Scale Anomaly as the Origin of Time

TL;DR

This paper proposes a novel approach to the problem of time in quantum gravity by linking scale anomaly and renormalization-group flow, leading to a time-dependent wavefunction in a scale-invariant model.

Contribution

It introduces a new mechanism where scale anomaly induces time evolution in quantum gravity models, providing a potential solution to the problem of time.

Findings

Wavefunction depends on RG flow scale, interpreted as time.

Application to three-body problem illustrates initial value setting.

Emergence of a time-dependent Schrödinger equation with cosmological implications.

Abstract

We explore the problem of time in quantum gravity in a point-particle analogue model of scale-invariant gravity. If quantized after reduction to true degrees of freedom, it leads to a time-independent Schr\"odinger equation. As with the Wheeler--DeWitt equation, time disappears, and a frozen formalism that gives a static wavefunction on the space of possible shapes of the system is obtained. However, if one follows the Dirac procedure and quantizes by imposing constraints, the potential that ensures scale invariance gives rise to a conformal anomaly, and the scale invariance is broken. A behaviour closely analogous to renormalization-group (RG) flow results. The wavefunction acquires a dependence on the scale parameter of the RG flow. We interpret this as time evolution and obtain a novel solution of the problem of time in quantum gravity. We apply the general procedure to the three-body problem, showing how to fix a natural initial value condition, introducing the notion of complexity. We recover a time-dependent Schr\"odinger equation with a repulsive cosmological force in the `late-time' physics and we analyse the role of the scale invariant Planck constant. We suggest that several mechanisms presented in this model could be exploited in more general contexts.