In this blog we have been critical of the claimed Unruh effect, where an accelerating observer would see empty space as "warm".
Detlev Buchholz et al. (2012, 2014) have analyzed Unruh radiation in two papers quite recently. They criticize the interpretation that the detector would see "heat".
Vladimir Belinski (1995) holds the opinion that Unruh or Hawking radiation do not exist.
Unruh radiation does not exist
Let us repeat our arguments against Unruh radiation.
1. We do not see a mechanism through which the accelerating machinery of a rocket could excite the radiation detector in the rocket.
2. If empty space would somehow heat up the detector on its own, energy conservation would be violated as heat would be generated from nothing.
3. Turning kinetic energy of the rocket into radiation is not possible because light-speed radiation cannot carry away all the momentum which the rocket loses if kinetic energy is turned into heat.
4. Unruh radiation would be a macroscopic effect: under very great acceleration, the rocket would heat up to, say, a billion degrees Kelvin. There is no such effect in classical physics. Thus, the classical limit of quantum physics would not hold.
5. A static radiation detector on Earth is under constant acceleration in the gravitational field of Earth. If the detector on its own gets excited, from where does the energy come? The Unruh effect would violate the equivalence principle of gravity and acceleration, or alternatively, energy conservation.
Quantum field theoretic derivation of the Unruh effect
wave W of frequency f
in the frame of the rocket
~~~~~~~~~~~~~~~~
● detector
|
<==
rocket
The derivation of Unruh radiation goes as follows.
Let us assume that a detector responds to photons of frequency f. Imagine a wave W of frequency f observed by a detector in the rocket in its accelerating frame.
1. The wave W would be a chirp in an inertial frame. That is, its frequency changes with time in an inertial frame.
2. The Fourier decomposition of a chirp contains a small amount of negative frequencies. We get the Fourier transform in an inertial frame with a Bogoliubov transformation of the wave in the accelerating frame.
3. Canonical quantization of a quantum field only contains positive frequencies.
4. The algebraic machinery of quantum field theory then claims that the negative frequencies in the inertial frame make a non-zero positive contribution to the expectation value of observing the wave W in the accelerating frame.
The error in the above derivation
The correct way to convert the wave W to another (accelerating) frame is to treat W as a classical electromagnetic wave. If W is circularly polarized, then the negative frequencies correspond to the opposite circular polarization. Thus, negative frequencies are very mundane, ordinary physics. There is nothing mysterious in them.
The algebraic machinery of quantum field theory, where only positive frequencies are assigned the status of real quanta, is a human invention. Why would Nature transform waves between different frames using the human-invented machinery? There is a more natural transformation using classical waves.
Another way to say this is that there is no way to quantize waves in an accelerating frame. The machinery of quantization was developed in inertial frames. There is no a priori reason why the machinery would work in an accelerating frame.
Our previous blog posting touched this issue. What is the ontology of quantum field theory? Is W a classical wave, or a quantum field object which transforms into an accelerating frame using some algebraic machinery?
The real Unruh effect: what happens to a laser beam if the laser or the observer is accelerated?
Suppose that a laser sends a right-handed circularly polarized wave. The laser is floating static in a Minkowski space.
An observer is in an accelerating rocket which speeds up toward the laser.
In the static frame, all the photons are right- handed.
But in the accelerating rocket, the observer sees a chirp, and if he has a detector which is only sensitive for left-handed photons, he may observe some.
If the observer absorbs a left-handed photon, he receives some left-handed angular momentum. But the laser beam only contained right-handed momentum. How is this possible?
Let the "laser" be a miniature rotating electric dipole. The dipole loses its angular momentum and radiates the angular momentum away in one or more right-handed photons.
If an observer absorbs a left-handed photon, how can angular momentum be conserved?
Acceleration of a frame is not a perturbative process: we have to quantize using a base of chirps, not plane waves?
● +
| ● antenna
| |
| <==
| rocket
● -
dipole
rotates
Let us make the rotating dipole macroscopic and analyze the classical electromagnetic wave. That is, we analyze the classical limit of the process. The dipole is floating freely in the Minkowski space. It rotates and produces right-handed polarized radiation toward the direction of a rocket.
On the rocket, it is very well possible to absorb left-handed angular momentum from right-handed photons, for example, by using a turbine which turns under the radiation pressure from the incoming radiation. Angular momentum is conserved because the turbine reflects, or scatters, incoming photons.
Let us then on the rocket have an antenna which can absorb a certain frequency. If we accelerate the antenna toward the rotating dipole, then the antenna sees a chirp, and absorbs some left-handed polarized radiation.
How can angular momentum be conserved?
We have a feeling that using the Fourier decomposition to "quantize" the chirp is not the right way to describe the process.
The blueshift of the wave when the rocket accelerates is not a perturbative process, since the wave gets shifted at a 100% probability.
In the classical limit, one can build a dipole antenna whose axis contains a ratchet, and the antenna might be able to extract left-handed angular momentum from the wave. In this case, the right-handed momentum would be absorbed by the rocket through ratchet.
If we have a dipole which can rotate either way, the classical wave will always make it to absorb right-handed angular momentum.
The right way to decompose the chirp to quanta might be to use a base of chirps.
If we have a free electron, then a plane wave is the natural way to decompose a wave packet.
But if the electron is flying under a static electric field, then a natural decomposition consists of bent orbits. Every electron will at a 100% probability fly along a bent orbit. It is not a perturbative phenomenon.
We need to think this in more detail. What is the relation between perturbative processes and a Fourier decomposition with plane waves? Should we decompose non-perturbative processes into chirps?
A path integral is about the orbits of particles. There is no problem if there is a static electric field which bends every orbit. It may be best to abandon plane wave Fourier decompositions in non-perturbative contexts and use a particle model and a path integral.
This realization may be a fatal blow to Steven Hawking's 1975 derivation of Hawking radiation. The derivation fully depends on decomposing a chirp to plane waves and finding negative frequencies in the decomposition. But the process of light passing through a collapsing star is non-perturbative. Using plane waves may be a wrong approach.
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