Exact Wavefunctions in a Noncommutative Field Theory

TL;DR

This paper derives exact wavefunctions and scattering amplitudes for a noncommutative field theory, revealing string-like features and providing insights into its quantum behavior and bound states.

Contribution

It introduces an exact solution approach for wavefunctions in a noncommutative field theory, connecting perturbative results with an explicit Schrödinger equation analysis.

Findings

Exact four-point scattering amplitude computed to all orders.

Wavefunctions exhibit two centers separated by a distance proportional to noncommutativity and momentum.

Identification of bound state wavefunction with string-like properties.

Abstract

We consider the nonrelativistic field theory with a quartic interaction on a noncommutative plane. We compute the four point scattering amplitude within perturbative analysis to all orders and identify the beta function and the running of the coupling constant. Since the theory admits an equivalent description via the N particle Schrodinger equation, we regain the scattering amplitude by finding an exact scattering wavefunction of the two body equation. The wave function for the bound state is also identified. These wave functions unusually have two center positions in the relative coordinates. The separation of the centers is in the transverse direction of the total momentum and grows linearly with the noncommutativity scale and the total momentum, exhibiting the stringy nature of the noncommutative field theory.