Dear Vladimir, On Mon, 14 Aug 2006, Vladimir Kisil wrote:
CD> the metric with up-indices is the inverse of the metric with CD> down-indices, we arive at the identity CD> e~mu e~alpha e.mu = (2 - dim) * e~alpha, CD> where dim is the number of row/columns of the metric.
I saw this rule in the diracgamma contraction but have doubts that it will be simple as that for generic Clifford units with metric diag_matrix(1,-1,0) for example.
Let us take that to be the metric for down-indices for definiteness. Now we are going to run into trouble if we ask what the result is if it is given up-indices. That should be the inverse... not such a good idea. Furthermore, from e.mu = g.mu.nu e~nu it follows that e.3 = 0, no matter what the e~mu are (it is actually rather unclear what e~3 would/should be in this case). Maybe it just wasn't your intention to take varidxes to mean the same as is common among physicists. This is why I also mentioned the possibility that clifford objects with a matrix as metric should take normal idxes instead of varidxes. Or even the possibility that matrices should only take normal indices and not varidxes. Best wishes, Chris