Explicit form for the most general Lorentz transformation revisited

Abstract

Explicit formulae for the $4\times 4$ 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 nonzero 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 $4\times 4$ real matrix $A$, where $A$ is a product of the diagonal matrix ${\rm diag}(+1, -1, -1, -1)$ and an arbitrary $4\times 4$ real antisymmetric matrix. The formula for $\exp A$ 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 $\eta^\dagger\overline{\sigma}^{\lower3pt\hbox{$\scriptstyle \mu$}}\chi$ transforms as a Lorentz four-vector, where $\chi$ and $\eta$ are two-component spinors. The advantage of the latter derivation is that the corresponding formula for a general Lorentz transformation $\Lambda$ reduces to the computation of the trace of a product of $2\times 2$ matrices. Both computations are shown to yield equivalent expressions for $\Lambda$.