arXiv subjects
Publications and source records attributed to M. G. Ivanov.
For QFT on a lattice of dimension d>=3, the vacuum energy (both bosonic and fermionic) is zero if the Hamiltonian is a function of the square of the momentum, and the calculation of the vacuum energy is performed in the ring of residue classes modulo N. This fact is related to a problem from number theory about the number of ways to represent a number as a sum of $d$ squares in the ring of residue classes modulo N.
To simulate the quantum systems at classical or quantum computers, it is necessary to reduce continuous observables (e.g. coordinate and momentum or energy and time) to discrete ones. In this work we consider the continuous observables represented in the positional systems as a series of powers of the radix mulitplied over the summands (``digits``), which turn out to be Hermitean operators with discrete spectrum. We investigate the obtained quantum mechanical operators of digits, the commutation relations between them and the effects of choice of numeral system on the lattices and representations. Furthermore, during the construction of the digital representation renormalizations of diverging sums naturally occur.
In frames of the nonlocal and nonpolynomial quantum theory of the one component scalar field in $D$-dimensional spacetime, stated by Gariy Vladimirovich Efimov, the expansion of the $\mathcal{S}$-matrix is revisited for different interaction Lagrangians and for some kinds of Gaussian propagators modified by different ultraviolet form factors $F$ which depend on some length parameter $l$. The expansion of the $\mathcal{S}$-matrix is of the form of a grand canonical partition function of some $D+N$-dimensional ($N\geq 1$) classical gas with interaction. The toy model of the realistic quantum field theory (QFT) is considered where the $\mathcal{S}$-matrix is calculated in closed form. Then, the functional Schwinger-Dyson and Schrödinger equations for the $\mathcal{S}$-matrix in Efimov representation are derived. These equations play a central role in the present paper. The functional Schwinger-Dyson and Schrödinger equations in Efimov representation do not involve explicit functional derivatives but involve a shift of the field which is the $\mathcal{S}$-matrix argument. The asymptotic solutions of the Schwinger-Dyson equation are obtained in different limits. Also, the solution is found in one heuristic case allowing us to study qualitatively the behavior of the $\mathcal{S}$-matrix for an arbitrary finite value of its argument. Self-consistency equations, which arise during the process of derivation, are of a great interest. Finally, in the light of the discussion of QFT functional equations, ultraviolet form factors and extra dimensions, the connection with functional (in terms of the Wilson-Polchinski and Wetterich-Morris functional equations) and holographic renormalization groups (in terms of the functional Hamilton-Jacobi equation) is made. In addition the Hamilton-Jacobi equation is formulated in an unconventional way.
To simulate a quantum system with continuous degrees of freedom on a quantum computer based on quantum digits, it is necessary to reduce continuous observables (primarily coordinates and momenta) to discrete observables. We consider this problem based on expanding quantum observables in series in powers of two and three analogous to the binary and ternary representations of real numbers. The coefficients of the series ("digits") are, therefore, Hermitian operators. We investigate the corresponding quantum mechanical operators and the relations between them and show that the binary and ternary expansions of quantum observables automatically leads to renormalization of some divergent integrals and series (giving them finite values).
Functional Hamilton-Jacobi (HJ) equation, the central equation of the holographic renormalization group (HRG), functional Schrödinger equation, and generalized Wilson-Polchinski (WP) equation, the central equation of the functional renormalization group (FRG), are considered in $D$-dimensional coordinate and abstract (formal) spaces. Instead of extra coordinates or an FRG scale, a holographic scalar field $\varLambda$ is introduced. The extra coordinate (or scale) is obtained as the amplitude of delta-field or constant field configurations of $\varLambda$. A rigorous derivation of corresponding integro-differential equation hierarchies for Green functions (GFs) as well as the integration formula for functionals are given. Using the integration formula, the functional (arbitrary configuration of $\varLambda$) solution for the translation-invariant two-particle GF is obtained. For the delta-field and the constant field configurations of $\varLambda$ this solution is studied in detail. Separable solution for two-particle GF is briefly discussed. Then, rigorous derivation of the quantum HJ and the continuity functional equations from the functional Schrödinger equation as well as the semiclassical approximation are given. An iterative procedure for solving the functional Schrödinger equation is suggested. Translation-invariant solutions for various GFs (both hierarchies) on delta-field configuration of $\varLambda$ are obtained. In context of continuity equation and open quantum field systems an optical potential is briefly discussed. Modes coarse graining growth functional for WP functional is analyzed. An approximation scheme is proposed for the generalized WP equation. With an optimized regulator translation-invariant solutions for two-particle and four-particle amputated GFs from approximated WP hierarchy are found analytically.
The nonlinear quantization of the domain wall (relativistic membrane of codimension 1) is considered. The membrane dust equation is considered as an analogue of the Hamilton-Jacobi equation, which allows us to construct its quantum analogue. The resulting equation has the form of a nonlinear Klein-Fock-Gordon equation. It can be interpreted as the mean field approximation for a quantum domain wall. Dispersion relations are obtained for small perturbations (in a linear approximation). The group speed of perturbations does not exceed the speed of light. For perturbations propagating along the domain wall, in addition to the massless mode (as in the classical case), a massive one appears. The result may be interesting in condensed matter theory and in membrane quantization in superstring and supergravity theories.
The dynamical production of low-mass X-ray binaries and brighter cataclysmic variables (CVs) in dense globular clusters is well-established. We investigate how the X-ray emissivity of fainter X-ray binaries (principally CVs and coronally active binaries) varies between different environments. We compile calculations (largely from the literature) of the X-ray emissivity of old stellar populations, including open and globular clusters and several galaxies. We investigate three literature claims of unusual X-ray sources in low-density stellar populations. We show that a suggested quiescent neutron star in the open cluster NGC 6819 is a foreground M dwarf. We show that the suggested diffuse X-ray emission from an old nova shell in the globular cluster NGC 6366 is actually a background galaxy cluster. And we show that a suggested population of quiescent X-ray binaries in the Sculptor Dwarf Galaxy is mostly (perhaps entirely) background galaxies. We find that above densities of $10^4$ M$_{\odot}$/pc$^3$, the X-ray emissivity of globular clusters increases, due to dynamical production of X-ray emitting systems. Below this density, globular clusters have lower X-ray emissivity than the other populations, and we do not see a strong dependence of X-ray emissivity due to density effects. We find significant correlations between X-ray emissivity and binary fraction, metallicity, and density. Sampling these fits via bootstrap techniques gives less significant correlations, but confirms the effect of metallicity on low-density populations, and that of density on the full globular cluster sample.
Nonlocal QFT of one-component scalar field $φ$ in $D$-dimensional Euclidean spacetime is considered. The generating functional (GF) of complete Green functions $\mathcal{Z}$ as a functional of external source $j$, coupling constant $g$, and spatial measure $dμ$ is studied. An expression for GF $\mathcal{Z}$ in terms of the abstract integral over the primary field $φ$ is given. An expression for GF $\mathcal{Z}$ in terms of integrals over the primary field and separable Hilbert space (HS) is obtained by means of a separable expansion of the free theory inverse propagator $\hat{L}$ over the separable HS basis. The classification of functional integration measures $\mathcal{D}\left[φ\right]$ is formulated, according to which trivial and two nontrivial versions of GF $\mathcal{Z}$ are obtained. Nontrivial versions of GF $\mathcal{Z}$ are expressed in terms of $1$-norm and $0$-norm, respectively. The definition of the $0$-norm generator $\varPsi$ is suggested. Simple cases of sharp and smooth generators are considered. Expressions for GF $\mathcal{Z}$ in terms of integrals over the separable HS with new integrands are obtained. For polynomial theories $φ^{2n},\, n=2,3,4,\ldots,$ and for the nonpolynomial theory $\sinh^{4}φ$, integrals over the separable HS in terms of a power series over the inverse coupling constant $1/\sqrt{g}$ for both norms ($1$-norm and $0$-norm) are calculated. Critical values of model parameters when a phase transition occurs are found numerically. A generalization of the theory to the case of the uncountable integral over HS is formulated. A comparison of two GFs $\mathcal{Z}$, one in the case of uncountable HS integral and one obtained using the Parseval-Plancherel identity, is given.
The present issue of the series < > represents the Proceedings of the Students Training Contest Olympiad in Mathematical and Theoretical Physics and includes the statements and the solutions of the problems offered to the participants. The contest Olympiad was held on May 21st-24th, 2010 by Scientific Research Laboratory of Mathematical Physics of Samara State University, Steklov Mathematical Institute of Russia's Academy of Sciences, and Moscow Institute of Physics and Technology (State University) in cooperation. The present Proceedings is intended to be used by the students of physical and mechanical-mathematical departments of the universities, who are interested in acquiring a deeper knowledge of the methods of mathematical and theoretical physics, and could be also useful for the persons involved in teaching mathematical and theoretical physics.
There are two different methods to describe membrane (string) fluids, which use different field content. The relation between the methods is clarified by construction of combined method. Dirac membrane field appears naturally in new approach. It provides a possibility to consider new aspects of electrodynamics-type theories with electric and magnetic sources. The membrane fluid models automatically prohibit simulatenos existence of electric and magnetic currents. Possible applications to the dark energy problem are mentioned.
String (membrane) theory could be considered as degenerate case of relativistic continuous media theory. The paper presents models of media, which are continuous distributions of interacting membranes, strings or particles.
The generally adopted approach in theory of relativistic strings and membranes, is similar to use of Lagrange coordinates in continious media mechanics. One can use an alternative approach, which is similar to use of Euler coordinates. Under such approach the consideration of thick (delocalized) membranes is natural. Membrane kinematics, which coorespond to Euler coordinates is constructed. Variables, similar to Hamiltonian variables, are introduced by means of Legander transformation. The case of free membranes appears to be degenerate. The examples of exact solutions of Einstein equations with delocalized membranes are presented.
New exact solutions of Einstein equations which describe black hole with radial cosmic strings are constructed in the paper. The case of infinitely thin strings and the case of delocalized strings are considered. The case of delocalized strings allows generalization to dimensions greater than 4.
A model considered in the paper generalizes supergravity type model to the case of delocalized membrane sources. A generalization of intersecting p-brane solution with delocalized membranes is presented.
A model considered in the paper generalizes membrane theory to the case of delocalized membranes. The model admits covariant formulation, which involves no constraints. It generalizes the notion of membrane to the case of smooth distribution of non-intersecting membranes. A generalization of p-brane solution with delocalized membranes is presented.
A globally consistent treatment of linearized gravity in the Randall-Sundrum background with matter on the brane is formulated. Using a novel gauge, in which the transverse components of the metric are non-vanishing, the brane is kept straight. We analyze the gauge symmetries and identify the physical degrees of freedom of gravity. Our results underline the necessity for non-gravitational confinement of matter to the brane.
Recently a realization of the four-dimensional gravity on a brane in five-dimensional spacetime has been discussed. Randall and Sundrum have shown that the equation for the longitudinal components of the metric fluctuations admit a normalizable zero mode solution, which has been interpreted as the localized gravity on the brane. We point out that the equation for the transverse components of the metric fluctuations has a solutions which is not localized on the brane. This indicates that probably the effective theory is unstable or, in other words, actually it is not four-dimensional but five-dimensional. Perhaps a modification of the proposal by using matter fields can lead to the trapping of gravity to the brane.
A simple derivation of the bound on entropy is given and the holographic principle is discussed. We estimate the number of quantum states inside space region on the base of uncertainty relation. The result is compared with the Bekenstein formula for entropy bound, which was initially derived from the generalized second law of thermodynamics for black holes. The holographic principle states that the entropy inside a region is bounded by the area of the boundary of that region. This principle can be called the kinematical holographic principle. We argue that it can be derived from the dynamical holographic principle which states that the dynamics of a system in a region should be described by a system which lives on the boundary of the region. This last principle can be valid in general relativity because the ADM hamiltonian reduces to the surface term.