The Stability and Instability of Relativistic Matter

  • Elliott H. Lieb
  • Horng-Tzer Yau


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 α2/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.


Density Matrix Ground State Energy Coulomb Potential Negative Eigenvalue Relativistic Matter 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  • Horng-Tzer Yau
    • 2
  1. 1.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA
  2. 2.School of MathematicsThe Institute for Advanced StudyPrincetonUSA

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