Pictures Paths Particles Processes

78 November 7, 2013
The continuum form of the action is
∫ [ 1
S[ϕ, J] =
2 m2 ϕ(⃗x) 2 + 1 2 ∇ϕ(⃗x)) 2 + λ ]
4
4! ϕ(⃗x)4 − J(⃗x)ϕ(⃗x)
d D x , (2.57)
The Feynman rules are seen to be
x
y
↔ Π(⃗x − ⃗y)
↔ − λ 4
¯h
x
↔ + J(⃗x)
¯h
Feynman rules, version 2.4 (2.58)
and also the SDe is a straightforward generalization of the one-dimensional case :
∫
{
φ(⃗x) = d D y Π(⃗x − ⃗y) × J(⃗y)
− λ (
4
φ(⃗y) 3 δ
+ 3¯hφ(⃗y)
6
δJ(⃗y) φ(⃗y) + δ 2 )}
¯h2 (δJ(⃗y)) 2 φ(⃗y)
The classical field equation for this case,
. (2.59)
m 2 ϕ(⃗x) − ⃗ ∇ 2 ϕ(⃗x) + λ 4
3! ϕ(⃗x)3 = J(⃗x) , (2.60)
can be obtained directly from the continuum action by the functional Euler-
Lagrange equation
(
)
δ
δϕ(⃗x) S[ϕ, J] − ∇ ⃗ δ
·
δ∇ϕ(⃗x) ⃗ S[ϕ, J] = 0 . (2.61)
It should be noted that the propagator only depends on |⃗x| and is therefore
rotationally invariant : this is a larger symmetry 16 than that of the original
lattice, that only allows rotations over multiples of π/2. The way in which the
relation between field values at two points depends on the coordinates of these
16 The increase in symmetry depends on an interplay between the lattice action and the form
of the continuum limit ; it is possible to construct actions in which the continuum symmetry
is not larger than that of the lattice theory.