Basic quantum mechanics for three Dirac equations in a curved spacetime
Autor(a) principal: | |
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Data de Publicação: | 2010 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Brazilian Journal of Physics |
Texto Completo: | http://old.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332010000200020 |
Resumo: | We study the basic quantum mechanics for a fully general set of Dirac matrices in a curved spacetime by extending Pauli's method. We further extend this study to three versions of the Dirac equation: the standard (Dirac-Fock-Weyl or DFW) equation, and two alternative versions, both of which are based on the recently proposed linear tensor representations of the Dirac field (TRD). We begin with the current conservation: we show that the latter applies to any solution of the Dirac equation, iff the field of Dirac matrices γµ satisfies a specific PDE. This equation is always satisfied for DFW with its restricted choice for the γµ matrices. It similarly restricts the choice of the γµ matrices for TRD. However, this restriction can be achieved. The frame dependence of a general Hamiltonian operator is studied. We show that in any given reference frame with minor restrictions on the spacetime metric, the axioms of quantum mechanics impose a unique form for the Hilbert space scalar product. Finally, the condition for the general Dirac Hamiltonian operator to be Hermitian is derived in a general curved spacetime. For DFW, the validity of this hermiticity condition depends on the choice of the γµ matrices. |
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Brazilian Journal of Physics |
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Basic quantum mechanics for three Dirac equations in a curved spacetimeDirac equationgravitationcurrent conservationHermitian Hamiltoniantensor representationWe study the basic quantum mechanics for a fully general set of Dirac matrices in a curved spacetime by extending Pauli's method. We further extend this study to three versions of the Dirac equation: the standard (Dirac-Fock-Weyl or DFW) equation, and two alternative versions, both of which are based on the recently proposed linear tensor representations of the Dirac field (TRD). We begin with the current conservation: we show that the latter applies to any solution of the Dirac equation, iff the field of Dirac matrices γµ satisfies a specific PDE. This equation is always satisfied for DFW with its restricted choice for the γµ matrices. It similarly restricts the choice of the γµ matrices for TRD. However, this restriction can be achieved. The frame dependence of a general Hamiltonian operator is studied. We show that in any given reference frame with minor restrictions on the spacetime metric, the axioms of quantum mechanics impose a unique form for the Hilbert space scalar product. Finally, the condition for the general Dirac Hamiltonian operator to be Hermitian is derived in a general curved spacetime. For DFW, the validity of this hermiticity condition depends on the choice of the γµ matrices.Sociedade Brasileira de Física2010-06-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersiontext/htmlhttp://old.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332010000200020Brazilian Journal of Physics v.40 n.2 2010reponame:Brazilian Journal of Physicsinstname:Sociedade Brasileira de Física (SBF)instacron:SBF10.1590/S0103-97332010000200020info:eu-repo/semantics/openAccessArminjon,MayeulReifler,Frankeng2010-06-23T00:00:00Zoai:scielo:S0103-97332010000200020Revistahttp://www.sbfisica.org.br/v1/home/index.php/pt/ONGhttps://old.scielo.br/oai/scielo-oai.phpsbfisica@sbfisica.org.br||sbfisica@sbfisica.org.br1678-44480103-9733opendoar:2010-06-23T00:00Brazilian Journal of Physics - Sociedade Brasileira de Física (SBF)false |
dc.title.none.fl_str_mv |
Basic quantum mechanics for three Dirac equations in a curved spacetime |
title |
Basic quantum mechanics for three Dirac equations in a curved spacetime |
spellingShingle |
Basic quantum mechanics for three Dirac equations in a curved spacetime Arminjon,Mayeul Dirac equation gravitation current conservation Hermitian Hamiltonian tensor representation |
title_short |
Basic quantum mechanics for three Dirac equations in a curved spacetime |
title_full |
Basic quantum mechanics for three Dirac equations in a curved spacetime |
title_fullStr |
Basic quantum mechanics for three Dirac equations in a curved spacetime |
title_full_unstemmed |
Basic quantum mechanics for three Dirac equations in a curved spacetime |
title_sort |
Basic quantum mechanics for three Dirac equations in a curved spacetime |
author |
Arminjon,Mayeul |
author_facet |
Arminjon,Mayeul Reifler,Frank |
author_role |
author |
author2 |
Reifler,Frank |
author2_role |
author |
dc.contributor.author.fl_str_mv |
Arminjon,Mayeul Reifler,Frank |
dc.subject.por.fl_str_mv |
Dirac equation gravitation current conservation Hermitian Hamiltonian tensor representation |
topic |
Dirac equation gravitation current conservation Hermitian Hamiltonian tensor representation |
description |
We study the basic quantum mechanics for a fully general set of Dirac matrices in a curved spacetime by extending Pauli's method. We further extend this study to three versions of the Dirac equation: the standard (Dirac-Fock-Weyl or DFW) equation, and two alternative versions, both of which are based on the recently proposed linear tensor representations of the Dirac field (TRD). We begin with the current conservation: we show that the latter applies to any solution of the Dirac equation, iff the field of Dirac matrices γµ satisfies a specific PDE. This equation is always satisfied for DFW with its restricted choice for the γµ matrices. It similarly restricts the choice of the γµ matrices for TRD. However, this restriction can be achieved. The frame dependence of a general Hamiltonian operator is studied. We show that in any given reference frame with minor restrictions on the spacetime metric, the axioms of quantum mechanics impose a unique form for the Hilbert space scalar product. Finally, the condition for the general Dirac Hamiltonian operator to be Hermitian is derived in a general curved spacetime. For DFW, the validity of this hermiticity condition depends on the choice of the γµ matrices. |
publishDate |
2010 |
dc.date.none.fl_str_mv |
2010-06-01 |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://old.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332010000200020 |
url |
http://old.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332010000200020 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
10.1590/S0103-97332010000200020 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
text/html |
dc.publisher.none.fl_str_mv |
Sociedade Brasileira de Física |
publisher.none.fl_str_mv |
Sociedade Brasileira de Física |
dc.source.none.fl_str_mv |
Brazilian Journal of Physics v.40 n.2 2010 reponame:Brazilian Journal of Physics instname:Sociedade Brasileira de Física (SBF) instacron:SBF |
instname_str |
Sociedade Brasileira de Física (SBF) |
instacron_str |
SBF |
institution |
SBF |
reponame_str |
Brazilian Journal of Physics |
collection |
Brazilian Journal of Physics |
repository.name.fl_str_mv |
Brazilian Journal of Physics - Sociedade Brasileira de Física (SBF) |
repository.mail.fl_str_mv |
sbfisica@sbfisica.org.br||sbfisica@sbfisica.org.br |
_version_ |
1754734865403609088 |