  In this paper, we develop the quantum theory of particles that has discrete
Poincar\'{e} symmetry on the one-dimensional Bravais lattice. We review the
recently discovered discrete Lorentz symmetry, which is the unique Lorentz
symmetry that coexists with the discrete space translational symmetry on a
Bravais lattice. The discrete Lorentz transformations and spacetime
translations form the discrete Poincar\'{e} 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 functions of the lattice
theory. The spacetime points where the Green's function is nonzero display a
lattice structure. During the propagation, the particles stay localized on a
single or a few sites to preserve the Lorentz symmetry.
