A Modified Method for Deriving Self-Conjugate Dirac Hamiltonians in Arbitrary Gravitational Fields and Its Application to Centrally and Axially Symmetric Gravitational Fields
Abstract
We have proposed previously a method for constructing self-conjugate Hamiltonians Heta in the eta-representation with a flat scalar product to describe the dynamics of Dirac particles in arbitrary gravitational fields. In this paper, we prove that, for block-diagonal metrics, the Hamiltonians Heta can be obtained, in particular, using "reduced" parts of Dirac Hamiltonians, i.e. expressions for Dirac Hamiltonians derived using tetrad vectors in the Schwinger gauge without or with a few summands with bispinor connectivities. Based on these results, we propose a modified method for constructing Hamiltonians in the eta-representation with a significantly smaller amount of required calculations. Using this method, here we for the first time find self-conjugate Hamiltonians for a number of metrics, including the Kerr metric in the Boyer-Lindquist coordinates, the Eddington-Finkelstein, Finkelstein-Lemaitre, Kruskal, Clifford torus metrics and for non-stationary metrics of open and spatially flat Friedmann models.