Symplectic structures and globally hyperbolic spacetimes
Romero Solha
TL;DR
The paper addresses the problem of constructing a symplectic structure on orientable globally hyperbolic Lorentzian 4-manifolds without relying on cotangent-bundle formalisms. It introduces a tangent-bundle decomposition into complex line bundles $N_\star$ and $N$, assigns Hermitian connections whose curvatures sum to a closed, nondegenerate two-form $\varpi$ on $M$, and analyzes the resulting symplectic foliation; Schwarzschild is worked out as a concrete nondegenerate example, while Minkowski and de Sitter yield $\varpi=0$. The work also proposes a geometric quantisation framework with the prequantum bundle $N_\star\otimes N$ and discusses a related Poisson structure arising from the symplectic foliation. A key contribution is the explicit global construction of a symplectic structure on spacetime from Lorentzian data, together with a potential deformation-quantisation viewpoint that connects gravity-inspired geometry to quantum formalism. Overall, the approach provides a novel bridge between 3+1 spacetime decomposition, symplectic geometry, and quantum gravity-inspired quantisation concepts.
Abstract
The aim of this note is to present a construction of symplectic structures on orientable globally hyperbolic 4-dimensional lorentzian manifolds. Said structures are defined on the manifold itself, not on its cotangent bundle. It also includes a discussion about their geometric quantisation.