Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Scientific Reports

Representation of nuclear magnetic moments via a Clifford algebra formulation of Bohm’s hidden variables

Ruggero María Santilli, Garret Sobczyk

Published Dec 18, 2022• 0 citations
Publisher

Abstract

In this paper, we outline the research conducted by the first named author and his associates on the axiom-preserving, thus isotopic completion of quantum mechanics into hadronic mechanics according to the historical legacy by A. Einstein, B. Podolsky and N. Rosen that quantum mechanics is not a complete theory and review the ensuing exact representation of the magnetic moment and spin of the Deuteron in its ground state thanks to the isotopic completion of Pauli's matrices with an explicit and concrete content of D. Bohm's hidden variable [Formula: see text]. We then outline the independent studies conducted by the second named author on the representation of the conventional Pauli's matrices via geometric Clifford algebras. We finally show that the combination of the two studies allows a mathematically rigorous, numerically exact and time invariant geometric representation of the magnetic moment, spin and hidden variable of the Deuteron in its ground state.

DOI
10.1038/s41598-022-24970-4

Related papers

The Clifford Algebra approach to Quantum Mechanics A: The Schroedinger and Pauli Particles

Basil J. Hiley, R. E. Callaghan

In this paper we show how all the quantum properties of Schroedinger and Pauli particles can be described entirely from within a Clifford algebra taken over the reals. There is no need to appeal to any `wave function'. To describe a quantum system, we define the Clifford density element [CDE] as a product of an element of a minimal left ideal and its Clifford conjugate. The properties of the system are then completely specified in terms of bilinear invariants of the first and second kind calculated using the CDE. Thus the quantum properties of a system can be completely described from within the algebra without the need to appeal to any Hilbert space representation. Furthermore we show that the essential bilinear invariants of the second kind are simply the Bohm energy and the Bohm momentum, entities that make their appearance in the Bohm interpretation. We also show how these parameters emerge from standard quantum field theory in the low energy, single particle approximation. There is no need to appeal to classical mechanics at any stage. This clearly shows that the Bohm approach is entirely within the standard quantum formalism. The method has enabled us to lay the foundations of an approach that can be extended to provide a complete relativistic version of Bohm model. In this paper we confine our attention to the details of the non-relativistic case and will present its relativistic extension in a subsequent paper.

arXiv:1011.4031•Nov 17, 2010•16 citations

The Clifford Algebra Approach to Quantum Mechanics B: The Dirac Particle and its relation to the Bohm Approach

Basil J. Hiley, R. E. Callaghan

In this paper we present for the first time a complete description of the Bohm model of the Dirac particle. This result demonstrates again that the common perception that it is not possible to construct a fully relativistic version of the Bohm approach is incorrect. We obtain the fully relativistic version by using an approach based on Clifford algebras outlined in two earlier papers by Hiley and by Hiley and Callaghan. The relativistic model is different from the one originally proposed by Bohm and Hiley and by Doran and Lasenby. We obtain exact expressions for the Bohm energy-momentum density, a relativistic quantum Hamilton-Jacobi for the conservation of energy which includes an expression for the quantum potential and a relativistic time development equation for the spin vectors of the particle. We then show that these reduce to the corresponding non-relativistic expressions for the Pauli particle which have already been derived by Bohm, Schiller and Tiomno and in more general form by Hiley and Callaghan. In contrast to the original presentations, there is no need to appeal to classical mechanics at any stage of the development of the formalism. All the results for the Dirac, Pauli and Schroedinger cases are shown to emerge respectively from the hierarchy of Clifford algebras C(13),C(30), C(01) taken over the reals as Hestenes has already argued. Thus quantum mechanics is emerging from one mathematical structure with no need to appeal to an external Hilbert space with wave functions.

arXiv:1011.4033•Nov 17, 2010•23 citations

Does the Wave Function Provide a Complete Description of Physical Reality?

Sheldon Goldstein, Joel L. Lebowitz

This is the introduction to the section on Quantum Mechanics in the centennial collection of noteworthy articles appearing in The Physical Review and Physical Review Letters through 1983, since it all began in 1893. The selections for this section are "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" by Einstein, Podolsky and Rosen; Bohr's response, with the same title; Bohm's first hidden variables paper: "A Suggested Interpretation of the Quantum Theory in Terms of `Hidden' Variables. I"; Aharonov and Bohm's "Significance of Electromagnetic Potentials in the Quantum Theory"; and "Experimental Test of Bell's Inequalities Using Time-Varying Analyzers" by Aspect, Dalibard and Roger.

arXiv:quant-ph/9512028•Dec 29, 1995•1 citation

A retrospective review of von Neumann’s analysis of hidden variables in quantum mechanics

Robert Golub, S. K. Lamoreaux

This article reviews the history of J. von Neumann’s analysis of hidden variables in quantum mechanics and the subsequent analysis by others. In his book The Mathematical Foundations of Quantum Mechanics, published in 1932, von Neumann performed an analysis of the consequences of introducing hidden parameters (hidden variables) into quantum mechanics. He arrived at two principal conclusions: first, hidden variables cannot be incorporated into the existing theory of quantum mechanics without major modifications, and second, if they did exist, the theory would have already failed in situations where it has been successfully applied. This analysis has been taken as an “incorrect proof” against the existence of hidden variables, possibly due to a mistranslation of the German word prufen. von Neumann’s so-called proof isn’t even wrong as such a proof does not exist, but it is an examination of the limitations imposed by internal consistency of the Hilbert space formulation of the theory. One of the earliest attempts to eliminate uncertainty, by D. Bohm, requires a major modification of quantum mechanics (observables are not represented by Hermitian operators), which supports von Neumann’s first principal conclusion. However, testing the Bohm theory requires constructing a physically impossible initial state. As such, the theory has no experimental consequences, so W. Pauli referred to it as an “uncashable check”. As there are no observable consequences, the Bohm theory is possibly a counterexample to von Neumann’s second conclusion that hidden variables in particular would have already led to a failure of the theory.

Academia quantum.•Aug 26, 2024•4 citations

On the Einstein–Podolsky–Rosen paradox

James S. Bell

Introduction The paradox of Einstein, Podolsky and Rosen was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality. In this note that idea will be formulated mathematically and shown to be incompatible with the statistical predictions of quantum mechanics. It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty. There have been attempts to show that even without such a separability or locality requirement no ‘hidden variable’ interpretation of quantum mechanics is possible. These attempts have been examined elsewhere and found wanting. Moreover, a hidden variable interpretation of elementary quantum theory has been explicitly constructed. That particular interpretation has indeed a grossly non-local structure. This is characteristic, according to the result to be proved here, of any such theory which reproduces exactly the quantum mechanical predictions. Formulation With the example advocated by Bohm and Aharonov, the EPR argument is the following. Consider a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern–Gerlach magnets, on selected components of the spins σ 1 and σ 2 .

Cambridge University Press eBooks•Jun 3, 2004•26 citations

Moment map and gauge geometric aspects of the Schrödinger and Pauli equations

Mauro Spera

In this paper we discuss various geometric aspects related to the Schrödinger and the Pauli equations. First we resume the Madelung–Bohm hydrodynamical approach to quantum mechanics and recall the Hamiltonian structure of the Schrödinger equation. The probability current provides an equivariant moment map for the group [Formula: see text] of volume-preserving diffeomorphisms of [Formula: see text] (rapidly approaching the identity at infinity) and leads to a current algebra of Rasetti–Regge type. The moment map picture is then extended, mutatis mutandis, to the Pauli equation and to generalized Schrödinger equations of the Pauli–Thomas type. A gauge theoretical reinterpretation of all equations is obtained via the introduction of suitable Maurer–Cartan gauge fields and it is then related to Weyl geometric and pilot wave ideas. A general framework accommodating Aharonov–Bohm and Aharonov–Casher effects is presented within the gauge approach. Furthermore, a kind of holomorphic geometric quantization can be performed and yields natural “coherent state” representations of [Formula: see text]. The relationship with the covariant phase space and density manifold approaches is then outlined. Comments on possible extensions to nonlinear Schrödinger equations, on Fisher-information theoretic aspects and on stochastic mechanics are finally made.

International Journal of Geometric Methods in Modern Physics•Feb 19, 2016•11 citations
In this paper, we outline the research conducted by the first named author and his associates on the axiom-preserving, thus isotopic completion of quantum mechanics into hadronic mechanics according to the historical legacy by A. Einstein, B. Podolsky and N. Rosen that quantum mechanics is not a complete theory and review the ensuing exact representation of the magnetic moment and spin of the Deuteron in its ground state thanks to the isotopic completion of Pauli's matrices with an explicit and concrete content of D. Bohm's hidden variable [Formula: see text]. We then outline the independent studies conducted by the second named author on the representation of the conventional Pauli's matrices via geometric Clifford algebras. We finally show that the combination of the two studies allows a mathematically rigorous, numerically exact and time invariant geometric representation of the magnetic moment, spin and hidden variable of the Deuteron in its ground state.
In this paper we show how all the quantum properties of Schroedinger and Pauli particles can be described entirely from within a Clifford algebra taken over the reals. There is no need to appeal to any `wave function'. To describe a quantum system, we define the Clifford density element [CDE] as a product of an element of a minimal left ideal and its Clifford conjugate. The properties of the system are then completely specified in terms of bilinear invariants of the first and second kind calculated using the CDE. Thus the quantum properties of a system can be completely described from within the algebra without the need to appeal to any Hilbert space representation. Furthermore we show that the essential bilinear invariants of the second kind are simply the Bohm energy and the Bohm momentum, entities that make their appearance in the Bohm interpretation. We also show how these parameters emerge from standard quantum field theory in the low energy, single particle approximation. There is no need to appeal to classical mechanics at any stage. This clearly shows that the Bohm approach is entirely within the standard quantum formalism. The method has enabled us to lay the foundations of an approach that can be extended to provide a complete relativistic version of Bohm model. In this paper we confine our attention to the details of the non-relativistic case and will present its relativistic extension in a subsequent paper.
In this paper we present for the first time a complete description of the Bohm model of the Dirac particle. This result demonstrates again that the common perception that it is not possible to construct a fully relativistic version of the Bohm approach is incorrect. We obtain the fully relativistic version by using an approach based on Clifford algebras outlined in two earlier papers by Hiley and by Hiley and Callaghan. The relativistic model is different from the one originally proposed by Bohm and Hiley and by Doran and Lasenby. We obtain exact expressions for the Bohm energy-momentum density, a relativistic quantum Hamilton-Jacobi for the conservation of energy which includes an expression for the quantum potential and a relativistic time development equation for the spin vectors of the particle. We then show that these reduce to the corresponding non-relativistic expressions for the Pauli particle which have already been derived by Bohm, Schiller and Tiomno and in more general form by Hiley and Callaghan. In contrast to the original presentations, there is no need to appeal to classical mechanics at any stage of the development of the formalism. All the results for the Dirac, Pauli and Schroedinger cases are shown to emerge respectively from the hierarchy of Clifford algebras C(13),C(30), C(01) taken over the reals as Hestenes has already argued. Thus quantum mechanics is emerging from one mathematical structure with no need to appeal to an external Hilbert space with wave functions.
This is the introduction to the section on Quantum Mechanics in the centennial collection of noteworthy articles appearing in The Physical Review and Physical Review Letters through 1983, since it all began in 1893. The selections for this section are "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" by Einstein, Podolsky and Rosen; Bohr's response, with the same title; Bohm's first hidden variables paper: "A Suggested Interpretation of the Quantum Theory in Terms of `Hidden' Variables. I"; Aharonov and Bohm's "Significance of Electromagnetic Potentials in the Quantum Theory"; and "Experimental Test of Bell's Inequalities Using Time-Varying Analyzers" by Aspect, Dalibard and Roger.
This article reviews the history of J. von Neumann’s analysis of hidden variables in quantum mechanics and the subsequent analysis by others. In his book The Mathematical Foundations of Quantum Mechanics, published in 1932, von Neumann performed an analysis of the consequences of introducing hidden parameters (hidden variables) into quantum mechanics. He arrived at two principal conclusions: first, hidden variables cannot be incorporated into the existing theory of quantum mechanics without major modifications, and second, if they did exist, the theory would have already failed in situations where it has been successfully applied. This analysis has been taken as an “incorrect proof” against the existence of hidden variables, possibly due to a mistranslation of the German word prufen. von Neumann’s so-called proof isn’t even wrong as such a proof does not exist, but it is an examination of the limitations imposed by internal consistency of the Hilbert space formulation of the theory. One of the earliest attempts to eliminate uncertainty, by D. Bohm, requires a major modification of quantum mechanics (observables are not represented by Hermitian operators), which supports von Neumann’s first principal conclusion. However, testing the Bohm theory requires constructing a physically impossible initial state. As such, the theory has no experimental consequences, so W. Pauli referred to it as an “uncashable check”. As there are no observable consequences, the Bohm theory is possibly a counterexample to von Neumann’s second conclusion that hidden variables in particular would have already led to a failure of the theory.
Introduction The paradox of Einstein, Podolsky and Rosen was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality. In this note that idea will be formulated mathematically and shown to be incompatible with the statistical predictions of quantum mechanics. It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty. There have been attempts to show that even without such a separability or locality requirement no ‘hidden variable’ interpretation of quantum mechanics is possible. These attempts have been examined elsewhere and found wanting. Moreover, a hidden variable interpretation of elementary quantum theory has been explicitly constructed. That particular interpretation has indeed a grossly non-local structure. This is characteristic, according to the result to be proved here, of any such theory which reproduces exactly the quantum mechanical predictions. Formulation With the example advocated by Bohm and Aharonov, the EPR argument is the following. Consider a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern–Gerlach magnets, on selected components of the spins σ 1 and σ 2 .
In this paper we discuss various geometric aspects related to the Schrödinger and the Pauli equations. First we resume the Madelung–Bohm hydrodynamical approach to quantum mechanics and recall the Hamiltonian structure of the Schrödinger equation. The probability current provides an equivariant moment map for the group [Formula: see text] of volume-preserving diffeomorphisms of [Formula: see text] (rapidly approaching the identity at infinity) and leads to a current algebra of Rasetti–Regge type. The moment map picture is then extended, mutatis mutandis, to the Pauli equation and to generalized Schrödinger equations of the Pauli–Thomas type. A gauge theoretical reinterpretation of all equations is obtained via the introduction of suitable Maurer–Cartan gauge fields and it is then related to Weyl geometric and pilot wave ideas. A general framework accommodating Aharonov–Bohm and Aharonov–Casher effects is presented within the gauge approach. Furthermore, a kind of holomorphic geometric quantization can be performed and yields natural “coherent state” representations of [Formula: see text]. The relationship with the covariant phase space and density manifold approaches is then outlined. Comments on possible extensions to nonlinear Schrödinger equations, on Fisher-information theoretic aspects and on stochastic mechanics are finally made.