nLab time-reversal symmetry

Idea

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For relativistc fermions

In the relativistic Dirac equation: Thaller 92, Thm. 3.10, cf. (1.26)

In solid state physics

Under the expected K-theory classification of topological phases of matter, time-reversal symmetry refers to that sector of twisted equivariant K-theory which is KR-theory, where the $\mathbb{Z}/2$ $\simeq \{\mathrm{E}, T\}$ acts

• on the Brillouin torus by momentum-reversal $T \,\colon\,k \mapstp -k$

and

• on the Hamiltonian and its related observables (specifically: the Fredholm operators $F$ whose space classifies K-theory) by complex conjugation $T \colon F \mapsto \overline{F}$.

Related concepts

• CPT theorem

• antiunitary operator

• Wigner's theorem

References

General

• Eugene P. Wigner, §26 in: Group theory: And its application to the quantum mechanics of atomic spectra, Academic Press (1959) [doi:978-0-12-750550-3]

In Relativistic Dirac theory

• Bernd Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer (1992) (doi:10.1007/978-3-662-02753-0)

In solid state physics

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(For the time being see the references at topological insulator etc.)