Ambiguity of Time Derivative of Superfunctions

Question

I think that there is an ambiguity for defining the time derivative of a superfunction on the phase space of pseudo-classical mechanics of Grassmann numbers.

Let $\xi$ be a Grassmann odd number. Its canonical conjugate variable is $p$. Let $f(p,\xi)$ be a superfunction defined on the phase space $(p,\xi)$ of "classical fermions". The time derivative can be defined in two ways:

1. $\dot{f}=\dot{\xi}\frac{\overrightarrow{\partial}}{\partial\xi}f+\dot{p}\frac{\overrightarrow{\partial}}{\partial p}f$,

2. $\dot{f}=f\frac{\overleftarrow{\partial}}{\partial\xi}\dot{\xi}+f\frac{\overleftarrow{\partial}}{\partial p}\dot{p}$.

Which definition should I use?

Answer 1

You can use either differentiation from the left or from the right. Their definitions are interrelated $$ \frac{\stackrel{\rightarrow}{\partial}_L}{\partial z}f~=~(-1)^{(|f|+1)|z|}~f\frac{\stackrel{\leftarrow}{\partial}_R}{\partial z},$$ so that they produce the same time derivative. See also e.g. my Phys.SE answers here & here.