«Wave Fields in Weyl Spaces and Conditions for the Existence of a Preferred Pseudo-Riemannian Structure J. Audretsch 1, F. Gahler2, and N. Straumann3 ...»
It should be clear that one obtains in all cases the Hamilton-Jacobi equation for the eikonal S. Furthermore, the gradient of S is always proportional to the zeroth order particle current, i.e., tangent to the trajectories of the WKB-limit. As for the Dirac and Klein-Gordon fields this leads to Dm = 0.
We have shown that this condition implies the existence of a preferred metric g in the conformal class  such that the Weyl connection of the bundle belonging to [g] is reducible to a connection in the bundle of orthonormal frames belonging In the language developed by Trautman  we could regard a mass function m of a nontrivial Weyl type as a Higgs field, and the equation Dm = 0 as the condition for spontaneous symmetry breaking of the group G down to L\. With these remarks, we would like to emphasize that our discussion of the reduction of a Weyl space to the pseudo-Riemannian structure of general relativity is analogous to the bundle reductions which occur in ordinary spontaneously broken gauge theories.
In his Banff lectures  Ehlers concludes the discussion of his work with
Pirani and Schild  with the following remarks:
"It must be admitted that the motivation of (the vanishing of the 'distance curvature') is not very satisfactory; it is an extraneous element of the theory. It seems that a deeper understanding of it could come from a better analysis of the Preferred Pseudo-Riemannian Structure 51 description of matter in a Weyl spacetime. It is quite possible that such an analysis (may be in terms of possible wave equations for massive particles) would show that a 'reasonable' description of matter is possible only if the Weyl space is, in fact, Riemannian."
We hope that our investigation has at least partially fulfilled the expectation expressed in these sentences.
Finally, we emphasize that the axiomatic scheme of  excludes from the very beginning a non-vanishing torsion. It should be clear that the conclusions of this paper also rest heavily on this assumption.
Acknowledgements. We thank the organizers and participants of the meeting on General Relativity on Schloss Ringberg (Sept. 1983) for critical and constructive remarks. Special thanks go to J. Ehlers for his interest in this work.
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(ed.). Dordrecht, Boston: Reidel 1973 Communicated by S. Hawking Received November 23, 1983