Regge calculus in the canonical form

(3+1) (continuous time) Regge calculus is reduced to the Hamiltonian form. For this purpose the tetrad-connection formulation of the Regge calculus is used: basic variables are connection matrices and antisymmetric area tensors with appropriate bilinear conditions imposed. In these variables the action can be made quasipolynomial (with arcsin as the only deviation from polynomiality). The constraints are classified, classical and quantum consequences are discussed.     

Area expectation values in quantum area Regge calculus

Regge calculus generalised to independent area tensor variables is considered. Continuous time limit is found and formal Feynman path integral measure corresponding to the canonical quantisation is written out. Quantum measure in the completely discrete theory is found which possesses the property to lead to the Feynman path integral in the continuous time limit whatever coordinate is chosen as the time. This measure can be well defined by passing to the integration over imaginary field variables (area tensors). Averaging with the help of this measure gives finite expectation values for areas.     

Discrete quantum gravity in the Regge calculus formalism

We discuss an approach to the discrete quantum gravity in the Regge calculus formalism that was developed in a number of our papers. The Regge calculus is general relativity for a subclass of general Riemannian manifolds called piecewise flat manifolds. The Regge calculus deals with a discrete set of variables, triangulation lengths, and contains continuous general relativity as a special limiting case where the lengths tend to zero. In our approach, the quantum length expectations are nonzero and of the order of the Plank scale, 10^?33 cm, implying a discrete spacetime structure on these scales.     

Parallelizable implicit evolution scheme for Regge calculus

The role of Regge calculus as a tool for numerical relativity is discussed, and a parallelizable implicit evolution scheme described. Because of the structure of the Regge equations, it is possible to advance the vertices of a triangulated spacelike hypersurface in isolation, solving at each vertex a purely local system of implicit equations for the new edge lengths involved. (In particular, equations of global “elliptic type” do not arise.) Consequently, there exists a parallel evolution scheme which divides the vertices into families of nonadjacent elements and advances all the vertices of a family simultaneously. The relation between the structure of the equations of motion and the Bianchi identities is also considered. The method is illustrated by a preliminary application to a 600-cell Friedmann cosmology. The parallelizable evolution algorithm described in this paper should enable Regge calculus to be a viable discretization technique in numerical relativity.     

Boundary terms in the action for the Regge calculus

The boundary terms in the action for Regge's formulation of general relativity on a simplicial net are derived and compared with the boundary terms in continuum general relativity.Supported in part by the National Science Foundation grants PHY 78-09620 and PHY 78-24275.     

Semiclassical regime of Regge calculus and spin foams

Recent attempts to recover the graviton propagator from spin foam models involve the use of a boundary quantum state peaked on a classical geometry. The question arises whether beyond the case of a single simplex this suffices for peaking the interior geometry in a semiclassical configuration. In this paper we explore this issue in the context of quantum Regge calculus with a general triangulation. Via a stationary phase approximation, we show that the boundary state succeeds in peaking the interior in the appropriate configuration, and that boundary correlations can be computed order by order in an asymptotic expansion. Further, we show that if we replace at each simplex the exponential of the Regge action by its cosine—as expected from the semiclassical limit of spin foam models—then the contribution from the sign-reversed terms is suppressed in the semiclassical regime and the results match those of conventional Regge calculus.     

Reggeon calculus rules for the triple Regge region

We show that Reggeon unitarity is sufficient to determine the Reggeon calculus rules for the one particle inclusive cross section in the triple Regge region. The central result is that only one vertex in each Reggeon diagram does not conserve E = 1 — angular momentum. The rules, which are most easily expressed using Raleigh-Schroedinger perturbation theory, lead to a simple closed expression for the inclusive cross section.     

Regge calculus and observations. II. Further applications

The method, developed in an earlier paper, for tracing geodesies of particles and light rays through Regge calculus space-times, is applied to a number of problems in the Schwarzschild geometry. It is possible to obtain accurate predictions of light bending by taking sufficiently small Regge blocks. Calculations of perihelion precession, Thomas precession, and the distortion of a ball of fluid moving on a geodesic can also show good agreement with the analytic solution. However difficulties arise in obtaining accurate predictions for general orbits in these space-times. Applications to other problems in general relativity are discussed briefly.     

Path integral measure in Regge calculus from the functional Fourier transform

The problem of fixing measure in the path integral for the Regge-discretised gravity is considered from the viewpoint of it is “best approximation” to the already known formal continuum general relativity (GR) measure. A rigorous formulation may consist in treating the measure as functional on the space of the metric functionals. We require coincidence of the measures for the discrete and continuous versions of the theory on some sufficiently large (dense) set of metric functionals which exist and admit exact definitions and calculation in the both versions. This set consists of generalisation of the usual finite-dimensional plane waves to the functional space so that the discrete measure follows by means of the functional Fourier transform. The possibility for such set to exist is due to the Regge manifold being a particular case of general Riemannian one (Regge calculus is a minisuperspace theory). Only a certain continuum measure among the local ones (the scale invariant Misner measure) is found to be reduciable in this way to the well defined Regge discretisation, and we find the two versions for the latter depending on what metric tensor, covariant or contravariant one, is taken as fundamental field variable. The closed expressions for the measure are obtained in the two simple cases of Regge manifold. These turn out to be quite reasonable one of them indicating to possibility of passing in backward direction when appropriately defined continuum limit of the Regge measure would reproduce the original continuum GR measure.     

An application of a first-order formulation of the Regge calculus

A numerical Regge spacetime is constructed by way of a new formulation of the Regge calculus [1,2,3]. The results obtained are compared with the known exact results [4]. It will be shown that the approximations suggested in [3] are, in this instance, very accurate.     

Factorization of topology changing amplitudes in the Regge calculus approach to quantum cosmology

We study the form of topology changing amplitudes within the Regge calculus approach to four-dimensional gravity. The four-dimensional simplicial complex is chosen to be a cone over the disjoint union of a number of topologically distinct lens spaces. By restricting attention to a simplicial minisuperspace, the analytic properties of the Regge action can be identified explicitly. The classical extrema and convergent steepest descent contours defining these amplitudes are determined, and a factorization property is established. In the cases studied, we find ground state wave functions which predict Lorentzian oscillatory behaviour in the late universe.     

Continuous moment sum rules and Regge amplitudes for pion-nucleon scattering

Using continuous moment sum rules and phase-shift data as the only input, Regge-parametrized amplitudes for pion-nucleon scattering are derived. These amplitudes can to great accuracy reproduce high-energy differential cross section and polarization data. They also agree in detail with other model amplitudes and reproduce fairly well the pion-nucleon s-channel helicity amplitudes at the energies where a complete analysis of pion-nucleon scattering has been performed. It turns out to be important that the integration of the amplitude over the unphysical range is treated with care.     

Neutrino in matter and external fields

A technique for describing various processes proceeding in matter and involving neutrinos and electrons is discussed. This technique is based on “the method of exact solutions,” which implies the use of solutions to proper Dirac equations for particle wave functions in matter. Exact solutions for the neutrino and the electron in the cases of uniform nonmoving and rotating matter are discussed. On studying relativistic neutrino motion and associated neutrino-energy quantization in rotating matter, a semiclassical interpretation of particle finite motion is developed. In the general case of neutrino and electron motion in matter with varying parameters, the corresponding effective force acting on the particles is determined. The possibility of electromagnetic-wave radiation by an electron that moves in a dense neutrino flux of varying density and which is accelerated by this kind of force is predicted.     

Neutrinos in matter and external fields

A brief survey of effects generated by the influence of the environment on neutrinos is presented. The issues considered here include flavor and spin oscillations of neutrinos in matter, in electromagnetic fields of various configurations, and in gravitational fields; the electromagnetic properties of the neutrinos and the environment-induced change in these properties; photoneutrino processes generated by the environment; urca processes in magnetic fields; various mechanisms that may be responsible for the asymmetry of neutrino radiation from neutron stars; quantum states of neutrinos in matter and the spin light of neutrinos in matter and external fields; and astrophysical and cosmological applications of the above processes and phenomena. The method that is employed to describe the effect of the environment on neutrinos (as well as on electrons) and which is based on the application of exact solutions to the corresponding modified Dirac equations for particles in matter is briefly discussed.