A. Zee, Quantum Field Theory in a Nutshell · Problem III.5.3
Field Theory Redux
Physics — QFT
Problem
Given a field theory we can compute the scattering amplitude of two particles in the nonrelativistic limit. We then postulate an interaction potential between the two particles and use nonrelativistic quantum mechanics to calculate the scattering amplitude, for example in Born approximation. Comparing the two scattering amplitudes we can determine . Derive the Yukawa and the Coulomb potentials this way. The application of this method to the interaction is slightly problematic since the delta function interaction is a bit singular, but it should be all right for determining whether the force is repulsive or attractive.Intuitive explanation
The key insight is that quantum field theory and nonrelativistic quantum mechanics must agree in their overlap regime—the nonrelativistic limit. By computing the scattering amplitude from Feynman diagrams (which encodes the underlying field interaction) and matching it to the Born approximation result from a classical potential , you can read off what potential reproduces the field theory's physics. This method reveals how fundamental interactions like particle exchange (Yukawa) or gauge forces (Coulomb) emerge as effective potentials between nonrelativistic particles, and lets you extract their range and strength directly from amplitudes.
Intuitive explanation
The key insight is that quantum field theory and nonrelativistic quantum mechanics must agree in their overlap regime—the nonrelativistic limit. By computing the scattering amplitude from Feynman diagrams (which encodes the underlying field interaction) and matching it to the Born approximation result from a classical potential , you can read off what potential reproduces the field theory's physics. This method reveals how fundamental interactions like particle exchange (Yukawa) or gauge forces (Coulomb) emerge as effective potentials between nonrelativistic particles, and lets you extract their range and strength directly from amplitudes.