The Uncertainty of Fluxes

TL;DR

This paper explores the quantum uncertainty in measuring electric and magnetic fluxes in curved space, develops a Hamiltonian framework for self-dual fields, and applies these concepts to string theory's Ramond-Ramond fields.

Contribution

It introduces a novel uncertainty principle for fluxes in quantum Maxwell theory on curved manifolds and develops a Hamiltonian theory for self-dual fields linked to Pontrjagin cohomology.

Findings

Fluxes modulo torsion are simultaneously measurable.

Self-dual fields are quantized via Pontrjagin self-dual cohomology.

The quantum Hilbert space contains both bosonic and fermionic states.

Abstract

In the ordinary quantum Maxwell theory of a free electromagnetic field, formulated on a curved 3-manifold, we observe that magnetic and electric fluxes cannot be simultaneously measured. This uncertainty principle reflects torsion: fluxes modulo torsion can be simultaneously measured. We also develop the Hamilton theory of self-dual fields, noting that they are quantized by Pontrjagin self-dual cohomology theories and that the quantum Hilbert space is Z/2-graded, so typically contains both bosonic and fermionic states. Significantly, these ideas apply to the Ramond-Ramond field in string theory, showing that its K-theory class cannot be measured.