Classical potential and particle creation I was reading up some references about QM/QFT and I came across this note (actually a problem set): https://www.classe.cornell.edu/~yuvalg/p4444/hw8-sol.pdf
In question 1, the author mentioned that "In the limit where particle production and annihilation can be neglected, one can use the idea of potentials. In QFT, this can be used to find how a potential is generated from particle exchange. In particular, the statement that “the photon is the carrier of the electric force” can be understood."
Can someone please elaborate more about this? In particular, why can we use the notion of potentials if particle production and annihilation can be neglected and how is a potential generated from particle exchange?
Thanks!
 A: there is a complete discussion of Weinberg's "Lectures on Quantum Mechanics". In Chapters 7 and 8, he discusses how the non-relativistic, elastic scattering, that is neglecting creation and annihilation, can be described given a Hamiltonian $H=H_0+V(\bf{x})$ obtaining in the Born approximation (initial and final states as free particles) that the scattering amplitude depends on the potential as the expression in the notes $f(q)=\int d^3x \exp{iqx} V(x) $ with q = k-k' (transferred momentum). In the case of the creation and annihilation of particles, you may use the formalism of S-matrix $S_{\beta\alpha}$ representing the transition rate between the asymptotic initial ($\alpha$) and final ($\beta$) states. If you assume an elastic scattering generated by a particle exchange (let's say e+N->e+N, with photon exchange) you will find the propagator of the exchanged particle. The particle exchanged is a formal mathematical representation of a perturbative series. It is also virtual since it has $q^2\neq0$ (which is not the case for a real photon). Finally, for the general case with creation and annihilation, you will use a more general formalism (Scattering amplitude) which recovers the scattering on the potential in the case of non-relativistic elastic scattering.
Hope it could be helpful
