nappropriateness of the Rindler quantizatio

It is argued that the Rindler quantization is not a correct approach to study the effects of acceleration on quantum fields. First, the "particle"-detector approach based on the Minkowski quantization is not equivalent to the approach based on the Rindler quantization. Second, the event horizon, which plays the essential role in the Rindler quantization, cannot play any physical role for a local noninertial observer.

There is a wide belief that the properties of a quantum field seen by an uniformly accelerated observer are correctly described by the Rindler quantization, i.e., by the quantization based on the Rindler coordinates. On the other hand, it is known that the Rindler quantization suffers from certain problems. For example, the Rindler quantization is unitarily inequivalent to the Minkowski quantization. However, this fact, being an artefact of the infinite volume @gerlach, is only a technical problem which is not really serious. Recently, it has been argued that the Rindler quantization suffers from a more serious problem @fedotov, namely, that the boundary condition on the horizon, required by the Rindler quantization, actually makes the Rindler quantization inconsistent outside the left and right wedges.

When quantization based od Rindler coordinates was discovered @full, it was argued that it could be appropriate to the physical situation of an impenetrable wall located on the horizon. However, in @unruh and most other papers that apply the Rindler quantization it is assumed that the horizon itself plays a physical role because it serves as a physical boundary that affects the properties of quantum fields. In this letter we argue that such an assumtion is groundless.

There is also a wide belief @bd that the Unruh effect @unruh, i.e., the thermal properties of the Minkowski vacuum seen by an uniformly accelerated observer, can be equivalently described with the Minkowski quantization, using a model of a "particle" detector. In this letter we show that the "particle"-detector approach to the Unruh effect based on the Minkowski quantization is not equivalent to the Rindler-quantization approach.

Let us start with the discussion of a "particle" detector. For definiteness, we use the model of a monopole detector described in @bd. Assuming that the detector and the field are in the ground state $|0,E_0\rangle$ initially, the first order of perturbation theory gives the amplitude for the transition to an excited state $|k,E\rangle$: $$\label{1}
A(k,\Delta E)=\bar{g} \int_{-\infty}^{\infty} d\tau \, e^{i\Delta E\,\tau}
\langle k|\phi(x(\tau))|0\rangle \; ,$$ where $\bar{g}=ig\langle E|m(0)|E_0\rangle$, $g$ is a real dimensionless coupling constant, $m(\tau)$ is the monopole moment operator, $x(\tau)$ is the trajectory of the detector, $\Delta E=E-E_0$, and $$\label{2}  
\langle k|\phi(x)|0\rangle =\frac{1}{\sqrt{(2\pi)^3 2\omega}} \,  
e^{i(\omega t -\bf{k}\cdot\bf{x})} \; .$$ We compare the predictions that can be obtained from this model based on the Minkowski quantization with the predictions that result from the Rindler quantization @unruh2 [@muller].

The Rindler-quantization approach predicts that the absorption of a Rindler particle by the accelerated atom will be seen by an inertial observer as an emission of a Minkowski particle. These are two descriptions of the same event seen by two different observers. According to this interpretation, the accelerated observer does not observe the emission of the Minkowski particle. On the other hand, in the approach based on (\[1\]), both the inertial and the accelerated observer can observe both the jump to a higher atom level <span>*and*</span> the emission of the Minkowski particle, as two different events. Obviously, the two approaches are not equivalent.

For a uniform acceleration, the two approaches agree in the prediction of a thermal distribution for $\Delta E$. However, even this partial agreement of the two approaches does not generalize when the uniform acceleration is replaced by a more complicated motion @padm3.

We now see that at least one of the two approaches must be wrong. Below we argue that it is the Rindler quantization that is wrong.

Even if one does not regard the mentioned technical problems @gerlach [@fedotov] with the Rindler quantization as really serious problems, one cannot deny that the event horizon plays the essential role for understanding the physical consequences of the Rindler quantization. Below we show that the event horizon does not correspond to any physical entity that could influence the properties of the fields seen by an accelerated observer, making the Rindler quantization physically meaningless.

Let $x'$ be the Fermi coordinates of an observer at $\mbox{\bf{x}}'=\bf{0}$ moving arbitrarily in flat spacetime. If the observer does not rotate, the corresponding metric is given by $g'_{ij}=-\delta_{ij}$, $g'_{0i}=0$, and @nels [@nikolic1] $$\label{1.1}
g'_{00}(t',\mbox{\bf{x}}')=(1+\mbox{\bf{a}}'(t')\cdot\mbox{\bf{x}}')^2 \; ,$$ where $\mbox{\bf{a}}'$ is the proper acceleration. From (\[1.1\]) we see that the Fermi coordinates of an accelerated observer possess a coordinate singularity at a certain $\bf{x}'$. However, in general, this coordinate singularity does not correspond to any physical boundary. Only $\bf{a}'(\infty)$, defining the event horizon, defines a physical boundary. However, in real life, acceleration never lasts infinitely long. And even if it does, the event horizon does not have any physical influence on a measuring procedure that lasts a finite time. On the other hand, a realistic measuring procedure always lasts a finite time. Therefore, the horizon cannot play any physical role.

Actually, the correct interpretation of the Fermi coordinates, and therefore also of the Rindler coordinates as their special case, is purely local @nikolic1 [@nikolic2], so they are not appropriate for quantization which requires a global approach to describe the EPR-like correlations.

We do not see a reason to doubt that the Minkowski quantization is the correct approach to quantization in a flat background. In particular, it is not in contradiction with the principle of general covariance. The only problem, not yet satisfactorily solved, is how to generalize it to a curved background.

Acknowledgment {#acknowledgment .unnumbered}
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This work was supported by the Ministry of Science and Technology of the Republic of Croatia under Contract No. 00980102.

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{'source': 'arXiv_out/gr-qc0103108'}