By Peeters K.

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B) Draw the 2 diagrams that contribute to the φφ correlator at one loop, and the 7 diagrams that contribute to the φφφ correlator at one loop. (c) What is the LSZ relation between the correlation functions defined above and the two and three particle “scattering” amplitudes Γ(2) and Γ(3) ? Give the mathematical expressions corresponding to these two amplitudes at up to and including one loop order. (d) How would you define the physical mass mphys and physical coupling constant λphys of this theory?

3 By shifting these new variables according to αn → µ µ αn − X˙ n (which we can do without changing the integral) the quadratic terms disappear, iδ 2¯h N ∑ n =1 iδ N µ µ − α2n + X˙ n X˙ n µ − m2 → 2αn X˙ n µ − (α2n + m2 ) . 13) µ All the Xn -dependence now sits in the first term inside the sum above. 11), these terms become i h¯ N ∑ α n ( X n − X n −1 ) = n =1 µ µ µ N −1 i µ i µ µ µ α N X N µ − α 1 X0 µ − ∑ α n +1 − α n X n µ . 14) 3 We are slightly cheating here because the argument of the exponent is imaginary rather than real; this can be repaired without changing the end result so we will ignore this issue here.

2 of M. Peskin and D. Schroeder, “An introduction to quantum field theory”, Perseus, 1995. 2. Correlation functions and Wick’s theorem Now that we have expressed the time evolution of the field φˆ (t, x ) entirely in terms of the field φˆ 0 (t, x ), we can go and compute correlation functions in the interacting theory at λ = 0. 39), but now with interactions. 15) and interpret this as the Feynman propagator in the presence of interactions. We have to be a bit careful with what we mean with |0 .