Weyl's Vision

Weyl's Vision and Semantic Unfolding in the Conception of Physical Geometry: Local Gauge Symmetry, Interaction Potentials, and the Articulation of Matter

The crucial insight of Weyl, from our perspective of semantic unfolding, is that the localization of an internal symmetry in the description pertaining to a matter form leads to the introduction of gauge field potentials in order that the physical description remains invariant. In this manner, the semantic unfolding in the conception of physical geometry is initiated by localizing global symmetries of material forms.

Then, the requirement of invariance in the physical description under this process of localization always leads to the introduction of gauge field potentials transferring some particular form of interaction. In other words, Weyl's fundamental realization was that the localization of matter symmetries was a necessary condition for expressing both, the notion of an interaction field via its gauge potentials, and the minimal coupling of these potentials with the matter sources.

Localization of Symmetry and Gauge Potential

The localization of a symmetry simply means that it may vary independently from point to point of the base space, within the range of ambiguity determined by the corresponding symmetry group. Thus, a copy of the same symmetry group is replicated from point to point of the base space, in such a manner that at each of these points, a symmetry element is subsumed totally independently from all the others at any other point.

It is precisely due to this freedom of independent symmetry variation from point to point that a proper means of following this variation from the infinitesimal to the global is required. These means establish the standards of congruence according to some rule of parallel transport along parameterized paths on the base space.

Congruence is expressed by the notion of a connection on a principal group sheaf of symmetry coefficients, or equivalently, by a connection on a vector sheaf of coefficients that assumes the role of the state representation space of this symmetry group sheaf.

A connection gives rise to a covariant means of differentiating sections of this sheaf. The physical interpretation is that a field is specified by its covariant differentiating effects on the vector sheaf state space, and thus it should be characterized locally by means of its gauge potentials to cause these internal variations.

In this way, gauge field potentials are introduced as necessary elements following the localization of a symmetry group, such that the physical description remains invariant with respect to the observable algebra sheaf of coefficients.

Cohomological Unfolding and Gauge Field Strength

The coboundary operator of infinitesimal cohomological unfolding should be twisted such that it becomes adapted to the localization of matter symmetry, and thus extended to the representation vector sheaf of states. As a result, the articulation of a point where some matter form is pertinent can be accounted for by the failure of this modified coboundary operator to extend to a differential de Rham complex.

This obstruction to extendibility of the modified coboundary operator to the next higher order is interpreted physically as the encoding trace of the field strength, mathematically being associated with the curvature of a connection on the vector sheaf of states. The curvature is a tensorial object with respect to the observable sheaf of coefficients, and thus expressed in commutative sectional terms.

Weyl's and Atiyah's Phraseology

But mass is a gravitational effect: it is the flux of the gravitational field through a surface enclosing the particle in the same sense that charge is the flux of the electric field. In a satisfactory theory it must therefore be as impossible to introduce a non-vanishing mass without the gravitational field as it is to introduce charge without electromagnetic field.

Gauge theory first appeared in physics in the early attempt by H. Weyl to unify general relativity and electro-magnetism. Weyl had noticed the conformal invariance of Maxwell's equations and sought to exploit this fact by interpreting the Maxwell field as the distortion of relativistic length produced by moving round a closed path. Weyl's interpretation was disputed by Einstein and never generally accepted. However, after the advent of quantum mechanics with its all-important complex wave-functions it became clear that phase rather than scale was the correct concept for Maxwell's equations, or in modern language that the gauge group was the circle rather than the multiplicative numbers. Unfortunately, while scale changes could be fitted into Einstein's theory by replacing the metric with a conformal structure, there was no room for phase to be incorporated into general relativity. Rather the gauge theory had to be superimposed as an additional structure on space-time and the unification sought by Weyl then disappeared.