On quantum solitons and their classical relatives: Reducible quantum fields and infinite constituent ‘‘elementary’’ systems

Abstract

We demonstrate that the emergence of translation modes in the quantization of some at least nonlinear field theory models (like, e.g., φ4 or the sine–Gordon systems) implies a specific structure of their state spaces namely this of the direct integral Hilbert space, which follows from the reducibility of the involved quantum field canonical commutation relations (CCR) algebras. As a special manifestation of this structure, one recovers infinite constituent ‘‘elementary’’ quantum systems living in the commutant of the CCR algebra, which appear as the Schrödinger or the two level ones. The corresponding Hamiltonians are derived. In addition, we propose a modification of the standard infrared Hilbert (photon field) space construction employed in quantum electrodynamics. We demonstrate that, in principle, Fermi (CAR) generators, carrying the spin–charge–momentum labels of Dirac particles, can be defined as operators in the electromagnetic (photon field) Hilbert space. The photon field (CCR) algebra is highly reducible, and in the present case fermions arise in the commutant of it, playing the role of intertwining operators.