Dirac brackets from magnetic backgrounds

TL;DR

This paper demonstrates how magnetic backgrounds in classical phase space, viewed as a generalized complex manifold, naturally lead to Dirac brackets, connecting magnetic terms with B-transformations in geometry.

Contribution

It establishes a geometric derivation of Dirac brackets from magnetic backgrounds using generalized complex geometry, linking physics and advanced mathematical structures.

Findings

Magnetic terms in symplectic forms arise naturally via B-transformations.

Dirac brackets can be derived from geometric transformations under magnetic fields.

The approach unifies magnetic interactions with generalized complex geometry.

Abstract

In symplectic mechanics, the magnetic term describing the interaction between a charged particle and an external magnetic field has to be introduced by hand. On the contrary, in generalised complex geometry, such magnetic terms in the symplectic form arise naturally by means of B-transformations. Here we prove that, regarding classical phase space as a generalised complex manifold, the transformation law for the symplectic form under the action of a weak magnetic field gives rise to Dirac's prescription for Poisson brackets in the presence of constraints.