Quantization of the open string on plane-wave limits of dS_n x S^n and non-commutativity outside branes

TL;DR

This paper performs canonical quantization of open strings in a plane-wave limit of dS_n x S^n with background fields, revealing non-commutative space structures extending beyond branes, and confirms the smooth Minkowski limit.

Contribution

It demonstrates the canonical quantization of open strings in a specific curved background, showing non-commutativity of string positions throughout the string, not just at endpoints.

Findings

String position operators are non-commutative for all world-sheet parameters.

Canonical quantization is effective due to a well-defined symplectic form.

Non-commutativity reduces to endpoints in the Minkowski limit.

Abstract

The open string on the plane-wave limit of $dS_n\times S^n $ with constant $B_2$ and dilaton background fields is canonically quantized. This entails solving the classical equations of motion for the string, computing the symplectic form, and defining from its inverse the canonical commutation relations. Canonical quantization is proved to be perfectly suited for this task, since the symplectic form is unambiguously defined and non-singular. The string position and the string momentum operators are shown to satisfy equal-time canonical commutation relations. Noticeably the string position operators define non-commutative spaces for all values of the string world-sheet parameter $\sig$, thus extending non-commutativity outside the branes on which the string endpoints may be assumed to move. The Minkowski spacetime limit is smooth and reproduces the results in the literature, in particular non-commutativity gets confined to the endpoints.