arXiv subjects

Vadim Belov

Publications and source records attributed to Vadim Belov.

3 recordsLinked to original sources

arXiv2019

On Geometry and Symmetries in Classical and Quantum Theories of Gauge Gravity

Spin Foam and Loop approaches to Quantum Gravity reformulate Einstein's theory of relativity in terms of connection variables. The metric properties are encoded in face bivectors/conjugate fluxes that are required to satisfy certain conditions, in order to allow for their geometric interpretation. We show that the (sub-)set of the so-called `volume simplicity constraints' is not implemented properly in the current EPRL-FK spinfoam vertex amplitude, if extended to arbitrary polyhedra. We then propose that a certain knot-invariant of the bivector geometry, induced on the boundary graph, encodes the missing conditions, allowing for reconstruction of a polytope from its two-dimensional faces. Implemented in the quantum amplitude, this leads to corrected semi-classical asymptotics for a hypercuboid, and is conjectured to be non-trivial in more general situations. The analysis of linear version of `volume simplicity' suggests to switch from hypersurface normals to edge lengths, that is from 3-forms directly to tetrads -- in the extended configuration space of the Plebanski constrained formulation. We then give the corresponding dual version of linear simplicity constraints, which prescribe 3d volume for the polyhedral faces in the boundary of a 4d polytope. We also analyse the status of metric/vielbein degrees of freedom and the role of local translations in the classical Einstein-Cartan gravity, viewed as a Poincare gauge theory. The relation with the diffeomorphism symmetry is established through the key concept of development, which generalizes parallel transport of vectors in the geometric theory of Cartan connections. We advocate the latter to be the natural gauge-theoretic framework for the theory of relativity.

gr-qc↗

arXiv2018

Poincare-Plebanski formulation of GR and dual simplicity constraints

We revise the classical continuum formulation behind the Spin Foam approach to the quantization of gravity. Based on the recent applications of the current EPRL-FK model beyond triangulations, we identify the tension with the implementation of the 'volume' part of simplicity constraints, required for the passage from the topological BF theory to gravity. The crucial role, played by 4d normals in the linear version of constraints, necessitates the extension of the configuration space, and we argue to switch from normal 3-forms directly to tetrads. The requirement of vanishing torsion leads to consider first an unconstrained extended Poincare BF theory, which we characterize fully both at the Lagrangian and Hamiltonian levels, paying special attention to its gauge symmetries. The simplicity constraints are introduced naturally, in the spirit of Plebanski formulation, and we give their tetradic version, dual to that of using 3-forms. This brings us much closer to the geometric content of General Relativity.

gr-qc↗

arXiv2017

On the volume simplicity constraint in the EPRL spin foam model

We propose a quantum version of the quadratic volume simplicity constraint for the EPRL spin foam model. It relies on a formula for the volume of 4-dimensional polyhedra, depending on its bivectors and the knotting class of its boundary graph. While this leads to no further condition for the 4-simplex, the constraint becomes non-trivial for more complicated boundary graphs. We show that, in the semi-classical limit of the hypercuboidal graph, the constraint turns into the geometricity condition observed recently by several authors.

gr-qc↗