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Mathematical Physics (math-ph) updates on the e-print archive

Published: 2018-02-20T20:30:00-05:00


Magnetic monopoles in noncommutative quantum mechanics 2. (arXiv:1802.06854v1 [math-ph])

In this paper we extend the analysis of magnetic monopoles in quantum mechanics in three dimensional rotationally invariant noncommutative space $\textbf{R}^3_\lambda$. We construct the model step-by-step and observe that physical objects known from previous studies appear in a very natural way. Nonassociativity became a topic of great interest lately, often in a connection with magnetic monopoles. We show that this model does not possess this property.

BFKL Spectrum of N=4 SYM: non-Zero Conformal Spin. (arXiv:1802.06908v1 [hep-th])

We developed a general non-perturbative framework for the BFKL spectrum of planar $\mathcal{N}=4$ SYM, based on the Quantum Spectral Curve (QSC). It allows one to study the spectrum in the whole generality, extending previously known methods to arbitrary values of conformal spin $n$. We show how to apply our approach to reproduce all known perturbative results for the Balitsky-Fadin-Kuraev-Lipatov (BFKL) pomeron dimension and get new predictions. In particular, we re-derived the Faddeev-Korchemsky Baxter equation for the Lipatov spin chain with non-zero conformal spin reproducing the corresponding BFKL kernel eigenvalue. We also get new non-perturbative analytic results for the pomeron dimension in the vicinity of $|n|=1,\;\Delta=0$ point and we obtained an explicit formula for the BFKL intercept function for arbitrary conformal spin up to the 3-loop order in the small coupling expansion and partial result at the 4-loop order. We also implemented the numerical algorithm of arXiv:1504.06640 as an auxiliary file to this arXiv submission. From the numerical result we managed to deduce an analytic formula for the strong coupling expansion of the intercept function for arbitrary conformal spin.

Leontovich boundary condition and associated surface waves. (arXiv:1802.06921v1 [math-ph])

We analyse waves that propagate along the boundary between a dielectric half-space and a half-space filled with a Lorentz material. We show that the corresponding interface condition has the form of a classical Leontovich condition on the boundary of a dielectric in a half-space problem. We study when this condition supports propagation of (dispersive) surface waves. We analyse the related dispersion relation for waves propagating along the boundary of a stratified half-space and determine the loss values at which is supports surface waves for a given frequency-wavenumber pair.

Double Field Theory and Membrane Sigma-Models. (arXiv:1802.07003v1 [hep-th])

We investigate geometric aspects of double field theory (DFT) and its formulation as a doubled membrane sigma-model. Starting from the standard Courant algebroid over the phase space of an open membrane, we determine a splitting and a projection to a subbundle that sends the Courant algebroid operations to the corresponding operations in DFT. This describes precisely how the geometric structure of DFT lies in between two Courant algebroids and is reconciled with generalized geometry. We construct the membrane sigma-model that corresponds to DFT, and demonstrate how the standard T-duality orbit of geometric and non-geometric flux backgrounds is captured by its action functional in a unified way. This also clarifies the appearence of noncommutative and nonassociative deformations of geometry in non-geometric closed string theory. Gauge invariance of the DFT membrane sigma-model is compatible with the flux formulation of DFT and its strong constraint, whose geometric origin is explained. Our approach leads to a new generalization of a Courant algebroid, that we call a DFT algebroid and relate to other known generalizations, such as pre-Courant algebroids and symplectic nearly Lie 2-algebroids. We also describe the construction of a gauge-invariant doubled membrane sigma-model that does not require imposing the strong constraint.

The damped wave equation with unbounded damping. (arXiv:1802.07026v1 [math.SP])

We analyze new phenomena arising in linear damped wave equations on unbounded domains when the damping is allowed to become unbounded at infinity. We prove the generation of a contraction semigroup, study the relation between the spectra of the semigroup generator and the associated quadratic operator function, the convergence of non-real eigenvalues in the asymptotic regime of diverging damping on a subdomain, and we investigate the appearance of essential spectrum on the negative real axis. We further show that the presence of the latter prevents exponential estimates for the semigroup and turns out to be a robust effect that cannot be easily canceled by adding a positive potential. These analytic results are illustrated by examples.

Existence, decay time and light yield for a reaction-diffusion-drift equation in the continuum physics of scintillators. (arXiv:1802.07053v1 [math-ph])

A scintillator is a material which converts incoming ionizing energy into visible light. This conversion process, which is a strongly nonlinear one, can be described by a Reaction-Diffusion-Drift equation we obtain from a model of continua with microstructure endowed with a suitable thermodynamics. For such an equation it can be show the global existence of renormalizable and weak solutions, and the solutions exponential decay estimates can be given; moreover we give also a mathematical definition for the light yield which is a measure of scintillation efficiency.

Extended $r$-spin theory and the mirror symmetry for the $A_{r-1}$-singularity. (arXiv:1802.07075v1 [math.AG])

By a famous result of K. Saito, the parameter space of the miniversal deformation of the $A_{r-1}$-singularity carries a Frobenius manifold structure. The Landau--Ginzburg mirror symmetry says that, in the flat coordinates, the potential of this Frobenius manifold is equal to the generating series of certain integrals over the moduli space of $r$-spin curves. In this paper we show that the parameters of the miniversal deformation, considered as functions of the flat coordinates, also have a simple geometric interpretation using the extended $r$-spin theory, first considered by T. J. Jarvis, T. Kimura and A. Vaintrob, and studied in a recent paper of E. Clader, R. J. Tessler and the author. We prove a similar result for the singularity $D_4$ and present conjectures for the singularities $E_6$ and $E_8$.

Comment on the Bekenstein bound. (arXiv:1802.07184v1 [math-ph])

We propose a rigorous derivation of the Bekenstein upper limit for the entropy/information that can be contained by a physical system in a given finite region of space with given finite energy. The starting point is the observation that the derivation of such a bound provided by Casini [6] is similar to the description of the black hole incremental free energy that had been given by the first named author [23]. The approach here is different but close in the spirit to [6]. Our bound is obtained by operator algebraic methods, in particular Connes' bimodules, Tomita-Takesaki modular theory and Jones' index are essential ingredients inasmuch as the von Neumann algebras in question are typically of type III. We rely on the general mathematical framework, recently set up in [26], concerning quantum information of infinite systems.

The anisotropic \lambda-deformed SU(2) model is integrable. (arXiv:1412.5181v4 [hep-th] UPDATED)

The all-loop anisotropic Thirring model interpolates between the WZW model and the non-Abelian T-dual of the anisotropic principal chiral model. We focus on the SU(2) case and we prove that it is classically integrable by providing its Lax pair formulation. We derive its underlying symmetry current algebra and use it to show that the Poisson brackets of the spatial part of the Lax pair, assume the Maillet form. In this way we procure the corresponding r and s matrices which provide non-trivial solutions to the modified Yang-Baxter equation.

Quantum Yang-Mills Theory in Two Dimensions: Exact versus Perturbative. (arXiv:1508.06305v6 [math-ph] UPDATED)

The standard Feynman diagrammatic approach to quantum field theories assumes that perturbation theory approximates the full quantum theory at small coupling even when a mathematically rigorous construction of the latter is absent. On the other hand, two-dimensional Yang-Mills theory is a rare (if not the only) example of a nonabelian (pure) gauge theory whose full quantum theory has a rigorous construction. Indeed, the theory can be formulated via a lattice approximation, from which Wilson loop expecation values in the continuum limit can be described in terms of heat kernels on the gauge group. It is therefore fundamental to investigate how the exact answer for 2D Yang-Mills compares with that of the continuum perturbative approach, which a priori are unrelated. In this paper, we provide a mathematically rigorous formulation of the perturbative quantization of 2D Yang-Mills, and we consider perturbative Wilson loop expectation values on $\mathbb{R}^2$ and $S^2$ in Coulomb gauge, holomorphic gauge, and axial gauge (on $\mathbb{R}^2$). We show the following equivalences and nonequivalences between these gauges: (i) Coulomb and holomorphic gauge are equivalent and are independent of the choice of gauge-fixing metric; (ii) both are inequivalent with axial-gauge. Additionally, we show that the asymptotics of exact lattice Wilson loop expectations on $S^2$ agree with perturbatively computed expectations in holomorphic gauge for simple closed curves to all orders. However, as a consequence of (ii), this result is necessarily false on $\mathbb{R}^2$. Our work therefore presents fundamental progress in the analysis of how continuum perturbation theory succeeds or fails in capturing the asymptotics of the continuum limit of the lattice theory.

The Bogoliubov free energy functional II. The dilute limit. (arXiv:1511.05953v3 [math-ph] UPDATED)

We analyse the canonical Bogoliubov free energy functional at low temperatures in the dilute limit. We prove existence of a first order phase transition and, in the limit $a_0\to a$, we determine the critical temperature to be $T_{\rm{c}}=T_{\rm{fc}}(1+1.49(\rho^{1/3}a))$ to leading order. Here, $T_{\rm{fc}}$ is the critical temperature of the free Bose gas, $\rho$ is the density of the gas, $a$ is the scattering length of the pair-interaction potential $V$, and $a_0=(8\pi)^{-1}\widehat{V}(0)$ its first order approximation. We also prove asymptotic expansions for the free energy. In particular, we recover the Lee-Huang-Yang formula in the limit $a_0\to a$.

Non-Archimedean Reaction-Ultradiffusion Equations and Complex Hierarchic Systems. (arXiv:1604.06471v2 [math-ph] UPDATED)

We initiate the study of non-Archimedean reaction-ultradiffusion equations and their connections with models of complex hierarchic systems. From a mathematical perspective, the equations studied here are the p-adic counterpart of the integro-differential models for phase separation introduced by Bates and Chmaj. Our equations are also generalizations of the ultradiffusion equations on trees studied in the 80's by Ogielski, Stein, Bachas, Huberman, among others, and also generalizations of the master equations of the Avetisov et al. models, which describe certain complex hierarchic systems. From a physical perspective, our equations are gradient flows of non-Archimedean free energy functionals and their solutions describe the macroscopic density profile of a bistable material whose space of states has an ultrametric structure. Some of our results are p-adic analogs of some well-known results in the Archimedean settting, however, the mechanism of diffusion is completely different due to the fact that it occurs in an ultrametric space.

On the reduced dynamics of a subset of interacting bosonic particles. (arXiv:1607.07211v2 [quant-ph] UPDATED)

The quantum dynamics of a subset of interacting bosons in a subspace of fixed particle number is described in terms of symmetrized many-particle states. A suitable partial trace operation over the von Neumann equation of an $N$-particle system produces a hierarchical expansion for the subdynamics of $M\leq N$ particles. Truncating this hierarchy with a pure product state ansatz yields the general, nonlinear coherent mean-field equation of motion. In the special case of a contact interaction potential, this reproduces the Gross-Pitaevskii equation. To account for incoherent effects on top of the mean-field evolution, we discuss possible extensions towards a second-order perturbation theory that accounts for interaction-induced decoherence in form of a nonlinear Lindblad-type master equation.

Stochastic Feynman Rules for Yang-Mills Theory on the Plane. (arXiv:1607.07463v3 [math-ph] UPDATED)

We analyze quantum Yang-Mills theory on $\mathbb{R}^2$ using a novel discretization method based on an algebraic analogue of stochastic calculus. Such an analogue involves working with "Gaussian" free fields whose covariance matrix is indefinite rather than positive definite. Specifically, we work with Lie-algebra valued fields on a lattice and exploit an approximate gauge-invariance that is restored when taking the continuum limit. This analysis is applied to show the equivalence between Wilson loop expectations computed using partial axial-gauge, complete axial-gauge, and the Migdal-Witten lattice formulation. As a consequence, we obtain intriguing Lie-theoretic identities involving heat kernels and iterated integrals.

Self-adjointness and spectral properties of Dirac operators with magnetic links. (arXiv:1701.04987v3 [math-ph] UPDATED)

We define Dirac operators on $\mathbb{S}^3$ (and $\mathbb{R}^3$) with magnetic fields supported on smooth, oriented links and prove self-adjointness of certain (natural) extensions. We then analyze their spectral properties and show, among other things, that these operators have discrete spectrum. Certain examples, such as circles in $\mathbb{S}^3$, are investigated in detail and we compute the dimension of the zero-energy eigenspace.

Bethe-Boltzmann Hydrodynamics and Spin Transport in the XXZ Chain. (arXiv:1702.06146v4 [cond-mat.stat-mech] UPDATED)

Quantum integrable systems, such as the interacting Bose gas in one dimension and the XXZ quantum spin chain, have an extensive number of local conserved quantities that endow them with exotic thermalization and transport properties. We discuss recently introduced hydrodynamic approaches for such integrable systems from the viewpoint of kinetic theory and extend the previous works by proposing a numerical scheme to solve the hydrodynamic equations for finite times and arbitrary locally equilibrated initial conditions. We then discuss how such methods can be applied to describe non-equilibrium steady states involving ballistic heat and spin currents. In particular, we show that the spin Drude weight in the XXZ chain, previously accessible only by rigorous techniques of limited scope or controversial thermodynamic Bethe ansatz arguments, may be evaluated from hydrodynamics in very good agreement with density-matrix renormalization group calculations.

Nondegeneracy, Morse Index and Orbital Stability of the Lump Solution to the KP-I Equation. (arXiv:1703.09879v2 [math.AP] UPDATED)

Let $Q(x,y)= 4 \frac{y^2-x^2+3}{ (x^2+y^2+3)^2}$ be the lump solution of the KP-I equation $$ \partial_x^2 (\partial_x^2 u-u + 3 u^2)-\partial_y^2 u=0.$$ We show that this solution is rigid in the following senses: the only decaying solutions to the linearized operator $$ \partial_x^2 (\partial_x^2 \phi -\phi + 6 Q \phi)-\partial_y^2 \phi=0$$ consist of the linear combinations of $ \partial_x Q$ and $ \partial_y Q$. Furthermore we show that the Morse index is exactly one and that it is orbital stable.

Analysis of zero modes for Dirac operators with magnetic links. (arXiv:1705.02959v3 [math-ph] UPDATED)

In this paper we provide a means to approximate Dirac operators with magnetic fields supported on links in $\mathbb{S}^3$ (and $\mathbb{R}^3$) by Dirac operators with smooth magnetic fields. We then proceed to prove that under certain assumptions, the spectral flow of paths along these operators is the same in both the smooth and the singular case. We recently characterized the spectral flow of such paths in the singular case. This allows us to show the existence of new smooth, compactly supported magnetic fields in $\mathbb{R}^3$ for which the associated Dirac operator has a non-trivial kernel. Using Clifford analysis, we also obtain criteria on the magnetic link for the non-existence of zero modes.

Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei. (arXiv:1706.00700v3 [math-ph] UPDATED)

We derive a classification of the self-adjoint extensions of the three-dimensional Dirac-Coulomb operator in the critical regime of the Coulomb coupling. Our approach is solely based upon the Kre{\u\i}n-Vi\v{s}ik-Birman extension scheme, or also on Grubb's universal classification theory, as opposite to previous works within the standard von Neumann framework. This let the boundary condition of self-adjointness emerge, neatly and intrinsically, as a multiplicative constraint between regular and singular part of the functions in the domain of the extension, the multiplicative constant giving also immediate information on the invertibility property and on the resolvent and spectral gap of the extension.

Gauss-Bonnet for matrix conformally rescaled Dirac. (arXiv:1707.04438v2 [math.QA] UPDATED)

We derive an explicit formula for the scalar curvature over a two-torus with a Dirac operator conformally rescaled by a globally diagonalizable matrix. We show that the Gauss-Bonnet theorem holds and extend the result to all Riemann surfaces with Dirac operators modified in the same way.

Causal Fermion Systems: A Primer for Lorentzian Geometers. (arXiv:1709.04781v2 [math-ph] UPDATED)

We give a brief introduction to causal fermion systems with a focus on the geometric structures in space-time.

Exact results for a fractional derivative of elementary functions. (arXiv:1711.07126v4 [math-ph] UPDATED)

We present exact analytical results for the Caputo fractional derivative of a wide class of elementary functions, including trigonometric and inverse trigonometric, hyperbolic and inverse hyperbolic, Gaussian, quartic Gaussian, and Lorentzian functions. These results are especially important for multi-scale physical systems, such as porous materials, disordered media, and turbulent fluids, in which transport is described by fractional partial differential equations. The exact results for the Caputo fractional derivative are obtained from a single generalized Euler's integral transform of the generalized hyper-geometric function with a power-law argument. We present a proof of the generalized Euler's integral transform and directly apply it to the exact evaluation of the Caputo fractional derivative of a broad spectrum of functions, provided that these functions can be expressed in terms of a generalized hyper-geometric function with a power-law argument. We determine that the Caputo fractional derivative of elementary functions is given by the generalized hyper-geometric function. Moreover, we show that in the most general case the final result cannot be reduced to elementary functions, in contrast to both the Liouville-Caputo and Fourier fractional derivatives. However, we establish that in the infinite limit of the argument of elementary functions, all three definitions of a fractional derivative - the Caputo, Liouville-Caputo, and Fourier- converge to the same result given by the elementary functions. Finally, we prove the equivalence between Liouville-Caputo and Fourier fractional derivatives.

Families of spectral triples and foliations of space(time). (arXiv:1711.07299v2 [math-ph] UPDATED)

We study a noncommutative analogue of a space(time) foliated by (spacelike) hypersurfaces. First, in the classical (commutative) case, we show that the canonical Dirac operator on the total space(time) (with either Riemannian or Lorentzian signature) can be reconstructed from the family of Dirac operators on the (spacelike) hypersurfaces. Second, in the noncommutative case, the same construction continues to make sense for an abstract family of spectral triples, and we prove that (in the case of Riemannian signature) the construction yields in fact a spectral triple, which we call a product spectral triple. In the case of Lorentzian signature, the corresponding 'Lorentzian spectral triple' can also be viewed as the 'reverse Wick rotation' of such product spectral triples. This construction of 'Lorentzian spectral triples' fits well into the Krein space approach to noncommutative Lorentzian geometry.

Integrable Floquet dynamics, generalized exclusion processes and "fused" matrix ansatz. (arXiv:1711.08884v2 [math-ph] UPDATED)

We present a general method for constructing integrable stochastic processes, with two-step discrete time Floquet dynamics, from the transfer matrix formalism. The models can be interpreted as a discrete time parallel update. The method can be applied for both periodic and open boundary conditions. We also show how the stationary distribution can be built as a matrix product state. As an illustration we construct a parallel discrete time dynamics associated with the R-matrix of the SSEP and of the ASEP, and provide the associated stationary distributions in a matrix product form. We use this general framework to introduce new integrable generalized exclusion processes, where a fixed number of particles is allowed on each lattice site in opposition to the (single particle) exclusion process models. They are constructed using the fusion procedure of R-matrices (and K-matrices for open boundary conditions) for the SSEP and ASEP. We develop a new method, that we named "fused" matrix ansatz, to build explicitly the stationary distribution in a matrix product form. We use this algebraic structure to compute physical observables such as the correlation functions and the mean particle current.

On the relation between Staruszkiewicz's quantum theory of the Coulomb field and the causal perturbative approach to QED. (arXiv:1712.06093v3 [math-ph] UPDATED)

At the classical level the electromagnetic field can be well identified at the spatial infinity. Staruszkiewicz pointed out that the quantization of the electromagnetic field at spatial infinity is essentially unique and follows from the two fundamental principles: 1) gauge invariance and 2) canonical commutation relations for canonically conjugated generalized coordinates, and constructed a simple and mathematically transparent quantum theory of the Coulomb field, predicting (among other things) a relation between the theory of unitary representations of the $SL(2, \mathbb{C})$ group and the fine structure constant. Until now this theory has stayed outside the main stream of the perturbative development in QED, mainly due to the unsolved infrared-type (IR) problems in the perturbative approach. Recently however there has been performed a more careful analysis of mass less free gauge fields, such as the electromagnetic potential field, their Wick and chronological products, which revealed the need for a more careful construction of these fields, and which opened a way to resolve IR problems (at least those which shows up at each order separately). In particular the more careful definition of these fields reveals the need for the analysis of the free and the interacting fields in QED at spatial infinity which have been unnoticed until now because the zero mass gauge fields have been treated with insufficient care. A need for comparison of the perturbatively constructed field at spatial infinity with the quantum phase field of the Staruszkiewicz theory arises, which leads to the proof of universality of the unit of charge. We give here a commentary on these facts.

Towards an M5-Brane Model I: A 6d Superconformal Field Theory. (arXiv:1712.06623v2 [hep-th] UPDATED)

We present an action for a six-dimensional superconformal field theory containing a non-abelian tensor multiplet. All of the ingredients of this action have been available in the literature. We bring these pieces together by choosing the string Lie 2-algebra as a gauge structure, which we motivated in previous work. The kinematical data contains a connection on a categorified principal bundle, which is the appropriate mathematical description of the parallel transport of self-dual strings. Our action can be written down for each of the simply laced Dynkin diagrams, and each case reduces to a four-dimensional supersymmetric Yang--Mills theory with corresponding gauge Lie algebra. Our action also reduces nicely to an M2-brane model which is a deformation of the ABJM model. While this action is certainly not the desired M5-brane model, we regard it as a key stepping stone towards a potential construction of the (2,0)-theory.

Laughlin states on higher genus Riemann surfaces. (arXiv:1712.09980v2 [cond-mat.str-el] UPDATED)

Considering quantum Hall states on geometric backgrounds has proved over the past few years to be a useful tool for uncovering their less evident properties, such as gravitational and electromagnetic responses, topological phases and novel geometric adiabatic transport coefficients. One of the transport coefficients, the central charge associated with the gravitational anomaly, appears as a Chern number for the adiabatic transport on the moduli spaces of higher genus Riemann surfaces. This calls for a better understanding of the QH states on these backgrounds. Here we give a detailed account of the construction of Laughlin states on Riemann surfaces of genus ${\rm g}>1$. We present a rigorous definition of the Laughlin states and their explicit construction, based on the path integral for the 2d bosonic field compactified on a circle, reproducing the conjectured $\beta^{\rm g}$-fold degeneracy for the filling fraction $\nu=1/\beta$. We also discuss the lowest Landau level, integer QH state and its relation to the bosonization formulas on higher genus Riemann surfaces.

Spectral analysis of the 2+1 fermionic trimer with contact interactions. (arXiv:1712.10209v4 [math-ph] UPDATED)

We qualify the main features of the spectrum of the Hamiltonian of point interaction for a three-dimensional quantum system consisting of three point-like particles, two identical fermions, plus a third particle of different species, with two-body interaction of zero range. For arbitrary magnitude of the interaction, and arbitrary value of the mass parameter (the ratio between the mass of the third particle and that of each fermion) above the stability threshold, we identify the essential spectrum, localise the discrete spectrum and prove its finiteness, qualify the angular symmetry of the eigenfunctions, and prove the increasing monotonicity of the eigenvalues with respect to the mass parameter. We also demonstrate the existence or absence of bound states in the physically relevant regimes of masses.

Boson-fermion duality in three dimensions. (arXiv:1802.06087v2 [cond-mat.str-el] UPDATED)

We study the 2+1 dimensional boson-fermion duality in the presence of background curvature and electromagnetic fields. The main players are, on the one hand, free massive complex scalar fields coupled to U(1) Maxwell-Chern-Simons gauge fields at Chern-Simons levels $\pm1$, representing relativistic composite bosons with one unit of attached flux, and on the other hand, free massive Dirac fermions. We prove, in a curved background and at the level of the partition function, that a doublet of relativistic composite bosons, in the infinite coupling limit, is dual to a doublet of Dirac fermions. The spin connection arises from the expectation value of the Wilson loop in the Chern-Simons theory, whereas a non-minimal coupling of bosons to the scalar curvature is necessary in order to obtain agreement between partition functions. Remarkably, we find that the correspondence does not hold in the presence of background electromagnetic fields, a pathology rooted to the coupling of electromagnetism to the spin angular momentum of the Dirac spinor, which can not be reproduced from minimal coupling in the bosonic side. The presence of framing and parity anomalies in the Chern-Simons and fermionic theories, respectively, poses a difficulty in realizing the duality as an exact mapping between partition functions. The existence of non matching anomalies is circumvented by the Dirac fermions coming in pairs, making the fermionic theory parity anomaly free, and by the inclusion of a Maxwell term in the bosonic side, acting as a regulator forcing the CS theory to be quantized in a non-topological way. The Coulomb interaction stemming from the Maxwell term is also of key importance to prevent intersections of worldlines in the path integral. An extension of the duality to the massless case fails if bosons and fermions are in a topological phase, but is possible when the mapping is between trivial theories.