Neutral and charged quommutators with and without symmetries

We present two sets of qualgebras involving operators which generalize creation and annihilation operators. These two groups of operators satisfy separately quommutation relations rather than commutation or anticommutation relations. The quommutators of the creation and annihilation operators generate new “neutral operators” which themselves are subjected to quommutation relations. Two solutions are presented. In the second one, some new symmetry relations are added to the system. In a certain sense these extra relations, rather than imposing new constraints on the parameters, increase their freedom.     

Baxter operators for arbitrary spin

We construct Baxter operators for the homogeneous closed XXX spin chain with the quantum space carrying infinite- or finite-dimensional s?₂ representations. All algebraic relations of Baxter operators and transfer matrices are deduced uniformly from Yang-Baxter relations of the local building blocks of these operators. This results in a systematic and very transparent approach where the cases of finite- and infinite-dimensional representations are treated in analogy. Simple relations between the Baxter operators of both cases are obtained. We represent the quantum spaces by polynomials and build the operators from elementary differentiation and multiplication operators. We present compact explicit formulae for the action of Baxter operators on polynomials.     

On the structure of the Galilean and translationally invariant operator algebra describing intrinsic properties of finite nuclear systems

The structure of the Galilean and translationally invariant operator algebra for finite systems of fermions is investigated. After performing the decomposition of the Fock space into Hilbert spaces for the center-of-mass motion and the intrinsic motion, “intrinsic” field operators are defined and their commutation relations established. These relations deviate in a certain particle number-dependent way from the usual fermion relations. It is shown that the operators corresponding to the intrinsic (e.g. nuclear) observables can be represented in the familiar way, the usual field operators being replaced by the intrinsic ones. In this theory the normal shell model calculations appear as the approximation performed by treating matrix elements of nuclear observables as if the intrinsic field operators were satisfying the exact Fermi commutation relations.     

On the relativistic spin projection operators

In the Rarita-Schwinger formalism, the relativistic spin projection operators are discussed with the help of the Pauli-Lubanski four-vector. It is shown that this approach is equivalent to the conventional one, but moreover, it enables one to derive recurrence relations for the spin projection operators. Such relations can be useful in practical applications.     

Dual squeezed states in an atom-photon cluster and their manifestations

The general kinetic equation for an isolated two-level atom and a high-Q cavity mode in a heat bath exhibiting quantum correlations (entangled bath) is applied to the analysis of the squeezed states of the collective system. Two types of collective operators are introduced for the analysis: one is based on bosonic commutation relations, and the other, on the commutation relations of the algebra obtained by a polynomial deformation of the angular momentum algebra. On the basis of these relations, formulas for observables are constructed that identify squeezed states in the system. It is shown that, under certain conditions, the collective system exhibits dual squeezing within the relations for boson operators, as well as for the operators constructed from the angular momentum algebra. Such squeezing is demonstrated under a projective measurement of an atom and for an entanglement swapping protocol. In the latter case, when measuring two initially independent atomic systems, depending on the type of measurement, two cavity modes collapse into a nonseparable state, which is described either by a nonseparability relation based on boson operators or by a relation based on the operators of the algebra of the quasimomentum of the collective system consisting of these two modes.     

Microscopic calculation of IBM parameters by using intrinsic states

The overlaps between intrinsic fermionic and bosonic wave functions are required to be the same. This provides relations between fermion and boson variables. These relations are used in conjunction with an OAI procedure for intrinsic states to map the shell-model space operators onto their equivalent boson space operators. As an example, a QQ interaction is mapped.     

Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum

In this work we apply the Dirac method in order to obtain the classical relations for a particle on an ellipsoid. We also determine the quantum mechanical form of these relations by using Dirac quantization. Then by considering the canonical commutation relations between the position and momentum operators in terms of curved coordinates, we try to propose the suitable representations for momentum operator that satisfy the obtained commutators between position and momentum in Euclidean space. We see that our representations for momentum operators are the same as geometric one.     

Baxter operators for arbitrary spin II

This paper presents the second part of our study devoted to the construction of Baxter operators for the homogeneous closed XXX spin chain with the quantum space carrying infinite or finite-dimensional s?₂ representations. We consider the Baxter operators used in Bazhanov et al. (1996, 1997, 1999, 2010) [1] and [2], formulate their construction uniformly with the construction of our previous paper. The building blocks of all global chain operators are derived from the general Yang-Baxter operators and all operator relations are derived from general Yang-Baxter relations. This leads naturally to the comparison of both constructions and allows to connect closely the treatment of the cases of infinite-dimensional representation of generic spin and finite-dimensional representations of integer or half-integer spin. We prove not only the relations between the operators but present also their explicit forms and expressions for their action on polynomials representing the quantum states.     

A BOSON-FERMION DESCIPTION OF THE NUCLEAR MANY-BODY PROBLEM

A new formalism of the nuclear many-body problem is established in which one ean work with various physical state vector spaces consisting of both Fermions and Bosons, all being equivalent to the original one consisting only of Fermions.With the help of the usual commutation relations and anticommutation relations between the annihilation and creation operators of Boson and Fermion, a generalized state vector space is established, which contains all the physical spaces each being equivalent to the original physical space in terms of pure Fermion operators. Transformation between the state vectors in various equivalent physical spaces are constructed. Basic operators in the original state space are transformed into effective operators in the new physical spaces.     

Comments on the properties of one- and two-body double tensor operators

Recently, the parity, charge, time and Hermitian conjugation properties of one-body tensor operators have been reexamined and the Racah algebra has been formally extended to two-body tensor operators. In this paper we consider several important aspects related to these problems. First, we present a proof of the communication relations for two-body operators which was previously postulated. Then, comparing the reduced matrix elements for one- and two-body operators we find the normalization constant for the latter. We subsequently show that the P- and T-conjugation relations for tensor operators are relativistically invariant. Finally, we analyze the question of Hermitian conjugation of double tensor operators and conclude that previously stated conditions on their ranks were too restrictive and resulted in omissions of some terms from theoretical analyses. We show examples of tensor operators which were customarily neglected but indeed should be retained in the effective Hamiltonian.     

原子模型势中任意算符矩阵元的递推关系

运用升降算子和超维里定理计算出原子模型势中任意算符Af^U2(r)的矩阵元〈n′1l′1|Af^U2(r)|n1l1〉的递推关系.     

Constructive approach to symmetrized two-particle expansions

A complete set of boundary conditions (exchange operators) which follow from the symmetrization postulate for a wave function of a bifermion cluster system is given. The commutation relations of bifermion operators and properties of exchange operators are investigated. For arbitrary bifermion operators the projected Hamiltonian is examined.     

Dobiński relations and ordering of boson operators

We introduce a generalization of the Dobiński relation, through which we define a family of Bell-type numbers and polynomials. Such generalized Dobiński relations are coherent state matrix elements of expressions involving boson ladder operators. This may be used in order to obtain normally ordered forms of polynomials in creation and annihilation operators, both if the latter satisfy canonical and deformed commutation relations.     

ON THE IRREDUCIBLE REPRESENTATIONS OF THE SIMPLE LIE GROUPS I——THE TENSOR BASIS FOR THE INFINITESIMAL GENERATORS OF THE CLASSICAL SIMPLE LIE GROUPS

In this paper, we analyse the commutation relations of the infinitesimal generatorsof all simple classical Lie groups and establish a new basis for these generators, calledthe tensor basis. In tensor basis, the infinitesimal, generators can be written as somescalar operators, some sets of angular momentum operators and some sets of irreducibletensor operators. The commutation relations, of these operators are very simple andhave many regularities. By means of the method that has been used in the earlier papers, "On the irre-ducible representations of the compact simple Lie groups of rank 2, I,II,III" and thetensor basis, all the irreducible representations of the classical simple Lie groups canbe calculated systematically.     

A Relation of the Noncommutative Parameters in Generalized Noncommutative Phase Space

We introduce the deformed boson operators which satisfy a deformed boson algebra in some special types of generalized noncommutative phase space.Based on the deformed boson algebra,we construct coherent state representations.We calculate the variances of the coordinate operators on the coherent states and investigate the corresponding Heisenberg uncertainty relations.It is found that there are some restriction relations of the noncommutative parameters in these special types of noncommutative phase space.     

Quantized Flag Manifolds and Irreducible *-Representations

In this paper we fill some gaps in the arguments of our previous papers [1,2]. In particular, we give a proof that the L operators of Conformal Field Theory indeed satisfy the defining relations of the Yang–Baxter algebra. Among other results we present a derivation of the functional relations satisfied by T and Q operators and a proof of the basic analyticity assumptions for these operators used in [1,2]. Received: 20 May 1998 / Accepted: 7 July 1998     

Inverse of Radiation Field Operators and the Generalized Jaynes-Cummings Model

We generalize the standard Jaynes-Cummings model (JCM) to a model Hamiltonian with the radiation field operators being the inverse of a harmonic oscillator's creation and annihilation operators. Some new commutative relations about the inverse operators are derived and the generalized JCM Hamiltonian's eigenstates are derived.     

The quantum groupSUq(2) andq-analog of angular momentum operators in quantum mechanics

We present three operators in quantum mechanics that obey the commutation relations of quantum groupSUq(2). These operators are nonlinear combinations of the conventional angular momentum operators and are called the quantumq-analog angular momentum operators. When the quantum deformation parameterr = Inq vanishes, these quantumq-analog angular momentum operators reduce to the usual angular momentum operators.     

Irrelevant operators and momentum-shell recursion relations ind=2 + ∈ dimensions

The momentum-shell recursion relations of Nelson and Pelcovits for ann-vector model near two dimensions are reexamined. The renormalization of the infinite set of relevant and marginal operators present in the system is studied. Ambiguities obtained in the ensuing recursion relations are shown to involve irrelevant operators only, thus justifying the procedure of Nelson and Pelcovits. The cases of finite external fieldh and finite spin anisotropyg are both considered.     

Topologies induced by representations of the canonical commutation relations

This paper is concerned with continuity properties of representations of the canonical commutation relations, and is mainly devoted to a detailed discussion of the topologies induced on the test function spaces. The notion of closability of a representation of the canonical commutation relations is introduced and studied. We also discuss the strong continuity of functions of self-adjoint operators, and use bounded functions to define an analogue of the strong operator topology on the set of all self-adjoint operators.