In an attempt to widen my own horizons I've decided to educate myself in Wheeler's Geometrodynamics.
In the so-called "already unified theory" one can essentially reproduce an electromagnetic field based solely upon the "footprint" it leaves upon spacetime (to put it crudely).
In seeking out generalizations I continually come across statements that geometrodynamics cannot be extended to fields with nonabelian group symmetries (ie.more general yang mills fields).
Why is this? Any explanation or direction towards an article/book explaining this would be greatly appreciated. Off the bat, I can't see why Rainich type conditions for other fields wouldn't exist.
Note, I'm only examining classical geometrodynamics at the moment.
For example, in this paper (László Szabó General Relativity and Gravitation, January 1982, Volume 14, Issue 1, pp 77–85), the author writes (on page 2, 2nd paragraph, emphasis mine):
In the case of non-abelian gauge theories the implementation of this programme* (referring to geometrization of the field as for electrodynamics) *is impossible since there is no unambiguous relation between field-variables and the space-time geometry as in the case of electrodynamics.
However, this statement doesn't clarify the reasoning at all.