Abstract
We have proposed previously a method for constructing self-conjugate Hamiltonians Hh in the h-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 Hh 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 h-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.
Highlights
In [1], we proposed a method for constructing self-conjugate Hamiltonians Hη in the η -representation with a flat scalar product to describe the dynamics of Dirac particles in arbitrary gravitational fields
Note that procedures for constructing self-conjugate Hamiltonians with a flat scalar product that could be used for studying the dynamics of spin 1/2 particles in gravitational fields of particular form have been proposed in literature more than once [14]-[17]
In [11], a self-conjugate Hamiltonian was obtained for the Painlevé-Gullstrand metric using tetrad vectors in the Schwinger gauge with a set of local Dirac matrices written in spherical coordinates
Summary
In [1], we proposed a method for constructing self-conjugate Hamiltonians Hη in the η -representation with a flat scalar product to describe the dynamics of Dirac particles in arbitrary gravitational fields. Note that procedures for constructing self-conjugate Hamiltonians with a flat scalar product that could be used for studying the dynamics of spin 1/2 particles in gravitational fields of particular form have been proposed in literature more than once [14]-[17]. These attempts are not general, but they produce correct results as applied to the choice of particular metrics and tetrad vectors. It turns out that for the block-diagonal metrics of the form (3) we can find the Hamiltonian Hη using the fairly simple formula (2)
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