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My guess it should look something like this:

$ c_\sigma = (\left|0\right>\left<\uparrow\right|+\left|\downarrow\right>\left<\downarrow\uparrow\right|)\delta_{\sigma,\uparrow}+(\left|0\right>\left<\downarrow\right|+\left|\uparrow\right>\left<\downarrow\uparrow\right|)\delta_{\sigma,\downarrow}$

where $\delta$ is a Kronecker delta and states $\left|0\right>,\left|\downarrow\right>,\left|\uparrow\right>,\left|\downarrow\uparrow\right>$ are orthonormal.

Now it behaves like annihilation operator

$c_{\downarrow}\left|0\right>=\left|0\right>, c_{\uparrow}\left|0\right>=\left|0\right>$

$c_{\downarrow}\left|\downarrow\right>=\left|0\right>, c_{\uparrow}\left|\downarrow\right>=\left|0\right>$

$c_{\downarrow}\left|\downarrow\uparrow\right>=\left|\uparrow\right>, c_{\uparrow}\left|\downarrow\uparrow\right>=\left|\downarrow\right>$

but anticommutator for example $[c_{\uparrow},c_{\downarrow}]_+$ isn't zero.

Is it possible to define it like that (in terms of basis states)?

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  • $\begingroup$ You will also encounter another problem with $[c_\downarrow,c_\downarrow]_+$. If you do, consider acting on $|\downarrow\rangle$ with this anticommutator. In addition to sign issue pointed out by Qmechanic, you have another inaccuracy. $\endgroup$ Apr 29, 2013 at 0:05

1 Answer 1

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Main point: You should allow the possibility of sign factors appearing into the definition of the Hilbert space representation of fermionic operators, cf. fermionic Fock space.

In more detail, consider the CAR algebra

$$\begin{align} \{c_{\sigma}, c_{\tau}\}_+~=~&0, \qquad \{c_{\sigma}, c^{\dagger}_{\tau}\}_+~=~\delta_{\sigma,\tau} {\bf 1}, \cr \{c^{\dagger}_{\sigma}, c^{\dagger}_{\tau}\}_+~=~&0, \qquad \sigma,\tau\in \{\uparrow,\downarrow\}. \end{align}\tag{1}$$

Next define

$$ \begin{align}c_{\sigma}\left|0\right>~:=~&0, \qquad \left|\sigma\right>~:=~ c^{\dagger}_{\sigma}\left|0\right>, \cr \left|\sigma\tau\right>~:=~& c^{\dagger}_{\sigma}\left|\tau\right>, \qquad \sigma,\tau\in \{\uparrow,\downarrow\}.\end{align}\tag{2}$$

Note that these definitions imply that

$$ \left|\sigma\tau\right> ~=~ -\left|\tau\sigma\right>, \qquad \sigma,\tau\in \{\uparrow,\downarrow\}.\tag{3}$$

In particular

$$ \left|\sigma\sigma\right> ~=~ 0, \qquad \sigma\in \{\uparrow,\downarrow\}. \tag{4}$$

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  • $\begingroup$ By the way, in a comment to swish previous question I mentioned that creation/annihilation operators for different fields may commute, in principle. This corresponds to taking the Fock space to be the ordinary tensor product, while it looks like the natural way is to take it as a graded tensor product. I wonder, at what point the bosonic tensor product turns ugly? I think that for spin projections it is near the rotational invariance, but what about two interacting but different fields? $\endgroup$ Apr 29, 2013 at 0:02
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    $\begingroup$ So I should change my states to |n>*|spin> in order to make proper operators? $\endgroup$
    – swish
    Apr 29, 2013 at 10:36
  • $\begingroup$ I updated the answer. $\endgroup$
    – Qmechanic
    May 1, 2013 at 19:04

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