Q&A: Infinite Probabilities, Relativity & Quantum Mechanics

[muppet]

In virtually every popsci book you read there's a discussion of a fundamental incompatibility between general relativity and quantum mechanics. Specifically, talk is often made of calculations that return infinite probabilities- which are obviously meaningless. I'm curious as to what it is specifically that we cannot calculate?
Also, are there any other areas of tension between the two formulations? The only other one I know of is that if one takes the collapse of the wavefunction to be "true" (i.e. to provide an ontological description of reality, rather than describing what we know about reality) then this cannot be made lorentz covariant (as it happens instantaneously, and simultaneity is a relative concept), so cannot be conceptually reconciled happily with special relativity (despite the fact that we have relativistic equations such as the Dirac and Klein-gordon equations). I'm a second year (UK) undergrad in maths and physics if that helps pitch an answer to the highest level I could understand.
Thanks in advance for your help!

 

[marcus]

the incompatibilities have been discussed at a technical level (as well as popsci) and someone here may have a good technical article to recommend to you.
Gen Rel is very different from Special. GR is background independent----the spacetime continuum has no fixed geometry. GR is developed without reference to any prior fixed geometry, and it has no absolute time.

Quantum theories are usually developed on a set geometry (like Minkowski or Euclidean geometry) fixed ahead of time. they often have some classical clock that gives meaning to the time variable.

In GR these things are not available and it seems very difficult to construct a quantum theory using the space and time of Gen Rel.

Also Gen Rel is diffeomorphism invariant ("general covariant") which means that no meaning is attached to a point of spacetime. there are no points of spacetime, only relationships between events. If you have one GR solution then you can moosh it around with a diffeomorphism and---in the view of GR---it is the SAME solution. It is often thought that the gravitational field in GR is represented by the spacetime metric (the distance function) but actually that isn't true, it is an equivalence class of metrics.
two metrics are equivalent if one can map into the other by a diffeomorphism
and a diffeomorphism is any smooth map with a smooth inverse

so GR has an idea of space and time that is radically different from the space and the time of any quantum theory
================================

another thing which people often do not mention when they discuss the incompatibility is that GR is a theory of GEOMETRY, dynamic spacetime geometry, and a quantum GR would have to have GEOMETRIC OBSERVABLES. For example lengths and volumes and areas and angles and stuff like that woud be observables.

And then the Heisenberg would apply and there would be INDETERMINACY of geometric quantities especially very small scale ones.
So inevitably the space geometry of any quantum GR would be CHAOTIC and foamy and fractally down at small scale.
So it wouldn't look like a usual smooth differentiable manifold at very small scale.

So GR offers Quantum a type of space and time it is not used to, for starters, like I discussed. But then if you get over that and DO get a quantum theory of GR then it would turn out to have a space that is not the smooth manifold continuum that GR is used to. So they BOTH have something difficult to assimilate.

... I'm curious as to what it is specifically that we cannot calculate?
...

The problem is deeper than that. There is not a well-defined mathematical context where you can say such and such well-defined thing should be calculated. So then if you could calculate it you would have solved the problem. it isn't like that.

Probably we need
1. a new model of spacetime different from the smooth manifold that Riemann gave us in 1850 ( and so far is the most general idea of the continuum that we have---with quantum field theory not even assimilated to that yet but only working on a selected rigid frame)

2. a new understanding of quantum mechanics and why it works

3. a new understanding of space time and matter---and why matter bends space---and how matter and space are just different aspects of the same thing-----that is: space and matter as EMERGENT properties of the same fundamental microscopic degrees of freedom

 

[muppet]

Thanks for your reply Marcus. So can I ask where the idea of infinite probabilities enters into it?