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Thermodynamic properties of an ideal Quark-Gluon plasma under quantum gravitational effects

Authors:
Djamel Eddine Zenkhri at University of Ouargla
Djamel Eddine Zenkhri
Abdelhakim Benkrane at University of Ouargla
Abdelhakim Benkrane
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Abstract

In this study, we investigate the thermodynamic properties of an ideal Quark-Gluon Plasma (QGP) at a vanishing chemical potential, under the influence of quantum gravitational effects, specifically incorporating the Linear-Quadratic Generalized Uncertainty Principle (LQGUP). We analyze the impact of LQGUP on key thermodynamic quantities, including the grand canonical potential, pressure, energy density, entropy, speed of sound, and the bulk viscosity’s response to changes in the speed of sound. Furthermore, we extend our analysis to examine the time evolution of the universe’s temperature in the presence of LQGUP effects.
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Eur. Phys. J. Plus (2025) 140:309
https://doi.org/10.1140/epjp/s13360-025-06250-y
Regular Article
Thermodynamic properties of an ideal Quark-Gluon plasma under quantum
gravitational effects
Djamel Eddine Zenkhria, Abdelhakim Benkraneb
Laboratoire LRPPS, Faculté des Mathématiques et des Sciences de la Matière, Université Kasdi Merbah Ouargla, Ouargla 30000, Algeria
Received: 8 October 2024 / Accepted: 24 March 2025
© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Abstract In this study, we investigate the thermodynamic properties of an ideal Quark-Gluon Plasma (QGP) at a vanishing chemical
potential, under the influence of quantum gravitational effects, specifically incorporating the Linear-Quadratic Generalized Uncer-
tainty Principle (LQGUP). We analyze the impact of LQGUP on key thermodynamic quantities, including the grand canonical
potential, pressure, energy density, entropy, speed of sound, and the bulk viscosity’s response to changes in the speed of sound.
Furthermore, we extend our analysis to examine the time evolution of the universe’s temperature in the presence of LQGUP effects.
1 Introduction
A deeper understanding of the universe depends on uncovering matter’s fundamental forces and behavior under extreme conditions.
According to Friedmann’s solution [1] of Einstein’s gravitational equation, the universe experienced an expansion from a singularity
point at time zero which has been confirmed by the formulation of Hubbel’s law for the redshift of distant galaxies [2]. At the dawn of
the universe, just microseconds after the Big Bang, a state of matter unlike anything observed today is thought to have existed: quark-
gluon plasma (QGP). This phase consists of unconfined quarks and gluons, the elementary building blocks of protons and neutrons,
which exist freely rather than confined within individual particles. It can be recreated in high-energy nuclear collisions, such as those
conducted at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC). Recent experimental results suggest
that these collisions have successfully produced a QGP with near-perfect fluid properties, making studying its thermodynamic and
transport characteristics increasingly pertinent. Throughout the universe’s evolution, the quark-gluon plasma (QGP) transitions into
hadronic matter, a process governed by quantum chromodynamics (QCD), which describes the strong interactions [3]. The key
distinction between the hadronic and QGP phases lies in the relative significance of short-range and long-range interactions among
their constituents across the expected phase transition. In the hadronic phase, short-range interactions among hadrons dominate,
which can be described by Boltzmann-Gibbs statistics. In contrast, the QGP phase is marked by a significant reduction in short-range
interactions due to “asymptotic freedom,” leading to a predominance of long-range interactions [4].
In recent years, the Generalized Uncertainty Principle (GUP) has garnered significant attention as an extension of the traditional
Heisenberg Uncertainty Principle (HUP), motivated by the need to reconcile quantum mechanics with gravitational effects [5–8].
The foundational theories, such as string theory, loop quantum gravity, deformed special relativity, and black hole physics, have
all contributed to the development of various forms of GUP, each characterized by a parameter β, which can be derived either
theoretically [9–12] or phenomenologically [13–16]. Kempf’s works are considered pioneering in the GUP [17–19], and one of his
main motivations was the necessity of a minimal length within the framework of quantum gravity and string theory.
The GUP model predicts a maximum observable momentum and a minimal measurable length. Accordingly, and via the Jacobi
identity xi,xjpi,pj0 results in [20]:
xi,pjiδij −βpδij +pipj
p+β2(p2δij +3pipj)(1)
where βα0/Mpcα0lp/.Mp4.34 ×10−9kg, lp10−35mandc3×108m/s are Planck mass and length and speed
of light, respectively. α0sets on the upper and lower bounds to α. Equation(1) leads to a minimal measurable length xmin ≈α0lp
suggesting that spacetime has a discrete nature [20], and maximum measurable momentum pmax ≈Mpc
α0. Given the fact that the
GUP leads to a minimum length, it is effective at microscopic scales and very high energies. As a significant consequence, we can
infer that the Generalized Uncertainty Principle plays a very important role in the early stages of the universe [21–23]. Recently, it
ae-mails: zenkhridjameleddine@gmail.com;dzenkhri@univ-ouargla.dz (corresponding author)
be-mail: abdelhakim.benkrane@univ-ouargla.dz
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