Klein Gordon Field: Define Density Matrix I want to calculate the Entropy of a KG Quantum Field which is in a mixed state.
Is it possible to define a density matrix as an analog to QM and therefore also a von Neumann Entropy?
Please refer me to appropriate introductory literature.
Thanks
 A: Yes, you can use the same methods as in regular quantum mechanics. The Fock space of a scalar field is separable and therefore well-behaved. One important thing is to recall that excited states with fixed momentum are not parts of Hilbert space, but state-valued distributions over momentum space - so you need to smear it to get a regular state. For example, you could compute the following density state:
\begin{align}
\rho=\frac{1}{2}\rho_{|0\rangle}+\frac{1}{2}\rho_{|f\rangle}\,.
\end{align}
The von-Neumann entropy of this state would clearly be $\log(2)$. This is because, the mixed state $\rho$ consists of a superposition of the vacuum state $\rho_{|0\rangle}=|0\rangle\langle 0|$ and a specific 1-particle state $\rho_{|f\rangle}=|f\rangle\langle f|$ with smearing function $f$ (such that $|f\rangle$ is normalized):
\begin{align}
|f\rangle=\int d^3k f(k) a_k^\dagger|0\rangle\,.
\end{align}
In a similar way, you can build arbitrary other mixed states. You only need to make sure that your resulting density operator $\rho: \mathcal{H}\to \mathcal{H}$ is a positive (self-adjoint) operator that is traceclass with $\mathrm{tr}\rho=1$.
