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Galileons are related to other implicitly integrable equations such as the first order nonlinear wave equation and the universal field equation, which is a Lagrangian for the Galileon.

We suggest that galileon theories should have an additional self-coupling of the fields to the trace of their own energy-momentum tensor. We explore the classical features of one such model, in flat 4D spacetime, with emphasis on solutions that are scalar analogues of gravitational geons. We discuss the stability of these scalar geons, and some of their possible signatures, including shock fronts., Comment: References added in v2

The 1+2 dimensional continuous Toda chain presents a formidable challenge to the construction of solutions. Two variable reductions of the equation are known, but up till now nothing more is known. In this paper a way to solve the equation is presented, and some solutions are explicitly constructed. The method depends upon connecting series solutions of the symmetry equation of the Toda chain with the Toda solutions, as is guaranteed by general theory. Such solutions of the symmetry equation are obtainable a series method. An explicit solution is constructed by this method, and a general proce- dure is given to realize further solutions, which, however are given only in implicit form.

A 3-bracket variant of the Virasoro-Witt algebra is constructed through the use of su(1,1) enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations., Comment: 6 pages, LateX

The set of points of a one-dimensional cut-and-project quasicrystal or model set, while not additive, is shown to be multiplicative for appropriate choices of acceptance windows. This leads to the definition of an associative additive graded composition law and permits the introduction of Lie algebras over such aperiodic point sets. These infinite dimensional Lie algebras are shown to be representatives of a new type of semi-direct product induced Lie algebras., Comment: 13 pages

This review describes the theory of internal supersymmetry as first introduced by Ne’eman and Fairlie independently in 1979 and its various developments since then including applications and predictions of the theory to elementary particle physics. In particular we give an explanation of the electroweak isospin and hypercharge spectrum in several generations and a prediction of the Weinberg angle and the Higgs mass derived from the embedding of su ( 2 ) × u ( 1 ) as the even part of the superalgebra su ( 2 / 1 ) . The formulas leading to these predictions are not su ( 2 / 1 ) invariant and we do not present a theory of su ( 2 / 1 ) symmetry breaking. Rather, we use the embedding of su ( 2 ) × u ( 1 ) in su ( 2 / 1 ) to give a geometric interpretation of the Higgs field as part of a superconnection and interpret the quartic polynomial entering into the Higgs mechanism as deriving from a super-Yang–Mills functional. Despite the fact that our formulas are not su ( 2 / 1 ) invariant, we find these geometrical structures interesting and suggestive of the presence of Lie superalgebras in nature.

We investigate the effects of given pressure gradients on hydrodynamic flow equations. We obtain results in terms of implicit solutions and also in the framework of an extra-dimensional formalism involving the diffusion/Schrodinger equation., Comment: Examples involving shocks and references added

It is shown how integrable non-linear equations of hydrodynamic type, which arise as continuations of Dirac-Born-Infeld equations to the case where the target space is of lower dimension than the base space, are associated with linear diffusion equations in a space of double the number of space co-ordinates.

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.

Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group. The simple explicit result exhibits connections between group theory, combinatorics, and Fourier analysis, especially in the large spin limit. Salient intuitive features of the formula are illustrated and discussed.

A relation between the classical Faulhaber and Bernoulli polynomials and the theory of the Korteweg–de Vries equation is established.

The general solution to the complex Bateman equation in a space of arbitrary dimensions is constructed.

The quantum mechanical transition between a free particle Lagrangian and the Klein–Gordon field description of a free particle (particle-wave duality) is conjectured to extend to an analogous construction of relativistically invariant wave equations associated with strings and branes. Electromagnetic interactions in the two systems are discussed. It is emphasized that all integrable free field theories, including those of Dirac–Born–Infeld type, are associated with Lagrangians equivalent to divergences on the space of solutions of the equations of motion.

Properties of the Dirac-Born-Infeld Lagrangian analogous to those of the Nambu-Goto String are analysed. In particular the Lagrangian is shown to be constant or zero on the space of solutions of the equations of motion if the Lagrangian is taken to any power other than 1/2., Comment: 9 pages, LaTeX

The Wigner-Weyl- Moyal approach to Quantum Mechanics is recalled, and similarities to classical probability theory emphasised. The Wigner distribution function is generalised and viewed as a construction of a bosonic object, a target space co-ordinate, for example, in terms of a bilinear convolution of two fermionic objects, e.g. a quark antiquark pair. This construction is essentially non-local, generalising the idea of a local current. Such Wigner functions are shown to solve a BPS generalised Moyal-Nahm equation., 7 pages, LaTeX, to appear in special issue of the J. of Chaos, Solitons and Fractals

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 give a series of integrable top equations associated with the projective geometry over as a -dimensional generalization of the three-dimensional Euler top equations. The general solution of the -dimensional top is shown to be given by an integration over a Riemann surface with genus .

The infinite limit of Matrix Theory in 4 and 10 dimensions is described in terms of Moyal Brackets. In those dimensions there exists a Bogomol'nyi bound to the Euclideanized version of these equations, which guarantees that solutions of the first order equations also solve the second order Matrix Theory equations. A general construction of such solutions in terms of a representation of the target space co-ordinates as non-local spinor bilinears, which are generalisations of the standard Wigner functions on phase space, is given., Comment: 10 pages, Latex, no figures. References altered, typos corrected

A large class of first-order partial nonlinear differential equations in two independent variables which possess an infinite set of polynomial conservation laws derived from an explicit generating function is constructed. The conserved charge densities are all homogeneous polynomials in the unknown functions which satisfy the differential equations in question. The simplest member of the class of equations is related to the Born–Infeld Equation in two dimensions. It is observed that some members of this class possess identical charge densities. This enables the construction of a set of multivariable equations with an infinite number of conservation laws.

The algebraic and Hamiltonian structures of the multicomponent dispersionless Benney and Toda hierarchies are studied. This is achieved by using a modified set of variables for which there is a symmetry between the basic fields. This symmetry enables formulae normally given implicitly in terms of residues, such as conserved charges and fluxes, to be calculated explicitly. As a corollary of these results the equivalence of the Benney and Toda hierarchies is established. It is further shown that such quantities may be expressed in terms of generalized hypergeometric functions, the simplest example involving Legendre polynomials. These results are then extended to systems derived from a rational Lax function and a logarithmic function. Various reductions are also studied., Comment: 29 pages, LaTeX

It is shown that each integrable mapping is connected with a hierarchical completely integrable sytem of equations of evolution type which are invariant with respect to the transformation described by this mapping.

The dual resonance model, which was a precursor of string theory was based upon the idea that two-particle scattering amplitudes should be expressible equivalently as a sum of contributions of an infinite number of $s$ channel poles each corresponding to a finite number of particles with definite spin, or as a similar sum of $t$ channel poles. The famous example of Veneziano \cite{ven} satisfies all these requirements, and is additionally ghost free.We recall other trajectories which provide solutions to the duality constraints, e.g. the general Mobi\"us trajectories and the logarithmic trajectories, which were thought to be lacking this last feature. We however demonstrate, partly empirically, the existence of a regime within a particular deformation of the Veneziano amplitude for logarithmic trajectories for which the amplitude remains ghost free., Comment: 18 pages, LaTeX ,DTP/94/19

In this paper all deformations of the general linear group, subject to certain restrictions which in particular ensure a smooth passage to the Lie group limit, are obtained. Representations are given in terms of certains sets of creation and annihilation operators. These creation and annihilation operators may belong to a generalisation of the $q$-quark type or $q$-hadronic type, of $q$-boson or $q$-fermion type. We are also led to a natural definition of $q$-direct sums of q-algebras., Comment: 15 pages

The recent progress in the study of Galileons, i.e. equations of second order with an action invariant under a Galilean transformation is related to work on `Universal Field Equations' \cite{dbfgov} which are second order equations arising by an iterative procedure from arbitrary Lagrangians of weight one in their first derivatives. It is pointed out that the Galileon is simply a Kaluza-Klein reduction of a Universal Field Equation. An implicit solution to the equation of motion is presented, and a class of explicit solutions pointed out. The multi-field extensions of both types of equations are derived from a first order formalism, which is simply the substantive derivative of fluid dynamics., Comment: 10 pages, no figures

Deformations of the canonical commutation relations which have the effect of altering the spectrum of a standard Hamiltonian, bilinear in creation and annihilation operators are described. The problem of going over from an eigenvalue situation, as is the case in the vast majority of papers in the literature, to a theory with time evolution is discussed, and a special example with deformation parameter an N th root of unity is constructed which possesses a consistent time evolution. This work is an account of some recent studies of associative deformations of the Heisenberg algebra of several creation and annihilation algebras, with Jean Nuyts of the University of Mons, Hainaut, together with some observations of my own concerning the difficulty of implementing time evolution in a quantum group context. It builds on earlier work with Cosmas Zachos (Argonne National Laboratory, USA), which in turn is re;ated to work of Manin, and Wess, Zumino and collaborators. The main idea is that, if quantum groups have any role in physics, then they must manifest themselves at the level of the basic rules of quantisation.

Finite Euler hierarchies of field theory Lagrangians leading to universal equations of motion for new types of string and membrane theories and for {\it classical} topological field theories are constructed. The analysis uses two main ingredients. On the one hand, there exists a generic finite Euler hierarchy for one field leading to a universal equation which generalises the Plebanski equation of self-dual four dimensional gravity. On the other hand, specific maps are introduced between field theories which provide a ``triangular duality'' between certain classes of arbitrary field theories, classical topological field theories and generalised string and membrane theories. The universal equations, which derive from an infinity of inequivalent Lagrangians, are generalisations of certain reductions of the Plebanski and KdV equations, and could possibly define new integrable systems, thus in particular integrable membrane theories. Some classes of solutions are constructed in the general case. The general solution to some of the universal equations is given in the simplest cases., 42 p., Plain TeX, replaces previous unprintable version corrupted by mailer

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 authors take a standard Hamiltonian, quadratic in a set of creation and annihilation operators, and demonstrate an alternative quantization by quommutators; i.e. a set of rules for transposition of these operators, different from the those of the usual canonical commutation relations which yields a new spectrum, of a type reminiscent of logarithmic trajectories.

The formalism of the Wigner distribution function is reviewed. In addition to the Liouville equation, which expresses the time rate of change of this function in terms of its Moyal bracket with the Hamiltonian, and its expression as a projection operator, a third equation is proposed with the aid of an auxiliary variable s, to which a formal solution is constructed in terms of known quantum-mechanical eigenfunctions and eigenvalues. In addition, an ab initio solution to the three equations in terms of an error function is found for the free particle in one dimension. Two views are advanced: the orthodox, that this new equation is merely a consistency requirement, and the speculative, that the measurement process has something to do with the choice of s.

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.

We explicitly deform classical oscillators to multiparameter quantum oscillators which do not commute with each other. These “anyonic” oscillators lead to the construction of a consistent quantum algebra GLq(N) with 1 2 (N−1)(N−2) dependent parameters, some of whose features and applications we discuss.

SynopsisSome disparate ideas in the literature are drawn together. The work of P. J. Olver and his associates on Lagrangians which vanish for arbitrary variations, the so-called null Lagrangians, is viewed as a parallel development to Witten's study of topological field theories. A theorem of Olver, that all hyperjacobians are expressible as divergences, and are thus candidates for the construction of null Lagrangians, is shown to follow directly from the observation that such entities appear in a power series development of the general associative product, and this technique facilitates the construction of multi-dimensional examples.

The results of minimizing the action for string-like systems on a simply-connected world sheet are shown to encode the Cartesian components of real null momentum four-vectors into co-ordinates on the world sheet. This identification arises consistently from different approaches to the problem., 9 pages, Latex2e