Quantum field theory on a discrete space and noncommutative geometry

TL;DR

This paper explores the quantization of a simple noncommutative quantum field theory model in zero dimensions, connecting it to four-dimensional Feynman diagram counting, and clarifies fundamental QFT concepts in a simplified setting.

Contribution

It provides a detailed analysis of noncommutative quantum field theory in zero dimensions, emphasizing the connection to four-dimensional diagram counting and clarifying basic QFT notions.

Findings

Explicit connection between zero-dimensional noncommutative models and four-dimensional Feynman diagrams

Clarification of quantum field theory concepts without divergence complications

Insights into symmetry breaking in noncommutative settings

Abstract

We analyse in detail the quantization of a simple noncommutative model of spontaneous symmetry breaking in zero dimensions taking into account the noncommutative setting seriously. The connection to the counting argument of Feyman diagrams of the corresponding theory in four dimensions is worked out explicitly. Special emphasis is put on the motivation as well as the presentation of some well-known basic notions of quantum field theory which in the zero-dimensional theory can be studied without being spoiled by technical complications due to the absence of divergencies.