Master Dissertation

Chapter 5
The Method of Epstein and
Glaser
5.1 Introduction
We want to express the S-matrix as a formal power series in R[[λ]], where
λ is the coupling constant λ = e, which is the unit of charge and R is the
ring
R = {Υ : D0 → D0|Υ is linear}.
We note that we are not concerned about the convergence of the series and
it might not have any convergence radius at all.
We begin from the expression.
S(g) = 1 +
∞
n=1
=: 1 + T
n 1
λ
n!
d 4 x1 · · · d 4 xnTn(x1, . . . , xn)g(x1) · · · g(xn)
We shall usually omit the λ in the notation. Further may assume that
Tn(x1, . . . , xn) is symmetric in x1, . . . xn. Otherwise we can always
symmetrize it. This is easy to see in 2 dimensions where we given
T2(x1, x2) can choose the symmetric map defined by
(T2(x1, x2) + T2(x2, x1))/2. The same way we can choose a symmetric map
in n-dimensions as
Tn(x1, x2, . . . , xn) + Tn(x2, x1, x3, . . . , xn) + · · · + Tn(xn, xn−1, . . . , x1)
.
n!
Since Tn(x1, . . . , xn) is symmetric we may occasionally use the short hand
notation Tn(x) where x = {xj ∈ M|j = 1, . . . , n} is disordered.
Along with the Wightman axioms we have to assume that each term of the
S-matrix is well-defined.
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