# Quantum statistical fluctuation of energy and its novel pseudo-gauge
dependence

Arpan Das  Institute of Nuclear Physics Polish Academy of Sciences, PL-31-342
Krakow, Poland<EMAIL_ADDRESS>
###### Abstract

We discuss the quantum statistical fluctuations of energy in subsystems of hot
relativistic gas for both spin-zero and spin half particles. We explicitly
show the system size dependence of the quantum statistical fluctuation of
energy. Our results show that with decreasing system size quantum statistical
fluctuations increase substantially. As the consistency of the framework, we
also argue that the quantum statistical fluctuations give rise to the known
result for statistical fluctuation of energy in the canonical ensemble if we
consider the size of the subsystem to be sufficiently large. For a spin-half
particle quantum fluctuations show some interesting novel features. We show
that within a small sub-system quantum statistical fluctuation of energy for
spin half particles depends on the various pseudo-gauge choices of the energy-
momentum tensor. Interestingly, for sufficiently large subsystems quantum
fluctuations obtained for different pseudo-gauge choices converge and we
recover the canonical-ensemble formula known for statistical fluctuations of
energy. Our calculation is very general and can be applied to any branch of
physics whenever one deals with a thermal system. As a practical application,
we argue that our results can be used to determine a coarse-graining scale to
introduce the concept of classical energy density or fluid element relevant
for the strongly interacting matter, in particular for small systems produced
in heavy-ion collisions.

  * January, 2022

Keywords: Quantum fluctuation, heavy-ion collision, pseudo-gauge dependence,
small systems.

##  1 Introduction

Relativistic heavy-ion collision experiments (RHICE), e.g., the Relativistic
Heavy Ion Collider (RHIC) at the Brookhaven National Laboratory (BNL) and the
Large Hadron Collider (LHC) at CERN gives us a unique opportunity to
investigate the properties of strongly interacting matter governed by Quantum
Chromodynamics (QCD), particularly in the non-perturbative regime. The physics
of strongly interacting matter so produced can be broadly separated into three
different stages, (i) description of the initial stage, (ii) bulk evolution of
the deconfined locally thermalized plasma and, (iii) hadronization of quark-
gluon plasma and freezeout of the hadrons. The relativistic dissipative
hydrodynamics has been successfully used to model the bulk evolution of the
strongly interacting matter. Modeling of the initial stage, bulk hydrodynamic
evolution, and the freezeout prescription of hadrons constitute the standard
model of the heavy-ion collisions [1, 2, 3, 4]. We should emphasize that
hydrodynamics is a macroscopic effective theory. Therefore from a microscopic
theory point of view, the notion of “coarse-graining” or the concept of “fluid
cell/element” is required. Further, hydrodynamic description assumes local
thermal equilibrium, where the well-defined concepts of energy density, number
density, etc. over a fluid element have been used. We do not argue on the
applicability or success of relativistic hydrodynamics. Rather, in this
article, we try to shed light on the concepts of energy density and how well
it is defined for a finite-size system. In this context, we study the quantum
statistical fluctuation of the time-time (“$tt$”) component of the energy-
momentum tensor ($\hat{\mathcal{T}}^{tt}$) within a quantum field theoretical
framework for spin-zero (scalar) as well as for spin-half (fermion) particles
[5, 6].

Statistical fluctuations of various thermodynamic quantities become relevant
in different fields of physics, as they encode possible information about
phase transitions [7, 8], dissipative phenomena [9], formation of structures
in the Early Universe [10, 11], etc. Fluctuations can encode the quantum
nature of the system arising due to quantum uncertainty relation or it can be
inherently present in thermodynamic systems due to its statistical nature (we
call such fluctuations as statistical fluctuations) [12]. The present study
combines quantum as well as statistical aspects of fluctuations, i.e. we study
quantum statistical fluctuations of energy in subsystems of hot relativistic
gas [5, 6]. We demonstrate that for small subsystem sizes due to the quantum
uncertainty relation quantum statistical fluctuations can be significantly
large. On the other hand for a large system, these quantum statistical
fluctuations give rise to standard statistical fluctuations for a
thermodynamic system (for a more detailed discussion see Refs. [5, 6]).

Furthermore, we argue that for spin-half particles quantum statistical
fluctuation of energy crucially depends on the form of the energy-momentum
tensor (EMT). Note that for a quantum field theoretical system Noether theorem
does not uniquely define the energy-momentum tensor (EMT). Mathematically, for
any conserved energy-momentum tensor $\hat{\mathcal{T}}^{\mu\nu}$, i.e.
$\partial_{\mu}\hat{\mathcal{T}}^{\mu\nu}=0$ one can construct an equivalent
energy-momentum tensor $\hat{\mathcal{T}}^{\prime\,\mu\nu}$ by adding the
divergence of an antisymmetric tensor, namely
$\hat{\mathcal{T}}^{\prime\,\mu\nu}=\hat{\mathcal{T}}^{\mu\nu}+\partial_{\lambda}\hat{\Phi}^{\nu\mu\lambda}$
[13, 14, 15]. Note that if
$\hat{\Phi}^{\nu\mu\lambda}=-\hat{\Phi}^{\nu\lambda\mu}$, then
$\partial_{\mu}\hat{\mathcal{T}}^{\mu\nu}=0$ implies
$\partial_{\mu}\hat{\mathcal{T}}^{\prime\mu\nu}=0$. Such different choices of
conserved energy-momentum tensor is also known as pseudo-gauge choices and the
tensor $\hat{\Phi}^{\mu\nu\lambda}$ is called the pseudo-gauge parameter. To
study the pseudo-gauge dependence of quantum statistical fluctuation of energy
we consider the canonical version of the energy-momentum tensor for spin-half
particle along with the Belinfante-Rosenfeld (BR) energy-momentum tensor [16,
17, 18], the de Groot-van Leeuwen-van Weert (GLW) energy-momentum tensor [19],
and the Hilgevoord-Wouthuysen (HW) energy-momentum tensor [20, 21]. Note that
in the context of hyperon polarization as observed in the heavy-ion collision
experiments, physical implications of pseudo-gauge transformation of the
energy-momentum tensor and the spin tensor have been extensively discussed
[14, 15, 22, 23, 24, 25].

For a qualitative and quantitative estimation of quantum statistical
fluctuation, we first derive a compact analytical formula for fluctuations in
a subsystem of hot relativistic gas. Then we use this analytical expression of
quantum statistical fluctuation for numerical estimation keeping in mind the
physical situations expected in relativistic heavy-ion collisions. Throughout
the manuscript we consider the Minkowski metric with mostly negative
signature, i.e. $g_{\mu\nu}=\hbox{diag}(+1,-1,-1,-1)$, three vectors are
represented with bold letters and scalar product for three-vectors and Lorentz
four vectors are denoted as, $A^{\mu}B_{\mu}=A\cdot
B=A^{0}B^{0}-\bi{A}\cdot\bi{B}$. The rest of the article is organized in the
following manner. We start with introducing the formalism in Sec. 2 for our
calculations. Then in Sec. 3, we present the results along with some
discussions. Finally, in Sec. 4 we conclude our findings with an outlook to
it.

##  2 Formalism

$\mathcal{S}_{V}$$\mathcal{S}$$\mathcal{S}_{V}$$\mathcal{S}_{a}$ Figure 1:
Pictorial representation of the physical system: $\mathcal{S}$ is a
closed/isolated system described by the microcanonical ensemble.
$\mathcal{S}_{V}$ is a subsystem of the closed system in equilibrium,
described by the canonical ensemble. $\mathcal{S}_{a}$ is a subsystem of
$\mathcal{S}_{V}$ which is described by a “Gaussian box”. We calculate the
quantum statistical fluctuation of energy within the subsystem
$\mathcal{S}_{a}$ [26].

In the present investigation, we look into the quantum statistical fluctuation
of energy within a subsystem $\mathcal{S}_{a}$ of the thermodynamic system
$\mathcal{S}_{V}$. The thermodynamic system $\mathcal{S}_{V}$ is described by
the canonical ensemble characterized by the temperature $T$ (or its inverse
$\beta=1/T$) (as depicted in Fig. (1)). We assume that the characteristic
volume of the sub-system $\mathcal{S}_{a}$ is less than the volume $V$ of the
system $\mathcal{S}_{V}$. As mentioned earlier to calculate the quantum
statistical fluctuation of energy we consider the time-time component of the
energy-momentum tensor ($\hat{\mathcal{T}}^{tt}$) of the scalar field as well
as the fermionic field. Following [5, 6, 13], we define a spatially smeared
operator $\hat{\mathcal{T}}^{tt}_{a}$ that describes the time-time component
of the energy-momentum tensor operator of a finite subsystem $\mathcal{S}_{a}$
placed at the origin of the coordinate system 111To remove any possible sharp
boundary effects the Gaussian box has been introduced in Eq. (1) to define a
smeared operator [5, 6, 13, 27]. Also note that the factor $(a\sqrt{\pi})^{3}$
in Eq. (1) has been introduced as normalization [13].,

$\displaystyle\hat{\mathcal{T}}^{tt}_{a}=\frac{1}{(a\sqrt{\pi})^{3}}\int
d^{3}x~{}\hat{\mathcal{T}}^{tt}(x)~{}\exp\left(-\frac{{\bi{x}}^{2}}{a^{2}}\right).$
(1)

In accordance with the general notion of quantum mechanics, we consider here
the variance 222For a general quantum field theory operator
$\hat{\mathcal{O}}$, a reasonable measure of the quantum spread in a given
state can be defined by the variance of the operator $\hat{\mathcal{O}}$
[13].,

$\displaystyle\sigma^{2}(a,m,T)=\langle:\hat{\mathcal{T}}^{tt}_{a}::\hat{\mathcal{T}}^{tt}_{a}:\rangle-\langle:\hat{\mathcal{T}}^{tt}_{a}:\rangle^{2}\,$
(2)

and the normalized standard deviation,

$\displaystyle\sigma_{n}(a,m,T)=\frac{(\langle:\hat{\mathcal{T}}^{tt}_{a}::\hat{\mathcal{T}}^{tt}_{a}:\rangle-\langle:\hat{\mathcal{T}}^{tt}_{a}:\rangle^{2})^{1/2}}{\langle:\hat{\mathcal{T}}^{tt}_{a}:\rangle}.$
(3)

as the measure of the fluctuation. In the definition of $\sigma^{2}$ and
$\sigma_{n}$ we have used the normal ordering of operators (denoted by the
symbol ‘: :’) to eliminate any zero-point energy contribution. Although, for a
quantum field theory operator $\hat{\mathcal{O}}$ one can define the normal
ordering prescription uniquely, i.e. $:\hat{\mathcal{O}:}$, but for a
composite operator, e.g. $\hat{\mathcal{O}}\hat{\mathcal{O}}$ the normal
ordering prescription to remove any divergent vacuum contribution is not
unique. In the present work, we define the normal ordering of composite
operator as $:\hat{\mathcal{O}}::\hat{\mathcal{O}}:$. Alternately one may also
define a normal ordered composite operator as
$:\hat{\mathcal{O}}\hat{\mathcal{O}}:$ [28], however, we argue in the
subsequent sections that our definition of the normal ordering, i.e.
$:\hat{\mathcal{O}}::\hat{\mathcal{O}}:$ reproduces correctly limiting results
for the energy fluctuation. Furthermore in Eqs. (2) and (3),
“$\langle\hat{\mathcal{O}}\rangle$” denotes the thermal average/thermal
expectation of an operator $\hat{\mathcal{O}}$. For the estimation of quantum
statistical fluctuations, we evaluate $\sigma^{2}$ and $\sigma_{n}$ for a
thermal system consisting of spin-zero as well as spin-half particles. For
this purpose, we need to find an appropriate expression of
$\hat{\mathcal{T}}^{tt}_{a}$ along with the prescription of thermal averaging.

###  2.1 Scalar field

To describe a system by a quantum scalar field in thermal equilibrium we
consider the Fourier space representation of a real scalar field [5, 13]

$\displaystyle\phi(t,{\bi{x}})=\int\frac{d^{3}p}{\sqrt{(2\pi)^{3}~{}2\omega_{\bi{p}}}}\left(\mathfrak{a}_{\bi{p}}^{{\phantom{\dagger}}}e^{-ip\cdot
x}+\mathfrak{a}_{\bi{p}}^{\dagger}e^{ip\cdot x}\right),$ (4)

here $\mathfrak{a}_{\bi{p}}^{{\phantom{\dagger}}}$ is the bosonic annihilation
operator and $\mathfrak{a}_{\bi{p}}^{\dagger}$ is the bosonic creation
operator satisfying the standard canonical commutation relations
$[\mathfrak{a}_{\bi{p}}^{{\phantom{\dagger}}},\mathfrak{a}_{\bi{p}^{\prime}}^{\dagger}]=\delta^{(3)}(\bi{p}-\bi{p}^{\prime})$.
$\omega_{\bi{p}}=\sqrt{{\bi{p}}^{2}+m^{2}}$ is the single particle dispersion
relation. For the scalar field as defined in Eq. (4) the time-time component
of the energy-momentum tensor operator is noting but the canonical Hamiltonian
density, i.e.,
$\hat{\mathcal{T}}^{tt}_{\phi}\equiv\mathcal{H}_{\phi}=(\dot{\phi}^{2}+({\bf{\nabla}\phi)}^{2}+m^{2}\phi^{2})/2$.
Here $\dot{\phi}$ and $\bf{\nabla}\phi$ represents time and space derivative
of the scalar field, respectively. For the scalar field to calculate the
variance $(\sigma_{\phi}^{2})$ and the normalized standard deviation
$(\sigma_{n,\phi})$ we have to perform the thermal averaging involving the
smeared operator $\hat{\mathcal{T}}^{tt}_{a,\phi}$ or $\mathcal{H}_{a,\phi}$.
For such thermal averaging it is sufficient to know the thermal expectation
values of the products of two and four creation and/or annihilation operators
[5, 29, 30, 31], i.e.,

$\displaystyle\langle\mathfrak{a}^{\dagger}_{{\bi{p}}}\mathfrak{a}_{{\bi{p}}^{\prime}}^{{\phantom{\dagger}}}\rangle=\delta^{(3)}({\bi{p}}-{\bi{p}}^{\prime})\mathfrak{f}_{b}(\omega_{{\bi{p}}}),$
(5)
$\displaystyle\langle\mathfrak{a}^{\dagger}_{{\bi{p}}}\mathfrak{a}^{\dagger}_{{\bi{p}}^{\prime}}\mathfrak{a}_{{\bi{k}}}^{{\phantom{\dagger}}}\mathfrak{a}_{{\bi{k}}^{\prime}}^{{\phantom{\dagger}}}\rangle=\bigg{(}\delta^{(3)}({\bi{p}}-{\bi{k}})~{}\delta^{(3)}({\bi{p}}^{\prime}-{\bi{k}}^{\prime})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\delta^{(3)}({\bi{p}}-{\bi{k^{\prime}}})~{}\delta^{(3)}({\bi{p}}^{\prime}-{\bi{k}})\bigg{)}\mathfrak{f}_{b}(\omega_{{\bi{p}}})\mathfrak{f}_{b}(\omega_{{\bi{p}}^{\prime}}).$
(6)

In these equations,
$\mathfrak{f}_{b}(\omega_{{\bi{p}}})=1/(\exp[\beta~{}\omega_{{\bi{p}}}]-1)$ is
the Bose–Einstein distribution function. Any other combinations of two and
four creation and/or annihilation operators which may appear in the
calculation of the thermal averaging of composite operators can be obtained
from Eqs. (5) and (6) using the commutation relation between
$\mathfrak{a}_{{\bi{p}}}^{{\phantom{\dagger}}}$ and
$\mathfrak{a}_{{\bi{p}}}^{\dagger}$.

###  2.2 Dirac field

To describe the system of spin-half particles in thermal equilibrium we
consider the Dirac field operator [6, 32]

$\displaystyle\psi(t,\bi{x})=\sum_{r}\int\frac{d^{3}p}{(2\pi)^{3}\sqrt{2\omega_{\bi{p}}}}\Big{(}\mathcal{U}_{r}^{{\phantom{\dagger}}}(\bi{p})a_{r}^{{\phantom{\dagger}}}(\bi{p})e^{-ip\cdot
x}+\mathcal{V}_{r}^{{\phantom{\dagger}}}(\bi{p})b_{r}^{\dagger}(\bi{p})e^{ip\cdot
x}\Big{)},$ (7)

here $a_{r}^{{\phantom{\dagger}}}(\bi{p})$ is particle annihilation operator
and $b_{r}^{\dagger}(\bi{p})$ is the antiparticle creation operator satisfying
anti-commutation relations,
$\\{a_{r}^{{\phantom{\dagger}}}(\bi{p}),a_{s}^{\dagger}(\bi{p}^{\prime})\\}=(2\pi)^{3}\delta_{rs}\delta^{(3)}(\bi{p}-\bi{p}^{\prime})$
and
$\\{b_{r}^{{\phantom{\dagger}}}(\bi{p}),b_{s}^{\dagger}(\bi{p}^{\prime})\\}=(2\pi)^{3}\delta_{rs}\delta^{(3)}(\bi{p}-\bi{p}^{\prime})$.
The index $r$ denotes the polarization degree of freedom. The Dirac spinors
$\mathcal{U}_{r}^{{\phantom{\dagger}}}(\bi{p})$ and
$\mathcal{V}_{r}^{{\phantom{\dagger}}}(\bi{p})$ are normalized as,
$\bar{\mathcal{U}}_{r}^{{\phantom{\dagger}}}(\bi{p})\mathcal{U}_{s}^{{\phantom{\dagger}}}(\bi{p})=2m\delta_{rs}$
and
$\bar{\mathcal{V}}_{r}^{{\phantom{\dagger}}}(\bi{p})\mathcal{V}_{s}^{{\phantom{\dagger}}}(\bi{p})=-2m\delta_{rs}$.
$\omega_{\bi{p}}=\sqrt{\bi{p}^{2}+m^{2}}$ is again the single particle energy.

Contrary to the energy-momentum tensor of a real scalar field, the canonical
energy-momentum tensor of spin half particles as obtained using the Noether
theorem is not symmetric. One can use a pseudo-gauge transformation to obtain
a symmetric energy-momentum tensor [14, 15, 22]. We should emphasize that the
symmetric form of the energy-momentum tensor naturally appears in the general
theory of relativity, where one obtains the energy-momentum tensor using the
variation of the action with respect to the space-time metric. In literature
various forms of symmetric energy-momentum tensor have been considered, e.g.,
Belinfante-Rosenfeld energy-momentum tensor [16, 17, 18], etc. To study any
effect of the form of various energy-momentum tensor or pseudo-gauge
transformation, we consider the canonical form of the energy-momentum tensor
along with the Belinfante-Rosenfeld energy-momentum tensor [16, 17, 18, 32],
de Groot-van Leeuwen-van Weert energy-momentum tensor [19] and Hilgevoord-
Wouthuysen energy-momentum tensor [20, 21] to obtain quantum statistical
fluctuation of energy.

####  2.2.1 Canonical EMT:

The canonical form of the energy-momentum tensor of a spin half field can be
shown to be [6, 32],

$\hat{\mathcal{T}}^{\mu\nu}_{\psi,Can}=\frac{i}{2}\bar{\psi}\gamma^{\mu}\mathcal{D}^{\nu}\psi,$
(8)

where
$\mathcal{D}^{\mu}\equiv\overrightarrow{\partial}^{\mu}-\overleftarrow{\partial}^{\mu}$.
In order to obtain the canonical energy-momentum tensor as given in Eq. (8)
one uses the Dirac equation for $\psi$ and $\bar{\psi}$ [32]. Note the
canonical energy momentum tensor is not manifestly symmetric under
$\mu\leftrightarrow\nu$ exchange. Using the “$tt$” component of the canonical
energy-momentum tensor we can obtain $\sigma^{2}_{\psi,Can}$ and
$\sigma_{n,\psi,Can}$ of the smeared operator
$\hat{\mathcal{T}}^{tt}_{\psi,Can,a}$ for the subsystem $\mathcal{S}_{a}$.

####  2.2.2 Belinfante-Rosenfeld EMT:

The Belinfante-Rosenfeld improved symmetric energy-momentum tensor has the
following form [6, 32]

$\displaystyle\hat{\mathcal{T}}^{\mu\nu}_{\psi,BR}=\frac{i}{2}\bar{\psi}\gamma^{\mu}\mathcal{D}^{\nu}\psi-\frac{i}{16}\partial_{\lambda}\Big{(}\bar{\psi}\Big{\\{}\gamma^{\lambda},\Big{[}\gamma^{\mu},\gamma^{\nu}\Big{]}\Big{\\}}\psi\Big{)}.$
(9)

Note that the Belinfante-Rosenfeld improved energy-momentum tensor as given in
Eq. (9) is also not manifestly symmetric under $\mu\leftrightarrow\nu$
exchange. However using the equation of motion for the Dirac fields one can
argue that $\hat{\mathcal{T}}^{\mu\nu}_{\psi,BR}$ is symmetric under
$\mu\leftrightarrow\nu$ exchange [32]. From Eqs. (8) and (9) it is clear that
$\hat{\mathcal{T}}^{tt}_{\psi,BR}=\hat{\mathcal{T}}^{tt}_{\psi,Can}$.
Therefore, thermal average of the normal ordered
$\hat{\mathcal{T}}^{tt}_{\psi,BR,a}$ and its fluctuation will be the same as
obtained in the canonical framework.

####  2.2.3 de Groot-van Leeuwen-van Weert EMT:

Another very important form of the symmetric energy-momentum tensor is the de
Groot-van Leeuwen-van Weert energy-momentum tensor for a massive spin-half
field. It can be expressed as [6, 19, 32],

$\displaystyle\hat{\mathcal{T}}^{\mu\nu}_{\psi,GLW}$
$\displaystyle=-\frac{1}{4m}\bar{\psi}\mathcal{D}^{\mu}\mathcal{D}^{\nu}\psi-g^{\mu\nu}\mathcal{L}_{D}$
(10)
$\displaystyle=\frac{1}{4m}\Big{[}-\bar{\psi}(\partial^{\mu}\partial^{\nu}\psi)+(\partial^{\mu}\bar{\psi})(\partial^{\nu}\psi)+(\partial^{\nu}\bar{\psi})(\partial^{\mu}\psi)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-(\partial^{\mu}\partial^{\nu}\bar{\psi})\psi\Big{]}.$

Here
$\mathcal{L}_{D}\equiv\frac{i}{2}\bar{\psi}\gamma^{\mu}\mathcal{D}_{\mu}\psi-m\bar{\psi}\psi$
is the Lagrangian density of a spin half field. The GLW symmetric energy-
momentum tensor is manifestly symmetric under $\mu\leftrightarrow\nu$
exchange. Using the “$tt$” component of the GLW energy-momentum tensor we can
obtain the fluctuation, i.e. $\sigma^{2}_{\psi,GLW}$ and $\sigma_{n,\psi,GLW}$
of the smeared operator $\hat{\mathcal{T}}^{tt}_{\psi,GLW,a}$ for the
subsystem $\mathcal{S}_{a}$.

####  2.2.4 Hilgevoord-Wouthuysen EMT:

Finally we also consider the Hilgevoord-Wouthuysen form of the symmetric
energy-momentum tensor [6, 20, 21, 32],

$\displaystyle\hat{\mathcal{T}}^{\mu\nu}_{\psi,HW}$
$\displaystyle=\hat{\mathcal{T}}^{\mu\nu}_{\psi,Can}+\frac{i}{2m}\bigg{(}\partial^{\nu}\bar{\psi}\sigma^{\mu\beta}\partial_{\beta}\psi+\partial_{\alpha}\bar{\psi}\sigma^{\alpha\mu}\partial^{\nu}\psi\bigg{)}$
(11)
$\displaystyle-\frac{i}{4m}g^{\mu\nu}\partial_{\lambda}\left(\bar{\psi}\sigma^{\lambda\alpha}\mathcal{D}_{\alpha}\psi\right),$

here $\sigma^{\mu\nu}\equiv(i/2)\left[\gamma^{\mu},\gamma^{\nu}\right]$. Here
also, using the “$tt$” component of the HW energy-momentum tensor we can
obtain the fluctuation, i.e. $\sigma^{2}_{\psi,HW}$ and $\sigma_{n,\psi,HW}$
of the smeared operator $\hat{\mathcal{T}}^{tt}_{\psi,HW,a}$ for the subsystem
$\mathcal{S}_{a}$.

Once we have defined different forms of the energy-momentum tensor of a spin-
half field now we discuss the thermal averaging prescription. Similar to the
scalar field here also to perform thermal averaging we only need to know the
thermal expectation values of the products of two and four creation and/or
annihilation operators, but for both particles and antiparticles [6, 29, 30,
31]. The thermal average of the products of two and four creation and/or
annihilation operators can be expressed as [6, 29, 30, 31],

$\displaystyle\langle
a_{r}^{\dagger}({\bi{p}})a_{s}^{{\phantom{\dagger}}}({\bi{p}}^{\prime})\rangle=(2\pi)^{3}\delta_{rs}\delta^{(3)}({\bi{p}}-{\bi{p}}^{\prime})\mathfrak{f}_{f}(\omega_{\bi{p}}),$
(12) $\displaystyle\langle
a^{\dagger}_{r}(\bi{p})a^{\dagger}_{s}(\bi{p}^{\prime})a_{r^{\prime}}^{{\phantom{\dagger}}}(\bi{k})a_{s^{\prime}}^{{\phantom{\dagger}}}(\bi{k}^{\prime})\rangle$
$\displaystyle=(2\pi)^{6}\Big{(}\delta_{rs^{\prime}}\delta_{r^{\prime}s}\delta^{(3)}(\bi{p}-\bi{k}^{\prime})~{}\delta^{(3)}(\bi{p}^{\prime}-\bi{k})$
$\displaystyle-\delta_{rr^{\prime}}\delta_{ss^{\prime}}\delta^{(3)}({\bi{p}}-\bi{k})~{}\delta^{(3)}({\bi{p}}^{\prime}-\bi{k}^{\prime})\Big{)}\mathfrak{f}_{f}(\omega_{{\bi{p}}})\mathfrak{f}_{f}(\omega_{{\bi{p}}^{\prime}}).$
(13)

In these equations $\mathfrak{f}_{f}(\omega_{{\bi{p}}})$ is the Fermi–Dirac
distribution function for particles. The operators associated with anti-
particles also satisfies similar equations as given in Eqs. (12) and (13). But
for antiparticles, the corresponding Fermi–Dirac distribution function differs
by the sign of the chemical potential associated with the conserved charge
[33]. In the present case, we do not consider any chemical potential.

##  3 Results and discussions

Once we have defined all the relevant operators and the rules for the thermal
averaging, now we present the analytical as well as the numerical estimation
of quantum statistical fluctuation of $\hat{\mathcal{T}}^{tt}_{a}$ for the
bosonic and fermionic system. We present our results to emphasize three
different aspects of quantum statistical fluctuations. These are, (i) scaling
or variation of fluctuation with the system size $(a)$, temperature ($T$) and
particle mass ($m$), (ii) dependence of quantum statistical fluctuation on the
form of the energy-momentum tensor operator or the pseudo-gauge dependence of
quantum statistical fluctuations and (iii) we also argue that in the large
volume limit the quantum statistical fluctuation obtained in a subsystem gives
rise to the statistical fluctuation already known from statistical physics.

###  3.1 Analytical formula

Before we go into any numerical analysis we first present analytical results.

####  3.1.1 For spin zero field:

As argued above using the expression of the canonical Hamiltonian density of a
real scalar field one can obtain the smeared operator $\mathcal{H}_{a,\phi}$
following the definition as given in Eq. (1). Using $\mathcal{H}_{a,\phi}$
along with the normal ordering and thermal averaging prescriptions one can get
the thermal expectation value of $:\mathcal{H}_{a,\phi}:$ and the
corresponding fluctuation $\sigma_{\phi}^{2}$ or $\sigma_{n,\phi}$. The
thermal expectation value of the operator $\mathcal{H}_{a,\phi}$ can be
expressed as [5],

$\langle:\mathcal{H}_{a,\phi}:\rangle=\int\frac{d^{3}{{p}}}{(2\pi)^{3}}~{}\omega_{{\bi{p}}}~{}\mathfrak{f}_{b}\left(\omega_{{\bi{p}}}\right)\equiv\varepsilon(T,m),$
(14)

which is energy density. This result is consistent with the result known from
the kinetic-theory considerations [12]. It is interesting to note that due to
the spatial uniformity of the system $\langle:\mathcal{H}_{a,\phi}:\rangle$ is
independent of scale $a$. Furthermore these results are also time independent.
Although $\langle:\mathcal{H}_{a,\phi}:\rangle$ is system size independent but
we show that the quantum fluctuation, i.e. $\sigma_{\phi}^{2}$ as defined in
Eq. (2) explicitly depends upon the scale ‘$a$’. Using the rules for the
thermal expectation values as given in Eqs. (5)-(6) and dropping all the
temperature independent vacuum energy terms, it can be shown that [5]

$\displaystyle\sigma^{2}_{\phi}(a,m,T)$ $\displaystyle=\int
dP~{}dP^{\prime}\mathfrak{f}_{b}(\omega_{{\bi{p}}})(1+\mathfrak{f}_{b}(\omega_{{\bi{p}}^{\prime}}))$
(15)
$\displaystyle\times\bigg{[}(\omega_{{\bi{p}}}\omega_{{\bi{p}}^{\prime}}+{\bi{p}}\cdot{\bi{p}}^{\prime}+m^{2})^{2}e^{-\frac{a^{2}}{2}({\bi{p}}-{\bi{p}}^{\prime})^{2}}$
$\displaystyle+(\omega_{{\bi{p}}}\omega_{{\bi{p}}^{\prime}}+{\bi{p}}\cdot{\bi{p}}^{\prime}-m^{2})^{2}e^{-\frac{a^{2}}{2}({\bi{p}}+{\bi{p}}^{\prime})^{2}}\bigg{]},$

where $dP=d^{3}{{p}}/((2\pi)^{3}2\omega_{{\bi{p}}})$. Eq. (15) represents the
result for the energy fluctuations in “Gaussian” subsystem $\mathcal{S}_{a}$
of the system $\mathcal{S}_{V}$. This formula also allows us to determine
variation of quantum statistical fluctuation for subsystem size ($a$),
temperature ($T$), and particle mass ($m$). The integrals as given in Eq. (15)
can be done analytically for a massless particle with Boltzmann statistics to
obtain an analytical formula for $\sigma_{\phi}^{2}$ [5]. We should note that
the particles can have internal degrees of freedom, e.g. isospin, color
charge, etc. These internal degrees of freedom can be included by introducing
the degeneracy factor ($g_{b}$). Technically this can be achieved by
replacements $\varepsilon\rightarrow g_{b}\varepsilon$ and
$\sigma_{\phi}^{2}\rightarrow g_{b}\sigma_{\phi}^{2}$ [5].

####  3.1.2 For spin-half field:

Analogous to the scalar field, for the spin-half field also, we obtain the
smeared operator $:\hat{\mathcal{T}}^{tt}_{\psi,a}:$. However, we should note
that for the fermionic system we have different choices of the energy-momentum
tensor. Using the prescription of the thermal average for the spin half field
we find [6],

$\displaystyle\langle:\hat{\mathcal{T}}^{tt}_{\psi,Can,a}:\rangle$
$\displaystyle=4\int\frac{d^{3}p}{(2\pi)^{3}}~{}\omega_{\bi{p}}~{}\mathfrak{f}_{f}(\omega_{\bi{p}})\equiv\varepsilon_{Can}(T,m)$
(16)
$\displaystyle=\langle:\hat{\mathcal{T}}^{tt}_{\psi,BR,a}:\rangle=\langle:\hat{\mathcal{T}}^{tt}_{\psi,GLW,a}:\rangle=\langle:\hat{\mathcal{T}}^{tt}_{\psi,HW,a}:\rangle.$

Although the exact expressions of energy-momentum tensor is pseudo-gauge
dependent but the thermal expectation value of
$:\hat{\mathcal{T}}^{tt}_{\psi,a}:$ does not depend on pseudo-gauge, i.e.,
$\varepsilon_{Can}(T,m)=\varepsilon_{BR}(T,m)=\varepsilon_{GLW}(T,m)=\varepsilon_{HW}(T,m)$.
One should note that energy density for spin-half field as obtained here is
also independent of the scale ‘$a$’. Now let us look into the expression of
the quantum fluctuation for different forms of the energy-momentum tensor [6],

$\displaystyle\sigma^{2}_{\psi,Can}(a,m,T)=2\int
dP~{}dP^{\prime}\mathfrak{f}_{f}(\omega_{{\boldsymbol{p}}})(1-\mathfrak{f}_{f}(\omega_{{\boldsymbol{p}}^{\prime}}))$
$\displaystyle\times\bigg{[}(\omega_{{\boldsymbol{p}}}+\omega_{{\boldsymbol{p}}^{\prime}})^{2}(\omega_{{\boldsymbol{p}}}\omega_{{\boldsymbol{p}}^{\prime}}+{\boldsymbol{p}}\cdot{\boldsymbol{p}}^{\prime}+m^{2})e^{-\frac{a^{2}}{2}({\boldsymbol{p}}-{\boldsymbol{p}}^{\prime})^{2}}$
$\displaystyle-(\omega_{{\boldsymbol{p}}}-\omega_{{\boldsymbol{p}}^{\prime}})^{2}(\omega_{{\boldsymbol{p}}}\omega_{{\boldsymbol{p}}^{\prime}}+{\boldsymbol{p}}\cdot{\boldsymbol{p}}^{\prime}-m^{2})e^{-\frac{a^{2}}{2}({\boldsymbol{p}}+{\boldsymbol{p}}^{\prime})^{2}}\bigg{]},$
(17)

$\displaystyle\sigma^{2}_{\psi,GLW}(a,m,T)=\frac{1}{2m^{2}}\int
dP~{}dP^{\prime}\mathfrak{f}_{f}(\omega_{{\boldsymbol{p}}})(1-\mathfrak{f}_{f}(\omega_{{\boldsymbol{p}}^{\prime}}))$
$\displaystyle\times\bigg{[}(\omega_{{\boldsymbol{p}}}+\omega_{{\boldsymbol{p}}^{\prime}})^{4}\left(\omega_{{\boldsymbol{p}}}\omega_{{\boldsymbol{p}}^{\prime}}-{\boldsymbol{p}}\cdot{\boldsymbol{p}}^{\prime}+m^{2}\right)e^{-\frac{a^{2}}{2}({\boldsymbol{p}}-{\boldsymbol{p}}^{\prime})^{2}}$
$\displaystyle-(\omega_{{\boldsymbol{p}}}-\omega_{{\boldsymbol{p}}^{\prime}})^{4}\left(\omega_{{\boldsymbol{p}}}\omega_{{\boldsymbol{p}}^{\prime}}-{\boldsymbol{p}}\cdot{\boldsymbol{p}}^{\prime}-m^{2}\right)e^{-\frac{a^{2}}{2}({\boldsymbol{p}}+{\boldsymbol{p}}^{\prime})^{2}}\bigg{]},$
(18)

and,

$\displaystyle\sigma^{2}_{\psi,HW}(a,m,T)=\frac{2}{m^{2}}\int
dP~{}dP^{\prime}\mathfrak{f}_{f}(\omega_{{\boldsymbol{p}}})(1-\mathfrak{f}_{f}(\omega_{{\boldsymbol{p}}^{\prime}}))$
$\displaystyle\times\Big{[}\left(\omega_{{\boldsymbol{p}}}\omega_{{\boldsymbol{p}}^{\prime}}+{\boldsymbol{p}}\cdot{\boldsymbol{p}}^{\prime}+m^{2}\right)^{2}\left(\omega_{{\boldsymbol{p}}}\omega_{{\boldsymbol{p}}^{\prime}}-{\boldsymbol{p}}\cdot{\boldsymbol{p}}^{\prime}+m^{2}\right)e^{-\frac{a^{2}}{2}({\boldsymbol{p}}-{\boldsymbol{p}}^{\prime})^{2}}$
$\displaystyle~{}~{}-(\omega_{{\boldsymbol{p}}}\omega_{{\boldsymbol{p}}^{\prime}}+{\boldsymbol{p}}\cdot{\boldsymbol{p}}^{\prime}-m^{2})^{2}(\omega_{{\boldsymbol{p}}}\omega_{{\boldsymbol{p}}^{\prime}}-{\boldsymbol{p}}\cdot{\boldsymbol{p}}^{\prime}-m^{2})e^{-\frac{a^{2}}{2}({\boldsymbol{p}}+{\boldsymbol{p}}^{\prime})^{2}}\Big{]},$
(19)

here $dP\equiv d^{3}{{p}}/((2\pi)^{3}2\omega_{{\bi{p}}})$. From Eqs. (17)-(19)
one can observe that the quantum fluctuation is indeed depends on the form of
the energy-momentum tensor or the pseudo-gauge choice. Here also the
degeneracy factor can be incorporated in the following manner:
$\varepsilon_{Can}\rightarrow g_{f}\varepsilon_{Can}$ and
$\sigma_{\psi}^{2}\rightarrow g_{f}\sigma_{\psi}^{2}$.

####  3.1.3 Large volume limit or the thermodynamic limit:

Since $\mathcal{S}_{a}$ is a subsystem of the larger system $\mathcal{S}_{V}$,
we expect that in the large volume limit $a\rightarrow\infty$, but still
keeping $a^{3}\ll V$, formula for quantum statistical fluctuation should
reproduce the known result from statistical mechanics [12]. This can be
checked using the Gaussian representation of the three dimensional Dirac delta
function, i.e.

$\delta^{(3)}({\bi{p}}-{\bi{p}^{\prime}})=\lim_{a\to\infty}\frac{a^{3}}{(2\pi)^{3/2}}e^{-\frac{a^{2}}{2}({\bi{p}}-{\bi{p}^{\prime}})^{2}}.$
(20)

In the large volume limit Eq. (15) boils down to [5],

$\sigma_{\phi}^{2}=\frac{g_{b}}{(2\pi)^{3/2}a^{3}}\int\frac{d^{3}{{p}}}{(2\pi)^{3}}~{}\omega_{{\bi{p}}}^{2}~{}\mathfrak{f}_{b}(\omega_{{\bi{p}}})(1+\mathfrak{f}_{b}(\omega_{{\bi{p}}})).$
(21)

The factor
$\mathfrak{f}(\omega_{{\bi{p}}})(1+\mathfrak{f}(\omega_{{\bi{p}}}))$ in the
above equation gives us the hint that $\sigma_{\phi}^{2}$ can be expressed in
term of the specific heat $C_{V}$. This in indeed true. Note that using the
expression of the energy density as given in Eq. (14) $C_{V}$ can be expressed
as,

$C_{V,\phi}=\frac{d\varepsilon}{dT}=\frac{g_{b}}{T^{2}}\int\frac{d^{3}{{p}}}{(2\pi)^{3}}~{}\omega_{{\bi{p}}}^{2}~{}\mathfrak{f}_{b}(\omega_{{\bi{p}}})(1+\mathfrak{f}_{b}(\omega_{{\bi{p}}})).$
(22)

Using Eqs. (21)-(22) we can conclude that in the large volume limit [5, 34],

$V_{a}\sigma_{n,\phi}^{2}=\frac{T^{2}C_{V,\phi}}{\varepsilon^{2}}=V\frac{\langle
E^{2}\rangle-\langle E\rangle^{2}}{\langle E\rangle^{2}}\equiv
V\sigma^{2}_{E},$ (23)

here $V_{a}=a^{3}(2\pi)^{3/2}$ may be considered as the volume of the
“Gaussian” subsystem $\mathcal{S}_{a}$ [5, 6]. Furthermore in the above
equation $E$ represents the energy of system $\mathcal{S}_{V}$ and
$V\sigma_{E}^{2}$ denotes the normalized energy fluctuation in the system
$\mathcal{S}_{V}$ [12]. Therefore we observe that in the large volume limit
the volume scaled quantum statistical fluctuation $V_{a}\sigma_{n,\phi}^{2}$
reproduces the correct thermodynamic fluctuation $V\sigma^{2}_{E}$ 333Since we
could reproduce the correct fluctuations in the large volume limit, this also
shows that the choice of the normal ordering prescription as considered here
is physically more appealing to obtain quantum statistical fluctuation.. Such
conclusion can also be obtained for a fermionic system. However for spin-half
particles something more interesting happens. Earlier we have shown that for
spin-half field quantum statistical fluctuation depends on the form of the
energy-momentum tensor, i.e. generically,
$\sigma^{2}_{\psi,Can}\neq\sigma^{2}_{\psi,GLW}\neq\sigma^{2}_{\psi,HW}$.
However if we consider the large volume limit then it can be argued that [6],

$\sigma_{\psi,Can}^{2}=~{}\frac{4~{}g_{f}}{(2\pi)^{3/2}a^{3}}\int\frac{d^{3}{{p}}}{(2\pi)^{3}}~{}\omega_{{\bi{p}}}^{2}~{}\mathfrak{f}_{f}(\omega_{{\bi{p}}})(1-\mathfrak{f}_{f}(\omega_{{\bi{p}}}))=\sigma_{\psi,GLW}^{2}=\sigma_{\psi,HW}^{2}.$
(24)

Hence in the large volume limit the energy-fluctuation does not depend on the
pseudo-gauge choice of the energy-momentum tensor. More explicitly444Here we
do not introduce different notations to represent statistical fluctuation of
energy for the system $\mathcal{S}_{V}$ to distinguish between bosonic and
fermionic systems. Rather, $V\sigma^{2}_{E}$ represents statistical
fluctuation within the system $\mathcal{S}_{V}$ as described by the canonical
ensemble for both spin-zero and spin half particles. Difference between the
expression of $V\sigma^{2}_{E}$ as given in Eq. (23) and Eq. (25) is
implicitly understood.,

$V_{a}\sigma_{n,\psi,Can}^{2}=V_{a}\sigma_{n,\psi,GLW}^{2}=\frac{T^{2}C_{V,\psi}}{\varepsilon_{Can}^{2}}=V\frac{\langle
E^{2}\rangle-\langle E\rangle^{2}}{\langle E\rangle^{2}}\equiv
V\sigma^{2}_{E}.$ (25)

###  3.2 Numerical results

Once we have discussed the analytical results in the previous sections, now we
present numerical results. In the left plot of Fig. (2) we show the variation
of normalized fluctuation $\sigma_{n,\phi}$ within the subsystem
$\mathcal{S}_{a}$ described by a real scalar field with the scale ‘$a$’ for
different temperature $(T)$ but fixed mass $(m)$ 555While presenting numerical
results we keep in mind the strongly interacting matter produced in heavy-ion
collision experiments. For the numerical estimation of fluctuation we consider
$100$ MeV $<T<400$ MeV, and particle masses: $0$ MeV $<m<1000$ MeV. We
consider the total degeneracy factor to be $40$, which is a typical degeneracy
factor for three flavor QCD matter. For a detailed discussion on the
degeneracy factor see Ref. [5]. Since we are taking the total degeneracy
factor to be $40$, we consider $g_{b}=40$ and $g_{f}=10$ for numerical
results. This is because contrary to $\sigma_{\phi}^{2}$ or $\sigma_{n,\phi}$,
$\sigma_{\psi}^{2}$ or $\sigma_{n,\psi}$ already includes the spin degeneracy
factor and particle-antiparticle degeneracy factor.. From this plot one may
observe that with temperature normalized fluctuation $\sigma_{n,\phi}$
decreases. Note that although $\sigma_{n,\phi}$ decreases with temperature,
but the variation of $\sigma_{n,\phi}$ with temperature is opposite to the
variation of $\sigma_{\phi}$ ( not shown here explicitly). Furthermore with
the system size ‘$a$’ quantum fluctuation decreases and for $a\rightarrow 0$
quantum fluctuation diverges. This is nothing but the manifestation of quantum
uncertainty principle. In the right plot of Fig. (2) we show the variation of
the normalized quantity $V_{a}\sigma_{n,\phi}^{2}/V\sigma^{2}_{E}$ with the
scale ‘$a$’ for different temperature $(T)$ but fixed mass $(m)$. As argued
previously that in the large volume limit or in the thermodynamic limit
$V_{a}\sigma_{n,\phi}^{2}$ should approach $V\sigma^{2}_{E}$, i.e.
$V_{a}\sigma_{n,\phi}^{2}/V\sigma^{2}_{E}$ should approach unity in the
thermodynamic limit. From this plot one may observe that for the range of
temperature and mass, for $a>1$ fm $V_{a}\sigma_{n,\phi}^{2}/V\sigma^{2}_{E}$
indeed approaches unity.

Figure 2: Left plot: variation of normalized fluctuation $\sigma_{n,\phi}$
within the subsystem $\mathcal{S}_{a}$ described by a real scalar field with
the scale ‘$a$’ for different temperature $(T)$ but fixed mass $(m)$. Right
plot: in this plot we demonstrate the large system size limit by plotting the
normalized quantity $V_{a}\sigma_{n,\phi}^{2}/V\sigma^{2}_{E}$ with the scale
‘$a$’ for different temperature $(T)$ but fixed mass $(m)$. For more
discussions see text and Ref. [5].

In the left plot of Fig. (3) we again show the variation of normalized
fluctuation $\sigma_{n,\phi}$ with the scale ‘$a$’ but for different mass
$(m)$ for a fixed temperature $(T)$. From this plot, we can conclude that with
an increase in mass ($m$) normalized fluctuation $\sigma_{n,\phi}$ increases.
Here also we emphasize that the variation of $\sigma_{n,\phi}$ with mass is
opposite to the variation of $\sigma_{\phi}$ ( not shown here). We also find
that for $a\rightarrow 0$ fluctuation diverges and with the system size ‘$a$’
quantum fluctuation decreases. In the right plot of Fig. (3) we show the
variation of the normalized quantity
$V_{a}\sigma_{n,\phi}^{2}/V\sigma^{2}_{E}$ with the scale ‘$a$’ for different
mass ($m$) but fixed temperature $(T)$. From this plot, we observe that in the
large volume limit or in the thermodynamic limit
$V_{a}\sigma_{n,\phi}^{2}/V\sigma^{2}_{E}$ approaches unity which is
consistent with the thermodynamic considerations. From Figs. (2) and (3) we
can conclude that the quantum statistical fluctuations become very important
at a small length scale (say $a\sim 0.1$ fm). This indicates that at such a
small length scale one can not naively use classical concepts of well-defined
energy density. Rather one also has to take into consideration possible
quantum effects.

Figure 3: Left plot: variation of normalized fluctuation $\sigma_{n,\phi}$
within the subsystem $\mathcal{S}_{a}$ described by a real scalar field with
the scale ‘$a$’ for different mass $(m)$ but fixed temperature $(T)$. Right
plot: in this plot we show the large system size limit by showing the
variation of the normalized quantity
$V_{a}\sigma_{n,\phi}^{2}/V\sigma^{2}_{E}$ with the scale ‘$a$’ for different
mass $(m)$ but fixed temperature $(T)$. For more discussions see text and Ref.
[5].

Once we have demonstrated the variation of quantum statistical fluctuation
with the size of the sub-system, temperature, and mass, now we explicitly show
the pseudo-gauge dependence of the quantum statistical fluctuation using
various forms of the energy-momentum tensor for a fermionic field. In the left
plot of Fig. (4) we show a comparison of the normalized standard deviation
($\sigma_{n,\psi}$) as obtained using different forms of the time-time
component of energy-momentum tensor for $T=0.14$ GeV but for two different
masses. In this plot the thick lines correspond to $m=1.0$ GeV and $T=0.14$
GeV. On the other hand thin lines correspond to $m=0.1$ GeV and $T=0.14$ GeV.
From this plot, it is evident that for small values of $a<0.6$ fm, the value
of $\sigma_{n,\psi}$ differs with various pseudo-gauge choices. This
difference in the value of $\sigma_{n,\psi}$ as obtained for various pseudo-
gauge choices increases for smaller values of $a$. It is also interesting to
note that for $m=1.0$ GeV the differences between different pseudo-gauges are
smaller as compared to the case of $m=0.1$ GeV. In the right plot of Fig. (4)
we show the variation of $\sigma_{n,\psi}$ for low mass ($m=0.02$ GeV) and
high temperature ($T=0.4$ GeV). From this plot, one may conclude that with
growing system size ($a$) the normalized standard deviation of fluctuations
($\sigma_{n,\psi}$) decreases for all pseudo-gauge choices. These figures also
show that fluctuations crucially depend on the ratio $m/T$.

Figure 4: Left plot: we show a comparison of $\sigma_{n,\psi}$ as obtained for
different pseudo-gauge choices of the energy-momentum tensor for $T=0.14$ GeV
but for two different masses. Right plot: we show a comparison of
$\sigma_{n,\psi}$ for small mass and high temperature as obtained for
different pseudo-gauge choices. Note that $\sigma_{n,\psi}$ is smallest for
the canonical framework among various pseudo-gauge choices. For more
discussions see text and Ref. [6].

Finally in Fig. (5) we demonstrate that in the large system size limit the
dimensionless measure of the fluctuation
$V_{a}\sigma_{n,\psi}^{2}/V\sigma^{2}_{E}$ approaches unity. In the left plot
and in the right plot of Fig. (5) we show the variation of
$V_{a}\sigma_{n,\psi}^{2}/V\sigma^{2}_{E}$ with the scale $a$ for $T=0.14$ GeV
but for $m=0.1$ GeV and $m=1.0$ GeV respectively. From this plots one may
observe that among various choices of pseudo-gauge, for the canonical
framework the decrease of $V_{a}\sigma_{n,\psi}^{2}/V\sigma^{2}_{E}$ with the
scale $a$ is fastest. Also note that with increasing $m/T$ ratio, the decrease
of $V_{a}\sigma_{n,\psi}^{2}/V\sigma^{2}_{E}$ with the scale ‘$a$’ is faster.

Figure 5: Left plot: variation of the normalized fluctuation
$V_{a}\sigma_{n,\psi}^{2}/V\sigma^{2}_{E}$ in the subsystem $\mathcal{S}_{a}$
with the length scale $a$ for $T=0.14$ GeV and $m=0.1$ GeV. Right plot:
variation of $V_{a}\sigma_{n,\psi}^{2}/V\sigma^{2}_{E}$ in the subsystem
$\mathcal{S}_{a}$ with the length scale $a$ but for $T=0.14$ GeV and $m=1.0$
GeV. From these plots we observe that in the large system size limit
$V_{a}\sigma_{n,\psi}^{2}/V\sigma^{2}_{E}$ approaches unity. Note that
$V_{a}\sigma_{n,\psi}^{2}/V\sigma^{2}_{E}$ is smallest for the canonical
framework among various pseudo-gauge choices and it also approaches unity
faster with the scale $a$. For more discussions see text and Ref. [6].

##  4 Conclusions

In this article, we have addressed the system size scaling of quantum
statistical fluctuation by considering a gas of spin-half as well as spin-zero
particles at finite temperature. We derived the formula for the quantum
statistical fluctuations of energy in subsystems of a relativistic gas at
finite temperature. A novel feature of such quantum statistical fluctuations
is that they depend on the form of the energy-momentum tensor. However, for a
sufficiently large system, different pseudo-gauges give rise to the same
expression for the fluctuations which agree with the canonical-ensemble
formula for energy fluctuation. Using explicit field theoretical calculations
we argued that irrespective of different characteristics of the systems, i.e.
bosonic or fermionic, and also irrespective of different pseudo-gauge choices
a generic feature of quantum fluctuations is that they decrease with the
system size and for a small length scale such fluctuations can be significant.
On a practical side, our calculations can be used for quantitative estimation
of fluctuations for given values of $T$, $m$, and degeneracy factor. If the
dimensionless measure of the quantum fluctuations (say, $\sigma_{n}$) are
small (smaller than 1) then only the classical picture of a well-defined
energy density can be considered safely. These analyses also give us a
practical way to determine the fluid cell or “coarse-graining” size. If the
“coarse-graining” scale is small so that we can not neglect the quantum
effects then quantum fluctuation should be combined in future works with
hydrodynamic fluctuations [35]. Moreover, these results can also be useful to
determine a scale of coarse-graining for which the pseudo-gauge choice becomes
irrelevant in the context of hydrodynamic modeling of high-energy collisions.
Using these results, for a given value of $T$, $m$ and the degeneracy factor
one can determine the scale $a_{0}$ so that above the scale $a_{0}$
fluctuations are small and pseudo-gauge independent. Then such a scale
$a>a_{0}$ can be safely used as the “coarse-graining” scale or scale for the
fluid element. What remains rather non-trivial to interpret is the pseudo-
gauge dependence of fluctuations for small systems. From an experimental point
of view, one can measure the expectation values of some combination of the
quantum field theoretical operators. However we have argued that such
expectation value can depend on the form of the operator or pseudo-gauge. Any
realistic measurements of such pseudo-gauge dependent quantities need to be
explored rigorously [36, 37]. Moreover these analysis contains important
results which may be relevant to a broader scientific community, in particular
with regards to the increased interest in small systems produced in heavy ion
collisions [38].

## Acknowledgments

I would like to thank the organizers of the “10th International Conference on
New Frontiers in Physics (ICNFP 2021) ” for giving me an opportunity to
present our recent works. I thank Wojciech Florkowski, Radoslaw Ryblewski, and
Rajeev Singh for a fruitful collaboration. My research work is supported by
the Polish National Science Centre Grant No. 2018/30/E/ST2/00432.

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# Genetic Algorithm to Optimize Design of Micro-Surgical Scissors

Fatemeh Norouziani1, Veerash Palanichamy2, Shivam Gupta3, and Onaizah Onaizah4
1,2,3,4The authors are with the Department of Computing & Software, McMaster
University, Ontario, Canada (emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Microrobotics is an attractive area of research as small-scale robots have the
potential to improve the precision and dexterity offered by minimally invasive
surgeries. One example of such a tool is a pair of micro-surgical scissors
that was developed for cutting of tumors or cancerous tissues present deep
inside the body such as in the brain. This task is often deemed difficult or
impossible with conventional robotic tools due to their size and dexterity.
The scissors are designed with two magnets placed a specific distance apart to
maximize deflection and generate cutting forces. However, remote actuation and
size requirements of the micro-surgical scissors limits the force that can be
generated to puncture the tissue. To address the limitation of small output
forces, we use an evolutionary algorithm to further optimize the performance
of the scissors. In this study, the design of the previously developed
untethered micro-surgical scissors has been modified and their performance is
enhanced by determining the optimal position of the magnets as well as the
direction of each magnetic moment. The developed algorithm is successfully
applied to a 4-magnet configuration which results in increased net torque.
This improvement in net torque is directly translated into higher cutting
forces. The new configuration generates a cutting force of 58 mN from 80
generations of the evolutionary algorithm which is a 1.65 times improvement
from the original design. Furthermore, the developed algorithm has the
advantage that it can be deployed with minor modifications to other
microrobotic tools and systems, opening up new possibilities for various
medical procedures and applications.

##  I INTRODUCTION

Minimally invasive surgery, often referred to as keyhole surgery, has become
increasingly prevalent in contemporary medical practices due to the numerous
advantages it offers [1]. It reduces trauma, minimizes scarring, reduces risks
associated with infection, and accelerates recovery time [2]. The robotic
instruments currently employed in minimally invasive surgical procedures lack
flexibility across various applications and are not designed to perform
effectively through small incisions [3]. The last decade has seen a lot of
advances in the field of small-scale robotics with sizes ranging from
micrometers to millimeters. These robots are useful as they can easily travel
through small incisions or can be deployed through injections and perform a
variety of operations such as targeted drug delivery [4], brachytherapy [5],
and biopsy [6].

Magnetically actuated tools hold significant promise for executing challenging
surgeries through small incisions, as they can be controlled wirelessly from
outside the body, offering high levels of dexterity and precision. One such
tool reported in the literature is an untethered pair of magnetically actuated
scissors [3]. These scissors are composed of titanium sheets, a nitinol wire
that acts as a restoring spring, and two NdFeB (Neodymium Iron Boron) Magnets.
The size of the scissors is 15 mm by 15 mm. The maximum cutting force
generated by scissors is proportional to the external magnetic flux density
applied. At 20 mT, the scissors can generate approximately 35 mN from a single
blade. These millimeter-size scissors have great potential for in-vivo tissue
cutting and conducting biopsies. However, the size of the scissors is
currently not suitable to carry out in-vivo procedures [3]. Moreover, the
cutting force and torques may not be large enough to cut through tissues. One
of the reasons is the restriction on the size of magnets for producing large
forces. Comparatively large magnets are required to have higher magnetic
moments as this parameter scales with the volume of magnetic material. Another
reason could be attributed to the friction present when the scissors have to
be closed and blades come in contact with each other. This can significantly
lower the force required for cutting, as a portion of the energy is otherwise
expended in overcoming friction [2, 7]. It is important to model the forces
used for cutting biological tissues [8, 9] as well as determine the position
of magnets for optimizing the cutting forces.

There are various studies in the literature where evolutionary algorithms are
used for optimizing robot design. In [10], Kim et al. used a genetic algorithm
to optimize design parameters for a transoral robotic system, while in [11],
Fei et al. used it to design a modular robot topology based on the specified
task. An evolutionary algorithm is to emulate biological evolution in which
multiple different solutions are evaluated against each other over multiple
iterations and the fittest solutions are kept over time. The set of solutions
is also modified through mutations and crossover between individuals,
eliminating some of the poorly performing solutions. The two main components
of implementing a genetic algorithm are the genome and the fitness function.
The genome is the encoding for each individual solution to our optimization
problem, and the fitness function determines the performance of our solution.
The use of genetic evolution for design optimization allows us to design an
algorithm that is adaptable for different problem scopes and parameters. The
algorithm designed in this paper can easily be modified to work with various
other types of microrobotic tools.

Figure 1: Schematic labeling the radii as well as interaction forces and
torques between the magnets.

In this study, we introduce an evolutionary algorithm to tackle the problem of
optimizing magnet position and orientation to generate a higher cutting force.
Specifically, we present a 4-magnet configuration instead of the original
2-magnet design to increase the total magnetic moment available on the scissor
while keeping the magnets the same size. Section II lays out the underlying
physical principles and modeling as well as algorithm design. Section III
discusses the results of the design optimization and presents the final
configuration. Section IV presents the conclusions and future directions for
the developed algorithm.

##  II Methods

The physical model of the scissors along with the algorithm design and fitness
function are presented in this section.

###  II-A Background

The physical principles of the scissors are modeled as a torque balance. The
force between two dipoles $i$ and $j$ having magnetic moment $\bm{m_{i}}$ and
$\bm{m_{j}}$ can be modelled as:

$\bm{F_{ij}}=\frac{3\mu_{0}}{4\pi r^{5}}[\bm{(m_{i}\cdot
r_{ij})m_{j}+(m_{j}\cdot r_{ij})m_{i}}\\\ +\bm{(m_{i}\cdot
m_{j})r_{ij}}-\frac{5(\bm{m_{i}\cdot r_{ij}})(\bm{m_{j}\cdot
r_{ij}})}{r^{2}}\bm{r_{ij}}]\\\ $ (1)

where $\bm{r_{ij}}$ is a vector from $\bm{m_{i}}$ to $\bm{m_{j}}$ and
$\bm{r_{ij}}$ is the magnitude of $\bm{r_{ij}}$. Figure 1 shows a schematic of
the interaction forces and torques between the magnets.

If $\bm{R_{i}}$ is a vector from the pivot point to any dipole $i$, then the
resulting torque from the interaction force given above in Equation 1 can be
calculated as:

$\bm{\tau_{fi}}=\bm{R_{i}\times F_{ij}}$ (2)

The torque generated by dipole $\bm{m_{i}}$ on $\bm{m_{j}}$ is given by:

$\bm{\tau_{ij}}=\frac{\mu_{0}}{4\pi r^{5}}[3\bm{m_{j}\times(m_{i}\cdot
r_{ij})r_{ij}}-r^{2}\bm{(m_{j}\times m_{i})}]$ (3)

When no external field is applied, the scissors are in an equilibrium which
can be represented as:

$\bm{\tau_{fi}+\tau_{ij}}-k\bm{p_{0}}=0$ (4)

Here, $k$ denotes the spring constant of the nitinol wire, $\mathbf{p_{0}}$
denotes the initial deflection of the spring when no external field is
applied.

If an external magnetic flux density $\bm{B}$ is applied then torque on a
magnet with magnetic moment $\bm{m_{i}}$ is given by:

$\bm{\tau_{m_{i}}}=\bm{m_{i}\times B}$ (5)

Therefore, the total torque on the scissors’ blade is given by:

$\bm{\tau_{net}}=\sum_{i,j=1}^{n}\bm{(\tau_{fi}+\tau_{ij}+\tau_{m_{i}})}$ (6)

Here, $i,j$ represents th $n$th magnet where $n$ is taken as 4 for this study.
Moreover, i $\neq$ j and any components of $\bm{\tau_{ij}}$ in the equation
above should be discarded for magnets on the same blade. For example, if
magnets 1 and 2 are on the same blade, the $\bm{\tau_{12}}$ is discarded from
the above equation as magnets on the same blade have no relative motion and
will not contribute to net torque on the blade.

###  II-B Representation

The parameters we are trying to optimize are the magnet position and the
orientation of each magnet which can be represented by a $3n$ genetic vector
of the shape:

$<x_{1},y_{1},M_{x_{1}},...,x_{n},y_{n},\theta_{n}>$

where $x_{i},y_{i}$ represent the x and y position of the ith magnet while
$\theta_{i}$ represents the orientation of the magnet from the positive
x-axis. Note that the magnetization vector $\bm{m}$ is calculated from the
angle $\theta_{i}$ using the trigonometric relationships which are then used
in the equations.

###  II-C Evaluation Function

In this case, the net torque is taken as the fitness function that will be
maximized and is shown in Equation 6. A penalty of value $p$ is subtracted
from the fitness score if an individual solution represents an invalid
configuration. Two ways for an invalid configuration to occur are when either
one or more of the magnets are outside the allowed region (blade geometry) or
if they are placed less than one magnet distance away from each other.

Since we want the scissors to be able to open to the same separation level as
the original design when a magnetic field is not present, we will have to make
sure the magnetic interaction torque ($\bm{\tau_{fi}+\tau_{ij}}$) is less than
the threshold torque that would result in having a smaller separation between
the scissor blades. This check is also done during fitness evaluation and
given a penalty if violated.

###  II-D Evolution

The initial population is generated randomly using a uniform distribution
while making sure they are within the boundaries of the scissors and do not
have magnets that are too close. The population for the next generation is
based on the previous generation. The population evolves solely based on the
initial individuals and the genetic operations applied to them. Within each
generation, individuals undergo various mutations and crossover, and it is
vital to ensure that the modified genes meet the restrictions mentioned above
that are controlled through the fitness function.

Specifically, we use an Evolutionary Algorithm (EA) with a ($\mu,\lambda$)
selection mechanism from the DEAP library (Python 3.11) to update the
population in each generation. $\mu$ represents the number of parents
selected, and $\lambda$ represents the number of offspring that form the next
generation. In the ($\mu,\lambda$) selection strategy, the $\mu$ best
offspring are chosen based on their fitness and form the next generation. The
parents are not part of the next generation, ensuring that it can avoid
getting trapped in a local minima [12]. The evolution parameters are specified
in Table I and worked well for this problem. The specific constants used for
the torque calculations that determine the fitness are given in Table II.

TABLE I: Evolution Parameters in the Code Parameter | Value
---|---
Population Size | 70
Number of Generations | 50
Crossover Probability | 0.7
Mutation Probability | 0.2
Differential Evolution Alpha Parameter | 0.5
Mutation Mean ($\mu$) | 0
Mutation Standard Deviation ($\sigma$) | 1
Selection Operator Tournament Size | 3
TABLE II: Constants used for fitness calculations Constant | Value
---|---
Magnetic Moment | 3.34E-2 $kA\cdot m^{2}$
Magnet Length (cube) | 31.8 $mm$
Applied Field | 20 $mT$
Minimum Magnet Separation Distance | 4.77 $mm$

##  III Results

Figure 2: Figure showing the optimal configuration of magnets found using
genetic evolution. The red dots represent the center of the magnets while the
arrows represent the magnetization direction

The generational convergence of the algorithm is tracked using maximum fitness
scores for each configuration which is shown in Figure 3. Most of the
evolution happens early, before 40 generations, where it follows a linear
progression. The algorithm seems to converge around 80 generations to an
optimal fitness of 5E-3 as shown in Figure 3. After this convergence, further
training is less likely to improve our fitness significantly. The maximum
fitness of 5.5E-3 represents a maximum torque $\bm{\tau_{max}}$ of 5.5 mN-m
for a single blade. Since this total torque does not include the restoring
torque from the nitinol wire in our fitness calculation, the resulting torque
can then be calculated using:

$\bm{\tau_{final}}=\bm{\tau_{max}}+\bm{\tau_{spring}}$ (7)

where $\bm{\tau_{spring}}=k\bm{p_{0}}$, and $k$ is the spring constant that
can be calculated from the specifications of the nitinol wire. Solving
Equation 7 gives us a torque of 4.6 mN-m, from which the force is calculated
for a single blade. The resulting cutting force is 58 mN which is 1.65 times
higher than the cutting force that was measured from the original design to be
35 mN for a single blade.

Figure 3: Graph showing the maximum fitness score throughout evolution. The
fitness function starts to saturate around 80 generations beyond which only
minimal improvements are produced.

The optimal configuration is shown in Figure 2. The algorithm prioritizes
putting the magnets as close to each other while satisfying the distance
constraint of not being within one magnet length of each other. Another
interesting assessment is that the magnet on the bottom left has a
magnetization vector with its y-component being negative. Although this
orientation may not allow for the maximum torque when a field is applied along
the positive y-axis, this orientation is necessary for the magnet
configuration to balance the magnetic interactions that could lead to the
scissors having more than our desired equilibrium deflection. Although we are
assuming magnets of the same size for our theoretical torque calculations and
in our evolutionary algorithm, smaller magnet sizes can be used in this
optimal configuration to generate the same cutting force as the original
design. This will help reduce the size of the microrobot and the weight of the
proposed design and prototype.

##  IV Conclusion

In this paper, we have shown that it is possible to use genetic evolution to
create an adaptable algorithm that can help generate design parameters for
microrobots. We specifically applied our methodology to generate magnet
position and orientation to improve the design of our previously developed
minimally invasive micro-surgical scissors. The new proposed configuration has
4 magnets compared to the 2 magnets seen in the original. This new design can
generate approximately 1.65 the cutting force of the original scissors at 58
mN. Our optimal configuration also allows the option for the device to be
scaled down while keeping similar cutting forces of 35 mN that are seen in the
original design which is an important and desirable characteristic of
microrobot design.

Future work will be done to redefine this problem using a multi-objective
evolutionary algorithm. Such an algorithm might be more suitable because the
scissors need to be able to open by themselves when a magnetic field is not
present in order to use only control input. Magnet configurations that improve
the maximum cutting torque negatively affect the restoring torque of the
scissors. This demands a 2-objective maximization problem which can provide a
more diverse solution set. Future research will also be done to extend our
algorithm to other types of devices [13, 14] that may need similar design
optimization.

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