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Volume 35, Issue 8, August 1994

Quantization of the nilpotent orbits in so(1,2)* and massless particles on (anti‐)de Sitter space–time
View Description Hide DescriptionThe nontrivial nilpotent orbits in so(1,2)*≂su(1,1)* are the phase spaces of the zero mass particles on the two‐dimensional (anti‐)de Sitter space–time. As is well known, the lack of global hyperbolicity (respectively, stationarity) for the anti‐de Sitter (respectively, de Sitter) space–time implies that the canonical field quantization of its free massless field is not uniquely defined. One might nevertheless hope to get the one‐particle quantum theory directly from an appropriate ‘‘first’’ quantization of the classical phase space. Unfortunately, geometric quantization (the orbit method) does not apply to the above orbits and a naive canonical quantization does not yield the correct result. To resolve these difficulties, we present a simple geometric construction that associates to them an indecomposable representation of SO_{0}(1,2) on a positive semidefinite inner product space. It is shown that quotienting out its one‐dimensional invariant subspace yields the first term of the holomorphic discrete series of representations of SO_{0}(1,2)≂SU(1,1)/Z _{2}. We interpret these results physically by showing that the above positive semidefinite inner product space is naturally isomorphic to a space of solutions of the conformally invariant zero‐mass Klein–Gordon equation on the (anti)‐de Sitter space–time, equipped with the usual Klein–Gordon inner product, obtained by integrating over a suitable spacelike hypersurface. As such, our version of geometric quantization selects in a natural way a quantization of the free massless particle. We show it is conformally [i.e., SO_{0}(2,2)] invariant and behaves correctly in the classical limit.

On certain quantum deformations of gl(N,R)
View Description Hide DescriptionIn this article all deformations of the general linear group, subject to certain restrictions which in particular ensure a smooth passage to the Lie group limit, are obtained. Representations are given in terms of certains sets of creation and annihilation operators. These creation and annihilation operators may belong to a generalization of the q‐quark type or q‐hadronic type with q‐boson or q‐fermion behavior. This leads to a natural definition of q‐direct sums of q‐algebras.

Order and spacings of energy levels for the Klein–Gordon equation
View Description Hide DescriptionOne can prove and extend some results about the order and spacings of energy levels for the Klein–Gordon equation with vector‐like and scalar‐like potentials.

Nonstandard scalar quantum fields
View Description Hide DescriptionThis article presents a theory of scalar quantum fields based on nonstandard analysis. Free scalar momentum fields and free Klein–Gordon fields are treated first. It is shown that the nonrigorous formalism of the standard theory can be replaced by a mathematically rigorous nonstandard theory. The second quantized energy–momentum operators are derived and the free quantum fields are diagonalized. Interacting scalar fields are considered and a nonstandard scattering operator is derived. A nonstandard counterpart to Wick’s theorem is developed and the φ^{4} interaction is briefly discussed.

Yang–Mills theory on a cylinder coupled to point particles
View Description Hide DescriptionA model of quantum Yang–Mills theory with a finite number of gauge invariant degrees of freedom is studied. The gauge field has only a finite number of degrees of freedom since it is assumed that space–time is a two‐dimensional cylinder. The gauge field is coupled to matter, modeled by either one or two nonrelativistic point particles. These problems can be solved without any gauge fixing, by generalizing the canonical quantization methods of S. G. Rajeev [Phys. Lett. B 212, 203 (1988)] to the case including matter. For this, the geometry of the space of connections is used, which has the structure of a principal fiber bundle with an infinite‐dimensional fiber. Both problems are reduced to finite‐dimensional, exactly solvable, quantum mechanics problems. In the case of one particle, it is found that the ground state energy will diverge in the limit of infinite radius of space, consistent with confinement. In the case of two particles, this does not happen if they can form a color singlet bound state (‘‘meson’’).

Compilation of two‐point and four‐point graphs in field theory in noninteger dimensions
View Description Hide DescriptionTwo‐point and four‐point graphs appearing in a three‐loop approximation in the field theory renormalization group scheme are calculated in general dimensions. Combining Feynman parameterization and direct integration loop integrals are represented in the form of the expressions, depending on the space dimension d as a parameter. The expressions obtained enables one to consider renormalization group functions directly at noninteger d.

Nodal‐surface conjectures for the convex quantum billiard
View Description Hide DescriptionStemming from known properties of one‐dimensional (1‐D) and 2‐D quantum billiards, it is conjectured that the nodal surface of the first‐excited state of the convex 3‐D quantum billiard intersects the billiard surface in a single simple closed curve. Examples of the validity of this conjecture are given for a number of elementary 3‐D billiard configurations. From these examples a second conjecture is introduced that addresses convex quantum billiards which are figures of rotation and contain one and only one plane of mirror symmetry normal to the axis of rotation. Two characteristic displacement parameters are defined which are labeled an axis length, L, and diameter, a. It is conjectured that a parameter κ≊1, exists, whose exact value depends on the properties of the billiard, such that for L≳κa (‘‘prolatelike’’) the nodal surface of the first‐excited state of a quantum billiard is a plane surface of mirror symmetry which divides the length of the billiard in half. For L<κa (‘‘oblatelike’’) the nodal surface of the first‐excited state is a plane surface of mirror symmetry which contains the rotation axis and divides the diameter of the billiard in half. Arguments are given in support of a third conjecture which addresses the regular polyhedra quantum billiards, termed ‘‘spherical‐like.’’ It is hypothesized that the nodal surface of the first‐excited state for any of these billiards is any plane of reflection symmetry of the given polyhedron.

Precise exponential estimates in adiabatic theory
View Description Hide DescriptionGeneral adiabatic evolutions associated to Hamiltonians, which admit a holomorphic extension with respect to the time variable in a complex strip, and whose spectrum satisfies a gap condition are studied. An explicit rate of exponential decay is given, which is related to simple geometric quantities associated to the spectrum of the Hamiltonian, for the transition probability between the two parts of the spectrum when the evolution is taken from −∞ to +∞.

An n‐dimensional interpretation of the δ ‐expansion for the Thomas–Fermi equation
View Description Hide DescriptionThe recent success of the δ‐expansion in the solution of the Thomas–Fermi equation is interpreted to be due to the selection of an expansion parameter that interpolates between the dimension of interest, n=3, and the dimension where the problem is exactly soluble,n=2.

Statistical conception of quantum field theory
View Description Hide DescriptionTwo equivalent modes of statistical description of stochastic particle production are considered, exclusive and inclusive. Information about particle production processes are distributed inside exclusive and inclusive descriptions in different ways. The exclusive description realizes an ‘‘all or nothing’’ investigation strategy, whereas the inclusive description realizes the ‘‘step by step’’ investigation strategy. Conventional quantum field theory is shown to use essentially an exclusive description. The possibility of a transition to a more reasonable investigation strategy (inclusive description) of particle production description is discussed.

Dirac’s equation on the quantum net
View Description Hide DescriptionThe quantum net model differs from other quantizations and discretizations of space‐time in being founded upon quantum‐logical principles. Fermions are envisaged as arising from defects in the net. Such defects are investigated and it is found that the attempt to vary them infinitesimally gives rise to a reticular version of the Dirac operator, which now has a simple interpretation in terms of quantum mechanical infinitesimal parallel transport. The analog of the usual massless Weyl equations are found to hold on the net, and algebraic solutions are deduced by exploiting its inherent logical structure. A correspondence principle then applies to show that these equations and their solutions correspond precisely to the ones expected in the continuum. (In the course of this, new basis‐independent expressions for the complex Minkowski‐space Dirac matrices are found.) Finally, an attempt is made to introduce gravity into the net, and the flat space DiracLagrangian is modified accordingly.

Phase space structure of non‐Abelian Chern–Simmons particles
View Description Hide DescriptionThe classical phase space structure of N SU(n+1) non‐Abelian Chern‐Simons (NACS) particles is investigated by first constructing the product space of associated SU(n+1) bundle with C P ^{ n } as the fiber. The Poisson bracket is calculated using the symplectic structure on the associated bundle and it is found that the minimal substitution in the presence of external gauge fields is equivalent to the modification of symplectic structure by the addition of field strength two form. Then, a direct product of the associated bundle is taken by the space of all connections and a specific connection is chosen by the condition of vanishing momentum map corresponding to the gauge transformation, thus recovering the quantum mechanical model of NACS particles in Lee and Oh [Phys. Rev. Lett. 72, 1141 (1994)].

Boundary integral equations for the scattering of electromagnetic waves by a homogeneous chiral obstacle
View Description Hide DescriptionThe time‐harmonic scattering of electromagnetic waves by a homogeneous chiral media are considered herein. The scattering problem is reduced using Bohren’s decomposition to an equivalent boundary integralequation which is shown to be uniquely solvable except for a discrete set of values of electromagnetic parameters of the scatterer. It should be noted that the boundary integral operators appearing in this equation are the same as for a dielectric scatterer, and hence their mapping properties are well known.

Resonance theory of elastic head waves in layered media. I. A fluid layer bottomed by an infinite solid
View Description Hide DescriptionIn the conventional theory of elastic wave in layered media, the head wave is believed to be the ‘‘critically refracted arrival.’’ However, there are many difficulties which remain unsettled. In order to clarify these questions, the corresponding boundary value problem is re‐treated by the standing wave method, in which the path of integration in the wave‐number plane is uniquely determined by the boundary conditions. It is found that the critical refraction does not exist. The physical mechanism of the head wave is shown to be a mode of resonance, and is a nonlinear effect of the elastic waveequation. The property of the head wave is discussed in detail. Some predictions are compared with observations.

Isotropic classical spin chains showing nearest neighbor correlated cationic distributions
View Description Hide DescriptionA model is proposed for calculating the magnetic susceptibility of isotropic classical spin chains showing nearest neighbor correlated cationic distributions. It allows one to obtain a general closed‐form expression: This expression formally looks like previous ones obtained in the case of random chains for which the neighboring cationic species are not correlated; but here the previous intervening scalar parameters are replaced by vectorial and matricial expressions.

Long time behavior of nonlinear stochastic oscillators: The one‐dimensional Hamiltonian case
View Description Hide DescriptionThe long time behavior of nonlinear, nondissipative systems, which are perturbed by a white noise force are discussed herein. Considering special nonlinear forces and an appropriate scaling, a stochastic convergence theorem is proven. In particular the convergence of the energy process of the system to a limit diffusion is discussed. This corresponds to convergence of the system to a stationary distribution. Furthermore, the limit process is investigated and an explicit formula for its transition probability density is given. An analytic approach to the convergence theorem in terms of a singular perturbation theorem for semigroups is also presented.

On a class of homogeneous nonlinear Schrödinger equations
View Description Hide DescriptionA class of homogeneous, norm conserving, nonlinear waveequations of the Schrödinger type is studied. It is shown that those equations which derive from a Lagrangian can be linearized, but have no regular confined solutions, whereas the equations which cannot be obtained from a local Lagrangian do admit such confined solutions. The latter however are unstable against small perturbations of the initial data.

Deformations preserving Huygens’ principle
View Description Hide DescriptionA new kind of deformation related to the Korteweg–de Vries hierarchy and its master symmetries is studied. The algebra of deformations is generated by trivial (point) transformations via a special recursion operator and proved to preserve Huygens’ principle for a certain family of hyperbolic linear equations. The corresponding evolution flows for Hadamard coefficients are presented explicitly. The connections with spectral deformations and Darboux transformations as well as recent works on the bispectral problem for a Schrödinger operator are discussed.

Lattice solitons directly by the bilinear method
View Description Hide DescriptionRecently a doubly periodic solution of the Kadomtsev–Petviashvili equation was deduced by summing over component solitons. The same solution was derived directly by the Hirota bilinear method. This alternate route enables one to obtain a new solution. Such a mechanism can also be applied to wavepacket dynamics, e.g., the Davey–Stewartson equation.

Hamiltonian properties of the canonical symmetry
View Description Hide DescriptionThe properties of the canonical symmetry are investigated. The densities of the local conservation laws for the nonlinear Schrödinger equation are shown to change under the action of the canonical symmetry by total space derivatives. It is also shown how the canonical symmetry can be used to look for the hierarchy of the Hamiltonian operators relevant to the system under consideration. It appears that only the invariance condition can be used to solve the problem.