Lorentz-Covariant Loop Quantum Gravity

We review the Lorentz-covariant approach to loop quantum gravity. This approach solves the Immirzi parameter problem occurring in the standard loop approach based on the SU(2) gauge group. We show that there exists a unique loop quantization preserving all the classical symmetries at the quantum level and that the results obtained with it, such as the area operator spectrum, are independent of the Immirzi parameter. The standard SU(2) approach violates the diffeomorphism invariance and is therefore an incorrect quantization of gravity.     

Instanton Effects and Quantum Spectral Curves

We study a spectral problem associated to the quantization of a spectral curve arising in local mirror symmetry. The perturbative WKB quantization condition is determined by the quantum periods, or equivalently by the refined topological string in the Nekrasov–Shatashvili (NS) limit. We show that the information encoded in the quantum periods is radically insufficient to determine the spectrum: there is an infinite series of instanton corrections, which are non-perturbative in \({\hbar}\), and lead to an exact WKB quantization condition. Moreover, we conjecture the precise form of the instanton corrections: they are determined by the standard or unrefined topological string free energy, and we test our conjecture successfully against numerical calculations of the spectrum. This suggests that the non-perturbative sector of the NS refined topological string contains information about the standard topological string. As an application of the WKB quantization condition, we explain some recent observations relating membrane instanton corrections in ABJM theory to the refined topological string.     

Bulk Correlation Functions in 2d Quantum Gravity

We compute the bulk three- and four-point tachyon correlators in the 2d Liouville gravity with a nonrational matter central charge c < 1, using and comparing two approaches. The continuous CFT approach exploits the action on the tachyons of the ground-ring generators deformed with Liouville and matter screening charges. We derive a general formula for the matter three-point OPE structure constants as a by-product. The discrete formulation of the theory is a generalization of the ADE string theories, in which the target space is the semi-infinite chain of points. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 132–145, January, 2006.     

Spectral Riemann surfaces of the Sturm—Liouville operator and quasiexactly solvable problems

Institute of Theoretical and Experimental Physics. State Committee on the Use of Atomic Energy. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 22, No. 2, pp. 92–94, April–June, 1988.     

Quantum dissipative systems. I. Canonical quantization and quantum Liouville equation

Sedov's variational principle, which is a generalization of the principle of least action to dissipative processes, is used to generalize canonical quantization and the von Neumann equations to dissipative systems. The example of a harmonic oscillator with friction is considered.Institute of Nuclear Physics, State University, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika Vol. 100, No. 3, pp. 402–417, September, 1994.     

About a Quantum Field Theory for 3D Gravity

We outline some aspects of a dilogarithmic quantum field theory (DQFT) for a 2+1 bordism category based on 3-manifolds equipped with principal flat bundles. We give some geometric evidence that it could be pertinent to 3D gravity, and we describe the building blocks of DQFT: the matrix dilogarithms.Lecture held in the Seminario Matematico e Fisico on November 24, 2003Received: April, 2004     

On the Spectral Gap of a Quantum Graph

We consider the problem of finding universal bounds of “isoperimetric” or “isodiametric” type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature.     

Spectral Gaps in the XXZ Quantum Spin Chain

We study the behavior of the finite-size energy spectrum of the HeisenbergIsing (XXZ) quantum spin chain in the momentum sector. Our numerical and perturbative results for the behavior of energy gaps are explained by considering the motion of string solutions of the Bethe-Ansatz equations. Bibliography: 13 titles.     

Spectral Analysis of an Effective Hamiltonian in Nonrelativistic Quantum Electrodynamics

We investigate spectral properties of an effective Hamiltonian which is obtained as a scaling limit of the Pauli–Fierz model in nonrelativistic quantum electrodynamics. The Lamb shift of a hydrogen-like atom is derived as the lowest order approximation (in the fine structure constant) of an energy level shift of the effective Hamiltonian.     

Some Results for Range of Random Walk on Graph with Spectral Dimension Two

Journal of Theoretical Probability - We consider the range of the simple random walk on graphs with spectral dimension two. We give a form of strong law of large numbers under a certain uniform...     

Coulomb Scattering of a Slow Quantum Particle in a Space of Arbitrary Dimension

We assume that a charged quantum particle moves in a space of dimension d = 2, 3,... and is scattered by a fixed Coulomb center. We derive and study expansions of the wave function and all radial functions of such a particle in integer powers of the wave number and in Bessel functions of a real order. We prove that finite sums of such expansions are asymptotic approximations of the wave functions in the low-energy limit.     

Extensions of Liouville theorems

The method of deriving Liouville's theorem for subharmonic functions in the plane from the corresponding Hadamard three-circles theorem is extended to a more general and abstract setting. Two extensions of Liouville's theorem for vector-valued holomorphic functions of several complex variables are also mentioned.     

Matrix Liouville equation

P. N. Lebedev Physics Institute, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 83, No. 1, pp. 41–50, April, 1990.     

Real Liouville Extensions

We characterize linear differential equations defined over a real differential field with a real closed field of constants C, which are solvable by real Liouville functions, as those having a differential Galois group whose identity component is solvable and C-split.     

Multipoint correlation functions in Liouville field theory and minimal Liouville gravity

We study (n+3)-point correlation functions of exponential fields in the Liouville field theory with n degenerate and three arbitrary fields and derive an analytic expression for these correlation functions in terms of Coulomb integrals. We consider the application of these results to the minimal Liouville gravity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 536–556, March, 2008.     

Toward an Analytic Perturbative Solution for the Abjm Quantum Spectral Curve

Theoretical and Mathematical Physics - We recently showed how nonhomogeneous second-order difference equations that appear in describing the ABJM quantum spectral curve can be solved using a Mellin...     

Multivariate Liouville distributions

A random vector (X₁, …, X_n), with positive components, has a Liouville distribution if its joint probability density function is of the formf(x₁ + … + x_n)x₁^a₁.1 … x_n^a_n.1 with thea_i all positive. Examples of these are the Dirichlet and inverted Dirichlet distributions. In this paper, a comprehensive treatment of the Liouville distributions is provided. The results pertain to stochastic representations, transformation properties, complete neutrality, marginal and conditional distributions, regression functions, and total positivity and reverse rule properties. Further, these topics are utilized in various characterizations of the Dirichlet and inverted Dirichlet distributions. Matrix analogs of the Liouville distributions are also treated, and many of the results obtained in the vector setting are extended appropriately.