Squeezing in Representation by Quadratic Combinations of Canonical Operators

The technique of integration within an ordered product of operators and the coherent-state representationare used to convert exponential operators of basis operators (p2, Q2, pQ + Qp) to those of the basis operators (a2, a+2,ara). The coherent state representation of unitary squeezing operators in the factorized form and their normal productform are thus derived. The squeezing engendered by operators of the general form exp{-i[αP2 + βQ2 + γ/2 (QP + PQ)] }is also obtained.     

General theory of renormalization of gauge invariant operators

We study the question of renormalization of gauge invariant operators in the gauge theories. Our discussion applies to gauge invariant operators of arbitrary dimensions and tensor structure. We show that the gauge noninvariant (and ghost) operators that mix with a given set of gauge invariant operators form a complete set of local solutions of a functional differential equation. We show that this set of gauge noninvariant operators together with the gauge invariant operators close under renormalization to all orders. We obtain a complete set of local solutions of the differential equation. The form of these solutions has recently been conjectured by Kluberg Stern and Zuber. With the help of our solutions, we show that there exists a basis of operators in which the gauge noninvariant operators “decouple” from the gauge invariant operators to all orders in the sense that eigenvalues corresponding to the eigenstates containing gauge invariant operators can be computed without having to compute the full renormalization metrix. We further discuss the substructure of the renormalization matrix.     

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.     

Effect of electron correlation on the forced electric dipole transition probabilities in fn systems

A method for the investigation of the effects of electron correlation on the forced electric dipole transition probabilities between levels of the fⁿ configuration is presented. The approach is based on double perturbation theory and on the application of effective operators. Effective operators describing the correlation effects are generated for various classes of perturbing configurations in terms of generalized unit tensor operators. A classification scheme for the possible operators is presented. In general, one- and two- particle operators are obtained. The latter do not appear in the standard Judd-Ofelt theory. One-particle operators dependent on unit tensor operators of odd rank are also new. The other one-particle operators are of the form considered in the Judd-Ofelt theory. However, the present results indicate that the interpretation of the empirical parameters of this theory should be modified to take into account electron correlation effects.     

Optimal multi-setting witness operators for two qudits

Quantum entanglement in bipartite systems is approached with the help of pseudo Bell operators, a relaxed form of Bell operators. Their connection to optimal witness operators is established. A method to construct pseudo Bell operators for pairs of qudits, based on multiple two-dimensional local measuring settings, is presented. Each of them provide two optimal (multi-setting) witness operators. They are rigorously proved to fulfill the conditions in the definition. We explain how to approximate the set of separable states and the set of entangled states. A class of Werner-like entangled states is identified. The spectral resolution of all multi-setting pseudo Bell operators is carried out in full.     

Symmetry properties of nonlocal quark bilinear operators on a Lattice(LP~3 Collaboration)

Using symmetry properties, we determine the mixing pattern of a class of nonlocal quark bilinear operators containing a straight Wilson line along a spatial direction. We confirm the previous study that mixing among the lowest dimensional operators, which have a mass dimension equal to three, can occur if chiral symmetry is broken in the lattice action. For higher dimensional operators, we find that the dimension-three operators will always mix with dimension-four operators, even if chiral symmetry is preserved. Also, the number of dimension-four operators involved in the mixing is large, and hence it is impractical to remove the mixing by the improvement procedure. Our result is important for determining the Bjorken-x dependence of the parton distribution functions using the quasi-distribution method on a Euclidean lattice. The requirement of using large hadron momentum in this approach makes the control of errors from dimension-four operators even more important.     

Neutrino mass operators of dimension up to nine in two-Higgs-doublet model

We study higher-dimensional neutrino mass operators in a low energy theory that contains a second Higgs doublet, the two Higgs doublet model. The operators are relevant to underlying theories in which the lowest dimension-five mass operators would not be induced. We list the independent operators with dimension up to nine with the help of Young tableau. Also listed are the lowest dimension-seven operators that involve gauge bosons and violate the lepton number by two units. We briefly mention some of possible phenomenological implications.     

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.     

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.     

On a Three-Body Problem in Relativistic Quantum Mechanics

Proceeding from the method of packing operators developed by Sokolov and from the “light front variables” technique, the explicit formulae for packing operators and auxiliary mass operators for a system of three particles with arbitrary spins are derived. It is shown that for the packing operators there exists an infinite number of solutions yielding different physical consequences. The problem of the theory substantiation is discussed; the arguments in favour of a certain choice of packing operators are produced.     

Constraints from conformal covariance on the mixing of operators of lowest twist

In massless QCD the operators which diagonalize the anomalous dimension matrix to first order in α_s form representations of the conformal group. This is proved by using anomalous Ward identities. Only a subgroup, the so-called collinear conformal group SU(1, 1) ? SO(2, 1), acts on the restricted class of operators of lowest twist. Towers of local operators bilinear and trilinear in the quark fields are constructed. They are considered as polynomials in the derivatives acting on the basic quark fields: bilinear operators are given by Jacobi polynomials, trilinear operators by Appell's polynomials.By using the explicit form of the anomalous dimension matrix we verify that the constructed bilinear collinear conformal operators are in fact eigenvectors. As the conformal trilinear operator basis is degenerate, the eigenvectors of the mixing matrix of trilinear operators are not determined by conformal covariance alone. However, by taking a conformal basis the number of independent operators is reduced considerably.     

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.     

Notes on the Elliptic Ruijsenaars Operators

We construct the spaces that the elliptic Ruijsenaars operators act on. It is shown that they are extensible to nonnegative selfadjoint operators on a space of square integrable functions, or preserve a finite dimensional vector space of entire functions. These facts are shown in terms of the R-operators which satisfy the Yang–lBaxter equation. The elliptic Ruijsenaars operators are considered as the elliptic analogues of the Macdonald operators or the difference analogues of the Lamé operators.     

A Non-Associative Quantum Mechanics

A non-associative quantum mechanics is proposed in which the product of three and more operators can be non-associative one. The multiplication rules of the octonions define the multiplication rules of the corresponding operators with quantum corrections. The self-consistency of the operator algebra is proved for the product of three operators. Some properties of the non-associative quantum mechanics are considered. It is proposed that some generalization of the non-associative algebra of quantum operators can be helpful for understanding of the algebra of field operators with a strong interaction.     

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.     

Representation of orthofermion spin operators

The algebraic expressions for the total spin operators expressed in terms of orthofermion creation and annihilation operators are combined into a single equation. The terms in this expressions are rearranged to provide representations of local spin operators. This task is essential for modelling a system of orthofermions in the presence of a magnetic field. By factorizing the orthofermion annihilation (creation) operators into charge and spin dependent parts, it is shown that the latter part suffices to represent spin number, raising and lowering operators. Finally a connection is provided between the spin systems and Greenberg’s infinite statistics.     

Transformation Operators for Sturm-Liouville Operators with Singular Potentials

We construct transformation operators for Sturm-Liouville operators with singular potentials from the space W ⁻¹ ₂(0,1) and show that these transformation operators naturally appear during factorisation of Fredholm operators of a special form. Some applications to the spectral analysis of Sturm-Liouville operators with singular potentials under consideration are also given. This revised version was published online in July 2006 with corrections to the Cover Date.     

Anomalous dimensions of gauge-invariant three fermion local operators of twist three

The matrices of anomalous dimensions of gauge-invariant three-fermion local operators of twist three are calculated in QCD with massless quarks. Furthermore we explicitly give the eigenvalues for operators containing up to eight derivatives, and the eigenvectors for operators containing up to three derivatives. The correspondence between eigenvectors of the anomalous dimension matrices and decuplet local baryon operators is derived by means of Fierz transformations.     

Quantum Gates and Hamilton Operators

Quantum gates are described by unitary operators. We discuss the construction of Hamilton operators from the unitary operators. Different techniques are applied.     

再谈相干态在算符编序中的应用

利用广义Weyl对应导出了相干态密度算符的S-编序表示.通过此表示并结合S-编序内的算符积分技术,导出了若干量子玻色算符的S-编序形式(包含了正规编序、Weyl编序和反正规编序).此方法进一步推广了相干态在算符编序中的应用.