In our previous blog post we noted that we must find out the roles of these two fields. This is a fundamental question of quantum field theory.
The Schrödinger equation is about the wave function of a massive particle. The particle may be an electron which moves under a classical electromagnetic field. The wave function is a complex-valued field, where the value is understood as the probability amplitude of the particle.
A radio transmitter and a receiver
Let us have an oscillating electric charge. It might be a radio transmitter. The charge produces classical electromagnetic waves.
Classical electromagnetic waves can be understood as the vectors E and B which obey Maxwell's equations or the electromagnetic wave equation. It is a vector field.
We believe that the radio transmitter sends many coherent photons, and those photons make up the classical electromagnetic wave.
The physical process which we observe is that the oscillator loses energy in packets of size hf, where h is the Planck constant and f is the frequency of the oscillation.
A radio receiver, in turn, absorbs energy in packets of size hf.
Quantum mechanics says that the "effect" of a classical electromagnetic wave can somehow "concentrate" in a small area and make a substantial change to the receiver. We can regard the "concentration" as a particle, a photon.
Classically, the energy in an electromagnetic wave dilutes rapidly when we move further away. Classically, the wave would make another oscillator to swing just a little bit. But in quantum mechanics, the other oscillator can absorb a sizable amount of energy, hf.
It is like somehow, the emitting oscillator would be placed very close to the receiving oscillator, so that the Coulomb field of the emitting oscillator charges can transfer a sizable amount of energy to the receiving oscillator.
Thus, one classical analogue of a photon is a magical process which suddenly moves the emitter very close to the receiver.
Does our classical model explain the collision of a photon and an electron? No. If we would bring the oscillator close to a random electron, the electron could fly to any direction. But a photon carries momentum p which always points away from the transmitter. A better classical analogue of a photon is a small electric dipole oscillator which carries the energy hf and the momentum hf (we assume c = 1).
In the case of a (classical) radio transmitter, we can work with the classical electromagnetic wave, and apply the quantum nature of the wave where needed. There is little need for abstract canonical quantization and creation and annihilation operators of photons.
Feynman diagrams and classical fields in QED
In his "Theory of positrons" paper, Feynman derives the electron propagator (= the Green's function) from the Dirac equation. In the link above, the author notes that the propagator resembles the Green's function for the massive Klein-Gordon equation, but the propagator for the electron is, of course, more complex as gamma matrices are involved.
In the link, the photon propagator is derived from Maxwell's equations. The propagator looks very similar to the massless Klein-Gordon Green's function.
Thus, Feynman works with two classical fields: the Dirac field and the electromagnetic field. The Green's function is the impulse response of the field equation to a Dirac delta source. It is like hitting the field, e.g., a drum skin, with a sharp hammer. Hitting with a hammer is a very concrete classical operation.
Where does the quantum nature of the fields show up in a Feynman diagram?
The input to the diagram is always whole quanta, that is, free particles. The output is free particles.
The virtual photon in Coulomb scattering is just an artifact of our Fourier decomposition of the classical 1 / r potential.
Klein and Nishina handle the absorption of a photon as a perturbation of the classical Dirac field by a classical electromagnetic wave.
Is it so that the quantum nature of things is only visible when the particles are far from each other, that is, in the input and output phases? And all the interaction can be handled with classical Dirac and electromagnetic waves?
Creation and annihilation operators
Let us have a quantum oscillator. We can feed energy into it in packets of size hf.
Adding a quantum of energy can be seen as an abstract "raising" operation. The emission is an abstract "lowering" operation.
The raising operation adds "more" oscillation in the resonance frequency of the oscillator.
In quantum field theory, we can introduce an abstract "creation" operator which adds one quantum of energy to the field. We may regard a photon as an oscillation in a "resonance frequency" of the field. Thus, a creation operator adds one unit of "more oscillation" to the field.
In momentum space, a photon of momentum p "raises" the oscillator associated with that momentum p.
The abstract concepts of creation and annihilation make sense. But are they needed when we study real-world processes? In the real world, working with classical fields seems to be the way.
Unruh/Hawking radiation and the Bogoliubov transformation
Let us have a scalar Klein-Gordon field. The Bogoliubov transformation tries to relate the quantum field states in an inertial frame and in an accelerating frame.
Some algebra with abstract creation and annihilation operators suggests that an accelerating observer should see empty space as containing quanta: Unruh radiation.
Emptiness in this means that an inertial observer sees no quanta.
In our blog posts in 2018 and 2019 we stressed that the existence of Unruh radiation would break conservation of momentum. Vladimir Belinski, Detlev Buchholz and several other authors have said that no Unruh radiation exists, and William Unruh has not been able to refute their arguments. Belinski says that there is no sense in making canonical quantization in an accelerated frame.
The canonical quantization of the scalar Klein-Gordon field is in literature done so that "positive frequencies" are associated with quanta of positive energy, and negative frequencies are forbidden, as they would be quanta of negative energy.
Suppose that our accelerated observer sees a wave packet of positive frequencies. In the frame of the inertial observer, it is a "chirp" and its Fourier decomposition contains also negative frequencies.
The algebra of creation and annihilation operators then tells us that the expectation value of the wave packet for the accelerated observer is > 0. That is, he must see quanta in empty space! Those quanta are Unruh radiation.
For the classical field, negative frequencies are not forbidden. They are solutions which satisfy the scalar Klein-Gordon equation. If the inertial observer sends a classical wave packet which contains just positive frequencies, the accelerated observer will see a classical chirp: its Fourier decomposition contains also negative frequencies.
The error of Unruh and many others is to think that the abstract Klein-Gordon field which we have built from creation and annihilation operators, and where negative frequencies are forbidden, is a real physical field which can be transformed between coordinate frames like a classical field.
And furthermore, that the rules about negative frequencies, which we physicists ourselves introduced to our abstract field, are honored by Nature!
Suppose that we have a laser beam of coherent circularly polarized photons. The photons are right-handed (positive frequency). An accelerated observer will see some of the photons left-handed (negative frequency). There is nothing mystical in those negative frequency photons. They do not pop out of empty space.
The transformation of one quantum of light into an accelerating frame
Suppose that observer A falls freely in the gravitational field of Earth, and sends one photon of right-handed circularly polarized light to observer B far away in space. The photon holds the energy hf, as measured by A.
Observer B sees the light sent by A as a chirp. B may see several different frequencies in that light, and even left-handed photons.
How do we ensure conservation of energy?
The experiment can be understood as a generalized scattering experiment where the 10^50 atoms of Earth interact through gravitation with the one photon sent by A. Like in Feynman diagrams, we may demand that energy is conserved. The scattering is a process of classical fields.
Feynman diagrams cannot describe the bending of a beam of electrons in an electric field because the perturbation approximation is too crude. The ascent of a photon in a gravitational field is comparable to the bending.
We conjecture that the correct way to describe the ascent of the photon is as a classical field, and when B measures individual photons in the chirp, he will see that the sum of their energies matches what it should be so that energy is conserved. In almost all cases, B will measure just a single redshifted photon.
A "quantum" only exists in a wave function collapse?
Above we have argued that all quantum processes can be described as interactions of classical waves.
But we do observe individual quanta in a camera, for example.
Maybe quanta are a result of a wave function collapse? We, as observers, are not waves but particles. We do not see the whole Fourier decomposition of a chirp. We see individual photons in the chirp - that is, particles.
Since there are no measurements done inside a Feynman diagram, all the phenomena in it concern classical fields. Quanta are only present in an observation - when we measure the output of the experiment.
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