On Wed, Sep 11, 2013 at 3:55 PM, F. B. <franz.bona...@gmail.com> wrote: > OK, now I have come to the generalized case. As of now we have two > algorithms in SymPy's master: > > Kahane's algorithm > gamma trace algorithm for simplified expressions (simplified = no Lorentz > contractions)
Excellent! > > There are some limitations, namely: > > in dimensional regularization, self-contracted Lorentz metric equals 4 - > epsilon instead of 4, where epsilon is the regularization parameter. > Kahane's algorithm works in four dimensions only. > gamma matrices can be represented in 2^n dimensions, same argument as before > applies. > if in a Feynman diagram there are distinct spinor lines, and Lorentz > contractions among gamma matrices of different spinor lines, Kahane's > algorithm as it is implemented now does not work. > > The Kennedy-Cvitanovic algorithm would solve these issues > > https://www.google.it/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CDEQFjAA&url=http%3A%2F%2Fwww.cns.gatech.edu%2Fgrouptheory%2Fversion9.0%2FGroupTheory.pdf&ei=Z-AwUtXEDqbE7Abl-4GIBg&usg=AFQjCNGJ8Xm-CdLNblQnPmYLj22tzAiJMA&sig2=22O-vRUWwNVYA-ii4JEuMA&bvm=bv.52109249,d.ZGU > > On chapter 11 there's the explanation. > > By the way, the birdtrack notation for tensors could be nice to implement > for general purposes in the tensor module (maybe a plotting facility too). +1 > > I'm going to write the connection topology of index contractions as a system > of Python dictionaries, after which it is necessary to: > > be able to apply the recoupling relation (which splits a spinor line in case > of an internal Lorentz lines). > see if it is worth to write a trace algorithm on partly symmetrized > expressions of gamma matrices. > > Point 2 is good for performance, but the existing trace algorithm should be > already enough if performance is not an issue. > > Do you have any suggestion about how to represent the birdtrack notation in > Python? Is it just for visualization or is it actually needed in the algorithm? > Any alternatives to a set of dictionary? I already wrote a draft > which stores contraction topology in two dictionaries: for Lorentz and > Spinor indices. That seems fine. If we figure out a better way, we can change it later. Ondrej -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to sympy+unsubscr...@googlegroups.com. To post to this group, send email to sympy@googlegroups.com. Visit this group at http://groups.google.com/group/sympy. For more options, visit https://groups.google.com/groups/opt_out.
