Riemann-Einstein Structure from Volume and Gauge Symmetry

It is shown how a metric structure can be induced in a simple way starting with a gauge structure and a preferred volume, by spontaneous symmetry breaking. A polynomial action, including coupling to matter, is constructed for the symmetric phase. It is argued that assuming a preferred volume, in the context of a metric theory, induces only a limited modification of the theory. §1 Attempts to Use a Gauge Principle for Gravity The fundamental laws of physics, as presently understood, largely follow from the two powerful symmetry principles: general covariance, and gauge invariance. There have been many attempts to unite these two principles. Indeed the very term, gauge invariance, originates from an early attempt [1] by Weyl to derive classical electromagnetism as a consequence of space-time symmetry, specifically symmetry under local changes of length scale. The modern understanding of gauge invariance, as a symmetry under transformations of quantum-mechanical wave functions, was reached by Weyl himself and also by London very shortly after the new quantum mechanics was first proposed. In this understanding of abelian gauge invariance, and in its nonabelian generalization [2], the space-time aspect is lost. The gauge transformations act only on internal variables. This formulation has had great practical success. Still, it is not entirely satisfactory to have two closely related, yet definitely distinct, fundamental principles, and several physicists have proposed ways to unite them. 1 One line of thought, beginning with Kaluza [3] and Klein [4], seeks to submerge gauge symmetry into general covariance. Its leading idea is that gauge symmetry arises as a reflection in the four familiar macroscopic space-time dimensions of general covariance in a larger number of dimensions, several of which are postulated to be small, presumably for dynamical reasons. Here we should take the opportunity to emphasize a point that is somewhat confused by the historically standard usages, but which it is vital to have clear for what follows. When physicists refer to general covariance, they usually mean the form-invariance of physical laws under coordinate transformations following the usual laws of tensor calculus, including the transformation of a given, preferred metric tensor. Without a metric tensor, one cannot form an action principle in the normal way, nor in particular formulate the accepted fundamental laws of physics, viz. general relativity and the Standard Model. From a purely mathematical point of view one might consider doing without the metric tensor; in that case general covariance becomes essentially the same concept as topological invariance. The existence of a metric tensor reduces the genuine symmetry to a much smaller one, in which space-times are required not merely to be topologically the same, but congruent (isometric), in order to be considered equivalent. In the Kaluza-Klein construction, for this reason, the gauge symmetries arise only from isometries of the compactified dimensions. Another line of thought proceeds in the opposite direction, seeking to realize general covariance – in the metric sense – as a gauge symmetry. A formal parallelism is readily perceived [5]. Indeed, it is quite helpful, in bringing spinors to curved space, to introduce a local gauge SO(3, 1) symmetry . The spin connection ω μ is the gauge potential of this symmetry. The lower index is a space-time index, and the upper indices are internal SO(3, 1) indices, in which ω is antisymmetric, as is appropriate for the gauge potential. In this construction one must, however, introduce an additional element that is not a feature of conventional gauge theories. That is, one postulates in addition the existence of a set of For definiteness I consider 3+1 dimensional space here.