Integration of odd variables III

We will proceed to describe how changes of variables effects the integration measure for odd variables. We will do this via a simple example rather than in full generality.

Integration measure with two odd variables
Let us consider the integration with respect to two odd variables, \(\{ \theta, \overline{\theta} \}\). Let us consider a change in variables of the form

\(\theta^{\prime} = a \theta + b \overline{\theta}\),
\( \overline{\theta}^{\prime} = c \theta + d \overline{\theta}\),

where a,b,c,d are real numbers (or complex if you wish).

Now, one of the basic properties of integration is that it should not depend on how you parametrise things. In other worlds we get the same result whatever variables we chose to employ. For the example at hand we have

\( \int D(\overline{\theta}^{\prime}, \theta^{\prime}) \theta^{\prime} \overline{\theta}^{\prime} = \int D(\overline{\theta}, \theta) \theta \overline{\theta}\).

Thus, we have

\(\int D(\overline{\theta}^{\prime}, \theta^{\prime}) (ad-bc)\theta \overline{\theta} = \int D(\overline{\theta}, \theta) \theta \overline{\theta}\).

In order to be invariant we must have

\(\int D(\overline{\theta}^{\prime}, \theta^{\prime})= \frac{1}{(ad-bc) }D(\overline{\theta}, \theta) \).

Note that the factor (ad-bc) is the determinant of a 2×2 matrix. However, note that we divide by this factor and not multiply in the above law. This is a general feature of integration with respect to odd variables, one divides by the determinant of the transformation matrix rather than multiply. This generalises to non-linear transformations that mix even and odd coordinates on a supermanifold. This is the famous Berezinian. A detailed discussion is outside the remit of this introduction.

Furthermore, note that the transformation law for the measure is really the same as the transformation law for derivatives. Thus, the Berezin measure is really a mixture of algebraic and differential ideas.

What next?
I think this should end our discussion of the elementary properties of analysis with odd variables. I hope it has been useful to someone!