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Mentor

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The problem is the renormalisation trick that works in other Quantum Field Theories does not work in Gravity:

https://quantumfrontiers.com/2013/06/18/we-are-all-wilsonians-now/

But as explained above we look on renormalisation differently these days. We think every theory has some kind of cutoff beyond which it is not necessarily correct in all aspects. These are called effective field theories. For example Quantum Electrodynamics at high energy becomes combined with the theory of weak interactions to become the Electroweak Theory:

https://en.wikipedia.org/wiki/Electroweak_interaction

Since this is an A level thread I would expect the OP to be familiar with all this. Because of this I have changed the level of the thread to B. If the OP thinks they are are at the A level contact a mentor to change it back, but then the answers will be more advanced and they may not be understood.

The specific answer to the question is using the effective field theory approach to quantum gravity they can be combined without any issues:

https://blogs.umass.edu/donoghue/research/quantum-gravity-and-effective-field-theory/

The real issue is, like all the other theories we have, we do not know what lies beyond that cutoff, or even where the cutoff actually is. It is thought to be about the Planck scale, but that is sort of an educated guess. All sorts of things could be going on before that extremely high energy level is reached.

Thanks

Bill

https://quantumfrontiers.com/2013/06/18/we-are-all-wilsonians-now/

But as explained above we look on renormalisation differently these days. We think every theory has some kind of cutoff beyond which it is not necessarily correct in all aspects. These are called effective field theories. For example Quantum Electrodynamics at high energy becomes combined with the theory of weak interactions to become the Electroweak Theory:

https://en.wikipedia.org/wiki/Electroweak_interaction

Since this is an A level thread I would expect the OP to be familiar with all this. Because of this I have changed the level of the thread to B. If the OP thinks they are are at the A level contact a mentor to change it back, but then the answers will be more advanced and they may not be understood.

The specific answer to the question is using the effective field theory approach to quantum gravity they can be combined without any issues:

https://blogs.umass.edu/donoghue/research/quantum-gravity-and-effective-field-theory/

The real issue is, like all the other theories we have, we do not know what lies beyond that cutoff, or even where the cutoff actually is. It is thought to be about the Planck scale, but that is sort of an educated guess. All sorts of things could be going on before that extremely high energy level is reached.

Thanks

Bill

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- #3

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So the field is working in darkness without much experimental guidance.

Instead the main guidance is analysing the constructing principles of each model in the pathwork, and to try to extract from what what the deeper principles are that has the potential to embrace the whole domain. Different researchers has different opinions (or educated guesses) on what the "right" constructing principles are, and which that are the fallacious ones. Another guiding principles is which approaches that adds more explanatory power, by for example reducing the number of free parametrs or the number of "ad hoc assumptions" going into the theories. Other approaches are to not think so much philosophical but just to try to look at the "tool space" and see what other possible mathematical solutions there is. All theories are based on assumptions. As it seems even "simplicity" is a relative concept.

/Fredrik

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QM requires constant time surfaces, which may not exist in curved spacetime?

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Concur. While I lack knowledge to comment on the apparent lack of experimental results, Fra Frederick's conclusion appears rigorous. Simplicity requires comparisons as distinguished from reductionism, if only to examine an emergent system beyond an arrangement of parts. Simplicity implies, even demands, the simplest solutions that{snip}

So the field is working in darkness without much experimental guidance.

Instead the main guidance is analysing the constructing principles of each model in the pathwork, and to try to extract from what what the deeper principles are that has the potential to embrace the whole domain. Different researchers has different opinions (or educated guesses) on what the "right" constructing principles are, and which that are the fallacious ones. Another guiding principles is which approaches that adds more explanatory power, by for example reducing the number of free parametrs or the number of "ad hoc assumptions" going into the theories. Other approaches are to not think so much philosophical but just to try to look at the "tool space" and see what other possible mathematical solutions there is. All theories are based on assumptions.As it seems even "simplicity" is a relative concept.

{italics added for emphasis}

Elegance inherent in simple working solutions tends to verify this conclusion despite the apparent subjective nature of 'elegance' and 'beauty' in mathematics.

- #6

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QM requires constant time surfaces, which may not exist in curved spacetime?

It's due to an old conundrum - does gravity really curve space-time or does treating the metric like a gravitational field (Guv) act on all objects to make it appear spacetime is curved. There is no way to tell the difference. It's easier to take the second view when doing quantum gravity.

Thanks

Bill

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Another take on this is that it comes down to what you think are "observables" in GR, they are not necessarily the same as what you get from observations. This is also one of the debates. Ie. can you just "decide" arbitrarily which "elements" in GR (in some reformulation) where to apply QM formalism? After all, you can not make an elementary "observation" of a global or even a patch of a metric, it is necessarily a compound observation.QM requires constant time surfaces, which may not exist in curved spacetime?

A tension is that, on one hand actual "real" observations often requires making an arbitrary choice. But the equivalence class that's remains when compensating for choices are sometimes thought of as are more "real" or observer invariant. (for good reasons). But one can also argue that the equivalence class is an emergent construction and thus may not be fundamental (although in a sense, more robust/real).

/Fredrik

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Maybe this link of Isham will help.

https://arxiv.org/pdf/gr-qc/9510063.pdf

It is not only for the advanced level; I think.

- #9

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General relativity is fundamentally different from any physical theory that has come before it. Pre-general relativistic theories have as part of their very formulation an a priori given metric. For example in the theory's action principle there appears an a priori given metric that is part of the very definition of the theory.

The Einstein-Hilbert action for classical GR, however, has no a priori given background metric that is part of the formulation of the theory. Roughly speaking there is no further container over which the dynamics of spacetime geometry can take place. This is a subtle point.

Now, if you wish the formulation of the quantum theory to also to not depend on any a priori background metric then you can make arguments that such a theory will be manifestly finite. Here is an argument given by Smolin:

“ A background independent operator must always be finite. This is because the regulator scale and the background metric are always introduced together in the regularization procedure. This is necessary, because the scale that the regularization parameter refers to must be described in terms of a background metric or coordinate chart introduced in the construction of the regulated operator. Because of this the dependence of the regulated operator on the cuttoff, or regulator parameter, is related to its dependence on the background metric. When one takes the limit of the regulator parameter going to zero one isolates the non-vanishing terms. If these have any dependence on the regulator parameter (which would be the case if the term is blowing up) then it must also have dependence on the background metric. Conversely, if the terms that are nonvanishing in the limit the regulator is removed have no dependence on the background metric, it must be finite. ”

If you want to quantise GR in a background independent manner then you can't use perturbation theory, for perturbation theory splits the metric into a flat non-dynamical part that serves as a background metric and a part that you*allow* to be dynamical which you then quantise (using standard techniques).

If you want to quantise GR in a background independent way you have to do a non-perturbative quantisation that does not depend on any a priori given metric. This would appear to be a big ask because you have to quantise Einstein's full field equations which are immensely complicated, and also because all standard quantisation techniques rely heavily on the existence of an a priori background metric, you have to come up with new quantisation techniques. There are serious technical problems to be overcome, not to mention serious conceptual problems that come up in attempting to quantise a fully dynamical theory of spacetime. Perturbation theory renders quantum gravity a standard QFT and so hides the serious conceptual problems that you must at some point tackle.

In loop quantum gravity there has been much progress in addressing the technical problems and also people working in LQG (for example Rovelli) have made much progress in addressing conceptual problems. But it is far from clear whether LQG or something that grows out of it will be the final theory.

The Einstein-Hilbert action for classical GR, however, has no a priori given background metric that is part of the formulation of the theory. Roughly speaking there is no further container over which the dynamics of spacetime geometry can take place. This is a subtle point.

Now, if you wish the formulation of the quantum theory to also to not depend on any a priori background metric then you can make arguments that such a theory will be manifestly finite. Here is an argument given by Smolin:

“ A background independent operator must always be finite. This is because the regulator scale and the background metric are always introduced together in the regularization procedure. This is necessary, because the scale that the regularization parameter refers to must be described in terms of a background metric or coordinate chart introduced in the construction of the regulated operator. Because of this the dependence of the regulated operator on the cuttoff, or regulator parameter, is related to its dependence on the background metric. When one takes the limit of the regulator parameter going to zero one isolates the non-vanishing terms. If these have any dependence on the regulator parameter (which would be the case if the term is blowing up) then it must also have dependence on the background metric. Conversely, if the terms that are nonvanishing in the limit the regulator is removed have no dependence on the background metric, it must be finite. ”

If you want to quantise GR in a background independent manner then you can't use perturbation theory, for perturbation theory splits the metric into a flat non-dynamical part that serves as a background metric and a part that you

If you want to quantise GR in a background independent way you have to do a non-perturbative quantisation that does not depend on any a priori given metric. This would appear to be a big ask because you have to quantise Einstein's full field equations which are immensely complicated, and also because all standard quantisation techniques rely heavily on the existence of an a priori background metric, you have to come up with new quantisation techniques. There are serious technical problems to be overcome, not to mention serious conceptual problems that come up in attempting to quantise a fully dynamical theory of spacetime. Perturbation theory renders quantum gravity a standard QFT and so hides the serious conceptual problems that you must at some point tackle.

In loop quantum gravity there has been much progress in addressing the technical problems and also people working in LQG (for example Rovelli) have made much progress in addressing conceptual problems. But it is far from clear whether LQG or something that grows out of it will be the final theory.

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https://arxiv.org/pdf/0907.4238.pdf

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