On `orbital' and `spin' angular momentum of light in classical and quantum theories -- a general framework

TL;DR

This paper develops a comprehensive framework for analyzing orbital and spin angular momentum of light in classical and quantum theories, clarifying their properties, quantization, and the paraxial approximation.

Contribution

It introduces a unified approach to distinguish and analyze orbital and spin angular momentum in classical and quantum light fields, including their algebraic properties and quantum operators.

Findings

Total angular momentum splits into orbital and spin parts in classical theory.

Quantum operators for orbital and spin angular momentum do not satisfy classical angular momentum algebra.

The total angular momentum operator satisfies the angular momentum algebra and generates the E(3) group.

Abstract

We develop a general framework to analyze the two important and much discussed questions concerning (a) `orbital' and `spin' angular momentum carried by light and (b) the paraxial approximation of the free Maxwell system both in the classical as well as quantum domains. After formulating the classical free Maxwell system in the transverse gauge in terms of complex analytical signals we derive expressions for the constants of motion associated with its Poincar\'{e} symmetry. In particular, we show that the constant of motion corresponding to the total angular momentum ${\bf J}$ naturally splits into an `orbital' part ${\bf L}$ and a `spin' part ${\bf S}$ each of which is a constant of motion in its own right. We then proceed to discuss quantization of the free Maxwell system and construct the operators generating the Poincar\'{e} group in the quantum context and analyze their algebraic properties and find that while the quantum counterparts $\hat{{\bf L}}$ and $\hat{{\bf S}}$ of ${\bf L}$ and ${\bf S}$ go over into bona fide observables, they fail to satisfy the angular momentum algebra precluding the possibility of their interpretation as `orbital' and `spin' operators at the classical level. On the other hand $\hat{{\bf J}}=\hat{{\bf L}}+ \hat{{\bf S}}$ does satisfy the angular momentum algebra and together with $\hat{{\bf S}}$ generates the group $E(3)$. We then present an analysis of single photon states, paraxial quantization both in the scalar as well as vector cases, single photon states in the paraxial regime. All along a close connection is maintained with the Hilbert space $\mathcal{M}$ that arises in the classical context thereby providing a bridge between classical and quantum descriptions of radiation fields.