Fermionic current on the circle ($J_n={\sum }_r:c_r^{\ast }{c}_{r+n}:$)

A Fourier mode of the normal-ordered fermion density is a fermionic current. On the finite-energy domain its defining sum has finitely many nonzero actions on each occupation vector. The modes satisfy $J_n^{\ast }=J_{-n}$ and $[J_n,J_m]=n{\delta }_{n+m,0}I$. The zero mode is charge; positive modes annihilate charge ground states and negative modes create the oscillators in the fermion-boson correspondence on the circle.