An Algebraic Approach to Light–Matter Interactions

• Published in: Advanced Physics Research Vol. 4; no. 1
• Main Author: Fernandez‐Corbaton, Ivan
• Format: Journal Article
• Language: English
• Published: Edinburgh John Wiley & Sons, Inc 01.01.2025; Wiley-VCH
• Subjects: Algebra; conformal symmetry; Electromagnetism; Hilbert space; Light; light–matter interactions; Operators (mathematics); Physics; Spacetime; Symmetry; Theoretical physics; Theory of relativity; T‐matrix
• ISSN: 2751-1200; 2751-1200
• DOI: 10.1002/apxr.202400088

A theoretical and computational framework for the study and engineering of light–matter interactions is reviewed in here. The framework rests on the invariance properties of electromagnetism, and is formalized in a Hilbert space whose conformally invariant scalar product provides connections to physical quantities, such as the energy or momentum of a given field, or the outcome of measurements. The light–matter interaction is modeled by the polychromatic scattering operator, which establishes a natural connection to a popular computational formalism, the transition matrix, or T‐matrix. This review contains a succinct yet comprehensive description of the main theoretical ideas, and illustrates some of the practical benefits of the approach. This is a review of an algebraic approach to light–matter interactions that is theoretically powerful and computationally friendly. Theoretical expressions can be developed and manipulated conveniently thanks to the generality of the basis on which the approach rests, and a compact notation. The tight connections to popular computational tools allow one to readily perform numerical calculations.