Majorana Representations of the Lorentz Group and Infinite-Component Fields

Abstract

A self-contained exposition is given of the theory of infinite-component fields with special emphasis on fields transforming under the Majorana representations of the Lorentz group, for which the scalar vertex function is written down explicitly for particles with arbitrary momenta and spins. The problem of spin and statistics for such fields is analyzed. A class of coupled representations of SL(2, C) is studied, containing unitary as well as nonunitary representations (including the Dirac 4-component spinors), for which invariant first-order equations can be written down for the free field. Most of the results are known (either from old or from recent publications), but are presented here in a unified way, their derivation sometimes being simplified. Among the few new points we mention: (1) The location of singularities of the matrix elements of some infinite-dimensional representations of SL(2, C) for complex values of the group parameters. (2) The construction of an infinite-component local Fermi field transforming under a unitary representation of SL(2, C). (3) the discussion of the quantization of Majorana fields with a proper account of the Fourier components with spacelike momenta.