On complex structures in physics

TL;DR

This paper reviews the role of complex structures in physics, focusing on their use in quantum theories and geometric frameworks like Hodge duality, charge conjugation, and Cauchy-Riemann structures in spacetime.

Contribution

It provides a detailed review of complex structures in physics, emphasizing their mathematical and physical significance in quantum mechanics and spacetime geometry.

Findings

Complex numbers are essential for Hermitian operators in quantum theories.

Complex structures relate to Hodge duality in vector and spinor spaces.

Charge conjugation and Cauchy-Riemann structures are analyzed in specific spacetime geometries.

Abstract

Complex numbers enter fundamental physics in at least two rather distinct ways. They are needed in quantum theories to make linear differential operators into Hermitian observables. Complex structures appear also, through Hodge duality, in vector and spinor spaces associated with space-time. This paper reviews some of these notions. Charge conjugation in multidimensional geometries and the appearance of Cauchy-Riemann structures on Lorentz manifolds with a congruence of null geodesics without shear are presented in considerable detail.