Integration Over Connections in the Discretized Gravitational Functional Integrals

Abstract

The result of performing integrations over connection type variables in the path integral for the discrete field theory may be poorly defined in the case of non-compact gauge group with the Haar measure exponentially growing in some directions. This point is studied in the case of the discrete form of the first order formulation of the Einstein gravity theory. Here the result of interest can be defined as generalized function (of the rest of variables of the type of tetrad or elementary areas) i. e. a functional on a set of probe functions. To define this functional, we calculate its values on the products of components of the area tensors, the so-called moments. The resulting distribution (in fact, probability distribution) has singular (δ-function-like) part with support in the nonphysical region of the complex plane of area tensors and regular part (usual function) which decays exponentially at large areas. As we discuss, this also provides suppression of large edge lengths which is important for internal consistency, if one asks whether gravity on short distances can be discrete. Some another features of the obtained probability distribution including occurrence of the local maxima at a number of the approximately equidistant values of area are also considered.

Simplicial Palatini action

“…Then the measure in such an extended configuration superspace is projected onto the hypersurface of true gravity by inserting δ-function factors ensuring the existence of the edge vectors and metric for which these tensors are bivectors, and the unambiguity (continuity) of the metric induced on the 3-dimensional faces of the 4-simplices. These factors can be chosen invariant with respect to an arbitrary redefinition of the tetrahedron edges [18] (ie, existence and continuity of metric are purely local properties not depending on the form of the simplex with the help of which these are formulated). To summarize, at small areas the phase volume is suppressed due to a sufficient number of volume elements d 6 v ab σ 2 |σ 4 together with the triangle inequalities for the edges which considerably limits the set of feasible configurations in the configuration superspace.…”

Defining Integrals Over Connections in the Discretized Gravitational Functional Integrals

“…In the previous paper [1] we have considered integration over the discrete connection type variable in the path integral for the discrete version of the first order formulation of Einstein gravity. Definition of the Euclidean version of the path integral requires special treatment because of unboundedness of the action [2].…”

“…Definition of the Euclidean version of the path integral requires special treatment because of unboundedness of the action [2]. If we are left in the Minkowsky region, we have to carefully define integrals over the exponentially growing on the Lorentz boosts Haar measure on the discrete SO (3,1) connections. These can be given sense of the generalized functions.…”

“…[8] we have suggested in Ref. [9] representation of the Einstein action 1 2 R √ −gd 4 x on the piecewise flat manifold in terms of selfdual and antiselfdual parts of area tensors and of finite rotation SO(4) (SO (3,1) in the Minkowsky case) matrices.…”

“…In the consistent path integral approach in the theory with independent area tensors the τ σ 2 s should be considered as fixed [1]. (These are the discrete analogs of the gauge fixed (non-dynamical) variables in the canonical formalism resulting in the continuous time limit.…”