# Requirement of Jordan-Wigner string in creation operator on Fock state

Our lecture notes described the action of the particle creation operator on a fermionic Fock state:

$$c_l^\dagger |n_1 n_2...\rangle = (-1)^{\sum_{j=1}^{l-1}n_j}|n_1 n_2 ... n_l+1 ...\rangle.$$

I am confused by why the $$(-1)^{\sum_{j=1}^{l-1}n_j}$$ factor is needed? Is it to maintain anti-symmetry of the wavefunction somehow?

Essentially yes. Recall the state for $$n_m=n_n=... =1$$ with all intermediate occupation numbers vanishing, $$|n_1 n_2...\rangle = c_m^\dagger c_n^\dagger ...|0\rangle.$$
If $$\ell$$ is one of the above occupied sites, $$n_\ell=1$$, nilpotence of the creation operators when $$c_\ell^\dagger$$ is applied on the state will net you 0, so you hardly care about about the phase multiplying it.
If not, then $$n_\ell+1=1$$, and, to get to that site, $$c_\ell^\dagger$$ has anticommuted with $$c_ m^\dagger, c_n^\dagger, ...$$ preceding it, to get to the $$\ell$$ site, so it has picked up $$\sum_{j=1}^{\ell-1}n_j$$ minus signs, multiplying each other, $$c_\ell^\dagger |n_1 n_2...\rangle = (-1)^{\sum_{j=1}^{\ell-1}n_j}|n_1 n_2 ... n_\ell+1 ...\rangle.$$