Weyl, “To understand nature, start with the group Γ of automorphisms and refrain from making the artificial logical distinction between basic and derived relations . . .”



Gauge transformations appear of primarily descriptive nature only if we consider them in their function as changes of local (in the mathematical sense) changes of trivializations. In this function they are comparable to the transformations of the coordinates in a differentiable manifold, which also seem to have a purely “descriptive” function. But the coordinate changes stand in close relation to (local) diffeomorphisms. Therefore the postulate of coordinate independence of natural laws, or of the Lagrangian density, can and is being restated in terms of diffeomorphism invariance in general relativity. Similarly, the local changes of trivializations may be read as local descriptions.

The question as to whether or not the automorphisms express crucial physical properties  has nothing to do with the specific gauge nature of the groups, but hinges on the more overarching question of physical adequateness and physical content of the theory. The question of whether or why gauge symmetries can express physical content is not much different from the Kretschmann question of whether or why coordinate invariance of the laws, respectively coordinate covariance description of a physical theory, can have physical content. In the latter case the answer to the question has been dealt with in the philosophy of physics literature in great detail. Weyl’s answer is contained in his thoughts on the distinction of physical and mathematical automorphisms.

Let us shed a side-glance at gravitational gauge theories not taken into account by Weyl. In Einstein-Cartan gravity, which later turned out to be equivalent to Kibble-Sciama gravity, the localized rotational degrees of freedom lead to a conserved spin current and a non-symmetric energy tensor. This is a structurally pleasing effect, fitting roughly into the Noether charge paradigm, although with a peculiar “crossover” of the two Noether currents and the currents feeding the dynamical equations, inherited from Einstein gravity and Cartan’s identification of translational curvature with torsion. The rotational current, spin, feeds the dynamical equation of translational curvature; the translational current, energy-momentum, feeds the rotational curvature in the (generalized) Einstein equation. It may acquire physical relevance only if energy densities surpass the order of magnitude 1038 times the density of neutron stars. By this reason the current cannot yet be considered a physically striking effect. It may turn into one, if gravitational fields corresponding to extremely high energy densities acquire empirical relevance. For the time being, the rotational current can safely be neglected, Einstein-Cartan gravity reduces effectively to Einstein gravity, and Weyl’s argument for the symmetry of the energy-momentum tensor remains the most “striking consequence” in the sense of  rotational degrees of freedom.

On the other hand, the translational degrees of freedom give a more direct expression for the Noether currents of energy-momentum than the diffeomorphisms. The physical consequences for the diffeomorphism degrees of freedom reduce to the invariance constraint for the Lagrangian density for Einstein gravity considered as a special case of the Einstein-Cartan theory (with effectively vanishing spin). Besides these minor shifts, it may be more interesting to realize that the approach of Kibble and Sciama agreed nicely with Weyl’s methodological remark that for understanding nature we better “start with the group Γ of automorphisms and refrain from making the artificial logical distinction between basic and derived relations . . . ”. This describes quite well what Sciama and Kibble did. They started to explore the consequences of localizing (in the physical sense) the translational and rotational degrees of freedom of special relativity. Their theory was built around the generalized automorphism group arising from localizing the Poincaré group.


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