Consequences of 't Hooft's equivalence class theory and symmetry by coarsegraining

According to 't Hooft [Class. Quantum. Grav. 16, 3263 (1999)], quantum grav ity can be postulated as a dissipative deterministic system, where quantum states at the "atomic scale" can be understood as equivalence classes of pr imordial states governed by a dissipative deterministic dynamics law at the "Planck scale." In this paper, it is shown that for a quantum system to ha ve an underlying deterministic dissipative dynamics, the time variable shou ld be discrete if the continuity of its temporal evolution is required. Bes ides, the underlying deterministic theory also imposes restrictions on the energy spectrum of the quantum system. It is also found that quantum symmet ry at the "atomic scale" can be induced from 't Hooft's coarse graining cla ssification of primordial states at the "Planck scale." (C) 2001 American I nstitute of Physics.