A Comment on Background Independence in Quantum Theory
In this communication we take up the significance and purpose of selecting the proper coordinate system from the flat space‐time of non‐relativistic theories to the quantum theoretic formulation of general relativity. The universal background problem is straight forwardly framed as a momentum‐energy...
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| Vydáno v: | Journal of the Chinese Chemical Society (Taipei) Ročník 63; číslo 1; s. 11 - 19 |
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| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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Weinheim
WILEY-VCH Verlag
01.01.2016
WILEY‐VCH Verlag Wiley Subscription Services, Inc |
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| ISSN: | 0009-4536, 2192-6549, 2192-6549 |
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| Abstract | In this communication we take up the significance and purpose of selecting the proper coordinate system from the flat space‐time of non‐relativistic theories to the quantum theoretic formulation of general relativity. The universal background problem is straight forwardly framed as a momentum‐energy portrait in nexus with its space‐time conjugates. The description is based on operator matrix algebra, where the related analogue of the secular equation yields a Klein‐Gordon type equation and the associated Minkowski eigentime element. The energy‐momentum and their conjugate partners are represented by spaces that have (+,−) signatures. The general theory implicates both non‐zero‐ and zero rest‐mass entities, and it is proved that the conjugate relationship between energy and time provide a simple derivation of the Schwarzschild line element for the case of a gravitational field outside a spherical non‐rotational uncharged mass. This result, indicating the appearance of a black hole as a true singularity in the energy‐time formulation, and obtained as a direct consequence of their conjugate relationship, manifests background independence in concert with Einstein’s equivalence principle. Inducing a reformulation of the Lorentz Transformation respecting the indefinite Minkowski metric, displays an interesting relation between complex dilations and indefinite metric spaces, validating the complex symmetric ansatz.
It is proven that a basic quantum theoretic ansatz of momentum‐energy and their conjugates, space time, impart the celebrated Schwarzschild metric, showing contrary to the classical case, the Schwarzschild radius to be a true theoretical singularity. An linear mapping of the Lorentz transformation in the Minkowski space displays an interesting relation between complex dilation and indefinite metric spaces. |
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| AbstractList | In this communication we take up the significance and purpose of selecting the proper coordinate system from the flat space‐time of non‐relativistic theories to the quantum theoretic formulation of general relativity. The universal background problem is straight forwardly framed as a momentum‐energy portrait in nexus with its space‐time conjugates. The description is based on operator matrix algebra, where the related analogue of the secular equation yields a Klein‐Gordon type equation and the associated Minkowski eigentime element. The energy‐momentum and their conjugate partners are represented by spaces that have (+,−) signatures. The general theory implicates both non‐zero‐ and zero rest‐mass entities, and it is proved that the conjugate relationship between energy and time provide a simple derivation of the Schwarzschild line element for the case of a gravitational field outside a spherical non‐rotational uncharged mass. This result, indicating the appearance of a black hole as a true singularity in the energy‐time formulation, and obtained as a direct consequence of their conjugate relationship, manifests background independence in concert with Einstein’s equivalence principle. Inducing a reformulation of the Lorentz Transformation respecting the indefinite Minkowski metric, displays an interesting relation between complex dilations and indefinite metric spaces, validating the complex symmetric ansatz.
It is proven that a basic quantum theoretic ansatz of momentum‐energy and their conjugates, space time, impart the celebrated Schwarzschild metric, showing contrary to the classical case, the Schwarzschild radius to be a true theoretical singularity. An linear mapping of the Lorentz transformation in the Minkowski space displays an interesting relation between complex dilation and indefinite metric spaces. In this communication we take up the significance and purpose of selecting the proper coordinate system from the flat space-time of non-relativistic theories to the quantum theoretic formulation of general relativity. The universal background problem is straight forwardly framed as a momentum-energy portrait in nexus with its space-time conjugates. The description is based on operator matrix algebra, where the related analogue of the secular equation yields a Klein-Gordon type equation and the associated Minkowski eigentime element. The energy-momentum and their conjugate partners are represented by spaces that have (+,-) signatures. The general theory implicates both non-zero- and zero rest-mass entities, and it is proved that the conjugate relationship between energy and time provide a simple derivation of the Schwarzschild line element for the case of a gravitational field outside a spherical non-rotational uncharged mass. This result, indicating the appearance of a black hole as a true singularity in the energy-time formulation, and obtained as a direct consequence of their conjugate relationship, manifests background independence in concert with Einstein's equivalence principle. Inducing a reformulation of the Lorentz Transformation respecting the indefinite Minkowski metric, displays an interesting relation between complex dilations and indefinite metric spaces, validating the complex symmetric ansatz. In this communication we take up the significance and purpose of selecting the proper coordinate system from the flat space‐time of non‐relativistic theories to the quantum theoretic formulation of general relativity. The universal background problem is straight forwardly framed as a momentum‐energy portrait in nexus with its space‐time conjugates. The description is based on operator matrix algebra, where the related analogue of the secular equation yields a Klein‐Gordon type equation and the associated Minkowski eigentime element. The energy‐momentum and their conjugate partners are represented by spaces that have (+,−) signatures. The general theory implicates both non‐zero‐ and zero rest‐mass entities, and it is proved that the conjugate relationship between energy and time provide a simple derivation of the Schwarzschild line element for the case of a gravitational field outside a spherical non‐rotational uncharged mass. This result, indicating the appearance of a black hole as a true singularity in the energy‐time formulation, and obtained as a direct consequence of their conjugate relationship, manifests background independence in concert with Einstein’s equivalence principle. Inducing a reformulation of the Lorentz Transformation respecting the indefinite Minkowski metric, displays an interesting relation between complex dilations and indefinite metric spaces, validating the complex symmetric ansatz. |
| Author | Brändas, Erkki J. |
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| Cites_doi | 10.1007/s10714-011-1210-x 10.1007/BF01877511 10.1002/qua.22097 10.1002/qua.24168 10.1007/978-94-007-5297-9_1 10.1016/S0010-4655(99)00449-X 10.1017/CBO9780511976186 10.1002/qua.22616 10.1142/3331 |
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| Copyright | Copyright © 2016 The Chemical Society Located in Taipei & Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim, Germany Copyright © 2016 The Chemical Society Located in Taipei & Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany Copyright Wiley Subscription Services, Inc. Jan 2016 |
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| References | Nicolaides, C. A.; Brändas, E. J. Unstable States in the Continuous Spectra, Part II: Interpretation, Theory and Applications; Elsevier: Amsterdam; Adv. Quant. Chem. 2012, 63, 1-348. Levitina, T.; Brändas, E. J. Comput. Phys. Commun. 2000, 126, 107-113. Brändas, E. J. Int. J. Quant. Chem. 2013, 113, 173-184. Belot, G. Gen. Relativ. Gravit. 2011, 43, 2865-2884. Nicolaides, C. A.; Brändas, E. J. Unstable States in the Continuous Spectra, Part I: Analysis, Concepts, Methods, and Results; Elsevier: Amsterdam; Adv. Quant. Chem. 2010, 60, 1-549. Brändas, E. J. Int. J. Quant. Chem. 2011, 111, 1321-1332. Schwarzschild, K. Sitzungsberichte der Königlich Preussischen Akademieder Wissenschaften 1916, 7, 189-196. Balslev, E.; Combes, J. M. Commun. Math. Phys. 1971, 22, 280-294. Brändas, E. J. Int. J. Quant. Chem. 2009, 109, 2860-2865. 1971; 22 2011; 43 2013; 113 1916; 7 2000; 126 2009; 109 2011; 111 e_1_2_2_3_2 e_1_2_2_13_2 e_1_2_2_4_2 e_1_2_2_12_2 e_1_2_2_11_2 e_1_2_2_10_2 e_1_2_2_1_2 e_1_2_2_2_2 Nicolaides C. A. (e_1_2_2_6_2) e_1_2_2_8_2 Nicolaides C. A. (e_1_2_2_5_2) e_1_2_2_9_2 Schwarzschild K. (e_1_2_2_7_2) 1916; 7 |
| References_xml | – reference: Levitina, T.; Brändas, E. J. Comput. Phys. Commun. 2000, 126, 107-113. – reference: Brändas, E. J. Int. J. Quant. Chem. 2009, 109, 2860-2865. – reference: Belot, G. Gen. Relativ. Gravit. 2011, 43, 2865-2884. – reference: Nicolaides, C. A.; Brändas, E. J. Unstable States in the Continuous Spectra, Part II: Interpretation, Theory and Applications; Elsevier: Amsterdam; Adv. Quant. Chem. 2012, 63, 1-348. – reference: Brändas, E. J. Int. J. Quant. Chem. 2011, 111, 1321-1332. – reference: Balslev, E.; Combes, J. M. Commun. Math. Phys. 1971, 22, 280-294. – reference: Nicolaides, C. A.; Brändas, E. J. Unstable States in the Continuous Spectra, Part I: Analysis, Concepts, Methods, and Results; Elsevier: Amsterdam; Adv. Quant. Chem. 2010, 60, 1-549. – reference: Schwarzschild, K. Sitzungsberichte der Königlich Preussischen Akademieder Wissenschaften 1916, 7, 189-196. – reference: Brändas, E. J. Int. J. Quant. Chem. 2013, 113, 173-184. – volume: 109 start-page: 2860 year: 2009 end-page: 2865 publication-title: Int. J. Quant. Chem. – volume: 7 start-page: 189 year: 1916 end-page: 196 publication-title: Sitzungsberichte der Königlich Preussischen Akademieder Wissenschaften – volume: 22 start-page: 280 year: 1971 end-page: 294 publication-title: Commun. Math. Phys. – start-page: 60 end-page: 549 publication-title: Unstable States in the Continuous Spectra, Part I: Analysis, Concepts, Methods, and Results; – volume: 113 start-page: 173 year: 2013 end-page: 184 publication-title: Int. J. Quant. Chem. – volume: 126 start-page: 107 year: 2000 end-page: 113 publication-title: Comput. Phys. Commun. – volume: 43 start-page: 2865 year: 2011 end-page: 2884 publication-title: Gen. Relativ. Gravit. – start-page: 63 end-page: 348 publication-title: Unstable States in the Continuous Spectra, Part II: Interpretation, Theory and Applications – volume: 111 start-page: 1321 year: 2011 end-page: 1332 publication-title: Int. J. Quant. Chem. – ident: e_1_2_2_11_2 – ident: e_1_2_2_2_2 doi: 10.1007/s10714-011-1210-x – volume: 7 start-page: 189 year: 1916 ident: e_1_2_2_7_2 publication-title: Sitzungsberichte der Königlich Preussischen Akademieder Wissenschaften – start-page: 63 ident: e_1_2_2_6_2 publication-title: Unstable States in the Continuous Spectra, Part II: Interpretation, Theory and Applications – ident: e_1_2_2_12_2 doi: 10.1007/BF01877511 – ident: e_1_2_2_13_2 doi: 10.1002/qua.22097 – ident: e_1_2_2_3_2 doi: 10.1002/qua.24168 – start-page: 60 ident: e_1_2_2_5_2 publication-title: Unstable States in the Continuous Spectra, Part I: Analysis, Concepts, Methods, and Results; – ident: e_1_2_2_8_2 doi: 10.1007/978-94-007-5297-9_1 – ident: e_1_2_2_1_2 doi: 10.1016/S0010-4655(99)00449-X – ident: e_1_2_2_4_2 doi: 10.1017/CBO9780511976186 – ident: e_1_2_2_10_2 doi: 10.1002/qua.22616 – ident: e_1_2_2_9_2 doi: 10.1142/3331 |
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| Snippet | In this communication we take up the significance and purpose of selecting the proper coordinate system from the flat space‐time of non‐relativistic theories... In this communication we take up the significance and purpose of selecting the proper coordinate system from the flat space-time of non-relativistic theories... |
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| SubjectTerms | Black holes Conjugate operators Conjugates Coordinates Equivalence principle General relativity Gravitational fields Lorentz transformations Mathematical analysis Matrix algebra Momentum Non-positive metrics Operators (mathematics) Quantum theory Relativistic theory Relativity |
| Title | A Comment on Background Independence in Quantum Theory |
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