Time-Reversal Operator

Operator

Time reversal is represented by an antiunitary operator $\mathsf T$. Its defining algebraic feature is

$$\mathsf T i\mathsf T^{-1} = -i.$$

For spinless wavefunctions in a real position representation, a common convention is complex conjugation:

$$\mathsf T=K, \qquad (K\psi)(\mathbf r)=\psi(\mathbf r)^*.$$

For spin $1/2$, a common convention is

$$\mathsf T = -i\sigma_yK, \qquad \mathsf T^2=-I.$$

Time reversal flips momentum and spin:

$$\mathsf T\mathbf p\mathsf T^{-1} = -\mathbf p, \qquad \mathsf T\mathbf S\mathsf T^{-1} = -\mathbf S.$$

Assumptions

• The representation and spin convention are specified.
• Antiunitary means antilinear and norm preserving.
• A Hamiltonian is time-reversal symmetric only when the appropriate transformed dynamics is invariant.
• Magnetic fields and other time-odd backgrounds must be transformed too when testing symmetry.

Common Mistakes

• Treating time reversal as an ordinary unitary operator.
• Forgetting complex conjugation when applying $\mathsf T$ to phases.
• Testing time-reversal symmetry while holding a time-odd external magnetic field fixed.
• Assuming $\mathsf T^2=+I$ for spin-$1/2$ systems.

Canonical Links

• Time Reversal
• Antiunitary Time Reversal
• Time Reversal for Spinless Particles
• Antiunitary Symmetries, First Look
• Antiunitary Symmetries
• Kramers Degeneracy
• Parity

References

• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.