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  Strange Grassmann double integration

+ 5 like - 0 dislike
1003 views

I can unterstand why because the integration over Grassman variables has to be translational invariant too, one has

dθ=0

and

dθθ=1

but I dont see where the rule for this double integration

d2θˉθθ=2i

comes from.

So can somebody explain to me how this is motivated and/or derived?

asked Apr 14, 2013 in Theoretical Physics by Dilaton (6,240 points) [ revision history ]
Is the last integral supposed to read something like d2θˉθθ?

This post imported from StackExchange Physics at 2014-03-09 16:25 (UCT), posted by SE-user Olof
@Olof yes, I just corrected the typo thanks.

This post imported from StackExchange Physics at 2014-03-09 16:25 (UCT), posted by SE-user Dilaton

1 Answer

+ 6 like - 0 dislike

As with anything that has to do with supersymmetry the details will be dependent on your exact conventions, but we can obtain the result as follows:

Assume we have two Grassman variables θ1 and θ2. By applying your first formula twice we find dθ1dθ2θ2θ1=1

Now combine these into θ=θ1+iθ2andˉθ=θ1iθ2.
We then have ˉθθ=2iθ2θ1
and hence dθ1dθ2ˉθθ=2i
which is exactly your second integral, if we identify the measure d2θ=dθ1dθ2.

This post imported from StackExchange Physics at 2014-03-09 16:25 (UCT), posted by SE-user Olof
answered Apr 14, 2013 by Olof (210 points) [ no revision ]
Ah thanks Olof, that is exactly what I needed.

This post imported from StackExchange Physics at 2014-03-09 16:25 (UCT), posted by SE-user Dilaton

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