Fermion annihilation operators from position and momentum

[haael]

Is it possible to express fermion annihilation operator as a function of position and momentum?

I've seen on Wikipedia the formula for boson annihilation operator:
[tex]
\begin{matrix} a &=& \sqrt{m\omega \over 2\hbar} \left(x + {i \over m \omega} p \right) \\ a^{\dagger} &=& \sqrt{m \omega \over 2\hbar} \left( x - {i \over m \omega} p \right) \end{matrix}
[/tex]

But what about fermions? Is it possible to get anticommutation relations from canonical relations alone, or is it necessary to postulate something else?

 

[tom.stoer]

But what about fermions? Is it possible to get anticommutation relations from canonical relations alone, or is it necessary to postulate something else?

You have to use Grassmann numbers or something like that. This can be seen by inspection of the (fermionic) anti-commutators

[tex]\{b, b\} = 2b^2 = 0[/tex]
[tex]\{b^\dagger, b^\dagger\} = 2{b^\dagger}^2 = 0[/tex]

which cannot be derived from commuting objects.