By Peeters K.

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This term in the expansion of the partition function contains n integrals over y1 , . . , yn , 1 n! iλ h¯ n d4 y1 d4 y2 · · · d4 yn δp δp δp ··· p p δJ (y1 ) δJ (y2 ) δJ (yn ) p n integrals × exp 1 2 d4 x1 d4 x2 J ( x1 ) GF ( x1 − x2 ) J ( x2 ) . 34) The variational derivatives bring down the exponent, and remove a d4 xi J ( xi ) from 53 . 3 Path integrals in field theory it, so that we get an expression of the form d4 y1 d4 y2 · · · d4 yn δp δp δp ··· p p δJ (y1 ) δJ (y2 ) δJ (y3 ) p exp (.

Any additional internal line will create a loop. So we have L = I − (v − 1) and thus the diagram has a power h¯ E−1+ L . 56) Thus, if we keep the structure of the diagram fixed (the number of external lines), then the power of h¯ corresponds to the number of loops. Summary: The LSZ formula expresses amplitudes between in and out states of particles with given momenta in terms of ‘amputated’ correlators of fields in position space. See also: The discussion of the four-point function has been adapted from M.

Odd number 3. Symmetry factors What are the symmetry factors for the following diagrams? 4. Feynman diagrams Consider the scalar field theory with both a φ4 and φ3 interaction term, S= 44 1 1 g λ d4 x − (∂µ φ)(∂µ φ) − m2 φ2 − φ3 − φ4 . 2 2 3! 4! 6 Exercises (a) Give the Feynman rules in coordinate space for this model. (b) Write down all the Feynman graphs for the amplitude for a 2-particle to 2-particle process, evaluated at zero-loop and one-loop order. Show by explicit counting of the powers of h¯ that all the graphs at the same loop order have the same power of h¯ .

### A guide to quantum field theory by Peeters K.

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