Twisting all the way: from Classical Mechanics to Quantum Fields

TL;DR

This paper explores how Drinfeld twist-induced noncommutative geometry affects classical and quantum theories, deforming products, symmetries, and quantization procedures on noncommutative spacetime.

Contribution

It systematically extends classical mechanics and quantum field theory to noncommutative spacetime using a geometric deformation approach.

Findings

Deformed Poisson brackets alter classical time evolution.

Derived noncommutative commutation relations for quantum operators.

Established a noncommutative correspondence principle for quantization.

Abstract

We discuss the effects that a noncommutative geometry induced by a Drinfeld twist has on physical theories. We systematically deform all products and symmetries of the theory. We discuss noncommutative classical mechanics, in particular its deformed Poisson bracket and hence time evolution and symmetries. The twisting is then extended to classical fields, and then to the main interest of this work: quantum fields. This leads to a geometric formulation of quantization on noncommutative spacetime, i.e. we establish a noncommutative correspondence principle from *-Poisson brackets to *-commutators. In particular commutation relations among creation and annihilation operators are deduced.