Lectures on Broken Symmetry and Goldstone Boson [MatSciRep:27]

Abstract

This report aims to obtain nonperturbative solutions, with the 'self-consistent' method of approximation. First an indefinite value is assumed for some physical quantity. Then this quantity is calculated using the given value, and the result is equated with the given value; This procedure gives an equation for a value of the quantity. Using this method, Nambu Model, Goldstone Model, Nishijima's theory of weak interaction, are some of the examples concerned with broken symmetry. The mechanism of violation of symmetry is the inequivalence between symmetric and asymmetric states. Inequivalent Representation itself is rather familiar in Quantum Field Theory. In the case of symmetry violation there arises an inequivalence, due to the limiting processes to infinity of normalization volume. Actual volume which contains the total system is finite but it is very big compared with the wavelength. The inner product vectors belonging to inequivalent subspace is non-zero but infinitesimal for large but finite normalization volume. It concludes that "Although it is possible to understand the violation of symmetry in terms of massless boson, it is quite improbable that such kind of violation is causing the actual broken symmetry, which can be found in the law of elementary particles".