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254 November 7, 2013
constant λ 4 to be small, but positive. If, on the other hand, it was small but
negative, perturbation theory would look very different ; in fact it would look
like nothing at all since for negative λ 4 the path integral is completely undefined.
Therefore, the perturbative expansion is not regular around λ 4 = 0, and in the
set of all ϕ 4 theories the point λ 4 = 0 constitutes an essential singularity.
All may not be lost, however. The method of Borel summation sometimes 2
enables us to assign a value to a sum with vanishing radius of convergence.
Suppose that a function of a positive variable x is given by the sum
f(x) = ∑ k≥0
c k x k , (10.3)
where the coefficients c k grow superexponentially. Clearly it is difficult to make
sense of such a sum ; but it may be possible to make sense of a related sum :
g(x) = ∑ k≥0
c k
k! xk , (10.4)
simply because the coefficients do not grow as rapidly. Let us suppose that this
is indeed the case. We then may employ the formula
∫ ∞
to arrive at the rule
0
dy exp(−y) (xy) n = n! x n , n = 0, 1, 2, . . . (10.5)
f(x) =
∫ ∞
0
dy e −y g(xy) . (10.6)
Notice that here, we have again interchanged summation and integration, thus
in a sense repairing the damage done when we arrived at the perturbation
expansion in the first place. This approach is called Borel summation. We can
illustrate this in a simple example. Let us take c k = 1, that is
we immediately find that
f(x) = ∑ k≥0
x k = 1
1 − x : (10.7)
g(x) = ∑ k≥0
x k
k! = ex , (10.8)
and indeed
∫ ∞
0
dy e −y e xy = 1
1 − x . (10.9)
2 In zero dimensions this will work. In four-dimensional Minkowski space things are not
nearly as simple. . .