Metaplectic quantization

Metaplectic Quantization

Hassan Jolany

Laboratoire Paul PainleveLaboratoire de Mathematiques,CNRS-UMR 8524Universite des Sciences et Technologies de Lille

12th March 2014

Hassan Jolany (Laboratoire Paul Painleve Laboratoire de Mathematiques,CNRS-UMR 8524 Universite des Sciences et Technologies de Lille )Does God believes in Quantization? 12th March 2014 1 / 67

Outline

1 IntroductionContraction between Physics and Mathematics and some JokesSociology and QuantizationSymmetry in Nature and QuantizationSymmetry in Art and Quantization

2 Representation theory and QuantizationLie groups and Lie algebraOrbit method and Coadjoint orbitsSymplectic manifoldsPre-QuantizationSymmetry in NatureQuantization of Coadjoint orbits

3 Metaplectic Quantization and Spin-GeometryHalf-forms


Introduction

Ancient philosophy about nature, Mathematics and Physics


Philosophy of nature

Philosophy of Galileo Galilei :)


Philosophy of Nature

Philosophy of Richard Feynman :)


Joke !!!

Philosophy of one week of God


Joke !!!

Philosophy of one week of God


Symmetry in Nature

There is a lot of symmetry in Nature. The word comes from Greek symand metria. It was associated to beauty by Greek and Romanphilosophers:

Vitruvius in De Architectura Libri Decem:The design of a temple depends on symmetry, the principles of which

must be carefully observed by the architect. They are due toproportion. Proportion is a correspondence among the measures of themembers of an entire work, and the whole to a certain part selected asstandard. From this result the principles of symmetry”

Most scientists and artists would agree that this is a description of”beauty” as it relates to their respective fields.


Symmetry in Nature


Symmetry in Nature


symmetry in Art

Leonardo da Vinci’s Vitruvian Man (ca. 1487) is often used as arepresentation of symmetry in the human body and, by extension,the natural universe.


Symmetry in :)


Symmetry in Mathematics

The Birth of Venus is a painting by Sandro Botticelli. ”Mostpeople perceive this painting as Symmetrical ..... Yet mostmathematicians will tell you that the arrangements of colors andforms are not symmetric in the Mathematical sense” [Mario Livio]


Symmetry in Mathematics

Hermann Weyl A thing is symmetrical if there is something youcan do to it so that after you have finished doing it it looks thesame as before.”

Mathematicians and scientists often used GROUP THEORY tostudy symmetry that is expressed by group transformationspreserving some structure. ”Evariste Galois [1811 - 1829] :

ax5 + bx4 + cx3 + dx2 + ex+ f = 0

Felix Klein ( Das Erlanger Programm, 1872)

Sophus Lie und Friedrich Engel (Theorie derTransformationsgruppen, 1888-1893)

Elie Cartan [Geometre Francais]


The Group of Symmetries of the Square

The square has eight symmetries - four rotations, two mirrorimages, and two diagonal flips:



These eight form a group under composition (do one, thenanother). Let’s give each one a color:



The Multiplication Table of D4 With Color


Orbit method

What is the Orbit Method?The Orbit Method is a method to determine all irreducible unitaryrepresentations of a Lie group

Why is it useful?Representation theory remains the method of choice forsimplifying the physical analysis of systems possesing symmetry.

The Orbit Method is entangled with its physical counterpartGeometric Quantization, which is an extension of the canonicalquantization scheme to curved manifolds

Definition (Representation)

A representation of a group G on a vector space V is a grouphomomorphism from G to GL(V ), i.e. a map ρ : G→ GL(V ) such thatρ(g1g2) = ρ(g1)ρ(g2) for all g1; g2 ∈ G.


Representation

Loosely speaking: a representation makes an identificationbetween abstract groups and more managable lineartransformations of vector spaces.


The Adjoint and Coadjoint Representation

Let G be a matrix group, i.e. a group of invertible matrices, and let Vbe its Lie algebra g. Then the adjoint representation Ad is defined byAd(g)X = gXg−1 for g ∈ G,X ∈ g, which is just matrix conjugation.

Let G be a Lie group. The coadjoint representation Ad∗ is thedual of the adjoint representation Ad, defined by

In case G is a matrix group then g ∼= g∗ and the coadjointrepresentation is just matrix conjugation again


Co-Adjoint Orbits

Definition: (coadjoint orbit OF )

Given F ∈ g∗. The coadjoint orbit O(F ) is the image of the mapκ : G→ g∗ defined by κ(g) = Ad∗(g)F .

Coadjoint orbits O(F ) are symplectic manifolds!Proof: exercise for the very motivated listener.

(Patrick Iglesias-Zemmour) Every connected Hausdorff symplecticmanifold is isomorphic to a coadjoint orbit of its group ofHamiltonian diffeomorphisms.Proof: exercise for Hassan!!!


Some Examples of Co-Adjoint Orbits



The Coadjoint orbits of SU(2),SU(3),SU(4) and SU(n)






Geometric Quantization

Now, we continue to define the theory of geometric quantization bysome axioms which are compatible with physical view.

Geometric quantization we associate to a symplectic manifold(M,ω) a Hilbert space H, and one associates to smooth functionsf : M → R skew-adjoint operators Of : H → H.

Paul Dirac introduced in his doctoral thesis, the ”method ofclassical analogy” for quantization which is now known as Diracaxioms as follows.

1] Poisson bracket of functions passes to commutator of operators:

O{f,g} = [Of ,Og]

2] Linearity condition must holds ,Oλ1f+λ2g = λ1Of + λ2Og forλ1, λ2 ∈ C3] Normalization condition must holds: 1 7→ i.I(Which I is identityoperator and i =

√−1)


Dirac Principal for Geometric quantization

Before to establish the axiom 4], we need to following definition.

Definition

Let (M,ω), be a symplectic manifold. A set of smooth functions {fj}is said to be a complete set of classical observables if and only if everyother function g such that {fi, g} = 0 for all {fj}, is constant. Also wesay that a family of operators is complete if it acts irreducibly on H

4] Minimality condition must holds: Any complete family of functionspasses to a complete family of operators. Moreover, if G be a groupacting on (M,ω) by symplectomorphisms and on H by unitarytransformations. If the G-action on M is trnsitive, then its action on Hmust be irreducible


Geometric Quantization

Now, we again recall the pre-quantization line bundle in formallanguage. In fact, we have two important method for GeometricQuantization;

1)Using line bundle. More precisely, In geometric quantization weconstruct the Hilbert space H as a subspace of the space ofsections of a line bundle L on a symplectic manifold M .

2)Without using line bundle: Using Spinc-structure instead of linebundle. One of advantage of this construction is better behaved ofphysical view but definig it is not so easy. We say (M,ω) is Spinc

prequantizable if and only if 12π [ω]− 1

2π [ω0] for some fixedcohomology class 1

2π [ω0] ∈ H2(M,R).


Pre-Quantum Line Bundle

Definition

In formal form. A pre-quantization line bundle for symplectic manifold(M,ω) is a complex line bundle L, such that the curvature class is thecohomology class [ω]. It is important to point out that complex linebundles are classified by H2(M,Z) via L 7→ c1(L) ,Therefore themanifold (M,ω) is prequantizable if and only if 1

2π [ω] be integral. i.e.,its integral on any closed 2-surface has to be an integer.



Definition

The second equivalent definition of pre-quantization of (M,ω), is aprincipal U(1)-bundle π : P →M and a connection form α on P withcurvature ω such that dα = −1

~π∗ω. Note that by this philosophy, we

can consider the pre-quantum line bundle as associated vector bundleL = P×U(1) C and also P = {v ∈ L :< v, v >= 1}

Theorem

Instead to working with pre-quantization datas(L, <,>,∇) we candirectly introduce prequantization datas by P and α, i.e., defineprequantization (P, α) with dα = −1

~π∗ω.



Now we the following theorem about classification of pre quantizationstructures on complex line bundle L.

Theorem

The prequantization structures on L are classified by

H1(M,R)

H1(M,Z)

Theorem

If the symplectic manifold M is simply connected then theprequantization structure on complex line bundle L is unique up togauge equivalnce.


Attempts at Quantization

As we saw in previous section, the prequantization is a firstattempt to get a Hilbert space out of a symplectic manifold (M,ω). Now, we try to find the quantum space as a subspace of Hilbertspace as follows

1. L is called prequantum line bundle and ∇ prequantumconnection.


2. The prequantum space

HprQ = {ψ ∈ Γ(L) :

∫Mh(ψ,ψ)ε <∞}

where

ε =ωn

n!

and is called Liouville form which is volume form in every symplecticmanifold and we have the following inner product on HprQ,

< ψ, φ >=

∫Mh(ψ, φ)ε

3. The observables are quantized via

f = −i~∇Xf + f = −i~[Xf +

i

~θ(Xf )

]+ f

here ω = −dθ where θ is local symplectic potential.4. The condition on the curvature of ∇ is clear form d( i~θ) = ω

i~ .Hassan Jolany (Laboratoire Paul Painleve Laboratoire de Mathematiques,CNRS-UMR 8524 Universite des Sciences et Technologies de Lille )Does God believes in Quantization? 12th March 2014 31 / 67

Polarization

Polarization is the second step of Geometric Quantization. Herewe explain the advantages of the notion of polarization inGeometric Quantization.

1) The space of sections obtained in prequantization is too largeand it contains functions of both position and momentum. Wewould like to end up with functions of just position. It is thenclear what we must do. We must pick the subspace of thefunctions which are independent of the momentum. In fact, wewould like to use of polarization as a way of selecting half of thedirections of M , and then select from the prequantum space thewave functions constant along those directions.

2) An advantage of restricting to a subspace of ”polarizedsections” is that the resulting prequantization may satisfy theminimality axiom 4 of Dirac principal.


Real and Complex Polarization

We consider two type of ”real” polarization and ”complex”polarization. In fact,the real case is well suited to cotangentbundles, and complex case is well suited for Kahler manifolds.

Definition

Let (M,ω) be a symplectic manifold. A real polarization on M is afoliation (i.e. an integrable distribution) D ⊂ TM on M which ismaximally isotropic, i.e. for all a ∈M

ωa(X,Y ) = 0,

for all X,Y ∈ Da and no large subspace of TaM which contains Da

properly has this property.


Complex Polarization

Moreover, despite the real polarization has more physicalinterpretation, but the one of problems of real polarization ingeometric quantization is defining a finite inner product of wavefunctions(polarized sections of L). So, for the purpose of geometricquantization we thus need a generalization of the notion of a realpolarization and we introduce here complex polarization.Moreover, one of the other advanteges of complex polarization isthat this type of polarization is very important in kahlerquantization. These type of polarization are of special interest forstablishing a bridge between geometric quantization and the theoryof irreducible unitary representations of Lie groups of symmetries.


Definition

A subbundle P ⊂ TMC of the complexified tangent bundle is called acomplex polarization if1. P is Lagrangian2. P involutive3. dimP ∩ P ∩ TM is constant

This definition shows that every complex polarization induces a realisotrpic distribution D := P ∩ P ∩ TM which is also involutive byFrobenius theorem. Moreover the complexification of distribution D isDC = P ∩ P and it is called isotropic distribution. Now we define thesubbundle E := (P + P ) ∩ TM and EC = P + P . Notice thatorthogonal symplectic complement of D is E, i.e., D⊥ = E and E iscalled coisotrpic distribution.Note that the subbundle P + P is stil not necessarly involutive.Imposing the following conditions on P ensures us that thepolarization is well behaved.


Real and Complex Polarization

There is a correspondence between Kahler polarization and Kahlerstructures, given by following theorem.

Theorem

Let (M,ω) be a symplectic manifold1. If (M,ω, J) is Kahler which J is complex structure, then P = T 1,0Mis a Kahler polarization and we call this the holomorphic Kahlerpolarization.2.If P is a Kahler polarization on M , then there exists a complexstructure J such that (M,ω, J) is a Kahler manifold.


Let (M,ω, J) be a compact Kahler manifold with positive-denitepolarization P , and (L,∇) be prequantum data. Let

Mquantum ={s ∈ Γ(L)|∇Xs = 0,∀X ∈ P

}Then Mquantum is fnite-dimensional.We define the quantization of (M,ω, J) to be Mquantum, which is thespace of holomorphic sections of the prequantum line bundle L.Here we introduce the space of polarized sections HP ⊂ H.

Definition

Let P be polarization and L→M be a prequantum Line bundle, wedefine HP ⊂ H to be the completion of square integrable sections ssuch that ∇Xs = 0 for all X ∈ P . We say to such sections as polarizedsections. Note that it can be shown that the space of square integrableholomorphic sections is closed and we can waive the word ”completion”in this definition.


Kahler polarization

Now by using of polarization in quantization we restrict the class ofquantizable observables as follows. In fact, we introduce a new class onquantum Hilbert space of polarized sections which we denote it byO(HP ) such that any operator which acts on the set of polarizedsections HP must map it to O(HP ).So, we will construct the new class of quantum observables.If Of be such operator, therefore for all X ∈ P and s ∈ HP we musthave

0 = ∇X(Ofs)

But by simple computation we have

∇X(Ofs) = −i~∇[X,Xf ]s.

So [X,Xf ] ∈ P , as Xf = Xf , then [X,Xf ] ∈ P


So we can define the new class of quantizable observables O(HP )as

O(HP ) = {Of |[X,Xf ] ∈ P,∀X ∈ P} .


Metaplectic Quantization and spin-Geometry

Metaplectic correction plays an important role in Geometricquantization. From physical point of view, metaplectic correctiongives the correct quantization of the harmonic oscillator. The factis that in quantum mechanics, we can not stop on the set ofholomorphic polarized sections and it can be observe that theenergy level for harmonic oscillator is wrong. Moreover, thedimensions of eigenspaces turn out to be wrong or shifted and wecan see this obstacle for Kepler problem in hydrogen atom.

In fact, let P be an arbitrary vertical polarization of (M,ω) with aprequantum line bundle L, so, we know that each leaf of thevertical polarization is a non-compact affine space, and moreoverour sections are constant along each leaf, then any non-vanishingsection is certainly not square integrable with respect to theL2-norm defined on the prequantum Hilbert space.


Metaplectic Quantization and spin-Geometry

The standard example is the case when M = T ∗N is a cotangentbundle over a manifold N with its usual symplectic form, and P isthe vertical polarization. Then the isotropic and coisotropicpolarizations agree (both equaling the vertical polarization), andtheir integral manifolds are the fibers of the bundle projectionT ∗N → N . For any polarized section, the integration with respectto ωn along these fibers will give an infinite contribution.

So, we must attempt to revise our set of holomorphic polarizedsections ΓP (L). To dealing with this, we must modify the linebundle L so that there is a naturally induced norm on covariantlyconstant sections ΓP (L).


Metaplectic correction

Now we need to constract L. But as we said before, the polarizedsections are covariantly constant along the leaves of D and if theseleaves are noncompact, then our polarized sections are not squaresquare integrable. A remedy to this obstacle is to integrate thepolarized sections not over symplectic manifold (M,ω) and instead ofit we integrate our polarized sections over M/D, or M/P . So, we tryto find the new line bundle L. As we saw before, our sections arelocally functions on M/P or M/D, and we are interested to have thefollowing L2-norm on M/P . By taking into account of this fact thatevery section s on any neighborhood U ⊂M , can be written as s = φswhich s is unit section,we have for some n-form µM/P

〈s, s〉 =

∫M/P

h(s, s)µM/P =

∫M/P

φφµM/P .



But, there is no natural measure µM/P on M/P or M/D. So toremedy of this problem, we must reconstruct the integrand h(s, s′) suchthat the integral makes sense on n-form µM/P and n-form µM/P be inform of measure. So we need to change s, s′ such that h(s, s′) define adensity form on M/P so that integral make sence. So, let s, s′ ∈ ΓP (L)and σ and σ are two section of some line bunde , then by constructingnew sections s = s⊗ σ and s′ = s′ ⊗ σ′ then h(s, s′) = h(s, s′)σσ′ so ifwe define the inner product as⟨

s, s′⟩

=

∫M/D

h(s, s′)σσ′

and σσ′ be a density form then our integral on M/D make sense. Sothe problem is now to defining this new space of sections of the form sbut before to try to answer to this obstacle we first need to half-formsand metalinear bundle as new geometrical structures which we explainhere.



The fact is that for integrating n-forms on the manifold M we requirethe choice of orientation. But if we apply the notion of density then wewill circumvent this need of orientation. First we need to followingdefinitions.

Definition

Let M be a manifold of dimension n. Then an n-form σ on M isdefined by a function which at x ∈M assigns to each basis{e1, e2, ..., en} of TxM a number that satisfies

σ(eg) = det(g).σ(e)

for all g ∈ GL(n,R) and e = e1 ∧ e2 ∧ ... ∧ en.

Now, we define the notion of frame bundle which is an essential tool inconstruction of half forms in metaplectic correction.



Definition

(Frame bundle) We recall that a frame bundle is a principal fiberbundle Fr(E) associated to any vector bundle E. The fiber of Fr(E)over a point x is the set of all ordered bases, or frames, for Ex. Thegeneral linear group acts naturally on Fr(E) via a change of basis,giving the frame bundle the structure of a principal GL(n,R)-bundle(where n is the rank E). More precisely let E →M be a real vectorbundle of rank n over a manifold M . A frame at a point x ∈ X is anordered basis for the vector space Ex. Equivalently, a frame can beviewed as a linear isomorphism f : Rn → Ex. The set of all frames atx, denoted Frx, has a natural right action by the general linear groupGL(n,R) of invertible k × k matrices: a group element g ∈ GL(n,R)acts on the frame f via composition to give a new framef ◦ g : Rn → Ex. The frame bundle of E, denoted by B(E) is thedisjoint union of all the Frx, i.e. B(E) =

∐x∈M Frx.



Also we need to α-density as a gemetrical tool in geometricquantization which we define here,

Definition

Let M be a manifold of dimension n. Then an α-density σ on M isdefined by an object which at x ∈M assigns to each basis{e1, e2, ..., en} of TxM a number that satisfies

σ(eg) = |det(g)|α .σ(e)

for all g ∈ GL(n,R) and e = e1 ∧ e2 ∧ ... ∧ en.

We define the half-form as as an object σ : B→ C, such that

σ(eg) = |det(g)|12 .σ(e)


Metaplectic Geometry

Now we come back to our quantization procedure. We would like todefine σ such that σ2 is a complex n-form and σσ is a complex density,so we need to such object σ, such that,

σ(eg) = |det(g)|12 .σ(e)

But again we face to an obstacle, and the trouble with this definition

σ(eg) = |det(g)|12 .σ(e) is that the root-square is not well defined on

GL(n,C). In order to remedy this we will have to find a lie grouprelated to GL(n,C) which have four component −R, +R, −iR, and+iR. So if we take double covering of GL(n,C) we have achived to

pass of this obstacle, because double covering ˜GL(n,C) has fourcomponent −R, +R, −iR, and +iR and is well defined.



Now, consider the following exact sequence of groups

0→ Z→ C× SL(n,C)→ GL(n,C)→ 0

which s : Z→ C× SL(n,C) and t : SL(n,C)→ GL(n,C) defined by

s(k) =

(2πik

n, e−

2πikn I

), t(m,B) = emB

We define the action of Z on C× SL(n,C) by

(k, (m,B)) 7→(m+

2πik

n, e−

2πikn B

)Also the map t is invariant under the action of Z and we will havefollowing isomorphism

(C× SL(n,C))

2Z' GL(n,C)


Metaplectic geometry

So, we can see the elements of GL(n,C) as

(m,B) =

{(m+

2πik

n, e−

2πikn B

): k ∈ Z,

}where B ∈ SL(n,C)



Now, in the same way, we can construct a new exact sequence

0→ 2Z→ C× SL(n,C)→ C× SL(n,C)

2Z→ 0

which s′ : 2Z→ C× SL(n,C), defined by s′(2k) =(

4πikn , e−

4πikn I)

The quotient ML(n,C) := (C× SL(n,C))/2Z is called as complexmetalinear group of dimension n. The elements of ML(n,C) can bewritten as following forms

(m,B) =

{(m+

4πik

n, e−

4πikn B

): k ∈ Z,

}where B ∈ SL(n,C)



So, we will have a covering map

ρ : ML(n,C)→ GL(n,C),

(m,B) 7→ emB

which following diagram commutes



It can be shown that the kernel of the covering mapρ : ML(n,C)→ GL(n,C) is just the set{

(0, I), (2πi

n, e−

2πin I)

}' Z2,

So ML(n,C) is the double cover of GL(n,C).We have following sequence

ML(n,C)→ GL(n,C)→ C∗

(m,B) 7→ emB 7→ det(emB) = enm detB = enm

So, we can introduce new holomorphic square root on ML(n,C), i.e.

χ : ML(n,C)→ C∗

(m,B) 7→ enm2


Metaplectic Geometry and Quantization

So, we can write χ2(A) = det(ρ(A)) and χ(A)χ(A) = |det(ρA)|. Now,in order to pass back to B(V ) and GL(n,C) we define the metalinearstructure and metalinear bundle. Note that we have the notationML(n,R) := ρ−1GL(n,R)



Definition

(Metalinear bundle) Let V be an n-dimensional vector space. Ametalinear MB(V ) is by definition a covering MB(V )→ B(V )together with an action MB(V )×ML(n,R)→MB(V ) and coveringρ : ML(n,C)→ GL(n,C) such that the diagram



Now, with this definition, for B ∈ML(n,R) and and b ∈ B(V ), wehave

σ(bB) = χ(B)σ(b),

so if σ1 and σ2 be two such forms then σ1σ2 gives an n−form on B(V )and moreover σ1σ2 gives complex density and we have

σ1(bB)σ2(bB) = det(ρ(B))σ1(b)σ2(b),

σ1(bB)σ2(bB) = |det(ρ(B))|σ1(b)σ2(b).

Note that there is no garantee that MB(V ) exists and even if it existsthen in general may not be unique. The existence condition is that theobstruction class in H2(M,Z2) must vanishes. Moreover the variouspossible chices for MB(V ) are parametrized by the cohomology groupH1(M,Z2).



We denote by Λ12 (V ) the space of half-forms and |Λ| (V ) the space of

the line of densities and we have the following bilinear pairing

Λ12 (V )× Λ

12 (V )→ Λn(V ),

(ρ1, ρ2) 7→ ρ1ρ2

and we have the following sesquilinear

Λ12 (V )× Λ

12 (V )→ |Λ| (V ),

(ρ1, ρ2) 7→ ρ1ρ2



Theorem

Let Λ12 (V ) denote the space of conjugate half-forms i.e. each element

satisfy σ(bB) = χ(b)σ(B) and Λ−12 (V ) denote the space of negative

half-forms which satisfy σ(bB) = χ(b)−1σ(B). A metalinear structureon V induces a metalinear structure on its dual bundle V ∗, such that

Λ−12 (V ) ∼= Λ

12 (V ∗)


Metaplectic Structure and Quantization

There are three main reason for introducing metaplectic structure inGeometric quantization which we explain here1) The sections s are on a bundle over M , while the sections of

Λ12T ∗C(M/P ) are over M/P . However for defining a tensor product of

sections s⊗ σ, we need both bundles to be over the same base space.2) The fact is that our construction M/P is not independent ofpolarization and it is dependent to polarization P and in order to havea correct geometrical picture of half-forms that could possibly allow forcomparison between different polarizations.3)In correct quantization we need to the choice of a metalinearstructure on each polarization P . More presicely, If we consider themetaplectic structure on symplectic manifold (M,ω) then we can put ametalinear structure on each Lagrangian subspace of TM and in turninduce a metalinear structure on the polarization P .



Now, if we embed the ML(n,R)-bundle MB(M/P )→M/P into aMetaplectic bundle Mp(M)→M then the half-forms defined in termsof their action on MB(M/P ) can be extended to frames over M bytaking them to be constant on metaplect transformations which don’tchange the embedding of MB(M/P ). Therefore, the half-form bundle

Λ12T ∗C(M/P )→M/P can be viewed as a bundle over M .



We need to define some primary definitions to construct metaplecticstructure. Firstly we define symplectic frame.

Definition

(Symplectic Frame) A symplectic frame at each x ∈M is an orderedbasis {e1, e2, ..., en, f1, f2, ..., fn} such that ω(ei, ej) = ω(fi, fj) = 0 andω(ei, fj) = δij for all i, j ≤ n. The collection of symplectic frames atx ∈M is equivalent to the symplectic group Sp(n,R).

From now on we denote the symplectic frame bundle by Bp(M)The fact is that because Sp(n,R) is diffeomorphic to the product of theunitary group U(n) and an Euclidean space. So, the fundamentalgroup of Sp(n,R) is Z. So as we explained before Sp(n,R) has uniquedouble covering, which we denote by Mp(n,R) and is calledMetaplectic group. Note also that, the metaplectic group Mp(2,R) isnot a matrix group, so metaplectic group is a little bit complicate.



To have a better picture of metaplectic group we give a generaldefenetion for it. Let (V, ω) be a symplectic vector space withdimV = 2n over F (here F is a nonarchimedean local field ofcharacteristic 0 and residual characteristic p) with associatedsymplectic group Sp(V ). The group Sp(V ) has a unique two-foldcentral extension Mp(V ) which is called the metaplectic group:

0→ {±1} →Mp(V )→ Sp(V )→ 0

So, we can write Mp(V ) = Sp(V )⊕ Z/2Z with group law given by

(g1, ε1).(g2, ε2) = (g1g2, ε1ε2c(g1, g2))

for some 2-cocycle c on Sp(V ) valued in {±1}.



Now, note that we have the natural embedding GL(n,R) ↪→ Sp(n,R)given by

A 7→(A 00 A∗−1

)So, GL(n,R) can be viewed as subgroup in Sp(2n,R) as the subgroupthat preserves the standard Lagrangian submanifold Rn ↪→ R2n.Restriction of the metaplectic group extension along this inclusiondefines the metalinear group Ml(n,R).



So, metaplectic structure on a symplectic manifold induces ametalinear structure for each Lagrangian submanifolds of TM andhence on each polarization of M .Now, as same as the definition of metalinear frame bundle we canconstruct the metaplectic frame bundle.



Definition

A metaplectic frame bundle Mp(M)→M is a principalMp(n,R)-bundle together with a covering map ρ : Mp(M)→ Bp(M)which makes the diagram

commutes. Note that here Mp(M)×Mp(n,R)→Mp(M) andBp(M)× Sp(n,R)→ Bp(M) are the natural group actions. Themetaplectic frame bundle Mp(M)→M is called metaplectic structureon M .



Now, by this definition we are ready to define metaplectic correction. Infact metaplectic correction is nothing else just the choice of metaplecticstructure. Note that Hence, (M,ω) admits metaplectic structures ifand only if the first Chern class c1(M) is even. For instance, (T ∗N, θ)where N is any orientable manifold has metaplectic structure. Asnatural example the complex projective spaces P2k+1C , k ∈ N0 hasunique metaplectic structure(because it is simply connected)



Now, we give the following important theorem. But first we need tothe definition of Bott connection

Definition

For M a smooth manifold and P ↪→ TM a foliation of M , incarnatedas a subbundle of the tangent bundle, the corresponding bundle

P⊥ ↪→ T ∗X

is the annihilator of P under the pairing of covectors with vectors. Thecorresponding Bott connection is the covariant derivative of vectorsX ∈ Γ(P ) on covectors ξ ∈ Γ(P⊥) given by the Lie derivative

∇X : ξ 7→ LXξ = iXdξ



Theorem

Let (M,ω) be a symplectic manifold and have a metaplectic structureand also assume P ⊂ TM be a strongly admissible real polarization.Let denote the annihilator bundle of P byP⊥ := {ξ ∈ T ∗M : ∀X ∈ P,< X, ξ >= 0} hence we can identify the

space of half-forms of Λ12 (M/D) with those sections of Λ

12 (P⊥) that

are P -constant by the Bott connection i.e. ∇Xξ := LXξ = 0 for anyX ∈ Γ(P ) and ξ ∈ Λ

12 (P⊥).



For construction of new sections s = s⊗ σ we said σ belong to someline bundle and we didn’t explain exatly this space of sections. Now bypervious theorem we can say σ ∈ Λ

12 (P⊥) for strongly admissible real

polarization P .So, from now on, the space of sections exactly is Γ(L⊗ Λ

12 (P⊥)).

Hence, we can define the polarized sections as follows.

Definition

We say a section s⊗ σ ∈ Γ(L⊗ Λ12 (P⊥)) is a polarized section if it

satisfy to ∇Xs = 0 and LXσ = 0 for all X ∈ Γ(P ). Note that here

Λ12 (P⊥) is the principal GL+(n)× Z4 bundle.


Inner product of Hilbert space for MetaplecticQuantization

Now we give an correct inner product on polarized sections ofΓ(L⊗ Λ

12 (P⊥)). Let s1 ⊗ σ1 and s2 ⊗ σ2 be such polarized sections.

So, h(s1, s2)σ1σ2 is a compactly supported smooth density on M/D,and we have the following Hermitian inner product.

< s1 ⊗ σ1, s2 ⊗ σ2 >=

∫M/D

h(s1, s2)σ1σ2


Conjecture

Conjecture: Does God believes in Quantization?
