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November 7, 2013 71
= ∑ [ 1
2 m2 ϕ(x) 2 + 1 ]
2 ϕ′ (x) 2 ∆
n
∫ [ 1
=
2 m2 ϕ(x) 2 + 1 ]
2 ϕ′ (x) 2 dx . (2.29)
The interaction and source terms in the path integral do not have a factor
∆ coming out naturally, but we may simply define the continuum limits by
redefining the objects in the action :
λ 4 → ∆λ 4 , J n → ∆J(x) , (2.30)
so that the continuum limit of the full action, including this time also the sources,
becomes 13
∫ [ 1
S[ϕ, J] =
2 m2 ϕ(x) 2 + 1 2 ϕ′ (x) 2 + λ ]
4
4! ϕ(x)4 − J(x)ϕ(x) dx . (2.31)
Note the notation with square brackets: the action is now no longer a number
depending on (a countably infinite set of) numbers, but rather on the functions
ϕ(x) and J(x) ; this is called a functional.
2.3.3 The continuum limit of the classical equation
For the discrete action, there is an obvious classical equation :
∂
∂ϕ n
S({ϕ}) = 0 ∀n , (2.32)
where, again, the source terms have been subsumed into the action. For the ϕ 4
model of Eq.(2.1), the classical equation is therefore
µϕ n − γ(ϕ n+1 + ϕ n−1 ) + ∆λ 4
ϕ 3 n = ∆J n (2.33)
3!
for all n, and the extra factor ∆ in the coupling constant and the sources have
been taken into account. The Weyl prescription leads us to write
µϕ n − γ(ϕ n+1 + ϕ n−1 ) ≈ m 2 ∆ϕ(x) − ∆ϕ ′′ (x) , (2.34)
so that the continuum limit of the classical field equation takes the form
m 2 ϕ(x) − ϕ ′′ (x) + λ 4
3! ϕ(x)3 = J(x) . (2.35)
This is precisely the Euler-Lagrange equation, that can also be obtained immediately
from the continuum form of the action by taking functional derivatives.
To see this, let us assume that the action of a theory can be written as
∫ ( )
S[ϕ] = F ϕ(x); ϕ ′ (x) dx . (2.36)
13 Strictly speaking, the Weyl ordering requires the replacement of J n not by ∆J(x) but by
∆J(x)+∆ 2 J ′ (x)/2. The additional term, however, vanishes in the continuum limit as ∆ → 0,
as do the higher powers of ∆ involved in the ϕ n 4 term.