# Optical Helicity and Chirality: Conservation and Sources
## 2 Review and Motivation
### Quantum Helicity Operator  
One reason to prefer the helicity over chirality as a measure of the degree of handedness of an optical field is its intuitive quantum-mechanical form. The operators for the electric and magnetic fields and the two vector potentials are given by [17]:  
\[\hat{\mathbf{E}} = \sqrt{\frac{\hbar ck}{2\epsilon_{0}V}}\sum_{\mathbf{k},\lambda}i \mathbf{e_{k},\lambda}\hat{a}_{\mathbf{k},\lambda}e^{i(\mathbf{k}\cdot\mathbf{ x}-\omega t)}+h.c,\] \[\hat{\mathbf{B}} = \sqrt{\frac{\hbar}{2\epsilon_{0}cKV}}\sum_{\mathbf{k},\lambda}i (\mathbf{k}\times\mathbf{e_{k},\lambda})\hat{a}_{\mathbf{k},\lambda}e^{i( \mathbf{k}\cdot\mathbf{x}-\omega t)}+h.c,\] \[\hat{\mathbf{A}} = \sqrt{\frac{\hbar}{2\epsilon_{0}cKV}}\sum_{\mathbf{k},\lambda} \mathbf{e_{k},\lambda}\hat{a}_{\mathbf{k},\lambda}e^{i(\mathbf{k}\cdot\mathbf{ x}-\omega t)}+h.c,\] \[\hat{\mathbf{C}} = \sqrt{\frac{\hbar\epsilon_{0}c}{2k^{3}V}}\sum_{\mathbf{k}, \lambda}(\mathbf{k}\times\mathbf{e_{k},\lambda})\hat{a}_{\mathbf{k},\lambda}e ^{i(\mathbf{k}\cdot\mathbf{x}-\omega t)}+h.c, \tag{5}\]  
where \(\lambda\) labels two orthogonal polarisation modes, \(\mathbf{e_{k},\lambda}\) and \(a_{\mathbf{k},\lambda}\) are respectively the polarisation vector and annihilation operator corresponding to a photon in mode \((\mathbf{k},\lambda)\), \(V\) is the quantisation volume and \(k\equiv\sqrt{\mathbf{k}^{2}}\). Using these field and vector potential operators, expanded in the basis of creation and annihilation operators for left- and right-circular polarisation modes, the quantum-mechanical version of the classical definition of the integrated helicity density (1) corresponds to the difference in the total number of left- and right-circularly polarised photons in the field [14; 15; 17]:  
\[\hat{\mathcal{H}}=\int\hat{h}\,d^{3}r=\hbar\sum_{\mathbf{k}}\left(\hat{N}_{ \mathbf{k},+}-\hat{N}_{\mathbf{k},-}\right). \tag{6}\]  
Here, \(\hat{N}_{\mathbf{k},+}\) and \(\hat{N}_{\mathbf{k},-}\) are the number operators for photons in the \((\mathbf{k},+)\) and \((\mathbf{k},-)\) modes, \(\hat{N}_{\mathbf{k},+}\equiv\hat{a}_{\mathbf{k},+}^{\dagger}\hat{a}_{\mathbf{ k},+}\) and \(\hat{N}_{\mathbf{k},-}\equiv\hat{a}_{\mathbf{k},-}^{\dagger}\hat{a}_{\mathbf{ k},-}\). This form of the helicity operator allows the electromagnetic definition of helicity to be connected to the concept of helicity in particle physics, where the helicity of a particle is defined as the projection of its spin angular momentum in the direction of propagation [19]. The angular momentum associated classically with circular polarisation is conventionally associated with the photon spin, meaning that each photon in a right- or left-handed mode contributes \(\pm\)\(\hbar\) of helicity. Parenthetically, it should be noted that the terminology of separating the total angular momentum into spin and orbital \"angular momenta\" must be approached with caution. There is a sense in which neither the spin, nor the orbital quantum operators correspond to angular momenta, as neither satisfy the commutation relations of an angular momentum [36; 37; 38; 39]. However, the quantities are physically distinct, as supported by experimental evidence [39], and are furthermore separately conserved under separate rotational transformations of the electromagnetic fields [36; 37; 38; 40].