Unifying gravity and quantum mechanics

[kent davidge]

I was sitting at my bed when I suddenly have an idea.

The unification of electromagnetism and gravity was made with General Relativity. For this to happen, one just need to write the energy due to electromagnetic field in "differential geometry" form, through the Electromagnetic tensor, in a way that it matches with the right hand side of the Einstein equation.

Then I wonder, for quantum-mechanics one just need to write the "quantized" energy of a particle in a manner to combine it with the stress-energy tensor and... uu-uhhh, we could FINALLY know what happens to particles in the singularity of a black hole.

Why would it not work?

I'm sorry for my bad poor English.

 

[PeterDonis]

for quantum-mechanics one just need to write the "quantized" energy of a particle in a manner to combine it with the stress-energy tensor

More precisely, you need to write the particle's energy, taking into account all of its quantum properties, in such a way that it is part of the stress-energy tensor.

However, this doesn't fix the key issue with unifying QM and GR, which is that in QM, the particle will not, in general, be in an eigenstate of energy--it will be in a superposition of states that have different energies. But each of these states will have different gravitational effects, i.e., different stress-energy tensors; so the particle will be in a superposition of stress-energy tensors. That will lead to different effects on the curvature of spacetime; so spacetime itself would need to be in a superposition of different curvatures. But GR is a classical theory: it only works if you have a single stress-energy tensor, not a superposition of different ones.

 

[mfb]

Then I wonder, for quantum-mechanics one just need to write the "quantized" energy of a particle in a manner to combine it with the stress-energy tensor and... uu-uhhh, we could FINALLY know what happens to particles in the singularity of a black hole.

People tried that as long as QFT exists. It is easy to write down such a theory, but it does not work. At least no one knows how it would lead to finite, meaningful results.

 

[kent davidge]

More precisely, you need to write the particle's energy, taking into account all of its quantum properties, in such a way that it is part of the stress-energy tensor.

However, this doesn't fix the key issue with unifying QM and GR, which is that in QM, the particle will not, in general, be in an eigenstate of energy--it will be in a superposition of states that have different energies. But each of these states will have different gravitational effects, i.e., different stress-energy tensors; so the particle will be in a superposition of stress-energy tensors. That will lead to different effects on the curvature of spacetime; so spacetime itself would need to be in a superposition of different curvatures. But GR is a classical theory: it only works if you have a single stress-energy tensor, not a superposition of different ones.

Where is the problem if we accept that the curvature of spacetime is probabilistic in small scales, such as near the singularity of a BH? And say that spacetime behave like a classical entity in larger scales, so that we can ignore ?

People tried that as long as QFT exists. It is easy to write down such a theory

I would like if you could show me some no-too-complicated equations.

 

[PeterDonis]

Where is the problem if we accept that the curvature of spacetime is probabilistic

The problem is that we don't have a workable theory that includes this. The obvious, and even not so obvious, ways of constructing such a theory don't work. That is why finding a workable theory of quantum gravity is still a hot research topic, more than half a century after the need for one was recognized.

 

[kent davidge]

The problem is that we don't have a workable theory that includes this. The obvious, and even not so obvious, ways of constructing such a theory don't work. That is why finding a workable theory of quantum gravity is still a hot research topic, more than half a century after the need for one was recognized.

Is it like the Coulomb force problem? That it predicts a infinite electric field strength at zero distance from the punctual source, but the reality is that quantum forces start acting up to repeal the two charges when they come close together, so the field is not infinite there, and if we try to put quantum equations in the problem, we get wrong predictions, because the Coulomb theory has a classical nature?

 

[PeterDonis]

Is it like the Coulomb force problem?

No. Quantum electrodynamics doesn't have the same issues as quantum gravity.

quantum forces start acting up to repeal the two charges when they come close together

Where are you getting this from?

if we try to put quantum equations in the problem, we get wrong predictions, because the Coulomb theory has a classical nature?

I don't know where you're getting this from either.

 

[kent davidge]

No. Quantum electrodynamics doesn't have the same issues as quantum gravity.
Where are you getting this from?
I don't know where you're getting this from either.

Well, suppose the two charged particles happen to be fermions. Then a repealing force would not allow the two fermions to occupy the same place.

 

[PeterDonis]

suppose the two charged particles happen to be fermions. Then a repealing force would not allow the two fermions to occupy the same place.

First, this is oversimplified. The correct statement is that fermion wave functions are antisymmetric under particle exchange. The wave functions include spin as well as position, so two fermions with opposite spins will not experience the "repelling force" you are talking about (thinking of it as a "repelling force" is also an oversimplification).

Second, saying that the antisymmetry of fermion wave functions "makes the field finite" at zero radius is not correct. The "infinite field at zero radius" problem still exists for fermions in QED. It is solved by renormalization. But that fix, which works for QED, doesn't work for quantum gravity, because unlike QED, quantum gravity is not renormalizable.

Third, the antisymmetry only applies to fermions, so your argument, even if it were correct for fermions, wouldn't explain why charged bosons (such as the weak bosons or the charged Higgs) can't produce an infinite Coulomb field.

 

[kent davidge]

First, this is oversimplified. The correct statement is that fermion wave functions are antisymmetric under particle exchange. The wave functions include spin as well as position, so two fermions with opposite spins will not experience the "repelling force" you are talking about (thinking of it as a "repelling force" is also an oversimplification).

Second, saying that the antisymmetry of fermion wave functions "makes the field finite" at zero radius is not correct. The "infinite field at zero radius" problem still exists for fermions in QED. It is solved by renormalization. But that fix, which works for QED, doesn't work for quantum gravity, because unlike QED, quantum gravity is not renormalizable.

Third, the antisymmetry only applies to fermions, so your argument, even if it were correct for fermions, wouldn't explain why charged bosons (such as the weak bosons or the charged Higgs) can't produce an infinite Coulomb field.

Oh very interesting thing to know. Can you recommend me lectures/textbooks that covers that subject? Like, antisymmetry of wave functions, etc. All introductory books I've found does not contain such subjects.

 

[PeterDonis]

All introductory books I've found does not contain such subjects.

For antisymmetry of wave functions, see, for example, Chapter 17 of Ballentine:

http://www-dft.ts.infn.it/~resta/fismat/ballentine.pdf

For the non-renormalizability of quantum gravity, you will probably need to look in the peer-reviewed literature.

 

[Nugatory]

Can you recommend me lectures/textbooks that covers that subject? Like, antisymmetry of wave functions, etc. All introductory books I've found does not contain such subjects.

There's a reason for that. This is quantum field theory, and there is no such thing as an "introductory" treatment of quantum field theory; it's something you encounter after you've started your PhD in theoretical physics. The closest to an introduction that I am aware of, accessible to someone who has completed a bachelor's degree in physics, is Lancaster and Blundell's "https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20".

Or if you want to get an idea what you're letting yourself in for without buying a fairly expensive book, you can find a prepublication version of Mark Srednicki's textbook here; pay particular attention to the introduction where he describes the physics you already have to know to get into it. (I'm not recommending this book because I think it's better than the other QFT texts out there, but because there's a free and unpirated edition online).

 

[Ibix]

pay particular attention to the introduction where he describes the physics you already have to know to get into it.

3/8.

 

[kent davidge]

Thanks

 

[mfb]

I would like if you could show me some no-too-complicated equations.

Something like this. You can write it down, but no one knows if that is reasonable, and even if it is, no one knows how to calculate things with that expression, because the tools developed for the other parts (everything apart from the "gravity" summand) don't work for the gravity part.

And as the others commented, you'll need a course in quantum field theory (and a lot of knowledge to understand this course) to understand what is going on in that equation.

 

[kent davidge]

Something like this. You can write it down, but no one knows if that is reasonable, and even if it is, no one knows how to calculate things with that expression, because the tools developed for the other parts (everything apart from the "gravity" summand) don't work for the gravity part.

And as the others commented, you'll need a course in quantum field theory (and a lot of knowledge to understand this course) to understand what is going on in that equation.

What is W in that equation?