A Class of Representations of the ∗-Algebra of the Canonical Commutation Relations over a Hilbert Space and Instability of Embedded Eigenvalues in Quantum Field Models

Abstract

In models of a quantum harmonic oscillator coupled to a quantum field with a quadratic interaction, embedded eigenvalues of the unperturbed system may be unstable under the perturbation given by the interaction of the oscillator with the quantum field. A general mathematical structure underlying this phenomenon is clarified in terms of a class of Fock space representations of the ?-algebra of the canonical commutation relations over a Hilbert space. It is also shown that each of the representations is given as a composition of a proper Bogolyubov (canonical) transformation and a partial isometry on the Fock space of the representation.     

The dynamical model and quantization of the Schwarzschild black hole

The mass of the Schwarzschild black hole, an observable quantity, is defined as a dynamical variable, while the corresponding conjugate is considered as a generalized momentum. Then a two-dimensional phase space is composed of the two variables. In the two-dimensional phase space, a harmonic oscillator model of the Schwarzschild black hole is obtained by a canonical transformation. By this model, the mass spectrum of the Schwarzschild black hole is firstly obtained. Further the horizon area operator, quantum area spectrum and entropy are obtained in the Fock representation. Lastly, the wave function of the horizon area is derived also. Supported by the National Natural Science Foundation of China (Grant No. 10773002) and the Natural Research Foundation of Heze University (Grant No. XY05WL02)     

谐振子产生算符和湮没算符的逆算符

本引入谐振子产生算符和湮没算符的逆算符，导出它们在Ｆｏｃｋ空间的表达式，并给出了一些简单应用。     

Parametric excitation of oscillatory states and symmetry in phase space

We study the excitation of nonlinear dissipative oscillator under influence of a monochromatic force at the level of a few quanta. With this purpose we consider an optical parametric oscillator combined with phase-modulation in which the oscillatory mode is excited through down-conversion process under a monochromatic laser field. The temporal Rabi oscillations of Fock states as well as the properties of oscillatory mode in phase space are studied with use of the Wigner functions.     

新纠缠态表象对耦合2谐振子系统的应用

我们研究一个耦合2谐振子系统的本征态问题。我们构造了由算子（x1+p2）和（x2+p1）的共同本征态组成的新纠缠态表象︱γ>。在︱γ>表象得到了系统哈密顿的本征值和本征态。同样的问题用二次量子化表象进行了研究。我们发现在Fock空间，二次量子化表象可以被用来推得本征态的正规积表示。特别是发现了系统基态为广义2模压缩态     

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).     

Normally-Ordered Time Evolution Operator for Mass-Varying Harmonic Oscillator and Wigner Function of Squeezed Number State

For investigating dynamic evolution of a mass-varying harmonic oscillator we constitute a ket-bra integration operator in coherent state representation and then perform this integral by virtue of the technique ofintegration within an ordered product of operators. The normally orderedtime evolution operator is thus obtained. We then derive the Wigner functionof$u(t)| n>, where | n> is a Fock state, which exhibits a generalized squeezing, the squeezing effect is related to the varying mass with time.     

Boson Realization of Representations for Quantum Groups and Its Differential Correspondence in Bargmann Space

By constructing the q-analogue of the Heisenberg-Weyl algebra in terms of usual creation and annihilation operators of boson states in the Fock space, the boson realization method recently suggested^[7-11] is generalized to obtain a class of representations of quantum group in the Fock space. The q-deformed differential realization of quantum groups proposed by Alvarez-Gaume is derived by making use of the boson realization in this paper.     

Irreducibility and Compositeness in q-Deformed Harmonic Oscillator Algebras

A study of the reducibility of the Fock space representation of the q-deformed harmonic oscillator algebra for real and root of unity values of the deformation parameter is carried out by using the properties of the Gauss polynomials. When the deformation parameter is a root of unity, an interesting result comes out in the form of a reducibility scheme for the space representation which is based on the classification of the primitive or nonprimitive character of the deformation parameter. An application is carried out for a q-deformed harmonic oscillator Hamiltonian, to which the reducibility scheme is explicitly applied.On leave from     

Permutation invariant algebras, a Fock space realization and the Calogero model

We study permutation invariant oscillator algebras and their Fock space representations using three equivalent techniques, i.e. (i) a normally ordered expansion in creation and annihilation operators, (ii) the action of annihilation operators on monomial states in Fock space and (iii) Gram matrices of inner products in Fock space. We separately discuss permutation invariant algebras which possess hermitean number operators and permutation invariant algebras which possess non-hermitean number operators. The results of a general analysis are applied to the -extended Heisenberg algebra, underlying the M-body Calogero model. Particular attention is devoted to the analysis of Gram matrices for the Calogero model. We discuss their structure, eigenvalues and eigenstates. We obtain a general condition for positivity of eigenvalues, meaning that all norms of states in Fock space are positive if this condition is satisfied. We find a universal critical point at which the reduction of the physical degrees of freedom occurs. We construct dual operators, leading to the ordinary Heisenberg algebra of free Bose oscillators. From the Fock-space point of view, we briefly discuss the existence of a mapping from the Calogero oscillators to the free Bose oscillators and vice versa. Received: 26 July 2001 / Revised version: 9 January 2002 / Published online: 12 April 2002     

An SU(1,1) Boson Realization Involved in Solving the Three Coupled Harmonic Oscillators' Problem

We find that a three-mode boson realization of the SU(1,1) Lie algebra is involved in solving the three coupled harmonic oscillators' problem. Tire unitary operator U, which is found to be able to transform the Fock space of a three-dimcrrsional isotropic Irarmonic oscillator into the space in which the Hamiltonian of three coupled oscillators is diagonized, is further decomposed as a quantum-mechanical rotation in three-mode Hilbert space followed by an SU(1,1) squeezing transformation. The coordinate representation of this SU(1,1) unitary operator is obtained.     

有限维q非谐振子广义相干态振幅N次方压缩

利用Zhang等人提出的单模辐射场的振幅N次方压缩理论,研究了有限维希尔伯特空间中单参数q变形非简谐振子广义相干态的振幅N次方压缩效应.结果发现:该量子光场的确存在场的振幅N次方压缩效应,其压缩条件分别与相位角Φ、维度参数s、压缩幂次N、平均光子数的方根r和变形参数q相关;这一结果与无限维希尔伯特空间单参数q变形广义相干态的情形截然不同.     

Generalized Bargmann Representation of Spin Coherent State

Based on the conclusion that the generalized Bargmann representation of a two-mode Fock state is a two-variable Hermite polynomial function [Hong-Yi Fan and Jun-hua Chen, Phys. Lett. A303 (2002) 311] we derive the generalized Bargmann representation of the spin coherent state and some new relations in the generalized function space.     

Quantum phases for a generalized harmonic oscillator

An effective Hamiltonian for the generalized harmonic oscillator is determined by using squeezed state wavefunctions. The equations of motion over an extended phase space are determined and then solved perturbatively for a specific choice of the oscillator parameters. These results are used to calculate the dynamic and geometric phases for the generalized oscillator with this choice of parameters.      

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.     

Universal and deterministic manipulation of the quantum state of harmonic oscillators: a route to unitary gates for Fock state qubits

We present a simple quantum circuit that allows for the universal and deterministic manipulation of the quantum state of confined harmonic oscillators. The scheme is based on the selective interactions of the referred oscillator with an auxiliary three-level system and a classical external driving source, and enables any unitary operations on Fock states, two by two. One circuit is equivalent to a single qubit unitary logical gate on Fock states qubits. Sequences of similar protocols allow for complete, deterministic, and state-independent manipulation of the harmonic oscillator quantum state.     

New Parametrized Entangled State Representations and Their Applications

We construct two new kinds of parametrized entangled states in two-mode Fock space. Using the technique of integration within an ordered product of operators, we prove that they span the complete space. Their applications to solving dynamic problems and finding new generalized squeezing are addressed.     

Solving the Schrödinger equation by reduction to a first-order differential operator through a coherent states transform

The Legendre transform expresses dynamics of a classical system through first-order Hamiltonian equations. We consider coherent state transforms with a similar effect in quantum mechanics: they reduce certain quantum Hamiltonians to first-order partial differential operators. Therefore, the respective dynamics can be explicitly solved through a flow of points in extensions of the phase space. This generalises the geometric dynamics of a harmonic oscillator in the Fock space. We describe all Hamiltonians which are geometrised (in the above sense) by Gaussian and Airy beams and write down explicit solutions for such systems.     

Periodic trajectories in the Hilbert space and generalized semi-classical bound states of many-body systems

Using a stationary phase approximation to calculate a functional integral defined on continuous overcomplete sets of vectors of the Hilbert space, one derives a generalized semi-classical quantization condition for periodic trajectories in the Hilbert space. This quantization condition is interpreted in terms of a variational principle. Application to the time dependent Hartree—Fock approximation is presented.     

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.