Fermion Operators -> Hubbard
Operators
Consider spin-1/2 particles:
First, the state 
is defined as
,
where 
is the
empty state,
is the
one-particle-with-spin-up state,
is the
one-particle-with-spin-down state,
is the
filled state, has two particles,
one
with spin up, another with spin down.
Second, define the filled state to be
,
( If the order of creating a particle
with spin down first then another with spin up is exchanged, the sign
will change, therefore the filled state must be defined in a certain
order. )
where 
is the
creation operator of one fermion with spin up,
is the
creation operator of one fermion with spin down.
then
where 
and 
follow the
fermion anticomutation relations.
