费米子动力学对称群的玻色子实现

将费米子动力学对称模型扩展为包含较高角动量费米子对,一般导致的动力学对称群是SO(4i+2)或者SP(4k+2),本文讨论它们精确的Dyson玻色子映像表示以及泡利原理在玻色子映像中的体现问题.     

A unified treatment of the groups SO(4) and SO(3,1)

The irreducible representations of the group SO(4) in which the SO(3) subgroup is reduced are studied by an explicit construction of the operators and the basis in the spinor representation. The basis function which is formally identical with that for the coupling of two angular momentaj ₁ andj ₂ is expressible in terms of a hypergeometric function and strongly resembles the one for the irreducible representations of the groups SO(3,1). For the Lorentz group, the bases for the unitary representations which require unphysical values ofj ₁ andj ₂ are found to be analytic continuation of those for SO(4). The realization of the unitary irreducible representations of the group SO(4) in the Hilbert space of these functions leads, for appropriate unphysical values ofj ₁,j ₂, to the Gelfand-Naimark formula for the principal and complementary series of the representations of SO(3;1). The matrix elements for finite transformations of SO(4) and SO(3,1) can be evaluated, in this approach, in a unified manner by using standard properties of the hypergeometric function. These turn out to be a finite sum of₃ F ₂-functions which, as expected, are polynomials for SO(4) and infinite series for SO(3,1). A number of special matrix elements are calculated from the general formula and these agree with the results obtained previously.The authors are deeply indebted to Professor S.Dutta Majumdar fo many important suggestions and clarifications.     

Subgroups of Hypercubic Group and Many Electron States in Crystals

The point subgroups of index 2 of hypercubic group and their irreducible representations are obtained. The elements of the hypercubic group are represented as rotation about two axis. Possible physical meaning of hypercubic group for electron states is investigated. The reduction relations for the representations of orthogonal group O ₄ on hypercubic group are obtained. These relations are used for additional classification of electron states in crystals.     

Differential Representations of SO（4） Dynamical Group

In this paper we present systematic differential representations for the dynamical group SO（4）. These representations include the left and the right differential representations and the left and the right adjoint differential representations in both the group parameter space and its coset spaces. They are the generalization of the differential representations of the SO（3） rotation group in the Euler angles. These representations may find their applications in the study of the physical systems with SO（4） dynamical symmetry.     

Elementary representations and intertwining operators for the group SU1(4)

Global realizations of all elementary induced representations (EIR) of the group SU¹(4), which is the double covering group of SO↑(5,1), are given. The Knapp-Stein intertwining operators are constructed and their harmonic analysis carried out. The invariant subspaces of the reducible EIR are introduced and the differential intertwining operators between partially equivalent EIR are defined. Invariant sequilinear forms on pairs of invariant subspaces are constructed. Differential identities between invariant sesquilinear forms on pairs of irreducible components of the reducible representations are derived. The results will be applied elsewhere to the nonpertubative analysis of Euclidean conformal invariant quantum field theory with fields of arbitrary spin.     

Quantum Algebra as Deformation Symmetry

The deformation maps as well as the general algebraic maps among algebras with three generators are systematically investigated in terms of symplectic geometry and geometric quantization on 2-D manifolds, from which the explicit Hamiltonian of Heisenberg model with SU_q(2) symmetry and arbitrary spin values are given. The deformation symmetries in differential dynamical systems and the q-deformed transformations of SO(3) group in usual R³ are also discussed.     

On the representation theory of quantum Heisenberg group and algebra

We show that the quantum Heisenberg groupH _q (1) and its *-Hopf algebra structure can be obtained by means of contraction from quantumSU _q (2) group. Its dual Hopf algebra is the quantum Heisenberg algebraU _q (h(1)). We derive left and right regular representations forU _q (h(1)) as acting on its dualH _q (1). Imposing conditions on the right representation, the left representation is reduced to an irreducible holomorphic representation with an associated quantum coherent state. Realized in the Bargmann-Hilbert space of analytic functions the unitarity of regular representation is also shown. By duality, left and right regular representations for quantum Heisenberg group with the quantum Heisenberg algebra as representation module are also constructed. As before reduction of group left representations leads to finite dimensional irreducible ones for which the intertwinning operator is also investigated.     

A Kohno–Drinfeld Theorem for the Monodromy of Cyclotomic KZ Connections

We compute explicitly the monodromy representations of “cyclotomic” analogs of the Knizhnik–Zamolodchikov differential system. These are representations of the type B braid group B_n¹{B_n^1} . We show how the representations of the braid group B _n obtained using quantum groups and universal R-matrices may be enhanced to representations of B_n¹{B_n^1} using dynamical twists. Then, we show how these “algebraic” representations may be identified with the above “analytic” monodromy representations.     

Dynamical symmetries in Kondo tunneling through complex quantum dots

Kondo tunneling reveals hidden SO(n) dynamical symmetries of evenly occupied quantum dots. As is exemplified for an experimentally realizable triple quantum dot in parallel geometry, the possible values n=3,4,5,7 can be easily tuned by gate voltages. Following construction of the corresponding o(n) algebras, scaling equations are derived and Kondo temperatures are calculated. The symmetry group for a magnetic field induced anisotropic Kondo tunneling is SU(2) or SO(4).     

On the use of the group SO(4, 2) in atomic and molecular physics

The dynamical non-invariance group SO(4, 2) for a hydrogen-like atom is derived through two different approaches. The first one is by an established traditional ascent process starting from the symmetry group SO(3). This approach is presented in a mathematically oriented original way with a special emphasis on maximally superintegrable systems, N-dimensional extension and little groups. The second approach is by a new symmetry descent process starting from the non-invariance dynamical group Sp(8, R) for a four-dimensional harmonic oscillator. It is based on the little known concept of a Lie algebra under constraints and corresponds in some sense to a symmetry breaking mechanism. This paper ends with a brief discussion of the interest of SO(4, 2) for a new group-theoretical approach to the periodic table of chemical elements. In this connection, a general ongoing programme based on the use of a complete set of commuting operators is briefly described. It is believed that the present paper could be useful not only to the atomic and molecular community but also to people working in theoretical and mathematical physics.     

Dynamical group for a ring potential

We show that the problem of a point particle moving in the ring potential of Hartmann possesses SO(2, 1) ⊗SO(2, 1) as a dynamical (or noninvariance) group.     

Free massless fermionic fields of arbitrary spin in d-dimensional anti-de sitter space

Free massless fermionic fields of arbitrary spins, corresponding to fully symmetric tensor-spinor irreducible representations of the flat little group SO(d−2), are described in d-dimensional anti-de Sitter space in terms of differential forms. Appropriate linearized higher-spin curvature 2-forms are found. Explicitly gauge invariant higher-spin actions are constructed in terms of these linearized curvatures.     

Chemical stability of hydroxysulphate green rust synthetised in the presence of foreign anions: carbonate, phosphate and silicate

Hydroxysulphate green rust species were precipitated in the presence of various anions. is stable at ∼pH 7 and is transformed into a mixture of magnetite and ferrous hydroxide when the pH raised at ∼12. In the presence of carbonate species, is partially transformed into a mixture of magnetite and siderite at ∼pH 8.5. This transformation is stopped when silicate anions are present in the solution. As already observed for phosphate anions, the adsorption of silicate anions on the lateral faces of the crystals may explain this stabilization effect. Sulphate anions are easily exchanged by carbonate species at ∼pH 10.5. In contrast, anionic exchange between sulphate and phosphate anions was not observed.     

Fermion mass spectrum in a generalized instanton background

The spectrum of the squared Dirac operator is studied in the background of symmetric instanton-like configurations on symmetric spaces G/H. The eigenvalues are expressed in terms of Casimir invariants of G and H. These eigenvalues determine the masses of fermions in Kaluza-Klein theories with compactification induced by the generalized instantons. The G-representations of the zero modes are determined for fermions in arbitrary representations of H on S_n and CP_n. For spheres in even dimensions a comparison with the index theorem reveals a remarkable relation between the Nth index (or anomaly) of SO(2N) and dimensions of SO(2N + 1) representations. Using the fundamental indices, we find the topologically stable symmetric solutions on S_2n for any gauge group, and compare with recent results on local stability.     

Fundamental Fermions Fit Inside One <Emphasis Type="Italic">su</Emphasis>(1|5) Irreducible Representation

The Lie superalgebra su(1|5) has irreducible representations of dimension 32, in which the 32 fundamental fermions of one generation (leptons and quarks, of left and right chirality, and their antiparticles) can be accommodated. The branching of these su(1|5) representations with respect to its subalgebra su(3)× su(2)× u(1) reproduces precisely the classification of these fundamental fermions according to the gauge group su(3)^c× su(2)^w× u(1)^w of the Standard Model. Furthermore, a simple construction of the relevant representations is given, and some consequences are discussed.     

The hidden SO(4) symmetry of general SU(2) thirring models

General four-fermion interactions in two dimensions with SU(2) invariance are shown to possess a hidden SO(4) symmetry. As a consequence physical states belong to irreducible representations of the two commuting O(3) subgroups and their interactions decouple accordingly. Two independnet stable trajectories of the renormalization group are shown to exist perturbatively and are consistently reproduced by abelian bosonization.     

Towards the Evaluation of the Relevant Degrees of Freedom in Nonlinear Partial Differential Equations

We investigate an operator renormalization group method to extract and describe the relevant degrees of freedom in the evolution of partial differential equations. The proposed renormalization group approach is formulated as an analytical method providing the fundamental concepts of a numerical algorithm applicable to various dynamical systems. We examine dynamical scaling characteristics in the short-time and the long-time evolution regime providing only a reduced number of degrees of freedom to the evolution process.     

The hidden symmetries of Taub-Nut and monopole scattering

The recently discovered hidden symmetries in the large-distance interactions of BPS monopoles with each other and with fluctuations around them are traced to the existence on the self-dual Taub-NUT metric of a Killing-Yano tensor. The global action on classical phase space of these symmetries is discussed. The quantum picture involving the “dynamical groups” SO(4), SO(4, 1) and SO(4, 2) is also given.     

Construction and classification of indecomposable finite-dimensional representations of the homogeneous Galilei group

We discuss finite-dimensional representations of the homogeneous Galilei group which, when restricted to its subgroup SO(3), are decomposed to spin 0, 1/2 and 1 representations. In particular we explain how these representations were obtained in our paper (M. de Montigny et al.: J. Phys. A39 (2006) 9365) via reduction of the classification problem to a matrix one admitting exact solutions, and via contraction of the corresponding representations of the Lorentz group. Finally, for discussed representations we derive all functional invariants.