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At least 883 records · Page 49Linked to original sources
arXiv2013

CMB and Random Flights: temperature and polarization in position space

The fluctuations in the temperature and polarization of the cosmic microwave background are described by a hierarchy of Boltzmann equations. In its integral form, this Boltzmann hierarchy can be converted from the usual Fourier-space base into a position-space and causal description. We show that probability densities for random flights play a key role in this description. The integral system can be treated as a perturbative series in the number of steps of the random flights, and the properties of random flight probabilities impose constraints on the domains of dependence. We show that, as a result of these domains, a Fourier-Bessel decomposition can be employed in order to calculate the random flight probability densities. We also illustrate how the H-theorem applies to the cosmic microwave background: by using analytical formulae for the asymptotic limits of these probability densities, we show that, as the photon distribution approaches a state of equilibrium, both the temperature anisotropies and the net polarization must vanish.

astro-ph.CO↗
arXiv2013

New reflection matrices for the U_q(gl(m|n)) case

We examine super symmetric representations of the B-type Hecke algebra. We exploit such representations to obtain new non-diagonal solutions of the reflection equation associated to the super algebra U_q(gl(m|n)). The boundary super algebra is briefly discussed and it is shown to be central to the super symmetric realization of the B-type Hecke algebra

math-ph↗
arXiv2012

Bethe ansatz for the Temperley-Lieb spin-chain with integrable open boundaries

In this paper we study the spectrum of the spin-1 Temperley-Lieb spin chain with integrable open boundary conditions. We obtain the eigenvalue expressions as well as its associated Bethe ansatz equations by means of the coordinate Bethe ansatz. These equations provide the complete description of the spectrum of the model.

nlin.SI↗
arXiv2013

Bethe Ansatz solution of the small polaron with nondiagonal boundary terms

The small polaron with generic, nondiagonal boundary terms is investigated within the framework of quantum integrability. The fusion hierarchy of the transfer matrices and its truncation for particular values of the anisotropy parameter are both employed, so that the spectral problem is formulated in terms of a TQ equation. The solution of this equation for generic boundary conditions is based on a deformation of the diagonal case. The eigenvalues of the model are extracted and the corresponding Bethe Ansatz equations are presented. Finally, we comment on the eigenvectors of the model and explicitly compute the eigenstate of the model which evolves into the Fock vacuum when the off-diagonal boundary terms are switched off.

math-ph↗
arXiv2013

Decoupling of Field Equations in Einstein and Modified Gravity

This paper is concerned with giving the proof that there is a general decoupling property of vacuum and nonvacuum gravitational field equations in Einstein gravity and $f(R,T)$-modifications. The constructions are possible in terms of geometric and physical objects adapted to certain nonholonomic 2+2 splitting with local fibred structure. This allows us to generate exact and/or parametric depending solutions with generic off-diagonal metrics and generalized, or Levi-Civita, connections. Different classes of modified spacetimes are determined by corresponding generating and integration functions depending, in general, on all space and time coordinates and may possess, or not, Killing symmetries. The initial data sets for the Cauchy problem and their global properties are analyzed. There are formulated the criteria of evolution with spacetime splitting and decoupling of fundamental field equations. Examples of exact solutions defining ellipsoid deformations of black hole metrics and solitonic configurations are provided.

physics.gen-ph↗
arXiv2011

Addition theorems for spin spherical harmonics. I Preliminaries

We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch-Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II.

math-ph↗
arXiv2011

Addition theorems for spin spherical harmonics. II Results

Based on the results of part I, we obtain the general form of the addition theorem for spin spherical harmonics and give explicit results in the cases involving one spin-$s'$ and one spin-$s$ spherical harmonics with $s',s=1/2$, 1, 3/2, and $|s'-s|=0$, 1. We obtain also a fully general addition theorem for one scalar and one tensor spherical harmonic of arbitrary rank. A variety of bilocal sums of ordinary and spin spherical harmonics are given in explicit form, including a general explicit expression for bilocal spherical harmonics.

math-ph↗
arXiv2011

Eigenvalue repulsion estimates and some applications for the one-dimensional Anderson model

We show that the spacing between eigenvalues of the discrete 1D Hamiltonian with arbitrary potentials which are bounded, and with Dirichlet or Neumann Boundary Conditions is bounded away from zero. We prove an explicit lower bound, given by $Ce^{-bN}$, where $N$ is the lattice size, and $C$ and $b$ are some finite constants. In particular, the spectra of such Hamiltonians have no degenerate eigenvalues. As applications we show that to leading order in the coupling, the solution of a nonlinearly perturbed Anderson model in one-dimension (on the lattice) remains exponentially localized, in probability and average sense for initial conditions given by a unique eigenfunction of the linear problem. We also bound the derivative of the eigenfunctions of the linear Anderson model with respect to a potential change.

cond-mat.dis-nn↗
arXiv2012

Fluctuation geometry: A counterpart approach of inference geometry

Starting from an axiomatic perspective, \emph{fluctuation geometry} is developed as a counterpart approach of inference geometry. This approach is inspired on the existence of a notable analogy between the general theorems of \emph{inference theory} and the the \emph{general fluctuation theorems} associated with a parametric family of distribution functions $dp(I|θ)=ρ(I|θ)dI$, which describes the behavior of a set of \emph{continuous stochastic variables} driven by a set of control parameters $θ$. In this approach, statistical properties are rephrased as purely geometric notions derived from the \emph{Riemannian structure} on the manifold $\mathcal{M}_θ$ of stochastic variables $I$. Consequently, this theory arises as an alternative framework for applying the powerful methods of differential geometry for the statistical analysis. Fluctuation geometry has direct implications on statistics and physics. This geometric approach inspires a Riemannian reformulation of Einstein fluctuation theory as well as a geometric redefinition of the information entropy for a continuous distribution.

math.ST↗
arXiv2011

Thermodynamic limit of random partitions and dispersionless Toda hierarchy

We study the thermodynamic limit of random partition models for the instanton sum of 4D and 5D supersymmetric U(1) gauge theories deformed by some physical observables. The physical observables correspond to external potentials in the statistical model. The partition function is reformulated in terms of the density function of Maya diagrams. The thermodynamic limit is governed by a limit shape of Young diagrams associated with dominant terms in the partition function. The limit shape is characterized by a variational problem, which is further converted to a scalar-valued Riemann-Hilbert problem. This Riemann-Hilbert problem is solved with the aid of a complex curve, which may be thought of as the Seiberg-Witten curve of the deformed U(1) gauge theory. This solution of the Riemann-Hilbert problem is identified with a special solution of the dispersionless Toda hierarchy that satisfies a pair of generalized string equations. The generalized string equations for the 5D gauge theory are shown to be related to hidden symmetries of the statistical model. The prepotential and the Seiberg-Witten differential are also considered.

math-ph↗
arXiv2012

Heat Kernels and Zeta Functions on Fractals

On fractals, spectral functions such as heat kernels and zeta functions exhibit novel features, very different from their behaviour on regular smooth manifolds, and these can have important physical consequences for both classical and quantum physics in systems having fractal properties.

math-ph↗
arXiv2013

Resonances in sinh- and sine-Gordon models and higher equations of motion in Liouville theory

The notion of operator resonances was introduced earlier by Al. Zamolodchikov within the framework of the conformal perturbation theory. The resonances are related to logarithmic divergences of integrals in the perturbation expansion, and manifest themselves in poles of the correlation functions and form factors of local operators considered as functions of conformal dimensions. The residues of the poles can be computed by means of some operator identities. Here we study the resonances in the Liouville, sinh- and sine-Gordon models, considered as perturbations of a massless free boson. We show that the well-known higher equations of motion discovered by Al. Zamolodchikov in the Liouville field theory are nothing but resonance identities for some descendant operators. The resonance expansion in the vicinity of a resonance point provides a regularized version of the corresponding operators. We try to construct the corresponding resonance identities and resonance expansions in the sinh- and sine-Gordon theories. In some cases it can be done explicitly, but in most cases we are only able to obtain a general form so far. We show nevertheless that the resonances are perturbatively exact, which means that each of them only appears in a single term of the perturbation theory.

hep-th↗
arXiv2013

A topological state sum model for fermions on the circle

A simple state sum model for fermions on a 1-manifold is constructed. The model is independent of the triangulation and gives exactly the same partition function as the Dirac functional integral with zeta-function regularisation. Some implications for more realistic physical models are discussed.

math-ph↗
arXiv2013

WKB Approximation to the Power Wall

We present a semiclassical analysis of the quantum propagator of a particle confined on one side by a steeply, monotonically rising potential. The models studied in detail have potentials proportional to $x^α$ for $x>0$; the limit $α\to\infty$ would reproduce a perfectly reflecting boundary, but at present we concentrate on the cases $α=1$ and 2, for which exact solutions in terms of well known functions are available for comparison. We classify the classical paths in this system by their qualitative nature and calculate the contributions of the various classes to the leading-order semiclassical approximation: For each classical path we find the action $S$, the amplitude function $A$ and the Laplacian of $A$. (The Laplacian is of interest because it gives an estimate of the error in the approximation and is needed for computing higher-order approximations.) The resulting semiclassical propagator can be used to rewrite the exact problem as a Volterra integral equation, whose formal solution by iteration (Neumann series) is a semiclassical, not perturbative, expansion. We thereby test, in the context of a concrete problem, the validity of the two technical hypotheses in a previous proof of the convergence of such a Neumann series in the more abstract setting of an arbitrary smooth potential. Not surprisingly, we find that the hypotheses are violated when caustics develop in the classical dynamics; this opens up the interesting future project of extending the methods to momentum space.

math-ph↗
arXiv2012

Canonical bifurcation in higher derivative, higher spin, theories

We present a non-perturbative canonical analysis of the D=3 quadratic-curvature, yet ghost-free, model to exemplify a novel, "constraint bifurcation", effect. Consequences include a jump in excitation count: a linearized level gauge variable is promoted to a dynamical one in the full theory. We illustrate these results with their concrete perturbative counterparts. They are of course mutually consistent, as are perturbative findings in related models. A geometrical interpretation in terms of propagating torsion reveals the model's relation to an (improved) version of Einstein-Weyl gravity at the linearized level. Finally, we list some necessary conditions for triggering the bifurcation phenomenon in general interacting gauge systems.

hep-th↗
arXiv2013

Quadrature Uncertainty and Information Entropy of Quantum Elliptical Vortex States

We study the quadrature uncertainty of the quantum elliptical vortex state using the associated Wigner function. Deviations from the minimum uncertainty states were observed due to the absence of the Gaussian nature. In our study of the entropy, we noticed that with increasing vorticity, entropy increases for both the modes. We further observed that, there exists an optimum value of ellipticity which gives rise to maximum entanglement of the two modes of the quantum elliptical vortex states. A further increase in ellipticity reduces the entropy thereby resulting in a loss of information carrying capacity. We check the validity of the entropic inequality relations, namely the subaddivity and the Araki-Lieb inequality. The later was satisfied only for a very small range of the ellipticity of the vortex while the former seemed to be valid at all values.

quant-ph↗
arXiv2013

Modified melting crystal model and Ablowitz-Ladik hierarchy

This paper addresses the issue of integrable structure in a modified melting crystal model of topological string theory on the resolved conifold. The partition function can be expressed as the vacuum expectation value of an operator on the Fock space of 2D complex free fermion fields. The quantum torus algebra of fermion bilinears behind this expression is shown to have an extended set of "shift symmetries". They are used to prove that the partition function (deformed by external potentials) is essentially a tau function of the 2D Toda hierarchy. This special solution of the 2D Toda hierarchy can be characterized by a factorization problem of $\ZZ\times\ZZ$ matrices as well. The associated Lax operators turn out to be quotients of first order difference operators. This implies that the solution of the 2D Toda hierarchy in question is actually a solution of the Ablowitz-Ladik (equivalently, relativistic Toda) hierarchy. As a byproduct, the shift symmetries are shown to be related to matrix-valued quantum dilogarithmic functions.

math-ph↗