Which equations give nonsense in the intersection of QM and GR?

[nomadreid]

One of the most frequent questions in this forum is why there is a contradiction between General Relativity and Quantum Mechanics. The most frequent answer is that, in high-energy conditions, some integrals diverge, giving nonsense. I could not find a mention of which integrals one is talking about. Could someone give me a corresponding link? Thanks.

Notes: (1) Other favorite answers concern, more or less in this order of frequency:
(a) that the combined theory turns out to be non-renormalizable (but since there are non-renormalizable theories which work, this in itself is not a condemnation).
(b) the continuity of GR and the discontinuity of QM (but it seems the consensus on that is that GR is an approximation to QM)
(c) the locality of GR and the non-locality of QM (but non-locality seems to have won out),
(d) the differences in symmetries
(e) the arrow of time
(f) Schrödinger's equation goes haywire (but no specifics were given in that answer)
(2) I was not sure whether this question should go under the Relativity Threads or the Quantum Mechanics Threads.

 

[atyy]

http://relativity.livingreviews.org/Articles/lrr-2002-5/fulltext.html sections 2.1 and 2.2

 

[Axe]

Nomareid
i have some comments for your query :
First of all, to make you very clear there is no such contradiction between GR and QM. It is just in the head of many, who are not clear on the subject.

Now for your other quick queries -
1) The combined theory turns out to be non-renormalizable. Just to make you clear we are not even sure if there is a combined renormalizable theory. But still as humans we expect everything to be single and elegant. It is every physicists wet dream to define the whole universe in one single equation since einstein did that.

2) the continuity of GR and the discontinuity of QM. Frankly i don't understand what are you trying to state by this.

3) the locality of GR and the non-locality of QM. There is no winning out. QM is elegant, pure and everlasting, GR is used because we can't deal with the large number of variables that appear in QM while dealing with massive object. And neways we don't even have data for analysing teh universe in QM. If we did, we could probably know the future.

4) the differences in symmetries. Again no idea what you are saying!

5) the arrow of time. What is your concern regarding this??

6) Schrödinger's equation goes haywire. Schrodinger equation can never go haywire. It can be understood as an axiom of QM.

Please revert back for further query if any. Thanks
Adarsh

 

[Mark M]

The problem is that trying to incorporate a graviton into standard quantum field theory in the same fashion as the photon has proven impossible.

We know gravity must be quantized in some form because:

(1) Gravitational waves exist
(2) One frequency of a classical system has an energy that is proportional to that system's frequency by the Planck Constant, [itex] h [/itex].

The evidence for the first point comes from Binary Pulsar 1913+16, which got the people who discovered it the 1993 Nobel Prize in Physics. The second point is a consequence of Planck's Law, [itex] E = hf [/itex] for the quanta of a field.

In quantum field theory, the interaction between two particles is represented by a Feynman Diagram. Because, say, a photon, can produce particle pairs, these must be incorporated in the form of loops in the diagram. In order to account for all possible loops, you must calculate the loop integral for different momenta, from zero to infinity (because particle interactions occur at zero distance, it follows from the uncertainty principle that momentum must be integrated to infinity). See this equation: [tex]I \sim \int_0^\infty p^{4J-8}d^{D}p[/tex] Since there are 4 extended spacetime dimensions, plug that in for D. J is the spin of the particle. For a spin [itex] \frac {1} {2} [/itex] fermion, the integral evaluates to a finite quantity. For a photon, however, it diverges. But, the divergences for loop diagrams in QED were limited to quantities such as the mass and charge of the electron, and the field normalization Z. Substituting the correct, finite, values for these parameters fixes the divergence in a process known as a re-normalization.

However, the loop integral for a spin 2 graviton goes to infinity rapidly - this is an ultraviolet divergence - re-normalization will not work.

There are other examples of divergences in Feynman Diagrams, this is just a simple one.

 

[Mark M]

Also, keep in mind the conflict isn't as much with quantum mechanics as it is with quantum field theory.

Nomareid
i have some comments for your query :
First of all, to make you very clear there is no such contradiction between GR and QM. It is just in the head of many, who are not clear on the subject.

This is false. Any attempt to quantize general relativity in the sense that classical electromagnetism was leads to ultraviolet divergences in loop integrals.

Now for your other quick queries -
1) Just to make you clear we are not even sure if there is a combined renormalizable theory. But still as humans we expect everything to be single and elegant. It is every physicists wet dream to define the whole universe in one single equation since einstein did that.

A theory of quantum gravity does not need to be a theory of everything. The only QG model that unifies the other forces is string theory. The canonical approaches make no attempt at unification. Also, even though we do not know what the correct model of QG is, we are fairly certain there must be one. Gravitational interaction between particles has been observed.

(b) the continuity of GR and the discontinuity of QM (but it seems the consensus on that is that GR is an approximation to QM)

Quantum mechanics holds that the energy levels a particular state may have are quantized into specific eigenstates, but it in no way implies that spacetime is fundamentally discrete. Any theory that discretizes spacetime is limited by the Fermi Satellite's result of no Lorentz violations.

(c) the locality of GR and the non-locality of QM (but non-locality seems to have won out),

There isn't a conflict over this. Quantum entanglement does not allow for faster-than-light transfer of information, and so it respects special relativity.

(f) Schrödinger's equation goes haywire (but no specifics were given in that answer)

It does not. The issue is providing a QFT description of gravity, the Schrodinger equations behave fine in a gravitational field.

 

[Axe]

@ . Any attempt to quantize general relativity in the sense that classical electromagnetism was leads to ultraviolet divergences in loop integrals.

Reply : i was not talking about quantum gravity. In adhering to differences, there is always a mathematical anamoly that still is unresolved, that being the reason for the contradiction between inflation theory and M-Theory.

@A theory of quantum gravity does not need to be a theory of everything.

Reply : Gravity + Quanta - Strings. As i see from my viewpoint i believe this is what a theory of everything basically means, combining all interactions in one.

@ we are fairly certain there must be one. Gravitational interaction between particles has been observed.

Reply - Please elaborate your comments on the observation of gravitational interaction. And as i know of we are not at all sure about any of the string theory to fulfill a real life standard model that can explain every observable practical phenomenons.

@ We know gravity must be quantized in some form because:

(1) Gravitational waves exist

Reply - As far as i know, we are not sure about existence of gravitational waves. In fact the existence of gravitational waves is the only mystery left that will resolve the conflict of universe being inflationary or cyclic. For reference please read "Endless Universe" by Steinhardt.

 

[Mark M]

What conflict between inflation and string theory? There is none, that isn't true. You are confusing string theory with the Ekpyrotic cyclic model. This model is based off of M-theory, but they are not intertwined. I do not know of any string theorists that support the idea, only Steinhardt and Turok.

That is not true, string theory is only one of many approaches to quantum gravity. It is the only one that attempts to unify all of the forces under one field theory. For example, the most prominent canonical approach is Loop Quantum Gravity. LQG makes no attempt to provide a description of anything outside of gravity.

We have observed the effect of gravity on neutron interference experiments. See here:

http://www.atomwave.org/rmparticle/ao%20refs/aifm%20refs%20sorted%20by%20topic/inertial%20sensing%20refs/gravity/COW75%20neutron%20gravity.pdf

This shows conclusively that there is a gravitational interaction between particles, ruling out models that avoid any form of quantum gravity. Again, you appear to mixing up string theory with quantum gravity in general. There are many approaches to quantum gravity, string theory is only one of them.

We have not 'detected' gravitational waves, hut we have 'observed' them. Did you read the link posted below it? Binary Pulsar 1913+16 conclusively demonstrated the existence of gravitational waves. Detecting them at, say, LIGO, is a whole other issue. Detailed analysis of gravitational waves from the early universe could shed light on problems such as the one you mentioned (keep in mind that inflation is supported by incredible amounts of evidence, but Steinhardt/Turok has absolutely none as of now. But, there is a difference between knowing they exist by observation, and taking detailed measurements on them at a detector. We've already done the former, the latter is the issue.

 

[Axe]

What conflict between inflation and string theory?

I Didnt say there was any. I mentioned the conflict between inflation and cyclic theories as mentioned by you.

And talking about Gravitational waves, maybe i was not updated about the detection of any gravitational waves recently. I Apologize for coming to conclusions without seeing the link you sent me.

And talking about gravitational waves, if they have been observed, i think that diminishes the possibility of the cyclic theory to be correct. However i have read their book and found their theory to be convincing of ideas on many levels. Not only are they convincing but they give profound solutions to many unanswered questions that cannot be anwered via inflationary theory like -

1) What was before the bang?
2) How did the energy appear from nowhere? (I know there's a quantum theory for this but that is still non-convincing).
3) Fate of the universe?
4) The mysterious nature of Dark Matter and Dark Energy.

That is the reason for me being convinced of their theory to be valid. Please elaborate if you have any point regarding this.

 

[Mark M]

Well, you said that there is a conflict between 'inflation and M-theory'. Keep in mind that M-theory refers to the non-pertubative string theory, the use dualities to relate the five superstring theories.

It doesn't diminish it whatsoever. Gravitational waves are a general relativistic effect, any legitimate cosmological model would obviously need to include them. Are you sure that they didn't say that their model made predictions regarding a 'gravitational wave background' from the early universe?

My issue with the model is simply that inflation is supported by evidence, and we know it can happen. The Ekpyrotic scenario seemsto take incredibly amount of fine-tuning to get two D3-branes to collide homogeneously in an 11 dimensional bulk.

 

[Axe]

@ Gravitational waves are a general relativistic effect, any legitimate cosmological model would obviously need to include them.

Reply - Not in the magnitude that they could be detected.

@ My issue with the model is simply that inflation is supported by evidence, and we know it can happen. The Ekpyrotic scenario seemsto take incredibly amount of fine-tuning to get two D3-branes to collide homogeneously in an 11 dimensional bulk.

Reply - Yeah but that definitely does not make it wrong. Anyways the evidence you quote for inflation is just the same for ekpyrotic model. The only major experimental difference between them is -

1) Polarization of cosmic microwave background radiation
2) Frequency distribution of gravitaional wave spectrum

According to the ekpyrotic model, the effect of gravitational waves in the current universe is so small that it is not practically possible to detect them via any instrument. Incidentally there is even a bet placed between Steinhardt and Hawkins about this.


@ The Ekpyrotic scenario seems to take incredibly amount of fine-tuning to get two D3-branes to collide homogeneously in an 11 dimensional bulk.

Reply - It is not that complicated, as you expect it to be. And anyways according to me a superluminal expansion is anyday a better explanation for all the energy and matter around us than the "ultimate free lunch" concept.