Geometric Quantization with Applications to Gromov-Witten Theory
This is an expository article on the techniques of quantization as they are applied to Gromov-Witten theory and related areas.
math.AG↗
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Parametrization of degenerate density matrices
This paper presents a parametrization of a degenerate density matrix. The problem needs to be approached first with a diagonalized form (the spectral representation) to deal with degeneracy. Such a form is useful for this parametrization in that the conditions to be a density matrix from a Hermitian matrix are applied only to a diagonal eigenvalue matrix, not a unitary matrix. Those conditions can be satisfied by parametrizing eigenvalues with squared spherical coordinates in dimension of the matrix. Degeneracy in eigenvalues brings symmetries between a eigenvalue matrix and a unitary matrix, which are realized in a form of a commuting unitary matrix, called a commutant. The associated redundant parameters in a unitary matrix have to be eliminated. It is realized in this paper that degrees of degeneracies can be defined as the total number of possible pairs of the same eigenvalues and one degree of degeneracy corresponds to one phase and one two dimensional rotation in a unitary matrix or a commutant. In this way all the degeneracies are identified and assigned to one phase-one rotation block. Therefore, a unitary matrix or a commutant is a product of these blocks and a general diagonal phase matrix. In physics a unitary matrix is often parametrized by rotation and phase matrices, called an angular representation here. There are many possible different phase configurations. It is often not a trivial matter whether a given parametrized unitary matrix is general. A simple diagram will be introduced to illustrate how to transform one phase configuration to another.
math-ph↗
The Thermal Aspects of Relativistic Quantum Field Theory as an Observational Window in a Deeper Layer of Quantum Space-Time or: Dirac's Revenge
In this paper we shall derive the thermal properties of the relativistic quantum vacuum from a more primordial underlying structure which shares some properties with the old Dirac-sea picture. We show in particular how the Tomita-KMS structure in RQFT is a consequence of the structure and dynamics of the underlying pattern of vacuum fluctuations. We explain the origin of the doubling phenomenon in thermofield theory and the duality symmetry between a local algebra of fields or observables and its commutant in RQFT and give an interpretation of the notion of thermal time.
hep-th↗
Algorithmic framework for group analysis of differential equations and its application to generalized Zakharov--Kuznetsov equations
In this paper, we explain in more details the modern treatment of the problem of group classification of (systems of) partial differential equations (PDEs) from the algorithmic point of view. More precisely, we revise the classical Lie--Ovsiannikov algorithm of construction of symmetries of differential equations, describe the group classification algorithm and discuss the process of reduction of (systems of) PDEs to (systems of) equations with smaller number of independent variables in order to construct invariant solutions. The group classification algorithm and reduction process are illustrated by the example of the generalized Zakharov--Kuznetsov (GZK) equations of form $u_t+(F(u))_{xxx}+(G(u))_{xyy}+(H(u))_x=0$. As a result, a complete group classification of the GZK equations is performed and a number of new interesting nonlinear invariant models which have non-trivial invariance algebras are obtained. Lie symmetry reductions and exact solutions for two important invariant models, i.e., the classical and modified Zakharov--Kuznetsov equations, are constructed. The algorithmic framework for group analysis of differential equations presented in this paper can also be applied to other nonlinear PDEs.
math-ph↗
Asymptotic formulas for spectral gaps and deviations of Hill and 1D Dirac operators
Let $L$ be the Hill operator or the one dimensional Dirac operator on the interval $[0,π].$ If $L$ is considered with Dirichlet, periodic or antiperiodic boundary conditions, then the corresponding spectra are discrete and for large enough $|n|$ close to $n^2 $ in the Hill case, or close to $n, \; n\in \mathbb{Z}$ in the Dirac case, there are one Dirichlet eigenvalue $μ_n$ and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $λ_n^-, \, λ_n^+ $ (counted with multiplicity). We give estimates for the asymptotics of the spectral gaps $γ_n = λ_n^+ - λ_n^-$ and deviations $ δ_n =μ_n - λ_n^+$ in terms of the Fourier coefficients of the potentials. Moreover, for special potentials that are trigonometric polynomials we provide precise asymptotics of $γ_n$ and $δ_n.$
math.SP↗
Revisiting Dirac and Schrödinger: A Proof Offered for the Non-Relativistic Time-Dependent Schrödinger Equation
In this paper, three plausible axioms together with two definitions are employed to build an axiomatic framework, and then with the help of the Dirac formalism, it is demonstrated that the time-dependent Schrödinger wave equation is no longer a postulate for the whole theory, but a theorem. Subsequently, a proof for the theorem is presented. The result implies that the other remaining axioms of the picture essentially involve the Schrödinger equation, and this consequence lets the whole theory become stronger.
quant-ph↗
Asymptotic stability for the Couette flow in the 2D Euler equations
In this expository note we discuss our recent work [arXiv:1306.5028] on the nonlinear asymptotic stability of shear flows in the 2D Euler equations of ideal, incompressible flow. In that work it is proved that perturbations to the Couette flow which are small in a suitable regularity class converge strongly in $L^2$ to a shear flow which is close to the Couette flow. Enstrophy is mixed to small scales by an almost linear evolution and is generally lost in the weak limit as t -> +/- infinity. In this note we discuss the most important physical and mathematical aspects of the result and the key ideas of the proof.
math.AP↗
Topological properties in topological insulators and superconductors
We give a self-contained and enriched review about topology properties in the rapidly growing field of topological states of matter (TSM). This review is mainly focus on the beautiful interplay of topology mathematics and condensed matter physics that issuing TSM. Fiber bundle theory is a powerful concept to describe the non-trivial topology properties underlying the physical system. So we briefly present some motivation of fiber bundle theory and following that several effective topological methods have been introduced to judge whether a fiber bundle is trivial or not. Next, we give some topological invariants that characterizes the non-trivial TSM in the non-interacting systems in all dimensions, which is called topological band theory. Following that, we review and generalize the topological response using topological field theory called Chern-Simons effective theory. Finally, the classification of free-fermion systems have been studied by loop space and K-theory.
math-ph↗
Symmetries of dynamical systems and convergent normal forms
It is shown that, under suitable conditions, involving in particular the existence of analytic constants of motion, the presence of Lie point symmetries can ensure the convergence of the transformation taking a vector field (or dynamical system) into normal form
math-ph↗
A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types
We study the Landau-Lifshitz model for the energy of multi-scale transition layers -- called "domain walls" -- in soft ferromagnetic films. Domain walls separate domains of constant magnetization vectors $m^\pm \in \mathbb{S}^2$ that differ by an angle $2α$. Assuming translation invariance tangential to the wall, our main result is the rigorous derivation of a reduced model for the energy of the optimal transition layer, which in a certain parameter regime confirms the experimental, numerical and physical predictions: The minimal energy splits into a contribution from an asymmetric, divergence-free core which performs a partial rotation in $\mathbb{S}^2$ by an angle $2θ$, and a contribution from two symmetric, logarithmically decaying tails, each of which completes the rotation from angle $θ$ to $α$ in $\mathbb{S}^1$. The angle $θ$ is chosen such that the total energy is minimal. The contribution from the symmetric tails is known explicitly, while the contribution from the asymmetric core is analyzed in [7]. Our reduced model is the starting point for the analysis of a bifurcation phenomenon from symmetric to asymmetric domain walls. Moreover, it allows for capturing asymmetric domain walls including their extended tails (which were previously inaccessible to brute-force numerical simulation).
math.AP↗
Magnetic Neumann Laplacian on a sharp cone
This paper is devoted to the spectral analysis of the Laplacian with constant magnetic field on a cone of aperture $α$ and Neumann boundary condition. We analyze the influence of the orientation of the magnetic field. In particular, for any orientation of the magnetic field, we prove the existence of discrete spectrum below the essential spectrum in the limit $α\to 0$ and establish a full asymptotic expansion for the $n$-th eigenvalue and the $n$-th eigenfunction.
math.AP↗
An Overview of the Relationship between Group Theory and Representation Theory to the Special Functions in Mathematical Physics
Advances in mathematical physics during the 20th century led to the discovery of a relationship between group theory and representation theory with the theory of special functions. Specifically, it was discovered that many of the special functions are (1) specific matrix elements of matrix representations of Lie groups, and (2) basis functions of operator representations of Lie algebras. By viewing the special functions in this way, it is possible to derive many of their properties that were originally discovered using classical analysis, such as generating functions, differential relations, and recursion relations. This relationship is of interest to physicists due to the fact that many of the common special functions, such as Hermite polynomials and Bessel functions, are related to remarkably simple Lie groups used in physics. Unfortunately, much of the literature on this subject remains inaccessible to undergraduate students. The purpose of this project is to research the existing literature and to organize the results, presenting the information in a way that can be understood at the undergraduate level. The primary objects of study will be the Heisenberg group and its relationship to the Hermite polynomials, as well as the Euclidean group in the plane and its relationship to the Bessel functions. The ultimate goal is to make the results relevant for undergraduate students who have studied quantum mechanics.
math-ph↗
Novel solutions to the tetrahedron equation
This is the English translation of the short note where the first nontrivial tetrahedron relation (solution of the Zamolodchikov tetrahedron equation) with variables on the edges was presented.
math-ph↗
Casimir Force of Fermions Coupled to Monopoles in Six Dimensional Spacetime
We calculate the Casimir force for a fermionic quantum field in a piston geometry with three parallel plates. The fermion satisfies bag boundary conditions on the plates and the spacetime is assumed to have compact extra dimensions. The calculation is performed in the cases where the extra space has toroidal and spherical topology. We are mainly interested in the case in which the fermion is coupled non-trivially to an extra dimensional defect, with a torus extra dimensional topological background. We found that in certain limits, the Casimir force corresponding to the defect-fermion system and to the sphere, has opposite sign, in reference to those corresponding to the toroidal extra dimensional spaces.
hep-th↗
The wave equation for level sets is not a Huygens' equation
Any surface can be foliated into equipotential hypersurfaces of the level sets. A current result is that the contours are the progressing wave fronts of a certain hyperbolic partial differential equation, a wave equation. It is connected with the gradient lines, as well as with a corresponding eikonal equation. The level of a surface point, seen as an additional coordinate, plays the central role in this treatment. A wave solution can be a sharp front. Here the validity of the Huygens' principle (HP) is of interest: there is no wake of the wave solutions in every dimension, if a special Cauchy initial value problem is posed. Additionally, there is no distinction into odd or even dimensions. To compare this with Hadamard's 'minor premise' for a strong HP, we calculate differential geometric objects like Christoffel symbols, curvature tensors and geodesic lines, to test the validity of the strong HP. However, for the differential equation for level sets, the main criteria are not fulfilled for the strong HP in the sense of Hadamard's 'minor premise'.
math.DG↗
A nonlocal connection between certain linear and nonlinear ordinary differential equations - Part II: Complex nonlinear oscillators
In this paper, we present a method to identify integrable complex nonlinear oscillator systems and construct their solutions. For this purpose, we introduce two types of nonlocal transformations which relate specific classes of nonlinear complex ordinary differential equations (ODEs) with complex linear ODEs, thereby proving the integrability of the former. We also show how to construct the solutions using the two types of nonlocal transformations with several physically interesting cases as examples.
nlin.SI↗
On nonlinear coherent states properties for electron-phonon dynamics
This work addresses a construction of a dual pair of nonlinear coherent states (NCS) in the context of changes of bases in the underlying Hilbert space for a model pertaining to the condensed matter physics, which obeys a $f$-deformed Heisenberg algebra. The existence and properties of reproducing kernel in the NCS Hilbert space are studied and discussed; the probability density and its dynamics in the basis of constructed coherent states are provided. A Glauber-Sudarshan $P$-representation of the density matrix and relevant issues related to the reproducing kernel properties are presented. Moreover, a NCS quantization of classical phase space observables is performed and illustrated in a concrete example of $q$-deformed coherent states. Finally, an exposition of quantum optical properties is given.
math-ph↗
Thermomechanics of hydrogen storage in metallic hydrides: modeling and analysis
A thermodynamically consistent mathematical model for hydrogen adsorption in metal hydrides is proposed. Beside hydrogen diffusion, the model accounts for phase transformation accompanied by hysteresis, swelling, temperature and heat transfer, strain, and stress. We prove existence of solutions of the ensuing system of partial differential equations by a carefully-designed, semi-implicit approximation scheme. A generalization for a drift-diffusion of multi-component ionized "gas" is outlined, too.
math-ph↗