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Second quantization
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Second quantization is a way of writing the quantum mechanics of 
indistinguishable particles in such a way that it makes statistical 
mechanics calculations easy and makes everything look like field theory.

One starts with a distinguished vacuum state |vac> and a family of
annihilation operators a(x) whith their adjoints, the creation 
operators a^*(x), satisfying the canonical commutation relations (CCR)
   [a(x),a(y)]=[a^*(x),a^*(y)]=0, 
   [a^(x),a^*(y)]=delta(x-y).
(This is for Bosons; for Fermions one has instead canonical 
anticommutation relations, CAR, and everything below gets additional 
minus signs in certain places.)

A pure (permutation symmetric) N-particle state with wave function 
psi(x_1:N) is written in second quantization as
   psi = integral dx_1:N psi(x_1:N) a^*(x_1:N) |vac>, 
hence the corresponding density matrix 
   rho = psi psi^* 
takes the form
   rho = integral dx_1:N dy_1:N rho(x_1:N,y_1:N),
where rho(x_1:N,y_1:N) is the rank one operator
   psi(x_1:N)psi^*(y_1:N)a^*(x_1:N)|vac><vac|a(y_1:N).
Using this correspondence, one can do in second quantization whatever 
one can do in first quantization (i.e., wave mechanics),
and match the results.

If f is a 1-particle operator given by an integral operator with
kernel f(x,y) (the general case follows by taking limits), so that
   (f psi)(x_1:N) 
   = sum_a integral dx f(x_a,x) psi(x_{1:a-1},x,x_{a+1:N}),
the formula 
    <f> = integral dx dy <x|Rho|y> f(x,y)
defines the 1-particle density matrix Rho. The form of f in second 
quantization is 
   f = integral dx dy f(x,y) a^*(x) a(y)
(exercise: check that it has indeed the desired action on an 
N-particle state!), hence one has 
  <f> = integral dx dy f(x,y) <a^*(y)a(x)>.
and comparison with the definition of Rho gives the formula 
  <x|Rho|y> = <a^*(y)a(x)> = trace a(x) rho a^*(y),
which can therefore be viewed as the definition of the 1-particle 
density matrix in second quantization. 

Authors who are afraid of integrals write instead similar formulas with
sums in place of integrals and discrete indices in place of the x,y.
Also, one can do the same in momentum space rather than position space,
which amounts to a change of basis but generally leads to 
computationally more tractable formulations.