Abstract
We consider the many-particle quantum mechanics of anyons, i.e. identical particles in two space dimensions with a continuous statistics parameter \({\alpha \in [0, 1]}\) ranging from bosons (α = 0) to fermions (α = 1). We prove a (magnetic) Hardy inequality for anyons, which in the case that α is an odd numerator fraction implies a local exclusion principle for the kinetic energy of such anyons. From this result, and motivated by Dyson and Lenard’s original approach to the stability of fermionic matter in three dimensions, we prove a Lieb-Thirring inequality for these types of anyons.
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Balinsky A.A.: Hardy type inequalities for Aharonov-Bohm magnetic potentials with multiple singularities. Math. Res. Lett. 10, 169–176 (2003)
Baker G.A., Canright G.S., Mulay S.B., Sundberg C.: On the Spectral Problem for Anyons. Commun. Math. Phys. 153, 277–295 (1993)
Dell’Antonio, G., Figari, R., Teta, A.: Statistics in Space Dimension Two. Lett. Math. Phys. 40, 235–256 (1997)
Dyson, J.: Stability of Matter. In: Statistical Physics, Phase Transitions and Superfluidity, Brandeis University Summer Institute in Theoretical Physics 1966, pp. 179–239. Gordon and Breach Publishers, New York (1968)
Dyson F.J., Lenard A.: Stability of Matter. I. J. Math. Phys. 8, 423–434 (1967)
Frank R.L., Seiringer R.: Lieb-Thirring Inequality for a Model of Particles with Point Interactions. J. Math. Phys. 53, 095201 (2012)
Fröhlich, J.: Quantum statistics and locality. In: Proceedings of the Gibbs Symposium (New Haven, CT, 1989), Providence, RI: Amer. Math. Soc., 1990, pp. 89–142
Goldin G.A., Menikoff R., Sharp D.H.: Representations of a local current algebra in nonsimply connected space and the Aharonov-Bohm effect. J. Math. Phys. 22, 1664–1668 (1981)
Hoffmann-Ostenhof M., Hoffman-Ostenhof T., Laptev A., Tidblom J.: Many-particle Hardy Inequalities. J. London Math. Soc. (2) 77, 99–114 (2008)
Khare, A.: Fractional Statistics and Quantum Theory. Second ed., Singapore: World Scientific, 2005
Laptev, A., Weidl, T.: Hardy inequalities for magnetic Dirichlet forms. In: Mathematical Results in Quantum Mechanics (Prague, 1998), Oper. Theory Adv. Appl. 108, Basel: Birkhäuser, 1999, pp. 299–305
Leinaas J.M., Myrheim J.: On the Theory of Identical Particles. Il Nuovo Cimento 37B, 1–23 (1977)
Lenard, A.: Lectures on the Coulomb Stability Problem. In: Statistical mechanics and mathematical problems, Battelle Rencontres, Seattle, Wash., 1971. Lecture Notes in Physics, vol. 20, pp. 114–135 (1973)
Lerda, A.: Anyons. Berlin–Heidelberg: Springer-Verlag, 1992
Lieb, E.H., Seiringer, R.: The stability of matter in quantum mechanics. Cambridge: Cambridge University Press, 2010
Lieb E.H., Thirring W.: Bound for the Kinetic Energy of Fermions which Proves the Stability of Matter. Phys. Rev. Lett. 35, 687–689 (1975)
Loss D., Fu Y.: Second Virial Coefficient of an Interacting Anyon Gas. Phys. Rev. Lett. 67, 294–297 (1991)
Lundholm, D.: Geometric extensions of many-particle Hardy inequalities. http://arxiv.org/abs/1101.2653v2 [math-ph], 2011
Lundholm, D., Solovej, J.P.: Local exclusion for intermediate and fractional statistics. http://arxiv.org/abs/1205.2520 [quant-ph], 2012
Lundholm, D., Solovej, J.P.: Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics. Ann. Henri Poincaré. http://arxiv.org/abs/1301.3436 [math-ph], (2013, to appear)
Melgaard M., Ouhabaz E.-M., Rozenblum G.: Negative discrete spectrum of perturbed multivortex Aharonov-Bohm Hamiltonians. Ann. Henri Poincaré 5, 979–1012 (2004)
Myrheim, J.: Anyons. In: Topological aspects of low dimensional systems (Les Houches, 1998), Les Ulis: EDP Sci., 1999, pp. 265–413
Payne L.E., Weinberger H.F.: An optimal Poincaré inequality for convex domains. Arch. Rat. Mech. Anal. 5, 286–292 (1960)
Rumin M.: Balanced distribution-energy inequalities and related entropy bounds. Duke Math. J. 160, 567–597 (2011)
Scott W.T.: Approximation to real irrationals by certain classes of rational fractions. Bull. Amer. Math. Soc. 46, 124–129 (1940)
Streater R.F., Wilde I.F.: Fermion states of a boson field. Nucl. Phys. B 24, 561–575 (1970)
Wilczek F.: Magnetic Flux, Angular Momentum, and Statistics. Phys. Rev. Lett 48, 1144–1146 (1982)
Wilczek F.: Quantum Mechanics of Fractional-Spin Particles. Phys. Rev. Lett 49, 957–959 (1982)
Wilczek, F.: Fractional Statistics and Anyon Superconductivity. Singapore: World Scientific, 1990
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Communicated by B. Simon
This work was partially supported by the Danish Council for Independent Research.
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Lundholm, D., Solovej, J.P. Hardy and Lieb-Thirring Inequalities for Anyons. Commun. Math. Phys. 322, 883–908 (2013). https://doi.org/10.1007/s00220-013-1748-4
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DOI: https://doi.org/10.1007/s00220-013-1748-4