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</div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Khatsymovsky%2C+V+M&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Khatsymovsky%2C+V+M&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Khatsymovsky%2C+V+M&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> </ul> </nav> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.00713">arXiv:2407.00713</a> <span> [<a href="https://arxiv.org/pdf/2407.00713">pdf</a>, <a href="https://arxiv.org/ps/2407.00713">ps</a>, <a href="https://arxiv.org/format/2407.00713">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> </div> <p class="title is-5 mathjax"> Soft synchronous gauge: principal value prescription </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.00713v2-abstract-short" style="display: inline;"> The synchronous gauge in gravity ($g_{0 位} = - 未_{0 位}$) is ill-defined due to the singularity at $p_0 = 0$ in the graviton propagator. Previously we studied "softening" this gauge by considering instead the gauge $n^位g_{位渭} = 0$, $n^位= (1, - \varepsilon (\partial^j \partial_j )^{- 1} \partial^k ) $ in the limit $\varepsilon \to 0$. We now explore the possibility of using a principal value prescri… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.00713v2-abstract-full').style.display = 'inline'; document.getElementById('2407.00713v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.00713v2-abstract-full" style="display: none;"> The synchronous gauge in gravity ($g_{0 位} = - 未_{0 位}$) is ill-defined due to the singularity at $p_0 = 0$ in the graviton propagator. Previously we studied "softening" this gauge by considering instead the gauge $n^位g_{位渭} = 0$, $n^位= (1, - \varepsilon (\partial^j \partial_j )^{- 1} \partial^k ) $ in the limit $\varepsilon \to 0$. We now explore the possibility of using a principal value prescription (not in the standard Cauchy sense), which amounts, roughly speaking, to replacing singularities $p_0^{-j} \Rightarrow [ (p_0 + i \varepsilon )^{-j} + (p_0 - i \varepsilon )^{-j} ] / 2$, which then behave like distributions. We show that such a propagator follows upon adding to the action a gauge-violating term of a general form, which reduces to $ \sim \int f_位螞^{位渭} f_渭\d^4 x $ with a constant operator $螞^{位渭}$ depending on $\partial$ and a metric functional $f_位$. The contribution of the ghost fields to the effective action is analysed. For the required intermediate regularization, the discrete structure of the theory at small distances is implied. It is shown that the ghost contribution can be disregarded in the limit $ \varepsilon \to 0$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.00713v2-abstract-full').style.display = 'none'; document.getElementById('2407.00713v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages, our previous paper arXiv:2312.17119 discusses a gravitational analogue of the {\it Landshoff} prescription; v2 adds consideration of some choice of free parameters that makes the required gauge fixing term non-singular (Eqs. (52-54))</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C45; 83C47 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.17119">arXiv:2312.17119</a> <span> [<a href="https://arxiv.org/pdf/2312.17119">pdf</a>, <a href="https://arxiv.org/ps/2312.17119">ps</a>, <a href="https://arxiv.org/format/2312.17119">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> </div> <p class="title is-5 mathjax"> Soft synchronous gauge in the perturbative gravity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2312.17119v2-abstract-short" style="display: inline;"> An attempt to directly use the synchronous gauge ($g_{0 位} = - 未_{0 位}$) in perturbative gravity leads to a singularity at $p_0 = 0$ in the graviton propagator. This is similar to the singularity in the propagator for Yang-Mills fields $A^a_位$ in the temporal gauge ($A^a_0 = 0$). There the singularity was softened, obtaining this gauge as the limit at $\varepsilon \to 0$ of the gauge… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.17119v2-abstract-full').style.display = 'inline'; document.getElementById('2312.17119v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.17119v2-abstract-full" style="display: none;"> An attempt to directly use the synchronous gauge ($g_{0 位} = - 未_{0 位}$) in perturbative gravity leads to a singularity at $p_0 = 0$ in the graviton propagator. This is similar to the singularity in the propagator for Yang-Mills fields $A^a_位$ in the temporal gauge ($A^a_0 = 0$). There the singularity was softened, obtaining this gauge as the limit at $\varepsilon \to 0$ of the gauge $n^位A^a_位= 0$, $n^位= (1, - \varepsilon (\partial^j \partial_j )^{- 1} \partial^k ) $. Then the singularities at $p_0 = 0$ are replaced by negative powers of $p_0 \pm i \varepsilon$, and thus we bypass these poles in a certain way. Now consider a similar condition on $n^位g_{位渭}$ in perturbative gravity, which becomes the synchronous gauge at $\varepsilon \to 0$. Unlike the Yang-Mills case, the contribution of the Faddeev-Popov ghosts to the effective action is nonzero, and we calculate it. In this calculation, an intermediate regularization is needed, and we assume the discrete structure of the theory at short distances for that. The effect of this contribution is to change the functional integral measure or, for example, to add non-pole terms to the propagator. This contribution vanishes at $\varepsilon \to 0$. Thus, we effectively have the synchronous gauge with the resolved singularities at $p_0 = 0$, where only the physical components $g_{j k}$ are active and there is no need to calculate the ghost contribution. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.17119v2-abstract-full').style.display = 'none'; document.getElementById('2312.17119v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">15 pages, added discussion of the importance of the effective ghost action being $O(\varepsilon^2)$ so that it can be ignored at $\varepsilon \to 0$</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C45; 83C47 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2306.11531">arXiv:2306.11531</a> <span> [<a href="https://arxiv.org/pdf/2306.11531">pdf</a>, <a href="https://arxiv.org/format/2306.11531">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217751X23501439">10.1142/S0217751X23501439 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the gravitational diagram technique in the discrete setup </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2306.11531v2-abstract-short" style="display: inline;"> This article is in the spirit of our work on the consequences of the Regge calculus, where some edge length scale arises as an optimal initial point of the perturbative expansion after functional integration over connection. Now consider the perturbative expansion itself. To obtain an algorithmizable diagram technique, we consider the simplest periodic simplicial structure with a frozen part of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2306.11531v2-abstract-full').style.display = 'inline'; document.getElementById('2306.11531v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2306.11531v2-abstract-full" style="display: none;"> This article is in the spirit of our work on the consequences of the Regge calculus, where some edge length scale arises as an optimal initial point of the perturbative expansion after functional integration over connection. Now consider the perturbative expansion itself. To obtain an algorithmizable diagram technique, we consider the simplest periodic simplicial structure with a frozen part of the variables ("hypercubic"). After functional integration over connection, the system is described by the metric $g_{位渭}$ at the sites. We parameterize $g_{位渭}$ so that the functional measure becomes Lebesgue. The discrete diagrams are free from ultraviolet divergences and reproduce (for ordinary, non-Planck external momenta) those continuum counterparts that are finite. We give the parametrization of $g_{位渭}$ up to terms, providing, in particular, additional three-graviton and two-graviton-two-matter vertices, which can give additional one-loop corrections to the Newtonian potential. The edge length scale is $\sim \sqrt{ 畏}$, where $畏$ defines the free factor $ ( - \det \| g_{位渭} \| )^{ 畏/ 2}$ in the measure and should be a large parameter to ensure the true action after integration over connection. We verify the important fact that the perturbative expansion does not contain increasing powers of $畏$ if its initial point is chosen close enough to the maximum point of the measure, thus justifying this choice. Discrete propagators depend on the Barbero-Immirzi parameter $纬$, which determines the ratio of timelike and spacelike elementary length scales. The existing estimates of $纬$ allow the propagator poles to have real energy for any (real) spatial momenta. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2306.11531v2-abstract-full').style.display = 'none'; document.getElementById('2306.11531v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 June, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">46 pages, 5 figures, typos corrected, refs added, typical diagrams added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 81S40 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Int. J. Mod. Phys. A, Volume No. 38, Issue No. 26n27, Article No. 2350143, Year 2023 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2212.13547">arXiv:2212.13547</a> <span> [<a href="https://arxiv.org/pdf/2212.13547">pdf</a>, <a href="https://arxiv.org/ps/2212.13547">ps</a>, <a href="https://arxiv.org/format/2212.13547">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217751X23500252">10.1142/S0217751X23500252 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the discrete version of the Kerr-Newman solution </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2212.13547v2-abstract-short" style="display: inline;"> This paper continues our work on black holes in the framework of the Regge calculus, where the discrete version (with a certain edge length scale $b$ proportional to the Planck scale) of the classical solution emerges as an optimal starting point for the perturbative expansion after functional integration over the connection, with the singularity resolved. An interest in the present discrete Ker… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.13547v2-abstract-full').style.display = 'inline'; document.getElementById('2212.13547v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2212.13547v2-abstract-full" style="display: none;"> This paper continues our work on black holes in the framework of the Regge calculus, where the discrete version (with a certain edge length scale $b$ proportional to the Planck scale) of the classical solution emerges as an optimal starting point for the perturbative expansion after functional integration over the connection, with the singularity resolved. An interest in the present discrete Kerr-Newman type solution (with the parameter $a \gg b$) may be to check the classical prediction that the electromagnetic contribution to the metric and curvature on the singularity ring is (infinitely) greater than the contribution of the $未$-function-like mass distribution, no matter how small the electric charge is. Here we encounter a kind of a discrete diagram technique, but with three-dimensional (static) diagrams and with only a few diagrams, although with modified (extended to complex coordinates) propagators. The metric (curvature) in the vicinity of the former singularity ring is considered. The electromagnetic contribution does indeed have a relative factor that is infinite at $b \to 0$, but, taking into account some existing estimates of the upper bound on the electric charge of known substances, it is not so large for habitual bodies and can only be significant for practically non-rotating black holes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.13547v2-abstract-full').style.display = 'none'; document.getElementById('2212.13547v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">31 pages, 3 figures, typos corrected, explanations added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 83C57 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Int. J. Mod. Phys. A, Volume No. 38, Issue No. 04n05, Article No. 2350025, Year 2023 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2112.13823">arXiv:2112.13823</a> <span> [<a href="https://arxiv.org/pdf/2112.13823">pdf</a>, <a href="https://arxiv.org/ps/2112.13823">ps</a>, <a href="https://arxiv.org/format/2112.13823">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217751X22500646">10.1142/S0217751X22500646 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the discrete version of the Reissner-Nordstr枚m solution </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2112.13823v1-abstract-short" style="display: inline;"> This paper generalizes our previous paper on the discrete Schwarzschild type solution in the Regge calculus, the simplicial electrodynamics earlier considered in the literature is incorporated in the case of the presence of a charge. Validity of the path integral approach is assumed, of which the only consequence used here is a loose fixation of edge lengths around a finite nonzero scale (we have… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2112.13823v1-abstract-full').style.display = 'inline'; document.getElementById('2112.13823v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2112.13823v1-abstract-full" style="display: none;"> This paper generalizes our previous paper on the discrete Schwarzschild type solution in the Regge calculus, the simplicial electrodynamics earlier considered in the literature is incorporated in the case of the presence of a charge. Validity of the path integral approach is assumed, of which the only consequence used here is a loose fixation of edge lengths around a finite nonzero scale (we have considered the latter earlier). In essence, the problem of determining the optimal background metric and electromagnetic field for the perturbative expansion generated by the functional integral is considered, for which the skeleton Regge and electrodynamic equations are analyzed. For the Regge equations, as we have earlier found, the Regge action on the simplest periodic simplicial structure and in the leading order over metric variations between 4-simplices can be substituted by a finite-difference form of the Hilbert-Einstein action (the piecewise constant metric there is defined by providing the vertices with coordinates). Thus we get the absence of the singularity inherent in the continuous solution. At the same time, the discrete solution is close to the continuum Reissner-Nordstr枚m one at large distances. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2112.13823v1-abstract-full').style.display = 'none'; document.getElementById('2112.13823v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 December, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 83C57 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Int. J. Mod. Phys. A, Volume No. 37, Issue No. 11n12, Article No. 2250064, Year 2022 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2103.05598">arXiv:2103.05598</a> <span> [<a href="https://arxiv.org/pdf/2103.05598">pdf</a>, <a href="https://arxiv.org/ps/2103.05598">ps</a>, <a href="https://arxiv.org/format/2103.05598">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217751X2150130X">10.1142/S0217751X2150130X <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the discrete version of the Kerr geometry </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2103.05598v1-abstract-short" style="display: inline;"> A Kerr type solution in the Regge calculus is considered. It is assumed that the discrete general relativity, the Regge calculus, is quantized within the path integral approach. The only consequence of this approach used here is the existence of a length scale at which edge lengths are loosely fixed, as considered in our earlier paper. In addition, we previously considered the Regge action on a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.05598v1-abstract-full').style.display = 'inline'; document.getElementById('2103.05598v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2103.05598v1-abstract-full" style="display: none;"> A Kerr type solution in the Regge calculus is considered. It is assumed that the discrete general relativity, the Regge calculus, is quantized within the path integral approach. The only consequence of this approach used here is the existence of a length scale at which edge lengths are loosely fixed, as considered in our earlier paper. In addition, we previously considered the Regge action on a simplicial manifold on which the vertices are coordinatized and the corresponding piecewise constant metric introduced, and found that for the simplest periodic simplicial structure and in the leading order over metric variations between 4-simplices, this reduces to a finite-difference form of the Hilbert-Einstein action. The problem of solving the corresponding discrete Einstein equations (classical) with a length scale (having a quantum nature) arises as the problem of determining the optimal background metric for the perturbative expansion generated by the functional integral. Using an one-complex-function ansatz for the metric, which reduces to the Kerr-Schild metric in the continuum, we find a discrete metric that approximates the continuum one at large distances and is nonsingular on the (earlier) singularity ring. The effective curvature $R_{位谓谓蟻}$, including where $R_{位渭} \neq 0$ (gravity sources), is analyzed with a focus on the vicinity of the singularity ring. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.05598v1-abstract-full').style.display = 'none'; document.getElementById('2103.05598v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">34 pages, 3 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 83C57 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Int. J. Mod. Phys. A, Volume No. 36, Issue No. 20, Article No. 2150130, Year 2021 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2101.07147">arXiv:2101.07147</a> <span> [<a href="https://arxiv.org/pdf/2101.07147">pdf</a>, <a href="https://arxiv.org/ps/2101.07147">ps</a>, <a href="https://arxiv.org/format/2101.07147">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0218271821500711">10.1142/S0218271821500711 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the Kerr metric in a synchronous reference frame </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2101.07147v1-abstract-short" style="display: inline;"> The Kerr metric is considered in a synchronous frame of reference obtained by using proper time and initial conditions for particles that freely move along a certain set of trajectories as coordinates. Modifying these coordinates in a certain way (keeping their interpretation as initial values at large distances), we still have a synchronous frame and the direct analogue of the Lemaitre metric, th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.07147v1-abstract-full').style.display = 'inline'; document.getElementById('2101.07147v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2101.07147v1-abstract-full" style="display: none;"> The Kerr metric is considered in a synchronous frame of reference obtained by using proper time and initial conditions for particles that freely move along a certain set of trajectories as coordinates. Modifying these coordinates in a certain way (keeping their interpretation as initial values at large distances), we still have a synchronous frame and the direct analogue of the Lemaitre metric, the singularities of which are exhausted by the physical Kerr singularity (the singularity ring). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.07147v1-abstract-full').style.display = 'none'; document.getElementById('2101.07147v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 January, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C15; 83C57 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Int. J. Mod. Phys. D, Volume No. 30, Issue No. 10, Article No. 2150071, Year 2021 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2008.13756">arXiv:2008.13756</a> <span> [<a href="https://arxiv.org/pdf/2008.13756">pdf</a>, <a href="https://arxiv.org/ps/2008.13756">ps</a>, <a href="https://arxiv.org/format/2008.13756">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.3390/universe6100185">10.3390/universe6100185 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the discrete version of the Schwarzschild problem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2008.13756v2-abstract-short" style="display: inline;"> We consider a Schwarzschild type solution in the discrete Regge calculus formulation of general relativity quantized within the path integral approach. Earlier, we found a mechanism of a loose fixation of the background scale of Regge lengths. This elementary length scale is defined by the Planck scale and some free parameter of such a quantum extension of the theory. Besides, Regge action was red… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.13756v2-abstract-full').style.display = 'inline'; document.getElementById('2008.13756v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2008.13756v2-abstract-full" style="display: none;"> We consider a Schwarzschild type solution in the discrete Regge calculus formulation of general relativity quantized within the path integral approach. Earlier, we found a mechanism of a loose fixation of the background scale of Regge lengths. This elementary length scale is defined by the Planck scale and some free parameter of such a quantum extension of the theory. Besides, Regge action was reduced to an expansion over metric variations between the tetrahedra and, in the main approximation, is a finite-difference form of the Hilbert-Einstein action. Using for the Schwarzschild problem a priori general non-spherically symmetrical ansatz, we get finite-difference equations for its discrete version. This defines a solution which at large distances is close to the continuum Schwarzschild geometry, and the metric and effective curvature at the center are cut off at the elementary length scale. Slow rotation can also be taken into account (Lense-Thirring-like metric). Thus we get a general approach to the classical background in the quantum framework in zero order: it is an optimal starting point for the perturbative expansion of the theory; finite-difference equations are classical, the elementary length scale has quantum origin. Singularities, if any, are resolved. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.13756v2-abstract-full').style.display = 'none'; document.getElementById('2008.13756v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 October, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 31 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">24 pages, 1 figure; Special issue "Selected Papers from the 17th Russian Gravitational Conference -- International Conference on Gravitation, Cosmology and Astrophysics (RUSGRAV-17)"; explanations and refs added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 83C57 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Universe, Vol. 6, No.10, 185 (2020) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1912.12626">arXiv:1912.12626</a> <span> [<a href="https://arxiv.org/pdf/1912.12626">pdf</a>, <a href="https://arxiv.org/ps/1912.12626">ps</a>, <a href="https://arxiv.org/format/1912.12626">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217751X2050058X">10.1142/S0217751X2050058X <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the discrete version of the black hole solution </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1912.12626v2-abstract-short" style="display: inline;"> A Schwarzschild type solution in Regge calculus is considered. Earlier, we considered a mechanism of loose fixing of edge lengths due to the functional integral measure arising from integration over connection in the functional integral for the connection representation of the Regge action. The length scale depends on a free dimensionless parameter that determines the final functional measure. For… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1912.12626v2-abstract-full').style.display = 'inline'; document.getElementById('1912.12626v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1912.12626v2-abstract-full" style="display: none;"> A Schwarzschild type solution in Regge calculus is considered. Earlier, we considered a mechanism of loose fixing of edge lengths due to the functional integral measure arising from integration over connection in the functional integral for the connection representation of the Regge action. The length scale depends on a free dimensionless parameter that determines the final functional measure. For this parameter and the length scale large in Planck units, the resulting effective action is close to the Regge action. Earlier, we considered the Regge action in terms of affine connection matrices as functions of the metric inside the 4-simplices and found that it is a difference form of the Hilbert-Einstein action in the leading order over metric variations between the 4-simplices. Now we take the (continuum) Schwarzschild problem in the form where spherical symmetry is not set a priori and arises just in the solution, take the difference form of the corresponding equations and get the metric (in fact, in the Lemaitre or Painlev茅-Gullstrand like frame), which is nonsingular at the origin, just as the Newtonian gravitational potential, obeying the difference Poisson equation with a point source, is cut off at the elementary length and is finite at the source. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1912.12626v2-abstract-full').style.display = 'none'; document.getElementById('1912.12626v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 December, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">24 pages, typos corrected, readability improved</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 83C57 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Int. Journ. Mod. Phys. A, Vol. 35, Nos. 11 & 12, 2050058 (2020) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1906.11805">arXiv:1906.11805</a> <span> [<a href="https://arxiv.org/pdf/1906.11805">pdf</a>, <a href="https://arxiv.org/ps/1906.11805">ps</a>, <a href="https://arxiv.org/format/1906.11805">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217751X19501860">10.1142/S0217751X19501860 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the discrete Christoffel symbols </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1906.11805v2-abstract-short" style="display: inline;"> The piecewise flat spacetime is equipped with a set of edge lengths and vertex coordinates. This defines a piecewise affine coordinate system and a piecewise affine metric in it, the discrete analogue of the unique torsion-free metric-compatible affine connection or of the Levi-Civita connection (or of the standard expression of the Christoffel symbols in terms of metric) mentioned in the literatu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.11805v2-abstract-full').style.display = 'inline'; document.getElementById('1906.11805v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1906.11805v2-abstract-full" style="display: none;"> The piecewise flat spacetime is equipped with a set of edge lengths and vertex coordinates. This defines a piecewise affine coordinate system and a piecewise affine metric in it, the discrete analogue of the unique torsion-free metric-compatible affine connection or of the Levi-Civita connection (or of the standard expression of the Christoffel symbols in terms of metric) mentioned in the literature, and, substituting this into the affine-connection form of the Regge action of our previous work, we get a second order form of the action. This can be expanded over metric variations from simplex to simplex. For a particular periodic simplicial structure and coordinates of the vertices, the leading order over metric variations is found to coincide with a certain finite difference form of the Hilbert-Einstein action. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.11805v2-abstract-full').style.display = 'none'; document.getElementById('1906.11805v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 November, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 June, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages, explanations and refs added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Int. Journ. Mod. Phys. A, Vol. 34, No. 30, 1950186 (2019) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1811.07160">arXiv:1811.07160</a> <span> [<a href="https://arxiv.org/pdf/1811.07160">pdf</a>, <a href="https://arxiv.org/ps/1811.07160">ps</a>, <a href="https://arxiv.org/format/1811.07160">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> </div> <p class="title is-5 mathjax"> Discrete Faddeev action for the tetrad fields strongly varying along different coordinates </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1811.07160v2-abstract-short" style="display: inline;"> Faddeev gravity using a $d$-dimensional tetrad (normally $d = 10$) is classically equivalent to general relativity (GR). The discrete Faddeev gravity on the piecewise flat spacetime normally assumes slowly varying metric and tetrad from vertex to vertex. Meanwhile, Faddeev action is finite (although not unambiguously defined) for discontinuous tetrad fields thus allowing, in particular, to consi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.07160v2-abstract-full').style.display = 'inline'; document.getElementById('1811.07160v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1811.07160v2-abstract-full" style="display: none;"> Faddeev gravity using a $d$-dimensional tetrad (normally $d = 10$) is classically equivalent to general relativity (GR). The discrete Faddeev gravity on the piecewise flat spacetime normally assumes slowly varying metric and tetrad from vertex to vertex. Meanwhile, Faddeev action is finite (although not unambiguously defined) for discontinuous tetrad fields thus allowing, in particular, to consider a surface as consisting of virtually independent elementary triangles, and its area spectrum as the sum of elementary area spectra. In the discrete connection form, area tensors are canonically conjugate to SO(10) connection matrices, and earlier we have found the elementary area spectrum, which is nonsingular just at large connection or the strongly varying fields constituting a kind of "antiferromagnetic" structure. We appropriately define discrete {\it connection} Faddeev action to unambiguously determine the discrete Faddeev action for the strongly varying fields, but weakly varying metric, equivalent in the continuum limit to the GR action with this metric. Previously, we considered large variations in only one direction, now we use an ansatz in some respects less common, but overall, probably the most common. A unified simplicial connection representation is written out (depending on an auxiliary connection) both for the discrete Faddeev action and for the Regge action. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.07160v2-abstract-full').style.display = 'none'; document.getElementById('1811.07160v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 February, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">12 pages, typos corrected, refs added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 53C05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1804.11212">arXiv:1804.11212</a> <span> [<a href="https://arxiv.org/pdf/1804.11212">pdf</a>, <a href="https://arxiv.org/ps/1804.11212">ps</a>, <a href="https://arxiv.org/format/1804.11212">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217751X18502202">10.1142/S0217751X18502202 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the non-perturbative graviton propagator </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1804.11212v4-abstract-short" style="display: inline;"> To reduce general relativity to the canonical Hamiltonian formalism and construct the path (functional) integral in a simpler and, especially in the discrete case, less singular way, one extends the configuration superspace, as in the connection representation. Then we perform functional integration over connection. The module of the result of this integration arises in the leading order of the ex… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1804.11212v4-abstract-full').style.display = 'inline'; document.getElementById('1804.11212v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1804.11212v4-abstract-full" style="display: none;"> To reduce general relativity to the canonical Hamiltonian formalism and construct the path (functional) integral in a simpler and, especially in the discrete case, less singular way, one extends the configuration superspace, as in the connection representation. Then we perform functional integration over connection. The module of the result of this integration arises in the leading order of the expansion over a scale of the discrete lapse-shift functions and has maxima at finite (Planck scale) areas/lengths and rapidly decreases at large areas/lengths, as we have mainly considered previously; the phase arises in the leading order (Regge action) of the stationary phase expansion. Now we consider the possibility of confining ourselves to these leading terms in a certain region of the parameters of the theory; consider background edge lengths as an optimal starting point for the perturbative expansion of the theory; estimate the background length scale and consider the form of the graviton propagator. In parallel with the general simplicial structure, we consider the simplest periodic simplicial structure with a part of the variables frozen ("hypercubic"), for which also the propagator in the leading approximation over metric variations can be written in a closed form. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1804.11212v4-abstract-full').style.display = 'none'; document.getElementById('1804.11212v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 December, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 April, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">28 pages; parallel consideration of the simplest periodic simplicial structure with a reduced set of variables is given</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 81S40 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Int. Journ. Mod. Phys. A, Vol. 33, No. 36, 1850220 (2018) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1708.02035">arXiv:1708.02035</a> <span> [<a href="https://arxiv.org/pdf/1708.02035">pdf</a>, <a href="https://arxiv.org/ps/1708.02035">ps</a>, <a href="https://arxiv.org/format/1708.02035">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217732317501814">10.1142/S0217732317501814 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> First order discrete Faddeev gravity at the strongly varying fields </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1708.02035v1-abstract-short" style="display: inline;"> We consider the Faddeev formulation of general relativity (GR), which can be characterized by a kind of $d$-dimensional tetrad (typically $d$=10) and a non-Riemannian connection. This theory is invariant w. r. t. the global, but not local, rotations in the $d$-dimensional space. There can be configurations with a smooth or flat metric, but with the tetrad that changes abruptly at small distances,… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1708.02035v1-abstract-full').style.display = 'inline'; document.getElementById('1708.02035v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1708.02035v1-abstract-full" style="display: none;"> We consider the Faddeev formulation of general relativity (GR), which can be characterized by a kind of $d$-dimensional tetrad (typically $d$=10) and a non-Riemannian connection. This theory is invariant w. r. t. the global, but not local, rotations in the $d$-dimensional space. There can be configurations with a smooth or flat metric, but with the tetrad that changes abruptly at small distances, a kind of "antiferromagnetic" structure. Previously, we discussed a first order representation for the Faddeev gravity, which uses the orthogonal connection in the $d$-dimensional space as an independent variable. Using the discrete form of this formulation, we considered the spectrum of (elementary) area. This spectrum turns out to be physically reasonable just on a classical background with large connection like rotations by $蟺$, that is, with such an "antiferromagnetic" structure. In the discrete first order Faddeev gravity, we consider such a structure with periodic cells and large connection and strongly changing tetrad field inside the cell. We show that this system in the continuum limit reduces to a generalization of the Faddeev system. The action is a sum of related actions of the Faddeev type and is still reduced to the GR action. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1708.02035v1-abstract-full').style.display = 'none'; document.getElementById('1708.02035v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 August, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">15 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C99; 53C05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Mod. Phys. Lett. A, Vol. 32, No. 35 (2017) 1750181 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1705.06654">arXiv:1705.06654</a> <span> [<a href="https://arxiv.org/pdf/1705.06654">pdf</a>, <a href="https://arxiv.org/ps/1705.06654">ps</a>, <a href="https://arxiv.org/format/1705.06654">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217732318500049">10.1142/S0217732318500049 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Simplicial Palatini action </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1705.06654v3-abstract-short" style="display: inline;"> 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… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1705.06654v3-abstract-full').style.display = 'inline'; document.getElementById('1705.06654v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1705.06654v3-abstract-full" style="display: none;"> 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1705.06654v3-abstract-full').style.display = 'none'; document.getElementById('1705.06654v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 October, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 May, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages, discussion of the finite diagram technique is added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 53C05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Mod. Phys. Lett. A, Vol. 33, No. 1 (2018) 1850004 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1612.06727">arXiv:1612.06727</a> <span> [<a href="https://arxiv.org/pdf/1612.06727">pdf</a>, <a href="https://arxiv.org/ps/1612.06727">ps</a>, <a href="https://arxiv.org/format/1612.06727">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> </div> <p class="title is-5 mathjax"> GL(4,R) representation of the gravity action on the piecewise flat spacetime </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1612.06727v1-abstract-short" style="display: inline;"> The gravity action on the piecewise flat Riemannian manifold is formulated using the discrete set of the nondegenerate 4$\times$4 matrices on the 3-simplices as some connection type variables. These variables are the discrete counterpart of the affine (Christoffel) connection used as independent variables in the Palatini form of the Einstein gravity action. Excluding these with the help of the equ… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.06727v1-abstract-full').style.display = 'inline'; document.getElementById('1612.06727v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1612.06727v1-abstract-full" style="display: none;"> The gravity action on the piecewise flat Riemannian manifold is formulated using the discrete set of the nondegenerate 4$\times$4 matrices on the 3-simplices as some connection type variables. These variables are the discrete counterpart of the affine (Christoffel) connection used as independent variables in the Palatini form of the Einstein gravity action. Excluding these with the help of the equations of motion we get the original discrete gravity action on the piecewise flat spacetime (Regge action). The discrete version of the diffeomorphisms and path integral are briefly discussed. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.06727v1-abstract-full').style.display = 'none'; document.getElementById('1612.06727v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 December, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">6 pages, report given at the International conference "Quantum Field Theory and Gravity 2016", August 1-7, 2016, TSPU, Tomsk, Russia</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 53C05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1607.01552">arXiv:1607.01552</a> <span> [<a href="https://arxiv.org/pdf/1607.01552">pdf</a>, <a href="https://arxiv.org/ps/1607.01552">ps</a>, <a href="https://arxiv.org/format/1607.01552">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217732316501145">10.1142/S0217732316501145 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Spectrum of area in the Faddeev formulation of gravity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1607.01552v2-abstract-short" style="display: inline;"> 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-sim… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1607.01552v2-abstract-full').style.display = 'inline'; document.getElementById('1607.01552v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1607.01552v2-abstract-full" style="display: none;"> 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 $纬_{\rm F}$ of the Barbero-Immirzi parameter $纬$. 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 $纬_{\rm F}$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1607.01552v2-abstract-full').style.display = 'none'; document.getElementById('1607.01552v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 July, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 July, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">20 pages. ref to our previous work arXiv:1206.5509 is added and discussed</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 53C05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Mod. Phys. Lett. A, Vol. 31, No. 19 (2016) 1650114 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1509.04974">arXiv:1509.04974</a> <span> [<a href="https://arxiv.org/pdf/1509.04974">pdf</a>, <a href="https://arxiv.org/ps/1509.04974">ps</a>, <a href="https://arxiv.org/format/1509.04974">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217732316500103">10.1142/S0217732316500103 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Affine connection form of Regge calculus </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1509.04974v1-abstract-short" style="display: inline;"> Regge action is represented analogously to how the Palatini action for general relativity (GR) as some functional of the metric and a general connection as independent variables represents the Einstein-Hilbert action. The piecewise flat (or simplicial) spacetime of Regge calculus is equipped with some world coordinates and some piecewise affine metric which is completely defined by the set of ed… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1509.04974v1-abstract-full').style.display = 'inline'; document.getElementById('1509.04974v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1509.04974v1-abstract-full" style="display: none;"> Regge action is represented analogously to how the Palatini action for general relativity (GR) as some functional of the metric and a general connection as independent variables represents the Einstein-Hilbert action. The piecewise flat (or simplicial) spacetime of Regge calculus is equipped with some world coordinates and some piecewise affine metric which is completely defined by the set of edge lengths and the world coordinates of the vertices. The conjugate variables are the general nondegenerate matrices on the 3-simplices which play a role of a general discrete connection. Our previous result on some representation of the Regge calculus action in terms of the local Euclidean (Minkowsky) frame vectors and orthogonal connection matrices as independent variables is somewhat modified for the considered case of the general linear group GL(4,R) of the connection matrices. As a result, we have some action invariant w. r. t. arbitrary change of coordinates of the vertices (and related GL(4,R) transformations in the 4-simplices). Excluding GL(4,R) connection from this action via the equations of motion we have exactly the Regge action for the considered spacetime. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1509.04974v1-abstract-full').style.display = 'none'; document.getElementById('1509.04974v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 September, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 53C05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Mod. Phys. Lett. A, Vol. 31, No. 01 (2016) 1650010 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1508.07573">arXiv:1508.07573</a> <span> [<a href="https://arxiv.org/pdf/1508.07573">pdf</a>, <a href="https://arxiv.org/ps/1508.07573">ps</a>, <a href="https://arxiv.org/format/1508.07573">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217732315501746">10.1142/S0217732315501746 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> First order minisuperspace model for the Faddeev formulation of gravity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1508.07573v1-abstract-short" style="display: inline;"> 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 some minisuperspace model for the Faddeev formulation where the tetrad fields are piecewise constant on the polytopes like 4-simplices or, say, cuboids int… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1508.07573v1-abstract-full').style.display = 'inline'; document.getElementById('1508.07573v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1508.07573v1-abstract-full" style="display: none;"> 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 some minisuperspace model for the Faddeev formulation where the tetrad fields are piecewise constant on the polytopes like 4-simplices or, say, cuboids into which ${\rm I \hspace{-3pt} R}^4$ can be decomposed. Now we study some representation of this (discrete) theory, an analogue of the Cartan-Weyl connection-type form of the Hilbert-Einstein action in the usual continuum GR. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1508.07573v1-abstract-full').style.display = 'none'; document.getElementById('1508.07573v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 August, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">12 pages, 1 figure, to appear in Mod. Phys. Lett. A</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 53C05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Mod. Phys. Lett. A, Vol. 30, No. 32 (2015) 1550172 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1412.7403">arXiv:1412.7403</a> <span> [<a href="https://arxiv.org/pdf/1412.7403">pdf</a>, <a href="https://arxiv.org/ps/1412.7403">ps</a>, <a href="https://arxiv.org/format/1412.7403">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> </div> <p class="title is-5 mathjax"> Faddeev gravity action on the piecewise constant fundamental vector fields </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1412.7403v1-abstract-short" style="display: inline;"> 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 com… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1412.7403v1-abstract-full').style.display = 'inline'; document.getElementById('1412.7403v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1412.7403v1-abstract-full" style="display: none;"> 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 $纬$ can be introduced in this formalism. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1412.7403v1-abstract-full').style.display = 'none'; document.getElementById('1412.7403v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 December, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">8 pages, reported on the conference "Quantum Field Theory and Gravity 2014" (July 28 - August 3 2014, Tomsk, Russia)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> TSPU Bulletin, No. 12(153), pp. 131-134, 2014 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1408.6375">arXiv:1408.6375</a> <span> [<a href="https://arxiv.org/pdf/1408.6375">pdf</a>, <a href="https://arxiv.org/ps/1408.6375">ps</a>, <a href="https://arxiv.org/format/1408.6375">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217732314501417">10.1142/S0217732314501417 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Some minisuperspace model for the Faddeev formulation of gravity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1408.6375v1-abstract-short" style="display: inline;"> 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 fi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1408.6375v1-abstract-full').style.display = 'inline'; document.getElementById('1408.6375v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1408.6375v1-abstract-full" style="display: none;"> 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1408.6375v1-abstract-full').style.display = 'none'; document.getElementById('1408.6375v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 August, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">14 pages, 3 figures, to appear in Mod. Phys. Lett. A</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Mod. Phys. Lett. A, Vol. 29, No. 27 (2014) 1450141 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1312.7116">arXiv:1312.7116</a> <span> [<a href="https://arxiv.org/pdf/1312.7116">pdf</a>, <a href="https://arxiv.org/ps/1312.7116">ps</a>, <a href="https://arxiv.org/format/1312.7116">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> </div> <p class="title is-5 mathjax"> On the Faddeev gravity on the piecewise flat manifold </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1312.7116v1-abstract-short" style="display: inline;"> We study the Faddeev formulation of gravity in which the metric is composed of vector fields. This system is reducible with the help of the equations of motion to the general relativity. The Faddeev action is evaluated for the piecewise flat ansatz for these fields when the metric corresponds to the flat interior of the 4-simplices of the general simplicial complex. Thereby an analogue of the Regg… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1312.7116v1-abstract-full').style.display = 'inline'; document.getElementById('1312.7116v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1312.7116v1-abstract-full" style="display: none;"> We study the Faddeev formulation of gravity in which the metric is composed of vector fields. This system is reducible with the help of the equations of motion to the general relativity. The Faddeev action is evaluated for the piecewise flat ansatz for these fields when the metric corresponds to the flat interior of the 4-simplices of the general simplicial complex. Thereby an analogue of the Regge action in the usual general relativity is obtained. A peculiar feature of the Faddeev gravity is finiteness of the action on the discontinuous fields, and this means possibility of the complete independence of the fields in the different 4-simplices or incoincidence of the 4-simplices on their common faces. The earlier introduced analogue of the Barbero-Immirzi parameter for the Faddeev gravity is taken into account. There is some freedom in defining the Faddeev action on the piecewise flat manifold, and the task is set to make use of this freedom to ensure that this discrete system be reducible with the help of the {\it discrete} equations of motion to the analogous discrete general relativity (Regge calculus). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1312.7116v1-abstract-full').style.display = 'none'; document.getElementById('1312.7116v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 December, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">10 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1212.0978">arXiv:1212.0978</a> <span> [<a href="https://arxiv.org/pdf/1212.0978">pdf</a>, <a href="https://arxiv.org/ps/1212.0978">ps</a>, <a href="https://arxiv.org/format/1212.0978">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> </div> </div> <p class="title is-5 mathjax"> On some feature and application of the Faddeev formulation of gravity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1212.0978v1-abstract-short" style="display: inline;"> In the Faddeev formulation of gravity, the metric is regarded as composite field, bilinear of $d = 10$ 4-vector fields. A unique feature is that this formulation admits the discontinuous fields. On the discrete level, when spacetime is decomposed into the elementary 4-simplices, this means that the 4-simplices may not coincide on their common faces, that is, be independent. We apply this to the pa… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1212.0978v1-abstract-full').style.display = 'inline'; document.getElementById('1212.0978v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1212.0978v1-abstract-full" style="display: none;"> In the Faddeev formulation of gravity, the metric is regarded as composite field, bilinear of $d = 10$ 4-vector fields. A unique feature is that this formulation admits the discontinuous fields. On the discrete level, when spacetime is decomposed into the elementary 4-simplices, this means that the 4-simplices may not coincide on their common faces, that is, be independent. We apply this to the particular problem of quantization of the surface regarded as that composed of virtually independent elementary pieces (2-simplices). We find the area spectrum being proportional to the Barbero-Immirzi parameter $纬$ in the Faddeev gravity and described as a sum of spectra of separate areas. According to the known in the literature approach, we find that $纬$ exists ensuring Bekenstein-Hawking relation for the statistical black hole entropy for arbitrary $d$, in particular, $纬= 0.39...$ for genuine $d = 10$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1212.0978v1-abstract-full').style.display = 'none'; document.getElementById('1212.0978v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 December, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">4 pages, Conference paper (Quantum Field Theory and Gravity 2012, July 31 - August 4 2012, Tomsk, Russia)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 53C05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> TSPU Bulletin, No. 13(128), pp. 76-80, 2012 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1206.5509">arXiv:1206.5509</a> <span> [<a href="https://arxiv.org/pdf/1206.5509">pdf</a>, <a href="https://arxiv.org/ps/1206.5509">ps</a>, <a href="https://arxiv.org/format/1206.5509">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> </div> <p class="title is-5 mathjax"> On area spectrum in the Faddeev gravity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1206.5509v1-abstract-short" style="display: inline;"> 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, indepe… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1206.5509v1-abstract-full').style.display = 'inline'; document.getElementById('1206.5509v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1206.5509v1-abstract-full" style="display: none;"> 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 $纬$ 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 $纬$ for arbitrary $d$, in particular, $纬= 0.39...$ for genuine $d = 10$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1206.5509v1-abstract-full').style.display = 'none'; document.getElementById('1206.5509v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 June, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">17 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 53C05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1201.0808">arXiv:1201.0808</a> <span> [<a href="https://arxiv.org/pdf/1201.0808">pdf</a>, <a href="https://arxiv.org/ps/1201.0808">ps</a>, <a href="https://arxiv.org/format/1201.0808">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> </div> <p class="title is-5 mathjax"> Faddeev formulation of gravity in discrete form </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1201.0808v1-abstract-short" style="display: inline;"> We study Faddeev formulation of gravity, in which the metric is composed of vector fields. We consider these fields constant in the interior of the 4-simplices of a simplicial complex. The action depends not only on the values of the fields in the interior of the 4-simplices but on the details of (regularized) jump of the fields between the 4-simplices. Though, when the fields vary arbitrarily slo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1201.0808v1-abstract-full').style.display = 'inline'; document.getElementById('1201.0808v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1201.0808v1-abstract-full" style="display: none;"> We study Faddeev formulation of gravity, in which the metric is composed of vector fields. We consider these fields constant in the interior of the 4-simplices of a simplicial complex. The action depends not only on the values of the fields in the interior of the 4-simplices but on the details of (regularized) jump of the fields between the 4-simplices. Though, when the fields vary arbitrarily slowly from the 4-simplex to 4-simplex, the latter dependence is negligible (of the next-to-leading order of magnitude). We put the earlier proposed in our work first order (connection) representation of the Faddeev action into the discrete form. We show that upon excluding the connections it is consistent with the above Faddeev action on the piecewise constant fields in the leading order of magnitude. Thus, using the discrete form of the connection representation of the Faddeev action can serve a way to fix the value of this action on the piecewise constant ansatz on simplices. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1201.0808v1-abstract-full').style.display = 'none'; document.getElementById('1201.0808v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 January, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 53C05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1201.0806">arXiv:1201.0806</a> <span> [<a href="https://arxiv.org/pdf/1201.0806">pdf</a>, <a href="https://arxiv.org/ps/1201.0806">ps</a>, <a href="https://arxiv.org/format/1201.0806">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/0264-9381/30/9/095006">10.1088/0264-9381/30/9/095006 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> First order representation of the Faddeev formulation of gravity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1201.0806v1-abstract-short" style="display: inline;"> We study Faddeev formulation of gravity, in which the metric is composed of vector fields or the tetrad of the ten-dimensional fields, $f^A_位$, where $位= 1, 2, 3, 4$ and $A = 1, ..., 10$ is vector index w. r. t. the Euclidean (or Minkowsky) ten-dimensional spacetime. We propose representation of the type of the Cartan-Weyl one. It is based on extending the set of variables by introducing the infin… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1201.0806v1-abstract-full').style.display = 'inline'; document.getElementById('1201.0806v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1201.0806v1-abstract-full" style="display: none;"> We study Faddeev formulation of gravity, in which the metric is composed of vector fields or the tetrad of the ten-dimensional fields, $f^A_位$, where $位= 1, 2, 3, 4$ and $A = 1, ..., 10$ is vector index w. r. t. the Euclidean (or Minkowsky) ten-dimensional spacetime. We propose representation of the type of the Cartan-Weyl one. It is based on extending the set of variables by introducing the infinitesimal SO(10) connection. Excluding this connection via equations of motion we reproduce the original Faddeev action. A peculiar feature of this representation is occurrence of the local SO(10) symmetry violating condition so that SO(10) symmetry is only global one in full correspondence with that the original Faddeev formulation just possesses SO(10) symmetry w. r. t. the global SO(10) rotation of the Euclidean ten-dimensional spacetime. We also consider analog of the Barbero-Immirzi parameter which can be naturally introduced in the considered representation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1201.0806v1-abstract-full').style.display = 'none'; document.getElementById('1201.0806v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 January, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C99; 53C05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Class.Quant.Grav.30:095006,2013 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1005.0061">arXiv:1005.0061</a> <span> [<a href="https://arxiv.org/pdf/1005.0061">pdf</a>, <a href="https://arxiv.org/ps/1005.0061">ps</a>, <a href="https://arxiv.org/format/1005.0061">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s10714-011-1227-1">10.1007/s10714-011-1227-1 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Gravity action on the rapidly varying metrics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1005.0061v1-abstract-short" style="display: inline;"> We consider a four-dimensional simplicial complex and the minisuperspace general relativity system described by the metric flat in the most part of the interior of every 4-simplex with exception of a thin layer of thickness $\propto \varepsilon$ along the every three-dimensional face where the metric undergoes jump between the two 4-simplices sharing this face. At $\varepsilon \to 0$ this jump wou… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1005.0061v1-abstract-full').style.display = 'inline'; document.getElementById('1005.0061v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1005.0061v1-abstract-full" style="display: none;"> We consider a four-dimensional simplicial complex and the minisuperspace general relativity system described by the metric flat in the most part of the interior of every 4-simplex with exception of a thin layer of thickness $\propto \varepsilon$ along the every three-dimensional face where the metric undergoes jump between the two 4-simplices sharing this face. At $\varepsilon \to 0$ this jump would become discontinuity. Since, however, discontinuity of the (induced on the face) metric is not allowed in general relativity, the terms in the Einstein action tending to infinity at $\varepsilon \to 0$ arise. In the path integral approach, these terms lead to the pre-exponent factor with \dfuns requiring that the induced on the faces metric be continuous, i. e. the 4-simplices fit on their common faces. The other part of the path integral measure corresponds to the action being the sum of independent terms over the 4-simplices. Therefore this part of the path integral measure is the product of independent measures over the 4-simplices. The result obtained is in accordance with our previous one obtained from the symmetry considerations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1005.0061v1-abstract-full').style.display = 'none'; document.getElementById('1005.0061v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 May, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2010. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">10 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 81S40 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Gen. Rel. Grav., Vol. 43, No. 11, pp. 3127-3138, 2011 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0912.1109">arXiv:0912.1109</a> <span> [<a href="https://arxiv.org/pdf/0912.1109">pdf</a>, <a href="https://arxiv.org/ps/0912.1109">ps</a>, <a href="https://arxiv.org/format/0912.1109">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217732310032548">10.1142/S0217732310032548 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Integration over connections in the discretized gravitational functional integrals </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0912.1109v1-abstract-short" style="display: inline;"> 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… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0912.1109v1-abstract-full').style.display = 'inline'; document.getElementById('0912.1109v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0912.1109v1-abstract-full" style="display: none;"> 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 ($未$-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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0912.1109v1-abstract-full').style.display = 'none'; document.getElementById('0912.1109v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 December, 2009; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2009. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 53C05; 81S40 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Mod. Phys. Lett. A, Vol. 25, No. 5 (2010) pp. 351-368 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0810.1630">arXiv:0810.1630</a> <span> [<a href="https://arxiv.org/pdf/0810.1630">pdf</a>, <a href="https://arxiv.org/ps/0810.1630">ps</a>, <a href="https://arxiv.org/format/0810.1630">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1063/1.3553185">10.1063/1.3553185 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Attributing sense to some integrals in Regge calculus </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0810.1630v2-abstract-short" style="display: inline;"> Regge calculus minisuperspace action in the connection representation has the form in which each term is linear over some field variable (scale of area-type variable with sign). We are interested in the result of performing integration over connections in the path integral (now usual multiple integral) as function of area tensors even in larger region considered as independent variables. To find… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0810.1630v2-abstract-full').style.display = 'inline'; document.getElementById('0810.1630v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0810.1630v2-abstract-full" style="display: none;"> Regge calculus minisuperspace action in the connection representation has the form in which each term is linear over some field variable (scale of area-type variable with sign). We are interested in the result of performing integration over connections in the path integral (now usual multiple integral) as function of area tensors even in larger region considered as independent variables. To find this function (or distribution), we compute its moments, i. e. integrals with monomials over area tensors. Calculation proceeds through intermediate appearance of $未$-functions and integrating them out. Up to a singular part with support on some discrete set of physically unattainable points, the function of interest has finite moments. This function in physical region should therefore exponentially decay at large areas and it really does being restored from moments. This gives for gravity a way of defining such nonabsolutely convergent integral as path integral. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0810.1630v2-abstract-full').style.display = 'none'; document.getElementById('0810.1630v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 July, 2009; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 October, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2008. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">14 pages, presentation improved</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C27; 53C05; 81S40 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Math. Phys., Vol. 52, No. 022502, pp. 1-14, 2011 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0808.1042">arXiv:0808.1042</a> <span> [<a href="https://arxiv.org/pdf/0808.1042">pdf</a>, <a href="https://arxiv.org/ps/0808.1042">ps</a>, <a href="https://arxiv.org/format/0808.1042">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Defining some integrals in Regge calculus </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0808.1042v1-abstract-short" style="display: inline;"> Regge calculus minisuperspace action in the connection representation has the form in which each term is linear over some field variable (scale of area-type variable with sign). We are interested in the result of performing integration over connections in the path integral. To find this function, we compute its moments, i. e. integrals with powers of that variable. Calculation proceeds through i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0808.1042v1-abstract-full').style.display = 'inline'; document.getElementById('0808.1042v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0808.1042v1-abstract-full" style="display: none;"> Regge calculus minisuperspace action in the connection representation has the form in which each term is linear over some field variable (scale of area-type variable with sign). We are interested in the result of performing integration over connections in the path integral. To find this function, we compute its moments, i. e. integrals with powers of that variable. Calculation proceeds through intermediate appearance of $未$-functions and integrating them out and leads to finite result for any power. The function of interest should therefore be exponentially suppressed at large areas and it really does being restored from moments. This gives for gravity a way of defining such nonabsolutely convergent integral as path integral. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0808.1042v1-abstract-full').style.display = 'none'; document.getElementById('0808.1042v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 August, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2008. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">7 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0808.1041">arXiv:0808.1041</a> <span> [<a href="https://arxiv.org/pdf/0808.1041">pdf</a>, <a href="https://arxiv.org/ps/0808.1041">ps</a>, <a href="https://arxiv.org/format/0808.1041">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Independent 4-tetrahedra connection representation of Regge calculus </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0808.1041v1-abstract-short" style="display: inline;"> We consider simplest piecewise flat manifold consisting of two identical 4-tetrahedra (call it bisimplex). General relativity action for arbitrary piecewise flat manifold can be expressed in terms of sum of the (half of) bisimplex actions. We use representation of each bisimplex action in terms of certain rotation matrices (connections). This gives representation of any minisuperspace piecewise… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0808.1041v1-abstract-full').style.display = 'inline'; document.getElementById('0808.1041v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0808.1041v1-abstract-full" style="display: none;"> We consider simplest piecewise flat manifold consisting of two identical 4-tetrahedra (call it bisimplex). General relativity action for arbitrary piecewise flat manifold can be expressed in terms of sum of the (half of) bisimplex actions. We use representation of each bisimplex action in terms of certain rotation matrices (connections). This gives representation of any minisuperspace piecewise flat gravity system in terms of connections which do not connect neighboring 4-tetrahedra (more appropriate would be call these self-connections). If Regge calculus with independent 4-tetrahedra is considered, i. e. when the length of an edge is not constrained to be the same for all the 4-tetrahedra containing this edge, self-connection representation leaves 4-tetrahedra independent also in connection matrices sector. Action remains sum of independent 4-tetrahedra terms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0808.1041v1-abstract-full').style.display = 'none'; document.getElementById('0808.1041v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 August, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2008. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0808.1039">arXiv:0808.1039</a> <span> [<a href="https://arxiv.org/pdf/0808.1039">pdf</a>, <a href="https://arxiv.org/ps/0808.1039">ps</a>, <a href="https://arxiv.org/format/0808.1039">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Gravity action on discontinuous metrics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0808.1039v1-abstract-short" style="display: inline;"> We consider minisuperspace gravity system described by piecewise flat metric discontinuous on three-dimensional faces (tetrahedra). There are infinite terms in the Einstein action. However, starting from proper regularization, these terms in the exponential of path integral result in pre-exponent factor with $未$-functions requiring vanishing metric discontinuities. Thereby path integral measure… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0808.1039v1-abstract-full').style.display = 'inline'; document.getElementById('0808.1039v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0808.1039v1-abstract-full" style="display: none;"> We consider minisuperspace gravity system described by piecewise flat metric discontinuous on three-dimensional faces (tetrahedra). There are infinite terms in the Einstein action. However, starting from proper regularization, these terms in the exponential of path integral result in pre-exponent factor with $未$-functions requiring vanishing metric discontinuities. Thereby path integral measure in Regge calculus is related to path integral measure in Regge calculus where length of an edge is not constrained to be the same for all the 4-tetrahedra containing this edge, i.e. in Regge calculus with independent 4-tetrahedra. The result obtained is in accordance with our previous one obtained from symmetry considerations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0808.1039v1-abstract-full').style.display = 'none'; document.getElementById('0808.1039v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 August, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2008. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">6 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0707.3331">arXiv:0707.3331</a> <span> [<a href="https://arxiv.org/pdf/0707.3331">pdf</a>, <a href="https://arxiv.org/ps/0707.3331">ps</a>, <a href="https://arxiv.org/format/0707.3331">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> </div> <p class="title is-5 mathjax"> On positivity of quantum measure and of effective action in area tensor Regge calculus </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0707.3331v1-abstract-short" style="display: inline;"> Because of unboundedness of the general relativity action, Euclidean version of the path integral in general relativity requires definition. Area tensor Regge calculus is considered in the representation with independent area tensor and finite rotation matrices. Being integrated over rotation matrices the path integral measure in area tensor Regge calculus is rewritten by moving integration cont… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0707.3331v1-abstract-full').style.display = 'inline'; document.getElementById('0707.3331v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0707.3331v1-abstract-full" style="display: none;"> Because of unboundedness of the general relativity action, Euclidean version of the path integral in general relativity requires definition. Area tensor Regge calculus is considered in the representation with independent area tensor and finite rotation matrices. Being integrated over rotation matrices the path integral measure in area tensor Regge calculus is rewritten by moving integration contours to complex plain so that it looks as that one with effective action in the exponential with positive real part. We speculate that positivity of the measure can be expected in the most part of range of variation of area tensors. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0707.3331v1-abstract-full').style.display = 'none'; document.getElementById('0707.3331v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 July, 2007; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2007. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages, 4 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/0612143">arXiv:gr-qc/0612143</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/0612143">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/0612143">ps</a>, <a href="https://arxiv.org/format/gr-qc/0612143">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.physletb.2007.06.039">10.1016/j.physletb.2007.06.039 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the possibility of finite quantum Regge calculus </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/0612143v3-abstract-short" style="display: inline;"> 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 cert… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0612143v3-abstract-full').style.display = 'inline'; document.getElementById('gr-qc/0612143v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/0612143v3-abstract-full" style="display: none;"> 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0612143v3-abstract-full').style.display = 'none'; document.getElementById('gr-qc/0612143v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 June, 2007; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 December, 2006; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2006. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">12 pages, plain LaTeX, readability improved, matches version to be published</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett.B651:388-393,2007 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/0602116">arXiv:gr-qc/0602116</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/0602116">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/0602116">ps</a>, <a href="https://arxiv.org/format/gr-qc/0602116">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.physletb.2006.05.002">10.1016/j.physletb.2006.05.002 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Path integral in area tensor Regge calculus and complex connections </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/0602116v3-abstract-short" style="display: inline;"> Euclidean quantum measure in Regge calculus with independent area tensors is considered using example of the Regge manifold of a simple structure. We go over to integrations along certain contours in the hyperplane of complex connection variables. Discrete connection and curvature on classical solutions of the equations of motion are not, strictly speaking, genuine connection and curvature, but… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0602116v3-abstract-full').style.display = 'inline'; document.getElementById('gr-qc/0602116v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/0602116v3-abstract-full" style="display: none;"> Euclidean quantum measure in Regge calculus with independent area tensors is considered using example of the Regge manifold of a simple structure. We go over to integrations along certain contours in the hyperplane of complex connection variables. Discrete connection and curvature on classical solutions of the equations of motion are not, strictly speaking, genuine connection and curvature, but more general quantities and, therefore, these do not appear as arguments of a function to be averaged, but are the integration (dummy) variables. We argue that upon integrating out the latter the resulting measure can be well-defined on physical hypersurface (for the area tensors corresponding to certain edge vectors, i.e. to certain metric) as positive and having exponential cutoff at large areas on condition that we confine ourselves to configurations which do not pass through degenerate metrics. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0602116v3-abstract-full').style.display = 'none'; document.getElementById('gr-qc/0602116v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 December, 2006; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 February, 2006; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2006. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">10 pages, plain LaTeX, matches the published version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett. B637 (2006) 350-355 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/0506072">arXiv:gr-qc/0506072</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/0506072">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/0506072">ps</a>, <a href="https://arxiv.org/format/gr-qc/0506072">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.physletb.2005.12.033">10.1016/j.physletb.2005.12.033 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the area expectation values in area tensor Regge calculus in the Lorentzian domain </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/0506072v2-abstract-short" style="display: inline;"> Wick rotation in area tensor Regge calculus is considered. The heuristical expectation is confirmed that the Lorentzian quantum measure on a spacelike area should coincide with the Euclidean measure at the same argument. The consequence is validity of probabilistic interpretation of the Lorentzian measure as well (on the real, i.e. spacelike areas). </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/0506072v2-abstract-full" style="display: none;"> Wick rotation in area tensor Regge calculus is considered. The heuristical expectation is confirmed that the Lorentzian quantum measure on a spacelike area should coincide with the Euclidean measure at the same argument. The consequence is validity of probabilistic interpretation of the Lorentzian measure as well (on the real, i.e. spacelike areas). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0506072v2-abstract-full').style.display = 'none'; document.getElementById('gr-qc/0506072v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 December, 2005; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 June, 2005; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2005. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">LaTeX, 7 pages, introduction and discussion given in more detail, references added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett.B633:653-656,2006 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/0506071">arXiv:gr-qc/0506071</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/0506071">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/0506071">ps</a>, <a href="https://arxiv.org/format/gr-qc/0506071">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1134/1.2103210">10.1134/1.2103210 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Discrete quantum gravity in the framework of Regge calculus formalism </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/0506071v2-abstract-short" style="display: inline;"> An approach to the discrete quantum gravity based on the Regge calculus is discussed which was developed in a number of our papers. Regge calculus is general relativity for the subclass of general Riemannian manifolds called piecewise flat ones. Regge calculus deals with the discrete set of variables, triangulation lengths, and contains continuous general relativity as a particular limiting case… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0506071v2-abstract-full').style.display = 'inline'; document.getElementById('gr-qc/0506071v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/0506071v2-abstract-full" style="display: none;"> An approach to the discrete quantum gravity based on the Regge calculus is discussed which was developed in a number of our papers. Regge calculus is general relativity for the subclass of general Riemannian manifolds called piecewise flat ones. Regge calculus deals with the discrete set of variables, triangulation lengths, and contains continuous general relativity as a particular limiting case when the lengths tend to zero. In our approach the quantum length expectations are nonzero and of the order of Plank scale $10^{-33}cm$. This means the discrete spacetime structure on these scales. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0506071v2-abstract-full').style.display = 'none'; document.getElementById('gr-qc/0506071v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 December, 2005; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 June, 2005; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2005. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">LaTeX, 16 pages, to appear in JETP</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/0406050">arXiv:gr-qc/0406050</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/0406050">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/0406050">ps</a>, <a href="https://arxiv.org/format/gr-qc/0406050">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.physletb.2004.09.039">10.1016/j.physletb.2004.09.039 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Modification of quantum measure in area tensor Regge calculus and positivity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/0406050v1-abstract-short" style="display: inline;"> A comparative analysis of the versions of quantum measure in the area tensor Regge calculus is performed on the simplest configurations of the system. The quantum measure is constructed in such the way that it reduces to the Feynman path integral describing canonical quantisation if the continuous limit along any of the coordinates is taken. As we have found earlier, it is possible to implement… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0406050v1-abstract-full').style.display = 'inline'; document.getElementById('gr-qc/0406050v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/0406050v1-abstract-full" style="display: none;"> A comparative analysis of the versions of quantum measure in the area tensor Regge calculus is performed on the simplest configurations of the system. The quantum measure is constructed in such the way that it reduces to the Feynman path integral describing canonical quantisation if the continuous limit along any of the coordinates is taken. As we have found earlier, it is possible to implement also the correspondence principle (proportionality of the Lorentzian (Euclidean) measure to $e^{iS}$ ($e^{-S}$), $S$ being the action). For that a certain kind of the connection representation of the Regge action should be used, namely, as a sum of independent contributions of selfdual and antiselfdual sectors (that is, effectively 3-dimensional ones). There are two such representations, the (anti)selfdual connections being SU(2) or SO(3) rotation matrices according to the two ways of decomposing full SO(4) group, as SU(2) $\times$ SU(2) or SO(3) $\times$ SO(3). The measure from SU(2) rotations although positive on physical surface violates positivity outside this surface in the general configuration space of arbitrary independent area tensors. The measure based on SO(3) rotations is expected to be positive in this general configuration space on condition that the scale of area tensors considered as parameters is bounded from above by the value of the order of Plank unit. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0406050v1-abstract-full').style.display = 'none'; document.getElementById('gr-qc/0406050v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 June, 2004; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2004. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">10 pages, plain LaTeX</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett. B601 (2004) 229-235 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/0406049">arXiv:gr-qc/0406049</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/0406049">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/0406049">ps</a>, <a href="https://arxiv.org/format/gr-qc/0406049">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.physletb.2004.09.038">10.1016/j.physletb.2004.09.038 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Feynman path integral in area tensor Regge calculus and correspondence principle </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/0406049v1-abstract-short" style="display: inline;"> The quantum measure in area tensor Regge calculus can be constructed in such the way that it reduces to the Feynman path integral describing canonical quantisation if the continuous limit along any of the coordinates is taken. This construction does not necessarily mean that Lorentzian (Euclidean) measure satisfies correspondence principle, that is, takes the form proportional to $e^{iS}$ (… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0406049v1-abstract-full').style.display = 'inline'; document.getElementById('gr-qc/0406049v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/0406049v1-abstract-full" style="display: none;"> The quantum measure in area tensor Regge calculus can be constructed in such the way that it reduces to the Feynman path integral describing canonical quantisation if the continuous limit along any of the coordinates is taken. This construction does not necessarily mean that Lorentzian (Euclidean) measure satisfies correspondence principle, that is, takes the form proportional to $e^{iS}$ ($e^{-S}$) where $S$ is the action. Requirement to fit this principle means some restriction on the action, or, in the context of representation of the Regge action in terms of independent rotation matrices (connections), restriction on such representation. We show that the representation based on separate treatment of the selfdual and antiselfdual rotations allows to modify the derivation and give sense to the conditionally convergent integrals to implement both the canonical quantisation and correspondence principles. If configurations are considered such that the measure is factorisable into the product of independent measures on the separate areas (thus far it was just the case in our analysis), the considered modification of the measure does not effect the vacuum expectation values. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0406049v1-abstract-full').style.display = 'none'; document.getElementById('gr-qc/0406049v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 June, 2004; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2004. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages, plain LaTeX</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett. B601 (2004) 222-228 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/0401053">arXiv:gr-qc/0401053</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/0401053">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/0401053">ps</a>, <a href="https://arxiv.org/format/gr-qc/0401053">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.physletb.2004.02.041">10.1016/j.physletb.2004.02.041 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the length expectation values in quantum Regge calculus </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/0401053v1-abstract-short" style="display: inline;"> 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 con… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0401053v1-abstract-full').style.display = 'inline'; document.getElementById('gr-qc/0401053v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/0401053v1-abstract-full" style="display: none;"> 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 {\it dynamically} in the originally discrete framework. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0401053v1-abstract-full').style.display = 'none'; document.getElementById('gr-qc/0401053v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 January, 2004; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2004. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">11 pages, plain LaTeX</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett. B586 (2004) 411-419 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/0304088">arXiv:gr-qc/0304088</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/0304088">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/0304088">ps</a>, <a href="https://arxiv.org/format/gr-qc/0304088">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> </div> <p class="title is-5 mathjax"> From areas to lengths in quantum Regge calculus </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/0304088v1-abstract-short" style="display: inline;"> 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… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0304088v1-abstract-full').style.display = 'inline'; document.getElementById('gr-qc/0304088v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/0304088v1-abstract-full" style="display: none;"> 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 $\det\|g_{位渭}\|$ 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0304088v1-abstract-full').style.display = 'none'; document.getElementById('gr-qc/0304088v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 April, 2003; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2003. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">10 pages, LaTeX</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/0304006">arXiv:gr-qc/0304006</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/0304006">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/0304006">ps</a>, <a href="https://arxiv.org/format/gr-qc/0304006">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.physletb.2003.06.020">10.1016/j.physletb.2003.06.020 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Regge calculus from discontinuous metrics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/0304006v1-abstract-short" style="display: inline;"> Regge calculus is considered as a particular case of the more general system where the linklengths of any two neighbouring 4-tetrahedra do not necessarily coincide on their common face. This system is treated as that one described by metric discontinuous on the faces. In the superspace of all discontinuous metrics the Regge calculus metrics form some hypersurface. Quantum theory of the discontin… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0304006v1-abstract-full').style.display = 'inline'; document.getElementById('gr-qc/0304006v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/0304006v1-abstract-full" style="display: none;"> Regge calculus is considered as a particular case of the more general system where the linklengths of any two neighbouring 4-tetrahedra do not necessarily coincide on their common face. This system is treated as that one described by metric discontinuous on the faces. In the superspace of all discontinuous metrics the Regge calculus metrics form some hypersurface. Quantum theory of the discontinuous metric system is assumed to be fixed somehow in the form of quantum measure on (the space of functionals on) the superspace. The problem of reducing this measure to the Regge hypersurface is addressed. The quantum Regge calculus measure is defined from a discontinuous metric measure by inserting the $未$-function-like phase factor. The requirement that this reduction would respect natural physical properties (positivity, well-defined continuum limit, absence of lattice artefacts) put rather severe restrictions and allows to define practically uniquely this phase factor. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0304006v1-abstract-full').style.display = 'none'; document.getElementById('gr-qc/0304006v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 April, 2003; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2003. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">8 pages, LaTeX</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett. B567 (2003) 288-293 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/0212110">arXiv:gr-qc/0212110</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/0212110">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/0212110">ps</a>, <a href="https://arxiv.org/format/gr-qc/0212110">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/S0370-2693(03)00439-8">10.1016/S0370-2693(03)00439-8 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Area expectation values in quantum area Regge calculus </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/0212110v3-abstract-short" style="display: inline;"> The Regge calculus generalised to independent area tensor variables is considered. The continuous time limit is found and formal Feynman path integral measure corresponding to the canonical quantisation is written out. The quantum measure in the completely discrete theory is found which possesses the property to lead to the Feynman path integral in the continuous time limit whatever coordinate i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0212110v3-abstract-full').style.display = 'inline'; document.getElementById('gr-qc/0212110v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/0212110v3-abstract-full" style="display: none;"> The Regge calculus generalised to independent area tensor variables is considered. The continuous time limit is found and formal Feynman path integral measure corresponding to the canonical quantisation is written out. The quantum measure in the completely discrete theory is found which possesses the property to lead to the Feynman path integral in the continuous time limit whatever coordinate is chosen as time. This measure can be well defined by passing to the integration over imaginary field variables (area tensors). Averaging with the help of this measure gives finite expectation values for areas. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0212110v3-abstract-full').style.display = 'none'; document.getElementById('gr-qc/0212110v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 March, 2003; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 26 December, 2002; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2002. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages, LaTeX, possible relation to quantisation of the usual length-based Regge calculus is discussed</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett. B560 (2003) 245-251 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/0206067">arXiv:gr-qc/0206067</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/0206067">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/0206067">ps</a>, <a href="https://arxiv.org/format/gr-qc/0206067">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/S0370-2693(02)02773-9">10.1016/S0370-2693(02)02773-9 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Area Regge calculus and continuum limit </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/0206067v1-abstract-short" style="display: inline;"> 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… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0206067v1-abstract-full').style.display = 'inline'; document.getElementById('gr-qc/0206067v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/0206067v1-abstract-full" style="display: none;"> 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0206067v1-abstract-full').style.display = 'none'; document.getElementById('gr-qc/0206067v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 June, 2002; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2002. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">10 pages, TeX</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett. B547 (2002) 321-327 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/0111063">arXiv:gr-qc/0111063</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/0111063">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/0111063">ps</a>, <a href="https://arxiv.org/format/gr-qc/0111063">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/S0370-2693(02)01322-9">10.1016/S0370-2693(02)01322-9 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Path integral measure in Regge calculus from the functional Fourier transform </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/0111063v2-abstract-short" style="display: inline;"> The problem of fixing measure in the path integral for the Regge-discretised gravity is considered from the viewpoint of it's "best approximation" to the already known formal continuum general relativity (GR) measure. A rigorous formulation may consist in comparing functional Fourier transforms of the measures, i.e. characteristic or generating functionals, and requiring these to coincide on som… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0111063v2-abstract-full').style.display = 'inline'; document.getElementById('gr-qc/0111063v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/0111063v2-abstract-full" style="display: none;"> The problem of fixing measure in the path integral for the Regge-discretised gravity is considered from the viewpoint of it's "best approximation" to the already known formal continuum general relativity (GR) measure. A rigorous formulation may consist in comparing functional Fourier transforms of the measures, i.e. characteristic or generating functionals, and requiring these to coincide on some dense set in the functional space. The possibility for such set to exist is due to the Regge manifold being a particular case of general Riemannian one (Regge calculus is a minisuperspace theory). The two versions of the measure are obtained depending on what metric tensor, covariant or contravariant one, is taken as fundamental field variable. The closed expressions for the measure are obtained in the two simple cases of Regge manifold. These turn out to be quite reasonable one of them indicating that appropriately defined continuum limit of the Regge measure would reproduce the original continuum GR measure. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0111063v2-abstract-full').style.display = 'none'; document.getElementById('gr-qc/0111063v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 December, 2001; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 November, 2001; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2001. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages, LaTeX, misprints in a formula (eq. (11)) removed</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett. B530 (2002) 251-257 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/0012097">arXiv:gr-qc/0012097</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/0012097">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/0012097">ps</a>, <a href="https://arxiv.org/format/gr-qc/0012097">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/S0370-2693(01)00296-9">10.1016/S0370-2693(01)00296-9 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the Faddeev-Popov determinant in Regge calculus </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/0012097v1-abstract-short" style="display: inline;"> The functional integral measure in the 4D Regge calculus normalised w.r.t. the DeWitt supermetric on the space of metrics is considered. The Faddeev-Popov factor in the measure is shown according to the previous author's work on the continuous fields in Regge calculus to be generally ill-defined due to the conical singularities. Possible resolution of this problem is discretisation of the gravit… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0012097v1-abstract-full').style.display = 'inline'; document.getElementById('gr-qc/0012097v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/0012097v1-abstract-full" style="display: none;"> The functional integral measure in the 4D Regge calculus normalised w.r.t. the DeWitt supermetric on the space of metrics is considered. The Faddeev-Popov factor in the measure is shown according to the previous author's work on the continuous fields in Regge calculus to be generally ill-defined due to the conical singularities. Possible resolution of this problem is discretisation of the gravity ghost (gauge) field by, e.g., confining ourselves to the affine transformations of the affine frames in the simplices. This results in the singularity of the functional measure in the vicinity of the flat background, where part of the physical degrees of freedom connected with linklengths become gauge ones. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0012097v1-abstract-full').style.display = 'none'; document.getElementById('gr-qc/0012097v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 December, 2000; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2000. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">5 pages, LaTeX</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett. B504 (2001) 359-361 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/0012095">arXiv:gr-qc/0012095</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/0012095">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/0012095">ps</a>, <a href="https://arxiv.org/format/gr-qc/0012095">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/S0370-2693(01)00295-7">10.1016/S0370-2693(01)00295-7 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Continuous matter fields in Regge calculus </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V M Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/0012095v1-abstract-short" style="display: inline;"> 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 th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0012095v1-abstract-full').style.display = 'inline'; document.getElementById('gr-qc/0012095v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/0012095v1-abstract-full" style="display: none;"> 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/0012095v1-abstract-full').style.display = 'none'; document.getElementById('gr-qc/0012095v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 December, 2000; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2000. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">4 pages, LaTeX</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett. B504 (2001) 356-358 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/9912112">arXiv:gr-qc/9912112</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/9912112">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/9912112">ps</a>, <a href="https://arxiv.org/format/gr-qc/9912112">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/S0370-2693(00)00201-X">10.1016/S0370-2693(00)00201-X <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The simplest Regge calculus model in the canonical form </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">Vladimir M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/9912112v1-abstract-short" style="display: inline;"> Dynamics of a Regge three-dimensional (3D) manifold in a continuous time is considered. The manifold is closed consisting of the two tetrahedrons with identified corresponding vertices. The action of the model is that obtained via limiting procedure from the general relativity (GR) action for the completely discrete 4D Regge calculus. It closely resembles the continuous general relativity action… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/9912112v1-abstract-full').style.display = 'inline'; document.getElementById('gr-qc/9912112v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/9912112v1-abstract-full" style="display: none;"> Dynamics of a Regge three-dimensional (3D) manifold in a continuous time is considered. The manifold is closed consisting of the two tetrahedrons with identified corresponding vertices. The action of the model is that obtained via limiting procedure from the general relativity (GR) action for the completely discrete 4D Regge calculus. It closely resembles the continuous general relativity action in the Hilbert-Palatini (HP) form but possesses finite number of the degrees of freedom. The canonical structure of the theory is described. Central point is appearance of the new relations with time derivatives not following from the Lagrangian but serving to ensure completely discrete 4D Regge calculus origin of the system. In particular, taking these into account turns out to be necessary to obtain the true number of the degrees of freedom being the number of linklengths of the 3D Regge manifold at a given moment of time. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/9912112v1-abstract-full').style.display = 'none'; document.getElementById('gr-qc/9912112v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 December, 1999; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 1999. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">LaTeX, 7 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett. B477 (2000) 248-252 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/9912111">arXiv:gr-qc/9912111</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/9912111">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/9912111">ps</a>, <a href="https://arxiv.org/format/gr-qc/9912111">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/S0370-2693(00)00617-1">10.1016/S0370-2693(00)00617-1 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Path integral in the simplest Regge calculus model </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">Vladimir M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/9912111v1-abstract-short" style="display: inline;"> The simplest (3+1)D Regge calculus model (with three-dimensional discrete space and continuous time) is considered which describes evolution of the simplest closed two-tetrahedron piecewise flat manifold in the continuous time. The measure in the path integral which describes canonical quantisation of the model in terms of area bivectors and connections as independent variables is found. It is s… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/9912111v1-abstract-full').style.display = 'inline'; document.getElementById('gr-qc/9912111v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/9912111v1-abstract-full" style="display: none;"> The simplest (3+1)D Regge calculus model (with three-dimensional discrete space and continuous time) is considered which describes evolution of the simplest closed two-tetrahedron piecewise flat manifold in the continuous time. The measure in the path integral which describes canonical quantisation of the model in terms of area bivectors and connections as independent variables is found. It is shown that selfdual-antiselfdual splitting of the variables simplifies the integral although does not admit complete separation of (anti-)selfdual sector. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/9912111v1-abstract-full').style.display = 'none'; document.getElementById('gr-qc/9912111v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 December, 1999; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 1999. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">LaTeX, 10 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett. B484 (2000) 160-166 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/9803027">arXiv:gr-qc/9803027</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/9803027">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/9803027">ps</a>, <a href="https://arxiv.org/format/gr-qc/9803027">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/S0370-2693(98)00448-1">10.1016/S0370-2693(98)00448-1 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Rotating vacuum wormhole </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/9803027v1-abstract-short" style="display: inline;"> We investigate whether self-maintained vacuum traversible wormhole can exist described by stationary but nonstatic metric. We consider metric being the sum of static spherically symmetric one and a small nondiagonal component which describes rotation sufficiently slow to be taken into account in the linear approximation. We study semiclassical Einstein equations for this metric with vacuum expec… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/9803027v1-abstract-full').style.display = 'inline'; document.getElementById('gr-qc/9803027v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/9803027v1-abstract-full" style="display: none;"> We investigate whether self-maintained vacuum traversible wormhole can exist described by stationary but nonstatic metric. We consider metric being the sum of static spherically symmetric one and a small nondiagonal component which describes rotation sufficiently slow to be taken into account in the linear approximation. We study semiclassical Einstein equations for this metric with vacuum expectation value of stress-energy of physical fields as the source. In suggestion that the static traversible wormhole solution exists we reveal possible azimuthal angle dependence of angular velocity of the rotation (angular velocity of the local inertial frame) that solves semiclassical Einstein equations. We find that in the macroscopic (in the Plank scale) wormhole case a rotational solution exists but only such that, first, angular velocity depends on radial coordinate only and, second, the wormhole connects the two asymptotically flat spacetimes rotating with angular velocities different in asymptotic regions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/9803027v1-abstract-full').style.display = 'none'; document.getElementById('gr-qc/9803027v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 March, 1998; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 1998. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">10 pages, uses LaTeX, submitted to Physics Letters B</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett. B429 (1998) 254-262 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/gr-qc/9604053">arXiv:gr-qc/9604053</a> <span> [<a href="https://arxiv.org/pdf/gr-qc/9604053">pdf</a>, <a href="https://arxiv.org/ps/gr-qc/9604053">ps</a>, <a href="https://arxiv.org/format/gr-qc/9604053">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/S0370-2693(96)01691-7">10.1016/S0370-2693(96)01691-7 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Ashtekar Constraint Surface as Projection of Hilbert-Palatini One </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Khatsymovsky%2C+V+M">V. M. Khatsymovsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="gr-qc/9604053v1-abstract-short" style="display: inline;"> The Hilbert-Palatini (HP) Lagrangian of general relativity being written in terms of selfdual and antiselfdual variables contains Ashtekar Lagrangian (which governs the dynamics of the selfdual sector of the theory on condition that the dynamics of antiselfdual sector is not fixed). We show that nonequivalence of the Ashtekar and HP quantum theories is due to the specific form (of the "loose rel… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/9604053v1-abstract-full').style.display = 'inline'; document.getElementById('gr-qc/9604053v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="gr-qc/9604053v1-abstract-full" style="display: none;"> The Hilbert-Palatini (HP) Lagrangian of general relativity being written in terms of selfdual and antiselfdual variables contains Ashtekar Lagrangian (which governs the dynamics of the selfdual sector of the theory on condition that the dynamics of antiselfdual sector is not fixed). We show that nonequivalence of the Ashtekar and HP quantum theories is due to the specific form (of the "loose relation" type) of constraints which relate self- and antiselfdual variables so that the procedure of (canonical) quantisation of such the theory is noncommutative with the procedure of excluding antiselfdual variables. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('gr-qc/9604053v1-abstract-full').style.display = 'none'; document.getElementById('gr-qc/9604053v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 April, 1996; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 1996. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages of LaTeX file</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett. B394 (1997) 57-61 </p> </li> </ol> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Khatsymovsky%2C+V+M&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Khatsymovsky%2C+V+M&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Khatsymovsky%2C+V+M&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> </ul> </nav> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div 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