# Optical Helicity and Chirality: Conservation and Sources
## 3 Microscopic Sources  
### Helicity in the Presence of Current and Charge  
We have introduced the helicity as a quantity associated with the free electromagnetic field, but it is also of interest as a way to describe the interaction of light with chiral or achiral matter. The presence of matter breaks dual-symmetry, as all known matter is made up of only electric charges, with no magnetic ones. In the presence of matter, therefore, helicity is not generally conserved. (We note parenthetically that the equations still remain invariant if we additionally \"rotate\" the electric charges into magnetic ones, introducing a charge and density and current density of magnetic charges. This symmetry of the equations means that it is in a sense a matter of convention that we speak of electric charges, rather than magnetic charges; the non-existence of magnetic monopoles can be rephrased as \"every particle has the same ratio of electric to magnetic charge\", and it is only a matter of convention that leads us to treat all charges as purely electric. See [41], Chapter 6 SS11.).  
One way of discussing helicity in the presence of charges is to treat the charges microscopically, with the fields described using the equations of free-space electromagnetism. In the presence of a current density \(\mathbf{j}\) and a charge density \(\rho\), it can be shown that the continuity equation becomes [17]:  
\[\partial_{t}h+\mathbf{\
abla}\cdot\mathbf{v}=\frac{1}{2}\sqrt{\frac{\mu_{0}}{ \varepsilon_{0}}}\Big{[}\mathbf{g}\cdot\left(\mathbf{\
abla}\times\mathbf{C} \right)+\mathbf{C}\cdot\left(\mathbf{\
abla}\times\mathbf{g}\right)\Big{]}, \tag{16}\]  
where \(\mathbf{g}\) is a vector field defined by the requirement \(\mathbf{\
abla}\times\mathbf{g}=\mathbf{j}^{\mathrm{T}}\), the transverse part of the current density. The continuity Equation (16) now expresses the non-conservation of helicity, with additional terms on the right-hand side showing how the matter acts as a source or sink of helicity in the field [17]. The terms \"source\" and \"sink\" must be treated with caution, as--unlike (for example) the case of energy--there is not a sense in which the matter \"gains\" or \"loses\" helicity when it absorbs or emits into the field.  
The chirality density obeys a similar continuity equation [30]:  
\[\partial_{t}\chi+\frac{1}{2\mu_{0}}\mathbf{\
abla}\cdot\left(\mathbf{E}\times \left(\mathbf{\
abla}\times\mathbf{B}\right)-\mathbf{B}\times\left(\mathbf{\
abla} \times\mathbf{E}\right)\right)=-\frac{1}{2}\left(\mathbf{j}\cdot\left(\mathbf{ \
abla}\times\mathbf{E}\right)+\mathbf{E}\cdot\left(\mathbf{\
abla}\times \mathbf{j}\right)\right). \tag{17}\]Comparison of the two continuity equations again demonstrates the close connection between chirality and helicity: the chirality is the quantity one would obtain by forming the helicity from the curl of the fields, rather than the fields [15]. This observation makes clear why a direct proportionality holds for monochromatic light, where taking the curl merely introduces a factor of \(i\omega\).  
It is clear that \(\mathbf{g}\) is acting like a magnetisation, as \(\mathbf{\
abla}\times\mathbf{M}=\mathbf{j}_{\text{free}}\). We also note that, as we only consider the transverse \(\mathbf{E}\) field, there is no analogous \"polarisation-like\" term to correspond with \(\mathbf{\
abla}\cdot\mathbf{P}=\rho\). This leaves the source term asymmetric in terms of electric and magnetic contributions.