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Additional resources for Field Theory, Quantization and Statistical Physics: In Memory of Bernard Jouvet

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3) and the Ward identity implies 0,07 0 oqJ t(X)qJ 0,07 + ab~(X)b = o. 6) . B o is non-degenerate implies that o of ~ j t(X)F = ar;r;(X)o. 7) Similarly, the other coefficients of 2? are covariant functions of 6. 2) is then a consequence of these covariance properties. 22 C. BECCHI ET AL. ApPENDIX B This appendix is devoted to the reduction ofthe Ward identity to canonical form. l. l) are fully reducible. This is automatic for X E[I', the semi-simple part ofCfJ. f! is Hermitian non-degenerate, which insures that X -+ t(X) is fully reducible.

4) = dx c5w«(x) is the infinitesimal generator of the global transformation. fapK:INCEN a the stability of N: implies iI TaK;;INC - f:pK~INC a K: K: iI = o. 8) Thus, one may choose to transform according to the regular representation. We shall now expand in increasing powers of {a:}. Let K~ be the term of K~ independent of {a:}. 9) where L/l is linear in {aa}. 10) where Q:(x) is now quadratic in {a:}, provided that 1 K~;;(x,y) 1 == c5 all(x)K;;(y) - I c5a/l(y)K~(x) = O. 11) 28 C. BECCHI ET AL. 2), we get 1 oXoY /J v K/JV(x ap' y) = 0.

Therefore for this class of integrals only technical difficulties arise for their evaluation. d. and factorization of polynomials and arithmetic over algebraic number fields. In all these domains significant progress has been made making it possible set up an efficient integrator for rational functions. For transcendental integrands of the logarithmic and exponential types the theoretical difficulty has been solved by Risch who has extended the theory of Liouville to this case. l8). The algorithm of Risch has been implemented for this class of functions, but some implementation problems still remain and this package is not ready for distribution yet.

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