Our "sharp hammer" hypothesis from last week suggests that the energy of a static Coulomb field is zero! The field consists exclusively of virtual photons which contain spatial momentum but no energy. Why would the field then contain any energy of its own at all?
The sharp hammer hypothesis is that the static Coulomb electric field consists of the Green's function of the electromagnetic field applied at the location of the point charge at infinitesimal time intervals.
| knock knock knock
|
v
#
#======= sharp hammer
v
_______________ drum skin
●
"point charge"
The classical analogue is that we hammer a drum skin at the same location at very short time intervals. A permanent, time-independent, depression is created in the drum skin. It is like a static Coulomb field.
In the classical case, there is energy stored in the static deformation of drum skin.
Classically, in newtonian mechanics, we can explain the movements of charges without assuming any energy stored in the electric field. It is just the particles that exert forces on each other.
Maybe the right quantum field model for the static electric field of an electron is that it is a pure spatial momentum field that does not contain any energy at all?
Energy is only present in Fourier components where the 4-momentum k contains energy. Since for a static field, those components have a total destructive interference, there is no energy in the static electric field.
This would solve the paradox which has been present since the discovery of the electron at the start of the 20th century: how can a point charge have a finite mass even though the energy of its static electric field is infinite?
An early attempt to "renormalize" the electron mass
A simple attempt to resolve the paradox was to assume that the electric field is cut off at some very short distance dr from the electron, and the mass of the "bare" electron has a large negative value -M. When we add -M to the very large energy E of the electric field, we get:
E - M = 511 keV,
that is, the well-known electron mass visible to the outside world.
The solution to the paradox is often viewed as an early example of renormalization, where we assume that the "bare" mass (or some other property of a particle) has a wildly different value from the "dressed" mass visible to the outside world.
The notion of the electron having a negative bare mass is strange. Negative masses behave in a strange way in classical physics.
Suppose that the renormalization scheme is correct. Then we could collide the electron with another at a very high energy. The electric field of the electron cannot react to the collision instantaneously because the speed of light is finite. The collision might reveal that the bare mass of the electron is -M, a negative value. That would be a strange observation.
The self-energy diagram of the electron
photon which the electron
sends to itself,
4-momentum k
~~~~~
/ \
electron e- -------------------------
4-momentum p
A B
The "self-energy" diagram of the electron is depicted above. The Feynman integral for the diagram contains the product of:
1. the electron propagator (4-momentum p - k) from A to B and
2. the photon propagator (4-momentum k) from A to B.
The integral diverges when we integrate over all 4-momenta k.
The standard way to solve the divergence problem is to put a cutoff to k and assume some "bare mass" -M for the electron, such that the electron mass visible to outside world becomes m = 511 keV.
However, our considerations about the conservation of the speed of the center of mass suggest that the diagram above is forbidden, or that the photon must be absorbed in zero time, and the the effect of the diagram is then actually null.
real photon, energy E
~~~~~~~~>
/
e- ●
<-
Let us analyze the diagram using the particle model. We may assume that the electron is initially static.
Suppose that the pointlike electron emits a real photon to the right, but after some finite time interval dt, reabsorbs it. During the time interval dt, the electron moves left.
The real photon carries some energy E to the right at the speed of light and is partially responsible for keeping the center of mass of the system static.
But then a miracle happens: the electron magically captures back the energy E carried by the photon. The electron is again static, but it has moved a little bit to the left. The center of mass has moved.
Could it be that at the time of absorption, the electron somehow jumps to the right to its original position? It might be, but in the momentum phase wave description of the process there is no such sudden jump in anything. When an electron absorbs a photon in the wave model, the electron does not suddenly jump to another place. How could the electron even know where it has to jump when it absorbs a photon?
What is going on here? If we think of classical particles, they only absorb other particles which are at their current location. They do not magically absorb particles which are far away. Working in momentum space in Feynman diagrams obfuscates this simple fact and people start to think that, in the plane wave representation of the momentum space, particles can actually absorb other particles which are located anywhere in the experiment area.
The misunderstanding comes from the fact that if we have independent particles at random locations of the experiment area, then they can meet at any location and one can absorb the other.
Is newtonian mechanics right about static forces between charges?
In newtonian mechanics, we can get rid of the energy of the electric field by assuming that signals travel infinitely fast. Then the positions of the charges explain everything, and there is no need to assume the existence of an electric field at all, let alone a field which would contain energy.
When we calculate a Feynman diagram where a photon only moves spatial momentum, no energy, then we do assume that the force acts instantaneously. We can drop the time dimension from the model. We are calculating using newtonian mechanics.
What about dynamically changing electromagnetic fields? They do contain energy in on-shell photons, or in virtual photons where the 4-momentum has the energy component non-zero.
An improved rubber plate model of the electromagnetic field
Two years ago, we were able to explain the flow of spatial momentum and energy in a radio transmitter by introducing the model where the electric field of a charge is an elastic "rubber plate".
^
| rubber plate
------------------●------------------
| charge
v waved up and down
When an external force waves an electric charge, the rubber plate attached to the charge oscillates and waves in the plate propagate towards infinity. The waves carry energy but little spatial momentum. The charge exchanges quite a lot of spatial momentum with the rubber plate, as the charge is waved. But the net spatial momentum flow from the charge is zero.
The large gross momentum flow between the plate and the charge explains how the kinetic energy of the charge can be transformed into on-shell photons which carry very little spatial momentum. The charge loses a lot of momentum when it loses its kinetic energy. Where does the extra momentum go? It cannot go to photons.
The answer is that the momentum is temporarily stored in the rubber plate and returned back to the charge when it swings back. The net flow of momentum is zero.
Two years ago we thought that the mass-energy of the static electric field of the charge temporarily absorbs the extra spatial momentum.
But our new hypothesis is that the mass-energy of a static Coulomb field is zero. What to do?
Suppose that the charge is initially static. Then we suddenly start to accelerate it up. Since the speed of light is finite, the plate has to stretch a little to accommodate the sudden movement of the charge. The rims of the plate are not moving yet but the center is moving up.
We do not need to assume that the plate has any rest mass at all. The stretching will create a force which resists the movement of the charge. The person who is pushing the charge up must give spatial momentum p upward to the plate.
Thus, the finite speed of light can be used to mimic inertial mass, even though the plate is massless.
The birth of electromagnetic waves, or real photons, is understood using Edward M. Purcell's diagram of the electric lines of force bending. See the top of the linked page for a picture how the field lines get wrinkled.
The wrinkle in the rubber plate moves outward. The wrinkle contains both an electric and a magnetic field. That is, the wrinkle is a dynamic electromagnetic field and contains energy.
In Purcell's diagram, the static electric fields can be regarded as containing zero energy. What matters is the energy in the wrinkle.
Scattering of colliding classical electromagnetic waves
A couple of weeks ago we tried to introduce a model of the electromagnetic field where a massless tense string is waved. The waves contain some mass-energy, and will cause inertia in the string. The problem in the model is that then energetic waves should propagate slower than low-energy waves because energetic waves contain a lot of mass-energy and inertia.
What about the rubber plate model? It looks plausible that waving a plate is harder if the plate contains a lot of energy in wrinkles. If there is little energy, then waves propagate precisely at the signal speed of special relativity, that is, c.
But if a wave contains a lot of energy, it should progress slower. Does this sound right?
If two energetic waves C and D collide, then there should probably be scattering of waves because inertia of the wave D disturbs the medium where the wave C propagates. This might be the root cause for the scattering of photons by photons? There would be scattering in classical electromagnetism, after all.
Electromagnetism is nonlinear in QED because energy can escape from the electromagnetic field into the Dirac field. Can we somehow harmonize this fact to the classical scattering of electromagnetic waves?
In Feynman diagrams, photon-photon scattering is modeled as a collision to a virtual pair.
photon
_____ _____
\ /
O virtual pair loop
_____/ \_____
photon
In our classical model, scattering of colliding waves is calculated from the electromagnetic coupling constant (~ tension of a unit electric field line).
In Feynman diagrams, scattering is probably determined by the electron charge / mass ratio and the coupling constant.
If these figures happen to agree, then we have found a formula which determines the electron mass from its charge and the electromagnetic coupling constant.
Yi Liang and Andrzej Czarnecki (2011) write that for low-energy photons, the cross section is proportional to the sixth power of the photon energy. That can hardly happen in our classical model. Conclusion: the classical model cannot explain quantum effects like the electron mass and photon-photon scattering.
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