The earlier treatments of Lorentz covariant harmonic oscillator have brought to light various difficulties, such as reconciling Lorentz symmetry with the full Fock space, and divergence issues with their functional representations. We present here a full solution avoiding those problems.The complete set of Fock states is obtained, together with the corresponding explicit wavefunction and their inner product integrals free from any divergence problem and the Lorentz symmetry fully maintained without additional constraints imposed. By a simple choice of the pseudo-unitary representation of the underlying symmetry group, motivated from the perspective of the Minkowski spacetime as a representation for the Lorentz group, we obtain the natural non-unitary Fock space picture commonly considered though not formulated and presented in the careful details given here. From a direct derivation of the appropriate basis state wavefunctions of the finite-dimensional irreducible representations of the Lorentz symmetry, the relation between the latter and the Fock state wavefunctions is also explicitly shown. Moreover, the full picture including the states with non-positive norm may give consistent physics picture as a version of Lorentz covariant quantum mechanics. Probability interpretation for the usual von Neumann measurements is not a problem as all wavefunctions restricted to a definite value for the 'time' variable are just like those of the usual time independent quantum mechanics. A further understanding from a perspective of the dynamics from the symplectic geometry of the phase space is shortly discussed.
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.
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