Berezin Integral - Change of Even and Odd Variables

Change of Even and Odd Variables

Let a coordinate transformation be given by, where are even and are odd polynomials of depending on even variables The Jacobian matrix of this transformation has the block form:

$\mathrm{J}=\frac{\partial\left( x,\theta\right) }{\partial\left(y,\xi\right) }=\left(\begin{array}{cc} A & B\\ C & D\end{array}\right) ,$

where each even derivative commutes with all elements of the algebra ; the odd derivatives commute with even elements and anticommute with odd elements. The entries of the diagonal blocks and are even and the entries of the offdiagonal blocks are odd functions, where mean right derivatives. The Berezinian (or the superdeterminant) of the matrix is the even function

defined when the function is invertible in Suppose that the real functions define a smooth invertible map of open sets in and the linear part of the map is invertible for each The general transformation law for the Berezin integral reads

$\int_{\Lambda^{m\mid n}}f\left( x,\theta\right) \mathrm{d}\theta\mathrm{d}x=\int_{\Lambda^{m\mid n}}f\left( x\left( y,\xi\right) ,\theta\left( y,\xi\right) \right) \varepsilon\mathrm{Ber~J~d}\xi\mathrm{d}y$

where is the sign of the orientation of the map The superposition is defined in the obvious way, if the functions do not depend on In the general case, we write where are even nilpotent elements of and set

where the Taylor series is finite.

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