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The Stability and Instability of Relativistic Matter

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The Stability of Matter: From Atoms to Stars

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Abstract

We consider the quantum mechanical many-body problem of electrons and fixed nuclei interacting via Coulomb forces, but with a relativistic form for the kinetic energy, namely p 2 /2m is replaced by (p 2 c 2 + m 2 c 4 ) 1/2 — mc 2 . The electrons are allowed to have q spin states (q = 2 in nature). For one electron and one nucleus instability occurs if Zα>2/π, where z is the nuclear charge and a is the fine structure constant. We prove that stability occurs in the many-body case if z α ≦ 2/π and α l/(47q). For small z, a better bound on a is also given. In the other direction we show that there is a critical αc (no greater than 128/15π) such that if α>αc then instability always occurs for all positive z (not necessarily integral) when the number of nuclei is large enough. Several other results of a technical nature are also given such as localization estimates and bounds for the relativistic kinetic energy.

Work partially supported by U.S. National Science Foundation grant PHY-85–15288-A02

The author thanks the Institute for Advanced Study for its hospitality and the U.S. National Science Foundation for support under grant DMS-8601978

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Author information

Authors and Affiliations

  1. Departments of Mathematics and Physics, Princeton University, P.O. Box 708, Princeton, NJ, 08544, USA

    Elliott H. Lieb

  2. School of Mathematics, The Institute for Advanced Study, Princeton, NJ, 08540, USA

    Horng-Tzer Yau

Authors
  1. Elliott H. Lieb
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  2. Horng-Tzer Yau
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Editors and Affiliations

  1. Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, A-1090, Wien, Austria

    Walter Thirring

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Lieb, E.H., Yau, HT. (2001). The Stability and Instability of Relativistic Matter. In: Thirring, W. (eds) The Stability of Matter: From Atoms to Stars. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04360-8_34

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  • DOI: https://doi.org/10.1007/978-3-662-04360-8_34

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-04362-2

  • Online ISBN: 978-3-662-04360-8

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