Consequences of 't Hooft's Equivalence Class Theory and Symmetry by Large Coarse Graining

TL;DR

This paper explores how 't Hooft's equivalence class theory and large coarse graining influence quantum gravity, emphasizing the necessity of discrete time for deterministic models and the emergence of quantum symmetry from primordial state classifications.

Contribution

It demonstrates that deterministic dissipative dynamics require discrete time and constrains the energy spectrum, linking primordial state classifications to quantum symmetry.

Findings

Discrete time is necessary for deterministic dissipative quantum systems.

Energy spectrum restrictions arise from underlying deterministic dynamics.

Quantum symmetry can be derived from coarse graining of primordial states.

Abstract

According to 't Hooft (Class.Quantum.Grav. 16 (1999), 3263), quantum gravity can be postulated as a dissipative deterministic system, where quantum states at the ``atomic scale''can be understood as equivalence classes of primordial states governed by a dissipative deterministic dynamics law at the ``Planck scale''. In this paper, it is shown that for a quantum system to have an underlying deterministic dissipative dynamics, the time variable should be discrete if the continuity of its temporal evolution is required. Besides, the underlying deterministic theory also imposes restrictions on the energy spectrum of the quantum system. It is also found that quantum symmetry at the ``atomic scale'' can be induced from 't Hooft's Coarse Graining classification of primordial states at the "Planck scale".