Quantization of some generally covariant model field theories

Abstract

This thesis reports a study of the quantization of generally covariant and nonlinear field theories. It begins by reviewing some existing general theories in Chapter 2 and Chapter 3, Chapter 2 deals with general classical theories while Chapter 3 examines various quantization schemes. The model field derived from the Lagrangian density

𝓵 = 1/4 ℇ[super]lkμλ (A [sub] k,t – A [sub] k,t)(A, {sub{ A[sub]μ,λ – A[sub]λ,μ).

is proposed in Chapter 4 especially for the study of general covariance. It is demonstrated that for this field general covariance alone does not appear to bring in anything physically new, a discussion is given on the differences between general covariance and Lorentz covariance. In subsequent chapters a generally covariant and nonlinear model field, a 4-surface of stationary 4-volume embedded in a 5-dimensional Pseudo-Euclidean space, is investigated. Firstly a manifestly covariant quantization programme is carried out. The model field is then examined in a special coordinate frame for the study of its nonlinearity. Various treatments of the intrinsic nonlinearity are examined starting with conventional perturbation theory in Chapter 6. The usual divergence problem in quantum field theory appears, in particular in the self-energy calculation of a one-particle state. A new variational method is proposed in Chapter 8 which is able to lead to finite results for one-particle states. The thesis is concluded with a chapter discussing some general problems involved and a chapter containing suggestions for further work.