Matrix D-brane Dynamics, Logarithmic Operators and Quantization of Noncommutative Spacetime

TL;DR

This paper explores the moduli space of D-particle couplings using logarithmic conformal field theory, deriving a non-abelian Born-Infeld action, and revealing spacetime quantization and uncertainty relations from string theory.

Contribution

It provides a novel analysis of D-particle dynamics via logarithmic operators, connecting moduli space geometry with spacetime uncertainties and quantum gravity effects.

Findings

Derives the non-abelian Born-Infeld action from conformal field theory.

Shows the Zamolodchikov metric encodes spacetime short-distance structure.

Reveals spacetime quantization and uncertainty relations from string interactions.

Abstract

We describe the structure of the moduli space of $\sigma$-model couplings for the worldsheet description of a system of $N$ D-particles, in the case that the couplings are represented by a pair of logarithmic recoil operators. We derive expressions for the canonical momenta conjugate to the D-particle couplings and the Zamolodchikov metric to the first few orders in $\sigma$-model perturbation theory. We show, using only very general properties of the operator product expansion in logarithmic conformal field theories, that the canonical dynamics on moduli space agree with the predictions of the non-abelian generalization of the Born-Infeld effective action for D-particles with a symmetrized trace structure. We demonstrate that the Zamolodchikov metric naturally encodes the short-distance structure of spacetime, and from this we derive uncertainty relations for the D-particle coordinates directly from the quantum string theory. We show that the moduli space geometry naturally leads to new forms of spacetime indeterminancies involving only spatial coordinates of target space and illustrate the manner in which the open string interactions between D-particles lead to a spacetime quantization. We also derive appropriate non-abelian generalizations of the string-modified Heisenberg uncertainty relations and the space--time uncertainty principle. The non-abelian uncertainties exhibit decoherence effects suggesting the interplay of quantum gravity in multiple D-particle dynamics.