Zero Mass Field Quantization and Kibble's Long - Force Criterion for the Goldstone Theorem.

The central theme of the dissertation is an investigation of the "long-range force" criterion used by Kibble in his discussion of the Goldstone Theorem. This investigation is broken up into the following sections:. I. Introduction. Spontaneous symmetry breaking, the Goldstone Theorem and the conditions under which it holds are discussed. The Higgs Mechanism is introduced as the chief method by which those conditions can be avoided. This motivates a discussion of Kibble's criterion for the Goldstone Theorem, as contrasted with the requirement of "manifest covariance" which is invoked in the theorem's more abstract proofs by spectral decomposition. II. Massless Wave Expansions. In order to make explicit calculations of the operator commutators used in applying Kibble's criterion, it is necessary to work out the operator expansions for a massless field. Unusual results are obtained which include operators corresponding to classical macroscopic field modes. III. The Kibble Criterion for Simple Models Exhibiting Spontaneously Broken Symmetries. The results of the previous section are applied to simple models with spontaneously broken symmetries, namely, the real scalar massless field and the Goldstone model without gauge coupling. It is shown that the commutators used by Kibble do not exhibit the characteristics expected of a long-range force. Nevertheless, it is shown that Kibble's criterion is satisfied, for reasons unrelated to the range of the force. A general argument is given to show that this phenomenon will occur in any model in which the Lagrangian does not contain derivative couplings. IV. The Higgs Mechanism in Classical Field Theory. The results of the previous section motivate a discussion of the applicability and physical meaning of Kibble's criterion to models which exhibit the Higgs Mechanism, (and which therefore cannot satisfy the requirements for the Goldstone Theorem). It is shown that the Higgs Mechanism has a simple interpretation in terms of classical field theory, namely, that it arises from a derivative coupling term between the Goldstone fields and the gauge fields. V. The Higgs Mechanism and Kibble's Criterion. This section draws together the material discussed in sections II - IV. Explicit calculations are made to evaluate Kibble's criterion on a Goldstone-Higgs type of model in the Coulomb gauge. It is found, as expected, that the criterion is not met, but not for reasons relating to the range of the mediating force. By referring to the findings of sections III and IV, it is concluded that the common denominator underlying both the Higgs Mechanism and the failure of Kibble's criterion is a structural aspect of the field equations: derivative coupling between fields.