# A Antisymmetry in fermionic Fock space

1. Aug 5, 2016

### IanBerkman

Dear all,

When we annihilate a particle at level $k$ in fermionic Fock space we use the relation
$$\hat{c}_k| \dots, 1_k, \dots \rangle = (-1)^{\sum_{i=1}^{k-1} n_i}|\dots,0_k,\dots\rangle.$$

Where the factor ($\pm1$) depends on the occupation numbers of all the levels below the level $k$.
However, I do not see what the link is between antisymmetry when permuting two electrons, i.e.
$$\hat{c}_n^\dagger\hat{c}_m^\dagger|0\rangle = -\hat{c}_m^\dagger\hat{c}_n^\dagger|0\rangle,$$
and how this is incorporated by counting the number of electrons below level $k$.

Ian

Last edited: Aug 5, 2016
2. Aug 10, 2016