Feynman Path Integral Formulation

6.12 Scalar Matter Fields 203
if and only if the curvature is locally zero, which is not true of any of the other
curvature squared terms.
6.12 Scalar Matter Fields
In the previous section we have discussed the construction and the invariance properties
of a lattice action for pure gravity. Next a scalar field can be introduced as the
simplest type of dynamical matter that can be coupled invariantly to gravity. In the
continuum the scalar action for a single component field φ(x) is usually written as
∫
I[g,φ]= 1 2
dx √ g[g μν ∂ μ φ∂ ν φ +(m 2 + ξ R)φ 2 ]+... (6.124)
where the dots denote scalar self-interaction terms. Thus for example a scalar field
potential U(φ) could be added containing quartic field terms, whose effects could
be of interest in the context of cosmological models where spontaneously broken
symmetries play an important role. The dimensionless coupling ξ is arbitrary; two
special cases are the minimal (ξ = 0) and the conformal (ξ = 1 6
) coupling case. In
the following we shall mostly consider the case ξ = 0. Also, it will be straightforward
to extend later the treatment to the case of an N s -component scalar field φ a
with a = 1,...,N s .
One way to proceed is to introduce a lattice scalar φ i defined at the vertices of
the simplices. The corresponding lattice action can then be obtained through a procedure
by which the original continuum metric is replaced by the induced lattice
metric, with the latter written in terms of squared edge lengths as in Eq. (6.3). For
illustrative purposes only the two-dimensional case will be worked out explicitly
here (Christ, Friedberg and Lee, 1982; Itzykson, 1983; Itzykson and Bander, 1983;
Jevicki and Ninomiya, 1985). The generalization to higher dimensions is straightforward,
and in the end the final answer for the lattice scalar action is almost identical
to the two dimensional form. Furthermore in two dimensions it leads to a natural
dicretization of the bosonic string action (Polyakov, 1989).
In two dimensions the simplicial lattice is built out of triangles. For a given triangle
it will be convenient to use the notation of Fig. 6.14, which will display more
readily the symmetries of the resulting scalar lattice action. Here coordinates will
be picked in each triangle along the (1,2) and (1,3) directions.
To construct a lattice action for the scalar field, one performs in two dimensions
the replacement
g μν (x) −→ g ij (Δ)
detg μν (x) −→ detg ij (Δ)
g μν (x) −→ g ij (Δ)
∂ μ φ∂ ν φ −→ Δ i φΔ j φ , (6.125)