Feynman Path Integral Formulation

6.13 Invariance Properties of the Scalar Action 209
6.13 Invariance Properties of the Scalar Action
In the very simple case of one dimension (d = 1) one can work out the details to any
degree of accuracy, and see how potential problems arise and how they are resolved.
Introduce a scalar field φ n defined on the sites, with action
I(φ) = 1 2
N
∑
n=1
( )
φn+1 − φ 2
n
V 1 (l n )
+ 1 N
l
2 ω n
∑ V 0 (l n ) φn 2 , (6.150)
n=1
with φ(N + 1) =φ(1). It is natural in one dimension to take for the “volume per
edge” V 1 (l n )=l n , and for the “volume per site” V 0 (l n )=(l n + l n−1 )/2. Here ω
plays the role of a mass for the scalar field, ω = m 2 . In addition one needs a term
λ L(l) =λ 0
N
∑
n=1l n , (6.151)
which is necessary in order to make the dl n integration convergent at large l. Varying
the action with respect to φ n gives
[
2 φn+1 − φ n
− φ ]
n − φ n−1
= ωφ n . (6.152)
l n−1 + l n l n l n−1
This is the discrete analog of the equation g −1/2 ∂g −1/2 ∂φ = ωφ. The spectrum of
the Laplacian of Eq. (6.152) corresponds to Ω ≡−ω > 0. Variation with respect to
l n gives instead
1
2l 2 n
(φ n+1 − φ n ) 2 = λ 0 + 1 4 ω (φ 2 n + φ 2 n+1) . (6.153)
For ω = 0 it suggests the well-known interpretation of the fields φ n as coordinates
in embedding space. In the following we shall only consider the case ω = 0, corresponding
to a massless scalar field.
It is instructive to look at the invariance properties of the scalar field action under
the continuous lattice diffeomorphisms defined in Eq. (6.102). Physically, these local
gauge transformations, which act on the vertices, correspond to re-assignments
of edge lengths which leave the distance between two fixed points unchanged. In
the simplest case, only two neighboring edge lengths are changed, leaving the total
distance between the end points unchanged. On physical grounds one would like to
maintain such an invariance also in the case of coupling to matter, just as is done in
the continuum.
The scalar nature of the field requires that in the continuum under a change of
coordinates x → x ′ ,
φ ′ (x ′ )=φ(x) , (6.154)