Abstract
An extensive group-theoretical treatment of linear relativistic field equations on Minkowski spacetime of arbitrary dimension D\geqslant 3D≥3 is presented. An exhaustive treatment is performed of the two most important classes of unitary irreducible representations of the Poincar'e group, corresponding to massive and massless fundamental particles. Covariant field equations are given for each unitary irreducible representation of the Poincar'e group with non-negative mass-squared.
Highlights
The representation cannot be faithful, so that it is trivial. (A different proof of the second part of the theorem may be found in the section 8.1.B of [2].). Another mathematical result which is of physical significance is the following theorem on unitary irreducible representations (UIRs) of compact Lie groups
Summary: On the one hand, the rules of quantum mechanics imply that quantum symmetries correspond to unitary representations of the symmetry group carried by the Hilbert space of physical states
If time translations constitute a one-parameter subgroup of the symmetry group, the Schrödinger equation for the time evolution of a state vector essentially is a unitary representation of this subgroup
Summary
Elementary knowledge of the theory of Lie groups and their representations is assumed (see e.g. the textbooks [1, 2] or the lecture notes [3]). The basic definitions of the Lorentz and Poincaré groups together with some general facts on the theory of unitary representations are reviewed in order to fix the notation and settle down the prerequisites
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