Quantum Field Theory

the S-matrix elements, where we were working primarily with momentum eigenstates,and ended up integrating over all of space. However, it’s rather simple to adapt theFeynman rules that we had earlier in momentum space to compute G (n) (x 1 . . .,x n ).For φ 4 theory, we have• Draw n external points x 1 , . . ., x n , connected by the usual propagators and vertices.Assign a spacetime position y to the end of each line.• For each line x∆ F (x − y).y from x to y write down a factor of the Feynman propagator• For each vertex y at position y, write down a factor of −iλ ∫ d 4 y.3.7.2 From Green’s Functions to S-MatricesHaving described how to compute correlation functions using Feynman diagrams, let’snow relate them back to the S-matrix elements that we already calculated. The firststep is to perform the Fourier transform,∫ [ n]∏˜G (n) (p 1 , . . .,p n ) = d 4 x i e −ip i·x iG (n) (x 1 , . . .,x n ) (3.105)i=1These are very closely related to the S-matrix elements that we’ve computed above. Thedifference is that the Feynman rules for G (n) (x 1 , . . .,x n ), effectively include propagators∆ F for the external legs, as well as the internal legs. A related fact is that the 4-momenta assigned to the external legs is arbitrary: they are not on-shell. Both of theseproblems are easily remedied to allow us to return to the S-matrix elements: we needto simply cancel off the propagators on the external legs, and place their momentumback on shell. We have〈p ′ 1 , . . .,p′ n ′|S − 1 |p ∏n ′n∏1 . . .,p n 〉 = (−i) n+n′ (p i ′ 2 − m 2 ) (p 2 j − m2 ) (3.106)i=1j=1× ˜G (n+n′) (−p ′ 1 , . . .,−p′ n ′, p 1, . . ., p n )Each of the factors (p 2 −m 2 ) vanishes once the momenta are placed on-shell. This meansthat we only get a non-zero answer for diagrams contributing to G (n) (x 1 , . . .,x n ) whichhave propagators for each external leg. You might think they all do, but it’s not true!Only diagrams that are fully connected, meaning each external point is connected toeach other external point, have this property. For example, of the diagrams that wewrote down which contribute to the four-point function 〈Ω| Tφ H (x 1 ) . . .φ H (x 4 ) |Ω〉, onlywill survive the multiplication by on-shell propagators in (3.106) to contribute tothe S-matrix for meson scattering in φ 4 theory.– 79 –