Group manifold analysis of the structure of relativistic quantum dynamics
In the paper a group law is given from which a relativistic quantum system with an invariant evolution parameter is obtained, the position operator belonging to the Lie algebra of the group. The motivation of the procedure is exposed in the introduction, mentioning the main difficulties involved. Section 2 presents a brief account of the relativistic quantum dynamics, without the rest mass energy, derived from the pseudoextensions P\({\tilde \otimes}U(1)\) of Poincaré group P by U(1). The breaking of the group structure by the introduction of the rest mass is analyzed. Section 3 gives the group \(G_{15}\) (including P) as a contraction of the conformal group, the additional parameters leading to the covariant position operators. Section 4 applies the quantization formalism to the central extension \(\tilde G_{16}\) of \(G_{15}\) by U(1). Section 5 studies the restriction to the mass shell when the quantization 1-form corresponding to the Klein-Gordon equation may be derived from the quantum group \(\tilde G_{16}\). An action of P is generated defining an orbit in the former group manifold. Section 6 contains final remarks on possible extensions of the work in the spirit of Klein's Erlangen program. An appendix summarizes the main formulae on P used in the text.
scientific article; zbMATH DE number 3944894
scientific article