|Logo of site by name loop insight. |
The insight to take away is that you have to
carefully look for those infinities
The Unruh effect is the predicted, but so-far not observed, particle production seen by an accelerated observer in flat space. It is a result obtained using quantum field theory and does not include gravity, and the particles are thermally distributed with a temperature that is proportional to the acceleration. The origin of the particle production is that the notion of particles, like the passage of time, is observer-dependent, and so what is Bob’s vacuum might be Alice’s thermal bath.The Unruh effect can be related to the Hawking effect, that is the particle production in the gravitational field of a black hole, by use of the equivalence principle. Neither of the two effects has anything to do with quantum gravity. In these calculations, space-time is treated as a fixed background field that has no quantum properties.
Loop Quantum Gravity (LQG) is an approach to quantum gravity that relies on a new identification of space-time degrees of freedom, which can then be quantized without running into the same problems as one does when quantizing perturbations of the metric. Or at least that’s the idea. The quantization prescription depends on two parameters, one is a length scale normally assumed to be of the order of the Planck length, and the other one is a parameter that everybody wishes wasn’t there and which will not be relevant in the following. The point is that LQG is basically a modification of the quantization procedure that depends on the Planck length.
In a recent paper now Hossain and Sadar from India claim that using the loop quantization method does not reproduce the Unruh effect
- Absence of Unruh effect in polymer quantization
Golam Mortuza Hossain, Gopal Sardar
The Unruh effect has not been measured yet, but experiments have been done for some while to measure the non-gravitational analog of the Hawking effect. Since the Hawking effect is a consequence of certain transformations in quantum field theory that also apply to other systems, it can be studied in the laboratory. There is some ongoing controversy whether or not it has been measured already, but in my opinion it’s really just a matter of time until they’ve pinned down the experimental uncertainties and will confirm this. It would be theoretically difficult to claim that the Unruh effect does not exist when the Hawking effect does. So, if it’s true what they claim in the paper, then Loop Quantum Gravity, or its quantization method respectively, would be pretty much ruled out, or at least in deep trouble.
What they do in the paper is that they apply the two quantization methods to quantum fields in a fixed background. As is usual in this calculation, the background remains classical. Then they calculate the particle flux that an accelerated observer would see. For this they have to define some operators as limiting cases because they don’t exist the same way for the loop quantization method. They find in the end that while the normal quantization leads the expected thermal spectrum, the result for the loop quantization method is just zero.
I kinda want to believe it, because then at least something would be happening in quantum gravity! But I see a big problem with this computation. To understand it, you first have to know that the result with the normal quantization method isn’t actually a nice thermal distribution, it is infinity. This infinity can be identified by a suitable mathematical procedure, in which case one finds that it is the zero of a delta function in momentum space. Once identified, it can be factored out, and the prefactor of the delta function is the thermal spectrum that you’ve been looking for. One can trace back the physical origin of this infinity to find it is, roughly speaking, that you’ve looked at the flux for an infinite volume.
These types of infinites appear in quantum field theory all over the place and they can be dealt with by a procedure called regularization that is the introduction of a parameter, the “regulator”, whose purpose is to capture the divergences so that they can be cleanly discarded of. The important thing about regularization is that you have to identify the divergences first before you can get rid of them. If you try to divide out an infinite factor from a result that wasn’t divergent, all you get is zero.
What the authors do in the paper is that they use a standard regularization method for the Unruh effect that is commonly used for the normal quantization, and apply this regularization also to the other quantization. Now the loop quantization in some sense already has a regulator, that’s the finite length scale that, when the quantization is applied to space-time, results in a smallest unit of area and volume. If this length scale is first set to zero, and then the regulator is removed, one gets the normal Unruh effect. If one first removes the regulator, the result is apparently zero. (Or so they claim in the paper. I didn’t really check all their approximations of special functions and so on.)
My suspicion therefore is that the result would have been finite to begin with and that the additional regularization is an overkill. The result is zero, basically, because they’ve divided out an infinity too much.
The paper however is very confusingly written and at least I don’t see at first sight what’s wrong with their calculation. I’ve now consulted three people who work on related things and neither of them saw an obvious mistake. I myself don’t care enough about Loop Quantum Gravity to spend more time on this than I already have. The reason I am telling you about this is because there has been absolutely no reaction to this paper. You’d think if colleagues go about and allegedly prove wrong the theory you’re working on, they’d be shouted down in no time! But everybody loop quantum just seems to have ignored this.
So if you’re working on loop quantum gravity, I would appreciate a pointer to a calculation of the Unruh effect that either confirms this result or proves it wrong. And the rest of you I suggest spread word that loop quantum gravity has been proved wrong, because then I’m sure we will get a clarification of this very very quickly ;)