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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

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

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.

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

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

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

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.

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.

't Hooft constructed a finite mass unit monopole by (a) embedding the electromagnetic U(1) gauge theory in an SU(2) theory via a Higgs triplet mechanism, (b) seeking solutions invariant under simultaneous rotations in ordinary and SU(2) space. We consider an SU(3) gauge theory broken down to U(2) by an octet Higgs vector and find two finite-mass stable solutions corresponding to the two ways of embedding SU(2) in SU(3). One solution is the pure unit monopole but the other, which promises to be lighter, has 1 2 unit of magnetic charge (thus violating Dirac's condition, naively applied) as well as being a source of isomagnetic flux (associated with the unbroken SU(2)).

We consider the possibility of satisfying the gauge field equations in dimensions greater than four by imposing linear relations amongst the components of the field strength tensor, F μν , generalising the idea of self-duality in four dimensions.

We investigate features of the infinite dimensional algebras we have previously introduced, which involve trigonometric functions in their structure constants. We find a realization for them which leads to a basis-independent formulation. A special family of them, the cyclotomic ones, contain SU ( N ) as invariant subalgebras. In this basis, it is evident by inspection that the algebra of SU(∞) is equivalent to the centerless algebra of SDiff 0 on two-dimensional manifolds. Gauge theories of SU(∞) are thus simply reformulated in terms of surface coordinates.

The generators of SU(1,1) and their Casimir invariant may be arrayed in a compact formula to construct all generators of the centreless Virasoro algebra which includes this SU(1,1) subalgebra. We illustrate properties of our construction through a simple differential operator realization of these algebras and comment on its usefulness.

By the introduction of a generating function for the class of transformations of two-dimensional chiral lagrangians recently shown to generate a Kac-Moody algebra, we show how structure of the transformations and the invariance proofs may be considerably clarified.

It is shown that all dual models derived maively from a recently discovered class of infinite graded Lie algebras with O(N) symmmetry posses ghosts if N > 2 with the possible exception of the quaternion case.

We compute the single loop non-planar fermion loops for external mesons in the model of Neveu-Schwarz-Ramond and Mandelstam, using functional methods for the entire calculation. We find a cancellation of the leading odd G parity pomeron, while the even G parity pomeron survives, when the contributions from fermion loops are included.

The recent general solution by Atiyah, Hitchin, Drinfeld and Manin of the self-duality equations for an arbitrary compact classical group is discussed. The Green function for a scalar field transforming as a vector under the group is shown to take an elegant form in the background field of the general self-dual solution. The massless solutions of the Dirac equation, for this representation of the group, are also explicitly exhibited.

SU(2) × U(1) gauge theories, in which the Higgs fields transform as doublets under SU(2) are interpreted as pure Yang-Mills theories in six dimensions, the components of the gauge potentials in the extra dimensions playing the role of the Higgs' fields. Two consistent theories are discovered: one in which SU(2) × U(1) is embedded in SU(3) and the vector bosons remain massless - and another where SU(2) × U(1) is embedded in the graded Lie algebra SU(2|1), the symmetry is spontaneously broken in a natural fashion and the theory is equivalent to that of Weinberg and Salam, with a specific value 30° for the Weinberg angle and a prediction of the Higgs' mass.

Novel infinite-dimensional algebras of the Virasoro/Kac-Moody/Floratos-Iliopoulos type are introduced, which involve trigonometric functions in their structure constants. They are then supersymmetrized, and relevant features of them are explored. An associated “lazy tongs” formulation of the SU (2) Kac-Moody algebra is also given.

Duality functions, i.e. constraints on the integration variables in a multichannel representation of the Veneziano form are developed for the multiparticle amplitude describing the emission (or absorption) of n elementary mesons by a baryon with an isobaric spectrum. The sequence of graphs related by duality consists of all distinct permutations of the order of absorption of the mesons by the baryon, and the resulting amplitude is expressible in terms of the resonance poles associated with any one of them. This is the generalisation of a model proposed by Burnett and Schwarz. This model has also recently been generalized in a completely different direction by Mandelstam.

A technique is presented for implementing the requirement of crossing symmetry and duality on an amplitude constructed from a maximal set of reasonable poles: An integral representation of the Veneziano form is found and new light is cast upon the integration paramaters for the multichannel extensions of the Veneziano form. A new example of a six-point function which is constructed from resonance poles and is dual, but has a different permutation symmetry than Chan's representation, is presented.

It is shown that integral representations may be written for the four- and five-point functions which are invariant under all 24 and 120 permutations of the external lines respectively and which incorporate all the distinct terms of the Veneziano form which constitute the amplitudes, provided certain trajectory constraints are satisfied.

An analogue model based upon the association of amplitudes for many particle processes with functional integrals over interaction regions of the exponential of the energy of momentum flow is shown to yield in first approximation the multiparticle Veneziano representation. The correction terms in the next approximation are shown to be just those given by the planar loop amplitude in the theory of Kikkawa, Sakita and Virasoro. This approach casts new light upon the divergence problem of the loop amplitudes in a dual theory but is does not provide, as yet, an unambiguous answer to it.

Schwinger's scheme of chiral dynamics is reviewed and elaborated. An explicit form of the generators of his chiral algebra is found, and with its help the relation of this work to the σ model is elucidated. The pion mass difference and the η-decay rate are calculated on the assumption that only the isospin subgroup of Schwinger's chiral group is broken by electromagnetic interactions.

We show that the Neveu-Schwarz model takes a simple form in the Koba-Nielsen formulation, the integrand taking the form of a Pfaffian. We find empirically that it is possible to improve the spectrum while retaining the Pfaffian formulation. Though the form obtained for the amplitude is suggestive of an underlying operational construction we have not succeeded in finding such a formulation. The Adler condition is implemented in this model in a non-trivial manner.

A model based on a separable potential is investigated and it is shown that the unstable states can be associated with poles of the continuation of the scattering amplitude into the second Riemann sheet.

In low energy meson baryon scattering a specific partial wave amplitude in the direct channel will contain contributions from the exchange of resonances in all partial waves in the crossed channel. These contributions, first considered by Carruthers for pion-nucleon scattering, are evaluated in the limit where the baryon kinetic energy is small compared to its mass, and expressed succinctly in terms of a 9 j symbol. Consequences for bootstrap theory are noted.

The model due to Ruijgrok and Van Hove is studied for a particular choice of coupling constants. The resulting theory displays an exact criterion for the existence of ghost states. In the limit as the number of intermediates states increases to simulate a realistic field theory the ghost behaviour disappears.

A set of evolution equations of the form d Y d t =Y 2 (for off-diagonal elements) where Y is an n × n symmetric matrix with vanishing diagonal elements, is shown to admit a class of integrable solutions, which are related to a generalisation of the Euler top.

A variant of the bosonic sector of N = 4 Yang-Mills theory is constructed which admits Bogomolny equations (i.e. first order equations of motion which minimize the action), but which does not, unlike similar cases hitherto known, posses a straightforward interpretation as a pure Yang-Mills field in extra dimensions.

Simple expressions for the ansatze of Atiyah and Ward are given in Yang's R gauge. Some non-linear transformations connecting solutions of the field equations are exhibited. These transformations are analogous to the Backlund transformations of the Sine-Gordon equation, etc., and provide an essentially algebraic inductive proof of the ansatze. Neither the sources for the anti-self-dual Maxwell fields used in the ansatze nor the reality conditions are discussed.

A previous work has shown that Feynman amplitudes have certain singularities, called pure second-type singularities, with positions that are independent of the masses of the internal particles. An attempt was made to prove that second type singularities never occur on the physical sheet. The error in this argument is pointed out, and a possible line of attack on the problem is suggested. (H.D.R.)