Length expectation values in quantum Regge calculus

Abstract

Regge calculus configuration superspace can be embedded into a more general superspace where the length of any edge is defined ambiguously depending on the 4-tetrahedron containing the edge. Moreover, the latter superspace can be extended further so that even edge lengths in each the 4-tetrahedron are not defined, only area tensors of the 2-faces in it are. We make use of our previous result concerning quantisation of the area tensor Regge calculus which gives finite expectation values for areas. Also our result is used showing that quantum measure in the Regge calculus can be uniquely fixed once we know quantum measure on (the space of the functionals on) the superspace of the theory with ambiguously defined edge lengths. We find that in this framework quantisation of the usual Regge calculus is defined up to a parameter. The theory may possess nonzero (of the order of Plank scale) or zero length expectation values depending on whether this parameter is larger or smaller than a certain value. Vanishing length expectation values means that the theory is becoming continuous, here dynamically in the originally discrete framework.

On the possibility of finite quantum Regge calculus

“…Let us make a simple scaling estimate. Consider estimation model used in [13]. The half of 36 components of the 6 antisymmetric tensors v ab λµ in a given 4-simplex are dynamical π ab λµ (λµ = 12, 23, 31), another half are τ ab λµ (λµ = 14, 24, 34, and a scale ε≪ 1 for τ σ 2 is chosen).…”

“…is made continuous by flattening the 4-simplices in the corresponding direction should coincide with Hamiltonian path integral (with that coordinate playing the role of time). Namely, in the Hamiltonian formalism absence of integration over area tensors of triangles which pick out some coordinate t (t-like ones) corresponds to some gauge fixing.Given the above (spontaneously arisen) asymmetry between the different area tensors we nevertheless can ask about maximally symmetrical form of the measure extended by inserting possible integrations[13]. We can integrate over d 6 τ σ 2 in a nonsingular way if δ-functions are inserted which fix the scale of these tensors, say, at the level ε ≪ 1.…”

Integration Over Connections in the Discretized Gravitational Functional Integrals

“…(the viewpoint is undertaken that the primed and unprimed edges are the same in the world coordinates, while tangential (affine) metric on the common face in the 4- [20,21] ∆g αβ = g αβ − g ′ αβ ). The product of (42) and…”

On the area expectation values in area tensor Regge calculus in the Lorentzian domain

“…Being defined in the extended configuration superspace with independent area tensors this measure then can be practically uniquely projected on the hypersurface in this superspace corresponding to the ordinary Regge calculus. The result is finite nonzero length expectation values [4,5].…”