William Unruh and several other authors have tried to apply quantum field theory to accelerating frames, but they have not done a careful analysis of conservation momentum and energy, the existence of a sensible classical limit, and some other cornerstones of traditional quantum field theory.
Actually, the term quantum field "theory" is misleading, because many central problems of the framework, like vacuum stability, remain open. It is not a theory in the sense of mathematics, but rather a toolpack of heuristic algorithms.
In the previous blog post we considered two thought experiments, one of which was:
Thought experiment 1. An electron statically supported in gravitational field by impinging photons.
e electron
↑ support
____________
Earth
We left open the question what a static observer will see.
Double Compton scattering
In Thought experiment 1 above, let us perform the scattering experiment in a freely falling frame, such that it is an inertial frame.
Definition 2. In a Feynman diagram, a virtual particle is any particle whose energy and momentum do not match any free particle in vacuum. A real particle is such that its energy and momentum in the diagram could match a free particle. Real particles are said to be on-shell and virtual particles are said to be off-shell.
A virtual particle can return to the "shell", for instance, by emitting as a photon the the extra energy it has.
photon A e photon B
》 / 《
《 / 》
》 /《
《 /~~~~~~~~
《 / | virtual photon
》 \ | momentum p
《 \ ~~~~~~
》 \
《 \
》 \
photon A e electron
Diagram 3.
In the Feynman diagram above, time flows upward.
An electron emits a virtual photon with momentum p, which takes the electron off-shell, that is, the electron is also virtual after the emission, its kinetic energy and momentum do not match. The virtual electron collides with photon A.
After the collision, the electron absorbs back the virtual photon. The electron is still off-shell because it has too much kinetic energy after the collision. To get back as a real electron, the electron has to emit the extra energy it has as photon B.
The above process is called the double Compton scattering. Ordinary Compton scattering does not produce photon B. The virtual photon in the diagram is sometimes called a "self-energy" photon, and the diagram above calculates a "self-energy" correction to probability amplitudes.
Classically, an electron that interacts with a wave in the electromagnetic field, makes the combined system electron & field nonlinear. The wave that scatters from the electron will have a very complex form.
Conjecture 4. We conjecture that at low energies, double Compton scattering and other similar quantum electrodynamical processes can be treated as fully classical phenomena. In the classical analysis, the electromagnetic wave replaces the wave function Ψ of the photon. The square of the electromagnetic field replaces the Born probability |Ψ|^2. At high energies, the electron has to be modeled with the Dirac relativistic wave equation.
The classical analysis of the scattering does not determine where the quantums of energy, that is, the photons, will be observed. We only obtain a probability distribution, or an interference pattern of electromagnetic waves. To make the process deterministic, one might use a de Broglie - Bohm type hidden variable interpretation, where markers that designate actual particles "sail" on the waves without affecting the dynamical behavior of the waves. In a future blog post we will elaborate on this idea.
If Conjecture 4 is true, that explains why optical phenomena, like the double slit experiment, can be handled fully classically besides the usual quantum mechanical wave function treatment.
The scattering matrix S for double Compton scattering has been calculated in several publications, e.g.:
Radiative Corrections to Compton Scattering
L. M. Brown and R. P. Feynman
Phys. Rev. 85, 231 – Published 15 January 1952
https://journals.aps.org/pr/abstract/10.1103/PhysRev.85.231
Let us now return to Thought experiment 1. An observer in the laboratory will see photons B as well as photons A scattering from the statically supported electron. What does a static observer on Earth see? Recall that the laboratory is falling freely, which means that the Earth static observer appears to accelerate upwards in the laboratory frame. Could it be that a static observer cannot detect photons A or B?
The question is how an accelerating observer will interpret the photons that an inertial observer sees to come out of the scattering process.
Photon in an accelerating frame
In Thought experiment 1, photons are emitted in the inertial laboratory frame. A photon can be modeled as a wave packet.
When the wave packets go far from the electron, they move under an essentially linear wave equation, and can be Fourier-decomposed into plane waves.
How does an accelerated observer see an electromagnetic wave packet?
Definition 5. An absorption detector of electromagnetic waves is a system where an electron is in a bound state and it has several metastable energy levels besides the ground state. A hydrogen atom, for example, is an absorption detector. A field detector is a free electric charge whose movement we can measure.
Classical electromagnetic waves are real-valued and contain the same amount of "positive" and "negative" frequencies. That is, at each moment t, the waves are of the form:
E(r, t) = ∫ f(⍵) * e^(i(k ⋅ r + ⍵ t)) + conjugate(f(⍵)) * e^(-i(k ⋅ r + ⍵ t)) d⍵,
where ⍵ is the angular velocity of the oscillation, k is the wave number vector, r is a vector in 3D space, t is time and the function conjugate(x) returns the complex conjugate of x. The sum above is real-valued because the sum is of two complex conjugate numbers. The function f is the Fourier decomposition of the wave into plane waves of different frequencies.
Definition 6. If we have an accelerating source of radiation, then a wave packet w(r, t) sent by this source will be distorted in an inertial frame.
We assume that the waveform as a function of the proper time of the source t, w(R, t) measured at a point R close and static relative to the source, will stay the same for an inertial source and an accelerating source. That is, the acceleration does not affect the output of the source.
Let w'(r, t) be the waveform in the inertial frame. We say that the distorted form w' is a result of the canonical transformation of w for this acceleration.
Conversely, if we have an accelerating observer, and a wave packet w(r, t) in an inertial frame, let w'(r, t) be the waveform that would have resulted if the source at the time of emission would have accelerated like the observer now, but in the opposite direction. Also in this case we say that w' is the result of the canonical transformation of w for this acceleration.
In some cases, Definition 6 is equivalent to the geometric optics approximation used in electrodynamics.
Definition 6 is very complicated, but still vague. It is very hard to do physics in an accelerated frame. The safe and most reliable way is to work in an inertial frame. In the case of wave packets, the Huygens principle makes their propagation complex. It is very hard to calculate the canonical transformation exactly. A rough approximation, however, is easy to calculate, if one just maps the proper time of the source to the proper time of the observer using a light speed signal.
Claim 7. An accelerated observer "sees" an electromagnetic wave packet as a classical real-valued electromagnetic wave. We can do a canonical transformation to the wave packet and use the Fourier transformation to decompose the real-valued wave packet into a spectrum of real-valued plane waves. An absorption detector carried by the observer will approximately detect this spectrum.
Note that Claim 7 again resurrects the "signals from the future" problem of Larmor radiation, because in order to do the Fourier transformation, we have to know the form of the whole wave packet, and that will we know only after we know the acceleration of the observer also in the future. The word approximately contains many sources of errors.
William Unruh, Stephen Hawking, and several others have claimed that photons should be modeled as purely positive frequency probability amplitude wave packets and that the packets should be manipulated with the Bogoliubov transformation when we switch to an accelerating frame:
https://en.m.wikipedia.org/wiki/Bogoliubov_transformation
A problem in the Unruh et al. approach is that when we do the canonical transformation, positive frequency wave packets become a sum of positive and negative frequencies. Negative frequencies come by because an accelerating observer sees a standard wave packet as a "chirp" due to the Doppler effect. A chirp in radar technology means an oscillation whose frequency goes up or down as time passes. One cannot build a chirp from purely positive frequencies.
In classical waves, there is no problem with the Doppler effect: real-valued waves will stay real-valued in an accelerating frame. On the other hand, if we model photons as purely positive frequency waves, we have no sensible representation for a chirp, that is, no classical limit that would be a chirp.
Based on Claim 7, we can now answer the question how a static observer sees photons A and B in Thought experiment 1: he will see them as real-valued wave packets that are chirps. He can calculate the spectrum with the Fourier transformation.
Particle in an accelerating frame
In a very simple case, we can describe a classical system completely at a time t with two real-valued parameters: the position of a particle on the x axis and its velocity v:
particle • --> v
0 ------------------------------> x axis
We can then attach a complex-valued probability amplitude to each classical configuration. The probability amplitude is a complex number of a fixed absolute value, and the value will rotate around the origin of the complex plane as time passes. This is the usual Feynman path integral way of thinking. The angular velocity of rotation depends on the total energy of the configuration. Since the energy is, by definition, always positive, the rotation will happen to the counterclockwise direction in the complex plane - in this sense, the "probability amplitude wave" is always positive frequency.
The probability amplitude wave lives in the abstract configuration space, with time t as an additional coordinate. The probability amplitude wave does not live in the simple spacetime coordinate space (x, t).
In the simple case above, we can usually work in quantum mechanics with a wave function which has just x and t as its parameters:
Ψ(x, t).
The wave function appears to live in the simple spacetime coordinate space (x, t).
Suppose that we prepare the system such that our particle is described by a wave packet. It is then a sum of infinitely many positive frequency plane waves.
If we try to switch to an accelerating frame and transform Ψ(x, t) with the canonical transformation, we end up with a chirp, which contains also negative frequency plane waves, not just positive. Does that mean that an accelerated observer may measure the particle to have a negative kinetic energy? That is nonsensical, which shows that we cannot do meaningful quantum mechanics in an accelerating frame by transforming a wave function with the canonical transformation.
We can do meaningful classical mechanics in an accelerating frame with the canonical transformation. If we have a classical electromagnetic wave packet, we can do the canonical transformation to it and we obtain its frequency spectrum as seen by an accelerated observer.
The error of Unruh, Hawking, and others, is that they have mixed the abstract configuration space & time t of quantum mechanics with the simple timespace (x, t) of classical mechanics and have applied the canonical transformation to a wave packet of quantum mechanics. The Bogoliubov method then shows that one can get negative frequency components of the transformed wave packet "for free", that is, the expectation value for the original wave packet is > 0.
That error is the origin of the strange claims that the vacuum would appear to contain particles to an accelerating observer, or that a black hole could evaporate in a non-unitary way.
Since the configuration space & time is an abstract framework, it actually makes no sense to talk about accelerating the whole framework. Accelerating in which coordinate, and what is accelerated?
As a parallel, we may think of a movie in a movie theater. It makes sense to say that a car in the movie is accelerating on a road, but what would it mean to accelerate the whole movie to some spatial direction? The movie is the framework, the car is an object in that framework for which "acceleration" is defined within that framework.
What if we accelerate an electron with electrons?
We can model this as a scattering experiment.
e e1 e
\ | /
\ | /
\| /
/| /
/ | /
/ |\
/ | \
/ | \
/ | \
/ | \
e e1 e
In the diagram above, time flows upward. Electron e1 is being accelerated by bombarding it with other electrons.
When an electron collides with another electron, photons may be emitted. Those photons can be interpreted as electron-electron bremsstrahlung:
https://en.m.wikipedia.org/wiki/Bremsstrahlung
Bremsstrahlung has a spectrum that is quite similar to a thermal spectrum. One could associate a temperature with bremsstrahlung. An electron in bremsstrahlung is able to convert some of it kinetic energy to a photon because another electron will absorb the extra momentum of the electron. Recall that a photon cannot carry away all the extra momentum.
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