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.     

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.     

Realisations of the representations of para-Fermi algebras—Part IV: Fock spaces of para-Bose and para-Fermi operators

The representations of the para-Fermi algebra in the Fock spaces of para-Bose and para-Fermi operators are constructed. The unitary equivalence of different representations is proved. The Bardeen-Cooper-Schrieffer pair creation and annihilation operators and the four fermion interaction appear as particular realisations of the para-Fermi algebra. The para-Fermi algebra representations in quantum mechanics are discussed.     

Quantization of U q [so(2n + 1)] with deformed para-Fermi operators

The observation thatn pairs of para-Fermi (pF) operators generate the universal enveloping algebra of the orthogonal Lie algebra so(2n + 1) is used in order to define deformed pF operators. It is shown that these operators are an alternative to the Chevalley generators. With this background U _q [so(2n + 1)] and its Cartan-Weyl generators are written down entirely in terms of deformed para-Fermi operators.     

Fock space representations of affine lie algebras and integral representations in the Wess-Zumino-Witten models

Fock space representations of affine Lie algebras are studied. Explicit forms of correction terms adding to the currentsF _i (z) are determined. It is proved that the Sugawara energy-momentum tensor on the Fock spaces is quadratic in free bosons. Furthermore, screening operators are constructed. This implies the existence of generalized hypergeometric integrals satisfying the Knizhnik-Zamolodchikov equation.     

Unitary Quantization and Para-Fermi Statistics of Order 2

We consider the relationship between the unitary quantization scheme and the para-Fermi statistics of order 2. We propose an appropriate generalization of Green’s ansatz, which has made it possible to transform bilinear and trilinear commutation relations for the creation and annihilation operators for two different para-Fermi fields φ_a and φ_b into identities. We also propose a method for incorporating para-Grassmann numbers ξ_k into the general unitary quantization scheme. For the parastatistics of order 2, a new fact has been revealed: the trilinear relations containing both para-Grassmann variables ξ_k and field operators ak and bm are transformed under a certain reversible mapping into unitary equivalent relations in which commutators are replaced by anticommutators, and vice versa. It is shown that this leads to the existence of two alternative definitions of the coherent state for para-Fermi oscillators. The Klein transformation for Green’s components of operators a_k and b_m is constructed in explicit form, which enabled us to reduce the initial commutation rules for the components to the normal commutation relations for ordinary Fermi fields. We have analyzed a nontrivial relationship between the trilinear commutation relations of the unitary quantization scheme and the so-called Lie supertriple system. The possibility of incorporating the Duffin–Kemmer–Petiau theory into the unitary quantization scheme is discussed briefly.     

Algebra of Screening Operators for the Deformed W n Algebra

We construct a family of intertwining operators (screening operators) between various Fock space modules over the deformed W _n algebra. They are given as integrals involving a product of screening currents and elliptic theta functions. We derive a set of quadratic relations among the screening operators, and use them to construct a Felder-type complex in the case of the deformed W ₃ algebra. Received: 3 March 1997 / Accepted: 20 May 1997     

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     

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.     

The Squeezing Effect of Three-Mode Operator as an Extension from Two-Mode Squeezing Operator

We theoretically study the squeezing effect in a 3-wave mixing process, generated by the operator S₃ o exp[m(a₁a₂-a₁^fa₂^f)+n(a₁a₃-a₁^fa₃^f)]S_{3}\equiv \exp[\mu(a_{1}a_{2}-a_{1}^{\dagger}a_{2}^{\dagger})+\nu(a_{1}a_{3}-a_{1}^{\dagger}a_{3}^{\dagger})]. The corresponding 3-mode squeezed vacuum state in Fock space and its uncertainty relation are presented. It turns out that S ₃ may exhibit enhanced squeezing. By virtue of integration within an ordered product (IWOP) of operators, we also give the S ₃’s normally ordered expansion. Finally, we calculate the Wigner function of 3-mode squeezed vacuum state by using the Weyl ordering invariance under similar transformations.     

Comments onq-algebras

A critical study of some elementary aspects ofq-algebras is presented. The results are: (i) theq-algebras are related to para-Bose (para-Fermi) algebras only when both reduce to the usual Bose (Fermi) case, (ii) after performing a linear transformation of the operatorsA andA that satisfy theq-algebra relation AA, a generalized version of Penney's theorem (in the sense that the new operators satisfy noncanonical commutation and anticommutation relations) is obtained, (iii) the spectrum of one of the Hamiltonians of the system is obtained from the correspondence principle, and (iv) a whole family ofq-algebra Hamiltonians is exhibited. This family has the property that the noncanonical commutation relation is stable.     

A remark on the integration of Schrödinger equation using quantum Itô's formula

When the potential is the Fourier transform of a totally finite complex-valued measure, a formula for the one-parameter unitary group generated by the Schrödinger operator in L ² (IR ⁿ ) is obtained entirely in terms of the basic field operators in a suitable Fock space by means of quantum stochastic calculus.     

Fock-type representation of the Lie superalgebraA(0, 1)

A Fock space of two pairs of generalized creation and annihilation operators is constructed. These operators belong to the odd part of the Lie superalgebraA(0, 1) and generate the whole algebra. The creation and annihilation operators define in the Fock space an infinite-dimensional irreducible representation of the algebraA(0, 1).     

Quantum Vertex Representations via Finite Groups¶and the McKay Correspondence

We establish a q-analog of our recent work on vertex representations and the McKay correspondence. For each finite group Γ we construct a Fock space and associated vertex operators in terms of wreath products of $Γ×ℂ^× and the symmetric groups. An important special case is obtained when Γ is a finite subgroup of SU ₂, where our construction yields a group theoretic realization of the representations of the quantum affine and quantum toroidal algebras of ADE type. Received: 17 August 1999 / Accepted: 2 December 1999     

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.     

The extremal microscopic coherent boson states

In a recent paper, the Fock-normal (=microscopic) fully-coherent statesL _L ⁽⁾ (L denotes the factorizing linear form) have been completely characterized, and the existence of nonpure elements in the extreme boundary of the weak-*-compact, convex setL _L has been shown. This letter is devoted to an analysis of the extremal fully-coherent states, especially those which are not pure states. An affine isomorphism is constructed betweenL _L ⁽⁾ and a certain convex subset of the normal completely positive maps on the bounded operators on Fock space. Then the extreme boundary ofL _L ⁽⁾ is determined by results and techniques from the theory of completely positive operators.     

Riesz Transforms on Deformed Fock Spaces

The Fock Von Neumann algebra , equipped with its canonical trace τ, is spanned by n hermitian operators acting on a Hilbert Fock space some commutation relations between and are defined by the n×n hermitian matrix A. We define a Riesz transform , where is the number operator, ∇^′ is aninner derivation (unbounded in general) and . Let 1<p<∞. We prove that is equivalent to for every with null trace, with constants which do not depend on n. Received: 24 November 1998 / Accepted: 2 March 1999     

A Kind of Four-Mode Nonlinear Entangled State Representation in Quantum Optics

Based on the technique of integration within an ordered product of nonlinear bosonic operators, we construct a new four-mode nonlinear entangled state | α,β,γ〉 _λ,μ in 4-mode Fock space, which can make up a complete set. Its properties and applications are discussed. A possible scheme to generate this state is also presented.     

Index formulas for generalized Wiener-Hopf operators and boson-fermion correspondence in 2N dimensions

The kernels of operators associated with special chiral gauge transformations (kinks) in the 2N-dimensional Dirac theory are explicitly determined. The result is used to obtain index formulas for Fredholm operators corresponding to continuous chiral gauge transformations. Moreover, the Fock space quadratic forms corresponding to the kinks are proved to converge to the Dirac field as the kink size goes to zero. It is also shown that forN 1, 2(mod 4) the Majorana field can be reached in a similar fashion.Work supported by the Netherlands Organisation for the Advancement of Research (NWO)     

On the three body contributions to the ground state energy of nuclear matter

The theory of nuclear matter is investigated by means of the method of unitary transformations in the special case of point transformations. The induced three body forces are constructed, their contributions to the ground state energy of nuclear matter are given in first order perturbation theory, and the connexion with Jastrow's procedure is shown. A first numerical estimate of the three body contributions to the energy per particle gives approximately 1 MeV in the physical density range. For higher values of the Fermi momentum (k _F ≈2 fm^?1) the contributions increase rapidly. Generally the induced three body forces cannot be neglected if one wants to calculate the correct saturation data.