Poincaré crystal on the one-dimensional lattice

In this paper, we develop the quantum theory that has discrete Poincaré symmetry on the one-dimensional Bravais lattice. We review the recently discovered discrete Lorentz symmetry which coexists with the discrete space translational symmetry on a Bravais lattice. The discrete Lorentz transformations and spacetime translations form the discrete Poincaré group, which are represented by unitary operators in a quantum theory. We find the conditions for the existence of representation, which are expressed as the congruence relation between quasi-momentum and quasi-energy. We then build the Lorentz-invariant many-body theory of indistinguishable particles by expressing both the unitary operators and Floquet Hamiltonians in terms of the field operators. Some typical Hamiltonians include the long-range hopping which fluctuates as the distance between sites increases. We calculate the Green's function of the theory. The spacetime points where the Green's function is nonzero display a lattice structure. During the propagation, the particle stays localized on a single or a few sites to preserve the Lorentz symmetry. (paper)