Covariant fields and new physics

[giulio_hep]

I assume that all the fundamental physics known - as of today - can be reduced to quantum general covariant fields (including spacetime itself to be seen as a field of those...).
Now, sorry if my question is quite abstract and based on tomorrow's hypothetical new physics, but would it be against GR should they discover - one day in the future - a new quantum field not being covariant? Is that either in principle ruled out as incompatible with GR or there is a theoretical possibility of extending the GR to new physics outside the domain of validity of the GR - let's say like an extension of the SM?

 

[bcrowell]

You refer to quantum fields, but is your question really quantum-mechanical? At least some of the fields of the standard model have a classical limit. Do we lose anything from your question if we make it purely classical?

It seems implausible to me that GR could be extended in the way you suggest, for the following reason. The Einstein field equations have a curvature tensor on one side and the stress-energy tensor on the other. The curvature tensor is tensorial simply because that's the way geometry works. Therefore it seems like the stress-energy tensor has to be tensorial. I don't see any obvious way to construct a tensorial stress-energy tensor from a non-tensorial field.

As an example, if something is a tensor, then it follows that if it vanishes in one coordinate system, then it vanishes in all others. Suppose we had something that violated this principle, implying that it wasn't a tensor. It vanishes in one coordinate system, so presumably its stress-energy is zero in that coordinate system. It doesn't vanish in some other coordinate system, so I would assume that its stress-energy would be nonzero in that system. But then the stress-energy is nontensorial as well, which seems incompatible with the field equations.

In general, it would be awfully strange if there were violations of diffeomorphism invariance. Diffeomorphism invariance just says that we can rename points in spacetime without affecting the laws of physics. If that weren't true, then it would be as though a daisy obeyed the laws of Mendelian genetics whenever you referred to it as a "daisy," but disobeyed them whenever you called it a "margarita."

If you look at extensions to GR that have already been described, such as Brans-Dicke or SME, I think they're usually framed in terms of introducing some field that is a tensor, but that acts like part of a geometrical background.

 

[giulio_hep]

Let me introduce an example from http://arxiv.org/abs/0905.1668 : they write "It is only the quantum mechanical wave
function that breaks the diffeomorphism invariance", "while keeping the invariance of the spatial three-geometry at the quantum mechanical level as well as at the classical level." and "There exist stringent experimental bounds on violation of Lorentz invariance at lower energies, but there is no observational evidence that Lorentz invariance is strictly maintained at the Planck energy.". My doubt is whether they're completely wrong or a tiny possibility of a "soft" violation does exist at very special conditions...

 

[bcrowell]

The paper is about quantum gravity, and seems extremely speculative. He proposes violation of conservation of energy and the introduction of a preferred time coordinate. I don't have the quantum gravity chops to be able to understand the paper, but one thing to realize is that in the classical regime we can't violate conservation of energy without somehow modifying the Einstein field equations, because they aren't self-consistent if the stress-energy tensor has a divergence.

 

[PeterDonis]

there is no observational evidence that Lorentz invariance is strictly maintained at the Planck energy

Well, of course not; we can't do experiments at the Planck energy. We can't even do experiments within many orders of magnitude of the Planck energy. So we have no way of ruling out the possibility that Lorentz invariance is violated at such scales. But any theory that suggests such a possibility has to explain why Lorentz invariance is not violated in the regime we can observe. (I haven't read the paper you linked in enough detail to see whether it attempts to do that.)

 

[bcrowell]

Moffat seems to be describing spontaneous symmetry breaking rather than breaking the symmetry of the underlying laws of physics. That seems like a completely different and much less radical thing to me. Even in a standard, well-tested model such as FLRW, we have spontaneous symmetry breaking that picks out a preferred time coordinate.