Bateman’s dual system (BDS) is a linear time-reversal invariant system that consists of damping and amplifying coordinate variables subjected to a harmonic oscillator potential. It is known that canonical quantization of the BDS in the underdamping region suffers from pathologies such as the non-stationary vacuum and the breakdown of Heisenberg’s uncertainty principle. In this paper, following a previous study that targeted the pathology-free massless BDS, the massive BDS is quantized by decomposing it into two independent effectively massless subsystems with reduced degrees of freedom. In each subsystem, the amplifying variable is the conjugate momentum of a damping variable that obeys a non-canonical quantization condition. By virtue of general scaling invariance including U(1), the variables in BDS are treated as non-self-adjoint without changing the total degrees of freedom. The physical states are constructed with time-reversal normalization and are shown to satisfy the time-independent uncertainty relation. The original massive BDS that fulfills the canonical quantization condition is reconstructed by superposing the two subsystems. The same method is also applied to the underdamped BDS to obtain the time-independent uncertainty relations. The expectation values of the coordinate operators represent the complex representation of the solution of the classical equation of motion. The scaling invariance may be broken by coupling with the environment.
We construct a new model of two-dimensional quantum spin systems that combines both symmetry-protected and intrinsic topological orders. It is referred as the bilayer Doubled Semion model (bDS) and is an instance of symmetry-enriched topological order. The model provides an explicit solution to the problem of combining symmetry-protected topological phases present in the Doubled Semion model with those in the Kitaev model. A bilayer lattice is introduced to this purpose, which allows us to combine a global spin-flip symmetry with a global spin-flavor symmetry. The bDS model exhibits non-trival braiding self-statistics of excitations and novel edge states protected by the
Z2×Z2 global symmetry. This construction makes compatible the methods of topological charge-fractionalization based on dualities and on string-flux mechanisms.