What is a virtual electron? A classical model

We say that an electron is on-shell or real if it satisfies the energy-momentum relation

       E^2 = p^2 + m^2,

where E is the total energy, p is the spatial momentum, and m is the electron mass 511 keV. We assume that c = 1.

What about an electron under a potential V? Last year we blogged about the inertial mass of an electron under a potential. Suppose that we have a plate with positive charge.

           e-     --->

_________________

 + + + + + + + + +

Let us lower the electron so close to the plate that the sum

       m + V

is small. But the inertial mass of the electron may be quite big in that case, too. The electron eats a hole into the electric field of the plate. If we move the electron around, the energy content in the electric field gets displaced: the inertial mass of the electron will appear substantial even though its total energy is very small.

If we allow the electron to sink into a negative energy zone, where m + V < 0, we may have the total energy of the electron E = 0, even though it has substantial momentum p.

An electron under the influence of a potential generally does not satisfy the energy-momentum relation - the electron is virtual!

Here we have a classical model for a virtual electron.

The electron in a hydrogen atom is virtual - and stays virtual - because it has sunk some 13.6 eV down in the Coulomb field of the proton. A real electron exists as a beautiful sine wave in empty space. It cannot have the complex waveform of an orbital in a hydrogen atom.

What about a virtual photon?

Close to any charged object, the electromagnetic field interacts with the charge. The charge acts as a source in the electromagnetic wave equation. The Coulomb potential -1 / r is what we get if we let the electromagnetic field assume its permanent form close to a charge.

A photon which interacts with a charge typically is virtual - it does not satisfy E = |p| like a real photon (we assume c = 1).

The (fictitious) photon which makes up the Coulomb potential in the Fourier decomposition of the potential, has zero energy but carries momentum p. The Fourier decomposition consists of time-independent waves of the type

       exp(-i (E t - p x)),

where E = 0.

In an earlier blog post we remarked that all interacting theories of two fields are nonlinear. The interaction brings us also this fact: if there is interaction, then a particle, or its wave, typically is off-shell.

The normal way for a particle to exist under an interaction is to be off-shell. The word "virtual" is a bad choice because it has the connotation of something imagined, not real. A better word might be off-shell, or simply an interacting particle.

Waves and "virtual" waves in water

We can consider waves in the middle of a lake - in open water - free. The waves have the familiar sine function waveform, and carry some energy and momentum which depends on the wavelength and the amplitude.

When we use an oar to row a boat, then the water "field" interacts strongly with the oar. The surface of water close to the oar will assume all kinds of forms. The short-lived "waves" near the oar are "virtual" and can carry a lot of momentum.

The word virtual is again a bad choice. Rowing a boat is hard work, not anything imagined or unreal. The water field is under a strong interaction with the oar. An interacting wave is a better name.