Re: Density operator in second quantization

erite423@xxxxxxxx wrote:
I have some questions concerning density operators and the second
quantization formalism:

1. What is the density operator for a many-electron system in second
quantization?

It's defined in the same way as in ordinary quantum mechanics. Pure
states are obtained by an outer product like |phi><phi|, while mixed
states are obtained by taking convex linear combinations and limits of
pure states. The difference now is that the states |psi> must be taken
from the multi-particle Fock space instead of the single particle
subspace.

However, I must warn you about a possible clash in terminology. When
dealing with many particle systems (which is where second quantization
comes in), there a physical observable that's called the "density
operator". It's defined as rho(x) = :psi(x)* psi(x):, which is the
normal ordered product of field operators. For example, when dealing
with a multi-electron system, it represents the total charge density at
point x. What you are talking about is still conventionally referred to
as a "density matrix", despite the fact that it's not really a matrix
in the textbook sense of the word.

2. How do I trace out degrees of freedom to form reduced density
operators?

Exactly the same way as you are used to. The first requirement for
taking a partial trace is the possibility of expressing the state space
as a tensor product. For example, when dealing with the quantization of
a linear field (or the second quantization of some single-particle
theory, which are equivalent), the state space can be expressed as a
tensor product of the states associated to short wavelength field modes
and long wavelength field modes (with an arbitrarily placed cutoff).
Then either the short- or longwavelength modes can be traced out.

3. Can the operator |vac> <vac|, mapping the vacuum state to itself
and all others to zero, be expressed in terms of creation and
annihilation operators?

Let N = sum_k a*_k a_k be the particle number operator. Its spectrum
consists of the natural numbers 0, 1, 2, ... . Take an analytic
function f(x) which satisfies f(0) = 1 and f(n) = 0, for integer n > 0.
Then P_vac = f(N) will be the projection operator you want. For
example, take f(x) = sin(Pi*x)/x = Pi - (Pi*x)^2/3! + (Pi*x)^4/5! - ...
.. Substitute N for x and you have the expression you wanted in terms of
the a_k and a*_k.

Hope this helps.

Igor

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