Some minisuperspace model for the Faddeev formulation of gravity

We consider Faddeev formulation of general relativity in which the metric is composed of ten vector fields or a $4 \times 10$ tetrad. This formulation reduces to the usual general relativity upon partial use of the field equations. A distinctive feature of the Faddeev action is its finiteness on the discontinuous fields. This allows to introduce its minisuperspace formulation where the vector fields are constant everywhere on ${\rm I \hspace{-3pt} R}^4$ with exception of a measure zero set (the piecewise constant fields). The fields are parameterized by their constant values {\it independently} chosen in, e. g., the 4-simplices or, say, parallelepipeds into which ${\rm I \hspace{-3pt} R}^4$ can be decomposed. The form of the action for the vector fields of this type is found. We also consider the piecewise constant vector fields approximating the fixed smooth ones. We check that if the regions in which the vector fields are constant are made arbitrarily small, the minisuperspace action and eqs of motion tend to the continuum Faddeev ones.Comment: 14 pages, 3 figures, to appear in Mod. Phys. Lett.

Spectrum of area in the Faddeev formulation of gravity

Faddeev formulation of general relativity (GR) is considered where the metric is composed of ten vector fields or a ten-dimensional tetrad. Upon partial use of the field equations, this theory results in the usual GR. Earlier we have proposed first-order representation of the minisuperspace model for the Faddeev formulation where the tetrad fields are piecewise constant on the polytopes like 4-simplices or, say, cuboids into which ${\rm R}^4$ can be decomposed, an analogue of the Cartan-Weyl connection-type form of the Hilbert-Einstein action in the usual continuum GR. In the Hamiltonian formalism, the tetrad bilinears are canonically conjugate to the orthogonal connection matrices. We evaluate the spectrum of the elementary areas, functions of the tetrad bilinears. The spectrum is discrete and proportional to the Faddeev analog $\gamma_{\rm F}$ of the Barbero-Immirzi parameter $\gamma$. The possibility of the tetrad and metric discontinuities in the Faddeev gravity allows to consider any surface as consisting of a set of virtually independent elementary areas and its spectrum being the sum of the elementary spectra. Requiring consistency of the black hole entropy calculations known in the literature we are able to estimate $\gamma_{\rm F}$.Comment: 20 pages. ref to our previous work arXiv:1206.5509 is added and discusse

On the possibility of finite quantum Regge calculus

The arguments were given in a number of our papers that the discrete quantum gravity based on the Regge calculus possesses nonzero vacuum expectation values of the triangulation lengths of the order of Plank scale $10^{-33}cm$. These results are considered paying attention to the form of the path integral measure showing that probability distribution for these linklengths is concentrated at certain nonzero finite values of the order of Plank scale. That is, the theory resembles an ordinary lattice field theory with fixed spacings for which correlators (Green functions) are finite, UV cut off being defined by lattice spacings. The difference with an ordinary lattice theory is that now lattice spacings (linklengths) are themselves dynamical variables, and are concentrated around certain Plank scale values due to {\it dynamical} reasons.Comment: 12 pages, plain LaTeX, readability improved, matches version to be publishe

A version of the connection representation of Regge action

We define for any 4-tetrahedron (4-simplex) the simplest finite closed piecewise flat manifold consisting of this 4-tetrahedron and of the one else 4-tetrahedron identical up to reflection to the present one (call it bisimplex built on the given 4-simplex, or two-sided 4-simplex). We consider arbitrary piecewise flat manifold. Gravity action for it can be expressed in terms of sum of the actions for the bisimplices built on the 4-simplices constituting this manifold. We use representation of each bisimplex action in terms of rotation matrices (connections) and area tensors. This gives some representation of any piecewise flat gravity action in terms of connections. The action is a sum of terms each depending on the connection variables referring to a single 4-tetrahedron. Application of this representation to the path integral formalism is considered. Integrations over connections in the path integral reduce to independent integrations over finite sets of connections on separate 4-simplices. One of the consequences is exponential suppression of the result at large areas or lengths (compared to Plank scale). It is important for the consistency of the simplicial description of spacetime.Comment: 20 page

Simplicial Palatini action

We consider the piecewise flat spacetime and a simplicial analog of the Palatini form of the general relativity (GR) action where the discrete Christoffel symbols are given on the tetrahedra as variables that are independent of the metric. Excluding these variables classically gives exactly the Regge action. This paper continues our previous work. Now we include the parity violation term and the analogue of the Barbero-Immirzi parameter introduced in the orthogonal connection form of GR. We consider the path integral and the functional integration over connection. The result of the latter (for certain limiting cases of some parameters) is compared with the earlier found result of the functional integration over connection for the analogous {\it orthogonal} connection representation of Regge action. These results, mainly as some measures on the lengths/areas, are discussed for the possibility of the diagram technique where the perturbative diagrams for the Regge action calculated using the measure obtained are finite. This finiteness is due to these measures providing elementary lengths being mostly bounded and separated from zero, just as finiteness of a theory on a lattice with an analogous probability distribution of spacings.Comment: 19 pages, discussion of the finite diagram technique is adde

Continuous matter fields in Regge calculus

We find that the continuous matter fields are ill-defined in Regge calculus in the physical 4D theory since the corresponding effective action has infinite terms unremovable by the UV renormalisation procedure. These terms are connected with the singular nature of the curvature distribution in Regge calculus, namely, with the presence in d>2 dimensions of the (d-3)-dimensional simplices where the (d-2)-dimensional ones carrying different conical singularities are meeting. Possible resolution of this difficulty is discretisation of matter fields in Regge background.Comment: 4 pages, LaTe

Faddeev gravity action on the piecewise constant fundamental vector fields

In the Faddeev formulation of gravity, the metric is regarded as composite field, bilinear of $d = 10$ 4-vector fields. We derive the minisuperspace (discrete) Faddeev action by evaluating the Faddeev action on the spacetime composed of the (flat) 4-simplices with constant 4-vector fields. This is an analog of the Regge action obtained by evaluating the Hilbert-Einstein action on the spacetime composed of the flat 4-simplices. One of the new features of this formulation is that the simplices are not required to coincide on their common faces. Also an analog of the Barbero-Immirzi parameter $\gamma$ can be introduced in this formalism.Comment: 8 pages, reported on the conference "Quantum Field Theory and Gravity 2014" (July 28 - August 3 2014, Tomsk, Russia

Area Regge calculus and continuum limit

Encountered in the literature generalisations of general relativity to independent area variables are considered, the discrete (generalised Regge calculus) and continuum ones. The generalised Regge calculus can be either with purely area variables or, as we suggest, with area tensor-connection variables. Just for the latter, in particular, we prove that in analogy with corresponding statement in ordinary Regge calculus (by Feinberg, Friedberg, Lee and Ren), passing to the (appropriately defined) continuum limit yields the generalised continuum area tensor-connection general relativity.Comment: 10 pages, Te

On area spectrum in the Faddeev gravity

We consider Faddeev formulation of gravity, in which the metric is bilinear of $d = 10$ 4-vector fields. A unique feature of this formulation is that the action remains finite for the discontinuous fields (although continuity is recovered on the equations of motion). This means that the spacetime can be decomposed into the 4-simplices virtually not coinciding on their common faces, that is, independent. This allows, in particular, to consider a surface as consisting of a set of virtually independent elementary pieces (2-simplices). Then the spectrum of surface area is the sum of the spectra of independent elementary areas. We use connection representation of the Faddeev action for the piecewise flat (simplicial) manifold earlier proposed in our work. The spectrum of elementary areas is the spectrum of the field bilinears which are canonically conjugate to the orthogonal connection matrices. We find that the elementary area spectrum is proportional to the Barbero-Immirzi parameter $\gamma$ in the Faddeev gravity and is similar to the spectrum of the angular momentum in the space with the dimension $d - 2$. Knowing this spectrum allows to estimate statistical black hole entropy. Requiring that this entropy coincide with the Bekenstein-Hawking entropy gives the equation, known in the literature. This equation allows to estimate $\gamma$ for arbitrary $d$, in particular, $\gamma = 0.39...$ for genuine $d = 10$.Comment: 17 page

From areas to lengths in quantum Regge calculus

Quantum area tensor Regge calculus is considered, some properties are discussed. The path integral quantisation is defined for the usual length-based Regge calculus considered as a particular case (a kind of a state) of the area tensor Regge calculus. Under natural physical assumptions the quantisation of interest is practically unique up to an additional one-parametric local factor of the type of a power of $\det\|g_{\lambda\mu}\|$ in the measure. In particular, this factor can be adjusted so that in the continuum limit we would have any of the measures usually discussed in the continuum quantum gravity, namely, Misner, DeWitt or Leutwyler measure. It is the latter two cases when the discrete measure turns out to be well-defined at small lengths and lead to finite expectation values of the lengths.Comment: 10 pages, LaTe