Feynman Path Integral Formulation

6.13 Invariance Properties of the Scalar Action 2096.13 Invariance Properties of the Scalar ActionIn the very simple case of one dimension (d = 1) one can work out the details to anydegree of accuracy, and see how potential problems arise and how they are resolved.Introduce a scalar field φ n defined on the sites, with actionI(φ) = 1 2N∑n=1( )φn+1 − φ 2nV 1 (l n )+ 1 Nl2 ω n∑ V 0 (l n ) φn 2 , (6.150)n=1with φ(N + 1) =φ(1). It is natural in one dimension to take for the “volume peredge” V 1 (l n )=l n , and for the “volume per site” V 0 (l n )=(l n + l n−1 )/2. Here ωplays the role of a mass for the scalar field, ω = m 2 . In addition one needs a termλ L(l) =λ 0N∑n=1l n , (6.151)which is necessary in order to make the dl n integration convergent at large l. Varyingthe 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−1This is the discrete analog of the equation g −1/2 ∂g −1/2 ∂φ = ωφ. The spectrum ofthe Laplacian of Eq. (6.152) corresponds to Ω ≡−ω > 0. Variation with respect tol n gives instead12l 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 coordinatesin embedding space. In the following we shall only consider the case ω = 0, correspondingto a massless scalar field.It is instructive to look at the invariance properties of the scalar field action underthe continuous lattice diffeomorphisms defined in Eq. (6.102). Physically, these localgauge transformations, which act on the vertices, correspond to re-assignmentsof edge lengths which leave the distance between two fixed points unchanged. Inthe simplest case, only two neighboring edge lengths are changed, leaving the totaldistance between the end points unchanged. On physical grounds one would like tomaintain such an invariance also in the case of coupling to matter, just as is done inthe continuum.The scalar nature of the field requires that in the continuum under a change ofcoordinates x → x ′ ,φ ′ (x ′ )=φ(x) , (6.154)