**footnotetext: e-mail<EMAIL_ADDRESS>address: I.I.S.
“G.Soleri–A.Bertoni” Saluzzo (CN), via Traversa del Quartiere n.2, Italy.
2020 Mathematics Subject Classification. Primary 53D50; Secondary 81S10,
81Q20.
These notes has been written during STRING MATH 2020 (a virtual Zoom event
during the Pandemic of Covid-19)

# Prequantization, Geometric quantization, Corrected Geometric Quantization

Simone Camosso

###### Abstract.

A comparison on some facts concerning the geometric quantization of symplectic
manifolds is presented here. Criticism and improvements on the sophisticated
theory of geometric quantization are presented touching briefly, all the
“salient points of the theory”. The unfamiliar reader can consider this as a
“soft” introduction to the topic.

###### Contents

  1. 1 Introduction
  2. 2 Kähler manifolds, classical mechanics and symmetries
    1. 2.1 Kähler manifolds
    2. 2.2 Kähler manifolds and classical mechanics
    3. 2.3 Kähler manifolds and classical mechanics with symmetries
  3. 3 Prequantization
    1. 3.1 Prequatization conditions
    2. 3.2 Examples
  4. 4 Geometric quantization
    1. 4.1 The Dirac axioms and quantum operators
    2. 4.2 Kähler polarizations
    3. 4.3 Kähler quantization, holomorphic sections and Szegö kernels
  5. 5 The corrected geometric quantization: the case of complex cotangent bundle
    1. 5.1 The square–root bundle, the half form hilbert space and the harmonic oscillator
    2. 5.2 The BKS pairing
    3. 5.3 The BKS pairing on the cotangent bundle, the time evolution and the Schrödinger equation
    4. 5.4 Reconstruction of Schrödinger equation on the cotangent bundle
    5. 5.5 The Klein–Gordon equation and the cotangent bundle
    6. 5.6 Considerations on the BKS pairing
    7. 5.7 The BKS pairing for one complex and one real polarization: the Segal–Bargmann transform
    8. 5.8 The BKS pairing between two complex polarizations: the Bogoliubov transformation
  6. 6 The Bohr–Sommerfeld (sub)manifolds
  7. 7 Conclusion

##  1\. Introduction

The intention of these pages is to offer a “distillation of ideas” on the
geometric quantization. The studies that have led to the birth of this line of
interest are principally due to these authors: [1], [2], [3], [4] and [5]. On
the contrary, the works that inspired this paper are principally two. The
first is a recent article of Carosso [6], where the author describes in a very
detailed way the procedure of the geometric quantization with relevant
attention and with professional criticism. The second is the celebrated book
of Woodhouse [7], considered as the basic text for “beginners”.

The first pages of this article are a review of basic facts on Kähler
manifolds and classical mechanics (especially from the mathematical point of
view). After introducing this fundamental formalism with a lot of examples,
the author proceeds the examination of the “quantization procedure”. The
section three is dedicated to the process of prequantization with the
description of all prequantum conditions adopted by different authors. In the
section four the process of geometric quantization is described introducing
the notion of polarization (real and complex) and examining the particular
case of compact Kähler manifolds. In the section five the square–root bundle
is introduced in order to pass to a corrected geometric quantization. BKS
pairing, used by different authors, is examined in three cases: in the
presence of two real polarizations, one real and one complex and, at the end,
two complex polarizations. The last section is dedicated to the Bohr-
Sommerfeld subvarieties with recent developments (see [8]).

In a subsection of the section five, the Schrödinger equation is derived for
the case of the cotangent bundle in a similar way to [6] with the difference
we used a result of Albeverio and Mazzucchi ([9]) instead the heat kernel.

The implicit question regarding the future of the geometric quantization is
legitimate and necessary. We will see, during these pages, different
criticisms on this rigorous procedure together different problems in the
construction of a quantization space that is sufficiently satisfactory.
Furthermore, at the present state of the art, an application for the
relativity theory would be desirable but not possible. What emerge in this
direction is that the machinery of geometric quantization seems to be not the
rigorous solution to the problem of the unification of the relativity and the
quantum theory (at least without future modifications). The problem of the
geometric quantization of the general relativity has been recently discussed
in [10] and [11]. An interesting variation is given by the deformation
quantization, not treated here, but much more suitable. Another possible way
in this direction is the method used by Feynman considering a “summation of
histories” or probability amplitudes. For this reason an interesting
development can be based on the relation between the geometric quantization
and the Feynman integral. A seed of this idea can be found in [12] where the
author shows how the Feynman integral can be view as a particular case of BKS
pairing. This possibility sure is not the end of the history, as other
approaches show this is not the only “via regia” (the literature regarding
other quantization methods is omitted here due to the large amount to be taken
into consideration). A last work that we cite is [13] where the author studied
the connection between the geometric quantization (GQ) process and the quantum
logic (QL). In this optics the geometric quantization can be a considered as a
“machinery” that produces Hilbert spaces with interesting properties.

##  2\. Kähler manifolds, classical mechanics and symmetries

###  2.1. Kähler manifolds

A Kähler manifold is represented by a quadruple $(M,\omega,J,K)$ where $M$ is
a complex manifold of complex dimension $n$, $\omega$ is the symplectic
structure locally given by:

$\omega\,=\,i\sum_{j,l=1}^{n}g_{jl}dz_{j}\wedge d\overline{z}_{l},$ (2.1)

where $g$ is the Riemannian metric on $M$, $J$ is the complex structure
associated to $M$ and $K$ is the Kähler potential associated to $\omega$:

$\omega\,=\,i\partial\overline{\partial}K.$

We recall that in a Kähler manifold $d\omega=0$ then locally $\omega$ admits a
potential $K$. Furthermore the metric $g$ is $J$-invariant and

$\omega(v,w)\,=\,g(Jv,w),$

for every $v,w\in TM$. Furthermore on $M$ it is possible to consider also an
hermitian metric, denoted by $h$ and defined by:

$h\,=\,g-i\omega.$

The complex structure $J:TM\rightarrow TM$ divides the tangent space in a
point $m\in M$, into a direct sum $T_{m}M\,=\,T_{m}M\oplus\overline{T_{m}M}$
where:

$T_{m}M\,=\,\\{v\in T_{m}M:Jv=iv\\},\ \ \overline{T_{m}M}\,=\,\\{v\in
T_{m}M:Jv=-iv\\}.$

The first
$T_{m}M=\mathop{}\\!\mathrm{span}\langle\frac{\partial}{\partial_{z_{j}}}\rangle_{j=1,\ldots,n}$
and the second
$\overline{T_{m}M}=\mathop{}\\!\mathrm{span}\langle\frac{\partial}{\partial_{\overline{z}_{j}}}\rangle_{j=1,\ldots,n}$.
Sometimes these operators are simply denoted by
$\partial=\left(\partial_{z_{1}},\ldots,\partial_{z_{n}}\right)$ and
$\overline{\partial}=\left(\partial_{\overline{z}_{1}},\ldots,\partial_{\overline{z}_{n}}\right)$.
Passing through the $2n$ real coordinates denoted by $\\{x_{j},y_{j}\\}$ (for
$j=1,\ldots,n$), we have:

$\frac{\partial}{\partial_{z_{j}}}\,=\,\frac{1}{2}\left(\frac{\partial}{\partial
x_{j}}-i\frac{\partial}{\partial y_{j}}\right),\ \
\frac{\partial}{\partial\overline{z}_{j}}\,=\,\frac{1}{2}\left(\frac{\partial}{\partial
x_{j}}+i\frac{\partial}{\partial y_{j}}\right)$

and respectively, the dual basis is $dz_{j}\,=\,dx_{j}+idy_{j}$ and
$d\overline{z}_{j}\,=\,dx-idy_{j}$.

######  Example 2.1.

An example of Kähler manifold is $\left(\mathbb{C}^{n},\omega,J,K\right)$,
where:

$\omega\,=\,\frac{i}{2}\sum_{j=1}^{n}dz_{j}\wedge d\overline{z}_{j},$

and the Kähler potential is given
by$K(z)\,=\,\sum_{j=1}^{n}z_{j}\overline{z}_{j}\,=\,\|z\|^{2}$. Here the
symplectic form is the standard form
$\omega_{0}\,=\,\sum_{j=1}^{n}dx_{j}\wedge dy_{j}$.

######  Example 2.2.

Another example of Kähler manifold is
$\left(\mathbb{PC}^{n}={\raisebox{1.99997pt}{$\mathbb{C}^{n}\setminus\\{0\\}$}\left/\raisebox{-1.99997pt}{$\mathbb{C}^{*}$}\right.},\omega_{\text{FS}},J,K\right)$,
where:

$\omega_{\text{FS}}\,=\,\frac{i}{2}\partial\overline{\partial}\log{\left(1+\|z\|^{2}\right)},$

is the Fubini–Study form and the Kähler potential is
$K(z)\,=\,\log{\left(1+\|z\|^{2}\right)}$ (here $\log$ denotes the natural
logarithm).

######  Example 2.3.

The unit disk $\mathbb{D}=\\{z\in\mathbb{C}:|z|^{2}<1\\}$ has the structure of
Kähler manifold $\left(\mathbb{D},\omega_{\text{HY}},J,K\right)$ where:

$\omega_{\text{HY}}\,=\,\frac{i}{\pi}\frac{dz\wedge
d\overline{z}}{\left(1-|z|^{2}\right)^{2}}$

is the hyperbolic form and
$K(z)\,=\,\frac{1}{\pi}\log{\left(1-|z|^{2}\right)}$.

######  Example 2.4.

A last example is the complex torus
$\mathbb{T}={\raisebox{1.99997pt}{$\mathbb{C}$}\left/\raisebox{-1.99997pt}{$\Lambda$}\right.}$
where $\Lambda$ is a lattice on $\mathbb{C}$.
$\left(\mathbb{T},\omega_{\text{flat}},J,K\right)$ has the structure of Kähler
manifold where:

$\omega_{\text{flat}}\,=\,\frac{i}{2}\partial\overline{\partial}H$

and $H:\mathbb{C}\times\mathbb{C}\rightarrow\mathbb{C}$ is a complex map
linear in the first factor and respectively antilinear in the second factor
such that $H(z,z)\geq 0$ and $H(z,z)=0$ if and only if $z=0$. In this case
$K=H$.

###  2.2. Kähler manifolds and classical mechanics

In this section we recall the relation between Kähler manifolds and classical
mechanics. Let $(M,\omega,J,K)$ be a Kähler manifold with
$\dim_{\mathbb{C}}{M}=n$ and $\omega$ its symplectic form, then it is possible
to interpret $M$ as a model of a classical system. For example $M=T^{\vee}C$
the cotangent space of a configuration space $C$ of dimension $n$. So in this
case $\omega$ is given in local symplectic coordinates and classical
observables are smooth functions $f$ on $M$. To each observable $f\in
C^{\infty}(M)$ there is an Hamiltonian vector field $X_{f}\in TM$ defined by
the equation:

$df\,=\,X_{f}\,\lrcorner\,\omega,$ (2.2)

we underline that this notation is equivalent to other common notations
$df\,=\,\omega(X_{f},\cdot)$ or $df\,=\,i_{X_{f}}\omega$. In local
coordinates:

$X_{f}\,=\,\partial_{p_{j}}\partial_{q_{j}}-\partial_{q_{j}}\partial_{p_{j}}.$
(2.3)

For any two observables $f,g$ we can define the Poisson bracket $\\{f,g\\}$
as:

$\\{f,g\\}\,=\,X_{g}(f)\,=L_{X_{g}}(f)\,=\,df(X_{g})\,=\,\omega\left(X_{f},X_{g}\right)\,=\,-\\{g,f\\},$
(2.4)

where $L_{X_{g}}$ is the Lie derivative along $X_{g}\in TM$. We recall that
the Lie derivative is defined by the flow
$\phi_{t}^{X_{g}}(m):\mathbb{R}\times M\rightarrow M$:

$L_{X_{g}}(f)\,=\,\frac{d}{dt}f(\phi_{t}^{X_{g}}(m)),$ (2.5)

where in the previous formula generally there is a tensorial field instead of
$f$.

An observable is conserved when $L_{X_{g}}(f)\,=\,0$. In particular there is a
smooth function $\mathcal{H}:M\rightarrow\mathbb{R}$, called Hamiltonian, such
that determines the trajectories of a classical system via the Hamilton’s
equations:

$\frac{d}{dt}\phi_{t}(m)\,=\,X_{\mathcal{H}}\phi_{t}(m),$ (2.6)

for a point $m\in M$.

The system $(M,\omega,X_{\mathcal{H}})$ is called Hamiltonian system. If
$\phi_{t}^{X_{\mathcal{H}}}$ is an integral curve of $X_{\mathcal{H}}$ then
the energy function $\mathcal{H}\left(\phi_{t}^{X_{\mathcal{H}}}\right)$ is
constant for all $t$ and
$\mathcal{H}\left(\phi_{t}^{X_{\mathcal{H}}}\right)\,=\,\mathcal{H}$.

Note that
$L_{X_{\mathcal{H}}}(\omega)=X_{\mathcal{H}}\,\lrcorner\,d\omega+d(X_{\mathcal{H}}\,\lrcorner\,\omega)=0+dd\mathcal{H}=0$.

Further information on Hamiltonian mechanics can be found in [14].

From the fact that $\omega$ is closed we have that locally admits a potential
$\theta\,=\,\sum_{j=1}^{n}p_{j}dq_{j}$. From the potential we can define the
lagrangian associate to an observable $f$ defined as:

$\mathcal{L}\,=\,X_{f}\,\lrcorner\,\theta-f.$ (2.7)

The lagrangian mechanics is associated to lagrangian submanifolds of $M$. When
the function $f$ is the Hamiltonian $\mathcal{H}$ then
$\mathcal{L}\,=\,X_{\mathcal{H}}\,\lrcorner\,\theta-\mathcal{H}$ is the
Legendre transform formula and by the Cartan’s formula we have that:

$L_{X_{\mathcal{H}}}\theta\,=\,X_{\mathcal{H}}\,\lrcorner\,d\theta+d\left(X_{\mathcal{H}}\,\lrcorner\,\theta\right)\,=\,d\mathcal{L}.$

We recall that $X_{\mathcal{H}}$ generates a diffeomorphism $\rho:M\rightarrow
M$ that preserve $\omega$. From the fact that $\omega\,=\,\rho^{*}\omega$ we
have that $d\theta-\rho^{*}d\theta\,=\,0$ so $\theta-\rho^{*}\theta$ is closed
and locally exact. Thus $\theta-\rho^{*}\theta\,=\,dS$, where $S$ is a
function on $M=T^{\vee}C$. Let $\gamma$ be an integral curve of
$X_{\mathcal{H}}$ with parametrization $t$, then we have that:

$(\theta(X_{\mathcal{H}})\circ\gamma)(w)|^{t}_{0}\,=\,(L\circ\gamma)(w)|^{t}_{0}$

and

$(\theta(X_{\mathcal{H}})\circ\gamma)(w)|^{t}_{0}\,=\,(\rho_{w}^{*}\theta)(\rho^{-1}_{w^{*}}X_{\mathcal{H}})(\gamma(0))|^{t}_{0}\,=\,(\rho_{w}^{*}\theta)(X_{\mathcal{H}})(\gamma(0))|^{t}_{0}$
$\,=\,\theta(X_{\mathcal{H}})(\gamma(0))|^{t}_{0}-(dS(X_{\mathcal{H}})\circ\gamma)(w)|^{t}_{0}$

because the pushforward of $X_{\mathcal{H}}$ along itself is the identity and,
the form $\theta-\rho_{w}^{*}\theta$ is exact. By integration:

$S\circ\rho_{w}\,=\,-\int_{0}^{t}\mathcal{L}\circ\gamma(w)dw+c,$ (2.8)

where $S\circ\rho_{w}$ is a local phase function (because $X_{\mathcal{H}}$ is
complete). We obtain thus a function $\mathcal{S}$ called the generating
function:

$\mathcal{S}(q,q^{\prime},t)\,=\,-\int_{0}^{t}\mathcal{L}\circ\gamma(w)dw+c,$
(2.9)

where $\gamma(t)\,=\,\rho_{t}(q,p)\,=\,(q^{\prime},p^{\prime})$ and $q(0)=q$
and $q^{\prime}=q(t)$. The function $\mathcal{S}$ generates a submanifold
$\Lambda\subset M$ where $\omega=0$ and where $\dim(M)=\dfrac{n}{2}$. Such
manifolds $\Lambda$ are called lagrangian submanifolds. The fiber coordinates
$p_{j}$ on $\Lambda$ are given by:

$p_{j}\,=\,\frac{\partial\mathcal{S}}{\partial q_{j}},$ (2.10)

for $j=1,\ldots,n$. In order to pass from $S(q,p(q^{\prime}),t)$ to
$\mathcal{S}(q,q^{\prime},t)$ we use the equations $(\ref{hamilton_J})$.

Details on lagrangian and hamiltonian mechanics are in [6] and [7].

###  2.3. Kähler manifolds and classical mechanics with symmetries

Let $G$ be a finite dimensional compact Lie group and $\mu^{G}:G\times
M\rightarrow M$ an action on the Kähler manifold $(M,\omega,J,K)$. Let
$\mathfrak{g}=\mathop{}\\!\mathrm{Lie}{(G)}$ be the Lie algebra associated to
$G$ and $\mathfrak{g}^{\vee}$ the dual space. If we have an $n$–form $\omega$
on $M$ it is associated a map
$\Psi:M\rightarrow\bigwedge^{n}\mathfrak{g}^{\vee}$ defined as:

$\Psi(m)\,=\,\mu_{m}^{G\vee}\omega,$ (2.11)

where $\mu_{m}^{G}:G\rightarrow M$ is $\mu_{m}^{G}(g)\,=\,g\cdot m$ (observe
that if the action of $G$ is symplectic then
$\mu_{m}^{G\vee}\omega\,=\,\omega$). Furthermore if $\omega$ is closed, then
also $\Psi$ ($\forall m\in M$) and $\Psi:M\rightarrow Z^{2}(\mathfrak{g})$. If
in addition $H^{1}(\mathfrak{g})\,=\,H^{2}(\mathfrak{g})\,=\,\\{0\\}$ then
$\forall m\in M$ there is a unique element $\Phi(m)$ in $\mathfrak{g}^{\vee}$
such that:

$d\Phi(m)\,=\,\Psi(m).$ (2.12)

There is a unique map $\Phi:M\rightarrow\mathfrak{g}$ such that:

$d\Phi\,=\,\Psi.$ (2.13)

We can be more explicit with $\Phi$. In fact, under our assumptions, there is
an homeomorphism between the Lie algebra $\mathfrak{g}$ and the hamiltonian
vector fields on $M$ (see [15]). In particular there is a unique homeomorphism
$\lambda:\mathfrak{g}\rightarrow C^{\infty}(M)$ such that
$\lambda(\xi)=f_{\xi}$ where:

$\xi_{M}\,\lrcorner\,\omega\,=\,df_{\xi}.$ (2.14)

The map $f_{\xi}$ depends linearly by $\xi$ so $\forall m\in M$ we can
consider $\Phi(m)\in\mathfrak{g}^{\vee}$ given by:

$\langle\Phi(m),\xi\rangle\,=\,f_{\xi}(m),$ (2.15)

where $\langle\cdot,\cdot\rangle$ is the pairing between $\mathfrak{g}$ and
$\mathfrak{g}^{\vee}$. In other words:

$d\langle\Phi(m),\xi\rangle\,=\,\xi_{M}\,\lrcorner\,\omega,$ (2.16)

for every $\xi\in\mathfrak{g}$ where $\Phi:M\rightarrow\mathfrak{g}^{\vee}$ is
called the moment map. The previous equation can be written in the following
form:

$\langle d_{m}\Phi(v),\xi\rangle\,=\,\omega_{m}\left(\xi_{M}(m),v\right),$
(2.17)

where $d_{m}\Phi:T_{m}M\rightarrow\mathfrak{g}^{\vee}$ for every $m\in M$ and
$v\in T_{m}M$.

The moment map has different properties. The first is that $d_{m}\Phi$ is the
transpose map of the valuation
$\mathop{}\\!\mathrm{val}:\mathfrak{g}\rightarrow T_{m}M$. We have that:

$\ker{(d_{m}\Phi)}\,=\,\mathfrak{g}_{M}(m)^{\perp},$ (2.18)

where $\mathfrak{g}_{M}(m)$ is the subspace of vectors $\xi_{M}(m)$. We have
that $G$ acts on $M$ transitively so $T_{m}M\,=\,\mathfrak{g}_{M}(m)$ for
every $m\in M$. We have also that:

$\mathop{}\\!\mathrm{Im}{(d_{m}\Phi)^{0}}\,=\,\\{\xi\in\mathfrak{g}:\xi_{M}(m)=0\\},$
(2.19)

is the stabilizator of $m\in M$. The stabilizator is discrete if and only if
$d_{m}\Phi$ is surjective. The last property is a conseguence of the fact that
$G$ acts on $\mathfrak{g}$ by the adjoint representation,
$\mathop{}\\!\mathrm{val}$ is a $G$–morphism, thus $\Phi$ is a $G$–morphism:

$\Phi(g\cdot m)=g\cdot\Phi(m),$ (2.20)

for every $g\in G$ and $m\in M$.

######  Example 2.5.

Let us consider $\mathbb{R}^{2}$ with $\omega=d\,x\wedge d\,y=rd\,r\wedge
d\,\theta$. Let us consider as $G$ the circle $S^{1}$ that acts with
rotations. The generator of the action is the field $x\frac{\partial}{\partial
y}-y\frac{\partial}{\partial x}=\frac{\partial}{\partial\theta}$. Thus
$\Phi^{S^{1}}=-\frac{1}{2}(x^{2}+y^{2})=\frac{1}{2}r^{2}$ is the moment map.

In general on $\left(\mathbb{C}^{n}\equiv\mathbb{R}^{2n},\omega\right)$ where
$\omega=\sum_{i=1}^{n}d\,x_{i}\wedge d\,y_{i}$, with the action of $S^{1}$ and
heights $m_{1},\ldots,m_{n}\in\mathbb{Z}$, the action has generator the vector
field $\sum_{i=1}^{n}m_{i}\frac{\partial}{\partial\theta_{i}}$ and moment map
given by:

$\Phi^{S^{1}}(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n})=-\frac{1}{2}\sum_{i=1}^{n}m_{i}\left(x_{i}^{2}+y_{i}^{2}\right).$

##  3\. Prequantization

###  3.1. Prequatization conditions

Recalling that a closed surface $\Sigma$ of $M$ is a surface that is compact
and without boundary, we state the prequantum condition PC1 as the following.

PC1 The integral of $\omega$ over any closed 2–surface $\Sigma\subset M$ is an
integral multiple of $2\pi\hbar$.

This prequantum condition PC1 is necessary for the existence of the hermitian
line bundle $L$ over $M$. In the case of $M$ simply connected PC1 is also
sufficient. Assume $M$ simply connected and $m_{0}\in M$ be a base point. Let
us consider the following set:

$\\{(m,z,\gamma):m\in M,z\in\mathbb{C},\gamma\ \ \text{a piecewise smooth path
from}\ \ m_{0}\ \ \text{to}\ \ m\\}$

and the equivalence relation $\sim$ defined as:

$(m,z,\gamma)\sim(m^{\prime},z^{\prime},\gamma^{\prime})\Leftrightarrow
m=m^{\prime}\wedge
z^{\prime}\,=\,ze^{\frac{i}{\hbar}\int_{\Sigma_{1}}\omega},$

where $\Sigma_{1}$ is any oriented 2–surface with boundary made up of $\gamma$
(from $m_{0}$ to $m$) and $\gamma^{\prime}$ (from $m$ to $m_{0}$). The surface
$\Sigma_{1}$ exists because $M$ is simply connected. We define the line bundle
$L$ as:

$L\,=\,{\raisebox{1.99997pt}{$\\{(m,z,\gamma):m\in M,z\in\mathbb{C},\gamma\ \
\text{a piecewise smooth path from}\ \ m_{0}\ \ \text{to}\ \
m\\}$}\left/\raisebox{-1.99997pt}{$\sim$}\right.}.$

We can define operations of addition and scalar multiplication between the
fibres:

$[(m,z,\gamma)]+[(m,z^{\prime},\gamma)]\,=\,[(m,z+z^{\prime},\gamma)],\ \
k[(m,z,\gamma)]\,=\,[(m,kz,\gamma)],$

with $k\in\mathbb{C}$ and $[(m,z,\gamma)],[(m,z^{\prime},\gamma)]\in L$.
Trivializations of $L$ are determined locally by symplectic potentials. In
fact let us assume that $\theta_{j}$ is a symplectic potential on a simply
connected open set $U_{j}\subset M$ of a collection $\\{U_{j}\\}_{j\in J}$.
Let us consider a point $m_{1}\in U_{j}$ and a curve $\gamma_{0}$ from $m_{0}$
to $m_{1}$. We define locally a section $s$ of $L$ in $U_{j}$ by

$s_{j}(m)\,=\,\left[\left(m,e^{-\frac{i}{\hbar}\int_{\gamma_{1}}\theta_{j}},\gamma\right)\right],$

where $\gamma_{1}$ is any curve from $m_{1}$ to $m$ in $U_{j}$ and $\gamma$ is
the curve from $m_{0}$ to $m$ obtained from $\gamma_{0}$ and $\gamma_{1}$. We
observe that a different choice of $\gamma_{1}$ gives the same value of
$s_{j}(m)$ and that a different choice of $m_{1}$ or $\gamma_{0}$ gives the
same section multiplied by a constant of modulus one. The effect of replacing
$\theta_{j}$ by $\theta_{k}\,=\,\theta_{j}+du_{kj}$, with $u_{kj}(m_{1})=0$,
is $s_{k}(m)=e^{-\frac{i}{\hbar}u_{kj}}s_{j}(m)$.

Now we can assume that the line bundle $L\rightarrow M$ exists. Let us
consider the parallel transport of a section $s$ respect to $\nabla$ around a
loop $\gamma$ of $m\in M$. Assume that $\gamma$ is the boundary of a
$2$–surface $\Sigma_{1}$ contained in the domain of $\theta$ a symplectic
potential. Solving the parallel transport equation $\nabla_{\dot{\gamma}}s=0$,
the result is equivalent to a linear transformation $L_{m}\rightarrow L_{m}$
given by the multiplication of $e^{\frac{i}{\hbar}\int_{\gamma}\theta}$ that,
by the Stokes’ theorem, is equivalent to
$e^{\frac{i}{\hbar}\int_{\Sigma_{1}}\omega}$. Now let us consider a second
surface $\Sigma_{2}$ with boundary $\gamma$ such that
$\Sigma\,=\,\Sigma_{1}\cup\Sigma_{2}$ is a closed $2$-surface in $M$ (in other
terms gluing together $\Sigma_{1}$ and $\Sigma_{2}$ in $\gamma$ we obtain the
closed 2–surface $\Sigma$). In a similar way the parallel transport gives a
linear transformation by the multiplication of
$e^{-\frac{i}{\hbar}\int_{\Sigma_{2}}\omega}$, where the minus is because the
boundary of $\Sigma_{2}$ is $-\gamma$. By the uniqueness of the solution of
the differential equation associated to the parallel trasport we have that:

$e^{\frac{i}{\hbar}\int_{\Sigma_{1}}\omega}\,=\,e^{-\frac{i}{\hbar}\int_{\Sigma_{2}}\omega},$

that is

$e^{\frac{i}{\hbar}\int_{\Sigma}\omega}\,=\,1,$

the last equation is equivalent to PC1 because $\int_{\Sigma}\omega\in
2\pi\hbar\mathbb{Z}$.

The prequantum condition is related to the existence of the hermitian line
bundle $L$ over $M$ also through the Weil theorem.

######  Theorem 3.1 (Weil, 1958, [16]).

Let $M$ be a smooth manifold and $\omega$ a real, closed 2–form whose
cohomology class $\left[\frac{\omega}{2\pi\hbar}\right]$ is integral. Then
there is a unique hermitian line bundle $L$ over $M$ with unitary connection
$\nabla$ so that $c_{1}(L)=\left[\frac{\omega}{2\pi\hbar}\right]$.

The converse is also true. In fact let $\\{U_{j}\\}$ be a contractible open
cover of $M$. By assumption there is a collection of $1$–forms
$\theta_{j}\in\Omega^{1}(U_{j})$ such that
$\frac{\omega}{2\pi\hbar}\,=\,d\theta_{j}$ on $U_{j}$. We can find $u_{jk}\in
C^{\infty}(U_{j}\cap U_{k})$ such that $du_{jk}\,=\,\theta_{j}-\theta_{k}$
whenever $U_{j}\cap U_{k}\,\not=\,\emptyset$. On $U_{j}\cap U_{k}\cap U_{l}$
(whenever $U_{j}\cap U_{k}\cap U_{l}\,\not=\,\emptyset$) we have that
$du_{jk}+du_{kl}+du_{lj}\,=\,0$, so the function
$u_{jkl}\,=\,u_{jk}+u_{kl}+u_{lj}$ is constant. They define a Cěch cohomology
class in $H^{2}(M,\mathbb{R})$. Now $\theta_{j}-\theta_{k}\,=\,d\log{g_{jk}}$
whenever $U_{j}\cap U_{k}\,\not=\,\emptyset$ and, the de Rham isomorphism send
$c_{1}(L)$ to $\left[\frac{\omega}{2\pi\hbar}\right]\in H^{2}(M,\mathbb{R})$
with:

$u_{jkl}\,=\,\frac{1}{2\pi
i\hbar}\left(\log{g_{jk}}+\log{g_{kl}}+\log{g_{lj}}\right).$

By assumption $g_{kl}$ are transition functions and must satisfy the cocycle
condition:

$g_{jk}g_{kl}g_{lj}\,=\,1,$

that is

$e^{2\pi i\hbar u_{jkl}}\,=\,1.$

So $\left[\frac{\omega}{2\pi\hbar}\right]\in H^{2}(M,\mathbb{Z})$.

We have a second version of the prequantization condition PC2.

PC2 $\left[\frac{\omega}{2\pi\hbar}\right]\in H^{2}(M,\mathbb{Z})$.

Note that in PC2 it is not required $M$ to be simply connected, so it is more
general than PC1.

Another prequantization of a symplectic manifold $(M,\omega)$ consist in a
$S^{1}$–bundle $(X,\alpha)$ with the projection $\pi:X\rightarrow M$ and
$\alpha$ an $S^{1}$ invariant 1–form such that satisfy the prequantization
condition PC3.

PC3 $d\alpha\,=\,\frac{1}{2\pi\hbar}\pi^{*}\omega$.

The $S^{1}$–bundle $(X,\alpha)$ it is also called the circle bundle and is
defined as:

$X\,=\,\\{(m,\lambda):m\in M,\lambda\in
L^{\vee}_{m},h(\lambda,\lambda)=1\\}\subset L^{\vee},$

where $h$ is the hermitian metric on $L$. We have that $(X,\alpha)$ is also
called a contact manifold.

The last way to state the prequantization condition derive from PC3 and
consists relating the curvature form of the connection $\nabla$ of the line
bundle $L$ and the symplectic form $\omega$. In other terms if $\Theta$ is the
curvature form of the unique covariant derivative $\nabla$ on $L$ compatible
with both the complex and hermitian structures, we have the prequantization
condition PC4.

PC4 $\Theta\,=\,-\frac{i}{\hbar}\omega$.

###  3.2. Examples

######  Example 3.2.

Let $H:\mathbb{C}\times\mathbb{C}\rightarrow\mathbb{C}$ be a complex linear
map in the first factor and antilinear in the second. Assume also that
$H(z,z)\geq 0$ and that is zero if and only if $v=0$. Let us define:

$\omega\,=\,\frac{i}{2}\partial\overline{\partial}H,$ (3.1)

then $\omega$ is a Kähler form for the complex torus
$\mathbb{T}\,=\,{\raisebox{1.99997pt}{$\mathbb{C}$}\left/\raisebox{-1.99997pt}{$\Lambda$}\right.}$
that is invariant under the action of the lattice $\Lambda$. Furthermore it is
possible to prove that the torus $\mathbb{T}$ is quantizable if and only if
$\mathop{}\\!\mathrm{Im}{H}(\Lambda\times\Lambda)\subset\mathbb{Z}$. This
condition it is in fact equivalent to the integrability condition PC4. We can
try to see this in one direction. Let us assume
$\mathop{}\\!\mathrm{Im}{H}(\Lambda\times\Lambda)\subset\mathbb{Z}$, then the
image of $\Lambda\times\Lambda$ through $H$ is in $\mathbb{Z}$. This condition
ensure the existence of a complex line bundle
$L=L(\chi,H)\cong\mathbb{C}\times\mathbb{T}$ where $\chi$ is a semicharacter
associated to $H$. On $L$ we can define an hermitian structure defined as:

$h(\theta(z),\theta(z))\,=\,F(z)|\theta(z)|^{2},$ (3.2)

where $\theta(v)$ is the holomorphic section of $L$, also called theta
function such that:

$\theta(z+\lambda)\,=\,A(\lambda,z)\theta(z),$ (3.3)

and $A$ is a factor of automorphy on $\Lambda\times\mathbb{C}$ (a good
reference is [17]). The second term in the definition of $h$ is $F(z)$ that is
a map $F:\mathbb{C}\rightarrow\mathbb{R}_{+}$ defined as:

$F(z)\,=\,e^{-\pi H(z,z)}.$ (3.4)

It is possible to see that with this definition, the function $h$ is invariant
under the action of $\Lambda$ and $h$ defines an hermitian structure on $L$.
Now analyzing the curvature form of the line bundle $L$ we find that it is
equal to
$-\partial\overline{\partial}\log{h}=-\partial\overline{\partial}\log{F}=\pi\partial\overline{\partial}H$.
This last term is equal to $-2\pi i\omega$ that is the condition PC4 with the
Planck constant equal to $1$. The converse of the proof is proved in [17].

######  Example 3.3.

Let $M=S^{2}$ be the sphere in $\mathbb{R}^{3}$ of radius $r$. The sphere is a
Kähler manifold with symplectic structure:

$\omega\,=\,r\sin{\varphi}d\varphi\wedge d\vartheta,$ (3.5)

that is a 2–form on $S^{2}$ where, $\vartheta\in[0,2\pi)$, $\varphi\in[0,\pi]$
and $r\in[0,+\infty)$. In order to be quantizable the integral:

$\int_{S^{2}}\omega\,=\,4\pi r,$

must be equal to $2\pi\hbar n$ for some $n\in\mathbb{Z}$. We conclude that not
all spheres satisfy the prequantization condition PC1 but only the spheres
with $r=\frac{n}{2}\hbar$.

######  Example 3.4.

Let $M=S^{2}$ be the sphere realized as section of the light-cone by the
hyperplane $x^{0}=ct=1$. It is show in [18] that $M$ can be quantized in the “
sense of Souriau”. The quantization is obtained as $S^{1}$–fiber bundle
$(S^{3},\pi)$ where $\pi$ are the so called “KS–transformations” (Kustaankeimo
and Stiefel transformations for the regularization of the Kepler problem)
which associates to each vector in the light–cone a one–index spinor. In
practice, in [18], it is described how from the KS–trasformations we obtain
the Hopf fibering of the sphere $S^{3}$. In this case the quantization
condition is the same of the previous example for $r=1$:

$\int_{S^{2}}\omega\,=\,4\pi.$

######  Example 3.5.

Let $M=\mathbb{D}$ be the unit disk. It is a Kähler manifold with the
hyperbolic form and, the prequantization condition PC1, is trivially
satisfied. The line bundle is trivial
$L=\mathbb{D}\times\mathbb{C}\rightarrow\mathbb{D}$ and
$h(z,t)\,=\,\left(1-|z|^{2}\right)^{2}|t|^{2}$ is the hermitian structure with
$(z,t)\in\mathbb{D}\times\mathbb{C}$. An analogue argument shows that the
complex space $\mathbb{C}^{n}$ is prequantizable, in the same way.

######  Example 3.6.

Let $M=\Sigma$ be a compact Riemann surface of genus $g\geq 2$. We can think
the Riemann surface as the quotient
${\raisebox{1.99997pt}{$\mathbb{D}$}\left/\raisebox{-1.99997pt}{$G$}\right.}$,
where $\mathbb{D}$ is the unit disk of $\mathbb{C}$ and $G$ is the subgroup of
$SU(1,1)$ of fractional linear transformation. An element $g\in G$ is
represented by a matrix:

$g\,=\,\left(\begin{array}[]{cc}a&b\\\
\overline{b}&\overline{a}\end{array}\right),$ (3.6)

such that $|a|^{2}-|b|^{2}\,=\,1$. The action on $\mathbb{D}$ is defined by
$g\cdot z\,=\,\frac{az+b}{\overline{b}z+\overline{a}}$. We have that $\Sigma$
with $\omega_{\text{HY}}$ is a Kähler manifold because $\omega_{\text{HY}}$ is
invariant by the action of $G$.

Let us consider $\\{(U_{j},z_{j})\\}_{j\in J}$ be a complex atlas on $\Sigma$
and $s_{j}(z)$ be holomorphic functions on $U_{j}$. We can define global
sections $s(z)\,=\,s_{j}(z)dz_{j}$ such that on $U_{j}\cap U_{k}$ we have that
$s_{j}(z)dz_{j}\,=\,s_{k}(z)dz_{k}$. Now from complex analysis we have that:

$\frac{dz_{k}}{dz_{j}}\cdot\frac{dz_{l}}{dz_{k}}\,=\,\frac{dz_{l}}{dz_{j}}$

that is, defining $\gamma_{jk}=\frac{dz_{k}}{dz_{j}}$, equivalent to
$\gamma_{jk}\gamma_{kl}\gamma_{lj}=1$ for every $z\in U_{j}\cap U_{k}\cap
U_{l}$. Then there exists a canonical bundle $K$. If we consider the
projection
$p:\mathbb{D}\rightarrow{\raisebox{1.99997pt}{$\mathbb{D}$}\left/\raisebox{-1.99997pt}{$G$}\right.}$
the pull-back $p^{*}(K)$ is holomorphically trivial and its holomorphic global
sections are of the form $s(z)dz$ on $\mathbb{D}$. We can think to global
holomorphic sections on $K$ as $1$–forms of type $(1,0)$ on $\mathbb{D}$
invariant under the action of $G$. They are forms $s(z)dz$ such that:

$s(g(z))\cdot\frac{1}{\left(\overline{b}z+\overline{a}\right)^{2}}dz\,=\,s(z)dz,$
(3.7)

for every $g\in G$. We can proceed as in the case of the torus defining an
hermitian structure:

$h(s(z),s(z))\,=\,(1-|z|^{2})^{2}|s(z)|^{2},$ (3.8)

that is invariant for every section of $K$ and $g\in G$.

At the end we see that the curvature form of the line bundle $K$ is equal to
$-\partial\overline{\partial}\log{h}=-\partial\overline{\partial}\log{(1-|z|^{2})^{2}}=\frac{2dz\wedge
d\overline{z}}{(1-|z|^{2})^{2}}$. This last term is equal to $-2\pi
i\omega_{\text{HY}}$ that is the condition PC4 with the Planck constant equal
to $1$.

######  Example 3.7.

A very remarkable example is the complex projective space $M=\mathbb{PC}^{n}$.
In this case we can start to show that $\mathbb{PC}^{1}$ satisfy the condition
PC2 with the convention that $h=1$ (in this case it is a Riemann surface and
the result it is true). To see this we can use these relations:

$\mathop{}\\!\mathrm{deg}{(L)}\,=\,\int_{\mathbb{PC}^{1}}c_{1}(L)\,=\,\frac{i}{2\pi}\int_{\mathbb{PC}^{1}}\partial\overline{\partial}\log{(1-|z|^{2})},$
(3.9)

where in this case $L\,=\,\mathcal{O}(1)$ is the hyperplane bundle. Now we can
directly perform the integration:

$\mathop{}\\!\mathrm{deg}{(\mathcal{O}(1))}\,=\,\frac{1}{\pi}\int_{\mathbb{R}^{2}}\frac{1}{(1+x^{2}+y^{2})^{2}}dxdy\,=\,\frac{1}{\pi}\int_{0}^{+\infty}\int_{0}^{2\pi}\frac{r}{(1+r^{2})^{2}}drd\vartheta\,=\,1.$
(3.10)

Thus we have that
$c_{1}(\mathcal{O}(1))\,=\,\left[\omega_{\text{FS}}\right]\in
H^{2}(\mathbb{PC}^{1},\mathbb{Z})$, that is PC2. The result is true for
general $n$ and to see this it is possible to use the chain of isomorphisms:
$H^{2}(\mathbb{PC}^{n},\mathbb{Z})\rightarrow
H^{2}(\mathbb{PC}^{n-1},\mathbb{Z})\rightarrow\ldots\rightarrow
H^{2}(\mathbb{PC}^{1},\mathbb{Z})$, in order to have
$\left[\omega_{\text{FS}}^{n}\right]\in H^{2}(\mathbb{PC}^{n},\mathbb{Z})$.

The result that the Kähler form $\omega_{\text{FS}}$ is integral is also
showed in [19].

######  Example 3.8.

Let $M=S^{2}\times S^{2}$ be the product of spheres with the same radius $r$.
The manifold $M$ is a Kähler manifold with symplectic structure:

$\omega\,=\,p_{1}^{*}\omega_{S^{2}}+p_{2}^{*}\omega_{S^{2}},$ (3.11)

where $p_{1},p_{2}$ are the projections on the two spheres with radius
$r\in[0,+\infty)$ and we define the symplectic structure as the pullback
respectively of the two Kähler structures on $S^{2}$. In order to be
quantizable we have that the integral:

$\int_{S^{2}\times\\{p^{\prime}\\}}\omega\,=\,4\pi r,$

for every $S^{2}\times\\{p^{\prime}\\}\subseteq S^{2}\times S^{2}$, must be
equal to $2\pi\hbar n$ for some $n\in\mathbb{Z}$. We conclude that
$r=\frac{n}{2}\hbar$, in this case the product of the two sphere is
prequantizable. We observe also that if the radii of the two spheres are
incommensurable then $S^{2}\times S^{2}$ is not prequantizable!

######  Example 3.9.

In this example we describe the prequantization of the Kepler manifold $X$
defined as:

$X\,=\,\\{(\xi,x)\in\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}:\xi\cdot\xi=1,x\cdot
x=0,x\not=0\\},$

where $\cdot$ is the usual Euclidean scalar product. Usually, the manifold $X$
is denoted also with $T^{\vee}(S^{n})\setminus\\{0\\}$. The Kepler manifold
has the following symplectic form:

$\omega_{\text{Kepler}}\,=\,\sum_{j=1}^{n+1}dx_{j}\wedge d\xi_{j}.$ (3.12)

We can think $X$ as a complex manifold with the identification of $X$ with the
complex light cone:

$C\,=\,\\{z\in\mathbb{C}^{n+1}:z\cdot z=0,z\not=0\\}.$

The identification can be realized using the map: $m:X\rightarrow C$ such that
$(\xi,x)\mapsto\|x\|\cdot\xi+ix$. We have that $(X,J,\omega)$ is a Kähler
manifold where $J$ is the pullback of the complex structure of $C^{n+1}$ and

$\omega_{\text{Kepler}}\,=\,2i\overline{\partial}\partial\sqrt{\frac{z\cdot\overline{z}}{2}}.$
(3.13)

The Kähler form is exact, trivially integral and there is a quantum line
bundle $L$. Moreover for $n\geq 3$ we have that $X$ is simply connected and
$L$ is holomorphically trivial over $X$ (details are in [20]).

######  Example 3.10.

Let $M=S^{1}\times\mathbb{R}$ be the cylinder. It can be identified with
$T^{\vee}S^{1}$. It is a Kähler manifold with $\omega=dp\wedge d\varphi$.
There is a symplectic potential $\theta=pd\varphi$ and $\omega=d\theta$ is
globally exact and the prequantum line bundle $L$ is trivial (but not
unique!).

##  4\. Geometric quantization

###  4.1. The Dirac axioms and quantum operators

In his work [21], P. Dirac defines the quantum Poisson bracket $[\cdot,\cdot]$
of any two variables $u$ and $v$ as:

$uv-vu=i\hbar[u,v]$ (4.1)

where $\hbar$ is the Plank constant $h$ over $2\pi$. The formula
$(\ref{dirac})$ is one of the basic postulates of quantum mechanics. We can
summarize these postulates as follows. To start we fix a symplectic manifold
$(M,\omega)$ of dimension $n$, with $\omega$ the corresponding symplectic
structure and an Hilbert space $\mathfrak{H}$. The quantization is a “way” to
pass from the classical system to the quantum system. In this case the
classical system (or phase space) is described by the symplectic manifold $M$
and the Poisson algebra of smooth function on $M$ denoted by
$(C^{\infty}(M),\\{\cdot,\cdot\\})$. The quantum system is described by
$\mathfrak{H}$. We define “quantization” a map $Q$ from the subset of the
commutative algebra of observables $C^{\infty}(M)$ to the space of operators
in $\mathfrak{H}$. Let $f\in C^{\infty}(M)$ be an observable we have that
$Q(f):\mathfrak{H}\rightarrow\mathfrak{H}$ is the corresponding quantum
operator. We can summarize the quantum axioms in this scheme:

  * 1.

Linearity: $Q(\alpha f+\beta g)=\alpha Q(f)+\beta Q(g)$, for every
$\alpha,\beta$ scalars and $f,g$ observables;

  * 2.

Normality: $Q(1)=\mathop{}\\!\mathrm{id}$, where $\mathop{}\\!\mathrm{id}$ is
the identity operator;

  * 3.

Hermiticity: $Q(f)^{*}=Q(f)$;

  * 4.

(Dirac) quantum condition: $[Q(f),Q(g)]=-i\hbar Q(\\{f,g\\})$;

  * 5.

Irreducibility condition: for a given set of observables $\\{f_{j}\\}_{j\in
J}$, with the property that for every other $g\in C^{\infty}(M)$, such that
$\\{f_{j},g\\}=0$ for all $j\in J$, then $g$ is constant. We can associate a
set of quantum operators $\\{Q(f_{j})\\}_{j\in J}$ such that for every other
operator $Q$ that commute with all of them is a multiple of the identity.

The last postulate states that in the case we consider a connected Lie group
$G$ we say that is a group of symmetries of the physical system if we have the
two following irreducible representation: one as symplectomorphisms acting on
$(M,\omega)$ and another as unitary transformations acting on $\mathfrak{H}$.
For many details about these postulates look [22].

######  Example 4.1 (Schrödinger quantization).

Let $M=\mathbb{R}^{2n}$ and $(q_{j},p_{j})$ be the canonical coordinates of
position and momentum. In this case $Q(q_{j})=q_{j}$ that acts as
multiplication and $Q(p_{j})=-i\hbar\frac{\partial}{\partial q_{j}}$. The
Hilbert space $\mathfrak{H}=L^{2}(\mathbb{R}^{n},dq_{j})$ and there are the
following relations of commutations:

$[Q(q_{k}),Q(q_{j})]=[Q(p_{k}),Q(p_{j})]=0,\ \ \ \
[Q(q_{k}),Q(p_{j})]=i\hbar\delta_{kj}\mathop{}\\!\mathrm{id}.$

Let us examine the form of these quantum operators. We can start considering
the case where $M=T^{\vee}C$ with $C$ the configuration space. Now $M$ is a
symplectic space with $\omega$ as symplectic form and is prequantizable with
an hermitian line bundle $(L,h)$. Proceeding in the choice of a potential
$\theta$ it is possible to construct $Q(f)$ for each observable $f\in
C^{\infty}(M)$ satisfying the quantum axioms. This operator has the following
form:

$Q(f)\,=\,-i\hbar X_{f}-X_{f}\,\lrcorner\,\theta+f.$ (4.2)

These operators $Q_{f}$ can be “glued” in order to form a global operator on
the sections of the corresponding line bundle $L$.

In general if we have a Kähler manifold $M$ that satisfies the prequantum
condition PC4, and if $s$ is a section of the line bundle $L$, for every $f\in
C^{\infty}(M)$, we have an Hamiltonian vector field $X_{f}$ and the operator
$\nabla_{X_{f}}$ that acts on $L^{2}(M,L,dq^{j})$. So if we define the
operator:

$Q(f)s=-i\hbar\nabla_{X_{f}}s+f\cdot s,$ (4.3)

now it is an hermitian linear operator and, when $f$ is constant, is only the
multiplication by $f$. The formula $(\ref{KonstantS})$ satisfy the Dirac
postulate 4.

###  4.2. Kähler polarizations

Polarizations of symplectic manifolds are introduced in order to have a
dependence of sections of $L$ (that are waves functions) by half the
coordinates of the configuration space.

A complex Kähler polarization of $M$ (of complex dimension $n$) is a smooth
complex distribution $D$ (subbundle of $TM$) that is a map that to each point
$m\in M$ assigns a linear subspace $D_{m}$ of $T_{m}M$, such that:

  * 1.

$\mathop{}\\!\mathrm{dim}{(D_{m})}=n$;

  * 2.

$D\cap\overline{D}=\emptyset$;

  * 3.

$(D\oplus\overline{D})_{m}=T_{m}M_{\mathbb{C}}$;

  * 4.

$D$ is involutive, $[D,D]\subset D$;

  * 5.

$D_{m}$ is isotropic, that is $\omega|_{D}=0$;

  * 6.

For every $\xi\not=0$ in $D$, $i\omega(\xi,\overline{\xi})>0$.

First note that if $D$ is a polarization, also $\overline{D}$ is a
polarization. Second by the Frobenius theorem $D$ is integrale: that is for
each point $m\in M$ there is an integrable submanifold $N$ (of $M$) whose
tangent space at $m$ is $D_{m}$. These $N$ are called leaves of $D$.

A section $s$ of the line bundle $L$ over $M$ is called polarized if:

$\nabla_{\xi}s\,=\,0,$ (4.4)

for all $\xi\in D$.

A polarization is real if $D\,=\,\overline{D}$.

A reference for Kähler polarization is [23]. Furthermore, if a Kähler manifold
has one Kähler polarization then it has many, that is changing polarization
the Hilbert space change. A good tool in order to consider possible relations
between these Hilbert spaces is the pairing we will see next.

######  Example 4.2 (Kähler polarization).

Let $(M,\omega,J,K)$ be a Kähler manifold, then a Kähler polarization $D$
consists in a submanifold spanned by vectors $\xi$ such that $\omega|_{D}=0$.
For example in the Boson–Fock space of dimension $1$, that is $\mathbb{C}$
with $\omega\,=\,\frac{i}{2}dz\wedge d\overline{z}$ and
$K(z)=\frac{1}{2}z\cdot\overline{z}$. The Kähler polarization
$\overline{D}\,=\,\langle\partial_{\overline{z}}\rangle$ is spanned by the
antiholomorphic basis. In this case the polarized sections are simply the
holomorphic functions. This example it is also called the holomorphic
quantization.

######  Example 4.3 (Real polarization).

Let $M=T^{\vee}C$ where $C$ is a configuration space. An example of real
polarization is the vertical polarization associated to the cotangent bundle.
The spann of $D$ is given by the momentum basis $\partial_{p_{j}}$. In this
case the wave functions depends only by the position and the condition of
preservation of $D$ on an observable $f$ is that:

$\frac{\partial^{2}}{\partial p_{j}\partial p_{k}}f\,=\,0,\ \ \forall
j,k=1,\ldots n.$

The vertical polarization is tipically associated to the Schödinger
representation of quantum mechanics. The holomorphic quantization brings to
the Fock–Segal–Bergmann representation.

###  4.3. Kähler quantization, holomorphic sections and Szegö kernels

In this subsection let us consider $(M,\omega,J,K)$ to be a compact Kähler
manifold of complex dimension $n$. Assume we have the prequantization
condition:

$\Theta\,=\,-2i\omega.$ (4.5)

######  Observation 4.4.

We must do some considerations. The first is that the condition $(\ref{PC5})$
is slightly different from PC4, this is due by a different definition of
$\omega$ but the geometrical essence is the same. Second, the condition
$(\ref{PC5})$ ensures that the Dirac postulate is satisfied and, the fact that
$\omega$ is integral, ensures the existence of the hermitian line bundle with
$(\ref{PC5})$ satisfied.

We call the triple $(L,\nabla,h)$ the prequantization bundle over
$(M,\omega)$. We observe that with the hermitian product $h$ and the volume
form $\omega^{n}=\omega\wedge\cdots\wedge\omega$, $n$ times defined on $M$, we
can consider the space of smooth holomorphic sections $s$ of $M$ such that:

$\|s\|^{2}=\int_{M}h(s,s)\frac{\omega^{n}}{n!}$ (4.6)

is finite. The $L^{2}$-completition of this space is an Hilbert space denoted
by $\mathfrak{H}$ or also $H^{0}(M,L)$.

In the compact Kähler case from $(\ref{PC5})$ we have that $L$ is a positive
line bundle and by the Kodaira embedding theorem there exists a positive
tensor power $L^{\otimes k}$ with $k\in\mathbb{N}$ and global holomorphic
sections $\\{s^{k}_{i}\\}_{i=0}^{n_{k}}$ that give the following embedding:

$\Phi:M\rightarrow\mathbb{P}^{n_{k}}(\mathbb{C}),$

where $\Phi(z)=[s^{k}_{0}(z):\cdots:s^{k}_{n_{k}}(z)]$. The set
$\\{s^{k}_{i}\\}_{i=0}^{n_{k}}$ is a basis for $H^{0}(M,L^{\otimes k})$ the
space of holomorphic sections of $L^{\otimes k}$. The
$\mathop{}\\!\mathrm{dim}{(H^{0}(M,L^{\otimes k}))}=1+n_{k}$.

In the contest of geometric quantization the parameter $k$ can be understood
as a quantum parameter in the sense that $k=\frac{1}{\hbar}$. If we imagine
that $\hbar\rightarrow 0$ then $k\rightarrow+\infty$ and we refind the
semiclassical limit. Let us consider the circle bundle $X$ of $L^{\vee}$
(defined previously) with $\pi:L^{\vee}\rightarrow M$ to $X$ that for
simplicity we denote in the same way. The circle bundle is the boundary of
$D=\\{(m,\lambda):m\in M,\lambda\in L_{m}^{\vee},h(\lambda,\lambda)\leq 1\\}$
that is a strictly pseudoconvex domain in $L$. We denote with $\|\cdot\|_{m}$
the induced norm by $h$ so we have that $D=\\{\rho>0\\}$ where
$\rho:L^{\vee}\rightarrow\mathbb{R}$ is defined as
$\rho(m,\lambda)=1-\|\lambda\|_{m}^{2}=1-a(m)^{-1}|\alpha|^{2}$ where we write
$\lambda=\alpha s^{\vee}_{L}$ and $a(m)=\|s^{\vee}_{L}\|_{m}^{2}$ is a smooth
function over a $U\subset M$ and $s^{\vee}_{L}$ is a local coframe over $U$.
We have a circle action on $X$ denoted by $r_{\theta}:S^{1}\times X\rightarrow
X$ with infinitesimal generator $\frac{\partial}{\partial\theta}$. As in [24]
we consider the holomorphic and respectively antiholomorphic subspaces
$T^{\prime}D,T^{\prime\prime}D\subset TD_{\mathbb{C}}$ and the correspondent
differentials
$d^{\prime}f=df|_{T^{\prime}},d^{\prime\prime}f=df|_{T^{\prime\prime}}$ for
$f$ smooth on $D$. $TD_{\mathbb{C}}=T^{\prime}D\oplus
T^{\prime\prime}D\oplus\mathbb{C}\frac{\partial}{\partial\theta}$ has a Cauchy
Riemann structure and, the vectors on $D$ that are elements of $T^{\prime}X$
(resp. $T^{\prime\prime}X$), are of the form
$\sum_{j}a_{j}\frac{\partial}{\partial\overline{z}_{j}}$ (resp.
$\sum_{j}a_{j}\frac{\partial}{\partial z_{j}}$). We can choose a basis for
these vector spaces and consider the Cauchy Riemann operator
$\overline{\partial}_{b}:\mathcal{C}^{\infty}(X)\rightarrow
C^{\infty}(X,(T^{\prime\prime\vee}D))$ defined as
$\overline{\partial}_{b}=df|_{T^{\prime\prime}}$. If we define
$\alpha=\frac{1}{i}d^{\prime}\rho|_{X}$ and the Volume form
$dV=\alpha\wedge(d\alpha)^{n}$ we have that $(X,\alpha)$ is a contact
manifold.

######  Definition 4.5.

We define the Hardy space
$H(X)=L^{2}(X)\cap\mathop{}\\!\mathrm{Ker}(\overline{\partial}_{b})$ that
admit the following decomposition:

$H(X)=\bigoplus_{k}H(X)_{k}$ (4.7)

where the subspaces

$H(X)_{k}=\\{f\in
C^{\infty}(X):f(e^{i\varphi}x)=e^{ik\varphi}f(x)\\}\cap\mathop{}\\!\mathrm{Ker}(\overline{\partial}_{b})$
(4.8)

are called $k$-Hardy spaces.

######  Observation 4.6.

On the Hardy spaces we have an hermitian product
$H(s_{1},s_{2})=\int_{M}h_{m}(s_{1}(m),s_{2}(m))dV_{M}(m)=\int_{X}\widehat{s_{1}}\overline{\widehat{s_{2}}}dV_{X}$
because there is an unitary isometry between sections
$|\widehat{s}(x)|^{2}=\|s(m)\|^{2}$ (between $H^{0}(M,L^{\otimes k})$ and
$H(X)_{k}$ with $m=\pi(x)$). The notation
$\widehat{s}=\langle\lambda,s(z)\rangle$ is used to denote the equivariant
smooth section defined on $L^{\vee}\setminus\\{0\\}$. In analogue way $s_{k}$
on $L^{\otimes k}$ determines a $\widehat{s}_{k}=\langle\lambda^{\otimes
k},s_{k}(z)\rangle$.

######  Definition 4.7 (Equivariant Szegö projector).

We define the equivariant Szegö projector $\Pi_{k}:L^{2}(X)\rightarrow
H(X)_{k}$ where $\forall f\in L^{2}(X)$ we have that:

$\Pi_{k}(f)=\sum_{j}\langle
f,\widehat{s}_{j}^{(k)}\rangle_{L^{2}(X)}\widehat{s}_{j}^{(k)}$

where $(\widehat{s}_{j}^{(k)})_{j=1}^{n_{k}}$ is an orthonormal basis of
$H(X)_{k}\cong H^{0}(M,L^{\otimes k})$.

Expanding
$\left(\Pi_{k}(f)\right)(x)=\int_{X}\sum_{j}\widehat{s}_{j}^{(k)}(x)\overline{\widehat{s}_{j}^{(k)}(y)}f(y)dV_{X}(y)$,
we have this other definition.

######  Definition 4.8 (Equivariant Szegö kernel).

The equivariant Szegö kernel is:

$\Pi_{k}(x,y)=\sum_{j}\widehat{s}_{j}^{(k)}(x)\overline{\widehat{s}_{j}^{(k)}(y)}.$

By a theorem of [25] it is possible to represent the Szegö kernel as a complex
Fourier integral operator (FIO representation).

######  Theorem 4.9.

Let $\Pi(x,y)$ be the Szegö kernel of $X$, the boundary of a strictly
pseudoconvex domain in $L$. Then there exists a symbol $s\in S^{n}(X\times
X\times\mathbb{R}_{+})$ that admit the following expansion:

$s(x,y,t)=\sum_{k=0}^{\infty}t^{n-k}s_{k}(x,y)$ (4.9)

so that

$\Pi(x,y)=\int_{0}^{+\infty}e^{it\psi(x,y)}s(x,y,t)dt,$ (4.10)

where $\psi\in C^{\infty}(D\times D)$ such that $\psi(x,x)=-i\rho(x)$ ($\rho$
define $D$), $d_{x}^{\prime\prime}\psi,d_{y}^{\prime}\psi$ vanish to infinite
order along the diagonal and $\psi(x,y)=-\overline{\psi(y,x)}$.

We have that $\Pi$ is a Fourier integral operator with complex phase $\psi$
and the canonical relation $\Lambda\subset(T^{\vee}X)\times(T^{\vee}X)$ is
generated by the phase $t\psi(x,y)$ on $X\times X\times\mathbb{R}_{+}$. In
fact the canonical relation $\Lambda$ is the lagrangian submanifold of
$(T^{\vee}X)\times(T^{\vee}X)$ that has as generating function the phase
function $t\psi$. The condition that must be true for the parametrization of
the lagrangian submanifold $\Lambda$ is that:

$\frac{d(t\psi)}{dt}=0$ (4.11)

that is when $\psi(x,y)=0$ and, on the diagonal $X\times X$ we have
$d_{x}{\psi}=-d_{y}\psi=-id^{\prime}\rho|_{X}$. Let
$\alpha=-id^{\prime}{\rho}$ and let
$\Sigma=\\{(x,r\alpha):r\in\mathbb{R}_{+}\\}$ be the symplectic cone generated
by the contact form $\alpha$, the real points of $\Lambda$ consist in the
diagonal $\Sigma\times\Sigma$. We say that $\Pi$ has a Toeplitz structure on
the symplectic cone $\Sigma$.

######  Observation 4.10.

The canonical relation $\Lambda$ cover an important rule in the theory of
quantization as Guillemin and Sternberg remark in [26] “the smallest subsets
of classical phase space in which the presence of a quantum mechanical
particle can be detected are its lagrangian submanifolds”.

Now we are ready to recall an important result due to [24] that uses the
method of stationary phase and the microlocal analysis of Szegö and Bergman
kernels.

######  Theorem 4.11 (Zelditch, Tian, Yau).

Let $M$ a compact complex projective manifold of dimension $n$, and let
$(A,h)$ a positive hermitian holomorphic line bundle. Let $g_{J}$ a Kähler
metric on $M$ and $-2i\omega=\Theta$ a Kähler form. For each $k\in\mathbb{N}$,
$h$ induces a hermitian metric $h_{k}$ on $L^{\otimes k}$. Let
$\\{s^{k}_{i}\\}_{i=0}^{n_{k}}$ be any orthonormal basis of
$H^{0}(M,L^{\otimes k})$ with $\mathop{}\\!\mathrm{dim}{(H^{0}(M,L^{\otimes
k}))}=1+n_{k}$. Then there exists a complete asymptotic expansion:

$\Pi_{k}(z,z)=\sum_{j}\|s_{j}^{(k)}(z)\|_{h_{k}}^{2}=a_{0}k^{n}+a_{1}(z)k^{n-1}+\cdots$
(4.12)

for some $a_{j}$ smooth with $a_{0}=1$.

The proof is on [24] but we recall briefly the scheme. First we observe that
$\Pi_{k}$ are Fourier coefficients of $\Pi$, so we have the following
expression:

$\Pi_{k}(x,y)=\int_{0}^{+\infty}\int_{S^{1}}e^{-ik\theta}e^{it\psi(r_{\theta}x,y,t)}s(r_{\theta}x,y,t)dtd\theta,$
(4.13)

where we used the FIO representation of $\Pi$ and $r_{\theta}$ denote the
$S^{1}$ action on $X$. Changing variables $t\mapsto kt$ we have an oscillating
integral:

$\Pi_{k}(x,y)=k\int_{0}^{+\infty}\int_{S^{1}}e^{ik\Psi(t,\theta,x,y)}s(r_{\theta}x,y,t)dtd\theta$
(4.14)

with phase $\Psi(t,\theta,x,y)=t\psi(r_{\theta}x,y)-\theta$. To simplify the
phase we consider an holomorphic coframe $s_{L}^{\vee}$ and
$a(z)=|s_{L}^{\vee}|_{z}^{-2}$. We write any $\sigma\in L^{\vee}_{z}$ as
$\sigma=\lambda s_{L}^{\vee}$ and for the coordinates
$(x,y)=(z,\lambda,w,\mu)\in X\times X$ we have
$\rho(z,\lambda)=a(z)|\lambda|^{2},\psi(z,\lambda,w,\mu)=-ia(z,w)\lambda\overline{\mu}$
with $a(z,w)$ is an almost analytic function on $M\times M$ such that
$a(z,z)=a(z)$. On $X$ we have $\lambda=a(z)^{-\frac{1}{2}}e^{i\phi}$ and
$\mu=a(w)^{-\frac{1}{2}}e^{i\phi^{\prime}}$. So for
$(x,y)=(z,\phi,w,\phi^{\prime})\in X\times X$ we have:

$\psi(z,\phi,w,\phi^{\prime})=-i\left(1-\frac{a(z,w)}{i\sqrt{a(z)}\sqrt{a(w)}}\right)e^{i(\phi-\phi^{\prime})}$
(4.15)

and on the diagonal we have
$\psi(r_{\theta}x,x)=-i\left(1-\frac{a(z,z)}{a(z)}e^{i\theta}\right)$ and
$\Psi(t,\theta,x)=-it(1-e^{i\theta})-\theta$. The critical points for $\Psi$
are $\theta=0$ and $t=1$, the Hessian $\Psi^{\prime\prime}$ at this point is
$\left(\begin{array}[]{cc}0&1\\\ 1&i\end{array}\right)$. Applying the
stationary phase method we find that:

$\Pi_{k}(x,x)\sim k\frac{\sqrt{2\pi
i}}{\sqrt{\det{(k\Psi^{\prime\prime}(1,0))}}}\sum_{j,k=0}^{+\infty}k^{n-k-j}L_{j}s_{k}(x,x)$

with $L_{j}$ differential operator of order $2j$ defined by:

$L_{j}s_{k}(x,x)=\sum_{\nu-\mu=j}\sum_{2\nu\geq
3\mu}i^{-j}2^{-\nu}\langle\Psi^{\prime\prime}(1,0)^{-1}D,D\rangle^{\nu}\frac{(g_{(t,\theta)}^{\mu}ts_{k}(r_{\theta}x,x))|_{(t=1,\theta=0)}}{\mu!\nu!}$

where
$\langle\Psi^{\prime\prime}(1,0)^{-1}D,D\rangle=2\frac{\partial^{2}}{\partial
t\partial\theta}-i\frac{\partial^{2}}{\partial t^{2}}$.

The hypothesis of the theorem 7.7.5 of [27] are satisfied because the phase
has nonnegative imaginary part and critical points are real and independent by
$x$. So we have that $\Pi_{k}$ has the following form:

$\Pi_{k}(x,x)=k^{n}C_{n}s_{0}(x,x)+k^{n-1}a_{1}(x,x)+\cdots$ (4.16)

The term $s_{0}(x,y)$ was expressed in [25] using that $\Pi$ is a projection
and on the diagonal one has that
$s_{0}(x,x)dV=\frac{1}{4\pi^{n}}\det{L_{X}}\|d\rho\|dx$ with $L_{X}$ the
restriction of the Levi form $L_{\rho}(z)=\sum\frac{\partial^{2}\rho}{\partial
z_{j}\partial\overline{z}_{k}}z_{j}\overline{z}_{k}$ to the maximal complex
subspace of $TX$. So:

$\Pi_{k}(x,x)dV=k^{n}C_{n}\alpha\wedge\omega^{n}+O(k^{n-1}).$ (4.17)

The main coefficient is a nonzero constant $a_{0}$ times $k^{n}$. Comparing
with the leading term of the Hirzebruch Riemann Roch polynomial we have that
$a_{0}=1$. This conclude the proof.

The method of the asymptotic expansion of the Szegö kernel in presence of
symmetries can be generalized in different ways. The interested reader can
consult the following literature [28], [29],[30], [31] and [32].

##  5\. The corrected geometric quantization: the case of complex cotangent
bundle

###  5.1. The square–root bundle, the half form hilbert space and the harmonic
oscillator

From now we restrict our attention for the case where $M=T^{\vee}Q$ with $Q$ a
configuration space of dimension $n$. Now $M=T^{\vee}Q$ is a Kähler manifold
with Kähler form $\omega$, complex structure $J$ and Kähler potential $K$. We
will consider associated to $M$ a Kähler polarization $D$.

In this subsection we recall some facts about determinant bundles,
“square–root bundles” and metaplectic correction.

######  Definition 5.1.

Let $D$ be a Kähler polarization of a Kähler manifold $(M,\omega,J,K)$. The
deteminant bundle $\mathfrak{K}$ is the complex line bundle for which the
sections are $n$–forms $\tau$ satisfying:

$\xi\,\lrcorner\,\tau\,=\,0,\ \ \forall\xi\in\overline{D}.$ (5.1)

A section $\tau$ is said polarized if

$\xi\,\lrcorner\,d\tau\,=\,0,\ \ \forall\xi\in\overline{D}.$ (5.2)

######  Definition 5.2.

By square–root of $\mathfrak{K}$ we will denote the complex line bundle
$\delta$ over $M$ such that $\delta\otimes\delta$ is isomorphic to
$\mathfrak{K}$.

On $\delta$ we have a partial connection $\nabla^{\delta}$ defined for
$\xi\in\overline{D}$ and given by $\nabla_{\xi}\tau=\xi\,\lrcorner\,d\tau$.
$\nabla^{\delta}$ descends from $\mathfrak{K}$ to $\delta$. So we have a
connection of $L$ denoted by $\nabla^{L}$ and a partial connection on $\delta$
denoted by $\nabla^{\delta}$. In conclusion we have a partial connection on
$L\otimes\delta$.

######  Proposition 5.3.

If $\tau$ is an $(n,0)$–form on $M$, then for each point $m$ in $M$ the
$2n$–form:

$(-1)^{\frac{n(n-1)}{2}}(-i)^{n}\overline{\tau}\wedge\tau,$ (5.3)

is a non negative multiple of the volume form
$dV_{M}\,=\,\frac{\omega^{n}}{n!}$. There is an unique hermitian structure on
$\delta$ such that for each section $s$ of $\delta$:

$h(s,s)\,=\,\|s\|^{2}\,=\,\sqrt{\frac{(-1)^{\frac{n(n-1)}{2}}(-i)^{n}\overline{(s\otimes
s)}\wedge(s\otimes s)}{2^{n}dV_{M}}}.$ (5.4)

$\mathop{}\\!\mathit{Proof}.$
Let $z_{1},\ldots,z_{n}$ be the holomorphic coordinates on $M$, then:

$\omega^{n}\,=\,n!\det{(ig)}dz_{1}\wedge d\overline{z_{1}}\wedge\ldots\wedge
dz_{n}\wedge d\overline{z_{n}},$ (5.5)

reordering differentials:

$\frac{\omega^{n}}{n!}\,=\,\det{(ig)}(-1)^{\frac{n(n-1)}{2}}d\overline{z_{1}}\wedge\ldots\wedge
d\overline{z_{n}}\wedge\ldots\wedge dz_{1}\wedge\ldots\wedge dz_{n},$ (5.6)

the second member is equivalent to $(\ref{half_f1})$ with
$\overline{\tau}\,=\,d\overline{z_{1}}\wedge\ldots\wedge d\overline{z_{n}}$
and $\tau\,=\,dz_{1}\wedge\ldots\wedge dz_{n}$. $\Box$

Both the hermitian structure on $L$ and on $\delta$ gives in a natural way an
hermitian structure on $L\otimes\delta$.

######  Definition 5.4.

We define the half form Hilbert space $\mathfrak{H}^{L\otimes\delta}$ as the
space of square integrable polarized sections of $L\otimes\delta$.

Before to see how observables are quantizable in this new Hilbert space we
recall some properties of sections of $L\otimes\delta$. First we can decompose
a section $s$ of $L\otimes\delta$ as $s\,=\,s^{L}\otimes s^{\delta}$. For
$s^{\delta}$ we have that:

$L_{\xi}(s^{\delta}\otimes s^{\delta})\,=\,s^{\delta}\otimes
L_{\xi}s^{\delta}+L_{\xi}s^{\delta}\otimes s^{\delta},\ \
\nabla_{\xi}(s^{\delta}\otimes
s^{\delta})\,=\,2\nabla_{\xi}(s^{\delta})\otimes s^{\delta},$ (5.7)

where $\xi$ is a vector field that preserve $\overline{D}$ and
$s^{\delta}\otimes
s^{\delta}\,=\,\left(s^{\delta}\right)^{2}\,=\,\tau\in\mathfrak{K}$.

We have that polarized sections $s\,=\,s^{L}\otimes s^{\delta}$ of
$L\otimes\delta$ are such that:

$\nabla_{\xi}s\,=\,\nabla_{\xi}s^{L}\otimes
s^{\delta}+s^{L}\otimes\nabla_{\xi}s^{\delta}\,=\,0,\ \
\forall\xi\in\overline{D}.$ (5.8)

Now if $f$ is an observable on $M$ such that $\xi_{f}$ preserves
$\overline{D}$ then the quantum operator is given by:

$Q(f)\,=\,\left(Q_{\text{preq}}s^{L}\right)\otimes s^{\delta}-i\hbar
s^{L}\otimes L_{X_{f}}(s^{\delta}),$ (5.9)

where $Q_{\text{preq}}$ is the prequantum operator and $s\,=\,s^{L}\otimes
s^{\delta}$ is a section of $L\otimes\delta$. Note that:

$\xi_{f}\,=\,2i(\partial_{\overline{z}}f\partial_{z}-\partial_{z}f\partial_{\overline{z}}),$
(5.10)

then

$L_{\xi_{f}}s^{\delta}\,=\,L_{\xi_{f}}\sqrt{\tau}\,=\,\frac{1}{2}\tau^{-\frac{1}{2}}L_{\xi_{f}}\sqrt{\tau}\,=\,\frac{1}{2}\tau^{-\frac{1}{2}}L_{2i\partial_{\overline{z}}f\partial_{z}}\tau\,=\,\frac{1}{2}\left(2i\partial_{z}\partial_{\overline{z}}f\right)\sqrt{\tau}.$

If we look for
$[\xi_{f},\partial_{\overline{z}}]\,=\,2i\partial_{\overline{z}}\partial_{z}f\partial_{\overline{z}}$
that is exactly the previous result. Then:

$Q(f)(s^{L}\otimes\sqrt{\tau})\,=\,\left(Q_{\text{preq}}s^{L}\right)\otimes\sqrt{\tau}-i\hbar
s^{L}\otimes\frac{1}{2}\left(2i\partial_{z}\partial_{\overline{z}}f\right)\sqrt{\tau}.$
(5.11)

######  Example 5.5.

In the case of $M=\mathbb{C}$, with hamiltonian
$\mathcal{H}=\frac{1}{2}z\overline{z}$, the prequantum operator was:

$Q_{\text{preq}}\,=\,\hbar z\partial_{z}$

with eigenfunctions (sections) $\psi_{n}\,=\,z^{n}$ and eigenvalues $\hbar n$.
These value of the energy are not the desired values. Let us consider a
polarization $D$ spanned by $\partial_{\overline{z}}$. Then

$2i\partial_{z}\partial_{\overline{z}}\mathcal{H}\,=\,i,$

and the corrected quantization gives:

$Q(f)\,=\,\hbar z\partial_{\overline{z}}+\frac{\hbar}{2}.$

This operators gives $\hbar\left(n+\frac{1}{2}\right)$, the correct
eigenvalues.

Details are in [33] and [7].

###  5.2. The BKS pairing

Let us consider two Hilbert spaces $\mathfrak{H}_{1}$ and $\mathfrak{H}_{2}$
given by two different polarizations $D_{1}$ and $D_{2}$ on the same line
bundle $L\rightarrow M$. The two Hilbert spaces are both part of the
prequantum Hilbert space $\mathfrak{H}$. Let
$P:\mathfrak{H}_{2}\rightarrow\mathfrak{H}_{1}$ be the orthogonal projection.
Let $\langle\cdot,\cdot\rangle$ be the inner product on $\mathfrak{H}_{1}$, we
define the pairing
$\langle\langle\cdot,\cdot\rangle\rangle:\mathfrak{H}_{1}\times\mathfrak{H}_{2}\rightarrow\mathbb{C}$
by:

$\langle\langle s_{1},s_{2}\rangle\rangle\,=\,\langle s_{1},Ps_{2}\rangle.$
(5.12)

The map is not unitary but if the flow of an observable $f\in C^{\infty}(M)$
preserves both polarizations then:

$\langle\langle Q(f)s_{1},s_{2}\rangle\rangle\,=\,\langle\langle
s_{1},Q(f)s_{2}\rangle\rangle.$ (5.13)

Note that when $D_{1}$ and $D_{2}$ are determinated by two complex structures
on the symplectic space then a multiple of the projector is in fact unitary.

###  5.3. The BKS pairing on the cotangent bundle, the time evolution and the
Schrödinger equation

In this subsection we enter in details examining the case of cotangent bundles
$M=T^{\vee}C$, for some configuration space $C$. In this case $M$ is a model
for classical mechanics, a canonical transformation $\rho$ preserves the
symplectic structure $\omega$. We can consider an Hamiltonian vector field
$X_{f}$ that generates a canonical flow $\rho_{t}$. This flow induces a time
evolution of observable via $f_{t}(m)=f(\rho_{t}(m))$ (pull-back for every
$m\in M$). Now this time evolution is lifted, through $Q_{\text{preq}}(f)$, at
the level of sections of an hermitian line bundle $L\rightarrow M$. In
particular the lifting flow is:

$\hat{\rho}_{t}s(m)\,=\,e^{-\frac{1}{\hbar}\int_{0}^{t}\mathcal{L}ds}s(\rho_{t}(m)),$
(5.14)

where $\mathcal{L}\,=\,X_{f}\,\lrcorner\,\theta-f$ (details are in [6]) and
this lift is unitary. Given an observable $f\in C^{\infty}(M)$ that preserve a
polarization $D$ then we describe the unitary time evolution by:

$Q_{\text{preq}}(f)s_{t}\,=\,\frac{d}{dt^{\prime}}(\hat{\rho}_{t^{\prime}-t}s_{t})|_{t^{\prime}=t}\,=\,-i\hbar\dot{s}_{t}.$
(5.15)

Considering the half–form construction, that is the line bundle
$L\otimes\delta$, we use a pull-back in order to define an evolution in $t$:

$\rho_{t}^{L\otimes\delta}s_{t}^{L\otimes\delta}\,=\,(\hat{\rho}_{t}s^{L})(s^{\delta}(\rho_{t})).$
(5.16)

The derivative at $t=0$ reproduce the quantum operator corrected. Now let us
assume on $M$ two transverse real polarizations $D_{1}$ and $D_{2}$. Then
$TM=D_{1}\oplus D_{2}$. We can write $M$ as the product of two configuration
spaces $C\times C^{\prime}$, one for position $q$ and the second as the
momentum space $q^{\prime}$. Then we have that:

$\omega=\partial_{q}\partial_{q^{\prime}}\mathcal{S}dq\wedge dq^{\prime},$
(5.17)

with $\mathcal{S}$ the generating function. We can define a pairing between
$\delta_{D_{1}}$ and $\delta_{D_{2}}$:

$(\sqrt{\tau_{1}},\sqrt{\tau_{2}})\,=\,\sqrt{\tau_{2}\wedge\overline{\tau_{1}}},$
(5.18)

where $\tau_{1},\tau_{2}$ are $n$-form of the determinant bundle. Then the BKS
pairing between sections of $\mathfrak{H}_{1}$ and $\mathfrak{H}_{2}$ is:

$\langle\langle
s_{1}^{L\otimes\delta_{D_{1}}},s_{2}^{L\otimes\delta_{D_{2}}}\rangle\rangle\,=\,\int_{M}(s_{1}^{L},s_{2}^{L})(\sqrt{\tau_{1}},\sqrt{\tau_{2}})dV_{M}.$
(5.19)

The Liouville measure is
$dV_{M}=\frac{\omega^{n}}{(2\pi\hbar)^{n}}=\frac{\det{\partial_{q}\partial_{q^{\prime}}\mathcal{S}}}{(2\pi\hbar)^{n}}(dq\wedge
dq^{\prime})^{n}$ and a computation shows that
$(\sqrt{\tau_{1}},\sqrt{\tau_{2}})dV_{M}\,=\,\frac{\sqrt{\det{\partial_{q}\partial_{q^{\prime}}\mathcal{S}}}}{(2\pi\hbar)^{n/2}}d^{n}q\wedge
d^{n}q^{\prime}$.

Let $\mathcal{S}=p\cdot q$ be the canonical generator, then the determinant
will be the identity, $q^{\prime}=p$ corresponds to the base space of $D_{2}$
whose leaves are surfaces of constant $q^{\prime}=p$ and, because the
polarization is transverse to $D_{1}$ the surfaces of constant $q$ are leaves
of $D_{1}$. At the end the pairing is given by:

$\langle\langle
s_{1}^{L\otimes\delta_{D_{1}}},s_{2}^{L\otimes\delta_{D_{2}}}\rangle\rangle\,=\,\frac{1}{(2\pi\hbar)^{n/2}}\int_{C}\overline{\psi}(q)\int_{C^{\prime}}\psi(p)e^{\frac{ip\cdot
q}{\hbar}}d^{n}pd^{n}q,$ (5.20)

where $\phi(q)$ and $\psi(p)e^{\frac{i}{\hbar}p\cdot q}$ are complex wave
functions. The projector corresponds effectively to the Fourier transform:

$Ps_{2}\,=\,\frac{1}{(2\pi\hbar)^{n/2}}\int_{C^{\prime}}\psi(p)e^{\frac{ip\cdot
q}{\hbar}}d^{n}p,$ (5.21)

with value in $\mathfrak{H}_{D_{1}}$.

###  5.4. Reconstruction of Schrödinger equation on the cotangent bundle

Let $s^{L\otimes\delta}=s^{L}\otimes\sqrt{\tau}$ be a $D_{1}$–polarized
section, $X_{f}$ an hamiltonian vector field and
$\hat{\rho}_{t}s^{L\otimes\delta}=\hat{\rho}_{t}s^{L}\rho_{t}^{*}\sqrt{\tau}$
the lift of the canonical flow in $D_{2}=\rho^{*}_{t}D_{1}$. Then:

$\langle\dot{s}^{L\otimes\delta},w^{L\otimes\delta}\rangle\,=\,-\frac{d}{dt^{\prime}}\left.\langle\langle\hat{\rho}_{t^{\prime}-t}s^{L\otimes\delta}_{t},w\rangle\rangle\right|_{t^{\prime}=0},$
(5.22)

this is for every $w\in\mathfrak{H}(D_{1})$ where $w=w^{L}\sqrt{\sigma}$ and
for $s^{L\otimes\delta}_{t}$ we assume equal to $s^{L\otimes\delta}_{0}$. We
have that:

$\langle\dot{s}^{L\otimes\delta},w^{L\otimes\delta}\rangle\,=\,-\lim_{h\rightarrow
0}\frac{\langle P\hat{\rho}_{h-t}s^{L\otimes\delta}_{t},w\rangle-\langle
P\hat{\rho}_{-t}s^{L\otimes\delta}_{t},w\rangle}{h}.$ (5.23)

We must to evaluate the following expression:

$\langle\langle\hat{\rho}_{t}s^{L\otimes\delta},w^{L\otimes\delta}\rangle\rangle\,=\,\int_{M}\overline{\psi(\rho_{t}(m))}w(q)e^{\frac{i}{\hbar}\int_{0}^{t}\mathcal{L}(\gamma(s))ds}(\rho_{t}^{*}\sqrt{\sigma},\sqrt{\sigma})dV_{M}.$
(5.24)

In our situation we consider the free particle on $M=\mathbb{R}^{2n}$, with
$\mathcal{L}=\mathcal{H}=\frac{p^{2}}{2m}$, $\rho_{t}$ the uniform motion on
flat space (curvature term is $0$) with equation: $q(t)=q+vt=q+\frac{p}{m}t$
and $p(t)=p$. We observe that:
$\int_{0}^{t}\mathcal{L}(\gamma(s))ds=\mathcal{H}t=\frac{p^{2}t}{2m}$. Let
$x:M\rightarrow\mathbb{R}^{2n}$ be the coordinates with
$x(\rho_{t}(m))=(q(t),p(t))$ and $dq^{n}=dq_{1}\wedge\ldots\wedge dq_{n}$.
Then:

$(\rho_{t}^{*}\sqrt{\sigma},\sqrt{\sigma})^{2}dV_{M}\,=\,d(x\circ\rho_{t})\,=\,d\left(q+\frac{t}{m}p\right)^{n}\wedge
dq^{n}\left(\frac{1}{(2\pi\hbar)^{n}}\right)=\left(\frac{t}{2\pi
m\hbar}\right)^{n}dp^{n}\wedge dq^{n}.$

That is:

$(\rho_{t}^{*}\sqrt{\sigma},\sqrt{\sigma})dV_{M}\,=\,\left(\frac{t}{2\pi
m\hbar}\right)^{\frac{n}{2}}dp^{n}\wedge dq^{n}.$

After these refinements we must to evaluate the integrals:

$\langle\langle\hat{\rho}_{t}s^{L\otimes\delta},w^{L\otimes\delta}\rangle\rangle\,=\,\int_{\mathbb{R}^{2n}}\overline{\psi(q(t),p(t))}w(q)e^{\frac{i}{\hbar}\frac{p^{2}t}{2m}}\left(\frac{t}{2\pi
m\hbar}\right)^{\frac{n}{2}}dp^{n}\wedge dq^{n}.$ (5.25)

In order to evaluate the integral we start as in [6] to expand the function
$\psi$ around the point $q$.

$\begin{multlined}\psi(q(t),p(t))\,=\,\psi\left(q+\frac{p}{m}t,p\right)\,=\,\psi(q)+\\\
+\frac{t}{m}\sum_{j}p_{j}\partial_{q_{j}}\psi(q)+\frac{t^{2}}{2m^{2}}\sum_{j,l}p_{j}p_{l}\partial_{q_{j}}\partial_{q_{l}}\psi(q)+\mathcal{O}(t^{3}).\end{multlined}\psi(q(t),p(t))\,=\,\psi\left(q+\frac{p}{m}t,p\right)\,=\,\psi(q)+\\\
+\frac{t}{m}\sum_{j}p_{j}\partial_{q_{j}}\psi(q)+\frac{t^{2}}{2m^{2}}\sum_{j,l}p_{j}p_{l}\partial_{q_{j}}\partial_{q_{l}}\psi(q)+\mathcal{O}(t^{3}).$
(5.26)

The contribution given by the second term in the expansion to the initial
integral is $0$ due by the parity of the integrand. Substituting inside the
integral:

$\int_{\mathbb{R}^{n}}w(q)\left(\frac{t}{2\pi
m\hbar}\right)^{\frac{n}{2}}\int_{\mathbb{R}^{n}}\left[\overline{\psi}(q)+\frac{t^{2}}{2m^{2}}\sum_{j,l}p_{j}p_{l}\partial_{q_{j}}\partial_{q_{l}}\overline{\psi}(q)\right]e^{\frac{i}{\hbar}\frac{p^{2}t}{2m}}dp^{n}dq^{n}+\mathcal{O}(t^{3}).$
(5.27)

Now we can evaluate the two integral respect the momentum variable $p$. The
first is:

$\int_{\mathbb{R}^{n}}e^{\frac{i}{2}\frac{t}{m\hbar}p^{2}}dp^{n}\,=\,\frac{(2\pi)^{\frac{n}{2}}}{\left(\frac{t}{m\hbar}\right)^{\frac{n}{2}}}e^{ic\frac{\pi}{4}},$
(5.28)

where $e^{ic\frac{\pi}{4}}$ is a phase that depends by the metaplectic
correction. This integral as been estimated using the stationary phase method
on multidimensional Fresnel integrals (the reference is [34]). The second
integral to estimate is the following:

$\int_{\mathbb{R}^{n}}p_{j}p_{l}e^{\frac{i}{2}\frac{t}{m\hbar}p^{2}}dp^{n}\,=\,\left(\int_{\mathbb{R}}p_{j}^{2}e^{\frac{i}{2}\frac{t}{m\hbar}p_{j}^{2}}dp_{j}\right)\cdot\left(\int_{\mathbb{R}^{n-1}}e^{\frac{i}{2}\frac{t}{m\hbar}\widehat{p}^{2}}d\widehat{p}^{n-1}\right),$
(5.29)

where the only terms different from zero inside the integral are when $j=l$,
so we can consider this integral as one dimensional integral with amplitude
$p_{j}^{2}$ (evaluating using the result of [9]) and a second $n-1$ integral
as before where $\widehat{p}$ is obtained by $p$ excluding the $j$ coordinate.
We find that:

$\int_{\mathbb{R}^{n}}p_{j}p_{l}e^{\frac{i}{2}\frac{t}{m\hbar}p^{2}}dp^{n}\,=\,e^{i\frac{\pi}{4}}\left(\frac{2m\hbar}{t}\right)^{\frac{1}{2}}\left[\frac{e^{i\frac{\pi}{2}}}{2}\left(\frac{2m\hbar}{t}\right)\frac{\sqrt{\pi}}{2}\cdot
2\right]\cdot\frac{(2\pi)^{\frac{n-1}{2}}}{\left(\frac{t}{m\hbar}\right)^{\frac{n-1}{2}}}e^{i\widetilde{c}\frac{\pi}{4}},$
(5.30)

where $c=\widetilde{c}+1$. After calculations and semplifications the integral
$(\ref{dynamics5})$ assume the following form:

$\int_{\mathbb{R}^{n}}w(q)e^{ic\frac{\pi}{4}}\left[\overline{\psi}(q)+\frac{it\hbar}{2m}\nabla\overline{\psi}(q)\right]dq^{n}+\mathcal{O}(t^{2}),$
(5.31)

that we can write as:

$\int_{\mathbb{R}^{n}}w(q)e^{ic\frac{\pi}{4}}\left[\overline{\psi}(q)-\frac{it}{\hbar}\mathcal{H}\overline{\psi}(q)\right]dq^{n}+\mathcal{O}(t^{2}),$
(5.32)

with $\mathcal{H}\,=\,-\frac{\hbar^{2}}{2m}\nabla^{2}$. Now after a
differentiation respect to $t$ at $t=0$ and taking the complex conjugate, we
find for every section of $L\otimes\delta$:

$i\hbar\frac{\partial}{\partial t}\psi(q)\,=\,\mathcal{H}\psi(q),$ (5.33)

the Schrödinger equation.

###  5.5. The Klein–Gordon equation and the cotangent bundle

Let $Y$ be a space–time manifold. Assuming that $Y$ is an orientable manifold,
we have that $T^{\vee}Y$ is a symplectic manifold with symplectic form:

$\omega_{e}\,=\,d\theta_{Y}+e\pi^{*}F,$ (5.34)

where $F$ is the electromagnetic field (a closed 2-form on $Y$), $\theta_{Y}$
is the symplectic potential, $e$ is the charge of a particle in an external
electromagnetic field and $\pi:T^{\vee}Y\rightarrow Y$.

The metric $g$ is the Lorentz metric with signature $(-,-,-,+)$. Let us
consider the function $N(x)\,=\,g(x,x)$ defined on $T^{\vee}Y$. The square
root of $N$ represents the mass of a particle in the classical state $x$. Let
us consider the complete hamiltonian vector field $\xi_{N}$ and the flux
$\phi_{t}^{N}$ generated by $\xi_{N}$.

In this case the prequantum condition reads as:

$\int_{\Sigma}eF\in h\mathbb{Z},$

for every compact oriented 2-surface $\Sigma$ on $Y$ with empty boundary.

We can consider a vertical polarization $D^{\mathbb{C}}$ on $T^{\vee}Y$ and
the associated Hilbert space $\mathfrak{H}_{D^{\mathbb{C}}}$ that is the
$L^{2}(Y,g)$.

Under the assumption that $D^{\mathbb{C}}$ and $T\phi_{t}^{N}(D^{\mathbb{C}})$
are transverse, it is possible to evaluate $Q_{\text{preq}}(N)s_{t}$. The
method is similar as before and is showed in [35] in the chapter 10. Let us
denote with $L_{e}$ the corrected quantum line bundle, then the BKS kernel
$BKS_{t}$ is given by:

$BKS_{t}(s_{1},s_{2})\,=\,\int_{T^{\vee}Y}\langle
s_{1},\phi_{t}^{N}s_{2}\rangle\left|\omega_{e}^{4}\right|,$ (5.35)

for local sections $s_{1}\,=\,\psi_{1}s_{1}^{L}\otimes
s_{1}^{\delta},s_{2}\,=\,\psi_{2}s_{2}^{L}\otimes s_{2}^{\delta}$.

What the author find at the end of calculations (in [35]), is the following
expression for the quantum operator:

$Q_{\text{preq}}(N)s\,=\,-\hbar^{2}\left[\Box-\frac{1}{6}R\right]s,$ (5.36)

where $\Box$ is the d’Alembert operator and $R$ is the scalar curvature of the
metric $g$. The operator $N$ is also called the “mass-squared operator” and,
the equation:

$-\Box\psi+\frac{1}{6}R\psi\,=\,0,$

is the Klein-Gordon equation and $\psi$ the wave function.

In [36] has been discovered that the critical metrics of the expectation value
of $Q(N)$ have to satisfy Einstein’s equation for suitable energy–momentum
tensor.

###  5.6. Considerations on the BKS pairing

If the first problem of the prequantization procedure was to have the
dependence on half variables by the wave functions (sections of the line
bundle), solved by introducing the concept of polarizations, the second
problem was the treatment of second order, or higher operators, as the free
particle and harmonic oscillator. This second problem has been solved using
the BKS construction with the half–forms bundle and the lift of the flow
$\rho_{t}$ that describes the dynamic of the system. As focused also by [6]
this seems the “exact” choice of the quantum operators, with a suitable
metaplectic correction and considering the Hamiltonian dynamics (so the
presence of symmetries). The final form of our observables $f\in
C^{\infty}(M)$, in the particular case of cotangent bundles for section
$s\sqrt{\tau}$ in the corrected bundle, seems to be:

$Q_{\text{preq}}(f)s_{t}\,=\,-i\hbar\frac{\partial}{\partial
t}P\hat{\rho}_{t}s|_{t=0},$ (5.37)

where $P$ is the usual Fourier transform. As observed always in [6]
generalization on operators of order bigger then $2$ is a problem. Generally
the Dirac axiom is not true, the time evolution can be not unitary, operators
are not always self–adjoint and operators can be not linear. These problems
have been showed not only in [6] but also in the deep study of [37].

###  5.7. The BKS pairing for one complex and one real polarization: the
Segal–Bargmann transform

Let us consider for $M$ a Kähler manifold and two polarizations: one $D_{1}$ a
real polarization and $D_{2}$ a Kähler polarization. Let us assume
$M=\mathbb{C}$, as in the example of the harmonic oscillator. We have that
$D_{2}$ is spanned by $\partial_{\overline{z}}$ and $D_{1}$ corresponds to
$q=$ constant. Let us denote the two Hilbert spaces by $\mathfrak{H}(D_{1})$
and $\mathfrak{H}(D_{2})$ and by $z=p+iq$ and $\overline{z}=p-iq$ the
respective coordinates. The following symplectic potential:

$\theta\,=\,\frac{i}{2}zd\overline{z}\,=\,\frac{pdq-qdp}{2},$ (5.38)

is associated to the Kähler form: $\omega=d\theta=dp\wedge dq$. The line
bundle $L$ over $M$ is trivial and the space $\mathfrak{H}(D_{2})$ corresponds
to the Fock space $\mathfrak{F}$ of holomorphic function with scalar product:

$(f,g)\,=\,\frac{1}{2\pi}\int_{\mathbb{C}}f(z)\overline{g(z)}e^{-\frac{|z|^{2}}{2}}dzd\overline{z}.$
(5.39)

A section $s$ of the line bundle has squared norm
$\|s\|^{2}\,=\,e^{-\frac{|z|^{2}}{2}}$ and $s$ have the following form:

$\phi(z)e^{-\frac{z\cdot\overline{z}}{4}},$ (5.40)

where $\phi(z)$ is an holomorphic function. A section for the real
polarization $D_{1}$ is the complex function:

$\psi(z)e^{-i\frac{pq}{2}}.$ (5.41)

Let us consider the half form bundle $\delta$ in order to ensure the
metaplectic correction. We can define
$P:\mathfrak{H}(D_{2})\rightarrow\mathfrak{H}(D_{1})$ and
$P^{\prime}:\mathfrak{H}(D_{1})\rightarrow\mathfrak{H}(D_{2})$ the projections
given by the BKS pairing. The pairing between the two hilbert spaces is given
by the following formula:

$\langle\langle\psi,\phi\rangle\rangle\,=\,\frac{C}{2\pi}\int_{\mathbb{R}^{2}}\overline{\psi}(q)\phi(z)e^{\frac{2ipq-p^{2}-q^{2}}{4}}dpdq,$
(5.42)

where $C$ is a constant we specify at the end. Now we can give and explicit
form for $P$ and $P^{\prime}$ using the reproducing property of holomorphic
function in the second hilbert space:

$\phi(z)\,=\,\frac{1}{2\pi}\int_{\mathbb{C}}\phi(w)e^{\frac{\overline{w}z-w\overline{w}}{2}}dwd\overline{w},$
(5.43)

where $w=x+iy$.

Now inserting this expression inside the BKS pairing
$(\ref{pairing__CRpolarization_ex})$, we can integrate respect $p$ on
$\mathbb{R}$ the expression in order to have:

$\langle\langle\psi,\phi\rangle\rangle\,=\,\frac{1}{2\pi}\int_{\mathbb{C}\times\mathbb{R}}\overline{\psi}(q)\phi(w)\overline{K}(q,w)e^{\frac{-w\overline{w}}{4}}dqdwd\overline{w},$
(5.44)

with

$K(q,w)\,=\,\frac{\overline{C}}{\sqrt{\pi}}e^{-\frac{q^{2}}{2}-iqw+\frac{w^{2}}{4}}.$
(5.45)

Then the expression for the two projections is:

$P(\phi)(q)\,=\,\frac{1}{2\pi}\int_{\mathbb{C}}\phi(w)\overline{K}(q,w)e^{\frac{-w\overline{w}}{2}}dwd\overline{w},$
(5.46)

and

$P^{\prime}(\psi)(w)\,=\,\frac{1}{2\pi}\int_{\mathbb{R}}\psi(q)\overline{K}(q,w)dq.$
(5.47)

We observe that functions $z^{m}$ in $\mathfrak{H}(D_{2})$ are mapped in
eigenfunctions of the harmonic oscillator. This is the Segal–Bargmann
transform and the ground state $1$ is mapped to:

$P(\phi)(q)\,=\,\frac{C}{\sqrt{\pi}}e^{-\frac{q^{2}}{2}},$ (5.48)

with $C=\pi^{\frac{1}{4}}$.

###  5.8. The BKS pairing between two complex polarizations: the Bogoliubov
transformation

The last case is the case of two complex polarizations $D_{1}$ and $D_{2}$
determined respectively by two complex structures $J_{1},J_{2}$. In this case
a multiple of the projector operator is unitary and the existence is guarantee
by the Stone Von Neumann theorem (see [7]). In this brief summary we consider
the case in one dimension with $M=\mathbb{C}$ and without metaplectic
correction. To the polarization $D_{1}$ we have the Hilbert space
$\mathfrak{H}(D_{1})$ that is the Fock space with holomorphic sections:

$\phi(z)\,=\,\psi(z)e^{-\frac{1}{4}z\overline{z}},$ (5.49)

where $\psi(z)$ is holomorphic respect $J_{1}$ and $z=p+iq$. The inner product
is:

$(\phi(z),\phi^{\prime}(z))\,=\,\frac{1}{2\pi}\int_{\mathbb{C}}\psi(z)\overline{\psi^{\prime}}(z)e^{-\frac{1}{2}z\overline{z}}dzd\overline{z}.$
(5.50)

Now let $J_{2}$ be a second complex structure that determines $D_{2}$ and a
second hilbert space $\mathfrak{H}(D_{2})$ (always a Fock space) with
sections:

$\phi^{\prime}(w)\,=\,\psi^{\prime}(w)e^{-\frac{1}{4}w\overline{w}},$ (5.51)

where $\psi^{\prime}(z)$ is holomorphic respect $J_{2}$ (the inner product is
the same of the previous case). The BKS pairing is given by:

$\langle\langle\phi(z),\phi^{\prime}(z)\rangle\rangle\,=\,\frac{1}{2\pi}\int_{\mathbb{C}^{2}}\psi(z)\overline{\psi^{\prime}}(w)e^{-\frac{1}{4}\left(z\overline{z}+w\overline{w}\right)}dzd\overline{z}dwd\overline{w}.$
(5.52)

The pairing determines a projection
$P:\mathfrak{H}(D_{2})\rightarrow\mathfrak{H}(D_{1})$ given by:

$P\psi^{\prime}(w)\,=\,\frac{1}{2\pi}\int_{\mathbb{C}}\overline{\psi}_{z}\psi^{\prime}(w)dwd\overline{w},$
(5.53)

where
$\psi_{z}(w)\,=\,e^{\frac{1}{2}(2\omega(z,(J_{1}+i)w)-\omega(z,J_{1}z)-\omega(w,J_{1}w))}$.
The projection of $\phi_{0}^{\prime}(w)=1\cdot e^{\frac{1}{4}w\overline{w}}$
(the ground state in $\mathfrak{H}(D_{1})$) is:

$\phi_{0}(z)\,=\,\left[\det{\frac{1}{2}(J_{1}+J_{2})}\right]^{-\frac{1}{2}}e^{\frac{\lambda(z)}{4}-\frac{z\overline{z}}{4}},$
(5.54)

where $\lambda(z)=2\omega(z,J_{1}Lz)-2i\omega(z,Lz)$ and
$L=(J_{1}+J_{2})^{-1}\cdot(J_{1}-J_{2})$. A good reference is [38] where also
the infinite dimensional case is explained. The same topic is present in [7].

##  6\. The Bohr–Sommerfeld (sub)manifolds

Let us consider a lagrangian submanifold $\Lambda$ of a cotangent space
$T^{\vee}Q$. It is possible to interpret the quantity $e^{i\frac{z}{\hbar}}$
as a covariantly constant section $s$ of the line bundle $L_{\Lambda}$. Now
there is a natural definition of square root of a volume form $\tau$ on
$\Lambda$ replacing $\tau$ by $\sqrt{\tau}$. The section $\sqrt{\tau}$ is a
section of $\sqrt{\Delta_{\Lambda}}$ where $\Delta_{\Lambda}$ is the line
bundle of the complex volume forms on $\Lambda$. We can consider the triple
$\left(\Lambda,s,\sqrt{\tau}\right)$ and to work with the machinery of
geometric quantization. Following [8], we can study the asymptotic expansion
of the squared norm of an “isotropic state” in particular conditions. An
isotropic state is a family of sections of an Hilbert space associated to the
lagrangian submanifold. Another problem is the study of finding a simultaneous
WKB eigenstate of a set of classical observables
$\mathcal{H}_{1},\ldots,\mathcal{H}_{n}$ in involution.

Assuming that $\Lambda$ must be of the form $h_{i}=E_{i}$ with $E_{i}$ that
are constant for every $i=1,\ldots,n$. We can consider the volume form $\tau$
of $\Lambda$. Assuming for simplicity that $\Lambda$ is simply connected, the
wave functions are well defined if $s\sqrt{\tau}$ is a single valued global
section of $L_{\Lambda}\otimes\sqrt{\Delta_{\Lambda}}$. The existences of this
section depends by the validity of the Bohr-Sommerfeld condition PC5. Assume
that $L=M\times\mathbb{C}$ and that $\theta$ is the symplectic potential, then
the corrected quantization condition is that:

$\left(e^{\frac{2i}{\hbar}\oint_{\gamma}\theta}\right)\tau$ (6.1)

a section of $\Delta_{L}$, admits a single values square root with values in
$\sqrt{\Delta_{L}}$. Here $\gamma$ is any loops in $\Lambda_{m}$ ($m\in M$).
There are other formulations of PC5, for example in [39] it is given by:

$\oint_{\gamma}\theta\,=\,2\pi\hbar(n+d),$ (6.2)

where $n\in\mathbb{Z}$, as in the first quantization condition PC1, with the
difference that there is the quantity $d$ associated to the holonomy from the
flat connection of the bundle of half-forms. Authors who worked in the setting
of geometric quantization of Bohr–Sommerfeld (sub)manifolds are [40],[21] and
[41] (where the last two concentrated the attention on the equivariant case).

##  7\. Conclusion

The real contribution of this paper consists of a new way to derive the
Schödinger equation, as in [6], with the difference we used a result of
Albeverio and Mazzucchi ([9]) on asymptotic expansion of Fresnel integrals
instead the heat kernel. All possible expressions of quantization conditions
have been examined in detail. These notes provide a complete, detailed summary
on the state of the art on the geometric quantization showing possible
connection between mathematics and physics.

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# An alternative proof of Gödel’s first incompleteness theorem

Zuhair Al-Johar

(August 2023)

##  1 Introduction

This proof of Gödel’s first incompleteness theorem doesn’t require
$\omega$-consistency, nor does it refer to codes of negated sentences as in
Rosser’s. It begins from where Gödel’s usual proof ends, and stalks it till it
ends proving it.

##  2 Proof Sketch

The proof is by negation. We assume that theory $T$ is first order,
consistent, complete, and yet fulfills Gödel’s criteria of representing all
computable functions. So generally we take $T$ to extend either $\sf PA$ or
$\sf Q$; then we show that these assumptions are contradictory. As a notation
$\operatorname{Proof}_{T}(x,y)$ is the usual Gödel provability, also
$\ulcorner.\urcorner$ refers to Gödel’s coding, and ${}^{o}n$ refers to
$n$-many repeated application of the successor operation over zero. The
negation function $``\operatorname{neg}"$ is defined after Rosser’s. Those
notations are present in [4] and [5]. We use $\vdash$ to stand for provability
in the usual metatheoretic sense. Generally when ”we have”, or provability
steps are written then they refer to metatheoretic provability in $T$.

Per those notations, we know from Gödel’s work that for any standard sentence
$s$ if $T\vdash s$, then there is some standard natural number $n$ such that
$\operatorname{Proof}_{T}(n,^{o}\ulcorner s\urcorner)$ and vise verse. Also if
we have $\operatorname{Proof}_{T}(n,^{o}\ulcorner s\urcorner)$ and $n$ is
standard, then $s$ is a standard sentence.

let $s$ be a standard sentence in the language of $T$. Let $E_{s}(v)$, called
extension of $s$, be the sentence form:

$\forall x\forall y\,(\operatorname{Proof}_{T}(x,v)\land[y=\min
z:\operatorname{Proof}_{T}(z,^{o}\ulcorner s\urcorner)]\to x\geq y)$

Now, let ”$D(E_{s}(v))$” stand for the diagonal sentence of the sentence form
$E_{s}(v)$ produced syntactically from $E_{s}(v)$ in an exact manner along
lines present in [5] (see section 4)

Accordingly; if $s$ is a standard sentence in the language of $T$, then we
have:

$\vdash_{T}\ D(E_{s}(v))\iff\forall x\forall
y\,(\operatorname{Proof}_{T}(x,^{o}\ulcorner D(E_{s}(v))\urcorner)\land[y=\min
z:\operatorname{Proof}_{T}(z,^{o}\ulcorner s\urcorner)]\to x\geq y)$

Now, Let $\operatorname{Dgn}(x,y)$ be an arithmetical formula such that if
there is a standard sentence $s$ and $y=\,^{o}\ulcorner s\urcorner$, then
$x=\,^{o}\ulcorner D(E_{s}(v))\urcorner$ [The exact definition is given at
section 4]

Define:

$\displaystyle remote(y)\iff\exists x\exists k\exists z:$
$\displaystyle\operatorname{Dgn}(x,y)\land$ $\displaystyle k=\min
n:\operatorname{Proof}_{T}(n,y)\land$ $\displaystyle z=\min
m:\operatorname{Proof}_{T}(m,x)\land$ $\displaystyle z<k$

So, from the above definitions if $T\vdash\neg D(E_{s}(v))$, then $T\vdash
remote(^{o}\ulcorner s\urcorner)$, while if $T\vdash D(E_{s}(v))$, then
$T\vdash\neg remote(^{o}\ulcorner s\urcorner)$.

Lemma 0: for every standard sentence $s$ in the language of $T$, for each
concrete natural $i$, we have:

$\vdash_{T}\ s\to\forall
n(\operatorname{Proof}_{T}(n,\operatorname{neg}(^{o}\ulcorner s\urcorner))\to
n\neq\ ^{o}i)$

In English: if theory $T$ proves sentence $s$ then it proves every internal
proof of the negation of $s$ to be non-standard.

Proof: if there is an internal standard proof of the negation of $s$, then per
Gödel’s work we’d have $T\vdash\neg s$, contradicting $T\vdash s$

Lemma I: for $n=0,1,2,..$

$\vdash_{T}\forall x[remote(x)\to\forall y(\operatorname{Proof}_{T}(y,x)\to
y\neq\ ^{o}n)]$

In English: all codes of proofs of a remote sentence are non-standard.

Proof: Suppose we have a proof of a remote sentence $s$ that is coded by a
standard natural (that is $x=^{o}\ulcorner s\urcorner$ and we have a standard
natural $y$ such that we have $\operatorname{Proof}_{T}(y,x)$), then per
Gödel’s we’d have $s$ is a standard sentence, so the diagonal sentence
$D(E_{s}(v))$ (see section 4) of its extension is a standard sentence and we
must have $T\vdash D(E_{s}(v))$; otherwise we’ll be having $T\vdash\neg
D(E_{s}(v))$ and so $T$ would prove the existence of a code of a proof of
$D(E_{s}(v))$ that is strictly smaller than the minimal code of a proof of
$s$, and so would be standardly coded, and accordingly we’ll have $T\vdash
D(E_{s}(v))$, a contradiction. So, we must have $T\vdash D(E_{s}(v))$ and so
$\min z:\operatorname{Proof}_{T}(z,^{o}\ulcorner D(E_{s}(v))\urcorner)\geq\min
k:\operatorname{Proof}_{T}(k,^{o}\ulcorner s\urcorner)$, thus $\neg
remote(^{o}\ulcorner s\urcorner)$.

Lemma II: for any standard sentence $s$, if $T\vdash\neg s$ and
$T\vdash\exists n:\operatorname{Proof}_{T}(n,^{o}\ulcorner s\urcorner)$, then
$T\vdash remote(^{o}\ulcorner s\urcorner)$

In English: if $T$ refutes a standard sentence and yet proves the existence of
an internal proof of that sentence, then that sentence is remote.

Proof: Since all codes of proofs of $s$ are non-standard [Lemma 0], then any
proof of the diagonal of the extension of $s$, that is $D(E_{s}(v))$, cannot
be standardly coded, because $D(E_{s}(v))$ is a standard sentence and it is
equivalent in $T$ to having every proof of it being not strictly less than the
minimal proof of $s$, and so non-standard; so we have $T\vdash\neg
D(E_{s}(v))$, thereby proving $remote(^{o}\ulcorner s\urcorner)$

Define:

$x=\mathfrak{K}\iff x=\min k:\exists
y\,(remote(y)\land\operatorname{Proof}_{T}(k,y))$

So, $\mathfrak{K}$ is the minimal code of a proof of a remote sentence.

From the fixed point lemma, take:

$\sigma\iff\forall
x\,(\operatorname{Proof}_{T}(x,^{o}\ulcorner\sigma\urcorner)\to
x\geq\mathfrak{K})$

Informally $\sigma$ is saying ”every proof of this sentence is, bigger than or
equal to, the minimal proof of remote sentences”.

Clearly we have $T\not\vdash\sigma$. So, we must have $T\vdash\neg\sigma$,
then there is a proof of ${}^{o}\ulcorner\sigma\urcorner$ in $T$ that is
strictly smaller than $\mathfrak{K}$. But $\sigma$ is a standard sentence. So
we must have $remote(^{o}\ulcorner\sigma\urcorner)$ [Lemma II]. Contradicting
the definition of $\mathfrak{K}$.

To see that; have $\sigma^{*}$ be the sentence $D(E_{\sigma}(v))$, then:

$\sigma^{*}\iff\forall x\forall
y(\operatorname{Proof}_{T}(x,^{o}\ulcorner\sigma^{*}\urcorner)\land[y=\min
z:\operatorname{Proof}_{T}(z,^{o}\ulcorner\sigma\urcorner)]\to x\geq y)$

Clearly $T\not\vdash\sigma^{*}$, and so we have $T\vdash\neg\sigma^{*}$, then
we’d have $remote(^{o}\ulcorner\sigma\urcorner)$, but from above there is a
proof of ${}^{o}\ulcorner\sigma\urcorner$ in $T$ that is strictly smaller than
$\mathfrak{K}$, a contradiction!

Thus $\sigma$ is undecidable in $T$. QED

The idea is that if $remote(y)$, then every code of a proof of $y$ cannot be
standard, since if otherwise, then we must have the minimal proof of the
diagonal of its extension being $\href
https://mathoverflow.net/a/451243/95347$ than the minimal proof of it. So
$\mathfrak{K}$, the minimal a proof of a remote sentence can be, must be non-
standard. The rest is clear!

The reason why $\operatorname{Dgn}$ is defined in arithmetic is because there
is an exact syntactical way of producing $E_{s}(v)$ from $s$ and of producing
$D(E_{s}(v))$ from $E_{s}(v)$, so there is an arithmetical formula such that
for any standard sentence $s$ we can input its Gödel code and get the Gödel
code of the diagonal of its extension. We know how this behaves exactly for
all standard codes of proofs, and so by definition of $remote$ sentences none
of those can have proofs coded with standard naturals. Since $T$ assumed to be
complete, then we can demonstrate there is at least one remote standard
sentence (e.g., the Gödel sentence), then we do have a minimal of their
proofs!

##  3 The proof without minimality

The above proof depends on the minimality principle which is provable by
induction. Here, this version won’t assume induction.

We need to change definitions of $E_{s}(v)$ and $remote(x)$ to the following:

$E_{s}(v)$ is the sentence form:

$\forall x\,(\operatorname{Proof}_{T}(x,v)\to\exists
y:\operatorname{Proof}_{T}(y,^{o}\ulcorner s\urcorner)\land x\geq y)$

Define:

$\displaystyle remote(y)\iff\exists x:$
$\displaystyle\operatorname{Dgn}(x,y)\land$ $\displaystyle\exists
z:\operatorname{Proof}_{T}(z,x)\land\forall k:\operatorname{Proof}_{T}(k,y)\to
z<k$

Now, the same state holds as of the old definition; that is, all above-
mentioned lemmas apply and in particular no proof (in $T$) of a remote
sentence can be standard!

From the fixed point lemma, take $\sigma$ to be the sentence defined by:

$\sigma\iff\forall
x(\operatorname{Proof}_{T}(x,^{o}\ulcorner\sigma\urcorner)\to\exists
y:remote(y)\land\operatorname{Proof}_{T}(x,y))$

Informally $\sigma$ says ”every proof of this sentence is a proof of a remote
sentence”.

Now, if $T\vdash\sigma$, then there is a standard proof of $\sigma$,
contradicting its remoteness asserted on the right hand. So, by completeness
of $T$, we must have $T\vdash\neg\sigma$. So, this says that there exists a
proof of ${}^{o}\ulcorner\sigma\urcorner$ in $T$ that is not a proof (in $T$)
of a remote sentence, so we have $\neg remote(^{o}\ulcorner\sigma\urcorner)$.
But, $\sigma$ is a standard sentence, and accordingly we’ll have
$remote(^{o}\ulcorner\sigma\urcorner)$ [Lemma II], a contradiction!

##  4 Defining Dgn

Here is the proper way of defining the relation $\operatorname{Dgn}$:

Let $h:\mathbb{N}\to\mathbb{N}$ be the function defined by:

$h(\ulcorner s\urcorner)=\ulcorner D(E_{s}(v))\urcorner$

for each sentence $s$ in the language of theory $T$, and $h(n)=0$ otherwise.

Define $E_{s}(v)$ to be the sentence form:

$\forall x\forall y\,(\operatorname{Proof}_{T}(x,v)\land[y=\min
z:\operatorname{Proof}_{T}(z,^{o}\ulcorner s\urcorner)]\to x\geq y)$

and $D(E_{s}(v))$ is the diagonal formula of the sentence form $E_{s}(v)$,
defined (along lines given here)) by:

$\forall v(\mathcal{G}_{f}(^{o}\ulcorner\forall v(\mathcal{G}_{f}(k,v)\to
E_{s}(v))\urcorner,v)\to E_{s}(v))$

Where $f:\mathbb{N}\to\mathbb{N}$ is the function defined by:

$f(\ulcorner\mathcal{A}\urcorner)=\,\ulcorner A(^{o}\ulcorner
A\urcorner)\urcorner$

, for any sentence form $A$ in the language of $T$, and $f(n)=0$ otherwise.
And of course $\mathcal{G}_{f}$ is the arithmetical formula representing $f$
in $T$.

Since $h$ is computable, then there would be a formula $\mathcal{G}_{h}(x,y)$
representing $h$ in theory $T$. Namely:

$\vdash_{T}\forall y\,(\mathcal{G}_{h}(^{o}\ulcorner s\urcorner,y)\to
y=\,^{o}\ulcorner D(E_{s}(v))\urcorner)$

Then we define:

$\operatorname{Dgn}(x,y)\iff\mathcal{G}_{h}(y,x)$

When minimality is not assumed, we do the same but on the respective
$E_{s}(v)$.

##  5 References

  1. 1.

Kurt Gödel, 1931, ”Über formal unentscheidbare Sätze der Principia Mathematica
und verwandter Systeme, I”, Monatshefte für Mathematik und Physik, v. 38 n. 1,
pp. 173–198. doi:10.1007/BF01700692

  2. 2.

—, 1931, ”Über formal unentscheidbare Sätze der Principia Mathematica und
verwandter Systeme, I”, in Solomon Feferman, ed., 1986. Kurt Gödel Collected
works, Vol. I. Oxford University Press, pp. 144–195. ISBN 978-0195147209. The
original German with a facing English translation, preceded by an introductory
note by Stephen Cole Kleene.

  3. 3.

Barkley Rosser (September 1936). ”Extensions of some theorems of Gödel and
Church”. Journal of Symbolic Logic. 1 (3): 87–91. doi:10.2307/2269028.

  4. 4.

Wikipedia. Rosser’s trick. Background. 13/08/2023

  5. 5.

Wikipedia. Diagonal Lemma. Proof. 13/08/2023

  6. 6.

Pudlák, Pavel, On the length of proofs of finitistic consistency statements in
first order theories, Logic colloq. ’84, Proc. Colloq., Manchester/U.K. 1984,
Stud. Logic Found. Math. 120, 165-196 (1986). ZBL0619.03037.

Citeseer pdf

  7. 7.

Joel David Hamkins (https://mathoverflow.net/users/1946/joel-david-hamkins),
Can this provide an example of incompleteness under the assumption of mere
consistency?, URL (version: 2023-07-21): https://mathoverflow.net/q/451243