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November 7, 2013 27
Note that, whereas the Green’s functions all have a perturbation expansion
starting with terms containing no λ 4 , the connected Green’s functions of increasing
order are also of increasingly high order in λ 4 : the higher connected
Green’s functions need more interactions than the lower ones.
1.2.6 The Schwinger-Dyson equation for the path integral
Although the path integral is, generally, a very complicated function of J, it
is nevertheless easy to find an equation describing it completely. This is the
Schwinger-Dyson equation (SDe), which we construct as follows. Let the action
be given by the general expression 10
S(ϕ) = ∑ k≥1
1
k! λ k ϕ k , (1.29)
where λ 2 = µ. Now, from the observation that
∂ p
∫
(∂J) p Z(J) = N exp ( − S(ϕ) + Jϕ ) ϕ p dϕ , p = 0, 1, 2, 3, . . . (1.30)
we immedately deduce that
⎡
⎤
⎣−J + ∑ λ k+1 ∂ k
⎦
k!
k≥0
(∂J) k Z(J) =
⎡
⎤
∫
= N exp ( − S(ϕ) + Jϕ ) ⎣−J + ∑ λ k+1
ϕ k ⎦ dϕ
k!
k≥0
∫
= N exp ( − S(ϕ) + Jϕ ) [ ]
S ′ (ϕ) − J dϕ = 0 , (1.31)
where in the last lemma we have used partial integration, and the fact that the
integrand vanishes at the endpoints at infinity. Symbolically, we may write the
SDe as ⌊ ( )
∂
∂
∂ϕ
⌋ϕ=∂/∂J
S(ϕ) Z(J) = S ′ Z(J) = JZ(J) . (1.32)
∂J
For a theory with K fields, we similarly have
⌊ ⌋
∂
S(ϕ 1 , ϕ 2 , . . . , ϕ K ) Z(J 1 , J 2 , . . . , J K ) = J n Z(J 1 , J 2 , . . . , J K ) .
∂ϕ n ϕ j=∂/∂J j
For our sample model, the ϕ 4 theory, the SDe reads 11
(1.33)
1
6 λ 4Z ′′′ (J) + µZ ′ (J) − JZ(J) = 0 . (1.34)
10 A constant, ϕ-independent term in the action is always immediately swallowed up by the
normalization factor N.
11 The SD equation is, in general, of higher than the first order. It therefore has several
independent solutions, only one of which corresponds to the usual perturbative expansion.
The nature of the other solutions is discussed in Appendix 2.