The Poincare Duality in Quantization of the Norm of Differential Forms

TL;DR

This paper develops new methodologies for Hodge decomposition and Poincare duality that do not rely on positive definite norms, enabling their application in pseudo-Riemannian manifolds like Minkowski space in physics.

Contribution

It introduces a framework based on linear independence of canonical classes for Hodge and Poincare duality, avoiding positive norm assumptions.

Findings

Hodge and Poincare duality expressed without positive definite norms

Quantization of field norm and action linked to topology and cohomology classes

Application to electromagnetic theory in Minkowski space

Abstract

The more important difference between Riemann and pseudo-Riemann manifolds is the metric signature and its theoretical consequences. The practical application for Physics Theories becomes often impossible due to the signature consequences. Eg., some of the rich results in Riemann Geometry and Topology become invalid for Physics if they are based on the concept of the positive definite norm; to avoid this problem, the proof machinery must avoid such assumption and must be based in other tools. This paper is a contribution to provide methodologies for Hodge decomposition and \poincare duality based on the concept of linear independence of canonical classes instead of the positive norm. As a result, the Hodge and norm decompositions are expressed based on continuous and discrete terms. When this result is applied to Classical Electromagnetic Theory, in pseudo-Riemann manifolds with minkowskian metric, magnitudes as the field norm and action have one discrete sum of terms. This result, as a quantization of the norm and action is a property of the Topology, in special of the Cohomology classes, that are sources of the field as well as the generators of action quantum.