Let us have a very heavy rotating electric dipole which falls freely in the gravitational field of Earth. We assume that the charge in the dipole is very small, so that it only emits one photon per second. The dipole sends right-handed photons to space.
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Earth rotating dipole
dipole in antenna
free fall in space
We have a dipole antenna far away in space. It receives some of the energy, momentum, and angular momentum sent by the free falling dipole.
The question is: how do we quantize the emission / absorption process? We believe that the absorber receives energy, momentum, and angular momentum in "packets". When are the packets formed and how do they behave in the gravitational field?
The length scale problem again
Distortion of electromagnetic waves is clearly related to the length scale problem. We have been blogging about the length scale problem in bremsstrahlung. How can a sharp classical wave whose form has features of size 10^-15 m appear as a nice sine wave of a wavelength 2 * 10^-12 m to the absorber of the photon?
We suggested that the uncertainty of the position of the electron causes destructive interference which wipes away all frequencies except the sine wave 2 * 10^-12 m frequency, and constructive interference moves the energy to the 2 * 10^-12 m frequency.
But now we realize that by making the electron very heavy and forcing it to make a turn of size 10^-15 m, we can eliminate the destructive interference. Does the spectrum of bremsstrahlung then contain waves whose wavelength is ~ 10^-15 m? Probably yes. A very heavy particle possesses a lot of kinetic energy it can afford to lose. We assumed that the electron in bremsstrahlung radiates away most of its kinetic energy. Since the electron did not have much kinetic energy, it could only produce a wavelength ~ 2 * 10^-12 m.
Are there different kinds of photons of the same energy and polarization?
In quantum mechanics we believe that photons of the same energy and polarization are identical. They behave in the same way regardless of their birth process. A 2 eV photon which is born in bremsstrahlung is identical to a 2 eV photon which is born in a state transition of an atom.
In the case of the falling dipole, naively the photon gets "stretched", and becomes a chirp. Could the stretching somehow show up at the absorber?
Classical aspects
Since the rotating dipole is very heavy, we can almost exactly know the phase of the wave sent by it. Also, we can know the location of the dipole very precisely.
The classical wave received by the absorber is a chirp where the frequency falls with time. Its Fourier decomposition contains all kinds of frequencies, and even "negative frequencies" which have left-handed angular momentum.
An accelerating observer meets an electromagnetic right hand polarized plane wave
The observer in a rocket will see the wave a as chirp. Its Fourier decomposition contains various "positive frequencies" which correspond to right hand polarized waves, and a small portion of "negative frequencies" which are left hand polarized.
Let us assume that the incoming plane wave has a very low intensity and contains just one photon per second.
How is it possible that an antenna held by the accelerating observer absorbs a left-handed photon, while the incoming wave only contains right-handed angular momentum?
Let us look at the classical limit. If an antenna somehow absorbs left-handed angular momentum, it either:
1. has to transfer right-handed angular momentum to the rocket, or
2. scatter the incoming wave in such a way that the scattered flux contains right-handed angular momentum (linear or in its spin).
The antenna might be a wind turbine which uses the radiation pressure to turn the turbine left. The right-handed momentum flows through its axis to the rocket, and right-handed momentum is also sent away in the reflected wave.
We suggest that one can calculate the response using classical wave equations. Though, of course, one has to use quantum mechanics to describe the absorption of a photon to an antenna.
The output wave can be decomposed into Fourier components. Each component may contain one or more output quanta. The quantization should be done in such a way that energy, momentum, and angular momentum are conserved.
How do we determine the probabilities for various collections of output quanta? We do not know. There might be a Feynman diagram like method for the calculation.
The wavelength of left-handed components is typically much larger than the main frequency of the right-handed components. If an antenna somehow is able to absorb a left-handed photon, then a lot of excess energy has to be scattered away. The outgoing wave may easily contain enough right-handed angular momentum, so that angular momentum is conserved.
In the previous blog posting we mentioned the "real Unruh effect", which is a paradoxical absorption of a left-handed photon from a beam of right-handed photons. It looks like the real Unruh effect occurs in nature. It happens in the classical limit and does probably happen also at the quantum level. The effect is extremely weak and we do not expect an experimental confirmation.
An accelerated emitter of a right hand polarized electromagnetic wave and an inertial observer
The Fourier decomposition of the emitted wave contains various frequencies and a small portion of negative frequencies.
If the observer absorbs a left-handed photon, it is typically of a much larger wavelength than the right-handed main component of the wave. We may assume that the emitting process somehow split a photon in two or more splinters, one of them having the left-handed angular momentum.
We suggest that one can calculate the response using classical wave equations, like in the previous section.
Deformation of an arbitrary electromagnetic planar wave under a gravitational field
Suppose that we aim a beam of laser light toward a large mass. The waveform is originally planar. The waveform gets distorted in a complex way by the gravitational field. Let the frequency of the laser be f.
The Fourier decomposition of the wave becomes very complex in the distortion process. The frequency f components have almost all energy, but they are mixed with longer wavelengths.
How do we interpret the longer wavelengths? Did photons collide with gravitons and split into smaller energy packets? Those smaller packets would have a longer wavelength.
How do we interpret the redshift of a laser beam sent up from the surface of a planet? A photon collides with a graviton, the graviton steals some energy and goes to build up the energy in the static gravitational field of the system?
Redshift is not a perturbational effect because every photon gets redshifted with a 100% probability. Maybe a non-perturbative process is better handled as a classical wave?
We might assume the following:
The appearance of a longer wavelength is associated with splitting of a photon in a "collision" with a "graviton". The rest of the energy in the original photon is contained in the other splinters of the photon.
We are not sure if there is a sensible way to explain a non-perturbative process in terms of collisions of quanta. That is why we put the quotation marks "" above. We suggest that the correct method is to calculate the distortion of the wave using classical wave equations like in the two previous sections.
The error of William Unruh and Stephen Hawking
We believe that the correct way to transform electromagnetic waves to an accelerating frame is to use classical field equations. Then "negative frequencies" is quite a mundane phenomenon of flipped handedness of a photon.
Unruh, Hawking, and several other researchers have thought that the quantum field, with its creation and annihilation operators, should be transformed to an accelerating frame. The algebra of those operators then leads to strange phenomena like Unruh and Hawking radiation.
In a Feynman diagram, "creation" of a photon happens by using the Green's function of the massless Klein-Gordon field. Annihilation is the time reverse of creation.
Creation is like hitting a drum skin with a sharp hammer. It is the impulse response. We need to analyze in detail if anything strange, like the Unruh effect, can happen under acceleration in the drum skin model. That is unlikely, since the drum skin should behave in the classical way.
Quantum field theory in curved spacetime
Daniel Kastner and Rudolf Haag developed a local quantum field theory approach since the 1980s. We have to compare our ideas to their framework.
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