Abstract
The Lorentz symmetry and the space and time translational symmetry are fundamental symmetries of nature. Crystals are the manifestation of the continuous space translational symmetry being spontaneously broken into a discrete one. We argue that, following the space translational symmetry, the continuous Lorentz symmetry should also be broken into a discrete one, which further implies that the continuous time translational symmetry is broken into a discrete one. We deduce all the possible discrete Lorentz and discrete time translational symmetries in 1+1-dimensional spacetime, and show how to build a field theory or a lattice field theory that has these symmetries.
Highlights
Symmetry plays an important role in modern physics
The crystals are classified by the point group and the space group according to their symmetry under rotation, translation and reflection[1], the Lorentz symmetry and its generalization the Poincare symmetry are the basic of the relativistic quantum field theory[2], the discovery of the violation of parity symmetry[3,4] improves our understanding of weak interaction, and the particle-hole, the time reversal and the chiral symmetry are used to classify different topological insulators and topological superconductors[5], to name just a few
We propose a theory about the Lorentz and Poincare symmetries in a spacetime with discrete space translational symmetry based on three hypotheses
Summary
Symmetry plays an important role in modern physics. It imposes a constraint on the physical laws and reduces the number of candidate theories describing nature. The continuous Poincare group consists of spatial and temporal translations of arbitrary distance and Lorentz transformations of arbitrary velocity[7], which is the symmetry group of a relativistic field theory but not the symmetry group of crystals. The continuous time translational symmetry should be spontaneously broken into a discrete one to be compatible with the discrete Lorentz symmetry. In 2012, Wilczek et al.[8,9] proposed a theory about the spontaneous breaking of the continuous time translational symmetry into a discrete one The matter with such a broken symmetry is dubbed a ”time crystal”.
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