A Fock space is the Hilbert direct sum of particle-number sectors built from a one-particle Hilbert space 
. The bosonic and fermionic versions
are
 |
and
 |
The zero-particle sector is 
and contains the vacuum. The symmetric and antisymmetric
tensor powers describe bosons and fermions, respectively. For a one-dimensional
bosonic space, normalized occupation vectors 
satisfy 
and 
. Thus 
counts particles, and 
. If 
, the fermionic sectors vanish for 
.
