三角Calogero-Moser模型的非动力学r矩阵结构

通过一定规范变换,构造了三角Calogero–Moser模型一种新的Lax算子,使其具有相应的非动力学r矩阵结构.同时发现该r矩阵结构与三角Ruijsenaars–Schneider模型的r矩阵完全相同.     

The Nondynamical r-Matrix Structure of the Elliptic Calogero–Moser Model

In this Letter, we construct a new Lax operator for the elliptic Calogero–Moser model with N=2. The nondynamical r-matrix structure of this Lax operator is also studied. The relation between our Lax operator and the Lax operator given by Krichever is also obtained.     

Gauge Transformation for BCr-KP Hierarchy and Its Compatibility with Additional Symmetry

The BCr-KP hierarchy is an important sub-hierarchy of the KP hierarchy. In this paper, the BCr-KP hierarchy is investigated from three aspects. Firstly, we study the gauge transformation for the BCr-KP hierarchy.Different from the KP hierarchy, the gauge transformation must keep the constraint of the BCr-KP hierarchy. Secondly,we study the gauge transformation for the constrained BCr-KP hierarchy. In this case, the constraints of the Lax operator must be invariant under the gauge transformation. At last, the compatibility between the additional symmetry and the gauge transformation for the BCr-KP hierarchy is explored.     

Integrable boundary conditions and modified Lax equations

We consider integrable boundary conditions for both discrete and continuum classical integrable models. Local integrals of motion generated by the corresponding “transfer” matrices give rise to time evolution equations for the initial Lax operator. We systematically identify the modified Lax pairs for both discrete and continuum boundary integrable models, depending on the classical r-matrix and the boundary matrix.     

The bi-Hamiltonian structure of fully supersymmetric Korteweg-de Vries systems

The bi-Hamiltonian structure of integrable supersymmetric extensions of the Korteweg-de Vries (KdV) equation related to theN=1 and theN=2 superconformal algebras is found. It turns out that some of these extensions admit inverse Hamiltonian formulations in terms of presymplectic operators rather than in terms of Poisson tensors. For one extension related to theN=2 case additional symmtries are found with bosonic parts that cannot be reduced to symmetries of the classical KdV. They can be explained by a factorization of the corresponding Lax operator. All the bi-Hamiltonian formulations are derived in a systematic way from the Lax operators.     

Minimal Coupling in Koopman-von Neumann Theory

Classical mechanics (CM), like quantum mechanics (QM), can have an operatorial formulation. This was pioneered by Koopman and von Neumann (KvN) in the 1930s. They basically formalized, via the introduction of a classical Hilbert space, earlier work of Liouville who had shown that the classical time evolution can take place via an operator, nowadays known as the Liouville operator. In this paper we study how to perform the coupling of a point particle to a gauge field in the KvN version of CM. So we basically implement at the classical operatorial level the analog of the minimal coupling of QM. We show that, differently than in QM, not only the momenta but also other variables have to be coupled to the gauge field. We also analyze in detail how the gauge invariance manifests itself in the Hilbert space of KvN and indicate the differences with QM. As an application of the KvN method we study the Landau problem proving that there are many more degeneracies at the classical operatorial level than at the quantum one. As a second example we go through the Aharonov-Bohm phenomenon showing that, at the quantum level, this phenomenon manifests its effects on the spectrum of the quantum Hamiltonian while at the classical level there is no effect whatsoever on the spectrum of the Liouville operator.     

Symmetry Reductions of a Lax Pair for （2＋1）-Dimensional Differential Sawada-Kotera Equation

Based on the symbolic computational system Maple, the similarity reductions of a Lax pair for the （2＋1 ）-dimensional differential Sawada Kotera （SK） equation by the classical Lie point group method, are presented. We obtain several interesting reductions. Comparing the reduced Lax pair＇s compatibility with the reduced SK equation under the same symmetry group, we find that the reduced Lax pairs do not always lead to the reduced SK equation. In general, the reduced equations are the subsets of the compatibility conditions of the reduced Lax pair.     

Gauge Transformation and Generalized Nonlocal Symmetries of the Korteweg-de Vries Equation

After extending the usual Lax pair of the Korteweg-de Vries (KdV) equation to a generalized form by using a gauge transformation, an adjoint Lax pair of the KdV equation is introduced. With the help of the spectral functions of the Lax pair and the adjoint Lax pair, a new nonlocal seed symmetry (which is gauge-invariant) is found and then a set of new infinitely many generalized nonlocal symmetries are obtained after establishing a general symmetry theory for an arbitrary nonlinear system.     

The CMS experiment at the LHC-status and physics potential

The most general momentum independent dynamical r-matrices are described for the standard Lax representation of the degenerate Calogero-Moser models based on gl _n and those r-matrices whose dynamical dependence can be gauge d away are selected. In the rational case, a non-dynamical r-matrix resulting from gauge transformation is given explicitly as an antisymmetric solution of the classical Yang-Baxter equation that belongs to the Frobenius subalgebra of gl _n consisting of the matrices with vanishing last row.     

A model of unified quantum chromodynamics and Yang-Mills gravity

Based on a generalized Yang-Mills framework, gravitational and strong interactions can be unified in analogy with the unification in the electroweak theory. By gauging T (4) × [SU (3)] color in flat space-time, we have a unified model of chromo-gravity with a new tensor gauge field, which couples universally to all gluons, quarks and anti-quarks. The space-time translational gauge symmetry assures that all wave equations of quarks and gluons reduce to a Hamilton-Jacobi equation with the same ’effective Riemann metric tensors’ in the geometric-optics (or classical) limit. The emergence of ef f ective metric tensors in the classical limit is essential for the unified model to agree with experiments. The unified model suggests that all gravitational, strong and electroweak interactions appear to be dictated by gauge symmetries in the generalized Yang-Mills framework.     

The bond-algebraic approach to dualities

An algebraic theory of dualities is developed based on the notion of bond algebras. It deals with classical and quantum dualities in a unified fashion explaining the precise connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of classical statistical mechanics (or (Euclidean) path integrals). Its range of applications includes discrete lattice, continuum field and gauge theories. Dualities are revealed to be local, structure-preserving mappings between model-specific bond algebras that can be implemented as unitary transformations, or partial isometries if gauge symmetries are involved. This characterization permits us to search systematically for dualities and self-dualities in quantum models of arbitrary system size, dimensionality and complexity, and any classical model admitting a transfer matrix or operator representation. In particular, special dualities such as exact dimensional reduction, emergent and gauge-reducing dualities that solve gauge constraints can be easily understood in terms of mappings of bond algebras. As a new example, we show that the ?₂ Higgs model is dual to the extended toric code model in any number of dimensions. Non-local transformations such as dual variables and Jordan–Wigner dictionaries are algorithmically derived from the local mappings of bond algebras. This permits us to establish a precise connection between quantum dual and classical disorder variables. Our bond-algebraic approach goes beyond the standard approach to classical dualities, and could help resolve the long-standing problem of obtaining duality transformations for lattice non-Abelian models. As an illustration, we present new dualities in any spatial dimension for the quantum Heisenberg model. Finally, we discuss various applications including location of phase boundaries, spectral behavior and, notably, we show how bond-algebraic dualities help constrain and realize fermionization in an arbitrary number of spatial dimensions.     

Hubbard model,conserved quantities,and computer algebra

The constants of motion of the half-filled four-point Hubbard model with cyclic boundary conditions are given in Wannier and Bloch representation. The total number operator and total spin operator are conserved and spin-reversal symmetry exists. In Wannier representation we have additionally the C_4v symmetry and in Bloch representation we have the total momentum operator which is conserved. The anticommutation relations for Fermi operators with spin are implemented using computer algebra. Using computer algebra, all the constants of motion are given. The one-dimensional Hubbard model admits a Lax representation. From the Lax pair we find a new constant of motion.     

Lax Equations, Singularities and Riemann–Hilbert Problems

The existence of singularities of the solution for a class of Lax equations is investigated using a development of the factorization method first proposed by Semenov-Tyan-Shanski? (Funct Anal Appl 17(4):259?C272, 1983) and Reymann and Semenov-Tian-Shansky (1994). It is shown that the existence of a singularity at a point t?=?t _i is directly related to the property that the kernel of a certain Toeplitz operator (whose symbol depends on t) be non-trivial. The investigation of this question involves the factorization on a Riemann surface of a scalar function closely related to the above-mentioned operator. Two examples are presented which show different aspects of the problem of computing the set of singularities of the solution to the system considered. The relation between the Riemann surfaces of the classical and Lax formulation is also considered.     

Generalized coulomb gauges for the free photon field

We point out that the classical notion of gauge transformations can be interpreted in two different ways in the quantum theory. Beside the transformation of the field operator appearing in the Gupta-Bleuler theory, the gauge transformations can be viewed as acting on the wave functions in a generalization of the Coulomb gauge.     

Lax pair formulation for the open boundary Osp(1∣2) spin chain

Based on the Lax pair formulation, we study the integrable conditions of the Osp(1∣2) spin chain with open boundaries. We consider both the non-graded and graded versions of the Osp(1∣2) chain. The Lax pair operators M_± for the boundaries can be induced by the Lax operator M_j for the bulk of the system. They correspond to the reflection equations (RE) and the Yang–Baxter equation, respectively. We further calculate the boundary K-matrices for both the non-graded and graded versions of the model with open boundaries. The double row monodromy matrix and transfer matrix of the spin chain have also been constructed. The K-matrices obtained from the Lax pair formulation are consistent with those from Sklyanin’s RE. This construction is another way to prove the quantum integrability of the Osp(1∣2) chain. We find that the Lax pair formulation has advantages in dealing with the boundary terms of the supersymmetric model.     

一个经典完全可积的非共形约束的SL(2,R)Wess-Zumino-Novikov-Witten模型(Ⅰ)

通过在ＳＬ（２，Ｒ）Ｗｅｓｓ－Ｚｕｍｉｎｏ－Ｎｏｖｉｋｏｖ－Ｗｉｔｔｅｎ（缩写为ＷＺＮＷ）模型中加入破坏共形对称性的约束，获得了一个新的经典完全可积的二维场论体系，它把著名的ｓｉｎｈ－Ｇｏｒｄｏｎ方程作为其特例。这个广义ｓｉｎｈ－Ｇｏｒｄｏｎ体系的特点是完全可积性和可超定域化，并且描写这些特点的ｒ矩阵是杨－Ｂａｘｔｅｒ方程（经典的）的一个解，它反对称，依赖于两个谱参数，但通过Ｌｏｏｐ代数的自同构变换和谱参数的重新定义后，此ｒ矩阵仍是依赖于一个谱参数的三角型ｒ矩阵。 关键词：     

Quantum field theory as effective BV theory from Chern–Simons

The general procedure for obtaining explicit expressions for all cohomologies of Berkovits' operator is suggested. It is demonstrated that calculation of BV integral for the classical Chern–Simons-like theory (Witten's OSFT-like theory) reproduces BV version of two-dimensional gauge model at the level of effective action. This model contains gauge field, scalars, fermions and some other fields. We prove that this model is an example of “singular” point from the perspective of the suggested method for cohomology evaluation. For arbitrary “regular” point the same technique results in AKSZ (Alexandrov, Kontsevich, Schwarz, Zaboronsky) version of Chern–Simons theory (BF theory) in accord with [N. Berkovits, Covariant quantization of the superparticle using pure spinors, JHEP 0109 (2001) 016, hep-th/0105050; N. Berkovits, ICTP lectures on covariant quantization of the superstring, hep-th/0209059; M. Movshev, A. Schwarz, On maximally supersymmetric Yang–Mills theories, Nucl. Phys. B 681 (2004) 324, hep-th/0311132; M. Movshev, A. Schwarz, Algebraic structure of Yang–Mills theory, hep-th/0404183].     

Hamiltonian Systems and Darboux Transformation Associated with a 3 × 3 Matrix Spectral Problem

Starting from a 3 × 3 matrix spectral problem, we derive a hierarchy of nonlinear equations. It is shown that the hierarchy possesses bi-Hamiltonian structure. Under the symmetry constraints between the potentials and the eigenfunctions, Lax pair and adjoint Lax pairs including partial part and temporal part are nonlinearied into two finitedimensional Hamiltonian systems (FDHS) in Liouville sense. Moreover, an explicit N-fold Darboux transformation for CDNS equation is constructed with the help of a gauge transformation of the spectral problem.     

A generalization of the Fermi-Breit equation to non-Coulombic spatial interactions

Starting from the relativistic invariance properties at classical level, we generalize the Darwin equation to the case of non-Coulombic spatial interactions. The relativistic correction terms for vector interactions are derived from a given nonrelativistic potential. We show that, for a Coulombic potential, the results coincide with those obtained in the Coulomb gauge. The results are adapted to the quantum theory obtaining a generalization of the Fermi-Breit equation. An Hermitian interaction operator is constructed. A critical comparison with other possible treatments of the retardation terms is performed also discussing the usual choice of the Coulomb gauge. Special attention is devoted to the construction of a model for quark interaction.