Realisations of the representations of para-Fermi algebra-Part III: Fock space of Fermi operators

Following the idea given by Kademova (1969b), Green's isomorphisms of the para-Fermi algebra F _2n ¹ into the space of all bilinear products of para-Fermi type operators are constructed. The induced transformations in Fock space of Fermi operators are considered. The difference between the induced transformations in Fock space of Bose operators and in Fock space of Fermi operators is discussed.     

Fock representations and BRST cohomology inSL(2) current algebra

We investigate the structure of the Fock modules overA ₁ ⁽¹⁾ introduced by Wakimoto. We show that irreducible highest weight modules arise as degree zero cohomology groups in a BRST-like complex of Fock modules. Chiral primary fields are constructed as BRST invariant operators acting on Fock modules. As a result, we obtain a free field representation of correlation functions of theSU(2) WZW model on the plane and on the torus. We also consider representations of fractional level arising in Polyakov's 2D quantum gravity. Finally, we give a geometrical, Borel-Weil-like interpretation of the Wakimoto construction.     

Irreducible highest weight representations of quantum groupsU q (gl(n,C))

Explicit recurrence formulas of canonical realization (boson representation) for quantum enveloping algebrasU _q (gl(n, C)) are given. Using them, irreducible highest weight representations ofU _q (gl(n, C)) are obtained as restriction of representation of Fock space to invariant subspace generated by vacuum as a cyclic vector.     

Realisations of the representations of para-Fermi algebra in Fock space of Bose operators: part I

Using the matrix realisations of para-Fermi operators we find isomorphic mappings with respect to the Green product of the para-Fermi algebra into second-order polynomials of creation and annihilation para-Bose operators with arbitrary order of parastatistics. In the Fock space ℋ ₂ ¹ of two Bose operators all the irreducible representations of the para-Fermi algebra are realised. The spaces ofn-particle Bose statesn=1,2,..., from which ℋ ₂ ¹ is constructed as a direct sum, can be interpreted as spaces of para-Fermi states of para-statisticsn.     

Hilbert space cocycles as representations of (3 + 1)-d current algebras

It is proposed that instead of normal representations, one should look at cocycles of group extensions valued in certain groups of unitary operators acting in a Hilbert space (e.g. the Fock space of chiral fermions), when dealing with groups associated to current algebras in gauge theories in 3 + 1 spacetime dimensions. The appropriate cocycle is evaluated in the case of the group of smooth maps from the physical three-space to a compact Lie group.The cocyclic representation of a componentX of the current is obtained through two regularizations, (1) a conjugation by a background potential dependent unitary operatorh _A, (2) by a subtraction-h _A ^-1 _xhA, where _x is a derivative along a gauge orbit. It is only the total operatorh _A ^-1 Xh _A -h _A ^-1 _xhA which is quantizable in the Fock space using the usual normal ordering subtraction.Supported by the Alexander von Humboldt Foundation     

Realisations of the representations of para-Fermi algebra in Fock space of Bose operators: part II

Using the method given in Part I of this series (this volume, pp. 109–114), Green's isomorphisms of the para-Fermi algebra F _2n ¹ with 2n generators into ∈ _2n+1 ^a(2) are constructed. All the representations of the para-Fermi algebra are realised in Fock space #x210B; _2n ¹ of 2ⁿ Bose opcrators.     

On some representations of the anticommutations relations

We study representations of the canonical anticommutation relations having the form: wherea(f),b*(f) and their adjoints are two basic anticommuting fields in a Fock Space.A complete determination of the type in terms of |K|=(K*K)^1/2 and a sufficient condition for quasi-equivalence are given.     

Formal GNS Construction and States in Deformation Quantization

On the level-1 Fock space modules of the algebra we define a level-0 action U ₀ of the , and an action of an Abelian algebra of conserved Hamiltonians commuting with the U ₀. An irreducible decomposition of the Fock space with respect to the level-0 action is derived by constructing a base of the Fock space in terms of the Non-symmetric Macdonald Polynomials. Received: 14 August 1996 / Accepted: 18 April 1997     

Multi-parameter deformations and multi-particle representations of the bosonic oscillator

In this paper, we consider a multi-parameter deformation of the bosonic oscillator algebra and determine the consistency conditions on the parameters. For a d-dimensional oscillator we find 2d parameters. Finally, we present the Fock representation of this oscillator. Received: 16 February 2000 / Published online: 11 May 2001     

Construction of the two-electron contribution to the Fock matrix by numerical integration

A novel method to numerically calculate the Fock matrix is presented. The Coulomb operator is re-expressed as an integral identity, which is discretized. The discretization of the auxiliary t dimension separates the x, y, and z dependencies transforming the two-electron Coulomb integrals of Gaussian-type orbitals (GTO) to a linear sum of products of two-dimensional integrals. The s-type integrals are calculated analytically and integrals of the higher angular-momentum functions are obtained using recursion formulae. The contributions to the two-body Coulomb integrals obtained for each discrete t value can be evaluated independently. The two-body Fock matrix elements can be integrated numerically, using common sets of quadrature points and weights. The aim is to calculate Fock matrices of enough accuracy for electronic structure calculations. Preliminary calculations indicate that it is possible to achieve an overall accuracy of at least 10^?12 E _h using the numerical approach.     

Highest weight representations of the Neveu-Schwarz and Ramond algebras

We construct a family of representationsK ^,w of the Neveu-Schwarz and Ramond algebras, which generalize the Fock representations of the Virasoro algebra. We show that the representationsK ^,w are intertwined by a vertex operator.The above results are used to give the proof of the conjectured formulas for the determinant of the contravariant form on the highest weight representations of the Neveu-Schwarz and Ramond algebras. Further results on the representation theory of the latter are derived from the determinant formulas.Partially supported by the National Science Foundation through the Mathematical Sciences Research InstitutePartially supported by NSF grant MCS-8201260     

Hilbert Modules and Stochastic Dilation of a Quantum Dynamical Semigroup on a von Neumann Algebra

A general theory for constructing a weak Markov dilation of a uniformly continuous quantum dynamical semigroup T _t on a von Neumann algebra ? with respect to the Fock filtration is developed with the aid of a coordinate-free quantum stochastic calculus. Starting with the structure of the generator of T _t , existence of canonical structure maps (in the sense of Evans and Hudson) is deduced and a quantum stochastic dilation of T _t is obtained through solving a canonical flow equation for maps on the right Fock module ?⊗Γ(L ²(ℝ₊,k ₀)), where k ₀ is some Hilbert space arising from a representation of ?^′. This gives rise to a *-homomorphism j _t of ?. Moreover, it is shown that every such flow is implemented by a partial isometry-valued process. This leads to a natural construction of a weak Markov process (in the sense of [B-P]) with respect to Fock filtration. Received: 15 June 1998/ Accepted: 4 March 1999     

A comparison of finite basis set and finite difference Hartree—Fock calculations for the open-shell (X 2Σ+) molecules BaF and YbF

A comparison is made of the accuracy with which the total electronic energy can be calculated by using the finite basis set approach (the algebraic approximation) and the finite difference method in calculations employing the Hartree—Fock model for the open shell ground (X ²Σ⁺) states of the fluorides BaF and YbF. The convergence of the calculations carried out within the algebraic approximation is monitored by employing systematically constructed basis sets of increasing size. The difference between the finite basis set and finite difference Hartree—Fock energies is 2.6μE _h for BaF and 2.8μE _h for YbF. Dipole moments determined within the algebraic approximation are also compared with the corresponding finite difference expectation values.     

Exponential representations of the canonical commutation relations

A class of representations of the canonical commutation relations is investigated. These representations, which are called exponential representations, are given by explicit formulas. Exponential representations are thus comparable to tensor product representations in that one may compute useful criteria concerning various properties. In particular, they are all locally Fock, and non-trivial exponential representations are globally disjoint from the Fock representation. Also, a sufficient condition is obtained for two exponential representations not to be disjoint. An example is furnished by Glimm's model for the :⁴: interaction for boson fields in three space-time dimensions.     

Models of Q algebra representations

This article gives a review of various straightforward models ofQ algebra representations. This is done using one and two variable function space models of theq-analogues of Lie enveloping algebras. The algebras considered are the quantum algebraU _q (su ₂ ) and aq analogue of the oscillator algebra. We present only the general framework and refer the reader to references of the joint work of the author and Willard Miller, Jr.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June 1992.     

Construction of cyclic representations of quantum algebras at q p =1 from their regular representations

Cyclic representations of maximal dimension of the quantum algebra U _q L associated with any finite-dimensional simple Lie algebra L are studied from its regular representation at q ^p =1, which is proved to be a quotient module of itself as a left module with respect to some submodules. The general theory is given after an instructive example U _q sl(2) is studied. Another explicit example U _q sl(3) is also presented.This work is supported in part by the National Natural Science Foundation of China. Author Fu is also supported by the Jilin Provincial Science and Technology Foundation of China     

Analytic Bethe ansatz for fundamental representations of Yangians

We study the analytic Bethe, ansatz in solvable vertex models associated with the YangianY(X _r ) or its quantum affine analogueU _q (X _r ⁽¹⁾ ) forX _r =B _r ,C _r andD _r . Eigenvalue formulas are proposed for the transfer matrices related to all the fundamental representations ofY(X _r ). Under the Bethe ansatz equation, we explicitly prove that they are pole-free, a crucial property in the ansatz. Conjectures are also given on higher representation cases by applying theT-system, the transfer matrix functional relations proposed recently. The eigenvalues are neatly described in terms of Yangian analogues of the semi-standard Young tableaux.     

Some monodromy representations of generalized braid groups

A flat connection on the trivial bundle over the complement inC ⁿ of the complexification of the system of the reflecting hyperplanes of theB _n,D _n Coxeter groups is built from a simple Lie algebra and its representation. The corresponding monodromy representations of the generalized braid groupsXB _n,XD _n are computed in the simplest case.     

Boson—Fermion Correspondence on the Circle via Quantum Stochastic Calculus

The quantum stochastic differential formula dB = (–1)dA, known to relateboson and fermion fields A and B, respectively, on the Fock space over L ²(R₊),is shown to hold in a modified form in a Fock space associated with the nontrivialcomplex line bundle over the circle S ¹.     

Visualization of deficiencies in approximate molecular wave functions: the orbital amplitude difference function for the matrix Hartree-Fock description of the ground state of the boron fluoride molecule

The orbital amplitude difference function is used to assess the quality of Hartree–Fock orbitals obtained by invoking the algebraic approximation for the BF ground-state. Systematic sequences of even-tempered, spherical-harmonic Gaussian-type basis functions are used to generate orbitals for which the corresponding total Hartree–Fock energy approaches the 1 μE _h level of accuracy. Exact orbitals are obtained from finite difference calculations using a grid based on spheroidal coordinates. The finite basis set approximations for the orbital are discretized. The accuracy of the discretization is assessed. For each occupied orbital a discretized representation of the orbital amplitude difference function is generated and analysed.