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.

Graßmannian integrals in Minkowski signature, amplitudes, and integrability

“…For this purpose, we need to introduce some basics of two classes of u(p, p|r+s) representations. In the special case of su(2, 2), these are well-known representations of the conformal algebra [41,42] and the two classes correspond to positive and negative energies. In general, they are equivalent to certain oscillator representations [43,44], which we will encounter in section 5.…”

Unitary representations of three dimensional Lie groups revisited: A short tutorial via harmonic functions

“…[41] for a suitable scalar product for this case). This will however not constitute a serious constraint, as we will show that the corresponding scalar product adapted to the suitable real form of S 3 C \ C2 × C2 will circumvent this difficulty.…”

“…[2,38,39,40] and references therein), and are characterised by two numbers ℓ0, ℓ1, whereas the pair [ℓ0, ℓ1] denotes the representation. Explicitly, they are given by the set of functions [41] …”

“…are orthogonal with respect to this scalar product [41]. The case of the complementary series is much more involved, having been analysed in detail in [8,33,34], where explicit constructions can be found.…”

Generalized harmonic functions and unitary representations of low dimensional Lie groups

“…This case will not be considered here (see [23] for the details). Irreducible representations of SL(2, C) have been studied by Gel'fand and can be given in terms of homogenous functions in C 2 [2,24,25,26,27] (see [28] for an English translation of [24]). Unitary representations are characterised by two numbers 0 and 1 and correspond either to 0 ∈ 1 2 N and 1 = iσ, σ ∈ R for the principal series or to 0 = 0 and 0 < 1 ≤ 1 for the complementary series.…”