ABSTRACT
In chapters 3 and 4 we wrote down the perturbation series for the partition function and for some correlation functions. We found that the coefficients of λn were given by a sum (d+1)n-dimensional integrals if the space dimension is d. Typically, some of these integrals diverge if the cutoffs of the theory are removed. This does not mean that something is wrong with the model, but merely means first of all that the function which has been expanded is not analytic if the cutoffs are removed. To this end we consider a small example. Let
Gδ(λ) := ∫∞ 0 dx
∫ 1 0 dk
e−x (8.1)
where δ > 0 is some cutoff and the coupling λ is positive. One may think of δ = T , the temperature, or δ = 1/L if Ld is the volume of the system. By explicit computation, using Lebesgue’s theorem of dominated convergence to interchange the limit with the integrals,
G0(λ) = lim δ→0
Gδ(λ) = ∫∞ 0 dx 2(
√ 1 + λx−√λx) e−x
= 2 + O(λ) −O( √
λ) (8.2)
Thus, the δ → 0 limit is well defined but it is not analytic. This fact has to show up in the Taylor expansion. It reads
Gδ(λ) = n∑
) ∫∞ 0 dx
∫ 1 0 dk
λj + rn+1 (8.3)
Apparently, all integrals over k diverge for j ≥ 1 in the limit δ → 0. Now, very roughly speaking, renormalization is the passage from the expansion (8.3) to the expansion
G0(λ) = n∑
) ∫∞ 0
dx 2x e−x λ − c√λ + Rn+1 (8.4)
by expanding the √
1 + λx term. One would say ‘the diverging integrals have been resumed to the nonanalytic term c
√ λ’. In the final expansion (8.4) all coefficients are finite and, for small
λ, the lowest order terms are a good approximation since (θλ ∈ [0, λ])
|Rn+1| = ∣ ∣ ∣2
∫∞ 0 dx
) 1
xn+1λn+1e−x ∣ ∣ ∣
≤ 2∣∣ (
)∣ ∣ ∫∞
n+1 n→∞∼ √2 (ne )n
λn+1 (8.5)
Here we used the Lagrange representation of the n+1’st Taylor remainder in the first line and Stirling’s formula, n! ∼ √2πn(n/e)n, in the last line. An estimate of the form (8.5) is typical for renormalized field theoretic perturbation series. The lowest order terms are a good approximation for weak coupling, but the renormalized expansion is only asymptotic, the radius of convergence of the whole series is zero. The approximation becomes more accurate if n approaches 1/λ, but then quickly diverges if n > e/λ. Or, for fixed n, the n lowest order terms are a good approximation as long as λ < 1/n.
