[Paper Review] Emergent Time in the Path Integral of Barbour and Bertotti's Timeless Mechanics

Sean Gryb|arXiv (Cornell University)|Apr 18, 2008

Black Holes and Theoretical Physics16 references11 citations

TL;DR

This paper investigates the emergence of time in a timeless quantum mechanical model using path integral quantization. It shows that while both timeless and absolute-time formulations yield equivalent results in the stationary phase approximation, the key difference lies in constraints imposed by boundary conditions—suggesting time can emerge when the approximation is accurate.

ABSTRACT

The problem of time is studied in a toy model for quantum gravity: Barbour and Bertotti's timeless formulation of non-relativistic mechanics simplified to the case where space is still absolute but time is relational. We quantize this timeless theory using path integrals and compare it to the path integral quantization of parameterized Newtonian mechanics, which contains absolute time. In general, we find that the solutions to the timeless theory are energy eigenstates, as predicted by the usual canonical quantization. Nevertheless, the path integral formalism brings new insight as it allows us to precisely determine the difference between the theory with and without time. This difference is found to lie in the form of the constraints imposed on the gauge fixing functions by the boundary conditions. In the stationary phase approximation, the constraints of both theories are equivalent. This suggests that a notion of time can emerge in systems for which the stationary phase approximation is either good or exact.

Motivation & Objective

• To investigate how time might emerge in a timeless formulation of non-relativistic mechanics.
• To compare path integral quantization of a relational time model with that of parameterized Newtonian mechanics containing absolute time.
• To identify the precise mathematical distinction between theories with and without time in the path integral formalism.
• To determine under what conditions a notion of time can emerge from a fundamentally timeless quantum theory.

Proposed method

• Quantizes Barbour and Bertotti's timeless mechanics using the path integral formalism.
• Compares the path integral formulation of the timeless theory to that of parameterized Newtonian mechanics with absolute time.
• Analyzes constraints on gauge-fixing functions imposed by boundary conditions in both theories.
• Applies the stationary phase approximation to compare solutions of the two theories.
• Identifies the role of boundary conditions in distinguishing theories with and without time.
• Uses canonical quantization results as a benchmark to validate path integral solutions.

Experimental results

Research questions

• RQ1 How does the path integral formalism reveal differences between a timeless theory and one with absolute time?
• RQ2 What role do boundary conditions play in distinguishing theories with and without time in the path integral approach?
• RQ3 Under what conditions does a notion of time emerge from a fundamentally timeless quantum system?
• RQ4 How do the constraints on gauge-fixing functions differ between the timeless and absolute-time formulations?
• RQ5 To what extent is the stationary phase approximation sufficient to recover time-like behavior in a timeless theory?

Key findings

• Solutions to the timeless theory are energy eigenstates, consistent with canonical quantization.
• The path integral formalism reveals that the difference between theories with and without time lies in the constraints on gauge-fixing functions due to boundary conditions.
• In the stationary phase approximation, the constraints of both theories become equivalent.
• The emergence of time is linked to the validity of the stationary phase approximation.
• When the stationary phase approximation is good or exact, a notion of time can emerge from a timeless system.
• The results suggest that time is not fundamental but can arise as an effective feature in quantum systems under specific conditions.