nLab time-reversal symmetry

Idea

We call something symmetric under time-reversal if, roughly speaking, it would look or behave the same way if time flowed backwards.

The precise mathematical formalization of this idea varies across the fields of classical and quantum physics, thermodynamics, probability theory, and other fields.

(…)

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 $T \colon F \mapsto \overline{F}$.

In other fields

For similar notions in probability theory, statistical physics and dynamical systems, see time-reversible stochastic process and detailed balance.

Related concepts

• CPT theorem

• antiunitary operator

• Wigner's theorem

• time-reversible stochastic process

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

(…)

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