74 research outputs found

### 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

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