论文标题
在（3+1）和（1+1）尺寸的主要粒子的波程上
On wave equations for the Majorana particle in (3+1) and (1+1) dimensions
论文作者
论文摘要
通常，被认为是数学上描述所谓的主要粒子的相对论波方程是具有真实Lorentz标量电势的Dirac方程以及所谓的Majorana条件。当然，取决于人使用的表示形式，所产生的微分方程会改变。它可能是耦合方程式的真实或复杂的系统，也可能是整个波函数的单个组件的单个复杂方程。这些方程式或方程系统中的任何一个都可以称为Majorana方程或方程式系统，因为它可用于描述Majoraana粒子。例如，在WEYL表示中，在（3+1）维度中，我们可以具有两个非等效的显式协变的复合式一阶方程。相反，在（1+1）维度中，我们具有复杂的耦合方程系统。在任何情况下，无论使用哪个方程式或方程式系统，描述（3+1）或（1+1）尺寸的主要粒子的波函数由四个或两个实际数量确定。本文的目的是从代数的角度研究和讨论所有这些问题，强调在（3+1）和（1+1）dirac，weyl和majorana表示的情况下，这些方程之间出现的相似性和差异。此外，为了加强这项任务，我们重新绘制并使用来自案例已经引入的过程的结果，以在（3+1）维度中获得两个组件的Majorana方程。同样，我们在（1+1）维度中首次介绍了一些类似的过程，然后使用我们获得的结果。
In general, the relativistic wave equation considered to mathematically describe the so-called Majorana particle is the Dirac equation with a real Lorentz scalar potential plus the so-called Majorana condition. Certainly, depending on the representation that one uses, the resulting differential equation changes. It could be a real or a complex system of coupled equations, or it could even be a single complex equation for a single component of the entire wave function. Any of these equations or systems of equations could be referred to as a Majorana equation or Majorana system of equations because it can be used to describe the Majorana particle. For example, in the Weyl representation, in (3+1) dimensions, we can have two non-equivalent explicitly covariant complex first-order equations; in contrast, in (1+1) dimensions, we have a complex system of coupled equations. In any case, whichever equation or system of equations is used, the wave function that describes the Majorana particle in (3+1) or (1+1) dimensions is determined by four or two real quantities. The aim of this paper is to study and discuss all these issues from an algebraic point of view, highlighting the similarities and differences that arise between these equations in the cases of (3+1) and (1+1) dimensions in the Dirac, Weyl, and Majorana representations. Additionally, to reinforce this task, we rederive and use results that come from a procedure already introduced by Case to obtain a two-component Majorana equation in (3+1) dimensions. Likewise, we introduce for the first time a somewhat analogous procedure in (1+1) dimensions and then use the results we obtain.
