M ,π ) , we define: α ( f ) = [ f η ] ∼, L for f η a representative of f .We have: ∀η, η � ∈ L � , ∀f η ∈ C ∞ ( M η , R ) , ∀f η � ∈ C ∞ ( M η � , R ) , f η ∼ L � f η � ⇒ f η ∼ L f η � ,hence α is well-defined as a map O ↓ ( L � , M ,π ) → O ↓ ( L , M ,π ) .Eq. (2.5. ) and eq. (2.5. ) hold because we can choose any representative we want to carry outthe evaluation or to compute the Poisson brackets.We now suppose that L is cofinal. Then, we can define:� σ : S ↓ ( L � , M ,π ) → S ↓ ( L , M ,π ) � � η � η∈ L � �→ �� � η � η∈ L ,where for η ∈ L , � � η = π η � →η � η � , with η � ∈ L � and η � � η . If η �� is an other element of L � such that η �� � η , there exists η ��� ∈ L � / η � � η ��� & η �� � η ��� ( L � is directed by hypothesis), hence: π η � →η � η � = π η � →η π η ��� →η � � η ��� = π η ��� →η � η ��� = π η �� →η π η ��� →η �� � η ��� = π η �� →η � η �� .If η ∈ L � , we can choose η = η � , so that � � η = � η , therefore σ ◦ � σ = id S ↓ ( L �, M ,π ) . On the other hand, ifthere exists an element � � η � η∈ L ∈ S ↓ ( L , M ,π ) , such that ∀η � ∈ L � , � η � = � η � , then � � η = π η � →η � η � = � η ,therefore � σ ◦ σ = id S ↓ ( L , M ,π ) .Then, for f = [ f η ] ∼, L ∈ O ↓ ( L , M ,π ) , we define:� α ( f ) = [ f η ◦ π η � →η ] ∼, L � ,for f η a representative of f and η � ∈ L � such that η � � η . If η �� is an other element of L � such that η �� � η , there exists η ��� ∈ L � / η � � η ��� & η �� � η ��� ( L � is directed by hypothesis), hence:( f η ◦ π η � →η ) ◦ π η ��� →η � = f η ◦ π η ��� →η = ( f η ◦ π η �� →η ) ◦ π η ��� →η �� ,so that f η ◦ π η � →η ∼ L � f η ◦ π η �� →η .If f κ is an other representative of f , there exists µ ∈ L / µ � η & µ � κ such that f η ◦ π µ→η = f κ ◦ π µ→κ . Since L � is cofinal in L , we can choose µ � ∈ L such that µ � � µ , and we have: f η ◦ π µ � →η = f η ◦ π µ→η ◦ π µ � →µ = f κ ◦ π µ→κ ◦ π µ � →µ = f κ ◦ π µ � →κ ,hence � α is well-defined as a map O ↓ ( L , M ,π ) → O ↓ ( L � , M ,π ) .If f η is a representative of f with η ∈ L � , we can choose η � = η , so that � α ( f ) = [ f η ] ∼, L � , therefore� α ◦ α = id O ↓ ( L �, M ,π ) . On the other hand, we have for all η ∈ L and all η � ∈ L with η � � η , f η ◦ π η � →η ] ∼, L = [ f η ] ∼, L , therefore α ◦ � α = id O ↓ ( L , M ,π ) . � We can now rewrite in terms of the concepts we have introduced the program that has beenfollowed in [6, 14, 15]. When considering a field theory constructed on an infinite dimensionalmanifold M ∞ , we will first, relying on our understanding of how physical effects are measured inpractice, undertake to identify what the elementary observables should be, and try to construct acorresponding collection of interconnected partial theories, where each partial theory M η will beassociated to a finite subset of elementary observables (those that can be defined on M η , ie. thatonly depend on the degrees of freedom retained by M η ). Since we have naturally a projection from M ∞ into each partial theory M η , we also immediately have a map from M ∞ into S ↓ ( L , M ,π ) .If the set of all elementary observables separate the points of M ∞ , then this map will be injective.Moreover, the projective limit of phase spaces will provide an extension of M ∞ in the sense that M ∞ can be identified with a dense subspace of S ↓ ( L , M ,π ) .Note that choosing a collection of elementary observables is not the same as choosing preferredcoordinates on M ∞ , for the construction here does not require the elementary observables to be independent , they can form an overdetermined system. This can make physically a crucial difference:to illustrate this point, one can think at the set of elementary observables as analogous to the set ofall the linear forms on a vector space, while preferred coordinates would correspond to the choiceof a basis (compare the two examples given in [10] for examples implementing a projective structurealong these lines; the model in [10, section 2] relies on a choice of basis, while the one in [10,section 3] does not). The set of all linear forms encodes nothing less but nothing more than thelinear structure of the vector space, and this structure might indeed have a deep physical relevance,while we probably want to avoid relying on a preferred basis, in order not to break the invarianceunder isomorphisms. Definition 2.6
We say that a (possibly infinite dimensional) symplectic manifold M ∞ is renderedby a projective system of phase spaces ( L , M , π ) ↓ if for all η ∈ L there exists an application π ∞→η : M ∞ → M η such that: ∀η ∈ L , π ∞→η is surjective and compatible with the symplectic structures; ∀η � η � ∈ L , π ∞→η = π η � →η ◦ π ∞→η � .Hence, we have a projective system of phase spaces ( L � {∞} , M , π ) ↓ , where we extend thepreorder of L to L � {∞} by requiring ∀η ∈ L , ∞ � η . From prop. 2.5, we have maps σ � : S ↓ ( L �{∞}, M ,π ) → S ↓ ( L , M ,π ) and σ − ∞ : S ↓ ( {∞}, M ,π ) → S ↓ ( L �{∞}, M ,π ) (since {∞} is cofinal in L � {∞} ), soby identifying S ↓ ( {∞}, M ,π ) with M ∞ , we define: σ ↓ := σ � ◦ σ − ∞ : M ∞ → S ↓ ( L , M ,π ) .Similarly, we have α � : O ↓ ( L , M ,π ) → O ↓ ( L �{∞}, M ,π ) and α − ∞ : O ↓ ( L �{∞}, M ,π ) → O ↓ ( {∞}, M ,π ) , so byidentifying O ↓ ( {∞}, M ,π ) with C ∞ ( M ∞ , R ), we define: α ↑ := α − ∞ ◦ α � : O ↓ ( L , M ,π ) → C ∞ ( M ∞ , R ) . roposition 2.7 With the notations of def. 2.6, σ ↓ � M ∞ � is dense in S ↓ ( L , M ,π ) . Proof
Let � � η � η∈ L ∈ S ↓ ( L , M ,π ) . For η ∈ L , we choose y η ∈ M ∞ such that π ∞→η ( y η ) = � η (this ispossible, since π ∞→η is surjective). We have: ∀η ∈ L , ∀η � � η, � σ ↓ � y η � �� η = π ∞→η � y η � � = π η � →η ◦ π ∞→η � � y η � � = π η � →η � � η � � = � η .Hence, the net � σ ↓ � y η � �� η � ∈ L converges in S ↓ ( L , M ,π ) to � � η � η∈ L , therefore � � η � η∈ L ∈ Im σ ↓ . � We close this subsection by mentioning the construction of a different kind of maps betweenprojective systems of phase spaces, that will be of interest when dealing with concrete examples.Indeed, we will often encounter the situation of having a projective system that has been originallyconstructed over a very large and complicated label set (in particular this can be a side-effect ofthe way we will handle constraints, as exhibited in section 3), but whose structure happens tobe considerably simpler, because we can group the labels into classes partitioning L , in such away that the projective consistency conditions force a symplectomorphic identification between themanifolds M η for all η belonging to the same class. Then, we probably want to define a label set L • by quotienting L according to those classes, and to identify the original projective system on L with an easier one built on L • .For example, suppose that the elements of L are pairs ( ε, θ ) , ordered in the product order(aka. ( ε, θ ) � ( ε � , θ � ) ⇔ ε � ε � & θ � θ ). Now, if it turns out that M ( ε,θ ) only depends on ε and π ( ε � ,θ � ) → ( ε,θ ) only on ε and ε � , then the projective condition on the states will actually impose � ( ε,θ ) = � ( ε,θ ) . Thus this projective limit is in reality just a projective limit on the set of all ε .This is a tool that we will use repeatedly in [10] (and also when proceeding to applications inquantum gravity). Proposition 2.8
Let L and L • be directed preordered sets and assume that we are given: a surjective map � : L → L • such that ∀η � η � ∈ L , � ( η ) � � ( η � ) ; a projective system of phase spaces ( L • , M • , π • ) ↓ on L • ; and for all η ∈ L , a symplectic manifold M η together with a symplectomorphism µ η : M η → M • � ( η ) .Then, defining for all η � η � ∈ L the projection: π η � →η := µ − η ◦ π • � ( η � ) →� ( η ) ◦ µ η � , (2.8. )( L , M , π ) ↓ is a projective system of phase spaces and the map: κ : S ↓ ( L • , M • ,π • ) → S ↓ ( L , M ,π ) � � • η • � η • ∈ L • �→ � µ − η � � • � ( η ) �� η∈ L , (2.8. )is bijective. Moreover, there exists a bijective map λ : O ↓ ( L , M ,π ) → O ↓ ( L • , M • ,π • ) such that: � • ∈ S ↓ ( L • , M • ,π • ) , ∀f ∈ O ↓ ( L , M ,π ) , λ ( f )( � • ) = f ( κ ( � • )) . (2.8. ) Proof
First, we check that κ is well-defined. Let � � • η • � η • ∈ L • ∈ S ↓ ( L • , M • ,π • ) and let η � η � ∈ L . Wehave � ( η ) � � ( η � ) and from eq. (2.8. ): π η � →η � µ − η � � � • � ( η � ) �� = µ − η ◦ π • � ( η � ) →� ( η ) � � • � ( η � ) � = µ − η � � • � ( η ) � ,hence � µ − η � � • � ( η ) �� η∈ L ∈ S ↓ ( L , M ,π ) .To prove that κ is bijective, we define:� κ : S ↓ ( L , M ,π ) → S ↓ ( L • , M • ,π • ) � � η � η∈ L �→ � � • η • � η • ∈ L • ,where ∀η • ∈ L • , � • η • := µ η � � η � for any η such that � ( η ) = η • (making use of the surjectivity of � ). � • η • does not depend on the choice of η ∈ � − �η • � ; indeed, if � ( η ) = � ( η � ), there exists η �� ∈ L suchthat η �� � η, η � , hence: µ η � � η � = µ η ◦ π η �� →η � � η �� � = π • � ( η �� ) →� ( η ) ◦ µ η �� � � η �� �= π • � ( η �� ) →� ( η � ) ◦ µ η �� � � η �� � = µ η � ◦ π η �� →η � � � η �� � = µ η � � � η � � .And by construction of � κ , we have κ ◦ � κ = id S ↓ ( L , M ,π ) as well as � κ ◦ κ = id S ↓ ( L •, M •,π• ) .Now, we define λ by: λ : O ↓ ( L , M ,π ) → O ↓ ( L • , M • ,π • ) [ f η ] ∼ �→ � f η ◦ µ − η � ∼ . λ is well-defined, for we have: ∀η, η � ∈ L , f η ∼ f η � ⇔ � ∃η �� � η � , η / f η ◦ π η �� →η = f η � ◦ π η �� →η � � ⇔ � ∃η �� � η � , η / f η ◦ µ − η ◦ π • � ( η �� ) →� ( η ) = f η � ◦ µ − η � ◦ π • � ( η �� ) →� ( η � ) � ⇒ � f η ◦ µ − η ∼ f η � ◦ µ − η � � .And by construction of λ , eq. (2.8. ) is fulfilled.Finally, to prove that λ is bijective, we construct a map � λ by:� λ : O ↓ ( L • , M • ,π • ) → O ↓ ( L , M ,π ) � f • η • � ∼ �→ [ f η ] ∼ ,where f η is defined for any η such that � ( η ) = η • by f η = f • η • ◦ µ η . To check that � λ is well-defined, let η, η � ∈ L such that there exist f • � ( η ) , f • � ( η � ) ∈ � f • η • � ∼ (note that this also covers the case � ( η ) = � ( η � ) = η • ). Then, there exists η • �� such that: f • � ( η ) ◦ π • η •�� →� ( η ) = f • � ( η � ) ◦ π • η •�� →� ( η � ) ,and, since � is surjective, there exists η �� ∈ L such that � ( η �� ) = η • �� . Next, using that L is a directed et, there exists η ��� ∈ L with η ��� � η, η � , η �� . Therefore, we have: f • � ( η ) ◦ π • � ( η ��� ) →� ( η ) = f • � ( η � ) ◦ π • � ( η ��� ) →� ( η � ) f • � ( η ) ◦ µ η ◦ π η ��� →η = f • � ( η � ) ◦ µ η � ◦ π η ��� →η � f • � ( η ) ◦ µ η ∼ f • � ( η � ) ◦ µ η � .And by construction of � λ , we have � λ ◦ λ = id O ↓ ( L , M ,π ) as well as λ ◦ � λ = id O ↓ ( L •, M •,π• ) . � Proposition 2.9
The previous result still holds if, instead of requiring � to be surjective, we simplyrequire � � L � to be a cofinal part of L • . Proof
This follows by combining prop. 2.8 with prop. 2.5. � For technical convenience, we will often specialize to a particular class of projective systemsof phase phases, namely the situation where for any η � η � , the symplectic manifold M η � can beidentified with the Cartesian product of M η with a symplectic manifold M η � →η (in other words thediscarded degrees of freedom can be collected into a phase space M η � →η ).This restriction is in fact not as radical as one could first think, for given a projection π : M → N as in def. 2.1, M can always be locally written as a Cartesian product of symplectic manifolds insuch a way that π correspond to the projection map on one factor of the product. Moreover, thereis only one (local) decomposition having this property.At the level of observables, writing M as a Cartesian product N ⊥ × N implies that the algebra O � of all observables over M is generated by O ∪ O ⊥ , with O the subalgebra of O � defined by theobservables over N and O ⊥ by the ones over N ⊥ . And asking the symplectic structure on M toagree with the symplectic structure on the Cartesian product moreover requires that any observablein O Poisson-commutes with any observable in O ⊥ . This is the reason why, at least locally, thesymplectic structure on M prescribes how to choose a subalgebra O ⊥ completing O : O ⊥ has to bethe set of all observables having vanishing Poisson brackets with any observable in O .To understand better why this factorization of M will not always hold globally, we can examinehow the proof of prop. 2.10 below is done: what we have is a foliation of M , of which each leaf islocally diffeomorphic to N via π . It is precisely when this local diffeomorphic identification fails tobe a global one, that we will not get a global factorization. This can happen at two different levels.First, the restriction of π to a given leaf is not necessarily a covering map, although it is locallydiffeomorphic: there can be ‘completeness’ issues, as exemplified by the ad hoc situation where M = �( � , � ; p , p ) �� |� | < exp( � )� ⊂ T ∗ ( R ) and N = �( � ; p )� ⊂ T ∗ ( R ) . Second, a coveringmap need not be bijective, unless N is simply-connected: for example, M being a symplectomorphiccovering of N provides a projection that is compatible with the symplectic structures in the senseof def. 2.1, but there is no corresponding factorization. nless otherwise stated, all manifolds considered in the present subsection will be finite dimen-sional manifolds. Proposition 2.10
Let M , N be finite dimensional symplectic manifolds and suppose that thereexists π : M → N satisfying def. 2.1.Then, for � ∈ M , there exist an open neighborhood U of � in M , an open neighborhood V of π ( � )in N , a manifold W and a symplectic structure Ω W on W such that there exists a diffeomorphismΦ : V × W → U satisfying ∀y ∈ V , ∀w ∈ W , π ◦ Φ( y, w ) = y and Φ ∗ Ω M = Ω N × Ω W .Moreover, Φ is unique in the following sense: if U � is an open subset of U , V � is a connectedopen subspace of V , W � is a symplectic manifold and Φ � : V � × W � → U � is a symplectomorphismsuch that ∀y ∈ V � , ∀z ∈ W � , π ◦ Φ � ( y, z ) = y , then there exists a symplectomorphism ψ : W � → W �� (with W �� an open subset of W ) such that ∀y ∈ V � , ∀z ∈ W � , Φ � ( y, z ) = Φ( y, ψ ( z )). Proof
Existence.
We call D = dim( M ), � = dim( N ) and � = D − � . For all � ∈ M , we define: W � = {w ∈ T � ( M ) | T � π ( w ) = 0 } and V � = {� ∈ T � ( M ) | ∀w ∈ W � , Ω M ,� ( �, w ) = 0 } .We have ∀υ ∈ T ∗π ( � ) ( N ) , ∀w ∈ W � , Ω M ,� ( π ∗ υ, w ) = υ ◦ T � π ( w ) = 0, hence ∀υ ∈ T ∗π ( � ) ( N ) , π ∗ υ ∈ V � .For � ∈ T � ( M ), we define υ � = Ω N ,π ( � ) ( T � π ( � ) , · ). Using eq. (2.1. ), we get ∀� ∈ T � ( M ) , T � π � π ∗ υ � �= υ � = T � π ( � ). So we can write � ∈ T � ( M ) as � = ( � − π ∗ υ � ) + π ∗ υ � with � − π ∗ υ � ∈ W � and π ∗ υ � ∈ V � .Hence, we have W � + V � = T � ( M ), and therefore W � ⊕ V � = T � ( M ), since dim( V � ) = dim( M ) − dim( W � ). Moreover, since eq. (2.1. ) implies that T � π is surjective, we have dim( W � ) = D − � anddim( V � ) = D − ( D − � ) = � .Now we choose � ∈ M and we consider a coordinate patch V on N containing π ( � ), withcoordinates y , � � � , y � . We define: X �,� � := π ∗ �y �,π ( � � ) = � ( y � ◦ π ) � � for all � � ∈ U := π − �V � . X , � � � , X � are vector fields on U such that ∀� � ∈ U , ( X ,� � , � � � , X �,� � ) is a basis of V � � . We calculatethe Lie brackets between two of these vector fields:� X � , X j � = � �y � ◦ π, �y j ◦ π � = � �� y � ◦ π, y j ◦ π �� = � �� y � , y j � ◦ π � = π ∗ � �� y � , y j ��where the second equality expresses the Lie brackets of two Hamiltonian vector fields and the thirdequality comes from prop. 2.2.Therefore, we have ∀� � ∈ U , � X � , X j � � � ∈ V � � . From Frobenius theorem [11, theorem 14.5], thereexist an open neighborhood U of � in U and coordinates � , � � � , � � , � � +1 , � � � , � D over U such that ∀� � ∈ U , ∂ � ,� � , � � � , ∂ � � ,� � is a basis of V � � . We define:�Φ : U → N × R � � � �→ π ( � � ) , ( � � +1 ( � � ) , � � � , � D ( � � )).We can now show that T � �Φ : T � ( M ) → T π ( � ) ( N ) × R � is bijective. Indeed, let � ∈ T � ( M )such that T � �Φ( � ) = 0. Then, in particular, we have T � π ( � ) = 0, so � ∈ W � . On the otherhand, we have �� k,� ( � ) = 0 for k = � + 1 , � � � , D , so � is a linear combination of ∂ � ,� , � � � , ∂ � � ,� ,hence � ∈ V � . From W � ⊕ V � = T � ( M ), � = 0. Therefore T � �Φ is injective, thus bijective, for im ( T � ( M )) = dim � T π ( � ) ( N ) × R � �.From the inverse function theorem [11, theorem 5.11], there exists an open neighborhood U of � in U such that �Φ��� U : U → �Φ � U � is a diffeomorphism. Hence there exist an open connectedneighborhood V of π ( � ) in N , an open subset W of R � , an open neighborhood U of � in U anda diffeomorphism Φ : V × W → U such that ∀y ∈ V , ∀z ∈ W , �Φ(Φ( y, z )) = ( y, z ). In particular, ∀y ∈ V , ∀z ∈ W , π ◦ Φ( y, z ) = y .At every point � � ∈ M and for every vector �, w ∈ T Φ − ( � � ) ( V × W ), T Φ − ( � � ) Φ( �, ∈ V � � and T Φ − ( � � ) Φ(0 , w ) ∈ W � � . In particular, we have Ω M ,� � � T Φ − ( � � ) Φ( �, , T Φ − ( � � ) Φ(0 , w )� = 0.We now consider z ∈ W , w, w � ∈ T z ( W ) and we define for all y ∈ V ,Ω y W ,z ( w, w � ) = Ω M , Φ( y,z ) ( T ( y,z ) Φ(0 , w ) , T ( y,z ) Φ(0 , w � )).Let � Y be a vector field on V , let � Z , � Z � be vector fields on W such that � Z z = w , � Z �z = w � . We define thevector fields Y = Φ ∗ �� Y , Z = Φ ∗ �0 , � Z � and Z � = Φ ∗ �0 , � Z � � on M . From [ Y , Z ] = [
Y , Z � ] = 0and � Ω M = 0, we have: Y �Ω M � Z , Z � �� = Z �Ω M � Y , Z � �� − Z � (Ω M ( Y , Z )) + Ω M �[ Z , Z � ] , Y �since Y � � ∈ V � � and Z � � , Z �� � , [ Z , Z � ] � � ∈ W � � , we have Y �Ω M � Z , Z � �� = 0. Therefore the differentialof y �→ Ω y W ,z ( w, w � ) is zero at every point y ∈ V , and V being connected, Ω y W ,z ( w, w � ) does notdepend on y . So, we define Ω W ,z ( w, w � ) = Ω y W ,z ( w, w � ).We now can check using eq. (2.1. ) and the definition of Ω W that Φ ∗ Ω M = Ω N × Ω W . ThereforeΩ W is a symplectic structure on W and Φ : V × W → U is a symplectomorphism. Uniqueness.
We consider symplectic manifolds V , W , and W � , a connected open subset V � of V ,and an application � ψ : V � × W � → W such that:Ψ : V � × W � → V × W y, z �→ y, � ψ ( y, z )induces a symplectomorphism V � × W � → Ψ � V � × W � � .For y ∈ V � , z ∈ W � , � ∈ T y ( V � ) , w ∈ T z ( W � ), we then have:0 = Ω V × W , Ψ( y,z ) � T ( y,z ) Ψ( �, , T ( y,z ) Ψ(0 , w )� = Ω W ,ψ ( y,z ) � T ( y,z ) � ψ ( �, , T ( y,z ) � ψ (0 , w )�.However, for Ψ to be a diffeomorphism, T ( y,z ) � ψ (0 , w ) should run through T ψ ( y,z ) ( W ) when w runsthrough T z ( W ). Therefore, we should have T ( y,z ) � ψ ( �,
0) = 0. Hence, V � being connected, � ψ ( y, z )cannot depend upon y . Accordingly, we define ψ ( z ) := � ψ ( y, z ), and Ψ | V � × W � → Ψ � V � × W � � being asymplectomorphism requires that ψ| W � →ψ� W � � should be a symplectomorphism. Note.
A more concise (albeit less instructive) proof of this result can be achieved by consideringthe closed 2-form σ := Ω M − π ∗ Ω N and applying a standard result of symplectic geometry [17,§ 5.24], telling us that the kernel of σ is an involutive distribution, and that σ defines a symplecticform on the quotient. � η �� X η �� →η � × X η � X η �� →η � × X η � →η × X η X η �� →η × X η φ η �� →η � φ η �� →η φ η � →η φ η �� →η � →η Figure 2.2 – Three-spaces consistency for factorizing systemsIn order to build a structure describing a collection of interconnected partial theories, wherethe relation between a more detailed partial theory M η � and a less detailed one M η is given bya factorization of M η � as M η � →η × M η , we also need to reformulate the three-spaces consistencycondition that we had for a projective system (fig. 2.1) in terms of a factorization requirement. Forthis, we ask for the symplectic manifold M η �� →η , that holds the degrees of freedom discarded whengoing directly from M η �� to M η , to decompose as the Cartesian product of M η �� →η � with M η � →η , where M η �� →η � holds the degrees of freedom discarded when going as a first step from M η �� to M η � , and M η � →η holds the ones discarded when going as a second step from M η � to M η (fig. 2.2).Having a factorizing system defined this way then provides us immediately with a projectivesystem as above. Reciprocally, if we give us a projective system of phase spaces in which anyprojection π η � →η can be understood as projecting on a factor of a Cartesian product (that is, if theresult of prop. 2.10 happens to hold globally and not just locally) and if moreover all the M η areconnected (which sounds physically sensible when speaking of phases spaces), we can construct acorresponding factorizing system of phase spaces. Definition 2.11
A factorizing system is a quintuple:� L , � X η � η∈ L , � X η � →η � η � η � , � φ η � →η � η � η � , � φ η �� →η � →η � η � η � � η �� �where: L is a preordered, directed set; � X η � η∈ L is a family of spaces indexed by L ; � X η � →η � η � η � is a family of spaces indexed by {η, η � ∈ L | η � η � } , such that, for all η ∈ L , X η→η has only one element; � φ η � →η � η � η � is a family of bijective maps φ η � →η : X η � → X η � →η ×X η indexed by {η, η � ∈ L | η � η � } such that φ η→η is trivial; . � φ η �� →η � →η � η � η � � η �� is a family of bijective maps φ η �� →η � →η : X η �� →η → X η �� →η � × X η � →η indexed by {η, η � , η �� ∈ L | η � η � � η �� } such that φ η �� →η � →η is trivial whenever two labels among η, η � , η �� are equal and: ∀η, η � , η �� ∈ L / η � η � � η �� , ( φ η �� →η � →η × id X η ) ◦ φ η �� →η = (id X η��→η� × φ η � →η ) ◦ φ η �� →η � . (2.11. )Whenever possible, we will use the shortened notation ( L , X , φ ) × instead of � L , � X η � η∈ L , � X η � →η � η � η � , � φ η � →η � η � η � , � φ η �� →η � →η � η � η � � η �� �. Definition 2.12
A factorizing system of phase spaces is a factorizing system ( L , M , φ ) × where: for all η ∈ L , M η is a symplectic manifold, and for all η � η � ∈ L , M η � →η is a symplecticmanifold, except if η � = η in which case M η→η is a set with just one element; for all η � η � ∈ L , φ η � →η is a symplectomorphism, and for all η � η � � η �� ∈ L , φ η �� →η � →η is asymplectomorphism. Proposition 2.13
If ( L , M , φ ) × fulfills def. 2.12 and if, for η � η � ∈ L , we define: � η � →η : M η � →η × M η → M η ( y, � ) �→ � and π η � →η = � η � →η ◦ φ η � →η (2.13. )then ( L , M , π ) ↓ is a projective system of phase spaces.Accordingly, we define the space of states by S × ( L , M ,φ ) := S ↓ ( L , M ,π ) (def. 2.3) and the space ofobservables by O × ( L , M ,φ ) := O ↓ ( L , M ,π ) (def. 2.4). Proof
We need to prove that ∀η � η � ∈ L , π η � →η is a surjective map compatible with the symplecticstructures, that ∀η ∈ L , π η→η = id M η , and that ∀η � η � � η �� ∈ L , π η �� →η = π η � →η ◦ π η �� →η � .For η ∈ L , we have π η→η = id M η (identifying M η and its trivial Cartesian product with aone-element set), so in particular it is a surjective map compatible with the symplectic structures.Let η ≺ η � ∈ L . M η � →η � = ∅ (as a manifold), hence � η � →η is surjective, therefore π η � →η is asurjective map.Let ( y, � ) ∈ M η � →η × M η and let υ ∈ T ∗� ( M η ). We have: ∀w, � ∈ T ( y,� ) ( M η � →η × M η ) ,υ ◦ � T ( y,� ) � η � →η � ( w, � ) = υ ( � ) = Ω M η ,� ( υ, � ) = Ω M η�→η × M η , ( y,� ) ((0 , υ ) , ( w, � )) ,so that � ∗η � →η υ = (0 , υ ), hence � T ( y,� ) � η � →η � ( � ∗η � →η υ ) = υ . Therefore � η � →η is compatible with the sym-plectic structures, and since φ η � →η is a symplectomorphism, π η � →η is compatible with the symplecticstructures.Let η � η � � η �� ∈ L and define: � η �� →η � →η : M η �� →η � × M η � →η × M η → M η ( z, y, � ) �→ � .We have: η �� →η � →η ◦ �id M η��→η� × φ η � →η � = � η � →η ◦ φ η � →η ◦ � η �� →η � ,and � η �� →η � →η ◦ � φ η �� →η � →η × id M η � = � η �� →η .Hence, composing eq. (2.11. ) to the right with � η �� →η � →η gives: � η �� →η ◦ φ η �� →η = � η � →η ◦ φ η � →η ◦ � η �� →η � ◦ φ η �� →η � ,so that we have π η �� →η = π η � →η ◦ π η �� →η � . � Proposition 2.14
Let ( L , M , π ) ↓ be a projective system of phase spaces and suppose that: for all η ∈ L , M η is connected; for all η ≺ η � ∈ L , there exist a symplectic manifold M η � →η and a symplectomorphism φ η � →η : M η � → M η � →η × M η such that π η � →η = � η � →η ◦ φ η � →η .Then, we can complete this input into a factorizing system ( L , M , φ ) × . Proof
For η ∈ L , we define M η→η to be a space with one element and φ η→η to be the trivialidentification.Let η � η � � η �� ∈ L . What we need to show is that there exists a symplectomorphism φ η �� →η � →η : M η �� →η → M η �� →η � × M η � →η such that eq. (2.11. ) is fulfilled. If two labels among η, η � , η �� are equal, we can choose φ η �� →η � →η to be the trivial identification, so we now consider the case η ≺ η � ≺ η �� .We define:Ψ := φ η �� →η ◦ φ − η �� →η � ◦ �id η �� →η � × φ η � →η � − : M η �� →η � × M η � →η × M η → M η �� →η × M η .Ψ is a symplectomorphism and satisfies: ∀ ( z, y, � ) ∈ M η �� →η � × M η � →η × M η , � η �� →η ◦ Ψ( z, y, � ) = � .Hence, applying the uniqueness part from the proof of prop. 2.10 (with V = V � = M η , W = M η �� →η and W � = M η �� →η � × M η � →η , using that M η is connected, as V � must be), there exists asymplectomorphism ψ : M η �� →η � × M η � →η → M η �� →η such that Ψ = ψ × id M η . Thus we define φ η �� →η � →η = ψ − . � If we have a family of finite dimensional symplectic manifolds, where each M η modeling a partialtheory can be written as a cotangent bundle on a configuration space C η , then a factorizing systembuilt over the family � C η � η∈ L can automatically be lifted as a factorizing system over the family � M η � η∈ L . Reciprocally, if we build a projective system of symplectic manifolds over this family,such that each projection can be understood as arising from a factorization of the underlyingconfiguration spaces, and if additionally all the configuration spaces are connected, then not onlycan the projective system of symplectic manifolds be put into a factorizing form (as follows fromprop. 2.14), but this factorizing form goes down to a factorizing system of the configuration spaces.It is important to note that, at the level of configuration spaces, a factorizing system containsmuch more input than a projective system does. The situation here is different than what wehave at the level of phase spaces, where projective and factorizing systems can, let aside global onsiderations, be matched unambiguously. The reason for this disparity is that the symplecticstructure on the phase spaces played a crucial role in the proof of prop. 2.10: when looking at aprojection between configuration spaces, that retains only a subset of the configuration variables,we have no additional structure that would allows us to select a preferred complementary set ofdiscarded variables. Definition 2.15
A factorizing system of smooth manifolds is a factorizing system ( L , C , φ ) × (def. 2.11;in particular eq. (2.11. ) holds) where: for all η ∈ L , C η is a smooth manifold, and for all η � η � ∈ L , C η � →η is a smooth manifold,except if η � = η in which case C η→η is a set with just one element; for all η � η � ∈ L , φ η � →η is a diffeomorphism, and for all η � η � � η �� ∈ L , φ η �� →η � →η is adiffeomorphism. Proposition 2.16
If ( L , C , φ ) × fulfills def. 2.15 and if: for all η ∈ L (resp. all η, η � ∈ L with η ≺ η � ), we define M η := T ∗ ( C η ) (resp. M η � →η := T ∗ ( C η � →η )), equipped with the canonical symplectic structure on a cotangent bundle; for all η, η � ∈ L with η ≺ η � (resp. all η, η � , η �� ∈ L with η ≺ η � ≺ η �� ), we naturally lift φ η � →η : C η � → C η � →η × C η (resp. φ η �� →η � →η : C η �� →η → C η �� →η � × C η � →η ) to a map � φ η � →η : M η � → M η � →η × M η (resp. � φ η �� →η � →η : M η �� →η → M η �� →η � × M η � →η ) between the cotangent bundles; for all η ∈ L , we define M η→η to be a set with one element, and for all η ∈ L (resp. all η, η � , η �� ∈ L with η � η � � η �� and at least two labels equals) we define � φ η→η (resp. � φ η �� →η � →η )to be the trivial identification;then ( L , M , � φ ) × is a factorizing system of phase spaces. Proof
We need to prove that ∀η ≺ η � , � φ η � →η is a symplectomorphism, that ∀η ≺ η � ≺ η �� , � φ η �� →η � →η isa symplectomorphism and that eq. (2.11. ) for the maps φ is lifted up to the corresponding equationfor the maps � φ .For η ∈ L , the symplectic structure on M η = T ∗ ( C η ) is defined by: ∀ ( �, p ) ∈ M η , ∀w, w � ∈ T ( �,p ) ( M η ) , Ω M η , ( �,p ) � w, w � � := w � ��� ( w ��� ) − w ��� � w � ��� � , (2.16. )where we define for w ∈ T ( �,p ) ( M η ), w ��� ∈ T � ( C η ) to be the horizontal projection of w , and w ��� ∈ T ∗� ( C η ) to be the vertical part of w defined using some local coordinate system around � (the map w �→ w ��� depends on this choice of local coordinates, however the anti-symmetrization ineq. (2.16. ) ensures that the definition of Ω M η , ( �,p ) is independent of this choice).For η ≺ η � ∈ L , the map � φ η � →η : M η � → M η � →η × M η is defined by: ∀ ( � � , p � ) ∈ M η � , � φ η � →η ( � � , p � ) := � � f η � →η ◦ φ η � →η ( � � ) , p � ◦ � T φ η�→η ( � � ) φ − η � →η � ( · , , � � η � →η ◦ φ η � →η ( � � ) , p � ◦ � T φ η�→η ( � � ) φ − η � →η � (0 , · )� � ,where f η � →η : C η � →η × C η → C η � →η and � η � →η : C η � →η × C η → C η are the projection maps of the artesian product. This map is bijective, because φ η � →η and � T φ η�→η ( � � ) φ − η � →η � are.Let ( �, p ) ∈ M η , ( y, � ) ∈ M η � →η and ( � � , p � ) = � φ η � →η �( y, � ) , ( �, p )� . From the definition of� φ η � →η , we have for all w ∈ T ( � � ,p � ) ( M η � ):�[ T � � ,p � � φ η � →η ] ( w )� ��� = �[ T � � f η � →η ◦ φ η � →η ] ( w ��� ) , [ T � � � η � →η ◦ φ η � →η ] ( w ��� )� .Now, we choose local coordinates around � in C η and around y in C η � →η , so we have localcoordinates around in ( y, � ) in C η � →η × C η that we can transport through φ − η � →η as local coordinatesaround � � = φ − η � →η ( y, � ) in C η � . Using these to define ( · ) ��� in T ( �,p ) ( M η ), T ( y,� ) ( M η � →η ) and T ( � � ,p � ) ( M η � ),we have for all w ∈ T ( � � ,p � ) ( M η � ):�[ T � � ,p � � φ η � →η ] ( w )� ��� = � w ��� ◦ � T y,� φ − η � →η � ( · , , w ��� ◦ � T y,� φ − η � →η � (0 , · )� .Therefore: ∀w, w � ∈ T ( � � ,p � ) ( M η � ) , Ω M η�→η × M η , (( y,� ) , ( �,p )) �[ T � � ,p � � φ η � →η ] ( w ) , [ T � � ,p � � φ η � →η ] ( w � )� == w � ��� ◦ � T y,� φ − η � →η � �[ T � � f η � →η ◦ φ η � →η ] w ��� ) ,
0� ++ w � ��� ◦ � T y,� φ − η � →η � �0 , [ T � � � η � →η ◦ φ η � →η ] ( w ��� )� − � w ↔ w � �= w � ��� ◦ � T y,� φ − η � →η � ◦ [ T � � φ η � →η ] ( w ��� ) − � w ↔ w � �= Ω M η� , ( � � ,p � ) � w, w � � .So � φ η � →η is a symplectomorphism, and in the same way we prove that for all η ≺ η � ≺ η �� , � φ η �� →η � →η is a symplectomorphism.Let η ≺ η � ≺ η �� ∈ L , eq. (2.11. ) for the maps φ implies: f η �� →η � →η ◦ φ η �� →η � →η ◦ f η �� →η ◦ φ η �� →η = f η �� →η � ◦ φ η �� →η � , � η �� →η � →η ◦ φ η �� →η � →η ◦ f η �� →η ◦ φ η �� →η = f η � →η ◦ φ η � →η ◦ � η �� →η � ◦ φ η �� →η � , & � η �� →η ◦ φ η �� →η = � η � →η ◦ φ η � →η ◦ � η �� →η � ◦ φ η �� →η � ,(where f η �� →η � →η : C η �� →η � × C η � →η → C η �� →η and � η �� →η � →η : C η �� →η � × C η � →η → C η �� →η are the projectionmaps of the Cartesian product), and, for all z, y, � ∈ C η �� →η � × C η � →η × C η :� T φ − η��→η�→η ( z,y ) ,� φ − η �� →η � ( · , ◦ � T z,y φ − η �� →η � →η � ( · ,
0) = � T z,φ − η�→η ( y,� ) φ − η �� →η � � ( · ,
0) ,� T φ − η��→η�→η ( z,y ) ,� φ − η �� →η � ( · , ◦ � T z,y φ − η �� →η � →η � (0 , · ) = � T z,φ − η�→η ( y,� ) φ − η �� →η � � (0 , · ) ◦ � T y,� φ − η � →η � ( · ,
0) , & � T φ − η��→η�→η ( z,y ) ,� φ − η �� →η � (0 , · ) = � T z,φ − η�→η ( y,� ) φ − η �� →η � � (0 , · ) ◦ � T y,� φ − η � →η � (0 , · ) ,therefore eq. (2.11. ) is fulfilled for the maps � φ . � Proposition 2.17
Let ( L , M , π ) ↓ be a projective system of phase spaces and suppose that: ∀η ∈ L , M η = T ∗ ( C η ) where C η is a smooth connected manifold; . ∀η ≺ η � ∈ L , there exist a smooth manifold C η � →η and a diffeomorphism φ η � →η : C η � → C η � →η × C η such that 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, we can complete this input into a factorizing system ( L , C , φ ) × . Proof
For η ∈ L , we define C η→η to be a space with one element and φ η→η to be the trivialidentification.Let η � η � � η �� ∈ L . What we need to show is that there exists a diffeomorphism φ η �� →η � →η : C η �� →η → C η �� →η � × C η � →η such that eq. (2.11. ) is fulfilled. If two labels among η, η � , η �� are equal, wecan choose φ η �� →η � →η to be the trivial identification, so we now consider the case η ≺ η � ≺ η �� .Let � �� , p �� ∈ T ∗ ( C η � ) and define:( y � , � � ; � � , p � ) = � φ η �� →η � ( � �� , p �� ) ,( y, � ; �, p ) = � φ η � →η ( � � , p � ) ,and ( z, � ; � • , p • ) = � φ η �� →η ( � �� , p �� ) .Now, from π η �� →η = π η � →η ◦ π η �� →η � , we have � = � • and p = p • , hence: � η � →η ◦ φ η � →η ◦ � η �� →η � ◦ φ η �� →η � ( � �� ) = � η �� →η ◦ φ η �� →η ( � �� ) ,and p �� ◦ � T y � ,� � φ − η �� →η � � (0 , · ) ◦ � T y,� φ − η � →η � (0 , · ) = p �� ◦ � T z,� φ − η �� →η � (0 , · ) ,thus, we get: � η �� →η � →η ◦ Ψ = � η �� →η ,and (0 , , · ) = [ T Ψ] (0 , · ) ,where Ψ := �id C η��→η� × φ η � →η � ◦ φ η �� →η � ◦ φ − η �� →η and � η �� →η � →η : C η �� →η � × C η � →η × C η → C η is theprojection on the third Cartesian factor.Finally, since C η is connected, there exists a diffeomorphism φ η �� →η � →η : C η �� →η → C η �� →η � × C η � →η such that Ψ = φ η �� →η � →η × id C η . � When we try to incorporate the dynamics in the formalism described in the previous section, wequickly realize that the intuitive picture we were relying on was quite oversimplified. For, althoughit should be true that we only need a finite dimensional truncation of the kinematical theory tohold the elementary kinematical observables associated to any given real experiment, in general wecannot write the dynamics in a closed form within such a truncation. s developed in appendix A, we take the point of view that from each kinematical observablearises a corresponding dynamical observable and, considering a family of functionally independentkinematical observables, it might be possible to write functional relations connecting the associateddynamical observables: here lies the predictive contents of the theory. However, such a functionalrelation can involve an infinite number of observables, and thus get silently dropped, if we neverlook at more than a finite number of observables at a time. When looking at a typical field theory,the interesting content of the dynamics lies precisely in those functional relations that can only bewritten over an infinite number of observables, and do not emerge from simpler relations withinfinite set of observables (a partial differential equation is mostly useless if we only dispose of adiscrete, finite set of initial values).On the other hand, if the the theory is to have any physically relevant predictivity, namelyif it is to be usable to formulate predictions for the output of some real experiments, it shouldat least be possible to approximate the dynamics with relations over finite sets of elementarydynamical observables (we do nothing else when elaborating numerical techniques to deal withpartial differential equations). In other words, although we may not be able to state exact predictionsfor any specific realistic experiment, we can restore predictivity in a weaker sense, by describinghow to refine an experiment and the associated approximate predictions to make them better andbetter.This concept of convergence is physically useful, notwithstanding the fact that we will not performthe infinite chain of experiments (that would again be a case of measuring an infinite number ofobservables, and we already mentioned that this is excluded in practice), because we can convertit into a notion of plausibility , by stating how to design an experimental protocol such that it willbe highly unlikely that the output lies outside some confidence domain.The object of this section is to formulate this raw idea more precisely, in order to develop aprocedure to solve constraints in a projective system of phase spaces. We begin by studying in detail under which conditions the dynamics actually can be formulatedstraightforwardly within a projective system of phase spaces, for this will be our building blockwhen addressing the generic case.Our aim here is the following: we want to write in each partial kinematical theory M ��� η aconstraint surface M ����� η , and to reassemble the resulting reduced phase spaces M �Y� η (see appendix A)into a new projective system of phase spaces. And we want to accomplish this in such a way thatwe can glue together the maps that, for each η , associate to the kinematical observables on M ��� η thecorresponding dynamical observables on M �Y� η , thus building a map from the set of all observableson the kinematical projective system into the set of all observables on the dynamical projectivesystem. For this map to accurately reproduce a given dynamics, it should give rise to functionalrelations between the dynamical observables that catch the full predictive power of the theory andit should account for the correct dynamical Poisson commutation relations.We start by looking at a symplectic manifold N ��� , that extracts, via a projection π ��� , specific π ��� γ π �Y� Here, we sketch a symplecticmanifold by a grid, each squareof which is to be thought as apoint in the manifold (and isemblematic for infinitely manyother points). M ��� M ����� M �Y� N ��� N ����� N �Y� Figure 3.1 – Phase space reductions on M ��� and N ��� , related by a projection π ��� degrees of freedom out of a bigger symplectic manifold M ��� (as were introduced in def. 2.1). Givena phase space reduction on M ��� , with reduced phase space M �Y� , we ask whether it is possible towrite closed equations, involving only the degrees of freedom retained in N ��� , and capturing allwhat the dynamics on M ��� has to say concerning these degrees of freedom.More precisely, we are looking for a phase space reduction on N ��� , but also for a projection π �Y� allowing to understand the reduced phase space N �Y� as a selection of dynamical degrees of freedomout of M �Y� (fig. 3.1). Indeed, if we consider an observable O ��� on N ��� , we can pull it back by π ��� into an observable O ��� � on M ��� . So using the dynamics on M ��� , we can obtain a correspondingdynamical observable O �Y� � on M �Y� . Now, if we can write the dynamics in closed form on N ��� , wecan also map directly O ��� to a dynamical observable O �Y� on N �Y� . The role of the projection π �Y� is then to ensure that the dynamics we have on N is actually consistent with the one on M , byrequiring O �Y� � to be precisely the pullback of O �Y� by π �Y� .If this is at all possible, both the reduction on N ��� and the projection π �Y� are uniquely determinedby the dynamics we choose on M ��� . Indeed, the constraint surface in N ��� has to be the projection by π ��� of the one in M ��� (for the constraint surface can be reconstructed if we know which kinematicalobservables are mapped to a vanishing dynamical observables), and we have from prop. A.6 (at leastin the finite dimensional case) that a reduction is completely determined by its constraint surface.Then, the uniqueness of π �Y� is enforced by requiring that it correctly makes the connection betweenthe dynamics on N ��� and the aforementioned map O ��� �→ O �Y� � (that only depends of π ��� and of thereduction on M ��� ). Definition 3.1
Let M ��� and N ��� be two symplectic manifolds and π ��� : M ��� → N ��� a surjectivemap compatible with the symplectic structures (def. 2.1). Let ( M �Y� , M ����� , δ ), resp. ( N �Y� , N ����� , γ ), bephase space reductions of M ��� , resp. N ��� (def. A.1). We say that these reductions are related by π ��� if: π ��� � M ����� � = N ����� ; there exists a surjective map π �Y� : M �Y� → N �Y� , compatible with the symplectic structures, suchthat: ∀� ∈ N ����� , ∀y � ∈ M �Y� , � ∃� � ∈ M ����� / δ ( � � ) = y � & π ��� ( � � ) = � � ⇔ � γ ( � ) = π �Y� ( y � )� .(3.1. ) Proposition 3.2
With the notations of def. 3.1, if π �Y� , and π �Y� , are two surjective maps satisfying q. (3.1. ), then π �Y� , = π �Y� , . Proof
Let y � ∈ M �Y� . Since δ is surjective, there exists � � ∈ M ����� such that δ ( � � ) = y � . Hence, π �Y� , ( y � ) = γ ◦ π ��� ( � � ) = π �Y� , ( y � ). � Proposition 3.3
We consider the same objects as in def. 3.1 and use the notations introduced indef. A.2. For f ∈ B ( N ��� ), we have f ◦ π ��� ∈ B ( M ��� ) and:( f ◦ π ��� ) �Y� = f �Y� ◦ π �Y� . Proof
Let y � ∈ M �Y� . Using eq. (3.1. ) into eq. (A.2. ), we have:( f ◦ π ��� ) �Y� ( y � ) = sup � f ◦ π ��� ( � � ) �� � � ∈ δ − �y � � �= sup � f ( � ) �� � ∈ γ − �π �Y� ( y � ) � � = f �Y� ◦ π �Y� ( y � ). � Why do we need to require eq. (3.1. ) for π �Y� instead of the seemingly more natural condition γ ◦ π ��� = π �Y� ◦ δ ? The physical reason behind eq. (3.1. ) is that we shall not look at themap δ but rather at δ − � · � , that sends a point in M �Y� to an orbit in M ����� (and similarly at γ − � · � instead of γ ), for this is the map that is dual to the application associating a kinematicalobservable to a dynamical one (in a way similar to π ��� being dual to the application that sendsan observable on N ��� into an observable on M ��� ). And, indeed, we can rewrite eq. (3.1. ) as π ��� � · � ◦ δ − � · � = γ − � · � ◦ π �Y� .That eq. (3.1. ) could fail in situations where γ ◦ π ��� = π �Y� ◦ δ does hold, can have local as wellas global causes, as illustrated by the examples below. It happens when the projection of an orbitin M ����� , though included in an orbit of N ����� , does not fill it. Proposition 3.4
If we replace in def. 3.1 the condition given by eq. (3.1. ) by the weaker assumption: γ ◦ π ��� = π �Y� ◦ δ , (3.4. )then the previous result (prop. 3.3) does not hold. Proof
As a counter example, we consider the following situation: M ��� = � R � , M �Y� = � R � , N ��� = � R � , N �Y� = R (with the standard symplectic structureon R : Ω R ( �, p ; � � , p � ) = � p � − � � p ); ∀ ( � � , p � ) �∈{ ,���, } ∈ M ��� , π ��� �( � � , p � ) �∈{ ,���, } � = ( � � , p � ) �∈{ , } ; M ����� = �( � � , p � ) �∈{ ,���, } �� p = 0 & � = � � and ∀ ( � � , p � ) �∈{ ,���, } ∈ M ����� , δ �( � � , p � ) �∈{ ,���, } � =( � � , p � ) �∈{ , } ; N ����� = �( � � , p � ) �∈{ , } �� p = 0� and ∀ ( � � , p � ) �∈{ , } ∈ N ����� , γ �( � � , p � ) �∈{ , } � = ( � , p ); ∀ ( � � , p � ) �∈{ , } ∈ M �Y� , π �Y� �( � � , p � ) �∈{ , } � = ( � , p ).We can check that ( M �Y� , M ����� , δ ) is a phase space reduction of M ��� and ( N �Y� , N ����� , γ ) is a hase space reduction of N ��� . π ��� and π �Y� are surjective maps compatible with the symplecticstructures, satisfying π ��� � M ����� � = N ����� and γ ◦ π ��� = π �Y� ◦ δ .However, if we consider f ∈ B ( N ��� ) defined by: ∀ ( � � , p � ) �∈{ , } ∈ N ��� , f �( � � , p � ) �∈{ , } � = �1 if � �
00 else ,we have f �Y� ◦ π �Y� ≡
1, but: ∀ ( � � , p � ) �∈{ , } ∈ M �Y� , ( f ◦ π ��� ) �Y� �( � � , p � ) �∈{ , } � = �1 if � �
00 else .Requiring local conditions in addition to eq. (3.4. ) would not help either, since even if everythingworks well locally, it may still goes wrong globally, as the following example shows: M ��� = � R � , M �Y� = � R � , N ��� = � R � , N �Y� = R ; ∀ ( � � , p � ) �∈{ ,���, } ∈ M ��� , π ��� �( � � , p � ) �∈{ ,���, } � = ( � � , p � ) �∈{ , } ; M ����� = �( � � , p � ) �∈{ ,���, } �� p = 0 , p = 0 & � = � + exp( � )� and ∀ ( � � , p � ) �∈{ ,���, } ∈ M ����� ,δ �( � � , p � ) �∈{ ,���, } � = ( � � , p � ) �∈{ , } ; N ����� = �( � � , p � ) �∈{ , } �� p = 0� and ∀ ( � � , p � ) �∈{ , } ∈ N ����� , γ �( � � , p � ) �∈{ , } � = ( � , p ); ∀ ( � � , p � ) �∈{ , } ∈ M �Y� , π �Y� �( � � , p � ) �∈{ , } � = ( � , p ).We can check that ( M �Y� , M ����� , δ ) is a phase space reduction of M ��� and ( N �Y� , N ����� , γ ) is aphase space reduction of N ��� . π ��� and π �Y� are surjective maps compatible with the symplecticstructures, satisfying π ��� � M ����� � = N ����� and γ ◦ π ��� = π �Y� ◦ δ .Moreover, eq. (3.1. ) holds at the linear level, namely: ∀� � ∈ M ����� , ∀� ∈ T π ��� ( � � ) ( N ����� ) , ∀w � ∈ T δ ( � � ) ( M �Y� ) , � ∃ � � ∈ T � � ( M ����� ) / T � � δ ( � � ) = w � & T � � π ��� ( � � ) = � � ⇔ � T π ��� ( � � ) γ ( � ) = T δ ( � � ) π �Y� ( w � )� ,for this reduces in the present example to: ∀� ∈ R , ∀� � ∈ R , ∀� w � ∈ R , � ∃ � � � , � � � ∈ R / � � � = � w � & � � � + exp( � ) � � � = � � � .However, if we consider the same f ∈ B ( N ��� ) as before, we have f �Y� ◦ π �Y� ≡
1, but: ∀ ( � � , p � ) �∈{ , } ∈ M �Y� , ( f ◦ π ��� ) �Y� �( � � , p � ) �∈{ , } � = �1 if � �
00 else . � Asking for the dynamics on M ��� to define a dynamics on N ��� in the sense above actually putsstrong restrictions (local as well as global ones) on what the constraint surface in M ��� can be.If we consider the special case where M ��� and N ��� are symplectic vector spaces, and π ��� is alinear map, the symplectic structure provides a natural decomposition of M ��� as P ��� ⊕ ( P ��� ) ⊥ , with P ��� = Ker π ��� and ( P ��� ) ⊥ ≈ N ��� (where the orthogonal subspace is defined with respect to the ymplectic structure; this is the linear version of prop. 2.10). What are the conditions for a vectorsubspace M ����� of M ��� to define a (linear) dynamics that will descend well through π ��� ? An obviousway of fulfilling this wish is to have a constraint surface M ����� that decomposes as M ����� = W ⊕ V where W and V are vector subsets of P ��� and ( P ��� ) ⊥ respectively: this would be a dynamics withno interaction between the degrees of freedom in N ��� and the ones in P ��� , so clearly we can writeseparately the dynamics on N ��� . However, a closer study of what is really needed shows that wehave an additional freedom to construct admissible constraint surfaces M ����� : instead of choosing V as a vector subset of ( P ��� ) ⊥ , it is enough for V to be included in W ⊥ , provided π ��� identifies therestriction to V of the symplectic structure Ω M ��� with the restriction to N ����� = π ��� �V � of Ω N ��� .This study of the linear case essentially translates to local necessary conditions in the genericcase. However, this holds only at the points in M ����� where the derivative of π ��� maps the tangentspace of the orbit of M ����� going through that point into the tangent space of an orbit of N ����� : for,although eq. (3.1. ) implies that π ��� should map an orbit into an orbit, this does not need to holdat the linear level (the derivative of a surjective map does not need to be surjective; nonethelessSard’s theorem [16] tells us, in a specific sense, that this ‘rarely’ fails). Proposition 3.5
Let M ��� and N ��� be two finite dimensional symplectic manifolds and π ��� : M ��� → N ��� a surjective map compatible with the symplectic structures. Let ( M �Y� , M ����� , δ ),resp. ( N �Y� , N ����� , γ ), be phase space reductions of M ��� , resp. N ��� . Assume these reductions arerelated by π ��� , and let � � ∈ M ����� , y � := δ ( � � ) ∈ M �Y� , � := π ��� ( � � ) ∈ N ����� and y := γ ( � ) ∈ N �Y� .Then, π ��� induces a surjective map δ − �y � � → γ − �y� .If moreover T � � π ��� � T � � � δ − �y � � �� = T � � γ − �y� �, then there exist V � � , W � � vector subspaces of T � � ( M ����� ) such that: T � � ( M ����� ) = V � � ⊕ W � � & Ω M ��� ,� � ( V � � , W � � ) = { } ; V � � ∩ Ker T � � π ��� = { } & W � � ⊂ Ker T � � π ��� ; π ��� ,∗� � Ω N ��� ,� | V �� = Ω M ��� ,� � | V �� . Proof
Let π �Y� be as in def. 3.1. . From eq. (3.1. ), we have γ ◦ π ��� = π �Y� ◦ δ , hence y = π �Y� ( y � ),and: ∀z ∈ N ����� , � z ∈ γ − �y� � ⇔ � γ ( z ) = π �Y� ( y � )� ⇔ � ∃ z � ∈ δ − �y � � / π ��� ( z � ) = z � ⇔ � z ∈ π ��� � δ − �y � � �� ,therefore π ��� � δ − �y � � � = γ − �y� .We now moreover assume that T � � π ��� induces a surjective linear map from Ker T � � δ = T � � � δ − �y � � �into Ker T � γ = T � � γ − �y� �. Then, there exist vector subspaces V �� � and W �� � of Ker T � � δ such that: W �� � = Ker T � � π ��� ∩ Ker T � � δ & Ker T � � δ = V �� � ⊕ W �� � ,and T � � π ��� induces a bijection V �� � → Ker T � γ .Next, we define the vector subspaces V y � and W y � of T y � ( M �Y� ) by: W y � := Ker T y � π �Y� & V y � := � W y � � ⊥ = � � ∈ T y � ( M �Y� ) �� Ω �Y� ,y � � �, W y � � = { } � , nd since π �Y� is compatible with the symplectic structures, we have T y � ( M �Y� ) = V y � ⊕ W y � and T y � π �Y� being surjective, it induces a bijection V y � → T y ( N �Y� ), such that: π �Y� ,∗y � Ω N �Y� ,y ��� V y� = Ω M �Y� ,y � �� V y� .Let � W � � := [ T � � δ ] − � W y � �. We have Ker T � � δ ⊂ � W � � , and from T � γ ◦ T � � π ��� �� W � � � = T y � π �Y� ◦T � � δ �� W � � � = { } and Ker T � γ = T � � π ��� � Ker T � � δ� ⊂ T � � π ��� �� W � � �, we also have T � � π ��� �� W � � � =Ker T � γ , hence there exists a vector subspace W � � of � W � � such that: W �� � ⊕ W � � = Ker T � � π ��� ∩ � W � � & � W � � = V �� � ⊕ W �� � ⊕ W � � ,and T � � δ � W � � � = T � � δ �� W � � � = W y � for T � � δ is surjective. Additionally, since T � � δ is surjective,there exists a vector subspace V � � of T � � ( M ����� ) such that: T � � ( M ����� ) = V �� � ⊕ W �� � ⊕ V � � ⊕ W � � ,with T � � δ inducing a bijective map V � � → V y � . So T � γ ◦ T � � π ��� = T y � π �Y� ◦ T � � δ induce a bijectivemap V � � → T y ( N �Y� ) = T � γ �T � ( N ����� ) � , therefore T � � π ��� induce a bijective map V �� � ⊕ V � � → T � ( N ����� ),such that, for all �, � ∈ V �� � ⊕ V � � :Ω N ��� ,� ( T � � π ��� ( � ) , T � � π ��� ( � )) = Ω N �Y� ,y ( T � γ ◦ T � � π ��� ( � ) , T � γ ◦ T � � π ��� ( � ))= Ω N �Y� ,y � T y � π �Y� ◦ T � � δ ( � ) , T y � π �Y� ◦ T � � δ ( � )�= Ω M �Y� ,y � ( T � � δ ( � ) , T � � δ ( � ))= Ω M ��� ,� � ( �, � ) .Finally, defining V � � := V �� � ⊕ V � � and W � � := W �� � ⊕ W � � , we have:Ω ��� ,� � ( V � � , W � � ) = Ω ��� ,� � � V � � , W � � � (for Ω ��� ,� � ( T � � ( M ����� ) , Ker T � � δ ) = { } )= Ω �Y� ,y � � V y � , W y � � (for Ω ��� ,� � | T �� ( M ����� ) = δ ∗� � Ω �Y� ,y � )= { } (for V y � = � W y � � ⊥ ),and W � � ⊂ Ker T � � π ��� , while Ker T � � π ��� ∩ V � � = { } . � Returning to the linear case previously mentioned, we can reformulate in terms of constraints thecondition we had for M ����� to define a closed dynamics on N ��� (through the straightforward dualitybetween the description of M ����� as a vector subspace and its description by linear constraints).This provides a specification of M ����� as characterized by three sets of constraints C P � , C N j , and C mix k , where the C P � only depend on the variables from P ��� and characterize in P ��� the projection P ����� of M ����� , similarly the C N j only depend on the variables from N ��� and characterize N ����� in N ��� , while the C mix k account for possible interactions. These interactions cannot be arbitrary: the ��� P ��� P ��X ,� P ��X ,� P ��X ,� ××× � � � N ����� M ��� = P ��� × N ��� M ����� = { ( y, � ) | � ∈ N ����� , y ∈ P ��X ,� } the P ��X ,� � are different gauge fixingsof a common constraint surface P ����� Figure 3.2 – A (rather broad but not exhaustive) way to construct an admissible dynamics on M ��� in the factorizing caserequirements on V discussed above prescribe that the constraints C mix ,�k , obtained on P ��� from the C mix k by fixing some � ∈ N ����� , should perform a partial gauge fixing of P ����� (prop. A.8).In the generic case of a symplectic manifold M ��� factorizing as M ��� = P ��� × N ��� (such asconsidered in subsection 2.3), the insight we gain from the linear case suggests a possibility,depicted in fig. 3.2, to design dynamics on M ����� that will project well on N ��� . This provides a muchbroader class of admissible dynamics than the trivial ones splitting into independent dynamics on P ��� and N ��� .Nevertheless, this procedure only corresponds to a sufficient condition for def. 3.1 to be fulfilled.Note that the gap between the necessary condition at the linear level supplied by prop. 3.5 andthe characterization of M ����� considered here does not solely arise from global considerations: for M ����� to be of this form, some additional integrability conditions (ie. requirements at the secondorder) need to hold, so that we can combine the prescriptions in the tangent space of each pointinto prescriptions in small open patches. Proposition 3.6
Let M ��� = P ��� × N ��� , where M ��� , P ��� and N ��� are finite dimensional symplecticmanifolds, and define: π ��� : M ��� → N ��� y, � �→ � .Let ( M �Y� , M ����� , δ ), ( P �Y� , P ����� , θ ), resp. ( N �Y� , N ����� , γ ), be phase space reductions of M ��� , P ��� ,resp. N ��� . Assume that there exist a submanifold P ��X of P ����� and a smooth map:Ψ : P ��X × N ����� → P ����� × N ����� y, � �→ ψ ( y, � ) , � ,such that: ImΨ = M ����� and Ψ | P ��X × N ����� → M ����� is a diffeomorphism; Ψ ∗ � Ω M ��� | T ( M ����� ) � = Ω P ��� | T ( P ��X ) × Ω N ��� | T ( N ����� ) ; . ∀� ∈ N ����� , P ��X ,� := ψ � P ��X × {�}� defines a partial gauge fixing of ( P �Y� , P ����� , θ ) (prop. A.8).Then, ( M �Y� , M ����� , δ ) and ( N �Y� , N ����� , γ ) are related by π ��� . Proof
From the definition of Ψ, we have π ��� � M ����� � = Im π ��� ◦ Ψ = N ����� .Let � ∈ N ����� . Using assumption 3.6. together with the definition of P ��X ,� , the map: ψ � : P ��X → P ��X ,� y �→ ψ ( y, � )is a diffeomorphism and, by 3.6. , it satisfies ψ �,∗ � Ω P ��� | T ( P ��X ,� ) � = Ω P ��� | T ( P ��X ) .Now, from prop. A.8, � P �Y� , P ��X ,� , θ| P ��X ,� � is a phase space reduction of P ��� . Hence, defining θ ��X ,� := θ| P ��X ,� ◦ ψ � , ( P �Y� , P ��X , θ ��X ,� ) is a phase space reduction of P ��� .Using 3.6. , we have for all � ∈ N ����� , y ∈ P ��X and for all � ∈ T � ( N ����� ), w ∈ T y ( P ��X ):0 = Ω M ��� ��� T ( y,� ) ψ � (0 , � ) , � � , �� T ( y,� ) ψ � ( w, , P ��� �� T ( y,� ) ψ � (0 , � ) , � T ( y,� ) ψ � ( w, ψ � is a diffeomorphism, � T ( y,� ) ψ � ( w,
0) runs through T ψ ( y,� ) ( P ��X ,� ) when w runsthrough T y ( P ��X ), so � T ( y,� ) ψ � (0 , � ) ∈ � T ψ ( y,� ) ( P ��X ,� )� ⊥ ∩ T ψ ( y,� ) ( P ����� ). As P ��X ,� defines a partial gaugefixing of ( P �Y� , P ����� , θ ), we have T ψ ( y,� ) ( P ����� ) = T ψ ( y,� ) ( P ��X ,� ) + K ψ ( y,� ) ( P ����� ), hence � T ( y,� ) ψ � (0 , � ) ∈K ψ ( y,� ) ( P ����� ). Therefore, ∂ � θ ��X ,� = 0.Without loss of generality, we can assume that N ����� is connected (otherwise M ����� is not con-nected either and we can consider each connected part of N ����� separately). Then, we can define θ ��X := θ ��X ,� .Using θ ��X , we define:� δ : M ����� → P �Y� × N �Y� y, � �→ ( θ ��X × γ ) ◦ � Ψ | P ��X × N ����� → M ����� � − ( y, � ) = � θ ( y ) , γ ( � )� .We want to prove that � P �Y� × N �Y� , M ����� , � δ � is a phase space reduction of M ��� .First, we need to show that � δ is surjective, that its derivative is surjective at each point, andtransports correctly the restriction to M ����� of the symplectic structure. Since � Ψ | P ����� × N ��X → M ����� � − is a diffeomorphism and transports the symplectic structure, we need only to check the correspondingproperties of θ ��X × γ . Now, since θ ��X and γ corresponds to phase space reductions, they indeedhave the required properties, and so does θ ��X × γ .Let ( y, � ) ∈ P ��X × N ����� . We choose a basis ( � � ) � � k of K � ( N ����� ) (with k := dim K � ( N ����� )) and wecomplete it into a basis ( � � ) � � � of T � ( N ����� ) (with � := dim T � ( N ����� )). We also choose a basis ( f j ) j � � of K y ( P ��X ) (with � := dim K y ( P ��X )) and complete it into a basis ( f j ) j � p of T y ( P ��X ) (with p := dim T y ( P ��X )).Then, we have: T Ψ( y,� ) ( M ����� ) = Vect ��� T ( y,� ) ψ � (0 , � � ) , � � � �� � � � � + Vect ��� T ( y,� ) ψ � ( f j , ,
0� �� j � p � .As proved above, we have ∀� ∈ T � ( N ����� ) , � T ( y,� ) ψ � (0 , � ) ∈ K ψ ( y,� ) ( P ����� ). Since ψ � is a diffeo-morphism P ��X → P ��X ,� , the � T ( y,� ) ψ � ( f j ,
0) for j � p span T ψ ( y,� ) ( P ��X ,� ). And since ψ � transports thesymplectic structure, we also have ∀w ∈ T y ( P ��X ) , � T ( y,� ) ψ � ( w, ∈ K ψ ( y,� ) ( P ��X ,� ) ⇔ w ∈ K y ( P ��X ). herefore, we have: K Ψ( y,� ) ( M ����� ) = Vect ��� T ( y,� ) ψ � (0 , � � ) , � � � �� � � k � + Vect ��� T ( y,� ) ψ � ( f j , ,
0� �� j � � � .Using K ψ ( y,� ) ( P ��X ,� ) ⊂ K ψ ( y,� ) ( P ����� ), we can now check that T Ψ( y,� ) � δ � K Ψ( y,� ) ( M ����� )� = { } . There-fore, the leaf of the foliation K ( M ����� ) that goes through Ψ( y, � ) is included in � δ − �{θ ��X ( y ) , γ ( � ) }� (as leaves of foliation are by definition connected).On the other hand, � δ − �{θ ��X ( y ) , γ ( � ) }� = Ψ � θ ��X ,− �{θ ��X ( y ) }� × γ − �{γ ( � ) }� � is connected asimage by a continuous map of a Cartesian product of connected spaces. And its tangent space atΨ( y, � ) is given by: T ( y,� ) Ψ � K y ( P ��X ) × K � ( N ����� )� = K Ψ( y,� ) ( M ����� ) (using 3.6. ).Therefore, � δ − �{θ ��X ( y ) , γ ( � ) }� is included in the leaf of the foliation K ( M ����� ) that goes throughΨ( y, � ).This concludes the proof that � P �Y� × N �Y� , M ����� , � δ � is a phase space reduction of M ��� . Now,using prop. A.6, there exists a symplectomorphism Φ : M �Y� → P �Y� × N �Y� such that Φ ◦ δ = � δ . π �Y� is then the projection corresponding to this factorization of M �Y� . � We are now ready to consider a projective system of phase spaces M ��� η , with a phase spacereduction of M ��� η for each η . As announced at the beginning of this subsection, we want toexamine the situation where the reduced phase spaces M �Y� η can be arranged into a new projectivesystem of phase spaces, in such a way that the maps, that translate the kinematical observablesinto dynamical ones for each η , are intertwined by the projections on both sides. Thus, we canassociate to an observable on the projective limit of the M ��� η an observable on the projective limitof the M �Y� η . In a dual way, to each state on this dynamical projective system of phase spacescorresponds a projective family of orbits in the constraint surfaces M ����� η (another option herewould be to consider projective family of probability measures, aka. statistical states, in whichcase we would map dynamical statistical states to on-shell supported, gauge invariant, kinematicalstatistical states).The previous study, examining a projection that relates the phase space reductions on two sym-plectic manifolds, is the key element for this construction. Indeed the requirement that the dynam-ical phase spaces should readily assemble into a new projective system can actually be enforced byasking, for each pair of index η � η � , that the reductions on M ��� η and M ��� η � should be related by π ��� η � →η . Definition 3.7
Let ( L , M ��� , π ��� ) ↓ be a projective system of phase spaces. An elementary reductionof ( L , M ��� , π ��� ) ↓ is a quadruple �� M �Y� η � η∈ L , � M ����� η � η∈ L , � π �Y� η � →η � η � η � , � δ η � η∈ L � such that: ( L , M �Y� , π �Y� ) ↓ is a projective system of phase spaces; ∀η ∈ L , ( M �Y� η , M ����� η , δ η ) is a phase space reduction of M ��� η ; . ∀η � η � ∈ L , π ��� η � →η � M ����� η � � = M ����� η and: ∀� η ∈ M ����� η , ∀y η � ∈ M �Y� η � , � ∃� η � ∈ M ����� η � / δ η � ( � η � ) = y η � & π ��� η � →η ( � η � ) = � η � ⇔ � δ η ( � η ) = π �Y� η � →η ( y η � )� .Whenever possible, we will use the shortened notation ( L , M , π, δ ) �Y� instead of �� M �Y� η � η∈ L , � M ����� η � η∈ L , � π �Y� η � →η � η � η � , � δ η � η∈ L �. Definition 3.8
We consider the same objects as in def. 3.7 and we define (in analogy to the definitionof S ↓ ( L , M ,π ) in def. 2.3) � S ↓ ( L , M ��� ,π ��� ) as:� S ↓ ( L , M ��� ,π ��� ) := ( D η ) η∈ L ∈ � η∈ L P ( M ��� η ) ������ ∀η � η � , π ��� η � →η �D η � � = D η ,where, for η ∈ L , P ( M ��� η ) is the set of subsets of M ��� η .Then, we define:Δ : S ↓ ( L , M �Y� ,π �Y� ) → � S ↓ ( L , M ��� ,π ��� ) ( y η ) η∈ L �→ � δ − η �{y η }� � η∈ L ,which is well-defined as a map S ↓ ( L , M �Y� ,π �Y� ) → � S ↓ ( L , M ��� ,π ��� ) , for we have ∀η � η � ∈ L , ∀y η � ∈ M �Y� η � , δ − η �� π �Y� η � →η ( y η � )�� = π ��� η � →η � δ − η � �y η � � �. Proposition 3.9
Let ( L , M ��� , π ��� ) ↓ be a projective system of phase spaces and let ( L , M , π, δ ) �Y� be an elementary reduction of ( L , M ��� , π ��� ) ↓ . We define (in analogy to def. 2.4) A ↓ ( L , M ��� ,π ��� ) as theset of equivalence classes in � η∈ L B ( M η ) for the equivalence relation defined by: ∀η, η � ∈ L , ∀f η ∈ B ( M ��� η ) , ∀f η � ∈ B ( M ��� η � ) ,f η ∼ ��� f η � ⇔ ( ∃ η �� ∈ L / η � η �� , η � � η �� & f η ◦ π ��� η �� →η = f η � ◦ π ��� η �� →η � ) ,and similarly A ↓ ( L , M �Y� ,π �Y� ) with the equivalence relation ∼ �Y� .Then, the map:( · ) �Y� : A ↓ ( L , M ��� ,π ��� ) → A ↓ ( L , M �Y� ,π �Y� ) [ f η ] ∼ ��� �→ � f �Y� η � ∼ �Y� is well-defined.For ( D η ) η∈ L ∈ � S ↓ ( L , M ��� ,π ��� ) and f = [ f η ] ∼ ��� ∈ A ↓ ( L , M ��� ,π ��� ) , we define:[ f η ] ∼ ��� �( D η ) η∈ L � := sup {f η ( � ) | � ∈ D η } ,(the definition of the equivalence relation ∼ ��� ensures that this is well-defined)Then, we have for all y ∈ S ↓ ( L , M �Y� ,π �Y� ) and all f ∈ A ↓ ( L , M ��� ,π ��� ) : �Y� ( y ) = f (Δ( y )) . (3.9. ) Proof
What we need to show is that for η, η � ∈ L , f η ∈ B ( M η ) and f η � ∈ B ( M η � ), � f η ∼ ��� f η � � ⇒ � f �Y� η ∼ �Y� f �Y� η � �. Indeed if there exist η �� ∈ L , with η �� � η , η �� � η � , and f η �� ∈ B ( M η �� ) such that f η ◦ π ��� η �� →η = f η � ◦ π ��� η �� →η � , then, from prop. 3.3: f �Y� η ◦ π �Y� η �� →η = � f η ◦ π ��� η �� →η � �Y� = � f η � ◦ π ��� η �� →η � � �Y� = f �Y� η � ◦ π �Y� η �� →η � .Then, we only need to check eq. (3.9. ) for a particular representative f η of f : f �Y� ( y ) = f �Y� η ( y η ) = sup �∈δ − η �{ y η }� f η ( � ) = sup �∈ ( δ �Y� ( y )) η f η ( � ) = f (Δ( y )) . � Proposition 3.10
Let ( L , M ��� , π ��� ) ↓ be a projective system of phase spaces. For all η ∈ L , we giveourselves a phase space reduction ( M �Y� η , M ����� η , δ η ) of M η . The following statements are equivalent: there exists a family of surjective maps � π �Y� η � →η � η � η � such that ( L , M , π, δ ) �Y� is an elementaryreduction of ( L , M ��� , π ��� ) ↓ ; ∀η � η � , ( M �Y� η � , M ����� η � , δ η � ) and ( M �Y� η , M ����� η , δ η ) are related by π ��� η � →η . Proof
By definition of an elementary reduction of ( L , M ��� , π ��� ) ↓ , we have 3.10. ⇒ .To prove the other direction, we need to show that the π �Y� η � →η induced by the π ��� η � →η satisfy thethree-spaces consistency condition: ∀η � η � � η �� ∈ L , π �Y� η �� →η = π �Y� η � →η ◦ π �Y� η �� →η � .For � η ∈ M ����� η , y η �� ∈ M �Y� η �� , we have (using def. 3.7. for η � η � and η � � η �� ):� δ η ( � η ) = π �Y� η � →η � π �Y� η �� →η � ( y η �� )�� ⇔ � ∃� η � ∈ M ����� η � / δ η � ( � η � ) = π �Y� η �� →η � ( y η �� ) & π ��� η � →η ( � η � ) = � η � ⇔ � ∃� η � ∈ M ����� η � , ∃� η �� ∈ M ����� η �� / δ η �� ( � η �� ) = y η �� & π ��� η �� →η � ( � η �� ) = � η � & π ��� η � →η ( � η � ) = � η � ⇔ � ∃� η �� ∈ M ����� η �� / δ η �� ( � η �� ) = y η �� & π ��� η � →η � π ��� η �� →η � ( � η �� )� = � η � .Hence, using π ��� η � →η ◦ π ��� η �� →η � = π ��� η �� →η , and applying prop. 3.2 with π �Y� η �� →η and π �Y� η � →η ◦ π �Y� η �� →η � , wehave π �Y� η �� →η = π �Y� η � →η ◦ π �Y� η �� →η � . � Recalling the discussion of subsection 2.2, regarding restrictions and extensions of the label set,we would like to understand how elementary reductions pass through these operations. It is quitestraightforward that everything will go smoothly if we restrict the label set.The interesting question occurs when we have an elementary reduction on a subset L � of L . Inparticular, if L � is cofinal in L , we can identify the kinematical spaces of states and observablesover the projective system restricted to L � with the ones over the original projective system on L (prop. 2.5), thus the transport of observables (from the kinematical to the dynamical theory) andstates (from the dynamical to the kinematical theory) arising from an elementary reduction on L � mmediately defines corresponding transport maps between the kinematical projective structure on L and the dynamical projective structure (which is then only defined for the label set L � ). In otherwords, we are still able to glue together the dynamical phase spaces M �Y� η ( η ∈ L � ) into a dynamicalprojective structure, to inherit observables on this structure and to project back its states. However,in general, there will not exist an elementary reduction on L , that would reproduce the sametransport maps (modulo the identification of the thus obtained dynamical structure on L with itsrestriction to L � ). This point will play a key role when moving to the regularization of a dynamicsthat does not break down well on the projective structure (subsection 3.2).What is lacking, when trying to extend to L an elementary reduction on L � , is the assurancethat there will exist phase space reductions of the M ��� η for η ∈ L \ L � , and that these reductionswill be compatible with each other, as well as with the given reductions on L � : specifically, forany pair of labels η � η � (one or both being in L \ L � ) the reductions should be related by π ��� η � →η (the elementary reduction on L � already accounts for the compatibility when both labels are in L � ).Prop. 3.12 shows slightly weaker hypotheses under which the extension is possible, provided the M ��� η for η ∈ L \ L � are finite dimensional. Proposition 3.11
Let ( L , M ��� , π ��� ) ↓ be a projective system of phase spaces and ( L , M , π, δ ) �Y� be an elementary reduction of ( L , M ��� , π ��� ) ↓ . If L � is a directed subset of L , � L � , M , π, δ � �Y� isan elementary reduction of � L � , M ��� , π ��� � ↓ and we have:� σ ��� L → L � ◦ Δ = Δ � ◦ σ �Y� L → L � ,where � σ ��� L → L � : � S ↓ ( L , M ��� ,π ��� ) → � S ↓ ( L � , M ��� ,π ��� ) , σ �Y� L → L � : S ↓ ( L , M �Y� ,π �Y� ) → S ↓ ( L � , M �Y� ,π �Y� ) are defined in analogyto prop. 2.5, while Δ : S ↓ ( L , M �Y� ,π �Y� ) → � S ↓ ( L , M ��� ,π ��� ) and Δ � : S ↓ ( L � , M �Y� ,π �Y� ) → � S ↓ ( L � , M ��� ,π ��� ) are defined asin def. 3.8.In addition, for any f ∈ A ↓ ( L � , M ��� ,π ��� ) , we have:� β ��� L ← L � ( f )� �Y� = β �Y� L ← L � ( f �Y� ) ,where β ��� / �Y� L ← L � : A ↓ ( L � , M ��� / �Y� ,π ��� / �Y� ) → A ↓ ( L , M ��� / �Y� ,π ��� / �Y� ) are defined in analogy to prop. 2.5, while( · ) �Y� : A ↓ ( L , M ��� ,π ��� ) → A ↓ ( L , M �Y� ,π �Y� ) and ( · ) �Y� : A ↓ ( L � , M ��� ,π ��� ) → A ↓ ( L � , M �Y� ,π �Y� ) are defined as in prop. 3.9. Proof
That � L � , M , π, δ � �Y� is an elementary reduction of � L � , M ��� , π ��� � ↓ can be immediatelychecked from def. 3.7.Let � y η � η∈ L ∈ S ↓ ( L , M �Y� ,π �Y� ) . We have:� σ ��� L → L � ◦ Δ �� y η � η∈ L � = � σ ��� L → L � �� δ − η �y η � � η∈ L � = � δ − η �y η � � η∈ L � = Δ � �� y η � η∈ L � � = Δ � ◦ σ �Y� L → L � �� y η � η∈ L � .Let f = [ f η ] ∼ ��� ∈ A ↓ ( L � , M ��� ,π ��� ) . We have:� β ��� L ← L � ( f )� �Y� = � f �Y� η � ∼ �Y� = β �Y� L ← L � ( f �Y� � ) . Proposition 3.12
Let ( L , M ��� , π ��� ) ↓ be a projective system of phase spaces and let L � be a cofinalsubset of L . We assume: that we are given an elementary reduction � L � , M , π, δ � �Y� of � L � , M ��� , π ��� � ↓ ; that for any η ∈ L \ L � , M ��� η is finite dimensional and we are given a phase space reduction� M �Y� η , M ����� η , δ η � of M ��� η ; that for any η ∈ L , and for any η � ∈ L � with η � � η , the reductions on M ��� η � and M ��� η are relatedby π ��� η � →η .Then, � L � , M , π, δ � �Y� can be completed into an elementary reduction ( L , M , π, δ ) �Y� of ( L , M ��� ,π ��� ) ↓ . Lemma 3.13
Let M ��� , N ��� and P ��� be symplectic manifolds and assume that N ��� and P ��� are finitedimensional. Let π ��� : M ��� → N ��� , π ��� : N ��� → P ��� , and π ��� : M ��� → P ��� be projections compatiblewith the symplectic structures, satisfying π ��� = π ��� ◦ π ��� . Let ( M �Y� , M ����� , δ ), ( N �Y� , N ����� , γ ) and( P �Y� , P ����� , η ) be phase space reductions of M ��� , N ��� and P ��� respectively.If the reductions on M ��� and N ��� are related by π ��� and the reductions on M ��� and P ��� arerelated by π ��� , then the reductions on N ��� and P ��� are related by π ��� . Proof
Applying def. 3.1. for π ��� and π ��� , we have: π ��� � N ����� � = π ��� �π ��� � M ����� �� = π ��� � M ����� � = P ����� .Let π �Y� : M �Y� → N �Y� and π �Y� : M �Y� → P �Y� be as in def. 3.1. . For any y, y � ∈ M �Y� such that π �Y� ( y ) = π �Y� ( y � ), there exists z ∈ γ − �π �Y� ( y ) � ⊂ N ����� (for γ is surjective from def. A.1. ) and, usingeq. (3.1. ), there exist �, � � ∈ M ����� such that: δ ( � ) = y , δ ( � � ) = y � and π ��� ( � ) = z = π ��� ( � � ).Therefore, π ��� ( � ) = π ��� ( z ) = π ��� ( � � ), so using again eq. (3.1. ), we have π �Y� ( y ) = η ◦ π ��� ( z ) = π �Y� ( y � ).Hence, π �Y� is constant on the level sets of π �Y� , so there exists a map π �Y� : N �Y� → P �Y� such that π �Y� = π �Y� ◦ π �Y� .Now, for z � ∈ P ����� and w ∈ N �Y� , there exists y ∈ M �Y� such that π �Y� ( y ) = w (for π �Y� issurjective) and we have:� η ( z � ) = π �Y� ( w )� ⇔ � η ( z � ) = π �Y� ( y )� ⇔ � ∃ � ∈ M ����� / δ ( � ) = y & π ��� ( � ) = z � � ⇔ � ∃ z ∈ N ����� / γ ( z ) = w & π ��� ( z ) = z � � ,where the last equivalence comes from setting z = π ��� ( � ) (for proving ‘ ⇒ ’) and using eq. (3.1. )with γ ( z ) = π �Y� ( y ) (for ‘ ⇐ ’). Hence, π �Y� fulfills eq. (3.1. ).In particular, we then have η ◦ π ��� = π �Y� ◦ γ . Thus, since N ��� , N �Y� and P �Y� are smooth finitedimensional manifolds, η ◦ π ��� is smooth and γ is surjective with surjective derivative at any point(def. A.1. ), the rank theorem implies [11, prop. 5.19] that π �Y� is smooth. inally, we need to show that π �Y� is a surjective map compatible with the symplectic structures.We have π �Y� � N �Y� � = π �Y� �π �Y� � M �Y� �� = π �Y� � M �Y� � = P �Y� . And, for any w ∈ N �Y� , there exists y ∈ M �Y� with π �Y� ( y ) = w , so that for any υ ∈ T ∗π �Y�2 ( w ) ( P �Y� ):[ T w π �Y� ] � υ ◦ [ T w π �Y� ]� = [ T w π �Y� ] ◦ � T y π �Y� � � υ ◦ [ T w π �Y� ] ◦ � T y π �Y� ��= � T y π �Y� � � υ ◦ � T y π �Y� �� = υ ,therefore π �Y� fulfills eq. (2.1. ). � Proof of prop. 3.12
Let η ∈ L and η � ∈ L \ L � , with η � � η . Since L � is a cofinal part of L , thereexists η �� ∈ L � such that η �� � η � � η . Using lemma 3.13 ( M ��� η � is finite dimensional, for η � ∈ L \ L � ,so M ��� η is finite dimensional, for η � η � ), the reductions on M ��� η � and M ��� η are related by π ��� η � →η .Hence, using prop. 3.10, there exists an elementary reduction ( L , M , � π , δ ) �Y� of ( L , M ��� , π ��� ),where ∀η � η � ∈ L , � π ��� η � →η = π ��� η � →η . And by prop. 3.2, ∀η � η � ∈ L � , � π �Y� η � →η = π �Y� η � →η , which suppliesthe desired result. � In practice, we will be interested in a kinematical projective structure that is a rendering, bya system of finite dimensional manifolds M ��� η , of an infinite dimensional symplectic manifold M ��� ∞ (def. 2.6). If the phase space reduction on M ��� ∞ satisfies the (admittedly very restrictive)requirement that it projects as a closed dynamics on M ��� η for all η , we will get an elementaryreduction of the kinematical projective structure, and the thus obtained dynamical projective systemwill automatically be a rendering of the physical phase space M �Y� ∞ (fig. 3.3).Moreover, the map turning observables on the kinematical projective structure into observableson the dynamical structure coincides with the one that can be defined directly from the phase spacereduction of M ��� ∞ (identifying the observables on the projective structures with functions on M ��� ∞ or M �Y� ∞ , as described in def. 2.6). It cannot be too much emphasized that this is a crucial point, fora physical theory is more than just a space of states: it is also a labeling of the observables overthis state space, that associates to the elementary observables a particular physical meaning. Thislabeling is the interface that allows us to make the connection between a given concrete measureprotocol and an observable of the theory, between the experimental world and the mathematicalformalism. Hence, from a physical point of view, a rendering of the physical phase space wouldbe useless if we do not tell at the same time how the elementary observables of our theory areconstructed in this rendering.As already mentioned above, we have, dual to the translation of observables, the possibility oftransporting dynamical states back to the kinematical theory (as projective families of orbits), andagain this transport reflects the map δ − ∞ � · � that sends a point in M �Y� ∞ to an orbit in M ����� ∞ . This isprobably not needed when the constraints are there to implement dynamics, since, as soon as wehave obtained the physical state space (and observables thereon!), the kinematical theory has playedits role and can be discarded. However, the same mathematical formalism of imposing constraintscan also describe the symmetry restriction of a theory. It has in this case an entirely differentphysical interpretation, and we are then not only interested in the symmetry restricted theory itself,but we also want to understand its states as special, symmetric states in the full theory (note thatthe constraints describing symmetry restriction being second class, we map a state on the restricted ∞ δ η � � � L � {∞} M ����� ∞ ⊂ M ��� ∞ ∞ M ����� η ⊂ M ��� η η π ��� ∞→η � Y � L � {∞} M �Y� ∞ ∞ M �Y� η η π �Y� ∞→η Figure 3.3 – Elementary reduction and renderingstate to a state on the unrestricted side: orbits are in this case just single points).
Proposition 3.14
Let ( L , M ��� , π ��� ) ↓ be a rendering of a (possibly infinite dimensional) symplecticmanifold M ��� ∞ (def. 2.6) and let ( M �Y� ∞ , M ����� ∞ , δ ∞ ) be a phase space reduction of M ��� ∞ . We supposethat, for all η ∈ L : M η is finite dimensional; we are given a phase space reduction � M �Y� η , M ����� η , δ η � of M ��� η that is related by π ��� ∞→η to thereduction of M ��� ∞ .Then, we have an elementary reduction ( L , M , π, δ ) �Y� of ( L , M ��� , π ��� ) ↓ and a rendering of M �Y� ∞ by ( L , M �Y� , π �Y� ) ↓ such that, for any y ∞ ∈ M �Y� ∞ :� σ ��� ↓ � δ − ∞ �y ∞ � � = Δ ◦ σ �Y� ↓ ( y ∞ ) , (3.14. )where � σ ��� ↓ : P ( M ��� ∞ ) → � S ↓ ( L , M ��� ,π ��� ) , σ �Y� ↓ : M �Y� ∞ → S ↓ ( L � , M �Y� ,π �Y� ) are defined in analogy to def. 2.6.Moreover, for any f ∈ A ↓ ( L , M ��� ,π ��� ) , we have:� β ��� ↑ ( f )� �Y� = β �Y� ↑ ( f �Y� ), (3.14. )where β ��� / �Y� ↑ : A ↓ ( L , M ��� / �Y� ,π ��� / �Y� ) → B ( M ��� / �Y� ∞ ) are defined in the same way as α ��� / �Y� ↑ : O ↓ ( L , M ��� / �Y� ,π ��� / �Y� ) →C ∞ ( M ��� / �Y� ∞ , R ) (def. 2.6). Proof
From prop. 3.12, we can complete the phase space reduction ( M �Y� ∞ , M ����� ∞ , δ ∞ ) of M ��� ∞ into anelementary reduction ( L � {∞} , M , π, δ ) �Y� of ( L � {∞} , M ��� , π ��� ) ↓ . In particular, ( L � {∞} , M �Y� ,π �Y� ) ↓ is a projective system of phase spaces; in other words, ( L , M �Y� , π �Y� ) ↓ is a rendering of M �Y� ∞ .Eq. (3.14. ) and eq. (3.14. ) then follow by applying twice the corresponding results from prop. 3.11(to go down from L � {∞} to both L and {∞} ), together with:� σ ��� ↓ = � σ ��� L �{∞}→ L ◦ � σ ��� ,− L �{∞}→{∞} & σ �Y� ↓ = σ �Y� L �{∞}→ L ◦ σ �Y� ,− L �{∞}→{∞} ,and: ��� ↑ = β ��� ,− L �{∞}←{∞} ◦ β ��� L �{∞}← L & β �Y� ↑ = β �Y� ,− L �{∞}←{∞} ◦ β �Y� L �{∞}← L . � We now turn to the general case, where, typically, the prerequisites of the previous section will not be satisfied.Recall that, as underlined above (prop. 3.12), these prerequisites will become milder and milder ifwe look for elementary reductions only defined on smaller and smaller cofinal subsets of the labelset L : the argument is that it’s easier to write closed dynamics over truncations of the theory ifwe consent to give up the coarsest truncations for lost and to only try to formulate such truncateddynamics in partial theories retaining enough elementary observables (and being thus able to exhibitfiner properties of the states). On the other hand, for what we are interested in (namely, defining aprojective structure for the dynamical theory, constructing on it the observables inherited from thekinematical theory, and, if need be, embedding its states in the initial projective structure), suchan elementary restriction restricted to a cofinal part of L is all we need.This observation motivates the following strategy: we will try to design an approximating scheme,indexed by a directed set E , that approaches the exact constraints (unadapted to the projectivestructure) by approximate constraints, projecting well on the M ��� η at least for all η ∈ L ε (where ε ∈ E parametrizes the level of approximation and L ε is a cofinal part of L that depends on ε ). Weexpect that the label subset L ε will get smaller and smaller (yet remaining cofinal), since formulatingmore accurate approximations of the dynamics will require deeper and deeper knowledge of theproperties of the states (such knowledge that is only accessible in partial theories with labels at thehigh end of L ).As an illustration of this idea, suppose that L consists of all possible finite subsets of points onthe real line (ordered by inclusion), take E to be the set of positive reals ε and define L ε ⊂ L toselect those subsets in which next neighbor points have a distance of at most ε . Thus a label η ∈ L will only qualify for belonging to L ε if, given a real function f , it can provide an approximation f | η with at least a resolution of ε (over the convex hull of η ). As ε gets smaller and smaller, weretain less and less labels η , yet L ε will keep cofinal. Notice that in this example we would useon E the reverse order ε � ε � ⇔ ε � ε � , because we think of the partial order on E in terms ofcoarser lattices being included in finer ones, rather than as an ordering of the lattice parameters(thus we will sometimes refer to the continuum limit as ε = ∞ , in the sense of having an infinitelyfine lattice, although in the present case ε = 0 would have been more intuitive).To make it more precise what we mean by approaching the exact constraints, we want theapproximation scheme to come with an additional input, namely a family of projections, goingfrom the space of exact solutions of the dynamics M �Y� into each space of approximate solutions M �Y� ,ε : it will tell us, for each level of approximation ε ∈ E , how to map the exacts orbits in M ����� to their approximate versions in M ����� ,ε . In other words, we will associate to each orbit in the exactconstraint surface a family (indexed by E ) of orbits intended to approach it, thus setting the stageto formulate a notion of convergence (this point will be examined more closely in the second halfof the present subsection). esides, it is sensible that the map from M �Y� to M �Y� ,ε does not retain all degrees of freedom,so that it only depends on the most distinctive properties of the dynamical states in M �Y� : theapproximation of an exact solution should drop those finest details, that can anyway not be handledcorrectly by the coarse dynamics underlying M �Y� ,ε . More precisely, we will require that the familyof approximated theories build on their own a projective system of phase spaces (with label set E ),and, in addition, we would like the approximating maps, bringing us from a finer approximateddynamical theory to a coarser one, to be expressible at the level of the truncated theories M �Y� ,εη , sothat we can assemble all M �Y� ,εη into a big projective system of phase spaces (whose label set will bea part of E × L ). The return of these quite restrictive requirements is that it supplies immediatelya dynamical projective structure, where we can represent the dynamical states and start doingcalculations with them, even before we have settled the question of convergence.Clearly, we are assuming here that we are provided with some non-trivial input, that will haveto come from a precise understanding of the system under study. The examples in [10], besidesdemonstrating that the procedure described here can indeed be put into practice in simple systems,also give some insights on how the needed input can be obtained, but it will require more extensiveinvestigations to develop systematic ways of constructing suitable approximating schemes in thesense above. Proposition 3.15
Let E , � and L , � be preordered, directed sets and suppose there exists for all ε ∈ E a cofinal part L ε of L such that: ∀ε � ε � , L ε ⊃ L ε � .We define EL := { ( ε, η ) | ε ∈ E , η ∈ L ε } and equip it with the preorder: ∀ ( ε, η ) , ( ε � , η � ) ∈ EL , ( ε, η ) � ( ε � , η � ) ⇔ � ε � ε � & η � η � �.Then EL , � is directed. Proof
Let ( ε, η ) , ( ε � , η � ) ∈ EL . Since E and L are directed, there exist ε �� ∈ E and � η ∈ L such that ε, ε � � ε �� and η, η � � � η . L ε �� being cofinal in L , there exists η �� ∈ L ε �� / � η � η �� . � Definition 3.16
Let ( L , M ��� , π ��� ) ↓ be a projective system of phase spaces. A regularized reductionof ( L , M ��� , π ��� ) ↓ is a sextuple:� E , ( L ε ) ε∈ E , � M �Y� ,εη � ( ε,η ) ∈ EL , � M ����� ,εη � ( ε,η ) ∈ EL , � π �Y� ,ε � →εη � →η � ( ε,η ) � ( ε � ,η � ) , � δ εη � ( ε,η ) ∈ EL �such that: E is a directed set indexing a family ( L ε ) ε∈ E of decreasing ( ∀ε � ε � , L ε ⊃ L ε � ), cofinal parts of L as in prop. 3.15; ∀ε ∈ E , �� M �Y� ,εη � η∈ L ε , � M ����� ,εη � η∈ L ε , � π �Y� ,ε→εη � →η � η � η � , � δ εη � η∈ L ε � is an elementary reduction of( L ε , M ��� , π ��� ) ↓ ; � EL , � M �Y� ,εη � ( ε,η ) ∈ EL , � π �Y� ,ε � →εη � →η � ( ε,η ) � ( ε � ,η � ) � is a projective system of phase spaces. henever possible, we will use the shortened notation ( L , M , π, δ ) �Y� , E instead of � E , ( L ε ) ε∈ E , � M �Y� ,εη � ( ε,η ) ∈ EL , � M ����� ,εη � ( ε,η ) ∈ EL , � π �Y� ,ε→εη � →η � ( ε,η ) � ( ε � ,η � ) , � δ εη � ( ε,η ) ∈ EL �. At that point we have written a projective structure for the dynamical theory, but as we em-phasized in the previous subsection, constructing the space of physical states is of little use if wedo not prescribe how to define on it the observables inherited from the kinematical theory. As afirst step in this direction, we will construct maps that transport kinematical observables into thedynamical theory at some level of approximation ε : given a particular kinematical observable, thedynamical observables constructed this way, for all possible ε , should be thought of as successiveapproximations of the exact dynamical version of this kinematical observable.Moreover, we can check that these maps transform well under restriction of the label sets L and E (provided the label subsets L � and E � considered are such that we still have a regularizedreduction after restricting ourselves to E � L � ). We will make use of this result at the end of thepresent subsection, when we will consider how regularized reductions interact with renderings. Definition 3.17
We consider the same objects as in def. 3.16. For ε ∈ E , we define Δ ε : S ↓ ( EL , M �Y� ,π �Y� ) → � S ↓ ( L , M ��� ,π ��� ) as:Δ ε := � σ ��� ,− L → L ε ◦ Δ ε L ε ◦ σ �Y� EL → L ε ,where σ �Y� EL → L ε : S ↓ ( EL , M �Y� ,π �Y� ) → S ↓ ( L ε , M �Y� ,ε ,π �Y� ,ε→ε ) is defined as in prop. 2.5 (for the directed part { ( ε, η ) | η ∈ L ε } of EL ), Δ ε L ε : S ↓ ( L ε , M �Y� ,ε ,π �Y� ,ε→ε ) → � S ↓ ( L ε , M ��� ,π ��� ) is defined as in def. 3.8 (for theelementary reduction ( L ε , M ε , π ε→ε , δ ε ) ↓ of ( L ε , M ��� , π ��� ) ↓ ), and � σ ��� L → L ε : � S ↓ ( L , M ��� ,π ��� ) → � S ↓ ( L ε , M ��� ,π ��� ) is defined in analogy to prop. 2.5 (for the cofinal part L ε of L ).Similarly, we define ( · ) ε : A ↓ ( L , M ��� ,π ��� ) → A ↓ ( EL , M �Y� ,π �Y� ) as:( · ) ε := β �Y� EL ← L ε ◦ ( · ) �Y� ,ε ◦ β ��� ,− L ← L ε ,where β �Y� EL ← L ε : A ↓ ( L ε , M �Y� ,ε ,π �Y� ,ε→ε ) → A ↓ ( EL , M �Y� ,π �Y� ) and β ��� L ← L ε : A ↓ ( L ε , M ��� ,π ��� ) → A ↓ ( L , M ��� ,π ��� ) aredefined as in analogy to prop. 2.5 (for the directed part { ( ε, η ) | η ∈ L ε } of EL and the cofinal part L ε of L ) and ( · ) �Y� ,ε : A ↓ ( L ε , M �Y� ,ε ,π �Y� ,ε→ε ) → A ↓ ( L ε , M ��� ,π ��� ) is defined as in prop. 3.9 (for the elementaryreduction ( L ε , M ε , π ε→ε , δ ε ) ↓ of ( L ε , M ��� , π ��� ) ↓ ).We have for all y ∈ S ↓ ( EL , M �Y� ,π �Y� ) and all f ∈ A ↓ ( L , M ��� ,π ��� ) : f ε ( y ) = f (Δ ε ( y )) . Proposition 3.18
Let ( L , M ��� , π ��� ) ↓ be a projective system of phase spaces and let ( L , M , π, δ ) �Y� , E be a regularized reduction of ( L , M ��� , π ��� ) ↓ . Let L � and E � be directed subsets of L and E respec-tively, such that, for all ε ∈ E � , L �ε := L ε ∩ L � is a cofinal part of L � .Then, � L � , M , π, δ � �Y� , E � is a regularized reduction of � L � , M ��� , π ��� � ↓ and, for any ε ∈ E � , wehave: σ ��� L → L � ◦ Δ ε = Δ �ε ◦ σ �Y� EL → E � L � , (3.18. )where � σ ��� L → L � : � S ↓ ( L , M ��� ,π ��� ) → � S ↓ ( L � , M ��� ,π ��� ) , σ �Y� EL → E � L � : S ↓ ( EL , M �Y� ,π �Y� ) → S ↓ ( E � L � , M �Y� ,π �Y� ) are defined inanalogy to prop. 2.5, while Δ ε : S ↓ ( EL , M �Y� ,π �Y� ) → � S ↓ ( L , M ��� ,π ��� ) and Δ �ε : S ↓ ( E � L � , M �Y� ,π �Y� ) → � S ↓ ( L � , M ��� ,π ��� ) are defined as in def. 3.17.In addition, for any f ∈ A ↓ ( L � , M ��� ,π ��� ) and any ε ∈ E � , we have:� β ��� L ← L � ( f )� ε = β �Y� EL ← E � L � ( f ε ) , (3.18. )where β ��� L ← L � : A ↓ ( L � , M ��� ,π ��� ) → A ↓ ( L , M ��� ,π ��� ) and β �Y� L ← L � : A ↓ ( E � L � , M �Y� ,π �Y� ) → A ↓ ( EL , M �Y� ,π �Y� ) are de-fined in analogy to prop. 2.5, while ( · ) ε : A ↓ ( L , M ��� ,π ��� ) → A ↓ ( EL , M �Y� ,π �Y� ) and ( · ) ε : A ↓ ( L � , M ��� ,π ��� ) → A ↓ ( E � L � , M �Y� ,π �Y� ) are defined as in def. 3.17. Proof L �ε being a cofinal part of L � for all ε ∈ E � ensures that def. 3.16. is fullfiled. Moreover, E � L � is then a directed subset of EL , hence def. 3.16. holds. Lastly, def. 3.16. follows from prop. 3.11,since, for any ε ∈ E � , L �ε is a directed part of L ε (as a cofinal part of the directed set L � ).Let ε ∈ E � . We have L ⊃ L � , L ε ⊃ L �ε , hence:� σ ��� L � → L �ε ◦ � σ ��� L → L � = � σ ��� L ε → L �ε ◦ � σ ��� L → L ε .And, from EL ⊃ E � L � , L ε ⊃ L �ε (identifying L ε with the subset { ( ε, η ) | η ∈ L ε } of EL and L �ε with the subset �( ε, η ) �� η ∈ L �ε � of E � L � as in def. 3.17), we also have: σ �Y� L ε → L �ε ◦ σ �Y� EL → L ε = σ �Y� E � L � → L �ε ◦ σ �Y� EL → E � L � .So, using the definition of Δ ε and Δ �ε from def. 3.17:� σ ��� L → L � ◦ Δ ε = � σ ��� L → L � ◦ � σ ��� ,− L → L ε ◦ Δ ε L ε ◦ σ �Y� EL → L ε = � σ ��� ,− L � → L �ε ◦ � σ ��� L ε → L �ε ◦ Δ ε L ε ◦ σ �Y� EL → L ε = � σ ��� ,− L � → L �ε ◦ Δ �ε L �ε ◦ σ �Y� L ε → L �ε ◦ σ �Y� EL → L ε (using prop. 3.11)= � σ ��� ,− L � → L �ε ◦ Δ �ε L �ε ◦ σ �Y� E � L � → L �ε ◦ σ �Y� EL → E � L � = Δ �ε ◦ σ �Y� EL → E � L � .Similarly, we have:( · ) ε ◦ β ��� L ← L � = β �Y� EL ← E � L � ◦ ( · ) ε . � Now, we would like to give a precise definition of the convergence we have been hinting atrepeatedly above. To ascertain convergence is crucial for ensuring that we will get consistentpredictions when refining the level of approximation at which we are conducting the calculations.Our unchanged goal is to make it possible to transport kinematical observables over to thedynamical theory, not only in an approximated fashion, but in such a way that we faithfully realizethe transport prescribed by the exact dynamics we are trying to implement (and, if the constraints K N ∞ N � �� N � � N � M Figure 3.4 – Convergence of a net of orbitswe are considering describe a symmetry restriction, we are also interested in the correct embeddingof the symmetric states in the full theory; as we already underlined in the previous subsection, themap between observables and the one between states are the two dual aspects of the bond betweenthe initial or kinematical theory and the restricted or dynamical one). Additionally, we would liketo be able to investigate the properties of this correct dynamics (at a certain level of precision) bymaking use of the approximated dynamics, where calculations will probably be more tractable.The straightforward course is to obtain the correct transport map as the limit of the net ofapproximated maps introduced previously. As illustrated in fig. 3.4, we begin by defining, givena family of orbits in a manifold, a notion of convergence, which is adjusted to our method fortransposing kinematical observables into dynamical ones (as explained in appendix A, this methodis itself motivated by taking the indicator functions as model observables).
Definition 3.19
Let M be a finite dimensional manifold and let ( N ε ) ε∈ E be a net of subsets of M .We say that the net ( N ε ) ε∈ E converges to the subset N ∞ if: ∀U open set ⊂ M such that N ∞ ∩ U � = ∅ , ∃ε ∈ E / ∀ε � � ε, N ε � ∩ U � = ∅ ; and ∀K compact set ⊂ M such that N ∞ ∩ K = ∅ , ∃ε ∈ E / ∀ε � � ε, N ε � ∩ K = ∅ . Proposition 3.20
Let M and ( N ε ) ε∈ E be as in def. 3.19 and let f ∈ C ∞� ( M , R ) (the space ofcompactly supported, smooth, real-valued functions on M ). We define: f ε := sup {f ( � ) | � ∈ N ε } & f ∞ := sup {f ( � ) | � ∈ N ∞ } .Then, lim ε∈ E , � f ε = f ∞ . Proof
Let δ >
0. We choose � ∈ N ∞ such that f ( � ) > f ∞ − δ/ f is smooth, so there exists anopen neighborhood U of � such that ∀� � ∈ U, f ( � � ) > f ∞ − δ . From def. 3.19. , there exists ε suchthat ∀ε � � ε , N ε � ∩ U � = ∅ . Hence, ∀ε � � ε , f ε � > f ∞ − δ .Let K := {� ∈ M | f ( � ) � f ∞ + δ} . Since f is compactly supported, K is compact. We have K ∩ N ∞ = ∅ , so, from def. 3.19. , there exists ε such that ∀ε � � ε , N ε � ∩ K = ∅ . Hence, ∀ε � � ε , f ε � � f ∞ + δ . E being directed, there exists ε � ε , ε . Then, ∀ε � � ε, �� f ∞ − f ε � �� � δ . � or any state over the dynamical projective system, and any η ∈ L , the framework laid at thebeginning of the present subsection allows to construct a net of approximated orbits representingthis dynamical state in M ��� η (def. 3.17). Thus we can define the space R of all dynamical states suchthat, for all η , the net of their approached projections on M ��� η converges in the sense above.Hopefully, R will be dense in the space of all dynamical states, but we do not require both spacesto coincide. It is in fact not really surprising that formulating the exact dynamics may requireto consider only states that are well-behaved enough (we can view this prescription on the samefooting as, for example, the routine requirement for fields to be smooth so that we can describetheir dynamics by partial differential equations).Reciprocally, we can also associate to any (sufficiently regular) kinematical observable its corre-sponding exact dynamical version, as an observable defined on R . At this point we can commenton the issue raised at the beginning of section 2, namely that even the Poisson-algebra generatedby finitely many observables could be too complicated to be represented on a finite dimensionalsymplectic manifold. We had argued that this problem should not arise when looking at kinemati-cal observables, yet it might (and generically will) occur for the dynamical observables. By defininga dynamical observable as the limit of a family of imperfect estimations, we escape this difficulty.On the one hand, each such estimation can be expressed over a sufficiently big partial theory, whilekeeping the partial theories finite dimensional, because the regularization allows us to keep undercontrol the algebra generated by finitely many of these approximated versions of the observables.On the other hand, an exact dynamical observable, being a limit, is allowed to depend on the fullprojective state � y εη � ( ε,η ) ∈ EL ∈ R . Definition 3.21
Let ( L , M ��� , π ��� ) ↓ be a projective system of finite dimensional phase spaces andlet ( Y ε ) ε∈ E be a net in � S ↓ ( L , M ��� ,π ��� ) . We say that the net ( Y ε ) ε∈ E converges to the element Y ∞ ∈ � S ↓ ( L , M ��� ,π ��� ) iff: ∀η ∈ L , the net ( Y εη ) ε∈ E converges to the subset Y ∞η of M ��� η in the sense of def. 3.19.If ( L , M , π, δ ) �Y� , E is a regularized reduction of ( L , M ��� , π ��� ) ↓ , we say that the regularizationconverges on a subset R of S ↓ ( EL , M �Y� ,π �Y� ) iff: ∀y ∈ R , the net (Δ ε ( y )) ε∈ E converges in � S ↓ ( L , M ��� ,π ��� ) . Proposition 3.22
Let ( L , M ��� , π ��� ) ↓ be a projective system of finite dimensional phase spaces. Wedefine: A �,↓ ( L , M ��� ,π ��� ) = � f ∈ A ↓ ( L , M ��� ,π ��� ) ��� ∃η ∈ L , ∃f η ∈ f / f η ∈ C ∞� ( M ��� η , R )� .If the net ( Y ε ) ε∈ E in � S ↓ ( L , M ��� ,π ��� ) converges to the element Y ∞ ∈ � S ↓ ( L , M ��� ,π ��� ) , then, for all f ∈ A �,↓ ( L , M ��� ,π ��� ) , the net ( f ( Y ε )) ε∈ E converges to f ( Y ∞ ).If ( L , M , π, δ ) �Y� , E is a regularized reduction of ( L , M ��� , π ��� ) ↓ such that the regularization con-verges on R ⊂ S ↓ ( EL , M �Y� ,π �Y� ) , then, for all f ∈ A �,↓ ( L , M ��� ,π ��� ) , we can define an application f �Y� on R by: y ∈ R , f �Y� ( y ) := lim ε∈ E , � f ε ( y ) . Proof
Let f ∈ A �,↓ ( L , M ��� ,π ��� ) and let η ∈ L , f η ∈ f such that f η ∈ C ∞� ( M ��� η , R ). For all ε ∈ E , wehave f ( Y ε ) = sup Y εη f η (for we can choose any representative of f to evaluate f on Y ε ). Now, the net( Y εη ) ε∈ E converges to the subset Y ∞η of M ��� η , hence, using prop. 3.20:lim ε∈ E , � f ( Y ε ) = sup Y ∞η f η = f ( Y ∞ ) .Now, if y ∈ R , we have from def. 3.21 that the net (Δ ε ( y )) ε∈ E converges in � S ↓ ( L , M ��� ,π ��� ) . Hence,the net ( f (Δ ε ( y ))) ε∈ E converges, but we have ∀ε ∈ E , f (Δ ε ( y )) = f ε ( y ) (def. 3.17). � Finally, we want to discuss how renderings (def. 2.6) can be incorporated in this procedure,and more specifically, how regularized reductions can be used to mirror phase space reductionsin infinite dimensional symplectic manifolds. Given a rendering of some infinite dimensionalsymplectic manifold M ��� ∞ , and a phase space reduction thereof, we will aim at constructing aregularized reduction whose dynamical projective system renders the dynamical phase space M �Y� ∞ .Additionally, we will require that the regularization converges (at least) on the dynamical statesthat are identified, through this rendering, with points in M �Y� ∞ , and that, for any such state, thefamily of orbits reflecting it in the kinematical structure can be identified with the correspondingorbit in M ��� ∞ .Then, besides being provided with a rendering of the dynamical theory, this last point will ensurethat the maps linking the kinematical side and the dynamical one are appropriately intertwined bythe identifications arising from the renderings on both sides.In prop. 3.24, we formulate more concise assumptions that are sufficient to bring forth thisoptimal setting. As illustrated in fig. 3.5, it relies on the successive approximations of the dynamicsbeing formulated as phase space reductions of M ��� ∞ , and the thus defined dynamical phase spaces M �Y� ,ε∞ building a rendering of the exact dynamical theory (denoted by M �Y� ,∞∞ ). Proposition 3.23
Let ( L , M ��� , π ��� ) ↓ be a projective system of finite dimensional phase spaces andlet ( L , M , π, δ ) �Y� , E be a regularized reduction of ( L , M ��� , π ��� ) ↓ . Assume that we have a symplecticmanifold M ��� ∞ and a phase space reduction ( M �Y� ∞ , M ����� ∞ , δ ∞ ) of M ��� ∞ such that: we have a rendering of M ��� ∞ by ( L , M ��� , π ��� ) ↓ and of M �Y� ∞ by ( EL , M �Y� , π �Y� ) ↓ ; for all y in M �Y� ∞ , the net �Δ ε ◦ σ �Y� ↓ ( y )� ε∈ E converges in � S ↓ ( L , M ��� ,π ��� ) to � σ ��� ↓ � δ − ∞ �{y}� �, where σ �Y� ↓ : M �Y� ∞ → S ↓ ( EL , M �Y� ,π �Y� ) is defined as in def. 2.6 and � σ ��� ↓ : P ( M ��� ∞ ) → � S ↓ ( L , M ��� ,π ��� ) is defined ina similar way.Then, the regularization converges on R := Im σ �Y� ↓ and for all y ∈ M �Y� ∞ , for all f ∈ A �,↓ ( L , M ��� ,π ��� ) ,we have: f �Y� ◦ σ �Y� ↓ ( y ) = � β ��� ↑ ( f )� �Y� ( y ) . ε∞ δ ε � ∞ δ ∞∞ � � � L � {∞}∞ L � {∞}∞ L � {∞}∞ � Y � L ε � {∞}∞ L ε � � {∞}∞ {∞}∞ π �Y� ,∞→ε � ∞→∞ π �Y� ,ε � →ε∞→∞ E � {∞} Figure 3.5 – Regularized reduction and rendering
Proof
Let y ∈ M �Y� ∞ and f ∈ A �,↓ ( L , M ��� ,π ��� ) . We have, using prop. 3.22: f �Y� ◦ σ �Y� ↓ ( y ) = lim ε∈ E , � f ε ◦ σ �Y� ↓ ( y ) = lim ε∈ E , � f ◦ Δ ε ◦ σ �Y� ↓ ( y ) = f ◦ � σ ��� ↓ � δ − ∞ �{y}� �= β ��� ↑ ( f ) � δ − ∞ �{y}� � = sup δ − ∞ �{y}� β ��� ↑ ( f ) = � β ��� ↑ ( f )� �Y� ( y ) . � Proposition 3.24
Let ( L , M ��� , π ��� ) ↓ be a projective system of finite dimensional phase spacesyielding a rendering of a symplectic manifold M ��� ∞ . Let E be a directed preordered set and assumethat: for any ε ∈ E � {∞} , we have a phase space reduction ( M �Y� ,ε∞ , M ����� ,ε∞ , δ ε∞ ) of M ��� ∞ ; for any ε ∈ E , we have a cofinal subset L ε of L and an elementary reduction�� M �Y� ,εη � η∈ L ε �{∞} , � M ����� ,εη � η∈ L ε �{∞} , � π �Y� ,ε→εη � →η � η � η � , � δ εη � η∈ L ε �{∞} � of ( L ε � {∞} , M ��� , π ��� ) ↓ ,arising from ( M �Y� ,ε∞ , M ����� ,ε∞ , δ ε∞ ); we have a rendering of M �Y� ,∞∞ by ( E , M �Y� ∞ , π �Y� ∞→∞ ) ↓ ; for any ε � ε � , L ε � ⊂ L ε and, for any η ∈ L ε � , we have a projection π �Y� ,ε � →εη→η : M �Y� ,ε � η → M �Y� ,εη ,compatible with the symplectic structures, and such that π �Y� ,ε→ε∞→η ◦ π �Y� ,ε � →ε∞→∞ = π �Y� ,ε � →εη→η ◦ π �Y� ,ε � →ε � ∞→η .Then, defining � L := L � {∞} and � E := E � {∞} (extending the preorders in such a way that ∞ is a greatest element), we can complete this input to build a regularized reduction �� L , M , π, δ � �Y� , � E f �� L , M ��� , π ��� � ↓ .If we moreover have: for any y ∈ M �Y� ,∞∞ , the net �� σ ��� ↓ � δ ε,− ∞ �π �Y� ,∞→ε∞→∞ ( y ) � �� ε∈ E converges in � S ↓ ( L , M ��� ,π ��� ) to � σ ��� ↓ � δ ∞,− ∞ �y� � ;then the hypotheses of prop. 3.23 are fulfilled. Proof
For any ε ∈ E we define � L ε := L ε � {∞} and we additionally define � L ∞ := {∞} . Withthis def. 3.16. is fulfilled.For any ε ∈ E , def. 3.16. comes from assumption 3.24. , and for ∞ ∈ � E , it reduces to( M �Y� ,∞∞ , M ����� ,∞∞ , δ ∞∞ ) being a phase space reduction of M ��� ∞ , which has been assumed in 3.24. .For any ε � ε � ∈ � E and any η ∈ � L ε , η � ∈ � L ε � with η � η � , we define π �Y� ,ε � →εη � →η := π �Y� ,ε→εη � →η ◦π �Y� ,ε � →εη � →η � . From assumption 3.24. , π �Y� ,ε→εη � →η : M �Y� ,εη � → M �Y� ,εη is a projection compatible with thesymplectic structures, and from assumption 3.24. (or 3.24. if η � = ∞ ), π �Y� ,ε � →εη � →η � : M �Y� ,ε � η � → M �Y� ,εη � is a projection compatible with the symplectic structures. Hence, π �Y� ,ε � →εη � →η : M �Y� ,ε � η � → M �Y� ,εη is aprojection compatible with the symplectic structures.Let ( ε, η ) � ( ε � , η � ) � ( ε �� , η �� ) ∈ � E � L . We have η � , η �� ∈ L ε � , hence, using points 3.24. and 3.24. : π �Y� ,ε→εη �� →η � ◦ π �Y� ,ε � →εη �� →η �� ◦ π �Y� ,ε � →ε � ∞→η �� = π �Y� ,ε→ε∞→η � ◦ π �Y� ,ε � →ε∞→∞ = π �Y� ,ε � →εη � →η � ◦ π �Y� ,ε � →ε � η �� →η � ◦ π �Y� ,ε � →ε � ∞→η �� .Since π �Y� ,ε � →ε � ∞→η �� is surjective, we then have: π �Y� ,ε→εη �� →η � ◦ π �Y� ,ε � →εη �� →η �� = π �Y� ,ε � →εη � →η � ◦ π �Y� ,ε � →ε � η �� →η � ,hence, using once more from 3.24. : π �Y� ,ε � →εη � →η ◦ π �Y� ,ε �� →ε � η �� →η � = π �Y� ,ε→εη � →η ◦ π �Y� ,ε→εη �� →η � ◦ π �Y� ,ε � →εη �� →η �� ◦ π �Y� ,ε �� →ε � η �� →η �� = π �Y� ,ε→εη �� →η ◦ π �Y� ,ε � →εη �� →η �� ◦ π �Y� ,ε �� →ε � η �� →η �� . (3.24. )Now, using repeatedly 3.24. , together with 3.24. , we have: π �Y� ,ε � →εη �� →η �� ◦ π �Y� ,ε �� →ε � η �� →η �� ◦ π �Y� ,ε �� →ε �� ∞→η �� = π �Y� ,ε→ε∞→η �� ◦ π �Y� ,ε � →ε∞→∞ ◦ π �Y� ,ε �� →ε � ∞→∞ = π �Y� ,ε �� →εη �� →η �� ◦ π �Y� ,ε �� →ε �� ∞→η �� and, since π �Y� ,ε �� →ε �� ∞→η �� is surjective: π �Y� ,ε � →εη �� →η �� ◦ π �Y� ,ε �� →ε � η �� →η �� = π �Y� ,ε �� →εη �� →η �� . (3.24. )Combining eq. (3.24. ) and eq. (3.24. ), we get: π �Y� ,ε � →εη � →η ◦ π �Y� ,ε �� →ε � η �� →η � = π �Y� ,ε→εη �� →η ◦ π �Y� ,ε �� →εη �� →η �� = π �Y� ,ε �� →εη �� →η ,therefore �� E � L , M �Y� , π �Y� � ↓ is a projective system of phase spaces, so def. 3.16. holds.Thus, using prop. 3.18 with assumption 3.24. , ( L , M , π, δ ) �Y� , E is a regularized reduction of L , M ��� , π ��� ) ↓ , while ( EL , M �Y� , π �Y� ) ↓ is a rendering of M �Y� ,∞∞ .We now assume that assumption 3.24. holds. Using eq. (3.18. ) for EL ⊂ � E � L , we have:� σ ��� � L → L ◦ �Δ ε = Δ ε ◦ σ �Y� � E � L → EL ,and using it for � E {∞} ⊂ � E � L :� σ ��� � L →{∞} ◦ �Δ ε = δ ε,− ∞ � · � ◦ σ �Y� � E {∞}→{ ( ε,∞ ) } ◦ σ �Y� � E � L → � E {∞} = δ ε,− ∞ � · � ◦ σ �Y� � E � L →{ ( ε,∞ ) } ,therefore:Δ ε ◦ σ �Y� ↓ = Δ ε ◦ σ �Y� � E � L → EL ◦ σ �Y� ,− E � L →{ ( ∞,∞ ) } = � σ ��� � L → L ◦ � σ ��� ,− L →{∞} ◦ δ ε,− ∞ � · � ◦ σ �Y� � E � L →{ ( ε,∞ ) } ◦ σ �Y� ,− E � L →{ ( ∞,∞ ) } = � σ ��� ↓ ◦ δ ε,− ∞ � · � ◦ π �Y� ,∞→ε∞→∞ .Hence, for all y ∈ M �Y� ,∞∞ , the net �Δ ε ◦ σ �Y� ↓ ( y )� ε∈ E converges in � S ↓ ( L , M ��� ,π ��� ) to � σ ��� ↓ � δ ∞,− ∞ �y� � . � At this point, the question that remains open is how to construct a rendering of M �Y� ,∞ by anet of reduced phase spaces M �Y� ,� , arising from constraint surfaces approaching M ����� ,∞ . In otherwords, we are lacking systematic recipes for setting up regularization schemes in the sense of theprocedure just described.Among the tools that are at our disposal is the gauge fixing/unfixing trick (taken from [1], whereit was however used in a completely different context), that would consist in first partially gaugefixing (prop. A.8) the original phase space reduction, and then gauge unfixing it in a slightlydifferent direction: thus we would deform the orbits (in the view of improving their projectability),and get an approximation of the dynamics that should be satisfactory in some neighborhood of thecommon gauge fixing surface (this technique is the one used in [10, section 3]). Another option,that might be in particular relevant when the gauge orbits are infinite dimensional, could be todrastically gauge fix them, before progressively lifting the gauge fixing conditions, thus approachinga given orbit by an increasing net of submanifolds inside it. In both cases, we get a naturalsymplectomorphism between M �Y� ,∞ and each M �Y� ,� , so we probably want to combine such methodswith projections from M �Y� ,∞ into symplectic submanifolds of it, to drop the degrees of freedom thatare disproportionately accurate at a given level of approximation.Also, there is presumably some link between the regularization procedure we are considering andvarious concepts developed in the context of Loop Quantum Gravity (often within a Lagrangian etting), exploring the interplay between discretization, coarse graining, diffeomorphism invariance,and the continuum limit [12]. Studying more precisely how these approaches are related to thestrategy proposed here could in particular help incorporate renormalization group ideas into thepicture. Acknowledgements
This work has been financially supported by the Université François Rabelais, Tours, France.This research project has been supported by funds to Emerging Field Project “Quantum Geometry”from the FAU Erlangen-Nuernberg within its Emerging Fields Initiative.
A Appendix: Classical constrained systems
To fix the notations and definitions, we summarize here some facts about constrained classicalsystems. We recall how a reduced phase space arises from a constraint surface in a symplecticmanifold [20, section 1.7], we introduce a notion of transport of observables to translate kinematicalobservables into dynamical ones (this facility is the main object of the physical discussion insection 3), and give a very brief account of partial gauge fixing [13].When considering a constraint surface M ����� in a symplectic manifold M ��� the pullback of thesymplectic structure ٠��� does not, in general, define a symplectic structure on M ����� : there mightbe directions in the tangent space of M ����� on which this pullback vanishes. These directionscorrespond to the gauge flow generated by first class constraints, and the gauge orbits need tobe quotiented out in order to get a reduced phase space M �Y� with a non-degenerate symplecticstructure ٠�Y� .Except for the first few definitions (which are tailored to match the needs of some results insection 3), this appendix focuses on finite dimensional manifolds: this is anyway the point of theformalism presented in the main text that we aim at describing a field theory in such a way thatwe can work mostly within the context of finite dimensional manifolds. In this appendix all manifolds will be smooth manifolds, all maps between them will be smoothand all submanifolds will be regular (ie. embedded) submanifolds. Where infinite dimensionalmanifolds are considered, these are Banach-modeled smooth manifolds, and symplectic structureson them are always strong symplectic structures [2, chap. VII].
Definition A.1
Let M ��� be a (possibly infinite dimensional) smooth symplectic manifold (withsymplectic structure Ω ��� ). A phase space reduction of M ��� is a triple ( M �Y� , M ����� , δ ) such that: M ����� is a submanifold of M ��� and M �Y� is a symplectic manifold (with symplectic structure Ω �Y� ); δ : M ����� → M �Y� is a surjective map and, for all y ∈ M �Y� , δ − �y� is connected; . for all � ∈ M ����� , Im( T � δ ) = T δ ( � ) ( M �Y� ) & Ω ��� ,� | T � ( M ����� ) = [ δ ∗ Ω �Y� ] � . For any bounded real-valued function f on M ��� , we define a corresponding dynamical observableon M �Y� by mapping to a point y in the reduced phase space the supremum of f on the correspondingorbit δ − �y� . The motivation for this definition is that we regard indicator functions as the mostfundamental observables: with the transport of observables defined this way, the indicator functionof some region in M ��� is mapped into the indicator function on the space of orbits that characterizewhether a given orbit crosses this region or not. In other words, the dynamical observable relatedto the indicator function of some region of M ��� will tell us whether the dynamical state of thesystem allows it to be measured in that region.Note that there can be relations between the dynamical observables f �Y� , � � � , f �Y� k arising fromfunctionally independent kinematical observables f , � � � , f k , or to state this more precisely we canhave dependencies:Im ( f �Y� × � � � × f �Y� k ) � = ( Im f �Y� ) × � � � × ( Im f �Y� k ) ,although the corresponding kinematical observables were independent:Im ( f × � � � × f k ) = ( Im f ) × � � � × ( Im f k ) .This is a crucial observation, since, indeed, the dynamical content of theory lies in such functionalrelations emerging between observables that were kinematically independent. Definition A.2
Let ( M �Y� , M ����� , δ ) be a phase space reduction of M ��� . We denote by B ( M ��� ) thespace of bounded, real-valued, functions on M ��� . For all f ∈ B ( M ��� ), we define f �Y� ∈ B ( M �Y� ) by: ∀y ∈ M �Y� , f �Y� ( y ) := sup � f ( � ) �� � ∈ δ − �y� �. (A.2. )For the rest of this appendix all manifolds will be finite dimensional manifolds. Definition A.3
Let M ��� be a smooth, finite dimensional, symplectic manifold (with symplecticstructure Ω ��� ). A pre-reduction of M ��� is a triple ( M �Y� , M ����� , δ ) such that: M ����� is a submanifold of M ��� and M �Y� is a manifold; the restriction of Ω ��� to T ( M ����� ) is of constant rank, thus defining a foliation K ( M ����� ) by ∀� ∈ M ����� , K � ( M ����� ) := � � ∈ T � ( M ����� ) ��� Ω ��� ,� ( �, · ) | T � ( M ����� ) = 0� ⊂ T � ( M ����� ); δ : M ����� → M �Y� is a surjective map and ∀� ∈ M ����� , Im( T � δ ) = T δ ( � ) ( M �Y� ) ; ∀y ∈ M �Y� , δ − �y� is a leaf of the foliation K ( M ����� ). Proposition A.4
Let M ��� be a smooth, finite dimensional, symplectic manifold (with symplecticstructure Ω ��� ) and let ( M �Y� , M ����� , δ ) be a phase space reduction of M ��� . Then, ( M �Y� , M ����� , δ ) isa pre-reduction of M ��� and we have: ∀� ∈ M ����� , K � ( M ����� ) = Ker T � δ = T � � δ − �δ ( � ) � � . (A.4. ) Proof
Defs. A.3. and A.3. are directly implied by def. A.1.Let y ∈ M �Y� and � ∈ δ − �y� . Since δ has surjective derivative at each point, we have as n implication of the rank theorem [11, theorem 5.22] that δ − �y� is a submanifold of M ����� withtangent space Ker T � δ ⊂ T � ( M ����� ) at � . Now, from def. A.1. , together with the non-degeneracy ofΩ �Y� (for M �Y� is a symplectic manifold), we have: ∀� � ∈ δ − �y� , Ker T � � δ = K � � ( M ����� ) .Hence, K ( M ����� ) has constant dimension, so def. A.3. is fulfilled.Additionaly, by maximality of the leaves, the connected submanifold δ − �y� is included in theleaf of the foliation K ( M ����� ) that goes through � . Reciprocally, since the leaf that goes through � is connected, and has tangent space K � � ( M ����� ) = Ker T � � δ at any point, δ is constant on it, henceit is included in δ − �y� . Thus, def. A.3. is fulfilled. � Proposition A.5
Let M ��� be a smooth, finite dimensional, symplectic manifold and let ( M �Y� , M ����� , δ )be a pre-reduction of M ��� . Then, there exists a symplectic structure Ω �Y� on M �Y� such that( M �Y� , M ����� , δ ) is a phase space reduction of M ��� . Proof
What we need to prove is that there exists a symplectic structure Ω �Y� on M �Y� such that: ∀� ∈ M ����� , Ω ��� ,� | T � ( M ����� ) = [ δ ∗ Ω �Y� ] � .The others points in def. A.1 are immediately fulfilled (in particular, for any y ∈ M �Y� , δ − �y� isconnected as a leaf of a foliation).Let � ∈ M ����� and let y := δ ( � ). Since δ has surjective derivative at each point, there exist bythe rank theorem [11, theorem 5.13] open neighborhoods U of � in M ����� , V of y in M �Y� and W of 0in R �−� (with � := dim M ����� and � := dim M �Y� ), and a diffeomorphism φ : V × W → U such that: ∀y � ∈ V , ∀z � ∈ W , δ ◦ φ ( y � , z � ) = y � .For any y � , z � ∈ V × W , we define Ω φ,z � �Y� ,y � by: ∀�, � � ∈ T y � ( M �Y� ) , Ω φ,z � �Y� ,y � � �, � � � = Ω ��� ,φ ( y � ,z � ) � T y � ,z � φ ( �, , T y � ,z � φ ( � � , � � := φ ( y � , z � ), Ω φ,z � �Y� ,y � satisfies: ∀�, � � ∈ T � � ( M ����� ) , Ω ��� ,� � � �, � � � = Ω φ,z � �Y� ,y � � T � � δ ( � ) , T � � δ ( � � )� ,for we have from def. A.3. :� T y � ,z � φ (0 , w ) | w ∈ T z � ( W )� = T � � � δ − �y � � � = K � � ( M ����� ) . (A.5. )Let � Y , � Y � be vector fields on V and � Z be a vector field on W . Defining Y := φ ∗ �� Y , Y � := φ ∗ �� Y � , Z := φ ∗ �0 , � Z �, we have [ Y , Z ] = [ Y � , Z ] = 0 and, from eq. (A.5. ):Ω ��� ( Z , · ) | T ( M ����� ) = 0 .Hence, we get, for any y � , z � ∈ V × W : � Ω ��� � Y , Y � , Z � φ ( y � ,z � ) = 0 (by definition of a symplectic form)= Z �Ω ��� ( Y , Y � )� φ ( y � ,z � ) � Z z � � z �� �→ Ω φ,z �� �Y� ,y � �� Y y � , � Y �y � �� .Now, for any � ∈ M ����� , we define Ω � �Y� : T δ ( � ) ( M �Y� ) × T δ ( � ) ( M �Y� ) → R by: ∀�, w ∈ T � ( M ����� ) , Ω � �Y� ( T � δ ( � ) , T � δ ( w )) = Ω ��� ,� ( �, w ) .That such an Ω � �Y� exists is established by the previous discussion and, since Im T � δ = T δ ( � ) ( M �Y� ),it is moreover unique. Thus, Ω � �Y� is well-defined.The previous argument also shows that, for any vector fields Y , Y � on M �Y� , � �→ Ω � �Y� � Y δ ( � ) , Y �δ ( � ) �is smooth and satisfies: ∀� ∈ M ����� , ∀w ∈ T � � δ − �δ ( � ) � � , T � � � � �→ Ω � � �Y� � Y δ ( � � ) , Y �δ ( � � ) �� ( w ) = 0 .The level sets of δ being connected, as underlined above, this allows us to define a smoothdifferential 2-form Ω �Y� satisfying: ∀� ∈ M ����� , Ω ��� ,� | T � ( M ����� ) = [ δ ∗ Ω �Y� ] � .Lastly, for any � ∈ M ����� , we also have:[ δ ∗ � Ω �Y� ] � = � Ω ��� ,� | T � ( M ����� ) = 0 ,and, from eq. (A.5. ):Ker T � δ = K � ( M ����� ) .Thus, T � δ being surjective, Ω �Y� is closed and non-degenerate, so it is indeed a symplectic form on M �Y� . � Proposition A.6
Let M ��� be a smooth, finite dimensional, symplectic manifold and let ( M �Y� , , M ����� , δ )and ( M �Y� , , M ����� , δ ) be two phase space reductions of M ��� arising from the same submanifold M ����� of M ��� . Then there exists a unique map ψ : M �Y� , → M �Y� , such that δ = ψ ◦ δ . Moreover, ψ is asymplectomorphism. Proof
From def. A.3. δ is constant on the level sets of δ and reciprocally, hence, as a consequence[11, prop. 5.21] of the rank theorem (using that both δ and δ are surjective and have surjectivederivative at each point, from def. A.3. ), there exists a unique diffeomorphism ψ : M �Y� , → M �Y� , such that δ = ψ ◦ δ .In particular, for � ∈ M ����� (with y := δ ( � )), we have T � δ = T y ψ ◦ T � δ , so that, using def. A.1. :[ δ ∗ Ω �Y� , ] � = Ω ��� ,� | T � ( M ����� ) = [ δ ∗ ψ ∗ Ω �Y� , ] � .Since T � δ and δ are surjective, ψ is a symplectomorphism. � Proposition A.7
Let M ��� be a smooth, finite dimensional, symplectic manifold and let ( M �Y� , M ����� , δ )be a phase space reduction of M ��� . Let f , g and {f , g} ��� ∈ C ∞ ( M ��� , R ) ∩ B ( M ��� ), and assume that: ∀� ∈ M ����� , X f,� , X g,� ∈ T � ( M ����� ) ,where the Hamiltonian vector field X f := �f is defined by Ω ��� ( X f , · ) = �f .Then f �Y� , g �Y� ∈ C ∞ ( M �Y� , R ) and {f �Y� , g �Y� } �Y� = � {f , g} ��� � �Y� . Proof
For all � ∈ M ����� , X f,� ∈ T � ( M ����� ), hence �f � �K � ( M ����� ) � = Ω ��� ,� � X f,� , K � ( M ����� )� ⊂ ��� ,� ( T � ( M ����� ) , K � ( M ����� )) = { } . The same holds for g . Therefore, f and g are constant onthe leaves of the foliation K ( M ����� ) on M ����� . As a consequence [11, prop. 5.20] of the rank theorem(using defs. A.3. and A.3. ), there exist smooth maps � f and � g : M �Y� → R such that f | M ����� = � f ◦ δ and g| M ����� = � g ◦ δ . Hence, f �Y� = � f and g �Y� = � g .In addition, we have for any � ∈ M ����� (with y := δ ( � )):[ δ ∗ �f �Y� ] � = �f � | T � ( M ����� ) = Ω ��� ,� | T � ( M ����� ) � X f,� ; · � (using X f,� ∈ T � ( M ����� ))= Ω �Y� ,y � T � δ � X f,� � ; T � δ ( · )� (using def. A.1. ).Thus, T � δ being surjective, X f �Y� ,y = T � δ � X f,� � and, similarly X g �Y� ,y = T � δ � X g,� � .Hence, we have: {f �Y� , g �Y� } �Y� ,y = Ω �Y� ,y � X g �Y� ,y , X f �Y� ,y �= Ω �Y� ,y � T � δ � X g,� � , T � δ � X f,� ��= Ω ��� ,� � X g,� , X f,� � (using def. A.1. and X f,� , X g,� ∈ T � ( M ����� ))= {f , g} ��� ,� .Since this holds for all � ∈ δ − �y� , this implies in particular that {f , g} ��� is constant on δ − �y� .Therefore � {f , g} ��� � �Y� ( y ) = {f , g} ��� ,� = {f �Y� , g �Y� } �Y� ,y . � Proposition A.8
Let ( M �Y� , M ����� , δ ) be a phase space reduction of M ��� and M ��X a submanifold of M ����� such that: for all � ∈ M ��X , T � ( M ����� ) = T � ( M ��X ) + K � ( M ����� ); the intersection of a leaf of the foliation K ( M ����� ) with M ��X is not void and is connected;We define δ ��X : M ��X → M �Y� by δ ��X := δ| M ��X . Then, ( M �Y� , M ��X , δ ��X ) is a phase space reduction of M ��� .Moreover, if f ∈ C ∞ ( M ��� , R ) ∩ B ( M ��� ) and ∀� ∈ M ����� , X f,� ∈ T � ( M ����� ), we have f �Y� = f ��X where: ∀y ∈ M �Y� , f �Y� ( y ) := sup � f ( � ) �� � ∈ δ − �y� � (def. A.2)and f ��X ( y ) := sup � f ( � ) �� � ∈ δ ��X ,− �y� �. Proof
Statements A.1. & A.1. . M ��X is a submanifold of M ����� , and M ����� is a submanifold of M ��� ,hence M ��X is a submanifold of M ��� . M �Y� is a symplectic manifold.The level sets of δ ��X are the intersection with M ��X of the leaves of the foliation K ( M ����� ) (usingdef. A.3. ), hence from assumption A.8. , δ ��X is surjective and its level sets are connected. Statement A.1. . Let � ∈ M ��X . We have T � δ ��X = T � δ| T � ( M ��X ) , hence: T � δ ��X �T � ( M ��X ) � = T � δ �T � ( M ��X ) � = T � δ �T � ( M ��X ) + Ker T � δ� T � δ �T � ( M ��X ) + K � ( M ����� ) � = T � δ �T � ( M ����� ) � = T δ ��X ( � ) ( M �Y� ) (using assump-tion A.8. , eq. (A.4. ) and def. A.1. for the phase space reduction ( M �Y� , M ����� , δ ) ).Next, we have:Ω ��� ,� | T � ( M ��X ) = [ δ ∗ Ω �Y� ] � | T � ( M ��X ) (using def. A.1. for the phase space reduction ( M �Y� , M ����� , δ )and T � ( M ��X ) ⊂ T � ( M ����� ))= �� δ| M ��X � ∗ Ω �Y� � � = [ δ ��X ,∗ Ω �Y� ] � . Observables.
Let f ∈ C ∞ ( M ��� , R ) ∩ B ( M ��� ) with ∀� ∈ M ����� , X f,� ∈ T � ( M ����� ). From the proof ofprop. A.7, f is then constant on the leaves of the foliation K ( M ����� ) on M ����� . Therefore, f �Y� = f ��X . � B References [1] Norbert Bodendorfer, Alexander Stottmeister, and Andreas Thurn. On a Partially Re-duced Phase Space Quantisation of General Relativity Conformally Coupled to a Scalar Field.
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