Non-Abelianizable First Class Constraints

We study the necessary and sufficient conditions on Abelianizable first class constraints. The necessary condition is derived from topological considerations on the structure of the gauge group. The sufficient condition is obtained by applying the implicit function theorem in calculus and studying the local structure of gauge orbits. Since the sufficient condition is necessary for the existence of proper gauge fixing conditions, we conclude that in the case of a finite set of non-Abelianizable first class constraints, the Faddeev-Popov determinant is vanishing for any choice of subsidiary constraints. This result is explicitly examined for the SO(3) gauge invariant model.Acknowledgement The financial support of Isfahan University of Technology (IUT) is acknowledged.     

Tensor Products of Hecke Algebras

Tensor product of irreducible representations of Hecke algebras are discussed. It is found that the tensor product of irreps of Hecke algebras generates representations of Birman-Wenzl algebra C_ƒ(γ, q) with γ = q³ or-q^-3. A procedure for the evaluation of tensor product coefficients (TPC's) of H_ƒ (q)oH_ƒ(q) ↓ C_ƒ(γ,q) is established when the representations of Cƒ(γ, q) remain irreducible. An example of deriving TPC's of H_ƒ (q)oH_ƒ(q) ↓ C_ƒ(γ, q) is given. It is also found that indecomposable representation of C₄(γ q) occurs in the tensor product [211]o[31].     

Quantum Algebras and Effective Interactions in Discrete Systems

It is shown that the interactions between spin-states and a radiation field can be described in terms of quantum algebras. As an example, we discuss discrete systems within the su _q (3) scheme.     

Orientational Order Parameters for Arbitrary Quantum Systems

The concept of quantum-mechanical nematic order, which is important in systems such as superconductors, is based on an analogy to classical liquid crystals, where order parameters are obtained through orientational expansions. This method is generalized to quantum mechanics based on an expansion of Wigner functions. This provides a unified framework applicable to arbitrary quantum systems. The formalism recovers the standard definitions for spin systems. For Fermi liquids, the formalism reveals the nonequivalence of various definitions of the order parameter used in the literature. Moreover, new order parameters for quantum molecular systems with low symmetry are derived, which cannot be properly described with the usual nematic tensors.     

On the Idempotents of Hecke Algebras

We give a new construction of primitive idempotents of the Hecke algebras associated with the symmetric groups. The idempotents are found as evaluated products of certain rational functions thus providing a new version of the fusion procedure for the Hecke algebras. We show that the normalization factors which occur in the procedure are related to the Ocneanu–Markov trace of the idempotents.      

On Ternary Quotients of Cubic Hecke Algebras

We prove that the quotient of the group algebra of the braid group introduced by Funar (Commun Math Phys 173:513–558, 1995) collapses in characteristic distinct from 2. In characteristic 2 we define several quotients of it, which are connected to the classical Hecke and Birman-Wenzl-Murakami quotients, but which admit in addition a symmetry of order 3. We also establish conditions on the possible Markov traces factorizing through it.     

Tighter Constraints of Quantum Correlations Among Multipartite Systems

International Journal of Theoretical Physics - We provide a characterization of multipartite systems constraints in terms of quantum correlations. By using the Hamming weight of the binary vectors...     

Hecke Symmetries and Characteristic Relations on Reflection Equation Algebras

We analyze the relation between the properties of Hecke symmetry (i.e., Hecke type R-matrix) and the algebraic structure of the corresponding reflection equation (RE) algebra. Analogues of the Newton relations and Cayley–Hamilton theorem for the matrix of generators of the RE algebra associated with a finite rank even Hecke symmetry are derived.     

Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry

Given a finite dimensional C ^*-Hopf algebra H and its dual Ĥ we construct the infinite crossed product and study its superselection sectors in the framework of algebraic quantum field theory. is the observable algebra of a generalized quantum spin chain with H-order and Ĥ-disorder symmetries, where by a duality transformation the role of order and disorder may also appear interchanged. If is a group algebra then becomes an ordinary G-spin model. We classify all DHR-sectors of – relative to some Haag dual vacuum representation – and prove that their symmetry is described by the Drinfeld double . To achieve this we construct localized coactions and use a certain compressibility property to prove that they are universal amplimorphisms on . In this way the double can be recovered from the observable algebra as a universal cosymmetry. Received: 4 September 1995\,/\,Accepted: 3 December 1996     

Intertwining Symmetry Algebras of Quantum Superintegrable Systems on Constant Curvature Spaces

A class of quantum superintegrable Hamiltonians defined on a hypersurface in a n+1 dimensional ambient space with signature (p,q) is considered and a set of intertwining operators connecting them are determined. It is shown that the intertwining operators can be chosen such that they generate the su(p,q) and so(2p,2q) Lie algebras and lead to the Hamiltonians through Casimir operators. The physical states corresponding to the discrete spectrum of bound states as well as the degeneration are characterized in terms of some particular unitary representations.     

脉冲控制量子系统的哈密顿量约化

探索了脉冲控制的含近简并能级的有限维量子系统的哈密顿量的约化．由一个非简并基态能级和几个近简并激发态能级组成的量子系统被一个短脉冲控制,目标是控制所有激发态的布居数之和．考虑了两个可以看成等价二能级系统的例子,当脉冲强度比较弱时,得到了原始系统和约化系统的简单关系;当脉冲强度比较强时,对于只含一个频率的脉冲,一阶近似的关系也是存在的．     

On a Class of Integrable Systems with a Cubic First Integral

A few years ago Selivanova gave an existence proof for some integrable models, in fact geodesic flows on two dimensional manifolds, with a cubic first integral. However the explicit form of these models hinged on the solution of a nonlinear third order ordinary differential equation which could not be obtained. We show that an appropriate choice of coordinates allows for integration and gives the explicit local form for the full family of integrable systems. The relevant metrics are described by a finite number of parameters and lead to a large class of models mainly on the manifolds \mathbb S²{{\mathbb S}^2} and \mathbb H²{{\mathbb H}^2} . Many of these systems are globally defined and contain as special cases integrable systems due to Goryachev, Chaplygin, Dullin, Matveev and Tsiganov.     

Parasupersymmetric Quantum Mechanics with Arbitrary p and N

Abstract

The generalization of parasupersymmetric quantum mechanics generated by an arbitrary number of parasupercharges and characterized by an arbitrary order of paraquantization is given. The relations for parasuperpotentials are obtained. It is shown that parasuperpotentials can be explicitly expressed via one arbitrary function.     

Quantum and Classical Implication Algebras with Primitive Implications

Join in an orthomodular lattice is obtained inthe same form for all five quantum implications. Theform holds for the classical implication in adistributive lattice as well. Even more, the definition added to an ortholattice makes it orthomodularfor quantum implications and distributive for theclassical one. Based on this result a quantumimplication algebra with a single primitive — andin this sense unique — implication is formulated. Acorresponding classical implication algebra is alsoformulated. The algebras are shown to be special casesof a universal implication algebra.     

Division Algebras and Quantum Theory

Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘three-fold way’. It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly ‘complex’ representations), those that are self-dual thanks to a symmetric bilinear pairing (which are ‘real’, in that they are the complexifications of representations on real Hilbert spaces), and those that are self-dual thanks to an antisymmetric bilinear pairing (which are ‘quaternionic’, in that they are the underlying complex representations of representations on quaternionic Hilbert spaces). This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. More generally, Hilbert spaces of any one of the three kinds—real, complex and quaternionic—can be seen as Hilbert spaces of the other kinds, equipped with extra structure.     

Quantum Groups and Fuss--Catalan Algebras

The categories of representations of compact quantum groups of automorphisms of certain inclusions of finite dimensional ℂ^*-algebras are shown to be isomorphic to the categories of Fuss–Catalan diagrams. Received: 9 March 2001 / Accepted: 12 November 2001     

Star Products and Quantum Algebras

I show explicitly that the star product on atriangular Poisson Lie group leads to a quantum algebrastructure (triangular Hopf algebra) on the quantizedenveloping algebra of the Lie algebra of the Lie group, and that equivalent star-productsgenerate isomorphic quantum algebras.     

Non-Commutative Algebras and Quantum Structures

We present a survey on pseudo-effect algebras and pseudo MV-algebras, which generalize effect algebras and MV-algebras by dropping the assumption on commutativity. A non-commutative logic is nowadays used even in programming languages. We show when a pseudo-effect algebra E is an interval in a unital po-group. This is possible, e.g. if E satisfies a Riesz-type decomposition property, i.e. another kind of distributivity with respect to addition. Every pseudo MV-algebra is an interval in a unital ℓ-group. We study a case when compatibility can be expressed by a pseudo MV-structure, i.e. when E can be covered by blocks being pseudo MV-algebras. Finally, we study the state space of such structures.