Bound States in Models of Asymtotic Freedom

Abstract

We describe a new formalism which expresses asymtotically free thories in a manifestly finite way, after renormalization and dimensional transmutation. The time evolution is NOT differentiable in these systems, so the hamiltonian does not exist. Instead, there is a new operator, the `Principal Operator', (which is roughly speaking the logarithm of the hamiltonian) which is finite (cut-off independent). We construct the Principal operator in several examples, including the Many body Problem of bosons in two dimensions with a short range attractive interaction. This allows us to estimate the ground state energy of a two-dimensional Bose condensate (with an attractive interaction). The ground state energy depends exponentially on the number of particles.