Article

# An "Accidental" Symmetry Operator for the Dirac Equation in the Coulomb Potential

Modern Physics Letters A (Impact Factor: 1.2). 07/2005; 20(30). DOI: 10.1142/S0217732305018505

Source: arXiv

**ABSTRACT**

On the basis of the generalization of the theorem about K-odd operators (K is the Dirac's operator), certain linear combination is constructed, which appears to commute with the Dirac Hamiltonian for Coulomb field. This operator coincides with the Johnson and Lippmann operator and is intimately connected to the familiar Laplace-Runge-Lenz vector. Our approach guarantees not only derivation of Johnson-Lippmann operator, but simultaneously commutativity with the Dirac Hamiltonian follows. Comment: 6 pages

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**ABSTRACT:**As is known, the so-called Dirac $K$-operator commutes with the Dirac Hamiltonian for arbitrary central potential $V(r)$. Therefore the spectrum is degenerate with respect to two signs of its eigenvalues. This degeneracy may be described by some operator, which anticommutes with $K$. If this operator commutes with the Dirac Hamiltonian at the same time, then it establishes new symmetry, which is Witten's supersymmetry. We construct the general anticommuting with $K$ operator, which under the requirement of this symmetry unambiguously select the Coulomb potential. In this particular case our operator coincides with that, introduced by Johnson and Lippmann many years ago. -
##### Article: The hidden symmetry of the Coulomb problem in relativistic quantum mechanics: From Pauli to Dirac

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**ABSTRACT:**Additional conserved quantities associated with an extra symmetry govern a wide variety of physical systems ranging from planetary motion to atomic spectra. We give a simple derivation of the hidden symmetry operator for the Dirac equation in a Coulomb field and show that this operator may be reduced to the one introduced by Johnson and Lippmann to include the spin degrees of freedom. This operator has been rarely discussed in the literature and has been neglected in recent textbooks on relativistic quantum mechanics and quantum electrodynamics. - [Show abstract] [Hide abstract]

**ABSTRACT:**The Dirac theory in the Euclidean Taub-NUT space gives rise to a large collection of conserved operators associated to genuine or hidden symmetries. They are involved in interesting algebraic structures as dynamical algebras or even infinite-dimensional algebras or superalgebras. One presents here the infinite-dimensional superalgebra specific to the Dirac theory in manifolds carrying the Gross-Perry-Sorkin monopole. It is shown that there exists an infinite-dimensional superalgebra that can be seen as a twisted loop superalgebra.