Moyal Brackets, Star Products and the Generalised Wigner Function

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.Comment: 7 pages, LaTeX, to appear in special issue of the J. of Chaos, Solitons and Fractal

Study of Quommutators of Quantum Variables and Generalized Derivatives

A general deformation of the Heisenberg algebra is introduced with two deformed operators instead of just one. This is generalised to many variables, and permits the simultaneous existence of coherent states, and the transposition of creation operators.Comment: 17 pages (Previous version was truncated in transmission

Lagrange Brackets and U(1) fields

The idea of a companion Lagrangian associated with $p$-Branes is extended to include the presence of U(1) fields. The Brane Lagrangians are constructed with $F_{ij}$ represented in terms of Lagrange Brackets, which make manifest the reparametrisation invariance of the theory; these are replaced by Poisson Brackets in the companion Lagrangian, which is now covariant under field redefinition. The ensuing Lagrangians possess a similar formal structure to those in the absence of an anti-symmetric field tensor.Comment: 7 pages, LaTeX, reference correcte

Moyal Brackets in M-Theory

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 correcte

Moyal Nahm Equations

Various aspects of the Nahm equations in 3 and 7 dimensions are investigated. The residues of the variables at simple poles in the 7-dimensional case form an algebra. A large class of matrix representations of this algebra is constructed. The large $N$ limit of these equations is taken by replacing the commutators by Moyal Brackets, and a set of non trivial solutions in a generalised form of Wigner distribution functions is obtained.Comment: 15 pages, latex, no figure

Morphing quantum mechanics and fluid dynamics

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 adde

Multi-Field Generalisations of the Klein-Gordon Theory associated with p-Branes

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.Comment: 13 pages, Late