  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.
