Volume 35, Issue 3, March 1994
Index of content:

Nonrenormalization theorem for the Abelian anomaly within dimensional regularization
View Description Hide DescriptionAn approach to the calculation of the Abelian axial anomaly within the dimensional regularization is proposed. This approach allows one to make all computations consistently in the framework of a perturbation theory to any number of loops by using the ordinary dimensional regularization. Handling of the γ_{5} matrix in the anomaly calculation requires only the possibility of cycle permutation under the trace sign. Our procedure leads to a normalization of the axial current that differs by a finite amount from that of the corresponding vector current. The question of normalization of the axial current in relation to the π^{0}→2γ decay amplitude is briefly considered.

Algebraic structure of parabose Fock space. I. The Green’s ansatz revisited
View Description Hide DescriptionA description of the Fock space of a parabose oscillator of order p based on commutation relations bilinear in creation and annihilation operators rather than trilinear ones is developed. A new statement of the well‐known Green’s ansatz is introduced to help understand the significance of the bilinear commutation relations. It is also applied to show how all parabose oscillators of even order (respectively, odd order) can be obtained as irreducible constituents of the Fock space associated with a Green’s ansatz of two (three) terms. Analogs of the form in which the parabose Green’s ansatz is presented are provided for one parafermi oscillator and for systems of many parabose and parafermi oscillators of the same order. Methods used in the paper allow various new results for q‐deformed parabose oscillators to be derived.

A Wigner‐function approach to (semi)classical limits: Electrons in a periodic potential
View Description Hide DescriptionA rigorous derivation of the semiclassical Liouville equation for electrons which move in a crystal lattice (without the influence of an external field) is presented herein. The approach is based on carrying out the semiclassical limit in the band‐structure Wigner equation. The semiclassical macroscopic densities are also obtained as limits of the corresponding quantum quantities.

Supersymmetry and the Atiyah–Singer index theorem. I. Peierls brackets, Green’s functions, and a proof of the index theorem via Gaussian superdeterminants
View Description Hide DescriptionThe Peierls bracket quantization scheme is applied to the supersymmetric system corresponding to the twisted spin index theorem. A detailed study of the quantum system is presented, and the Feynman propagator is exactly computed. The Green’s function and functional integral methods provide a direct derivation of the index as a single universal superdeterminant.

Supersymmetry and the Atiyah–Singer index theorem. II. The scalar curvature factor in the Schrödinger equation
View Description Hide DescriptionThe quantization of the superclassical system used in the proof of the index theorem results in a factor of (ℏ^{2}/8)R in the Hamiltonian. The path integral expression for the kernel is analyzed up to and including the two‐loop order. The existence of the scalar curvature term is confirmed by comparing the linear term in the heat kernel expansion with the two‐loop order terms in the loop expansion. In the operator formalism this term arises from the fermionic sector whereas in the path integral formulation it comes from the bosonic sector.

Coherent state approach to electron nuclear dynamics with an antisymmetrized geminal power state
View Description Hide DescriptionA formulation of the complete dynamics of electrons and nuclei is presented. The dynamical equations are derived using the time‐dependent variational principle (TDVP). The approximate electronic state vectors are antisymmetrized geminal power (AGP) states parameterized as projected coherent states, while the nuclei are treated as classical point particles. This leads to a formulation of time‐dependent AGP theory that generalizes time‐dependent Hartree–Fock (TDHF) theory and explicitly includes the dynamics of the nuclei. The linear approximation to the evolution equations (the classical harmonic approximation) which corresponds to a generalized random phase approximation (RPA) based on an AGP electronic reference state and which explicitly includes the dynamics of the nuclei, is studied and presented in this paper. The equations are formulated in terms of the primitive nonorthogonal electronic atomic basis thus avoiding any transformation to orthonormal molecular orbitals during the evolution.

Thermodynamics of vortices in the plane
View Description Hide DescriptionThe thermodynamics of vortices in the critically coupled Abelian Higgs model, defined on the plane, are investigated by placing Nvortices in a region of the plane with periodic boundary conditions: a torus. It is noted that the moduli space for Nvortices, which is the same as that of N indistinguishable points on a torus, fibrates into a C P _{ N−1} bundle over the Jacobi manifold of the torus. The volume of the moduli space is a product of the area of the base of this bundle and the volume of the fiber. These two values are determined by considering two 2‐surfaces in the bundle corresponding to a rigid motion of a vortex configuration, and a motion around a fixed center of mass. The partition function for the vortices is proportional to the volume of the moduli space, and the equation of state for the vortices is P(A−4π N)=NT in the thermodynamic limit, where P is the pressure, A the area of the region of the plane occupied by the vortices, and T the temperature. There is no phase transition.

Path integral for the damped harmonic oscillator coupled to its dual
View Description Hide DescriptionThe time independent Bateman model which describes the damped harmonic oscillator coupled to its dual is studied using the path integral approach. The related propagator is calculated by means of the matrix representation.

Elliptic Baker–Akhiezer functions and an application to an integrable dynamical system
View Description Hide DescriptionAn ansatz for elliptic Baker–Akhiezer functions is used to find new solutions for various spectral problems—the Sturm–Liouville problem with four and five‐gap Lamé potentials and two third‐order problems. One of the third‐order problems is the classical Halphen equation with a spectral parameter. The eigenfunctions are used to construct elliptic solutions for integrable cases of a generalized Hènon–Heiles system.

A geometric criterion for adiabatic chaos
View Description Hide DescriptionChaos in adiabatic Hamiltonian systems is a recent discovery and a pervasive phenomenon in physics. In this work, a geometric criterion is discussed based on the theory of action from classical mechanics to detect the existence of Smale horseshoe chaos in adiabatic systems. It is used to show that generic adiabatic planar Hamiltonian systems exhibit stochastic dynamics in large regions of phase space. To illustrate the method, results are obtained for three problems concerning relativistic particle dynamics,fluid mechanics, and passage through resonance, results which either could not be obtained with existing methods, or which were difficult and analytically impractical to obtain with them.

An integrable three‐particle system
View Description Hide DescriptionA study of the existence of some integrable systems with nonlinear constants of motion is presented using the approach of the theory of generalized (dynamical or hidden) symmetries. Two Lagrangians are considered, both obtained by modifying the Toda Lagrangian. First a two‐particle system is studied and then the results are generalized to a three‐particle system. It is shown that in both cases the Lagrangians possess nonlinear constants of motion in involution and, thus, they are integrable.

Super‐Toda lattices
View Description Hide DescriptionThe Lax formalism, as described by Oevel et al. and in an earlier and more fundamental form by Semenov, Kostant, Symes, and Adler, can easily be generalized to the case where anticommuting variables are involved, the so‐called supercase. In this article this super‐Lax formalism is applied to the well‐known associative superalgebra G=Mat(m,n,Λ). Subspaces of G to which the super‐Poisson structures can be restricted arise in a natural way. Taking L in one of these subspaces formally leads to superextensions of the hierarchy of nonrelativistic Toda lattices. In the simplest case, where only nearest‐neighbor interaction is involved, the equations are explicitly solved. Furthermore, the relevant two super‐Hamiltonian structures are explicitly calculated. Finally a superextension of the relativistic Toda lattice with a super‐Hamiltonian structure is described herein.

p‐adic superanalysis. III. The structure of p‐adic super‐Lie groups
View Description Hide DescriptionThis work is the third one of a series of articles in which the foundations of a p‐adic supermanifold field theory are proposed. In this article p‐adic super‐Lie groups are defined and their properties studied; finally, the p‐adic groups of matrices are analyzed.

Topological symmetry breaking in self‐interacting theories on toroidal space–time
View Description Hide DescriptionThe possibility of topological mass generation through symmetry breaking in a simple model consisting of a self‐interacting (massive or massless) λφ^{4}scalar field on the space–time T ^{ N }×R ^{ n }, n, N∈N _{0}−T ^{ N } being a general torus—is investigated. The nonrenormalized effective potential is calculated and the specific dependences of the generated mass on the compactification lengths and on the initial mass of the field are determined. Later, in order to obtain the renormalized topologically generated mass, the analysis is restricted to n+N=4 dimensions. It is shown that if the field is massive no symmetry breaking can occur. On the contrary, when it is massless, for n=1 and n=0 and for values of the vector of compactification lengths belonging to some specific domain of R ^{ N }, symmetry breaking does actually take place. Explicit values of the mass topologically generated in this way are obtained.

Affine projection tensor geometry: Decomposing the curvature tensor when the connection is arbitrary and the projection is tilted
View Description Hide DescriptionThis paper constructs the geometrically natural objects which are associated with any projection tensor field on a manifold with any affine connection. The approaches to projection tensor fields which have been used in general relativity and related theories assume normal projection tensors of codimension one and connections which are metric compatible and torsion‐free. These assumptions fail for projections onto lightlike curves or surfaces and other situations where degenerate metrics occur as well as projections onto two‐surfaces and projections onto space–time in the higher dimensional manifolds of unified field theories. This paper removes these restrictive assumptions. One key idea is to define two different ‘‘extrinsic curvature tensors’’ which become equal for normal projections. In addition, a new family of geometrical tensors is introduced: the cross‐projected curvature tensors. In terms of these objects, projection decompositions of covariant derivatives, the full Riemann curvature tensor and the Bianchi identities are obtained and applied to perfect fluids, timelike curve congruences, string congruences, and the familiar 3+1 analysis of the spacelike initial value problem of general relativity.

Five‐dimensional axisymmetric stationary solutions as harmonic maps
View Description Hide DescriptionThe complete scheme of the application of one‐ and two‐dimensional subspaces and the subgroups method to five‐dimensional gravity with a G _{3} group of motion are presented here in space–time and in potential space formalisms. From this method one obtains the Kramer, Belinsky–Ruffini, Dobiasch–Maison, Clément, Gross–Perry–Sorkin solutions, etc., as special cases.

Complex analytic realizations for quantum algebras
View Description Hide DescriptionA method for obtaining complex analytic realizations for a class of deformed algebras based on their respective deformation mappings and their ordinary coherent states is introduced. Explicit results of such realizations are provided for the cases of the qoscillators (q‐Weyl–Heisenberg algebra) and for the su_{ q }(2) and su_{ q }(1,1) algebras and their coproducts. They are given in terms of a series in powers of ordinary derivative operators which act on the Bargmann–Hilbert space of functions endowed with the usual integration measures. In the q→1 limit these realizations reduce to the usual analytic Bargmann realizations for the three algebras.

Becchi–Rouet–Stora–Tyutin structure of polynomial Poisson algebras
View Description Hide DescriptionThe Becchi–Rouet–Stora–Tyutin (BRST) structure of polynomial Poisson algebras is investigated. It is shown that Poisson algebras provide nontrivial models where the full BRST recursive procedure is needed. Quadratic Poisson algebras may already be of arbitrarily high rank. Explicit examples are provided, for which the first terms of the BRST generator are given. The calculations are cumbersome but purely algorithmic, and have been treated by means of the computer algebra system R E D U C E. Our analysis is classical (=nonquantum) throughout.

An inequality for Legendre polynomials
View Description Hide DescriptionThe following inequality is established: ‖P _{ n }(cos ϑ)‖< [√1+(π^{4}/16)(n+1/2)^{4} sin^{4} ϑ]^{−1}, 0<ϑ<π, n=1,2,..., where P _{ n }(x) denotes the Legendre polynomial of degree n. The relation P ^{2} _{ n }(cos ϑ) + (4/π^{2})× Q ^{2} _{ n }(cos ϑ) < [√1+(π^{4}/16)(n+1/2)^{4} sin^{4} ϑ]^{−1}, n=1,2,..., on [θ_{ n1},θ_{ n,n+1}], is proven where Q _{ n }(x) denotes the Legendre function of second kind, cos θ_{ n1} the largest zero of Q _{ n }(x), and cos θ_{ n,n+1}=−cos θ_{ n1}. Similarly we obtain the inequalities ‖J _{0}(x)‖ < [√1+(π^{4}/16)x ^{4}]^{−1}, x≠0, and J ^{2} _{0}(x) + Y ^{2} _{0}(x)< [√1+(π^{4}/16)x ^{4}]^{−1}, x≥y _{1}, where y _{1}=0.893577... is the first positive zero of Y _{0}(x), and J _{0}(x), Y _{0}(x) denote the Bessel functions of the first and second kind, respectively. The results of the present paper arise out of some problems of nuclear and particle physics.

Convergence of an infinite product of Lie transformations
View Description Hide DescriptionSufficient conditions are given for convergence of a product of Lie transformations generated by homogeneous polynomials of increasing order. It is shown that if the coefficients of the polynomials do not grow more rapidly with order than exponential functions, then there exists a finite domain around the origin for which the infinite product is a convergent transformation.