Lorentz transformations in the light front

Abstract

In undergraduate and graduate courses, one usually studies the Lorentz's transformation defined in four-dimensional space: three spatial dimensions and one temporal. In this work we present an alternative way for the Lorentz's transformations using the light front coordinates via Minkowski space-time and its properties. Dirac in one of his works, published in 1949, introduced three different forms of relativistic dynamics: the usual or most common one, called instant form, characterized by the hypersurface specified by the boundary conditions defined at $t=0$ and time evolution for the system at instances $t>0$; the point form, whose surface is a hyperboloid, and are described by the initial conditions $x^{\mu }{x}_{\mu }=a{}^2$ with the constant $a$ and $t>0$ and the light front form, whose hypersurface is tangent to the light cone, being defined by the initial conditions at $x^{+}=\frac{c}{\sqrt{2}}(t+\frac{z}{c})=0$ and the “time evolution” for “instances $x^{+}>0$”. Therefore, the variable $x^{+}$ plays the role of light-front “time”. The results we present here are connected to the study of Lorentz's transformations using light-cone coordinates and its consequences in the outcome expressions for diverse physical quantities of interest.

Keywords:
Lorentz Transform; Minkowski Space; Light Front