What happens when you try to join QM and GR?

[gk007]

What are the actual problems of joining these two theories?

(I've heard various people say that the equations 'fail' or 'blow-up', but I want to know exactly what happens...)

 

[marcus]

What are the actual problems of joining these two theories?

(I've heard various people say that the equations 'fail' or 'blow-up', but I want to know exactly what happens...)

Part of the problem is what to use for spacetime. To build a conventional quantum theory you need some preestablished spacetime geometry to put the fields/particles on. You can imagine choosing: it could be a classical 3D space with a separate time variable, or the standard flat (minkowski) space of special relativity, or some other prearranged geometric set-up.

But GR does not like to start with some predefined 4D geometry. The theory is about how geometry itself arises---and interacts dynamically with matter. So if you wanted to build a general relativistic QFT you would immediately confront this essential problem at the level of foundations.

There is more to say. Here's an introductory overview by one of the main people involved.
http://arxiv.org/abs/gr-qc/0604045 The PDF is free online.
This is a non-technical essay which was selected to be Chapter 1 in a book treating the problem you ask about---merging QM and GR.
The book includes chapters by string theorists as well as chapters by proponents of various non-string quantum gravity approaches--thirty-some experts in all. It was published in 2009 by Cambridge University Press, and is called Approaches to Quantum Gravity: Towards a New Understanding of Space Time, and Matter

The book is an interesting document. If you know some string theory names you will recognize 4 of the authors: Joe Polchinski, Tom Banks, Gary Horowitz, Wati Taylor. Another of the chapters was written by Gerard 't Hooft. The authors are supposed to be considering the problem you mentioned: a merger of QM+GR, a truly general relativistic quantum field theory. In some sense this book should contain the answer to your question: "why is it hard". And it should contain the various ways that those who wrote the book are trying to overcome the problems.

But all I would especially recommend reading is Chapter 1, which is free online at that link I gave.
http://arxiv.org/abs/gr-qc/0604045
I think the author, Rovelli, presents the clearest understanding of the QG problem--although in this chapter he does not offer any specific solution.

 

[tom.stoer]

To be a bit more specific: in order to construct quantum field theories one uses the metric to define spacelike and timelike distances. This is required e.g. to define commutation relations between field operators. If you want to construct field operators for the gravitational field which will define distances in some classical limit you would have to use a pre-defined geometry.

There is much more to say about that, but it may explain one specific problem and why the construction principles of QFT need to be adjusted for QG.

 

[Finbar]

It's worth noting that one can consistently join QM and GR within the frame work of quantum field theory as long as one only considers energy scales which are well below the Planck scale(the scale at which gravity becomes strong). This means treating GR as an effective field theory.

Here it is still possible to treat gravity as small perturbations on space-time. A graviton is then the quanta corresponding to these perturbations analogous to a photon being a small perturbation of the EM field.
The theory of gravitons will then give predictions such as corrections to Newton's inverse squared law.



The problems arise when gravity becomes strong and these perturbations around a fixed spacetime are no longer a valid approximation. Then the problems Marcus and Tom mentioned arise and one needs a non-perturbative/background independent formulation of the theory.

 

[tom.stoer]

It's worth noting that one can consistently join QM and GR within the frame work of quantum field theory as long as one only considers energy scales which are well below the Planck scale(the scale at which gravity becomes strong). This means treating GR as an effective field theory.

Are you referring to Asymptotic Safety or to a non-renormalizable theory.

 

[Finbar]

Are you referring to Asymptotic Safety or to a non-renormalizable theory.

What I said applies for both cases where gravity is AS or if its non-renomalizable. In either case if we treat GR as an effective field at scales below the Planck scale we can make predictions regardless of its UV behavior. If this low energy effective theory is formulated perturbatively it must break down at the Planck scale. However if you can formulate the theory non-perturbatively i.e. not as small perturbations around a fixed metric then you may be able to push the theory beyond the Planck scale a la AS or CDT. On the other hand you may need to introduce new degrees of freedom e.g. string theory.

 

[gk007]

Thanks