Unified field theory from Hamilton cotangent bundle geometry - The Einstein-Maxwell system

TL;DR

This paper proposes a geometric unification of gravity and electromagnetism by treating a scalar field on the cotangent bundle $T^*M$—the point-particle Hamiltonian—as the fundamental object. By constructing a diffeomorphism-invariant action on phase space within Hamiltonian geometry, the authors derive a field equation for the Hamilton function whose expansion in momenta reveals spacetime equations. Specifically, using $H_{G}$ recovers Einstein vacuum, while $H_{EM}$—with carefully chosen parameter relations—yields the Einstein–Maxwell system, with Maxwell's equations emerging from cubic momentum terms and the Einstein equations from quadratic momentum terms. The framework encodes interactions directly in the phase-space geometry, requiring no extra dimensions or additional fields, and offers a path to include further fields and higher-spin dynamics via more general Hamiltonians.

Abstract

The unification of all physical fields into one mathematical object and the derivation of all physical field equations from that object in one framework is a long-lasting endeavor in fundamental physics. We suggest a new approach to achieve this goal by encoding physical fields into the geometry of the 1-particle phase space on spacetime (the cotangent bundle) through Hamilton geometry. The fundamental field, which contains information about all physical fields in spacetime and defines the phase space geometry, is a scalar field in phase space that is interpreted as a point-particle Hamiltonian. We construct an action principle for scalar fields in phase space and derive the corresponding scalar field equation. By choosing a specific scalar field, namely the Hamiltonian describing a charged particle in curved spacetime with an electromagnetic field, we show that this phase-space scalar field equation is equivalent to the coupled Einstein-Maxwell equations in spacetime, thus providing a geometric unification of gravity and electromagnetism. We further discuss how this approach differs from previous unification attempts and its potential for describing further physical fields and their dynamics in a unified manner in terms of phase-space geometry.