Is the existence of unbounded operators a problem for quantum mechanics? In response to Carcassi, Calderon, and Aidala

Abstract

In this paper I argue against Carcassi, Calderon, and Aidala's recent claim that the Hilbert spaces are unphysical and should be replaced with the Schwartz spaces in quantum mechanics, since Hilbert spaces include states with infinite expectation values for certain observables. I also review and discuss issues regarding unbounded operators in quantum mechanics raised by Streater and Wightman, Heathcote, and Lemos. I argue that the existence of infinite expectation values does not cause problems in quantum mechanics. On the other hand, replacing the Hilbert spaces with the Schwartz spaces would cause more issues, as it would exclude a class of meaningful Hamiltonian evolutions. I also discuss the question in literature whether reformulating quantum mechanics with essentially self-adjoint operators instead of self-adjoint operators may cause problems. I further analyse the hierarchies of the notions of "physicality" and possibility in fundamental physics, and suggest that "physicality" is a vague concept. Finally, I connect the problem raised by Carcassi, Calderon, and Aidala with the problem of the Hadamard condition in quantum field theory.