A Modified Method for Deriving Self-Conjugate Dirac Hamiltonians in Arbitrary Gravitational Fields and Its Application to Centrally and Axially Symmetric Gravitational Fields
Affiliation(s)
Russian Federal Nuclear Center, All-Russian Research Institute of Experimental Physics, Sarov, Russia.
Russian Federal Nuclear Center, All-Russian Research Institute of Experimental Physics, Sarov, Russia.
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.
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.
KEYWORDS
Self-Conjugate Hamiltonian, Dirac Particle, Arbitrary Gravitational Field, Schwinger Gauge, Kerr Metric
Self-Conjugate Hamiltonian, Dirac Particle, Arbitrary Gravitational Field, Schwinger Gauge, Kerr Metric
Cite this paper
Gorbatenko, M. and Neznamov, V. (2015) A Modified Method for Deriving Self-Conjugate Dirac Hamiltonians in Arbitrary Gravitational Fields and Its Application to Centrally and Axially Symmetric Gravitational Fields. Journal of Modern Physics, 6, 303-326. doi: 10.4236/jmp.2015.63034.
Gorbatenko, M. and Neznamov, V. (2015) A Modified Method for Deriving Self-Conjugate Dirac Hamiltonians in Arbitrary Gravitational Fields and Its Application to Centrally and Axially Symmetric Gravitational Fields. Journal of Modern Physics, 6, 303-326. doi: 10.4236/jmp.2015.63034.
References
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http://dx.doi.org/10.1103/PhysRevD.80.064044 [17] Andretsch, J. and Schafer, G. (1978) General Relativity and Gravitation, 9, 243, 489. [18] Arminjon, M. arxiv:1107.4556v2 [gr-qc]. [19] Gorbatenko, M.V. and Neznamov, V.P. (2010) Physical Review D, 82, Article ID: 104056.
http://dx.doi.org/10.1103/PhysRevD.82.104056 [20] Foldy, L.L. and Wouthuysen, S.A. (1950) Physical Review, 78, 29-36.
http://dx.doi.org/10.1103/PhysRev.78.29 [21] Neznamov, V.P. and Silenko, A.J. (2009) Journal of Mathematical Physics, 50, Article ID: 122302.
http://dx.doi.org/10.1063/1.3268592 [22] Gorbatenko, M.V. and Neznamov, V.P. arxiv: 1105.4709 [gr-qc].
[1] Gorbatenko, M.V. and Neznamov, V.P. (2011) Physical Review D, 83, Article ID: 105002.
http://dx.doi.org/10.1103/PhysRevD.83.105002 [2] Kerr, R.P. (1963) Physical Review Letters, 11, 237.
http://dx.doi.org/10.1103/PhysRevLett.11.237 [3] Boyer, R.H. and Lindquist, R.W. (1967) Journal of Mathematical Physics, 8, 265-281.
http://dx.doi.org/10.1063/1.1705193 [4] Eddington, A.S. (1924) Nature, 113, 192.
http://dx.doi.org/10.1038/113192a0 [5] Finkelstein, D. (1958) Physical Review, 110, 965-967.
http://dx.doi.org/10.1103/PhysRev.110.965 [6] Painlevè, P. (1921) C. R. Acad. Sci. (Paris), 173, 677-680. [7] Gullstrand, A. (1922) Arkiv for Matematik, Astronomi och Fysik, 16, 1-15. [8] Kruskal, M. (1960) Physical Review, 119, 1743-1745.
http://dx.doi.org/10.1103/PhysRev.119.1743 [9] Novikov, I.D. (1963) AJ, 40, 772. [10] Fiziev, P.P. arxiv:1012.3520. [11] Lasenby, A., Doran, C., Pritchard, J., Caceres, A. and Dolan, S. (2005) Physical Review D, 72, Article ID: 105014.
http://dx.doi.org/10.1103/PhysRevD.72.105014 [12] De-Donder, Th. (1921) La gravifique Einshtenienne. Gauthier-Villars, Paris. [13] Fock, V.A. (1939) Journal of Physics, 1, 81-116. [14] Obukhov, Y.N. (2001) Physical Review Letters, 86, 192.
Obukhov, Y.N. (2002) Fortschritte der Physik, 50, 711-716.
http://dx.doi.org/10.1002/1521-3978(200205)50:5/7<711::AID-PROP711>3.0.CO;2-Z [15] Leclerc, M. (2006) Classical and Quantum Gravity, 23, 4013-4019.
http://dx.doi.org/10.1088/0264-9381/23/12/001 [16] Obukhov, Y.N., Silenko, A.J. and Teryaev, O.V. (2009) Physical Review D, 80, Article ID: 064044.
http://dx.doi.org/10.1103/PhysRevD.80.064044 [17] Andretsch, J. and Schafer, G. (1978) General Relativity and Gravitation, 9, 243, 489. [18] Arminjon, M. arxiv:1107.4556v2 [gr-qc]. [19] Gorbatenko, M.V. and Neznamov, V.P. (2010) Physical Review D, 82, Article ID: 104056.
http://dx.doi.org/10.1103/PhysRevD.82.104056 [20] Foldy, L.L. and Wouthuysen, S.A. (1950) Physical Review, 78, 29-36.
http://dx.doi.org/10.1103/PhysRev.78.29 [21] Neznamov, V.P. and Silenko, A.J. (2009) Journal of Mathematical Physics, 50, Article ID: 122302.
http://dx.doi.org/10.1063/1.3268592 [22] Gorbatenko, M.V. and Neznamov, V.P. arxiv: 1105.4709 [gr-qc].