Bargmann Fock Representation Research Articles

Superalgebraic structure of Lorentz transformations

We constructed superalgebraic representation of the algebra of second quantization of spinors. It is a fermionic analog of the Bargmann-Fock representation and is based on the use of the density of Grassmann variables in impulse space and their derivatives. We showed that Dirac matrices and Lorentz transformation generators can be expressed in terms of such densities. Lorentz transformations are considered as Bogolyubov transformations of creation and annihilation operators. We constructed superalgebraic form of the Dirac equation and the vacuum state vector. We showed that in the superalgebraic form of the complex Clifford algebra generators corresponding to Dirac gamma matrices are not equivalent. Clifford vector corresponding to diagonal matrix γ0 annihilates the vacuum, and the remaining ones give nonzero values. This means that there is asymmetric direction corresponding to the time axis.

The Quantization of the E ⊗ e Jahn-Teller Hamiltonian.

The E ⊗ e Jahn-Teller Hamiltonian in the Bargmann-Fock representation gives rise to a system of two coupled first-order differential equations in the complex field, which may be rewritten in the Birkhoff standard form. General leapfrog recurrence relations are derived, from which the quantized solutions of these equations can be obtained. The results are compared to the analogous quantization scheme for the Rabi Hamiltonian.

Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups

In a first part, we are concerned with the relationships between polynomials in the two generators of the algebra of Heisenberg—Weyl, its Bargmann—Fock representation with differential operators and the associated one-parameter group.Upon this basis, the paper is then devoted to the groups of Riordan matrices associated to the related transformations of matrices (i.e., substitutions with prefunctions). Thereby, various properties are studied arising in Riordan arrays, in the Riordan group and, more specifically, in the "striped" Riordan subgroups; further, a striped quasigroup and a semigroup are also examined. A few applications to combinatorial structures are also briefly addressed in the Appendix.

An exactly solvable system from quantum optics

We investigate a generalisation of the Rabi system in the Bargmann–Fock representation. In this representation the eigenproblem of the considered quantum model is described by a system of two linear differential equations with one independent variable. The system has only one irregular singular point at infinity. We show how the quantisation of the model is related to asymptotic behaviour of solutions in a vicinity of this point. The explicit formulae for the spectrum and eigenfunctions of the model follow from an analysis of the Stokes phenomenon. An interpretation of the obtained results in terms of differential Galois group of the system is also given.

Q-analytic functions, fractals and generalized analytic functions

We introduce a new class of complex functions of complex argument which we call q-analytic functions. These functions satisfy q-Cauchy–Riemann equations and have real and imaginary parts as q-harmonic functions. We show that q-analytic functions are not the analytic functions. However some of these complex functions fall in the class of generalized analytic functions. As a main example we study the complex q-binomial functions and their integral representation as a solution of the D-bar problem. In terms of these functions the complex q-analytic fractal, satisfying the self-similar q-difference equation is derived. A new type of quantum states as q-analytic coherent states and corresponding q-analytic Fock–Bargmann representation are constructed. As an application, we solve quantum q-oscillator problem in this representation, and show that the wave functions of quantum states are given by complex q-binomials.

PROBABILITY MEASURES ON ℂ ARISING FROM THE JACOBI–SZEGÖ PARAMETERS FOR CONTINUOUS DUAL HAHN POLYNOMIALS

In this paper, we shall construct probability measures on ℂ, which can be expressed by the Mellin convolution product of the modified Bessel functions. Our measures on ℂ are related to the Jacobi–Szegö parameters for the continuous dual Hahn polynomials Sn(x2, a, b, c) under the special choice of parameters a, b, c. This paper contains new materials which go further than our previous results in Ref.7. The most interesting thing is that the Mellin convolution of two modified Bessel functions can again be expressed in terms of the modified Bessel functions, by choosing parameters a, b, c appropriately. The origin of our research in this direction goes back to the Bargmann–Fock representation of the classical non-Gaussian random variables.5–8 Our results would have a potential to be related to the higher powers of creation and annihilation operators acting on a new class of interacting Fock spaces.

Coherent-State Path Integrals and Brownian Motions

A stochastic approach to the rigorous foundation of the coherent-state (phase-space) path integral is given. Stochastic integrals and some generalizations of the Feynman–Kac theorem are used for this purpose. In this approach, quantum mechanics is described in terms of the Fock–Bargmann representation; a classical Hamiltonian is related to the corresponding quantum Hamiltonian on the Fock–Bargmann space, seen as a Hilbert subspace of \(L^{2}({\bf R}^{2})\). The coherent-state path integral is realized as a conditional expectation of a stochastic process defined by the exponential of the Fisk–Stratonovich integral of the fundamental 1-form along a path of Brownian motion on the phase space \({\bf R}^{2}\).

Some aspects of a weak Weyl–Heisenberg algebra deformation

In the weak deformation (WD) approximation of the Weyl–Heisenberg algebra, the corresponding generalized coherent states and displacement operator are constructed. It is shown that those states, and contrary to the non-deformed Weyl-Heisenberg algebra, are not eigenstates of the annihilation operator. Moreover, and as an alternative to the Chaïchian et al. Q-deformed path integral approach (where Q is the deformation parameter), using the Bargmann Fock representation, we propose in the WD approximation, a general simple formalism. As an application, we calculate the propagator and the wave function of the harmonic oscillator.PACS Nos.: 03.65.Fd, 31.15.Kb

Macroscopic entangled states

The interaction between an ensemble of two-level identical atoms and three-mode light when one of the modes is classical and its Rabi frequency far exceeds the effective Rabi frequencies of the remaining quantized modes is considered. An effective Hamiltonian that describes a process like non-degenerate down conversion is found and the statistics of the quantized modes is discussed with the use of an exact solution of the Schrodinger equation for a field in the Fock–Bargmann representation. It is found that the occupation number of each quantized mode can depend exponentially on the number of atoms and the light state may be squeezed. The squeezing obtained is nearly perfect and results in a continuous analogue of the maximally entangled Einstein–Podolski–Rosen state of the light with a macroscopically large number of photons.

Characterization of Multidimensional Euclidean Landau States by Coherent State Transformations

We deal with a family of multidimensional generalized coherent states obtained using the Fock–Bargmann representation of the Heisenberg group. We prove that the ranges of the corresponding coherent state transformations coincide with the spaces of bound states of an even-dimensional analogue of a charged particle in a uniform magnetic field. This provides a new characterization of Euclidean Landau states.

Generalized intelligent states of the [formula omitted] algebra

Schrödinger–Robertson uncertainty relation is minimized for the quadrature components of Weyl generators of the algebra su ( N ) . This is done by determining explicit Fock–Bargmann representation of the su ( N ) coherent states and the differential realizations of the elements of su ( N ) . New classes of coherent and squeezed states are explicitly derived.

Algebraization of Spectral Problems in the Bargmann—Fock Representation

Examples of quasi-exactly solvable hamiltonians in the Bargmann—Fock representation are given. The existence of an invariant subspace is studied as a result of hidden symmetry of the spectral problem.

Z3-graded Grassmann variables, parafermions and their coherent states

A relation between the Z 3-graded Grassmann variables and parafermions is established. Coherent states are constructed as a direct consequence of such a relationship. We also give the analog of the Bargmann–Fock representation in terms of these Grassmann variables.

Microscopic calculation of nuclear systems involving nonspherical clusters

The algebraic version of the resonating-group method is extended to cover cases where one of the clusters has an open p shell. A basis of Pauli-allowed states labeled with the symmetry indices of irreducible representations of the SU(3) group is constructed in the Fock-Bargmann representation. Dynamical variables that describe nonspherical degrees of freedom of the cluster are introduced. The matrix elements of the nucleon-nucleon interaction are analytically found by using the operator-representation method. The nuclear system 9Li+n is taken as an example.

Generalized intelligent states for an arbitrary quantum system

Generalized Intelligent States (coherent and squeezed states) are derived for an arbitrary quantum system by using the minimization of the so-called Robertson-Schr\"odinger uncertainty relation. The Fock-Bargmann representation is also considered. As a direct illustration of our construction, the P\"oschl-Teller potentials of trigonometric type will be shosen. We will show the advantage of the Fock-Bargmann representation in obtaining the generalized intelligent states in an analytical way. Many properties of these states are studied.

Fisher Information Matrix of Husimi Distribution

We calculate the Fisher information matrix of Husimi distribution in the Fock–Bargmann representation. It turns out that the Fisher information of the position and that of the momentum move in opposite directions, and that a weighted trace of the Fisher information matrix is a constant independent of the Husimi distribution. This may be interpreted as a kind of uncertainty relation (in the spirit of Heisenberg uncertainty principle) from the statistical inference point of view.

A simple proof of Wehrl's conjecture on entropy

Based on the hypercontractivity of the free harmonic oscillator Hamiltonian and the canonical commutation relation, we present a simple proof of Wehrl's conjecture on the Shannon entropy of phase space probability densities in the Fock-Bargmann representation.

Multimode q-Oscillator Algebras with q2(k+1) = 1 and Their Bargmann–Fock Representations

A countably infinite class of multimode q-oscillator algebrasAk;m (k = 1, 2, ...,∞ m = 1, 2, ...) is obtained with the aid of the R-matrix method in quantumgroup theory for q2(k+1) = 1. The related Fock spaces are given and they showthat the q-particle systems described by Ak;m obey a generalized Pauli exclusionprinciple. The algebras Ak;m are represented on a kind of q-holomorphic functionspaces B(¯η)k;m which are generalizations of the usual Bargmann–Fock spaceswith many Grassmann variables and have Hilbert space structures with the scalarproduct given by an algebraically defined integral. When taking k = 1 or k →∞, all of the above are reduced to the corresponding results for the usual multimodefermion and boson systems, respectively.

SQUEEZED STATES AND TWO-PARAMETER QUANTUM GROUPS

We investigate two-parameter deformed Weyl–Heisenberg algebra in the Fock–Bargmann representation. The commutator [aqp,āqp] acts like squeezing operator on the space of the entire analytic functions. We find that the second parameter gets absorbed into the first parameter, yielding the same result for squeezing as in the case of single parameter study.

Functional Representations for Fock Superalgebras

The Fock space of bosons and fermions and its underlying superalgebra are represented by algebras of functions on a superspace. We define Gaussian integration on infinite-dimensional superspaces, and construct super-analogs of the classical function spaces with a reproducing kernel — including the Bargmann–Fock representation — and of the Wiener–Segal representation. The latter representation requires the investigation of Wick ordering on Z2-graded algebras. As application we derive a Mehler formula for the Ornstein–Uhlenbeck semigroup on the Fock space.