Charge Conjugation Symmetry in the Finite Basis Approximation of the Dirac Equation

• Main Authors: Maen Salman, Trond Saue
• Format: Article
• Language: English
• Published: MDPI AG 2020-07-01
• Series: Symmetry
• Subjects: Dirac equation; finite basis approximation; charge conjugation symmetry; kinetic balance
• Online Access: https://www.mdpi.com/2073-8994/12/7/1121

Four-component relativistic atomic and molecular calculations are typically performed within the no-pair approximation where negative-energy solutions are discarded. These states are, however, needed in QED calculations, wherein, furthermore, charge conjugation symmetry, which connects electronic and positronic solutions, becomes an issue. In this work, we shall discuss the realization of charge conjugation symmetry of the Dirac equation in a central field within the finite basis approximation. Three schemes for basis set construction are considered: restricted, inverse, and dual kinetic balance. We find that charge conjugation symmetry can be realized within the restricted and inverse kinetic balance prescriptions, but only with a special form of basis functions that does not obey the right boundary conditions of the radial wavefunctions. The dual kinetic balance prescription is, on the other hand, compatible with charge conjugation symmetry without restricting the form of the radial basis functions. However, since charge conjugation relates solutions of opposite value of the quantum number <inline-formula> <math display="inline"> <semantics> <mi>κ</mi> </semantics> </math> </inline-formula>, this requires the use of basis sets chosen according to total angular momentum <i>j</i> rather than orbital angular momentum <i>ℓ</i>. As a special case, we consider the free-particle Dirac equation, where opposite energy solutions are related by charge conjugation symmetry. We show that there is additional symmetry in that solutions of the <i>same</i> value of <inline-formula> <math display="inline"> <semantics> <mi>κ</mi> </semantics> </math> </inline-formula> come in pairs of opposite energy.