Volume 48, Issue 2, February 2007
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Bogoliubov Hamiltonians and oneparameter groups of Bogoliubov transformations
View Description Hide DescriptionOn the bosonic Fock space, a family of Bogoliubov automorphisms corresponding to a strongly continuous oneparameter group of symplectic maps is considered. We give conditions that guarantee it to be implemented by a strongly continuous oneparameter group of unitary operators. The generator of such will be called a Bogoliubov Hamiltonian. Given , a Bogoliubov Hamiltonian is defined up to an additive constant. We introduce two kinds of Bogoliubov Hamiltonians: type I, characterized by vanishing of the expectation value at the vacuum, and type II, characterized by the fact that its infimum equals zero. We give conditions so that they are well defined. We show that there exist cases when only is well defined, even though the classical Hamiltonian is positive (which may be interpreted as a kind of an infrared catastrophe), and when only is well defined (which means that one needs to introduce an infinite counterterm in the formula for the Hamiltonian).

Neumark operators and sharp reconstructions: The finite dimensional case
View Description Hide DescriptionA commutative positive operator valued (POV) measure with real spectrum is characterized by the existence of a projection valued measure (the sharp reconstruction of ) with real spectrum such that can be interpreted as a randomization of . This paper focuses on the relationships between this characterization of commutative POV measures and Neumark’s extension theorem. In particular, we show that in the finite dimensional case there exists a relation between the Neumark operator corresponding to the extension of and the sharp reconstruction of . The relevance of this result to the theory of nonideal quantum measurement and to the definition of unsharpness is analyzed.

Least uncertainty principle in deformation quantization
View Description Hide DescriptionDeformation quantization generally produces families of cohomologically equivalent quantizations of a single physical system. We conjecture that the physically meaningful ones (i) allow enough observable energy distributions, i.e., ones for which no pure quantum state has negative probability, and (ii) reduce the uncertainty in the probability distribution of the resulting quantum states. For the simple harmonic oscillator this principle selects the classic GroenewoldMoyal (or Weyl) product on phase space while for the free particle, in which there is no real quantization, all cohomologically equivalent quantizations are equally good.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Ideal basis sets for the Dirac Coulomb problem: Eigenvalue bounds and convergence proofs
View Description Hide DescriptionBasis sets are developed for the Dirac Coulomb Hamiltonian for which the resulting numerical eigenvalues and eigenfunctions are proved mathematically to have all the following properties: to converge to the exact eigenfunctions and eigenvalues, with necessary and sufficient conditions for convergence being known; to have neither missing nor spurious states; to maintain the Coulomb symmetries between eigenvalues and eigenfunctions of the opposite sign of the Dirac quantum number ; to have positive eigenvalues bounded from below by the corresponding exact eigenvalues; and to have negative eigenvalues bounded from above by . All these properties are maintained using functions that may be analytic or nonanalytic (e.g., Slater functions or splines); that match the noninteger power dependence of the exact eigenfunctions at the origin, or that do not; or that extend to as do the exact eigenfunctions, or that vanish outside a cavity of large radius (convergence then occurring after a second limit, ). The same basis sets can be used without modification for potentials other than the Coulomb, such as the potential of a finite distribution of nuclear charge, or a screened Coulomb potential; the error in a numerical eigenvalue is shown to be second order in the departure of the potential from the Coulomb. In certain bases of Sturmian functions the numerical eigenvalues can be related to the zeros of the Pollaczek polynomials.

Nonperturbative AdlerBardeen theorem
View Description Hide DescriptionThe AdlerBardeen theorem has been proven only as a statement valid at all orders in perturbation theory, without any control on the convergence of the series. In this paper we prove a nonperturbative version of the AdlerBardeen theorem in by using recently developed technical tools in the theory of Grassmann integration. The proof is based on the assumption that the boson propagator decays fast enough for large momenta. If the boson propagator does not decay, as for Thirring contact interactions, the anomaly in the WI (Ward Identities) is renormalized by higher order contributions.

Simple fivedimensional wave equation for a Dirac particle
View Description Hide DescriptionA firstorder relativistic wave equation is constructed in five dimensions. Its solutions are eightcomponent spinors, interpreted as singleparticle fermion wave functions in fourdimensional spacetime. Use of a “cylinder condition” (the removal of explicit dependence on the fifth coordinate) reduces each eightcomponent solution to a pair of degenerate fourcomponent spinors. It is shown that, when the cylinder condition is applied, the results obtained from the new equation are the same as those obtained from the Dirac equation. Without the cylinder condition, on the other hand, the equation implies the existence of a scalar potential, and for zeromass particles it leads to a fourdimensional fermionic equation analogous to Maxwell’sequation with sources.

Left and righthanded neutrinos and BaryonLepton masses
View Description Hide DescriptionThe selfdual and antiselfdual parts of the electromagnetic field tensor satisfying the vacuum Maxwell equations are shown to be related in a covariant manner to a lefthanded and righthanded twocomponent Weyl neutrino and , respectively. A simple quantum mechanical analysis of a composite system with a certain interaction shows that such a model can exhibit a twofold branching and defect in the total energy of the system, which could then be interpreted as Baryon and Lepton mass formation.

Soliton stability in some knot soliton models
View Description Hide DescriptionWe study the issue of stability of static solitonlike solutions in some nonlinear field theories which allow for knotted field configurations. Concretely, we investigate the AratynFerreiraZimerman model [Phys. Lett. B456, 162 (1999);Phys. Rev. Lett.83, 1723 (1999)], based on a Lagrangian quartic in first derivatives with infinitely many conserved currents, for which infinitely many soliton solutions are known analytically. For this model we find that sectors with different (integer) topological charges (Hopf index) are not separated by an infinite energy barrier. Further, if variations which change the topological charge are allowed, then the static solutions are not even critical points of the energy functional. We also explain why soliton solutions can exist at all, in spite of these facts. In addition, we briefly discuss the Nicole model [J. Phys. G4, 1363 (1978)], which is based on a sigmamodeltype Lagrangian. For the Nicole model we find that different topological sectors are separated by an infinite energy barrier.

Super PicardFuchs equation and monodromies for supermanifolds
View Description Hide DescriptionFollowing, Aganagic and Vafa (eprint hepth/0403192) and Hori and Vafa (eprint hepth/0002222), we discuss the PicardFuchs equation for the super LandauGinsburg mirror to the super CalabiYau in (using techniques of Greene and Lazaroiu [Nucl. Phys. B604, 181 (2001),eprint hepth/0001025] and Misra [Fortschr. Phys.52, 831 (2004),eprint hepth/0311186]), Meijer basis of solutions, and monodromies (at 0,1 and ) in the large and small complex structure limits, as well as obtain the mirror hypersurface, which in the large Kähler limit turns out to be either a bidegree(6,6) hypersurface in or a ( singular) bidegree(6,12) hypersurface in .
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 GENERAL RELATIVITY AND GRAVITATION


Variational techniques in general relativity: A metricaffine approach to Kaluza’s theory
View Description Hide DescriptionA new variational principle for general relativity based on an action functional involving both the metric and the connection as independent, unconstrained degrees of freedom is presented. The extremals of are seen to be pairs in which is a Ricci flat metric and is the associated Riemannian connection. An application to Kaluza’s theory [Sitz. Preuss. Akad. Wiss., 966 (1921)] of interacting gravitational and electromagnetic fields is discussed.

Scattering for massive Dirac fields on the Kerr metric
View Description Hide DescriptionStarting with the Dirac equation outside the event horizon of a nonextreme Kerr black hole, we develop a timedependent scattering theory for massive Dirac particles. The explicit computation of the modified wave operators at infinity is done by implementing a timedependent logarithmic phase shift from the free dynamics to offset the long range term in the full Hamiltonian due to the presence of the gravitational force. Analytical expressions for the wave operators are also given.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Support of the logarithmic equilibrium measure on sets of revolution in
View Description Hide DescriptionFor sets of revolution in , we investigate the limit distribution of minimum energy point masses on that interact according to the logarithmic potential , where is the Euclidean distance between points. We show that such limit distributions are supported only on the “outmost” portion of the surface (e.g., for a torus, only on that portion of the surface with positive curvature). Our analysis proceeds by reducing the problem to the complex plane where a nonsingular potential kernel arises, the level lines of which are ellipses.

Spinor extended Lorentzforcelike equations as consequence of a spinorial structure of spacetime
View Description Hide DescriptionAs previously shown, the special relativistic dynamical equation of the Lorentz force type can be regarded as a consequence of a succession of spacetime dependent infinitesimal Lorentz boosts and rotations. This insight indicates that the LorentzForcelike equation has a fundamental meaning in physics. We show how this result may be spinorially obtained starting out from the application of an infinitesimal element of to the individual spinors, which are regarded here as being more fundamental objects than fourvectors. In this way we get a set of new dynamical spinor equationsinducing the extended LorentzForcelike equation in the Minkowski spacetime and geometrically obtain the spinor form of the electromagnetic field tensor. The term extended refers to the dynamics of some additional degrees of freedom that may be associated with the intrinsic spin, namely, with the dynamics of three spacelike mutually orthogonal fourvectors, all of them orthogonal to the linear fourmomentum of the object under consideration that finally, in the particle’s proper frame, are identified with the generators of .

New solution family of the Jacobi equations: Characterization, invariants, and global Darboux analysis
View Description Hide DescriptionA new family of skewsymmetric solutions of the Jacobi partial differential equations for finitedimensional Poisson systems is characterized and analyzed. Such family has some remarkable properties. Firstly, it is defined for arbitrary values of the dimension and the rank. Secondly, it is described in terms of arbitrary differentiable functions, namely, it is not limited to a given degree of nonlinearity. Additionally, it is possible to determine explicitly the fundamental properties of those solutions, such as their Casimir invariants and the algorithm for the reduction to the Darboux canonical form, which have been reported only for a very limited sample of finitedimensional Poisson structures. Moreover, such analysis is carried out globally in phase space, thus improving the usual local scope of the Darboux theorem.
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 STATISTICAL PHYSICS


Random point fields for paraparticles of any order
View Description Hide DescriptionRandom point fields which describe gases consisting of paraparticles of any order are given by means of the canonical ensemble approach. The analysis for the cases of the parafermion gases are discussed in full detail and it is shown that the partition functions are power of that of the usual (i.e., ) fermion. The same is true for parabosons.

Ground state energy of the low density Hubbard model: An upper bound
View Description Hide DescriptionWe derive an upper bound on the ground state energy of the threedimensional (3D) repulsive Hubbard model on the cubic lattice agreeing in the low density limit with the known asymptotic expression of the ground state energy of the dilute Fermi gas in the continuum. As a corollary, we prove an old conjecture on the low density behavior of the 3D Hubbard model, i.e., that the total spin of the ground state vanishes as the density goes to zero.

Spectral gap and decay of correlations in U(1)symmetric lattice systems in dimensions
View Description Hide DescriptionWe consider manybody systems with a global U(1) symmetry on a class of lattices with the (fractal) dimension and their zero temperature correlations whose observables behave as a vector under the U(1) rotation. For a wide class of the models, we prove that if there exists a spectral gap above the ground state, then the correlation functions have a stretched exponentially decaying upper bound. This is an extension of the McBryanSpencer method at finite temperatures to zero temperature. The class includes quantum spin and electron models on the lattices, and our method also allows finite or infinite (quasi)degeneracy of the ground state. The resulting bounds rule out the possibility of the corresponding magnetic and electric longrange orders.

Fluctuationdissipation theorems in an expanding universe
View Description Hide DescriptionThe recently constructed minimal extension of the special relativistic OrnsteinUhlenbeck process to curved spacetime cannot be used to model realistically diffusions on cosmological scales because it does not take into account the evolution of the thermodynamical state of the matter with cosmological time. We therefore introduce a new class of nonminimal extensions characterized by timedependent friction and noise coefficients. These new processes admit timedependent Jüttner distributions as possible measures in momentum space; associated fluctuationdissipation theorems are also proved and discussed.
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 METHODS OF MATHEMATICAL PHYSICS


Approximate solutions to the quantum driftdiffusion model of semiconductors
View Description Hide DescriptionApproximate solutions to the quantum driftdiffusion model for semiconductors are found with the aid of the symmetry group analysis. The system is written as singularly perturbed equations which are solved up to first order in the scaled reduced Planck constant. An example of solution of a nanoscale junction is provided and the difference with the classical case is shown.

Weak quantum Borcherds superalgebras and their representations
View Description Hide DescriptionWe define a weak quantum Borcherds superalgebra , which is a weak Hopf superalgebra. We also discuss the basis and the highest weight modules of . Then we study the weak form and the classical limit of .
