metaplectic quantization on coadjoint orbits

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Metaplectic Quantization
Hassan Jolany
Laboratoire Paul Painlevé
Laboratoire de Mathématiques,CNRS-UMR 8524
Université des Sciences et Technologies de Lille
12th March 2014
Hassan Jolany (Laboratoire Paul Painlevé Laboratoire de Mathématiques,CNRS-UMR 8524 Université des Scien
Does God believes in Quantization? 12th March 2014 1 / 67

Outline
1 Introduction
Contraction between Physics and Mathematics and some Jokes
Sociology and Quantization
Symmetry in Nature and Quantization
Symmetry in Art and Quantization
2 Representation theory and Quantization
Lie groups and Lie algebra
Orbit method and Coadjoint orbits
Symplectic manifolds
Pre-Quantization
Symmetry in Nature
Quantization of Coadjoint orbits
3 Metaplectic Quantization and Spin-Geometry
Half-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

Symmetry in Nature
There is a lot of symmetry in Nature. The word comes from Greek sym
and metria. It was associated to beauty by Greek and Roman
philosophers:
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 to
proportion. Proportion is a correspondence among the measures of the
members of an entire work, and the whole to a certain part selected as
standard. 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 Art
Leonardo da Vinci’s Vitruvian Man (ca. 1487) is often used as a
representation 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. ”Most
people perceive this painting as Symmetrical ..... Yet most
mathematicians will tell you that the arrangements of colors and
forms are not symmetric in the Mathematical sense” [Mario Livio]

Symmetry in Mathematics
Hermann Weyl A thing is symmetrical if there is something you
can do to it so that after you have finished doing it it looks the
same as before.”
Mathematicians and scientists often used GROUP THEORY to
study symmetry that is expressed by group transformations
preserving 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 der
Transformationsgruppen, 1888-1893)
Elie Cartan [Geometre Francais]

The Group of Symmetries of the Square
The square has eight symmetries - four rotations, two mirror
images, and two diagonal flips:

These eight form a group under composition (do one, then
another). 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 unitary
representations of a Lie group
Why is it useful?
Representation theory remains the method of choice for
simplifying the physical analysis of systems possesing symmetry.
The Orbit Method is entangled with its physical counterpart
Geometric Quantization, which is an extension of the canonical
quantization scheme to curved manifolds
Definition (Representation)
A representation of a group G on a vector space V is a group
homomorphism from G to GL(V ), i.e. a map ρ : G → GL(V ) such that
ρ(g1g2) = ρ(g1)ρ(g2) for all g1; g2 ∈ G.

Representation
Loosely speaking: a representation makes an identification
between abstract groups and more managable linear
transformations of vector spaces.

The Adjoint and Coadjoint Representation
Let G be a matrix group, i.e. a group of invertible matrices, and let V
be its Lie algebra g. Then the adjoint representation Ad is defined by
Ad(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 the
dual of the adjoint representation Ad, defined by
In case G is a matrix group then g ∼
= g∗ and the coadjoint
representation 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 map
κ : 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 symplectic
manifold is isomorphic to a coadjoint orbit of its group of
Hamiltonian 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 by
some 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 functions
f : M → R skew-adjoint operators Of : H → H.
Paul Dirac introduced in his doctoral thesis, the ”method of
classical analogy” for quantization which is now known as Dirac
axioms 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 for
λ1, λ2 ∈ C
3] Normalization condition must holds: 1 7→ i.I(Which I is identity
operator 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 every
other function g such that {fi, g} = 0 for all {fj}, is constant. Also we
say that a family of operators is complete if it acts irreducibly on H
4] Minimality condition must holds: Any complete family of functions
passes to a complete family of operators. Moreover, if G be a group
acting on (M, ω) by symplectomorphisms and on H by unitary
transformations. If the G-action on M is trnsitive, then its action on H
must be irreducible

Geometric Quantization
Now, we again recall the pre-quantization line bundle in formal
language. In fact, we have two important method for Geometric
Quantization;
1)Using line bundle. More precisely, In geometric quantization we
construct the Hilbert space H as a subspace of the space of
sections of a line bundle L on a symplectic manifold M.
2)Without using line bundle: Using Spinc-structure instead of line
bundle. One of advantage of this construction is better behaved of
physical view but definig it is not so easy. We say (M, ω) is Spinc
prequantizable if and only if 1
2π [ω] − 1
2π [ω0] for some fixed
cohomology 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 the
cohomology class [ω]. It is important to point out that complex line
bundles are classified by H2(M, Z) via L 7→ c1(L) ,Therefore the
manifold (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 a
principal U(1)-bundle π : P → M and a connection form α on P with
curvature ω such that dα = −1
~ π∗ω. Note that by this philosophy, we
can consider the pre-quantum line bundle as associated vector bundle
L = P ×U(1) C and also P = {v ∈ L :< v, v >= 1}
Theorem
Instead to working with pre-quantization datas(L, <, >, ∇) we can
directly introduce prequantization datas by P and α, i.e., define
prequantization (P, α) with dα = −1
~ π∗ω.

Now we the following theorem about classification of pre quantization
structures 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 the
prequantization structure on complex line bundle L is unique up to
gauge equivalnce.

Attempts at Quantization
As we saw in previous section, the prequantization is a first
attempt to get a Hilbert space out of a symplectic manifold (M, ω)
. Now, we try to find the quantum space as a subspace of Hilbert
space as follows
1. L is called prequantum line bundle and ∇ prequantum
connection.

2. The prequantum space
HprQ
= {ψ ∈ Γ(L) :
Z
M
h(ψ, ψ) ∞}
where
=
ωn
n!
and is called Liouville form which is volume form in every symplectic
manifold and we have the following inner product on HprQ,
ψ, φ =
Z
M
h(ψ, φ)
3. The observables are quantized via
ˆ
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~ .

Polarization
Polarization is the second step of Geometric Quantization. Here
we explain the advantages of the notion of polarization in
Geometric Quantization.
1) The space of sections obtained in prequantization is too large
and it contains functions of both position and momentum. We
would like to end up with functions of just position. It is then
clear what we must do. We must pick the subspace of the
functions which are independent of the momentum. In fact, we
would like to use of polarization as a way of selecting half of the
directions of M, and then select from the prequantum space the
wave functions constant along those directions.
2) An advantage of restricting to a subspace of ”polarized
sections” is that the resulting prequantization may satisfy the
minimality 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 cotangent
bundles, and complex case is well suited for Kähler manifolds.
Definition
Let (M, ω) be a symplectic manifold. A real polarization on M is a
foliation (i.e. an integrable distribution) D ⊂ TM on M which is
maximally 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 physical
interpretation, but the one of problems of real polarization in
geometric quantization is defining a finite inner product of wave
functions(polarized sections of L). So, for the purpose of geometric
quantization we thus need a generalization of the notion of a real
polarization and we introduce here complex polarization.
Moreover, one of the other advanteges of complex polarization is
that this type of polarization is very important in kähler
quantization. These type of polarization are of special interest for
stablishing a bridge between geometric quantization and the theory
of irreducible unitary representations of Lie groups of symmetries.

Definition
A subbundle P ⊂ TMC of the complexified tangent bundle is called a
complex polarization if
1. P is Lagrangian
2. P involutive
3. dimP ∩ P̄ ∩ TM is constant
This definition shows that every complex polarization induces a real
isotrpic distribution D := P ∩ P̄ ∩ TM which is also involutive by
Frobenius theorem. Moreover the complexification of distribution D is
DC = P ∩ P̄ and it is called isotropic distribution. Now we define the
subbundle E := (P + P̄) ∩ TM and EC = P + P̄. Notice that
orthogonal symplectic complement of D is E, i.e., D⊥ = E and E is
called coisotrpic distribution.
Note that the subbundle P + P̄ is stil not necessarly involutive.
Imposing the following conditions on P ensures us that the
polarization is well behaved.

Real and Complex Polarization
There is a correspondence between Kähler polarization and Kähler
structures, given by following theorem.
Theorem
Let (M, ω) be a symplectic manifold
1. If (M, ω, J) is Kähler which J is complex structure, then P = T1,0M
is a Kähler polarization and we call this the holomorphic Kähler
polarization.
2.If P is a Kähler polarization on M, then there exists a complex
structure J such that (M, ω, J) is a Kähler manifold.

Let (M, ω, J) be a compact Kähler manifold with positive-denite
polarization P, and (L, ∇) be prequantum data. Let
Mquantum =

s ∈ Γ(L)|∇Xs = 0, ∀X ∈ P̄
Then Mquantum is fnite-dimensional.
We define the quantization of (M, ω, J) to be Mquantum, which is the
space 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, we
define HP ⊂ H to be the completion of square integrable sections s
such that ∇X̄s = 0 for all X ∈ P. We say to such sections as polarized
sections. Note that it can be shown that the space of square integrable
holomorphic sections is closed and we can waive the word ”completion”
in this definition.

Kähler polarization
Now by using of polarization in quantization we restrict the class of
quantizable observables as follows. In fact, we introduce a new class on
quantum Hilbert space of polarized sections which we denote it by
O(HP ) such that any operator which acts on the set of polarized
sections 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 must
have
0 = ∇X̄(Of s)
But by simple computation we have
∇X̄(Of s) = −i~∇[X̄,Xf ]s.
So [X̄, Xf ] ∈ P̄, 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 Geometric
quantization. From physical point of view, metaplectic correction
gives the correct quantization of the harmonic oscillator. The fact
is that in quantum mechanics, we can not stop on the set of
holomorphic polarized sections and it can be observe that the
energy level for harmonic oscillator is wrong. Moreover, the
dimensions of eigenspaces turn out to be wrong or shifted and we
can see this obstacle for Kepler problem in hydrogen atom.
In fact, let P be an arbitrary vertical polarization of (M, ω) with a
prequantum line bundle L, so, we know that each leaf of the
vertical polarization is a non-compact affine space, and moreover
our sections are constant along each leaf, then any non-vanishing
section is certainly not square integrable with respect to the
L2-norm defined on the prequantum Hilbert space.

Metaplectic Quantization and spin-Geometry
The standard example is the case when M = T∗N is a cotangent
bundle over a manifold N with its usual symplectic form, and P is
the vertical polarization. Then the isotropic and coisotropic
polarizations agree (both equaling the vertical polarization), and
their integral manifolds are the fibers of the bundle projection
T∗N → N. For any polarized section, the integration with respect
to ωn along these fibers will give an infinite contribution.
So, we must attempt to revise our set of holomorphic polarized
sections ΓP (L). To dealing with this, we must modify the line
bundle L so that there is a naturally induced norm on covariantly
constant sections ΓP (L̂).

Metaplectic correction
Now we need to constract L̂. But as we said before, the polarized
sections are covariantly constant along the leaves of D and if these
leaves are noncompact, then our polarized sections are not square
square integrable. A remedy to this obstacle is to integrate the
polarized sections not over symplectic manifold (M, ω) and instead of
it we integrate our polarized sections over M/D, or M/P. So, we try
to find the new line bundle L̂. As we saw before, our sections are
locally functions on M/P or M/D, and we are interested to have the
following L2-norm on M/P. By taking into account of this fact that
every section s on any neighborhood U ⊂ M, can be written as s = φs̃
which s̃ is unit section,we have for some n-form µM/P
hs, si =
Z
M/P
h(s, s)µM/P =
Z
M/P
φ̄φµM/P .

But, there is no natural measure µM/P on M/P or M/D. So to
remedy of this problem, we must reconstruct the integrand h(s, s0) such
that the integral makes sense on n-form µM/P and n-form µM/P be in
form of measure. So we need to change s, s0 such that h(s, s0) define a
density form on M/P so that integral make sence. So, let s, s0 ∈ ΓP (L)
and σ and σ are two section of some line bunde , then by constructing
new sections s̃ = s ⊗ σ and s̃0 = s0 ⊗ σ0 then h(s̃, s̃0) = h(s, s0)σ̄σ0 so if
we define the inner product as
s̃, s̃0
=
Z
M/D
h(s, s0
)σ̄σ0
and σ̄σ0 be a density form then our integral on M/D make sense. So
the problem is now to defining this new space of sections of the form s̃
but before to try to answer to this obstacle we first need to half-forms
and metalinear bundle as new geometrical structures which we explain
here.

The fact is that for integrating n-forms on the manifold M we require
the choice of orientation. But if we apply the notion of density then we
will circumvent this need of orientation. First we need to following
definitions.
Definition
Let M be a manifold of dimension n. Then an n-form σ on M is
defined 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 in
construction of half forms in metaplectic correction.

Definition
(Frame bundle) We recall that a frame bundle is a principal fiber
bundle 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. The
general 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 vector
bundle of rank n over a manifold M. A frame at a point x ∈ X is an
ordered basis for the vector space Ex. Equivalently, a frame can be
viewed as a linear isomorphism f : Rn → Ex. The set of all frames at
x, denoted Frx, has a natural right action by the general linear group
GL(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 frame
f ◦ g : Rn → Ex. The frame bundle of E, denoted by B(E) is the
disjoint union of all the Frx, i.e. B(E) =
`
x∈M Frx.

Also we need to α-density as a gemetrical tool in geometric
quantization which we define here,
Definition
Let M be a manifold of dimension n. Then an α-density σ on M is
defined 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)|
1
2 .σ(e)

Metaplectic Geometry
Now we come back to our quantization procedure. We would like to
define σ such that σ2 is a complex n-form and σ̄σ is a complex density,
so we need to such object σ, such that,
σ(eg) = |det(g)|
1
2 .σ(e)
But again we face to an obstacle, and the trouble with this definition
σ(eg) = |det(g)|
1
2 .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 group
related 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 four
component −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−2πik
n I

, t(m, B) = em
B
We define the action of Z on C × SL(n, C) by
(k, (m, B)) 7→

m +
2πik
n
, e−2πik
n B

Also the map t is invariant under the action of Z and we will have
following 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−2πik
n 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 s0 : 2Z → C × SL(n, C), defined by s0(2k) =

4πik
n , e−4πik
n I

The quotient ML(n, C) := (C × SL(n, C))/2Z is called as complex
metalinear group of dimension n. The elements of ML(n, C) can be
written as following forms
(m, B) =

m +
4πik
n
, e−4πik
n 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→ em
B
which following diagram commutes

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

(0, I), (
2πi
n
, e−2πi
n 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→ em
B 7→ det(em
B) = enm
det B = enm
So, we can introduce new holomorphic square root on ML(n, C), i.e.
χ : ML(n, C) → C∗
(m, B) 7→ e
nm
2

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 metalinear
structure and metalinear bundle. Note that we have the notation
ML(n, R) := ρ−1GL(n, R)

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

Now, with this definition, for B ∈ ML(n, R) and and b ∈ B(V ), we
have
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 exists
then in general may not be unique. The existence condition is that the
obstruction class in H2(M, Z2) must vanishes. Moreover the various
possible chices for MB(V ) are parametrized by the cohomology group
H1(M, Z2).

We denote by Λ
1
2 (V ) the space of half-forms and |Λ| (V ) the space of
the line of densities and we have the following bilinear pairing
Λ
1
2 (V ) × Λ
1
2 (V ) → Λn
(V ),
(ρ1, ρ2) 7→ ρ1ρ2
and we have the following sesquilinear
Λ
1
2 (V ) × Λ
1
2 (V ) → |Λ| (V ),
(ρ1, ρ2) 7→ ρ1ρ2

Theorem
Let Λ
1
2
(V ) denote the space of conjugate half-forms i.e. each element
satisfy σ(bB) = χ(b)σ(B) and Λ−1
2 (V ) denote the space of negative
half-forms which satisfy σ(bB) = χ(b)−1
σ(B). A metalinear structure
on V induces a metalinear structure on its dual bundle V ∗, such that
Λ−1
2 (V ) ∼
= Λ
1
2 (V ∗
)

Metaplectic Structure and Quantization
There are three main reason for introducing metaplectic structure in
Geometric quantization which we explain here
1) The sections s are on a bundle over M, while the sections of
Λ
1
2 T∗
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 of
polarization and it is dependent to polarization P and in order to have
a correct geometrical picture of half-forms that could possibly allow for
comparison between different polarizations.
3)In correct quantization we need to the choice of a metalinear
structure on each polarization P. More presicely, If we consider the
metaplectic structure on symplectic manifold (M, ω) then we can put a
metalinear structure on each Lagrangian subspace of TM and in turn
induce a metalinear structure on the polarization P.

Now, if we embed the ML(n, R)-bundle MB(M/P) → M/P into a
Metaplectic bundle Mp(M) → M then the half-forms defined in terms
of their action on MB(M/P) can be extended to frames over M by
taking them to be constant on metaplect transformations which don’t
change the embedding of MB(M/P). Therefore, the half-form bundle
Λ
1
2 T∗
C(M/P) → M/P can be viewed as a bundle over M.

We need to define some primary definitions to construct metaplectic
structure. Firstly we define symplectic frame.
Definition
(Symplectic Frame) A symplectic frame at each x ∈ M is an ordered
basis {e1, e2, ..., en, f1, f2, ..., fn} such that ω(ei, ej) = ω(fi, fj) = 0 and
ω(ei, fj) = δij for all i, j ≤ n. The collection of symplectic frames at
x ∈ 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 the
unitary group U(n) and an Euclidean space. So, the fundamental
group of Sp(n, R) is Z. So as we explained before Sp(n, R) has unique
double covering, which we denote by Mp(n, R) and is called
Metaplectic group. Note also that, the metaplectic group Mp(2, R) is
not a matrix group, so metaplectic group is a little bit complicate.

To have a better picture of metaplectic group we give a general
defenetion for it. Let (V, ω) be a symplectic vector space with
dimV = 2n over F (here F is a nonarchimedean local field of
characteristic 0 and residual characteristic p) with associated
symplectic group Sp(V ). The group Sp(V ) has a unique two-fold
central 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, 1).(g2, 2) = (g1g2, 12c(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 0
0 A∗−1

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

So, metaplectic structure on a symplectic manifold induces a
metalinear structure for each Lagrangian submanifolds of TM and
hence on each polarization of M.
Now, as same as the definition of metalinear frame bundle we can
construct the metaplectic frame bundle.

Definition
A metaplectic frame bundle Mp(M) → M is a principal
Mp(n, R)-bundle together with a covering map ρ̃ : Mp(M) → Bp(M)
which makes the diagram
commutes. Note that here Mp(M) × Mp(n, R) → Mp(M) and
Bp(M) × Sp(n, R) → Bp(M) are the natural group actions. The
metaplectic frame bundle Mp(M) → M is called metaplectic structure
on M.

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

Now, we give the following important theorem. But first we need to
the definition of Bott connection
Definition
For M a smooth manifold and P ,→ TM a foliation of M, incarnated
as a subbundle of the tangent bundle, the corresponding bundle
P⊥
,→ T∗
X
is the annihilator of P under the pairing of covectors with vectors. The
corresponding Bott connection is the covariant derivative of vectors
X ∈ 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 structure
and also assume P ⊂ TM be a strongly admissible real polarization.
Let denote the annihilator bundle of P by
P⊥ := {ξ ∈ T∗M : ∀X ∈ P, X, ξ = 0} hence we can identify the
space of half-forms of Λ
1
2 (M/D) with those sections of Λ
1
2 (P⊥) that
are P-constant by the Bott connection i.e. ∇Xξ := LXξ = 0 for any
X ∈ Γ(P) and ξ ∈ Λ
1
2 (P⊥).

For construction of new sections s̃ = s ⊗ σ we said σ belong to some
line bundle and we didn’t explain exatly this space of sections. Now by
pervious theorem we can say σ ∈ Λ
1
2 (P⊥) for strongly admissible real
polarization P.
So, from now on, the space of sections exactly is Γ(L ⊗ Λ
1
2 (P⊥)).
Hence, we can define the polarized sections as follows.
Definition
We say a section s ⊗ σ ∈ Γ(L ⊗ Λ
1
2 (P⊥)) is a polarized section if it
satisfy to ∇Xs = 0 and LXσ = 0 for all X ∈ Γ(P). Note that here
Λ
1
2 (P⊥) is the principal GL+(n) × Z4 bundle.

Inner product of Hilbert space for Metaplectic
Quantization
Now we give an correct inner product on polarized sections of
Γ(L ⊗ Λ
1
2 (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 =
Z
M/D
h(s1, s2)σ1σ2

Conjecture
Conjecture: Does God believes in Quantization?
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