Optical chirality — the handedness of an electromagnetic field — is conventionally described by a single scalar: the chirality density, proportional to the inner product of the electric field and the curl of the electric field (plus the magnetic analogue). This scalar works in isotropic media where the field's handedness is uniform in all directions.
Smagin and Dyakov show that in anisotropic and structured media, the scalar is insufficient. They extend the Lipkin zilch formalism — a set of conserved quantities of the free electromagnetic field discovered in 1964 — to construct a full tensor description of chirality. The tensor has multiple independent components, each measuring chirality along a different axis or in a different polarization basis.
In anisotropic media, the field's handedness can vary with direction. A circularly polarized beam propagating along one crystal axis may have different chirality from the same beam along a different axis. The scalar density averages over these directional variations, losing information that the tensor preserves.
The tensor framework reveals complementary measures of chirality that arise naturally in structured electromagnetic environments but are invisible to the scalar description. These measures have direct physical relevance: they determine how the field interacts with chiral matter (enantioselective molecules, handed metamaterials) in ways that depend on the field's orientation relative to the material's anisotropy.
One number for chirality was enough in vacuum. In structured media, chirality is a tensor — it has direction as well as magnitude.