Uniqueness of the Hamiltonian in quantum field theories

TLDR: In this article, it was shown that for one such method of obtaining convergence, the "limit" operator is not unique, and that the cutoff operator also converges to H+R, where R is an arbitrary bounded positive operator.

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.