Symmetries of Relativistic Hydrogen Atom
Keywords:Dirac equation, Coulomb interaction, hydrogen atom, relativistic quantum mechanics, symmetry
The Dirac equation in the external Coulomb field is proved to possess the symmetry determined by 31 operators, which form the 31-dimensional algebra. Two different fermionic realizations of the SO(1,3) algebra of the Lorentz group are found. Two different bosonic realizations of this algebra are found as well. All generators of the above-mentioned algebras commute with the operator of the Dirac equation in an external Coulomb field and, therefore, determine the algebras of invariance of such Dirac equation. Hence, the spin s = (1, 0) Bose symmetry of the Dirac equation for the free spinor field, proved recently in our papers, is extended here for the Dirac equation interacting with an external Coulomb field. A relativistic hydrogen atom is modeled by such Dirac equation. We are able to prove for the relativistic hydrogen atom both the fermionic and bosonic symmetries known from our papers in the case of a non-interacting spinor field. New symmetry operators are found on the basis of new gamma matrix representations of the Clifford and SO(8) algebras, which are known from our recent papers as well. Hidden symmetries were found both in the canonical Foldy–Wouthuysen and covariant Dirac representations. The found symmetry operators, which are pure matrix ones in the Foldy–Wouthuysen representation, become non-local in the Dirac model.
V. Fock. Zur Theorie des Wasserstoffatoms. Z. Phys. 98, 145 (1935). https://doi.org/10.1007/BF01336904
V. Bargmann. Zur Theorie des Wasserstoffatoms. Bemerkungen zur gleichnamigen Arbeit von V. Fock. Z. Phys. 99, 576 (1936). https://doi.org/10.1007/BF01338811
P.A.M. Dirac. The quantum theory of the electron. Proc. Roy. Soc. Lond. A. 117, 610 (1928). https://doi.org/10.1098/rspa.1928.0023
M.H. Johnson, B.A. Lippmann. Relativistic Kepler problem. Phys. Rev. 78, 329 (1950).
E. De Groot. The virial theorem and the Dirac H atom. Am. J. Phys. 50, 1141 (1982). https://doi.org/10.1119/1.12917
A.A. Stahlhofen. Algebraic solutions of relativistic Coulomb problems. Helv. Phys. Acta 70, 372 (1997).
J-L. Chen, D-L. Deng, M-G. Hu. SO(4) symmetry in the relativistic hydrogen atom. Phys. Rev. A. 77, 034102 (2008). https://doi.org/10.1103/PhysRevA.77.034102
A.A. Stahlhofen. Comment on "SO(4) symmetry in the relativistic hydrogen atom". Phys. Rev. A. 78, 036101 (2008). https://doi.org/10.1103/PhysRevA.78.036101
W. Pauli. On the conservation of the lepton charge. Nuovo Cim. 6, 204 (1957). https://doi.org/10.1007/BF02827771
F. G¨ursey. Relation of charge independence and baryon conservation to Pauli's transformation. Nuov. Cim. 7, 411 (1958). https://doi.org/10.1007/BF02747705
I.Yu. Krivsky, V.M. Simulik. The Dirac equation and spin 1 representations, a connection with symmetries of the Maxwell equations. Theor. Math. Phys. 90, 265 (1992). https://doi.org/10.1007/BF01036532
A.G. Nikitin. Superalgebras of symmetry operators for Coulomb and Aharonov-Bohm-Coulomb systems. In: Photon and Poincar'e group (Nova Sci., 1999) [ISBN:9781560727187].
Th.W. Ruijgrok. On the relativistic hydrogen atom. Acta Phys. Pol. 87 43 (1976).
V.M. Simulik, I.Yu. Krivsky. Clifford algebra in classical electrodynamical hydrogen atom model. Adv. Appl. Cliff. Algebras 7, 25 (1997). https://doi.org/10.1007/BF03041213
V.M. Simulik, I.Yu. Krivsky, I.L. Lamer. Bosonic symmetries, solutions and conservation laws for the Dirac equation with nonzero mass. Ukr. J. Phys. 58, 523 (2013). https://doi.org/10.15407/ujpe58.06.0523
V.M. Simulik. On the gamma matrix representations of SO(8) and Clifford algebras. Adv. Appl. Cliff. Algebras 28, 93 (2018). https://doi.org/10.1007/s00006-018-0906-3
V.M. Simulik, I.Yu. Krivsky. On the extended real Clifford-Dirac algebra and new physically meaningful symmetries of the Dirac equation with nonzero mass. Dopov. NAN Ukr. No. 5, 82 (2010) (in Ukrainian).
I.Yu. Krivsky, V.M. Simulik. Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra. Cond. Matt. Phys. 13, 43101 (2010). https://doi.org/10.5488/CMP.13.43101
V.M. Simulik, I.Yu. Krivsky, I.L. Lamer. Application of the generalized Clifford-Dirac algebra to the proof of the Dirac equation Fermi-Bose duality. TWMS J. App. Eng. Math. 3, 46 (2013). https://doi.org/10.1109/MMET.2012.6331206
V.M. Simulik, I.Yu. Krivsky. Bosonic symmetries of the Dirac equation. Phys. Lett. A. 375, 2479 (2011). https://doi.org/10.1016/j.physleta.2011.03.058
B. Wybourne. Classical Groups for Physicists (Wiley, 1974) [ISBN: 978-0471965053].
J. Elliott, P. Dawber. Symmetry in Physics (Macmillian Press, 1979), Vol. 1 [ISBN: 978-0333382707]. https://doi.org/10.1007/978-1-349-07635-2_1
L.L. Foldy, S.A. Wouthuysen. On the Dirac theory of spin 1/2 particles and its non-relativistic limit. Phys. Rev. 78, 29 (1950). https://doi.org/10.1103/PhysRev.78.29
V.M. Simulik, I.O. Gordiyevich. On the symmetry of relativistic hydrogen atom and the Foldy-Wouthuysen representation. In: Abstracts of the Reports of the Intern. Conference of Young Scientists and Post-Graduates (Institute of Electron Physics, 2013).
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