Advances in High Energy Physics

Volume 2019, Article ID 2729352, 7 pages

https://doi.org/10.1155/2019/2729352

## Klein–Gordon Oscillator in a Topologically Nontrivial Space-Time

Departamento de Física, CFM, Universidade Federal de Santa Catarina, CP 476, 88.040-900 Florianópolis, SC, Brazil

Correspondence should be addressed to L. C. N. Santos; rb.csfu@sotnas.siul

Received 10 January 2019; Revised 2 March 2019; Accepted 10 April 2019; Published 13 May 2019

Academic Editor: Diego Saez-Chillon Gomez

Copyright © 2019 L. C. N. Santos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

In this study, we analyze solutions of the wave equation for scalar particles in a space-time with nontrivial topology. Solutions for the Klein–Gordon oscillator are found considering two configurations of this space-time. In the first one, the space is assumed where the metric is written in the usual inertial frame of reference. In the second case, we consider a rotating reference frame adapted to the circle . We obtained compact expressions for the energy spectrum and for the particles wave functions in both configurations. Additionally, we show that the energy spectrum of the solution associated with the rotating system has an additional term that breaks the symmetry around .

#### 1. Introduction

The local and the global structures of space-time play an important role in the behavior of quantum systems. In this aspect, it is believed that global features of space-time may be directly related to the shift of energy levels of quantum particles. In the particular case of the (time)(space) space-time, which is locally flat but which has a nontrivial topology, one may consider the effect of periodic boundary conditions in one spatial direction. In this space-time we have one compactified spacelike dimension; thus it is expected that despite the flat geometry, measurable effects occur in observable quantities. Nontrivial space-times have been studied extensively in literature; interesting applications are found in the study of atomic Bose-Einstein condensates [1] with toroidal optical dipole traps. In the context of quantum field theory, vacuum polarization in a nonsimply connected space-time with the topology of is considered in [2]. It was found that the vacuum energy for a free spinor field in twisted and untwisted configurations is different in space.

On the other hand, in quantum mechanics the harmonic oscillator is one of the most significant systems to be studied. In recent years, the relativistic version of the harmonic oscillator has been considered in several studies [3–16]. This important potential has been introduced as a linear interaction in the Klein–Gordon equation [17]. In the case of the Dirac equation, the so-called Dirac oscillator has been introduced as an instance of a relativistic potential such that its nonrelativistic limit leads to the harmonic oscillator plus a strong spin-orbit coupling [18]; this result is similar to the one that is obtained for the Klein–Gordon oscillator when the spin-orbit is absent. Most recently, the relativistic harmonic oscillator has been studied in the context of the Kaluza–Klein theory [19], where the Klein–Gordon oscillator coupled to a series of cosmic strings in five dimensions has been considered. In [20, 21] the author considers the effect of such kind of topological defect on scalar bosons described by the Duffin–Kemmer–Petiau (DKP) formalism. The Klein–Gordon oscillator in a noncommutative phase space under a uniform magnetic field has been studied in [22]. In this paper the authors conclude that the Klein–Gordon oscillator in a noncommutative space with a uniform magnetic field has behavior similar to the Landau problem in the usual space-time.

Another aspect of interest in our work is the influence of noninertial effects on quantum systems. As in classical physics, quantum mechanics is sensitive to the use of noninertial reference systems. These effects can be taken into account through an appropriate coordinate transformation. Previous research reported in literature [23, 24] shows that rotating frames in the Minkowski space-time can play the role of a hard-wall potential. Recently these ideas have been applied to the case of spaces with nontrivial topology. In particular, a rotating system was proposed recently in [25] where a scalar field on a circle (topology ) with a Dirichlet cut has been considered. In [26], a similar study was carried out in the case of a five-dimensional space-time.

Therefore, in this contribution, we will study bosons in the space-time by considering the scalar wave equation for the Klein–Gordon oscillator. In fact, solutions of wave equations in curved spaces and nontrivial topology have been explored in various contexts [16, 26–37]. We will examine the combination of the Klein–Gordon oscillator and a space with nontrivial topology. Afterwards, a rotating frame in the space-time will be considered. We will show that the oscillator potential can form bound states for the Klein–Gordon equation in this space-time, and beyond that the momentum associated with the nontrivial topology is discrete. This is an expected result, since the topology of space is associated with the periodicity of the boundary conditions. In the case of a rotating frame in the space-time, we will see interesting results associated with noninertial effects: the energy levels are shifted and the region of the space-time where the particle can be placed is restricted.

This work is organized as follows: In Section 2, we will study the space-time metric with a nontrivial topology and define a coordinate transformation that connects it to a rotating frame. In Section 3, we will derive the Klein–Gordon (KG) equation with a potential of the harmonic oscillator type and solve the associated differential equation. Similarly, we will solve again the KG equation in Section 4 but we will consider a noninertial frame. Finally, we will present our conclusions in Section 5.

#### 2. Nontrivial Space-Time Topology and Noninertial Reference Frame

In this section, we define the line element that describes the space-time geometry in agreement with the proposal of this work. We want to study the behavior of massive scalar fields (zero spin particles) under the influence of a gravitational field generated by a space-time with the nontrivial topology . In this geometry, represents the usual uncompactified space-time directions, and is an compactified dimension. We discuss the relationship between and the effects for a rotational frame inserted in that scenario. Figure 1 shows a representation of this space-time where the temporal coordinate is absent.