Advances in Mathematical Physics
Volume 2009 (2009), Article ID 987524, 21 pages
Research Article

The Meaning of Time and Covariant Superderivatives in Supermechanics

Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de San Luis Potosí, 78290, Mexico

Received 23 January 2009; Accepted 21 April 2009

Academic Editor: José F. Cariñena

Copyright © 2009 Gil Salgado and José A. Vallejo-Rodríguez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 ) must be replaced by the simplest linear supermanifold 1|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.