Integration of odd variables I

Before we consider odd variables, let us describe how to algebraically define integration of functions over the circle.

Functions on the circle
Recall the Fourier expansion. It is well known that any continuous function on the circle is of the form

\(f(x) = \frac{a_{0}}{2} + \sum_{n=1}^{\infty}\left( a_{n} \cos(nx) + b_{n}\sin(nx) \right) \),

with the a’s and b’s being constants, i.e. independent of the variable x.

The fundamental theorem of calculus
The fundamental theorem of calculus states that

\(\int_{S^{1}} dx \: \frac{\partial f(x)}{\partial x } = 0 \),

as functions on the circle are periodic.

Integration of functions
It turns out that integration of functions over the circle can be defined algebraically up to a choice in measure. To see this observe

\(\int_{S^{1}} dx f(x) = \int_{S^{1}} dx \frac{a_{0}}{2} + \int_{S^{1}} dx \sum_{n=1}^{\infty}\left( a_{n} \cos(nx) + b_{n}\sin(nx) \right)\)

Then we can write

\(\int_{S^{1}} dx f(x) = \frac{a_{0}}{2} \int_{S^{1}} dx + \int_{S_{1}} dx \frac{\partial }{\partial x} \sum_{n=1}^{\infty} \left ( \frac{a_{n}}{n}\sin(nx) + \frac{- b_{n}}{n} \cos(nx) \right)\)

to get via the fundamental theorem of calculus

\(\int_{S^{1}} dx f(x) = \frac{a_{0}}{2} \int_{S^{1}} dx\).

So we have just about defined integration completely algebraically from the fundamental theorem of calculus. All we have to do is specify the normalisation

\(\int_{S^{1}} dx \).

The standard choice would be

\(\int_{S^{1}} dx = 2 \pi\),

to get back to our usual notion of integration of periodic functions. Though it would be quite consistent to consider some other normalisation, say to unity.

Anyway, up to a normalisation the integration of functions over the circle selects the “constant term” of the corresponding Fourier expansion.

What next?
So, the above construction demonstrates that integration of functions over a domain without boundaries can be defined algebraically, up to a normalisation. This served as the basis for Berezin who defined the notion of integration of odd variables.

Recall that odd variables have no topology and no boundaries. The integration with respect to such variables cannot be in the sense of Riemann. However, thinking of functions of odd variables in analogy to periodic functions integration can be defined algebraically. We will describe this next time.