Eine phasenraumdarstellung für spinsysteme

We start with the definition of two mapping operators, one of them is the projection operator onto coherent spin states. With the help of these operators we derive a mapping theorem which defines a correspondence between the operators in spin space andc-number functions of a certain class. It is shown that this correspondence is one-to-one. The quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase space average of classical statistical mechanics. We also derive a product theorem which allows us to transcribe the equations of motion for operators into equivalent equations for thec-number functions. As an illustration of the theory, some examples are discussed.     

Fluctuations of a macro-spin in a superradiant system

Fluctuation phenomena in a superradiating atomic system are investigated in the framework of the generalized phase space method. An operator form of the superradiant master equation is mapped onto ac-number space. Evolution of the most probable path and small deviations from the path are determined. Fluctuations around the path are solved in a closed form. A remarkable enhancement of fluctuation is observed and this is recognized as a sort of anomalous fluctuation around an unstable point.     

The Heisenberg ferromagnet in spin coherent state representation

A new approach to Heisenberg ferromagnet using the spin coherent state representation is developed. The differential operator representation of spin angular momentum operators is used to derive thec-number analogs of the basic quantum mechanical equations, viz., the Schrödinger, Bloch and Liouville equations for the Heisenberg ferromagnet. As an important illustration of our formulation, which has noad hoc assumptions and does not use any boson representation, the excitation spectrum for one, two and three spin waves is obtained. In these cases it is also shown that eigenvalue spectrum can be obtained by completely ignoring the kinematical interactions.     

Green Functions of the Stationary Schrödinger Equation and Tomographic Representation of Quantum Mechanics

Invertible maps of operators of quantum observables onto functions of c-number arguments and their associative products are reviewed. In particular, the symplectic tomography map is discussed and an expression connecting an arbitrary operator and its tomographic symbol is written down. This formula is applied to obtain explicit expressions for tomographic symbols, which are symplectic tomograms of Green functions of stationary and nonstationary Schrödinger equations written for the case of harmonic oscillator. The connection between the so-called classical propagator (X,,,t,X,,,0) and the tomographic symbol of the evolution operator of nonstationary Schrödinger equation is found. The spin tomography is presented as a map of operators acting in spinor space onto functions of c-arguments. As an example, the spin located in a magnetic field is considered and the tomographic symbol of resolventa is obtained. Tomographic symbols of hermitian conjugate operators are shown to be complex conjugate functions.     

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.     

Theory of high-temperature superconductivity in cuprates

A microscopic theory of electronic spectrum and superconducting pairing in the high-temperature cuprate superconductors is presented. The theory is based on consideration of strong electron correlations within the Bogolyubov polar model. The Dyson equation is derived by using the equation of motion method for the thermodynamic Green functions in terms of the Hubbard operators. The self-energy is evaluated in the noncrossing approximation for electron scattering on spin and charge fluctuations induced by kinematic interaction. The theory demonstrates that a strong Coulomb repulsion results in the anomalous electronic spectrum and unconventional (d-wave) superconducting pairing with high T _c mediated by the antiferromagnetic exchange and spin fluctuations.     

Symmetries of theq-difference heat equation

The symmetry operators of aq-difference analog of the heat equation in one space dimension are determined. They are seen to generate aq-deformation of the semidirect product of sl(2, ) with the three-dimensional Weyl algebra. It is shown that this algebraic structure is preserved if differentq-analogs of the heat equation are considered. The separation of variables associated to the dilatation symmetry is performed and solutions involving discreteq-Hermite polynomials are obtained.     

Diagonalization of the spin part of the exchange hamiltonian in the second-quantization method for composite particles

Using the atomic second-quantization procedure an explicit expression is derived for the operator of exchange interaction between two atoms a and b in a state of any value of their atomic orbital and spin momenta, in the approximation of one and two pairs of exchanging electrons and without assuming the orthogonality between the orbitals of a and b. The obtained exchange interaction operator diagonalized in spin space is a second-order polynomial in the scalar product ?_a · ?_b. The coefficients of the polynomial are given operators operating in orbital space only. The possible physical implications are discussed.     

Gluon condensation and the bag model of hadrons

Migdal's scheme for gluon condensation is applied to a model where gluons are dynamically confined by coloured Higgs fields and are also subjected to ac-number colour field due to external “semiclassical” quarks. We show that, no matter how strong the external field becomes, stabilization of the gluonic ground state is guaranteed by the formation of an inhomogeneous gluon condensate. The contribution to the total energy from the condensate is computed semiclassically using a variational approximation. A physical interpretation of the results obtained is given in terms of a renormalization of the MIT bag model input parameters in particular the zero point energy coefficientZ and the bag constantB.

     

Systematic classical continuum limits of integrable spin chains and emerging novel dualities

We examine certain classical continuum long wave-length limits of prototype integrable quantum spin chains. We define the corresponding construction of classical continuum Lax operators. Our discussion starts with the XXX chain, the anisotropic Heisenberg model and their generalizations and extends to the generic isotropic and anisotropic gln$g{l}_n$ magnets. Certain classical and quantum integrable models emerging from special “dualities” of quantum spin chains, parametrized by c-number matrices, are also presented.     

Renormalized Perturbation Expansion in Kinetic Theory

Ladder summation techniques are applied to the recently developed c-number diagram expansion for kinetic equations describing the relaxation of quantum fluids. By way of the resummation, the kinetic equations are reformulated in terms of a T-matrix which describes the scattering of two particles influenced by the remaining particles of the system via the particle statistics (Bose or Fermi). At sufficiently high temperatures (Maxwell-Boltzmann statistics), the T-matrix introduced coincides with the usual T-matrix of conventional two-body scattering theory in free space. In low-temperature Fermi systems, the T-matrix differs from the reaction matrix of the Brueckner-Goldstone theory because the “healing” property of the two-body wave function does not obtain.     

The Supersymmetry and Eigen Energy Spectrum of a Charged Dirac Particle in a Uniform Constant Magnetic Field

Based on the opinion that the γ-matrices in Dirac equation have structure and are decomposable, we decompose the γ-matrices into the direct product of the operators in the spin space and the particle-antiparticle space. By using this method, we attain a complete set of commutative operators, a set of quantum numbers and the correspondingly eigen solutions of the Hamiltonian for a charged Dirac particle moving in a uniform constant magnetic field. In addition, the dynamic supersymmetry of the Hamiltonian is unveiled. Spin symmetry breaking and particle-antiparticle symmetry breaking are discussed, and the supersymmetric group operator of the degenerate spin subspace resulting from the spin residual supersymmetry is found.     

Theq-deformed virasoro algebra

Theq-deformed Virasoro algebra is studied in the context of the string theory. It is shown that the anomaly term is not ac-number and depends strongly on the deformation parameterq. Supported by the Algerian Ministry of High Education and Research under contract N^oD2501/17/01/93. Permanent address     

Exact generalized Fokker-Planck equation for arbitrary dissipative and nondissipative quantum-systems

We start from a density matrix equation in its most general form. It comprises the action of external fields on the system, internal interactions, as well as the action of dissipative mechanisms (heat-baths or reservoirs), which may be Markoffian or non-Markoffian. We then define a distribution function of a type introduced previously byHaken, Risken, Weidlich for atoms. This distribution function,f, which is now formulated quite generally with aid of projection operators,P _ik , establishes a connection between theP _ik 's and classical variablesv _ik . By means off it is possible to exactly calculate all quantum mechanical expectation values by purec-number procedures. If the basic density matrix equation is Markoffian, it is even possible to calculate all time-ordered multitime averages byc-number procedures usingf, as had been demonstrated byHaken, Risken andWeidlich. In the present paper we derive in an explicit way an exactc-number partial differential equation forf. It contains derivatives of arbitrarily high order. In important classes of problems, it can be reduced to an ordinary FokkerPlanck equation, however. Our new equation has many applications, e.g. in the quantum theory of lasers, nonlinear quantum optics, spinresonance, and spin-wave-theory, as will be demonstrated in forthcoming papers. We wish to thank Prof. W.Weidlich and Dipl. Phys. H.Vollmer for several valuable discussions. In addition, H.Vollmer has kindly checked our calculations.     

Representations of aq-deformed Heisenberg algebra

We extend the symmetric operators of theq-deformed Heisenberg algebra to essentially self-adjoint operators. On the extended domains the product of the operators is not defined. To represent the algebra we had to enlarge the representation and we find a Hilbert space representation of the deformed Heisenberg algebra in terms of essentially self-adjoint operators. The respective diagonalization can be achieved by aq-deformed Fourier transformation.     

Operator orderings and functional formulations of quantum and stochastic dynamics

We study the connection between operator ordering schemes and thec-number formulations of quantum mechanics, which are based on generating functionals and functional integrals. We show by explicit construction that different operator ordering schemes are related to different functional and functional integral formulations of quantum mechanics. The results of these considerations are applied to classical non-linear stochastic dynamics by using the formal analogy between the Fokker-Planck equation and the Schrödinger equation.     

Functional quantum statistics of light propagation in a two-level system

The functional Fokker-Planck formalism developed in a preceding paper is applied to the problem of a radiation field propagating in a medium, which contains resonant two-level atoms. Besides the electromagnetic field also the medium is described by continuous space dependent fields. We give the masterequation and transform it into ac-number functional differential equation for a characteristic functional. This equation is reduced considerably by the projection onto one dimension and the introduction of the diffusion approximation. It forms a solid basis for the study of all types of light propagation in resonant media including classical and quantum noise. We give an approximate solution of this equation by considering the problem of an externally pumped optical transmission line, in the case that saturation effects are absent. The spectral function of the electric field strength is obtained which describes a statistical mixture of photons with the quasiparticles of the polarization field. It shows the onset of a condensation of the quasiparticles into a single state. Self excitation of the transmission line is obtained at a certain threshold of the atomic inversion. This threshold is characterized by a finite occupation number of one single quasiparticle state. The influence of a finite length of the transmission line is briefly considered.     

Static,c-number solutions of a two dimensional sine-Gordon-like field equation

A sine-Gordon type scalar field with a \(\cos ^4 [(\sqrt {\bar \lambda } \phi )/m]\) potential in two dimensions is studied and a static,c-number, stable, finite energy solution obtained. The intersoliton potential is evaluated for this model and is shown to be long range, attractive and ‘strong’ in the weak coupling regime. A positronium-like spectrum of bound states of the soliton-antisoliton pair is derived using this potential. The coupling with a massless quark field is discussed and an exact, stable and analytic solution of the spinor field exhibiting confinement in one space dimension is obtained.