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258 November 7, 2013
10.2 More on symmetry factors
10.2.1 The origin of symmetry factors
In this section we shall return to the ‘simple’ world of zero dimensions, since
symmetry factors do not depend on the dimensionality of the theory. Let us
consider again the ϕ 3/4 theory, with a path integrand
exp ( −S(ϕ) + Jϕ ) (
= exp − µ 2! ϕ2 − λ )
3
∑
(
= e −µϕ2 /2! 1 −λ3 ϕ 3
n
n 3 ! 3!
1,3,4≥0
3! ϕ3 − λ 4
4! ϕ4 + Jϕ
) n3 ( 1 −λ4 ϕ 4
n 4 ! 4!
) n4
1
n 1 !
(
Jϕ
) n1
(10.20)
We see that a diagram with n 3 three-vertices, n 4 four-vertices and n 1 source
vertices carries an a priori factor of
1
n 1 ! n 3 ! n 4 ! (3!) n3 (4!) n4 .
10.2.2 Explicit computation of symmetry factors