  A powerful tool for studying the behavior of classical field theories is
Derrick's theorem: one may rule out the existence of localized inhomogeneous
stable field configurations (solitons) by inspecting the Hamiltonian and making
scaling arguments. For example, the theorem can be used to rule out compact
domain wall configurations for the classic $\phi^4$ theory in $3+1$ dimensions
and greater. We argue no such obstruction to constructing solitons exists in
the framework of algebraic/axiomatic quantum field theory (AQFT), and that
states like the example given lie in topologically trivial superselection
sectors of the Hilbert space. A proof is presented making use of the relative
entropy, and the implications are explored for a few common models of scalar
fields and the pure Yang-Mills theory.
