Renormalizability of higher-spin theories

Abstract

We shall consider here the theories of elementary particles with spin S ⩾ 1. If we construct the theory of the particle with higher spint and use the representation (i, j) of the Lorentz group, where i ≠ 0 and j ≠ 0 we have to introduce subsidiary conditions, singling out the components having definite spin. Very good examples are given by the Fierz theory of integer spin 5, based on the representation (S/2, S/2) and the Rarita-Schwinger theory of particles with half-integer spin S + ½, based on the representation {(½, 0) ⊕ (0, ½)} ⊗ (S/2, S/2). The components of the representation (S/2, S/2) are described uniquely by the components of symmetric traceless tensor Uμ1μ2…μs: $$\Psi _{AB} = \tau _{AB}^{\mu _{1}...\mu _{s}}U_{\mu _{1}...\mu _{s}}$$ where μ i = 1, 2, 3, 4; A, B = (S/2), (S/2) − 1, …, − (S/2); and \(\tau _{AB}^{\mu _{1}...\mu _{s}}\) is a generalization of well-known Pauli matrices σ μ to arbitrary spin.1