Virtual Photon Exchange and Charge Sign
Most recent answer: 10/22/2007
Q:
Is there a difference in the kind of virtual photons emitted or absorbed by charged particles that would explain why like charges repel, while dissimilar charges attract?
- Steven (age 20)
M.S.U
- Steven (age 20)
M.S.U
A:
No, there isn’t really any difference in the virtual photons which are
exchanged between like and unlike charges, but the answer is that
photons are called "vector" particles -- they have spin 1 and their
propagators in relativistic quantum field theory have 4x4 components
(only four of which are not zero, but that’s enough). Furthermore, the
interaction vertex of a photon and an electron, for example, has an
associated Feynman rule which is sensitive to each of the components of
the photon field.
In a calculation of the scattering matrix element of two like-signed particles in relativistic quantum electrodynamics, a minus sign appears in the matrix element relative to the same calculation with two oppositely-signed particles, for a single photon exchange.
It’s been noted that this isn’t enough to generate an observable difference in the scattering of like- and unlike-sign particles. After all, quantum mechanical matrix elements are squared in order to compute probabilities of observing outcomes. The last ingredient is to notice that the amplitude for the scattering process interferes with the amplitude for no scattering -- you add the matrix elements for the two processes and you square the result. No scattering corresponds to a matrix element of +1, and the sign of the scattering amplitude relative to that now makes a difference.
Two observations: If the energy of the scattering is high, then the initial and final states are very different, and the contribution to the scattered final state from an unscattered initial state becomes very small. In this limit, attractive and repulsive Coulomb forces give the same measurable behavior. Rutherford scattering gives the same differential cross section without regard to whether the force is attractive or repulsive. Only if the energy of the scattering is low and you can watch the charged particles move towards or away from each other can you determine the relative sign.
The second observation is that the interference effect mentioned above and which is advertised as the big difference between attraction and repulsion is an incomplete argument without the flipped sign of the scattering matrix element. The reason for this is that the exchange of scalar particles (for example, pions, whose exchange holds protons and neutrons together in a nucleus) is always attractive, even when the participating particles are exchanged for their antiparticles. Pions don’t have enough components to their fields, and the interaction vertex of a pion and a fermion like a proton doesn’t involve any components. This interaction is always attractive, even though the interference argument above holds for it too.
Sources: Peskin and Schroeder, An Introduction to Quantum Field Theory, and Schiff, Quantum Mechanics.
In a calculation of the scattering matrix element of two like-signed particles in relativistic quantum electrodynamics, a minus sign appears in the matrix element relative to the same calculation with two oppositely-signed particles, for a single photon exchange.
It’s been noted that this isn’t enough to generate an observable difference in the scattering of like- and unlike-sign particles. After all, quantum mechanical matrix elements are squared in order to compute probabilities of observing outcomes. The last ingredient is to notice that the amplitude for the scattering process interferes with the amplitude for no scattering -- you add the matrix elements for the two processes and you square the result. No scattering corresponds to a matrix element of +1, and the sign of the scattering amplitude relative to that now makes a difference.
Two observations: If the energy of the scattering is high, then the initial and final states are very different, and the contribution to the scattered final state from an unscattered initial state becomes very small. In this limit, attractive and repulsive Coulomb forces give the same measurable behavior. Rutherford scattering gives the same differential cross section without regard to whether the force is attractive or repulsive. Only if the energy of the scattering is low and you can watch the charged particles move towards or away from each other can you determine the relative sign.
The second observation is that the interference effect mentioned above and which is advertised as the big difference between attraction and repulsion is an incomplete argument without the flipped sign of the scattering matrix element. The reason for this is that the exchange of scalar particles (for example, pions, whose exchange holds protons and neutrons together in a nucleus) is always attractive, even when the participating particles are exchanged for their antiparticles. Pions don’t have enough components to their fields, and the interaction vertex of a pion and a fermion like a proton doesn’t involve any components. This interaction is always attractive, even though the interference argument above holds for it too.
Sources: Peskin and Schroeder, An Introduction to Quantum Field Theory, and Schiff, Quantum Mechanics.
(published on 10/22/2007)