The No-Boundary Wave Function
Simplicial Approximation
The no-boundary wave function of the universe (NBWF) is defined by functional integrals over certain four-dimensional geometries and matter field configurations [NBWF-def]. Analogous integrals can be studied in minisuperspace models where the infinite number of degrees o
f freedom of the continuum are truncated to a finite number. The papers in this section concern models based on piecewise linear (simplicial) approximations to geometries using the methods of the Regge calculus.
A simplicial geometry is defined by a piecewise linear manifold (like a geodesic dome but in four-dimensions) and an assignment of lengths to its edges. Better and better approximations to a given continuum geometry can be defined by finer and finer triangulations with more and more edges. For a given manifold the simplical NBWF is defined by an integral of exp(-action) over some of the edge lengths. Difficult functional integrals become tractable multiple integrals.
Simplicial Quantum Gravity [66]
A brief overview of the issues in the simplical approximation to sum-over-histories defining the NBWF, its semiclassical approximation, sums over topology, etc. These are developed in detail in the papers on this page and the next.
Simplicial Minisuperspace I -- General Discussion [62]
This paper introduces the notion of simplicial superspace and the simplicial approximation to the NBWF which lives on this superspace. Explicit formulae are given for the elements of the wave function's construction in terms of the edge lengths of the simplical geometry. The semiclassical approximation is discussed. It's shown how the diffeomorphism invariance of general relativity might be recovered in the limit of finer and finer triangulations. Various ideas for implementing a sum over topologies as a sum over piecewise linear manifolds are discussed.
Simplicial Minisuperspace II -- Classical Solutions for Simple Triangulations [67]
The semiclassical approximation to the NBWF in the simplicial approximation is determined by the saddle points of the Regge action on simplicial geometries. This paper calculated these saddle points numerically for the simplest triangulations of closed four-dimensional manifolds with topologies S4, CP2 and S2 X S2.
Simplical Minisuperspace III -- Integration Contours in a Five-Simplex Model [83]
This paper calculates a semiclassiclassical approximation to the NBWF in the simplest simplicial minisuperspace model assuming just pure gravity and a positive cosmological constant. The simplicial manifold consists of five four-simplices joined to make a five simplex from which one four-simplex face has been removed. Further symmetries are assumed to restrict the simplicial minisuperspace to just two edge length variables. The contour of integration defining the NBWF is explicitly discussed, and the behavior of the wave function exhibited.
Boundary Terms in the Action for the Regge Calculus [48]
(with R. Sorkin) The continuum action for general relartivity has a surface term that is the integral of the trace of the extrinsic curvature. This paper derives the surface term for the (Regge) action for a simplicial geometry.
Solutions of the Regge Equations on Some Triangulations of CP2 [110]
(with Z. Perjés) This continues the program started in Simplicial Minisuperspace II (above) of exploring the saddle points of the Regge action on simplicial manifolds with various topologies and various levels of refinement. In particular it considers the solutions of the Regge equations with positive cosmological constant on a family of triangulations of CP2. The saddle point action is computed both analytically and numerically and compared to the action of the most symmetric continuum saddle point (Fubini-Study). The results show that merely increasing the refinement of the triangulation does not ensure a steady approach to the continuum.
Numerical Quantum Gravity [73]
This paper sketches a program of numerical computation of the predictions of the wave function of the universe by evaluating sums over simplicial geometries. Mostly a review of discussion in the preceding papers in more general terms.
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