LQG, polymer quantization and Unruh Effect

Does LQG predict Unruh Effect? Short Answer: There's no reason as yet to believe it doesn't.

Unruh Effect is the name for the phenomenon established by Willium Unruh in 1976 which says that an observer moving with constant acceleration in flat spacetime will observe particle production/ radiation. Moreover, this observer will see blackbody radiation with a fixed temperature which is proportional to the acceleration they are moving with. The result follows from quantum field theory. The basic idea is that the vacuum state, although Lorentz invariant, is not invariant under transformations to non-inertial frames. This means that an inertial and an accelerated observer would differ on what is the vacuum. Consequently where an inertial observer will see a vacuum, his accelerating friend will see a bunch of particles.

Last December, Hossain and Sardar had written a paper  titled 'Absence of Unruh Effect in Polymer Quantization', where it was argued that  the above mentioned radiation completely disappears for scalar fields in Minkowski Space if one quantizes said field by 'polymer quantization' instead of the usual Fock quantization. . Now this 'polymer quantization' is sort of related Loop Quantum Gravity (we'll shortly see how exactly) and the corollary claim seemed to be that this is a prediction from LQG . This would be a striking deviation from semi-classical physics (that is, physics where you take gravity to be classical but other fields to be quantum), and would make LQG's claim as a candidate theory of Quantum Gravity rather suspect. Any observation of Unruh Effect (which however is a tricky issue ) would serve to rule out LQG. 

That would be a pretty huge step for Quantum Gravity, and this paper created a bit of a ripple in the blogosphere. String Theory blogger Lubos Motl and QG phenomenologist Sabine Hossenfelder blogged about the paper. Dr. Hossenfelder suggested that the zero Unruh Radiation that the authors obtained may well have been  the result of dividing by one infinity too many.  LQG founder Carlo Rovelli wrote a paper questioning the claim of Hossain and Sardar. Rovelli argued that the energy scale of Quantum Gravity was too high to affect a 'low energy' effect like Unruh Radiation. He did not however address the question whether polymer quantizing the fields instead of Fock quantizing them could change the 'low energy'  physics sufficiently to make Unruh radiation disappear. 

 I wrote a paper on this recently. But before going into my results, let us understand why the results of Hossain and Sardar, right or wrong, don't put LQG in any sort of jeopardy.  First, I must shed some light on this mysterious 'polymer quantization' I've been referring to and its relation with Loop Quantum Gravity.  

 First, there's a bit of a nomenclatural issue here - the name 'polymer quantization' has been used to describe two entirely different quantizations of scalar fields. The one we are talking about here was first developed in this paper by Hossain, Husain and Seahra. For reasons that will become clear soon, we'll call this the Momentum Space Polymer Quantization (MSPQ). A different quantization for scalar fields, introduced by Ashtekar et al in this paper, is also known in literature as Polymer Quantization. We'll call this Position Space Polymer Quantization (PSPQ).


What do any of these have to do with (Canonical) Loop Quantum Gravity? Unlike String Theory, LQG is not a 'theory of everything' - a theory of some fundamental entity from which all fields spring. No, LQG is really just a quantization prescription which tries to get you a quantum theory for the gravitational field.  It does not as such have anything to say about other matter fields. Except that the usual Fock quantization that we employ for matter fields depends crucially on some background geometry to be present. In other words, our usual quantization method for matter fields requires that the field lives in some spacetime, and the quantum theory carries information about the spacetime these fields live on (For instance representations of the symmetries of the spacetime should be through unitary operators). But if gravity itself is quantized as in LQG, there is no longer any fixed background spacetime. So what LQG is telling us about matter fields is that we need to find a different quantization procedure for matter fields as well, a procedure that does not require a background geometry to be present. That is basically all that LQG demands from matter fields - that they should be quantized in a 'background independent' manner. This is achievable, following strategies similar to the one LQG employs for gravity. For the scalar field, this is achieved by the PSPQ.  Here one constructs a Hilbert Space without taking any information about the geometry of the spacetime.  In technical terms, the inner product in this Hilbert Space is diffeomorphism invariant. The bottomline here is that PSPQ is perfectly compatible with LQG.

MSPQ is an entirely different quantization. Here one makes use of the polymer particle representation, a loop-like quantization for systems with finite degrees of freedom. This is the quantization employed in loop quantum cosmology. Ashtekar et al quantized the non-relativistic particle using this quantization and studied it as a toy model for LQG. A characteristic of this quantization is that to define basic observables one needs to define a length scale, which we will call the polymer scale. MSPQ starts with a free scalar field on a Minkowski spacetime. Mode expanding, one gets a system of uncoupled harmonic oscillators. Now each harmonic oscillator is quantized using the polymer particle representation mentioned above. Hossain, Hosain and Seahra obtained a propagator for scalar fields this way and showed that it violates Lorentz invariance, the violation depending upon the polymer scale. When the polymer scale is taken to zero, the propagator converges to the usual Feynman propagator.

It should however be clear that this quantization is not independent of the background geometry. The mode expansion does depend on it. The only connection of this quantization with LQG is in that all the harmonic oscillators are quantized in a loop-like way.  Importantly, there is no reason at all to take a violationg of MSPQ as a violation of LQG. However,  MSPQ can be treated as an interesting Lorentz violating field theory in its own right.

To summarize the story so far, the field theory which predicted the absence of Unruh Effect is only inspired by LQG but in no way derived from it , which means you shouldn't take a prediction of this theory as a true blue LQG prediction. However it is an interesting field theory on its own.  We'll now drop the ugly 'MSPQ' and refer to this simply as polymer quantization.

Hossain and Sardar had studied Unruh Effect in this theory using the method of Bogoliubov transformation, which is how William Unruh had originally derived it. There's another way of deriving the same result, also by Unruh, where one studies a moving quantum mechanical detector coupled to a scalar field in a vacuum state. Unruh showed that if the detector moved with constant velocity, it can never make a transition from a lower energy state to a higher energy state. In other words it will never absorb a field quantum. That is expected, as for this detector the field is in the vacuum state and therefore in no position to lend energy to anyone.  However if it accelerates, there's a non-zero probability that the detector will absorb a field quantum. Moreover, Unruh showed for a detector moving with constant acceleration that for the detector to be at equilibrium with the field (that is, when the probability of absorbing and emitting field quanta are equal) the distribution of the energy eigenstates of the detector will be according to the Boltzmann distribution. This is the distribution associated with a body at a constant temperature. So the detector will notice a temperature which, Unruh showed, would be proportional to its acceleration.

So I decided to check whether Unruh Effect did vanish for polymer quantization using this derivation, now coupling the detector to a polymer quantized field. I managed to show that it doesn't generally vanish. However I didn't manage to estimate the probability of absorption and emission. This is due to my lack of expertise in numerical methods. However, it should be a straightforward exercise for someone with expertise in this area. It would be interesting to see what exactly the distribution of energy eigenstates of the detector will be in this case. It is unlikely that it will be Boltzmann, and my guess would be that the detector observes radiation which is non-thermal.

 So, nothing too weird there. What's more interesting is that even inertial observers will detect radiation. This is not surprising when you recall that the theory breaks Lorentz invariance. The Unruh effect arises because inertial and non-inertial observers cannot agree on the vacuum. What's vacuum to one is full of particles to the other. Similarly when a theory is not invariant under Lorentz boosts, inertial observers moving with different velocities will also disagree on the vacuum. In this case the quantization chooses out a preferred frame. If the detector is in this preferred frame, it won't ever absorb field quanta. But if it is moving with constant speed then it can in principle see radiation. What I show in the paper is that it indeed happens for the polymer quantized scalar field - it does detect radiation.

What's more interesting is that the radiation does not disappear even if you keep making the polymer scale smaller and smaller. This means that the polymer quantized theory gives a result that is quite widely divergent from the usual field theory result which should in principle be testable even at low energies! It is quite possible that the present results have already ruled out this polymer quantization. In any case, this will be a strong test for the polymer quantized field theory.

For more details, check out the paper

Update: There's a recent paper by Hosain and Louko on the same issue with pretty much identical approach. I haven't gone through it carefully as yet, but their result supports mine and they have done significantly more numerical work which should make contact with phenomenology possible.