Explicit Form for the Most General Lorentz Transformation Revisited

Abstract

Explicit formulae for the (Formula presented.) Lorentz transformation matrices corresponding to a pure boost and a pure three-dimensional rotation are very well known. Significantly less well known is the explicit formula for a general Lorentz transformation with arbitrary non-zero boost and rotation parameters. We revisit this more general formula by presenting two different derivations. The first derivation (which is somewhat simpler than previous ones appearing in the literature) evaluates the exponential of a (Formula presented.) real matrix A, where A is a product of the diagonal matrix (Formula presented.) and an arbitrary (Formula presented.) real antisymmetric matrix. The formula for (Formula presented.) depends only on the eigenvalues of A and makes use of the Lagrange interpolating polynomial. The second derivation exploits the observation that the spinor product (Formula presented.) transforms as a Lorentz four-vector, where (Formula presented.) and (Formula presented.) are two-component spinors. The advantage of the latter derivation is that the corresponding formula for a general Lorentz transformation (Formula presented.) reduces to the computation of the trace of a product of (Formula presented.) matrices. Both computations are shown to yield equivalent expressions for (Formula presented.).