Volume 31, Issue 4, April 1990
Index of content:

Defining equations for supergroup orbits in super Clifford modules
View Description Hide DescriptionThe defining equations for group orbits of a maximal subgroup of Gl_{ p‖q } in certain highest weight representations of the Lie super algebra gl_{ p‖q } are discussed.

Some aspects of quantum groups and supergroups
View Description Hide DescriptionSome features of Manin’s construction of quantum groups are developed and extended to supergroups.

Classification of unitary highest weight representations for noncompact real forms
View Description Hide DescriptionUsing Jakobsen theorems, unitarizability in Hermitian symmetric spaces is discussed. The set of all missing highest weights is explicitly calculated and the construction of their corresponding highest weights vectors is studied.

Maximum entropy formalism and analytic extrapolation
View Description Hide DescriptionThe maximum entropy principle is used to obtain a new analytic extrapolation method just complementary to the Padé‐type method which leads to rigorous upper and lower bounds on the extrapolated function on the cut complex plane. Among the large class of functions that could equally well represent an analytical function in the experimental region a choice is made of the unique function that maximizes the entropy functional associated to this set of functions. The result is the least biased function compatible with the actual experimental data. This extrapolation method is applied to kaon–nucleon experimental data in order to obtain the most reasonable values for the KNY coupling constants compatible with the available experimental data and analytical constraints.

A simple Lagrangian for integrable systems
View Description Hide DescriptionA simple Lagrangian for integrable models that can describe the entire hierachy of equations associated with the system is proposed. In addition, the Lax equation, the recursion relation between the conserved quantities, as well as the vanishing of the Nijenhuis torsion tensor can also be derived from the system of equations following from this Lagrangian.

Generating functions, bi‐Hamiltonian systems, and the quadratic‐Hamiltonian theorem
View Description Hide DescriptionThe concept of bi‐Hamiltonian systems and its connection with the theory of canonoid transformations is shown from a geometrical viewpoint. The relations between symplectic and canonoid diffeomorphisms are studied using an approach based on the theory of generating functions. These results are used for obtaining a new theorem which will represent an intrinsic and coordinate‐free generalization of the ‘‘quadratic Hamiltonian theorem.’’

Conformal transformations and the effective action in the presence of boundaries
View Description Hide DescriptionThe conformal properties of the heat kernel expansion are used to determine the local form of the coefficients in a manifold with boundary. The conformal transformation of the effective action is obtained. A novel derivation of the boundary term in the Gauss–Bonnet–Chern theorem is detailed.

Sectional curvature and tidal accelerations
View Description Hide DescriptionSectional curvature is related to tidal accelerations for small objects of nonzero rest mass. Generically, the magnification of tidal accelerations due to high speed goes as the square of the magnification of energy. However, some space‐times have directions with bounded increases in tidal accelerations for relativistic speeds. These investigations also yield a characterization of null directions that fail to satisfy the generic condition used in singularitytheorems. For Ricci flat four‐dimensional space‐times, tidally nondestructive directions are characterized as repeated principal null directions.

Block‐diagonalization in second quantization
View Description Hide DescriptionThe unitary matrix that brings a Hermitian H into block‐diagonal form can be uniquely determined under very simple and transparent conditions. Int his work the block‐diagonalization problem is investigated in the framework of the second quantization formalism. Starting with an operator Ḩ which in any n‐particle Fock space has a well‐defined matrix representation an attemt was made to answer the question whether the transformation matrices T which can be separately given in the various n‐particle spaces can be considered matrix representations of the same operator. T̂. Interestingly, the very important result was reached that the block‐diagonalization operator T̂ exists and is unique. As a particular example, attention was concentrated on the case of an operator Ĥ given by a one‐particle operator. In this case the block‐diagonalization operator can be constructed and given in explicit from. This approach is applied to the theory of Green’s functions where the block‐diagonalization of the Hamiltonian has interesting consequences that are illustrated in some details.

Path integral solution for a particle confined in a region
View Description Hide DescriptionThe propagator relative to a particle constrained to move in a finite region of space is calculated in the framework of path integrals. This region of the three‐dimensional space is delimited through a sector of opening angle α, and also through the action of two attractive harmonic potentials, one being central and located in the 0x y plan, and the other directed along the z axis, with respective pulsations ω and ω_{0}. It is shown that for α=π/2 and π the propagator is the sum of propagators evaluated on classical paths. The important case of the edge (α=2π) is considered.

Criteria for the Kato smoothness with respect to a dispersive N‐body Schrödinger operator
View Description Hide DescriptionThe operators of the form H=ω(D)+∑v _{ a }(π^{ a } x), which are a natural generalization of N‐body Schrödinger operators, are studied. Certain criteria are presented that allow one to verify if a given pseudodifferential operator is Kato smooth with respect to H.

The probability operator in quantum theory and quantization of a system of free fields
View Description Hide DescriptionA theoretical framework for quantization, defined by the normalized positive‐definite probability operator establishing dynamical correspondence between classical and quantum Poisson brackets, is presented. The resulting quantum theory, unlike the conventional one, admits consistent probabilistic interpretation. It is shown that, in the nonrelativistic case, quantization based on the probability operator leads to the theory known as ‘‘quantum mechanics with a non‐negative quantum distribution function.’’ A generalization of the proposed framework to the case of the relativistic theory of fields is attempted. Four auxiliary problems of constructing probability operators of one‐dimensional field oscillators in Bose and Fermi algebras are formulated and solved. On the basis of these solutions it is concluded that spinor fields are not quantizable in the Bose algebra with the help of the probability operator (the analog of Pauli’s theorem). An equation for the probability operator of a system of free fields is derived from the principles of dynamical correspondence and translational invariance. The physical meaning of the operators corresponding to classical field amplitudes, such as annihilation and creation operators of field quanta with definite energy‐momentum, is shown to emerge as a consequence of this equation. It is shown that the quantization of a system consisting only of tensor fields or only of spinor fields in the formalism of the probability operator leads to difficulties. It is shown further that these difficulties can be removed by considering quantization of a system containing both tensor and spinor fields. As an illustration, quantization of a system consisting of a massive vector field and a massive spinor field is considered and it is found that a noncontradictory quantization requires the mass of the vector particle to be less than that of the spinor particle. The probability operator thus acts as a mechanism of selection of the fields to be quantized already at the level of free fields.

The causal geometry of twistor space
View Description Hide DescriptionThe problem of classifying the nature of the vector connecting a pair of points in Minkowski space is examined within the twistor theoretic framework. Two approaches are considered, one algebraic and the other geometric. The latter of the two is studied in some detail, providing some insight into the relation between the causal structure of Minkowski space and the geometry of projective twistor space.

An example of a nontrivial causally simple space‐time having interesting consequences for boundary constructions
View Description Hide DescriptionAn example is given of a causally simple space‐time that may serve as a counterexample for various purposes, such as showing that for a general causally simple space‐time the chronological common past of a terminal indecomposable future set need not be an indecomposable set.

Initial value problem for colliding gravitational plane waves. III
View Description Hide DescriptionThe development of a homogeneous Hilbert problem (HHP) approach to the initial value problem (IVP) for colliding gravitational plane waves with noncollinear polarization that began in two earlier papers [I. Hauser and F. J. Ernst, J. Math. Phys. 3 0, 872 (1989) and 3 0, 2322 (1989)] is continued. After formulating the HHP, the description of how one can apply it to generate a new family of solutions of the colliding wave problem that generalizes a three‐parameter family constructed by Ernst, García, and Hauser [J. Math. Phys. 2 9, 681 (1988)] using a double‐Harrison transformation is given. Then the proof that the solution of the new HHP indeed solves the IVP that is posed is presented. A matrix Fredholm equation of the second kind that is equivalent to the HHP is also deduced. This will be used in a sequel to complete the proof of existence of solutions of the HHP and the proof that certain assumed differentiability hypotheses are in fact valid.

Complete integrability of two‐dimensional gravity with dynamical torsion
View Description Hide DescriptionThe most general Lagrangian for two‐dimensional gravity with dynamical torsion is considered. A general solution of the system of nonlinear equations of motion is found. Also found is a global solution of the Cauchy problem.

Renormalized equations for linear transport in stochastic media
View Description Hide DescriptionThe master equation is used to obtain a model describing the ensemble‐averaged intensity corresponding to linear particle transport in randomly mixed immiscible fluids. An asymptotic limit corresponding to small amounts of opaque fluids admixed with large amounts of transparent fluids is employed to reduce the complexity of the description. In the limit of a single transparent fluid, a renormalized transport equation is obtained, involving an effective source and effective interaction coefficients that account, in a simple way, for the statistical nature of the problem in this asymptotic limit.

Propagators for relativistic systems with non‐Abelian interactions
View Description Hide DescriptionThe relativistic evolution of a system of particles in the proper‐time Schwinger–DeWitt formalism is investigated. For a class of interactions that can be represented as Fourier transforms of bounded complex matrix‐valued measures, a Dyson series representation of the propagator is obtained. This class of interactions is non‐Abelian and includes both external electromagnetic and Yang–Mills fields. The study of the relativistic problem is facilitated by embedding the original quantum evolution into a larger class of evolution problems that result if one makes an analytic continuation of the metric tensorg _{μν}. This continuation is chosen so that the extended propagator shares (for all signatures of g _{μν} ) the Gaussian decay properties typical of heat kernels. Estimates of the nth‐order Dyson iterate kernels are found that ensure the absolute convergence of the perturbation series. In this fashion a number of analytic and smoothness properties of the propagator are determined. In particular, it is demonstrated that the convergent Dyson series representation constructs a fundamental solution of the equations of motion.

Finite energy solutons for (1+1)‐dimensional σ models
View Description Hide DescriptionVarious properties of finite energy solutions of the (1+1)‐dimensional sigma models are studied. It is shown that the energy densities of these solutions exhibit some extended lump‐like structures that cross each other without interaction, and that the procedure of adding a soliton to a given solution developed by Harnad, Schnider, and Saint‐Aubin [Commun. Math. Phys. 9 2, 329 (1984); 4 3, 33 (1984)] mixes the matrix elements of this solution in a complicated way, but it does not modify its energy momentum tensor density. The one parameter family of conserved currents discovered by Eichenherr, Forger, and Pohlmeyer [Nucl. Phys. B 1 5 5, 381 (1979); 1 6 4, 528 (1980); Commun. Math. Phys. 4 6, 207 (1976)] must be considered to distinguish between such solutions. Finally the Uhlenbeck method of construction of Euclidean solutions modified to make it applicable to the construction of Minkowskian solutions for some classes of sigma models. It is shown that this method does not modify the energy momentum tensor density either.

Fourier transforms on A/G and knot invariants
View Description Hide DescriptionFourier transformation on A/G leads to some elementary insight into Witten’s expression for the Jones polynomial.