#  $q$-generalized Tsallis thermostatistics in Unruh effect for mixed fields

Giuseppe Gaetano Luciano111email<EMAIL_ADDRESS>(corresponding
author)${}^{\hskip 0.28453pt1,2}$ and Massimo Blasone222email:
<EMAIL_ADDRESS>0.28453pt1,2}$ 1Dipartimento di Fisica,
Università di Salerno, Via Giovanni Paolo II, 132 I-84084 Fisciano (SA),
Italy.
2INFN, Sezione di Napoli, Gruppo collegato di Salerno, Via Giovanni Paolo II,
132 I-84084 Fisciano (SA), Italy.

###### Abstract

It was shown that the particle distribution detected by a uniformly
accelerated observer in the inertial vacuum (Unruh effect) deviates from the
pure Planckian spectrum when considering the superposition of fields with
different masses. Here we elaborate on the statistical origin of this
phenomenon. In a suitable regime, we provide an effective description of the
emergent distribution in terms of the nonextensive $q$-generalized statistics
based on Tsallis entropy. This picture allows us to establish a nontrivial
relation between the $q$-entropic index and the characteristic mixing
parameters $\sin\theta$ and $\Delta m$. In particular, we infer that $q<1$,
indicating the superadditive feature of Tsallis entropy in this framework. We
discuss our result in connection with the entangled condensate structure
acquired by the quantum vacuum for mixed fields.

Field mixing, Unruh effect, $q$-generalized statistics, Tsallis entropy

##  I Introduction

The phenomenon of quantum mixing, i.e. the superposition of particle states
with different masses, is among the most challenging topics in Particle
Physics. In the Standard Model, it appears in the quark sector through
Kobayashi-Maskawa matrix KM , a three-family generalization of Cabibbo mixing
matrix between $d$ and $s$ quarks Cabi . On the other hand, convincing
evidences of flavor mixing and oscillations in the neutrino sector have been
provided in recent years by Super-Kamiokande SKK  and SNO experiments SNO ,
confirming Pontecorvo’s pioneering idea Pont  and opening a window into
physics beyond the Standard Model.

Recently, the relevance of mixing transformations has prompted their study
from a more fundamental field-theoretical (QFT) perspective. QFT effects on
flavor mixing have been analyzed both for Dirac fermions BV95  and bosons
BlasCap . This has uncovered the limits of the original quantum mechanical
approach by pointing out the orthogonality between the vacuum for fields with
definite flavor and that for fields with definite mass, the former becoming a
condensate of particle-antiparticle pairs. The properties of flavor vacuum
have been further explored in Cabo , where it has been shown that the Fock
space for flavor fields cannot be obtained by the direct product of the Fock
spaces for massive fields. Therefore, the nontrivial nature of mixing appears
as a genuine QFT feature boiling down to the nonfactorizability of the flavor
states in terms of those with definite mass, including the vacuum state
(flavor vacuum).

All of the above studies have been developed in Minkowski spacetime. The QFT
approach to mixing has been extended to Rindler (uniformly accelerated) metric
in Luciano ; NonTN  and to curved background in Quaranta . In particular, in
Luciano ; NonTN  it has been found that the vacuum condensate detected by the
Rindler observer due to Unruh effect Unruh  deviates from the Planckian
density profile in the presence of mixed fields, the departure being dependent
on the mass difference and the mixing angle. Such a result has been originally
interpreted as a breakdown of the thermality of Unruh radiation for mixed
fields. In passing, we mention that unconventional behaviors of Unruh effect
are not entirely unusual in the literature, see for instance Dop ; Hammad ;
deformed1 ; deformedPet  and possible implications for particle decays Decay1
; Decay2 ; Decay3 .

In its traditional form, the particle number spectrum of Unruh condensate
follows the rules of Boltzmann-Gibbs statistics. However, in Tsallis1 ;
Tsallis2 ; Tsallis3 ; Tsallis4  it has been argued that systems exhibiting
long-range interactions and/or spacetime entanglement, either on quantum or
classical grounds, require a generalization of Boltzmann-Gibbs theory to the
so called nonextensive Tsallis $q$-thermostatistics. This occurs through a
suitable (nonadditive) redefinition of the entropy, which still recovers
Boltzmann-Gibbs formula in the $q=1$ limit. The $q$-generalized statistical
mechanics proposed by Tsallis has provided encouraging results in describing a
broad class of complex systems, such as self-gravitating stellar systems App1
; App3 , black holes Tsallis3 , the cosmic background radiation App7 ; App8 ,
low-dimensional dissipative systems Tsallis4 , solar neutrinos App11 , polymer
chains Polch  and modified cosmological models App13 ; App14 , among others.
Furthermore, it has paved the way for an intensive study of alternative
statistical models within the framework of information theory Ren .

Starting from the above premises, in this work we feature the entangled
condensate structure of the vacuum for mixed fields in the language of
nonextensive Tsallis statistics. We consider Unruh effect as a specific
playground. In this context, it is shown that the modified Unruh distribution
for mixed fields can be described by an appropriately $q-$generalized
distribution based on Tsallis entropy. This allows us to establish an
effective connection between the nonextensive $q$-entropic index and the
characteristic mixing parameters $\sin\theta$ and $\Delta m$ in a suitable
approximation. We find that $q<1$, which corresponds to a superadditive
Tsallis entropy. To make our analysis as transparent as possible, we deal with
a simplified model involving only two scalar fields. The case of fermion
mixing will be shortly addressed at the end with similar results.

The remainder of this paper is structured as follows: in Sec. II we analyze
the canonical quantization of a massive scalar field for the Rindler observer
using the Bogoliubov transformation method. This leads us to derive the Unruh
effect in a natural way. Section III contains the study of the QFT formalism
of flavor mixing in Rindler spacetime. Exploiting these tools, in Sec. IV we
introduce the basics of nonextensive Tsallis thermostatistics and investigate
the connection with field mixing at the level of Unruh vacuum condensate. We
discuss our result in relation with the complex condensate structure acquired
by the quantum vacuum for mixed fields. Conclusions and outlook are summarized
in Sec. V. The work ends with an Appendix devoted to review the theory of
field mixing in Minkowski background.

Throughout all the manuscript, we adopt the mostly negative signature for the
$4$-dimensional metric and natural units $\hslash=c=k_{\mathrm{B}}=1$.
Furthermore, we use the notation

$x=\\{t,\textbf{x}\\},\qquad{\bf{x}}=\\{x^{1},\vec{x}\\},\qquad\vec{x}=\\{x^{2},x^{3}\\}\,,$
(1)

for $4$-, $3$\- and $2$-vectors, respectively.

##  II QFT in Rindler spacetime and Unruh effect

In this Section we briefly review the quantization of a massive scalar field
for a uniformly accelerated (Rindler) observer. Without loss of generality, we
assume the acceleration to be along the $x^{1}$-axis. In this setting, it is
useful to introduce the new coordinates $-\infty<\eta,\xi<\infty$, such that
$t=\xi\sinh\eta$, $x^{1}=\xi\cosh\eta$, while leaving $\vec{x}$ unchanged. In
terms of these coordinates, Minkowski metric takes the form

$ds^{2}={(dt)}^{2}-{(dx^{1})}^{2}-\sum_{j=2}^{3}{(dx^{j})}^{2}\underset{\tiny{Rindler\;coord.}}{\longrightarrow}ds^{2}=\xi^{2}d\eta^{2}-d\xi^{2}-\sum_{j=2}^{3}{(dx^{j})}^{2}\,,$
(1)

which admits $B=\partial/\partial{\eta}$ as a time-like Killing vector.
Henceforth, we refer to the coordinates $\\{t,x^{1}\\}$ and $\\{\eta,\xi\\}$
as Minkowski and Rindler coordinates, respectively. Accordingly, the metric
(1) shall be named Rindler metric.

Let us now consider the world-line of fixed spatial coordinates, i.e.

$\xi(\tau)=a^{-1}=\mathrm{const}\,,\quad\vec{x}(\tau)=\mathrm{const},$ (2)

where $\tau$ is the proper time measured along the line. Substitution of Eq.
(2) into the metric (1) yields $\eta(\tau)=a\tau$, i.e., the proper time
$\tau$ for an observer moving along the line (2) is the same as the Rindler
time $\eta$, up to the scale factor $a$.

In Minkowski coordinates Eq. (2) becomes

$t(\tau)=a^{-1}\sinh a\tau,\quad x^{1}(\tau)=a^{-1}\cosh
a\tau,\quad\vec{x}(\tau)=\mathrm{const}\,,$ (3)

which is an hyperbola with asymptotes $t=\pm x^{1}$ in the $(t,x^{1})$ plane.
In special relativity, it is well known that the hyperbolic motion generalizes
the concept of Newtonian uniformly accelerated motion, with $|a|$ being the
magnitude of the proper acceleration Mukhanov . In particular, for $a>0$ the
observer moves along the branch of hyperbola in the right wedge
$R_{+}=\\{x|x^{1}>|t|\\}$, while for $a<0$ the motion occurs in the left wedge
$R_{-}=\\{x|x^{1}<-|t|\\}$ (see Fig. 1). This reveals the peculiar features of
the causal structure of Rindler spacetime: since a uniformly accelerated
observer in $R_{+}$ cannot receive (send) any signal from (to) the future
(past) wedge $t>|x^{1}|$ $(t<-|x^{1}|)$, the asymptote $t=|x^{1}|$
($t=-|x^{1}|$) appears to him as a future (past) event horizon. Notice that
the time ordering of the two horizons is reversed in $R_{-}$, since the
Killing vector $B$ is past oriented in this wedge. Accordingly, a Rindler
observer in $R_{+}$ turns out to be causally separated from $R_{-}$, and vice-
versa.

Figure 1: Causal structure of Rindler metric in $1+1$ dimensions. We are
assuming that Rindler observer accelerates along the $x^{1}$-axis, while the
transverse dimensions $x^{2},x^{3}$ are kept constant during the motion. The
arrow indicates the direction of motion.

Bearing in mind the causal structure of the metric (1), let us deal with the
quantization of a charged scalar field $\phi(x)$ of mass $m$ for the Rindler
observer333Where there is no ambiguity, we shall denote by $x$ both the sets
of Minkowski and Rindler coordinates.. In Rindler coordinates, the Klein-
Gordon equation reads

$\bigg{\\{}{\bigg{(}\frac{\partial}{\partial
t}\bigg{)}}^{2}-\sum_{j=1}^{3}{\bigg{(}\frac{\partial}{\partial
x^{j}}\bigg{)}}^{2}+m^{2}\bigg{\\}}\hskip
0.28453pt\,\phi(x)=0\underset{Rindler\;coord.}{\longrightarrow}\bigg{\\{}\frac{1}{\xi^{2}}\frac{\partial^{2}}{\partial\eta^{2}}-\frac{\partial^{2}}{\partial\xi^{2}}-\frac{1}{\xi}\frac{\partial}{\partial\xi}-\sum_{j=2}^{3}{\bigg{(}\frac{\partial}{\partial
x^{j}}\bigg{)}}^{2}+m^{2}\bigg{\\}}\hskip 0.28453pt\,\phi(x)=0\,,$ (4)

which has the modes

$u_{\kappa}^{\,(\sigma)}(x)=\frac{\theta(\sigma\xi)}{2\pi\Omega}\frac{1}{\Gamma(i\Omega)^{2}\sqrt{\sinh(\pi\Omega)}}\hskip
0.56905ptK_{i\Omega}(\mu_{k}\xi)\hskip
0.28453pte^{i\left(\vec{k}\cdot\vec{x}-\sigma\Omega\eta\right)}\,,$ (5)

as solutions of frequency $\Omega>0$ with respect to the time coordinate
$\eta$. Following Takagi , here we have introduced the shorthand notation
$\kappa\equiv(\Omega,\vec{k})$. The Heaviside step function
$\theta(\sigma\xi)$ restricts the support of $u_{\kappa}^{\,(\sigma)}$ to only
one of the two Rindler wedges. Specifically, $\sigma=+$ refers to the right
wedge $R_{+}$, while $\sigma=-$ to the left wedge. The function
$\Gamma(i\Omega)$ is the usual Euler’s gamma, while $K_{i\Omega}(\mu_{k}\xi)$
denotes the modified Bessel function of second kind. The reduced Minkowski
frequency is given by $\mu_{k}={(m^{2}+|\vec{k}|^{2})}^{1/2}$.

It is easy to verify that the Rindler modes (5) form a complete and
orthonormal set with respect to the Klein-Gordon product in Rindler
coordinates. This allows us to take the following expansion for the scalar
field Takagi

$\phi(x)=\sum_{\sigma,\hskip 1.13809pt\Omega}\,\int
d^{2}k\,\Big{\\{}{b^{\,(\sigma)}_{\kappa}}\,u_{\kappa}^{\,(\sigma)}(x)+{\bar{b}^{\,(\sigma)\dagger}_{\kappa}}\,u_{\kappa}^{\,(\sigma)*}(x)\Big{\\}},$
(6)

where $\sum_{\sigma,\hskip 1.13809pt\Omega}$ stands for
$\sum_{\sigma=\pm}\int_{0}^{\infty}d\Omega\hskip 0.56905pt$. The ladder
operators ${b^{\,(\sigma)}_{\kappa}}$ (${\bar{b}^{\,(\sigma)}_{\kappa}}$) are
assumed to be canonical. They act as annihilators of Rindler particles
(antiparticles) of frequency $\Omega$ and transverse momentum $\vec{k}$ in the
wedge $R_{\sigma}$. Rindler vacuum $|0_{\mathrm{R}}\rangle$ is defined by
$b_{\kappa}^{\,(\sigma)}|0_{\mathrm{R}}\rangle=\bar{b}_{\kappa}^{\,(\sigma)}|0_{\mathrm{R}}\rangle=0$,
$\forall\sigma,\kappa\,$. On the other hand,
${b^{\,(\sigma)\dagger}_{\kappa}}$ (${\bar{b}^{\,(\sigma)\dagger}_{\kappa}}$)
creates a Rindler particle (antiparticle) with the same quantum numbers as
defined above.

We now focus on the relation between the quantization (6) and the Minkowski
plane-wave expansion

$\phi(x)=\int
d^{3}{k}\,\Big{\\{}a_{\textbf{k}}\,U_{\textbf{k}}(x)+{\bar{a}_{\textbf{k}}}^{\dagger}\,U_{\textbf{k}}^{\hskip
0.28453pt*}(x)\Big{\\}},$ (7)

where
$U_{\bf{k}}(x)={[{2\omega_{{\bf{k}}}{(2\pi)}^{3}}]}^{-1/2}e^{i\left({\bf{k}}\cdot{\bf{x}}-\omega_{{\bf{k}}}t\right)}$
are the plane-waves of frequency
$\omega_{\textbf{k}}=({m^{2}+|\textbf{k}|^{2}})^{1/2}$. Here, $a_{{\bf{k}}}$
($\bar{a}_{{\bf{k}}}$) denote the canonical annihilators of Minkowski
particles (antiparticles) with momentum ${\bf{k}}$ and frequency
$\omega_{{\bf{k}}}$, such that
$a_{\bf{k}}|0_{\mathrm{M}}\rangle=\bar{a}_{\bf{k}}|0_{\mathrm{M}}\rangle=0$,
$\forall{\bf{k}}$, where $|0_{\mathrm{M}}\rangle$ is the Minkowski vacuum. As
before, the rôle of creation operators is played by the adjoint
$a^{\dagger}_{\bf{k}}$ ($\bar{a}^{\dagger}_{\bf{k}}$). Since this formalism
holds for the set of inertial observers in Minkowski spacetime, in what
follows we indifferently refer to the expansion (7) as either Minkowski or
inertial field quantization.

To derive the Bogoliubov transformation between the two field representations
introduced above, we compare the expansions (6) and (7) on a space-like
hypersurface $\Sigma$ which lies in the Rindler manifold $R_{\pm}$. By
multiplying both sides by the Rindler mode $u^{(\sigma)}_{\kappa}$, we obtain
Takagi

$b_{\kappa}^{\,(\sigma)}=\int
d^{3}k^{\prime}\left\\{a_{\textbf{k}^{\prime}}\hskip
0.28453pt(u^{(\sigma)}_{\kappa},U_{\textbf{k}^{\prime}})\,+\,\bar{a}^{\dagger}_{\textbf{k}^{\prime}}\hskip
0.28453pt(u^{(\sigma)}_{\kappa},U^{*}_{\textbf{k}^{\prime}})\right\\}\,.$ (8)

The explicit expressions of the Bogoliubov coefficients
$(u^{(\sigma)}_{\kappa},U_{\textbf{k}^{\prime}})$ and
$(u^{(\sigma)}_{\kappa},U^{*}_{\textbf{k}^{\prime}})$ are rather awkward to
exhibit. They are given in Takagi . However, Eq. (8) can be cast in a more
transparent form by introducing the following superposition of
$a_{\textbf{k}}$-operators

$d_{\kappa}^{\,(\sigma)}=\int
dk_{1}\,\frac{1}{\sqrt{2\pi\omega_{\bf{k}}}}\,{\bigg{(}\frac{\omega_{{\bf{k}}}+k_{1}}{\omega_{{\bf{k}}}-k_{1}}\bigg{)}}^{i\sigma\Omega/2}\,a_{{\bf{k}}},$
(9)

(similarly, the definition of $\bar{d}_{\kappa}^{\,(\sigma)}$ is obtained by
replacing $a_{{\bf{k}}}$ with $\bar{a}_{{\bf{k}}}$). In Takagi , it has been
shown that these new operators still obey the canonical commutator.
Furthermore, they share a common vacuum with the $a_{\textbf{k}}$’s, since
they are linear combinations of these annihilators only. In other terms, the
$d$-operators provide an alternative representation for the scalar field,
which is unitarily equivalent to the plane-wave quantization from the
perspective of inertial observers444At first glance, the physical meaning of
$d$-operators may appear quite unclear. However, in Takagi  it has been shown
that they diagonalize the generator of Lorentz boosts along the $x^{1}$-axis.
Therefore, the field quantization in this representation exploits the symmetry
of Minkowski spacetime under boost transformations, just as the plane-wave and
spherical-wave quantizations rely on the symmetry under spacetime translations
and rotations, respectively.. This has been discussed in more detail in
Luciano .

In terms of the operators (9), Eq. (8) can be rewritten as

$b_{\kappa}^{\,(\sigma)}=\sqrt{1+N_{\mathrm{BE}}(\Omega)}\hskip
0.28453ptd_{\kappa}^{\,(\sigma)}\,+\,\sqrt{N_{\mathrm{BE}}(\Omega)}\hskip
0.28453pt\bar{d}_{\tilde{\kappa}}^{\,(-\sigma)\,\dagger}\,,$ (10)

where $\tilde{\kappa}\equiv(\Omega,-\vec{k})$ and

$N_{\mathrm{BE}}(\Omega)=\frac{1}{e^{2\pi\Omega}-1}\,.$ (11)

is the Bose-Einstein distribution function.

Next, by resorting to Eqs. (9) and (10), we can evaluate the expected number
spectrum of Rindler particles in the Minkowski vacuum, obtaining

$\langle 0_{\mathrm{M}}|b_{\kappa^{\prime}}^{\,(\sigma^{\prime})\dagger}\hskip
0.56905ptb_{\kappa}^{\,(\sigma)}|0_{\mathrm{M}}\rangle=N_{\mathrm{BE}}(\Omega)\hskip
0.56905pt\delta_{\sigma\sigma^{\prime}}\hskip
0.56905pt\delta^{3}(\kappa-\kappa^{\prime})\,,$ (12)

(similarly for $\bar{b}_{\kappa}^{\,(\sigma)}$). Since the proper energy of
quanta detected by an observer moving with uniform acceleration $a$ is

$E=a\Omega\,,$ (13)

it is more appropriate to rewrite the distribution (11) in the form

$N_{\mathrm{BE}}(\Omega)=\frac{1}{e^{E/T_{\mathrm{U}}}-1}\,,$ (14)

where $T_{\mathrm{U}}=a/(2\pi)$ is the Unruh temperature Unruh . Thus, we
recover the well-known Unruh result that Minkowski vacuum appears as a Bose-
Einstein thermal distribution of Rindler particles, with temperature
$T_{\mathrm{U}}$ being proportional to the magnitude of the proper
acceleration Unruh .

As discussed in Takagi , the inherent properties of the condensate (12) can be
explored in depth by expressing the Minkowski vacuum $|0_{\mathrm{M}}\rangle$
in terms of the Rindler one $|0_{\mathrm{R}}\rangle$. In so doing, we infer
that the inertial vacuum acquires the structure of a coherent state of
pairwise-correlated Rindler particles. Specifically, an excitation in the
positive wedge $R_{+}$ is correlated to an excitation of opposite spatial
momentum in the negative region $R_{+}$, and vice-versa. Since the two wedges
are causally disconnected, this turns out to be an EPR-like correlation
between space-like separated quanta.

The spectrum (12) diverges for any fixed $\kappa=\kappa^{\prime}$. This is due
to the fact that the creation operators $a_{{\bf{k}}}^{\dagger}$,
$b_{\kappa}^{\dagger}$ (and the corresponding operators for antiparticles) do
not produce normalizable states when applied on the respective vacua. To avoid
conceptual difficulties we shall otherwise encounter later, we introduce the
following set of functions Hawk1995

$f_{nl}(k)=\chi_{n}(k)\exp^{-2\pi ilk/\varepsilon}\,,$ (15)
$\chi_{n}(k)=\left\\{\begin{array}[]{rcl}&&\varepsilon^{-1/2}\,\,\,\,\,\,\,\,\,\mathrm{for}\left(n-\frac{1}{2}\right)\varepsilon<k<\left(n+\frac{1}{2}\right)\varepsilon\\\\[8.53581pt]
&&0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{otherwise}\end{array}\right.\,,$
(16)

where $\varepsilon$ is a positive constant of dimension of inverse length. By
exploiting the completeness and orthonormality of this set Takagi , we define
the Minkowski wave packet by

$U_{nl\vec{k}}(x)=\int dk_{1}f_{nl}(k_{1})U_{{\bf{k}}}(x)\,,$ (17)

where $n$ and $l$ run over all the integers.

On the other hand, to form the Rindler wave packet, we restrict the subscript
$n$ of $\\{f_{nl}\\}$ to positive integers. Notice that this does not affect
the orthonormality nor the completeness of the set (15), provided that the
argument $k$ is now replaced by the Rindler frequency $\Omega$ and
$\varepsilon$ is assumed to be dimensionless. We then define

$u_{nl\vec{k}}(x)=\int_{0}^{\infty}d\Omega
f_{nl}(\Omega)u^{(\sigma)}_{\kappa}(x)\,.$ (18)

Two comments are in order here: first, the wave packets (17) and (18) satisfy
a box-like normalization and are complete, due to the definition (15) of the
smearing function. Furthermore, in the above construction we have left the
reduced momentum $\vec{k}$ untouched, as it does not enter the Bose-Einstein
distribution function (14) explicitly. More properly, we should extend the
wave packet formalism to this quantum number as well, resulting in a new pair
of subscripts in place of each component of $\vec{k}$. However, since this
procedure would burden the notation without providing any conceptual
advantage, we continue to use the symbol $\vec{k}$, taking care of this
aspect.

We can now repeat the computation of the number spectrum of Rindler particles
in the Minkowski vacuum. By choosing the parameter $\varepsilon$ in the
definition of $f_{nl}(k_{1})$ much less than the reduced Minkowski frequency
$\mu_{k}$ and the $\varepsilon$ in $f_{nl}(\Omega)$ much less than unity, we
are led to Takagi

$\langle 0_{\mathrm{M}}|\hskip
0.56905ptb_{n^{\prime}l^{\prime}\vec{k}^{\prime}}^{\,(\sigma^{\prime})\dagger}\hskip
0.28453ptb_{nl\vec{k}}^{\,(\sigma)}\hskip
0.56905pt|0_{\mathrm{M}}\rangle=N_{\mathrm{BE}}({\Omega_{n}})\delta_{\sigma\sigma^{\prime}}\delta_{nn^{\prime}}\delta_{ll^{\prime}}\delta_{\vec{k}\vec{k^{\prime}}}\,,$
(19)

where $\Omega_{n}=n\varepsilon$. Of course, for fixed values of
$\sigma=\sigma^{\prime},n=n^{\prime},l=l^{\prime},m=m^{\prime}$ and
$\vec{k}=\vec{k}^{\prime}$, this gives

$\mathcal{N}(\Omega_{n})\equiv\langle 0_{\mathrm{M}}|\hskip
0.56905ptb_{nl\vec{k}}^{\,(\sigma)\dagger}\hskip
0.28453ptb_{nl\vec{k}}^{\,(\sigma)}\hskip
0.56905pt|0_{\mathrm{M}}\rangle=N_{\mathrm{BE}}({\Omega_{n}})\,,$ (20)

As expected, the use of properly normalizable wave packets results into a
regularization of the Unruh spectrum (12). In the next Section, we shall see
how the distribution (20) is modified when dealing with the superposition of
fields with different masses.

##  III QFT of flavor mixing in Rindler spacetime

The QFT treatment of flavor mixing, originally developed for Dirac neutrinos
in Minkowski background BV95  and later extended to mesons, such as the
$K^{0}-\bar{K}^{0}$, $B^{0}-\bar{B}^{0}$ and $\eta-\eta^{\prime}$ systems
BlasCap ; Ji , has revealed a series of nontrivial features that are totally
missed by quantum mechanics. These aspects are reviewed in the Appendix, with
particular emphasis on the issue of the unitary inequivalence between the Fock
space for fields with definite flavor and the Fock space for fields with
definite mass. Following Luciano , here we generalize the quantization of
mixed fields to the Rindler metric. We consider the mixing transformations in
a simplified two-flavor model with charged scalar fields555Strictly speaking,
for bosons we should refer to the mixing of quantum numbers such as the
strangeness or isospin, rather than flavor. However, in what follows we
improperly label such a quantum number as flavor and the corresponding fields
as definite flavor fields. On the other hand, the fields with definite mass
will be referred to as mass fields.. Denoting by $\chi=A,B$ ($i=1,2$) the
flavor (mass) label, these transformations read

$\displaystyle\phi_{A}(x)$ $\displaystyle=$
$\displaystyle\phi_{1}(x)\,\cos\theta+\phi_{2}(x)\,\sin\theta\,,$ (21)
$\displaystyle\phi_{B}(x)$ $\displaystyle=$
$\displaystyle-\phi_{1}(x)\,\sin\theta+\phi_{2}(x)\,\cos\theta\,,$ (22)

where $\theta$ is the mixing angle and $\phi_{i}$ ($i=1,2$) are two free
charged scalar fields of masses $m_{i}$, such that $m_{2}\neq m_{1}$. For
definiteness, we set $m_{2}>m_{1}$. Let
$\pi_{i}=\partial_{t}\phi_{i}^{\dagger}$ be the conjugate momenta.

In the canonical quantization formalism, it is known that

$\qquad\big{[}\phi_{i}(x),\,\pi_{j}(x^{\prime})\big{]}_{t=t^{\prime}}=\big{[}\phi^{\dagger}_{i}(x),\,\pi^{\dagger}_{j}(x^{\prime})\big{]}_{t=t^{\prime}}=i\,\delta^{3}(x-x^{\prime})\,\delta_{ij}\,,\qquad
i,j=1,2\,,$ (23)

with all other equal-time commutators vanishing. In the Appendix we discuss
the algebraic structure of Eqs. (21) and (22), showing that each of them
appears as a rotation combined with a Bogoliubov transformation when seen at
level of ladder operators. Notice that this peculiar structure, which is
absent in quantum mechanics, arises from the necessity to take account of the
antiparticle degrees of freedom intrinsically built in QFT. As a result, the
vacuum state for flavor fields becomes a condensate of massive particle-
antiparticle pairs (see Eq. (60)).

To find out how the phenomenon of mixing appears to the Rindler observer, we
retrace the same steps leading to Eq. (6) and consider the following free
field-like expansions for mixed fields Luciano

$\phi_{\ell}(x)=\sum_{\sigma,\hskip 1.13809pt\Omega}\,\int
d^{2}k\,\Big{\\{}{b^{\,(\sigma)}_{\kappa,\ell}}(\eta)\,u_{\kappa,j}^{\,(\sigma)}(x)+{\bar{b}^{\,(\sigma)\dagger}_{\kappa,\ell}}(\eta)\,u_{\kappa,j}^{\,(\sigma)*}(x)\Big{\\}},\quad(\ell,j)=\left\\{(A,1),(B,2)\right\\},$
(24)

where we have used the shorthand notation
$b^{\,(\sigma)}_{\kappa,\ell}(\eta)\equiv
b^{\,(\sigma)}_{\kappa,\ell}(\theta,\eta)$ for the ladder operators (similarly
for $\bar{b}^{\,(\sigma)}_{\kappa,\ell}(\eta)$). By comparison with Eq. (54),
it is clear that these operators provide the Rindler counterpart of Minkowski
flavor annihilators given in Eq. (55). It is a matter of calculations to show
that they obey the (equal-time) canonical commutators.

As remarked above, the mixing relations at level of ladder operators hide a
Bogoliubov transformation between the flavor and mass bases. At the same time,
the field quantizations for Minkowski and Rindler observers are connected to
each other by the Bogoliubov transformation (10) responsible for the thermal
Unruh effect. Overall, we expect that the Rindler annihilators in the flavor
representation are related to the corresponding Minkowski operators in the
mass basis by a combination of these two Bogoliubov transformations. To
analyze such an interplay, we compare the expansions (24) and (54). By using
Eqs. (21)-(22) and the transformation (55), after some tedious but
straightforward calculations, we obtain Luciano

$\displaystyle b_{\kappa,A}^{\,(\sigma)}$ $\displaystyle=$
$\displaystyle\sqrt{1+N_{\mathrm{BE}}(\Omega)}\,\,\Big{[}\cos\theta\,d_{\kappa,1}^{\,(\sigma)}\,+\,\sin\theta\sum_{\hskip
1.70717pt\sigma^{\prime},\hskip
0.28453pt\Omega^{\prime}}\Big{(}d_{(\Omega^{\prime},\vec{k}),2}^{\,(\sigma^{\prime})}\,\;{\cal
A}_{(\Omega,\Omega^{\prime}),\,\vec{k}}^{(\sigma,\sigma^{\prime})}\,+\,\bar{d}_{(\Omega^{\prime},-\vec{k}),2}^{\,(\sigma^{\prime})\dagger}\,\;{\cal
B}_{(\Omega,\Omega^{\prime}),\,\vec{k}}^{(\sigma,\sigma^{\prime})}\,\Big{)}\Big{]}$
$\displaystyle+$
$\displaystyle\,\sqrt{N_{\mathrm{BE}}(\Omega)}\,\,\Big{[}\cos\theta\,\bar{d}_{\tilde{\kappa},1}^{\,(-\sigma)\dagger}\,+\,\sin\theta\sum_{\hskip
1.70717pt\sigma^{\prime},\hskip
0.28453pt\Omega^{\prime}}\Big{(}\bar{d}_{(\Omega^{\prime},-\vec{k}),2}^{(-\sigma^{\prime})\dagger}\,\;{\cal
A}_{(\Omega,\Omega^{\prime}),\,\vec{k}}^{(\sigma,\sigma^{\prime})}\,+\,d_{(\Omega^{\prime},\vec{k}),2}^{(-\sigma^{\prime})}\,\;{\cal
B}_{(\Omega,\Omega^{\prime}),\,\vec{k}}^{(\sigma,\sigma^{\prime})}\,\Big{)}\Big{]},$

where we have omitted for simplicity the time-dependence of
$b_{\kappa,A}^{\,(\sigma)}$ (a similar expression hold true for
$b_{\kappa,B}^{\,(\sigma)}$ as well). We stress that the $d$-operators are an
equivalent way of rewriting the standard Minkowski annihilators appearing in
Eq. (7) (see Eq. (9)). From the above relation, we can clearly distinguish the
action of the thermal Bogoliubov transformation (encoded by the coefficients
${(1+N_{\mathrm{BE}})^{1/2}}$ and ${N^{1/2}_{\mathrm{BE}}}$) from the action
of the mixing Bogoliubov transformation (which appears through the
coefficients ${\cal
A}_{(\Omega,\Omega^{\prime}),\,\vec{k}}^{(\sigma,\sigma^{\prime})}$ and ${\cal
B}_{(\Omega,\Omega^{\prime}),\,\vec{k}}^{(\sigma,\sigma^{\prime})}$). The
Bogoliubov coefficients related to mixing are given by Luciano

$\displaystyle{\cal
A}_{(\Omega,\Omega^{\prime}),\,\vec{k}}^{(\sigma,\sigma^{\prime})}=\int^{+\infty}_{-\infty}\frac{dk_{1}}{4\pi}\left(\frac{1}{\omega_{{\bf{k}},1}}+\frac{1}{\omega_{{\bf{k}},2}}\right){\left(\frac{\omega_{{\bf{k}},1}+k_{1}}{\omega_{{\bf{k}},1}-k_{1}}\right)}^{i\sigma\Omega/2}{\left(\frac{\omega_{{\bf{k}},2}+k_{1}}{\omega_{{\bf{k}},2}-k_{1}}\right)}^{-i\sigma^{\prime}\Omega^{\prime}/2}e^{i(\omega_{{\bf{k}},1}-\,\omega_{{\bf{k}},2})t}\,,$
(26) $\displaystyle{\cal
B}_{(\Omega,\Omega^{\prime}),\,\vec{k}}^{(\sigma,\sigma^{\prime})}=\int^{+\infty}_{-\infty}\frac{dk_{1}}{4\pi}\left(\frac{1}{\omega_{{\bf{k}},2}}-\frac{1}{\omega_{{\bf{k}},1}}\right){\left(\frac{\omega_{{\bf{k}},1}+k_{1}}{\omega_{{\bf{k}},1}-k_{1}}\right)}^{i\sigma\Omega/2}{\left(\frac{\omega_{{\bf{k}},2}+k_{1}}{\omega_{{\bf{k}},2}-k_{1}}\right)}^{-i\sigma^{\prime}\Omega^{\prime}/2}e^{i(\omega_{{\bf{k}},1}+\,\omega_{{\bf{k}},2})t}\,.$
(27)

Despite the nontrivial structure of ${\cal
A}_{(\Omega,\Omega^{\prime}),\,\vec{k}}^{(\sigma,\sigma^{\prime})}$ and ${\cal
B}_{(\Omega,\Omega^{\prime}),\,\vec{k}}^{(\sigma,\sigma^{\prime})}$,
interesting implications of Eq. (III) can still be derived for $t=\eta=0$ and
in the reasonable approximation of small difference between the masses of the
two fields, i.e. $|\Delta
m^{2}|/{m_{1}^{2}}=|m_{2}^{2}-m_{1}^{2}|/m_{1}^{2}\ll 1$.666Notice that the
assumption $|\Delta m^{2}|/m^{2}\ll 1$ makes even more physical sense if
considered for mixing of neutrino fields, the study of which is reserved for
future investigation. Indeed, if we evaluate the number spectrum of mixed
particles detected by the Rindler observer in the inertial vacuum, we get to
the leading order Luciano

$\langle
0_{\mathrm{M}}|\,b_{\kappa,\chi}^{\,(\sigma)\dagger}(0)\,b_{\kappa^{\prime},\chi}^{\,(\sigma)}(0)\,|0_{\mathrm{M}}\rangle\,=\,N_{\mathrm{BE}}(\Omega)\,\delta^{3}(\kappa-\kappa^{\prime})-\,\frac{|\Delta
m^{2}|}{8\,\mu_{k,1}^{\hskip
0.28453pt2}}\sin^{2}\theta\,\frac{\sigma\,(\Omega^{\prime}+\Omega)\,G(\Omega,\Omega^{\prime})}{\sinh\left[\frac{\pi}{2}\,\sigma\,(\Omega^{\prime}+\Omega)\right]}\,\delta(\vec{k}-\vec{k}^{\prime})\,+\,\mathcal{O}\left(\frac{|\Delta
m^{2}|}{\mu^{2}_{k,1}}\right)^{2}\,,$ (28)

for $\chi=A,B$, where

$G(\Omega,\Omega^{\prime})\equiv\sqrt{1+N_{\mathrm{BE}}(\Omega)}\,\sqrt{N_{\mathrm{BE}}(\Omega^{\prime})}\,+\,\sqrt{N_{\mathrm{BE}}(\Omega)}\,\sqrt{1+N_{\mathrm{BE}}(\Omega^{\prime})}\,.$
(29)

Thus, the standard Bose-Einstein distribution of Unruh vacuum condensate (12)
turns out to be spoilt in the presence of mixed fields. Notice that, for
$\theta\rightarrow 0$, Eq. (28) recovers the usual result, as expected in the
absence of mixing. Similar considerations hold true for $m_{1}\rightarrow
m_{2}$ and in the relativistic limit $|\vec{k}|^{2}\gg m_{1}^{2}+m_{2}^{2}$,
since the parameter ${|\Delta m^{2}|}/{\mu^{2}_{k,i}}$ becomes increasingly
small.

As argued at the end of Sec. II, in order to avoid unphysical divergencies in
the spectrum, it is convenient to revisit the above formalism by employing
wave packets. With the aid of Eqs. (15)-(18), the density (20) of Rindler
mixed particles with quantum numbers $\Omega_{n},l,\vec{k}$ then becomes

$\mathcal{N}_{\theta,\Delta m}(\Omega_{n})\,\equiv\,\langle
0_{\mathrm{M}}|\hskip 0.56905ptb_{nl\vec{k}}^{\dagger}(0)\hskip
0.28453ptb_{nl\vec{k}}(0)|0_{\mathrm{M}}\rangle\,=\,N_{\mathrm{BE}}({\Omega_{n}})\,-\,\frac{|\Delta
m^{2}|}{4\,\mu_{k,1}^{\hskip
0.28453pt2}}\sin^{2}\theta\,\frac{\Omega_{n}\,G(\Omega_{n},\Omega_{n})}{\sinh\left({\pi}\Omega_{n}\right)}\,+\,\mathcal{O}\left(\frac{|\Delta
m^{2}|}{\mu^{2}_{k,1}}\right)^{2}\,,$ (30)

where we have dropped the indices $\sigma$ and $\chi$ from $\langle
0_{\mathrm{M}}|\hskip 0.56905ptb_{nl\vec{k}}^{\dagger}(0)\hskip
0.28453ptb_{nl\vec{k}}(0)|0_{\mathrm{M}}\rangle$, since the r.h.s. of Eq (28)
is in fact insensitive to them. By resorting to the definition (29) of
$G(\Omega,\Omega)$, we are finally led to

$\mathcal{N}_{\theta,\Delta
m}(\Omega_{n})\,=\,N_{\mathrm{BE}}({\Omega_{n}})\,-\,\frac{|\Delta
m^{2}|}{4\,\mu_{k,1}^{\hskip
0.28453pt2}}\sin^{2}\theta\,\Omega_{n}\operatorname{csch}^{2}(\pi\Omega_{n})\,+\,\mathcal{O}\left(\frac{|\Delta
m^{2}|}{\mu^{2}_{k,1}}\right)^{2}\,,$ (31)

where $\operatorname{csch}(x)=1/\sinh(x)$.

In Luciano  the modified distribution (28) (or, equivalently, (31)) has been
interpreted as being due to a breakdown of the thermality of Unruh radiation
induced by field mixing. We further elaborate on the physical meaning of the
result (31) in the next Section.

##  IV Flavor Mixing and q-generalized Tsallis statistics

In the standard Boltzmann-Gibbs thermodynamics, it is well known that entropy
is an additive quantity, which means that, given two probabilistically
independent systems $A$ and $B$ with entropies $S_{\mathrm{BG}}(A)$ and
$S_{\mathrm{BG}}(B)$, respectively, the total entropy is simply
$S_{\mathrm{BG}}(A+B)=S_{\mathrm{BG}}(A)+S_{\mathrm{BG}}(B)$. At the
statistical level, the Boltzmann-Gibbs entropy of a system in an equilibrium
macrostate can be expressed in terms of the corresponding microscopic
configurations as

$S_{\mathrm{BG}}=-\sum_{i=1}^{W}p_{i}\log p_{i}\,,$ (32)

for a set of $W$ discrete microstates, where $\\{p_{i}\\}$ is the set of
probability distribution with the condition $\sum_{i=1}^{W}p_{i}=1$. If
probabilities are all equal, this takes the well-known form
$S_{\mathrm{BG}}=\log W$. It is immediate to check that $S_{\mathrm{BG}}$
satisfies the additivity property as defined above.

Despite the wide range of applicability of the Boltzmann-Gibbs theory, for
complex systems exhibiting long-range interactions and/or spacetime
entanglement, it has been argued that the standard Boltzmann-Gibbs theory
should be generalized to a nonextensive statistical mechanics based on the
nonadditive Tsallis entropy Tsallis1 ; Tsallis2 ; Tsallis3 ; Tsallis4

$S_{q}\,=\,\frac{1-\sum_{i=1}^{W}p_{i}^{q}}{q-1}\,=\,\sum_{i=1}^{W}p_{i}\log_{q}\frac{1}{p_{i}}\,,$
(33)

with

$\log_{q}z\equiv\frac{z^{1-q}-1}{1-q},\quad(\log_{1}z=\log z)\,.$ (34)

Note that $S_{q}$ recovers Boltzmann-Gibbs entropy $S_{\mathrm{BG}}$ in the
$q\rightarrow 1$ limit. Furthermore, by considering again two
probabilistically independent systems such that
$p_{ij}^{A+B}=p_{i}^{A}p_{j}^{B},\,\forall{(i,j)}$, the definition (33) leads
to

$S_{q}(A+B)=S_{q}(A)+S_{q}(B)+(1-q)S_{q}(A)S_{q}(B)\,,$ (35)

indicating that $S_{q}$ is superadditive or subadditive, depending on whether
$q<1$ or $q>1$. Thus, the dimensionless index $q\in\mathbb{R^{+}}$ quantifies
the departure of Tsallis entropy from Boltzmann-Gibbs one. For this reason, it
is named nonextensive Tsallis parameter. Paradigmatic examples of systems
obeying the generalized statistics (33) are the strongly gravitating black
holes Tsallis3 , albeit in recent years Tsallis thermostatistics has found
applications in a variety of physical scenarios App1 ; App3 ; App7 ; App8 ;
App11 ; App13 ; App14 .

Now, within an approximation called factorization approach, it has been shown
that the Tsallis entropy (33) can be used to derive the following generalized
Bose-Einstein distribution Buyu ; Buyu2 ; Buyu3 ; Buyu4 ; Chen

$N_{q}(\epsilon_{n})=\frac{1}{\left[1+(q-1)\beta\epsilon_{n}\right]^{1/(q-1)}-1}\,,$
(36)

where $\epsilon_{n}$ is the energy of the $n$-th state of the system and
$\beta=1/T$. Clearly, for $q\rightarrow 1$, Eq. (36) gives back the
conventional Bose-Einstein distribution. By definition, the generalized
distribution $N_{q}$ must be non-negative. This gives rise to the following
constraints

$\left\\{\begin{array}[]{rcl}&&0\leq\epsilon_{n}\leq\left[(1-q)\beta\right]^{-1}\,\,\,\,\,\,\,\,\,\
\mathrm{for}\,\,\,q<1,\\\\[8.53581pt] &&\epsilon_{n}\geq 0\hskip
85.35826pt\mathrm{for}\,\,\,q>1\,.\end{array}\right.$ (37)

For the sake of clarity, it must be said that Eq. (36) can only be regarded as
an approximation AppBE . Indeed, the exact generalized distribution cannot be
derived analytically for arbitrary values of $q$. However, for systems with a
relatively large total number of particles (such as fields), the difference
between the exact and approximated expressions turns out to be fairly
negligible at very low temperatures (see AppBE  for more detailed numerical
estimations). Hence, since typical values of Unruh temperatures are expected
to be extremely small (we recall that an acceleration $a\simeq
10^{20}\,\mathrm{m/s^{2}}$ is barely enough to reach a temperature of
$1\,{\mathrm{K}}$), we safely fall within the regime of validity of Eq. (36),
which can then be considered as the starting point of our next computations.

In the previous Section we have emphasized that Unruh spectrum for mixed
fields loses its characteristic Planckian profile, the deviation being
proportional to the mixing parameters (see Eq. (31)). Given the complex
entangled structure induced by mixing in the vacuum state, the question
naturally arises as to whether such an effect can be explained in mechanical
statistical terms by resorting to the nonextensive Tsallis framework. Of
course, since the correction in Eq. (31) slightly affects the Bose-Einstein
spectrum at both high and low energy regimes, it is reasonable to expand the
generalized distribution (36) for tiny departures of $q$ from unity. To the
leading order, we obtain

$\displaystyle
N_{q}(\epsilon_{n})=\frac{1}{e^{\beta\epsilon_{n}}-1}\,+\,\frac{1}{8}\left(\beta\epsilon_{n}\right)^{2}\,\operatorname{csch}^{2}\left(\frac{\beta\epsilon_{n}}{2}\right)\left(q-1\right)\,+\,\mathcal{O}\left(q-1\right)^{2}.$
(38)

To compare with the distribution function (31), we resort to Eq. (13) and set
$\epsilon_{n}=a\Omega_{n}$, $\beta=1/T_{\mathrm{U}}=2\pi/a$, where the Unruh
temperature $T_{\mathrm{U}}$ has been defined after Eq. (14). By plugging into
$N_{q}(\epsilon_{n})$, this becomes

$N_{q}(\Omega_{n})=N_{\mathrm{BE}}(\Omega_{n})\,+\,\frac{\pi^{2}}{2}\,\Omega_{n}^{2}\,\operatorname{csch}^{2}\left(\pi\hskip
0.85358pt\Omega_{n}\right)\left(q-1\right)\,+\,\mathcal{O}\left(q-1\right)^{2}\,,$
(39)

where the zeroth-order term $N_{\mathrm{BE}}$ is the distribution function
(14). Therefore, within Tsallis thermostatistics, the distribution which
extremizes the entropy (33) according to the maximum entropy principle can be
expanded around $q=1$ as above. At a conceptual level, we notice that
significant deviations from the Bose-Einstein spectrum still arise at the
lowest order, since the extra term depends on the energy scale in a nontrivial
way.

To show the correspondence between the modification induced by flavor mixing
and the $q$-generalized distribution based on Tsallis entropy, let us now
compare Eqs. (31) and (39). A straightforward calculation gives (to the
leading order)

$\mathcal{N}_{\theta,\Delta
m}(\Omega_{n})=N_{q}(\Omega_{n})\,\,\,\Longrightarrow\,\,\,q=1-\frac{|\Delta
m^{2}|}{2\pi^{2}\mu_{k,1}^{\hskip 0.28453pt2}\hskip
0.28453pt\Omega_{n}}\sin^{2}\theta\,,$ (40)

which implies $q<1$, $\forall\,\Omega_{n}$. This means that we are in the
superadditive regime of Tsallis entropy, as it can be seen from Eq. (35).

Thus, the thermostatistical properties of Unruh condensate for mixed particles
can be effectively described in terms of the nonextensive Tsallis statistics,
the entropic $q$-index satisfying the condition (40). As expected, the
deviation of $q$ from unity depends on the mixing angle and the mass
difference in such a way that, for $\theta\rightarrow 0$ and/or $\Delta
m\rightarrow 0$, the usual Boltzmann-Gibbs theory with $q=1$ is recovered.

The above considerations provide us with an alternative way of interpreting
the modified distribution (31). Indeed, in Luciano  mixing was seen as the
origin of a breakdown of the thermality of Unruh effect via the appearance of
exotic terms in the spectrum. On the other hand, the present result shows that
one can still maintain the standard thermal picture, provided that the
underlying statistics is assumed to obey Tsallis’s prescription. In passing,
we point out that a similar analysis has been developed in Shababi  in the
context of deformations of Heisenberg uncertainty principle (Generalized
Uncertainty Principle). Even in that case it has been argued that GUP
corrections to Unruh effect for a gas of relativistic massive particles can be
mimicked by a Tsallis-like statistics with a modified ($q$-dependent) formula
for Unruh temperature. Connections between deformed uncertainty relations and
generalized entropies in the framework of Unruh effect have also been
discussed in deformed2 .

A remarkable property to comment on is the running behavior of $q$ as a
function of the energy scale $\Omega_{n}$. Although not envisioned by Tsallis
in his original approach, this should not be entirely surprising for quantum
field theoretical or quantum gravity systems when renormalization group is
applied App13 . A similar scenario with a varying nonextensive parameter has
been recently discussed in App13  in the context of modified cosmological
models.

In this regard, one might spot a pathological behavior of Eq. (40) in the
limit of vanishing Rindler frequency $\Omega_{n}$. Actually, it must be
stressed that modes with frequency below a certain threshold lie outside the
domain of validity of the approximation (38), and thus of our analysis.
Indeed, for $\Omega_{n}\rightarrow 0$ the $q$ index strongly deviates from
unity, which _a posteriori_ would invalidate the series expansion (39). In
order to keep our formalism self-consistent, the condition $|q-1|\ll 1$ must
be satisfied, which in turn implies the cutoff $\Omega_{n}\gg|\Delta
m^{2}|\sin^{2}\theta/(2\pi^{2}\mu^{\hskip 0.28453pt2}_{k,1})$. This means that
the more accurate the approximation of small mass difference between the mixed
fields, the higher the number of $\Omega_{n}$-frequency modes that fit with
the $q$-generalized Bose-Einstein distribution (36). In the limit $|\Delta
m^{2}|/\mu^{\hskip 0.28453pt2}_{k,1}\ll 1$, the entire spectrum of Rindler
modes is approximately spanned. For instance, for sample values characteristic
of relativistic maximally mixed (i.e. $\theta=\pi/4$) particles with squared
mass difference $10^{-3}\,\mathrm{eV}^{2}$ and typical energy
$1\,\mathrm{GeV}$,777These are typical values for atmospheric neutrinos RPDG .
the lower bound on $\Omega_{n}$ takes the value $\Omega^{(min)}_{n}=|\Delta
m^{2}|\sin^{2}\theta/(2\pi^{2}\mu^{\hskip 0.28453pt2}_{k,1})\simeq 10^{-23}$.
On the other side, the departure from extensive thermodynamics becomes
increasingly negligible for large $\Omega_{n}$, restoring the Boltzmann-Gibbs
theory in the limit $\Omega_{n}\rightarrow\infty$. This is consistent with the
fact that, the higher the energy of the state, the lower the average number of
particles that can be stored, with both the standard and $q$-generalized
distributions approaching zero as $\Omega_{n}$ increases. As a consequence,
the difference between the two spectra is expected to shrink as
$\Omega_{n}\rightarrow\infty$.

Now, the relation (40) states that $q<1$ within our analysis. To see whether
the constraint (37) is fulfilled, let us employ Eqs. (13) and (40). A direct
substitution in the upper condition yields

$0\leq a\hskip 0.28453pt\Omega_{n}\leq\frac{\pi\mu^{\hskip
0.28453pt2}_{k,1}}{|\Delta m^{2}|\hskip 0.56905pt\sin^{2}\theta}\,a\hskip
0.28453pt\Omega_{n}\hskip 8.53581pt\Longrightarrow\hskip 8.53581pt0\leq
1\leq\frac{\pi\mu^{\hskip 0.28453pt2}_{k,1}}{|\Delta m^{2}|\hskip
0.56905pt\sin^{2}\theta}\,,$ (41)

which is indeed satisfied in the approximation $|\Delta m^{2}|/\mu^{\hskip
0.28453pt2}_{k,1}\ll 1$.

The connection between the perturbed spectrum (31) and the $q$-generalized
Bose-Einstein distribution (38) can be explained in terms of the entangled
structure acquired by the Minkowski vacuum for mixed fields. As discussed in
the Appendix, the vacuum for definite flavor fields becomes a condensate of
entangled particle-antiparticle pairs having both equal and different masses
BV95 ; BlasCap . Consequently, while the standard Unruh effect arises from
one-type fluctuations popping out near the Rindler horizon (one element of
which crossing the horizon, the other escaping in the form of Unruh
radiation), for mixed fields it can be generated by different types of
entangled pairs (see Fig. 2). This further degree of freedom results into an
increase of the total entropy of the system, which in turn alters the
characteristic number spectrum of particles. As shown above, such an effect
can be described by modeling the new vacuum distribution according to Tsallis
$q$-thermodynamics rather than Boltzmann one, the departure being proportional
to the mixing parameters and the energy scale (see Eq. (40)). From Eq. (35),
we indeed notice that having $q<1$ amounts to saying that the entropy function
associated to the vacuum condensate of $\phi_{A}$-quanta (e.g. the blue-blue
pairs) and $\phi_{B}$-quanta (the red-red pairs) is higher than the sum of the
entropies associated to the condensates of the two fields separately, due to
the presence of hybrid (blue-red and red-blue) particle-antiparticle pairs. We
stress that this is a peculiar field theoretical effect boiling down to the
nonfactorizability of Fock space for flavor fields, including the vacuum state
Cabo  (see also Appendix).

Figure 2: Pictorial interpretation of Unruh spectrum for mixed fields.
Different (online) dot colors correspond to different particle-antiparticle
flavors. Unruh effect originates from quantum vacuum fluctuations close to
Rindler horizon. In the absence of mixing, vacuum contains a single type of
particle-antiparticle pairs (either blue-blue or red-red). Conversely, for
mixed fields hybrid types of pairs (red-blue and blue-red) do appear as well.
This spoils the characteristic particle spectrum, which can be approximately
identified with the $q$-generalized Bose-Einstein condensate based on Tsallis
entropy.

From the above considerations, we infer that the correlations induced by
mixing do spoil the macroscopic properties of Unruh thermal condensate by
affecting the statistical behavior of its microscopic configurations. This
gives rise to an entangled condensate structure of both equal and hybrid
particle-antiparticle pairs that obey the nonadditive Tsallis entropy law in a
suitable limit. Remarkably, we notice that nonextensive statistics based on
Tsallis entropies have been largely used in the study of entanglement TSENT ,
such as the relative entropy and the Peres criterion.

The above result is quite general, since it is not confined to bosons solely.
In NonTN  the Unruh effect has been investigated in the case of mixing of
(Dirac) fermions, and in particular of neutrinos, showing that the vacuum
condensate of Rindler particles should be modified as (see Eq. (23) of NonTN )

$\mathcal{N}_{\theta,\Delta
m}(\Omega_{n})\,=\,N_{\mathrm{FD}}(\Omega_{n})\,+\,\sin^{2}\theta\,\,\mathcal{O}\left(\frac{|\Delta
m^{2}|}{\mu^{2}_{k,1}}\right)\,=\,\frac{1}{e^{2\pi\Omega_{n}}+1}\,+\,\sin^{2}\theta\,\,\mathcal{O}\left(\frac{|\Delta
m^{2}|}{\mu^{2}_{k,1}}\right),$ (42)

where the zeroth-order term is now given by the Fermi-Dirac distribution
function, while the higher-order corrections $\mathcal{O}\left({|\Delta
m^{2}|}/{\mu^{2}_{k,1}}\right)$ depend on the convolution integral of the
condensation density of mass vacuum in such a way that the Pauli principle is
still satisfied. In particular, we have NonTN

$\displaystyle\hskip 75.39963pt\mathcal{O}\left(\frac{|\Delta
m^{2}|}{\mu^{2}_{k,1}}\right)\,=\,\frac{e^{\pi\Omega_{n}}}{2\hskip
0.28453pt\cosh\left(\pi\Omega_{n}\right)}\hskip
2.56073ptN_{F,F}(\Omega_{n})\,-\,\frac{e^{-\pi\Omega_{n}}}{2\hskip
0.28453pt\cosh\left(\pi\Omega_{n}\right)}\hskip
2.56073ptN_{G,G}(\Omega_{n})\,,$ (43) $\displaystyle
N_{F,F}(\Omega_{n})=\sum_{r=1,2}\int\,F^{\hskip
0.56905pt*}_{r}(k_{1},\Omega_{n})\,F_{r}(k_{1},\Omega_{n})\,{|V_{\bf{k}}|}^{2},\quad\,\,N_{G,G}(\Omega_{n})=\sum_{r=1,2}\int\,G^{\hskip
0.56905pt*}_{r}(k_{1},\Omega_{n})\,G_{r}(k_{1},\Omega_{n})\,{|V_{\bf{k}}|}^{2}.$
(44)

Here $F_{r}$ and $G_{r}$ are proper combinations of Dirac modes of spin $r$ in
Minkowski and Rindler quantizations NonTN  and

$V_{\bf{k}}(t)=\left(\frac{\omega_{{\bf k},1}+m_{1}}{2\omega_{{\bf
k},1}}\right)^{\frac{1}{2}}\left(\frac{\omega_{{\bf k},2}+m_{2}}{2\omega_{{\bf
k},2}}\right)^{\frac{1}{2}}\left(\frac{|{\bf k}|}{\omega_{{\bf
k},2}+m_{2}}-\frac{|{\bf k}|}{\omega_{{\bf
k},1}+m_{1}}\right)\,e^{i(\omega_{{\bf k},2}+\omega_{{\bf k},1})\,t},$ (45)

is the analogue of the Bogoliubov coefficient ${\lambda}^{\bf{k}}_{12}(t)$ in
Eq. (57) for fermion mixing BV95  (we have assumed for simplicity
$\textbf{k}=(k_{1},0,0))$. Once more, it is easy to check that Eq. (46)
reproduces the standard result for $\theta\rightarrow 0$ and/or
$m_{1}\rightarrow m_{2}$, consistently with the absence of mixing in both
cases. The same holds in the quantum mechanical limit of large momenta with
respect to the mass difference, since the condensation density
$|V_{\bf{k}}|^{2}\rightarrow 0$.

At the same time, the $q$-modified Fermi-Dirac distribution in Tsallis
thermostatistics can be approximately written as Buyu3

$N_{q}(\epsilon_{n})=\frac{1}{\left[1+(q-1)\beta\epsilon_{n}\right]^{1/(q-1)}+1}\,.$
(46)

Therefore, by expanding for small deviations of $q$ from unity and following
the same reasoning as earlier, we arrive at a relation akin to Eq. (40) for
fermions. Clearly, such an extension deserves careful attention, since
neutrinos are the most abundant and emblematic example of mixed particles.

##  V Conclusions and Outlook

The Unruh effect predicts that a uniformly accelerated observer measures a
Planck emission distribution in Minkowski vacuum. However, for quantum fields
exhibiting entanglement correlations induced by mixing, this result turns out
to be nontrivially spoilt Luciano ; NonTN . Here we have discussed this
phenomenon from a statistical point of view. Working in the approximation of
small mass difference between the mixed fields, we have shown that the
modified vacuum distribution can be modeled by the $q$-generalized Bose-
Einstein distribution based on the nonadditive Tsallis entropy. In this
effective description, the deviation from Planckianity is found to be
quantified by the mixing angle $\theta$ and the mass difference $\Delta m$.
Furthermore, the $q$-entropic index exhibits a running behavior, which is
reasonably expected for QFT systems as discussed in App13 . The outcome that
$q<1$ indicates that we are in the superadditive regime of Tsallis statistics,
consistently with the appearance of both equal and hybrid particle-
antiparticle pairs in the vacuum.

Apart from more formal aspects, we remark that the above picture allows us to
extend the peculiar thermal features of Unruh effect to mixed fields. Indeed,
in Luciano ; NonTN  flavor mixing was seen as responsible for the emergence of
nonthermal contributions in the Unruh spectrum. Here we have proved that the
origin of these extra terms can be explained in terms of a departure of the
vacuum distribution from Boltzmann-Gibbs statistics. In turn, this phenomenon
is attributable to the complex structure acquired by the vacuum state for
mixed fields, which becomes a condensate of entangled particle-antiparticle
pairs of different species. In other words, we can still identify a
temperature for the vacuum distribution, provided that we work in the
framework of Tsallis’s thermostatistics. Nevertheless, following AbePla  we
point out that the new physical temperature would be different from Unruh
temperature by a factor depending on the nonextensivity $q$-parameter and the
modified entropy $S_{q}$, in such a way that the usual result is still
recovered for $q\rightarrow 1$. In this regard, we mention that a similar
$q$-dependent expression for Unruh temperature in Tsallis’s theory has been
obtained in Shababi  in the context of the Generalized Uncertainty Principle.
Possible connections between the two results need further consideration and
will be addressed elsewhere.

In passing, we highlight that a nonthermal behavior of Unruh effect has been
recently exhibited in Dop  even for the case of a single (i.e. unmixed)
massive field. In that case, it has been found that, contrary to what happens
with a linear dispersion relation characteristic of massless fields, the
thermality of Unruh condensate would be lost for more general dispersion
relations including a mass term, unless one defines a varying apparent Unruh
temperature depending on both the acceleration and the degree of departure
from linearity. Therefore, it would be interesting to investigate whether such
a result interfaces with our reformulation of Unruh effect in Tsallis’s
language. For this purpose, however, a formalism based on the relativistic
Doppler shift method is required Dop , since the Bogoliubov transformation
approach is insensitive to the mass of the field when computing Unruh vacuum
distribution Takagi .

Beyond the above issues, several other aspects remain to be analyzed. To avoid
unnecessary technicalities, we have focused on a simplified model involving
only two scalar fields, noticing that similar considerations can be extended
to fermions quite straightforwardly. Furthermore, our perturbative analysis
relies on the leading-order approximation of small difference between the
masses of mixed fields. The question thus arises as to how the connection (40)
between the nonextensive $q$-index and the mixing parameters would appear for
arbitrary mass differences, as well as in the case of three flavor
generations. Another extension is to apply the above formalism to the best-
known Hawking radiation, which has been largely studied within the framework
of nonextensive corrected-entropies in recent years Barrow .

From a more phenomenological perspective, it would be challenging to test
possible experimental implications of our result. As well known, direct
evidences of Unruh effect have not yet been obtained, the obvious reason being
the fact that the Unruh temperature is extremely small even for huge
accelerations. However, there have been many proposals in the literature to
bypass hindrances arising from technical limitations by focusing on analogues
of Unruh effect, even at the classical level. For instance, feasible tests are
being analyzed by simulating vacuum fluctuations of Minkowski spacetime
through gravity waves on the surface of water subject to white noise Leona .
Attempts to detect indirect traces of Unruh radiation have also been carried
out in graphene graphene  and metamaterials Smoly , where the effects of
Rindler-like horizons are mimicked by means of photons waveguides. Thus, such
analog models provide the only test bench for probing the Unruh effect and any
possible deviation from the standard behavior to date.

Finally, one more direction to explore is whether Tsallis statistics and mixed
particles are intertwined on a more fundamental level that goes beyond the
specific framework of Unruh effect. In this vein, we emphasize the recent
proposal to solve the long-standing problem of abundance of primordial
${}^{7}\mathrm{Li}$, which is affected by the neutrino interactions and
primordial magnetic field, by investigating the impact of Big Bang
nucleosynthesis predictions of adopting a Tsallis distribution for the nucleon
energies Litium . Work along the above research lines is presently under
active consideration.

###### Acknowledgements.

One of the authors (GGL) is grateful to Costantino Tsallis (Centro Brasileiro
de Pesquisas Fisicas, Brazil) and Gaetano Lambiase (Università degli Studi di
Salerno, Italy) for helpful conversations.

##  Appendix A QFT of flavor mixing in Minkowski spacetime

We review the QFT formalism of flavor mixing for the simplest case of two
scalar fields in Minkowski background BlasCap . Toward this end, we introduce
the algebraic generator of mixing

$G_{\theta}(t)=\exp\left[-i\theta\int d^{3}x\hskip
0.56905pt\Big{(}\pi_{1}(x)\phi_{2}(x)\,-\,\pi_{2}(x)\phi_{1}(x)\Big{)}\right],$
(47)

where the notation has already been set up in Sec. III. In terms of this
operator, the mixing transformations in Eqs. (21) and (22) can be cast as

$\phi_{\ell}(x)=G_{\theta}^{-1}(t)\phi_{j}(x)G_{\theta}(t)\,,$ (48)

where $(\ell,j)=\left\\{(A,1),(B,2)\right\\}$. One can prove that
$G_{\theta}(t)$ belongs to $SU(2)$ group, the algebra of which is closed by
the operators BlasCap

$S_{+}(t)=-i\int d^{3}x\hskip 1.99168pt\pi_{1}(x)\phi_{2}(x)\,,\quad
S_{-}(t)=-i\int d^{3}x\hskip 1.99168pt\pi_{2}(x)\phi_{1}(x)\,,$ (49)
$S_{3}=-\frac{i}{2}\int d^{3}x\hskip
1.99168pt\Big{(}\pi_{1}(x)\phi_{1}(x)-\pi_{2}(x)\phi_{2}(x)\Big{)}\,,\quad
S_{0}=-\frac{i}{2}\int d^{3}x\hskip
1.99168pt\Big{(}\pi_{1}(x)\phi_{1}(x)+\pi_{2}(x)\phi_{2}(x)\Big{)}\,.$ (50)

Within the framework of QFT mixing, the generator (47) plays a pivotal rôle,
as it provides the dynamical map between the Fock space $\mathcal{H}_{A,B}$
for the fields with definite flavor and the Fock space $\mathcal{H}_{1,2}$ for
the fields with definite mass888For brevity, henceforth $\mathcal{H}_{A,B}$
and $\mathcal{H}_{1,2}$ are simply referred to as “flavor” and “mass” Fock
space, respectively.. Indeed, let us consider the generic matrix element of
$\mathcal{H}_{1,2}\,=\mathcal{H}_{1}\otimes\mathcal{H}_{2}$, i.e.
${}_{1,2}\langle a|\phi_{i}(x)|b\rangle_{1,2}$ ($i=1,2$), where
$|a\rangle_{1,2}$ and $|b\rangle_{1,2}$ are arbitrary states in
${\mathcal{H}}_{1,2}$. By inverting Eq. (48) with respect to $\phi_{j}$, we
get

${}_{1,2}\langle a|G_{\theta}(t)\hskip 0.56905pt\phi_{\ell}(x)\hskip
0.56905ptG^{-1}_{\theta}(t)|b\rangle_{1,2}=\hskip 0.28453pt_{1,2}\langle
a|\phi_{j}(x)|b\rangle_{1,2}\,,$ (51)

which in fact shows that $G_{\theta}^{-1}(t)|b\rangle_{1,2}$ is a vector in
$\mathcal{H}_{A,B}$. Therefore, we can write

$G^{-1}_{\theta}(t):{\mathcal{H}}_{1,2}\mapsto{\mathcal{H}}_{A,B}\,.$ (52)

In particular, for the vacuum state
$|0_{\mathrm{M}}\rangle_{1,2}\equiv|0\rangle_{1,2}=|0\rangle_{1}\otimes|0\rangle_{2}$,
this gives

$|0(\theta,t)\rangle_{A,B}=G^{-1}_{\theta}(t)|0\rangle_{1,2}\,.$ (53)

The above relation allows us to define the time-dependent flavor vacuum
$|0(\theta,t)\rangle_{A,B}$ in terms of the corresponding mass vacuum
$|0\rangle_{1,2}$.

A comment is in order here. For quantum mechanical systems (i.e. systems with
finite number of degrees of freedom), $G_{\theta}(t)$ is a unitary operator
that preserves the canonical commutation relations. This is ensured by Stone-
von Neuman theorem SvN ; Stone , which states that any two irreducible
representations of the canonical commutators are unitarily equivalent in
Quantum Mechanics. Accordingly, mass and flavor representations give rise to
physically equivalent descriptions of mixing. On the other hand, in QFT the
transformation (47) is found to be nonunitary in the infinite volume limit,
which means that the vacua $|0\rangle_{1,2}$ and $|0(t)\rangle_{A,B}$ become
mutually orthogonal and the related Fock spaces unitarily inequivalent. This
is quite different from the conventional perturbation theory, where the vacuum
of the interacting theory is expected to be essentially the same as that of
the free theory (up to a phase factor) BogoCit . Clearly, such an
inequivalence and its implications disappear for $\theta=0$ and/or
$m_{2}=m_{1}$, consistently with the fact that there is no mixing in both
cases.

To find out how the mapping (53) affects the structure of the flavor vacuum,
we now focus on the derivation of ladder operators in the flavor basis. By
using the standard plane-wave quantization (7) for both $\phi_{1}$ and
$\phi_{2}$, Eq. (48) leads to the following expansions for the flavor fields

$\phi_{\ell}(x)=\int d^{3}{k}\hskip
0.56905pt\Big{\\{}a_{\textbf{k},\ell}(\theta,t)\,U_{\textbf{k},j}(x)+\bar{a}_{\textbf{k},\ell}^{\dagger}(\theta,t)\,U_{\textbf{k},j}^{\hskip
0.28453pt*}(x)\Big{\\}},\,\,\quad(\ell,j)=\left\\{(A,1),(B,2)\right\\},$ (54)

where

$a_{\textbf{k},\ell}(\theta,t)\,\equiv\,G^{-1}_{\theta}(t)\hskip
0.56905pta_{\textbf{k},j}\hskip 0.56905ptG_{\theta}(t)\,,$ (55)

is the annihilator of a quantum with definite flavor $\ell$ (for simplicity,
we refer to this operator as flavor annihilator and use the handier notation
$a_{\textbf{k},\ell}(\theta,t)\equiv a_{\textbf{k},\ell}(t)$). From the above
relation, we obtain

$a_{{{\bf{k}}},A}(t)=\cos\theta\,a_{{{\bf{k}}},1}\,+\,\sin\theta\,\left({\rho}^{\bf{k}\,*}_{12}(t)\,a_{{{\bf{k}}},2}\,+\,{\lambda}^{\bf{k}}_{12}(t)\,\bar{a}^{\dagger}_{-{{\bf{k}}},2}\right),$
(56)

(similarly for $a_{\textbf{k},B}(\theta,t))$. Therefore, the flavor
annihilator is related to the corresponding ladder operators in the mass basis
via of a Bogoliubov transformation (the terms in the brackets) nested into a
rotation. The Bogoliubov coefficients are defined as

${\rho}^{\bf{k}}_{12}(t)=|{\rho}^{\bf{k}}_{12}|\,e^{i\left(\omega_{{\bf{k}},2}-\omega_{{\bf{k}},1}\right)t}\,,\quad{\lambda}^{\bf{k}}_{12}(t)=|{\lambda}^{\bf{k}}_{12}|\,e^{i\left(\omega_{k,1}+\omega_{k,2}\right)t}\,,$
(57)

where

$|{\rho}^{\bf{k}}_{12}|\equiv\frac{1}{2}\left(\sqrt{\frac{\omega_{{\bf{k}},1}}{\omega_{{\bf{k}},2}}}\,+\,\sqrt{\frac{\omega_{{\bf{k}},2}}{\omega_{{\bf{k}},1}}}\right),\qquad|{\lambda}^{\bf{k}}_{12}|\equiv\frac{1}{2}\left(\sqrt{\frac{\omega_{{\bf{k}},1}}{\omega_{{\bf{k}},2}}}\,-\,\sqrt{\frac{\omega_{{\bf{k}},2}}{\omega_{{\bf{k}},1}}}\right).$
(58)

It is easy to verify that

${|{\rho}^{\bf{k}}_{12}|}^{2}\,-\,{|{\lambda}^{\bf{k}}_{12}|}^{2}=1\,,$ (59)

which ensures that the flavor operator (55) and its conjugate are still
canonical (at equal times).

The mapping (53) induces a physically nontrivial structure in the flavor
vacuum, which becomes an entangled $SU(2)$ coherent state made up by particle-
antiparticle pairs both of the same and different masses BV95 . In turn, this
inequivalence affects the well-known oscillation formula to include the
antiparticle degrees of freedom BlasPlb . The condensation density of flavor
vacuum is given by

${}_{A,B}\langle 0(t)|a_{{{\bf{k}}^{\prime}},i}^{\dagger}\,a_{{\bf
k},j}|0(t)\rangle_{A,B}=\sin^{2}\theta\;{|{\lambda}^{\bf{k}}_{12}|}^{2}\delta_{ij}\delta^{3}({\textbf{k}-\textbf{k}^{\prime}})\,.$
(60)

Clearly, by exploiting the symmetric structure of Eq. (55), one can reverse
the above reasoning and analyze the properties of mass vacuum, which appears
as a condensate of particle-antiparticle pairs having both equal and different
flavors. In line with our previous considerations on the disappearance of the
inequivalence for vanishing mixing, the condensation density (60) goes to zero
for $\theta=0$ (since $a_{{\bf{k}},A}\rightarrow a_{{\bf{k}},1}$ and
$a_{{\bf{k}},B}\rightarrow a_{{\bf{k}},2}$) and/or $m_{2}=m_{1}$ (since the
Bogoliubov coefficients reduce to ${\rho}^{\bf{k}}_{12}(t)=1$ and
${\lambda}^{\bf{k}}_{12}(t)=0$, which in turn implies that $a_{{\bf{k}},A}$
and $a_{{\bf{k}},B}$ are simple superpositions of $a_{{\bf{k}},1}$ and
$a_{{\bf{k}},2}$). Notice that the same behavior occurs for
${|{\bf{k}}|}^{2}\gg\frac{{m_{1}}^{2}+{m_{2}}^{2}}{2}$, thus allowing to
recover the standard quantum mechanical description of mixing in the
relativistic approximation.

The complex structure of the flavor vacuum $|0(t)\rangle_{A,B}$ has been
recently studied in Cabo , where it has been established that the Fock space
for flavor fields cannot be obtained by the direct product of the spaces for
massive fields. This strengthen the result that entanglement properties for
mixed fields already emerge at the level of vacuum state. As a remark, the
observation that flavor mixing can be associated with (single-particle)
entanglement traces back to Dimauro  and has inspired a series of studies on
violations of Bell, Leggett-Garg and Mermin-Svetchlichny inequalities,
nonlocality, gravity-acceleration degradation effects and other similar
phenomena SimPhe1 ; SimPhe2 ; SimPhe3 ; SimPhe4 ; SimPhe5 . The entanglement
content of the flavor vacuum has been explicitly quantified in Vacent  in the
limit of small mass difference and/or mixing angle.

## References

  * (1)
  * (2)

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