How Large are Dissipative Effects in Non-Critical Liouville String Theory?

TL;DR

This paper analyzes the magnitude of dissipative effects in non-critical Liouville string theory, arguing they are of order E^2/M_P, and discusses the role of nonlinear terms in the density matrix evolution.

Contribution

It clarifies why dissipative effects in non-critical Liouville strings are expected to be small and highlights the importance of nonlinear terms for complete positivity analysis.

Findings

Dissipative effects are of order E^2/M_P in the theory.

Energy conservation is only statistical, not operator-level.

Nonlinear terms are crucial for analyzing complete positivity.

Abstract

In the context of non-critical Liouville strings, we clarify why we expect non-quantum-mechanical dissipative effects to be of order E^2/M_P, where E is a typical energy scale of the probe, and M_P is the Planck scale. In Liouville strings, energy is conserved {\it at best} only as a statistical average, as distinct from Lindblad systems, where it is {\it strictly} conserved at an operator level, and the magnitude of dissipative effects could only be much smaller. We also emphasize the importance of nonlinear terms in the evolution equation for the density matrix, which are important for any analysis of complete positivity.