Qns about Fock Space and 2nd Quantizatio

  • Thread starter bobbytkc
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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 [tex]|n_{\mu}\rangle[/tex] and that the [tex]n_{\mu}[/tex] are quantum numbers representing the number of particles in the [tex]\mu[/tex] state. Is this in distinction with the Hilbert space? In my impression, the Hilbert space is the vector space representing the states of an [tex]n[/tex] 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 [tex]a_{\mu}^{\dagger}|0\rangle[/tex] and a 2 particle system is given by [tex]a_{\mu}^{\dagger} a_{\nu}^{\dagger} |0\rangle[/tex]. We know in first quantization however, that for indistinguishable particles in 2 orthogonal states [tex]|\mu\rangle[/tex] and [tex]|\nu\rangle[/tex] the 2 particle state is is simply the singlet state given by [tex]\frac{1}{\sqrt{2}}\left\{|\mu\rangle\otimes|\nu\rangle-|\nu\rangl\otimes|\mu\rangle\right\}[/tex]. Am I right in understanding that writing [tex]a_{\mu}^{\dagger} a_{\nu}^{\dagger} |0\rangle[/tex] in the fock space formalism is equivalent to writing the singlet state [tex]\frac{1}{\sqrt{2}}\left\{|\mu\rangle\otimes|\nu\rangle-|\nu\rangl\otimes|\mu\rangle\right\}[/tex]?


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 [tex]\frac{1}{\sqrt{6}}\left\{|123\rangle+|312\rangle+|231\rangle-|321\rangle-|132\rangle-|213\rangle\right\}[/tex] 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.
 

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