The stability and instability of relativistic matter

Elliott H. Lieb, Horng Tzer Yau

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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 p2/2 m is replaced by (p2c2+m2c4)1/2-mc2. 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 α is the fine structure constant. We prove that stability occurs in the many-body case if zα≦2/π and α<1/(47 q). For small z, a better bound on α 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.

Original languageEnglish (US)
Pages (from-to)177-213
Number of pages37
JournalCommunications In Mathematical Physics
Issue number2
StatePublished - Jun 1988

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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