The uncertainty of fluxes
The development of quantum electrodynamics by Dirac (1927) produced rules for the equal time commutator of the electric and magnetic fields as operators while the components of the electric field commute among themselves, and so do the components of the magnetic field. Subsequently, Bohr and Rosenfeld (1933) started discussing thought experiments to verify that electric and magnetic fields cannot be measured simultaneously with arbitrary precision. The present article extends this discussion in a purely mathematical manner to a situation where the underlying Euclidean space is replaced by a compact oriented Riemannian 3-manifold. The fields under discussion are still free, i.e.\ electric and magnetic currents are set to zero. The uncertainty discovered this way reflects torsion. Not only does the argument apply to abelian gauge fields in general, but also to the so-called \(B\)-field and the Ramond-Ramond field in 10-dimensional superstring theory. Here the quantization is expressed in terms of topological \(K\)-theory and it is shown here that the Hilbert space of states is graded by \(K\)-theory modulo torsion, hence is \(\mathbb Z/2\mathbb Z\)-graded.
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