Quantum Field Theory

momenta, we have the amplitudes given bye − e − ∼1(p − p ′ ) 2 andµ −e−e+ µ +∼1(p + q) 2 (6.95)µ − µ −6.6.1 The Coulomb PotentialWe’ve come a long way. We’ve understood how to compute quantum amplitudes ina large array of field theories. To end this course, we use our newfound knowledge torederive a result you learnt in kindergarten: Coulomb’s law.To do this, we repeat our calculation that led us to the Yukawa force in Sections3.5.2 and 5.7.2. We start by looking at e − e − → e − e − scattering. We havep,sq,r/p,s // /q,r= −i(−ie) 2 [ū(⃗p ′ )γ µ u(⃗p)] [ū(⃗q ′ )γ µ u(⃗q)](p ′ − p) 2 (6.96)Following (5.49), the non-relativistic limit of the spinor is u(p) → √ m(ξξ). Thismeans that the γ 0 piece of the interaction gives a term ū s (⃗p)γ 0 u r (⃗q) → 2mδ rs , whilethe spatial γ i , i = 1, 2, 3 pieces vanish in the non-relativistic limit: ū s (⃗p)γ i u r (⃗q) → 0.Comparing the scattering amplitude in this limit to that of non-relativistic quantummechanics, we have the effective potential between two electrons given by,∫U(⃗r) = +e 2 d 3 p e i⃗p·⃗r(2π) 3 |⃗p| = + e2(6.97)2 4πrWe find the familiar repulsive Coulomb potential. We can trace the minus sign thatgives a repulsive potential to the fact that only the A 0 component of the intermediatepropagator ∼ −iη µν contributes in the non-relativistic limit.For e − e + → e − e + scattering, the amplitude isp,sq,r/p,s // /q,r= +i(−ie) 2 [ū(⃗p ′ )γ µ u(⃗p)] [¯v(⃗q ′ )γ µ v(⃗q)](p ′ − p) 2 (6.98)– 147 –