Decomposition of the Fock space in two-dimensional triangle and honeycomb lattice systems

We consider the symmetry group inherent in two-dimensional triangle and honeycomb lattice systems. We find analytically and numerically the character of the reducible representation for the corresponding Fock space. Using the irreducible characters and the reducible character of the representation, we decompose the Fock space explicitly. For example, we calculate the multiplicity of each irreducible representation contained in the Fock space.     

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).     

A Vanishing Theorem for Operators in Fock Space

We consider the bosonic Fock space over the Hilbert space of transversal vector fields in three dimensions. This space carries a canonical representation of the group of rotations. For a certain class of operators in Fock space, we show that rotation invariance implies the absence of terms which either create or annihilate only a single particle. We outline an application of this result in an operator theoretic renormalization analysis of Hamilton operators, which occur in non-relativistic qed.     

On the implementation of Sμ U(2) action in the irreducible representations of twisted canonical commutation relations

The natural S U(2) action on the creation and annihilation operators satisfying twisted canonical commutation relations (TCCR) is investigated. It is shown that the Fock representation is the only covariant irreducible representation of TCCR.Partially supported by the program RP.1.10.     

A converse to a theorem by Friedrichs

It is proved that the requirement of implementability of a group of canonical transformations defines a class of irreducible representations of the CAR. As a corollary a converse to Friedrichs' theorem about canonical transformations implementable in the Fock representation is obtained.     

The Two-Variable Coherent States of the Quantum Algebra suq(l,1)

By means of the k-order q-boson realizations of su_q(1,1), a new kind of q-coherent states, the two-variable coherent states, are introduced for the quantum algebra su_q(l,1).They are relative to a definite irreducible representation of su_q(1,1) and shown to satisfy a completeness relation in a subspace of q-boson's Fock space. With the help of these coherent states the z-representation, representation in functional space, for su_q(1,1) is also 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.     

Quadratic Bosonic and Free White Noises

We discuss the meaning of renormalization used for deriving quadratic bosonic commutation relations introduced by Accardi [ALV] and find a representation of these relations on an interacting Fock space. Also, we investigate classical stochastic processes which can be constructed from noncommutative quadratic white noise. We postulate quadratic free white noise commutation relations and find their representation on an interacting Fock space. Received: 23 August 1999 / Accepted: 8 December 1999     

On the 1-cohomology of Lie groups

Given a continous representation of a Lie group in a Banach space we study its 1-cohomology. We prove that the computation of the 1-cocycles can be reduced to that of the 1-cocycles of the differential of the representation. When the group is semi-simple and the representation is K-finite, we prove that the cohomology is equivalent to the cohomology of the Lie algebra representation on K-finite vectors. We prove, using Casimir operators, that there exist only a finite number of irreducible representation of a semi-simple Lie group with a non-trivial cohomology. Exemples of such representations are given.     

Bosonization of Admissible Representation of Uq(sl2) at level -1/2

We construct a representation of Uq(sl2) at level -1/2 by using the bosonic Fock spaces. The irreducible modules are obtained as the kernel of a certain operator, in contrast to the construction by Feingold and Frenkel for q = 1 where such a procedure is not necessary. We also bosonize the q-vertex operators associated with the vector representation.     

Fock space construction of the massless dipole field

We construct the Fock space representation of the free massless scalar dipole field in terms of creation and annihilation operators for the eigenvectors of the momentum operator. The Poincaré group is implemented unitarily only on a subspace of the full (positive metric) Hilbert space. The subspace possesses a hermitean, local, irreducible scalar field constructed out of the (non-hermitean) dipole field. Thus this subspace is a perfect candidate for a physical subspace of observable particles. We show that this possibility is however excluded by the fact that these particles interact with an external c-number source in a manner that violates unitarity. We illustrate our construction by applying it to the linearized Higgs model with external c-number source and examine the (non-trivial) dynamics of the dipole degrees of freedom in this case. An explicit separation of the physical degrees of freedom from the unphysical ones is presented for this interacting model.     

从二维正态分布密度函数的角度研究量子力学基本表象

利用二维正态分布密度函数和有序算符内的积分技术,简捷地得到了坐标本征态、动量本征态、坐标-动量中介表象和相干态在Fock表象中的表达式.     

Symmetrized Wavefunctions in a Quantum N-Body System

A general method for symmetrizing the generalized radial functions in a quantum N-body system to belong to the irreducible representation of the proper point symmetry group of the system is presented, based on irreducible bases in the point group space.     

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     

Connes distance of 2D harmonic oscillators in quantum phase space

We study the Connes distance of quantum states of two-dimensional (2D) harmonic oscillators in phase space. Using the Hilbert-Schmidt operatorial formulation, we construct a boson Fock space and a quantum Hilbert space, and obtain the Dirac operator and a spectral triple corresponding to a four-dimensional (4D) quantum phase space. Based on the ball condition, we obtain some constraint relations about the optimal elements. We construct the corresponding optimal elements and then derive the Connes distance between two arbitrary Fock states of 2D quantum harmonic oscillators. We prove that these two-dimensional distances satisfy the Pythagoras theorem. These results are significant for the study of geometric structures of noncommutative spaces, and it can also help us to study the physical properties of quantum systems in some kinds of noncommutative spaces.     

A driven damped harmonic oscillator in the ket-vector representation of the density operator

By virtue of the entangled-state basis and the ket-vector representation of the density operator, we solve the master equation of a driven damped harmonic oscillator. In this representation, the density operators are mapped to vectors of a two-mode Fock space whose first mode is the system mode and the second mode is a fictitious one. We derive the Glauber–Sudarshan P function of the quantum state.     

An alternative realization of 't Hooft's quantum determinism

We present an alternative to 't Hooft's mechanism relating a deterministic system to a 1D quantum oscillator. We argue that it is possible to map 't Hooft's ontological space onto an irreducible representation of a 2D isotropic oscillator. A direct group contraction procedure can then be applied to achieve the correspondence.     

Symplectic Group Representation of the Two-Mode Squeezing Operator in the Coherent State Basis

We find that the coherent state projection operator representation of the two-mode squeezing operator constitutes a loyal group representation of symplectic group, which is a remarkable property of the coherent state. As a consequence, the resultant effect of successively applying two-mode squeezing operators are equivalent to a single squeezing in the two-mode Fock space. Generalization of this property to the 2n-mode case is also discussed.     

Symplectic Group Representation of the Two-Mode Squeezing Operator in the Coherent State Basis

We find that the coherent state projection operator representation of the two-mode squeezing operator constitutes a loyal group representation of symplectic group, which is a remarkable property of the coherent state. As a consequence, the resultant effect of successively applying two-mode squeezing operators are equivalent to a single squeezing in the two-mode Fock space. Generalization of this property to the 2n-mode case is also discussed.     

Gauge invariant Lagrangian construction for massive higher spin fermionic fields

We formulate a general gauge invariant Lagrangian construction describing the dynamics of massive higher spin fermionic fields in arbitrary dimensions. Treating the conditions determining the irreducible representations of Poincaré group with given spin as the operator constraints in auxiliary Fock space, we built the BRST charge for the model under consideration and find the gauge invariant equations of motion in terms of vectors and operators in the Fock space. It is shown that like in massless case [I.L. Buchbinder, V.A. Krykhtin, A. Pashnev, Nucl. Phys. B 711 (2005) 367, hep-th/0410215], the massive fermionic higher spin field models are the reducible gauge theories and the order of reducibility grows with the value of spin. In compare with all previous approaches, no off-shell constraints on the fields and the gauge parameters are imposed from the very beginning, all correct constraints emerge automatically as the consequences of the equations of motion. As an example, we derive a gauge invariant Lagrangian for massive spin 3/2 field.