# All possible contractions 52 in this context a

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(all possible contractions) : . (52) In this context, a contraction is a line drawn between two fields in the normal-ordered product; the line represents a propagator between the two corresponding spacetime points. For example, T { φ 0 ( x 1 ) φ 0 ( x 2 ) } = : φ 0 ( x 1 ) φ 0 ( x 2 ) + φ 0 ( x 1 ) φ 0 ( x 2 ) : = : φ 0 ( x 1 ) φ 0 ( x 2 ) + D F ( x 1 , x 2 ) : (53) and, in shorthand, T { φ 1 φ 2 φ 3 } = : φ 1 φ 2 φ 3 + φ 1 φ 2 φ 3 + φ 1 φ 2 φ 3 + φ 1 φ 2 φ 3 : (54) = : φ 1 φ 2 φ 3 + D 12 φ 3 + D 13 φ 2 + D 23 φ 1 : . (55) One more example: T { φ 1 φ 2 φ 3 φ 4 } = : φ 1 φ 2 φ 3 φ 4 + φ 1 φ 2 φ 3 φ 4 + (permutations) + φ 1 φ 2 φ 3 φ 4 + (permutations) : (56) = : φ 1 φ 2 φ 3 φ 4 + D 12 φ 3 φ 4 + (permutations) + D 12 D 34 + (permutations) : . (57) The proof of Wick’s theorem involves verifying the formula for two fields, then proceeding by induction. The only terms on the right hand side of Wick’s theorem that are relevant for computing correlation functions are the fully contracted terms; the rest of the terms vanish inside h 0 | · | 0 i . That is, h 0 | T { φ 0 ( x 1 ) · · · φ 0 ( x n ) }| 0 i = (all full contractions) . (58) So, for example, h 0 | T { φ 1 φ 2 φ 3 φ 4 }| 0 i = h 0 | : φ 1 φ 2 φ 3 φ 4 + φ 1 φ 2 φ 3 φ 4 + φ 1 φ 2 φ 3 φ 4 : | 0 i (59) = D 12 D 34 + D 13 D 24 + D 14 D 23 . (60) 7
Feynman diagrams We are now ready to compute correlation functions using equation (46) and Wick’s theorem. Let us compute the 2-point function h Ω | T { φ ( x 1 ) φ ( x 2 ) }| Ω i assuming the interaction L int = λ φ 4 / 4! . First the numerator: (numerator) = 0 T φ 0 ( x 1 ) φ 0 ( x 2 ) exp i Z d 4 x L int ( φ 0 ) 0 (61) = h 0 | T { φ 0 ( x 1 ) φ 0 ( x 2 ) }| 0 i + - i λ 4! Z d 4 x h 0 | T { φ 0 ( x 1 ) φ 0 ( x 2 ) φ 4 0 ( x ) }| 0 i + O ( λ 2 ) (62) = D 12 + - i λ 4! Z x 3 h φ 1 φ 2 φ x φ x φ x φ x i + 4 · 3 h φ 1 φ 2 φ x φ x φ x φ x i ! + O ( λ 2 ) (63) = D 12 + - i λ 8 Z x D 12 D 2 xx + - i λ 2 Z x D 1 x D 2 x D xx + O ( λ 2 ) . (64) The factor of 3 in the third line is due to the 3 different ways of contracting the φ x ’s. The factor of 4 · 3 is due to the 4 ways to contract φ 1 with φ x and the 3 ways to contract φ 2 with one of the remaining φ x ’s. This result can be represented by a series of diagrams 1 1 1 2 2 2 x x + + O ( λ 2 ) + (65) with symmetry factors 1 , 1 / 8 , and 1 / 2 respectively. More complicated symmetry factors There are many ways to do the contractions at O ( λ 2 ) . Here is one way: 1 2 - i λ 4! 2 Z x,y h φ 1 φ 2 φ x φ x φ x φ x φ y φ y φ y φ y i · 4 · 4 · 3 · 2 · 1 · 2 (66) There is one factor of 4 because φ 1 has 4 choices of φ x to contract with. Similarly, φ 2 has 4 choices of φ y . Then the first remaining φ x has 3 choices of φ y , while the second has 2, and the last has only 1. There is another factor of 2 because the same Feynman diagram would result if we instead contracted φ 1 with φ y and φ 2 with φ x ; this would be equivalent to swapping the vertices x y . Thus, the Feynman diagram has a symmetry factor of 1 / 6 and looks like: (67) Another graph at O ( λ 2 ) looks like (68) and corresponds to the expression 1 2 - i λ 4!