Mathematical physicists are familiar with a large set of tools designed for dealing with linear operators, which are so common in both the classical and quantum theories; but many of those tools are useless with nonlinear equations of motion. In this work a general algebra and calculus is developed for working with nonlinear operators: The basic new tool being the ‘‘slash product,’’ defined by A(1+eB) = A+eA/B + O(e^2). For a generic time development equation, the propagator is constructed and then there follows the formal version of time dependent perturbation theory, in remarkable similarity to the linear situation. A nonperturbative approximation scheme capable of producing high accuracy computations, previously developed for linear operators, is shown to be applicable as well in the nonlinear domain. A number of auxiliary mathematical properties and examples are given.

We consider the motion of tachyons (faster-than-light particles) in the framework of General Relativity. An important feature is the large contribution of low energy tachyons to the energy-momentum tensor. We also calculate the gravitational field produced by tachyons in particular geometric arrangements; and it appears that there could be self-cohering bundles of such matter. This leads us to suggest that such theoretical ideas might be relevant to major problems (dark matter and dark energy) in current cosmological models.

The algebraic formulation of large N matrix mechanics recently developed by Halpern and Schwartz leads to a practical method of numerical computation for both action and Hamiltonian problems. The new technique posits a boundary condition on the planar connected parts X_w , namely, that they should decrease rapidly with increasing order. This leads to algebraic and/or variational schemes of computation which show remarkably rapid convergence in numerical tests on some many-matrix models. The method allows the calculation of all moments of the ground state, in a sequence of approximations, and excited states can be determined as well. There are two unexpected findings: a large d expansion and a new selection rule for certain types of interactions.