A relativistic toy model for back-reaction

TL;DR

This paper models the back-reaction of a quantum scalar field on a collapsing circular string, suggesting it might prevent collapse but highlighting the limitations of semi-classical analysis at critical points.

Contribution

It introduces a semi-classical model incorporating quantum back-reaction on a dynamic string, revealing potential prevention of collapse and limitations of the semi-classical approach.

Findings

Back-reaction may prevent the string's collapse to zero radius

Semi-classical analysis fails at the bounce point

Full quantum treatment is needed for post-bounce dynamics

Abstract

We consider a quantized massless and minimally coupled scalar field on a circular closed string with a time-dependent radius $R(t)$, whose undisturbed dynamics is governed by the Nambu-Goto action. Within the semi-classical treatment, the back-reaction of the quantum field onto the string dynamics is taken into account in terms of the renormalized expectation value of the energy-momentum tensor including the trace anomaly. The results indicate that the back-reaction could prevent the collapse of the circle $R\downarrow0$ -- however, the semi-classical picture fails to describe the string dynamics at the turning point (i.e., possible bounce) at finite values of $R$ and $\dot R$. The fate of the closed string after that point (e.g., oscillation or eternal acceleration) cannot be determined within the semi-classical picture and thus probably requires the full quantum treatment. PACS: 04.62.+v, 03.70.+k, 11.15.Kc, 04.60.-m.