  This paper extends the tools of C*-algebraic strict quantization toward
analyzing the classical limits of unbounded quantities in quantum theories. We
introduce the approach first in the simple case of finite systems. Then we
apply this approach to analyze the classical limits of unbounded quantities in
bosonic quantum field theories with particular attention to number operators
and Hamiltonians. The methods take classical limits in a
representation-independent manner and so allow one to compare quantities
appearing in inequivalent Fock space representations.
