Scalar Casimir Eﬀects in a Lorentz Violation Scenario Induced by the Presence of Constant Vectors E. R. Bezerra de Mello1and M. B. Cruz2y

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Scalar Casimir Eﬀects in a Lorentz Violation Scenario Induced by the

Presence of Constant Vectors

E. R. Bezerra de Mello1, ∗and M. B. Cruz2, †

1Departamento de Física, Universidade Federal da Paraíba

Caixa Postal 5008, 58051-970, João Pessoa, Paraíba, Brazil

2Centro de Ciências Exatas e Sociais Aplicadas, Universidade Estadual da Paraíba

CEP 58706-550, Patos, Paraíba, Brazil

Abstract

In this work, we consider a theoretical model that presents a violation of Lorentz symmetry in the

approach of quantum ﬁeld theory. The theoretical model adopted consists of a real massive scalar

quantum ﬁeld conﬁned in the region between two large parallel plates. The violation of Lorentz symmetry

is introduced by a CPT-even, aether-like approach, considering a direct coupling between the derivative

of the scalar ﬁeld with two orthogonal constant vectors. The main objective of this paper is to analyze the

modiﬁcation of the Casimir energy and pressure caused by the anisotropy of space-time as a consequence

of these couplings. The conﬁnement of the scalar quantum ﬁeld between the plates is implemented by

imposing boundary conditions on them.

Keywords: Scalar ﬁelds, Lorentz violation, Casimir eﬀect, Boundary conditions, Constant vectors.

∗Electronic address: emello@ﬁsica.ufpb.br

†Electronic address: messiasdebritocruz@gmail.com

arXiv:2210.09243v2 [hep-th] 11 May 2023

I. INTRODUCTION

In 1948, H. B. Casimir proposed a theoretical model to investigate the structure of the quan-

tum vacuum associated with electromagnetic ﬁelds [1]. Speciﬁcally, he assumed that the presence

of two large, parallel, and isolated plates could be considered for this investigation. From a clas-

sical point of view, the vector wave number parallel to the plates suﬀers no restrictions; however,

the vector orthogonal to them must be discretized. In his original work, Casimir predicted that

the quantum ﬂuctuations of the electromagnetic ﬁeld also satisfy these conditions. By subtract-

ing the zero-point energy for photons in the presence and absence of two parallel ﬂat plates, he

obtained the result that the plates attract each other with a force per unit area given by:

A=−π2~c

240a4,(I.1)

where Ais the area of the plates and ais the distance between them. This eﬀect was experimen-

tally observed ten years later by M. J. Sparnaay [2]; however, only in the 1990s were experiments

able to conﬁrm the Casimir eﬀect with a high degree of accuracy [3]. This eﬀect is one of the most

direct manifestations of the existence of vacuum quantum ﬂuctuations. Initially little studied,

from the 1970s, the scientiﬁc community from several areas began to explore the phenomenon,

especially in Quantum Field Theory (QFT).

In general, we can say that the Casimir eﬀect is a consequence of boundary conditions imposed

on quantum ﬁelds. These boundaries can be material media, interfaces between two phases of

the vacuum, or even space-time topologies. The simplest way to theoretically study the Casimir

eﬀect is through the presence of two parallel plates placed in a vacuum. However, other situations

have been studied. For example, the analysis of the Casimir force for a piston conﬁguration in

R3, with one dimension being slightly curved and the other two being inﬁnite, was performed in

[4]. Moreover, the Casimir energy associated with massless Majorana and bosonic vector ﬁelds

conﬁned between two parallel plates has been investigated in [5].

Since QFT is based on the theory of Special Relativity, Lorentz symmetry is generally assumed

to be preserved. However, there are other theories that propose models where this symmetry is

violated [6], such as the Horava-Lifshitz (H-L) model, which introduces an anisotropy between

space and time. Consequently, the space-time anisotropy in a given QFT model can modify the

spectrum of the Hamiltonian operator. The violation of Lorentz symmetry (LV) has also been

questioned under both theoretical and experimental contexts. V. A. Kostelecky and S. Samuel

[8] presented a mechanism in string theory that allows for the violation of Lorentz symmetry at

the Planck energy scale. This mechanism results in a non-vanishing vacuum expectation value

of some vector or tensor components, which implies preferential directions in space-time. Other

possible mechanisms for the violation of Lorentz symmetry include space-time non-commutativity

[9–13], variation of coupling constants [14–16], and modiﬁcations of quantum gravity [17, 18].

As mentioned earlier, the violation of Lorentz symmetry has been under question since the

results obtained in [8]. If we accept this violation, an immediate question arises: What is the

inﬂuence of this violation on a given theoretical model from a quantum point of view? It is

expected that this violation will certainly aﬀect the entire structure of the theory. One of the

most important consequences of the violation concerns the energy spectrum of the modiﬁed

Hamiltonian operator. In fact, a QFT model is physically sensible only if it possesses a vacuum

state. Therefore, an important question to be answered is: How is the vacuum structure modiﬁed

by the presence of Lorentz symmetry violation? This question cannot be answered solely on

theoretical grounds and requires experimental veriﬁcation. Although the violation of Lorentz

symmetry occurs at large-scale energy, some remnants of this violation can be present in low-

energy phenomena [19]. Due to the high precision in the measurement of the Casimir force, the

corresponding experiment is a good candidate to detect the remnants left by Lorentz symmetry

breaking.

In this sense, the study of Lorentz symmetry violation has become of great theoretical and

experimental interest. Due to the high precision in the measurement of the Casimir force,

the corresponding experiment can be used as a laboratory to study possible traces left by the

LV. The ﬁrst studies on the Casimir eﬀect in the context of Lorentz-violating theories were

conducted in [20, 21] for diﬀerent extensions of the Quantum Electrodynamics (QED) that

break Lorentz symmetry. An analysis of the Casimir eﬀect associated with massless scalar and

fermionic quantum ﬁelds conﬁned between two large plates, considering space-time anisotropy

in the Horava-Lifshitz formalism, was developed in [22] and [23], respectively. Following the

same line of investigation, more recently, the Casimir eﬀect associated with a massive real scalar

ﬁeld was studied in [24]. The analysis of Casimir eﬀects associated with massive real scalar and

fermionic ﬁelds, considering CPT-even Lorentz symmetry breaking in an aether-like scenario

through the direct coupling between the ﬁeld’s derivative and an arbitrary constant four-vector,

was investigated in [26] and [27], respectively. The analysis of the Casimir energy and topological

mass associated with a massive scalar ﬁeld in the LV scenario was considered in [28]. In [29],

the inﬂuence of the presence of a constant magnetic ﬁeld on the Casimir eﬀect in the Lorentz-

violating scalar ﬁeld was considered. Moreover, the thermal eﬀect on the Casimir energy and

pressure caused by the Lorentz-violating scalar ﬁeld was investigated in [30].

In this paper, we aim to continue the investigation carried out in the papers mentioned in the

preceding paragraph. However, we will now consider the presence of two orthogonal constant

vectors that interact with a real scalar quantum ﬁeld conﬁned between two large parallel plates.

In this context, we will impose speciﬁc boundary conditions on the ﬁelds that should be satisﬁed

under the plates. This analysis can be seen as a contribution to the existing literature. The

presence of these two orthogonal constant vectors will introduce Lorentz violation, making the

analysis of the Casimir eﬀects more complex.

This work is organized as follows. In Section II, we provide a brief description of the theoretical

model for the real scalar ﬁeld with terms responsible for space-time anisotropy. In Section III, we

focus on calculating the Casimir energies and pressures while considering three diﬀerent boundary

conditions obeyed by the quantum ﬁelds on the plates. In Section IV, we summarize our most

important results. We will adopt natural units, ~=c= 1, and a metric signature of −2

II. MODEL SETUP

In this section, we will introduce the theoretical model that we want to analyze. This model is

composed of a massive real scalar quantum ﬁeld whose dynamics are governed by the lagrangian

density below:

L=1

2ηµν ∂µφ∂νφ+λ1(u·∂φ)2+λ2(v·∂φ)2−µ2φ2.(II.1)

The violation of Lorentz symmetry is represented by the presence of two arbitrary constant

vectors uµand vµcoupled to the scalar ﬁeld through its derivative. The dimensionless parameters

λi, for i= 1,2, are considered much smaller than unity in this analysis and are introduced to

investigate the inﬂuence of LV on the physical results obtained in this paper. Moreover, the

quantity µin Eq. (II.1) is deﬁned in terms of the mass in natural units (µ=mc/~).

In this formalism, the modiﬁed Klein-Gordon equation (KG) reads,

+λ1(u·∂)2+λ2(v·∂)2+µ2φ(x)=0.(II.2)

The energy-momentum tensor can be obtained from Eq. (II.1) using the standard deﬁnition,

which is given by:

Tµν =∂L

∂(∂µφ)(∂νφ)−ηµν L.(II.3)

So, it reads,

Tµν = (∂µφ)(∂νφ) + λ1uµ(∂νφ)(u·∂φ) + λ2vµ(∂νφ)(v·∂φ)−ηµν L.(II.4)

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摘要：

ScalarCasimirEectsinaLorentzViolationScenarioInducedbythePresenceofConstantVectorsE.R.BezerradeMello1,andM.B.Cruz2,y1DepartamentodeFísica,UniversidadeFederaldaParaíbaCaixaPostal5008,58051-970,JoãoPessoa,Paraíba,Brazil2CentrodeCiênciasExataseSociaisAplicadas,UniversidadeEstadualdaParaíbaCEP58706-55...

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Scalar Casimir Eﬀects in a Lorentz Violation Scenario Induced by the Presence of Constant Vectors E. R. Bezerra de Mello1and M. B. Cruz2y

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