Spectral discontinuities in constrained dynamical models

As examples of models having interesting constraint structures, we derive a quantum mechanical model from the spatial freezing of a well-known relativistic field theory-the chiral Schwinger model. We apply the Hamiltonian constraint analysis of Dirac (1964 Lectures on Quantum Mechanics (New York: Belfer Graduate School of Science)) and find that the nature of constraints depends critically on a c-number parameter present in the model. Thus, a change in the parameter alters the number of dynamical modes in an abrupt and non-perturbative way. We have obtained new real energy levels for the quantum mechanical model as we explore complex domains in the parameter space. These were forbidden in the parent chiral Schwinger field theory where the analogue Jackiw-Rajaraman parameter is restricted to be real. We explicitly show the existence of modes that satisfy the higher derivative Pais-Uhlenbeck form of dynamics (Pais and Uhlenbeck 1950 Phys. Rev. 79 145). We also show that the Cranking model (Valatin 1956 Proc. R. Soc. A 238 122), well known in nuclear physics, can be interpreted as a spatially frozen version of another well-studied relativistic field theory in (2 + 1)-dimensions-the Maxwell-Chern-Simons-Proca model (Deser et al 1982 Phys. Rev. Lett. 48 975, Deser et al 1982 Ann. Phys. 140 372).