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The multi-field generalisation of the Bateman equation arises from considerations of the continuation of String and Brane equations to the case where the base space is of higher dimension than the target space. The complex extension of this equation possesses a remarkably large invariance group, and admits a very simple implicit form for its general solution, in addition to the special case of holomorphic and anti-holomorphic explicit solutions. A class of inequivalent Lagrangians for this equation is discovered.

The general solution to the complex Bateman equation is constructed. It is given in implicit form in terms of a functional relationship for the unknown function. The known solution of the usual Bateman equation is recovered as a special case.

We present two sets of qualgebras involving operators which generalize creation and annihilation operators. These two groups of operators satisfy separately quommutation relations rather than commutation or anticommutation relations. The quommutators of the creation and annihilation operators generate new “neutral operators” which themselves are subjected to quommutation relations. Two solutions are presented. In the second one, some new symmetry relations are added to the system. In a certain sense these extra relations, rather than imposing new constraints on the parameters, increase their freedom.

The Algebra of Weyl symmetrised polynomials in powers of Hamiltonian operatorsP andQ which satisfy canonical commutation relations is constructed. This algebra is shown to encompass all recent infinite dimensional algebras acting on two-dimensional phase space. In particular the Moyal bracket algebra and the Poisson bracket algebra, of which the Moyal is the unique one parameter deformation are shown to be different aspects of this infinite algebra. We propose the introduction of a second deformation, by the replacement of the Heisenberg algebra forP, Q with aq-deformed commutator, and construct algebras ofq-symmetrised Polynomials.

Deformations of the infiniteN limit of the ZamolodchikovWN algebra are discussed. A recent one, due to Pope, Romans and Shen with non-zero central extensions for every conformal spin is shown to be formally renormalisable to one representable in Moyal bracket form. Another deformation is discovered which, like the algebra of Pope et al. possesses automatic closure, but has non-zero central extension only in the Virasoro subalgebra.

The purpose of this article is to initiate a study of a class of Lorentz invariant, yet tractable, Lagrangian Field Theories which may be viewed as an extension of the Klein-Gordon Lagrangian to many scalar fields in a novel manner. These Lagrangians are quadratic in the Jacobians of the participating fields with respect to the base space co-ordinates. In the case of two fields, real valued solutions of the equations of motion are found and a phenomenon reminiscent of instanton behaviour is uncovered; an ansatz for a subsidiary equation which implies a solution of the full equations yields real solutions in three-dimensional Euclidean space. Each of these is associated with a spherical harmonic function., 13 pages, Latex

Ignoring the problem of sources and singularities, explicit expressions are constructed for the ansatze of Atiyah and Ward. These take an especially simple form in theR gauge of Yang. Some non-linear transformation properties of the self-duality equations in this gauge provide an inductive proof of the ansatze. There is a six-parameter family of these Backlund transformations. They take real SU(2) gauge fields into real SU(1, 1) gauge fields and vice versa.

It is shown that the entire Virasoro, Ramond and Neveu-Schwarz algebras can each be constructed from a finite number of well-chosen generators satisfying a small number of conditions. The most economical sets consist of just two starting generators in all cases, subject to eight conditions for the Virasoro case, five conditions for the Ramond case, and nine conditions for the Neveu-Schwarz case.

The iterative solution of the Schrodinger integral equation (for zero energy) associated with certain singular potentials containing logarithmic singularities of the form lognr/r2 fails to converge unless a cut-off is introduced. The solution for these potentials is shown to be renormalizable by a multiplicative factor dependent upon the cut-off. The existence of a perturbation-series solution with divergent terms as an indication that the closed solution possesses an essential singularity in the expansion parameter is examined for a soluble case, and a functional relationship between this parameter and the cut-off is found which explicitly exhibits the essential singularity in the perturbation series. This phenomenon is conjectured to occur in a more general situation.

The infinite sequence of conserved quantities which were recently discovered in free-field theories by Kibble, are shown to be a consequence of translation invariance and the quadratic nature of the Lagrangian. This observation leads to a natural extension of Kibble’s work by exploiting the full Lorentz invariance of the free-field equations, leading to the construction of further new conservation laws. The question whether the existence of a conservation law implies an invariance of the Lagrangian is raised and answered in the negative.