Symplectic Microgeometry IV: Quantization
aa r X i v : . [ m a t h . S G ] J u l SYMPLECTIC MICROGEOMETRY IV:QUANTIZATION
ALBERTO S. CATTANEO, BENOIT DHERIN, AND ALAN WEINSTEIN
Abstract.
We construct a special class of semiclassical Fourierintegral operators whose wave fronts are the symplectic micromor-phisms of [8]. These operators have very good properties: theyform a category on which the wave front map becomes a functorinto the cotangent microbundle category, and they admit a totalsymbol calculus in terms of symplectic micromorphisms enhancedwith half-density germs. This new operator category encompassesthe semi-classical pseudo-differential calculus and offers a functo-rial framework for the semi-classical analysis of the Schrödingerequation. We also comment on applications to classical and quan-tum mechanics as well as to a functorial and geometrical approachto the quantization of Poisson manifolds.
Contents
1. Introduction 2Outline of the paper 52. Categories of Fourier integral operators 62.1. Categories of canonical relations 72.2. Generating families 82.3. Half-densities 102.4. Fourier integral operators 122.5. The semiclassical limit 143. Quantization of cotangent lifts 173.1. Canonical identifications 193.2. Generating families, tubular neighborhoods and microexponentials 193.3. The canonical half-density ρ B Q h ( a, T ∗ φ )
5. Applications and further directions 395.1. Quantization functors 395.2. The energy monoid 405.3. Classical flows as symplectic micromorphisms 415.4. Classical symmetries 425.5. Generalized symmetries 425.6. Quantization of the energy monoid 435.7. Quantization of Poisson manifolds 435.8. Quantization of symmetries 445.9. Quantization of the classical flow 45References 46
Acknowledgement.
The research of A.S.C. was partially supported bythe NCCR SwissMAP, funded by the Swiss National Science Foun-dation, and by SNF Grant No. 200020_192080. B.D. acknowledgespartial support from the NWO Grant 613.000.602 and the SNF GrantPA002-113136. A.W. acknowledges partial support from NSF grantDMS-0707137 and the UC Berkeley Committee on Research. We wouldlike to thank David Barrett, Ivan Contreras and Pedro de M. Rios foruseful comments on this paper and discussions on generating families,and Ted Voronov for extensive correspondence concerning his work onthick morphisms. The latter began around 2014, when we had prepareda preliminary version of the present paper, and Voronov had posted onthe ArXiv preprints of work (see citations in the introduction below)published several years later.1.
Introduction
From an analytical point of view, symplectic geometry is the ge-ometry underlying the calculus of Fourier integral operators (FIO’s)[2, 15, 16, 18, 27]. The present article is concerned with developing asimilar calculus in the context of microsymplectic geometry [8]. Sincethe latter enjoys much better functorial properties than its macro ana-log, it is not too surprising that these good properties persist in theassociated operator calculus.The geometry of canonical relation composition in the symplectic“category” is central to symplectic geometry itself in many ways [32].This “geometric calculus” also plays an important role in the calculusof FIO’s through a canonical relation associated to each FIO: its wavefront. The geometry of the wave front usually contains a lot of infor-mation about the class of FIO’s associated to it. The reader will find
YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 3 detailed treatments of this relationship in the standard literature onFIO’s [15, 16, 18, 19, 27]). Here, we are mostly concerned with func-torial and categorical aspects of this relationship between the FIO’sand their wave fronts. Namely, just as for canonical relations, the com-position of two FIO’s usually fails to be a FIO. Now, a well-behavedcomposition of the wave front canonical relations, if some restrictionis imposed on them, can guarantee a well-behaved composition of theoperators in the corresponding FIO classes. In this sense, the symplec-tic geometry of the wave fronts controls the categorical and functorialaspects of the FIO calculus.In [8], we constructed a category of particularly well-behaved germsof canonical relations: the symplectic micromorphisms. This articlestudies the corresponding class of (semi-classical) FIO’s, which inher-its, for the most part, the good properties of the symplectic micromor-phism composition. It turns out that this class of operators encom-passes and extends the whole calculus of pseudo-differential operatorsin the semi-classical limit. Moreover, the hamiltonian flows of classicaldynamics can be described, for asymptotically small times, in terms ofsome symplectic micromorphisms satisfying purely algebraic relationsmimicking time translations at a purely categorical level. Upon quan-tization, we recover the Schrödinger flow of quantum mechanics in thesemi-classical limit in terms of our semi-classical FIO’s. This approachto quantization is very related to the considerations surrounding thenotion of “thick morphism" developed by Voronov in [28, 29, 30] andby Khudaverdian and Voronov in [21]. Their work, done in the con-text of supermanifolds, was originally aimed at an understanding of L ∞ morphisms between homotopy structures, but many of their results areparallel to ours when restricted to the case of ordinary manifolds. Inaddition, many of our results have also been used in [11, 12, 13] by Men-cattini and one of the authors to quantize momentum maps as well asthe underlying group actions. In [14], it has been used by Wagemannand one of the authors to quantize Leibniz algebras.One defining trait of Fourier integral operators is that they admitexplicit representations in terms of oscillatory integrals, once given agenerating family for their wave fronts. Symplectic micromorphismspossess canonical generating families (up to a choice of an exponentialmap germ on the smooth manifolds) with very good properties (statedin paragraph 2.4.2) which allow one to define a total symbol calculusfor their corresponding semi-classical FIO’s. These canonical gener-ating families are obtained as deformations of generating families for YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 4 cotangent lifts. Locally, such a semiclassical FIO Q ~ ( a, f, φ ) : L ~ ( R m ) −→ L ~ ( R n ) may be written in integral form similar to that for pseudo-differentialoperators(1.1) (cid:0) Q ~ u (cid:1) ( y ) := (2 π ~ ) − m + n Z a ( p, y ) e i ~ (cid:0) h p,φ ( y ) − x i + f ( p,y ) (cid:1) u ( x ) dp dx, where φ : R n → R m is a smooth map, a is an ~ -dependent smoothfunction (on the pullback bundle φ ∗ ( T ∗ R m ) ) called the symbol of theoperator, and the phase is the generating function of a symplecticmicromorphism from T ∗ R m to T ∗ R n with core map φ . (See section2.5.1 for a precise definition of the semiclassical intrinsic Hilbert space L ~ ( R n ) .) We recover pseudo-differential operators when m = n , φ = id and f = 0 , in which case the corresponding symplectic micromorphismis the identity.In this paper, we will mostly focus on the semi-classical limit of suchoperators, using the integral symbol "s.c. R " instead of " R " to remindus of this fact. The semi-classical limit is concerned with equivalenceclasses of such operators that have the same asymptotic behavior whenthe parameter ~ in the phase of the oscillatory integral goes to zero. De-pending on the problem at hand, one may be interested in asymptoticsof order ~ N for some fixed N ; here, we will be interested in asymp-totics modulo ~ ∞ , which roughly means that we consider two FIO’sequivalent if they have the same complete asymptotic expansions in ~ as ~ → .An application of the stationary phase principle, Theorem (1.1) showsthat the semi-classical limit of an FIO is controlled by the germ of itswave front WF( Q ~ ) and of its symbol around gr φ (the graph of thesymplectic micromorphism core map). The symplectic microgeometryterminology introduced in [8] is a very convenient language to deal withthe semi-classical limit of these FIO’s.Let us recall some basic definitions of microgeometry as introducedin [8]. A microfold [ M, A ] (called a “local manifold pair" in [31]) isan equivalence class of manifold pairs ( M, A ) , where A is a closed sub-manifold of M such that two pairs ( M , A ) and ( M , A ) are equivalentif there exists a third one ( M , A ) for which M is simultaneously anopen submanifold of both M and M . A symplectic microfold isa microfold [ M, A ] such that M is a symplectic manifold and A is alagrangian submanifold; a lagrangian submicrofold [ L, S ] of [ M, A ] is a submicrofold (i.e. a microfold such that L ⊂ M and S ⊂ A ) suchthat L is a lagrangian submanifold. YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 5
This paper initiates the study of Fourier integral operators for whichthe semi-classical limit is controlled by a lagrangian submicrofold of itswave front that satisfies the transversality condition in [8], or, in otherwords, by a symplectic micromorphism.
Outline of the paper.
In Section 2, we give a brief introduction tothe calculus of semi-classical Fourier integral operators. In particular,we review the central notion of generating families for lagrangian sub-manifolds. Example 2 recalls a standard construction for generatingfamilies of conormal bundles which depends only on a choice of a tubu-lar neighborhood. This is the central construction on which we aregoing to build on in this paper. We also stress the functorial aspects ofthe FIO calculus, and we introduce the notion of a “Fourier category”associated to any always-well-composing collection of wave fronts. Inparagraph 2.4.2, we isolate special conditions for the wave fronts interms of their generating families guaranteeing a very convenient inte-gral representation for their associated FIO’s.In Section 3, we focus on semi-classical FIO’s whose wave frontsare cotangent lifts of smooth maps. These canonical relations form acategory, and so do their associated FIO’s. We show how to use thestandard generating family construction for conormal bundles in thecontext of cotangent lifts using exponential map germs to constructthe tubular neighborhood required by the construction. We show thatthe domain of such a generating family admits a canonical vertical half-density. This fact is very important to us, since it reduces the usualambiguity of the FIO integral representation to only one thing: thechoice of exponential germs on the underlying manifolds.In Section 4, we extend the class of allowed wave fronts from cotan-gent lifts to all symplectic micromorphisms. To do so, we first intro-duce a notion of deformation for conormal bundles and their generatingfamilies. Then, we show that symplectic micromorphisms are in one-to-one correspondence with deformations of cotangent lifts generatingfamilies once an exponential map germ has been fixed. This allowsus to extend the integral representation of Section 3 to FIO’s whosewave fronts are general symplectic micromorphisms. We study thelocal theory of these operators, with several examples; in particular,we show that semi-classical pseudo-differential operators fall into ourclass. Moreover, we give explicit formulas for the operator composi-tion in this local setting; this involves a composition formula for thewave front generating families as well as for the total symbols of theseoperators.
YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 6
The presence of a canonical exponential in the local setting allowsus to identify our category of FIO’s with the category of enhancedsymplectic micromorphisms between cotangent bundles of R n for n =1 , , . . . , where an enhanced symplectic micromorphism is a symplecticmicromorphism carrying near its core a half-density germ correspond-ing to the total symbol of the operator. Namely, in this context, we havetwo inverse functors: the quantization functor Q ~ that associates to anenhanced symplectic micromorphism a semi-classical FIO through theintegral representation as in (1.1) and the total symbol functor σ thatassociates to the operator (1.1) its wave front (in the form of its gen-erating function f ) enhanced with the total symbol a (identified in thelocal case with a smooth function germ on φ ∗ ( T ∗ R m ) around the zerosection).In Section 5, we comment on applications and further directions. Inparticular, we explain how to extend the quantization and total sym-bol functors, which are present in the local setting, to a semi-classicalFIO calculus over any smooth manifold category enriched with someadditional geometric structures sufficient to allow the construction ofexponential map germs on the manifolds in a canonical way. We con-tinue by relating the calculus developed here to the quantization ofPoisson manifolds via oscillatory integrals and symplectic groupoids[20, 33, 34]. (In particular the generating function of the formal sym-plectic groupoid integrating a Poisson structure obtained in [7] canbe understood as the jet of the generating function of the quantizingFIO in this setting.) Finally, we show how the small-time asymp-totics of classical hamiltonian mechanics can be expressed in a purelycategorical way in the framework of symplectic microgeometry. In par-ticular, we show how classical flows can be modeled as the action ofa special monoid in the microsymplectic category, the energy monoid,on the hamiltonian system’s phase-space; enhancement and quantiza-tion of the action symplectic micromorphism recovers the usual unitarySchrödinger flows of quantum mechanics. We also explain how sym-metries can be modeled very naturally in this framework, both at theclassical and quantum level.2. Categories of Fourier integral operators
In this section, we give a brief presentation of the theory of Fourierintegral operators. We focus mostly on the categorical and geometricalaspects of this calculus since they are the main concern for us in thispaper, referring the reader to standard texts [15, 16, 18, 19, 27] on thesubject for the more analytical aspects.
YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 7
Categories of canonical relations.
The symplectic “category”.
Let M i , i = 1 , , , be three symplec-tic manifolds, and let L i ( i = 1 , ) be a canonical relation from M i to M i +1 , i.e., a closed lagrangian submanifold of the symplectic manifoldproduct M i × M i +1 , where M denotes the symplectic manifold with op-posite symplectic form − ω M . One can compose L with L as binaryrelations yielding the subset L ◦ L of M × M . This compositionfails in general to be a lagrangian submanifold, or even a submanifold;however, there are many examples for which it does. For instance, wehave the well know proposition: Proposition 1.
A sufficient condition for the set-theoretic compositionof the canonical relations L and L to be a canonical relation is thatthe intersection in M × M × M × M of L × L with M × ∆ M × M (where ∆ M denotes the diagonal in M × M ) is transversal andproperly embedded in M × M via the canonical factor projection. Inthis case, we say that the canonical relations have strongly tranversecomposition . We denote by
Sympl ext the (extended) symplectic “category” whoseobjects are cotangent bundles and whose morphisms are taken to becanonical relations between them. The quotation marks are there tostress that this is not a category in the usual sense since compositionis not always defined. However, this “category” contains a honest sub-category formed by the cotangent lifts as described below.2.1.2.
Schwartz transform and cotangent lifts.
The canonical relationsfrom T ∗ M to T ∗ N can be put in one-to-one correspondence with thelagrangian submanifolds of T ∗ ( M × N ) via the Schwartz transform (see[2]), S : T ∗ M × T ∗ N −→ T ∗ ( M × N ) , which is the symplectomorphism that sends (cid:0) ( p , x ) , ( p , x ) (cid:1) to ( − p , p , x , x ) .Now, to any smooth map M φ ← N , we can associate a special canonicalrelation, its cotangent lift, which we will consider going in the oppositedirection to φ , T ∗ φ : T ∗ M −→ T ∗ N, YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 8 by pulling back the conormal bundle of gr φ , seen as a submanifold of T ∗ ( M × N ) , via the Schwartz transform: T ∗ φ := n(cid:16)(cid:0) p , φ ( x ) (cid:1) , (cid:0) ( T x φ ) ∗ p , x (cid:1)(cid:17) : ( p , x ) ∈ φ ∗ ( T ∗ M ) o . The collection C of all cotangent lifts is a subcategory of Sympl ext which is a true category; namely, we always have that T ∗ φ ◦ T ∗ φ = T ∗ ( φ ◦ φ ) . Generating families.
Generating functions. An exact lagrangian embedding λ :Σ ֒ → T ∗ X is a lagrangian embedding for which λ ∗ θ = dS for some S ∈ C ∞ (Σ) , where θ is the Liouville 1-form on T ∗ X .Composing λ with the bundle projection π : T ∗ X → X , we obtain amap π Σ from Σ to X . Antecaustic points are elements of Σ at which T π Σ is not an isomorphism. When π Σ is a diffeomorphism, we say that the lagrangian submani-fold λ (Σ) is projectable , in which case the differential dS Σ : X → T ∗ X of S Σ := S ◦ π − parametrizes the lagrangian submanifold λ (Σ) . Thefunction S is called a generating function of the lagrangian sub-manifold. (It is well-defined up to a constant on each component of Σ . ) Conversely, to any function S ∈ C ∞ ( X ) , we can associate theprojectable lagrangian submanifold Im dS ⊂ T ∗ X that has S as agenerating function; explicitly, Im dS := (cid:8) ( dS ( x ) , x ) : x ∈ X (cid:9) . There are, however, many interesting non-projectable lagrangiansubmanifolds. For instance, the conormal bundle N ∗ C of a (non-open)submanifold C ⊂ X is highly non-projectable in the sense that allpoints are antecaustic points: if we consider the lagrangian embeddinggiven by the inclusion ι C : N ∗ C −→ T ∗ X, we see that the preimage of π ◦ ι C at any point c ∈ C consists ofthe whole fiber N ∗ c C . As a consequence cotangent lifts are also non-projectable, since they are conormal bundles to the graph of the un-derlying map. Recall that the conormal bundle N ∗ S of a submanifold S ⊂ X is the lagrangiansubmanifold of T ∗ X consisting of the covectors to X based along S and vanishingon T S . We have chosen this term because the images of these points under π Σ are knownas caustic points. (Although the antecaustic points play a key role in symplecticgeometry, we have not found another concise term for them in the literature.) YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 9
For these lagrangian submanifolds with antecaustic points, there isstill a notion of generating function. The price to pay, however, isthe introduction of additional variables for the generating functionsthrough a fibration p : B → X that “unfolds” the lagrangian sub-manifold at antecaustic points. This leads to the notion of generatingfamily.2.2.2. Generating families.
A submersion p : B → X together with asmooth function S ∈ C ∞ ( B ) defines two lagrangian submanifolds: thelagrangian submanifold Im dS in T ∗ B whose generating function is S (we regard it as a canonical relation from the point to T ∗ B ) and thecotangent lift T ∗ p , which we can see as a canonical relation from T ∗ B to T ∗ X . If these lagrangian submanifolds have a strongly transversalcomposition their composition is a lagrangian submanifold of T ∗ X . A generating family for a lagrangian submanifold L in T ∗ X is a triple ( B, p, S ) as above such that L = T ∗ p ◦ Im dS. Given a function S ∈ C ∞ ( B ) and a fibration p : B → X , the canonicalrelations Im dS and T ∗ p have a strongly transversal composition if andonly if the first factor projection π B of T ∗ p on T ∗ B is transversal to Im dS and the second factor projection π X of T ∗ p on T ∗ X becomesa proper embedding when restricted to the points y ∈ T ∗ p such that π B ( y ) ∈ Im dS . Observe that the intersection Im π B ∩ Im dS consistsof the points such that(2.1) ( dS ( b ) , b ) = ( T ∗ b p ( η ) , b ) , for some η ∈ T ∗ π ( b ) X . Since Im π B is the annihilator of the verticalbundle of the fibration p (i.e. all covectors vanishing on the subbundle ker p ∗ of T B ), a point of ( dS ( b ) , b ) is in Im π B iff the vertical part of dS vanishes at b . In the case of strongly transversal composition, theset Σ of all points in B where this happens is a submanifold, calledthe fiber critical submanifold of the generating family. The smooth Recall that the composition is called transversal if the product Im dS × T ∗ p ofthese canonical relations has transversal intersection with the diagonal ∆ T ∗ B × T ∗ X in T ∗ B × T ∗ B × T ∗ X (we ignore the point in the definition of Im dS ), and stronglytransversal if the image of this transversal intersection embeds properly in T ∗ X under the natural projection. When the composition is transversal, S is also calleda Morse family of functions over X . i ∗ ◦ dS ( b ) = 0 , where i ∗ is the dual of the inclusion i of the vertical subbundle ker p ∗ into T B . YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 10 map λ : Σ −→ T ∗ Xb ( η, p ( b )) with η defined by equation (2.1) is a lagrangian embedding whose imageis exactly T ∗ p ◦ Im dS .Since the standard construction of a generating family for a conormalbundle N ∗ C out of a tubular neighborhood of C ⊂ X , is central for us,we spell it out in the following example ([2]). Example 2.
To begin, we fix a tubular neighborhood of C , thatis, the data ( V, Ψ) of a neighborhood V of C in X equipped with adiffeomorphism Ψ from a neighborhood U of the zero section of thenormal bundle N C into V that maps the zero section identically to C . We denote by U c the restriction of U to the fiber N c C and, corre-spondingly, by Ψ c the restriction of Ψ to U c . This allows us to mapa neighborhood of the zero section of the bundle N ∗ C ⊕ N C over C diffeomorphically into an open submanifold of N ∗ C × X as follows: ( p, v, c ) ( p, Ψ c ( v )) , where p ∈ N ∗ c C and v ∈ N c C . This gives us a generating fam-ily ( B Ψ C , p C , S C ) for N ∗ C , where B Ψ C is defined as the image of thismapping. We will denote by ( p, x, c ) points in B Ψ C . The fibration p C : B Ψ C → V is the projection ( p, x, c ) x , and the generating func-tion is given by the canonical pairing S C ( p, x, c ) = (cid:10) p, Ψ − c ( x ) (cid:11) . The critical submanifold Σ C of S C then consists of the points of theform ( p, c, c ) where c ∈ C and p ∈ N c C . This yields an embedding τ C : N ∗ C −→ B Ψ C , whose image is exactly Σ C , which has an obvious retraction r C ( p, x, c ) =( p, c ) . Composing τ C with the lagrangian embedding λ C : Σ C −→ T ∗ X, generated by the generating family, we obtain the canonical inclusion ι C of N ∗ C into T ∗ X .2.3. Half-densities.
YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 11 α -densities and the intrinsic Hilbert space. An α -density for α ∈ C on a n -dimensional real vector space V is a map ρ from the space offrames F ( V ) (a frame is an ordered basis e = ( e , . . . , e n ) ) to C suchthat ρ ( e · A ) = | det A | α ρ ( e ) for any matrix A ∈ GL ( n ) . We denote by | V | α the one-dimensionalcomplex vector space of α -densities on V .Given a finite dimensional vector bundle E over a smooth manifold X , we denote by | E | α the complex line bundle over M whose fiber at x ∈ X is | E x | α . We will reserve the notation | Ω | α ( X ) for the space ofsmooth sections of | T M | α ; its subspace of compactly supported sectionswill be denoted by | Ω | αc ( X ) . For general density bundles | E | α → X , wewill use the standard notation Γ( X, | E | α ) to denote the section space.An exact sequence of vector bundles −→ A −→ B −→ C −→ over a manifold X induces a canonical isomorphism between the den-sity bundles | B | α ≃ | A | α ⊗ | C | α as well as their corresponding sectionspaces. In particular, we can identify | Ω | ( M ) ⊗ | Ω | ( M ) with thespace | Ω | ( M ) of -density bundle sections on M . These sections canbe integrated, which allows us to give to the vector space | Ω | c ( M ) ofcompactly supported half-densities the structure of a pre-Hilbert spacewith the symmetric bilinear form h µ, ν i := Z M µν, where ¯ µ is the complex conjugate of the half-density µ . The completionof this pre-Hilbert space is usually called the intrinsic Hilbert space of M (see [2] for more details), and we will denote it by H ( M ) .2.3.2. Integrating half-densities over fibrations .
Let p : B → X be afibration. There is a notion of “pushforward” from half-densities on B to half-densities on X that requires some extra data in the formof a section of | ker p ∗ | → X , where ker p ∗ the vertical bundle of thefibration. It goes as follows: First observe that the exact sequence −→ ker p ∗ −→ T B −→ p ∗ ( T X ) −→ , of vector bundles over B induces the canonical isomorphism | Ω | ( B ) ≃ Γ( B, | ker p ∗ | ) ⊗ Γ( B, | p ∗ ( T X ) | ) . (2.2)Now, suppose that we are given a smooth family of half-densities ρ ( x ) ∈ | Ω | ( p − ( x )) compactly supported on the fibers of p . The data YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 12 of ρ allows us to map half-densities on B to half-densities of M byintegrating them on the fibers of p . Namely, ρ can be regarded as asection of the half density bundle | ker p ∗ | → X , and, in the light of(2.2), we can regard the tensor product µ ⊗ ρ for any half-density µ on B as living in Γ( B, | ker p ∗ | ) ⊗ Γ( B, | p ∗ ( T X ) | ) . The restriction of µ ⊗ ρ to the fibers of p gives thus a family of densi-ties on the fiber p − ( x ) with values in the fixed vector space | T x X | .Therefore, we can integrate µ ⊗ ρ on this fibers and obtain θ ( x ) := Z p − ( x ) µ ⊗ ρ ∈ | T x X | , which is a half-density on X .2.4. Fourier integral operators.
We describe here special classesof FIO that are of concern for us. For a more general presentation,we refer the reader to the standard references [16, 15, 19]. We beginby outlining the main ingredients out of which FIO’s are made andby commenting on the general problem of FIO composition, whichparallels the ill-defined composition of canonical relations.2.4.1.
The FIO “category”.
Given a canonical relation L from T ∗ X to T ∗ Y , one can associate a class of operators Four ~ ( L ; X, Y ) , theFourier integral operators with wave front L , from the intrinsic Hilbertspace H ( X ) to the intrinsic Hilbert space H ( Y ) . More precisely, anoperator Q in this class is a family Q = { Q ~ : ~ ∈ (0 , } of operatorsdepending smoothly on a parameter ~ in a way that will be made clearlater on. We will be mostly concerned with the asymptotics of theseoperators in the semiclassical limit, that is, when ~ → . For us, a semiclassical FIO will mean an equivalence class of FIO’s that havethe same expansion in ~ at all orders. The space of all FIO’s between X and Y will be denoted by Four ~ ( X, Y ) and the space of semiclassicalFIO’s by SFour ( X, Y ) . We will come back to the latter at the end ofthis section.An FIO can be defined explicitly in terms of an oscillatory integralrepresentation. For this, we need to fix a generating family for thewave front as well as some “vertical” half-density on the total spaceof the generating family. We refer the reader to [18] for the generaldescription of these representations.In general, the composition of two FIO’s fails to be a FIO and,therefore, the collection Four ~ of all FIO’s only form a “category” in YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 13 the same sense as
Sympl ext does. Actually, this is more than an mereanalogy. Namely, we can define the map
W F : Four ~ ( X, Y ) −→ Sympl ext ( T ∗ X, T ∗ Y ) that associates to an FIO T its wave front WF( T ) . Now, it is a wellknow-result ([18]) that when the wave fronts of two FIO’s T and T have strongly transversal composition, then the composition of the op-erators themselves yields a FIO whose wave front is given by the com-position of the wave fronts: WF( T ◦ T ) = WF( T ) ◦ WF( T ) . Moreover, the wave front generating families also compose: if ( B i , p i , S i ) is a wave front generating family for T i ∈ Four ~ ( M i , M i +1 ) ( i = 1 , ,then the fibration B × M B −→ M × M together with the generating function ( S + S )( b , b ) = S ( b ) + S ( b ) is the generating family for WF( T ◦ T ) . Therefore, whenever we canfind a subcategory L of canonical relations in Sympl ext with stronglytransversal composition, we obtain an honest category
Four L ~ of op-erators formed by the FIO’s having canonical relations in L as wavefront. In this case the wave front map becomes a functor WF :
Four L ~ −→ L between the corresponding categories. Example 3.
For instance, the category C of cotangent lifts has anassociated category of FIO’s, for which the "kernel" of the wave frontfunctor WF assigns to each identity morphism T ∗ id X in C the algebraof pseudodifferential operators on X .2.4.2. Integral representation.
We now give a more descriptive presen-tation of a FIO class whose wave fronts have generating families withvery good properties. In particular, this class encompasses the FIO’son cotangent lifts as we will see in details in the next section.
Assumption.
Let ( B, p, S ) be a generating family for a canonical re-lation from T ∗ X to T ∗ Y . We denote by p X and p Y the compositionsof the generating family fibration p : B → X × Y with the canonicalprojections on the corresponding factors. From now on, we will assumethat: YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 14 (1) The manifold B is fibered over its critical submanifold π Σ : B → Σ , and any fiber π − ( s ) can be identified with a neighborhood of p X ( s ) in X .(2) The generating function S : B → R has a unique non-degeneratecritical point on the fiber p − Y ( y ) for each y ∈ Y . The set Z of all thesecritical points is a submanifold of Σ . This assumption implies in particular that a half-density Ψ on X induces a family Ψ( s ) ∈ | Ω | ( π − ( s )) by considering the restrictions of Ψ to neighborhood of p X ( s ) in X . Thus, given another half-density a ∈ | Ω | ( L ) on the canonical relation L generated by ( B, p, S ) , that we transportwith the lagrangian embedding λ to the critical submanifold Σ itself,we can identify the tensor product Ψ ⊗ λ ∗ a with a half-density on B .Now, suppose we are given a section ρ of the half-density vertical bundle | ker( p Y ) ∗ | → B . With this extra data, we can define an operator Q ~ ( a, B ) : | Ω | ( X ) → | Ω | ( Y ) , by fiber integration as explained in paragraph 2.3.2:(2.3) (cid:0) Q ~ ( a, B )Ψ (cid:1) ( y ) := Z p − Y ( y ) λ ∗ a ⊗ Ψ ⊗ ρ e i ~ S . This integral representation can be taken as a definition for the FIO’sin
Four ~ ( L ; X, Y ) provided that the wave front L has a generatingfamily with the properties as above. Note that any other choice of ρ would yield the same class of FIO.This integral representation makes it clear that, once a section ρ hasbeen fixed, we have a map, called the total symbol map , σ B,ρ : Four ~ ( L ; X, Y ) → | Ω | ( L ) , that associates to the operator T its total symbol , that is, the half-density a on its wave front L . Remark . Note that the total symbol is not an invariant of the FIOsince it depends on a choice of a generating family as well as on thechoice of the vertical half-density section ρ .2.5. The semiclassical limit.
The semiclassical intrinsic Hilbert space.
We say that a smoothmap ~ f ~ from the parameter space (0 , to a normed vector space V is of order ~ ∞ (and we will write f ~ ∈ O ( h ∞ )) if, for each positiveinteger N , there is a real positive constant C N such that k f ( ~ ) k ≤ YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 15 C N ~ N . Two paths are equivalent if their difference is of order ~ ∞ . Wedenote by V ~ the quotient space. Whenever V is finite dimensional,the Borel summation Theorem allows us to identify V ~ with the space V [[ ~ ]] of formal power series in ~ with coefficients in V by taking theTaylor series of f ~ at . In the particular cases when V = C or R , wewill mostly prefer the interpretation of the asymptotic spaces C ~ and R ~ in terms of the corresponding rings of formal power series R [[ ~ ]] and C [[ ~ ]] .Now let a ~ and b ~ be sections of a finite dimensional vector bundle E → X depending smoothly on a parameter ~ ∈ (0 , . We will saythat a ~ and b ~ are equivalent modulo ~ ∞ if the difference between theirlocal representations at any point as well as those of all their derivativesare of order ~ ∞ . In the case of the line bundle of α -densities on X , wewill write | Ω ~ | α ( X ) for the resulting quotient space.The inner product on | Ω | ( X ) yields an inner product on | Ω ~ | ( X ) with values in C [[ ~ ]] in the the sense of [5]. We will call the resultinginner product space the semiclassical Hilbert space of the manifold X ,and we will denote it by H ~ ( X ) , though of course it is not an Hilbertspace in the usual sense. Elements in H ~ ( X ) can be seen as classesof ~ -dependent half-densities on X that have the same semiclassicallimit, that is, the same O ( ~ ∞ ) asymptotics when ~ → . Upon Taylorexpansion, we can also regard H ~ ( X ) as the space | Ω | ( X )[[ ~ ]] of formalpower series in ~ with coefficients in the half-densities.2.5.2. Oscillatory integrals on microfolds.
Let µ ~ ∈ | Ω | ( B ) be a com-pactly supported density on B depending smoothly on ~ ∈ (0 , , andlet S : B → R be a smooth function whose critical points form asubmanifold Z of B . The stationary phase theorem tells us that theasymptotics modulo O ( ~ ∞ ) of the oscillatory integral Q ~ ( µ, S ) = Z B µ ~ e i ~ S , depends only on the behavior of µ ~ and S in a neighborhood of Z . Tobe more precise, we introduce the following definition: Definition 5.
Let B be a manifold and Z ⊂ B a submanifold. Acut-off function χ : B → R for Z is a smooth function such that thereexist neighborhoods U and V of Z in B such that U ⊂ V and suchthat χ | U ≡ and χ | V c ≡ where V c is the complement of V in B .The stationary phase theorem tells us that Q ~ ( µχ ) and Q ~ ( µχ ′ ) areequivalent modulo ~ ∞ for any two cut-off functions χ, χ ′ : B → R for Z . For this reason, any two densities on X having the same germ on YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 16 Z will have equal oscillatory integrals modulo ~ ∞ . This fact allows usto define semi-classical integrals on manifold germs or microfolds (seethe definition in the Introduction). Definition 6. An α -density on [ B, Z ] is an α -density germ [ µ ] on B around Z . We will use the notation | Ω | α ([ B, Z ]) . Let [ µ ] ∈ | Ω | ([ B, Z ]) be a density germ around Z and let S : B → R be as above. We definethe semiclassical oscillatory integrals.c. Z µe i ~ S to be the equivalence class modulo ~ ∞ of Q ~ ( χµ ) , where µ ∈ [ µ ] and χ is a cut-off function for Z in B .2.5.3. Semiclassical Fourier integral operators.
We are now interestedin the semiclassical limit of FIO’s whose defining generating fami-lies satisfy the assumptions in paragraph 2.4.2. Therefore, instead ofconsidering Q ~ ( a, B ) in (2.3) as a collection of operators indexed by ~ ∈ (0 , , we will see it as single operator Q ~ ( a, B ) : H ~ ( X ) −→ H ~ ( Y ) acting on the semiclassical intrinsic Hilbert spaces. Moreover, becauseof our assumption that S has a single critical point on each of thefibers p − Y ( y ) and that these critical points form a submanifold Z of B , we see, from the last paragraph, that generating families ( B, p, S ) having the same germ around at Z and half-densities a having thesame germ around λ ( Z ) ⊂ L will yield operators Q ~ ( a, B ) with thesame asymptotics modulo O ( ~ ∞ ) . In other words, these operatorswill coincide when looked upon as acting on the semiclassical intrinsicHilbert spaces.From the considerations above, it makes sense to introduce germsof lagrangian submanifolds and generating families to start with. Thismotivates the following definition: Definition 7.
A generating family for a lagrangian submicrofold [ L, C ] of a cotangent bundle T ∗ X is a triple ([ B, Z ] , [ S ] , [ p ]) for which there isa representative ( B, S, p ) that is a generating family for a representative L of the lagrangian submicrofold and such that(1) the critical submanifold Σ contains Z ,(2) the lagrangian embedding λ : Σ → T ∗ X maps Z diffeomorphi-cally onto C . Example 8.
As an example of this last definition, let us consider thecase of the conormal bundle N ∗ C seen as the lagrangian submicrofold YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 17 [ N ∗ C, C ] of [ T ∗ X, X ] . The generating family ( B Ψ C , p C , S C ) of Exam-ple 4.1 induces a generating family for the microbundle N ∗ C by tak-ing germs appropriately. Namely, one easily sees that the informationneeded to generate the lagrangian embedding germ [ ι C ] : N ∗ C −→ [ T ∗ X, C ] is contained in the microfold [ B C , Z C ] , where Z C is the submanifold ofpoints of the form (0 , c, c ) with c ∈ C , along with the germs [ p C ] : [ B C , Z C ] −→ [ T ∗ X, C ] , [ S C ] : [ B C , Z C ] −→ [ R , . Since Z C is contained in Σ C , we can take the germ [Σ C , Z C ] , which wewill call the critical submicrofold of the generating function germ [ S C ] . As before, we have the embedding [ τ C ] : N ∗ C −→ [ B C , Z C ] , whose image is [Σ C , Z C ] . This data produces the lagrangian embeddinggerms [ λ C ] : [Σ C , Z C ] −→ [ T ∗ X, C ] , [ ι C ] : N ∗ C −→ [ T ∗ X, C ] , where [ ι C ] = [ λ C ◦ τ C ] as before.Forgetting now everything but the O ( ~ ∞ ) asymptotics of our FIO’s,we can directly start from the data of a generating family ([ B, Z ] , [ p ] , [ S ]) of a lagrangian submicrofold [ L, C ] in T ∗ ( X × Y ) for which a represen-tative satisfies the assumptions in paragraph (2.4.2) and half-densitygerms [ a ] on the lagrangian submicrofold [ L, C ] . In this way, we obtainoperators Q ~ ([ a ] , [ B, Z ]) from H ~ ( X ) to H ~ ( Y ) by replacing the inte-gral in (2.3) by its semiclassical version introduced in Definition 6. Wecall these operators the semiclassical Fourier integral operators ,and we denote their space by SFour ( X, Y ) . Note that the wave frontmap WF now takes its values in lagrangian submicrofolds and the totalsymbol map in half-density germs.3. Quantization of cotangent lifts
In this section, we study the semiclassical FIO’s associated to thecategory C of cotangent lifts. For this, we need to introduce the notionof micro (and local) exponential: From now on, we will write simply T ∗ X for [ T ∗ X, X ] when it is clear from thecontext that we are interested in the microfold YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 18
Definition 9.
Let M be a smooth manifold. A micro exponential is a diffeomorphism germ [Ψ] : [ T M, Z M ] → M that sends the zerosection Z M to M and such that, for each x , T Ψ x = id , where Ψ x isthe restriction of Ψ to T x M . A local exponential is a representative Ψ ∈ [Ψ] of a micro exponential. We will usually denote the domain of Ψ x by U x and its range by V x .We next show that cotangent lifts are very special in the followingsense: Proposition 10.
Let T ∗ φ : T ∗ M → T ∗ N be a cotangent lift to asmooth map φ from N to M and let Ψ : U ⊂ T M → M be a localexponential on M as in Definition 9. Then, there is a canonical gen-erating family ( B Ψ φ , p φ , S φ ) , depending only on the choice of the localexponential, such that(1) the assumptions in paragraph 2.4.2 hold;(2) the image of the critical points Z φ of S φ on the p N -fibers by thelagrangian embedding is the graph of φ seen as a submanifold of T ∗ φ ;(3) there is a canonical section ρ B : N → | ker( p N ) ∗ | Let us see some immediate consequences of these good properties.First of all, the O ( ~ ∞ ) asymptotics of FIO’s from H ~ ( M ) to H ~ ( N ) whose wave front are the cotangent lifts and with integral representa-tion(3.1) (cid:16) Q ~ ( a, T ∗ φ ) u (cid:17) ( x ) := s.c. Z p − N ( x ) u ⊗ a ⊗ ρ B e i ~ S φ , where u ∈ H ~ ( M ) are completely determined by (1) the lagrangiansubmicrofolds [ T ∗ φ, gr φ ] , (2) the corresponding generating family germ (cid:16) [ B Ψ φ , Z φ ] , [ p φ ] , [ S φ ] (cid:17) , and (3) the half-density germs [ a ] ∈ | Ω ~ | ([ T ∗ φ, gr φ ]) . Moreover, thewhole calculus depends only on the germs of the local exponentialsaround the zero sections, that is, the micro exponentials [Ψ] : [ T ∗ M, Z M ] → M of Definition 9.Second of all, the class of semiclassical FIO’s on cotangent lifts hasvery good functorial properties. Since the cotangent lifts form a cat-egory C , their associated semiclassical FIO’s are always composablewhen sources and targets are compatible, and so the collection SFour C ~ of these FIO’s also forms a category, and we have a wave front functor WF :
SFour C ~ −→ C , YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 19 as explained in the previous section.The rest of this section is devoted to the proof of Proposition 10.Throughout, M and N will be two smooth manifolds as above whosepoints will be denoted by x and x respectively. Correspondingly, wewill write ( p , x ) and ( v , x ) to denote points in T ∗ M and T M and ( p , x ) and ( v , x ) for points in T ∗ N and T N .3.1.
Canonical identifications.
Let φ : N → M be a smooth map.The conormal bundle of its graph N ∗ (gr φ ) can be identified with thepullback bundle φ ∗ ( T ∗ M ) via the the lagrangian embedding ι φ : φ ∗ ( T ∗ M ) −→ T ∗ ( M × N ) , given by ι φ ( p , x ) = (cid:16)(cid:0) p , − dφ ∗ p (cid:1) , (cid:0) φ ( x ) , x (cid:1)(cid:17) . (3.2)Let us list here several obvious but crucial identifications, which willbe used extensively in what follows: gr φ ≃ N, (3.3) N ∗ gr φ ≃ φ ∗ ( T ∗ M ) , (3.4) N gr φ ≃ φ ∗ ( T M ) , (3.5)and, therefore, the vector bundle N ∗ gr φ ⊕ N gr φ over gr φ can beidentified with the pullback vector bundle φ ∗ ( T ∗ M ⊕ T M ) over N .From now on, we will not make a strict distinction between N ∗ gr φ and the cotangent lift T ∗ φ , and similarly for the other identifications.3.2. Generating families, tubular neighborhoods and micro ex-ponentials.
Example 2 shows how to construct a generating family fora general conormal bundle once a tubular neighborhood for the sub-manifold has been given. Since cotangent lifts are essentially conormalbundles, this construction works also for them. However, we want todiscuss a special class of tubular neighborhoods in the context of cotan-gent lifts, which will prove to be very convenient for us.
Remark . A connection ∇ on M gives rise to a micro exponentialby taking the germ of the connection’s exponential map exp ∇ . If wereplace germs by jets in the previous definition, we obtain what is calleda formal exponential in [17] in the context of Fedosov star-products.A micro exponential produces a “micro linearization” of the manifold M , that is, a germ [ l ] : [ M × M, ∆ M ] −→ [ T ∗ M, Z M ] , YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 20 given in representatives by l ( x, y ) = v , where Ψ x ( v ) = y . The notionof manifold linearization has been introduced in [4] in order to extendthe pseudo-differential operator calculus on R n to general manifolds.Micro exponentials are also related to Milnor’s construction of tangentmicrobundles [24].Returning to cotangent lifts, we can now easily construct a tubularneighborhood for the graph of φ : N → M out of the extra data of amicro exponential [Ψ] on M only. For the neighborhood itself, we take V φ := [ x ∈ N Ψ( U φ ( x ) ) × φ − (Ψ( U φ ( x ) )) , where U x is the domain of Ψ x for a fixed representative Ψ ∈ [Ψ] . Now,using the identification of the graph normal bundle with the pull back φ ∗ ( T M ) , we obtain the tubular neighborhood diffeomorphism germfrom N gr φ to V φ given explicitly by ( v , x ) (cid:0) Ψ φ ( x ) ( v ) , x (cid:1) . We will denote this map again by Ψ in order to keep the notation simpleand to acknowledge the micro exponential dependence in the tubularneighborhood notation ( V φ , Ψ) .Now, repeating the generating family construction for conormal bun-dles given in Example 4.1 with the tubular neighborhood ( V φ , Ψ) , weobtain a generating family ( B Ψ φ , p φ , S φ ) for the cotangent lift T ∗ φ . Invery explicit terms, we have that B Ψ φ = n ( p , x , x ) : p ∈ T ∗ φ ( x ) M, x ∈ ImΨ φ ( x ) , x ∈ N o ,p φ ( p , x , x ) = ( x , x ) ,S φ ( p , x , x ) = (cid:10) p , Ψ − φ ( x ) ( x ) (cid:11) , Σ φ = n ( p , φ ( x ) , x ) : p ∈ T ∗ φ ( x ) M, x ∈ N o . Again, we denote by τ φ the embedding of φ ∗ ( T ∗ M ) into B Ψ φ and by λ φ the lagrangian embedding of Σ φ into T ∗ ( M × N ) . The composition ι φ := λ φ ◦ τ φ yields the usual inclusion (3.2).Let us check now that this generating family satisfies the assumptionsof paragraph 2.4.2. First of all, observe that we have a retraction π Σ from B Ψ φ onto the critical submanifold Σ given by π Σ : ( p , x , x ) ( p , φ ( x ) , x ) , the fiber of which is the neighborhood V x of x in M determined bythe exponential Ψ . This shows the first part of the assumption. Now, YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 21 if we consider the restriction of S φ on the fiber p − N ( x ) = T ∗ φ ( x ) M × Ψ φ ( x ) ( U φ ( x ) ) , and if compute its critical point there, we see that the vanishing of ∂S φ ∂p ( p , x , x ) = Ψ − φ ( x ) ( x ) on the fiber implies that Ψ − φ ( x ) ( x ) = 0 , which is equivalent to x = φ ( x ) . In turn, the vanishing of ∂S φ ∂x ( p , x , x ) = h p , T x Ψ − φ ( x ) i at x = φ ( x ) implies that p = 0 since the derivative T φ ( x ) Ψ − φ ( x ) isthe identity by definition of the exponential. This shows that S φ has asingle critical point on p − N ( x ) given by (0 , φ ( x ) , x ) . We see that thecollection of all these critical points is the following submanifold Z φ := n (0 , φ ( x ) , x ) : x ∈ N o , which is contained in the (usual) critical submanifold Σ φ of the gener-ating family. Finally, we see that the lagrangian embedding λ φ carries Z φ to the graph of φ , seen as a submanifold of the cotangent lift T ∗ φ .We see then that the semiclassical limit of an FIO whose wave frontis a cotangent lift is controlled by the lagrangian microfold [ T ∗ φ, gr φ ] (which we will continue to denote simply by T ∗ φ ) and its generatingfamily germ(3.6) (cid:16) [ B Ψ φ , Z φ ] , [ p φ ] , [ S φ ] (cid:17) . The canonical half-density ρ B . The fibration p N : B Ψ φ → N has a section, given by s N ( x ) = (0 , φ ( x ) , x ) , whose image is Z φ . Wewill exhibit a canonical section ρ B of the half-density bundle | ker( p N ∗ ) | −→ B Ψ φ (3.7)associated to the vertical bundle of this projection.Let us introduce the shorthand T M for the vector bundle T ∗ M ⊕ T M over M . As a first step, observe that the half-density bundle | T M | hasa canonical section ρ M obtained fiberwise from the canonical symplecticform on T x M , seen as a symplectic vector space for each x ∈ M . Thissection ρ M produces a new section, which we will continue to denoteby ρ M , on the pullback bundle ( φ ◦ p N ) ∗ ( T M ) −→ B Ψ φ . YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 22
Our strategy now is to construct a vector bundle isomorphism A be-tween ( φ ◦ p N ) ∗ ( T M ) and ker( p N ) ∗ and to use it in order to carry ρ M to a section of (3.7), which will be our canonical half-density ρ B . Inorder to produce A , we can first observe that the tangent space of B ψφ at z = ( p , x , x ) can be decomposed into the direct sum T z B Ψ φ = T ∗ φ ( x ) M ⊕ T x M ⊕ T x N. Thus, the vertical bundle fiber at z is ker z ( p N ∗ ) = T ∗ φ ( x ) M ⊕ T x M since it is the tangent space to the fiber p − N ( x ) at z . At this point, wecan write down the desired vector bundle isomorphism A by using thederivative of a micro exponential. Fiberwise, A ( z ) is the linear mapfrom ( φ ◦ p N ) ∗ ( T M ) z = T ∗ φ ( x ) M ⊕ T φ ( x ) M to ker z ( p N ∗ ) = T ∗ φ ( x ) M ⊕ T x M given in matrix form by A ( z ) = id T ∗ φ ( x M ∂ v Ψ φ ( x ) (cid:0) Ψ − φ ( x ) ( x ) (cid:1) ! . Finally, we can set ρ B ( z )( e ) := π ~ ) m + n ρ M ( z )( e · A ( z ) − ) , where m = dim M and n = dim N . Example 12.
Let us compute ρ B in the case where M = R m , N = R n and Ψ x : R m → R m is a global diffeomorphism of for all x ∈ R n . Inthis case, we have that B Ψ φ = R mp × R mx × R nx , and we can identify the fibers of both ( φ ◦ p N ) ∗ ( T M ) and ker( p N ) ∗ withthe vector space V := ( R m ) ∗ ⊕ R m . The canonical section ρ M is theconstant symplectic half density ρ M ( x ) = p dp p dx on the symplectic vector space V . Hence, we obtain that ρ B (cid:0) ( p , x , x ) (cid:1) = (cid:12)(cid:12)(cid:12) det ∂ v Ψ φ ( x ) (cid:0) Ψ − φ ( x ) ( x ) (cid:1)(cid:12)(cid:12)(cid:12) − √ dp √ dx (2 π ~ ) m , since ρ M ( x )( e · A ( z ) − ) = | det A ( z ) − | ρ M ( x ) .In particular, in the case of the global exponential coming from thecanonical affine connection on R m , Ψ x ( v ) = x + v , we have that (cid:12)(cid:12) det ∂ v Ψ φ ( x ) ( v ) (cid:12)(cid:12) = 1 in the above formula for ρ B ( z ) . YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 23
The Fourier integral operator Q h ( a, T ∗ φ ) . In this paragraph,we put everything together to construct the semi-classical Fourier in-tegral operator Q ~ ( a, T ∗ φ ) : H ~ ( M ) −→ H ~ ( N ) associated to a half-density a ∈ | Ω ~ | ( φ ∗ ( T ∗ M )) . It goes as follows:we start with u ∈ H ~ ( M ) . Now, the restriction of the half-densityproduct u ⊗ a ∈ | Ω ~ | (cid:0) M × φ ∗ ( T ∗ M ) (cid:1) to the submanifold B Ψ φ ⊂ ( M × φ ∗ ( T ∗ M ) (cid:1) produces a half-density on B ψφ , which we will still write as u ⊗ a . We are now in the case of Sec-tion 2.3.2, with a half-density u ⊗ a ⊗ e i ~ S φ on B Ψ φ and the canonicalhalf-density section ρ B on the fibers of the projection p N : B Ψ φ → N .This allows us to define our operator pointwise as the semi-classical os-cillatory integral (3.1) since, as already shown, S φ has a unique criticalpoint in the fiber p − N ( x ) , namely at s N ( x ) = (0 , φ ( x ) , x ) . Example 13.
We continue Example 12, in which M = R m , N = R n ,the micro exponential is a global diffeomorphism (as for instance theone coming from the canonical affine connection, i.e., Ψ x ( v ) = x + v ).In coordinates, we have S φ ( p , x , x ) := (cid:10) p , Ψ − φ ( x ) ( x ) (cid:11) ,a := a ( p , x ) p dp p dx ,u := u ( x ) p dx . Using the formula for ρ B computed in Example 12, we obtain that auρ B = (cid:0) a ( p , x ) u ( x ) (cid:12)(cid:12)(cid:12) det ∂ v Ψ φ ( x ) (cid:0) Ψ − φ ( x ) ( x ) (cid:1)(cid:12)(cid:12)(cid:12) − p dx (cid:1) dp dx (2 π ~ ) ym , is a density on p − N ( x ) with values in | T ∗ x N | . Hence, (cid:0) Q ~ ( a, T ∗ φ ) u (cid:1) ( x ) coincides with (cid:18) s.c. Z a ( p , x ) u ( x ) (cid:12)(cid:12)(cid:12) det ∂ v Ψ φ ( x ) (cid:12)(cid:12)(cid:12) − e i ~ h p , Ψ − φ ( x ( x ) i dp dx (2 π ~ ) m (cid:19) p dx . In the case, we take the canonical affine connection Ψ x ( v ) = x + v ,we obtain the semi-classical version of pseudo-differential operators.4. Quantization of symplectic micromorphisms
In this section, we extend the semiclassical FIO calculus on cotan-gent lifts developed in the previous section to a more general class ofwave fronts. We do so by deforming the canonical generating families
YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 24 of cotangent lifts. It turns out that these new wave fronts belong toa special class of canonical relation germs: the class of symplectic mi-cromorphisms which are the morphisms of the cotangent microbundlecategory
Cot extmic as constructed in [8]. As is the case for cotangentlifts, symplectic micromorphisms always compose well, and we obtaina new category of semiclassical Fourier integral operators together witha wave front functor
WF :
SFour
Cot extmic → Cot extmic . This new category encompasses well known examples of semiclassicalFIO’s, the main example of which is the class of pseudo-differentialoperators.Since our deformations of cotangent lift generating families have ameaning for general conormal bundles, we start with this case.4.1.
Conormal microbundle deformations.
Our goal here is to de-scribe a class of lagrangian deformations of conormal bundles along atubular neighborhood and show how to obtain their generating familiesas deformations of the standard generating family of Example 2.A tubular neighborhood ( V, Ψ) of C in X fibers the neighborhood V of C into slices V c := Ψ( U c ) over the points c ∈ C . The conormalbundle N ∗ V c of each of these slices is a lagrangian submanifold of T ∗ X ,which is transversal to the conormal bundle of C . Moreover, N ∗ C and N ∗ V c intersect only at c . The distribution Λ over C given by thecollection of tangent spaces to N ∗ V c for each c ∈ C is thus a lagrangiandistribution over C , which is transversal to N ∗ C , hence we have thelagrangian splitting T ( T ∗ X ) = T ( N ∗ C ) ⊕ Λ of the tangent space to T ∗ X along C . Definition 14. A deformation of the lagrangian microfold N ∗ C alonga tubular neighborhood ( V, Ψ) of C in M is a lagrangian submicrofold [ L, C ] of T ∗ X that is transversal to the lagrangian distribution Λ de-fined above.Our goal is to show that all deformations of N ∗ C along a giventubular neighborhood are obtained from the standard conormal bundlegenerating family by deforming its generating function of the conormalbundle as follows: Definition 15.
Let [ S C ] be the generating function of the conormalbundle to C . We call a deformation of [ S C ] a function germ of the form YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 25 [ S fC ] = [ S C ] − [ r ∗ C f ] . where [ f ] ∈ C ∞ ( N ∗ C, C ) s.t. ∂ p f (0 , x ) = 0 . Proposition 16.
The triple (cid:0) [ B Ψ C , Z C ] , [ p C ] , [ S fC ] (cid:1) generates a deforma-tion [ L Ψ f , C ] of N ∗ C along the tubular neighborhood ( V, Ψ) . Conversely,all such deformations arise this way. Moreover, the critical submicro-fold [Σ fC , Z C ] is given, in representatives, by all the points z ∈ B Ψ C ofthe form z = (cid:0) p, Ψ c ( ∂ p f ( p, c ) , c (cid:1) , where ( p, c ) is taken in an appropriate neighborhood of the zero sec-tion of N ∗ C . The corresponding lagrangian embedding germ [ λ fC ] :[Σ fC , Z C ] → [ T ∗ X, C ] is given explicitly by λ fC ( z ) = (cid:16) T ∗ v f Ψ c ( p ) , Ψ c ( c ) (cid:17) , where v f = ∂ p f ( p, c ) .Remark . A consequence of Proposition 16 is that the critical sub-microfolds for all deformations arise as follows: for each deformation [ f ] , there is a diffeomorphism germ [ χ f ] : [Σ C , Z C ] −→ [Σ fC , Z C ] , that fixes the core. It is given in representatives by χ f ( p, c, c ) = (cid:16) p, c, Ψ c ( ∂ p f ( p, c )) (cid:17) . Observe also that we obtain a family of lagrangian embeddings [ ι fC ] : N ∗ C → [ T ∗ X, C ] deforming the canonical inclusion by composition: [ i fC ] = [ λ fC ◦ χ f ◦ τ C ] . Explicitly, we have ι fC ( p, c ) = (cid:16) T ∗ v f Ψ c ( p ) , Ψ c ( c ) (cid:17) . Proof.
Let us first prove that the triple (cid:0) [ B Ψ C , Z C ] , [ p C ] , [ S fC ] (cid:1) is a gen-erating family. This amounts to showing that there is a representative S fC ∈ [ S fC ] that is non-degenerate. In is enough to work locally. In localcoordinates, we have that S fC ( p, x, c ) = h p, Ψ − c ( x ) i − f ( p, c ) , and that the critical set Σ fC is the locus of points such that H ( p, x, c ) =0 with H ( p, x, c ) := Ψ − c ( x ) − ∂ p f ( p, c ) . YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 26
Observe now that, for all (0 , c, c ) ∈ Z C , we have H (0 , c, c ) = 0 and ∂H∂x (0 , c, c ) = id . Hence, an application of the implicit function theorem tells us that,for each ( p, c ) with p small enough, there is an unique x ( p, c ) suchthat H ( p, x ( p, c ) , c ) = 0 , and thus the solutions of this equation form asubmanifold Σ fC containing Z C . Taking the germ [Σ fC , Z C ] yields thus asubmicrofold: the critical submicrofold of [ S fC ] . Actually, the equation H ( p, x, c ) = 0 is explicitly solvable since f is independent of x : foreach ( p, c ) sufficiently close to the zero section, we can take x ( p, c ) = Ψ c ( ∂ p f ( p, c )) , which gives the form of the critical submicrofold in representatives. Itfollows that (cid:0) [ B Ψ C , Z C ] , [ p C ] , [ S fC ] (cid:1) is a generating family generating a la-grangian submicrofold, which we denote by [ L Ψ f , C ] . A straightforwardcomputation now gives the form of the lagrangian embedding germ λ fc .It remains to see that [ L Ψ f , C ] is a deformation of N ∗ C and thatall deformations arise this way. For this, let us identify the cotangentbundle T ∗ N ∗ C with N ∗ C ⊕ N C and introduce the diffeomorphism germ g : [ T ∗ N ∗ C, C ] −→ [ B Ψ C , Z C ] , that maps ( p, v, c ) to ( p, Ψ( v ) , c ) . It produces an equivalent generatingfamily (cid:0) [ T ∗ N ∗ C, C ] , p C ◦ g , S C ◦ g (cid:1) whose critical submicrofold is the conormal microbundle N ∗ C . Thelagrangian embedding germ of this new generating family can be con-veniently described as the restriction to the critical submicrofold of asymplectomorphism germ [ χ ] : [ T ∗ N ∗ C, C ] −→ [ T ∗ X, C ] , that sends the vertical distribution V ( T ∗ N ∗ C ) over C to the lagrangiandistribution Λ . (The existence of such a germ [ χ ] is guaranteed byTheorem 7.1 in [31].) Now, in this new description, the generatingfunction deformations read ( S fC ◦ g )( p, v, c ) = h p, v i − f ( p, c ) . The corresponding critical submicrofolds are nothing but the lagrangiansubmicrofolds [gr df, C ] and the lagrangian embedding germs are givenby the restriction of [ χ ] to them. Since lagrangian submicrofolds ofthe form [gr df, C ] are exactly the lagrangian submicrofolds through C that are transversal to the vertical distribution in T ∗ N ∗ C , then the YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 27 generated lagrangian submicrofolds [ χ (gr df ) , C ] are exactly all the la-grangian submicrofolds through C that are transversal to Λ , that is,all the deformations of N ∗ C along the tubular neighborhood. (cid:3) Symplectic micromorphisms as deformed cotangent lifts.
In micro-geometry, we can define a canonical relation from T ∗ M to T ∗ N to be a lagrangian submicrofold [ V, gr φ ] ⊂ T ∗ M × T ∗ N whose core is the graph of a smooth map φ : N → M . Althoughcanonical relations do not compose well in general, it is possible tosingle out a class for which they do. This class is the class of symplecticmicromorphisms as introduced in [8]. To distinguish them, we will usethe special notation ([ V ] , φ ) : T ∗ M −→ T ∗ N, instead of [ V, gr φ ] . They can be characterized in several ways, the mostuseful for us now being the following: Definition 18.
A symplectic micromorphism ([ V ] , φ ) from T ∗ M to T ∗ N is a canonical relation [ V, gr φ ] that is transversal to the lagrangiandistribution Λ : =
T Z M ⊕ V ( T ∗ N ) . (4.1)over gr φ .Cotangent lifts are symplectic micromorphisms, but not all symplec-tic micromorphism arise this way. However, any symplectic micromor-phism can be realized as a cotangent lift deformation in the sense ofparagraph 4.1. More precisely, given a micro exponential [Ψ] on M ,one can deform the associated generating family (cid:0) [ B Ψ φ , Z φ ] , [ p φ ] , [ S φ ] (cid:1) of the cotangent lift T ∗ φ : T ∗ M → T ∗ N by adding a function germ toit:(4.2) S fφ ( p , x , x ) = h p , Ψ − φ ( x ) ( x ) i − f ( p , x ) , where [ f ] is a function germ on the conormal bundle N ∗ (gr φ ) (iden-tified, as usual, with the pullback bundle φ ∗ ( T ∗ M ) ) that is identicallynull on the zero section and whose derivative in the fiber directionvanishes. This yields a lagrangian submicrofold [ L f Ψ , gr φ ] , which is adeformation of T ∗ φ along the tubular neighborhood ( V φ , Ψ) in the senseof Definition 14. YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 28
Now, as explained in paragraph 4.1, [ L f Ψ , gr φ ] is transversal to thelagrangian distribution Λ Ψ given by tangent spaces at points of gr φ tothe conormal bundles N ∗ V ( φ ( x ) ,x ) of the tubular neighborhood slices V ( φ ( x ) ,x ) := Ψ φ ( x ) ( U φ ( x ) ) × { x } . As is easily checked, the lagrangian distribution Λ Ψ is independent ofthe micro exponential and coincides with the lagrangian distribution(4.1) defining symplectic micromorphisms. Therefore, an applicationof Proposition 16 to this case immediately yields: Proposition 19.
Once a micro exponential [Ψ] on M is fixed, thereis a one-to-one correspondence between the symplectic micromorphismsfrom T ∗ M to T ∗ N with core map φ and the deformations [ L f Ψ , gr φ ] ofthe core map cotangent lift along the tubular neighborhood ( V φ , Ψ) . Enhancements and quantization.
Our goal here is to realizethe class of semiclassical FIO’s on symplectic micromorphisms in termsof a two-step construction performed on them: enhancement and quan-tization.
Definition 20. An enhancement of a symplectic micromorphism (cid:0) [ V ] , φ (cid:1) : T ∗ M → T ∗ N is a half-density germ [ a ] ∈ | Ω | (cid:0) [ V, gr φ ] (cid:1) . The triple ([ a ] , [ V ] , φ ) : T ∗ M → T ∗ N will be called an enhancedsymplectic micromorphism .Let us fix a micro exponential [Ψ] on M and let ([ B Ψ φ , Z φ ] , p φ , S fφ ) be the corresponding generating family of ([ V ] , φ ) : T ∗ M → T ∗ N as inProposition 19. An enhancement [ a ] of the symplectic micromorphism ([ V ] , φ ) yields a half density germ [ a Ψ ] := [( ι fφ ) ∗ a ] ∈ | Ω | (cid:0) φ ∗ ( T ∗ M ) (cid:1) , where [ ι fφ ] : φ ∗ ( T ∗ M ) → T ∗ ( M × N ) is the lagrangian embedding germgiven by the generating family. Now, we can associate to the triple T = ([ a ] , [ V ] , φ ) a semiclassical Fourier integral operator Q ~ ( T ) : H ~ ( M ) −→ H ~ ( N ) exactly as we did for cotangent lifts, except that we replace S φ by itsdeformed version S fφ and a by a Ψ in the semiclassical integral (3.1). Thecrucial point is that both S φ and S fφ have a the same unique critical YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 29 point in the integration fiber p − N ( x ) , namely (0 , φ ( x ) , x ) . Moreover,the deformed family (cid:16) [ B Ψ φ , Z φ ] , [ p φ ] , [ S fφ ] (cid:17) continues to satisfy the assumptions of paragraph 2.4.2.4.4. Local theory.
Quantization of symplectic micromorphisms.
We are interestedhere in the enhancement and the quantization of symplectic micromor-phisms ([ L ] , φ ) : T ∗ U −→ T ∗ V between cotangent bundles of some open subsets U ⊂ R k and V ⊂ R l ,whose canonical global coordinates we denote by ( p , x ) and ( p , x ) respectively. As already noticed in Examples 12 and 13, this case hasthe special special feature of having a global exponential Ψ x ( v ) : = x + v for each open subset U ⊂ R n . Hence, we can canonically associate agenerating family to any symplectic micromorphism ([ L ] , V ) from T ∗ U to T ∗ V . Namely, Proposition 19 shows that there is a unique functiongerm [ f ] ∈ C ∞ (cid:16) φ ∗ ( T ∗ U ) , Z (cid:17) with ∂ p f (0 , x ) = 0 such that (cid:16) [ B ( U, V ) , Z φ ] , [ p φ ] , [ S fφ ] (cid:17) is a generating family for ([ L ] , φ ) , where B ( U, V ) = ( R k ) ∗ × U × V,Z φ = { } × gr φ,S fφ ( p , x , x ) = (cid:10) p , φ ( x ) − x (cid:11) + f ( p , x ) . Remark . The generating function S fφ above differs by a minus signfrom our previous definition in (4.2). This sign is irrelevant, and wechoose to write the generating function this way here in order to betteragree with the usual sign convention for the phase of pseudo-differentialoperators. Notation . At times, it will prove useful to collect the terms in S fφ that do not depend on x into the single term F ( p , x ) : = h p , φ ( x ) i + f ( p , x ) . (4.3) YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 30
The use of the upper case letter F in the generating function notation S Fφ instead of the lower case f will mean that we consider the generatingfunction S Fφ ( p , x , x ) = −h p , x i + F ( p , x ) , in which the upper case F is related to the lower case f via the relation(4.3). We write S φ in case F ( p , x ) = h p , φ ( x ) i , that is for thegenerating function of the cotangent lifts. Since, the generating familyis canonical in the local setting, we will also use the notation [ S Fφ ] todenote the corresponding symplectic micromorphism ([ L F ] , φ ) .The critical microfold of S Fφ is Σ Fφ = (cid:26)(cid:16) p , ∂ p F ( p , x ) , x (cid:17) : ( p , x ) ∈ W (cid:27) , where W is a suitable neighborhood of the zero of φ ∗ ( T ∗ U ) . This yieldsthe explicit formula i Fφ ( p , x ) = (cid:18)(cid:16) − ∂ x S Fφ ( z ) , x (cid:17) , (cid:16) ∂ x S Fφ ( z ) , x (cid:17)(cid:19) , = (cid:18)(cid:16) p , ∂ p F ( p , x ) (cid:17) , (cid:16) ∂ x F ( p , x ) , x (cid:17)(cid:19) , where z is the image of ( p , x ) in Σ Fφ , for the embedding of [ φ ∗ ( T ∗ U ) , Z φ ] into T ∗ U × T ∗ V .In the local case, an enhancement of [ S Fφ ] can be identified with agerm [ a ] of semiclassical square integrable function a ∈ L ~ (( R m ) ∗ × V ) around the { }× V , and the semiclassical intrinsic Hilbert space H ~ ( U ) with L ~ ( U ) in the notation of paragraph 2.5.1. The quantization ofthe enhanced symplectic micromorphism ([ a ] , S Fφ ) is the semiclassicalFourier integral operator from L ~ ( U ) to L ~ ( V ) given by the formula(4.4) (cid:16) Q ~ (cid:0) [ a ] , S fφ (cid:1) Ψ (cid:17) ( x ) := s.c. Z R m × U a ( p , x )Ψ( x ) e i ~ S fφ ( p ,x x ) dp dx (2 π ~ ) m , where S fφ ( p , x , x ) = h p , φ ( x ) − x i + f ( p , x ) . Note that theseoperators comprise the class of semiclassical pseudo-differential oper-ators. Namely, when both U and V are R n , the core map φ is theidentity and f = 0 , we obtain the well-know integral representation for YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 31 semiclassical pseudo-differential operators: (cid:0) Q ~ ([ a ] , S id )Ψ (cid:1) ( x ) = π ~ ) n s.c. Z R n a ( p, y ) f ( y ) e i ~ h p,x − y i dpdy, where a is the total symbol of the operator. Hence, this defines a map Op ~ : [ a ] Q ~ ([ a ] , S ) from function germs on T ∗ R n around the the zero section to opera-tors acting on L ~ ( R n ) . This is known as the standard quantization ofthe symbol a , which satisfies the correspondence principle of quantummechanics: Op ~ ([ x ])Ψ( x ) = ~ i ∂ Ψ ∂x ( x ) , Op ~ ([ p ])Ψ( x ) = x Ψ( x ) , where [ p ] denotes the germ of the projection ( p, x ) p and [ x ] thegerm of the projection ( p, x ) x around the zero section.Here are other examples which are not pseudo-differential operators.The first example describes the class of semiclassical FIO’s whosewave fronts are the symplectic micromorphisms from T ∗ R n to the point. Example 23.
Consider a micromorphism ([ L ] , φ ) from T ∗ R n to thecotangent bundle of the point, which we denote by ⋆ . The core map φ : { ⋆ } → R n is completely determined by the choice of a point x ∈ R n ,and [ L ] is a lagrangian submanifold germ through x transversal to thezero section. Since φ ∗ ( T ∗ R n ) is the fiber T ∗ x R n , the generating functionis a function germ [ S fx ] : [ T ∗ x R n , → [ R , of the form S fx ( p ) = −h p, x i + f ( p ) . A enhancement [ a ] in this case is also a function germ on T ∗ x U at theorigin, and the corresponding quantization Q ~ ([ a ] , S fx ) : L ~ ( R n ) −→ C [[ ~ ]] is a distribution on R n . When f = 0 , we obtain that Q ~ ([ a ] , S x )Ψ = (cid:16) F − ~ (cid:0) a F h (Ψ) (cid:1)(cid:17) ( x ) , where F ~ is the asymptotic Fourier transform on R n . In particular,this yields the derivatives of the delta function concentrated at x : Q ~ ( p α , S x ) = ( − i ~ ) | α | ∂ α δ ( x − x ) , where α ∈ N n is a multi-index.The second example describes the class of semiclassical FIO’s whosewave fronts are the unique symplectic micromorphism from the pointto T ∗ R n . YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 32
Example 24.
There is only one symplectic micromorphism e U from T ∗ E (where E = { ⋆ } is the point) to T ∗ U . Explicitly, e U is given by (cid:0) [ { } × U ] , pr (cid:1) , where pr is the projection of U to the unique pointof E . The generating function of e U is the zero function S ( x ) = 0 .An enhancement of e U is, therefore, a function f ∈ L ~ ( U ) , and itsquantization Q ~ ( f, e U ) : C [[ ~ ]] −→ L ~ ( U ) can be identified with itself: Q ~ ( f, e U )Ψ = f Ψ .4.4.2. Generating function composition formula.
Consider U , V , and W open subsets of euclidean spaces, and let ( B ( U, V ) , S Fφ ) and ( B ( V, W ) , S Gψ ) be the generating families generating the symplectic micromorphisms T ∗ U ([ L ] ,φ ) −→ T ∗ V ([ L ] ,ψ ) −→ T ∗ W, and let (cid:0) B ( U, W ) , S G ◦ Fφ ◦ ψ (cid:1) be the local generating family of the compo-sition (cid:0) [ L ◦ L ] , φ ◦ ψ (cid:1) . Obviously, symplectic micromorphism compo-sition induces a composition operation on local generating families S Gψ ◦ S Fφ := S G ◦ Fφ ◦ ψ . This composition is associative in the sense that it produces a categoryin which objects are open subsets U of R n for some n and a morphismfrom U to V is a local generating family (cid:0) B ( U, V ) , S Fφ (cid:1) . The identitymorphism on U is the local generating family (cid:0) B ( U, U ) , S I id (cid:1) , where thegenerating function is the usual phase of pseudo-differential operators S I id ( p , x , x ) = I ( p , x ) − p x , where I ( p , x ) = p x . This category inherits a symmetric monoidalstructure from the microsymplectic category. The unit object is thepoint R = { ⋆ } , the tensor product on objects is the usual cartesianproduct U × V , and the tensor product of local generating families (cid:16) B ( U , V ) , S F φ (cid:17) ⊗ (cid:16) B ( U , V ) , S F φ (cid:17) is the local generating family (cid:16) B (cid:0) U × U , V × V (cid:1) , S F φ ⊗ S F φ (cid:17) , where the tensor product of the generating functions is given by (cid:0) S Fφ ⊗ S Gψ (cid:1) ( p , ˜ p , x , x , ˜ x , ˜ x ) := S Fφ ( p , x , x ) + S Gψ (˜ p , ˜ x , ˜ x ) . The unit object R is initial: There is only one morphism from R to any other open subset U . Namely, B ( R , U ) = U and the onlygenerating function in G ( R , U ) is the zero function that we denote by S U ( x ) = 0 . YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 33
We now derive an explicit formula for the generating function S G ◦ Fφ ◦ ψ .We will use two general facts on composition and reduction of gener-ating families: Composition . Let p i : B i → M i × M i +1 be a generating family forthe canonical relation L i ⊂ T ∗ M i × T ∗ M i +1 with generating function S i for i = 1 , . Suppose that the composition of L and L is transversal.Then, the fibration B × M B → M × M together with the generatingfunction ( S + S )( b , b ) = S ( b ) + S ( b ) is a generating family forthe canonical relation L ◦ L . Reduction.
Let p : B → M be a generating family with gen-erating function S . Suppose that we can factor p as a composition B π −→ B ′ p ′ −→ M of two fibrations such that the restriction of S toeach fiber π − ( b ′ ) has the unique critical point γ ( b ′ ) . Then, the twogenerating families ( B, p, S ) and ( B ′ , p ′ , S ◦ γ ) generate the same la-grangian submanifold of T ∗ M .With this in mind, we obtain that the fibration B ( U, V ) × V B ( V, W ) p V −→ U × W with generating function (cid:0) S Fφ ⊗ V S Gψ (cid:1) ( p , p , x , x , x , x ) := S Fφ ( p , x , x ) + S Gψ ( p , x , x ) , is a generating family for [ L Gψ ◦ L Fφ ] . Now, we can factor the fibration p V into B ( U, V ) × V B ( V, W ) π −→ B ( U, W ) π U × π W −→ U × W. Assuming that the restriction of S Fφ ⊗ V S Gψ to each fiber π − ( p , x , x ) has a unique critical point γ ( p , x , x ) , we obtain a formula for thegenerating function composition S Gψ ◦ S Fφ := ( S Fφ ⊗ V S Gψ ) ◦ γ. The following lemma guarantees the existence of a unique critical pointon each fiber.
Lemma 25.
In the notation as above, we have that the restrictionof S Fφ ⊗ V S Gψ to the fiber π − ( p , x , x ) has a unique non-degeneratecritical point (¯ p , ¯ x ) = γ ( p , x , x ) which is given as the unique solution of system p = ∂ x F ( p , x ) , (4.5) ¯ x = ∂ p G ( p , x ) . (4.6) YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 34
Proof.
By definition, we have: ( S Fφ ⊗ V S Gψ )( p , x , x , p , x ) = p x + F ( p , x ) + G ( p , x ) − p x . Now, the critical points along π − ( p , x , x ) = ( R lV ) ∗ × V are the points (¯ p , ¯ x ) in the fiber satisfying the equation ∂ (cid:0) S Fφ ⊗ V S Gψ (cid:1) ∂ ( p , x ) (cid:0) p , x , ¯ x , ¯ p , x (cid:1) = 0 , which is exactly equivalent to the system (4.5)-(4.6), since ∂ (cid:0) S Fφ ⊗ V S Gψ (cid:1) ∂ ( p , x ) (cid:0) p , x , ¯ x , ¯ p , x (cid:1) = (cid:18) ¯ p − ∂ x F ( p , ¯ x )¯ x − ∂ p G (¯ p , x ) (cid:19) . Thanks to the fact that F (0 , x ) = 0 and ∂ p G (0 , x ) = ψ ( x ) , we get ∂ (cid:0) S Fφ ⊗ V S Gψ (cid:1) ∂ ( p , x ) (cid:0) , x , ψ ( x ) , , x (cid:1) = 0 for all x , meaning that γ (0 , x , x ) = (0 , ψ ( x )) . In particular, this istrue for x = φ ◦ ψ ( x ) . Now, the Hessian at this point is ∂ (cid:0) S Fφ ⊗ V S Gψ (cid:1) ∂ ( p , x ) (cid:0) , φ ◦ ψ ( x ) , ψ ( x ) , , x (cid:1) = (cid:18) id 0 − ∂ p ∂ x G (0 , x ) id (cid:19) , which is invertible. Now, for small p , the implicit function theoremgives us a section of critical points (cid:2) B ( V, W ) , Z φ ◦ ψ (cid:3) γ −→ h ( B ( U, V ) × V B ( V, W ) , Z φ × V Z ψ i . (cid:3) Definition 26.
Given a function f ∈ C ∞ ( R k ) with only one criticalpoint on R k , we denote by Stat ( x ) { f } the value of f at this point x where ∂ x f ( x ) = 0 . If f also depends on a variable y in R l , we denote by Stat ( x ) { f } ( y ) the function depending on y defined by f ( x ( y ) , y ) where x ( y ) is the implicit function solution of the equation ∂ x f ( x ( y ) , y ) = 0 .We now immediately have the following proposition: Proposition 27. ( G ◦ F )( p , x ) = Stat (¯ p, ¯ x ) n F ( p , ¯ x ) + G (¯ p, x ) − ¯ p ¯ x o . Moreover, F = I ◦ F = F ◦ I, where I ( p, x ) = h p, x i . YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 35
Enhancement composition formula.
We denote by E ( K ) the spaceof enhancements of a symplectic micromorphism K : T ∗ U → T ∗ V .In the local setting, the micromorphism K is associated to a uniquegenerating family ( B ( U, V ) , S Fφ ) . Suppose now that we have anothersymplectic micromorphism L : T ∗ V → T ∗ W with generating family ( B ( V, W ) , S Gψ ) . We want to define a composition for enhancements ◦ : E ( K ) × E ( L ) −→ E ( L ◦ K ) . In the previous paragraph, we have seen that we have a fibration B ( U, V ) × V B ( V, W ) π −→ B ( U, W ) on whose fiber the generating function S Fφ ⊗ V S Gψ has a single nondegenerate critical point. The evaluation of S Fφ ⊗ S Gψ at this criticalpoint is exactly S Gψ ◦ S Fφ . For [ a ] ∈ E ( K ) and [ b ] ∈ E ( L ) , we define theenhancement composition as(4.7) (cid:0) a ◦ b (cid:1) ( p , x ) := s.c. Z π − ( p ,φ ◦ ψ ( x ) ,x ) a ( p , x ) b ( p , x ) e i ~ Θ( F,G ) dp dx (2 π ~ ) l , where Θ( F, G ) is defined as the difference Θ( F, G ) = S Fφ ⊗ V S Gψ − π ∗ ( S Gψ ◦ S Fφ ) . Remark . Observe that the leading term in ~ of the stationary phaseasymptotic expansion of a ◦ b is proportional to a ( p , x ) b ( p , x ) where (¯ p , ¯ x ) is the critical point of S Fφ ⊗ S Gψ on the fiber π − ( p , ψ ◦ φ ( x ) , x ) .This is the usual composition of half-densities carried by composablecanonical relations. Proposition 29.
In the notation above, we have that ([ a ] , S Fφ ) = ([1] , S I id ) ◦ ([ a ] , S Fφ )= ([ a ] , S Fφ ) ◦ ([1] , S I id ) for any enhanced symplectic micromorphism ([ a ] , S Fφ ) .Proof. We will show only the first equality, since the second is provensimilarly. Recall that I ( p, x ) = px and that I ◦ F = F = F ◦ I. Wehave that (cid:0) Θ( I, F ) (cid:1) ( z ) = p φ ( x ) + F ( p , x ) + I ( p , x ) − p x − p φ ( x ) − (cid:0) F ◦ I (cid:1) ( p , x )= F ( p , x ) + p ( x − x ) − F ( p , x ) , YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 36 where z = ( p , φ ( x ) , x , p , x , x ) . Plugging this into the enhancementcomposition formula (4.7), we obtain (cid:0) ◦ a (cid:1) ( p , x ) = e − i ~ F ( p ,x ) (2 π ~ ) l Z dx a ( p , x ) e i ~ F ( p ,x ) Z dp e i ~ p ( x − x ) . = e − i ~ F ( p ,x ) Z dx a ( p , x ) e i ~ F ( p ,x ) δ x ( x )= a ( p , x ) , where δ x is the delta function concentrated at x . (cid:3) Proposition 30.
Let E and E be any two composable enhanced sym-plectic micromorphisms. Then: Q ~ ( E ) ◦ Q ~ ( E ) = Q ~ ( E ◦ E ) mod O ( ~ ∞ ) . Moreover, if Q ~ ([ a ] , L ) = 0 , then [ a ] = 0 mod O ( ~ ∞ ) Proof.
Suppose we have enhanced symplectic micromorphisms E =([ a ] , S Fφ ) and E = ([ b ] , S Gψ ) in the situation T ∗ U E −→ T ∗ V E −→ T ∗ W. A straightforward computation yields I = (cid:16) Q ~ ( E ) ◦ Q ~ ( E )Ψ (cid:17) ( x )= Z R l × V dp dx (2 π ~ ) l b ( p , x ) (cid:0) Q ~ ( E )Ψ (cid:1) ( x ) e i ~ S G Ψ ( p ,x ,x ) , = Z R l × V dp dx (2 π ~ ) l Z R k × U dp dx (2 π ~ ) k a ( p , x ) b ( p , x )Ψ( x ) e i ~ (cid:0) S G Ψ ( p ,x ,x )+ S Fφ ( p ,x ,x ) (cid:1) . Now, interchanging the integrals and writing 1 as e i ~ (cid:0) S Gψ ◦ S Fφ (cid:1) ( p ,φ ◦ ψ ( x ) ,x ) e − i ~ (cid:0) S Gψ ◦ S Fφ (cid:1) ( p ,φ ◦ ψ ( x ) ,x ) , we obtain that I = Z R k × U dp dx (2 π ~ ) k ( a ◦ b )( p , x )Ψ( x ) e i ~ (cid:0) S Gψ ◦ S Fφ (cid:1) ( p ,φ ◦ ψ ( x ) ,x ) , = Q ~ ( E ◦ E ) . Of course, all the computations should be understood modulo O ( ~ ∞ ) and with the appropriate cut-off functions thrown in. The fact that Q ~ ([ a ] , L ) = 0 implies that [ a ] = 0 mod ~ ∞ is obvious. (cid:3) Corollary 31.
Enhanced local symplectic micromorphisms form a cat-egory.
YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 37
Proof.
Let ([ a i ] , T i ) , i = 1 , , , be composable enhanced local symplec-tic micromorphisms, and set Q i := Q ~ ([ a i ] , T i ) be their quantization.We need to prove that [ a ◦ ( a ◦ a )] = [ a ◦ ( a ◦ a )] . We know that associativity holds at the level of quantization: Q ( Q Q ) = ( Q Q ) Q . Now, we also have that Q ( Q Q ) = Q ~ (cid:16) [( a ◦ ( a ◦ a )] , T ◦ T ◦ T (cid:17) , ( Q Q ) Q = Q ~ (cid:16) [( a ◦ a ) ◦ a )] , T ◦ T ◦ T (cid:17) , and, therefore, [ a ◦ ( a ◦ a )] = [ a ◦ ( a ◦ a )] mod ~ ∞ . (cid:3) Example 32.
We have see that the quantization Q ~ ([ a ] , S id ) of theenhanced symplectic micromorphism ([ a ] , T ∗ id) : T ∗ R n −→ T ∗ R n coincides with the standard quantization Op ~ ( a ) of a seen as a sym-bol (germ) in C ∞ ( T ∗ R n ) . Therefore, the enhancement compositiondegenerates in this case to the star product ⋆ qp defined by the identity Op ~ ( a ) ◦ Op ~ ( b ) = Op ~ ( a ⋆ qp b ) , giving the composition of symbols in the qp-ordering for pseudodiffer-ential operators.The next example shows that one can recover the quantization out ofthe enhancement composition; so, in our case, enhancing is quantizing. Example 33.
Enhancements (Ψ , e U ) of the unique symplectic micro-morphism e U : E → T ∗ U can be identified with the space Ψ ∈ L ~ ( U ) .Let ([ a ] , T ) be a symplectic micromorphism from T ∗ U to T ∗ U . Aneasy computation shows that composition degenerates into quantiza-tion; namely, we have that [ a ] ◦ Ψ = Q ~ ([ a ] , T )Ψ . States and costates.
We call a state on U an enhanced sym-plectic micromorphism (Ψ , e U ) from the cotangent microbundle of thepoint E to T ∗ U . As is clear from Example 24, the set of states on U can be identified with the Hilbert space Q ~ ( T ∗ U ) = L ~ ( U ) , and we willuse the Dirac “ket” notation (Ψ , e U ) ! | Ψ i , YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 38 for states. An enhanced symplectic micromorphism A = ([ a ] , S Fφ ) from T ∗ U to T ∗ V induces an operator ˆ A from the states on U to the stateson V by composition ˆ A | Ψ i := A ◦ (Ψ , e U ) . The following proposition tells us that the quantization functor stemsfrom the composition of enhancements:
Proposition 34.
Let A : T ∗ U → T ∗ V be an enhanced symplecticmicromorphism, the n ˆ A | Ψ i = | Q ~ ( A )Ψ i , where Ψ is a state on U . Proof.
Set A = ([ a ] , S Fφ ) . Then, we have by definition that ˆ A | Ψ i = ( a ◦ Ψ , e V ) . Now, a straightforward computation yields that Θ(0 , F ) = S Fφ ,which in turns gives us that a ◦ Ψ( x ) = s.c. Z Ψ( x ) a ( p , x ) e i ~ S Fφ ( p ,x ,x ) dp dx (2 π ~ ) l = Q ~ ( A )Ψ( x ) . (cid:3) A costate on U is an enhanced symplectic micromorphism from T ∗ U to E . It is completely determined by the data of a point x ∈ U , a germof a lagrangian submanifold [ L ] around x in T ∗ U transversal to thezero section and an enhancement [ a ] : T ∗ x U → R as in Example 23. Wewill use the Dirac “bra” notation (cid:0) [ a ] , S Fx (cid:1) ! h a, F, x | to denote costates, where S Fx is the generating function of [ V ] . Notethat the set of costates does not itself form a vector space; however, onemay consider the free vector space generated by costates. Using thecomposition of enhanced symplectic micromorphisms, we can define a“pairing” between states and costates on U : (cid:10) a, F, x (cid:12)(cid:12) Ψ (cid:11) := ([ a ] , S Fx ) ◦ (Ψ , S ) is a symplectic micromorphism from E to E and, thus, a power seriesbelonging to C [[ ~ ]] . Example 35.
If we denote by h x | the special costate (1 , , x ) on R , weobtain that h x | Ψ i = Ψ( x ) , YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 39 and if we denote by ˆ H the operator on states associated to the enhance-ment ([ H ] , S ) : R → R of the identity micromorphism, we obtain that h x | ˆ H | Ψ i = F − ~ (cid:0) H F ~ (Ψ) (cid:1) ( x ) , and in particular h x | ˆ p | Ψ i = − i ~ ∂ Ψ ∂x ( x ) , h x | ˆ x | Ψ i = x Ψ( x ) . In view of Example 32, we have that ˆ f ◦ ˆ g = \ f ⋆ M g. Applications and further directions
In this section, we describe in an informal way some of the appli-cations of symplectic micromorphism enhancements and quantization.Roughly, symplectic microgeometry and its quantized version offer aframework in which dynamics can be expressed in a purely categoricalway both at the classical and quantum level. There is a special monoidin the microsymplectic category, the energy monoids whose action oncotangent microbundles represent the Hamiltonian flows of classicalmechanics. Symmetries of space are implemented by the action of gen-eral monoids in the microsymplectic category and they correspond tothe presence of a Poisson structure on the core of the monoid. The en-hancement and quantization of this picture recovers the semi-classicalversion of the Schrödinger evolution. Let us start with some consider-ation on the functoriality of our constructions, very closely related thework of Khudaverdian and Voronov cited in the Introduction.5.1.
Quantization functors.
In paragraph 4.4, we restricted the cat-egory of manifolds we started with to the category E formed by theopen subsets of the Euclidean spaces R n endowed with the canonicalEuclidean metric. This extra data of a metric made it possible to pro-duce a canonical (global even!) exponential Ψ x ( v ) = v + x for eachobject in E since Ψ is exactly the exponential associated to the affineconnection generated by the metric. (Note that the only piece of extradata we need for our construction on the top of the manifold is theexponential Ψ ; we do not make direct use of the metric or the affinestructure.) This allowed us to identify, in a non ambiguous way, theclass of semiclassical Fourier integral operators SFour ~ ( E ) with wavefronts in the category Cot extmic ( E ) formed by the symplectic micromor-phisms between the cotangent bundles on the object of E with the space YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 40 of symplectic micromorphism enhancements. In symbols, we had theidentification
SFour ~ ([ V ] , φ ) = | Ω ~ | ([ V ] , φ ) , thanks to the canonical way we could associate to ([ V ] , φ ) a generatingfunction S fφ and to a semiclassical FIO with wave front ([ V ] , φ ) theintegral representation (4.4). This allowed us further to endow thecorresponding collection E ( E ) of enhanced symplectic micromorphismswith the structure of a category by pulling back the FIO compositionto their total symbols as in paragraph 4.4.3. This situation can besummarized by the following commutative diagram of functors SFour ~ ( E ) σ ✲✛ Q ~ E ( E ) Cot extmic ( E ) ✛ U W F ✲ where σ is the total symbol functor, WF the wave front functor, Q ~ the quantization functor, and U the forgetful functor. Observe that σ is an isomorphism of categories with inverse functor given by thequantization functor Q ~ .We conclude this discussion by observing that the functorial con-structions above apply in the very exact same way to any other cat-egory E of manifolds carrying some extra data that make it possibleto assign in a canonical way a micro exponential to each of the man-ifolds. For instance, we may take E to be the category of riemannianmanifolds and the canonical micro exponentials given by the germ ofthe Levi-Civita connection exponential. Note also that the functor Q ~ is strictly monoidal with respect to the obvious monoidal structures onthe source and target categories.In the sequel, we will comment on possible applications of our con-struction, sometimes assuming implicitly a suitable category E of man-ifolds endowed with the appropriate extra data.5.2. The energy monoid.
The Lie algebra T of the time translationgroup ( R , +) is the abelian Lie algebra on R . Its dual, which we denoteby E , is thus the Poisson manifold, with zero Poisson structure.We call T the Lie algebra of time and E the Poisson manifold ofenergy. Its cotangent bundle T ∗ E = T × E is a trivial symplectic mi-crogroupoid with source and target coinciding with the bundle projec-tion; the composable pair space is T ∗ E ⊕ T ∗ E and the groupoid product YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 41 is just the addition of times in a fiber of constant energy: m E (cid:0) ( t , E ) , ( t , E ) (cid:1) = ( t + t , E ) . One verifies that the graph of the groupoid product is a symplecticmicromorphism µ E := (cid:0) gr[ m E ] , ∆ E ) : T ∗ E ⊗ T ∗ E −→ T ∗ E , where the core map ∆ E is the diagonal map on E . It is easy to see that µ E satisfies the following associativity and unitality equations µ E ◦ ( µ E ⊗ id) = µ E ◦ (id ⊗ µ E ) ,µ E ◦ ( e E ⊗ id) = id = µ E ◦ (id ⊗ e E ) , where e E is the unique symplectic micromorphism from the cotangentbundle of the point to T ∗ E . In other words, ( T ∗ E , µ E ) is a monoidobject in the microsymplectic category (see [10] for a systematic studyof these monoids).5.3. Classical flows as symplectic micromorphisms.
Consider aclassical hamiltonian system H : T ∗ Q → R . The time evolution Ψ t generated by H produces a lagrangian submanifold W H := (cid:26)(cid:16)(cid:0) t, H (Ψ t ( z )) (cid:1) , z, Ψ t ( z ) (cid:17) : t ∈ I, z ∈ T ∗ Q (cid:27) , of T ∗ E × T ∗ Q × T ∗ Q , where I is the maximal interval on which theflow Ψ t is defined. It gives a symplectic micromorphism ρ H := (cid:0) [ W H ] , H | Z Q × id Q (cid:1) : T ∗ E ⊗ T ∗ Q −→ T ∗ Q. Proposition 36.
Let H ∈ C ∞ ( T ∗ Q ) be a hamiltonian. Then that ρ H ◦ ( e E ⊗ id) = id ,ρ H ◦ ( µ E ⊗ id) = ρ H ◦ (id ⊗ ρ H ) . In other words, this turns T ∗ Q into a T ∗ E -module in the microsym-plectic category.Proof. A straightforward composition of canonical (micro) relationsyields ρ H ◦ (id ⊗ ρ H ) = (cid:26)(cid:18) t , H (cid:0) Ψ t ◦ Ψ t ( z ) (cid:1) , t , H (cid:0) Ψ t ( z ) (cid:1) , z, Ψ t ◦ Ψ t ( z ) (cid:19) : z, t (cid:27) ,ρ H ◦ ( µ E ⊗ id) = (cid:26)(cid:18) t , H (cid:0) Ψ t + t ( z ) (cid:1) , t , H (cid:0) Ψ t + t ( z ) (cid:1) , z, Ψ t + t ( z ) (cid:19) : z, t (cid:27) . We see that these microfolds coincide for time independent hamiltoni-ans, since Ψ t ◦ Ψ t = Ψ t + t and H (Ψ t ( z )) = H ( z ) for all t , t and t .One gets the unitality axiom in a similar way. (cid:3) YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 42
The Hamilton-Jacobi formulation of classical mechanics tells us that,in Darboux coordinates, the hamiltonian flow ( p t , x t ) = Ψ Ht ( p, x ) admits, for short times, a generating function S satisfying ∂ t S ( t, p, x t ) = H t (cid:0) p t , x t (cid:1) ,∂ p S ( t, p, x t ) = x,∂ x S ( t, p, x t ) = p t , with the initial condition S (0 , p, x ) = h p, x i . This is precisely the gen-erating function of the symplectic micromorphism ρ H .5.4. Classical symmetries.
It is possible to generalize the previousscheme to a general Hamiltonian action of a Lie group G on T ∗ P withmomentum map J : T ∗ P → G ∗ , where G is the Lie algebra of G . Inthis case, we define the symmetry submanifold to be W G := (cid:26)(cid:16)(cid:0) v, J (exp( v ) z ) (cid:1) , z, exp( v ) z ) (cid:17) : v ∈ U, z ∈ T ∗ Q (cid:27) , where U is the maximal neighborhood of in the Lie algebra G onwhich the exponential mapping exp : G → G is defined. Taking thegerm of W G around the graph of J | Q × id Q , yields a symplectic micro-morphism ρ G from T ∗ G ∗ ⊗ T ∗ Q (where ⊗ denotes the tensor producton objects defined by the Cartesian product) to T ∗ Q . Now, thanksto the exponential mapping, we can define a generating function germfrom T ∗ G ∗ ⊕ T ∗ G ∗ to R via the formula S G ( v, w, µ ) := D µ, exp − (cid:0) exp( v ) exp( w ) (cid:1)E , where h , i is the canonical paring between the Lie algebra and its dual.This generating function germ defines a symplectic micromorphism µ G from T ∗ G ∗ ⊗ T ∗ G ∗ to T ∗ G ∗ . One can show that ( T ∗ G ∗ , µ G ) is a monoidand that ( T ∗ Q, ρ G ) is a T ∗ G ∗ -module.5.5. Generalized symmetries.
We can consider more general sym-metries in microgeometry by allowing arbitrary monoids ( T ∗ P, µ ) toact on phase spaces T ∗ Q . It turns out that a general monoid induces aPoisson structure on its core P , and, conversely, any Poisson manifold ( P, Π) induces a monoid ( T ∗ P, µ ) by considering the local symplecticgroupoid integrating P and by taking µ to be the germ of the groupoidproduct around the graph of the diagonal map on P ([10]). Moreover,one can show that a T ∗ P -module, ρ : T ∗ P ⊗ T ∗ Q −→ T ∗ Q, YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 43 induces a momentum map germ, i.e., a Poisson map germ [ J ] from thecotangent microbundle T ∗ Q to the Poisson manifold P . Such situationsmay arise for instance if we start with a Lie groupoid acting in anhamiltonian way on a phase space.5.6. Quantization of the energy monoid.
A straightforward, butlengthy, computation would show that the following enhancement ([1] , µ E ) of the energy monoid ( T ∗ E , µ E ) yields a monoid in the category of en-hanced symplectic microfolds. More interesting is the quantization ofthis enhanced monoid: it should produce a associative product on H ~ E ,that is on C ∞ ( E )[[ ~ ]] . Let us compute it: Q ~ (([1] , µ E ) (cid:0) f ⊗ g (cid:1) ( E ) = s.c. Z ˆ f ( t )ˆ g ( t ) e − i ~ S ( t ,t ,E ) dt dt (2 π ~ ) , where ˆ f and ˆ g are the asymptotic Fourier transforms of f and g , andwhere S is the generating function of the energy monoid. Since S =( t + t ) E , we see that the quantization of ([1] , µ E ) yields the extensionof the usual product of functions on C ∞ ( E )[[ ~ ]] .5.7. Quantization of Poisson manifolds.
Suppose we have a monoid ( T ∗ P, µ ) with induced Poisson structure Π on P . Since the core mapof µ is the diagonal map ∆ on P , an enhancement of µ is given by anyhalf-density germ [ a ] ∈ | Ω h | ( T ∗ P ⊕ T ∗ P ) around the zero-section. Given any such enhancement of µ , we obtainthus an operator ⋆ ~ := Q ~ ([ a ] , µ ) : H ~ ( P ) ⊗ H ~ ( P ) −→ H ~ ( P ) . The functoriality of Q ~ implies that ⋆ ~ is an associative product if andonly if the enhancement is associative, in which case the associativealgebra ( H ~ ( P ) , ⋆ ~ ) should be considered as the quantization of thePoisson manifold ( P, Π) . In this context, the quantization of Poissonmanifolds becomes equivalent to the existence of associative enhance-ments. When P is an open subset of R n we have the identification of H ~ ( P ) with C ∞ ( P )[[ ~ ]] , and we can write the product as f ⋆ ~ g ( x ) = s.c. Z ˆ f ( p )ˆ g ( p ) a ( p , p , x ) e i ~ S ( p ,p ,x ) dp dp (2 π ) m , where S is the generating function of µ . Note that when the Poissonstructure is the zero Poisson structure, or, equivalently, when µ is thegraph of vector bundle addition in T ∗ P , we have that the generatingfunction is S = h p + p , x i as in the energy monoid case. The associa-tive enhancement ([1] , µ ) gives us back the usual product of functions. YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 44
General associative enhancements of general monoids ( T ∗ P, µ ) corre-sponding to non-zero Poisson structures should yield star-products on C ∞ ( P )[[ ~ ]] upon asymptotic expansion. The first non trivial case isgiven by the integral representation of the Moyal product on R n en-dowed with its canonical symplectic form.This is to be related to a quantization program for Poisson manifoldsrelying on asymptotic integrals of the Moyal type and on the symplecticgroupoid of the Poisson manifold (see [20, 33, 34]). In this approach,a star-product should be obtained from the asymptotic expansion ofa semiclassical Fourier integral operator whose phase is the generatingfunction of the symplectic groupoid. In the special case of symplecticsymmetric spaces, similar integrals were worked out in [3, 6, 26, 25]. Forthe linear Poisson structures, the paper [1] gives a integral version ofKontsevich’s star-product, where the generating function is the Baker-Campbell-Hausdorff formula of the associated Lie algebra. In [7], ithas been shown that this latter generating function is exactly the gen-erating function of the corresponding symplectic groupoid and that,more generally, generating functions of the local symplectic groupoidfor Poisson structures on open subsets of R n , can be extracted fromthe tree-level part of Kontsevich star-product ([22]).In the same spirit, Wagemann and one of the authors [14] start byshowing that the Gutt star-product can be understood as the quan-tization in the above sense of a symplectic micromorphism, and thenthey extend this construction to quantize Leibniz algebras. Both in[14] and [13], formal deformation quantizations are obtained from sym-plectic micromorphism quantization by taking Feynman expansions ofthe corresponding oscillatory integrals.5.8. Quantization of symmetries.
Suppose now that we have a gen-eral symmetry ρ : T ∗ P ⊗ T ∗ Q −→ T ∗ Q, where ( T ∗ P, µ ) is a general monoid with induced Poisson structure Π on P , acting on T ∗ Q . One can show that the core map of ρ is J | Z Q × id Q ,where J : T ∗ Q → P is a Poisson map germ around the zero section of T ∗ Q , which can be considered as the momentum map of the the action.An enhancement of ρ is thus a half-density germ [ b ] ∈ | Ω ~ | (cid:0) J ∗| Q ( T ∗ P ) ⊕ T ∗ Q (cid:1) around the zero section. Quantizing this data, we obtain an operator Q ~ ([ b ] , ρ ) : H ~ P ⊗ H ~ Q −→ H ~ Q , YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 45 which is a representation of the quantum algebra ( H ~ P , ⋆ ~ ) on H Q provided that [ b ] ◦ ([1] ⊗ [ b ]) = [ b ] ◦ ([ a ] ⊗ [1]) . This approach has been used in [13], [11], and [12] to quantize groupactions through their cotangent lifts as well as to quantize their mo-mentum maps.5.9.
Quantization of the classical flow.
We want to quantize thesymplectic micromorphism ρ H associated to the classical flow Ψ Ht on T ∗ R n of a hamiltonian H : T ∗ R n → R as described in paragraph 5.3.Since, in general, Q ~ ( T ∗ R n ) can be identified with the space of L -functions on R n , we obtain, after quantization, a linear operator Q ~ ([ a ] , ρ H ) : L ( E ) ⊗ L ( R n ) −→ L ( R n ) for any enhancement a = a ( t, p , x ) of the symplectic micromorphism.Now, we may define the following the operator on L ( R n ) by U at := Q ~ ([ a ] , ρ H )( u t ⊗ · ) , where u t ( E ) = e − i ~ t E is the state with “definite time” t on the spaceof energies. An explicit computation yields ( U at Φ)( x ) := 1(2 π ~ ) n s.c. Z Φ( x ) a ( t , p , x ) e i ~ p x − S ( t ,p ,x ) dp dx , where S is the generating function of phase flow Ψ tH solution of theHamilton-Jacobi equation as explained in paragraph 5.3. Then, a clas-sical result of semi-classical analysis can now be reformulated in termsof symplectic micromorphisms and their enhancements: For any Hamil-tonian H : T ∗ R n → R , there exists an enhancement of ([ a ] , ρ H ) suchthat U at is the propagator, modulo ~ ∞ , of the Schrödinger equation withquantum hamiltonian given by the semi-classical pseudo-differential op-erator ˆ H = Q ~ ([ H ] , id T ∗ R n ) , where the germ [ H ] is understood as an enhancement of the identitymap on T ∗ R n seen as symplectic micromorphism. Moreover, it is easyto show that, U a = id and U at ◦ U at = U at + t (when defined) is equivalentto ([ a ] , ρ H ) being an action of energy monoid enhanced as in paragraph5.6 on T ∗ Q thanks to the fact that u t + t = Q ~ ([1] , µ E )( u t ⊗ u t ) , YMPLECTIC MICROGEOMETRY IV: QUANTIZATION 46 and the module axioms: ( U at ◦ U at )Φ = (cid:16) Q ~ ([ a ] , ρ H ) ◦ (cid:0) id ⊗ Q ~ ([ a ] , ρ H ) (cid:1)(cid:17) ( u t ⊗ u t ⊗ Φ) , = Q ~ ([ a ] , ρ H ) ◦ (cid:16)(cid:0) Q ~ ([1] , µ E )( u t ⊗ u t ) (cid:1) ⊗ Φ) (cid:17) , = U at + t Φ . References [1] Andler, M., Dvorsky, A., and Sahi, S., Kontsevich quantization and invariantdistributions on Lie groups.
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