Quantizing Gravity: Uncovering Mathematical Incompatibilities

[Parmenides]

Hello,

Near the end of undergraduate physics, we are often told about the difficulty of quantizing the gravitational field and the absurdities that arise from it. However, I've yet to see a mathematical demonstration of where the incompatibilities of QFT and GR arise. Does anybody know where I can find some mathematics behind these claims or could anyone demonstrate it?

 

[bhobba]

Such claims are now outdated:
http://arxiv.org/abs/grqc/9512024
http://arxiv.org/abs/1209.3511

The real issue is we don't have a quantum theory of gravity valid to all energies - the EFT elucidated in the above papers is only valid up to about the Plank scale. But we are pretty sure all our usual QFT theories such as QED (ie except String Theory, LQG etc) break down there, so it's not really that big a deal. Of course we want to remove that - but its no different to QED etc.

Thanks
Bill

 

[andrien]

Hello,

Near the end of undergraduate physics, we are often told about the difficulty of quantizing the gravitational field and the absurdities that arise from it. However, I've yet to see a mathematical demonstration of where the incompatibilities of QFT and GR arise. Does anybody know where I can find some mathematics behind these claims or could anyone demonstrate it?

Well,if you will take the Hilbert action for gravity,you will see that R/G should have dimension of mass +4.You can also easily see by looking at the riemann curvature tensor that scalar curvature involves two powers of derivative and thus has mass dimension +2.So G^-1 should have mass dimension +2 so as to make the action dimensionless.So you have G with negative power of mass which indicates that resulting theory is nonrenormalizable,so you can at best go for an effective field theory description.You can also get the mass dimension of G by comparing Newton's gravitation law with coulomb law.There fine structure is dimensionless,hence G has mass dimension -2.

 

[bhobba]

So you have G with negative power of mass which indicates that resulting theory is nonrenormalizable,so you can at best go for an effective field theory description.

Yes - all true.

But the modern view is renormalizabilty is not that big a deal since theories that are renormalizable like QED are not fundamental so need modifications at higher energies eg QED is replaced by the electroweak theory ie one can do calculations to all orders but since it breaks down at higher energies its not really a worry if it wasn't - as long as it's valid up to where it breaks down.

Thanks
Bill

 

[bapowell]

The primary conceptual problem is that the metric of the space, used as a background on which to define fields in other theories, is itself a dynamical field in general relativity. This makes defining things like canonical commutation relations difficult (how do you define a causality constraint on fields according to some metric when the metric itself is fluctuating?)

Related to this is the problem of time. Poincare invariance singles out a unique time parameter suitable for studying the evolution of fields in non-gravitational quantum field theories. In GR, time is a geometric quantity determined by the metric; since the metric is a dynamical field, the flow of time is intermixed with the dynamical evolution of gravitational systems. The loss of Poincare symmetry in general relativity also introduces ambiguities in our definitions for quantities like mass and spin.