Index of content:
Volume 27, Issue 9, September 1986

Nonlinear equations with superposition formulas and the exceptional group G_{2}. I. Complex and real forms of g_{2} and their maximal subalgebras
View Description Hide DescriptionIn order to study nonlinear ordinary differential equations with superposition principles, related to the exceptional simple Lie groupG _{2}, the complex and real forms of its Lie algebra are examined and their maximal subalgebras are summarized. In particular the parabolic subalgebras of the noncompact real form g ^{NC} _{2}(R) are determined. Explicit matrix realizations of the fundamental representation D(1,0) are used and studied in connection with invariant subspaces in a seven‐dimensional (complex or real) vector space. The results are collected in three tables of specific interest for the study of nonlinear differential equations, which will be developed in Paper II of this series.

Eulerian parametrization of Wigner’s little groups and gauge transformations in terms of rotations in two‐component spinors
View Description Hide DescriptionA set of rotations and Lorentz boosts is presented for studying the three‐parameter little groups of the Poincaré group. This set constitutes a Lorentz generalization of the Euler angles for the description of classical rigid bodies. The concept of Lorentz‐generalized Euler rotations is then extended to the parametrization of the E(2)‐like little group and the O(2,1)‐like little group for massless and imaginary‐mass particles, respectively. It is shown that the E(2)‐like little group for massless particles is a limiting case of the O(3)‐like or O(2,1)‐like little group. A detailed analysis is carried out for the two‐component SL(2,c) spinors. It is shown that the gauge degrees of freedom associated with the translationlike transformation of the E(2)‐like little group can be traced to the SL(2,c) spins that fail to align themselves to their respective momenta in the limit of large momentum and/or vanishing mass.

Generating relations for reducing matrices. II. Corepresentations
View Description Hide DescriptionThe auxiliary group approach developed in Paper 1 [R. Dirl, P. Kasperkovitz, M. I. Aroyo, J. N. Kotzev, and M. Angelova‐Tjurkedjieva, J. Math. Phys. 2 7, 37 (1986)] is generalized for the case of corepresentations of antiunitary groups. It allows us to reduce the multiplicity problem and to derive consistent generating relations for the elements of the reducing matrices for coreps. Two examples are worked out to illustrate the general scheme.

Analysis on generalized superspaces
View Description Hide DescriptionThe analysis over σ‐commutative algebras (generalized supercommutative algebras), that is, differentiation and integration for functions defined on superspace over a σ‐commutative algebra, is studied.

Linearization and Painlevé property of Liouville and Cheng equations
View Description Hide DescriptionIt is demonstrated that the Liouville equation and the Cheng equation (describing a chemical reaction) are free from movable critical manifolds and possess the Painlevé property. The associated linearizing transformations and general solutions follow naturally from the Painlevé analysis.

The anomaly structure of theories with external gravity
View Description Hide DescriptionThe cohomology problem of the overall local symmetry group of theories with external gravity, including diffeomorphisms, local Lorentz, and gauge transformations, is studied, in order to determine all possible anomalies. To this end the nontrivial cohomology classes of the coupled system of two coboundary operators are classified in the abstract. Using this result and a technical assumption the nontrivial cohomology classes of the coboundary operator associated with diffeomorphisms are determined. These possible anomalies split in any dimension into two distinct families. Both are calculated (the second only in four dimensions). Using known results about gauge and local Lorentz anomalies, the possible anomalies of the overall local symmetry group are determined.

A path‐integral–Riemann‐space approach to the electromagnetic wedge diffraction problem
View Description Hide DescriptionA path integral constructed over a particular Riemann space is developed and applied to two‐dimensional wedge problems. This path‐integral–Riemann‐space (PIRS) approach recovers the exact solutions of the heat conduction and the corresponding electromagnetic wedge problems. A high‐frequency asymptotic evaluation of the PIRS electromagnetic wedge solution returns the standard geometrical theory of diffraction(GTD) results. Ramifications of this approach and its relationships with known path‐integral methods are examined.

Group‐related coherent states
View Description Hide DescriptionCoherent states defined with respect to an irreducible r a y representation u: g→u _{ g }, g∈G, of an arbitrary locally compact separable group G are examined. It is shown that the following conditions (a)–(d) are equivalent: (a) u admits coherent states, (b) u is square integrable, (c) the W*‐system implemented by u is integrable, and (d) u is a subrepresentation of the left regular c‐representation, where c is the respective multiplier of u. Furthermore, the group theoretical background of what is called the ‘‘P‐representation of observables’’ associated with coherent states is investigated: It is shown that the P‐representation (which corresponds to a covariant semispectral measure) fulfills a certain maximality requirement. The P‐representation can be used to represent the quantum system in question on the Hilbert spaceL ^{2}(G,d g) of square‐integrable functions (with respect to Haar measured g) on the kinematical group G.

Evolution loops
View Description Hide DescriptionThe problem of manipulating Schrödinger’s particle by time‐dependent external fields is discussed. New solutions of the evolution problem, called e v o l u t i o n l o o p s, are found. They correspond to the ‘‘particle memory’’ in the sense of Brewer and Hahn [Sci. Am. 2 5 1(12), 50 (1984)]. A technique of generating the unitary operations by perturbing the evolution loops is outlined.

Variational processes and stochastic versions of mechanics
View Description Hide DescriptionThe dynamical structure of any reasonable stochastic version of classical mechanics is investigated, including the version created by Nelson [E. Nelson, Q u a n t u m F l u c t u a t i o n s (Princeton U.P., Princeton, NJ, 1985); Phys. Rev. 1 5 0, 1079 (1966)] for the description of quantum phenomena. Two different theories result from this common structure. One of them is the imaginary time version of Nelson’s theory, whose existence was unknown, and yields a radically new probabilistic interpretation of the heat equation. The existence and uniqueness of all the involved stochastic processes is shown under conditions suggested by the variational approach of Yasue [K. Yasue, J. Math. Phys. 2 2, 1010 (1981)].

Large‐N solution of the Klein–Gordon equation
View Description Hide DescriptionAn iterative 1/N expansion scheme is developed to solve the Klein–Gordon equation to obtain the energy spectrum of a scalar particle in a spherically symmetric potential. For the Coulomb potential, this approach is shown to yield the exact results.

Conformal transformations and viscous fluids in general relativity
View Description Hide DescriptionIt is shown that viscous fluid solutions can be obtained by performing conformal transformations of vacuum solutions of Einstein’s field equations. The solutions obtained by such a procedure can be matched, under certain conditions, to their respective original vacuum metrics.

A class of perfect fluid metrics with flat three‐dimensional hypersurfaces
View Description Hide DescriptionThe class of perfect fluid and vacuum space‐times with a family of flat three‐slices and a tensor of exterior curvature covariantly constant within these slices is examined and the corresponding solutions are found. It is shown that this class contains the class of metrics with three commuting Killing vectors. Therefore, e.g., all known stationary metrics with cylindrical or plane symmetry are generalized. An instruction is given for constructing perfect fluid metrics with this symmetry and a connection to a vacuum across surfaces p=0. Thereby the equation of state of the interior rotating perfect fluid can be arbitrarily chosen and the positivity of density and pressure can be forced. A geometric criterion of the interior metric with rotating matter is found that decides whether the exterior solution is stationary or static. Besides solutions with three symmetries, inhomogeneous metrics also are found. Among them is a solution with one symmetry and rotating, expanding, shearing, and accelerating perfect fluid. All resulting vacuum solutions are already known.

About vacuum solutions of Einstein’s field equations with flat three‐dimensional hypersurfaces
View Description Hide DescriptionThe class of vacuum space‐times with a family of flat three‐slices and a traceless tensor of exterior curvature K _{ a b } is examined. Metrics without symmetry and solutions describing gravitational radiation are obtained. It turns out that there is a correlation between rank (K _{ a b }) and the Petrov type. Although the resulting solutions are already known, the richness of the class of space‐times with flat slices becomes obvious. An example is given of a metric with one‐parameter manifold of families of flat slices.

An example of affine collineation in the Robertson–Walker metric
View Description Hide DescriptionAn affine collineation for the Robertson–Walker metric is found. It implies a condition on the metric that is compatible with Einstein’s equations for a perfect fluid satisfying the Hawking–Ellis energy conditions. It is shown how the geodesics of the metric are obtained from the constant of motion associated to the affine collineation.

Gravitational solutions, including radiation, for a perturbed light beam
View Description Hide DescriptionLinearized field equations and solutions are derived for a perturbed sheet beam of light. The work is based on an exact solution of a collimated beam in the geometrical limit. The linearized field changes of the initially curved background metric can be put, with the help of the harmonic conditions, into a normal coordinate form. These six normal coordinates satisfy six linearized, inhomogeneous, field equations in three variables. Stationary solutions include divergent beams. Gravitational waves propagating opposite to the beam’s flux are found to be confined to a region about the propagation axis of the beam, much as is experienced in wave guides. Radiative cases can be produced by large angle scattering of light and are discussed in terms of their effect on an ideal optical antenna. The effect is one that grows linearly with time. The growth time is prohibitively long for the most energetic systems that can be realistically considered in the foreseeable future.

Three remarks on Powers’ theorem about irreducible fields fulfilling CAR
View Description Hide DescriptionFirst it is shown that within a relativistic Fermi field theory, a bound ∥Ψ_{ k }( f,t)∥ ≤C∥ f ∥_{2} already implies canonical anticommutation relations (CAR). Then under Powers’ assumptions a linear, first‐order differential equation for the fields ψ_{ k }(x,t) is derived. This shows that in the set of generalized free fields fulfilling CAR only the free fields are irreducible at time zero. Finally Fermi fields in two space‐time dimensions are considered. It is shown that only four‐fermion interaction might be compatible with CAR and a bound on the coupling strength is derived.

Four‐dimensional boson field theory. II. Existence
View Description Hide DescriptionThe existence of the continuum, quantum field theory found by Baker and Johnson [G. A. Baker, Jr. and J. D. Johnson, J. Phys. A 1 8, L261 (1985)] to be nontrivial is proved rigorously. It is proved to satisfy all usual requirements of such a field theory, except rotational invariance. Currently known information is consistent with rotational invariance however. Most of the usual properties of other known Euclidean boson quantum field theories hold here, in a somewhat weakened form. Summability of the sufficiently strongly ultraviolet cutoff bare coupling constant perturbation series is proved as well as a nonzero radius of convergence for high‐temperature expansions of the corresponding continuous‐spin Ising model. The description of the theory by these two series methods is shown to be equivalent. The field theory is probably not asymptotically free.

Formal power series solutions of supersymmetric (N=3) Yang–Mills equations
View Description Hide DescriptionFormal power series solutions of the linear system with one spectral parameter associated to the constraint equations for Yang–Mills superconnections on N‐extended super‐Minkowski space are considered. For N=3 the integrability equations reduce to the supersymmetric field equations. The method of approach is identical to that used by Takasaki for the self‐dual equations, based upon formal power series in a spectral parameter and in spatial variables. The problem is reduced to a linear system of equations for a superfield with values in an ∞‐dimensional Grassmann manifold. The formal solution is expressed in terms of data on a (3‖2N)‐dimensional superhypersurface. However, a difficulty arises with respect to the Cauchy problem, which becomes formally solvable only for an extended system, breaking the relativistic invariance through introduction of additional superfields.

On the formulation of the positive‐energy theorem in Kaluza–Klein theories
View Description Hide DescriptionThe positive‐energy theorem is formulated in the context of Kaluza–Klein theories. Different cases are considered, including the situation in which no symmetry is assumed. This work offers a new technique for stability considerations in Kaluza–Klein theories.