Phase-space approach to relativistic quantum mechanics. II. Geometrical aspects

Abstract

The formalism introduced in Paper I [J. Math. Phys. 18, 952 (1977)] is made manifestly covariant by including as an admissible phase space any 2n-dimensional submanifold of the forward tube 𝒯={x−iy∈Cn+1‖ y0≳‖y‖} which is of the ’’product’’ form σ=S−iΩλ, where Ωλ ={y∈Rn+1‖ y0= (λ2+y2)1/2}, λ≳0, and S is any space-or-lightlike submanifold of space–time Rn+1. The σ’s have natural symplectic structures covariant with respect to the Poincaré group, and a norm ∥¨∥σ on the space 𝒦 of solutions is defined by integrating with respect to the Liouville measure on σ. This automatically gives ∥f∥2σ as the total flux of a conserved space–time vector field, implying that ∥f∥σ is independent of σ. Some inconsistencies encountered in the space–time theory of Klein–Gordon particles appear to be resolved in the phase-space framework.