Dyson Maps and Unitary Evolution for Maxwell Equations in Tensor Dielectric Media

TL;DR

This paper reformulates Maxwell equations in complex media as a quantum-like Dirac equation with unitary evolution, introducing Dyson maps and a new representation suitable for quantum computing simulations.

Contribution

It presents a novel reformulation of Maxwell equations in tensor dielectric media as a Dirac-like equation with unitary evolution, avoiding the RSW vector and enabling quantum computing applications.

Findings

Derived a pseudo-Hermitian dynamics for Maxwell equations in complex media.

Constructed a Dyson map to recover Hermiticity and unitary evolution.

Provided an explicit example with a uniaxial tensor dielectric medium.

Abstract

A reformulation of Maxwell equations for an inhomogeneous, anisotropic, passive and non-dispersive medium results in a quantum-like Dirac equation that admits unitary time evolution. In contrast to other approaches, there is no a-priori introduction of the Riemann-Silberstein-Weber (RSW) vector but the Maxwell equations are considered in their standard fields, with given constitutive relations. From the electromagnetic conservation quantities a pseudo-Hermitian dynamics is found together with a Dyson map that recovers the full Hermicity of the dynamics in an extended Hilbert space that describes the physical notion of unitary evolution. As an example, a uniaxial tensor dielectric medium is considered, with the explicit Dyson map yielding an optimal representation in a set of generalized RSW vectors. In this newly discovered form, a Quantum Computing (QC) implementation for simulation of electromagnetic wave propagation in complex media can be made, with further extension into plasmas.