Novel Phenomena in Noncommutative Field Theory: Emergent Geometry

Abstract: Noncommutative field theory (NCFT) is an extension of quantum field theory (QFT) that redefines spacetime, replacing commuting coordinates with a noncommutative structure. This shift fundamentally alters the way fields, interactions, and symmetries are understood. NCFT uniquely integrates with supersymmetry, making it a natural framework for unifying quantum mechanics and gravity. It also provides a consistent mechanism for spontaneous supersymmetry breaking. Unlike conventional QFT, which quantizes fields on a fixed spacetime, NCFT begins by quantizing spacetime itself. This perspective reveals novel phenomena, such as ultraviolet-infrared mixing and a natural transition from discrete to continuous geometries. It offers insights into quantum gravity at the Planck scale. Mathematically, NCFT bridges quantum mechanics and geometry through operator algebras, enabling the exploration of new theories unattainable in traditional frameworks. This dual role cements NCFT as a cornerstone of modern theoretical physics. This work is published as Chapter 18, "Novel phenomena in noncommutative field theory: emergent geometry", in A Modern Course in Quantum Field Theory, Volume 2 (Second Edition), IOP Publishing (2025). ISBN 978-0-7503-5834-7. DOI 10.1088/978-0-7503-5834-7. The published chapter is available at the publisher's website: https://iopscience.iop.org/book/mono/978-0-7503-5834-7/chapter/bk978-0-7503-5834-7ch18