Differential calculus of odd variables.

Here we will define the notion of differentiation with respect to an odd variable and examine some basic properties.

Differentiation with respect to an odd variable is completely and uniquely defined via the following rules:

  1. \(\frac{ \partial \theta^{\beta} }{\partial \theta^{\alpha}} = \delta_{\alpha}^{\beta} \).
  2. Linearity:
    \(\frac{\partial}{\partial \theta }(a f(\theta)) = a \frac{\partial}{\partial \theta } f(\theta)\).
    \(\frac{\partial}{\partial \theta }( f(\theta) + g(\theta)) = \frac{\partial}{\partial \theta }f(\theta) + \frac{\partial}{\partial \theta } g(\theta)\).
  3. Leibniz rule:
    \(\frac{\partial}{\partial \theta }( f(\theta)g(\theta)) = \frac{\partial f(\theta)}{\partial \theta } + (-1)^{\widetilde{f}} f \frac{\partial g(\theta)}{\partial \theta } \).

The operator \(\frac{\partial }{\partial \theta }\) is odd, that is it changes the parity of the function it acts on. This must be taken care of when applying Leibniz’s rule.

Elementary properties
It is easy to see that

\(\frac{\partial}{\partial \theta^{\alpha}}\frac{\partial}{\partial \theta^{\beta}}+ \frac{\partial}{\partial \theta^{\beta}}\frac{\partial}{\partial \theta^{\alpha}}=0\),

in particular

\(\left( \frac{\partial}{\partial \theta} \right)^{2}=0\).

\(\frac{\partial}{\partial \theta} (a + \theta b+ \overline{\theta}c + \theta \overline{\theta} d ) = b + \overline{\theta}d\).

\(\frac{\partial}{\partial \overline{\theta}} (a + \theta b+ \overline{\theta}c + \theta \overline{\theta} d ) = c- \theta d\).

Changes of variables
Under changes of variable of the form \(\theta \rightarrow \theta^{\prime}\) the derivative transforms as standard

\(\frac{\partial}{\partial \theta^{\prime}} = \frac{\partial\theta}{\partial \theta^{\prime}} \frac{\partial}{ \partial \theta}\).

We will have a lot more to say about changes of variables (coordinates) another time.

What next?
We now know how to define and use the derivative with respect to an odd variable. Note that this was done algebraically with no mention of limits. As the functions in odd variables are polynomial the derivative was simple to define.

Next we will take a look at integration with respect to an odd variable. We cannot think in terms of boundaries, limits or anything resembling the Riemann or Lebesgue notions of integration. Everything will need to be done algebraically.

This will lead us to the Berezin integral which has the strange property that integration and differentiation with respect to an odd variable are the same.