Pictures Paths Particles Processes

Chapter 10
Appendices
10.1 Convergence issues in perturbation theory
Let us reinspect Eq.(1.25), taking µ = 1 for simplicity :
G 2n = H 2n /H 0 ,
H 2n = ∑ (4k + 2n)!
2 5k+n 3 k (2k + n)! k! (−λ 4) k ,
k≥0
H 0 = ∑ (4k)!
2 5k 3 k (2k)! k! (−λ 4) k . (10.1)
k≥0
Although we have treated the expressions for the H’s as if they were well-defined
objects, in fact these series do not converge ! For large k and fixed n the k th
term in H 2n contains the numerical coefficient
(4k + 2n)!
2 5k+n 3 k (2k + n)! k!
which increases superexponentially 1 with k : which implies that the series has a
radius of convergence equal to zero. The procedure of taking the ratio H 2n /H 0 ,
while it mixes terms of different order in λ 4 , does not help to repair this ; a
simple numerical study shows that
G 2 = ∑ k≥0
σ k (−λ 4 ) k , σ k ∼ k! (2/3) k , (10.2)
so that also G 2 (and, it can be checked, the higher G’s) are described by series
with vanishing radius of convergence. This should not come as a surprise. For,
in the discussion of the perturbation expansion we have assumed the coupling
1 This means that the coefficient increases with k faster than A k for any A : roughly
speaking, it grows like (k!) .
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