Pictures Paths Particles Processes
Pictures Paths Particles Processes
Pictures Paths Particles Processes
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November 7, 2013 37<br />
If we wish, we may treat the λ 2 term as an interaction, described by a vertex<br />
with two legs. the SDe is then seen to be<br />
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= +<br />
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. (1.61)<br />
corresponding to<br />
φ(J) = J µ − λ 2<br />
µ φ(J) − λ 4<br />
6µ<br />
(<br />
)<br />
φ(J) 3 + 3φ(J) ∂<br />
∂J φ(J) +<br />
∂2<br />
(∂J) 2 φ(J)<br />
. (1.62)<br />
Multiplying the equation by µ and transposing the λ 2 term to the left, we obtain<br />
φ(J) =<br />
J λ 4<br />
−<br />
µ + λ 2 6(µ + λ 2 )<br />
(<br />
)<br />
φ(J) 3 + 3φ(J) ∂<br />
∂J φ(J) +<br />
∂2<br />
(∂J) 2 φ(J)<br />
, (1.63)<br />
precisely what we woud have obtained by taking the combination (µ + λ 2 ) as<br />
the kinetic part from the start. This procedure, by which the effect of two-point<br />
(effective) vertices is subsumed in a redefinition of the kinetic part, is called<br />
Dyson summation. In the present example, the summation is of course trivial ;<br />
but we shall see that two-point interactions can also arise from more complicated<br />
Feynman diagrams corresponding to higher orders in perturbation theory. The<br />
manner in which Dyson summation is usually treated is by explicitly writing<br />
out the propagator, ‘dressed’ with two-point vertices in all possible ways :<br />
+ + + + · · ·<br />
= 1 µ − 1 µ λ 1<br />
2<br />
µ + 1 µ λ 1<br />
2<br />
µ λ 1<br />
2<br />
µ − 1 µ λ 1<br />
2<br />
µ λ 1<br />
2<br />
µ λ 1<br />
2<br />
µ + · · ·<br />
= 1 ∑<br />
(<br />
− λ ) k<br />
2<br />
µ µ<br />
= 1 µ<br />
k≥0<br />
1<br />
1 + λ 2 /µ = 1<br />
, (1.64)<br />
µ + λ 2<br />
where it should come as no surprise that we cheerfully ignore all issues about<br />
convergence, in the spirit of perturbation theory. Every propagator line can<br />
(and must !) be dressed in this way once any two-point vertex (elementary<br />
of effective, that is, as the result of a collection of closed loops with two legs<br />
sticking out) is at hand.
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