# Coherent states for generalized uncertainty relations as Tsallis probability
amplitudes: new route to non-extensive thermostatistics

Petr Jizba<EMAIL_ADDRESS>FNSPE, Czech Technical University in Prague,
Břehová 7, 115 19, Prague, Czech Republic  ITP, Freie Universität Berlin,
Arnimallee 14, D-14195 Berlin, Germany  Gaetano Lambiase<EMAIL_ADDRESS>Dipartimento di Fisica, Università di Salerno, Via Giovanni Paolo II, 132
I-84084 Fisciano (SA), Italy  INFN, Sezione di Napoli, Gruppo collegato di
Salerno, Italy  Giuseppe Gaetano Luciano<EMAIL_ADDRESS>Applied Physics Section of Environmental Science Department, Escola
Politècnica Superior, Universitat de Lleida, Av. Jaume II, 69, 25001 Lleida,
Spain  Luciano Petruzziello<EMAIL_ADDRESS>INFN, Sezione di Napoli,
Gruppo collegato di Salerno, Italy  Dipartimento di Ingegneria, Università di
Salerno, Via Giovanni Paolo II, 132 I-84084 Fisciano (SA), Italy  Institut für
Theoretische Physik, Albert-Einstein-Allee 11, Universität Ulm, 89069 Ulm,
Germany

###### Abstract

We study coherent states associated to a generalized uncertainty principle
(GUP). We separately analyze the cases of positive and negative deformation
parameter $\beta$, showing that the ensuing probability distribution is a
Tsallis distribution whose non-extensivity parameter $q$ is monotonically
related to $\beta$. Moreover, for $\beta<0$ (corresponding to $q<1$), we
reformulate the GUP in terms of a one-parameter class of Tsallis entropy-power
based uncertainty relations, which are again saturated by the GUP coherent
states. We argue that this combination of coherent states with Tsallis entropy
offers a natural conceptual framework allowing to study quasi-classical regime
of GUP in terms of non-extensive thermodynamics. We substantiate our claim by
discussing generalization of Verlinde’s entropic force and ensuing
implications in the late-inflation epoch. Corresponding dependence of the
$\beta$ parameter on cosmological time is derived for the reheating epoch. The
obtained $\beta$ is consistent with values predicted by both string-theory
models and the naturalness principle. Further salient issues, including
derivation of new $\beta$-dependent expressions for the lowest possible value
of the spin and Immirzi parameter in Loop Quantum Gravity, and connection of
our proposal with the Magueijo–Smolin doubly special relativity are also
discussed. This article provides a more extended and comprehensive treatment
of our recent letter [Phys. Rev. D 105, L121501 (2022)].

##  I Introduction

The Heisenberg Uncertainty Principle (HUP) – the cornerstone of quantum
mechanics – provides an intrinsic limitation to the simultaneous knowledge of
position and momentum of any quantum system. While working successfully in
low-energy regime, it is expected to be modified approaching the Planck scale,
due to quantum gravitational effects possibly coming into play. Several models
of quantum gravity, such as String Theory, Loop Quantum Gravity, Quantum
Geometry and Doubly Special Relativity, have converged on the idea that the
HUP should be generalized so as to account for the emergence of a minimal
length at Planck scale. The ensuing uncertainty relations are typically
referred to as Generalized Uncertainty Principles (GUPs).

The simplest version of GUP can be obtained by adding a term quadratic in the
momentum uncertainty over the standard Heisenberg limit VenezGrossMende ;
Gross ; Ciafa ; Koni ; MM ; Kempf ; FS ; Adler2 ; Capoz ; MS1 ; MS2 ; Bojo ,
i.e.,

$\displaystyle\delta x\,\delta
p\,\geq\,\frac{\hbar}{2}\left(1\,+\,\beta\,\frac{\delta
p^{2}}{m_{p}^{2}}\right)\,,$ (1)

where $m_{p}=\sqrt{\hbar c/G}\approx 2.2\times 10^{-8}$ Kg is the Planck mass
and with $c=1$. The (dimensionless) $\beta$ parameter quantifies the departure
from HUP. Note that such term is not fixed by the theory, albeit it is
generally assumed to be of order unity VenezGrossMende ; Gross ; Ciafa ; Koni
. However, many theoretical Das ; Brau ; Pedram ; ScardCas ; Bosso ; QC ;
Scardigli:2018jlm ; Petroz ; DaRocha ; Buoninf ; CGgrav ; Chung:2019raj ;
fabian ; LucBosFri  and experimental Bruk ; AliTest ; Bawaj ; Pendu ; GravBar
; BossoLigo ; LucLuc ; Primord  studies are being developed to infer the
magnitude of $\beta$, as well as its sign Kempf ; MS1 ; Bojo ; CGgrav ; JKS ;
KLVY ; Ong ; illu ; illu2 . Clearly, the traditional quantum mechanical limit
is recovered for $\beta\rightarrow 0$ and/or $\delta p\ll m_{p}$.

The symbol $\delta$ appearing in Eq. (1) denotes uncertainty of a given
observable and it does not need to be a priori related to the standard
deviation. In fact, in the original Heisenberg relation $\delta$ can represent
Heisenberg’s “ungenauigkeiten” (i.e., error-disturbance uncertainties caused
by the back-reaction in simultaneous measurement of $x$ and $p$) or $\delta
p=\langle\psi||p||\psi\rangle\equiv\langle|p|\rangle_{\psi}$, see, e.g.
Maggiore:93 . Nevertheless, in cases when $\delta$ is identified with the
standard deviation (henceforth denoted by $\Delta$), the generalized
uncertainty principle (1) directly follows from the deformed commutation
relation (DCR)

$\displaystyle\left[\hat{x},\hat{p}\right]\,=\,i\hbar\left(1+\beta\,\frac{\hat{p}^{2}}{m_{p}^{2}}\right)\,,$
(2)

via the Cauchy–Schwarz inequality Schrodinger ; Robertson ; Muk , provided one
restricts the attention to mirror symmetric states satisfying
$\langle\hat{p}\rangle_{\psi}=0$. Commutator (2) is typically supplemented
with the commutators

$\displaystyle\left[\hat{x},\hat{x}\right]\,=\,\left[\hat{p},\hat{p}\right]\,=\,0\,,$
(3)

which, together with (2), satisfy Jacobi identities (up to the lowest-order
approximation in $\beta\delta p^{2}/m^{2}_{p}$) and determine the whole
symplectic structure of the model.

So far, the quadratic GUP (1) has been largely used to study phenomenology of
quantum gravity in many sectors, ranging from quantum mechanics Bernardo ;
Panella ; Ijmpd ; bpw  to particle physics Das ; Husain ; BossoLuciano  and
cosmology JizSc ; DFLV ; Giardino . Applications have also been studied in
non-linear optics and condensed matter Tawfik ; Braidotti ; iorio , where the
mass parameter $m_{p}$ should actually be identified with the effective mass
scale in the quantum description at hand. As an example, we might mention a
1$D$ lattice where the usual Weyl-Heisenberg algebra $W_{1}$ for $\hat{p}$ and
$\hat{x}$ operators is deformed to the Euclidean algebra $E(2)$, which entails
GUP with $m_{p}$ being related to an inverse lattice spacing at small $p$
(momenta deep inside the Brillouin zone). Conversely, for large momenta (near
the border of the Brillouin zone), such GUP drifts away from the quadratic GUP
and becomes linear in $p$ JKS .

On the other hand, comparably less attention has been devoted to the analysis
of the quasi-classical domain of the GUP. This regime has, however, relevant
and potentially observable implications in the early Universe cosmology and
ensuing astrophysics Kiefer ; Kiefer2 . In order to probe physics in the
quasi-classical domain, it is customary to rely on coherent states (CSs). CSs
are, in a sense, privileged quantum states in the description of quantum-to-
classical transition, as they are the only states that remain pure in the
decoherence process Dutra ; Matacz . Since CSs are pure, they allow for
maximal resolution in phase-space, thus appearing as the closest quantum
counterparts of classical points. Additionally, the CS formalism offers a
convenient description which can draw upon developments in quantum optics
Mandel .

To grasp the core aspects of GUP phenomenology in the quasi-classical quantum
regime, it should be emphasized that the deformed commutator (2) significantly
affects the phase-space structure of any quantum system, yielding non-trivial
implications at microscopic level. For instance, in Chang  it has been shown
that Eq. (2) modifies the elementary cell volume of each quantum state, which
becomes momentum-dependent. This motivates a reformulation of quantum
statistical mechanics when considering GUP. Preliminary attempts to derive the
statistics that emerges from the phase-space cell volume implied by the GUP
have been conducted in Shababi ; LucianoTs , where the generalized statistics
with a quadratic correction over Gaussian profile emerges quite naturally if
one assumes Eq. (2) along with the condition of invariance of the total phase-
space volume. The question thus arises as to how the interconnection between
quantum and statistical properties can be rigorously formalized in the quasi-
classical regime of GUP.

In order to tackle the aforementioned issues, we investigate in this work
possible observational effects of GUP systems in their decoherence domain. For
this purpose, we first introduce the Schrödinger–Nieto-type of minimum-
uncertainty CSs Schrodinger ; Nieto:78  associated with GUP. Subsequently, we
show that, in momentum representation, these states coincide with probability
amplitudes derived in Tsallis statistics, which is a non-extensive
generalization of Boltzmann–Gibbs statistics based on a non-additive
redefinition of the entropy Tsallis1 ; Tsallis2 ; Tsallis3 ; jizba-korbel:19 ;
Abe1 ; JKZ ; Caratheodory ; CaratheodoryII . Furthermore, by using the Bekner-
Babenko inequality, we recast the GUP for $\beta<0$ in terms of a one-
parameter class of Tsallis entropy-power-based uncertainty relations (EPUR),
which are also saturated by the GUP CSs. By invoking the Maximum Entropy
Principle (MEP), which states that the thermodynamic entropy is the
statistical entropy evaluated at the maximal entropy distribution, we infer
that the combination of GUP CSs with Tsallis entropy naturally allows us to
describe the quasi-classical domain of GUP in terms of non-extensive Tsallis
thermodynamics (NTT). To elucidate our point, we discuss three pertinent
examples from cosmology: a) the GUP generalization of Verlinde’s entropic
gravity force Verlinde  and its connection with conformal gravity (CG)
mannheim ; mannheim_a ; mannheim_b ; mannheim_c ; Kazanas  and early Universe
cosmology; b) Magueijo–Smolin doubly special relativity (DSR) and finally; c)
we derive new $\beta$-dependent expressions for the lowest possible value of
the spin and Immirzi parameter in Loop Quantum Gravity (LQG), discussing their
relevance for the gauge group structure of the spin networks in this theory.

The remainder of the work is organized as follows: in Sec. II we derive
coherent states for the quadratic GUP with both $\beta>0$ and $\beta<0$. A
large portion of the section is dedicated to exploring the properties of these
CSs. In Section III, we present some fundamentals of non-extensive Tsallis
thermodynamics. In particular, we pay a particular attention to an integrating
factor for the heat one-form and show that in contrast to conventional
thermodynamics it factorizes into thermal and entropic part. The connection
between GUP and Tsallis entropy-power based uncertainty relations is
investigated in Sec. IV, while in Sec. V we discuss some illustrative examples
from the early Universe cosmology. Conclusions and outlook are finally
summarized in Sec. VI. For the sake of clarity, we relegate some more
technical considerations to three appendices.

##  II Coherent states for GUP

We shall start with a short outline of the basic steps that lead from the DCR
(2) to GUP (1). To this end, we will assume that $\delta$ in (1) is identified
with the standard deviation computed with respect to some density matrix
$\varrho$.

For the variance, (i.e., square of the standard deviation) of an observable
$\mathcal{\hat{A}}$ we can write

$\displaystyle(\Delta\mathcal{\hat{A}})^{2}_{{\varrho}}$ $\displaystyle\equiv$
$\displaystyle\mbox{Tr}(\mathcal{\hat{A}}^{2}\hat{\varrho})-\mbox{Tr}(\mathcal{\hat{A}}\hat{\varrho})^{2}\
$ (4) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\left(\lambda-\langle\mathcal{\hat{A}}\rangle^{2}_{\varrho}\right)\
\\!d{\mbox{Tr}}\\!\left(E_{\lambda}^{(\mathcal{\hat{A}})}\\!\hat{\varrho}\right)\,,$

where $E_{\lambda}^{(\mathcal{\hat{A}})}$ represents the projection–valued
measure of $\mathcal{\hat{A}}$ corresponding to spectral value $\lambda$. If
we now confine ourselves to the canonical observables $\hat{x}$ and $\hat{p}$,
the passage from the DCR (2) to GUP (1) is as follows: we first set
$\mathcal{\hat{A}}=\hat{x}-\langle\hat{x}\rangle_{\varrho}$ and
$\mathcal{\hat{B}}=\hat{p}-\langle\hat{p}\rangle_{\varrho}$ so that
$(\Delta{{x}})^{2}_{\varrho}=\langle\mathcal{\hat{A}}^{2}\rangle_{\varrho}$,
$(\Delta{{p}})^{2}_{\varrho}=\langle\mathcal{\hat{B}}^{2}\rangle_{\varrho}$
and
$[\hat{x},\hat{p}]\hat{\varrho}=[\mathcal{\hat{A}},\mathcal{\hat{B}}]\hat{\varrho}$;
then, for an arbitrary vector $\psi\in{\mbox{Ran}}\hat{\varrho}$ and a generic
$\gamma\in\mathbb{R}$, we have

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle|\\!|(\mathcal{\hat{B}}-i\gamma\mathcal{\hat{A}})\psi|\\!|^{2}$
(5) $\displaystyle=$
$\displaystyle\langle\psi|\mathcal{\hat{B}}^{2}|\psi\rangle+i\gamma\langle\psi|[\mathcal{\hat{A}},\mathcal{\hat{B}}]|\psi\rangle+\gamma^{2}\langle\psi|\mathcal{\hat{A}}^{2}|\psi\rangle\,.$

Thus,

$\displaystyle\mbox{Tr}(\mathcal{\hat{B}}^{2}\hat{\varrho})+i\gamma\mbox{Tr}([\mathcal{\hat{A}},\mathcal{\hat{B}}]\hat{\varrho})+\gamma^{2}\mbox{Tr}(\mathcal{\hat{A}}^{2}\hat{\varrho})\
\geq\ 0\,.$ (6)

The LHS is smallest for
$\gamma=i\mbox{Tr}([\mathcal{\hat{B}},\mathcal{\hat{A}}]\hat{\varrho})/(2\mbox{Tr}(\mathcal{\hat{A}}^{2}\hat{\varrho}))$,
which brings (6) to the form

$\displaystyle\mbox{}\mbox{Tr}(\mathcal{\hat{A}}^{2}\hat{\varrho})\mbox{Tr}(\mathcal{\hat{B}}^{2}\hat{\varrho})=(\Delta{{x}})^{2}_{\varrho}(\Delta{{p}})^{2}_{\varrho}\
\geq\ \frac{1}{4}\mbox{Tr}(i[\hat{x},\hat{p}]\hat{\varrho})^{2}.$ (7)

This is nothing but the quantum mechanical version of the covariance
inequality known from probability calculus Muk . By employing (2), we now
obtain

$\displaystyle(\Delta{{x}})_{\varrho}(\Delta{{p}})_{\varrho}\ \geq\
\frac{\hbar}{2}\left(1+\beta\,\frac{(\Delta{{p}})^{2}_{\varrho}+\langle\hat{p}\rangle^{2}_{\varrho}}{m_{p}^{2}}\right).$
(8)

If the density matrix $\varrho$ is mirror symmetric (i.e.,
$\langle\hat{p}\rangle_{\varrho}=0$), then the inequality (8) indeed coincides
with the GUP (1).

To find $\hat{\varrho}$ that saturates the GUP (8), we observe from (5) that
the inequality is saturated if and only if for all
$\psi\in{\mbox{Ran}}\hat{\varrho}$ the equation
$(\mathcal{\hat{B}}-i\gamma\mathcal{\hat{A}})|\psi\rangle=0$ holds. If this
equation has more than one solution for given $\gamma$,
$\langle\hat{x}\rangle_{\varrho}$ and $\langle\hat{p}\rangle_{\varrho}$, the
corresponding $\varrho$ with the minimum uncertainty is a mixture of CSs
(i.e., pure GUP-saturating states). Unless otherwise specified, we will
further assume that $\langle\hat{x}\rangle_{\varrho}=x_{0}=0$, since $x_{0}$
will only affect the phase factor in CSs (see Section II.7).

It is apparent that on the class of mirror symmetric $\varrho$’s, the equation

$\left(\hat{p}-i\gamma\hat{x}\right)|\psi\rangle\ =\ 0\,,$ (9)

admits only one solution for $\psi\in L^{2}(\mathbb{R})$ [cf., e.g., Eq.
(12)], so that the minimum–uncertainty $\hat{\varrho}$ is a pure (coherent)
state. It is convenient to seek the solution to (9) in the momentum
representation, _i.e_ , $|\psi\rangle\mapsto\psi(p)=\langle p|\psi\rangle$. In
the momentum space, $\hat{x}$ and $\hat{p}$ satisfying DCR can be represented
as Kempf

$\displaystyle\hat{p}\,\psi(p)\ =\ p\,\psi(p)\,,$
$\displaystyle\hat{x}\,\psi(p)\ =\
i\hbar\left(1+\beta\,\frac{p^{2}}{m_{p}^{2}}\right)\frac{d}{dp}\psi(p)\,.$
(10)

However, in doing so, the non-symmetric nature of $\hat{x}$ would produce an
inconsistent variance for the ensuing CS. For this reason, we resort to
another representation of $\hat{x}$ and $\hat{p}$ complying with (2), namely

$\displaystyle\mbox{}\hat{p}\,\psi(p)\ =\ p\,\psi(p)\,,$
$\displaystyle\mbox{}\hat{x}\,\psi(p)\ =\
i\hbar\left(\frac{d}{dp}+\frac{\beta}{2\,m_{p}^{2}}\,\
\left\\{p^{2},\frac{d}{dp}\right\\}\right)\psi(p)\,,$ (11)

with $\\{~{},~{}\\}$ representing the anticommutator (more details on the
self-adjointness of this operator are discussed in the accompanying letter
JLLP ). With this, we can cast (9) into an equivalent form

$\frac{d}{dp}\,\psi(p)\ =\
-\frac{\left(1+\frac{\beta\gamma\hbar}{m_{p}^{2}}\right)}{\gamma\hbar\left(1+\beta\frac{p^{2}}{m_{p}^{2}}\right)}\,p\,\psi(p)\,,$
(12)

whose generic solution is

$\displaystyle\psi(p)\ =\
N\left[1+({\beta\,p^{2}})/{m_{p}^{2}}\right]_{+}^{-\frac{m_{p}^{2}}{2\beta\gamma\hbar}-\frac{1}{2}}\,,$
(13)

which represents a two-parameter class of CSs with parameters $\beta$ and
$\gamma$. Here, $[z]_{+}=\max\\{z,0\\}$, which guarantees that the wave
functions (13) are single-valued. At this stage, it is important to
distinguish two qualitatively different situations, namely $\beta>0$ and
$\beta<0$.

###  II.1 Positive $\beta$ case

The coefficient $N$ in (13) ensures that $\int|\psi(p)|^{2}dp=1$ and for
$\beta>0$

$N_{{}_{>}}\ =\
\sqrt{\sqrt{\frac{\beta}{m^{2}_{p}\pi}}\,\frac{\Gamma\left(\frac{m_{p}^{2}}{\beta\gamma\hbar}+1\right)}{\Gamma\left(\frac{m_{p}^{2}}{\beta\gamma\hbar}+\frac{1}{2}\right)}}\,,$
(14)

with $\Gamma(x)$ being the conventional Euler’s gamma function.

$\delta x$$\delta p$allowed region ($\beta>0$)$(\delta x)_{\rm{min}}$

Figure 1: For $\beta>0$ the GUP implies a “minimal length” $(\delta x)_{\rm
min}=\hbar\sqrt{\beta}/m_{p}$.

It should be stressed that the GUP inequality (1) with $\beta>0$ predicts the
existence of a unique minimal $\delta x$, such that

$\displaystyle(\delta x)_{\rm min}\ =\ \hbar\sqrt{\beta}/m_{p}\ =\
\sqrt{\beta}\ \\!\ell_{p}\,,$ (15)

where $\ell_{p}=\sqrt{\hbar G/c^{3}}\approx 10^{-33}$cm is the Planck length
(see Fig. 1). As argued in Ref. Kempf , a minimal uncertainty in position
means that the position operator is no longer essentially self-adjoint, but
only symmetric. As a matter of fact, one might appropriately restrict the
definition region of $\hat{x}$ so that the deficiency index is $(0,0)$, which
consequently leads to a one-parameter class of self-adjoint extensions of the
operator $\hat{x}$ on $L^{2}(\mathbb{R})$, cf. also Ref. JLLP . Unfortunately,
the spectrum of such $\hat{x}$ operators is discrete, which is not compatible
with the $\beta>0$ case. Indeed, the expectation value of the deformed
commutation relation (2) with respect to any eigenstate of $\hat{x}$ gives a
zero left-hand side of (2), while the right-hand side is always non-zero for
$\beta>0$. Such a situation would not happen should both $\hat{p}$ and
$\hat{x}$ have a continuous spectrum, because the corresponding eigenstates do
not belong to the domain of the commutator. On the other hand, the expectation
value of the right-hand side of Eq. (2) can be zero but only if $\beta<0$,
which however would contradict our original assumption $\beta>0$. A rescue
strategy that is often employed in this context Kempf  is to keep $\hat{x}$
being merely symmetric (which ensures that all expectation values are real)
while giving up on self-adjointness. In turn, this opens the way for the
introduction of minimal positional uncertainties.

The aforementioned mismatch between a discrete spectrum of the operator
$\hat{x}$ and the existence of $(\delta x)_{\rm min}$ leads to yet another
interesting point. In contrast to conventional lattice discretization of
space, the aforementioned $(\delta x)_{\rm min}$ does not introduce any UV
cutoff scale in the momentum space that would be proportional to the inverse
lattice spacing. In other words, the GUP with $\beta>0$ does not provide any
universal upper bound for $\delta p$, which instead naturally appears in the
GUP with negative $\beta$ (see Fig. 2).

###  II.2 Negative $\beta$ case

In this case, the CS (13) can be rewritten in the following, equivalent form

$\displaystyle\psi(p)\ =\
N_{{}_{<}}\left[1-({|\beta|\,p^{2}})/{m_{p}^{2}}\right]_{+}^{\frac{m_{p}^{2}}{2|\beta|\gamma\hbar}-\frac{1}{2}}\,,$
(16)

where

$\displaystyle N_{{}_{<}}\ =\
\sqrt{\sqrt{\frac{|\beta|}{m^{2}_{p}\pi}}\,\frac{\Gamma\left(\frac{1}{2}+\frac{m_{p}^{2}}{|\beta|\gamma\hbar}\right)}{\Gamma\left(\frac{m_{p}^{2}}{|\beta|\gamma\hbar}\right)}}.$
(17)

The physical scenario with $\beta<0$ has been less explored in literature than
the $\beta>0$ case, though the related GUP has a number of relevant
implications in cosmology JKS ; CGgrav , astrophysics Ong  and DSR MS1 ; MS2
and it is also mathematically better behaved. Note that for $\beta<0$, Eq.
(13) involves noninteger power of negative reals, which generally leads to
multi-valued CS. Because wave functions must be single-valued, CS has to have
bounded support, which means that $\hat{p}$ must be bounded with spectrum
$|\sigma(\hat{p})|\leq m_{p}/\sqrt{|\beta|}$. The ensuing operator $\hat{x}$
corresponding to the formal differential expression (11) is self-adjoint and
has a continuous spectrum JLLP . Consequently, none of the eigenvectors of
$\hat{x}$ belong to the Hilbert space
$L^{2}((-m_{p}/\sqrt{|\beta|},m_{p}/\sqrt{|\beta|}))$ (just like none of the
eigenvectors of $-i\hbar\nabla$ belong to the Hilbert space
$L^{2}(\mathbb{R}^{d})$). Instead, they belong to the space
${{\mathcal{S}}}^{\prime}((-m_{p}/\sqrt{|\beta|},m_{p}/\sqrt{|\beta|}))$,
namely the space of complex-valued tempered distributions. Hence, there is a
spectral transition from discrete to continuous spectrum when $\beta$ becomes
negative.

$\delta x$$\delta p$allowed region ($\beta<0$)$(\delta p)_{\rm{max}}$

Figure 2: For $\beta<0$ the GUP implies a “maximal momenta” $(\delta p)_{\rm
max}=m_{p}/\sqrt{|\beta|}$.

In passing, we pinpoint that, since eigenstates are now outside of the Hilbert
space, one can avoid subtleties with deformed commutation relations
encountered in the $\beta>0$ case.

As a final remark, let us observe that as $\beta\rightarrow 0$, both (13) and
(16) reduce to the usual minimum-uncertainty Gaussian wave-packet (Glauber
coherent state) associated with the conventional Heisenberg uncertainty
relations.

###  II.3 Physical meaning of $\gamma$ parameter

While the role of $\beta$ as the GUP deformation parameter is quite clear, the
role of $\gamma$ is less obvious. In order to find a meaning for $\gamma$, we
note that (9) implies

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\langle\psi|\left(\hat{p}+i\gamma\hat{x}\right)\left(\hat{p}-i\gamma\hat{x}\right)|\psi\rangle$
(18) $\displaystyle=$ $\displaystyle(\Delta
p)^{2}+i\gamma\langle\psi|\left[\hat{x},\hat{p}\right]|\psi\rangle+\gamma^{2}(\Delta
x)^{2}$ $\displaystyle=$ $\displaystyle(\Delta p)^{2}-\gamma\
\\!|\langle\psi|\left[\hat{x},\hat{p}\right]|\psi\rangle|+\gamma^{2}\frac{|\langle\psi|\left[\hat{x},\hat{p}\right]|\psi\rangle|^{2}}{4(\Delta
p)^{2}}\,,~{}~{}~{}~{}$

where in the last line we used the fact that (8) is saturated. The above
expression has a single solution for $\gamma$, which with the help of (2) can
be cast as

$\displaystyle\gamma\ =\ \frac{2(\Delta p)^{2}}{\hbar\left[1\ +\
\beta\,{(\Delta p)^{2}}/{m_{p}^{2}}\right]}\,.$ (19)

By realizing that (19) is valid for CSs, we can rewrite it in a more succinct
form as

$\displaystyle\gamma^{2}\ =\ \frac{(\Delta p)^{2}_{CS}}{(\Delta
x)^{2}_{CS}}\,.$ (20)

We observe that $\gamma$ can be defined also in the limit $\Delta
p\rightarrow\infty$, even though GUP (8) is in such a case meaningless. Note
also that CSs (13) are indeed mirror symmetric states and they satisfy
$\langle\hat{p}\rangle_{CS}=\langle\hat{x}\rangle_{CS}=0$. The relation (19)
will play an important role in Section II.6.

###  II.4 Another look at the CS (13)

To better understand the structure of (13), we rephrase the two-parameter
class of CSs in terms of another two parameters (say $q$ and $b$) so that

$\displaystyle q\ =\
\frac{\beta\gamma\hbar}{m^{2}_{p}+\beta\gamma\hbar}+1\,,\qquad b\ =\
\frac{2m_{p}}{\gamma\hbar}+\frac{2\beta}{m_{p}}\,.$ (21)

This relation is supposed to be valid for $\beta\lessgtr 0$. In particular, we
can observe with the help of (19) that, for fixed variance $\Delta p$, the $q$
parameter is a monotonically increasing function of $\beta$ provided
$\beta\neq-m_{p}^{2}/[3(\Delta p)^{2}]$, in which case the value of $q$ is
undefined. Should we have no deformation (i.e., $\beta=0$), then $q=1$.

The previous reparametrization allows us to set (13) in the form

$\displaystyle\psi(p)\ =\
{N}_{{}_{\lessgtr}}\left[1-b\,(1-q)\frac{p^{2}}{2m_{p}}\right]^{\frac{1}{2(1-q)}}_{+}\,.$
(22)

This is nothing but the probability amplitude for the Tsallis distribution of
a free non-relativistic particle

$\displaystyle\mbox{}q_{{}_{T}}(p|q,b)=|\psi(p)|^{2}=\frac{1}{Z}\left[1-b\,(1-q)\frac{p^{2}}{2m_{p}}\right]^{\frac{1}{1-q}}_{+}\\!\\!,$
(23)

with $Z={N}_{{}_{\lessgtr}}^{-2}$ being the “partition function”.

###  II.5 Toward a relativistic generalization

It is interesting to point out that Eq. (13) can be brought also into a
relativistic-like form. To see this, we perform another substitution, namely

$\displaystyle q\ =\
\frac{\beta\gamma\hbar}{m^{2}_{p}+\beta\gamma\hbar}+1\,,\;\;\;\tilde{b}\ =\
\frac{b\,m_{p}}{1+b\,(1-q)\,m_{p}}\,.$ (24)

This choice yields

$\displaystyle\mbox{}\psi(p)$ $\displaystyle=$
$\displaystyle{\tilde{N}}_{1}\left[1-\tilde{b}(1-q)\left(\frac{p^{2}}{2m^{2}_{p}}+1\right)\right]^{\frac{1}{2(1-q)}}\,.$
(25)

The last relation is particularly useful, since we can identify it with the
leading-order approximation of the function

$\displaystyle\mbox{}\psi(p)$ $\displaystyle\sim$
$\displaystyle\tilde{N}_{2}\left[1-\tilde{b}\,(1-q)\sqrt{1+\frac{p^{2}}{m^{2}_{p}}}\right]^{\frac{1}{2(1-q)}}$
(26) $\displaystyle\stackrel{{\scriptstyle q\rightarrow
1}}{{\longrightarrow}}$
$\displaystyle\tilde{N}_{3}\exp\left(-\frac{1}{2\theta}\,\sqrt{1+\frac{p^{2}}{m^{2}_{p}}}\right),$

for $p^{2}\ll m_{p}^{2}$. The corresponding probability distribution

$\displaystyle\mbox{}q_{J}(p|\theta)$ $\displaystyle=$
$\displaystyle|\psi(p)|^{2}$ (27) $\displaystyle\approx$
$\displaystyle\frac{1}{2\,m_{p}K_{1}(1/\theta)}\
\\!\exp\left(-\frac{1}{\theta}\,\sqrt{1+\frac{p^{2}}{m^{2}_{p}}}\right)\\!.$

is nothing but the Maxwell–Jüttner distribution for relativistic particles,
with $\theta\ =\ {kT}/{m_{p}}\ =\ \gamma\hbar/{(2m_{p}^{2})}$ being the
temperature in Planck units. Here and in subsequent reasonings we adopt the
convention that $c=1$. At this point, we note that:

  * •

should we have considered 3D momentum, then the normalization factor would
involve the Bessel function $K_{2}$ instead of $K_{1}$;

  * •

should we have required the Maxwell–Jüttner probability amplitude to be an
exact coherent state of the GUP, we would need to modify Eq. (1). The present
form of the GUP is only the leading-order approximation towards
Maxwell–Jüttner type of coherent states.

In the following, we will focus only on the form of GUP CSs (23). GUP with
exact Maxwell–Jüttner type of CSs will be discussed in our future work (see
also the discussion in Sec. VI for more comments on this issue).

###  II.6 Some essentials about Tsallis distribution

A few remarks concerning (23) are now in order. Tsallis distribution of this
type is also known as $q$-Gaussian distribution and denoted as
$\exp_{q}(-b\,p^{2}/2\,m_{p})$. In the limit $q\rightarrow 1$,
$\exp_{q}(-b\,p^{2}/2\,m_{p})\rightarrow\exp(-b\,p^{2}/2\,m_{p})$. We have
already pointed this out, since $q\rightarrow 1$ is equivalent to
$\beta\rightarrow 0$. In addition, because the momentum is unbounded for
$q\geq 1$ (i.e. $\beta\geq 0$), the distribution (23) is normalizable only for
values of $1\leq q<3$. For values $q<1$ (which means $\beta<0$), the
$q$-Gaussian distribution needs to be set to zero for
$p>\sqrt{2\,m_{p}/b\,(1-q)}$, as only in such cases the argument in the square
brackets of Eq. (23) is not negative. Therefore, for $\beta<0$ the
corresponding coherent states must have a finite support.

Moreover, for $q\geq 5/3$ the variance of (23) is undefined (infinite) and
thus the GUP cannot even be formulated. However, when $q<5/3$, then $(\Delta
p)^{2}=2\,m_{p}/b\,(5-3q)$ (see for example Ref. Thistleton ). The latter
identity implies that the parameter $\gamma$ and $(\Delta p)^{2}$ must be
related, namely

$\displaystyle\mbox{}(\Delta p)^{2}=\frac{2\,m_{p}}{b\,(5-3q)}\ \Rightarrow\
\gamma=\frac{2(\Delta p)^{2}}{\hbar\left[1\ +\ \beta{(\Delta
p)^{2}}/{m_{p}^{2}}\ \\!\right]}\,,$ (28)

which precisely coincides with the result (19). This, in turn, justifies our
choice of the representation of $\hat{x}$ and $\hat{p}$ operators. Should we
have started with the representation (10) instead, then we would have arrived
at the Tsallis-type coherent states with slightly different parameters than in
(13). The ensuing variance would have then implied

$\displaystyle\gamma\ =\ \frac{2(\Delta p)^{2}}{\hbar\left[1\ +\
3\beta{(\Delta p)^{2}}/{m_{p}^{2}}\ \\!\right]}\,,$ (29)

which is clearly incompatible with (19).

Furthermore, it should be stressed that the mean value does not exist for
$q>2$, so such coherent states cannot be mirror symmetric as assumed. In the
upcoming considerations, we will thus restrict our attention to the physical
domain of interest $q<5/3$ with a particular focus on the negative $\beta$,
which entails $q<1$.

In passing, we emphasize that the Tsallis distribution is equivalent to the
Student-$t$ distribution, which is used in statistical hypothesis testing, but
it also arises naturally in situations with small sample sizes (namely
$2/(1-q)$) or samples with unknown standard deviations.

###  II.7 The connection of states (22) with generalized coherent states

Canonical (or Glauber’s) CSs known from the (multimode) quantum harmonic
oscillator satisfy three basic properties Mandel ; BJV ; perelomov : 1) they
are eigenstate of lowering operators, 2) they are generated via translation
(or displacement) operators and 3) they are minimum uncertainty states. There
exists a huge variety of the so called generalized CSs, which maintain only
some of the above three conditions. For instance, generalizations based on
translation operators were developed mostly by Perelomov perelomov  for
systems with group-related fundamental commutation relations (hence the name
group-related or Perelomov CSs). Coherent states which are eigenstates of
lowering operators were generalized mostly by Barut and Girardello BG , again
in the group-theoretic context (hence the name Barut–Girardello’s CSs).
Finally, CSs that saturate uncertainty relations were generalized e.g. by
Nieto et al. nieto  by following an analogy with Schrödinger’s original
definition of CSs (hence they are called Nieto’s or Schrödinger-Nieto’s CSs).
Accordingly, our CSs belong conceptually to the same class of generalized CSs
as the Schrödinger–Nieto CSs.

From a mathematical-physics point of view, any set of would-be CSs qualifies
as a family of (generalized) coherent states if it meets the following two
properties KlauderIIa :

  * •

all elements of the set are strongly continuous functions of their label
variables;

  * •

there exists a measure on the label space such that the unit operator admits
the resolution of unity.

We will now demonstrate that the Tsallis probability amplitudes (13) can be
incorporated into a set of generalised CSs that saturate the uncertainty
relation (8).

Without assuming mirror symmetry and the auxiliary constraint
$\langle\hat{x}\rangle_{\psi}=x_{0}=0$, we observe that (8) is saturated by
states satisfying

$\displaystyle(\mathcal{\hat{B}}-i\gamma\mathcal{\hat{A}})|\psi\rangle\ =\
\left[\hat{p}-p_{0}-i\gamma(\hat{x}-x_{0})\right]|\psi\rangle\ =\ 0\,,$ (30)

where $p_{0}$ and $x_{0}$ are expectation values of $\hat{p}$ and $\hat{x}$ in
the state $|\psi\rangle$, respectively. The corresponding differential
equation has the solution (for concreteness we focus on the $\beta<0$ case)

$\displaystyle\mbox{}\psi(p;p_{0},x_{0})\ \equiv\ \langle
p|\psi,p_{0},x_{0}\rangle$ $\displaystyle\mbox{}=Ne^{-\frac{\varpi
m_{p}}{\hbar\gamma\sqrt{|\beta|}}\mbox{\footnotesize{arctanh}}(p\sqrt{|\beta|}/m_{p})}\left[1-\frac{{|\beta|\,p^{2}}}{m_{p}^{2}}\right]_{+}^{\frac{m_{p}^{2}}{2|\beta|\gamma\hbar}-\frac{1}{2}}\\!,~{}$
(31)

where $\varpi=i\gamma x_{0}-p_{0}$ and $N$ is a normalization constant.

Two comments are now in order: a) the actual value of $x_{0}$ in (31) does not
affect the probability distribution (23), as it only appears in the phase
factor, and b) the parameter $\gamma$ in (30)-(31) is now explicitly dependent
on $p_{0}$. As for the last point, by following the same procedure as in
Section II.3, we arrive at

$\displaystyle\gamma=\frac{2(\Delta p)^{2}}{\hbar\left\\{1\ +\
\beta{\left[(\Delta p)^{2}+p_{0}^{2}\right]}/{m_{p}^{2}}\ \\!\right\\}}\ =\
\frac{\Delta p}{\Delta x}\,.$ (32)

The latter can be written formally as $(\Delta p)_{CS}/(\Delta x)_{CS}$, which
emphasises that the variances have been calculated with respect to states that
satisfy the GUP (8).

When we adhere to the usual HUP and fix $(\Delta p)_{CS}$ and $(\Delta
x)_{CS}$, we can see that $\gamma$ is fixed while the expectation value of
momenta, $\langle\hat{p}\rangle_{\psi_{CS}}=p_{0}$, is unrestricted (it does
not appear explicitly in HUP). In the present case when we fix $(\Delta
p)_{CS}$ and $(\Delta x)_{CS}$, we still get fixed $\gamma$ but due to (8)
$p_{0}$ is not free anymore — it is fixed as well. This makes it difficult to
define measure on the label space $\\{x_{0},p_{0}\\}$ that would in the limit
$|\beta|\rightarrow 0$ converge to Glauber’s measure. So, rather than fixing
$(\Delta p)_{CS}$ and $(\Delta x)_{CS}$, we fix instead directly $\gamma$
(which fixes only fraction of $(\Delta p)_{CS}$ and $(\Delta x)_{CS}$ but not
their respective values) in which case the GUP (8) does not restrict $p_{0}$
to some specific value. In fact, it will be shortly seen that this strategy
will allow us to define a measure on the label space $\\{x_{0},p_{0}\\}$ with
a correct limiting behavior.

We note, in passing, that in the limit $|\beta|\rightarrow 0$, Eq. (31)
reduces to the usual minimal uncertainty Schrödinger wave packet

$\displaystyle\mbox{}\psi(p;p_{0},x_{0})|_{\beta\rightarrow 0}\ \propto\
\exp\left[\frac{ix_{0}(p_{0}-p)}{\hbar}-\frac{(p-p_{0})^{2}}{2\gamma\hbar}\right]$
$\displaystyle\mbox{\hskip 14.22636pt}=\
\exp\left[\frac{ix_{0}(p_{0}-p)}{\hbar}-\frac{(p-p_{0})^{2}}{4(\Delta
p)^{2}}\right],~{}~{}~{}~{}~{}~{}$ (33)

where in the second line we used the fact that $\gamma|_{|\beta|\rightarrow
0}=2(\Delta p)^{2}/\hbar\equiv 2(\Delta p)^{2}_{CS}/\hbar$ [cf., Eq. (32)].

When $\gamma$ is fixed, it is not difficult to see that states (31) are
strongly continuous functions of the label variables $x_{0}$ and $p_{0}$. A
strong continuity means KlauderIIa  that for every convergent label set such
that $\\{p_{0}^{\prime},x_{0}^{\prime}\\}\rightarrow\\{p_{0},x_{0}\\}$, the
distance between the two corresponding quantum states
$|\\!||\psi,p_{0}^{\prime},x_{0}^{\prime}\rangle-|\psi,p_{0},x_{0}\rangle|\\!|\rightarrow
0$, with $|\\!||\psi\rangle|\\!|=\sqrt{\langle\psi|\psi\rangle}$.

Now, since

$\displaystyle\mbox{}|\\!||\psi,p_{0}^{\prime},x_{0}^{\prime}\rangle-|\psi,p_{0},x_{0}\rangle|\\!|^{2}$
$\displaystyle=\
|\\!||\psi,p_{0}^{\prime},x_{0}^{\prime}\rangle|\\!|^{2}+|\\!||\psi,p_{0},x_{0}\rangle|\\!|^{2}$
$\displaystyle\mbox{\hskip
8.53581pt}-2\mbox{Re}\langle\psi,p_{0}^{\prime},x_{0}^{\prime}|\psi,p_{0},x_{0}\rangle\,,$
(34)

the strong continuity of the vectors is thus a simple consequence of the
continuity of the matrix element of any two CSs in their label variables. In
the present case we have

$\displaystyle\mbox{}\langle\psi,p_{0}^{\prime},x_{0}^{\prime}|\psi,p_{0},x_{0}\rangle$
$\displaystyle\mbox{}=\ |N|^{2}\int_{-m/\sqrt{|\beta|}}^{m/\sqrt{|\beta|}}dp\
\\!e^{-\frac{(\varpi^{\prime*}+\varpi)m_{p}}{\hbar\gamma\sqrt{|\beta|}}\
\\!\mbox{\footnotesize{arctanh}}(p\sqrt{|\beta|}/m_{p})}$
$\displaystyle\mbox{}\times\left[1-\frac{{|\beta|\,p^{2}}}{m_{p}^{2}}\right]_{+}^{\frac{m_{p}^{2}}{|\beta|\gamma\hbar}-1}$
$\displaystyle\mbox{}=\
|N|^{2}\frac{m_{p}}{\sqrt{|\beta|}}\int^{\infty}_{-\infty}dz\
\\!e^{-\frac{(\varpi^{\prime*}+\varpi)m_{p}}{\hbar\gamma\sqrt{|\beta|}}z}\
\\!\left(\cosh z\right)^{-a}\\!.~{}~{}~{}~{}~{}$ (35)

Here $|N|^{2}$ is a shorthand notation for $N(p_{0})N^{*}(p_{0}^{\prime})$ and
$a=\frac{2m_{p}^{2}}{\hbar\gamma|\beta|}$. Since $\left(\cosh z\right)^{-a}$
is absolutely integrable over ${\mathbb{R}}$ and decreases at infinity faster
than any power of $|z|^{-1}$ [in fact faster than $\exp(-|z|a)$], we may
differentiate under the integral sign any number of times. By virtue of the
dominated convergence theorem, we can write for
$|p_{0}|,|p_{0}^{\prime}|<m_{p}/\sqrt{|\beta|}$

$\displaystyle\mbox{}\langle\psi,p_{0}^{\prime},x_{0}^{\prime}|\psi,p_{0},x_{0}\rangle$
$\displaystyle=$
$\displaystyle|N|^{2}\frac{m_{p}}{\sqrt{|\beta|}}e^{i\frac{(p_{0}^{\prime}+p_{0})m_{p}}{\hbar\gamma\sqrt{|\beta|}}\frac{\partial}{\partial
y}}\left.\int_{-\infty}^{\infty}dz\ \\!e^{iyz}\ \\!\left(\cosh
z\right)^{-a}\right|_{y=\frac{(x_{0}^{\prime}-x_{0})m_{p}}{\hbar\sqrt{|\beta|}}}$
(36) $\displaystyle=$
$\displaystyle|N|^{2}\frac{m_{p}}{\sqrt{|\beta|}}e^{i\frac{(p_{0}^{\prime}+p_{0})m_{p}}{\hbar\gamma\sqrt{|\beta|}}\frac{\partial}{\partial
y}}\left.\left(\frac{2^{a-1}}{\Gamma(a)}\left|\Gamma\left(\frac{a}{2}+i\frac{y}{2}\right)\right|^{2}\right)\right|_{y=\frac{(x_{0}^{\prime}-x_{0})m_{p}}{\hbar\sqrt{|\beta|}}}$
$\displaystyle=$
$\displaystyle\frac{\Gamma\left(\frac{a}{2}-\frac{(p_{0}^{\prime}+p_{0})m_{p}}{2\hbar\gamma\sqrt{|\beta|}}+i\frac{(x_{0}^{\prime}-x_{0})m_{p}}{2\hbar\sqrt{|\beta|}}\right)\Gamma\left(\frac{a}{2}+\frac{(p_{0}^{\prime}+p_{0})m_{p}}{2\hbar\gamma\sqrt{|\beta|}}-i\frac{(x_{0}^{\prime}-x_{0})m_{p}}{2\hbar\sqrt{|\beta|}}\right)}{\sqrt{\Gamma\left(\frac{a}{2}-y\right)\Gamma\left(\frac{a}{2}+y\right)}\sqrt{\Gamma\left(\frac{a}{2}-y^{\prime}\right)\Gamma\left(\frac{a}{2}+y^{\prime}\right)}}\,.~{}~{}~{}$

Here $y=\frac{p_{0}m_{p}}{\hbar\gamma\sqrt{|\beta|}}$ and
$y^{\prime}=\frac{p_{0}^{\prime}m_{p}}{\hbar\gamma\sqrt{|\beta|}}$. In the
second line we have used the Ramanujan formula Ramanujan .

We now employ the fact that for $\mbox{Re}(\zeta)>0$, the gamma function
$\Gamma(\zeta)$ is a continuous function of its argument and satisfies
$[\Gamma(\zeta)]^{*}=\Gamma(\zeta^{*})$. This, in turn, implies that the
matrix elements of any two CSs are continuous in their label variables. The
latter is particularly true for a subclass of states with $p_{0}=0$, i.e.,
mirror symmetric states.

To prove the existence of the integral measure $\mu(x_{0},p_{0})$ for the
resolution of unity, we need to show that there exists $\mu(x_{0},p_{0})$ such
that

$\displaystyle\int_{-\infty}^{\infty}dx_{0}\int_{-m_{p}/\sqrt{|\beta|}}^{m_{p}/\sqrt{|\beta|}}dp_{0}\
\\!\mu(x_{0},p_{0})\ \\!\langle
p^{\prime}|\psi,p_{0},x_{0}\rangle\langle\psi,x_{0},p_{0}|p\rangle\ =\
\delta(p^{\prime}-p)\,.$ (37)

The LHS of (37) can be explicitly written in the form

$\displaystyle\int_{-\infty}^{\infty}dx_{0}\int_{-m_{p}/\sqrt{|\beta|}}^{m_{p}/\sqrt{|\beta|}}dp_{0}\
\\!\mu(x_{0},p_{0})\ \\!|N(p_{0})|^{2}\
\\!e^{i\frac{x_{0}m_{p}}{\hbar\sqrt{|\beta|}}\
\\!\left[\mbox{\footnotesize{arctanh}}(p^{\prime}\sqrt{|\beta|}/m_{p})-\mbox{\footnotesize{arctanh}}(p\sqrt{|\beta|}/m_{p})\right]}$
$\displaystyle\mbox{\hskip 28.45274pt}\times\
e^{\frac{p_{0}m_{p}}{\hbar\gamma\sqrt{|\beta|}}\
\\!\left[\mbox{\footnotesize{arctanh}}(p^{\prime}\sqrt{|\beta|}/m_{p})+\mbox{\footnotesize{arctanh}}(p\sqrt{|\beta|}/m_{p})\right]}\
\\!\left[\left(1-\frac{{|\beta|\,p^{\prime
2}}}{m_{p}^{2}}\right)\left(1-\frac{{|\beta|\,p^{2}}}{m_{p}^{2}}\right)\right]^{\frac{m_{p}^{2}}{2|\beta|\gamma\hbar}-\frac{1}{2}}\,,$
(38)

where

$\displaystyle|N(p_{0})|^{2}\ =\
\frac{\sqrt{|\beta|}}{m_{p}}\frac{\Gamma\left(a\right)}{2^{a-1}}\
\\!\left[\Gamma\left(\frac{a}{2}-y\right)\Gamma\left(\frac{a}{2}+y\right)\right]^{-1}\\!.~{}~{}$
(39)

The latter goes to zero when $p_{0}$ reaches its bounding values $\pm
m_{p}/\sqrt{|\beta|}$. In passing we note that Stirling’s approximation
implies the correct limiting behavior

$\displaystyle|N(p_{0})|^{2}|_{|\beta|\rightarrow
0}=\frac{\exp\left(-\frac{p_{0}^{2}}{\gamma\hbar}\right)}{\sqrt{\pi\gamma\hbar}}=\frac{\exp\left(-\frac{p_{0}^{2}}{2(\Delta
p)^{2}}\right)}{\sqrt{2\pi(\Delta p)^{2}}}\,.~{}~{}~{}$ (40)

Since (38) equals to $\delta(p^{\prime}-p)$, $\mu(x_{0},p_{0})$ must depend
only on $p_{0}$. This allows to perform the integration over $x_{0}$ and
rewrite (38) in the form

$\displaystyle\mbox{}2\pi\hbar\int_{-m_{p}/\sqrt{|\beta|}}^{m_{p}/\sqrt{|\beta|}}dp_{0}\
\\!\mu(p_{0})\ \\!|N(p_{0})|^{2}\
\\!e^{\frac{2p_{0}m_{p}}{\hbar\gamma\sqrt{|\beta|}}z}$ $\displaystyle\times\
\left(\cosh z\right)^{-\frac{2m_{p}^{2}}{\hbar\gamma|\beta|}}\\!\
\delta(p^{\prime}-p)$ $\displaystyle\mbox{}=\
\frac{2\pi\hbar^{2}\gamma\sqrt{|\beta|}}{m_{p}}\int_{-a/2}^{a/2}dy\
\\!{\tilde{\mu}(y)}\ \\!|\tilde{N}(y)|^{2}\ \\!e^{2yz}$ $\displaystyle\times\
\left(\cosh z\right)^{-a}\ \\!\delta(p^{\prime}-p)\,.$ (41)

Here we have set $z=\mbox{{arctanh}}(p\sqrt{|\beta|}/m_{p})$,
$\tilde{\mu}(y)=\mu(p_{0})$ and $\tilde{N}(y)=N(p_{0})$. So, the measure
$\mu(p_{0})$ can be obtained by taking the inverse of the finite Laplace
transform of $(\cosh z)^{a}$. After some analysis (cf. Appendix C for
technical details) we obtain

$\displaystyle\mbox{}\mu(x_{0},p_{0})\ =\ \mu(p_{0})\ =\ \tilde{\mu}(y)$
$\displaystyle\mbox{}=\
\frac{1}{2\pi\hbar}\frac{1}{\left[1-\left(\frac{2y}{a}\right)^{\\!2}\right]}\
\\!\sum_{k=0}^{\infty}\delta(\pm y+a/2-k)\,,$ (42)

with $y\in(-a/2,a/2)$. Both sign conventions are admissible. So, the measure
is discrete in the $p_{0}$ variable while continuous in the $x_{0}$ variable.
Note that in (38) the Cauchy principal value integral should be utilised in
the $p_{0}$ integration in order to see that the endpoint singularities in
(42) are integrable. In Appendix C we show that in the limit
$|\beta|\rightarrow 0$ we regain the conventional (Glauber’s) CS measure.

Let us close this subsection by noting that the UR-saturating CSs belong to
the class of so-called pointer states, i.e. those states that are least
affected by the interaction with the environment zurek1 ; zurek2 ; zurek3 . In
fact, Schrödinger’s CSs maximize Shannon–Gibbs entropy subject to prior data
one possesses about a system, namely the first two moments of the position and
momentum variables. Since the latter indirectly reflect system’s environment,
the probability distribution assigned is the least prone to losing quantum
coherence, compared to other pure-state distributions that fulfill the same
prior data. Such CSs are particularly pertinent in the quasi-classical domain
of quantum theory, as they are maximally predictable (or robust) despite of
decoherence zurek3 ; zurek4 .

##  III Non-extensive thermodynamics relations

It is clear that the probability distributions (23) and (25) decay
asymptotically following a power law rather than an exponential law. If we
keep variance and mean as the only statistical observables, power-law type
distributions are incompatible with the conventional MEP applied to
Shannon–Gibbs entropy. On the other hand, distributions (23) and (25) are
maximizers for Tsallis (differential) entropy (TE) Tsallis2 ; JKZ , i.e., for
entropic functional of the form

$\displaystyle{\mathcal{S}}_{q}^{T}(\mathcal{F})$ $\displaystyle=$
$\displaystyle\frac{k_{T}}{(1-q)}\left(\int_{\mathbb{R}}d{p}\,\mathcal{F}^{q}(p)-1\right),$
(43)

where $\mathcal{F}$ is a probability density function. The constant $k_{T}$
may in general depend on $q$, and becomes the Boltzmann constant $k_{B}\approx
1.380649\times 10^{23}$JK-1 in the limit $q\rightarrow 1$. Moreover, it can
easily be deduced (by L’Hôpital’s rule) that in the limit $q\rightarrow 1$,
the TE tends to the Shannon–Gibbs’ entropy

$\displaystyle{\mathcal{S}}^{S}(\mathcal{F})$ $\displaystyle=$ $\displaystyle-
k_{B}\int_{\mathbb{R}}d{p}\ \\!\mathcal{F}(p)\ln\mathcal{F}(p)\,.$ (44)

The coherent state distribution $\exp_{q}\left(-bp^{2}/m_{p}\right)$ is the
maximizer of the entropy ${\mathcal{S}}_{2-q}^{T}(\mathcal{F})$ subject to a
second moment constraint Tsallis1 ; Tsallis2 ; JKZ  and it tends to a Gaussian
distribution for $q\rightarrow 1$.

Given the importance of TE and NTT in following sections, we now derive the
prerequisite NTT relations. Although these results have already been addressed
in the literature (e.g. Ref. Tsallis2 ), it s instructive to derive them in an
alternative and more systematic manner via Caratheodory’s theorem.

We first observe that ${\mathcal{S}}_{q}^{T}$ satisfies the following non-
additivity law for two independent subsystems (say $A$ and $B$) Tsallis1 ;
Tsallis2 :

$\displaystyle\mbox{}{\mathcal{S}}_{q}^{T}(A,B)\\!$ $\displaystyle=$
$\displaystyle\\!{\mathcal{S}}_{q}^{T}(A)\ +\ {\mathcal{S}}_{q}^{T}(B)\ +\
\frac{1-q}{k_{T}}\ \\!{\mathcal{S}}_{q}^{T}(A){\mathcal{S}}_{q}^{T}(B)$ (45)
$\displaystyle=$
$\displaystyle\\!{k_{T}}\\!\left[\left(\frac{1}{k_{T}}{\mathcal{S}}_{q}^{T}(A)\right)\oplus_{q}\left(\frac{1}{k_{T}}{\mathcal{S}}_{q}^{T}(B)\right)\right],~{}~{}$

where the symbol $\oplus_{q}$ denotes the $q$-deformed “sum”, which is defined
as in Ref. Tsallis2

$\displaystyle x\oplus_{q}y\ =\ x\ +\ y\ +\ (1-q)xy\,.$ (46)

In (45) we have employed the short-hand notation
${\mathcal{S}}_{q}^{T}(A)\equiv{\mathcal{S}}_{q}^{T}(\mathcal{F}_{A})$ and
${\mathcal{S}}_{q}^{T}(A,B)\equiv{\mathcal{S}}_{q}^{T}(\mathcal{F}_{A}\cdot\mathcal{F}_{B})$,
with $\mathcal{F}_{X}$ being a probability density associated with the
(continuous) random variable describing the system $X$.

Let us now briefly outline the passage to thermodynamics based on Tsallis
entropy. In particular, we will derive the connection between heat one-form
and Tsallis entropy (analogue of the Clausius relation for reversible heat
exchange) that will be needed in Section V.2. To this end, we start by
considering two systems ($A$ and $B$) in contact (both thermal and mechanical)
with each other. Suppose that these have volumes $V(A)$ and $V(B)$ and
internal energies $U_{q}(A)$ and $U_{q}(B)$, and that each volume and number
of particles (as well as the combined internal energy and total volume) are
fixed.

In thermodynamic equilibrium, the total entropy ${\mathcal{S}}_{q}^{T}(A,B)$
must be maximal. By using (45) we thus have

$\displaystyle\mbox{}0\ =\ d{\mathcal{S}}_{q}^{T}(A,B)$ $\displaystyle=$
$\displaystyle\left[\left(1+\frac{1-q}{k_{T}}{\mathcal{S}}_{q}^{T}(B)\right)\left(\frac{\partial{\mathcal{S}}_{q}^{T}(A)}{\partial
U_{q}(A)}\right)_{V(A)}-\left(1+\frac{1-q}{k_{T}}{\mathcal{S}}_{q}^{T}(A)\right)\left(\frac{\partial{\mathcal{S}}_{q}^{T}(B)}{\partial
U_{q}(B)}\right)_{V(B)}\right]dU_{q}(A)$ (47) $\displaystyle+$
$\displaystyle\left[\left(1+\frac{1-q}{k_{T}}{\mathcal{S}}_{q}^{T}(B)\right)\left(\frac{\partial{\mathcal{S}}_{q}^{T}(A)}{\partial
V(A)}\right)_{U_{q}(A)}-\left(1+\frac{1-q}{k_{T}}{\mathcal{S}}_{q}^{T}(A)\right)\left(\frac{\partial{\mathcal{S}}_{q}^{T}(B)}{\partial
V(B)}\right)_{U_{q}(B)}\right]dV(A)\,,$

where we have employed the fact that the total internal energy and volume are
fixed

$\displaystyle U_{q}(A,B)\ =\ U_{q}(A)\ +\ U_{q}(B)\ =\ \mbox{const.}\,,$ (48)
$\displaystyle V(A,B)\ =\ V(A)\ +\ V(B)\ =\ \mbox{const.}\,.$ (49)

We have also assumed that TE ${\mathcal{S}}_{q}^{T}$ is expressed in terms of
entropy’s natural state variables, i.e. $U_{q}$ and $V$.

From (47), we obtain the two identities, which reflect the fact that when a
system is in thermodynamic equilibrium, then it is simultaneously in thermal
and mechanical equilibrium. The first identity one can be written in the form

$\displaystyle\frac{k_{T}\beta(A)}{1+((1-q)/k_{T}){\mathcal{S}}_{q}^{T}(A)}$
$\displaystyle=$
$\displaystyle\frac{k_{T}\beta(B)}{1+((1-q)/k_{T}){\mathcal{S}}_{q}^{T}(B)}$
(50) $\displaystyle=$ $\displaystyle k_{T}\beta^{*}\,,$

where (by analogy with conventional extensive thermodynamics) we have defined

$k_{T}\beta\ =\ \left(\frac{\partial{\mathcal{S}}_{q}^{T}}{\partial
U_{q}}\right)_{V}\,.$ (51)

In connection with (50), it should be emphasized that the physical temperature
is not $(k_{T}\beta)^{-1}$ (as would be the case at $q\rightarrow 1$), but
rather

$\displaystyle\vartheta\ =\ \frac{1}{k_{T}\beta^{*}}\ =\
\left(1+\frac{1-q}{k_{T}}{\mathcal{S}}_{q}^{T}\right)\frac{1}{k_{T}\beta}\,.$
(52)

Equation (50) basically represents the 0th law of thermodynamics, which
ensures that one may assign the same observable empirical temperature
$\vartheta$ to all sub-systems in thermal equilibrium. We will shortly see
that the empirical temperature $\vartheta$ can be identified with the
“absolute” temperature $T_{\rm{phys}}$.

The second identity can be cast as

$\displaystyle\frac{\left({\partial{\mathcal{S}}_{q}^{T}(A)}/{\partial
V(A)}\right)_{U_{q}(A)}}{1+((1-q)/k_{T}){\mathcal{S}}_{q}^{T}(A)}$
$\displaystyle=$
$\displaystyle\frac{\left({\partial{\mathcal{S}}_{q}^{T}(B)}/{\partial
V(B)}\right)_{U_{q}(B)}}{1+((1-q)/k_{T}){\mathcal{S}}_{q}^{T}(B)}$ (53)
$\displaystyle=$ $\displaystyle\frac{p_{\rm{phys}}}{\vartheta}\,,$

Eq. (53) reflects the fact that when two systems are in mechanical equilibrium
their pressures are the same. This allows to identify physical pressure
$p_{\rm{phys}}$ as

$\displaystyle p_{\rm{phys}}\ =\
\frac{\vartheta}{1+((1-q)/k_{T}){\mathcal{S}}_{q}^{T}}\left(\frac{\partial{\mathcal{S}}_{q}^{T}}{\partial
V}\right)_{\\!U_{q}}\,.$ (54)

Note that in the limit $q\rightarrow 1$ the conventional relation is
recovered.

Let us now see what is a thermodynamic variable conjugate to
${\mathcal{S}}_{q}^{T}$. In fact, Caratheodory theorem Caratheodory ;
CaratheodoryII  ensures that heat one-form has an integration factor, but
since the entropy is not additive one cannot use the conventional Carnot cycle
argument Huang  in the proof of Clausius equality to simply identify the
integration factor with the inverse temperature.

Let us dwell a bit more on this last point. Let us first assume that heat
$Q_{q}$ is additive in the same way as internal energy [cf. relation (48)]. We
also pass from entropy’s natural variables to general state variables
$\\{{\bf{a}},\vartheta\\}$. Here ${\bf{a}}$ will represent a collection of
relevant state variables and $\vartheta$ is the empirical temperature whose
existence is guaranteed by the 0th law of thermodynamics [see Eq. (52)].

Since, by Caratheodory theorem the exact differential associated with the heat
one-form is the entropy, one can write

$\displaystyle d{\mathcal{S}}_{q}^{T}({\bf{a}},\vartheta)\ =\
\mu({\bf{a}},\vartheta)\delta Q_{q}({\bf{a}},\vartheta)\,,$ (55)

We now divide the analyzed system into two subsystems $A$ and $B$, which are
respectively described by state variables $\\{{\bf{a}}_{A},\vartheta\\}$ and
$\\{{\bf{a}}_{B},\vartheta\\}$. Consequently

$\displaystyle\delta Q_{q}({\bf{a}}_{A},\vartheta)\ =\
\frac{1}{\mu_{A}({\bf{a}}_{A},\vartheta)}\
\\!d{\mathcal{S}}_{q}^{T}({\bf{a}}_{A},\vartheta)\,,$ $\displaystyle\delta
Q_{q}({\bf{a}}_{B},\vartheta)\ =\ \frac{1}{\mu_{B}({\bf{a}}_{B},\vartheta)}\
\\!d{\mathcal{S}}_{q}^{T}({\bf{a}}_{B},\vartheta)\,.$ (56)

So, for the whole system

$\displaystyle\delta Q_{q}(A,B)\ =\ \delta Q_{q}(A)\ +\ \delta Q_{q}(B)\,,$
(57)

with

$\displaystyle\delta Q_{q}(A,B)\ \equiv\ \delta
Q_{q}({\bf{a}}_{A},{\bf{a}}_{B},\vartheta)\ =\
\frac{1}{\mu_{A+B}({\bf{a}}_{A},{\bf{a}}_{B},\vartheta)}\
\\!d{\mathcal{S}}_{q}^{T}({\bf{a}}_{A},{\bf{a}}_{B},\vartheta)\,,~{}~{}~{}~{}$
(58)

and we can write

$\displaystyle d{\mathcal{S}}_{q}^{T}({\bf{a}}_{A},{\bf{a}}_{B},\vartheta)\ =\
\frac{\mu_{A+B}({\bf{a}}_{A},{\bf{a}}_{B},\vartheta)}{\mu_{A}({\bf{a}}_{A},\vartheta)}\
\\!d{\mathcal{S}}_{q}^{T}({\bf{a}}_{A},\vartheta)\ +\
\frac{\mu_{A+B}({\bf{a}}_{A},{\bf{a}}_{B},\vartheta)}{\mu_{B}({\bf{a}}_{B},\vartheta)}\
\\!d{\mathcal{S}}_{q}^{T}({\bf{a}}_{B},\vartheta)\,.$ (59)

Let us now assume that there is only one state variable (apart from
$\vartheta$), so that ${\bf{a}}=a$. If there were more state variables, our
following argument would hold true as well, but we would need to employ more
than two subsystems. Assuming we can invert
${\mathcal{S}}_{q}^{T}({{a}}_{A},\vartheta)$ and
${\mathcal{S}}_{q}^{T}({{a}}_{B},\vartheta)$, we can express $a_{A}$ and
$a_{B}$ as

$\displaystyle a_{A}\ =\
a_{A}({\mathcal{S}}_{q}^{T}(A),\vartheta)\;\;\;\mbox{and}\;\;\;a_{B}\ =\
a_{B}({\mathcal{S}}_{q}^{T}(B),\vartheta)\,.$ (60)

With this, Eq. (59) can be cast into form

$\displaystyle
d{\mathcal{S}}_{q}^{T}({\mathcal{S}}_{q}^{T}(A),{\mathcal{S}}_{q}^{T}(B),\vartheta)\
$ $\displaystyle=$ $\displaystyle\
\frac{\mu_{A+B}({\mathcal{S}}_{q}^{T}(A),{\mathcal{S}}_{q}^{T}(B),\vartheta)}{\mu_{A}({\mathcal{S}}_{q}^{T}(A),\vartheta)}\
\\!d{\mathcal{S}}_{q}^{T}(A)\ +\
\frac{\mu_{A+B}({\mathcal{S}}_{q}^{T}(A),{\mathcal{S}}_{q}^{T}(B),\vartheta)}{\mu_{B}({\mathcal{S}}_{q}^{T}(B),\vartheta)}\
\\!d{\mathcal{S}}_{q}^{T}(B)\ +\ 0\ \\!d\vartheta\,.~{}~{}$ (61)

Since $d{\mathcal{S}}_{q}^{T}$ (for all considered systems, i.e. $A$, $B$ and
$A+B$) must be an exact differential, so that ${\mathcal{S}}_{q}^{T}$ is a
proper state function, integrability conditions give the following set of
equations

$\displaystyle\frac{\partial\log(\mu_{A}({\mathcal{S}}_{q}^{T}(A),\vartheta))}{\partial\vartheta}\
=\
\frac{\partial\log(\mu_{B}({\mathcal{S}}_{q}^{T}(B),\vartheta))}{\partial\vartheta}\
=\
\frac{\partial\log(\mu_{A+B}({\mathcal{S}}_{q}^{T}(A),{\mathcal{S}}_{q}^{T}(B),\vartheta))}{\partial\vartheta}\,,$
(62)
$\displaystyle\frac{1}{\mu_{A}({\mathcal{S}}_{q}^{T}(A),\vartheta)}\frac{\partial\mu_{A+B}({\mathcal{S}}_{q}^{T}(A),{\mathcal{S}}_{q}^{T}(B),\vartheta)}{\partial{\mathcal{S}}_{q}^{T}(B)}\
=\
\frac{1}{\mu_{B}({\mathcal{S}}_{q}^{T}(B),\vartheta)}\frac{\partial\mu_{A+B}({\mathcal{S}}_{q}^{T}(A),{\mathcal{S}}_{q}^{T}(B),\vartheta)}{\partial{\mathcal{S}}_{q}^{T}(A)}\,.$
(63)

Note that the expression (62) implies that the derivatives cannot depend on
entropy, but only on $\vartheta$. By denoting the RHS of (62) as
$-w(\vartheta)$, we might resolve (62) in the form

$\displaystyle\mbox{}\mu_{A}({\mathcal{S}}_{q}^{T}(A),\vartheta)\ =\
\psi_{A}({\mathcal{S}}_{q}^{T}(A))\ \\!\exp\left(-\int
w(\vartheta)d\vartheta\right)\,,$
$\displaystyle\mbox{}\mu_{B}({\mathcal{S}}_{q}^{T}(B),\vartheta)\ =\
\psi_{B}({\mathcal{S}}_{q}^{T}(B))\ \\!\exp\left(-\int
w(\vartheta)d\vartheta\right)\,,$
$\displaystyle\mbox{}\mu_{A+B}({\mathcal{S}}_{q}^{T}(A),{\mathcal{S}}_{q}^{T}(B),\vartheta)$
$\displaystyle\mbox{}=\
\psi_{A+B}({\mathcal{S}}_{q}^{T}(A),{\mathcal{S}}_{q}^{T}(B))\
\\!\exp\left(-\int w(\vartheta)d\vartheta\right),~{}~{}$ (64)

where $\psi$ are some arbitrary functions of the entropy. It is worth
stressing $\mu$ factorized into purely entropic and purely temperature-based
parts. In addition, the temperature part of $\mu$ is not explicitly
$q$-dependent.

Let us now observe from (45) that

$\displaystyle\mbox{}d{\mathcal{S}}_{q}^{T}({\mathcal{S}}_{q}^{T}(A),{\mathcal{S}}_{q}^{T}(B),\vartheta)\
=\ d{\mathcal{S}}_{q}^{T}(A,B)$ $\displaystyle=\ \left(1+\frac{1-q}{k_{T}}\
\\!{\mathcal{S}}_{q}^{T}(B)\right)\ \\!d{\mathcal{S}}_{q}^{T}(A)$
$\displaystyle+\ \left(1+\frac{1-q}{k_{T}}\
\\!{\mathcal{S}}_{q}^{T}(A)\right)\ \\!d{\mathcal{S}}_{q}^{T}(B)\,.$ (65)

By comparing this equation with (61) and (64), we can make the identification
$\psi({\mathcal{S}}_{q}^{T})=c\left(1+\frac{1-q}{k_{T}}\
\\!{\mathcal{S}}_{q}^{T}\right)$, with $c$ being an arbitrary constant. At
this point, one can easily check that also the second integrability condition
(63) is satisfied. In conventional thermodynamics, $\psi$ would be only a
constant, and thus the inverse integration factor could be identified with a
genuine absolute temperature. In the context of non-additive entropy
${\mathcal{S}}_{q}^{T}$, we see that this is no longer the case. Fortunately,
$\mu$ has a simple factorized form, which allows us to rephrase (55) as

$\displaystyle d{\mathcal{S}}_{q}^{T}\ =\ \left(1+\frac{1-q}{k_{T}}\
\\!{\mathcal{S}}_{q}^{T}\right)\frac{\delta Q_{q}}{T_{\rm{phys}}}\,.$ (66)

Here we, have denoted the temperature part of $\mu$ in (64) with
$1/T_{\rm{phys}}$. Therefore,

$\displaystyle T_{\rm{phys}}\ =\ \frac{1}{c}\ \\!\exp\left(\int
w(\vartheta)d\vartheta\right)\,,$ (67)

plays the role of an absolute temperature in Tsallis thermostatistics. Note
that by writing [as in Eqs. (52) and (54)]

$\displaystyle d{\mathcal{S}}_{q}^{T}$ $\displaystyle=$
$\displaystyle\left(\frac{\partial{\mathcal{S}}_{q}^{T}}{\partial
U_{q}}\right)_{V}dU_{q}\ +\
\left(\frac{\partial{\mathcal{S}}_{q}^{T}}{\partial V}\right)_{U_{q}}dV$ (68)
$\displaystyle=$ $\displaystyle\
\left(1+\frac{1-q}{k_{T}}{\mathcal{S}}_{q}^{T}\right)\frac{1}{\vartheta}\
\\!dU_{q}$ $\displaystyle+$ $\displaystyle\
\left(1+\frac{1-q}{k_{T}}{\mathcal{S}}_{q}^{T}\right)\frac{p_{\rm{phys}}}{\vartheta}\
\\!dV\,,$

we obtain

$\displaystyle\frac{\vartheta}{T_{\rm{phys}}}\ \\!\delta Q_{q}\ =\ dU_{q}\ +\
p_{\rm{phys}}dV\,.$ (69)

The first law of thermodynamics (energy conservation) appears when we identify
the empirical temperature introduced in (52) and (54) with $T_{\rm{phys}}$. In
turn, this implies $w(\vartheta)=1/\vartheta$. Additionally, when
$T_{\rm{phys}}$ has the same units of the temperature, then $c=1$. Relations
(66) and (69) were first derived in Ref. Abe1  using a different approach.

Tsallis (or non-extensive) thermodynamics, which is implied by (66) and (69),
is expected to be instrumental in the characterization of statistical systems
with long-range interactions Abe1  and/or strong correlations jizba-korbel:19
.

Before concluding this section, we point out that (66) can be equivalently
rewritten as

$\displaystyle\delta Q_{q}\ $ $\displaystyle=$ $\displaystyle\
\frac{k_{T}T_{\rm{phys}}}{1-q}\ \\!d\ln\left(1+\frac{1-q}{k_{T}}\
\\!{\mathcal{S}}_{q}^{T}\right)$ (70) $\displaystyle=$ $\displaystyle
T_{\rm{phys}}\ \\!d{\mathcal{S}}_{q}^{R}\,,$

where ${\mathcal{S}}_{q}^{R}$ is the so-called Rényi entropy Renyi

$\displaystyle{\mathcal{S}}_{q}^{R}(\mathcal{F})$ $\displaystyle=$
$\displaystyle\frac{k_{T}}{(1-q)}\ln\left(\int_{\mathbb{R}}d{p}\,\mathcal{F}^{q}(p)\right)\,,$
(71)

that plays an important role in both classical and quantum information theory
JD:16 ; JDJ ; JDP . Unless otherwise specified, we will set in the following
$k_{T}=1$.

##  IV GUP and Tsallis entropy-power based uncertainty relations

###  IV.1 Shannon entropy power

By relying on the notion of entropic functional (in this case Tsallis entropic
functional) that is extremized by CSs, one can reformulate the variance-based
GUP (1) in terms of entropic uncertainty relations. This can be done in line
with conventional quantum mechanics Hirschman , where the key aspect is that
Glauber CSs saturate Heisenberg’s uncertainty relation (UR) and at the same
time maximize Shannon’s entropy. The most compact form of the ensuing entropic
UR can be phrased in terms of (Shannon) entropy power JD:16 ; JDJ ; JDP .

Entropy power (EP) was originally introduced by C. Shannon in order to solve a
number of information-theoretic problems related to continuous random
variables Shannon . In its essence, EP describes the variance of a would-be
Gaussian random variable with the same Shannnon entropy as the random variable
under investigation. Hence, by denoting Shannon’s EP of a continuous
$D$-dimensional random variable $\mathcal{X}$ as $N(\mathcal{X})$, EP should
satisfy the defining relation

$\displaystyle{\mathcal{S}}^{S}(\mathcal{X})\ =\
{\mathcal{S}}^{S}(\sqrt{N(\mathcal{X})}\cdot{\mathcal{Z}}^{{}_{G}})\,,$ (72)

where $\\{{\mathcal{Z}}_{i}^{{}_{G}}\\}$ represents a $D$-dimensional Gaussian
random vector with zero mean and unit covariance matrix. Equation (72) has the
unique solution Shannon

$\displaystyle N(\mathcal{X})\ =\ \frac{1}{2\pi
e}\exp\left(\frac{2}{D}{\mathcal{S}}^{S}(\mathcal{X})\right)\,.$ (73)

Conventional canonical commutation relation $[\hat{x},\hat{p}]=i\hbar$ implies
that eigenstates of $\hat{x}$ in momentum representation or $\hat{p}$ in
position representation are plane waves. This, in turn, dictates that the $x$
and $p$-representation wave functions $\psi(x)$ and $\hat{\psi}(p)$,
respectively, must be related via Fourier transform (here $\hat{\psi}(p)$
should not be mistaken with an operator). If we now apply the Beckner–Babenko
inequality for Fourier-transform duals Beckner1975 ; Babenko1962 , i.e.

$\displaystyle\mbox{}\left[\left(\frac{q^{\prime}}{2\pi\hbar}\right)^{D}\right]^{1/q^{\prime}}|\\!||\psi|^{2}|\\!|_{q^{\prime}/2}$
$\displaystyle\mbox{\hskip 28.45274pt}\leq\
\left[\left(\frac{q}{2\pi\hbar}\right)^{D}\right]^{1/q}|\\!||\hat{\psi}|^{2}|\\!|_{q/2}\,,$
(74)

(where $|\\!|X|\\!|_{p}$ is the $p$-norm, and $1/q+1/q^{\prime}=1$ with
$q\in{\mathbb{R}}^{+}$, i.e., $q^{\prime}$ and $q$ are Hölder conjugates), and
set $q=q^{\prime}=2$, we obtain the entropy power inequality

$\displaystyle N(|\hat{\psi}|^{2})N(|{\psi}|^{2})\ \geq\
\frac{\hbar^{2}}{4}\,.$ (75)

It is important to note that this saturates only for CSs JDJ ; JDP ; Hirschman
; BB . Moreover, it can be shown JD:16  that (75) automatsubsumes the
variance-based Heisenberg uncertainty relation. The fact that uncertainty
relation (75) is phrased entirely in terms of Shannon’s entropy is, in part, a
reason why Shannon’s entropy plays such a key role in conventional quantum
mechanics and quantum information theory.

###  IV.2 Entropy powers based on Tsallis distribution

When dealing with GUP saturated by Tsallis probability amplitude states, it is
convenient to work with EP based on Tsallis distribution (23). By emulating
the procedure outlined in the previous subsection, we define the entropy
powers associated with Tsallis entropies as solutions of the equation

$\displaystyle{\mathcal{S}}_{q}^{T}\left({\mathcal{X}}\right)\ =\
{\mathcal{S}}_{q}^{T}\left(\sqrt{M_{q}^{T}(\mathcal{X})}\cdot{\mathcal{Z}}^{T}\right)\,,$
(76)

where $\\{{\mathcal{Z}}_{i}^{T}\\}$ represents a Tsallis random vector with
zero mean and unit covariance matrix. Such a vector is distributed with
respect to the $q$-Gaussian probability density function that extremizes
${\mathcal{S}}_{q}^{T}$. To solve Eq. (76), we first use the scaling relation
for Tsallis entropy, namely

$\displaystyle{\mathcal{S}}_{q}^{T}(a\mathcal{X})\ $ $\displaystyle=$
$\displaystyle\ {\mathcal{S}}_{q}^{T}(\mathcal{X})\ \oplus_{q}\
\ln_{q}|a|^{D}\,,$ (77)

where $a\in\mathbb{R}$ and the $q$-deformed logarithm (or simply
$q$-logarithm) is defined as in Tsallis2

$\ln_{q}x\,=\,\frac{x^{1-q}-1}{1-q}\,.$ (78)

The relation (77) follows directly from the chain of identities

$\displaystyle{\mathcal{S}}_{q}^{T}(a\mathcal{X})$ $\displaystyle=\
\frac{1}{1-q}\left[\int d^{D}{\bf y}\left(\int d^{D}{\bf x}\ \\!\delta({\bf
y}-a{\bf x}){\mathcal{F}}({\bf{x}})\right)^{\\!q}-1\right]$ $\displaystyle=\
|a|^{D(1-q)}\ \\!{\mathcal{S}}_{q}^{T}(\mathcal{X})\ +\ \ln_{q}|a|^{D}$
$\displaystyle=\ {\mathcal{S}}_{q}^{T}(\mathcal{X})\ \oplus_{q}\
\ln_{q}|a|^{D}\,.$ (79)

In the second step, we make use of

$\displaystyle{\mathcal{S}}_{q}^{T}\left({\mathcal{Z}}^{T}\right)\ $
$\displaystyle=$ $\displaystyle\
\ln_{q}\left[\left(\frac{\pi}{\mathfrak{b}(1-q)}\right)^{\frac{D}{2}}\
\\!\frac{\Gamma\left(\frac{1}{1-q}-\frac{D}{2}\right)}{\Gamma\left(\frac{1}{1-q}\right)}\
\\!\left(1-\frac{D}{2q}(1-q)\right)^{\frac{1}{(q-1)}}\right]\,,$ (80)

with $\mathfrak{b}=[2q-D(1-q)]^{-1}$. Combining (76), (77) and (80), we arrive
at the Tsallis EP

$\displaystyle M_{q}^{T}(\mathcal{X})$ $\displaystyle=$ $\displaystyle A_{q}\
\\!\left[\exp_{q}({\mathcal{S}}_{q}^{T}(\mathcal{X}))\right]^{2/D}\ =\ A_{q}\
\\!\exp_{1-(1-q)D/2}\left({\frac{2}{D}\
\\!{\mathcal{S}}_{q}^{T}(\mathcal{X})}\right)\,,$ (81)

where the constant $A_{q}$ is defined as

$A_{q}\ =\ \left[\left(\frac{\pi}{\mathfrak{b}(1-q)}\right)^{\frac{D}{2}}\
\\!\frac{\Gamma\left(\frac{1}{1-q}-\frac{D}{2}\right)}{\Gamma\left(\frac{1}{1-q}\right)}\
\\!\left(1-\frac{D}{2q}(1-q)\right)^{\frac{1}{(q-1)}}\right]^{-{2}/{D}}.$ (82)

In the above derivation, we employed the sum rule for the $q$-deformed
calculus: $\ln_{q}x\oplus_{q}\ln_{q}y=\ln_{q}xy$, as well as the definition of
the $q$-exponential $e_{q}^{x}=[1+(1-q)x]^{1/(1-q)}$ and the fact that
$e_{q}^{\ln_{q}x}=\ln_{q}(e_{q}^{x})=x$.

As a consistency check, we might notice that in the $q\rightarrow 1$ limit
Shannon’s EP is recovered:

$\displaystyle\lim_{q\rightarrow 1}M_{q}^{T}(\mathcal{X})\ =\ \frac{1}{2\pi
e}\exp\left(\frac{2}{D}{\mathcal{S}}^{S}(\mathcal{X})\right)\ =\ \
N(\mathcal{X})\,.~{}~{}$ (83)

###  IV.3 Entropy power inequalities for GUP transformations

By analogy with the conventional commutation relations, where the ensuing
entropic URs are of the form (75), one can derive entropic URs also for the
DCR (2). In this latter case it can be expected that the role of Shannon’s
entropy will be overtaken by Tsallis’ entropy (43). The actual rationale
behind this fact is not difficult to understand. First, we notice (cf.
Appendix A) that for $\beta<0$ the eigenstate of the position operator in the
momentum representation is

$\displaystyle\psi_{x}(p)\,=\,B_{x}\
\\!\frac{e^{-ixm_{p}{\mbox{\scriptsize{arctanh}}}\left(p\sqrt{|\beta|}/m_{p}\right)/\hbar\sqrt{|\beta|}}}{\sqrt{m^{2}_{p}\,-\,p^{2}|\beta|}}\,,$
(84)

with $B_{x}=\sqrt{m_{p}^{2}/2\pi\hbar}$. This implies that the position and
momentum representations of a wave function are related not via conventional
Fourier transform but, instead, via Abel transform (cf. Appendix B)

$\displaystyle\mbox{}\psi(x)$
$\displaystyle\mbox{}=\int_{\frac{-m_{p}}{\sqrt{|\beta|}}}^{\frac{m_{p}}{\sqrt{|\beta|}}}\frac{dp}{\sqrt{2\pi\hbar}}\
\\!\frac{e^{ixm_{p}{\mbox{\scriptsize{arctanh}}}\left(p\sqrt{|\beta|}/m_{p}\right)/\hbar\sqrt{|\beta|}}}{\sqrt{1-p^{2}|\beta|/m_{p}^{2}}}\
\\!\tilde{\psi}(p)\,,$ (85)

(to distinguish it from the HUP-based QM, the conjugate wave function of
$\psi(x)$ is denoted as $\tilde{\psi}(p)$ here). This does not allow to use
the Beckner–Babenko inequality directly, but we might observe that after the
substitution

$\displaystyle z\ =\ m_{p}\
\\!\mbox{{arctanh}}\left(p\sqrt{|\beta|}/m_{p}\right)/\sqrt{|\beta|}\,,$ (86)

we can rewrite (85) as

$\displaystyle\mbox{}\psi(x)\ =\ \int_{\mathbb{R}}\frac{dz}{\sqrt{2\pi\hbar}}\
\\!e^{ixz/\hbar}\bar{\psi}(z)\,,$ (87)

where

$\bar{\psi}(z)\ =\
\frac{\tilde{\psi}(m_{p}\tanh(z\sqrt{|\beta|}/m_{p})/\sqrt{|\beta|})}{{\cosh(z\sqrt{|\beta|}/m_{p})}}.$
(88)

This allows the use the Beckner–Babenko inequality in the form

$\displaystyle\mbox{}\left[\left(\frac{q^{\prime}}{2\pi\hbar}\right)^{D}\right]^{1/q}|\\!||\bar{\psi}|^{2}|\\!|_{q^{\prime}/2}$
$\displaystyle\mbox{\hskip 36.98866pt}\leq\
\left[\left(\frac{q}{2\pi\hbar}\right)^{D}\right]^{1/q^{\prime}}|\\!||\psi|^{2}|\\!|_{q/2}\,,$
(89)

where $q^{\prime}$ and $q$ are Hölder conjugates with
$q^{\prime}\in[2,\infty)$, so that $q\in[1,2]$. It can be checked numerically
that (89) is saturated by the CSs $\tilde{\psi}_{CS}(p)$ with the non-
extensivity index $2-q^{\prime}/2$ (i.e., Tsallis distributions
$|\tilde{\psi}|^{2}_{CS}(p)=q_{T}(p|2-q^{\prime}/2,b)$) and associated
$\psi_{CS}(x)$ with the non-extensivity index $2-q/2$ [see Eq. (128)]. The
analytical proof can be readily done for the cases $q=1$ and $q=2$.

Now, we can follow the same steps as in the case of Shannon EPURs JLLP . After
some algebra we arrive at

$\displaystyle
M_{q/2}^{T}(|{\psi}|^{2})M_{q^{\prime}/2}^{T}(|\tilde{\psi}|^{2})$
$\displaystyle\geq$
$\displaystyle\frac{\hbar^{2}}{4}\frac{q^{2}}{(3q/2-1)(3q^{\prime}/2-1)}$ (90)
$\displaystyle=$ $\displaystyle\frac{\hbar^{2}}{4}f(q)\,.$

The function $f(q)$ is positive and monotonically increasing for $q\in[1,2]$
with $\max f(q)=1$. It is important to stress that $f(q)$ depends only on $q$,
while no other GUP parameters are present. We may now use the identity

$\displaystyle f(q)$ $\displaystyle=$
$\displaystyle\left[\frac{2}{(|2/q-1|+1)(3q/2-1)}\right]$ (91)
$\displaystyle\times\left[\frac{2}{(|2/q^{\prime}-1|+1)(3q^{\prime}/2-1)}\right]$
$\displaystyle=$ $\displaystyle\phi(q/2)\phi(q^{\prime}/2)\,,$

and rewrite the EPUR (90) in the form

$\displaystyle\tilde{M}_{q/2}^{T}(|{\psi}|^{2})\tilde{M}_{q^{\prime}/2}^{T}(|\tilde{\psi}|^{2})\
\geq\ \frac{\hbar^{2}}{4}\,.$ (92)

where we have defined the “rescaled entropy power”
$\tilde{M}_{x}^{T}=\phi^{-1}(x)M_{x}^{T}$. This EPUR clearly emulates the form
of Shannon’s EPUR (75) by having the irreducible universal lower bound.

Since the Tsallis entropy ${\mathcal{S}}_{q^{\prime}/2}^{T}$ is maximized by
the Tsallis distribution with the non-extensivity parameter $2-q^{\prime}/2$,
i.e. by $q_{T}(p|2-q^{\prime}/2,b)$ and because for $\beta<0$ the non-
extensivity parameter $2-q^{\prime}/2<1$, we have the conditions
$q^{\prime}>2$ and $q<2$. Note that both these conditions are also required by
Beckner–Babenko inequality, cf. Ref JLLP . The formulation of uncertainty
relation (92) in terms of Tsallis’ entropy suggests that Tsallis’ entropy
should be essential to GUP quantum mechanics and its associated quantum
information theory, much in the same way that Shannon’s entropy is essential
to conventional quantum mechanics.

##  V Cosmological and gravitational applications

###  V.1 Generalization of DCR to 3 dimensions and associated CS

In order to discuss prospective cosmological implications, it is important to
generalize our 1-dimensional formalism to 3 dimensions. This can be done, for
instance, by extending the algebra (2)-(3) as (see e.g. Refs. Kempf ; Tawfik )

$\displaystyle\left[\hat{\bf{x}}_{i},\hat{\bf{p}}_{j}\right]\ =\
i\hbar\delta_{ij}\left(1\ +\ \beta\frac{{\bf{p}}^{2}}{m_{p}^{2}}\right)\,,$
$\displaystyle\left[\hat{\bf{p}}_{i},\hat{\bf{p}}_{j}\right]\ =\ 0\,,$ (93)

where the ensuing commutator $\left[\hat{\bf{x}}_{i},\hat{\bf{x}}_{j}\right]$
will be chosen after the explicit representation of $\hat{\bf{x}}_{i}$ is
deduced from (93). In particular, in the momentum space, $\hat{\bf{x}}_{i}$
and $\hat{\bf{p}}_{i}$ satisfying DCR (93) can be represented by [cf. Eqs.
(11)]

$\displaystyle\mbox{}\hat{\bf{p}}_{i}\,\psi({\bf{p}})\ =\
{\bf{p}}_{i}\,\psi({\bf{p}})\,,$
$\displaystyle\mbox{}\hat{\bf{x}}_{i}\,\psi({\bf{p}})\ =\
i\hbar\left(\frac{d}{d{\bf{p}}_{i}}+\frac{\beta}{2\,m_{p}^{2}}\
\\!\left\\{{\bf{p}}^{2},\frac{d}{d{\bf{p}}_{i}}\right\\}\right)\psi({\bf{p}})\,,$
(94)

thus implying that

$\displaystyle\left[\hat{\bf{x}}_{i},\hat{\bf{x}}_{j}\right]\ =\
2i\hbar\frac{\beta}{m_{p}^{2}}\left(\hat{\bf{x}}_{i}\hat{\bf{p}}_{j}-\hat{\bf{x}}_{j}\hat{\bf{p}}_{i}\right)\,.$
(95)

As a consistency check, one can verify that the commutators (93) and (95)
satisfy the Jacobi identity.

By analogy with Section II we can deduce corresponding URs. These read

$\displaystyle(\Delta{\bf{x}}_{i})_{\varrho}(\Delta{\bf{p}}_{j})_{\varrho}\
\geq\
\frac{\hbar}{2}\delta_{ij}\left(1+\beta\,\frac{(\Delta{\bf{p}})^{2}_{\varrho}+\langle\hat{\bf{p}}\rangle^{2}_{\varrho}}{m_{p}^{2}}\right)\,,$
$\displaystyle(\Delta{\bf{x}}_{i})_{\varrho}(\Delta{\bf{x}}_{j})_{\varrho}\
\geq\
\hbar\frac{\beta}{m_{p}^{2}}\left|\left\langle\left(\hat{\bf{x}}_{i}\hat{\bf{p}}_{j}-\hat{\bf{x}}_{j}\hat{\bf{p}}_{i}\right)\right\rangle_{\varrho}\right|\,,$
$\displaystyle(\Delta{\bf{p}}_{i})_{\varrho}(\Delta{\bf{p}}_{j})_{\varrho}\
\geq\ 0\,.$ (96)

In the following we will again consider only a mirror symmetric density matrix
$\varrho$ satisfying $\langle\hat{\bf{p}}\rangle_{\varrho}=0$, so as to attain
the GUP (1). For simplicity, we will concentrate only on isotropic density
matrices, for which $\langle\hat{\bf{x}}\rangle_{\varrho}=0$. This will be
fully satisfactory for our subsequent reasoning.

As will be shown shortly, on the class of mirror symmetric $\varrho$’s, the
equation [cf. Eq. (9)]

$\left(\hat{\bf{p}}_{k}-i\gamma_{k}\hat{\bf{x}}_{k}\right)|\psi\rangle\ =\
0\,,\;\;\;k=1,2,3\,,$ (97)

admits only one solution for $\psi\in L^{2}(\mathbb{R})$, so that the
minimum–uncertainty $\hat{\varrho}$ is a pure (coherent) state. To see this,
let us first introduce the generator of rotations111It is worth observing
that, in the context of the GUP, the definition of the angular momentum might
not coincide with the generator of the rotations. For further details on this
aspect, the interested reader can consult Refs. angmom ; angmom2 .. Bearing in
mind the DCR (93) and (95), we can define Kempf

$\displaystyle\hat{\ \\!\bf{L}}_{k}\ =\
\frac{1}{1+\frac{\beta}{m_{p}^{2}}\hat{\bf{p}}^{2}}\ \\!\epsilon_{klm}\
\\!\hat{\bf{x}}_{l}\hat{\bf{p}}_{m}\,,$ (98)

where $\epsilon_{klm}$ is the Levi-Civita antisymmetric symbol. These
operators satisfy the standard _so_(3) algebra, i.e.

$\displaystyle\left[\hat{\ \\!\bf{L}}_{k},\hat{\ \\!\bf{L}}_{l}\right]\ =\
i\hbar\ \\!\epsilon_{klm}\ \\!\hat{\ \\!\bf{L}}_{m}\,,$ (99)

alongside with the other familiar commutation relations

$\displaystyle\left[\hat{\bf{x}}_{k},\hat{\ \\!\bf{L}}_{l}\right]\ =\ i\hbar\
\\!\epsilon_{klm}\ \\!\hat{\bf{x}}_{m}\,,$
$\displaystyle\left[\hat{\bf{p}}_{k},\hat{\ \\!\bf{L}}_{l}\right]\ =\ i\hbar\
\\!\epsilon_{klm}\ \\!\hat{\bf{p}}_{m}\,.$ (100)

With the operator $\hat{\ \\!\bf{L}}_{k}$ at hand, we rewrite
$(\Delta{\bf{x}}_{i})_{\varrho}(\Delta{\bf{x}}_{j})_{\varrho}$ in (96) in the
form

$\displaystyle(\Delta{\bf{x}}_{i})_{\varrho}(\Delta{\bf{x}}_{j})_{\varrho}\ $
$\displaystyle\geq$
$\displaystyle\frac{\beta}{m_{p}^{2}}\left|\left\langle\left(1+\frac{\beta}{m_{p}^{2}}{\hat{\bf{p}}}^{2}\right)\\!\left[\hat{\
\\!\bf{L}}_{i},\hat{\ \\!\bf{L}}_{j}\right]\right\rangle_{\\!\\!\varrho}\
\\!\right|$ (101) $\displaystyle\mbox{}=\
\frac{\beta}{m_{p}^{2}}\left|\mbox{Tr}\\!\left[\left(1+\frac{\beta}{m_{p}^{2}}{\hat{\bf{p}}}^{2}\right)\\!\hat{\
\\!\bf{L}}_{i}\left[\hat{\ \\!\bf{L}}_{j},\hat{\varrho}\right]\right]\right|$
$\displaystyle\mbox{}=\
\frac{\beta}{m_{p}^{2}}\left|\mbox{Tr}\\!\left[\left(1+\frac{\beta}{m_{p}^{2}}{\hat{\bf{p}}}^{2}\right)\\!\hat{\
\\!\bf{L}}_{j}\left[\hat{\ \\!\bf{L}}_{i},\hat{\varrho}\right]\right]\right|$
$\displaystyle\mbox{}=\ 0\,.$

The last line is a consequence of the assumed isotropy of $\hat{\varrho}$.

To find the CS, we return back to Eq. (97). Because the isotropy of
$\hat{\varrho}$ implies $\hat{\bf{L}}_{j}|\psi\rangle=0$ for $j=1,2,3$, we
obtain from (97) that

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\hat{\bf{L}}_{j}\left(\hat{\bf{p}}_{k}-i\gamma_{k}\hat{\bf{x}}_{k}\right)|\psi\rangle\
\\!=\
\\!\left([\hat{\bf{L}}_{j},\hat{\bf{p}}_{k}]-i\gamma_{k}[\hat{\bf{L}}_{j},\hat{\bf{x}}_{k}]\right)|\psi\rangle$
(102) $\displaystyle=$
$\displaystyle-i\hbar\varepsilon_{jkl}\left(\hat{\bf{p}}_{l}-i\gamma_{k}\hat{\bf{x}}_{l}\right)|\psi\rangle\,.$

This, for instance, gives

$\displaystyle\gamma_{1}\hat{\bf{x}}_{l}|\psi\rangle$ $\displaystyle=$
$\displaystyle\gamma_{2}\hat{\bf{x}}_{l}|\psi\rangle\ =\
\gamma_{3}\hat{\bf{x}}_{l}|\psi\rangle\,,$

for any $l=1,2,3$, which ensures that $\gamma_{k}$ is $k-$independent
($|\psi\rangle$’s satisfying (97) cannot be all zero-eigenvalue states of
$\hat{\bf{x}}_{l}$ as $(\Delta{\bf{x}}_{l})_{\varrho}\neq 0$). With this, Eq.
(97) can be rewritten in the form

$\displaystyle\frac{\partial}{\partial{\bf{p}}_{k}}\,\psi({\bf{p}})\ =\
-\frac{\left(1+\frac{\beta\gamma\hbar}{m_{p}^{2}}\right)}{\gamma\hbar\left(1+\frac{\beta}{m_{p}^{2}}{\bf{p}}^{2}\right)}\,{\bf{p}}_{k}\,\psi({\bf{p}})\,,$
(103)

This system of 3 equations admits only one (normalized) solution for $\psi\in
L^{2}(\mathbb{R}^{3})$, namely

$\displaystyle\psi({\bf{p}})\ =\ N\left[1\ +\
({\beta\,{\bf{p}}^{2}})/{m_{p}^{2}}\right]_{+}^{-\frac{m_{p}^{2}}{2\beta\gamma\hbar}-\frac{1}{2}}\,,$
(104)

with the normalization constant

$\displaystyle\mbox{}N_{{}_{>}}^{2}\ =\ \frac{\beta^{3/2}}{2\pi m_{p}^{3}\
\\!B(5/2,{m_{p}^{2}}/{\beta\gamma\hbar}-3/2)}\,,\;\;{\mbox{for}}\;\;\
\beta>0\,,$ $\displaystyle\mbox{}N_{{}_{<}}^{2}\ =\ \frac{\beta^{3/2}}{2\pi
m_{p}^{3}\ \\!B(5/2,{m_{p}^{2}}/{|\beta|\gamma\hbar})}\,,\;\;{\mbox{for}}\;\;\
\beta<0\,,$ (105)

where $B(x,y)$ is the beta function. The uniqueness of the solution (104)
ensures that the minimum–uncertainty $\hat{{\varrho}}$ is again a pure CS.

Note that the CS (104) is indeed as close as we can get to the classical
situation. Namely, the irreducible non-zero lower bound is saturated and all
other uncertainty relations are bigger than zero, which is also true in
classical physics.

Among all pointer states in the would-be GUP driven Universe, only CSs (104)
(similarly to their 1D counterparts (13)) saturate both the
“${\bf{x}}$-${\bf{p}}$” GUPs and ensuing Tsallis EPURs. Moreover, in Section
IV.3 we have seen that the very existence of Tsallis EPUR indicates that TE
should be a relevant entropy functional in the GUP context. If we couple this
observation with the fact that CSs (104) extremize Tsallis entropy in 3D
space, we might invoke (similarly as in conventional statistical physics) MEP
but this time with Tsallis entropy (in place of Shannon–Gibbs entropy) to
discuss a statistical physics of an ensemble of non-interacting GUP-governed
particles in their quasi-classical regime. In this respect, non-extensive
thermodynamics of Tsallis Tsallis1 ; Tsallis2  provides the necessary
mathematical framework that can be utilized to explore the quasi-classical
domain of a GUP Universe.

###  V.2 Verlinde’s entropic gravity

In Ref. JLLP  we have used the entropy one-form (66) to show that Verlinde’s
entropic-gravity force Verlinde  defined by the relation $F\delta x\ =\
T\delta S$, ($S$ is the holographic entropy obeying the area-law scaling —
basically the Bekenstein–Hawking entropy) generalizes in the present context
and yields the gravitational potential

$\displaystyle V(r)\ =\ \frac{r_{s}}{2}\left[-\frac{1}{r}\ +\ (1-q)\kappa_{2}\
\\!r\right]\,,$ (106)

where $\kappa_{2}=\omega_{2}/\ell_{p}^{2}$ ($\ell_{p}=\hbar/m_{p}\approx
1.6\times 10^{-35}$m is the Planck length, $\omega_{2}=\pi$ is the second
Hill’s coefficient JLLP ; Hill ) and $r_{s}=2M$ is the Schwarzchild radius
(here and in subsequent reasonings we adopt the convention that the
gravitational constant $G=1$). The gravitational potential (106) coincides
with the Mannheim–Kazanas gravitational potential of a static, spherically
symmetric source of mass $M$ in conformal Weyl gravity (CWG) mannheim_a ;
mannheim_b ; mannheim_c . There, a parameter in front of the linear term is
identified with the inverse Hubble length $R_{H}$ (more precisely with
$1/(2R_{H})$) Kazanas . What is quite intriguing here is that, for present
macroscopic scales (i.e., $R_{H}\sim 10^{26}$m), the Mannheim–Kazanas solution
has been successful in fitting more than two hundred galactic rotation curves
with no adjustable parameters (other than the galactic mass-to-light ratios)
and with no need for dark matter or other exotic modifications of gravity
mannheim_a ; mannheim_b ; mannheim_c . Despite the fact that macroscopic-scale
gravity does not fall within the assumed quasi-classical regime, the idea that
the coefficient in front of a linear term in (106) should be associated with
the inverse Hubble length is valid even in the early Universe cosmology. This
is because the argument of CWG leading to this result is independent of an
actual Universe epoch MK2 .

In conventional cosmology, it is expected that a quasi-classical (decoherence)
description becomes pertinent at the late-inflation epoch (after the first
Hubble radius crossing) and perhaps even after its end during reheating
KieferII ; Burges . So, in this period the NTT should be a suitable framework
for the description of an “inflaton gas”. For instance, by viewing the
“inflaton gas” as an ideal gas, the NTT predicts that the inflaton pressure
should satisfy for $0<q<1$ a polytrope relation Du:04 ; Abe4

$\displaystyle p\ =\ \frac{2\pi\hbar^{2}}{m_{i}\ \\!e^{5/3}}\
\\!\rho^{5/3}\,,$ (107)

where $\rho=N/V$ is the particle density and $m_{i}$ the mass of the inflaton.
In this connection, it should be stressed that the relation (107) holds for
$0<q<1$ but not in the limit $q\rightarrow 1$, see, e.g. Abe4 . In fact, at
$q=1$ one has the familiar pressure relation $p\propto\rho_{E}$. So, the NTT
and extensive limits are not interchangeable. The polytrope relation of the
type (107) often appears in phenomenological studies on late inflation, see,
e.g. Bar:16 ; Bar:18 .

In order to gain information about $\beta$, we employ the CWG observation that
cosmologically viable linear term in (106) should have its parameter
associated with $1/R_{H}$. According to CWG the Newtonian potential (106)
should dominate on short scales, while the linear one becomes prominent at
large scales. Both potentials get equal at $R_{H}$, which in our case implies
that $q=1-\ell_{p}^{2}/(\pi R_{H}^{2})$. Note that this is compatible with the
condition that $r_{s}=R_{H}$. By combining the latter expression for $q$ with
(19) and (21), we obtain $|\beta|\simeq
m_{p}^{2}\,\ell_{p}^{2}/(2\pi\left(\Delta p\right)^{2}_{\psi}R_{H}^{2})$.

To see how such $\beta$ explicitly depends on a cosmological time $t$, we
first write $R_{H}(t)=H^{-1}(t)=a(t)/\dot{a}(t)$, where $H$ is the Hubble
parameter and $a(t)$ is the scale factor. The latter can be evaluated, e.g.,
from the Vilenkin–Ford inflationary model VF , where
$a(t)=A\sqrt{\mathrm{sinh}\left(B\,t\right)}$, with $B=2\sqrt{\Lambda/3}$
($\Lambda$ is the cosmological constant). We then use the relativistic
equipartition theorem $\left(\Delta p\right)^{2}_{\psi}\simeq
12\left(k_{B}T\right)^{2}$, cf. Ref JLLP . After simple algebraic
manipulations, we obtain

$\displaystyle|\beta|\ \equiv\ |\beta(t)|\ =\
\frac{m_{p}^{2}\,\ell_{p}^{2}\,\Lambda}{72\pi\left(k_{B}T\right)^{2}\mathrm{tanh}^{2}\left({2t\sqrt{{\Lambda}/{3}}}\right)}\,.$
(108)

For concreteness’ sake, let us consider the reheating epoch, i.e. time scale
$t\simeq 10^{-33}$s. By taking $m_{i}=10^{12}\div 10^{13}$GeV, $T$ of the
order of the reheating temperature $T_{R}\simeq 10^{7}\div 10^{8}$GeV and the
presently known value of the cosmological constant $\Lambda\simeq
10^{-52}$m-2, we obtain $|\beta|\sim 10^{-2}\div 1$, which is in agreement
with the values predicted by string-theory models, cf. e.g. VenezGrossMende ;
Ciafa ; Koni . This result is also consistent with the naturalness principle
that dictates that not so far from the Planck scale the $\beta$ should not be
too large nor too small. Let us finally re-emphasize that the above connection
with the CWG exists only when $\beta<0$, as otherwise the linear term in (106)
would have an erroneous sign.

###  V.3 Loop Quantum Gravity

Loop Quantum Gravity (LQG) is a non-perturbative and background-independent
theory of quantum gravity, characterized by quantum operators for areas and
volumes that exhibit discrete spectra Rovelli . A basic postulate is that the
spacetime structure is formed by finite loops nested into extremely fine
networks - the spin networks. These are graphs with edges having labels
$j=0,1/2,1,3/2\dots$. As shown in Rovelli2 , the area element carried by a
given surface punctured by the spin network edge $j$ is

$a(j)=8\pi\ell_{p}^{2}\gamma_{LQG}\sqrt{j(j+1)}\,,$ (109)

where $\gamma_{LQG}$ is the Immirzi parameter Immirzi , which is a positive,
real-valued number that measures the size of a quantum of area in Planck
units.

####  V.3.1 Black hole quasinormal modes

While providing a fundamental prediction of LQG, Eq. (109) is beset by the
ambiguity that $\gamma_{LQG}$ is in principle undetermined. A possible way to
fix it is by connecting the relation between the area and mass of a
Schwarzschild black hole to the area produced by the spin network through the
definition of quasinormal modes QNM , which are a set of damped oscillations
satisfying the perturbation equations of the Schwarzschild geometry ReW . In
so doing, the minimum value $j_{min}=1$ has been obtained in QNM , which in
turn fixes the Immirzi parameter to

$\gamma_{LQG}=\frac{\log 3}{\pi\sqrt{8}}\,.$ (110)

Also, we emphasize that the result $j_{min}=1$ has been interpreted in QNM  by
assuming that the effective gauge group of the spin networks to consider in
LQG is SO(3) (whose unitary representations are in fact labeled by integers)
instead of its covering (and normally adopted) group SU(2).

The above considerations apply to the case where the standard Boltzmann-Gibbs
statistics is used for black holes. However, non-trivial conclusions on the
gauge structure of LQG can be reached within Tsallis’ framework, as recently
suggested in AbreuQN . To show how non-extensivity - and, consequently, the
GUP - affects black holes quasinormal modes, let us recall that quasinormal
mode frequencies $\omega_{n}$ for large damping are limited by Damp

$M\omega_{n}=0.04371235+\frac{i}{4}\left(n+\frac{1}{2}\right),$ (111)

with $M$ and $n$ being the mass of the black hole and a non-negative integer,
respectively. Moreover, the real part $w_{n}\equiv Re[\omega_{n}]$ of these
frequencies obeys $w_{n}=\log 3/(8\pi M)$ Hod .

Based on Hod , one can assume that the variation $\Delta M$ in the mass of the
black hole equals the energy of a quantum with frequency $w_{n}$, i.e. $\Delta
M=\hbar w_{n}=\hbar\log 3/(8\pi M)$. Combining this equation with the standard
mass/area relation $A=16\pi M^{2}$ for a Schwarzschild black hole, we get the
corresponding surface change $\Delta A=4\hbar\log 3$. This result can be used
to relate the Immirzi parameter to the value of $j_{min}$. According to Bohr’s
correspondence principle, the oscillatory frequency of a classical system
should correspond to the transition frequency of the analogue quantum system.
In the framework of LQG, the most natural description of a black hole
“transition” is in terms of the appearance or disappearance of a puncture with
spin $j_{min}$. The ensuing change in the area of the black hole can be
quantified by Eq. (109) with $j=j_{min}$, yielding in the end

$\gamma_{LQG}=\frac{\log 3}{2\pi\sqrt{j_{min}(j_{min}+1)}}\,.$ (112)

To infer the dependence of $j_{min}$ in Tsallis statistics, we should first
evaluate the number of configurations in a punctured surface, taking into
account the multiplicity of each state $j$. By resorting to Tsallis entropy,
the following expression is then obtained for the case of a microcanonical
ensemble AbreuQN

$j_{min}=\frac{1}{2}\left\\{\left[1+(1-q)\frac{A}{4\ell_{p}^{2}}\right]^{\frac{\log
3}{(1-q)\frac{A}{4\ell_{p}^{2}}}}-1\right\\}.$ (113)

Thus, non-extensivity introduces an extra degree of freedom, resulting into a
different spectrum for $j_{min}$. Clearly, the standard result $j_{min}=1$ is
recovered in the $q\rightarrow 1$ limit, which in turn gives the value (110)
of $\gamma_{LQG}$. Also, Eq. (113) enables to constrain departure from
extensivity $1-q$. Toward this end, we notice that the lowest (nonzero)
allowed spin $j_{min}=1/2$ is obtained, provided that $1-q\approx
1.37\,\frac{4\ell_{p}^{2}}{A}$, while $j_{min}\rightarrow\infty$ for
$1-q\rightarrow-4\ell_{p}^{2}/A$.

Using Eq. (21), we can now convert the $q$-dependence of $j_{min}$ into a
$\beta$-dependence to see how GUP interfaces with LQG. After some algebra, we
are led to

$\displaystyle\mbox{}j^{(\beta)}_{min}\ =$
$\displaystyle\frac{1}{2}\left\\{\left[\frac{1+\beta\Delta
p^{2}\left(\frac{3}{m_{p}^{2}}-\frac{A}{2\hslash^{2}}\right)}{1+3\beta\frac{\Delta
p^{2}}{m_{p}^{2}}}\right]^{-\frac{2\hslash^{2}\log
3}{A}\left(\frac{3}{m_{p}^{2}}+\frac{1}{\beta\Delta
p^{2}}\right)}-1\right\\}.$ (114)

In Fig. 3 we plot this expression as a function of $\beta$ for a micro black
hole of area $A=16\pi\ell_{p}^{2}$ and for $A=32\pi\ell_{p}^{2}$ (grey dashed
lines are at the intersection with the usually allowed (positive half-
integers) values of the spin $j_{min}=1/2,1,3/2,\dots$). Furthermore, we set
the energy scale to $\Delta p\simeq m_{p}$. For $A=16\pi\ell_{p}^{2}$,
constraints on non-extensivity discussed above turn into the condition
$|\beta|\sim\mathcal{O}(10^{-2})$, which is consistent with the result in Sec
V.2 (see below Eq. (108)). Notice that this estimate is slightly reduced in
the case of $A=32\pi\ell_{p}^{2}$. In fact, the $\beta$ range can be shortened
by increasing the area of the black hole sufficiently. This implies that, for
any physically realizable black hole, a small departure from extensivity/HUP-
based QM is likely to account for a minimum spin $j_{min}\neq 1$. However, it
must be observed that only the $\beta<0$ scenario can accommodate
$j_{min}=1/2$, since $j_{min}>1$ as far as one considers positive values of
$\beta$ (see Fig. 3). Therefore, contrary to the result of QNM , we conclude
that both SU(2) and SO(3) are possible gauge groups of the spin networks of
LQG in the presence of GUP.

Figure 3: Values of $j_{min}$ as a function of $\beta$ for
$A=16\pi\ell_{p}^{2}$ (black solid line) and $A=32\pi\ell_{p}^{2}$ (blue
dashed line). We set the energy scale to $\Delta p\simeq m_{p}$ (online
colors).

####  V.3.2 Immirzi parameter

Based on Eqs. (112) and (114), let us also exhibit the GUP-corrected
expression of Immirzi parameter

$\displaystyle\gamma^{(\beta)}_{LQG}$ $\displaystyle=$
$\displaystyle\frac{\log 3}{\pi}\times$
$\displaystyle\left\\{\left[1-\beta\,\frac{A\Delta
p^{2}\,m_{p}^{2}}{2\hslash^{2}\left(m_{p}^{2}+3\beta\Delta
p^{2}\right)}\right]^{-\frac{4\hslash^{2}\log 3\,\left(m_{p}^{2}+3\beta\Delta
p^{2}\right)}{\beta\,A\Delta
p^{2}\,m_{p}^{2}}}\\!\\!-1\right\\}^{-\frac{1}{2}}\\!\\!\\!\\!\\!.$

The behavior of $\gamma^{(\beta)}_{LQG}$ versus $\beta$ is plotted in Fig. 4
for $A=16\pi\ell_{p}^{2}$ and $A=32\pi\ell_{p}^{2}$. Some comments are in
order here. Though seemingly counter-intuitive, we find that the modified
Immirzi parameter somehow depends on the properties of the state through
$\Delta p$. A possible explanation is that since in our framework Tsallis
statistics affects quantum mechanics in a state-dependent way (see Eq. (21)),
it is then quite natural to expect that the counting of micro-states for a
given system is such. In turn, given that the Immirzi parameter is fixed by
matching the semiclassical black hole entropy and the counting of micro-states
in LQG, one gets the conclusion. Furthermore, we notice that
$\gamma^{(\beta)}_{LQG}$ correctly equals the expected limit (110) for
$\beta\rightarrow 0$. Finally, the value $\gamma^{(\beta)}_{LQG}=\frac{\log
3}{\pi\sqrt{3}}$, corresponding to $j_{min}=1/2$, is obtained for small
(negative) deviation of $\beta$ from zero for any realizable black hole,
consistently with what is stated above.

Figure 4: Plot of $\gamma^{(\beta)}_{LQG}$ as a function of $\beta$ for
$A=16\pi\ell_{p}^{2}$ (black solid line) and $A=32\pi\ell_{p}^{2}$ (blue
dashed line). We set the energy scale to $\Delta p\simeq m_{p}$ (online
colors).

###  V.4 Magueijo–Smolin DSR

It is interesting to note that the deformed commutation relation (2) with
$\beta<0$ and magnitude (108) complies nicely with Magueijo–Smolin DSR MS1 ;
MS2 . In a nutshell, DSR is a theory that coherently tries to implement a
second invariant (namely $m_{p}$ or equivalently $\ell_{p}$) besides the speed
of light into the transformations among inertial reference frames. The
Magueijo–Smolin DSR model predicts that the DCR should vanish at Planck scale
(thus physics should become deterministic there), while at low energies it
approaches the conventional canonical commutator. In our case, we indeed see
from (108) that, for allowed cosmological times, the deformation parameter
$|\beta(t)|$ monotonically decreases with increasing $t$. This is consistent
with the expectation of the Magueijo–Smolin DSR model that $\beta$ should grow
at high-energy scales so that at Planck scale the right-hand-side of the
commutation relation (2) vanishes. Although such a behavior is quite
encouraging, it is only indicative, as we cannot extend the validity of our
formalism up to the Planck scale, since this would be at odds with the
presumed quasi-classical domain of validity.

Nevertheless, a quantitative comparison can still be done by employing the DCR
predicted by the Magueijo–Smolin model. To keep our reasoning simple, it
suffices to focus on the one-dimensional problem, since in higher spatial
dimensions the DSR-inspired commutation relations are compatible with spatial
commutativity. To make the generalization introduced in Sec. V.1 consistent
with the above scenario, one should, in principle, account for an extra term
in Eq. (93) which still preserves rotational invariance and complies with a
precise prescription (as discussed, e.g., in Ref. angmom2 ). However, for the
purpose of the upcoming considerations, such a technical aspect can be safely
neglected, as it would not affect the validity of our result.

Let us introduce the Magueijo–Smolin deformed commutation relation MS2

$\left[\hat{x},\hat{p}\right]\ =\ i\left(1-\ell_{p}\hat{E}\right)\,,$ (116)

with $\hat{E}$ denoting the energy operator. A straightforward comparison with
Eq. (2) and $\beta<0$ shows that the two DCRs can be related. Indeed, by
reconstructing the energy of a non-relativistic system in Eq. (2) by suitably
introducing the mass of the analyzed quantum system, we immediately deduce
that the value of the deformation parameter providing the exact match of the
prediction associated to the two distinct pictures is given by

$\beta\equiv\beta(m)\ =\ \frac{m_{p}}{2m}\,.$ (117)

One can also achieve this result by examining the non-relativistic limit of
the DSR-inspired commutation relations coraddu , and then comparing it to the
GUP case.

Eq. (117) encodes an interesting outcome. As a matter of fact, in the
framework of DSR, the invariance of the scale (i.e., Planck scale) at which
quantum gravitational effects are deemed to become relevant requires a
fundamental deformation of the action of Lorentz transformations on momentum
space. In particular, the ensuing modified composition laws acquire a non-
linear character MS1 ; MS2 , which renders the whole analysis highly non-
trivial. A consequence of this occurrence that is commonly encountered in
several quantum gravity candidate models is related to the possibility of
witnessing a non-vanishing curvature in momentum space (see for instance Refs.
curved1 ; curved2 ; curved3 ; curved4 ; curved5 ; curved6 ; curved7 ), which
is thus liable to be treated with geometric tools stemming, e.g., from general
relativistic methods. In light of this observation, a first resemblance
between the Magueijo–Smolin DSR and GUP can already be drawn at this stage.
Indeed, in a series of recent works fab1 ; fab2  it was shown that GUP-
inspired deformations of the canonical Heisenberg algebra can be reinterpreted
in terms of a non-vanishing curvature in momentum space, thereby making the
aforesaid approach compatible with DSR also from this standpoint.

On the other hand, the most compelling subject paired with Eq. (117) and the
DSR theoretical scheme is represented by the so-called “soccer ball” problem
sb1 ; sb2 ; sb3 ; sb4 ; sb5 ; sb6 . In short, the issue lies in the non-linear
behavior of momentum composition law, which in principle does not prohibit the
enhancement of quantum gravitational corrections when composite systems are
accounted for; the more macroscopic the considered system, the more pronounced
this effect is expected to become. However, since the macroscopic world we
experience everyday exhibits no trace of quantum gravitational signatures,
there should be an explanation that motivates the suppression of such a
phenomenon. In this sense, Eq. (117) might be viewed as a potential way out,
since the inverse proportionality between $\beta$ and $m$ conveys a reduction
in the magnitude of the deformations to the standard Heisenberg algebra when
the mass of the studied system increases (i.e., when the
mesoscopic/macroscopic regime is approached). This “inverse soccer ball”
tendency is not entirely new in the context of GUP, as it has already been
pointed out in Ref. camel . Therefore, the problem originated by the non-
linear momentum composition law might be solved by requiring consistency
between the predictions of the two distinct deformations of the standard
Heisenberg algebra. In so doing, we would manage to achieve a foreseeable
scaling dependence on the size of the considered system regardless of the non-
triviality of the underlying momentum space, which would thus be left
untouched. More details on this topic require further investigation and will
be presented elsewhere.

##  VI Discussion and Conclusions

In this paper, we have unified two seemingly unrelated concepts, namely the
generalized uncertainty principle and Tsallis (thermo)statistics. On the one
hand, the GUP strives to explore the consequences of the existence of a
minimal length scale (be it fundamental or emergent). For this reason, it has
a large number of both theoretical and experimental implications, which range
from early Universe cosmology and astrophysics, to condensed matter theory and
quantum optics. On the other hand, Tsallis statistics is a theoretical concept
that accounts for systems with long-range correlations or long-time memory,
for which the conventional central limit theorem does not apply. Again, it has
a large number of both theoretical and experimental implications that span
such fields as statistical physics, thermodynamics and complexity theory, with
applications ranging from condensed matter theory, fluid dynamics and social
sciences to quantum information. A merger of the two concepts presented here
is intriguing from a conceptual perspective for the following reasons:

  * •

Modifications of quantum mechanics caused by the existence of a minimal length
scale (be it fundamental or effective) is in the semi-classical (decoherence)
limit necessarily equivalent to Tsallis statistics. This, in turn, provides a
new methodology for the study of GUP systems and, at the same time, a new
arena for Tsallis statistics.

  * •

Quantum mechanical systems in the decoherence limit represent an ongoing field
of intense research in quantum information theory because decoherence is the
main impediment to the realization of devices for quantum information
processing, e.g., quantum computers. It is thus important to get a handle on
how prospective GUP modifications to quantum mechanics would influence
decoherence-borne errors in quantum information processing and sensing.

  * •

Decohered quantum regime is also known to be pertinent in observational
cosmology (decoherence description is supposed to be valid at the late-
inflation period — after the first Hubble radius crossing, and perhaps even
after its end during reheating). Prospective GUP corrections should be then
relevant at these early epochs of the Universe.

  * •

In Sec. II.5 we have noticed that the form (2) of the GUP provides the
leading-order approximation towards Maxwell–Jüttner (i.e. relativistic) type
deformation of the uncertainty relation. However, it must be said that
Maxwell–Jüttner distribution only represents the first naive approach to
develop relativistic statistical mechanics, as it arises from
Maxwell–Boltzmann distribution with the classical energy-velocity relation
being replaced by its relativistic counterpart. On the other hand, a self-
consistent relativistic statistical theory is one built upon the Kaniadakis
probability density function and ensuing entropy Kaniadakis1 ; Kaniadakis2 ;
Kaniadakis3 . The latter represents a one-parameter modification of
Boltzmann–Gibbs–Shannon entropy and it is naturally imposed by Lorentz
transformations (see also RevLucia  for a recent review of gravitational and
cosmological applications in Kaniadakis statistics). Based on our result,
Kaniadakis’ probability amplitude should then coincide with coherent states
associated with a fully relativistic version of GUP. Since GUP models in
relativistic theories are still controversial, we expect that the recipe
presented here could pave the way for the correct formulation of relativistic
GUP when applied to the Kaniadakis statistics.

At the same time, advances prompted by the present analysis are also expected
in more practical/experimental contexts, since:

  * •

Tsallis entropy with its entropy power are measurable quantities (there is a
number of coding theorems and communication protocols for them) and they are
indeed routinely used both in classical and quantum information theory and in
quantum optics. So, the Tsallis entropy-power uncertainty relations discussed
here are experimentally accessible.

  * •

Inflation-based considerations discussed here not only restrict a numerical
value of the GUP deformation parameter $\beta$, but the GUP semi-classical
regime predicts, e.g., a very specific polytrope state equation for inflaton
field. A specific imprint of this should be observed in the cosmic microwave
background radiation Bar:16 ; Bar:18 .

  * •

Postulating a similar GUP commutator between the canonical variables of the
electromagnetic field in quantum optics, one can evaluate corrections to the
radiation pressure noise and shot noise in various optomechanical systems in
their semi-classical regime, e.g., Michelson–Morley type interferometers.
These corrections might be experimentally observed, e.g., in future advanced
LIGO detectors.

To substantiate our point, we have employed here the NTT to generalize
Verlinde’s entropic force. Apart from obtaining a modified Newtonian
(basically Mannheim–Kazanas) potential, we have argued that such a
generalization should be phenomenologically pertinent at the late-inflation
epoch. The corresponding dependence of the GUP $\beta$ parameter on
cosmological time $t$ has also been derived for the reheating epoch. The
$\beta$ parameter inferred in this way is consistent both with values
predicted by string-theory models and with the naturalness principle.
Moreover, we have shown that the dependence of $\beta$ on $t$ is compatible
with the Magueijo–Smolin doubly special relativity scenario. Moreover, a more
precise comparison has revealed more similarities between the two approaches,
such as the common prediction of an underlying curved momentum space.
Interestingly, these contact points might potentially represent the solution
to a phenomenological issue called “soccer ball” problem plaguing DSR, without
sacrificing the non-trivial geometry of momentum space. Finally, within the
context of the NTT, we have derived new $\beta$-dependent expressions for the
lowest possible value of the spin and Immirzi parameter in Loop Quantum
Gravity. We have shown that the $\beta<0$ choice can easily accommodate
$j_{min}=1/2$, while $j_{min}>1$ provided one works with positive values of
$\beta$. This implies that both SU(2) (which has been normally adopted) and
SO(3) (which has been claimed in QNM ) are possible gauge groups of LQG spin
networks in the presence of GUP.

###### Acknowledgements.

We thank John Klauder, Vladimir Lotoreichik and Pavel Exner for useful
discussions and comments on the manuscript. P.J. was in part supported by the
FNSPE CTU grant RVO14000. G.G.L. acknowledges the Spanish “Ministerio de
Universidades” for the awarded Maria Zambrano fellowship and funding received
from the European Union - NextGenerationEU. L.P. is grateful to the “Angelo
Della Riccia” foundation for the awarded fellowship received to support the
study at Universität Ulm. G.G.L. and L.P. acknowledge networking support by
the COST Action CA18108.

##  Appendix A Eigenstates of the position operator

Since the generalized commutator between position and momentum takes the form
(2), the plane waves are no longer eigenstates of the operator $\hat{x}$ in
the momentum representation and of $\hat{p}$ in the position representation.
This in turn means that wave functions in position and momentum representation
are not connected via a Fourier transform. To see how they are related, let us
first consider the eigenstates of the operator $\hat{x}$ in the momentum
representation. These are given by solving the following eigenvalue equation

$\hat{x}\langle
p|x\rangle\,\equiv\,x\psi_{x}(p)\,=\,i\hbar\left(\frac{d}{dp}+\frac{\beta}{m^{2}_{p}}\,p^{2}\frac{d}{dp}+\frac{\beta}{m^{2}_{p}}p\right)\psi_{x}(p)\,,$
(118)

which can be equivalently rewritten as

$\displaystyle\frac{d}{dp}\
\\!\psi_{x}(p)\,=\,\frac{\left(x\,-\,i\hbar\frac{\beta}{m^{2}_{p}}p\right)}{i\hbar\left(1\,+\,\frac{\beta}{m^{2}_{p}}p^{2}\right)}\
\\!\psi_{x}(p)\,.$ (119)

The solution is of the form

$\displaystyle\psi_{x}(p)\,=\,A_{x}\
\\!\frac{e^{-ixm_{p}\arctan\left(p\sqrt{\beta}/m_{p}\right)/\hbar\sqrt{\beta}}}{\sqrt{m^{2}_{p}\,+\,p^{2}\beta}}\,,$
(120)

for positive $\beta$, and

$\displaystyle\psi_{x}(p)\,=\,B_{x}\
\\!\frac{e^{-ixm_{p}{\mbox{\scriptsize{arctanh}}}\left(p\sqrt{|\beta|}/m_{p}\right)/\hbar\sqrt{|\beta|}}}{\sqrt{m^{2}_{p}\,-\,p^{2}|\beta|}}\,,$
(121)

for negative $\beta$ (i.e., $\beta=-|\beta|$).

In the following, we discuss the two cases $\beta>0$ and $\beta<0$ separately.

### Negative $\beta$ case

In this case, $\psi_{x}(p)$ is not quadratically integrable, since

$\displaystyle|\\!|\psi_{x}|\\!|^{2}$ $\displaystyle=$ $\displaystyle\langle
x|x\rangle\,=\,|B_{x}|^{2}\int_{-m_{p}/\sqrt{|\beta|}}^{m_{p}/\sqrt{|\beta|}}\frac{dp}{m_{p}^{2}-p^{2}|\beta|}$
(122) $\displaystyle=$ $\displaystyle\\{z=m_{p}\
\\!\mbox{{arctanh}}\left(p\sqrt{|\beta|}/m_{p}\right)/\sqrt{|\beta|}\\}$
$\displaystyle=$
$\displaystyle\left(\frac{|B_{x}|}{m_{p}}\right)^{2}\int_{-\infty}^{\infty}dz\
=\ \infty\,.$

The ensuing scalar product for two eigenstates is

$\displaystyle\langle x^{\prime}|x\rangle$
$\displaystyle={|B_{x}|}^{2}\int_{-m_{p}/\sqrt{|\beta|}}^{m_{p}/\sqrt{|\beta|}}dp\
\\!\frac{e^{-i(x-x^{\prime})m_{p}{\mbox{\scriptsize{arctanh}}}\left(p\sqrt{|\beta|}/m_{p}\right)/\hbar\sqrt{|\beta|}}}{m_{p}^{2}-p^{2}|\beta|}$
$\displaystyle=\left(\frac{|B_{x}|}{m_{p}}\right)^{2}\int_{-\infty}^{\infty}dz\
e^{-i(x-x^{\prime})z/\hbar}$
$\displaystyle=\left(\frac{|B_{x}|}{m_{p}}\right)^{2}2\pi\hbar\
\\!\delta(x-x^{\prime})\,.$ (123)

Therefore, we can set $B_{x}=\sqrt{m_{p}^{2}/2\pi\hbar}$. In Ref JLLP  it was
shown that the $\hat{x}$ operator is self-adjoint when an appropriate (and in
a sense natural) domain is chosen.

In passing, we should note that, although continuous observables such as
$\hat{x}$ or $\hat{p}$ are routinely employed in quantum theory, they are
really unphysical idealizations: the set of possible outcomes in any realistic
measurement is always countable, since the state space of any apparatus with
finite spatial extent has a countable basis. In turn, our reasoning related to
$\beta<0$ should thus be understood in this mathematically idealized sense —
as done with conventional Heisenberg $p$-$x$ uncertainty relations.

##  Appendix B Connection between wave functions in momentum and position
representation

In this Appendix, we discuss the connection between wave functions in momentum
and position representation in the presence of the modified commutator (2).

### Negative $\beta$ case

In this case the position and momentum representations of a wave function are
related via the relation

$\displaystyle\psi(x)$ $\displaystyle=$
$\displaystyle\int_{\frac{-m_{p}}{\sqrt{|\beta|}}}^{\frac{m_{p}}{\sqrt{|\beta|}}}\frac{dp}{\sqrt{2\pi\hbar}}\
\\!\frac{e^{ixm_{p}{\mbox{\scriptsize{arctanh}}}\left(p\sqrt{|\beta|}/m_{p}\right)/\hbar\sqrt{|\beta|}}}{\sqrt{1-p^{2}|\beta|/m_{p}^{2}}}\
\\!\tilde{\psi}(p)$ (124) $\displaystyle=$
$\displaystyle\\{z=m_{p}\mbox{{arctanh}}\left(p\sqrt{|\beta|}/m_{p}\right)/\sqrt{|\beta|}\\}$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}}\frac{dz}{\sqrt{2\pi\hbar}}\
\\!e^{ixz/\hbar}\
\\!\frac{\tilde{\psi}(m_{p}\tanh(z\sqrt{|\beta|}/m_{p})/\sqrt{|\beta|})}{\cosh(z\sqrt{|\beta|}/m_{p})}$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}}\frac{dz}{\sqrt{2\pi\hbar}}\
\\!e^{ixz/\hbar}\bar{\psi}(z)\,,$

where

$\bar{\psi}(z)\ =\
\frac{\tilde{\psi}(m_{p}\tanh(z\sqrt{|\beta|}/m_{p})/\sqrt{|\beta|})}{{\cosh(z\sqrt{|\beta|}/m_{p})}}.$
(125)

Note that this formula is valid only for $D=1$ dimension. In passing we can
easily check that the analogue of Parseval–Plancherel theorem holds, namely

$\displaystyle\int_{\mathbb{R}}dx\ \\!|\psi(x)|^{2}$ $\displaystyle=$
$\displaystyle\int_{-m_{p}/\sqrt{|\beta|}}^{m_{p}/\sqrt{|\beta|}}{dp}\
\\!|\tilde{\psi}(p)|^{2}$ (126) $\displaystyle=$ $\displaystyle\
\int_{\mathbb{R}}{dz}\ \\!|\bar{\psi}(z)|^{2}\,,$

or equivalently $|\\!|\psi|\\!|_{2}\ =\ |\\!|\tilde{\psi}|\\!|_{2}\ =\
|\\!|\bar{\psi}|\\!|_{2}$.

Note also that from the last line in (124), one can also easily deduce that
the momentum operator in the position representation has the form

$\displaystyle\hat{p}^{(x)}\ =\ m_{p}\tanh\left(-i\hbar\sqrt{|\beta|}/m_{p}\
\\!\frac{d}{dx}\right)/\sqrt{|\beta|}\,.$ (127)

It is straightforward to verify that the operator satisfies the canonical
commutation relation (2).

There is yet another interesting consequence of Eq. (124), namely one can
directly compute from it the corresponding position-space coherent state. In
particular, by using the Tsallis probability amplitude (13) (i.e., momentum-
space coherent state) we can write for the corresponding position-space
coherent state $\psi_{CS}(x)$ that

$\displaystyle\psi_{CS}(x)$ $\displaystyle=\
N\int_{\mathbb{R}}\frac{dz}{\sqrt{2\pi\hbar}}\ \\!e^{ixz/\hbar}$
$\displaystyle\times\,\frac{\left[m^{2}_{p}-m^{2}_{p}(\tanh(z\sqrt{|\beta|}/m_{p}))^{2}\right]^{m_{p}^{2}/(2|\beta|\gamma\hbar)-1/2}}{\cosh\left(z\sqrt{|\beta|}/m_{p}\right)}$
$\displaystyle=\
Nm_{p}^{m_{p}^{2}/(|\beta|\gamma\hbar)-1}\int_{\mathbb{R}}\frac{dz}{\sqrt{2\pi\hbar}}\
\\!e^{ixz/\hbar}$
$\displaystyle\times\,{\cosh\left(z\sqrt{|\beta|}/m_{p}\right)^{-m_{p}^{2}/(|\beta|\gamma\hbar)}}$
$\displaystyle=\
\tilde{N}\left|\Gamma\left(\frac{m^{2}_{p}/(|\beta|\gamma\hbar)}{2}+i\frac{xm_{p}}{2\sqrt{|\beta|}\hbar}\right)\right|^{2}$
$\displaystyle=\ \sqrt{\frac{m_{p}\
\\!\Gamma(2m^{2}_{p}/(|\beta|\gamma\hbar))}{4\pi\sqrt{|\beta|}\hbar\
\\!\Gamma^{4}(m^{2}_{p}/(|\beta|\gamma\hbar))}}$
$\displaystyle\times\,\left|\Gamma\left(\frac{m^{2}_{p}/(|\beta|\gamma\hbar)}{2}+i\frac{xm_{p}}{2\sqrt{|\beta|}\hbar}\right)\right|^{2}\,.$
(128)

In the derivation we used the Ramanujan formula Ramanujan

$\displaystyle\mbox{}\int_{-\infty}^{\infty}e^{-i\xi s}\
\\!\left|\Gamma(a+is)\right|^{2}\ \\!ds$ $\displaystyle\mbox{\hskip
8.53581pt}=\ \frac{2\pi}{2^{2a}}\Gamma(2a)\left[\cosh(\xi/2)\right]^{-2a}\,,$
(129)

that is valid for $a\in(-1,0)\cup(0,\infty)$. The normalization factor on the
last line of (128) was obtained by employing the Mellin–Barnes beta integral
(cf. e.g. Refs. distributions ; Ober )

$\displaystyle\int_{-\infty}^{\infty}\ \\!\left|\Gamma(a+ibs)\right|^{4}\
\\!ds\ =\ \frac{2\pi}{b}\frac{\Gamma^{4}(2a)}{\Gamma(4a)}\,.$ (130)

In passing we note that the state $\psi_{CS}(x)$ is even-parity state (as
required) and, in addition, it belongs to the Schwartz class, i.e., it decays
rapidly at infinity along with all derivatives.

For consistency we can now check that $\psi_{CS}(x)$ from Eq. (128) provides a
correct positional variance, which together with the momentum variance deduced
from $\tilde{\psi}_{CS}(p)$ (cf. Eq. (29)) saturates the GUP (1). To this end
one can use the formula for the Fourier transform of
$\left|\Gamma(a+ibs)\right|^{4}$ (see Ober , Eq. (274), p. 46) to show that

$\displaystyle\mbox{}\frac{b\Gamma(4a)}{2\pi\Gamma^{4}(2a)}\int_{-\infty}^{\infty}\
\\!s^{2}\ \\!\left|\Gamma(a+ibs)\right|^{4}\
\\!ds=\frac{a^{2}}{b^{2}(1+4a)}\,.$ (131)

If we now use from (130) that $a=m_{p}^{2}/(2|\beta|\gamma\hbar)$ and
$b=m_{p}/(2\sqrt{|\beta|}\hbar)$ we get

$\displaystyle(\Delta x)^{2}_{CS}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\ \\!x^{2}\ \\!\psi_{CS}(x)\ \\!dx$ (132)
$\displaystyle=$ $\displaystyle\frac{\hbar
m_{p}^{2}}{2m_{p}^{2}\gamma+\hbar|\beta|\gamma^{2}}\ =\ \frac{(\Delta
p)_{CS}^{2}}{\gamma^{2}}\,,$

where the last identity results from (20) and (28). This is equivalent to the
saturated GUP.

##  Appendix C Derivation of the measure $\mu(x_{0},p_{0})$

Here we derive the measure for the resolution of unity (42). To this end we
realize that the expression

$\displaystyle F(a/2,2z)\ \equiv\ \int_{-a/2}^{a/2}dy\ \\!{\tilde{\mu}(y)}\
\\!|\tilde{N}(y)|^{2}\ \\!e^{2yz}\,,$ (133)

is the finite Laplace transform. The inversion formula for the finite Laplace
transform is given by the principal value integral Laplace-transf

$\displaystyle\mbox{}{\tilde{\mu}(y)}\ \\!|\tilde{N}(y)|^{2}$ $\displaystyle=\
\frac{1}{2\pi i}\lim_{R\rightarrow\infty}\int_{c-iR}^{c+iR}e^{-sy}\
\\!F(a/2,s)\ \\!ds\,.$ (134)

Here the integral is taken over any open contour $\Gamma$ joining the points
$c-iR$ and $c+iR$ in the finite complex $s$ plane as $R\rightarrow\infty$. The
arbitrariness of $\Gamma$ stems from the fact that $F(a/2,s)$ is an entire
function of $s$. In practice, the integration is typically done along the
vertical line $Re(s)=c$ in the complex plane. In contrast to the usual inverse
Laplace transform, the value of $c$ can be freely chosen. So, in particular,
(134) is not necessarily related to the Bromwich integral.

If we now demand that

$\displaystyle F(a/2,2z)$ $\displaystyle=$
$\displaystyle\frac{m_{p}}{2\pi\hbar^{2}\gamma\sqrt{|\beta|}}\ \\!(\cosh
z)^{a}\,,$ (135)

we should obtain, by inserting (135) to (134), the desired measure
$\mu(x_{0},p_{0})$. The actual integral that needs to be evaluated in order to
obtain the measure is

$\displaystyle\frac{1}{2\pi
i}\lim_{R\rightarrow\infty}\int_{c-iR}^{c+iR}e^{-sy}\
\\!\left(\cosh\mbox{$\frac{s}{2}$}\right)^{a}\ \\!ds\,.$ (136)

By choosing $c=0$, we can rewrite (136) in the form

$\displaystyle\mbox{}\frac{1}{\pi}\lim_{R\rightarrow\infty}\int_{-R}^{R}e^{-i2sy}\
\\!\left(\cos{s}\right)^{a}\ \\!ds$
$\displaystyle\mbox{}=\frac{1}{\pi}\lim_{n\rightarrow\infty}\sum_{k=-n}^{n}\int_{(k-1/2)\pi}^{(k+1/2)\pi}e^{-i2sy}\
\\!\left(\cos s\right)^{a}\ \\!ds$
$\displaystyle\mbox{}=\frac{1}{\pi}\lim_{n\rightarrow\infty}\sum_{k=-n}^{n}(-1)^{ka}e^{-i2k\pi
y}\int_{-\pi/2}^{\pi/2}e^{-i2sy}\ \\!\left(\cos s\right)^{a}\ \\!ds$
$\displaystyle\mbox{}=\frac{1}{2^{a}}\lim_{n\rightarrow\infty}\sum_{k=-n}^{n}e^{-ik\pi(2y\pm
a)}\frac{\Gamma(a+1)}{\Gamma(1+\mbox{$\frac{a}{2}$}+y)\Gamma(1+\mbox{$\frac{a}{2}$}-y)}$
$\displaystyle\mbox{}=\frac{1}{2^{a-1}}\\!\sum_{k=-\infty}^{\infty}\delta(\mbox{$\pm
2y+a-2k$})\frac{\Gamma(a+1)}{\Gamma(1+\mbox{$\frac{a}{2}$}+y)\Gamma(1+\mbox{$\frac{a}{2}$}-y)}$
$\displaystyle\mbox{}=\frac{1}{2^{a-1}}\sum_{k=0}^{\infty}\
\\!\delta(\mbox{$\pm
2y+a-2k$})\frac{\Gamma(a+1)}{\Gamma(1+k)\Gamma(1+a-k)},~{}~{}~{}~{}$ (137)

where on the second line we have set $R=(n+1/2)\pi$, on the fourth line we
have used the identity ramanujan_b

$\displaystyle\mbox{}\int_{-\pi/2}^{\pi/2}e^{-isy}\ \\!\left(\cos
s\right)^{a}\ \\!ds$ $\displaystyle=\
\frac{\pi}{2^{a}}\frac{\Gamma(a+1)}{\Gamma[1+\mbox{$\frac{1}{2}$}(a+y)]\Gamma[1+\mbox{$\frac{1}{2}$}(a-y)]}$
$\displaystyle=\ \frac{\pi}{2^{a}}\binom{a}{\mbox{$\frac{1}{2}$}(a+y)}\,,$
(138)

(the last line represents the generalised binomial coefficient), on the fifth
line we have employed Poisson’s summation formula vladimirov

$\displaystyle\sum_{k=-\infty}^{\infty}\delta(x-kT)\ =\
\frac{1}{T}\sum_{k=-\infty}^{\infty}\exp\left(\pm i\frac{2\pi}{T}kx\right)\,,$
(139)

and finally, on the last line, we have taken advantage of the fact that
$\Gamma(1+k)$ has poles when $k$ is a negative integer.

By inserting (137) into (133), we can verify that the desired $(\cosh z)^{a}$
term is indeed the solution of the integral. To see this explicitly, consider
the upper sign in (137). We first notice that (137) can be rewritten as

$\displaystyle\frac{1}{2^{a}}\sum_{k=0}^{\infty}e^{(a-2k)\mbox{$\frac{\partial}{\partial
2y}$}}\binom{a}{k}\
\\!\delta(y)=\left[\cosh\left(\mbox{$\frac{1}{2}\frac{\partial}{\partial
y}$}\right)\right]^{a}\delta(y)\,.~{}~{}~{}$ (140)

By coupling this result with the identity vladimirov

$\displaystyle
e^{2zy}\left[\cosh\left(\mbox{$\frac{1}{2}\frac{\partial}{\partial
y}$}\right)\right]^{a}\ \\!\delta(y)$ $\displaystyle=$
$\displaystyle\delta(y)\ \\!e^{zy}\ \\!(\cosh z)^{a}$ (141) $\displaystyle=$
$\displaystyle\delta(y)\ \\!(\cosh z)^{a}\,,$

[which can be checked term-by-term by Taylor expanding the exponent in (140)],
we can write (133) as

$\displaystyle\mbox{}\int_{-a/2}^{a/2}dy\ \\!{\tilde{\mu}(y)}\
\\!|\tilde{N}(y)|^{2}\ \\!e^{2yz}$ $\displaystyle\mbox{}=\
\frac{m_{p}}{i(2\pi)^{2}\hbar^{2}\gamma\sqrt{|\beta|}}\int_{-a/2}^{a/2}\\!dy\
\\!e^{2yz}$ $\displaystyle\mbox{\hskip 2.84526pt}\times\
\lim_{R\rightarrow\infty}\int_{c-iR}^{c+iR}\\!ds\ \\!e^{-sy}\
\\!\left(\cosh\mbox{$\frac{s}{2}$}\right)^{a}$ $\displaystyle\mbox{}=\
\frac{m_{p}}{2\pi\hbar^{2}\gamma\sqrt{|\beta|}}\int_{-a/2}^{a/2}dy\
\\!\delta(y)\ \\!(\cosh z)^{a}$ $\displaystyle\mbox{}=\
\frac{m_{p}}{2\pi\hbar^{2}\gamma\sqrt{|\beta|}}\ \\!(\cosh z)^{a}\,,$ (142)

which is indeed $F(a/2,2z)$. We would get the same result if we had begun with
the lower sign in (137).

In view of the explicit form of $|\tilde{N}(y)|^{2}$, see Eq. (39), we get for
the measure

$\displaystyle\mbox{}\mu(x_{0},p_{0})\ =\ \mu(p_{0})\ =\ \tilde{\mu}(y)$
$\displaystyle\mbox{}=\
\frac{1}{2\pi\hbar}\frac{1}{\left[1-\left(\frac{2y}{a}\right)^{\\!2}\right]}\
\\!\sum_{k=0}^{\infty}\delta(\pm y+a/2-k)\,,$ (143)

with $y\in(-a/2,a/2)$. In (143), any sign convention can be used. The
resolution of unity thus reads

$\displaystyle\mbox{}\mathds{1}\ =\
\int_{\mathds{R}}dx_{0}\int_{-m_{p}/\sqrt{|\beta|}}^{m_{p}/\sqrt{|\beta|}}dp_{0}\
\\!\mu(p_{0})\ \\!|\psi,p_{0},x_{0}\rangle\langle\psi,x_{0},p_{0}|$
$\displaystyle\mbox{}=\
\frac{\hbar\gamma\sqrt{|\beta|}}{m_{p}}\int_{\mathds{R}}dx_{0}\int_{-a/2}^{a/2}dy\
\\!\tilde{\mu}(y)\ \\!|\psi,y,x_{0}\rangle\langle\psi,x_{0},y|$
$\displaystyle\mbox{}=\ \int_{\mathds{R}}dx_{0}\sum_{k=0}\
\\!\frac{\hbar\gamma\sqrt{|\beta|}}{2\pi\hbar\ \\!m_{p}}\
\\!\frac{\left(\frac{a}{2}\right)^{2}}{k(a-k)}\
\\!|\psi,k,x_{0}\rangle\langle\psi,x_{0},k|$ $\displaystyle\mbox{}=\
\int_{\mathds{R}}dx_{0}\sum_{p_{0}}\ \\!\frac{\Delta p_{0}}{2\pi\hbar}\
\frac{|\psi,p_{0},x_{0}\rangle\langle\psi,x_{0},p_{0}|}{\left[1-\frac{p_{0}^{2}|\beta|}{m_{p}^{2}}\right]}\,.$
(144)

Since $\delta(\pm y+a/2-k)$ implies that only contributing values of $p_{0}$
are $\mp m_{p}/\sqrt{|\beta|}\pm k\gamma\hbar\sqrt{|\beta|}/m_{p}$, we have
that the difference between two neighbouring values of $p_{0}$ is

$\displaystyle\Delta p_{0}\ =\ \Delta
k\frac{\gamma\hbar\sqrt{|\beta|}}{m_{p}}\ =\
\frac{\gamma\hbar\sqrt{|\beta|}}{m_{p}}\,.$ (145)

With this we may write

$\displaystyle\sum_{k}\dots\ =\sum_{k}\Delta k\dots=\ \sum_{p_{0}}\Delta
p_{0}\ \\!\frac{m_{p}}{\gamma\hbar\sqrt{|\beta|}}\dots\,,$ (146)

where $p_{0}$ is discrete with values
$-m_{p}/\sqrt{|\beta|}+k\gamma\hbar\sqrt{|\beta|}/m_{p}$. This fact was used
on the last line in (144). It is important to bear in mind that the Cauchy
principal value integral should be utilised in the $p_{0}$ integral in (144)
in order to see that the endpoint singularities are integrable.

Note that in the $|\beta|\rightarrow 0$ limit (144) reduces to

$\displaystyle\mbox{}\mathds{1}\ =\
\int_{\mathds{R}}dx_{0}\int_{\mathds{R}}dp_{0}\ \\!\frac{1}{2\pi\hbar}\
\\!|\psi,p_{0},x_{0}\rangle\langle\psi,x_{0},p_{0}|\,,~{}~{}~{}$ (147)

and hence

$\displaystyle\mu(p_{0})|_{|\beta|\rightarrow 0}\ =\
\frac{1}{2\pi\hbar}\,,~{}~{}~{}~{}$ (148)

which is the conventional value for (normalized) Schrödinger wave packets.

Should we have worked with states that are not normalized to unity (which is
typical, e.g. for Glauber CSs BJV ), we could assimilate the normalization
factor to the measure, in which case we would obtained [cf. (39) and (143)]

$\displaystyle\mbox{}\mu(x_{0},p_{0})_{\rm{NN}}\ =\ \mu(p_{0})_{\rm{NN}}\ =\
\tilde{\mu}(y)_{\rm{NN}}$ $\displaystyle\mbox{}\propto\
\frac{\sqrt{a}}{2^{a}}\ \\!\frac{\Gamma\left(a+1\right)\
\\!\sum_{k=0}^{\infty}\delta(\pm
y+a/2-k)}{\Gamma(1+\frac{a}{2}+y)\Gamma(1+\frac{a}{2}-y)}\,,$ (149)

which in the $|\beta|\rightarrow 0$ limit reduces to

$\displaystyle{\mu}(p_{0})_{\rm{NN},|\beta|\rightarrow 0}\ \propto\
\exp\left(-\frac{p_{0}^{2}}{\gamma\hbar}\right)\,,$ (150)

as one would expect.

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