The Meaning of Time and Covariant Superderivatives in Supermechanics

We present a review of the basics of supermanifold theory (in the sense of Berezin-Kostant-Leites-Manin) from a physicist's point of view. By considering a detailed example of what does it mean the expression "to integrate an ordinary superdifferential equation" we show how the appearance of anticommuting parameters playing the role of time is very natural in this context. We conclude that in dynamical theories formulated whithin the category of supermanifolds, the space that classically parametrizes time (the real line R) must be replaced by the simplest linear supermanifold R-1 vertical bar 1. This supermanifold admits several different Lie supergroup structures, and we analyze from a group-theoretic point of view what is the meaning of the usual covariant superderivatives, relating them to a change in the underlying group law. This result is extended to the case of N-supersymmetry. Copyright (C) 2009 G. Salgado and J. A. Vallejo-Rodriguez.