In the latest two blog posts we showed that a classical electromagnetic field might be adequate for explaining Coulomb scattering, as well as the real photons produced in it.
But classical electromagnetic theory does not explain production of electron-positron pairs in a collision.
In classical electromagnetism we have no rule about how an electromagnetic field could "induce" a wave in the Dirac field. We do have rules about how particles described by the Dirac equation produce electromagnetic fields, but no converse rule.
How an electromagnetic field A "induces" a Dirac field in the Feynman model
The Feynman diagram model does tell us how the converse rule might work: any electromagnetic field at every spacetime point constantly produces Dirac waves, using a Green's function for the Dirac field. The waves are mostly (?) virtual. In some cases, these waves can gain enough energy to become real particles.
The QED lagrangian contains the interaction term
-e ψ-bar γ^μ A_μ ψ.
It is a reasonable guess that the interaction term acts as a source for the free Dirac equation. Any electromagnetic field A produces a Dirac field wave which always contains both an electron and a positron, virtual or real.
For each spacetime point, the particles are produced as an impulse response to a Dirac delta source. That is, a Green's function produces those Dirac field waves.
Dirac field waves can escape as real particles if they somehow gain the right amount of energy relative to their momentum. They must then obey the energy-momentum relation:
E^2 = p^2 + m_e^2,
where m_e is the electron mass 511 keV.
If the waves obey that relation, they "resonate" with the free Dirac field, and can exist indefinitely.
If the waves do not obey the energy-momentum relation, they are "malformed" waves and can only exist for a very short time, or across a very short distance. The energy and the momentum that were put into malformed Dirac waves quickly return back to the electromagnetic field A. Malformed waves do not "resonate" with the Dirac field.
What original energy and momentum do the virtual pairs acquire when they are produced by an electromagnetic field? If the field A has a Fourier component whose form is something like
exp(-i (E t - p x)),
then a reasonable guess is that the virtual pair "inherits" the energy E and the momentum p from the electromagnetic field. Another way to same the same thing is that the pair "absorbs" a photon from the Fourier component.
A time-dependent electromagnetic field A produces virtual pairs with non-zero energy
Let an electric field A produce a virtual pair. Let us treat the pair with a particle model
In an earlier blog post we noted that if A is time-independent, then the energy of its Fourier components is zero, and the electron will have some (e.g., positive) mass-energy m, and the positron a negative mass energy -m. An electric field accelerates the particles to the exact same direction, since the masses differ just by a sign. The particle with a negative mass-energy -m cannot gain any kinetic energy: its kinetic energy -m v^2 grows even more negative if it accelerates.
In a collision, the field A is time-dependent, and it has Fourier components where the energy E is positive. Then the combined energy of the virtual pair is non-zero. The electric field accelerates the particles in a different way. For example, if both particles have positive energy, then an electric field accelerates the electron and the positron to opposite directions. Both particles gain energy. They can become real particles.
A semiclassical model for virtual pair production
distance L
e- ● ● e+
A classical model for a virtual pair is simply an electron and a positron which are static and close to each other. The distance L has to be > 1.4 * 10^-15 m for the system energy to be positive.
A strong external electric field can pull the electron and the positron apart. They become real particles.
We have now a semiclassical model for the process:
1. We determine the Fourier decomposition of the classical field A.
2. Each Fourier component of A can at each spacetime point (t, x) produce a virtual pair, and the pair inherits the energy E and the momentum p from the Fourier component. The probability of production at (t, x) depends on the coupling constant α as well as the strength of the Fourier component.
3. The particles in the pair move like classical point particles thereafter. But they do not observe the fields of each other until they have some distance between them? Maybe we can assume that they are born with enough energy to lift them to the distance 1.4 * 10^-15 m from each other?
If the particles cannot gain enough energy, they will collide and annihilate, producing a photon whose energy is E and momentum p.
4. How should we modify A when it lost (E, p) to the new pair? In a Feynman diagram, the new pair gets its (E, p) from a virtual photon sent by one of the colliding particles. Obviously, we must subtract (E, p) from an incoming particle.
5. How do we return (E, p) back to A if the particles annihilate? If the pair did not interact with other particles, then we probably must return the (E, p) like nothing would have happened.
Item 4 above suggests that the virtual pair is produced by a collision of an incoming particle to one or both of the components of a classical virtual pair which originally has zero energy and momentum. The Fourier component of A in this case is a Lorentz-transformed component of the Coulomb electric field of the incoming particle. The component is a longitudinal wave.
The model is probabilistic: we get probabilities for the new electron and the new positron departing with certain momenta. In that sense, the model is not classical - it is a quantum model.
Classical electromagnetism is founded on:
1. point particles, and
2. a field, the electromagnetic field.
We see that it is impossible to make a deterministic classical model for pair production. Pairs would be produced by the field, which makes the end result necessarily probabilistic. Classical electromagnetism is a mixture of quanta (electrons) and a wave (the electromagnetic field).
How can we reconcile different length scales?
The length scale of a 1 MeV collision is just 3 * 10^-15 m, while the Compton wavelength is much larger, 2 * 10^-12 m. How such a tiny length scale can produce Dirac waves whose wavelength is almost 1000-fold?
It is probably because we do not know the position of the collision better than the Compton wavelength. The Dirac waves are the combined effect of many different paths of particles. Individual paths have features whose size is only 3 * 10^-15 m. When we sum the effect of all paths, the resulting wave is of the scale 2 * 10^-12 m.
A similar thing happens with the Schrödinger equation of two electrons. The Coulomb field has a feature size 3 * 10^-15 m or smaller. The de Broglie wavelength of the electrons is larger than 2 * 10^-12 m.
No comments:
Post a Comment