Letters in Mathematical Physics最新文献

Given a homomorphism (tau ) from a suitable finite group ({mathsf {Gamma }}) to (textsf{SU}(4)) with image ({mathsf {Gamma }}^tau ), we construct a cohomological gauge theory on a non-commutative resolution of the quotient singularity (mathbbm {C}^4/{mathsf {Gamma }}^tau ) whose BRST fixed points are ({mathsf {Gamma }})-invariant tetrahedron instantons on a generally non-effective orbifold. The partition function computes the expectation values of complex codimension one defect operators in rank r cohomological Donaldson–Thomas theory on a flat gerbe over the quotient stack ([mathbbm {C}^4/,{mathsf {Gamma }}^tau ]). We describe the generalized ADHM parametrization of the tetrahedron instanton moduli space and evaluate the orbifold partition functions through virtual torus localization. If ({mathsf {Gamma }}) is an abelian group the partition function is expressed as a combinatorial series over arrays of ({mathsf {Gamma }})-coloured plane partitions, while if ({mathsf {Gamma }}) is non-abelian the partition function localizes onto a sum over torus-invariant connected components of the moduli space labelled by lower-dimensional partitions. When ({mathsf {Gamma }}=mathbbm {Z}_n) is a finite abelian subgroup of (textsf{SL}(2,mathbbm {C})), we exhibit the reduction of Donaldson–Thomas theory on the toric Calabi–Yau four-orbifold (mathbbm {C}^2/,{mathsf {Gamma }}times mathbbm {C}^2) to the cohomological field theory of tetrahedron instantons, from which we express the partition function as a closed infinite product formula. We also use the crepant resolution correspondence to derive a closed formula for the partition function on any polyhedral singularity.

A Lorentz-covariant system of wave equations is formulated for a quantum-mechanical three-body system in one space dimension, comprised of one photon and two identical massive spin one-half Dirac particles, which can be thought of as two electrons (or alternatively, two positrons). Manifest covariance is achieved using Dirac’s formalism of multi-time wave functions, i.e., wave functions (Psi ({textbf {x}}_{text {ph}},{textbf {x}}_{text {e}_1},{textbf {x}}_{text {e}_2})) where ({textbf {x}}_{text {ph}},{textbf {x}}_{text {e}_1},{textbf {x}}_{text {e}_2}) are generic spacetime events of the photon and two electrons, respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifolds ({{textbf {x}}_{text {ph}}={textbf {x}}_{text {e}_1}}) and ({{textbf {x}}_{text {ph}}={textbf {x}}_{text {e}_2}}) compatible with conservation of probability current. The corresponding initial-boundary value problem is shown to be well-posed, and it is shown that the unique solution can be represented by a convergent infinite sum of Feynman-like diagrams, each one corresponding to the photon bouncing between the two electrons a fixed number of times.

In this work, we analyze the spectral (zeta )-function associated with the self-adjoint extensions, (T_{A,B}), of quasi-regular Sturm–Liouville operators that are bounded from below. By utilizing the Green’s function formalism, we find the characteristic function, which implicitly provides the eigenvalues associated with a given self-adjoint extension (T_{A,B}). The characteristic function is then employed to construct a contour integral representation for the spectral (zeta )-function of (T_{A,B}). By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the (zeta )-function to a larger region of the complex plane. We also present a method for computing the value of the spectral (zeta )-function of (T_{A,B}) at all positive integers. We provide two examples to illustrate the methods developed in the paper: the generalized Bessel and Legendre operators. We show that in the case of the generalized Bessel operator, the spectral (zeta )-function develops a branch point at the origin, while in the case of the Legendre operator it presents, more remarkably, branch points at every nonpositive integer value of s.

This paper continues the study of explicit asymptotic formulas for standing coastal trapped waves, focusing on the spectral properties of the operator (langle nabla , D(x)nabla rangle ), which is the spatial component of the wave operator with a degenerating wave propagation velocity. We aim to construct spectral series—pairs of asymptotic eigenvalues and formal asymptotic eigenfunctions—corresponding to the high-frequency regime, where the eigenvalue is (varvec{omega }rightarrow infty ). Extending earlier results, this study addresses the nearly integrable case, providing a more detailed asymptotic behavior of eigenfunctions. Depending on their domain of localization, these eigenfunctions can be expressed in terms of Airy functions and their derivatives or Bessel functions. In addition, we introduce a canonical operator with violated (imprecisely satisfied) quantization conditions.

Motivated by the simplest case of tt*-Toda equations, we study the large and small x asymptotics for ( x>0 ) of real solutions of the sinh-Godron Painlevé III((D_6)) equation. These solutions are parametrized through the monodromy data of the corresponding Riemann–Hilbert problem. This unified approach provides connection formulae between the behavior at the origin and infinity of the considered solutions.

We use the Hamiltonian reduction method to construct the Ruijsenaars dual systems to generalized Toda chains associated with the classical Lie algebras of types (B, C, D). The dual systems turn out to be the B, C and D analogues of the rational goldfish model, which is, as in the type A case, the strong coupling limit of rational Ruijsenaars systems. We explain how both types of systems emerge in the reduction of the cotangent bundle of a Lie group and provide the formulae for dual Hamiltonians. We compute explicitly the higher Hamiltonians of goldfish models using the Cauchy–Binet theorem.

The Stone–von Neumann theorem is a fundamental result which unified the competing quantum-mechanical models of matrix mechanics and wave mechanics. In this article, we continue the broad generalization set out by Huang and Ismert and by Hall, Huang, and Quigg, analyzing representations of locally compact quantum-dynamical systems defined on Hilbert modules, of which the classical result is a special case. We introduce a pair of modular representations which subsume numerous models available in the literature and, using the classical strategy of Rieffel, prove a Stone–von Neumann-type theorem for maximal actions of regular locally compact quantum groups on elementary C*-algebras. In particular, we generalize the Mackey–Stone–von Neumann theorem to regular locally compact quantum groups whose trivial actions on (mathbb {C}) are maximal and recover the multiplicity results of Hall, Huang, and Quigg. With this characterization in hand, we prove our main result showing that if a dynamical system ((mathbb {G},A,alpha )) satisfies the multiplicity assumption of the generalized Stone–von Neumann theorem, and if the coefficient algebra A admits a faithful state, then the spectrum of the iterated crossed product (widehat{mathbb {G}}^textrm{op}ltimes (mathbb {G}ltimes A)) consists of a single point. In the case of a separable coefficient algebra or a regular acting quantum group, we further characterize features of this system, and thus obtain a partial converse to the Stone–von Neumann theorem in the quantum group setting. As a corollary, we show that a regular locally compact quantum group satisfies the generalized Stone–von Neumann theorem if and only if it is strongly regular.

There exist several good reasons why one may wish to add a total derivative to an interaction in quantum field theory, e.g., in order to improve the perturbative construction. Unlike in classical field theory, adding derivatives in general changes the theory. The analysis whether and how this can be prevented is presently limited to perturbative orders (g^n), (nle 3). We drastically simplify it by an all-orders formula, which also allows to answer some salient structural questions. The method is part of a larger program to (re)derive interactions of particles by quantum consistency conditions, rather than a classical principle of gauge invariance.

In this paper, we mainly investigate Lax structure and tau function for the large BKP hierarchy, which is also known as Toda hierarchy of B type. Firstly, the large BKP hierarchy can be derived from fermionic BKP hierarchy by using a special bosonization, which is presented in the form of bilinear equation. Then from bilinear equation, the corresponding Lax equation is given, where in particular the relation of flow generator with Lax operator is obtained. Also starting from Lax equation, the corresponding bilinear equation and existence of tau function are discussed. After that, large BKP hierarchy is viewed as sub-hierarchy of modified Toda (mToda) hierarchy, also called two-component first modified KP hierarchy. Finally by using two basic Miura transformations from mToda to Toda, we understand two typical relations between large BKP tau function (tau _n(textbf{t})) and Toda tau function (tau _n^textrm{Toda}(textbf{t},-textbf{t})), that is, (tau _n^{textrm{Toda}}(textbf{t},-{textbf{t}})=tau _n(textbf{t})tau _{n-1}(textbf{t})) and (tau _n^{textrm{Toda}}(textbf{t},-{textbf{t}})=tau _n^2(textbf{t})). Further, we find (big (tau _n(textbf{t})tau _{n-1}(textbf{t}),tau _n^2(textbf{t})big )) satisfies bilinear equation of mToda hierarchy.