Feynman Path Integral Formulation

Chapter 6Lattice Regularized Quantum Gravity6.1 The Lattice TheoryThe following sections are based on the lattice discretized description of gravityknown as Regge calculus, where the Einstein theory is expressed in terms of asimplicial decomposition of space-time manifolds. Its use in quantum gravity isprompted by the desire to make use of techniques developed in lattice gauge theories(Wilson, 1973), 1 but with a lattice which reflects the structure of space-timerather than just providing a flat passive background (Regge, 1961). It also allowsone to use powerful nonperturbative analytical techniques of statistical mechanicsas well as numerical methods. A regularized lattice version of the continuum fieldtheory is also usually perceived as a necessary prerequisite for a rigorous study ofthe latter.In Regge gravity the infinite number of degrees of freedom in the continuum isrestricted by considering Riemannian spaces described by only a finite number ofvariables, the geodesic distances between neighboring points. Such spaces are takento be flat almost everywhere and are called piecewise linear (Singer and Thorpe,1967). The elementary building blocks for d-dimensional space-time are simplicesof dimension d. A 0-simplex is a point, a 1-simplex is an edge, a 2-simplex is atriangle, a 3-simplex is a tetrahedron. A d-simplex is a d-dimensional object withd + 1 vertices and d(d + 1)/2 edges connecting them. It has the important propertythat the values of its edge lengths specify the shape, and therefore the relative angles,uniquely.A simplicial complex can be viewed as a set of simplices glued together in sucha way that either two simplices are disjoint or they touch at a common face. Therelative position of points on the lattice is thus completely specified by the incidencematrix (it tells which point is next to which) and the edge lengths, and this in turninduces a metric structure on the piecewise linear space. Finally the polyhedronconstituting the union of all the simplices of dimension d is called a geometrical1 As an example of a state-of-the-art calculation of hadron properties in the lattice formulation ofSU(3) QCD see (Aoki et al, 2003).169