Heisenberg current algebra of a fermion on the circle ($[J_n,J_m]=n{\delta }_{n+m,0}I$)

For integer modes filled at indices below zero, set $E_{rs}=c_r^{†}{c}_s-{\delta }_{rs}{1}_{r<0}$ and $J_n={\sum }_r{E}_{r,r+n}$ on the finite-energy domain. The canonical anticommutation relations give a central term ${\delta }_{st}{\delta }_{ur}(1_{r<0}-1_{s<0})$ in $[E_{rs},E_{tu}]$. Summing across the filled-empty boundary gives $[J_n,J_m]=n{\delta }_{n+m,0}I$, with $J_n^{\ast }=J_{-n}$. The current's level is the coefficient of this central term.

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.