A Modified Method for Deriving Self-Conjugate Dirac Hamiltonians in Arbitrary Gravitational Fields and Its Application to Centrally and Axially Symmetric Gravitational Fields

We have proposed previously a method for constructing self-conjugate Hamiltonians H_eta 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 H_eta 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.


Introduction
In [1], we proposed a method for constructing self-conjugate Hamiltonians H  in the  -representation with a flat scalar product to describe the dynamics of Dirac particles in arbitrary gravitational fields.
Using the algorithm proposed in [1], we calculated Hamiltonians in the  -representation for the Schwarzschild and Friedmann-Robertson-Walker cosmological model metrics. However, application of the algorithm to the Kerr metric necessitated a large amount of calculations to find Christoffel symbols, bispinor connectivities etc., and cumbersome algebraic transformations of arising expressions.
We made attempts to simplify the algorithm [1]. First, we proved the theorem, according to which a Hamiltonian in the  -representation for an arbitrary gravitational field, including a time-dependent one, is a Hermitian part of the initial Dirac Hamiltonian H  derived using tetrad vectors in the Schwinger gauge 2 .
Then, for block-diagonal metrics, using Eq. (1), we proved the second theorem, according to which the Hamiltonians H  and H   in Eq. (1) can be replaced by their "reduced" parts without or with a few summands with bispinor connectivities: Block-diagonal metrics are understood to be metric tensors of the form . ( Apparently, the cases belong to the same kind as (3) when 01 0 g  , and also when 02 g or 03 g are used instead of 01 g .
In Eq. (2), red H  is part of the initial Dirac Hamiltonian, which contains only the mass term and terms with momentum operator components (i.e. with coordinate derivatives).
The summand H   in (2) One can see that H   is a fairly simple expression, which in some cases differs from zero for the blockdiagonal metrics with 0 0 k g  . For example, the Kerr metric in Boyer-Lindquist coordinates [2], [3] belongs to such case. Of course, application of Eq. (2) makes the procedure of deriving self-conjugate Hamiltonians in the representation much less complicated. Eqs. (1) and (2) are proven in Sects. 3, 4 of this paper.
In the Conclusions, we discuss the outcome of this study and the results of applying the developed algorithm to the evolution of bound atomic and quark states in the expanding universe.

Hamiltonian in the  -representation
Let us recall the line of corresponding reasoning and introduce the notation. Tetrad vectors are defined by the relation We assume that the quantum mechanical motion of particles is described by the Dirac equation, which is written in the units of Here, m is the particle mass,  is a four-component "column" bispinor, and   are 4 4  Dirac matrices, The round parentheses in (7) contain a covariant bispinor derivative,    : Eq. (9) for    contains the bispinor connectivity   , for finding which one should fix some system of tetrad vectors H   defined as (5). Upon that, the quantity   can be expressed through "Christoffel" vector derivatives in the following way (the "Christoffel" derivatives are denoted by a semicolon): The expression for S  in (10) is defined below, see (14). The bispinor connectivity   given by (10) It follows from (11), (8), In terms of the matrices   , the Dirac equation (7) can be written as follows: It is convenient (but not necessary) to choose the quantities   so that they have the same form for all local frames of reference. Both systems   and   can be used to construct a full system of 4 4  matrices. The full system is, for example, the system 4 Any system of Dirac matrices provides for several discrete automorphisms. We restrict ourselves to the automorphism The matrix D will be called anti-Hermitizing.
It follows from (7) that the initial Hamiltonian is given by the following expression: The operator H (16) has a meaning of the evolution operator of the wave function of a Dirac particle in the chosen global frame of reference.
Ref. [1] formulates the rules of finding a Hamiltonian in the  -representation for a Dirac particle in an arbitrary gravitational field. A-priori information, which is assumed to be known, is information about the metric tensor   In order to find n m H  , we introduce a tensor, mn f , with components 0 0 00 m n mn mn The tensor mn f satisfies the condition As n m H  we can use any triplet of three-dimensional vectors, satisfying the relation In what follows, the quantities, dependent on the choice of tetrad vectors, are denoted by a tilde, if they are calculated in the system of tetrad vectors in the Schwinger gauge.
2) In accordance with (16), we write a general expression for the Hamiltonian H  . Here: ; ; 3) The expression for the Hamiltonian H  equals It follows from (39), (40) that Considering Eq. (37), we obtain Let us introduce the following notation: Some transformations give the following expressions for Y ;0 and for Z : where   In (50), 5 is a totally antisymmetric third-rank tensor.
Then, we calculate Y and Z using the relations (3), (5), (17), (18), and diagonal representation of Thus, it turns out that for the block-diagonal metrics of the form (3) we can find the Hamiltonian H  using the fairly simple formula (2).

Centrally symmetric gravitational field
This section presents Hamiltonians in the  -representation for Dirac particles in centrally symmetric gravitational fields, when the metrics are written in various coordinates.

The Schwarzschild metric
Writing the Schwarzschild solution in the coordinates In Eq. (54), 0 r is the gravitational radius   The resulting expression for H  derived in [1] and revised to include (26) is In Eq. (55), It is easy to verify that Eq. (55) can be found in a comparatively straightforward manner using formula (2), if we take into account that In Refs. [14], [1], the authors also derived a Hamiltonian for the Schwarzschield metric in isotropic coordinates The expression for H  can be easily derived from (2) using (59)

Eddington-Finkelstein metric
The Eddington-Finkelstein solution ( [4], [5]) in the coordinates is given by The inverse tensor has the following form: The Hamiltonian in the  -representation is calculated using (2) given that for the metric of interest. We obtain:

Painlevé-Gullstrand metric
In this section, we find a self-conjugate Hamiltonian H  for a Dirac particle in a spherically symmetric gravitational field described by the Painlevé-Gullstrand metric. The Hamiltonian H  for this metric is calculated first using the algorithm of [1] and then using (1) and (2).
These transformations give The operator  for the Painlevé-Gullstrand metric equals Thus, as applied to the Painlevé-Gullstrand metric, the same Hamiltonian H  was obtained both by the standard algorithm and in a simpler manner using (2). 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.
The Hamiltonian from [11] can be written as The Hamiltonians (76) and (81) are physically equivalent, because they are related through a unitary transformation, 1 1 , Generally speaking, all Hamiltonians in the Schwinger gauge are connected with each other by physically equivalent matrices of spatial rotation. This is what we meant [1] speaking about the uniqueness of Hamiltonians in the  -representation (see the comments by M.Arminjon in [18]).

Finkelstein-Lemaitre metric
It is of independent interest to study the motion of a Dirac particle in the nonstationary Finkelstein-Lemaitre metric [5], because the time coordinate in this metric coincides with the proper time.
The determinants equal Non-zero components of tetrad vectors in the Schwinger gauge: For this metric, in (2), 0 H    . "Reduced" Hamiltonian: We insert (86) into (2) and obtain The Hamiltonian (87) is self-conjugate with a fairly complex time dependence.

Hamiltonian in the  -representation for Dirac particles in the Kruskal gravitational field
The Kruskal metric [8] is a further development of the Lemaitre-Finkelstein metric to build the most complete frame of reference for a point-mass field. The formula below, in which the frame of reference is synchronous, has been developed by I.D. Novikov [9]. In the   The determinants equal     1 cos sin , 16 Eqs. (88), (89) show that the metric (89) is related to the radial coordinate R and proper time  through the parameter  .
"Reduced" Hamiltonian: According to (2), with H 0    , we have The derivative 1 1 H R    in the last summand of (93) should allow for the dependence   R,   (see (89)).

Kerr metric in the Boyer-Lindquist coordinates
The Kerr solution in the Boyer-Lindquist coordinates [3]    The inverse tensor has the following form: Here,

Tetrad vectors in the Schwinger gauge
We will need expressions for tetrad vectors in the Schwinger gauge. The results of calculating the components of tetrad vectors H    are presented in Table 2. Table 3 shows the components of vectors.
The Hamiltonian H  is calculated using (2).
We put the tetrad vector components  

Conclusions
This study develops the algorithm proposed in [1] for constructing self-conjugate Hamiltonians H  in the  -representation with a flat scalar product to describe the dynamics of Dirac particles in arbitrary gravitational fields. We prove that a Hamiltonian in the  -representation for any gravitational field, including a time-dependent field, is a Hermitian part of the initial Dirac Hamiltonian H  derived using tetrad vectors in the Schwinger gauge. We also prove that for the block-diagonal matrices like (3), the Hamiltonian H  can be calculated by the formula (2) using "reduced" parts of the Hamiltonians H  and H   without or with a small number of summands with bispinor connectivities. Using this method, we for the first time find self-conjugate Hamiltonians H  for the Kerr metric in the Boyer-Lindquist form and for the Eddington-Finkelstein, Finkelstein-Lemaitre, Kruskal, Clifford torus metrics and also for non-stationary metrics of open and spatially flat Friedmann models.
In this paper, we also prove physical equivalence of Dirac Hamiltonians in a weak Kerr field in harmonic Cartesian and Boyer-Lindquist coordinates. We point at the necessity of using harmonic Cartesian coordinates for clear physical interpretation of individual terms in the Hamiltonians.
In [22], the algorithm for deriving self-conjugate Dirac Hamiltonians in the  -representation is extended to the electromagnetic case. The Hamiltonian derived is applied to the case when the nonstationary gravitational field describes the spatially flat Friedmann model, and the electromagnetic field is an extension of the Coulomb potential to the case of this model. Following other authors [17], we demonstrate that energy levels in atomic systems are invariable in cosmological time.
Spectral lines of atoms in the spatially flat Friedmann model are identical at different points of cosmological time, and redshift is attributed completely to the growth of the wavelength of photons in the expanding universe.
At the same time, we observed that interaction forces and physical dimensions of atomic and quark bound systems vary with universe expansion.
The expressions for Hamiltonians H  , derived in this paper, can also be employed to study the behavior of Dirac particles in the vicinity of black holes, and scattering and absorption of such particles by black holes.