Volume 31, Issue 9, September 1990
Index of content:

Uniqueness of the Virasoro algebra
View Description Hide DescriptionIt is shown that the structure constants of the Virasoro algebra are uniquely specified, up to rescaling of the generators, by the requirement that they be nonvanishing. Permitting some of the structure constants to vanish leads to other Lie algebras, including the Witt algebras. Given the grading of the Virasoro algebra, nonvanishing of the structure constants is both necessary and sufficient for the algebra to be the Virasoro algebra.

A unitary representation of SL(2,R)
View Description Hide DescriptionA given unitary representation of the group SL(2,R), belonging to the discrete series, is shown to involve necessarily some special functions (in particular, Laguerre and Hardy–Pollaczek polynomials). Various realizations of this representation are investigated, including the coherent states one. More generally, it is shown that the representations of the discrete series of the universal covering of SL(2,R) involves generalized Laguerre and Pollaczek polynomials. The Riemann zeta function is shown to be concerned with these representations.

Branching rules for the Weyl group W(D _{ n })
View Description Hide DescriptionReduction theorems for the decomposition of induced and irreducible characters of W(D _{ n }) in terms of induced and irreducible characters of W(D _{ n−1}), respectively, are given.

Point symmetries of conditionally integrable nonlinear evolution equations
View Description Hide DescriptionThe Lie point symmetries of the first two equations in the Kadomtsev–Petviashvili (KP) hierarchy, introduced by Jimbo and Miwa, are investigated. The first is the potential KP equation, the second involves four independent variables and is called the Jimbo–Miwa (JM) equation. The joint symmetry algebra for the two equations is shown to have a Kac–Moody–Virasoro structure, whereas the symmetry algebra of the JM equation alone does not. Subgroups of the joint symmetry group are used to perform symmetry reduction and to obtain invariant solutions.

The two‐dimensional magnetic field problem revisited
View Description Hide DescriptionThe Atiyah–Patodi–Singer index theorem is used to relate the analytic and topological indices of a spin 1/2 ‐charged particle in a two‐dimensional magnetic field.

Hosotani breaking of E _{6} to a subgroup of rank five
View Description Hide DescriptionOn multiply connected manifolds, it is possible to construct vacuum gauge configurations with nontrivial holonomy groups. This is the basis of the Hosotani mechanism. This naturally suggests a ‘‘Hosotani inverse problem’’: If we wish to break a gauge group G to a subgroup H, what are the possible finite holonomy groups having this effect, and what can one say about the fundamental groups of the underlying manifolds? Usually, this problem is too difficult to solve, but we show that, for G=E _{6} and H locally isomorphic to the rank five group SU(3)×SU(2)×U(1)×U(1), a complete solution is possible. It is hoped that the results will aid a search for examples of Calabi–Yau manifolds leading to a low‐energy gauge group of rank five.

A new approach to global stability analysis of one‐dimensional continuous dissipative systems
View Description Hide DescriptionIt is proved by a direct construction that symmetry‐breaking instabilities in one‐dimensional dissipative continuous systems can be studied in terms of a suitable function G whose minima correspond to stable steady states of the system, while other extremal points correspond to unstable states. The existence of such a function is proved for a large class of continuous dissipative physicochemical systems in one spatial dimension. The function G is exact, in the sense that it is obtained by taking into account all nonlinear terms in the evolution equations of the system. Since in some respect this function has properties similar to the Ginzburg–Landau function widely used in second‐order phase transitions, it is called the nonlinear Ginzburg–Landau function (NLGLF) of the system. The NLGLF may be useful for studying the global stability under symmetry‐breaking conditions as well as the character (first and second order) of the transitions between different steady states. The construction of the NLGLF usually requires simple numerical or analytical calculations. A specific example, taken from the physics of liquid crystals, has been worked out analytically in the present paper.

Treatment of higher‐order Lagrangians via the construction of dynamically equivalent first‐order Lagrangians
View Description Hide DescriptionFor a given, in general, singular Lagrangian containing higher‐order time derivatives, a dynamically equivalent Lagrangian with only first‐order time derivatives is constructed. A Hamiltonian structure for this first‐order Lagrangian is then found with the use of the Dirac theory of constraints. It is shown that in the case of a nonsingular higher‐order Lagrangian, the Ostrogradsky dynamics is derived in this way. Further, it is shown that ambiguities characteristic of higher‐order Lagrangian systems do not appear when using this construction. In particular, it is shown that the addition of a total time derivative term to the higher‐order Lagrangian can only induce a time‐independent canonical transformation, even in the case of a singular Lagrangian.

Lagrangian and Hamiltonian many‐time equations
View Description Hide DescriptionThe Lagrangian and Hamiltonian many‐time equations are derived for a finite‐dimensional system with an arbitray number of primary and secondary first‐class constraints. Assuming that all the secondary constraints are generators of gauge transformations, the general form of the Lagrangian gauge algebra is given.

Balance laws and centro velocity in dissipative systems
View Description Hide DescriptionStarting with a density that is conserved for a dynamical system when dissipation is ignored, a local conservation law is derived for which the total flux (integrated over the spatial domain) is unique. When dissipation is incorporated, the conservation law becomes a balance law. The contribution due to dissipation in this balance law is split in a unique way in a part that is proportional to the density and in a divergence expression that adds to the original (conservative) flux density; the total additional flux is uniquely defined. It is shown that these total fluxes appear in the expression for the centro velocity, i.e., in the velocity of the center of gravity of the density, which shows that this velocity can be defined in a unique way (in contrast to a local velocity). Applications to the Korteweg–de Vries–Burgers equations and to the incompressible Navier–Stokes equations are given.

Stability analysis of the Yagle–Levy multidimensional inverse scattering algorithm
View Description Hide DescriptionA layer‐stripping algorithm suggested by Yagle and Levy for solving a multidimensional inverse scattering problem is analyzed. It is shown that the unfiltered version of the algorithm is unstable; for the filtered version, it is proven that the reconstruction depends continuously on the data.

The zero curvature formulation of the sKdV equations
View Description Hide DescriptionThe fermionic extensions of the KdV equation are derived from the zero curvature condition associated with the superalgebra OSp(2‖1). This derivation clarifies why there are only two such extensions possible and why only one of them is supersymmetric. A Lenard type of derivation of these equations is also presented.

Bäcklund transformations for the isospectral and nonisospectral AKNS hierarchies
View Description Hide DescriptionBy converting the usual Lax pairs for the isospectral and nonisospectral AKNS hierarchies into Lax pairs in Riccati forms, a unified explicit form of Bäcklund transformations for these hierarchies of nonlinear evolution equations can be obtained. In the isospectral case it is an auto‐Bäcklund transformation; however, in the nonisospectral case it is not an auto‐Bäcklund transformation.

Classical, linear, electromagnetic impedance theory with infinite integrable discontinuities
View Description Hide DescriptionThe impedance theory is formulated for classical, linear electromagneticscattering from a compact obstacle with a finite number of nonintersecting boundaries. The boundaries are allowed to support infinite, integrable discontinuities in electromagnetic response and the compact regions can depend on space and time. The direct scattering problem is discussed, generalizing recent results by Sabatier and collaborators for the scalar impedance acoustic problem to classical electromagnetism. A chain of Maxwellscatteringequations are derived for the direct scattering problem. Two kinds of ambiguities of electromagnetism at a fixed angle of incidence are found to arise, one from discontinuities in electromagneticmaterial properties, and the other is from time dispersion. Cases are mentioned when parts of the scattering medium are allowed to have time‐dependent motions. This is in contrast to the case of scalar acoustics where ambiguities are intrinsic to certain infinite families of values of Young’s modulii.

On the modification of the Stark states of hydrogen by a weak ac electric field
View Description Hide DescriptionA derivation is presented for the compact integral representation that describes the first‐order perturbed wavefunction of an electron in a fixed Coulomb field and in a harmonic uniform electric field, for the case of an initial stationary Stark state with the symmetry axis along the electric field.

Properties of Leach–Flessas–Gorringe polynomials
View Description Hide DescriptionA generating function is obtained for the polynomials recently introduced by Leach, Flessas, and Gorringe [J. Math. Phys. 3 0, 406 (1989)], and is then used to relate the Leach–Flessas–Gorringe (or LFG) polynomials to Hermite polynomials. The generating function is also used to express a number of integrals involving the LFG polynomials as finite sums of parabolic cylinder functions.

Inverse scattering problem for the 3‐D Schrödinger equation and Wiener–Hopf factorization of the scattering operator
View Description Hide DescriptionSufficient conditions are given for the existence of a Wiener–Hopf factorization of the scattering operator for the 3‐D Schrödinger equation with a potential having no spherical symmetry. A consequence of this factorization is the solution of a related Riemann–Hilbert problem, thus providing a solution of the 3‐D inverse scattering problem.

Scattering for step‐periodic potentials in one dimension
View Description Hide DescriptionQuantum scattering is developed for impurities in potentials that tend to a periodic function in one direction and a constant in the other. Two new technical results are obtained for Hill’s equation. Analytic, asymptotic, and spectral properties are established for solutions of the Schrödinger equation for step‐periodic potentials, with and without impurity. The properties have all been used in Marchenko–Newton inverse scattering. Results apply feasibly to electron, photon, and phonon propagation in layered media.

A Clifford algebra quantization of Dirac’s electron positron field
View Description Hide DescriptionThe quantum field theory of free Dirac particles (four‐component massive spin‐ 1/2 particles) is ‘‘derived’’ by a Segal quantization procedure. First, details are given on how the spinor space of Dirac is actually a minimal left ideal of the Clifford algebra associated with a Lorentz inner product space (+,−,−,−), and how the homogeneous group actions break the natural two‐component quaternion structure to give familiar four‐component complex spinors. Second, Wigner’s procedure for constructing unitary representations of the Poincaré group is used to construct the appropriately induced infinite‐dimensional representation of the inhomogeneous group starting from the above four‐dimensional nonunitary representation. Third, and finally, Segal’s procedure for quantizing classical Fermion fields is adapted to this infinite‐dimensional Hilbert space to obtain the Clifford algebra of annihilation–creation operators for spin‐ 1/2 particles. The familiar Fock space appears as a minimal left ideal in this second Clifford algebra.

Dirac quantization of massive spin‐one particles in an external symmetrical tensor field
View Description Hide DescriptionDirac’s method is used to quantize massive spin‐one particles interacting with an external symmetric tensor field. It is shown that Dirac equations of motion are identical to Euler–Lagrange equations and that φ^{0}, the 0th component of the unknown field, is the only component that depends on the external field. Furthermore, Dirac commutators of the field and the Lorentz generators are calculated. It is shown that the field components except φ^{0} transform like components of a free field, and that φ^{0} transforms like a field component in an external potential.