Covariant Quantum Mechanics and Quantum Spacetime

We present in the article the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg-Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors under the Lorentz symmetry. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group $C^*$-algebra giving the algebra of observables. The key feature of the formulation is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product. The associated covariant harmonic oscillator Fock state basis has all the standard properties in exact analog to those of a harmonic oscillator with Euclidean position and momentum operators of any `dimension'. Galilean limit of the Lorentz symmetry and the classical limit of the Lorentz covariant framework are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space, given both in terms of real/complex number coordinates and noncommutative operator coordinates. The latter gives an explicit picture of the (projective) Hilbert space as a quantum/noncommutative spacetime.
Comment: 35 pages