Why are theories of QM and relativity not possible to combine?

[PrincePhoenix]

Why are theories of Quantum Mechanics and Relativity not possible to combine? I read this on wikipedia and heard this on a documentary on TV.

 

[Demystifier]

It IS possible to combine them, but in more than one way. Moreover, each of the ways abandons some of the common prejudices about relativity, or about quantum theory, or about both.

 

[tom.stoer]

One basic problem in combining GR and QM is that according to QM space must be quantized. But naive quantization methods applied and well-established in quantum field theories for other interactions (like QED, QCD, ...) seem to fail for gravity. One reason is that constructing these quantum field theories uses space-time as a background with a fixed structure which defines certain mathematical properties of the quantum fields. But in GR there is no fixed spacetime structure on top of which the theory can be defined; the spacetime itself becomes dynamical and subject to quantum fluctuations; this is where naive quantization methods break down.

In the last decades a coupe approaches have been developed which allow a quantizatio of spacetime. Each approach has certain benefits and certain intrinsic problems and shortcomings. All approaches differ in detail, but agree on some generic results which seem to be quite robust against details of the approach.

That means we seem to know some tips of iceberg, but he whole iceberg is still unexplored.

Well developed approaches are loop quantum gravity (LQG), string theory (which is mkore than just quantum gravity), causal dynamical triangulation (CDT) and asymptotic safety which stays as close as possible to standard field theory methods.

 

[inflector]

One basic problem in combining GR and QM is that according to QM space must be quantized.

(bold is mine)

What is it about QM in particular that makes it so that as you said: "according to QM space must be quantized?"

 

[JustinLevy]

tom.stoer is trying to answer a question for someone unfamiliar with the math and techniques, so I think that summary is okay.

He's not necessarily saying continuous spacetime is replaced by a lattice -- which is the possible inference you seem to be reacting to. How about this re-wording to get a bit more precise:

In GR the metric of spacetime is considered a dynamic field, so when trying to quantize GR then according to QM the excitations of this field ("spacetime") must be quantized.


That could probably use some work too, but is hopefully closer. Although I had to refer to the metric.

 

[tom.stoer]

I am sorry for the confusion.

With "must be quantized" I do not mean that there are discrete quanta of space, but that one must apply a quantization procedure. Discretization and quantization are two different things, unfortunately often confused.

Let's make an example: quantizing momentum means replacing p with -id/dx. It does not necessarily mean that momentum becomes discrete.

The Einstein equation read

G = T

where G = G[g] is a mathematical object depending on the spacetime metric g, whereas T = T[Matter, g] is the energy-momentum tensor and depends on the mater fields (and on g, too). As the matter fields are "quantized", g must be quantized as well. As the matter fields live in a Hilbert space (or are integrated over in a path integral) the same applies to g as well.

I hope that it became clear that quantization refers only to this mathematical procedure required to describe matter fields and spacetime. Of course there are approaches to quantum gravity which result in spacetime descreteness - but I prefer approaches where this is a result (e.g. LQG) instead of using it as input, as a calculational tool (e.g. CDT).

 

[JustinLevy]

With "must be quantized" I do not mean that there are discrete quanta of space, but that one must apply a quantization procedure. Discretization and quantization are two different things, unfortunately often confused.

Ah, that is a much better way of describing it. Thanks.

 

[tom.stoer]

You're welcome

 

[inflector]

Yes, thanks for the clarification. I'm just trying to make sure that I didn't miss something important in my understanding.