# Qns about Fock Space and 2nd Quantizatio

hey guys, I am only now just starting to familiarize myself with the Fock Space formalism by making my way through Asher Peres' excellent text on Quantum Theory. I am deeply confused by certain parts though, and I'd greatly appreciate any help, as I am pretty much self learning this subject. Some are very elementary questions on the subject, I hope you don't mind.

1) I am given to understand that Fock space is the vector space spanned the states $$|n_{\mu}\rangle$$ and that the $$n_{\mu}$$ are quantum numbers representing the number of particles in the $$\mu$$ state. Is this in distinction with the Hilbert space? In my impression, the Hilbert space is the vector space representing the states of an $$n$$ particle system for fixed number of particles. I would like to know whether this is correct.

2)For fermions, the state representing a single particle system is given by $$a_{\mu}^{\dagger}|0\rangle$$ and a 2 particle system is given by $$a_{\mu}^{\dagger} a_{\nu}^{\dagger} |0\rangle$$. We know in first quantization however, that for indistinguishable particles in 2 orthogonal states $$|\mu\rangle$$ and $$|\nu\rangle$$ the 2 particle state is is simply the singlet state given by $$\frac{1}{\sqrt{2}}\left\{|\mu\rangle\otimes|\nu\rangle-|\nu\rangl\otimes|\mu\rangle\right\}$$. Am I right in understanding that writing $$a_{\mu}^{\dagger} a_{\nu}^{\dagger} |0\rangle$$ in the fock space formalism is equivalent to writing the singlet state $$\frac{1}{\sqrt{2}}\left\{|\mu\rangle\otimes|\nu\rangle-|\nu\rangl\otimes|\mu\rangle\right\}$$?

3) In this formalism, what is the equivalent way to doing a partial trace of a density matrix? Right now, I have a 3 particle system, given in first quantization as $$\frac{1}{\sqrt{6}}\left\{|123\rangle+|312\rangle+|231\rangle-|321\rangle-|132\rangle-|213\rangle\right\}$$ and would like to trace out two particles to get the reduced density matrix for a single particle. What is the procedure to do this is 2nd quantization?

Any help is much appreciated. Thanks.