arXiv subjects
Publications and source records attributed to Leonid Perlov.
We consider General Relativity as a limit case of the Scalar-Tensor theory with Barbero-Immirzi field when the field tends to a constant. We use Shapiro time delay experimental limit of $1/w = (2.1 \pm 2.3)10^{-5}$ provided by the Cassini spacecraft to find the Barbero-Immirzi parameter value.
The form of Hamiltonian and Diffeomorphism constraints in Sen-Ashtekar-Barbero-Immirzi variables is well known for the spacelike 3+1 ADM foliation. It is also known that Sen-Ashtekar-Barbero-Immirzi connection can be introduced only in 3 dimensional space and does not work for $D > 3$. The reason it works in $D = 3$ is due to existence of isomorphism between $so(3)$ algebra and $R^3$ space with vector product. It turns out that similar isomorphism exists between $so(2,1)$ algebra and $R^3_{2,1}$ space algebra with respect to its vector product. By using this isomorphism we find both analog of Sen-Ashtekar-Barbero-Immirzi connection for timelike 3+1 foliation and corresponding forms of Gauss, Diffeomorphism and Hamiltonian constraints. We then combine spacelike and timelike foliation constraints into the generalized form of the Hamiltonian and Diffeomorphism constrains using generalized Sen-Ashtekar-Barbero-Immirzi connection variables. We prove that Immirzi parameter is covariant with respect to timelike-spacelike ADM foliation change as in both cases in self-dual Ashtekar case it disappears in Hamiltionian constraint keeping it polynomial.
In this paper we first provide the proof of $SU(2)$ Y-Map convergence. Then, by using $SU(1,1)$ LQG simplicity constraints we define $SU(1,1)$ Y-Map from infinitely differentiable with a compact support functions on $SU(1,1)$ to the functions (not necessarily square integrable) on $SL(2,C)$, and prove its convergence as well.
We introduce 3+1 timelike foliation of the four dimensional Lorentz manifold to derive the 3+1 Sen-Ashtekar-Barbero-Immirzi formalism in case of $SO(2,1)$ rotation gauge group, which is possible due to the existence of the $so(2,1)$ algebra isomorphism to $R^3_{2,1}$ algebra with respect to the vector product. We prove that the newly introduced flux and extrinsic curvature variables preserve the symplectic structure of the original variables. We then introduce the modified rotational constraint and succeed to write it as a Gauss constraint of a newly obtained connection. The newly obtained connection is slightly different from the classical 3+1 spacelike Sen-Ashtekar-Barbero-Immirzi connection as it contains in addition the Minkowski metric $\eta_{ij}$ as a coefficient. Our result has a very simple form and clearly shows how $so(2,1)$ connection is different from $so(3)$ one. Also it is the first time that the key-stone fact that makes the whole formalism work in timelike 3+1 case, i.e. $so(2,1) \simeq R^3_{2,1}$ isomorphism and its relation to the $so(2,1)$ connection has been researched.
In this paper we introduce the n-dimensional hypersurface quantum volume operator by using the n-dimensional holonomy variation formula. Instead of trying to construct the n-dimensional hypersurface volume operator by using the n-1 dimensional hypersufrace volume operators, as it is usually done in 3d case, we introduce the n-dimensional volume operator directly. We use two facts - first, that the area of the n-dimensional hypersurface of the n+1 dimensional manifold is the volume of the n dimensional induced metric and secondly that the holonomy variation formula is valid for the n-dimensional hypersufrace in the n+1 manifold with connection values in any Lie algebra.
In this paper we derive the uncertainty principle for the Loop Quantum Cosmology homogeneous and isotropic FLWR model with the holonomy-flux algebra. The uncertainty principle is between the variables $c$, with the meaning of connection and $μ$ having the meaning of the physical cell volume to the power $2/3$, i.e $v^{2/3}$ or a plaquette area. Since both $μ$ and $c$ are not operators, but rather the random variables, the Robertson uncertainty principle derivation that works for hermitian operators, can not be used. Instead we use the Wigner-Moyal-Groenewold phase space formalism. The Wigner-Moyal-Groenewold formalism was originally applied to the Heisenberg algebra of the Quantum Mechanics. One can derive from it both the canonical and path integral QM as well as the uncertainty principle. In this paper we apply it to the holonomy-flux algebra in case of the homogeneous and isotropic space. Another result is the expression for the Wigner function on the space of the cylindrical wave functions defined on $R_b$ in $c$ variables rather than in dual space $μ$ variables.
In this paper we research all possible finite-dimensional representations and corresponding values of the Barbero-Immirzi parameter contained in EPRL simplicity constraints by using Naimark's fundamental theorem of the Lorentz group representation theory. It turns out that for each non-zero pure imaginary with rational modulus value of the Barbero-Immirzi parameter $γ= i \frac{p}{q}, p, q \in Z, p, q \ne 0$, there is a solution of the simplicity constraints, such that the corresponding Lorentz representation is finite dimensional. The converse is also true - for each finite-dimensional Lorentz representation solution of the simplicity constraints $(n, ρ)$, the associated Barbero-Immirzi parameter is non-zero pure imaginary with rational modulus, $γ= i \frac{p}{q}, p, q \in Z, p, q \ne 0$. We solve the simplicity constraints with respect to the Barbero-Immirzi parameter and then use Naimark's fundamental theorem of the Lorentz group representations to find all finite-dimensional representations contained in the solutions.
In this paper we naturally obtain the values of the Barbero-Immirzi parameter as the solution of the simplicity constraints rather than setting it a priori. Particularly the Main theorem shows that if $γ= \pm i$ then the simplicity constraints require that the corresponding Lorentz group representations be necessary finite dimensional and therefore non-unitary.
The most important part of the new spin-foam loop quantum gravity formulation is the map $Y$: $H^{SU(2)} \rightarrow H^{SL(2,C)}$. It was only recently shown that the Y-Map is convergent in spite of the fact that the classical Peter-Weyl theorem is not applicable to it, as Lorentz group is not compact. In this paper we provide an alternative map $\tilde{Y}$. The $\tilde{Y}$ map has an advantage of preserving the Lorentz covariance, which gets broken in the case of Y-Map. The image of a new map $\tilde{Y}$ contains the weighted infinite sum of $SL(2,C)$ matrix coefficients. The sum is convergent and its limit is the square integrable functions of $SL(2,C)$ with the measure $L^2(g, e^{-|Y|^2/\hbar}η(g) du \,dY )$ according to the Holomorphic Huebschmann-Peter-Weyl theorem, which is applicable to the rational representations of the non-unitary groups, particularly non-unitary finite Lorenz representations. Since in LQG the unitary evolution is not mandatory as it does not follow from the Wheeler-DeWitt dynamics equation, the choice of the non-unitary representation is valid. As it was stated in the original LQG formulation: "there is no sense in which conventional unitarity is necessary in the theory".
In quantum gravity the unitary evolution does not follow from the Wheeler-DeWitt dynamics equation as it follows from the Schrödinger equation in non-relativistic quantum mechanics. Therefore we can define a spin-foam model based on SL(2,C) spinor finite non-unitary representations. The recently discovered holomorphic Peter-Weyl theorem \cite{Huebschmann} made it possible to decompose the delta function of a non-compact Lorentz group into the convergent sum of the matrix coefficients. We calculate the vertex amplitude with the help of that theorem and obtain a simple expression for our model. The $SL(2,C)$ Hilbert space is defined from $SU(2)$ Hilbert space by Huebschmann-Kirillov transform \cite{Huebschmann}. A new transform is simpler than the well known Hall transform as it does not contain a heat kernel convolution. We do not set Barbero-Immirzi constant $γ$ a priori, instead we obtain it as a solution of the diagonal and off-diagonal simplicity constraints being $γ= \frac{-in}{(|n| + 2p)}$ where $p$ is a non-negative half-integer. When $p=0$ the solution corresponds to the Ashtekar's self-dual connections. We point out that the Barbero-Immirzi becomes real when one chooses a unitary representation. It is complex when the representation is non-unitary principal series or non-unitary spinor representation.
The expansion of a square integrable function on $SL(2,C)$ into the sum of the principal series matrix coefficients with the specially selected representation parameters was recently used in the Loop Quantum Gravity $\cite{RovelliBook2}$, $\cite{Rovelli2010}$. In this paper we prove that the sum $\sum\limits_{j=1}^{\infty}\sum\limits_{|m| \le j}\sum\limits_{|n| \le j} \frac{D^{(j, \tau j)}_{jm, jn}(g)}{j^k}$, where $ j, m, n \in Z, \tau \in C$ is convergent to a square integrable function on $SL(2,C)$. We also prove that for each fixed m: $\sum\limits_{j=1}^{\infty}\frac{D^{(j, \tau j)}_{jm, jm}(g)}{j^k}$ is convergent and that the limit is a square integrable function on $SL(2,C)$. We then prove convergence of the sums $\sum\limits_{j=|p|}^{\infty}\sum\limits_{|m| \le j}\sum\limits_{|n| \le j} d^{\frac{j}{2}}_{pm} D^{(j, \tau j)}_{jm, jn}(g)$, where $d^{\frac{j}{2}}_{|p|m} = (j+1)^{\frac{1}{2}}\int\limits_{SU(2)}\phi(u)\overline{ D^{\frac{j}{2}}_{|p|m}(u)} \; du \; $ is $\phi(u)$'s Fourier transform and $ p, j, m, n \in Z, \tau \in C, u \in SU(2), g \in SL(2,C)$, thus establishing the map between the square integrable functions on $SU(2)$ and the space of the functions on $SL(2,C)$. Such maps were first used in $\cite{RovelliBook2}$.
The main result of this paper is the 4-dimensional supermetric version of the Wheeler-DeWitt equation, that uses only one time variable for the both roles - as internal time and for the ADM split, as Hamiltonian evolution parameter. We study the ADM split with respect to the scalar massless field serving as internal time. The 4-dimensional hyper-surfaces $Σ_{ϕ= const}$ span the 5-dimensional space with the scalar field being the fifth coordinate. As a result we obtain the analog of the Wheeler-DeWitt equation for the 4-dimensional supermetric. We compare the ADM action with the non-compactified Kaluza-Klein action for the same physical space and obtain the equation for the extrinsic curvature and the scalar massless field.