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Physics Letters A

Volume 450, 31 October 2022, 128371
Physics Letters A

Supersymmetric relativistic quantum mechanics in time-domain

https://doi.org/10.1016/j.physleta.2022.128371Get rights and content

Highlights

  • •

    A supersymmetric theory in a temporal domain can be consistently formulated for relativistic quantum particles.

  • •

    This is a new idea that relies in the possible time-dependent mass of particles.

  • •

    We apply this theory to the problem of two neutrino oscillations, using a time dependence of neutrino mass.

  • •

    It is reproduced the well-known non-relativistic result for such oscillation, with correct numerical observational values.

Abstract

A supersymmetric relativistic quantum theory in the temporal domain is developed for bi-spinor fields satisfying the Dirac equation. The simplest time-domain supersymmetric theory can be postulated for fields with time-dependent mass, showing an equivalence with the bosonic supersymmetric theory in time-domain. Solutions are presented and they are used to produce probability oscillations between mass states. As an application of this idea, we study the two-neutrino oscillation problem, showing that flavour state oscillations may emerge from the supersymmetry originated by the time-dependence of the unique mass of the neutrino.

Introduction

Supersymmetry is one of the cornerstones of theoretical physics [1], being ubiquitous in almost every branch of physics [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34]. The standard way to proceed is to construct supersymmetric theories in a space-domain, where the superpartners and supercharge operators take into account the spatial variations of, for example, external potentials. In this work, it is not our aim to focus in those spatial supersymmetries, but instead to inquire if a temporal version of such theories is possible for massive fields in relativistic quantum mechanics.

Recently, in Refs. [33], [34], it was introduced the concept of supersymmetry in time-domain (T-SUSY) for Maxwell equations. This is a supersymmetry occurring in the temporal part of the massless field dynamics, completely uncoupled from the spatial evolution of the field. In Ref. [34] is studied the applications of T-SUSY in the realm of optics for dispersive media, showing the novel capabilities of this theory to introduce hypothetical materials with new optical features. Besides, they showed that this time-domain supersymmetry applies to any field described, in principle, by a d'Alambertian equation. In other words, they developed the bosonic version of the T-SUSY theory.

Supersymmetric theories for purely bosonic systems have been extensively developed [20], [21], [22]. On the contrary, it is the purpose of this manuscript to show that a T-SUSY theory can also be obtained in relativistic quantum mechanics for fields described by the Dirac equation, thus complementing the usual spatial supersymmetry theory [35]. In this case, we show that the simplest T-SUSY theory may be constructed for time-dependent massive fields satisfying Dirac equation, finding solutions for different possible time-dependent masses. Also, this theory is equivalent to its bosonic partner, allowing us to obtain a massive particle field behaviour that is analogue to a light-like one.

Besides, this theory produces probability states oscillations as a consequence of its supersymmetry. Thus, it can be used to study the neutrino oscillation problem. In this way, we give a different perspective to the origin of neutrino oscillations through supersymmetry in time-domain.

Section snippets

T-SUSY for Dirac equation

Let us consider a bi-spinor field Ψ satisfying the Dirac equation in flat spacetime for a field with mass m,iγμ∂μΨ=mΨ, where ∂μ are the covariant derivatives. Here, γμ are the gamma matrices in the Dirac representation, fulfilling γμγν+γνγμ=2ημν14×4, with the flat spacetime metric ημν=diag(1,−1,−1,−1), and with the identity matrix 14×4. The cases with external potential are discussed in the last section, but for now it is enough to consider the T-SUSY theory for Eq. (1).

In the following

Simple solutions

We can get simple solutions for particle fields with time-dependent mass by first considering k⋘m(t). For this case, from Eq. (16), we findE±(t)≈∓m(t). Therefore, the two mass states have different behaviour stemming from just one time-dependent mass.

Similarly, another simple solution can be obtained in the light-mass (ultra-relativistic) case, when k≫m(t). For this case, k corresponds (approximately) to the energy of the particle. In this case, from Eq. (16), we can find an approximated form

Oscillations in T-SUSY

The above T-SUSY theory allows now to consider the phenomenon of oscillation of states in a different fashion. These T-SUSY oscillations have their origin in the subjacent supersymmetry due to the temporal dependence of the mass. In order to obtain these physical states, let us return to the physical time t from Eq. (1), through an inverse Wick rotation applied to the previous solutions.

Considering the bi-spinor (2), let us now define a bi-spinor with mixed statesΦ=(ΦaΦb)=(cos⁡θ12×2sin⁡θ12×2−sin

Discussion

When the mass of a relativistic quantum mechanical field is time-dependent, then a supersymmetric in time-domain theory may be constructed. This T-SUSY is the time analogue of any relativistic supersymmetric quantum theory. Along this work, we have presented solutions for different time-dependent masses. A remarkable outcome is that this theory contains solutions that mimic the light-like behaviour studied for the bosonic T-SUSY theory.

One of the main results extracted here is the possibility

CRediT authorship contribution statement

Felipe A. Asenjo: Writing – review & editing, Writing – original draft, Visualization. Sergio A. Hojman: Writing – review & editing, Writing – original draft, Methodology. Héctor M. Moya-Cessa: Writing – review & editing, Writing – original draft, Methodology. Francisco Soto-Eguibar: Writing – review & editing, Writing – original draft, Methodology.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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