# Quantum models à la Gabor for space-time metric

Gilles Cohen-Tannoudjia, Jean-Pierre GAZEAUb,
Célestin Habonimanac, and Juma Shabanid a Laboratoire de recherche sur les
sciences de la matière, LARSIM CEA, Université Paris-Saclay, F-91190 Saint-
Aubin, France
<EMAIL_ADDRESS>b Université Paris Cité, CNRS, Astroparticule
et Cosmologie, F-75013 Paris, France
<EMAIL_ADDRESS>c Ecole Normale Supérieure, Université du Burundi,
Bujumbura, Burundi
<EMAIL_ADDRESS>d Université du Burundi, Bujumbura, Burundi
<EMAIL_ADDRESS>
###### Abstract.

As an extension of Gabor signal processing, the covariant Weyl-Heisenberg
integral quantization is implemented to transform functions on the eight-
dimensional phase space $\left(x,k\right)$ into Hilbertian operators. The
$x=\left(x^{\mu}\right)$’s are space-time variables and the
$k=\left(k^{\mu}\right)$’s are their conjugate wave vector-frequency
variables. The procedure is first applied to the variables $\left(x,k\right)$
and produces canonically conjugate essentially self-adjoint operators. It is
next applied to the metric field $g_{\mu\nu}(x)$ of general relativity and
yields regularised semi-classical phase space portraits
$\check{g}_{\mu\nu}(x)$. The latter give rise to modified tensor energy
density. Examples are given with the uniformly accelerated reference system
and the Schwarzschild metric. Interesting probabilistic aspects are discussed.

Keywords: Covariant Weyl-Heisenberg integral quantization, time-frequency,
position-wave vector, space-time metric, general relativity, geometry of
information

MSC: 46L65, 81S05, 81S30, 83C45, 81R30

###### Contents

  1. 1 Introduction
  2. 2 From signal analysis and reconstruction to integral quantization
  3. 3 Gabor quantization of functions of time-frequency variables
    1. 3.1 Gabor quantization
    2. 3.2 Gabor semi-classical portrait
  4. 4 Beyond Gabor: Weyl-Heisenberg covariant integral quantization of the time-frequency plane
  5. 5 Probabilistic aspects
  6. 6 Eight-dimensional case and general relativity
  7. 7 The example of the uniformly accelerated reference system
  8. 8 The example of the Schwarzschild metric field
  9. 9 Final discussion

##  1\. Introduction

One of the difficulties to accept arbitrary change of space-time coordinates
in general relativity lies in the primordial learning of our environment. For
instance, in the case of the Schwarzschild metric,
$\mathrm{d}s^{2}=(1-2m/r)\mathrm{d}t^{2}-(1-2m/r)^{-1}\mathrm{d}r^{2}-r^{2}\mathrm{d}\theta^{2}-r^{2}\sin^{2}\theta\mathrm{d}\phi^{2}$,
the radial coordinate appears as the most natural and easily understandable in
terms of basic physics, namely a mass and a radius, while numerous coordinates
(Gullstrand–Painlevé, Kruskal-Szekeres, Lemaître, Eddington–Finkelstein …)
used to examine the continuation of the Schwarzschild solution, are not so
immediate.

From the early stages of development our senses undergo temporal and spatial
dimensions of this environment. This training can be understood through the
most basic physical laws like conservation of energy, momentum, angular
momentum, through myriads of tiny collisions at all scales between any living
matter and the rest of the world. In particular, the essentially Galilean
perception of our tempo-spatial environment might be in mathematical terms
compared with Fourier reconstruction and analysis of tempo-spatial signals
$s(x)\,,x:=(t,\boldsymbol{r})$ in the Hilbert space
$L^{2}(\mathbb{R}^{4},\mathrm{d}^{4}x=\mathrm{d}t\,\mathrm{d}\boldsymbol{r})\equiv\mathcal{H}$.

(1.1)
$\begin{split}s(x)&=\frac{1}{4\pi^{2}}\int_{\mathbb{R}^{4}}\mathrm{d}^{4}k\,\hat{s}(k)\,e^{\mathrm{i}k\cdot
x}\,,\quad k:=(\omega,\boldsymbol{k})\,,\quad k\cdot x:=\omega
t-\boldsymbol{k}\cdot\boldsymbol{r}\,,\\\
\hat{s}(k)&=\frac{1}{4\pi^{2}}\int_{\mathbb{R}^{4}}\mathrm{d}^{4}x\,s(x)\,e^{-\mathrm{i}x\cdot
k}\,.\end{split}$

The minkowskian inner product $k\cdot x$ has been chosen for convenience, but,
in the present context, the Fourier exponentials in the integrals (1.1) are
not to be interpreted as a travelling wave. They are just the basic
ingredients of the Fourier transform for determining the frequency $\omega$
and wave vector $\boldsymbol{k}$ content of the signal $s(x)$. The latter have
inverse dimension of time and space vector respectively, so no additional
constants appear in the Fourier transform. Of course, they can appear in the
expressions of the signals, and they can also be introduced into (1.1) to give
more physical flavor to the above formula in terms of energy and momentum.

As is now well established in signal or image processing, Fourier techniques
are advantageously replaced by Gabor reconstruction and analysis. One chooses
a probe signal, or window, $\psi$, which is well localized in time-space and
frequency-wave-vector at once, and which is normalized in $\mathcal{H}$,
$\|\psi\|^{2}=\int_{\mathbb{R}^{4}}\mathrm{d}^{4}x\,|\psi(x)|^{2}$=1. This
probe is then translated and modulated as

(1.2) $\psi(x)\rightarrow\psi_{y,l}(x)=e^{\mathsf{i}l\cdot x}\,\psi(x-y)\,.$

We then get the reconstruction of the signal $s(x)$ in $\mathcal{H}$ in terms
of its Gabor transform $S(x,k)$:

(1.3)
$\begin{split}s(x)&=\frac{1}{(2\pi)^{4}}\int_{\mathbb{R}^{8}}\mathrm{d}^{4}y\,\mathrm{d}^{4}l\,S(y,l)\,\psi_{y,l}(x),\\\
S(x,k)&=\langle\psi_{x,k}|s\rangle=\int_{\mathbb{R}^{4}}\mathrm{d}^{4}y\,\overline{\psi_{x,k}(y)}\,s(y)\,.\end{split}$

Like for Fourier conservation of the energy (in the signal sense) holds:

(1.4)
$\int_{\mathbb{R}^{4}}\mathrm{d}^{4}x\,|s(x)|^{2}=\|s\|^{2}=\frac{1}{(2\pi)^{4}}\int_{\mathbb{R}^{8}}\mathrm{d}^{4}x\,\mathrm{d}^{4}k\,|S(x,k)|^{2}:=\|S\|^{2}\,.$

This _Plancherel_ formula results from the resolution of the identity operator
$\mathbbm{1}$ in $\mathcal{H}$ provided by the continuous non-orthogonal
family of coherent states $\\{\psi_{x,k}\,,\,(x,k)\in\mathbb{R}^{8}\\}$
(overcompleteness):

(1.5)
$\mathbbm{1}=\frac{1}{(2\pi)^{4}}\int_{\mathbb{R}^{8}}\mathrm{d}^{4}x\,\mathrm{d}^{4}k\,\left|\psi_{x,k}\right\rangle\left\langle\psi_{x,k}\right|\,.$

Like the Fourier transform is based on the one-dimensional unitary irreducible
representations (UIR) of the abelian translation group in $\mathbb{R}^{4}$,
the above Gabor transform is based on the infinite-dimensional von Neumann UIR
of the Weyl-Heisenberg group $\mathrm{H}_{4}$ [1]. This UIR is nothing else
than the non-commutative extension of the translation group in
$\mathbb{R}^{8}$ when the latter is combined with a one-parameter cocycle.
Quantum mechanics for one Galilean elementary system is itself based on the
UIR of $\mathrm{H}_{3}$.

In this article, we take advantage of the resolution of the identity in the
Hilbert space of signals $\mathcal{H}$ to linearly map functions $f(x,k)$ on
the phase space $\\{(x,k)\\}\sim\mathbb{R}^{8}$, and, more generally,
distributions [2], to operators $A_{f}$ in $\mathcal{H}$. The procedure is
called Weyl-Heisenberg covariant integral quantization, or more simply, Gabor
quantization. According to the choice of the probe $\psi$ it can yield
regularisations of the original $f$ through spectral properties of the
operator $A_{f}$ and expected values or lower symbols or semi-classical phase
space portraits
$\check{f}(x,k):=\left\langle\psi_{x,k}\right|A_{f}\left|\psi_{x,k}\right\rangle$
of $A_{f}$. The mathematical content of the present work is, of course, not
original. It has been comprehensively developed over the past decades, with
the noticeable contributions by Werner [3], Feichtinger and Kozek [4], Luef
and Skrettingland [5]. In this context, we should also remind the pioneer
works by Klauder [6, 7], Berezin [8], Daubechies and Grossmann [9, 10] on the
integral quantization based on standard coherent state, as particular cases of
Gabor quantization with a quantum mechanics flavour. Concerning the genesis of
specific aspects of our approach, see for instance the recent [11, 12, 13]
[14, 15, 16] for more explanations on the motivations and illustrations. The
maps $f\mapsto A_{f}\mapsto\check{f}$ have probabilistic features which have
to be exploited within some informational interpretation and offer new
perspectives, particularly in cosmology. Moreover, restricting the
quantization to scalar fields $g(x)$, like metric fields of general
relativity, leads to interesting outcomes. Of course the extension to
tensorial fields is straightforward notwithstanding the increase of
technicalities.

The two following sections are heavily inspired from the recent [15], and
should be considered as a necessary reminder in view of extending our time-
frequency ($\mathbb{R}^{2}$) approach to the time-space-frequency-wavevector
$\mathbb{R}^{8}$. In Section 2 is presented the integral quantization under
its general form. In Sections 3 and 4 we give an outline of the Gabor
quantization and its generalisations for the simplest case of the time-
frequency phase space $\\{(t,\omega)\\}\sim\mathbb{R}^{2}$. In Section 5 we
proceed with the analysis of the procedure in terms of observational and
probabilistic aspects. In Section 6 we extend this material to the eight-
dimensional case for which the symmetry group is $\mathrm{H}_{4}$. However,
for a sake of simplicity, we restrict our procedure to the Gabor quantization
based on the choice of probe functions $\psi$. In Section 7, motivated by the
1935 Einstein-Rosen paper [17] the Gabor quantization is applied to the most
elementary (and singular!) metric field of general relativity, namely the
uniformly accelerated reference system. We show how the regularising constant
(introduced by hand by the authors) naturally emerges from our procedure. In
Section 8 we also consider another historical metric, the Schwarzschild
solution, and show how the Gabor quantization and its phase space portraits
lead to appealing regularisations, at the price of breaking the rotational
symmetry. With Section 9 we end this article by sketching some interesting
perspectives.

##  2\. From signal analysis and reconstruction to integral quantization

Let $(X,\mu)$ be a measure space and $\mathcal{H}$ be a (separable) Hilbert
space. An operator-valued function

(2.1) $X\ni x\mapsto\mathsf{M}(x)\ \mbox{acting in}\ \mathcal{H}\,,$

resolves the identity operator $\mathbbm{1}$ in $\mathcal{H}$ with respect the
measure $\mu$ if

(2.2) $\int_{X}\mathsf{M}(x)\,\mathrm{d}\mu(x)=\mathbbm{1}$

holds in a weak sense, i.e.,
$\int_{X}\langle\psi|\mathsf{M}(x)|\psi^{\prime}\rangle\,\mathrm{d}\mu(x)=\langle\psi|\psi^{\prime}\rangle$
for any $\psi,\psi^{\prime}$ in the common domain of the $\mathsf{M}(x)$’s,
a.e. $x$.

As we have in (1.3), analysis and reconstruction of a signal are described by
the action of (2.2) on a vector in $\mathcal{H}$

(2.3) $\mathcal{H}\ni|s\rangle\overset{\mbox{{\scriptsize
reconstruction}}}{=}\int_{X}\overset{\mbox{{\scriptsize
analysis}}}{\overbrace{\mathsf{M}(x)|s\rangle}}\,\mathrm{d}\mu(x)\,.$

On the other hand, integral quantization is the linear map of a function on
$X$ to an operator in $\mathcal{H}$ which is defined by

(2.4) $f(x)\mapsto\int_{X}f(x)\mathsf{M}(x)\,\mathrm{d}\mu(x)=A_{f}\,,\quad
1\mapsto\mathbbm{1}\,.$

If the operators $\mathsf{M}(x)$ in (2.2) are nonnegative and bounded, i.e.,
$\langle\phi|\mathsf{M}(x)|\phi\rangle\geq 0$ for (almost) all $x\in X$ and
all $\phi\in\mathcal{H}$, one says that they form a (normalised) positive
operator-valued measure (POVM) on $X$. If they are further unit trace-class,
i.e. $\mathrm{tr}(\mathsf{M}(x))=1$ for all $x\in X$, i.e., if the
$\mathsf{M}(x)$’s are density operators, then the map

(2.5)
$f(x)\mapsto\check{f}(x):=\mathrm{tr}(\mathsf{M}(x)A_{f})=\int_{X}f(x^{\prime})\,\mathrm{tr}(\mathsf{M}(x)\mathsf{M}(x^{\prime}))\,\mathrm{d}\mu(x^{\prime})$

is a local averaging of the original $f(x)$ (which can very singular, like a
Dirac !) with respect to the probability distribution on $X$,

(2.6) $x^{\prime}\mapsto\mathrm{tr}(\mathsf{M}(x)\mathsf{M}(x^{\prime}))\,.$

This averaging, or semi-classical portrait of the operator $A_{f}$, is in
general a regularisation, depending of course on the topological nature of the
measure space $(X,\mu)$ and the functional properties of the
$\mathsf{M}(x)$’s.

Now, consider a set of parameters $\boldsymbol{\kappa}$ and corresponding
families of POVM $\mathsf{M}_{\boldsymbol{\kappa}}(x)$ solving the identity

(2.7)
$\int_{X}\mathsf{M}_{\boldsymbol{\kappa}}(x)\,\mathrm{d}\mu(x)=\mathbbm{1}\,,$

One says that the classical limit $f(x)$ holds at $\boldsymbol{\kappa}_{0}$ if

(2.8)
$\check{f}_{{\boldsymbol{\kappa}}}(x):=\int_{X}f(x^{\prime})\,\mathrm{tr}(\mathsf{M}_{\boldsymbol{\kappa}}(x)\mathsf{M}_{\boldsymbol{\kappa}}(x^{\prime}))\,\mathrm{d}\mu(x^{\prime})\to
f(x)\quad\mbox{as}\quad\boldsymbol{\kappa}\to\boldsymbol{\kappa}_{0}\,,$

where the convergence $\check{f}\to f$ is defined in the sense of a certain
topology.

Otherwise said,
$\mathrm{tr}(\mathsf{M}_{\boldsymbol{\kappa}}(x)\mathsf{M}_{\boldsymbol{\kappa}}(x^{\prime}))$
tends to

(2.9)
$\mathrm{tr}(\mathsf{M}_{\boldsymbol{\kappa}}(x)\mathsf{M}_{\boldsymbol{\kappa}}(x^{\prime}))\to\delta_{x}(x^{\prime})$

where $\delta_{x}$ is a Dirac measure with respect to $\mu$,

(2.10)
$\int_{X}f(x^{\prime})\,\delta_{x}(x^{\prime})\,\mathrm{d}\mu(x^{\prime})=f(x)\,.$

Actually, nothing guarantees the existence of such a limit on a general level.
Nevertheless if the semi-classical $\check{f}_{{\boldsymbol{\kappa}}}$ might
appear as more realistic and easily handled than the original $f$, the
question is to evaluate the range of acceptability of the parameters
$\boldsymbol{\kappa}$.

##  3\. Gabor quantization of functions of time-frequency variables

In this section we present the material in the basic two-dimensional time-
frequency case in order to progressively make the reader familiar to our
quantization method.

###  3.1. Gabor quantization

In order to manage time-frequency functions $f(b,\omega)$, we first start from
the resolution of the identity provided by the Gabor POVM. The latter is built
as follows.

(3.1)
$\mathbbm{1}=\int_{\mathbb{R}^{2}}\frac{\mathrm{d}b\,\mathrm{d}\omega}{2\pi}|\psi_{b\omega}\rangle\langle\psi_{b\omega}|\,,\quad\langle\psi_{b\omega}|\psi_{b^{\prime}\omega^{\prime}}\rangle\neq\delta(b-b^{\prime})\,\delta(\omega-\omega^{\prime})$

where the functions

(3.2) $\langle\delta_{t}|\psi_{b\omega}\rangle:=e^{\mathsf{i}\omega
t}\psi(t-b)$

are the modulated-transported unit-norm probe-vectors in
$L^{2}(\mathbb{R},\mathrm{d}t)$ and where
$\mathbb{R}\supset\Delta\mapsto\int_{\Delta}\frac{\mathrm{d}b\,\mathrm{d}\omega}{2\pi}|\psi_{b\omega}\rangle\langle\psi_{b\omega}|$
is the corresponding normalised POVM on the plane. From (2.4) the quantization
of $f(b,\omega)$ is given by:

(3.3) $f\mapsto
A_{f}=\int_{\mathbb{R}^{2}}\frac{\mathrm{d}b\,\mathrm{d}\omega}{2\pi}\,f(b,\omega)\,|\psi_{b\omega}\rangle\langle\psi_{b\omega}|\,.$

The latter maps a function (or tempered distribution) $f(b,\omega)$ on the
time-frequency plane to the integral operator $A_{f}$ acting in the Hilbert
space of finite-energy signals as

(3.4)
$(A_{f}s)(t)=\langle\delta_{t}|A_{f}|s\rangle=\int_{-\infty}^{+\infty}\mathrm{d}t^{\prime}\,\mathcal{A}_{f}(t,t^{\prime})\,s(t^{\prime})\,,$

with integral kernel given by

(3.5)
$\mathcal{A}_{f}(t,t^{\prime})=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\mathrm{d}b\,\widehat{f}_{\omega}(b,t^{\prime}-t)\,\psi(t-b)\,\overline{\psi(t^{\prime}-b)}$

Here $\widehat{f}_{\omega}(b,y)$ is the partial Fourier transform with respect
to the variable $\omega$:

(3.6)
$\widehat{f}_{\omega}(b,y)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\mathrm{d}\omega\,f(b,\omega)\,e^{-\mathsf{i}\omega
y}\,.$

The outcome of the construction of the operator $A_{f}$ through (3.4) should
be analysed from the quantum measurement viewpoint. Within the framework of
quantum physics, a physically relevant operator of the above form $A_{f}$ is a
self-adjoint operator whose expectation value is the “unsharp” representation
[18, 19, 20]

(3.7)
$\displaystyle\mathrm{tr}\left(\rho_{m}A_{f}\right)=\int_{X}f(x)\,\mathrm{tr}(\rho_{m}\rho(x))\,\mathrm{d}\nu(x)\,,$

where $\rho_{m}$ is a density operator ($\sim$ mixed quantum state) describing
the way the physical system under observation has been prepared. In (3.3) a
real function $f(b,\omega)$ is mapped to a symmetric operator and if $f$ is
semi-bounded then $A_{f}$ is self-adjoint through the Friedrichs extension
[21] of its associated semi-bounded quadratic form. For unbounded real $f$,
self-adjointness is not guarantee and should be individually investigated.

Let us go through some specific situations. The Gabor quantization of
separable functions $f(b,\omega)=u(b)v(\omega)$ yields the integral kernel

(3.8)
$\mathcal{A}_{uv}(t,t^{\prime})=\frac{\hat{v}(t^{\prime}-t)}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\mathrm{d}b\,u(b)\,\psi(t-b)\,\overline{\psi(t^{\prime}-b)}\,,$

where $\hat{v}$ is the Fourier transform of $v$. Therefore the action on a
signal $s(t)$ reads as the combination of convolution and multiplication

(3.9)
$(A_{uv}s)(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\mathrm{d}b\,\psi(b)\,u(t-b)\,\left(\overline{\tilde{\psi}}_{b}\hat{\tilde{v}}\ast
s\right)(t)\,,\quad\psi_{b}(t):=\psi(t-b)\,,$

where $\tilde{\psi}(t):=\psi(-t)$. In the monovariable case $f(b,\omega)=u(b)$
the integral kernel (3.5) reads as

(3.10)
$\mathcal{A}_{u}(t,t^{\prime})=\delta(t^{\prime}-t)\int_{-\infty}^{+\infty}\mathrm{d}b\,u(b)\,|\psi(t-b)|^{2}=\delta(t^{\prime}-t)\,\left(|\psi|^{2}\ast
u\right)(t)\,,$

and one gets the multiplication operator

(3.11) $(A_{u(b)}s)(t)=\left(u\ast|\psi|^{2}\right)(t)\,s(t)\,.$

For the other monovariable case $f(b,\omega)=v(\omega)$ one gets the integral
kernel

(3.12)
$\mathcal{A}_{v(\omega)}(t,t^{\prime})=\frac{\hat{v}(t^{\prime}-t)}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\mathrm{d}b\,\psi(b)\,\overline{\tilde{\psi}}(t-t^{\prime}-b)=\frac{\hat{v}(t^{\prime}-t)}{\sqrt{2\pi}}\,R_{\psi\psi}(t-t^{\prime})$

where

(3.13)
$R_{\psi\psi}(t):=\int_{-\infty}^{+\infty}\mathrm{d}t^{\prime}\,\psi(t^{\prime})\,\overline{\psi(t^{\prime}-t)}=\left(\psi\ast\overline{\tilde{\psi}}\right)(t)=\overline{\widetilde{R}}_{\psi\psi}(t)$

is the autocorrelation of the probe, i.e. the correlation of the probe with a
delayed copy of itself as a function of delay. Note that

(3.14)
$\frac{1}{\sqrt{2\pi}}R_{\psi\psi}(t)=\mathcal{F}^{-1}\left[|\hat{\psi}|^{2}\right](t)\,.$

We eventually get the convolution operator acting on the signal:

(3.15)
$(A_{v(\omega)}s)(t)=\frac{1}{\sqrt{2\pi}}\left[\left(R_{\psi\psi}\hat{\tilde{v}}\right)\ast
s\right](t)$

The application of these formulae to time and frequency variables yields the
centered time and frequency operators:

(3.16) $\displaystyle A_{b}$ $\displaystyle=T-\langle
T\rangle_{\psi}\,\mathbbm{1}\,,\quad(Ts)(t)=ts(t)\,,$ (3.17) $\displaystyle
A_{\omega}$
$\displaystyle=\Omega-\langle\Omega\rangle_{\psi}\,\mathbbm{1}\,,\quad(\Omega
s)(t)=-\mathsf{i}\,\partial_{t}s(t)\,,$

where $\langle A\rangle_{\psi}$ is the mean value of the operator $A$ in the
state $\psi$, as it is usually denoted within the quantum framework. The
choice of an even probe $\psi$ allows to get rid of these constants. It is the
case with the standard choice of the normalised centred Gaussian with width
$\sigma$

(3.18) $\psi(t)\equiv
G_{\sigma}(t)=\frac{1}{\pi^{1/4}\sqrt{\sigma}}e^{-\dfrac{t^{2}}{2\sigma^{2}}}\,.$

Note that its autocorrelation is also a (not normalised) Gaussian:

(3.19) $R_{G_{\sigma}G_{\sigma}}(t)=e^{-\frac{t^{2}}{4\sigma^{2}}}\,.$

The operators $T$ and $\Omega$, like $A_{b}$ and $A_{\omega}$, are essentially
self-adjoint and obey the “canonical” commutation rule (no $\hbar$ here!):

(3.20) $T\,\Omega-\Omega\,T\equiv[T,\Omega]=\mathsf{i}\,\mathbbm{1}\,,$

with its immediate Fourier uncertainty consequence

(3.21)
$\Delta_{s}T\,\Delta_{s}\Omega\geq\frac{1}{2}\,,\quad\Delta_{s}A:=\sqrt{\langle
s|A^{2}|s\rangle-(\langle s|A|s\rangle)^{2}}$

Now, one should keep in mind that the CCR $[A,B]=\mathrm{i}\mathbbm{1}$ for a
self-adjoint ($A$, $B$) pair, with common domain, holds true only if both have
continuous spectrum $(-\infty,+\infty)$. The expression of the CCR in terms of
the respective unitary operators reads as

(3.22) $e^{\mathrm{i}\sigma\Omega}\,e^{\mathrm{i}\tau
T}=e^{\mathrm{i}\sigma\tau}e^{\mathrm{i}\tau
T}\,e^{\mathrm{i}\sigma\Omega}\,,\quad\mbox{(Weyl relations)}\,.$

von Neumann proved (1931) that up to multiplicity and unitary equivalence the
Weyl relations have only one solution (see [22] for the proof). In the present
formalism, time is elevated to the status of a quantum observable. This is
absolutely not the case for the time operator of quantum mechanics (see for
instance the review [23]). Indeed, in quantum mechanics the Hamiltonian is
bounded below and so the quantum time, although symmetric, is not self-adjoint
and has no self-adjoint extension.

Concerning the square of time and frequency variables we obtain:

(3.23) $\displaystyle A_{b^{2}}$ $\displaystyle=(T-\langle
T\rangle_{\psi})^{2}+\Delta^{2}_{\psi}T\,\mathbbm{1}\,,$ (3.24) $\displaystyle
A_{\omega^{2}}$
$\displaystyle=(\Omega-\langle\Omega\rangle_{\psi})^{2}+\Delta^{2}_{\psi}\Omega\,\mathbbm{1}\,.$

With the choice of Gaussian probe (3.18) these operators read:

(3.25) $A_{b^{2}}=T^{2}+\frac{\sigma^{2}}{2}\,\mathbbm{1}\,,\quad
A_{\omega^{2}}=\Omega^{2}+\frac{\sigma^{2}}{2}\,\mathbbm{1}\,.$

###  3.2. Gabor semi-classical portrait

The Gabor semi-classical phase-space portrait of $A_{f}$ is given by

(3.26)
$\check{f}(b,\omega)=\int_{\mathbb{R}^{2}}\frac{\mathrm{d}b^{\prime}\,\mathrm{d}\omega^{\prime}}{2\pi}\,f(b^{\prime},\omega^{\prime})\,|\langle\psi_{b\omega}|\psi_{b^{\prime}\omega^{\prime}}\rangle|^{2}\,.$

The overlap $\langle\psi_{b\omega}|\psi_{b^{\prime}\omega^{\prime}}\rangle$ is
given by

(3.27)
$\begin{split}\langle\psi_{b\omega}|\psi_{b^{\prime}\omega^{\prime}}\rangle&=\int_{\mathbb{R}}\mathrm{d}t\,e^{-\mathrm{i}(\omega^{\prime}-\omega)t}\,\overline{\psi(t-b^{\prime})}\,\psi(t-b)\\\
&=e^{-\mathrm{i}(\omega^{\prime}-\omega)b}\left(\mathcal{F}\left[\mathtt{t}_{b^{\prime}-b}\bar{\psi}\right]\ast\mathcal{F}\left[\psi\right]\right)(\omega^{\prime}-\omega)\,,\end{split}$

where $\mathcal{F}[f]:=\hat{f}$ (Fourier transform),
$(\mathtt{t}_{b}f)(t):=f(t-b)$, and

$(b^{\prime},\omega^{\prime})\mapsto|\langle\psi_{b\omega}|\psi_{b^{\prime}\omega^{\prime}}\rangle|^{2}/2\pi$

is a probability distribution on the phase space equipped with the Lebesgue
measure $\mathrm{d}b^{\prime}\,\mathrm{d}\omega^{\prime}$. Hence the
regularity properties of $\check{f}$ depend on those of the probe. The
function $\check{f}(b,\omega)$ is an elaborate combination a partial Fourier
transform and multi-convolutions:

(3.28)
$\check{f}(b,\omega)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\mathrm{d}\tau\,e^{\mathrm{i}\omega\tau}\int_{\mathbb{R}}\mathrm{d}b^{\prime}\,\widehat{f}_{\omega}(b-b^{\prime},\tau)\,\left(\left(\overline{\tilde{\psi}}\,\widetilde{\mathtt{t}_{b^{\prime}}\psi}\right)\ast\left(\overline{\mathtt{t}_{-b^{\prime}}\psi}\,\psi\right)\right)(\tau)\,.$

In the monovariable case $f(b,\omega)=u(b)$ this formula simplifies to

(3.29)
$\check{u}(b,\omega)\equiv\check{u}(b)=\left(u\ast\left(|\psi|^{2}\ast|\widetilde{\psi}|^{2}\right)\right)(b)=\left(u\ast
R_{|\psi|^{2}|\psi|^{2}}\right)(b)\,.$

where we note the appearance of the autocorrelation of the probability
distribution $t\mapsto|\psi(t)|^{2}$ on the temporal axis. For the simplest
cases time and time squared we get

(3.30) $\check{b}=b-\langle
b\rangle_{R_{|\psi|^{2}|\psi|^{2}}}\,,\quad\check{b^{2}}=\left(b-\langle
b\rangle_{R_{|\psi|^{2}|\psi|^{2}}}\right)^{2}+\sigma^{2}_{R_{|\psi|^{2}|\psi|^{2}}}(b)\,,$

where $\langle s\rangle_{p}$ stands for the expected value of the signal $s$
with respect to the probability distribution $p$, and $\sigma^{2}_{p}(s(t))$
is its variance with respect to $p$. An analogous formula holds (in the
Fourier side) for the monovariable case $f(b,\omega)=v(\omega)$:

(3.31)
$\check{v}(b,\omega)\equiv\check{v}(\omega)=\overline{\mathcal{F}}\left[\hat{v}\left(\overline{\tilde{\psi}}\ast\psi\right)^{2}\right](\omega)\,,\quad\overline{\mathcal{F}}:=\mathcal{F}^{-1}\,.$

With the choice of Gaussian probe (3.18) the overlap (3.27) is given by

(3.32) $\langle
G_{\sigma,b^{\prime},\omega^{\prime}}|G_{\sigma,b,\omega}\rangle=e^{-\mathrm{i}(\omega^{\prime}-\omega)\dfrac{b+b^{\prime}}{2}}\,e^{-\dfrac{(b-b^{\prime})^{2}}{4\sigma^{2}}}\,e^{-\dfrac{\sigma^{2}(\omega-\omega^{\prime})^{2}}{4}}\,,$

and the semi-classical portrait of the operator $A_{f}$ is the double Gaussian
convolution:

(3.33)
$\check{f}(b,\omega)=\int_{\mathbb{R}^{2}}\frac{\mathrm{d}b^{\prime}\,\mathrm{d}\omega^{\prime}}{2\pi}\,f(b^{\prime},\omega^{\prime})\,e^{-\dfrac{(b-b^{\prime})^{2}}{2\sigma^{2}}}\,e^{-\dfrac{\sigma^{2}(\omega-\omega^{\prime})^{2}}{2}}\,.$

As a consequence we observe that no classical limit holds at $\sigma\to 0$ or
$\sigma\to\infty$. This is just an illustration of the time-frequency
uncertainty principle. Separable functions $f(b,\omega)=u(b)v(\omega)$ remain
separable. From Eq. (3.29) and the convolution of two Gaussian, one finds:

(3.34) $\check{f}(b,\omega)=\left(u\ast
G^{2}_{\sqrt{2}\sigma}\right)(b)\,\left(v\ast
G^{2}_{\sqrt{2}/\sigma}\right)(\omega)\,,$

and in particular

(3.35) $\check{u}(b)=\left(u\ast
G^{2}_{\sqrt{2}\sigma}\right)(b)\,,\quad\check{v}(\omega)=\left(v\ast
G^{2}_{\sqrt{2}/\sigma}\right)(\omega)\,.$

Here the classical limit exists separately, and the convergence is simple at
least for regular enough $u$ and $v$:

(3.36) $\check{u}(b)\underset{\sigma\to
0}{\to}u(b)\,,\quad\check{v}(\omega)\underset{\sigma\to\infty}{\to}v(\omega)\,.$

For the simplest cases,

(3.37) $\displaystyle\check{b}$
$\displaystyle=b\,,\quad\check{b^{2}}=b^{2}+\sigma^{2}\,,$ (3.38)
$\displaystyle\check{\omega}$
$\displaystyle=\omega\,,\quad\check{\omega^{2}}=\omega^{2}+\frac{1}{\sigma^{2}}\,.$

##  4\. Beyond Gabor: Weyl-Heisenberg covariant integral quantization of the
time-frequency plane

Gabor signal analysis and quantization are the simplest ones among a vast
amount of possibilities, all of them being based on the unitary dual of the
Weyl-Heisenberg group. Let us remind the most important features of this group
that we use in our approach to quantization. More details are given in the
pedagogical [11].

We recognize in the construction of the Gabor family (1.2) the combined
actions of the two unitary operators introduced in (3.22), with respective
generators the self-adjoint time and frequency operators

(4.1)
$L^{2}(\mathbb{R},\mathrm{d}t)\ni\psi(t)\mapsto\psi_{b,\omega}(t)=\left(e^{\mathrm{i}\omega
T}e^{-\mathrm{i}b\Omega}\psi\right)(t)\,,$

Two alternative forms of the action (4.1) are provided by the Weyl formulae
(3.22) combined with the Baker-Campbell-Hausdorff formula:

(4.2)
$\psi_{b,\omega}(t)=e^{\mathrm{i}b\omega}\left(e^{-\mathrm{i}b\Omega}e^{\mathrm{i}\omega
T}\psi\right)(t)=e^{\mathrm{i}\frac{b\omega}{2}}\left(e^{\mathrm{i}(\omega
T-b\Omega)}\psi\right)(t)\,.$

In the above appears the Weyl or displacement operator up to a phase factor

(4.3) $e^{\mathrm{i}(\omega
T-b\Omega)}=\mathcal{D}(b,\omega)\,,\quad\psi_{b,\omega}(t)=e^{\mathrm{i}\frac{b\omega}{2}}\left(\mathcal{D}(b,\omega)\psi\right)(t)\,.$

The appearance of this phase factor like that one appearing in the composition
formula (3.22) indicates that the map $(b,\omega)\mapsto\mathcal{D}(b,\omega)$
is a projective unitary representation of the time-frequency abelian plane.
Dealing with a true representation necessitates the introduction of a third
degree of freedom to account for this phase factor. Hence we are led to work
with the Weyl-Heisenberg group $\mathrm{G}_{\rm WH}$.

(4.4)
$G_{\mathrm{WH}}=\\{g=(\varsigma,b,\omega)\,,\,\varsigma\in{\mathbb{R}}\,,\,(b,\omega)\in{\mathbb{R}}^{2}\\}\,,$

with neutral element: $(0,0,0)$, and

(4.5)
$g_{1}g_{2}=\left(\varsigma_{1}+\varsigma_{2}+\frac{1}{2}(\omega_{1}b_{2}-\omega_{2}b_{1})\,,b_{1}+b_{2},\;\omega_{1}+\omega_{2}\right)\,,\quad
g^{-1}=(-\varsigma,-b,-\omega)\,.$

The Weyl-Heisenberg group symmetry underlying the Gabor transform is
understood through its unitary irreducible representation (UIR). As a result
of the von-Neumann uniqueness theorem, any infinite-dimensional UIR, $U$, of
$\mathrm{G}_{\rm WH}$ is characterized by a real number $\lambda\neq 0$ (there
is also the degenerate, one-dimensional, UIR corresponding to $\lambda=0$). If
the Hilbert space carrying the UIR is the space of finite-energy signals
$\mathcal{H}=L^{2}({\mathbb{R}},\mathrm{d}t)$, the representation operators
are defined by the action similar to (4.1) (with the choice $\lambda=1$) and
completed with a phase factor:

(4.6) $U(\varsigma,b,\omega)=e^{\mathrm{i}\varsigma}e^{-\mathrm{i}\omega
b/2}\,e^{\mathrm{i}\omega
T}\,e^{-\mathrm{i}b\Omega}=e^{\mathrm{i}\varsigma}\,\mathcal{D}(b,\omega)\,.$

We now pick a bounded traceclass operator $\mathfrak{Q}_{0}$ on $\mathcal{H}$
with

(4.7) $\mathrm{tr}\,\mathfrak{Q}_{0}=1\,.$

Its unitary Weyl-Heisenberg transport yields the continuous family of bounded
traceclass operators

(4.8)
$\mathfrak{Q}(b,\omega)=U(\varsigma,b,\omega)\mathfrak{Q}_{0}U(\varsigma,b,\omega)^{{\dagger}}=\mathcal{D}(b,\omega)\mathfrak{Q}_{0}\mathcal{D}(b,\omega)^{{\dagger}}\,.$

Applying the Schur Lemma to the irreducible projective unitary representation
$(b,\omega)\mapsto\mathcal{D}(b,\omega)$ allows to prove the resolution of the
identity obeyed by the operator-valued function $\mathfrak{Q}(b,\omega)$ on
the time-frequency plane

(4.9)
$\int_{\mathbb{R}^{2}}\mathfrak{Q}(b,\omega)\,\frac{\mathrm{d}b\,\mathrm{d}\omega}{2\pi}=\mathbbm{1}\,.$

It ensues the Weyl-Heisenberg integral quantization in its most general
formulation:

(4.10) $f(b,\omega)\mapsto
A_{f}=\int_{\mathbb{R}^{2}}f(b,\omega)\,\mathfrak{Q}(b,\omega)\,\frac{\mathrm{d}b\,\mathrm{d}\omega}{2\pi}\,.$

If $\mathfrak{Q}_{0}$ is symmetric, a real function $f(b,\omega)$ is mapped to
a symmetric operator $A_{f}$. Moreover, if $\mathfrak{Q}_{0}$ is non-negative,
i.e., is a density operator, then a real semi-bounded function $f(b,\omega)$
is mapped to a self-adjoint operator $A_{f}$ as it was already stated about
the Gabor quantization (3.4).

Due to the homogeneity of the phase space, i.e. the choice of the origin is
arbitrary, we have translational covariance in the sense that the quantization
of the translated of $f$ is unitarily equivalent to the quantization of $f$:

(4.11)
$\begin{split}&U(\varsigma,b_{0},\omega_{0})\,A_{f}\,U(\varsigma,b_{0},\omega_{0})^{{\dagger}}=A_{\mathcal{T}(b_{0},\omega_{0})f}\,,\\\
&\mbox{where}\
\left(\mathcal{T}(b_{0},\omega_{0})f\right)(b,\omega):=f\left(b-b_{0},\omega-\omega_{0}\right)\,.\end{split}$

Note that the Gabor quantization (3.3) corresponds to the choice of the
orthogonal projector $\mathfrak{Q}_{0}=|\psi\rangle\langle\psi|$.

There exists an equivalent form to the quantization (4.10). This form is
expressed in terms of the displacement operator $\mathcal{D}$, the symplectic
Fourier transform of $f(b,\omega)$,

(4.12)
$\mathfrak{f}_{s}[f](b,\omega):=\int_{\mathbb{R}^{2}}e^{-\mathrm{i}(b\omega^{\prime}-b^{\prime}\omega)}\,f(b^{\prime},\omega^{\prime})\,\frac{\mathrm{d}b^{\prime}\,\mathrm{b}\omega^{\prime}}{2\pi}\,,\quad\mathfrak{f}_{s}^{2}=\mathbbm{1}\,,$

and the Weyl transform of the operator $\mathfrak{Q}_{0}$ defined as the
“apodization” function on the time-frequency plane

(4.13)
$\Pi(b,\omega):=\mathrm{tr}\left(\mathcal{D}(-b,-\omega)\mathfrak{Q}_{0}\right)\,,\quad\Pi(0,0)=1\,.$

The inverse of the latter transform is given by

(4.14)
$\mathfrak{Q}_{0}=\int_{\mathbb{R}^{2}}\mathcal{D}(b,\omega)\,\Pi(b,\omega)\,\frac{\mathrm{d}b\,\mathrm{d}\omega}{2\pi}\,,$

which means that $\mathfrak{Q}_{0}$ is the quantization of
$f(b,\omega)=2\pi\delta(b,\omega)$ such that
$\overline{\mathfrak{f}_{s}}[f](b,\omega)=1$.

One obtains

(4.15)
$A_{f}=\int_{\mathbb{R}^{2}}\mathcal{D}(b,\omega)\,\overline{\mathfrak{f}_{s}}[f](b,\omega)\,\Pi(b,\omega)\,\frac{\mathrm{d}b\,\mathrm{d}\omega}{2\pi}\,.$

From this form is obtained the action $A_{f}$ as the integral operator

(4.16) $\mathcal{L}^{2}(\mathbb{R},\mathrm{d}t)\ni
s(t)\mapsto(A_{f}s)(t)=\int_{-\infty}^{+\infty}\mathrm{d}t^{\prime}\,\mathcal{A}_{f}(t,t^{\prime})\,s(t^{\prime})\,,$

with kernel given by

(4.17)
$\mathcal{A}_{f}(t,t^{\prime})=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\mathrm{d}b\,\widehat{f}_{\omega}(b,t^{\prime}-t)\,\widehat{\Pi}_{\omega}\left(t-t^{\prime},b-\frac{t+t^{\prime}}{2}\right)\,,$

where the partial Fourier transforms $\widehat{f}_{\omega}$ and
$\widehat{\Pi}_{\omega}$ are defined in (3.6).

The semi-classical portrait (2.5) of $A_{f}$ now reads:

(4.18)
$\check{f}(b,\omega)=\mathrm{tr}(\mathfrak{Q}(b,\omega)A_{f})=\int_{\mathrm{\mathbb{R}^{2}}}f(b^{\prime},\omega^{\prime})\,\mathrm{tr}\left(\mathfrak{Q}(b,\omega)\mathfrak{Q}(b^{\prime},\omega^{\prime})\right)\,\frac{\mathrm{d}b^{\prime}\,\mathrm{d}\omega^{\prime}}{2\pi}\,.$

Equivalently:

(4.19)
$\begin{split}\check{f}(b,\omega)&=\int_{\mathbb{R}^{2}}\mathfrak{f}_{s}\left[\Pi\,\widetilde{\Pi}\right](b-b^{\prime},\omega-\omega^{\prime})\,f(b^{\prime},\omega^{\prime})\,\frac{\mathrm{d}b^{\prime}\,\mathrm{d}\omega^{\prime}}{2\pi}\\\
&=\int_{\mathbb{R}^{2}}\mathfrak{f}_{s}\left[\Pi\right]\ast\mathfrak{f}_{s}\left[\widetilde{\Pi}\right](b-b^{\prime},\omega-\omega^{\prime})\,f(b^{\prime},\omega^{\prime})\,\frac{\mathrm{d}b^{\prime}\,\mathrm{d}\omega^{\prime}}{4\pi^{2}}\,,\end{split}$

where $\widetilde{\Pi}(b,\omega)=\Pi(-b,-\omega)$.

##  5\. Probabilistic aspects

Let us develop the probabilistic content of the quantization and smoothing
procedure described in the previous section.

With a true probabilistic content, i.e., with a choice of operator
$\mathfrak{Q}_{0}$, or, equivalently, with a choice of function $\Pi$ in
(4.13), such that

(5.1)
$\mathrm{tr}\left(\mathfrak{Q}(b,\omega)\mathfrak{Q}(b^{\prime},\omega^{\prime})\right)\geq
0\quad\forall\,(b,\omega)\ \mathrm{and}\ (b^{\prime},\omega^{\prime})\
\mathrm{a.e.}\,$

the meaning of the convolution appearing in the integral (4.19)

(5.2)
$\frac{1}{2\pi}\mathfrak{f}_{s}\left[\Pi\right]\ast\mathfrak{f}_{s}\left[\widetilde{\Pi}\right]$

is clear: it is the probability distribution (w.r.t. the measure
$\mathrm{d}b\,\mathrm{d}\omega/2\pi$) for the difference of two vectors in the
time-frequency plane, viewed as independent random variables, and thus is
adapted to the abelian and homogeneous structure of the latter (choice of
origin is arbitrary!).

Let us examine in more details the relationship between $\mathfrak{Q}_{0}$ and
its Weyl transform $\Pi$. First we note the important property issued from
(4.13). If $\mathfrak{Q}_{0}$ is bounded symmetric, i.e. self-adjoint, then
its Weyl transform is “CP” symmetric:

(5.3) $\overline{\Pi(b,\omega)}=\Pi(-b,-\omega)\,.$

Second, due to the unit trace condition on $\mathfrak{Q}_{0}$, the symplectic
Fourier transform of $\Pi$ is a quasi-probability distribution,

(5.4)
$\int_{\mathrm{\mathbb{R}^{2}}}\mathfrak{f}_{s}\left[\Pi\right](b,\omega)\frac{\mathrm{d}b\,\mathrm{d}\omega}{2\pi}=1\,.$

The trivial case corresponds to $\Pi(b,\omega)=1$, of course. Then
$\mathfrak{f}_{s}\left[\Pi\right](b,\omega)=2\pi\delta(b,\omega)$ is a
probability distribution like its squared convolution (5.1), despite the fact
that $\mathfrak{Q}_{0}$ is not a density operator,
$\mathfrak{Q}_{0}=2\mathrm{P}$, where $\mathrm{P}$ is the parity operator
(whose the trace can be consistently defined as
$\mathrm{tr}(\mathrm{P})=1/2$). This no filtering choice yields the popular
Weyl-Wigner integral quantization equivalent to the standard ($\sim$
canonical) quantization.

In the case of Gabor quantization with a probe $\psi$, we already established
in (3.26) that (5.2) is equal to
$|\langle\psi_{b\omega}|\psi_{b^{\prime}\omega^{\prime}}\rangle|^{2}$. The
corresponding $\Pi_{\psi}$ is given by

(5.5)
$\begin{split}\Pi_{\psi}(b,\omega)&=\langle\psi|\mathcal{D}(-b,-\omega)\psi\rangle=\langle\mathcal{D}(b,\omega)\psi|\psi\rangle\\\
&=\sqrt{2\pi}e^{-\mathrm{i}\frac{b\omega}{2}}\left(\mathcal{F}[\overline{\psi}]\ast\mathcal{F}[\mathrm{t}_{-b}\psi]\right)(\omega)\,,\end{split}$

where $\mathcal{F}$ is the ordinary Fourier transform and
$\mathcal{F}[\mathrm{t}_{-b}\psi](\omega)=e^{\mathrm{i}b\omega}\mathcal{F}[\psi](\omega)$.

We are specially interested in building self-adjoint $A_{f}$, from physically
relevant real functions or distributions $f(b,\omega)$ through (4.10), that
also yield the probability distribution (5.2). An appealing choice for the
operator $\mathfrak{Q}_{0}$ is that it belongs to the class of density
operators for which (5.1) holds true, or, equivalently, that the quantity
$\mathfrak{f}_{s}\left[\Pi\right]\ast\mathfrak{f}_{s}\left[\widetilde{\Pi}\right]/2\pi$,
built from the corresponding $\Pi$, is a probability distribution on the time-
frequency plane equipped with the measure
$\mathrm{d}b\,\mathrm{d}\omega/2\pi$. The latter form of the condition is
important because there is no guarantee that
$\mathfrak{f}_{s}\left[\Pi\right]$ itself is positive and so defines a
probability distribution. For instance, in the Gabor case,
$\mathfrak{f}_{s}\left[\Pi_{\psi}\right]$ is equal to

(5.6)
$\begin{split}\mathfrak{f}_{s}[\Pi](b,\omega)&=\int_{-\infty}^{\infty}\mathrm{d}t\,e^{-\mathrm{i}\omega
t}\,\overline{\psi\left(\frac{t}{2}-b\right)}\,\psi\left(-\frac{t}{2}-b\right)\\\
&\equiv 2\pi\mathcal{W}_{\psi}(-b,-\omega)\,,\end{split}$

where $\mathcal{W}_{\psi}$ is the Wigner function [24, 25] of $\psi$, and it
is well known that the latter is a probability distribution only if $\psi(t)$
has the form

(5.7) $\psi(t)=\mathrm{Cst}\,e^{-at^{2}+bt+c}\,,\quad
a,b,c\in\mathbb{C}\,,\quad\mathrm{Re}(a)>0.$

Let us now examine the standard Gaussian examples of functions
$\Pi=\widetilde{\Pi}_{G}$ having probabilistic symplectic Fourier transform

(5.8)
$\begin{split}\Pi_{G}(b,\omega)=e^{-\frac{b^{2}}{2\sigma^{2}}}e^{-\frac{\omega^{2}}{2\tau^{2}}}\,,&\quad\mathfrak{f}_{s}\left[\Pi_{G}\right](b,\omega)=\sigma\tau
e^{-\frac{\tau^{2}b^{2}}{2}}e^{-\frac{\sigma^{2}\omega^{2}}{2}}\\\
\mathfrak{f}_{s}\left[\Pi_{G}\right]^{\ast
2}(b,\omega)&=\frac{\sigma\tau}{2}e^{-\frac{\tau^{2}b^{2}}{4}}e^{-\frac{\sigma^{2}\omega^{2}}{4}}\,.\end{split}$

Depending on the pair $(\sigma,\tau)$ the above functions do or do not yield
non-negative operators $\mathfrak{Q}_{0}$. They do if
$\sigma=\tau:=\sqrt{2\tanh\frac{1}{\Theta}}<\sqrt{2}$. We then obtain the
Boltzman-Planck like density operator [2]:

(5.9)
$\mathfrak{Q}_{0}=\left(1-e^{-\frac{1}{\Theta}}\right)\sum_{n=0}^{\infty}e^{-\frac{n}{\Theta}}|n\rangle\langle
n|\equiv\rho(\Theta)\,,$

in the Hilbertian basis made of eigenvectors $|n\rangle$ of the harmonic
operator

(5.10)
$\left(\frac{\Omega^{2}}{2\omega_{0}^{2}}+\frac{T^{2}}{2t_{0}^{2}}\right)|n\rangle=\frac{1}{\omega_{0}t_{0}}\left(n+\frac{1}{2}\right)|n\rangle\,.$

For convenience one choose units $\omega_{0}=1=t_{0}$.

In this context, the parameter $\Theta$ plays the role of a temperature (in
suitable units). More generally, from the inverse Weyl transform (4.14) we
obtain the matrix elements of $\mathfrak{Q}_{0}$ in the above basis
$\\{|n\rangle\\}$ in terms of those of the displacement operator:

(5.11) $\left(\mathfrak{Q}_{0}\right)_{mn}:=\langle
m|\mathfrak{Q}_{0}|n\rangle=\int_{\mathbb{R}^{2}}\mathcal{D}_{mn}(b,\omega)\,\Pi(b,\omega)\,\frac{\mathrm{d}b\,\mathrm{d}\omega}{2\pi}\,.$

The latter involve associated Laguerre polynomials $L^{(\alpha)}_{n}(t)$ [33]:

(5.12)
$\mathcal{D}_{mn}(b,\omega)=\sqrt{\dfrac{n!}{m!}}\,e^{-|z|^{2}/2}\,z^{m-n}\,L_{n}^{(m-n)}(|z|^{2})\,,\quad\mbox{for}\
m\geq n\,,$

with $L_{n}^{(m-n)}(u)=\frac{m!}{n!}(-u)^{n-m}L_{m}^{(n-m)}(u)$ for $n\geq m$,
and in which $z:=\frac{1}{\sqrt{2}}(b+\mathrm{i}\omega)$. Therefore, if the
function $\Pi$ is isotropic, i.e. $\Pi(b,\omega):=w(|z|^{2})$, then
$\mathfrak{Q}_{0}$ is the unit-trace diagonal operator

(5.13) $\mathfrak{Q}_{0}=\sum_{n=0}^{\infty}\mathcal{L}_{n}(w)|n\rangle\langle
n|\,,$

where

(5.14)
$\mathcal{L}_{n}(w)=\left(\mathfrak{Q}_{0}\right)_{nn}=\int_{0}^{+\infty}e^{-u/2}\,L_{n}(u)\,w(u)\,\mathrm{d}u$

is a particular Laguerre transform [34, 35] with

$L_{n}(u)\equiv
L^{(0)}_{n}(u)=\sum_{m=0}^{n}(-1)^{m}\binom{n}{m}\frac{u^{m}}{m!}\,.$

Since the Laguerre polynomials form an orthonormal basis of the Hilbert space
$L^{2}(\mathbb{R}^{+},e^{-u}\mathrm{d}u)$, the transform (5.14) is defined for
all $n$ and for all $w(u)$ such that $e^{u/2}w(u)$ belongs to this space.
Furthermore, if $w$ is such that $\mathcal{L}_{n}(w)\geq 0$ for all $n$, then
$\mathfrak{Q}_{0}$ is a density operator. Let us give some hint in the search
of solutions to this interesting problem by viewing $w$ as the Laplace
transform of a nonnegative function or distribution $\ell$ with support in
$[1/2,\infty)$:

(5.15) $w(u)=\int_{\frac{1}{2}}^{+\infty}e^{-ut}\,\ell(t)\,\mathrm{d}t\,.$

Then from the integral formula [33],

(5.16) $\begin{split}\int_{0}^{\infty}\,e^{-\nu
x}\,x^{\lambda}\,L_{n}^{\alpha}(x)\mathrm{d}x&=\frac{\Gamma(\lambda+1)\Gamma(\alpha+n+1)}{n!\,\Gamma(\alpha+1)}\nu^{-\lambda-1}\\\
&\times{}_{2}F_{1}(-n,\lambda+1;\alpha+1;\nu^{-1})\,,\end{split}$

and assuming that the Frobenius theorem is valid here, we derive

(5.17)
$\mathcal{L}_{n}(w)=\frac{1}{2}\int_{0}^{+\infty}e^{-(n+1/2)\Theta}\,\ell(t(\Theta))\,\frac{\mathrm{d}\Theta}{\Theta^{2}}\,,\quad
t(\Theta)=\frac{1}{2}\coth\left(\frac{1}{2\Theta}\right)\,.$

There results the convex integral of the Boltzmann-Planck-like density
operators $\rho(\Theta)$ defined in (5.9):

(5.18)
$\mathfrak{Q}_{0}=\frac{1}{2}\int_{0}^{+\infty}\rho(\Theta)\,e^{-\frac{1}{2\Theta}}\ell(t(\Theta))\,\frac{\mathrm{d}\Theta}{\Theta^{2}}\,.$

##  6\. Eight-dimensional case and general relativity

Hereafter we restrict our quantization procedure to the ones based on the
choice of a probe $\psi$. We remind that the Gabor quantization of functions
or distributions $f(x,k)$ on the phase space $\\{(x,k)\in\mathbb{R}^{8}\\}$ is
based on the overcompleteness (1.5) of the family of translated and modulated
unit-norm probes $\psi_{x,k}(y)=e^{\mathrm{i}k\cdot y}\psi(y-x)$:

(6.1) $f(x,k)\mapsto
A_{f}=\frac{1}{(2\pi)^{4}}\int_{\mathbb{R}^{8}}\mathrm{d}^{4}x\,\mathrm{d}^{4}k\,f(x,k)\,\left|\psi_{x,k}\right\rangle\left\langle\psi_{x,k}\right|\,.$

$A_{f}$ acts in the Hilbert space of finite-energy signals
$s(x)\in\mathcal{H}=L^{2}(\mathbb{R}^{4},\mathrm{d}^{4}x)$ as the integral
operator

(6.2)
$s(x)\mapsto(A_{f}s)(x)=\langle\delta_{x}|A_{f}|s\rangle=\int_{\mathbb{R}^{4}}\mathrm{d}^{4}x^{\prime}\,\mathcal{A}_{f}(x,x^{\prime})\,s(x^{\prime})\,,$

with integral kernel given by

(6.3)
$\mathcal{A}_{f}(x,x^{\prime})=\frac{1}{4\pi^{2}}\int_{\mathbb{R}^{4}}\mathrm{d}^{4}y\,\widehat{f}_{k}(y,x^{\prime}-x)\,\psi(x-y)\,\overline{\psi(x^{\prime}-y)}\,.$

Here $\widehat{f}_{k}(x,y)$ is the partial Fourier transform with respect to
the four-vector variable $k$:

(6.4)
$\widehat{f}_{k}(x,y)=\frac{1}{4\pi^{2}}\int_{\mathbb{R}^{4}}\mathrm{d}^{4}k\,f(x,k)\,e^{-\mathsf{i}k\cdot
y}\,.$

Remember the non-conventional sign in the Fourier exponential due to our
choice of space-time metric. The Gabor semi-classical phase-space portrait of
$A_{f}$ is given by

(6.5)
$\check{f}(x,k)=\int_{\mathbb{R}^{8}}\frac{\mathrm{d}^{4}x^{\prime}\,\mathrm{d}^{4}k^{\prime}}{(2\pi)^{4}}\,f(x^{\prime},k^{\prime})\,|\langle\psi_{x,k}|\psi_{x^{\prime},k^{\prime}}\rangle|^{2}\,.$

Explicit forms of $\check{f}$ are analogous to the time-frequency case (3.28)
apart from the fact that their writing might become quite cumbersome. Here we
are essentially concerned with the quantization and semi-classical portraits
of fields on space-time, like the metric field of GR. Therefore the Gabor
quantization of the field $u(x)$ yields the multiplication operator

(6.6) $(A_{u}s)(x)=\left(u\ast|\psi|^{2}\right)(x)\,s(x)\,,$

and its semi-classical portrait is given by

(6.7)
$\check{u}(x)=\left(u\ast\left(|\psi|^{2}\ast|\widetilde{\psi}|^{2}\right)\right)(x)=\left(u\ast
R_{|\psi|^{2}|\psi|^{2}}\right)(x)\,.$

Here convolution and autocorrelations are for functions on $\mathbb{R}^{4}$
equipped with the its Lebesgue measure $\mathrm{d}^{4}x$:

(6.8) $\displaystyle(f\ast g)(x)$
$\displaystyle=\int_{\mathbb{R}^{4}}\mathrm{d}^{4}y\,f(x-y)\,g(y)\,,$ (6.9)
$\displaystyle R_{\psi\psi}(x)$
$\displaystyle=\int_{\mathbb{R}^{4}}\mathrm{d}^{4}x^{\prime}\,\psi(x^{\prime})\,\overline{\psi(x^{\prime}-x)}=\left(\psi\ast\overline{\tilde{\psi}}\right)(x)\,,$

where we remind that $\tilde{\psi}(x)=\psi(-x)$. Repeating (3.16) and (3.30),
the space-time variables acquire the status of quantum observables acting on
the space of signals:

(6.10) $\displaystyle x^{\mu}\mapsto A_{x^{\mu}}$
$\displaystyle=X^{\mu}-\langle X^{\mu}\rangle_{\psi}\,\mathbbm{1}\,,\quad
X^{\mu}(s)(x)=x^{\mu}s(x)\,,$ (6.11) $\displaystyle\check{x^{\mu}}$
$\displaystyle=x^{\mu}-\langle x^{\mu}\rangle_{R_{|\psi|^{2}|\psi|^{2}}}\,.$

With the choice of the Gaussian probe

(6.12)
$G_{\boldsymbol{\sigma}}(x)=\prod_{\mu=0,1,2,3}G_{\sigma_{\mu}}(x^{\mu})\,,$

the above formula assume the simple expressions:

(6.13) $\displaystyle(A_{u}s)(x)$ $\displaystyle=\left(u\ast
G^{2}_{\boldsymbol{\sigma}}\right)(x)\,s(x)\,,\quad\sqrt{\boldsymbol{\sigma}}:=\prod_{\mu=0,1,2,3}\sqrt{\sigma_{\mu}}\,,$
(6.14) $\displaystyle\check{u}(x)$
$\displaystyle=\left(u\ast\left(G^{2}_{\boldsymbol{\sigma}}\ast
G^{2}_{\boldsymbol{\sigma}}\right)\right)(x)=\left(u\ast
G^{2}_{\sqrt{2}\boldsymbol{\sigma}}\right)(x)\,.$

The above formalism allows us to give a quantum version of the metric fields
$(g_{\mu\nu}(x))$ of general relativity,

(6.15) $g_{\mu\nu}(x)\mapsto A_{g_{\mu\nu}}\,,$

and to give a regularised version depending of the choice of probe,

(6.16) $\check{g}_{\mu\nu}(x)\,.$

As a first example, let us examine the Minkowskian metric in Cartesian
coordinates, $(g_{\mu\nu})=\mathrm{diag}(1,-1,-1,-1)$. We trivially get
$\left(A_{g_{\mu\nu}}\right)=\mathrm{diag}(\mathbbm{1},-\mathbbm{1},-\mathbbm{1},-\mathbbm{1})$.
Let us now examine what happens if one quantizes the same metric as is
expressed in cylindric coordinates
$(g^{\mathrm{cyl}}_{\mu\nu}(x^{0},\rho,\theta,x^{3})=\mathrm{diag}(1,-1,-\rho^{2},-1)$.
This metric is singular in the sense that its determinant cancels on the third
axis $\rho=0$. Due to the rotational symmetry about this axis, we naturally
choose $\boldsymbol{\sigma}=(\sigma_{0},\sigma,\sigma,\sigma_{3})$. We then
find the new metric operator and its semi-classical portrait:

(6.17)
$\begin{split}\left(A_{g_{\mu\nu}}\right)&=\mathrm{diag}(\mathbbm{1},-\mathbbm{1},-(\hat{\rho}^{2}+\sigma^{2}\mathbbm{1}),-\mathbbm{1})\,,\quad\hat{\rho}^{2}=\left(X^{1}\right)^{2}+\left(X^{2}\right)^{2}\,,\\\
\mathrm{d}s^{2}&=\mathrm{d}{x_{0}}^{2}-\mathrm{d}\rho^{2}-(\rho^{2}+2\sigma^{2})\mathrm{d}\theta^{2}-\mathrm{d}{x_{3}}^{2}\,.\end{split}$

Both expressions have become regular on the third axis.

In the case of isotropic space spherical coordinates for the spatial part are
privileged, i.e.,
$(g^{\mathrm{spher}}_{\mu\nu}(x^{0},r,\theta,\phi)=\mathrm{diag}(1,-1,-r^{2},-r^{2}\sin^{2}\theta)$.
This metric is singular as its determinant cancels at the origin $r=0$.
Choosing $\boldsymbol{\sigma}=(\sigma_{0},\sigma,\sigma,\sigma)$. we obtain
for the metric operator and its semi-classical portrait:

(6.18)
$\begin{split}\left(A_{g_{\mu\nu}}\right)&=\mathrm{diag}\left(\mathbbm{1},-\mathbbm{1},-\left(\hat{r}^{2}+\frac{3}{2}\sigma^{2}\mathbbm{1}\right),-(\hat{\rho}^{2}+\sigma^{2})\mathbbm{1}\right)\,,\\\
\mathrm{d}s^{2}&=\mathrm{d}{x_{0}}^{2}-\mathrm{d}r^{2}-\left(r^{2}+3\sigma^{2}\right)\mathrm{d}\theta^{2}-(\rho^{2}+2\sigma^{2})\mathrm{d}\phi^{2}\,,\end{split}$

with $\hat{\rho}^{2}=\left(X^{1}\right)^{2}+\left(X^{2}\right)^{2}$. Both
expressions are now regular at the origin.

Be aware that simply adding a positive constant to $\rho^{2}$ (cylindrical
case) and to $\rho^{2}$ and $r^{2}$ (spherical case) creates non-trivial
changes to the Euclidean geometry of the spatial part of the metric. One shows
that deformations of geodesics from straight lines involve elliptic functions
[36].

Next, if the metric fields are known solutions of the Einstein equations for a
given tensor energy density

(6.19) $R_{\mu\nu}-\frac{1}{2}R=\kappa\,T_{\mu\nu}\,,\quad\kappa=\frac{8\pi
G}{c^{4}}\,,$

then their respective $\psi$ regularised versions give rise to the modified
tensor energy density $\check{T}_{\mu\nu}$ through the equations:

(6.20)
$\check{T}_{\mu\nu}=\frac{1}{\kappa}\left(\check{R}_{\mu\nu}-\frac{1}{2}\check{R}\right)\,.$

This procedure offers the opportunity to work with smooth versions of the
metric field, where the smoothing has a probabilistic content. Moreover, the
notion of empty space becomes mathematically totally idealistic, since any
quantization of the above Gabor type introduces (fictitious? real?) matter,
however minute it may be. This matter arises from the lack of information
which is encoded in the parameters of the probe $\psi$.

The important point to be examined is the physical meaning of the set of
parameters $\boldsymbol{\kappa}$ introduced in Section 2, or in more
illustrative terms, the set $\boldsymbol{\sigma}$ of gaussian widths
introduced in (6.12). The probabilistic nature of these parameters should
reflect our inability to reach exactness in terms of the information about the
observed system through data and interpretative model. There exist limit
values assumed by these parameters beyond which the mathematical model loses
its physical relevance, i.e. where no measurement or observation makes sense.
Those limit values should be viewed as based on or originate in the existence
of a unique quantum of information, existing besides other elementary quanta
as they were listed by one of us in [26, 27]. It would be fruitful to revisit
the present approach from a more informational viewpoint. See for instance
[28] for a survey of the employed concepts, and [29, 30, 31, 32] for a more
specific à la Souriau viewpoint based on the symmetry of considered manifolds,
the simplest one being the phase space $\mathbb{R}^{8}$. We already tackled
the subject with the expressions (5.9) and (5.18).

##  7\. The example of the uniformly accelerated reference system

In their illuminating 1935 article [17], Einstein and Rosen state that

> Every field, in our opinion, must therefore adhere to the fundamental
> principle that singularities of the field are to be excluded.

Needless to say that we agree with this statement, and the first example they
provide gives us the opportunity to show how Gabor quantization of the metric
field cures the singularity problem. Following the Principle of Equivalence
one can view a uniformly accelerated reference system as at rest provided that
there exists a homogeneous gravitational field in space with respect to it.
The corresponding metric field is given by

(7.1)
$\mathrm{d}s^{2}=\alpha^{2}x_{1}^{2}\mathrm{d}x_{0}^{2}-\mathrm{d}x_{1}^{2}-\mathrm{d}x_{2}^{2}-\mathrm{d}x_{3}^{2}\,.$

The singularity holds on the hyperplane $x_{1}=0$. For $x_{1}\neq 0$ this
leads to the Einstein field equations

(7.2) $R_{\mu\nu}=0\,.$

The restriction $x_{1}\neq 0$ is necessary since the Ricci tensor is
indeterminate on the hyperplane $x_{1}=0$. In view of regularisation the
authors of [17] propose to modify the metric (7.1) with the introduction of a
small constant $\varsigma$ as

(7.3)
$\mathrm{d}s^{2}=\left(\alpha^{2}x_{1}^{2}+\varsigma\right)\mathrm{d}x_{0}^{2}-\mathrm{d}x_{1}^{2}-\mathrm{d}x_{2}^{2}-\mathrm{d}x_{3}^{2}\,.$

This ansatz gives rise to a regular metric and a new stress-energy tensor
$\check{T}_{\mu\nu}$ through the gravitational equation with source

(7.4)
$\check{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\,\check{R}=\kappa\,\check{T}_{\mu\nu}\,,\quad\kappa=\frac{8\pi
G}{c^{4}}\,,$

whose the nonvanishing components read

(7.5)
$\check{T}_{22}=\check{T}_{33}=-\frac{\alpha^{2}\varsigma}{\kappa}\left(\alpha^{2}x_{1}^{2}+\varsigma\right)^{-2}\,,$

which cancels at $\varsigma=0$, as expected. This fictitious or not stress-
energy tensor is a pure pressure in the directions 2 and 3.

Now the Einstein-Rosen ansatz is not an ansatz any longer if we apply the
Gabor quantization procedure to the metric field in (7.1) and the resulting
semi-classical portraits. This is the reason why we checked the quantities
appearing in (7.4). To simplify we choose the Gaussian probe (6.12). From
(3.25) we get for the metric operators:

(7.6)
$\left(A_{g_{\mu\nu}}\right)=\mathrm{diag}\left(\alpha^{2}X_{1}^{2}+\alpha^{2}\frac{\sigma_{1}^{2}}{2}\mathbbm{1},-\mathbbm{1},-\mathbbm{1},-\mathbbm{1}\right)\,,\quad
X_{1}s(x)=x_{1}s(x)\,,$

and for their semi-classical portrait,

(7.7)
$\left(\check{g}_{\mu\nu}\right)=\mathrm{diag}\left(\alpha^{2}x_{1}^{2}+\alpha^{2}\sigma_{1}^{2},-1,-1,-1\right)\,.$

Thus the Einstein-Rosen parameter can be identified as

(7.8) $\varsigma=\alpha^{2}\sigma_{1}^{2}$

and is interpreted as proportional to the Gaussian variance for the variable
$x_{1}$. Hence, (7.9) becomes

(7.9)
$\check{T}_{22}=\check{T}_{33}=-\frac{\sigma_{1}^{2}}{\kappa}\left(x_{1}^{2}+\sigma_{1}^{2}\right)^{-2}\,,$

The regularisation parameter $\sigma_{1}$ conveys our degree of ignorance of
what is the “exact” geometric modelling of the above homogeneous gravitational
field in space.

##  8\. The example of the Schwarzschild metric field

We now turn our attention to the Schwarzschild solution for the static
spherically symmetric field produced by a spherical symmetric body at rest:

(8.1)
$\begin{split}\mathrm{d}s^{2}&=\left(1-\frac{2m}{r}\right)\mathrm{d}t^{2}-\left(1-\frac{2m}{r}\right)^{-1}\mathrm{d}r^{2}-r^{2}\mathrm{d}\theta^{2}-r^{2}\sin^{2}\theta\mathrm{d}\phi^{2}\\\
&\equiv
U\mathrm{d}t^{2}-V\mathrm{d}r^{2}-r^{2}\mathrm{d}\theta^{2}-r^{2}\sin^{2}\theta\mathrm{d}\phi^{2}\,,\end{split}$

with appropriate units. Note that the metric for the de Sitter space-time with
horizon has the same form, just replace $2m/r$ with $\Lambda r^{2}/3$ in
$g_{00}$ and $g_{rr}$, where $\Lambda$ is the positive cosmological constant.
The cosmological horizon holds at $r=\sqrt{3/\Lambda}$.

Since the Schwarzschild solution holds outside the surface of the body that is
producing the field, i.e. for $2m<r<\infty$, and since $r=0$ and $r=2m$ are
singularities, pseudo for the latter since the determinant of the metric does
not cancel at $r=2m$, we have to examine if our approach is able to regularise
these singularities to a certain extent. Let us pick a regular isotropic
$\psi$ such that $|\psi|^{2}(x)\equiv|\psi|^{2}(t,r)$ and, consequently,
$R_{|\psi|^{2}|\psi|^{2}}(r)$ are probability distributions. Applying (6.6)
and (6.7) yields the metric multiplication operators

$\displaystyle(A_{U}s)(x)$
$\displaystyle=\left(U\ast|\psi|^{2}\right)(x)\,s(x)=\left(1-\frac{2m}{r}+2m\left\langle
Y_{r}(r^{\prime})\left(\frac{1}{r}-\frac{1}{r^{\prime}}\right)\right\rangle_{|\psi|^{2}}\right)s(x)$
(8.2) $\displaystyle\equiv\widetilde{U}_{|\psi|^{2}}(r)\,s(x)$
$\displaystyle(A_{V}s)(x)$
$\displaystyle=\left(V\ast|\psi|^{2}\right)(x)\,s(x)=\left(1+\frac{2m}{r}+\frac{2m}{r}\left\langle\frac{m}{r^{\prime}}\ln\frac{|r+r^{\prime}-2m|}{|r-r^{\prime}-2m|}\right\rangle_{|\psi|^{2}}+\right.$
$\displaystyle\left.+2m\left\langle
Y_{r}(r^{\prime})\left(\frac{1}{r^{\prime}}-\frac{1}{r}+\frac{m}{rr^{\prime}}\ln\frac{|r-r^{\prime}-2m|}{|r^{\prime}-r-2m|}\right)\right\rangle_{|\psi|^{2}}\right)s(x)$
(8.3)
$\displaystyle\equiv\widetilde{V}_{|\psi|^{2}}(r)\,s(x)\equiv\left(2-\widetilde{U}_{|\psi|^{2}}(r)+L_{|\psi|^{2}}(r)\right)\,s(x)$
(8.4) $\displaystyle(A_{r^{2}}s)(x)$
$\displaystyle=\left(r^{2}\ast|\psi|^{2}\right)(x)\,s(x)=\left(r^{2}+\langle{r^{\prime}}^{2}\
\rangle_{|\psi|^{2}}\right)s(x)\,,$ (8.5)
$\displaystyle(A_{r^{2}\sin^{2}\theta}s)(x)$
$\displaystyle=\left(r^{2}\sin^{2}\theta\ast|\psi|^{2}\right)(x)\,s(x)=\left(r^{2}\sin^{2}\theta+\frac{2}{3}\langle{r^{\prime}}^{2}\
\rangle_{|\psi|^{2}}\right)s(x)\,,$

after regularisation of logarithmic singularity through Cauchy principal value
in (8.3). We have primed the integration variable in the computation of
expected values in order to avoid the confusion with the external variable
$r=\|\boldsymbol{x}\|$, and we have introduced the Heaviside function:
$Y_{r}(r^{\prime}):=Y(r^{\prime}-r)$. Note that the $2/3$ in (8.5) is the
average of $\sin^{2}\theta$ on the sphere. Remind that the notation $\langle
f\rangle_{p}$ stands for the expected value of the function $f$ with respect
to the probability distribution $p$.

The corresponding semi-classical portraits have similar expressions provided
that we replace $|\psi|^{2}$ with $R_{|\psi|^{2}|\psi|^{2}}$:

(8.6) $\displaystyle\check{U}(x)$ $\displaystyle=\left(U\ast
R_{|\psi|^{2}|\psi|^{2}}\right)(x)=\widetilde{U}_{R_{|\psi|^{2}|\psi|^{2}}}(r)\,,$
(8.7) $\displaystyle\check{V}(x)$ $\displaystyle=\left(V\ast
R_{|\psi|^{2}|\psi|^{2}}\right)(x)=\widetilde{V}_{R_{|\psi|^{2}|\psi|^{2}}}(r)\,,$
(8.8) $\displaystyle\check{r^{2}}(x)$ $\displaystyle=\left(r^{2}\ast
R_{|\psi|^{2}|\psi|^{2}}\right)(x)=r^{2}+\langle{r^{\prime}}^{2}\
\rangle_{R_{|\psi|^{2}|\psi|^{2}}}\,,$ (8.9)
$\displaystyle\check{r^{2}\sin^{2}\theta}(x)$
$\displaystyle=\left(r^{2}\sin^{2}\theta\ast\mathbb{R}_{|\psi|^{2}|\psi|^{2}}\right)(x)=r^{2}\sin^{2}\theta+\frac{2}{3}\langle{r^{\prime}}^{2}\
\rangle_{R_{|\psi|^{2}|\psi|^{2}}}\,,$

Let us analyse the functions $\widetilde{U}_{p}(r)$ and
$\widetilde{V}_{p}(r)$, where $p=p(t,r)$ denotes a spatially isotropic
probability distribution on $\mathbb{R}^{4}$, as are $|\psi|^{2}$ or
$R_{|\psi|^{2}|\psi|^{2}}$. Let us write their expressions after rearranging
some terms:

(8.10) $\displaystyle\widetilde{U}_{p}(r)$
$\displaystyle=1-\frac{2m}{r}+2m\left\langle
Y_{r}(r^{\prime})\left(\frac{1}{r}-\frac{1}{r^{\prime}}\right)\right\rangle_{p}=1-\frac{2m}{r}\left\langle\mathds{1}_{[0,r]}(r^{\prime})\right\rangle_{p}-2m\left\langle
Y_{r}(r^{\prime})\frac{1}{r^{\prime}}\right\rangle_{p}\,,$ (8.11)
$\displaystyle\widetilde{V}_{p}(r)$
$\displaystyle=2-\widetilde{U}_{p}(r)+L_{p}(r)\,,$

with

(8.12) $\displaystyle L_{p}(r)$
$\displaystyle=\frac{2m^{2}}{r}\left\langle\frac{1}{r^{\prime}}\left[\ln\frac{|r^{\prime}+(r-2m)|}{|r^{\prime}-(r-2m)|}+Y_{r}(r^{\prime})\ln\frac{|r^{\prime}-(r-2m)|}{|r^{\prime}-(r+2m)|}\right]\right\rangle_{p}$
(8.13)
$\displaystyle=\frac{2m^{2}}{r}\left\langle\frac{1}{r^{\prime}}\left[\mathds{1}_{[0,r]}(r^{\prime})\ln\frac{|r^{\prime}+(r-2m)|}{|r^{\prime}-(r-2m)|}+Y_{r}(r^{\prime})\ln\frac{|r^{\prime}+(r-2m)|}{|r^{\prime}-(r+2m)|}\right]\right\rangle_{p}$

### Regularisation at Schwarzschild radius value

We note the regularisation of the metric at the classical singularity $r=2m$:

(8.14) $\displaystyle\widetilde{U}_{p}(2m)$ $\displaystyle=\left\langle
Y_{2m}(r^{\prime})\left(1-\frac{2m}{r^{\prime}}\right)\right\rangle_{p}>0\,,$
(8.15) $\displaystyle\widetilde{V}_{p}(2m)$
$\displaystyle=1+\left\langle\mathds{1}_{[0,2m]}(r^{\prime})\right\rangle_{p}+\left\langle
Y_{2m}(r^{\prime})\left(\frac{2m}{r^{\prime}}+\frac{m}{2r^{\prime}}\ln\frac{r^{\prime}}{|r^{\prime}-4m|}\right)\right\rangle_{p}>0\,.$

### Behaviour at large $r$

Trivially we have

(8.16)
$\widetilde{U}_{p}(r)\underset{r\to\infty}{\to}1\,,\quad\widetilde{V}_{p}(r)\underset{r\to\infty}{\to}1\,.$

### Behaviour at $r=0$

Less trivially, we have

(8.17) $\widetilde{U}_{p}(r)\underset{r\to
0}{\to}1-\left\langle\frac{2m}{r^{\prime}}\right\rangle_{p}\,,\quad\widetilde{V}_{p}(r)\underset{r\to
0}{\to}1+\left\langle\frac{2m}{r^{\prime}}\right\rangle_{p}+4m^{2}\left\langle\frac{1}{r^{\prime}(r^{\prime}-2m)}\right\rangle_{p}\,,$

where the last mean value should be understood in the sense of Cauchy
principal value, more precisely Hilbert transform, for the integral

(8.18)
$\left\langle\frac{1}{r^{\prime}(r^{\prime}-2m}\right\rangle_{p}=4\pi\int_{-\infty}^{+\infty}\mathrm{d}t^{\prime}\int_{0}^{+\infty}\mathrm{d}r^{\prime}\frac{r^{\prime}p(t^{\prime},r^{\prime})}{r^{\prime}-2m}=-4\pi^{2}\int_{-\infty}^{+\infty}\mathrm{d}t^{\prime}\,\operatorname{H}(Y(\cdot)(\cdot)p(t^{\prime},(\cdot))(2m)\,,$

with the notation

(8.19)
$\displaystyle\operatorname{H}(u(\cdot))(t)={\frac{1}{\pi}}\,\operatorname{p.v.}\int_{-\infty}^{+\infty}{\frac{u(\tau)}{t-\tau}}\;\mathrm{d}\tau\,.$

Now, the positiveness of the derivative of $\widetilde{U}_{p}$,

(8.20)
$\frac{\mathrm{d}\widetilde{U}_{p}}{\mathrm{d}r}(r)=\frac{2m}{r^{2}}\,\left\langle\mathds{1}_{[0,r]}(r^{\prime})\right\rangle_{p}>0\,,$

entails that $\widetilde{U}_{p}(r)$ is monotone increasing from
$U_{\mathrm{min}}=1-\left\langle\dfrac{2m}{r^{\prime}}\right\rangle_{p}$ (at
which the slope is infinite) to $1$. Hence one deals with two cases.

  1. (1)

$U_{\mathrm{min}}>0$, i.e.,
$\left\langle\dfrac{2m}{r^{\prime}}\right\rangle_{p}<1$. Then the temporal
term of the Schwarzschild metric is completely regularised.

  2. (2)

$U_{\mathrm{min}}\leq 0$, i.e.,
$\left\langle\dfrac{2m}{r^{\prime}}\right\rangle_{p}\geq 1$. Then there exists
a smaller “Schwarzschild radius” $r_{s0}\in(0,2m)$ for the temporal part,
defined by the equation

(8.21)
$r_{s0}=\frac{2m\left\langle\mathds{1}_{[0,r_{s0}]}(r^{\prime})\right\rangle_{p}}{1-2m\left\langle
Y_{r_{s0}}(r^{\prime})\frac{1}{r^{\prime}}\right\rangle_{p}}\,.$

The situation is less obvious for the radial metric term
$\widetilde{V}_{p}(r)$. Nevertheless, through a suitable choice of probability
distribution one can expect a full regularisation too.

A more developed analysis with explicit examples together with the
determination of the geodesics and of the stress-energy tensor
$\check{T}_{\mu\nu}$ resulting of this regularisation and its interpretation
will be addressed in a forthcoming work.

##  9\. Final discussion

Let us initiate the discussion by quoting the authors of [37]:

> … infinities in General relativity come in the form of singularities [38]
> and they point to the breakdown of our current understanding of gravity. The
> standard view in the community is that quantum gravity should be able to
> resolve this issue by smoothing out singularities. Nonetheless, despite the
> many existing approaches to quantum gravity, there is no consensus about
> what it is and how one should construct a quantum theory of spacetime, thus
> a proof of principle for the singularity avoidance is yet to be found.

The present work should be viewed as a partial contribution to this program.
We have presented a non-orthodox quantization which transforms functions on
the eight-dimensional phase space (time-space, frequency-wavevector) into
operators in the Hilbert space of signals on space-time, or equivalently, in
the Hilbert space of their Fourier transforms. Non-orthodox means that there
is no Planck constant involved in our approach, and so there is no associated
particle, no mass, no energy, no momentum associated with the resulting
quantum objects. We remind that with the Planck constant one enters the domain
of quantum mechanics and quantum field theory with its Fock space formalism.
Its fundamental role is to bridge two worlds, for instance the world of
classical waves with its phase space (time-space, frequency-wave vector) and
the quantum mechanics built from the classical phase space (space, momentum).

As a first exploration, we have considered the metric field of general
relativity, but we could as well examine the Maxwell electromagnetic field.
These fields are functions of space-time coordinates, but nothing prevents us
to extend their variable domain with the inclusion of frequencies and wave
vectors. The latter could be viewed as phenomenological quantities, analogous
to those introduced in various studies, e.g., waves in plasmas.

Concerning general relativity, we have shown that the quantization of a metric
field that is solution of the Einstein equation in empty space gives rise to
some regularised version of it and to a stress-energy tensor. Although the
latter could be thought as “fictitious”, we believe on the contrary that its
existence is unescapable due to our ignorance of an “exact” mathematical model
for describing gravitation. In this regard, non-emptiness of space-time is a
type of attribute which might appear to physicists as idealistic or
superfluous while it is unescapable as a completion of any mathematical model.

Finally, one could think that our approach of transforming space-time points
and metric fields into operators is parent of non-commutative geometry, as is
nicely presented, for instance, in [39]. Actually, our quantization of the
classical geometry of space-time yields a (formal) commutative algebra of
multiplication operators, formal meaning that we make abstraction of domain
considerations in dealing with products of such operators. On the other hand,
the algebra becomes non-commutative if we quantize extensions of the metric
field to functions depending on frequency and wave-vector.

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