Supersymmetric quantum field theories from induced representations

Abstract

This thesis investigates the application of the method of induced representations in supersymmetric quantum field theories. First, the main features of the theory of induced representations of Lie groups, Lie algebras and Lie superalgebras are presented. The procedure for obtaining irreducible representations in the important case of Lie groups or algebras with an invariant, Abelian subgroup or subalgebra is described. This procedure is then applied to the Poincar6 group for arbitrary dimensions of space-time. The representations obtained are used to construct free quantum fields, without the use of a Lagrangian. The usual characteristics of these fields, such as the field equations, are shown to be consequences of the representation theory. The induced representation procedure for algebras is demonstrated by the construction of the irreducible supermultiplets for the N = 1 Poincaré superalgebra (in those dimensions for which it exists). Again, the construction proceeds from the representations, not from a Lagrangian. Finally, a mixture of the group and algebra versions of the procedure developed in the preceding parts of the thesis is applied to the inhomogeneous orthosymplectic superalgebra. This superalgebra is relevant to BRST covariant quantisation of gauge fields. A consequence of the systematic application of the representation theory is the derivation of the Parisi-Sourlas mechanism in pseudo-Euclidean space.