de Broglie, General Covariance and a Geometric Background to Quantum Mechanics
What is striking about de Broglie’s foundational work on wave–particle dualism is the role played by pseudo-Riemannian geometry in his early thinking. While exploring a fully covariant description of the Klein–Gordon equation, he was led to the revolutionary idea that a variable rest mass was essent...
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MDPI AG
2024-01-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/16/1/67 |
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| author | Basil Hiley Glen Dennis |
| author_facet | Basil Hiley Glen Dennis |
| author_sort | Basil Hiley |
| collection | DOAJ |
| description | What is striking about de Broglie’s foundational work on wave–particle dualism is the role played by pseudo-Riemannian geometry in his early thinking. While exploring a fully covariant description of the Klein–Gordon equation, he was led to the revolutionary idea that a variable rest mass was essential. DeWitt later explained that in order to obtain a covariant quantum Hamiltonian, one must supplement the classical Hamiltonian with an additional energy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mo>ℏ</mo><mn>2</mn></msup><mi mathvariant="script">Q</mi></mrow></semantics></math></inline-formula> from which the quantum potential emerges, a potential that Berry has recently shown also arises in classical wave optics. In this paper, we show how these ideas emerge from an essentially geometric structure in which the information normally carried by the wave function is contained within the algebraic description of the geometry itself, within an element of a minimal left ideal. We establish the fundamental importance of conformal symmetry, in which rescaling of the rest mass plays a vital role. Thus, we have the basis for a radically new theory of quantum phenomena based on the process of mass-energy flow. |
| first_indexed | 2024-03-08T10:34:04Z |
| format | Article |
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| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-08T10:34:04Z |
| publishDate | 2024-01-01 |
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| spelling | doaj.art-0f2ec415a84741f8bf2530c48906c0652024-01-26T18:38:48ZengMDPI AGSymmetry2073-89942024-01-011616710.3390/sym16010067de Broglie, General Covariance and a Geometric Background to Quantum MechanicsBasil Hiley0Glen Dennis1Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UKDepartment of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UKWhat is striking about de Broglie’s foundational work on wave–particle dualism is the role played by pseudo-Riemannian geometry in his early thinking. While exploring a fully covariant description of the Klein–Gordon equation, he was led to the revolutionary idea that a variable rest mass was essential. DeWitt later explained that in order to obtain a covariant quantum Hamiltonian, one must supplement the classical Hamiltonian with an additional energy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mo>ℏ</mo><mn>2</mn></msup><mi mathvariant="script">Q</mi></mrow></semantics></math></inline-formula> from which the quantum potential emerges, a potential that Berry has recently shown also arises in classical wave optics. In this paper, we show how these ideas emerge from an essentially geometric structure in which the information normally carried by the wave function is contained within the algebraic description of the geometry itself, within an element of a minimal left ideal. We establish the fundamental importance of conformal symmetry, in which rescaling of the rest mass plays a vital role. Thus, we have the basis for a radically new theory of quantum phenomena based on the process of mass-energy flow.https://www.mdpi.com/2073-8994/16/1/67quantum foundationsquantum mechanics in pseudo-Riemannian spacesgeometric algebra approachrest mass and conformal rescaling |
| spellingShingle | Basil Hiley Glen Dennis de Broglie, General Covariance and a Geometric Background to Quantum Mechanics Symmetry quantum foundations quantum mechanics in pseudo-Riemannian spaces geometric algebra approach rest mass and conformal rescaling |
| title | de Broglie, General Covariance and a Geometric Background to Quantum Mechanics |
| title_full | de Broglie, General Covariance and a Geometric Background to Quantum Mechanics |
| title_fullStr | de Broglie, General Covariance and a Geometric Background to Quantum Mechanics |
| title_full_unstemmed | de Broglie, General Covariance and a Geometric Background to Quantum Mechanics |
| title_short | de Broglie, General Covariance and a Geometric Background to Quantum Mechanics |
| title_sort | de broglie general covariance and a geometric background to quantum mechanics |
| topic | quantum foundations quantum mechanics in pseudo-Riemannian spaces geometric algebra approach rest mass and conformal rescaling |
| url | https://www.mdpi.com/2073-8994/16/1/67 |
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