Volume 39, Issue 11, November 1998
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Twoform gauge fields in the Becchi–Rouet–Stora–Tyutin superspace
View Description Hide DescriptionWe consider a geometrical description for tensorgauge fields. Based on this geometrical treatment, we develop the theory involving one and twoform gauge fields by means of the Becchi–Rouet–Stora–Tyutin (BRST) superfield formalism. This permits us to directly obtain invariant Lagrangians for both BRST and antiBRST transformations and we shall see that all the ingredients of the formalism (ghosts, ghostforghosts and all the auxiliary fields) naturally occur. We introduce collective fields to construct the field–antifield quantum action in a generic gauge. We deal with both Abelian and nonAbelian cases. In this last case, the BRST superspace formulation sheds more light on this still open problem.

Expansion of the hydrogenlike wave functions in the inverse principal quantum number, for the discrete and continuous spectra
View Description Hide DescriptionAn expansion with respect to the inverse principal quantum number has been derived recently for the hydrogenlike wave functions of the discrete spectrum. The basic functions of this expansion were Bessel functions multiplied by a power of the distance r to the origin. We show that this expansion remains valid for the continuous spectrum, with minor changes. For the wave function corresponding to the energy a.u. of the continuum, the pure imaginary number (positive real ) must be substituted for the integer n in the expansion for the discrete spectrum, and a normalizing factor must be introduced.

Complex Hamiltonian evolution equations and field theory
View Description Hide DescriptionThe form of the complex Hamiltonian evolution equations (linear and nonlinear) is deduced here from that of the classical Hamiltonian systems. Their properties and those of the associated continuous Poisson bracket are studied. It is shown that if any spacelocalized solution of the above type equations is expanded in any Hilbert space basis the expansion coefficients are canonically conjugate variables and obey Hamiltonian systems of complex equations. The last is proved also for sets of “canonically conjugate” functionals of general type. As an application, commutation relations which are equivalent in form to Dirac’s quantization rules (the constant in place of ℏ is undetermined) are obtained. The question of how a nonlinear field theory may contain quantum mechanics as a special case is discussed in view of this paper’s results.

Canonical and quasicanonical realizations of the superalgebra and the Coulomb problem in superspaces
View Description Hide DescriptionCanonical and quasicanonical realizations of the superalgebra are presented. In these realizations there is only one independent Casimir operator: The quadratic one C, the eigenvalues of which allow specification of the involved representations. Pauli’s method for solving the spectrum of the Coulomb problem through quantization of the Runge–Lenz vector is extended to an arbitrary dimension superspace with D bosonic coordinates and d (even) fermionic coordinates. The symmetry algebra of the Hamiltonian is the superalgebra It is shown how the energy spectrum is related to the Casimir operator of in its quasicanonical realization. Then the energy levels are given in terms of the eigenvalues of the operator

Wick’s theorem for nonsymmetric normal ordered products and contractions
View Description Hide DescriptionWe consider arbitrary splits of field operators into two parts; , and use the corresponding definition of normal ordering introduced earlier [T. S. Evans and D. A. Steer, Nucl. Phys. B 474, 481 (1996)]. In this case the normal ordered products and contractions have none of the special symmetry properties assumed in existing proofs of Wick’s theorem. Despite this, we prove that Wick’s theorem still holds in its usual form as long as the contraction is a cnumber. Wick’s theorem is thus shown to be much more general than existing derivations suggest, and we discuss possible simplifying applications of this result.

Algebraic description of invariant oscillator states
View Description Hide DescriptionA grouptheoretical description of identical harmonic oscillators on a line is presented. It provides a scheme for labeling the energy eigenstates that are invariant under the permutation group and for obtaining the symmetric operators that transform these degenerate eigenfunctions among themselves. The symmetry algebra that these generators form is in general polynomial. The 2 and 3particle cases are considered in detail. For the simple 2body problem the invariance algebra is found to be the cubic SU(2) algebra: In the 3body case, the permutational invariant states are characterized with the help of the subgroup chain The labeling and step operators are obtained from determining an integrity basis for the scalar in Generating functions techniques are used to that end; an eightdimensional basis is found whose elements span the symmetry algebra of the three identical oscillator problem. These constants of motion are seen to generate a nonlinear algebra whose representation on the symmetric states is provided.

Rational conformal field theory extensions of in terms of bilocal fields
View Description Hide DescriptionThe rational conformal field theory extensions of at are in onetoone correspondence with onedimensional integral lattices Each extension is associated with a pair of oppositely charged “vertex operators” of charge square Their product defines a bilocal field whose expansion in powers of gives rise to a series of (neutral) local quasiprimary fields (of dimension ). The associated bilocal exponential of a normalized current generates the algebra spanned by the (and the unit operator). The extension of this construction to higher (integer) values of the central charge is also considered. Applications to a quantum Hall system require computing characters (i.e., chiral partition functions) depending not just on the modular parameter τ, but also on a chemical potential ζ. We compute such a ζ dependence of orbifold characters, thus extending the range of applications of a recent study of affine orbifolds.

A histories approach to quantum mechanics
View Description Hide DescriptionA histories approach to quantum mechanics is formulated without the consistency requirement. Plausible, physically motivated axioms for a history structure are presented. In this structure, the consistency requirement is unnecessary because the joint sequential distributions are already probability measures. It is shown that the logic of a history structure is a temporal effect algebra and conversely, any temporal effect algebra is isomorphic to the logic of a history structure. When this general framework is specialized to a Hilbert space context, it reduces to a previously studied formalism.

A number of quasiexactly solvable Nbody problems
View Description Hide DescriptionWe present several examples of quasiexactly solvable Nbody problems in one, two, and higher dimensions. We study various aspects of these problems in some detail. In particular, we show that in some of these examples the corresponding polynomials form an orthogonal set, and many of their properties are similar to those of the Bender–Dunne polynomials. We also discuss QES problems where the polynomials do not form an orthogonal set.

SU(2) quantum kinematics: Rotationobservable versus angularmomentum generalized commutation relations
View Description Hide DescriptionThe canonical commutation relations of quantum mechanics are generalized to the case where appropriate dynamical variables are angularmomentum, rotationangle, and rotationaxis observables. To this end, SU(2) is “quantized” on the compact group manifold, according to the standard procedure of nonAbelian quantum kinematics. Quantumkinematic invariant operators are introduced, and their commutation relations with the rotation variables are found in an explicit manner. The quantumkinematic invariants yield superselection rules in the form of eigenvalue equations of an isotopic structure (which one should solve in the applications, in order to get multiplets that carry the irreducible representations of the underlying quantum kinematic models). A wide range of applicability of SU(2) quantum kinematics is suggested.

Unified treatment of the Coulomb and harmonic oscillator potentials in D dimensions
View Description Hide DescriptionQuantum mechanical models and practical calculations often rely on some exactly solvable models like the Coulomb and the harmonic oscillator potentials. The D dimensional generalized Coulomb potential contains these potentials as limiting cases, thus it establishes a continuous link between the Coulomb and harmonic oscillator potentials in various dimensions. We present results which are necessary for the utilization of this potential as a model and practical reference problem for quantum mechanical calculations. We define a Hilbert space basis, the generalized Coulomb–Sturmian basis, and calculate the Green’s operator on this basis and also present an SU(1,1) algebra associated with it. We formulate the problem for the onedimensional case, too, and point out that the complications arising due to the singularity of the onedimensional Coulomb problem can be avoided with the use of the generalized Coulomb potential.

Sigma models coupled with Abelian gauge fields
View Description Hide DescriptionTwo distinct Skyrme terms for the model are defined, only one of which preserves the equivalence with the corresponding O(3) model. A gauging prescription in which the composite connection of the model is replaced by the U(1) gauge connection is defined. A U(1) gauged model and two distinct U(1) gauged models with Skyrme terms are defined. The existence of topologically stable vortices in each of the three U(1) gauged models is proved analytically.

Highest weight irreducible representations of the quantum algebra
View Description Hide DescriptionA class of highestweight irreducible representations of the algebra the quantum analog of the completion and central extension of the Lie algebra is constructed. It is considerably larger than the class of representations known so far. Within each module a basis is introduced and the transformation relations of the basis under the action of the Chevalley generators are explicitly given. The verification of the quantum algebra relations is shown to reduce to a set of nontrivial qnumber identities. All representations are restricted in the terminology of S. Levendorskii and Y. Soibelman [Commun. Math. Phys. 140, 399–414 (1991)].

Symmetries of history quantum theories and decoherence functionals
View Description Hide DescriptionRecently, Schreckenberg investigated symmetries in the context of history quantum theories. In the case that the space of histories is given by the set of projectors on some finite dimensional Hilbert space he obtained, first, a complete characterization of all physical symmetries of history quantum theories — an analog of Wigner’s theorem — and, second, a complete mathematical characterization of symmetries of single decoherence functionals. In this paper we extend Schreckenberg’s results to the case where the underlying space of histories is given by the set of projectors on some infinite dimensional Hilbert space.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


A version of Thirring’s approach to the Kolmogorov–Arnold–Moser theorem for quadratic Hamiltonians with degenerate twist
View Description Hide DescriptionWe give a proof of the Kolmogorov–Arnold–Moser (KAM) theorem on the existence of invariant tori for weakly perturbed Hamiltonian systems, based on Thirring’s approach for Hamiltonians that are quadratic in the action variables. The main point of this approach is that the iteration of canonical transformations on which the proof is based stays within the space of quadratic Hamiltonians. We show that Thirring’s proof for nondegenerate Hamiltonians can be adapted to Hamiltonians with degenerate twist. This case, in fact, drastically simplifies Thirring’s proof.

On periodical trajectories of the billiard systems within an ellipsoid in and generalized Cayley’s condition
View Description Hide DescriptionThe analytical condition for a trajectory of the billiard system within an ellipsoid of any dimension is given. An interesting property is obtained—every periodical trajectory with the period less than the dimension of the space is placed in a coordinate hyperplane.

Electromagnetic fields in air of travelingwave currents above the earth
View Description Hide DescriptionThe problem of the electromagnetic field created by a thin, straight conductor of infinite length carrying a forward travelingwave current with a complex propagation constant γ above a homogeneous and isotropic planar earth of wave number is formulated in terms of contour integrals. In the limit where γ becomes equal to the freespace wave number the component of the magnetic field in air normal to both conductor and interface is evaluated in closed form in terms of known special functions while the remaining components of the field in air are expressed as series expansions in via the application of a contour integration technique. The new analytical formulas involve familiar transcendental functions and are valid at any distance from the source. The analysis sheds light on the intricate nature of lowfrequency electromagnetic fields generated by transmission lines in the presence of a conducting or a dielectric halfspace.

Point vortices on a sphere: Stability of relative equilibria
View Description Hide DescriptionIn this paper we analyze the dynamics of point vortices moving on a sphere from the point of view of geometric mechanics. The formalism is developed for the general case of vortices, and the details are worked out for the (integrable) case of three vortices. The system under consideration is SO(3) invariant; the associated momentum map generated by this SO(3) symmetry is equivariant and corresponds to the moment of vorticity.Poisson reduction corresponding to this symmetry is performed; the quotient space is constructed and its Poisson bracket structure and symplectic leaves are found explicitly. The stability of relative equilibria is analyzed by the energy–momentum method. Explicit criteria for stability of different configurations with generic and nongeneric momenta are obtained. In each case a group of transformations is specified, modulo which one has stability in the original (unreduced) phase space. Special attention is given to the distinction between the cases when the relative equilibrium is a nongreat circle equilateral triangle and when the vortices line up on a great circle.

Symmetries, separability and volume forms
View Description Hide DescriptionIn order to clarify certain misconceptions in the literature, we discuss the details of the way determining equations for general symmetries of a Lagrangian system differ from determining equations for Noether symmetries, and establish the minimal set of extra conditions which have to be imposed in practical situations for the former to reduce to the latter. We further derive properties by which, in situations where the system is integrable through quadratic integrals in involution, the components of the corresponding Noether symmetries themselves can be used to compute the separation variables, if they exist, for the Hamilton–Jacobi equation.

Relativistic eccentric anomalies
View Description Hide DescriptionWe use Kepler’s eccentric anomaly in order to get explicit equations for the planetary orbits in a solar system. By successively modifying the corresponding Lagrangian, we build a Lagrangian for a relativistic twobody problem, the trajectories of which are conics in Minkowski space. The equations of these trajectories, projected on a threespace, exhibit new anomalies.
