Remarks on the Algebraic Structure of Spontaneous Symmetry Breaking in Unified Gauge Theories

Abstract

One of the most interesting recent developments in physics, and indeed in many other branches of science too, has been the gradual realization of the importance and universality of spontaneous symmetry breaking. For example, three out of four of this mornings lectures dealt with the topic of spontaneous symmetry breaking in three different areas--mathematics, physics and biophysics. Within Physics, spontaneous symmetry breaking plays an important role in many phenomena--magnetism, crystallography and superconductivity for example are just macroscopic manifestations of spontaneous symmetry breaking for microscopic systems--and there are even examples in classical physics such as the Jacobi ellipsoids.¹ However, the subject of this talk will be the appearance of spontaneous symmetry breaking in particle physics, in particular in the context of unified gauge theories.² This context is, perhaps, particularly appropriate for an Einstein symposium because unified gauge theory represents a major step towards the realization of Einstein’s dream of unifying the fundamental interactions by means of a single universal principle. It is also, perhaps, appropriate to note that 1979 is not only the centenary of Einstein’s birth but is also the fiftieth anniversary of the publication of a paper by Hermann Weyl which was the forerunner of unified gauge theory.³ In this paper Weyl showed that the Maxwell-Lorentz theory of electromagnetism was not only gauge-invariant but could be derived directly and simply from the principles of gauge theory, using the gauge group U(1). Present-day unified gauge theories are essentially the extension of Weyl’s principles to arbitrary compact Lie gauge groups--together with spontaneous symmetry breaking.