For some reason, diffeomorphism invariance seems to be treated like a second-cla
ID: 2049429 • Letter: F
Question
For some reason, diffeomorphism invariance seems to be treated like a second-class citizen in the land of symmetries. In nonrelativistic quantum mechanics, we consider Galilean invariance so important that we form our Hilbert space operators from irreducible representations of the Galilei group. In relativistic quantum mechanics, I think we can do the same thing with representations of the Poincare group. (Could someone back me up on that? How do operators work in Fock space?) But when it comes time to consider quantum gravity, we do not grant diffeomorphism invariance an analogous role, citing the fact that it is a gauge symmetry and thus indicating a mere superfluousness in our mathematical description of physical states.I have a few issues with that. First of all, gauge symmetries can be quite important; it is the gauge invariance of Maxwell's equation that gives rise via Noether's theorem to (local) conservation of electric charge. What's the Noether charge for diffeomorphism invariance? Second of all, it seems to me that diffeomorphism invariance is more than just a gauge symmetry. Any statement that has rotational invariance, translational invariance, Lorentz invariance etc. (all locally) as it's implications surely has some physical significance. Can't we easily imagine a universe in which the laws of physics looked profoundly different in different coordinate systems?
Has there been any work in building quantum gravity from the representation theory of the diffeomorphism group?
Any help would be greatly appreciated.
Explanation / Answer
siddharth: General covariance (or diffeomorphism covariance if you must call it that) can be thought of as the 'local' generalization of the translation group. That is, in place of global translations, xµ ? xµ + eaµ, where aµ = const, an infinitesimal coordinate transformation may be written as xµ ? xµ + e?µ where ?µ = ?(x)µ. This is analogous to the gauge groups of electromagnetism we have gauge transformations Aµ ? Aµ + ?µ which are global or local depending on whether or not ? is a constant.