The Fedosov ∗-product in Mathematica

Abstract

The computer program ‘Fecom.nb’ implementing the Fedosov ∗-product in Darboux coordinates is presented. It has been written in Mathematica 6.0 but it can be easily modified to be run in some earlier version of Mathematica. To optimize computations elements of the Weyl algebra are treated as polynomials. Several procedures which order the terms are included. New version program summary Program title: Fecom.nb Catalogue identifier: AEBU_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEBU_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 8107 No. of bytes in distributed program, including test data, etc.: 42 175 Distribution format: tar.gz Programming language: Mathematica v.6.0 Computer: All Operating system: All RAM: Sufficient for installation of Mathematica Classification: 17.16 Catalogue identifier of previous version: AEBU_v1_0 Journal reference of previous version: Comput. Phys. Comm. 179 (2008) 924 Does the new version supersede the previous version?: Yes Nature of problem: Computing the ∗-product of the Weyl type in the Fedosov formalism in a Darboux chart. Solution method: Inputting the dimension of the phase space, coefficients of the symplectic connection, the range of approximation and the functions to be multiplied; computing the Abelian connection and the flat sections of the Weyl bundle representing the multiplied functions; calculating the projection of the ○-product of these flat sections on the phase space. Reasons for new version: Optimization and including the trivial cases – deformations of the 0th and the 1st order. Summary of revisions: In case we calculate the ∗-product up to the odd power h k , it is necessary to know the Abelian connection and the flat sections up to the degree 2 k − 1 . But for the even power h k , it is sufficient to know the Abelian connection and the flat sections up to the degree 2 k − 2 . In the new version of the program we use this observation. Now the running time of the program for even powers h k is shorter than in the previous version. The 0th and the 1st order of the deformation are trivial – the pointwise product of functions and the Poisson bracket of them respectively. But to make the program complete we added the possibility of calculating also these trivial situations. The new version of the program works then also for the maximal power of h equal 0 and 1. Running time: The test run, provided with the distribution, took approximately 2 minutes to run using Mathematica 7.0 on a Windows XP machine.