Abstract
By the Pauli exclusion principle, no quantum state can be occupied by more than one electron. One can state this as a constraint on the one electron density matrix that bounds its eigenvalues by 1. Shortly after its discovery, the Pauli principle was replaced by anti-symmetry of the multi-electron wave function. In this paper we solve a longstanding problem about the impact of this replacement on the one electron density matrix, that goes far beyond the original Pauli principle. Our approach uses Berenstein and Sjamaar’s theorem on the restriction of an adjoint orbit onto a subgroup, and allows us to treat any type of permutational symmetry.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berenstein A., Sjamaar R.: Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion. J. Amer. Math. Soc. 13(2), 433–466 (2000)
Bernstein I., Gelfand I., Gelfand S.: Schubert cells and cohomology of the space G/P. Russ. Math. Survey 28(3), 1–26 (1973)
Borland R.E., Dennis K.: The conditions on the one-matrix for three-body fermion wavefunctions with one-rank equal to six. J. Phys. B: Atom Molec. Phys. 5, 7–15 (1972)
Cohen, A.M., van Leeuwen, M., Lisser, B.: LiE, a software package for Lie group theoretical computations, available at http://www-mathlabo.univ-poitiers.fr/~maavl/LiE/
Coleman A.J.: Structure of Fermion Density Matrices. Rev. Mod. Phys. 35, 668–686 (1963)
Coleman A.J., Yukalov V.I.: Reduced density matrices: Coulson’s challenge. Springer, Berlin (2000)
Fulton W., Harris J.: Representation theory. Springer, New York (1991)
Fulton, W.: Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, 1997
Fulton W.: Schubert varieties and degeneracy loci. Springer, Berlin (1998)
Dadok J., Kac V.: Polar representations. J. Algebra 92(2), 504–524 (1985)
Duck, T., Sudarshan, E.C.G.: Pauli and the spin-statistics theorem. Singapore: World Scientific, 1997
Franz M.: Moment polytopes of projective G-varieties and tensor products of symmetric group representations. J. Lie Theory 12, 539–549 (2002)
Franz, M.: Convex, a Maple package for convex geometry, available at http://www-fourier.ujf-grenoble.fr/~franz/convex/
Grudziński, H., Hirsch, J.: Search for new conditions for fermion N-representability. http://arXiv.org/list/math-ph/0311026, 2003
Klyachko A.: Stable bundles, representation theory, and Hermitian operators. Selecta Math. 4, 419–445 (1998)
Klyachko, A.: Vector bundles, Linear representations, and Spectral problems. Proc. Int. Congress of Math. Beijing 2002, Invited Lectures, Vol. II, Beijing: Higher Edication Press, 2003, pp. 599–614
Klyachko, A.: Quantum marginal problem and representations of the symmetric group. http://arXiv.org/list/quant-ph/0409113, 2004
Klyachko A.: Quantum marginal problem and N-representability. J. Phys. Conf. Series 36, 72–86 (2006)
Klyachko, A.: Dynamic symmetry approach to entnglement. In: Proc. NATO Advanced Study Inst., Cargese, Corsica, France, 2005, J.-P. Gazeau et. al. eds., Amsterdam: IOS Press, 2007, pp. 25–54
Lascoux A.: Classes de Chern d’un produit tensoriel. C. R. Acad. Sci. Paris 286, 385–387 (1978)
Lascoux A., Schützenberger M.-P.: Symmetry and flag manifolds. Lecture Notes in Mathematics 25, 159–198 (1974)
Lascoux A., Schützenberger M.-P.: Polyôme de Schubert. C. R. Acad. Sci. Paris 294, 447–450 (1982)
Liu Y.-K., Christandl M., Verstraete V.: N-representability is QMA-complete. Phys. Rev. Lett. 98, 110503 (2007)
Macdonald I.G.: Schubert polynomials. London Math. Soc. Lecture Notes 166, 73–99 (1991)
Macdonald I.G.: Symmetric functions and Hall polynomials. Clarendon Press, Oxford (1995)
Mazziotti, D.A. (eds): Reduced density matrix mechanics with application to many electron atoms and molecules. John Wiley and Sons, New York (2007)
Müller C.W.: Sufficient conditions for pure state N-representability. J. Phys. A: Math. Gen. 32, 4139–4148 (1999)
Ness L.: A stratification of the null cone via moment map. Amer. J. Math. 106, 1281–1329 (1984)
Peltzer C.P., Brandstatter J.J.: Studies in the theory of generalized density operators III. J. Math. Anal. Appl. 33, 263–277 (1971)
Perelomov A.M.: Generalized coherent states and their applications. Springer, Berlin (1986)
Ruskai M.B.: N-representability problem: Particle-hole equivalence. J. Math. Phys. 11, 3218–3224 (1970)
Ruskai M.B.: Comments on Peltzer–Brandstatter papers: Two counterexamples. J. Math. Anal. Appl. 44, 131–135 (1973)
Ruskai M.B.: Connecting N-representability to Weyl’s problem: The one particle density matrix for N = 3 and R = 6. J. Phys. A: Math. Theor. 40, F961–F967 (2007)
Vinberg, E., Popov, V.: Invariant theory. In: “Algebraic Geometry IV”, A.N. Parshin, I. Shafarevich, eds., Berlin: Springer, 1992
Weyl H.: The theory of groups and quantum mechanics. Dover, New York (1931)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M.B. Ruskai
Electronic Supplementary Material
Rights and permissions
About this article
Cite this article
Altunbulak, M., Klyachko, A. The Pauli Principle Revisited. Commun. Math. Phys. 282, 287–322 (2008). https://doi.org/10.1007/s00220-008-0552-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-008-0552-z