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.

Read more about this topic:  Berezin Integral

Famous quotes containing the words change of, change, odd and/or variables:

    “We are such docile creatures, normally, that it takes a virus to jolt us out of life’s routine. A couple of days in a fever bed are, in a sense, health-giving; the change in body temperature, the change in pulse rate, and the change of scene have a restorative effect on the system equal to the hell they raise.”
    —E.B. (Elwyn Brooks)

    “I change my opinions often, but not my way of thinking.”
    —Mason Cooley (b. 1927)

    “I don’t think he has any idea he’s a dog, not really. Of course, he thinks he has a rather odd figure for a man.”
    —Dodie Smith, and Lewis Allen. Pamela Fitzgerald (Ruth Hussey)

    “The variables are surprisingly few.... One can whip or be whipped; one can eat excrement or quaff urine; mouth and private part can be meet in this or that commerce. After which there is the gray of morning and the sour knowledge that things have remained fairly generally the same since man first met goat and woman.”
    —George Steiner (b. 1929)