Abstract
Pizzetti’s formula explicitly shows the equivalence of the rotation invariant integration over a sphere and the action of rotation invariant differential operators. We generalize this idea to the integrals over real, complex, and quaternion Stiefel manifolds in a unifying way. In particular, we propose a new way to calculate group integrals and try to uncover some algebraic structures which manifest themselves for some well-known cases like the Harish-Chandra integral. We apply a particular case of our formula to an Itzykson–Zuber integral for the coset \({{{\rm SO}(4)/[{\rm SO}(2)\times {\rm SO}(2)]}}\). This integral naturally appears in the calculation of the two-point correlation function in the transition of the statistics of the Poisson ensemble and the Gaussian orthogonal ensemble in random matrix theory.
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References
Akemann, G., Baik, J., Di Francesco, P. (eds.): The oxford handbook of random matrix theory, 1st edn. Oxford University Press, Oxford (2011)
Altland, A., Zirnbauer, M.R.: Novel symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55, 1142 (1997). arXiv:cond-mat/9602137
Berezin, F.A.: Introduction to superanalysis, 1st edn, D. Reidel Publishing Company, Dordrecht (1987)
Berezin F.A., Karpelevich F.I.: Zonal spherical functions and Laplace operators on some symmetric spaces. Doklady Akad. Nauk SSSR 118, 9 (1958)
Bergère, M., Eynard, B.: Some properties of angular integrals. J. Phys. A 42, 265201 (2009). arXiv:0805.4482 [math-ph]
Collins, B., Śniady, P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264, 773 (2006). arXiv:math-ph/0402073
Coulembier, K.: The orthosymplectic superalgebra in harmonic analysis. J. Lie Theory 23, 55 (2013). arXiv:1208.3827 [math.RT]
De Bie, H., Eelbode, D., Sommen, F.: Spherical harmonics and integration in superspace II. J. Phys. A Math. Theor. 42, 245204 (2009). arXiv:0905.2092 [math-ph]
Forrester, P.J.: Log-Gases and random matrices, 1st edn. Princeton University Press, Princeton (2010)
Gorin, T.: Integrals of monomials over the orthogonal group. J. Math. Phys. 43, 3342 (2002). arXiv:math-ph/0112012
Guhr, T., Kohler, H.: Recursive construction for a class of radial functions. I. Ordinary space. J. Math. Phys. 43, 2707 (2002).arXiv:math-ph/0011007
Guhr, T., Kohler, H.: Recursive Construction for a class of radial functions II—Superspace. J. Math. Phys. 43, 2741 (2002). arXiv:math-ph/0012047
Guhr, T., Müller-Groeling, A., Weidenmüller, H.A.: Random-matrix theories in quantum physics: Common concepts. Phys. Rep. 299,189 (1998)
Haake, F.: Quantum signatures of chaos, 2nd edn. Springer Verlag, Berlin (2001)
Harish-Chandra.: Differential operators on a semisimple Lie algebra. Am. J. Math. 79, 87 (1957)
Huckleberry, A., Puettmann, A., Zirnbauer, M.R.: Haar expectations of ratios of random characteristic polynomials. (2007). arXiv:0709.1215 [math-ph]
Howe R.: Remarks on classical invariant theory. Trans. Am. Math. Soc. 313, 539 (1989)
Ingham A.E.: An integral which occurs in statistics. Proc. Camb. Phil. Soc. 29, 271 (1933)
Itzykson C., Zuber J.B.: The planar approximation. J. Math. Phys. 21, 411 (1980)
Ivanov, D.: The supersymmetric technique for random-matrix ensembles with zero eigenvalues. J. Math. Phys. 43, 126 (2002). arXiv:cond-mat/0103137
Kieburg, M., Verbaarschot, J.J.M., Zafeiropoulos, S.: Dirac spectra of 2-dimensional QCD-like theories. (2014). arXiv:1405.0433 [hep-lat]
Kieburg, M., Grönqvist, J., Guhr, T.: Arbitrary rotation invariant random matrix ensembles and supersymmetry: orthogonal and unitary-symplectic case. J. Phys. A 42, 275205 (2009). arXiv:0905.3253 [math-ph]
Kohler, H.: Group integrals in chaotic quantum systems, Phd thesis, University Heidelberg. (2009). http://archiv.ub.uni-heidelberg.de/volltextserver/1406/
Leutwyler H., Smilga A.: Spectrum of Dirac operator and role of winding number in QCD. Phys. Rev. D 46, 5607 (1992)
Littelmann, P., Sommers, H.-J., Zirnbauer, M.R.: Superbosonization of invariant random matrix ensembles. Math. Phys. 283, 343 (2008). arXiv:0707.2929 [math-ph]
Louck, J.D.: Unitary symmetry and combinatorics. World Scientific Publication Co. Pte. Ltd., London (2008)
Lysik G.: Mean-value properties of real analytic functions. Arch. Math. (Basel) 98, 61 (2012)
Mehta, M.L.: Random matrices, 3rd edn. Academic, New York (2004)
Muirhead R.J.: Aspects of multivariate statistical theory. Wiley, New York (1982)
Okounkov, A., Olshanski, G.: Shifted jack polynomials, binomial formula, and applications. Math. Res. Lett. 4, 69 (1997). arXiv:q-alg/9608020
Pizzetti P.: Sulla media dei valori che una funzione dei punti dello spazio assume alla superlicie di una sfera. Rend. Reale Accod. Lincei 18, 182 (1909)
Schlittgen, B., Wettig, T.: Generalizations of some integrals over the unitary group. J. Phys. A 36, 3195 (2003). arXiv:math-ph/0209030
Siegel C.L.: über die analytische Theorie der quadratischen Formen. Ann. Math. 36, 527 (1935)
Shatashvili S.L.: Correlation functions in the Itzykson-Zuber model. Commun. Math. Phys. 154(2), 421–432 (1993)
Shuryak, E.V., Verbaarschot, J.J.M.: Random matrix theory and spectral sum rules for the Dirac operator in QCD. Nucl. Phys. A 560, 306 (1993). arXiv:hep-th/9212088
Sommers, H.-J.: Superbosonization. Act. Phys. Pol. B 38, 1001 (2007). arXiv:0710.5375 [cond-mat.stat-mech]
Verbaarschot, J.J.M.: The spectrum of the QCD Dirac operator and chiral random matrix theory: the threefold way. Phys. Rev. Lett. 72, 2531 (1994). arXiv:hep-th/9401059
Verbaarschot, J.J.M., Wettig, T.: Random matrix theory and chiral symmetry in QCD. Ann. Rev. Nucl. Part. Sci. 50, 343 (2000). arXiv:hep-ph/0003017
Verbaarschot, J.J.M., Zahed, I.: Random matrix theory and QCD3. Phys. Rev. Lett. 73, 2288 (1994). arXiv:hep-th/9405005
Zirnbauer, M.R.: Riemannian symmetric superspaces and their origin in random matrix theory. J. Math. Phys. 37, 4986 (1996). arXiv:math-ph/9808012
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Coulembier, K., Kieburg, M. Pizzetti Formulae for Stiefel Manifolds and Applications. Lett Math Phys 105, 1333–1376 (2015). https://doi.org/10.1007/s11005-015-0774-x
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DOI: https://doi.org/10.1007/s11005-015-0774-x