Feynman Path Integral Formulation

6.12 Scalar Matter Fields 203if and only if the curvature is locally zero, which is not true of any of the othercurvature squared terms.6.12 Scalar Matter FieldsIn the previous section we have discussed the construction and the invariance propertiesof a lattice action for pure gravity. Next a scalar field can be introduced as thesimplest type of dynamical matter that can be coupled invariantly to gravity. In thecontinuum the scalar action for a single component field φ(x) is usually written as∫I[g,φ]= 1 2dx √ 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 fieldpotential U(φ) could be added containing quartic field terms, whose effects couldbe of interest in the context of cosmological models where spontaneously brokensymmetries play an important role. The dimensionless coupling ξ is arbitrary; twospecial cases are the minimal (ξ = 0) and the conformal (ξ = 1 6) coupling case. Inthe following we shall mostly consider the case ξ = 0. Also, it will be straightforwardto extend later the treatment to the case of an N s -component scalar field φ awith a = 1,...,N s .One way to proceed is to introduce a lattice scalar φ i defined at the vertices ofthe simplices. The corresponding lattice action can then be obtained through a procedureby which the original continuum metric is replaced by the induced latticemetric, with the latter written in terms of squared edge lengths as in Eq. (6.3). Forillustrative purposes only the two-dimensional case will be worked out explicitlyhere (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 identicalto the two dimensional form. Furthermore in two dimensions it leads to a naturaldicretization of the bosonic string action (Polyakov, 1989).In two dimensions the simplicial lattice is built out of triangles. For a given triangleit will be convenient to use the notation of Fig. 6.14, which will display morereadily the symmetries of the resulting scalar lattice action. Here coordinates willbe 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 dimensionsthe replacementg 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)