Graded Exterior Differential Calculus @ Mathematica

Description

This package is a modification of EDC (Exterior Differential Calculus) to enable Mathematica to carry out calculations with expressions containing Grassmann variables and Grassmann (graded) differential forms. It is intended for Superalgebra calculations.

The user is assumed to be familiar with EDC. Unlike EDC, where any symbols can be used to denote differential forms or matrices, in Graded Exterior Differential Calculus (superEDC) a specific notation for Grassmann variables must be used: all Grassmann variables must have Head "gv" (e.g., gv[1], gv[k], gv[{f[1]}, x, y]). The Wedge operator is used to denote products of Grassmann variables, as well as products of forms or matrices. The exterior derivative operator d can now handle expressions containing Grassmann variables / forms. Two sets of new functions are introduced:

those whose names begin with "wedge" (wedgePower, wedgeDet, wedgeInverse, wedgeD) which generalize the corresponding Mathematica functions (Power, Det, Inverse, D) to allow for the presence of Grassmann variables in their arguments. As in EDC, Wedge itself generalizes Times and Dot.
those whose names begin with "super" (superWedge, superTrace, superTranspose, superInverse) which are specific to (homogeneous, square) supermatrices. The definitions of these functions agree with those given in the Wikipedia Supermatrix article: http://en.wikipedia.org/wiki/Supermatrix.

The notebook superEDCmanual.nb contains the definitions of the new functions and examples of their use, while superEDC.m is the initialization code that must be loaded first. The notebook superEDCpalette.nb is a palette for entering the new functions defined in superEDC. All files are included in the compressed archives below.

The package is compatible with all Mathematica versions 3.0 or later.

Version History

Version 1.1.7 is the initial release.

Version 1.2.1 enables wedgeD to differentiate w.r.t. Grassmann variables also.

Version 1.2.4 allows the user to define symbolic Grassmann variables with non-zero differentiable-form degree.

Version 1.3.6 corrects an error in the previous versions that did not allow zeros in the odd sectors of a supermatrix, introduces three new fuctions and modifies superTranspose so it can act on matrices representing tensors with different index positions. The superEDCmanual is rewritten and contains, as an example, a non-trivial calculation.

Version 1.3.8 allows negative / fractional powers in wedgePower, introduces definitions for body / soul and includes a palette for entering all functions defined in superEDC. Also, the material in the manual has been reorganized in a, hopefully, more readable form.

Version 1.3.9 corrects the trace error (in ordinary EDC) and eliminates a false warning, printed under Mathematica versions >5.

Version 1.4.3 improves the basic routine for recognizing even/odd variables. In some cases, the old routine gave wrong results. Also, Wedge in superEDC now allows multiplication of a symbolic with an explicit matrix. Finally, a new function (wedgeAdjoint) is defined.

Version 1.4.5 corrects some minor bugs and adds four new auxiliary functions introduced in EDC version 3.8.0.

Version 1.4.8 corrects some minor bugs.

The present version 1.5.0 adds new warning messages and the function FormCoef, which was introduced in EDC version 3.8.7.

Download (Version 1.5.0 - May 2013)

Graded Exterior Differential Calculus: superEDC150.zip, superEDC150.sit.