[Paper Review] What is Dynamics in Quantum Gravity?

Przemysław Małkiewicz|arXiv (Cornell University)|May 18, 2015

Noncommutative and Quantum Gravity Theories8 references3 citations

TL;DR

This paper investigates how the choice of internal clock in quantum gravity affects quantum dynamics, proposing a unique quantization map for Dirac observables across different reduced phase spaces. It demonstrates that semiclassical corrections—such as those altering minimal volume, maximal curvature, and bounce count—depend critically on the internal clock, revealing that key quantum cosmological features are not absolute but clock-dependent.

ABSTRACT

The appearance of Hamiltonian constraint in the canonical formalism for general relativity reflects the lack of a fixed external time. The dynamics of general relativistic systems can be expressed with respect to an arbitrarily chosen internal degree of freedom, the so called internal clock. We investigate the way in which the choice of internal clock determines the quantum dynamics and how much different quantum dynamics induced by different clocks are. We develop our method of comparison by extending the Hamilton-Jacobi theory of contact transformations to include a new type of transformations which transform both the canonical variables and the internal clock. We employ our method to study the quantum dynamics of the Friedmann-Lemaitre model and obtain semiclassical corrections to the classical dynamics, which depend on the choice of internal clock. For a unique quantisation map we find the abundance of inequivalent semiclassical corrections induced by quantum dynamics taking place in different internal clocks. It follows that the concepts like minimal volume, maximal curvature and the number of quantum bounces, often used to describe quantum effects in cosmological models, depend on the choice of internal clock.

Motivation & Objective

• To address the 'multiple choice problem' in quantum gravity, where the lack of a fixed external time leads to ambiguity in defining dynamics via different internal clocks.
• To develop a consistent framework for comparing quantum dynamics across different internal clocks in Hamiltonian constraint systems.
• To ensure a unique quantum representation of Dirac observables (constants of motion) regardless of the chosen internal clock, removing quantization ambiguities.
• To investigate how different internal clocks induce inequivalent semiclassical dynamics in the Friedmann-Lemaitre model.
• To demonstrate that physically significant features—like minimal volume, maximal curvature, and number of bounces—depend on the choice of internal clock.

Proposed method

• Extends Hamilton-Jacobi theory to include contact transformations that transform both canonical variables and the internal clock.
• Introduces pseudocanonical transformations to relate different reduced phase spaces corresponding to different internal clocks.
• Applies an extended quantization procedure that enforces a unique quantum representation of Dirac observables across all internal clock choices.
• Uses the Klauder semiclassical portrait method to compare quantum dynamics by analyzing coherent state representations in phase space.
• Imposes the condition that the operator measure $ M(q,p) $ must be invariant under clock redefinitions to ensure consistency in quantum representation.
• Employs a phase space portrait approach to visualize and compare semiclassical dynamics across different internal clocks in the Friedmann-Lemaitre model.

Experimental results

Research questions

• RQ1 How does the choice of internal clock affect the resulting quantum dynamics in a canonical quantum gravity framework?
• RQ2 To what extent do semiclassical corrections in cosmological models depend on the choice of internal clock?
• RQ3 Can a unique quantum representation of Dirac observables be maintained across different internal clocks, and how does this affect dynamics?
• RQ4 Are physically significant features like minimal volume, maximal curvature, and number of bounces invariant under different internal clock choices?
• RQ5 What is the role of pseudocanonical transformations in relating quantum dynamics defined on different reduced phase spaces?

Key findings

• Different choices of internal clock lead to inequivalent semiclassical corrections in the Friedmann-Lemaitre model, even with a unique quantization map.
• The minimal volume, maximal curvature, and number of quantum bounces in the model are not absolute but depend on the choice of internal clock.
• The requirement of a unique quantum representation for Dirac observables forces the operator measure $ M(q,p) $ to be invariant under clock redefinitions, ensuring consistency across phase spaces.
• Semiciclassical dynamics derived from different internal clocks are inequivalent, as shown by distinct phase space portraits using the Klauder method.
• The extended quantization procedure ensures that differences in dynamics arise solely from the clock choice, not from quantization ambiguities.
• The paper concludes that the concept of quantum dynamics in quantum gravity is inherently dependent on the choice of internal clock, challenging the assumption of universality in quantum cosmological features.