Light-Cone Physics

Abstract

In classical physics differential equations are used to describe the evolution of a physical system. A well-defined problem must then include a set of initial values. In the quantum case, the corresponding information is provided when canonical commutators are specified. In a non-relativistic case these are specified on a surface of equal time, since constant time surfaces are the only ones that every particle trajectory crosses exactly once. In the relativistic case the existence of an upper bound to velocities means that the concept of simultaneity becomes ambiguous. Through each space-time point we can draw a hypercone (the light-cone); events occuring o’utside the light-cone cannot influence, or be influenced by an event at the tip of the cone. It is then a matter of definition how the surface of simultaneity is drawn. Any space-like surface may be used to set the initial data (or canonical commutators). Dirac1) observed that initial data can be given on various surfaces, leading to different forms of dynamics. The conventional formulation is to specify a surface at x⁰ = 0, but here I will use another choice of Dirac, namely a hypersurface tangent to the lightcone, defined by x⁰, where we use the notation x± = $\frac{1}{{\sqrt 2}}$ (x° ± x^d-1).