Space-time Symmetries

Abstract

This chapter is devoted to the perhaps simplest kind of symmetry. The manner of deriving QM from CM presented above allows to equal the space-time symmetries of the quantum-mechanical state functions to that of the classical ones. On this basis, spatial and temporal symmetries of the wave functions are dealt with, (i), in general (Wigner’s theorem), (ii), w.r.t. space inversion (parity), (iii) external time periodicity (Floquet theory), (iv), spatial periodicity (Bloch’s theorem), (v), internal time periodicity (time crystals), (vi), combined space-time periodicity (choreographic crystals). The symmetry breaking of homogeneity of space during freezing and the corresponding Nambu-Goldstone modes1 are only mentioned here.

Keywords: Bloch’s theorem, Floquet theory, Hamiltonian, Limiting function, Localization, Parity symmetry, Periodically driven systems, Spatially periodic potential, Stationary-state function, Symmetry, Time crystal, Wave function, Weight function, Wigner’s theorem.