Quantum Theory Of A Massless Relativistic Surface And A Two Dimensional Bound State Problem

by Jens Hoppe

MIT 1982

Part 1:

A massless relativistic surface is defined in a Lorentz invariant way by letting its action be proportional to the volume swept out in Minkowski space. The system is described in light cone coordinates and by going to a Hamiltonian formulation one sees that the dynamics only depends on the transverse coordinates x and y. The Hamiltonian is invariant under the group of area preserving reparametrizations whose Lie Algebra can be shown to correspond in some sense to the large N-Limit of SU(N). Using this one arrives at a SU(N) invariant, large N-two-matrix model with a quartic interaction [X,Y]^2.

Part 2:

The problem of N particles with nearest neighbors delta-function interactions is defined by regularizing the 2 body problem and deriving an eigenvalue integral equation that is equivalent to the Schrödinger equation (for bound states). The 3 body problem is discussed extensively and it is argued to be free of irregularities, in contrast with the known results in 3 dimensions. The crucial role of the dimension is displayed in looking at the limit of a short range potential.