Quotient Construction of 't Hooft's Quantum Equivalence Classes

Most recently ‘t Hooft has postulated (G ‘t Hooft, Class. Quant. Grav. 16 (1999) 3263-3279) that quantum states at the atomic scale''can be understood as equivalence classes of primordial states governed by a dissipative deterministic theory underlying quantum theory at the Planck scale’’. Defining invariant subspaces clearly for primordial states according to a given evolution, we mathematically re-formulate ‘t Hooft’s theory as a quotient space construction with the time-reversible evolution operator induced naturally. With this observation and some analysis, ‘t Hooft’s theory is generalized beyond his case where the evolution at the ``Planck scale’’ is periodic or the time is discrete. We also give a novel illustration that the Fock space of quantum oscillator could follow from the quotient space construction for certain primordial states obeying non-reversible evolution governed by a non-Hermitian Hamiltonian.