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What is this Hilbert space? Is it the complex Hilbert space of wavefunctionals spanned by using the spinor-field configurations as the basis vectors?

I know that the wavefunctional space carries a representation of the commutator quantisation $$[\phi (x), \pi (y) ]=i\delta (x-y) .$$ But does the wavefunctional space carry a representation of the anti-commutator quantisation?

If the wavefunctional space isn't the right one, then we could also define this Hilbert space using a representation of $$\{ a_r(p_1), a_s^{\dagger} (p_2) \}=\delta ^r_s\delta (p_1-p_2).$$ The representation of this is well-known : four basis vectors $|0\rangle _r$ and $|1\rangle _r$ at each $p$, with $r=1, 2$

Is this all that's known about the Hilbert space of the quantised spinor field? Is there no analogue of the wavefunctional space here?

Also, are we working with some-sort of generalised Hilbert space which allows for Grassman number coefficients of the basis vectors?

If yes, why are we doing this? In elementary linear algbera, it is proved that complex numbers are sufficient to solve the eigenvalue problem of any operator. Then why are we generalising the notion of a complex Hilbert space? Is this not against a postulate of quantum mechanics?

Qmechanic
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Ryder Rude
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  • Possible duplicates: https://physics.stackexchange.com/q/612252/2451 , https://physics.stackexchange.com/q/269699/2451 , https://physics.stackexchange.com/q/714474/2451 – Qmechanic Dec 13 '22 at 08:38
  • @Qmechanic I think I can write an answer. Please remove the duplicate. The other posts don't address the first half of this post. – Ryder Rude Dec 13 '22 at 10:34
  • @Qmechanic This is embarassing. I realise my answer to the first half isn't correct... – Ryder Rude Dec 13 '22 at 14:16
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    There is no "the Hilbert space". The point of Haag's theorem is that there are infinitely many inequivalent representations of the canonical (anti-)commutation relations, so these relations do not suffice to fix the Hilbert space. – ACuriousMind Dec 14 '22 at 11:20
  • @ACuriousMind Is the wavefunctional space one of the possible spaces which can carry a representation of this? The path integral makes it tempting to think that there's a wavefunctional basis. – Ryder Rude Dec 14 '22 at 11:22
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    see https://physics.stackexchange.com/a/721726/50583 for everything I know about the formalization of the wave-functional approach – ACuriousMind Dec 14 '22 at 11:25
  • @ACuriousMind Do you think that the Hilbert space representations become a huge burden in QFT? Can doing everything directly using the algebra save us? I wish we could just use the algebra. – Ryder Rude Dec 14 '22 at 11:29

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