Volume 27, Issue 10, October 1986
Index of content:

Classification of ten‐dimensional kinematical groups with space isotropy
View Description Hide DescriptionAll the abstract ten‐dimensional real Lie algebras that contain as a subalgebra the algebra of the three‐dimensional rotation group (generators J) and decompose under the rotation group into three three‐vector representation spaces (J itself, K, and P) and a scalar (generator H) are classified. In all cases, the existence of a homogeneous space of dimension 4 is shown.

Nahm’s equations, singular point analysis, and integrability
View Description Hide DescriptionA singular point analysis (Painlevé test) for certain special cases of Nahm’s equations is performed. It is shown that there are cases in which the equations do not pass the test. The Laurent expansion does not contain the right number of arbitrary expansion coefficients. Nevertheless the systems under consideration are completely integrable.

On the integration of the self‐similar equations, and the meaning of the Cole–Hopf transformation
View Description Hide DescriptionThe existence of a first integral of the self‐similar equations associated with a given partial differential system, in one space dimension, is shown, whenever there exists a conserved quantity Q whose scale remains time independent. That result is most conveniently derived through the introduction of Y≡exp Q as a new unknown. In the case of the Burgers equation, the velocity potential Φ has the required property, and the transformation to the new variable exp Φ is precisely the Cole–Hopf transform. The latter, as is well known, linearizes the Burgers equation.

On the Bäcklund transformation and Hamiltonian properties of superevaluation equations
View Description Hide DescriptionBäcklund and Darboux–Bäcklund transformations are deduced for the superevaluation equations recently deduced by Kupershmidt [B. Kuperschmidt, ‘‘A super‐K–d V equation—An integrable system,’’ preprint UTSI‐Tullahoma, 1984; Phys. Lett. A 1 0 2, 213 (1983); J. Phys. A 1 7, L 863 (1984)] and Gürses [H. Gürses and O. Oguz, Phys. Lett. A 1 0 8, 437 (1985)]. By a extension of the technique of BPT [M. Boiti, F. Pempinelli, and G. Z. Tu, Nuovo Cimento B 7 9, 231 (1984)] to anticommuting variables the bi‐Hamiltonian structure and hence the form of the recursion operator for the Lie–Bäcklund symmetry for such equations are deduced. Incidentally some explicit forms of the Lie–Bäcklund symmetry are also deduced.

Smoothness of the action of the gauge transformation group on connections
View Description Hide DescriptionThe NLF–Lie group structure of the group G of the gauge transformations, defined as the group of sections of the bundle P[G] associated to the principal bundle P(M,G), is discussed. Other current definitions of the group of gauge transformations are shown to admit a nontrivial smooth structure only in the case of compact G. The space C of principal connections, as well, is given the structure of local affine NLF‐manifold, after identifications of connections with sections of a convenient vector bundle on M. Finally, the smoothness of the action of G on C is proved in general. In the case of compact M, the group G becomes a tame Fréchet–Lie group and the action a tame smooth action.

Approach to equilibrium for random walks on graphs and for stochastic infinite particle processes
View Description Hide DescriptionAssociated with a finite connected graph g is a doubly stochastic (Laplacian) matrix G(g). A lower bound on the ‘‘mass gap’’ for G(g) is obtained, i.e., the first nonzero eigenvalue of G(g). This estimate is then used to estimate the mass gap for some infinite particle stochastic generators with corresponding processes that are closely related to Monte Carlo methods employed in statistical mechanics calculations.

Sites and googly twistor spaces. I. Vacuum
View Description Hide Description(Anti‐) self‐dual Yang–Mills fields may be described by twistors of the same or opposite handedness as the fields. These are called the leg‐break and googly descriptions, respectively. The leg‐break twistor space is a complex manifold; the Yang–Mills field is given by a vector bundle over this manifold, and massless fields minimally coupled to the Yang–Mills field are given by elements of certain sheaf cohomology groups on the manifold. In this paper, the structure of the googly twistor space when no Yang–Mills field is present is elucidated. It is shown that the googly twistor space is a s i t e. Sites are generalizations of topological spaces, in which the primitive concept is that of an open set rather than that of a point. The massless fields on space‐time are given by the elements of a sheaf cohomology group on the site. Also, this site is isomorphic to a leg‐break site, consisting of a family of open sets in the leg‐break manifold. This provides a strong link between the googly and the leg‐break spaces. The following paper treats the case where a Yang–Mills field is present.

Sites and googly twistor spaces. II. Yang–Mills fields
View Description Hide Description(Anti‐) self‐dual Yang–Mills fields may be described by twistors of the same or opposite handedness as the fields. These are called the leg‐break and googly descriptions, respectively. The leg‐break twistor space is a complex manifold; the Yang–Mills field is given by a vector bundle over this manifold; and massless fields minimally coupled to the Yang–Mills field are given by elements of certain sheaf cohomology groups on the manifold. In the previous paper, we analyzed the structure of the googly twistor space when no Yang–Mills field is present, and showed that it was a s i t e. (Sites are generalizations of topological spaces, in which the primitive concept is that of an open set rather than that of a point.) In this paper, we treat the case where a gauge field is present. We show that the field is represented by a vector bundle over the site, and that massless fields minimally coupled to the Yang–Mills field are given by the elements of a sheaf cohomology group on the site. Also, this vector bundle is isomorphic to one over a leg‐break twistor site. This provides a strong link between the googly and the leg‐break spaces.

Comment on a paper by Espindola, Teixeira, and Espindola [J. Math. Phys. 2 7, 151 (1986)]
View Description Hide DescriptionIt is shown that the criterium to decide whether two Hamiltonians are equivalent given by Espíndola e t a l. [O. Espindola, N. L. Teixeira, and M. L. Espindola, J. Math. Phys. 2 7, 151 (1986)] is incorrect.

Canonical transformations and the equivalence problem
View Description Hide DescriptionSeveral equivalence relations for Hamiltonian systems are studied. The relationship to the theory of canonical transformations is analyzed. In the hyperregular case, the results are transformed into the Lagrangian formulation. The gauge group of Lagrangian mechanics is obtained by looking at the generating functions for canonical fiber invariant transformations. An intrinsic proof of a theorem of Henneaux [M. Henneaux, Ann. Phys. (NY) 1 4 0, 45 (1982)] is given.

Dirac’s theory of constraints in field theory and the canonical form of Hamiltonian differential operators
View Description Hide DescriptionA simple algorithm for constructing the canonical form of Hamiltonian systems of evolution equations with constant coefficient Hamiltonian differential operators is given. The result of the construction is equivalent to the canonical system derived using Dirac’s theory of constraints from the corresponding degenerate Lagrangian.

A Dyson‐like expansion for solutions to the quantum Liouville equation
View Description Hide DescriptionGiven a Hamiltonian of the form H=h+λv, the convergence of a Dyson‐like expansion (in λ) is constructed and shown for the Wigner distribution function that solves the quantum Liouville equation that corresponds to H. Here, h is a quadratic polynomial in p, q; its coefficients may depend continuously on time. The potential v is a function of p and t as well as q; roughly speaking, it is the Fourier transform of a time‐dependent measure.

Conformal collineations and anisotropic fluids in general relativity
View Description Hide DescriptionRecently, Herrera e t a l. [L. Herrera, J. Jimenez, L. Leal, J. Ponce de Leon, M. Esculpi, and V. Galino, J. Math. Phys. 2 5, 3274 (1984)] studied the consequences of the existence of a one‐parameter group of conformal motions for anisotropic matter. They concluded that for special conformal motions, the stiff equation of state (p=μ) is singled out in a unique way, provided the generating conformal vector field is orthogonal to the four‐velocity. In this paper, the same problem is studied by using conformal collineations (which include conformal motions as subgroups). It is shown that, for a special conformal collineation, the stiff equation of state is not singled out. Non‐Einstein Ricci‐recurrent spaces are considered as physical models for the fluid matter.

Collision‐free gases in static space‐times
View Description Hide DescriptionCollision‐free gases in static space‐times are analyzed by developing previous work in static spherically symmetric space‐times and extending the analysis to include the cases of planar and hyperbolic symmetry. By assuming that the distribution function of the gas inherits the space‐time symmetries, distribution solutions to the Einstein–Liouville equations, which are without expansion, rotation, shear, and heat flow, but which have an anisotropic stress are found. The conditions for the gas to behave like a perfect fluid are considered and the relation between equations of state and the distribution function are investigated. In particular, distribution functions that generate the γ‐law equation of state are found. The solutions are extended to find invariant Einstein–Maxwell–Liouville solutions for a charged gas, subject to a consistency condition on the invariant electromagnetic potential. Finally, the general solution of Liouville’s equation in the static space‐times is obtained and a particular nonstatic solution is considered, which can be shown to lead to a self‐gravitating gas with expansion, shear, and heat flow.

Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric. II
View Description Hide DescriptionAn alternate treatment of the results of paper I is given. As in that paper, the Unruh boundary condition is formulated, the Unruh vacuum is defined as a state satisfying this boundary condition and the thermal character of the state is exhibited. The present work differs in that it uses the double‐wedge region of the Kruskal manifold and defines and uses a precise notion of distinguished modes.

Linear transport theory in a random medium
View Description Hide DescriptionThe time‐independent linear transport problem in a purely absorbing (no scattering) random medium is considered. A formally exact equation for the ensemble averaged distribution function 〈Ψ〉 is derived. Under the assumption of a two‐fluid statistical mixture, with the transition from one fluid to the other assumed to be determined by a Markov process, an exact solution to this equation for 〈Ψ〉 is obtained. In the source‐free case, this solution is shown to agree with the result obtained by ensemble averaging simple exponential attenuation. Several approximations to the exact equation for 〈Ψ〉 are considered, and numerical results given to assess the accuracy of these approximations.

Charge operators without local commutativity
View Description Hide DescriptionA Goldstone‐type theorem is proved for quantum field theories in any n≥2 space‐time dimensions, without assuming local commutativity.

A simple proof of duality for local algebras in free quantum field theory
View Description Hide DescriptionNew and simple proofs of duality for local von Neumann algebras in free‐scalar field models associated with a general class of regions in Minkowski space are presented. The proofs are given for both the massive and massless cases and an abstract result of Araki [H. Araki, J. Math. Phys. 4, 1343 (1963)] is assumed. The properties of the local algebras are analyzed using the associated real linear manifolds. Duality is proved in the massive models using elementary properties of Sobolev spaces and in the massless model using dilatation covariance. A proof of the factor property and the cyclicity and separability of the vacuum for these local algebras is also given.

Yang–Mills cohomology in four dimensions
View Description Hide DescriptionThe local polynomial cohomology space of the Yang–Mills BRS operator in four dimensions is computed. In order to simplify the analysis, without omitting the physically interesting cases, the investigation is limited to polynomials whose Fadeev–Popov charge and UV naive dimensions have upper bounds. Furthermore the results are used to compute, á l a Stora, the local functional Yang–Mills anomalies, from which the uniqueness of the Adler–Bardeen–Jackiw anomaly follows.

Traveling‐wave solutions and the coupled Korteweg–de Vries equation
View Description Hide DescriptionSome coupled nonlinear equations are considered for studying traveling‐wave solutions. By introducing a stream function Ψ it is shown that if one of the solutions is of the form v≡v(x−c t), the other also must be of the form u≡u(x−c t). In addition, the possibility of including cubic nonlinear terms has been considered and such a system, assuming that the solutions are of the traveling‐wave type, has been solved.