Fundamental Constants and the Problem of Time

TL;DR

This paper explores how certain constants in parametrized theories become irrelevant classically and proposes a reinterpretation at the quantum level to address the 'problem of time' in quantum gravity, with implications for the nature of Planck's constant.

Contribution

It introduces a reinterpretation of classically irrelevant constants as an unconstrained Hamiltonian, offering a potential resolution to the quantum 'problem of time' in gravity.

Findings

The ratio of kinetic to potential terms can serve as an unconstrained Hamiltonian.

A physical state can obey a Poincaré algebra and be an approximate eigenstate of 3-geometry.

Solutions for minisuperspace models demonstrate the approach's viability.

Abstract

We point out that for a large class of parametrized theories, there is a constant in the constrained Hamiltonian which drops out of the classical equations of motion in configuration space. Examples include the mass of a relativistic particle in free fall, the tension of the Nambu string, and Newton's constant for the case of pure gravity uncoupled to matter or other fields. In the general case, the classically irrelevant constant is proportional to the ratio of the kinetic and potential terms in the Hamiltonian. It is shown that this ratio can be reinterpreted as an {\it unconstrained} Hamiltonian, which generates the usual classical equations of motion. At the quantum level, this immediately suggests a resolution of the "problem of time" in quantum gravity. We then make contact with a recently proposed transfer matrix formulation of quantum gravity and discuss the semiclassical limit. In this formulation, it is argued that a physical state can obey a (generalized) Poincar\'e algebra of constraints, and still be an approximate eigenstate of 3-geometry. Solutions of the quantum evolution equations for certain minisuperspace examples are presented. An implication of our proposal is the existence of a small, inherent uncertainty in the phenomenological value of Planck's constant.