Index of content:
Volume 36, Issue 4, April 1995

Covariant algebraic calculation of the one‐loop effective potential in non‐Abelian gauge theory and a new approach to stability problem
View Description Hide DescriptionThe recently proposed algebraic approach is used for calculating the heat kernel on covariantly constant background to study the one‐loop effective action in the non‐Abelian gauge theory. The general case of arbitrary space‐time dimension, arbitrary compact simple gauge group and arbitrary matter is considered and a covariantly constant gauge field strength of the most general form, which has many independent color and space‐time invariants, and covariantly constant scalar fields as a background (Savvidy type chromomagnetic vacuum) is assumed. The explicit formulas for all the needed heat kernels and zeta‐functions are obtained. A new method is proposed to study the vacuum stability and it will be shown that the background field configurations with covariantly constant chromomagnetic fields can be stable only in the case when the number of independent field invariants is greater than one and the values of these invariants differ little from each other. The role of space‐time dimension is analyzed in this connection and it is shown that this is possible only in space‐times of dimension greater than four.

Overcomplete basis for one‐dimensional Hamiltonians
View Description Hide DescriptionIt is demonstrated for one‐dimensional Hamiltonians, and under a special hypothesis on the ground state wave function, that it is possible to build an analytical overcomplete basis that generalizes the coherent state basis of the harmonic oscillator, without any reference to group theory. The semi‐classical consequences of this formalism are developed and the usual ideas of normal and antinormal expansions for operators are extended.

Poincaré algebra in ordinary and Grassmann space and supersymmetry
View Description Hide DescriptionIn this theory scalars, spinors, and vectors are described as particles living in a d‐dimensional ordinary and a d‐dimensional Grassmann space, with d≥5. Operators of translations and the Lorentz transformations in both spaces form the super‐Poincaré algebra. It is the super‐Pauli–Ljubanski vector of an odd Grassmann character, which generates spinors. The theory offers a new insight into quantum theory of particles and their supersymmetric nature.

Path integral for spinning particle in magnetic field via bosonic coherent states
View Description Hide DescriptionUsing the coherent states representation and the Schwinger boson model, the propagator of a spinning particle in interaction with an arbitrary magnetic field is derived in the framework of the path integral formalism. To show the relevance of the formalism to some applications, the Green function of spinning particle s=1/2 in special configuration of magnetic field is calculated. The transition amplitude and Berry’s phase for an arbitrary spin and time dependent magnetic field are deduced. The case of half integer spin is also considered.

The Calogero bound for nonzero angular momentum
View Description Hide DescriptionIt is shown that the ‘‘Calogero’’ bound on the number of bound states in an attractive monotonous potential is not optimal for a strictly positive angular momentuml and a new bound including an extra additive term is proposed. It is N _{l}(V)<(2/π)∫_{0} ^{∞}√‖V(r)‖dr+1−√1 +(2/π)^{2}l(l+1). From this new bound it is possible to obtain a bound on the total number of bound states for arbitrary angular momentum. The situation for −1/2≤l<0 is investigated and a bound under the condition that r ^{2} V(r) has a single extremum is given. Consequences for zero angular momentumbound states in two dimensions are discussed.

A variational proof of the Thomas effect
View Description Hide DescriptionA simple variational proof of the Thomas effect is provided, which shows clearly why it is related to the Efimov effect. The proof is also valid in the case of three equal masses a generalization needed in the briefly commented application to triviality of (λφ^{4})_{ d } in the nonrelativistic limit.

Phase‐space formalism: Generalized Hermite polynomials, orthogonal functions and creation–annihilation operators
View Description Hide DescriptionHermite polynomials with many variables and many indices play a crucial role within the framework of phase‐space formulation of classical or quantum mechanics. A generalized operational formalism useful to handle these polynomials is discussed and also a new set of creation–annihilation operators associated to the phase‐space harmonic oscillator orthogonal functions is introduced. The Fourier transform of these orthogonal functions is studied by discussing the analogies with the ordinary case. Finally, the integral representation of the generalized Hermite polynomials and a simple technique to deal with the generalized heat equation are discussed.

Integral formulas for wave functions of quantum many‐body problems and representations of gl _{ n }
View Description Hide DescriptionExplicit integral formulas are derived for eigenfunctions of quantum integrals of the Calogero–Sutherland–Moser operator with a trigonometric interaction potential. In particular, explicit formulas for Jack’s symmetric functions are derived. To obtain such formulas, one must use the representation of these eigenfunctions by means of traces of intertwining operators between certain modules over the Lie algebragl _{ n }, and the realization of these modules on functions of many variables.

Solutions of A _{∞} Toda equations based on noncompact group SU(1,1) and infinite‐dimensional Grassmann manifolds
View Description Hide DescriptionSpecial solutions of A _{∞} Toda equations are constructed based on noncompact group SU(1,1). These solutions are derived from the holomorphic construction of infinite‐dimensional nonlinear Grassmann σ models. It is also shown that the solutions give a family of solutions of self‐dual Einstein equations in a certain continuum limit.

Path integral over the generalized coherent states
View Description Hide DescriptionA path integral written in terms of the group theoretic coherent states by using the Kähler structure of the coherent statemanifold with the particular emphasis on the boundary‐fixing term derivation is considered herein. The path integral for a propagator of the system with Hamiltonian linear in the SU(2)/SU(1,1) generators is shown to be diagonalized by an appropriate motion in the phase space.

Partial order of quantum effects
View Description Hide DescriptionThe set of effects is not a lattice with respect to its natural order. Projection operators do have the greatest lower bounds (and the least upper bounds) in that set, but there are also other (incomparable) effects which share this property. However, the coexistence, the commutativity, and the regularity of a pair of effects are not sufficient for the existence of their infima and suprema. The structure of the range of an observable (as a normalized POV measure) can vary from that of a commutative Boolean to a noncommutative non‐Boolean subset of effects.

The q‐deformation of quantum mechanics of one degree of freedom
View Description Hide DescriptionThe q‐quantum mechanics of the one degree of freedom is studied. Among others the holomorphic representation of q‐deformed Heisenberg–Weyl algebra and its realization by covariant Berezin symbols is described.

Further uses of the quantum mechanical path integral in quantum field theory
View Description Hide DescriptionIt has been shown how matrix elements of the form 〈x‖exp(−iHt)‖y〉 which arise when using operator regularization to do perturbative calculations in quantum field theory can be evaluated using the quantum mechanical path integral (QMPI). This technique has the advantage of eliminating loop momentum integrals and algebraically complicated vertices in gauge theories. A similar (but distinct) approach of Polyakov and Strassler has been applied to one loop processes with external vector particles and is related to the string based methods of Bern and Kosower. In this article, several features of the QMPI technique are examined. First, it is demonstrated how the path ordering in the QMPI can be handled by considering a model of three interacting scalar fields, each with a distinct mass. Next, it is shown how the QMPI can be used when the external wave function is not a plane wave field. The particular case of having an exponentially damped wave function is considered. Next, a discussion of the difference between the approach of Polyakov and Strassler and that employed here is given. Finally, it is demonstrated how the QMPI can be used to considerably simplify calculations in quantum gravity.

Scattering theory for mesoscopic quantum systems with non‐trivial spatial asymptotics in one dimension
View Description Hide DescriptionBasic results which are needed for the formulation of a quantitative theory of charge transport in mesoscopic quantum‐interference devices are derived. In particular, orthogonality and proper normalization of scattering states for one‐dimensional quantum systems with nonzero and periodic potential asymptotics are discussed. Properties of the S‐matrix are investigated. Results are obtained within the framework of ordinary linear differential equations by investigation of the spectral resolution of the identity and, alternatively, directly from asymptotic properties of Jost solutions and the theory of generalized functions. Based on the S‐matrix and properties of the scattering states, an independent‐particle model for the current response of mesoscopic (quasi‐) one‐dimensional electronic devices may be formulated.

Unification of the Dirac and Einstein Lagrangians in a tetrad model
View Description Hide DescriptionIn a recent article all classical tensor systems which admit Fermi quantization are characterized as those having unitary Lie–Poisson brackets. Examples include Euler’s tensorequation for a rigid body and Dirac’s equation in tensor form. In this article it is shown that the tensor form of the DiracLagrangian can be derived from a tetrad formulation of a Kaluza–Klein model, which unifies the Dirac and Einstein Lagrangians. In this formulation, the isometric modes of the tetrad propagate as fermions, whereas the self‐adjoint modes propagate as gravitons. An analogy is made with the rigid and elastic modes of a deformable body, where the rigid modes are Fermi quantized and the elastic modes are Bose quantized. However, unlike a deformable body for which the Euclidean metric is positive definite, fermion and graviton modes are not generally separable, unless the gravitational fluctuations are limited by a certain bound. It is shown that this bound applies whenever bispinors are defined in a general relativistic framework. That is, the bound does not depend on whether bispinors or tensors are used to describe the fermion modes.

On the number of states bound by one‐dimensional finite periodic potentials
View Description Hide DescriptionBound states and zero‐energy resonances of one‐dimensional finite periodic potentials are investigated, by means of Levinson’s theorem. For finite range potentials supporting no bound states, a lower bound for the (reduced) time delay at threshold is derived.

Perturbation method for the energy‐momentum tensor of an optical soliton
View Description Hide DescriptionElectromagnetic fields of solitons, which are expansions in a small parameter measuring the bandwidth of the radiation over the carrier frequency are studied herein. Conservation laws for energy and momentum are derived from Maxwell’sequations using perturbation expansions in this small parameter. In this derivation, densities are averaged over half a period of the carrier wave, and certain symmetries of the susceptibility are exploited. The general procedure is outlined and the leading terms are explicitly calculated.

Determination of the permittivity and conductivity in R ^{3} using wave splitting of Maxwell’s equations
View Description Hide DescriptionTime domain wave‐splitting of Maxwell’sequations is applied to the inverse problem of determining the permittivity and conductivity in three dimensions (where the reflected field is produced by an impulse dipole, exterior to the scattering medium). The structure of the fundamental solution is analyzed, and the transport equation is used to derive a condition for the reconstruction. A numerical scheme to reconstruct the permittivity and conductivity simultaneously is given, using the tangential reflection fields due to a dipole with a fixed position and orientation.

Multifold Kepler systems—Dynamical systems all of whose bounded trajectories are closed
View Description Hide DescriptionAccording to the Bertrand theorem, the Kepler problem and the harmonic oscillator are the only central force dynamical systems that have closed orbits for all bounded motions. In this article, an infinite number of dynamical systems having such a closed orbit property are found on T*(R ^{3}−{0}) by applying a slightly modified Bertrand’s method to a spherical symmetric Hamiltonian with two undetermined functions of the radius. Actually, for any positive rational number ν, there exists a Hamiltonian system with the closed orbit property just mentioned, which system will be called the ν‐fold Kepler system. Each of the systems is completely integrable and further allows the explicit expression of trajectories. The bounded trajectories in the configuration space R ^{3}−{0} may have self‐intersection points. Moreover, the ν‐fold Kepler system is reducible to a two‐degrees‐of‐freedom system, which is completely integrable and gives rise to flows on the two‐torus for bounded motions. If ν is allowed to take irrational numbers, any flow is shown to be dense in the torus. In conclusion, on the analogy of the Kepler problem, the Runge–Lenz‐like vector for the ν‐fold Kepler system is touched upon.

The high‐temperature expansion of the hierarchical Ising model: From Poincaré symmetry to an algebraic algorithm
View Description Hide DescriptionIt is shown that the hierarchical model at finite volume has a symmetry group which can be decomposed into rotations and translations as the familiar Poincaré groups. Using these symmetries, it is shown that the intricate sums appearing in the calculation of the high‐temperature expansion of the magnetic susceptibility can be performed, at least up to the fourth order, using elementary algebraic manipulations which can be implemented with a computer. These symmetries appear more clearly if we use the two‐adic fractions to label the sites. Then the new algebraic methods are applied to the calculation of quantities having a random walk interpretation. In particular, it is shown that the probability of returning to the starting point after m steps has poles at D=−2,−4,...,−2m, where D is a free parameter playing a role similar to the dimensionality in nearest neighbor models.