LQG predicts the Unruh Effect. Comment to the paper "Absence of Unruh effect in polymer quantization" by Hossain and Sardar
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Comment to the paper “Absence of Unruh effect in polymerquantization” by Hossain and Sardar
Carlo Rovelli
CPT, Aix-Marseille Universit´e, Universit´e de Toulon, CNRS,Samy Maroun Center for Time, Space and the Quantum.Case 907, F-13288 Marseille, France. (Dated: December 30, 2014)A recent paper claims that loop quantum gravity predicts the absence of the Unruh effect. I showthat this is not the case, and take advantage of this opportunity to shed some light on some relatedissues.
The Unruh effect is the fact that a standardthermometer moving at constant acceleration a in the vacuum state of a quantum field onflat space measures a temperature T = a ~ / π ,in units where c = k Boltzaman = 1. The paper[1] claims that Loop Quantum Gravity (LQG)predicts the absence of this effect as a conse-quence of the fact that LQG modifies a quan-tum field theory at high-energy (short dis-tance). The claim is wrong, because the Un-ruh effect is a low-energy (large distance) phe-nomenon, not a high-energy (short distance)one. An accelerated local detector interactingwith a quantum field field tests the proper-ties of the field only at length scales of order L ∼ /a . The phenomenon is not sensible tothe behaviour of the field at shorter scales.In fact, there are a number of derivationsof the Unruh effect, and some of these in-volve only the low-energy sector of the theory.Other derivations, like Unruh’s original one, gothough the high-energy sector in a convolutedmanner, and require an infinite renormaliza-tion. The need for an infinite subtraction is aweakness of the derivation method, not a fea-ture of the phenomenon itself, which can bederived in a straightforward manner. Below,for completeness, I recall one manner to derivethe Unruh effect which is explicitly insensibleto the short-scale behavior of the field.In the paper [1], the authors recognisethat LQG modifies only the short-distance be-haviour of the field, not its large-distance be-haviour. This implies that the prediction ofthe Unruh effect cannot be modified, at leastuntil the accelerations reaches the scale where quantum gravitational effects become relevant,namely the Planck acceleration a ∼ / √ ~ G , ascale where, as argued in [2], it is accelerationitself to be bound by LQG effects. So, I take a ≪ / √ ~ G below.I give here a brief derivation of the UnruhEffect from LQG (in part following [3], see also[4]). A thermometer can be modelled by atwo-state system with an energy gap ǫ betweenits two energy eigenstates ( | i , | i ), interactingwith a generic self-adjoint observable A of asystem via a simple interaction term such as V ( t ) = g ( | ih | + | ih | ) A ( t ) . (1)where g is a small coupling constant. The am-plitude for the thermometer to have moven upat time t , with the system going from an initialtime independent state | i i to a final state | f i is given to first order in g by Fermi’s goldenrule: W + ( t ) = i ~ g Z t ∞ dt ′ h , f | A ( t ) | , i i (2)= i ~ g Z t ∞ dt ′ e i ( t − t ′ ) ǫ h f | e − iHt A ( t ′ ) | i i . The probability is the square of the amplitudesummed over the final states, which gives P + ( t ) = g ~ Z t ∞ dt ′ Z t −∞ dt ′′ e i ( t ′ − t ′′ ) ǫ h i | A ( t ′ ) A ( t ′′ ) | i i . Assuming time independence, the integranddepends only on the difference s = t ′′ − t ′ P + ( t ) = g Z t ∞ dT Z T −∞ ds e isǫ h i | A ( s ) A (0) | i i . and the probability to move up per unit timeis therefore, in the large-time limit, p + = P + ( t ) dt = g Z ∞−∞ dt e isǫ h i | A ( s ) A (0) | i i , namely the value in ǫ of the Fourier transform˜ f AA of f AA ( t ) ≡ h i | A ( t ) A (0) | i i . (3)Repeating the same calculation for the proba-bility to go down, we obtain the value of ˜ f AA in − ǫ . Now, if the interaction does not dis-turb the system much and if the ratio of thetwo happens to be p + p − = e − βǫ , (4)then the equilibrium distribution for the detec-tor will be Boltzmannian, at inverse tempera-ture β . Therefore we find that the conditionfor the thermometer to measure the inversetemperature β is that˜ f AA ( − ǫ ) = e − βǫ ˜ f AA ( ǫ ) . (5)In Fourier transform, this reads f AA ( − t ) = f AA ( t − iβ ) . (6)which can be recognised as a form of the cel-ebrated KMS condition, which characterisesequilibrium [5].Armed with this general tool, it is now easyto derive the Unruh effect. If the detectormoves in a quantum field in the vacuum state,following a trajectory x ( s ), where s is theproper time along the trajectory, and A is thefield operator, then f AA ( s ) = h i | A ( s ) A (0) | i i = h | φ ( x ( s )) φ ( x (0)) | i (7)is simply the two-point function of the fieldtheory, along the trajectory. The trajectory ofa detector moving at constant acceleration inMinkowski space is x ( s ) = (cid:18) a sinh( as ) , a cosh( as ) , , (cid:19) . (8) Therefore it follows from the previous generaldiscussion that a constantly accelerated detec-tor sees a temperature β if the two-point func-tion of the quantum field along the trajectory f ( s ) = h | φ ( x ( s )) φ ( x (0)) | i (9)satisfies the KMS condition f ( s ) = f ( − s + iβ ) . (10)or equivalently, in Fourier transform,˜ f ( − ǫ ) = e − βǫ ˜ f ( ǫ ) . (11)Observe that this is a condition on the Fouriercomponents of the two point function at theenergy scale at which the thermometer works.Short distance physics plays no role here.For a free massless field, the two point func-tion is proportional to the square of the inverseof the 4-distance. h | φ ( x ) φ (0)) | i ∼ | x | (12)Along the trajectory: f ( s ) = h | φ ( x ( s )) φ ( x (0))) | i ∼ | x ( s ) − x (0) | (13)This can be easily computed from (8), giving f ( s ) = 2 a (cosh( as ) − , (14)which satisfies the KMS condition (10) withthe Unruh temperature a/ π . This provesthat an accelerated detector in the vacuum of afree massless theory measures the Unruh tem-perature.In LQC the two-point function at the scaleof the acceleration is not affected by quantumgravitational corrections, since these becomerelevant only at the Planck scale. In particu-lar, the two-point function for the perturbativeexcitations of the gravitational field has beencomputed in covariant quantum gravity for theLorentzian theory in [6–8]. The result of theseworks is that at the lowest order the two-pointfunction converges to the free one, in the largedistance limit. Here “large” means large withrespect to the Planck scale. Therefore it isclear that LQG has no effect whatsoever onthe prediction of the Unruh effect.What therefore went wrong in [1]? The an-swer is interesting. As observed by SabineHossenfelder [9], trying to compute a quantityin a finite theory like LQG by means of aninfinite renormalisation is, besides being per-verse, also a doubtful procedure. One risks tosubtract “one infinity too much” [9]. By com- plicating a simple low-energy effect, expressingit in terms of the difference between divergentsums, one gets things wrong.This is probably a general lesson in quan-tum gravity. The theory is finite, becauseof the Planck scale cut off provided by thephysical discreteness of the geometry. Infiniterenormalisation calculation, in this context,obscure, rather than clarifying, the physics. [1] G. M. Hossain and G. Sardar, “Absence ofUnruh effect in polymer quantization,” arXiv:1411.1935 .[2] C. Rovelli and F. Vidotto, “Evidence forMaximal Acceleration and SingularityResolution in Covariant Loop QuantumGravity,” Physical Review Letters (2013)091303, arXiv:1307.3228 .[3] E. Bianchi, “Entropy of Non-Extremal BlackHoles from Loop Gravity,” , arXiv:1204.5122 .[4] C. Rovelli and F. Vidotto,
Introduction tocovariant loop quantum gravity . CambridgeUniversity Press, 2014.[5] R. Haag,
Local Quantum Physics: Fields, Particles, Algebras . Springer, Aug., 1996.[6] E. Bianchi and Y. Ding, “Lorentzian spinfoampropagator,”
Phys.Rev.
D86 (2012) 104040, arXiv:1109.6538 .[7] M. Han and M. Zhang, “Asymptotics ofSpinfoam Amplitude on Simplicial Manifold:Lorentzian Theory,”
Class.Quant.Grav. (2013) 165012, arXiv:1109.0499 .[8] M. Han, “Covariant Loop Quantum Gravity,Low Energy Perturbation Theory, andEinstein Gravity,” Phys. Rev. D (2014)124001, arXiv:1308.4063 .[9] S. Hossenfelder, “Backreaction.” http://backreaction.blogspot.ithttp://backreaction.blogspot.it