Physics Question #582
Karl-Alexander Berg Brisebois, a 13 year old male from Dunham asks on January 22, 2002,
Conservation of momentum can explain the exchange of a virtual photon between two electrons and the repulsive force that results. But it cannot explain the attractive force that exists between an electron and a positron. Can you explain this force to me?
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Unfortunately this question has arisen out of a popularised version of quantum field theory, which seems to explain something, but does not.
In ordinary classical electrodynamics, a charge is surrounded by an electromagnetic field. That field has as its dominant part a so called Coulomb part, which is a radial electric field. Charges interact with that field by feeling a force of eE, where e is their charge and E is the electric field. Exactly the same is true in the quantized version of the theory. A charge is surrounded by an electromagnetic field, and the other particle feels that electomagnetic field, with a force of eE + evxB (x=cross product).
In quantizing the electromagnetic field, it is often convenient to regard the interactions of that field with charges as though the energy in the field were actually carried by particles, sometimes called photons. But in other situations that is a terrible way to think of the electromagnetic field, and in particular in the case of the Coulomb field. While the Coulomb field can be expressed in terms of a Fourier transform, as though it were made up of a huge sum of plane waves, this is almost always a bad way of thinking about it, since interference effects between these plane waves can be crucial (as they are in this case). That is, you cannot simply regard these plane waves as though they were each independent momentum carriers.
answered on June 21, 2002
A colleague of Dr. Unruh's, Dr. Brewer adds the following in an attempt to clarify this rather difficult concept in quantum physics.
When people talk about the electromagnetic force being "mediated" by exchange of photons, they are pushing a metaphor beyond its intended limits. The electromagnetic FIELD is the agent in quantum field theory as much as in classical electrodynamics, and trying to reduce the field to "single photon exchange" is like trying to describe human thought in terms of 2 neurons firing. Feynman got a Nobel prize for inventing a one-to-one correspondence between big complicated path integrals and simple cartoon-like diagrams, a graphical representation that gave the right hemisphere better purchase on the concepts involved. But people tend to treat the cartoons as if they were the real thing, with predictable results. Yes, if you scatter an electron off another electron, or off a positron, at high energy, the first-order diagram (which gives a pretty good approximation to the right answer) can be represented as a single photon exchange between the particles. But this diagram does not translate into a simple classical exchange, no matter how appealing the cartoon may be. It is the nature of the EM field that unlike charges attract, and that empirical fact is built into the corresponding field in quantum electrodynamics. So your question starts with two incorrect statements, unless you are using the word "explain" to mean something much more vague than a Physics "explanation".
Einstein said (something like), "Everything should be explained as simply as possible, but NOT MORE simply."
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