Abstract
We present 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. 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 is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. The language of pseudo-Hermitian quantum mechanics is adopted for a clear illustration of the aspect, with a metric operator obtained as really the manifestation of the Minkowski metric on the space of the state vectors. 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. Galilean limit and the classical limit are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space.
Highlights
The language of pseudo-Hermitian quantum mechanics is adopted for a clear illustration of the aspect, with a metric operator obtained as really the manifestation of the Minkowski metric on the space of the state vectors
Our group had implemented a, quantum relativity symmetry, group theoretical formulation of the full dynamical theory of the familiar quantum mechanics with rigorous classical limit given as the Newtonian theory, obtained through a contraction of the relativity symmetry applied to the specific representation [1]
A major part of that is needed to formulate the c → ∞ contraction. Each such component is shown to give essentially the same physical theory of covariant quantum mechanics we present in detail on the coherent state basis, in the abstract form and in wavefunctions, with the Lorentz invariant indefinite inner product
Summary
Our group had implemented a, quantum relativity symmetry, group theoretical formulation of the full dynamical theory of the familiar quantum mechanics with rigorous classical limit given as the Newtonian theory, obtained through a contraction of the relativity symmetry applied to the specific representation [1]. A major part of that is needed to formulate the c → ∞ contraction Each such component is shown to give essentially the same physical theory of covariant quantum mechanics we present in detail on the coherent state basis, in the abstract form and in wavefunctions, with the Lorentz invariant indefinite inner product. The latter is first devoted to the WWGM framework or the observable algebra, focusing on the symmetry transformations and the dynamics as a specific case of such a symmetry flow, with the real parameter characterizing transformation corresponding to an evolution parameter, which is taken as the proper time in the case.
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