Uniqueness of the Hamiltonian in quantum field theories

Abstract

In most quantum field theories, one defines the Hamiltonian (energy) operatorH as a limit of “cutoff” operators$H_s :H = \mathop {\lim }\limits_{s \to \infty } H_s$. (The operatorH _s would be the correct Hamiltonian for a world in which all momenta are smaller thans.) Since the cutoff operators seldom converge in any of the standard operator topologies, it is often necessary to invent more subtle notions of “convergence”. For some of the these, it is not obvious that the “limit” operatorH is unique. In this note we point out that for one such method of obtaining convergence, the “limit” operator isnot unique. In fact, (under mild assumptions about the operatorsH _s ), ifH _s converges toH, thenH _s also converges toH+R, whereR is an arbitrary bounded positive operator.