- #1
Stalafin
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Homework Statement
Show that [tex]\int d\theta e^{\theta(\xi-\eta)}=\delta(\xi-\eta)[/tex],
where all of the above variable are Grassmann numbers. All this is in the holomorphic representation, where for some generic function:
[tex]f(\theta)=f_0 + f_1\theta[/tex]
Homework Equations
How do I arrive at that bloody delta? As explained below, I see the analogy where theta all by itself acts like a delta function. But from there on...?
The Attempt at a Solution
Obviously, by expanding to first order according to the Grassmann algebra (all orders above the first vanish):
[tex]=\int d\theta 1+\theta(\xi-\eta) = \xi-\eta[/tex]
What I do get is that the theta all by itself acts like a delta-function around 0 (taking the generic function above):
[tex]\int d\theta \theta f(\theta) = \int d\theta (f_1 + \theta f_2) = f_1 = f(0)[/tex]
This we could have also written as:
[tex]\int d\theta f(\theta)\delta(\theta - 0) = f(0)[/tex]
Where is the analogy?