Composition of Lorentz Transformations in Terms of Their Generators

TL;DR

This paper derives a covariant, general formula for composing Lorentz transformations using their generators, simplifying calculations for specific subalgebras in Minkowski space-time.

Contribution

It provides a covariant, explicit BCH formula for Lorentz transformations in terms of their generators, including simplified forms for subalgebras.

Findings

Derived a covariant BCH formula for Lorentz transformations.

Simplified the formula for subalgebras generated by a single pair of generators.

Provided explicit expressions for all subalgebras.

Abstract

Two-forms in Minkowski space-time may be considered as generators of Lorentz transformations. Here, the covariant and general expression for the composition law (Baker-Campbell-Hausdorff formula) of two Lorentz transformations in terms of their generators is obtained. Every subalgebra of the Lorentz algebra of such generators, up to one, may be generated by a sole pair of generators. When the subalgebra is known, the above BCH formula for the two two-forms simplifies. Its simplified expressions for all such subalgebras are also given.