Time inversion, Self-similar evolution, and Issue of time

TL;DR

This paper explores the concept of time inversion and fractal time, revealing how stochastic fluctuations with 1/f spectrum can emerge in linear systems, with implications for quantum gravity and multi-scale evolution.

Contribution

It introduces a novel framework where time inversions and fractal time lead to stochastic behavior in linear systems, connecting to quantum gravity issues.

Findings

Time inversions can induce late-time stochastic fluctuations.

Fractal time satisfies dx=dt and exhibits 1/f noise spectrum.

The approach links time inversion to quantum gravity considerations.

Abstract

We investigate the question, "how does time flow?" and show that time may change by inversions as well. We discuss its implications to a simple class of linear systems. Instead of introducing any unphysical behaviour, inversions can lead to a new multi- time scale evolutionary path for the linear system exhibiting late time stochastic fluctuations. We explain how stochastic behaviour is injected into the linear system as a combined effect of an uncertainty in the definition of inversion and the irrationality of the golden mean number. We also give an ansatz for the nonlinear stochastic behaviour of (fractal) time which facilitates us to estimate the late and short time limits of a two-time correlation function relevant for the stochastic fluctuations in linear systems. These fluctuations are shown to enjoy generic 1/f spectrum. The implicit functional definition of the fractal time is shown to satisfy the differential equation dx=dt. We also discuss the relevance of intrinsic time in the present formalism, study of which is motivated by the issue of time in quantum gravity.