I have developed a Mathematica program that computes the traces of
gamma-matrices, simplifies sums of delta-functions (Einstein summation
convention), and evaluates certain types of Feynman integrals, namely the
ones in Itzykson and Zuber "QFT", pp. 419-422. I took much trouble to
simplify the intermediate expressions in a useful, controlled and general
way, this is roughly 80% of the code.
Though the code has been debugged, I do not get the results in I-Z;
perhaps they are incorrect. An indication of this is Eq. (8-119) in that
book. It is not manifestly symmetric in rho and sigma, and in fact the trace
of the gamma matrices is not symmetric in rho and sigma at all. Theory says
that \Gamma in (8.115) is symmetric in rho and sigma, but that's only after
the integration is performed. I do not get their result whether or not I
symmetrize (8-119). The situation in I-Z is delicate anyway, as analysis
shows that one of the integrals on p. 421 is divergent, so there is some
question of what an "exact" calculation should give.
I'd be interested in hearing from others who have attempted the
calculations in I-Z, and suggest this as a rather stringent test case.
(Incidentally, it is outrageous of I-Z to have included such a brutish
calculation in what is supposed to be a textbook, without even indicating
that the calculations were computer-aided.)
I have two files, a run file and a code file. Five or six pages of
analysis are also available (power-counting to keep track of divergences,
organization of the calculations).
PS. It would be good to have software to automatically evaluate integrals
like (5.23) in Abrikosov et al. "Methods of QFT in Stat. Phys." See paper by
Belyakov (Sov. Phys. JETP, Vol. 13, No. 4) . The problem is to automate the
handling of the domains of integration; perhaps this is too difficult though
it can be done by hand in this special case. Has anyone obtained Belyakov's
results using Mathematica?
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