Transforming antiunitary symmetries to a normal form

Abstract

We look at explicit ways to bring one or two antiunitary symmetries into a standard form via unitary conjugation. We carefully reproduce Wigner's proof in two special cases, where the antiunitary operators square to $+I$, or to $-I$. Wigner's method is constructive and we show how leads to two algorithms to compute the needed unitaries in small examples. We then adapt these algorithms to deal with two such antiunitary matrices that commute up to a sign. This leads to a proof a finite-dimensional version of the well-known ten-fold way of topological physics. This will allow physicists to perform a change of basis on any finite-dimensional model in one of the Altland-Zirnbauer symmetry classes to a standardized version of that symmetry class in which time-reversal and and particle-hole symmetry are standard operations that can be implemented efficiently in standard numerical software. This will also make it easier to find formulas for some key isomorphisms between graded real $C^{*}$-algebras.