Quantizing Gravity: Uncovering Mathematical Incompatibilities

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In summary, the incompatibilities between QFT and GR arise because GR treats space and time as dynamical fields, while QFT only considers them as structures in the quantum field. This leads to problems with canonical commutation relations and the definition of mass and spin. However, renormalizability is not that big of a deal since theories that are renormalizable like QED are not fundamental.
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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?
 
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Such claims are now outdated: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
 
  • #3
Parmenides said:
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.
 
  • #4
andrien said:
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
 
  • #5
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.
 

1. What is quantizing gravity?

Quantizing gravity is the process of trying to reconcile the theories of general relativity and quantum mechanics, which are currently two of the most successful theories in physics, into one unified theory. This is important because both theories are incompatible when applied to extreme conditions, such as at the center of a black hole or during the Big Bang.

2. Why is quantizing gravity important?

Quantizing gravity is important because it would provide a better understanding of the fundamental forces and particles that make up the universe. It would also help us to better understand the behavior of matter and energy at extreme conditions, such as in the early universe or in the presence of a black hole.

3. What are the challenges in quantizing gravity?

The main challenge in quantizing gravity is the incompatibility between general relativity, which describes gravity in terms of the curvature of spacetime, and quantum mechanics, which describes the behavior of particles at a subatomic level. This creates mathematical inconsistencies that need to be resolved in order to develop a unified theory.

4. How are scientists approaching the problem of quantizing gravity?

Scientists are approaching the problem of quantizing gravity through various approaches, such as string theory, loop quantum gravity, and causal dynamical triangulation. These theories attempt to reconcile the incompatibilities between general relativity and quantum mechanics by introducing new concepts and mathematical frameworks.

5. What are the potential implications of successfully quantizing gravity?

If scientists are able to successfully quantize gravity, it would have a significant impact on our understanding of the universe and potentially lead to new technologies and advancements. It could also help to solve some of the biggest mysteries in physics, such as the nature of dark matter and dark energy, and provide a deeper understanding of the fundamental laws of nature.

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