The Grassmann analytic continuation principle says that a real function f(x,ξ) that depends on n real variables x=(x1,…,xn) and n nilpotent Grassmann numbers ξ=(ξ1,…,ξn) has the following Taylor expansion containing only a finit (2n?) number of terms
f(x+ξ)=∑αξαα!∂α∂xαf(x)
where the ξj∈Gn which is the fermionic part of the supermanifold at hand are given by
ξj=∑|ϵ|>0ajϵθϵ
The θi are the generators of Gn, α=(α1,…,αn)∈Nn, and
ξα=ξα11…ξαnn.
How is this related to expanding a superfield in Grassmann coordinates?
For example, a a scalar superfiled F(x,θ1,θ2) that depends on the ordinary
spacetime coordinates as well as on two Grassmann coordinates can be expanded as
F(x,θ1,θ2)=A(x)+B(x)θ1+C(x)θ2+D(x)θ1θ2
Comparing this expansion to the Grassmann analytic continuation principle above I would think that in this specific case we have n=2, α=(α1,α2) and ξ=ξα11ξα22 as there are two Grassmann coordinates in addition to conventional spacetiem.
The first thing I dont understand is why in the Grassmann analytic continuation principle the function is not expanded directly in the θi but the ξ are used instead which makes it hard for me to see what is going on.
Otherwise, I would have guessed that from setting α1=0 and α2=0 the coefficient A(x) would correspond to f(x), from setting α=(2,1) one obtains B(x)=∂f∂θ1, from setting α=(1,2) C(x)=∂f∂θ2 and from α=(1,1) one has D(x)=∂2f∂θ1∂θ2