  The theory of circuit quantum electrodynamics has successfully analyzed
superconducting circuits on the basis of the classical Lagrangian, and the
corresponding quantized Hamiltonian, describing these circuits. In many
simplified versions of these networks, the modeling involves a Lagrangian that
is singular, describing an inherently constrained system. In this work, we
demonstrate the failure of the Dirac-Bergmann theory for the quantization of
realistic, nearly singular superconducting circuits, both reciprocal and
nonreciprocal. The correct treatment of nearly singular systems involves a
perturbative Born-Oppenheimer analysis. We rigorously prove the validity of the
corresponding perturbation theory using Kato-Rellich theory. We find that the
singular limit of this regularized analysis is, in many cases, completely
unlike the singular theory. Dirac-Bergmann, which uses the Kirchhoff's (and
Tellegen's) laws to deal with constraints, predicts dynamics that depend on the
detailed parameters of nonlinear circuit elements, e.g., Josephson inductances.
By contrast, the limiting behavior of the low-energy dynamics obtained from the
regularized Born-Oppenheimer approach exhibits a fixed point structure, flowing
to one of a few universal fixed points as parasitic capacitance values go to
zero.
