To be continued
Second Quantization:
Roughly speaking, when the idea of operators introduced from
the beginning, the physical world has been transformed from
classical to quantum.
This is implied from the first quantization:
The classical momentum, which is just a vector, transfromed into a
derivative of space/position operator times Plank constant and an
imaginary number in quantum mechanics.
While momentum becomes an operator, the wave functions are still
vectors in mathematical representation as in classical mechanics.
The second quantization is the transformation of the the wave
functions, from vectors to operators, the field operators.
The field operators can act on state vectors in occupation-number
space, which means that the state vectors are simply imply the number of
particles in each eigenstates. Creation operators and destruction
operators, similar to those creation/destruction operators in simple
harmonic oscillation, are used to describe the increase or decrease of
the number of particles.
As mentioned in still
warming up, the wave functions are antisymmetric for fermions, but
symmetric for bosons, so the field operators have the same property.
Therefore, the creation/destruction operators are different for bosons
and for fermions to satisfy this property. They are called the boson
operators and the fermion operators.
Boson operators satisfy the commutation relation,
and have the properties,
,
where d ( d-dagger ) are the destruction ( construction ) operators
for bosons, which destroy ( create ) a boson in state i. 
,
from 0 to infinity, is the number of particles in state i.
Fermion operators satisfy the anticommutation relation,
and have the properties,
,
where c ( c-dagger ) are the destruction ( construction ) operators
for fermions, which destroy ( create ) a fermion with spin
in state i or site i.
For more details, please look up the references
or quantum mechanics and statistical mechanics books.
