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We know Grassmann numbers are complex numbers. Hence Grassmann integrals are also complex. How can we convert a Grassmann integral into real one, ie is there any transformation of converting complex Grassmann numbers to real grassmann numbers?

Qmechanic
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Jasmine
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  • Welcome to Physics Stack Exchange. Please note that clarity helps people understand your question. Part of good clarity is proper English punctuation, such as using a space between sentences. I edited this question to fix the punctuation. Please do pay attention to these important details in future posts. – DanielSank Aug 07 '15 at 00:56
  • In what sense are Grassman numbers complex numbers? I'm pretty sure these are not the same thing. I think Grassman numbers are more like the Fermionic $a^\dagger$ operator. Perhaps you can convert path integrals involving Grassman numbers into expressions involving complex numbers. – DanielSank Aug 07 '15 at 00:58
  • This is way above me, but on the off chance it helps, (and you probably are aware) Wikipedia calls them c-numbers, which was very confusing to me till I read that this was Dirac's notation, I would have automatically taken c-numbers to mean complex numbers until I read that Dirac meant classical numbers. Wikipedia is not well written in this section, imo. –  Aug 07 '15 at 01:23
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    Have you looked at this? https://en.wikipedia.org/wiki/Grassmann_number

    Grassmann numbers are more like matrices than actual numbers.

     –  Aug 07 '15 at 01:54
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    The question is unclear: Grassmann number aren't real number, Grassmann integrals aren't actually integrals (complex or real), so what are you actually trying to ask? – ACuriousMind Aug 07 '15 at 13:06

1 Answers1

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Comments to the question (v3):

  1. A Grassmann-odd number is not a complex number. It is a complex supernumber $z=x+iy$, which can be decomposed in real and imaginary supernumbers, cf. e.g. this and this Phys.SE posts.

  2. The Berezin integral $\int\! d\theta~f(\theta)$ over supernumbers is an ordinary complex number $c=a+ib\in\mathbb{C}$, which can be decomposed in real and imaginary numbers.

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