Volume 17: Pages 393-411, 2004

Uniqueness of Lorentz‐Covariant World‐Space Representations

V. Arunasalam

The uniqueness of the Lorentz‐covariant 4‐vector representations of the classical‐dynamical variables and the associated quantum‐mechanical Hermitian matrix operators in the world‐space or the Minkowski space or the (space‐time continuum) 4‐space is examined in some detail. Also, the intrinsic connections or links that exist among the world‐space, the associated canonically conjugate (momentum‐energy continuum) 4‐space, the corresponding (wave vector frequency continuum) Fourier transform 4‐space, and the (space‐time continuum) linear displacement differential operator 4‐space are also examined in sufficient detail both for zero and nonzero rest mass particles. The mapping among these four mathematically distinct 4‐spaces is one‐to‐one. The main emphasis here is on the manifestly Lorentz‐covariant formalisms rather than on the commonly used Lorentz‐invariant ones. The relativistic momentum‐energy (or, equivalently, the de Broglie wave vector frequency) 4‐vector “dispersion relations” of both the zero and nonzero rest mass particles clearly demonstrates the self‐consistency between wave mechanics and relativity.

Keywords: uniqueness of 4‐vectors, uniqueness of world vectors, Lorentzcovariant representations, space‐time continuum, momentum‐energy continuum, Fourier wave vector frequency continuum, relativistic momentum‐energy dispersion relation, relativistic de Broglie wave dispersion relations, zero and nonzero rest mass particles, self‐consistency between wave mechanics and relativity

Received: February 17, 2004; Published online: December 15, 2008