# Fermion creation op anticommutator relations

1. Nov 10, 2007

### WarnK

1. The problem statement, all variables and given/known data
2. Relevant equations
Given is
$$c_p = \sum_{n_i} (-1)^{N_p}|...,n_p=0,...><...,n_p=1,...|$$
$$c_p^{\dagger} = \sum_{n_i} (-1)^{N_p}|...,n_p=1,...><...,n_p=0,...|$$
$$N_p = \sum_{i=1}^{p-1}n_i$$

Verify
$$\{c_p,c_q\} = \{c_p^{\dagger},c_q\} = \{c_p^{\dagger},c_q^{\dagger}\} = 0$$
$$\{c_p^{\dagger},c_p\} = 1$$

3. The attempt at a solution

If we let the commutator $$\{c_p,c_q\}$$ act on some state, any state where $$n_p$$ or $$n_q$$ are zero will just give zero back. If $$n_p$$ and $$n_q$$ are one, then both $$c_p c_q$$ and $$c_q c_p$$ puts zeros there and give a factor $$(-1)^{N_p+N_q}$$ in front. I can't see where any sign change comes from.