Fundamentals on the Noncommutative Plane

TL;DR

This paper explores the behavior of charged matter in noncommutative gauge theories, deriving Feynman rules, analyzing divergence structures, and highlighting differences from commutative theories, especially regarding bound states and momentum conservation.

Contribution

It introduces Feynman rules and divergence analysis for charged matter in noncommutative Yang-Mills and QED, revealing unique features like bound-state wavefunctions and momentum nonconservation.

Findings

Charged particles behave similarly to the commutative case.

Bound-state wavefunctions resemble those in magnetic fields.

No reduction in divergence degree for charged fermion loops.

Abstract

We consider the addition of charged matter (``fundametals'') to noncommutative Yang-Mills theory and noncommutative QED, derive Feynman rules and tree-level potentials for them, and study the divergence structure of the theory. These particles behave very much as they do in the commutative theory, except that (1) they occupy bound-state wavefunctions which are essentially those of charged particles in magnetic fields, and (2) there is slight momentum nonconservation at vertices. There is no reduction in the degree of divergence of charged fermion loops like that which affects nonplanar noncommutative Yang-Mills diagrams.