Problems between Quantum Mechanics and General Relativity

[PabloAMC]

I have read several times that general relativity has some problems with quantum mechanics and they are not compatible. However, special relativity can be introduced in quantum mechanics mainly by Dirac equations (so I am pretty sure that the problem of passing from a frame where the parameter is time to one where the parameter is the proper time τ, and spacetime is 4 dimensional should not be a problem. I have had two degree subjects in quantum mechanics and one in general relativity. In this last one it was stated that in order to change from special relativity to general relativity no curvature terms should be added, but we should change the partial derivatives, by covariant derivatives (using Christoffel symbols).
So, taking all into account, does anyone know any mathematically expressed explanation of where is this problem? (It sounds to me something about renormalization, but I cannot say)
Thanks in advance

 

[Paul Colby]

Almost all of the machinery of quantum field theory is based on a flat space-time. This can be taken over to curved space-time "background" but doesn't lead to a quantum theory of gravity. A true theory is expected to be background independent. I think no one knows how to do this. Attempts based on a weak or linearized theory have issues with too many (an infinite number) of renormalization constants. I'm kind of a fossil in this regard and haven't kept up with the literature so things may be different now.

 

[PabloAMC]

A true theory is expected to be background independent.

Thanks a lot. However, unless I am wrong, I think that one of the main approaches (string theory) is not background independent

 

[Paul Colby]

With like 10^27 (not an accurate number and based on hearsay) different vacuum states it's unlikely they can really make the claim of background independence. I have a real hard time referring to string theory as a theory.

 

[haushofer]

Thanks a lot. However, unless I am wrong, I think that one of the main approaches (string theory) is not background independent

Its current formulation isn't. But that doesn't kill a theory. E.g., Fierz-Pauli theory is also background dependent, but you can iterate it to GR, which is background independent.

 

[haushofer]

The problem is indeed that if you regard GR as a field theory and quantize it, it is not renormalizable. This means it can only be trusted up to a certain energy scale, and hence is 'only' effective. But there are other problems in the quantization of GR. E.g., Zee's book on GR treats them somewhere at the end of the book.

 

[PabloAMC]

The problem is indeed that if you regard GR as a field theory and quantize it, it is not renormalizable.

Do you know where I can find a text describing that?

But there are other problems in the quantization of GR. E.g., Zee's book on GR treats them somewhere at the end of the book.

The book you are referring to is Einstein Gravity in a nutshell, isn't it?

 

[haushofer]

Yes. That book, and zee's qft book, are good resources.

 

[PabloAMC]

Thanks a lot. I'll take a look

 

[sridhar10chitta]

While on the topic, It may be useful to look into Relational Mechanics as well. See Relational Mechanics and Implementation of Mach’s Principle with Weber’s Gravitational Force by Andre Koch Torres Assis

 

[ohwilleke]

Another way that the Standard Model and general relativity are not compatible is that the Standard Model treats all of its fundamental particles as point masses which create black hole singularities in GR.

 

[kodama]

Another way that the Standard Model and general relativity are not compatible is that the Standard Model treats all of its fundamental particles as point masses which create black hole singularities in GR.

do you have any ideas on how to solve this? i.e modify SM to give particles spatial extent or modify GR to remove singularities?

 

[ohwilleke]

If you retain classical GR, one solution would be to smear particles across their Compton wavelength or Heisenberg uncertainty under QM so that they would cease to be point particles; if you moved to a graviton formulation of GR, there wouldn't be any true singularities in GR.