Study of algorithm of Dirac algebra

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We have developed GRACE system, an automatic computation system in particle physics. In this system, the expressions of Feynman amplitudes using Dirac algebra are eliminated in the symbolic manipulation way, and the numerical result of these expressions will be computed.In the symbolic manipulation package (REDUCE), the straightforward eliminations don't work due to lack of memory. After the study of algorithm of Dirac algebra, many complicated and sophisticated techniques have been introduced, so that the calculations of tree level and one-loop lever becomes reliable in the framework of standard model and super-symmetric model.Thus it is shown that these calculation will be successfully done. This fact means that 1-loop calculations will be automatically possible in principle. In order to check the symbolic calculation, two methods are studied and tested in the following way :・ Comparison with another symbolic manipulation had been done. In general,the expressions produced by REDUCE are quite different from those of another system (FORM). Then numerical comparison had been successfully done.・ The introduction of additional gauge parameters is very useful. The in-dependency of gauge parameter is found.Both methods are quite important to confirm the correctness of algorithm of symbolic manipulation, because the algorithm is rather complicated.Concerning the physical application, several processes in standard model and super-symmetric model are calculated.We should try the physical processes such a e+ e- → 4 fermion processes, which are important for the LEP-II experiment at CERN (European Organization for Nuclear Research).As the next step, the study of algorithm for helicity amplitude and the symbolical optimization should be considered to reduce the computation time for the symbolic step and numerical step.