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In mathematics, the Duistermaat–Heckman formula, due to Duistermaat and Heckman (1982), states that the pushforward of the canonical (Liouville) measure on a symplectic manifold under the moment map is a piecewise polynomial measure. Equivalently, the Fourier transform of the canonical measure is given exactly by the stationary phase approximation.

Berline & Vergne (1982) and, independently, Atiyah & Bott (1984) showed how to deduce the Duistermaat–Heckman formula from a localization theorem for equivariant cohomology.
References

Berline, Nicole; Vergne, Michele (1982), "Classes caracteristiques equivariantes. Formule de localisation en cohomologie equivariante", Comptes rendus de l'Académie des sciences
Atiyah, Michael Francis; Bott, Raoul (1984), "The moment map and equivariant cohomology", Topology. an International Journal of Mathematics 23 (1): 1–28, doi:10.1016/0040-9383(84)90021-1, MR 721448
Duistermaat, J. J.; Heckman, G. J. (1982), "On the variation in the cohomology of the symplectic form of the reduced phase space", Inventiones Mathematicae 69 (2): 259–268, doi:10.1007/BF01399506, MR 674406

Mathematics Encyclopedia

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