Lorentz Transformations

TL;DR

This paper provides a clear derivation of Lorentz group properties, including generators for rotations and boosts, and explores their mathematical structure and implications for transformations and spinor representations.

Contribution

It introduces a transparent derivation of Lorentz transformations using exponential maps and explores their properties without relying solely on commutation relations.

Findings

Derived explicit forms of Lorentz transformation matrices for finite rotations and boosts.

Showed how to infer higher order generators from lower order calculations.

Presented field representations and spinor transformation matrices for finite parameters.

Abstract

This paper describes a particularly didactic and transparent derivation of basic properties of the Lorentz group. The generators for rotations and boosts along an arbitrary direction, as well as their commutation relations, are written as functions of the unit vectors that define the axis of rotation or the direction of the boost (an approach that can be compared with the one that in electrodynamics, works with the electric and magnetic fields instead of the Maxwell stress tensor). For finite values of the angle of rotation or the boost's velocity, collectively denoted by V, the existence of an exponential expansion for the coordinate transformation's matrix, M (in terms of GV where G is the generator) requires that the matrix's derivative with respect to V, be equal to GM. This condition can only be satisfied if the transformation is additive as it is indeed the case for rotations, but not for velocities. If it is assumed, however, that for boosts such an expansion exists, with V = V(v), v being the velocity, and if the above condition is imposed on the boost's matrix then its expression in terms of hyperbolic cosh(V) and sinh(V} is recovered, and the expression for V(= arc tanh(v)) is determined. A general Lorentz transformation can be written as an exponential containing the sum of a rotation and a boost, which to first order is equal to the product of a boost with a rotation. The calculations of the second and third order terms show that the equations for the generators used in this paper, allow to reliably infer the expressions for the higher order generators, without having recourse to the commutation relations. The transformationmatrices for Weyl spinors are derived for finite values of the rotation and velocity, and field representations, leading to the expression for the angular momentum operator, are studied.