UPDATE Sept 15, 2018. The analysis below is flawed about a "real phonon". A real phonon is like a pressure wave. There have to be one or more cycles of pressure to talk about real phonons. If we use a rod to implement a single pull or push operation, then it is just half a cycle. It is not a real phonon. We can call half a cycle a "virtual phonon".
----
We could simulate the static electric pull or push between charges using an infinitely rigid rod between them. Just put the appropriate tension on the rod and there you go.
But in special relativity, there cannot be infinitely rigid rods, as they would permit faster-than-light communication. All rods have to be elastic.
In our previous blog post, one of the mysteries was why a Feynman "virtual photon" causes the correct phase shift on the electron wave function.
A possible explanation is that the pull between the nucleus and the electron is actually implemented using an elastic rod. The phase shift is due to the fact that there are degrees of freedom in the elastic rod, and, consequently, the phase of the pull phonon in the rod evolves as the phonon travels along the rod.
The rod is between the nucleus and the electron. The nucleus pulls on the rod to implement the electric attraction.
electron ------->
| elastic rod
|
nucleus Z ----->
The rod hypothesis makes the backreaction on the electric field sensible when we switch on the interaction in some spacetime patch containing the electron. The reaction on the electron is to be pulled by the rod, and the backreaction of the rod is to get stretched by the electron.
In the QED lagrangian, it is unclear what is the backreaction on the electric field if we temporarily switch on the interaction. The effect on the electron Dirac wave function makes sense, but what is the backreaction on the electric field? If the electric field is rigidly determined by other charges in the system and does not have degrees of freedom of its own, then the backreaction is directly on those other charges. The electric field has no role in this.
Electromagnetic waves do have degrees of freedom but how could we implement a pull or a push with them? A low-energy photon which comes from the side and is absorbed by the nucleus-electron system can produce a pull or push. A problem in this model is where is the photon produced? Why would it cause the phase shift given in the Feynman formula?
If the electric field is an elastic rod, what are electromagnetic waves then? Adjacent rods interact. Electromagnetic waves are transverse waves that travel from a rod to an adjacent one. When an electron-nucleus system absorbs a photon, that means the phonon in the rod gives up its elastic energy to the kinetic energy of the electron and the nucleus.
Our blog post from spring 2018 stated that a photon is always created and absorbed by two charges of opposite sign. Contrary to the Larmor formula, an isolated accelerating charge cannot emit electromagnetic radiation. Conservation of momentum requires that a system of opposite charges must create or absorb a photon. The rod model clarifies this point.
Why we have not observed longitudinal electromagnetic waves? The wave is between two charges. Since the electric field goes as 1 / r^2, the energy of longitudinal waves probably falls very fast. Electromagnetic waves, on the other hand, are an efficient method for transmitting energy over great distances.
The rod hypothesis may also clarify momentum conservation in a QED system.
Our hypothesis has similarity to the rubber band model of QCD:
https://en.m.wikipedia.org/wiki/Color_confinement
How do real and virtual pairs come into the picture in the rod model? What is the reaction of the Dirac field to a phonon in the rod? Feynman lets a virtual photon to transform temporarily into a virtual pair. In QCD, the rubber band may break and its endpoints are new quarks.
The rod model can be used in the non-relativistic Schrödinger equation, too. There the force is mediated instantly and we do have an infinitely rigid rod which mediates the force instantly and does not contribute to the phase shift of the electron.
When we have an elastic rod, then the system electron + rod in a path integral gains a phase shift from both the electron and the rod (or, rather the phonon), since the path integral is the product of individual integrals of the rod and the electron. This shows that the non-relativistic Schrödinger equation gives a wrong prediction of the phase shift of the electron. The Feynman formula is right.
UPDATE: A fellow physicist at the University of Helsinki raised the question of what is the rod made of? If we have a rod made of iron atoms, it is the rest mass of iron atoms which makes it possible to transmit momentum and energy along the rod.
We have three ideas about the structure of the rod:
1. The rod contains some of the rest mass of the static electric field. But does the phonon move at the speed of light then?
2. The rod has no rest mass and its mass-energy comes exclusively from the mass-energy of the phonon. It is kind of a Münchhausen trick: the phonon creates the rod that it travels along. But if the rod has a zero mass-energy, then the uncertainty about the position of the rod is infinite, which does not make sense.
3. A path integral considers and calculates processes of classical systems like point particles. We can assume those particles are connected with classical objects like infinitely narrow rods. It is the probability amplitudes and their interference which gives rise to quantum phenomena. Thus, there is no problem in assuming a narrow rod. What about the mass of the rod? We certainly can use a zero mass rod to push or pull in classical mechanics. The elasticity of the rod comes from special relativity. What is characteristic of rods is that we can transmit a high amount of momentum with just a little bit of energy. A real phonon is not bound by the mass-shell rule of a photon. Thus, a phonon is a model for a virtual photon.
Alternative 2 can be seen as a limiting case where we let the atom rest mass go to zero in the rod and let the strength of their interaction go infinite.
Alternative 3 is probably the right solution to the problem.
We can study the rod hypothesis experimentally by letting two charged objects, say marbles, interact through the electric push or pull. In theory, we could measure the speed at which the electric force propagates and measure elastic properties of the interaction.
TODO: calculate the mass-energy of a suitable "quantum mechanical" rod if we want to use the rod to simulate the electric force between the nucleus and the electron. Is the mass-energy close to the mass-energy of the electric field?
No comments:
Post a Comment