Non-compact symmetries in field theories with indefinite metric

Abstract

We motivate the study of non-compact internal symmetries appearing in some renormalizable field theories by the indefinite metric method. The use of the indefinite metric formalism in constructing a positive-definite physical Hilbert space is illustrated for the non-compact free fields and the non-compact sigma models. We discuss the problems associated with non-compact symmetries occuring in these theories. The notion that positivity of the quantum hamiltonian for a non-compact theory must be based on its classical field theory analogue must be abandoned. For the non-compact O(1, N)/O( N) sigma model we show how to obtain a unitary S-matrix in the physical subspace from a pseudo-unitary S-matrix in the overall Hilbert space, as well as the feature of dynamical mass generation and asymptotic freedom behavior. For the non-compact CP N sigma model we discuss the appearance of a dynamically generated composite gauge boson without external input. The prospect of composite gauge bosons for supergravity theories is assessed. Whenever possible, we compare and discuss the features of compact versus non-compact symmetries in a field theory.