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arXiv:2608.27112

Kramers pseudospin and the quantum number proposed for the many-electron Dirac?Coulomb Hamiltonian

Shadan Ghassemi Tabrizi

quant-phphysics.chem-phSubmitted Aug 27, 2026• Announced Aug 28, 2026• 0 citations
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Abstract

The square of a sum of single-electron time reversals has been proposed as a conserved quantity with an integer eigenvalue spectrum for the many-electron Dirac-Coulomb Hamiltonian [Phys. Rev. A 94, 052104 (2016)], and its expectation value is in use as a diagnostic of Kramers contamination. This work identifies the operator behind the construction, a component of a pseudospin attached to the chosen Kramers pairs, whose angular-momentum algebra reproduces the reported spectrum and yields the eigenvectors in closed form. The label $k$ entering the proposed quantum number $-k^2$ is twice the magnitude of a pseudospin projection onto an axis fixed by the basis and changes when the Kramers pairs are rephased. The commutation with the Hamiltonian is unfounded, and the proposed quantum number does not stand.

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21 pages

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The square of a sum of single-electron time reversals has been proposed as a conserved quantity with an integer eigenvalue spectrum for the many-electron Dirac-Coulomb Hamiltonian [Phys. Rev. A 94, 052104 (2016)], and its expectation value is in use as a diagnostic of Kramers contamination. This work identifies the operator behind the construction, a component of a pseudospin attached to the chosen Kramers pairs, whose angular-momentum algebra reproduces the reported spectrum and yields the eigenvectors in closed form. The label kkk entering the proposed quantum number −k2-k^2−k2 is twice the magnitude of a pseudospin projection onto an axis fixed by the basis and changes when the Kramers pairs are rephased. The commutation with the Hamiltonian is unfounded, and the proposed quantum number does not stand.
In a recent paper [M. Sato, K. Hasebe, K. Esaki, and M. Kohmoto, Prog. Theor. Phys. 127, 937 (2012)] Sato and his collaborators established a generalization of the Kramers degeneracy structure to pseudo-Hermitian Hamiltonian systems, admitting even time-reversal symmetry, |T2=1T^{2}=1T2=1|⁠. This extension is achieved using the mathematical structure of split-quaternions instead of quaternions, usually adopted in the case of Hermitian Hamiltonians with odd time-reversal symmetry, |T2=−1T^{2}=-1T2=−1|⁠. Here we find that the metric operator for the pseudo-Hermitian Hamiltonian |HHH| that allows the realization of the generalized Kramers degeneracy is necessarily indefinite. We show that such |HHH| with real spectrum also possesses odd antilinear symmetry induced from the existing odd time-reversal symmetry of its Hermitian counterpart |hhh|⁠, so that the generalized Kramers degeneracy of |HHH| is in fact crypto-Hermitian Kramers degeneracy. We study in greater detail a new example of the pseudo-Hermitian split-quaternionic four-level Hamiltonian system, which admits an indefinite metric operator and time-reversal symmetry and, as a consequence, a generalized Kramers degeneracy structure. We provide a complete solution of the eigenvalue problem, construct pseudo-Hermitian ladder operators closing the normal and abnormal pseudo-fermionic algebras, and show that this system fulfills a crypto-Hermitian degeneracy.
Solutions of the Dirac equation for an electron in the Coulomb potential are obtained using operator invariants of the equation, namely the Dirac, Johnson-Lippmann and recently found new invariant. It is demonstrated that these operators are the spin invariants. The generalized invariant is constructed and the exact general solution of the Dirac equation are found. In particular, the explicit expressions of the bispinors corresponding to the three complete sets of the invariants, their eigenvalues and quantum numbers are calculated. It is shown that the general solution of one center Coulomb Dirac equation contains free parameters. Changing one or more of these parameters, one can transform one solution of the Dirac equation into any other. It is shown for the first time that these invariants determine electron spatial probability amplitude and spin polarization in each quantum state. Electron probability densities and spin polarizations are explicitly calculated in the general form for several electron states in the hydrogen-like energy spectrum. Spatial distributions of these characteristics are shown to depend essentially on the invariant set, demonstrating, in spite of the accidental degeneracy of energy levels, physical difference of the states corresponding to different spin invariants.
In this paper it is shown that Maxwell's theory of the electromagnetic field in vacuum can be stated in a form closely parallel to Dirac's theory of the electron. The electromagnetic field is described by a three-by-one column matrix whose elements are linear combinations of the three independent spinor components of the field. The Maxwell equations take a form similar to the Dirac equation for a free electron but involving three-by-three matrices. In terms of a wave function normalized so that the integral-square is the number of photons, the classical expressions for the energy, momentum, and angular momentum in the field are related to expected values of the Hamiltonian, momentum, and angular momentum operators. The angular momentum operator consists of an orbital part plus a three-by-three spin-one matrix part. As in Dirac's theory, the contributions from the states of negative Hamiltonian eigenvalue must be subtracted---these are states of opposite circular polarization.
The recent two papers have successively proposed the concepts and more precise expressions of strict conservation or statistical conservation of physical quantities, which are different from the expression of a conserved quantity in quantum mechanics. The Hamiltonian operator of the system discussed in this paper are independent of time, so it is commutative with the operators of an angular momentum itself, its component and its square, so that all these quantities are conserved quantities. In an isolated composite system, one of the angular momentum components of the total system are strictly conserved, while the angular momentum components of the subsystems can only be conserved statistically, both of which can be expressed in terms of eigenvalues and eigenfunctions. Generally, angular momentum itself does not have eigenvalues and eigenfunctions, so it cannot be expressed as strict conservation in terms of these concepts. Although there are eigenvalues and eigenfunctions for the square of angular momentum, they cannot express strict conservation either. And they can only be expressed as statistical conservation.
With respect to the principal operators including the sixteen 4-4 matrices of the Dirac electron, two covariant forms of the extended Heisenberg picture can be presented with the “two covariant Hamiltonians HI and HII” of the first and second order, respectively. One form with HII involving the proper electromagnetic moment corresponds perfectly to the Frenkel-Kramers model which was recently treated by the author; the other form with HI containing the Zitterbewegung corresponds partly to the so-called Weyssenhoff-like model. That an amalgamation of the former form and the Zitterbewegung brings immediately the latter is explicitly shown by virtue of a decomposition of the operator of the Dirac electron into two corresponding parts. This decomposition is derived from the author's extension of the Feynman-Bunge-Corben formula for the position operator of the electron and, in case without external fields, it corresponds to the covariant form of the so-called ‘gerade-ungerade’ decomposition. In the case of the free state mixed with positive and negative energy, the Zitterbewegung of operators derived from HI can properly be realized by the expectation values of the operators. In this process, the author's complex hyperbolic metric of velocity which represents a new negative energy state plays an essential role. These results give an important clue to solve the puzzling problem of the construction of a consistent classical model for the Dirac electron, and also may be useful for investigation of ‘internal structure’ of elementary particles as well as their spin.