Volume 31, Issue 10, October 1990
Index of content:

Classification of gauge groups in terms of algebraic structure of first class constraints
View Description Hide DescriptionProperties of gauge transformations for singular Lagrangians are investigated to classify types of gauge groups. A general method of the classification is proposed based on properties of structure functions of the Poisson brackets (or the commutators) of first class constraints. A remarkable result is that the algebraic structure of the gauge group is essentially determined by the first class constraints of the final step of constraint series which are required successively from the stationarity conditions of the constraints. Owing to this consequence, the classification of gauge groups is made simple and transparent. The structure and property of the gauge group can be characterized in terms of the algebraic structure functions among the final step constraints and the number of the steps of the constraints series. The formulation proposed will give a clue to find new types of gauge groups.

Symmetrized powers of the fundamental irrep of E _{6}
View Description Hide DescriptionThe resolution of all possible antisymmetrically symmetrized powers of the fundamental irrep (2 7) is given. This reduces the problem of plethysms of the fundamental irrep to the evaluation of ordinary Kronecker products of E _{6} irreps.

Algebras for the two‐sphere and the three‐sphere groups of compact simple Lie groups
View Description Hide DescriptionThe infinite‐dimensional Lie algebras corresponding to the Lie groups of smooth maps from two and three spheres to compact simple Lie groups are studied. The problem of their central extension is solved. The problem of the existence of semidirect sum algebras containing these and the algebras of the groups of diffeomorphisms on the two and three spheres is treated.

Rough surfaces and the renormalization group
View Description Hide DescriptionPerturbative techniques are used to derive a renormalization group equation for a free scalar field in a domain with rough boundaries and mixed boundary conditions. It is found that to second order in the surface height the boundary conditions scale not only as the perturbed area, but also with a nonlinear term that arises from matching orders of surface height in the perturbation series for the field amplitude.

Hearing the shape of a general doubly connected domain in R ^{3} with impedance boundary conditions
View Description Hide DescriptionThe basic problem in this paper is that of determining the geometry of a general doubly connected domain in R ^{3} together with an impedance condition on its inner bounding surface and another impedance condition on its outer bounding surface, from the complete knowledge of the eigenvalues {λ_{ j }}^{∞} _{ j=1} for the three‐dimensional Laplacian using the asymptotic expansion of the spectral function θ(t)=∑^{∞} _{ j=1} exp(−tλ_{ j }) for small positive t.

Anomalies from geometric quantization of fermionic field theories
View Description Hide DescriptionGeometric quantization on (infinite‐dimensional) graded symplectic manifolds is elaborated for a restricted class of phase spaces. The formalism includes the treatment of Fermionic field theories. The chiral anomaly [U(1)‐anomaly] as well as the non‐Abelian (covariant) anomaly of D‐dimensional non‐Abelian gauge theories is calculated in this framework.

On identically closed forms locally constructed from a field
View Description Hide DescriptionLet M be an n‐dimensional manifold with derivative operator ∇_{ a } and let B(M) be an arbitrary vector bundle over M, equipped with a connection. A cross section of B defines a field φ on M. Let α be a p‐form on M (with p<n) which is locally constructed from φ and finitely many of its derivatives (as well as, possibly, some ‘‘background fields’’ ψ and their derivatives) such that dα=0 for all cross sections φ. Suppose further that α=0 for the zero cross section, φ=0. It is proven here that there exists a (p−1)‐form β that also is a local function of φ,ψ and finitely many of their derivatives, such that α=dβ. A number of applications of this result are described. In particular, gauge invariance is established for the charges and the total fluxes derived from gauge‐dependent conserved currents, and severe limitations are established on the the possibilities for gravitational analogs of magnetic charges.

Schrödinger processes and large deviations
View Description Hide DescriptionFor a large system of independent diffusing particles, each of which is killed at a certain space‐time dependent rate, the conditional distribution of surviving trajectories in a bounded time interval is computed, given the approximate form of the initial and final empirical distribution of surviving particles. This generalizes a result for the Brownian case without killing, which was first obtained by Schrödinger [Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl. 1 9 3 1, 144].

Solution of the multivariate Fokker–Planck equation by using a maximum path entropy principle
View Description Hide DescriptionThis paper proposes an approach via the maximum entropy principle in order to determine the nonstationary solutions of the Fokker–Planck equation with time varying coefficients. The constraints are not the state moments (as usual) but their dynamic equations. The maximum entropy principle herein utilized is a slight extension of Jaynes’ principle, which involves the ‘‘path entropy’’ of the stochastic process.

Symmetries in predictive relativistic mechanics
View Description Hide DescriptionSymmetries of a two‐body relativistic harmonic oscillator and a two‐body relativistic Coulomb system are considered. It is shown that, in the harmonic case, the Lie algebra of first integrals includes Poincaré algebra and u(3). In the Coulomb case, the Lie algebra of first integrals includes Poincaré algebra and one of the algebras so(1, 3), so(4), or the algebra corresponding to the group of rigid motions in R^{3}. In both cases, the algebra generated by internal symmetry together with the complete space‐time symmetry is infinite dimensional.

On the mixing‐enhancing structure of a class of quantum dynamical semigroups
View Description Hide DescriptionA sufficient condition for a dissipative evolution to give rise to an ever more chaotic state is obtained and the structure of the corresponding density matrix is studied. As a byproduct, the equation of motion in the Schrödinger picture is, in some cases, explicitly solved.

On the complete system of observables in quantum mechanics
View Description Hide DescriptionThis paper contains a series of remarks about the concept of Complete System of Observables (CSO) in quantum mechanics and a discussion of two definitions of CSO, one given by Jauch [Helv. Phys. Acta 3 3, 711 (1960)] and the other by Prugovecki [Can. J. Phys. 4 7, 1083 (1968)].

Applications of the differentiability of eigenvectors and eigenvalues to a perturbed harmonic oscillator
View Description Hide DescriptionThe potential interaction λx ^{2}/(1+g x ^{2}), g>0, of the harmonic oscillator H _{0}=−d ^{2}/d x ^{2}+x ^{2} considered as an operator in the space L _{2}(−∞, ∞) is bounded. This together with the nondegeneracy of the eigenvalues implies that the eigenvectors of the perturbed harmonic oscillator as functions of the parameters λ and g are strongly differentiable. The eigenvalues are therefore differentiable functions for every real λ and every real g>0. In particular, the first eigenvalue E _{1}(λ) as a function of λ is strictly concave ( E ^{‘} _{1}(λ)<0). This paper, exploiting the above properties, aims at several inequalities for the eigenvalues of H _{0}+λx ^{2}/(1+g x ^{2}), g>0. Emphasis is given to the inequality that follows from the strict concavity of the function E _{1}(λ).

Factorizations of the S matrix
View Description Hide DescriptionThe S matrix of the Schrödinger equation, regarded as a function on the real axis with values in the group of unitary operators L ^{2}(S^{2})^{]} L ^{2}(S^{2}), where S^{2} is the unit sphere in R^{3}, is factorized in two different ways. One of these is a standard Wiener–Hopf factorization with respect to the real line. The other is the kind of factorization that defines the Jost function and which has been found to be a useful tool for the solution of the inverse scattering problem. A number of results are given that relate the two factorizations, their existence as well as the indices they give rise to. Some known theorems on the standard factorization lead to new results for the three‐dimensional inverse scattering problem for the Schrödinger equation with a noncentral potential; in particular, a characterization of admissible S matrices is obtained.

Anisotropic homogeneous cosmologies with perfect fluid and electric field
View Description Hide DescriptionA new three‐parameter family of cosmological models is found, which are solutions of Einstein–Maxwell equations in a space‐time filled with electrically neutral stiff matter. They generalize an anisotropic homogeneous cosmology without electromagnetic field by Vajk and Eltgroth [J. Math. Phys. 1 1, 2212 (1970)] and support a conjecture about proportionality of electromagnetic four‐potential of an Einstein–Maxwell solution and the Killing vector of a corresponding space‐time with stiff matter. This conjecture turns out to be the clue to a new solution‐generating method of Einstein–Maxwell fields with sources.

Curvature‐squared cosmology in the first‐order formalism
View Description Hide DescriptionVariation of the R+αR ^{2} action with respect to independent metric and connection fields is shown to be equivalent to the metric‐compatible fourth‐order gravity coupled to a vector defined as a function of the trace of the energy‐momentum tensor. The field equations are second order. The Friedmann cosmology based on this model is studied and it is shown to include nonsingular solutions at t=0.

Shear‐free perfect fluids in general relativity II. Aligned, Petrov type III space‐times
View Description Hide DescriptionPetrov type III, shear‐free, perfect fluid solutions of the Einstein field equations, with a barotropic equation of statep=p(w) satisfying w+p≢0, are investigated. It is shown that if the acceleration of the fluid is orthogonal to the two‐spaces spanned by the repeated principal null direction of the Weyl tensor and the fluid four‐velocity, or if the fluid four‐velocity lies in the two‐spaces spanned by the principal null directions of the Weyl tensor, then the fluid’s volume expansion is zero.

Reliability of perturbation theory in general relativity
View Description Hide DescriptionThe relation between perturbation theory and exact solutions in general relativity is tackled by investigating the existence and properties of smooth one‐parameter families of solutions. On the one hand, the coefficients of the Taylor expansion (in the parameter) of any given smooth family of solutions necessarily satisfy the hierarchy of perturbation equations. On the other hand, it is the converse question (does any solution of the perturbation equations come from Taylor expanding some family of exact solutions ?) which is of importance for the mathematical justification of the use of perturbation theory. This converse question is called the one of the ‘‘reliability’’ of perturbation theory. Using, and completing, recent results on the characteristic initial value problem, the local reliability of perturbation theory for general relativity in vacuum is proven very generally. This result is then generalized to the Einstein–Yang–Mills equations (and therefore, in particular, to the Einstein–Maxwell ones). These local results are then partially extended to global ones by: (i) proving the existence of semiglobal vacuum space‐times (respectively, Einstein–Yang–Mills solutions) which are stationary before some retarded time u _{0}, and radiative after u _{0}, and which admit a smooth conformal structure at future null infinity; and (ii) constructing smooth one‐parameter families of such solutions whose Taylor expansions are of the ‘‘multipolar post‐Minkowskian’’ type which has been recently used in perturbation analyses of radiative space‐times.

Exact model for a relativistic star
View Description Hide DescriptionAssuming that the physical three‐space in a relativistic superdense star has the geometry of a three‐spheroid, a static spherically symmetric model based on an analytic closed‐form solution of Einstein’s field equations is presented. Assuming the density of the order of 2×10^{14} g cm^{−3}, estimates of the total mass and size of the stars of the model are obtained for various values of a density‐variation parameter that is suitably defined. The total mass and the boundary radius of each of these models are of the order of the mass and size of a neutron star.

Initial value formulation for the spherically symmetric dust solution
View Description Hide DescriptionAn initial value formulation for the dust solution with spherical symmetry is given explicitly in which the initial distributions of dust and its velocity on an initial surface are chosen to be the initial data. As special cases, the Friedmann universe, the Schwarzschild solution in comoving coordinates, and a spherically symmetric and radially inhomogeneous cosmological model are derived.