Pictures Paths Particles Processes

74 November 7, 2013
− λ 4
6
(
φ(y) 3 δ
+ 3¯hφ(y)
δJ(y) φ(y) + δ 2 )}
¯h2 (δJ(y)) 2 φ(y)
. (2.46)
On this basis, we can now formulate Feynman rules for the continuum limit :
x y
↔ Π(x − y)
↔ − λ 4
¯h
x ↔ +J(x) ¯h
Feynman rules, version 2.3 (2.47)
This comes with the understanding that the positions of all vertices are to be
integrated over, and that the field function φ is now a functional of the source
J. For a free theory there are no interactions, and we find
∫
φ(x) = dy Π(x − y) J(y) . (2.48)
We see that the free field is the sum of its responses to the source, weighted
by the correlation between the position where the field is measured and that
of the strength of the source at all points. It is this property that establishes
the propagator as the ‘differential-equation’ Green’s function; but note that this
correspondence is only valid for non-interacting theories.
2.3.5 Field configurations in one dimension
Before entering spaces of more dimensions, we may have a look at the field
variables. The zero-dimensional variable ϕ, with its integration element, is in
the discrete one-dimensional formulation replaced by the whole set ϕ, for which
the path integration element reads, of course,
Dϕ = ∏ n
dϕ n
The continuum limit of this object is defined to be the continuum-formulation
path integration element, however badly defined this may be. The assigning a
functional value S[ϕ] to a given field ϕ(x) is not problematic ; rather it is the
prescription of how all field configurations are to be summed over that makes
it so hard to define path integrals rigorously 14 . It is instructive to consider the
14 In fact, the mathematical definition of continuum path integrals relies on the discrete
formulation !