Abstract
In this paper, we outline the research conducted by the first named author and his associates on the axiompreserving, 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 \(\lambda\). 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.
Similar content being viewed by others
Introduction
In this paper, we combine the studies by the first named author, the physicist R. M. Santilli, and the independent studies by the second named author, the mathematician G. Sobczyk, to present a mathematically rigorous, numerically exact and time invariant representation of the magnetic moment and spin of the Deuteron in its ground state.
We begin with an outline of the inability by quantum mechanics to achieve an exact representation of nuclear magnetic moments under the assumption of the tabulated values of the magnetic moments of the proton and of the neutron in vacuum^{1}
where \(\mu _N\) is the unit called nuclear magneton.
As an example, the magnetic moment predicted by quantum mechanics (qm) for the Deuteron under value (1) is given by
while the experimentally measured value is given by
resulting in a deviation of the theoretical prediction in excess of about \(3\%\) of the experimental value,
with larger deviations for heavier nuclei (Fig. 1).
Additionally, quantum mechanics has been unable to achieve a consistent representation of the spin \(S_d = 1\) of the Deuteron in its ground state, since the only stable bound state predicted by quantum mechanics at short distance for a proton and a neutron is the singlet with antiparallel spins and null total angular momentum \(J_d = 0\). As a result of this occurrence, the spin \(S_d = 1\) of the Deuteron is represented via a collection of orbital contributions yielding \(J_d = 1\)^{3} that cannot exist in the ground state; and other unresolved aspects.
Fermi^{4}, Weisskopf^{3} and other founders of nuclear physics formulated the hypothesis that in the transition from isolated particles in vacuum to members of a nuclear structure, protons and neutrons experience a deformation of their extended charge distribution with consequential change of their magnetic moments (1) while conserving their spin 1/2 (see the statement at the top of p. 31 of Refs.^{3,5} for a preliminary experimental verification). Despite its simplicity, the above FermiWeisskopf hypothesis has profound mathematical and theoretical implications because it requires the transition from the quantum mechanical representation of particles as nondeformable points to a covering representation of particles as extended, therefore deformable.
In fact, an important axiomatic limitation of quantum mechanics is precisely its strict locality, that is, the sole possibility of characterizing particles as being pointlike in vacuum, thus solely admitting actionatadistance potential interactions (hereon called Hamiltonian interactions).
By contrast, experimental measurements have established that nuclear volumes are generally smaller than the sum of the volume of the nucleon constituents. Consequently, nuclei are composed by extended nucleons in condition of partial mutual penetration of their dense charge distributions, resulting in Hamiltonian as well as new interactions expected to be nonlinear (in the wave function), nonlocal (because defined over volumes) and not representable with a Hamiltonian (hereon called nonHamiltonian interactions).
Additional experiments have established deviations of the predictions of quantum mechanics from experimental values for: the anomalous magnetic moment of the muons^{6}; atoms in condensed matter^{7}; heavy ions physics^{8}; and other fields. The above insufficiencies of quantum mechanics supports:

(1)
The historical 1935 argument by A. Einstein, B. Podolsky and N. Rosen that “quantum mechanics is not a complete theory” (EPR argument)^{2};

(2)
The historical completions of quantum mechanics, such as the nonlinear completion by Heisenberg^{9}, the nonlocal completion by de Broglie and Bohm^{10}, and the completion via hidden variables by Bohm^{11};

(3)
The studies over the past fifty years conducted by R. M. Santilli and other scholars on the EPR completion of quantum mechanics into the axiompreserving hadronic mechanics^{12,13,14,15} for the representation of the dimension, shape and density of hadrons with Hamiltonian and nonHamiltonian interactions (see Ref.^{14} for detailed studies, Refs.^{16,17,18,19,20} for recent accounts, Ref.^{21} for a recent overview, proceedings^{22} on the 2020 International Teleconference on the EPR argument, and independent works^{23,24,25,26,27,28,29,30}).
The above studies have achieved a numerically exact and time invariant representation of the magnetic moment and spin of the Deuteron in its true ground state (that with null orbital contributions \(J_d = 0\)) thanks to an axiompreserving, thus isotopic completion of Pauli’s matrices with an explicit and concrete realization of Bohm’s hidden variable \(\lambda\), which completion was first introduced in Eq. (6.28.20), p. 354, Vol. II of Ref.^{14}, used in various applications such as the verification of the EPR argument^{16}, and then applied to the representation of the experimental data of the Deuteron in its ground state, resulting in the numeric value \(\lambda = 2.65557\) (see the latest Ref.^{31} and prior references quoted therein).
Independently from the above studies, the mathematician G. Sobczyk has conducted systematic studies on the representation of the conventional Pauli matrices, via geometric Clifford algebras (see Refs.^{32,33,34,35,36} and references quoted therein).
In this paper we show that, by joining the studies by R. M. Santilli and G. Sobczyk, geometric Clifford algebras can provide a mathematically rigorous, numerically exact and time invariant representation of the magnetic moment, spin and hidden variable of the Deuteron in its ground state.
Outline of isotopic methods
In this section, we outline the isomathematical and isomechanical branch of hadronic mechanics^{14} (see^{18,19,20} for a recent review). Let \(\xi : \{(a, b, \ldots ), \times , I\}\) be the universal enveloping associative algebra of quantum mechanics with generic elements a, b product \(ab = a\times b\) and conventional multiplicative unit \(I = 1\). The central assumption of isotopic methods, first introduced in p. 71, Vol. II^{13}, is the EPR completion of \(\xi : \{(a, b, ..), \times , I\}\) into the isoassociative algebra \({\hat{\xi }}:\{(a, b\ldots ), \star , {\hat{I}}\}\) characterized by the axiompreserving product called isoproduct
where the quantity \({\hat{T}}\), called the isotopic element, is positivedefinite but otherwise possesses an arbitrary functional dependence on local variables such as relative coordinates r, momenta p, wavefunctions \(\psi\), etc. hereon tacitly implied.
Completion (5) of the product evidently implies the corresponding compatible completion of the multiplicative unit 1 into the isounit of \({\hat{\xi }}: \{(a, b,\ldots ), \star , {\hat{I}}\}\)^{37}
with realizations of the type^{15}
where: \(n_{k\rho }^2, k = 1, 2, 3\) represents the nucleon semiaxes which are hereon assumed to be the same for all nucleons, with normalization for the sphere
where \(n_3^2\) is directed along the spin axis; \(n_{4\rho }^2\) represents the nucleon density also assumed for simplicity to be the same for all nucleons; and \(\Gamma\) represents nonlinear, nonlocal and non potential interactions.
Consequently, the representation of extended particles via isotopic methods requires two operators, the conventional Hamiltonian H for the representation of Hamiltonian interactions, and the isotopic element \({\hat{T}}\) for the representation of the dimension, shape and density of particles as well as their nonHamiltonian interactions.
Via the use of isomathematics and isomechanics, Santilli achieved in 1994^{38} (see also Ref.^{39} and the review/update^{20}) an exact representation of the magnetic moment of the Deuteron and of other stable nuclei via a quantitative representation of the Fermi–Weisskopf hypothesis^{4,3} permitted by realizations (7).
Since, by main assumption, the spin of nucleons remains \(S = 1/2\), and the magnetic moment of nucleons is linked to the spin via the known gyromagnetic factor g
Santilli^{38} reached the following completion of the gyromagnetic factor for individual nucleons (called isorenormalization because it is characterized by nonHamiltonian interactions)
The knowledge of the nucleon density \(n_4\), from the fit of particle physics experiments^{40,41} and of deviation (4) for the deformation of \(n_3\), allowed a numeric representation of the magnetic moment of the Deuteron and of other nuclei.
The representation of the spin \(S_d = 1\) of the Deuteron as a stable twobody problem of spin 1/2 particles in the ground state (i.e., with null value of the angular momentum) was first reached in Section 8.2.5, p. 90 on of Ref.^{15}, and then studied in detail in Ref.^{31} jointly with the first known achievement of the numeric value of Bohm’s hidden variable \(\lambda\) for the Deuteron, thanks to the axial triplet coupling permitted by hadronic mechanics with the proton and the neutron coupled along their symmetry axis with parallel spin^{42} (Figs. 2 and 3).
Isotopic methods also allowed a representation of the spin \(S_d = 1\) of the Deuteron in its ground state thanks to the axial triplet coupling of the proton and the neutron^{42} (see Section 2.7.2 of Ref.^{20}, Section 8.2.5 of Ref.^{15} and the latest Ref.^{31}).
(iii) Accurate fits without the conjecture of unknown parameters or particles of the BoseEinstein correlation^{40,41}, the behavior of the mean lives of unstable hadrons with speed^{43,44}, the anomalous magnetic moment of the muons^{45}, and new vistas for the possible achievement of controlled nuclear fusions^{15,42}.
Subsequently, Santilli^{16} achieved in 1998 a verification of the EPR argument via the Lieisotopic completion of the Lie algebras of quantum mechanics (see Refs.^{13,14}, independent study^{24} and the recent update^{46}) applied to the isotopies \(\hat{SU}(2)\) of the SU(2)spin algebra (see Section 3 of Ref.^{19} for a recent review). The main results of this study are:

(I)
The confirmation of the validity of Bell’s inequality^{47} for pointlike particles under electromagnetic interactions and its inapplicability (rather than its violation) for extended particles under strong interactions with Hamiltonian and nonHamiltonian interactions and consequential recovering of the classical picture for systems of extended spin 1/2 particles under strong interactions^{16}.

(II)
The progressive recovering of Einstein’s determinism^{2} with the increase of the density of strongly interacting systems, and its full recovering at the limit of Schwartzschild horizon^{17}.

(III)
The conversion of quantum mechanical strongly divergent perturbative series into strongly convergent isotopic series, evidently in view of the very small value of the isotopic element \({\hat{T}}\) of Eq. (9), sandwiched in between all products. Apparently, this result sets up the foundations for the elimination of divergencies in particle physics (see Chapters 11 and 12, Vol. II of Ref.^{14} and upgrade^{15}). As an example isotopic methods achieved the first known attractive force between the identical electrons of valence bonds (see Chapter 4 on of^{48}) whose resulting strong valance bond permitted the first known exact representation of experimental data for the hydrogen^{49} and water^{50} molecules without divergent perturbative series.
Representation of nuclear magnetic moments and spins via geometric Clifford algebras
Consider the fundamental representation of the SU(2)spin algebra on \(\mathbb {R}^3\) given by the celebrated Pauli’s matrices \(\sigma _k\), here recalled with the symbols of Clifford’s geometric algebra^{32,33,34,35,36}
(where \(e_3\) refers hereon to the spin direction) with commutation rules
and enveloping associative algebra \(\xi\) with products of the type \(e_i e_j\). The spin \(S = 1/2\) can be characterized by the eigenvalue equations on a Hilbert space \(\mathbb {H}\) with basis \( b>\)
Santilli introduced in the 1983 Vol. II of^{13} the EPR completion \({\hat{\xi }}\) of the universal enveloping algebra \(\xi\) of Lie algebras, including that of SU(2) via nonunitary transformations
of the associative product, resulting in the isoproduct of \({\hat{\xi }}\)
with ensuing nonunitary transformation of the multiplicative unit I of SU(2) into the isounit of \(\hat{SU}(2)\)
Santilli then introduced in the 1995 Vol. II of^{14} the realization of the isotopic element \({\hat{T}}\) and isounit \({\hat{I}} = 1/{\hat{T}}\) in terms of Bohm’s hidden variables \(\lambda\)^{11}
resulting in the following isoPauli matrices first introduced in Eq. (6.8.20), p. 254, Vol. II^{14,16}
The classification of all possible isotopies of Pauli’s matrices was subsequently provided in Section 3, Ref.^{19}.
It should be noted that Santilli derived isoPauli matrices (18) from the isotopy in \((3+1)\)dimensions of Dirac’s gamma matrices thanks to the isotopy of the spinorial covering of the Poincaré symmetry \(\mathcal{{\hat{P}}}(3.1)\) Ref.^{52}.
Consequently, isoPauli matrices (18) are invariant over time when formulated on the HilbertMyungSantilli isospace \(\mathcal{{\hat{H}}}\)^{53} over the isofield of isocomplex isonumbers \(\mathcal{{\hat{C}}}\)^{37}.
Note also the isomorphism between the LieSantilli isoalgebra \(\hat{SU}(2)\) and the conventiional Lie algebra SU(2) which is established by the preservation under isotopy of the SU(2) structure constants in the isocommutation rules
It is important to note that, despite their seemingly generalized structure (18), the isoPauli matrices do indeed characterize the spin \(S = 1/2\) of nucleons when formulated via isomathematics^{14,51} on isospace \(\mathcal{{\hat{H}}}\) with isobasis \( {\hat{b}}>\)
The primary meaning of Santilli’s isoPauli matrices (18) is therefore that of introducing the new notion of hadronic spin whose numeric values are conventional yet possessing a novel degree of freedom given by the explicit and concrete realization of Bohm hidden variable \(\lambda\).
Independently from the above studies, Sobczyk^{32,33,34,35,36} studied the representations of the conventional Pauli matrices (11) in terms of Clifford’s algebra \(\mathbb {G}_3=\mathbb {G}_3(\mathbb {R}^3 )\) with standard basis
where \(\textbf{e}_1, \textbf{e}_2,\textbf{e}_3\) are interpreted as unit vectors along the x, y, zaxes of the coordinates space of te Euclidean 3space \(\mathbb {R}^3\), with properties
We then interpret the quantities
to be unit bivectors defining the oriented planar directions of the xy, xz, and yzplanes, respectively, with squares
The unit trivector \(i=\textbf{e}_{123}=\textbf{e}_1\textbf{e}_2\textbf{e}_3\), in the center of the algebra \(\mathbb {G}_3\), also has square \(1\).
The geometric algebra \(\mathbb {G}_3\) is isomorphic to the familiar \(2\times 2\) complex matrix algebra \({{{\mathcal {M}}}}_2(\mathbb {C})\), which defines the coordinate matrices of \(\mathbb {G}_3\) with respect to the spectral basis,
where \(u_{\pm }:=\frac{1}{2}(1\pm \textbf{e}_3 )\) are mutually annihilating idempotents satisfying the rules
as can be easily verified.
The complex \(2\times 2\) matrix
where \(i\equiv \textbf{e}_{123}\), is the coordinate matrix of the geometric number \(A \in \mathbb {G}_3\) with explicit form,
According to the above realization of our Clifford geometric algebra, Pauli matrices (11) are the coordinate matrices of the unit vectors \(\textbf{e}_1,\textbf{e}_2,\textbf{e}_3\) defined by
where again \(i = \textbf{e}_1 \textbf{e}_2 \textbf{e}_3\) is the unit trivector in the geometric algebra \(\mathbb {G}_3\). Thus, the so called Pauli vectors \(e_k\equiv \textbf{e}_k, k=1,2,3\), are identified with the unit vectors \(\textbf{e}_k\) along the xyzaxes (for details, see Refs.^{32,33,34,35,36}.
By upgrading and extending Ref.^{31}, we now study the representation of nuclear magnetic moments via the reformulation of isoPauli matrices (18)^{16} in terms of geometric algebra (21)^{33}. For this purpose, we note that when formulated on the associative envelope \(\xi\), the isoPauli matrices satisfy all algebraic properties of the conventional Pauli matrices. Consequently, we here introduce the representation of isoPauli matrices (18) in terms of Clifford geometric algebra \(\tilde{\mathbb {G}}_3 = \tilde{\mathbb {G}}_3(\mathbb {R}^3 )\) with the isobasis
and main properties equivalent to (22)–(29), including their interpretation in terms of vectors, bivectors and trivectors,
The standard basis of unit isovectors \(\{{\hat{\textbf{e}}}_1, {\hat{\textbf{e}}}_2, {\hat{\textbf{e}}}_3\}\) define the the x, y, z isocoordinate axes, respectively. The isospectral basis is
where \({\hat{u}}_{\pm }:=\frac{1}{2}(1\pm {\hat{\textbf{e}}}_3 )\) are mutually annihilating isoidempotents. In the standard isobasis of \({\hat{\mathbb {G}}}_3\),
where
is the unit isotrivector of the associative geometric algebra \(\tilde{\mathbb {G}}_3\). It must be remembered that for the isobasis vectors \({\hat{\textbf{e}_k}}\),
for \(k=1,2,3\), where the \(\star\) denotes the isogeometric product. The products in Eqs. (35) and (36) are also isoproducts.
We now show that the hidden variable \(\lambda\) of Ref.^{16} can provide a second representation of the deformation of the magnetic moment of nucleons of Refs.^{38,39} with consequential exact representation of nuclear magnetic moments^{31}.
By introducing the realization of the hidden variable \(\lambda\)
with respect to the standard basis (21), the isoPauli unit \({\hat{I}}\), the isoreciprocal \({\hat{T}}\), and the isovector basis \(\{ {\hat{\textbf{e}}}_k\}\), expressed in the geometric algebra \(\mathbb {G}_3\), are given by
which are a different expression of, but equivalent to, Eqs. (16) and (17). Consequently
which expression is equivalent to the second part of Eq. (15).
By recalling that \(e_3\) characterizes the nucleon spin \(S = 1/2\), we reach the important result that the replacement of the standard basis of the geometric Clifford algebra \(\mathbb {G}_3\) of Pauli’s matrices (11) with their isoPauli form (18), implies the EPR completion of \(e_3\) into the expression defined on the isobasis \( {\hat{b}}>\) of the isoPauli matrices
Recall that the quantum mechanical (qm) relationship between magnetic moments \(\mu\) and spins S occurs via the gyromagnetic factor g of Eq. (9), and that the corresponding relation for the isotopic branch of hadronic mechanics (hm) is given by an expression of the type^{38}
where K is an isorenormalization constant of the gyronamgentic factor g created by the the new notion of spin 1/2 represented by isoPauli matrices (18) with Bohm’s hidden variable \(\lambda\)^{14}.
By using property (41), we reach the relation
Recall also that: 1) Bohm’s hidden variable \(\lambda\) is associated with the spin of a particle according to Eq. (18); 2) The proton and the neutron have the same spin 1/2 and essentially the same mass, thus being characterized by the same \(\lambda\); and 3) The quantum mechanical representation of the magnetic moment of the Deuteron is in excess of about \(3\%\) according to Eq. (4). By also selecting the value for conformity with the selected spin orientation (Fig. 3)
we can write the expression per each nucleon
from which we obtain the isorenormalized values of the magnetic moment of the proton and of the neutron
with corresponding value for the magnetic moment of the deuteron
From this, we obtain the numeric value
with consequential confirmation of the numeric value of Bohm’s hidden variable for the Deuteron first achieved in Ref.^{31}
Its invariance over tine follows from the derivation of isoPauli matrices (18) from the LieSantilli isosymmetry \(\mathcal{{\hat{P}}}(4.1)\)^{16,52}.
The representation of the magnetic moment of \(4He2\) is a consequence (Fig. 3), but that for other stable nuclei requires the still missing consistent representation of nuclear spins which was initiated in Ref.^{54} and will be studied in a forthcoming paper.
We should finally note that we have not used the full isotopic Clifford algebra \(\hat{\mathbb {G}}\) introduced by da Rocha and Vaz Jr. (see^{55} with important applications to flavor quark theories), since we have merely introduced the conventional geometric formulation \(\tilde{\mathbb {G}}_3\) in terms of isoPauli matrices (18). This is due to the fact that the full geometric isotopy \(\hat{\mathbb {G}}_3\) of \(\mathbb {G}_3\) would have required the use of isoproduct (5) with the isotopic element \({\hat{T}} = e^{\phi e_3} = 1/{\hat{I}}\), and the consequential lack of representation in Eq. (41) of the magnetic moment of the Deuteron (3) for spin \(S_d = 1\) in the ground state.
The understanding is however that the full isoClifford isogeometric isoalgebra \(\hat{\mathbb {G}}_{3N}\) is expected to be important for the numerically exact and time invariant representation of the spins and magnetic moments of nuclei with \(N \ge 2\) nucleons.
Concluding remarks
Various experiments have established that the quantum mechanical prediction of the magnetic moment of the Deuteron is about \(3\%\) in excess of the experimental value, Eq. (4), with bigger deviations for bigger nuclei (Fig. 1)^{1}.
The basic event studied in this paper to achieve an exact representation of nuclear magnetic moments consists of the fact that a rotating charge distribution creates a magnetic field along the rotation axis. When the charge distribution is deformed into a prolate ellipsoid, the magnetic field decreases, and when it is deformed into an oblate ellipsoid it increases (Fig. 2).
This evidence led Fermi^{4}, Weisskopf^{3} and other founders of nuclear physics, to suggest the hypothesis that the difference between quantum mechanical predictions of nuclear magnetic moments and their experimental values may be due to deformations of the charge distributions of protons and neutrons under strong nuclear interactions.
The study of this historical hypothesis required the completion of quantum mechanics according to the 1935 historical argument by Einstein, Podolsky and Rosen, that quantum mechanics is not a complete theory^{2}, because quantum mechanics can only characterize pointlike particles that, as such, cannot be deformed.
Extensive studies over the past half a century conducted by R. M. Santilli and other scholars^{12,13,14,15,16,17,18,19,20,21,22} (see Refs.^{23,24,25,26,27,28,29,30} for independent reviews) have shown that the extended character of nucleons, their deformations and their consequential nonHamiltonian interactions can be represented via the axiompreserving, nonunitary completion of the universal enveloping associative algebra of quantum mechanics in terms of associative isoproduct \(a {\hat{T}} b\) of Eq. (5) where \({\hat{T}}\), called the isotopic element, is positivedefinite, yet possesses an arbitrary functional dependence on any needed local variable.
The dimension, shape and density of nucleons as well as their deformations and nonHamiltonian interactions are then represented with the new isotopic element \({\hat{T}}\) according to realizations of type (7).
These studies lead to the construction of the axiompreserving nonunitary completion of quantum mechanics into hadronic mechanics^{13,14,15} (see Refs.^{18,19,20} for a recent review) which coincides with quantum mechanics at the abstract, realizationfree level. Via the use of the isomathematical and isomechanical branch of hadronic mechanics, Santilli^{38} achieved in 1994 the first known numerically exact and time invariant representation of the magnetic moment of the Deuteron and other nuclei according to the Fermi Weisskopf hypothesis via a mere 1.5% prolate deformation of the charge distribution of protons and neutrons when they are members of a nuclear structure.
In subsequent studies, Santilli^{16} introduced the broader notion of hadronic spin for extended particles under strong interactions, which spin is illustrated by isoPauli matrices (18), possess conventional spin values, and exhibits an explicit and concrete realization of Bohm’s hidden variable \(\lambda\)^{11} in term of the isotopic element \({\hat{T}}\), thus being hidden in the axiom of associativity.
In this paper we have shown, apparently for the first time, that the indicated notion of hadronic spin with Bohm’s hidden variable \(\lambda\) allows a second numerically exact and time invariant representation of the magnetic moment of the Deuteron and of other nuclei. The representation has been possible thanks to the reformulation of Santilli’s isotopy of Pauli matrices (first introduced in p. 254, Vol. II of Ref.^{14} and used in Ref.^{16} for the verification of the EPR argument) in terms of the conventional Clifford’s algebras, Refs.^{32,33,34,35,36}.
In a nutshell, we can say that the Copenhagen interpretation of quantum mechanics deals with the simplest possible realization of quantum axioms, while the EPR completion of quantum into hadronic mechanics deals with progressively broader realizations of the same axioms for systems with progressively increasing complexity.
Data availability
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
References
Vonsovsk, S. Magnetism of Elementary Particles (Mir Publishers, Moscow, 1975).
Einstein, A., Podolsky, B. & Rosen, N. Can quantummechanical description of physical reality be considered complete?. Phys. Rev. 47, 777–791 (1935).
Blatt, J. M. & Weisskopf, V. F. Theoretical Nuclear Physics (Wiley, Hoboken, 1952).
Fermi, E. Nuclear Physics (University of Chicago Press, Chicago, 1949).
Rauch, H. & Zeilinger, A. Demonstration of SU(2) symmetry by neutron interferometers. Hadron. J. 4, 1280–1285 (1981).
Miller, J. P., de Rafael, E. & Roberts, B. L. Muon (g2): Experiment and theory. Rep. Prog. Phys. 70, 795–881 (2007).
Fadel, M., Zibold, F. T., Decamps, B. & Treutlein, Ph. Spatial entanglement patterns and Einstein–Podolsky–Rosen steering in Bose–Einstein condensates. Science 360, 409 (2018).
Schukraft, J. Heavyion physics with the ALICE experiment at the CERN Large Hadron Collider. Trans. R. Soc. A 370, 917–932 (2012).
Heisenberg, W. Lee model and quantization of non linear field equation. Nachr. Akad. Wiss. Gottingen IIa, 111 (1953).
Goldstein, S. Bohmian (de BroglieBohm) Mechanics (Stanford Encyclopedia of Philosophy, 2021).
Bohm, D. A suggested interpretation of the quantum theory in terms of hidden variables. Phys. Rev. 85, 166–182 (1952).
Santilli, R. M. Embedding of Liealgebras into Lieadmissible algebras. Nuovo Cimento 51, 570 (1967).
Santilli, R. M. Foundation of Theoretical Mechanics, Vol. I (1978) and Vol. II (1982), Springer, Germany. http://www.santillifoundation.org/docs/Santilli209.pdf, http://www.santillifoundation.org/docs/santilli69.pdf
Santilli, R. M. Elements of Hadronic Mechanics, Vols. I (1995), II (1995), III (2016), Ukraine Academy of Sciences, Kiev. http://www.ibr.org/ElementsHadrfonicMechanics.htm
Santilli, R. M. Overview of the Einstein–Podolsky–Rosen Argument That ’Quantum Mechanics is not a Complete Theory’ APAV  Accademia Piceno Aprutina dei Velati, Pescara, Italy (2021). http://www.santillifoundation.org/eproverview2021.pdf
Santilli, R. M. Isorepresentation of the Lieisotopic SU(2) algebra with application to nuclear physics and local realism. Acta Appl. Math. 50, 177–190 (1998).
Santilli, R. M. Studies on the classical determinism predicted by A. Einstein, B. Podolsky and N. Rosen. Ratio Math. 37, 5–23 (2019).
Santilli, R. M. Studies on A. Einstein, B. Podolsky and N. Rosen prediction that quantum mechanics is not a complete theory, I: Basic methods. Ratio Math. 38, 5–69 (2020).
Santilli, R. M. Studies on A. Einstein, B. Podolsky and N. Rosen prediction that quantum mechanics is not a complete theory, II: Apparent proof of the EPR argument. Ratio Math. 38, 71–138 (2020).
Santilli, R. M. Studies on A. Einstein, B. Podolsky and N. Rosen prediction that quantum mechanics is not a complete theory, III: Illustrative examples and applications. Ratio Math. 38, 139–222 (2020) http://eprdebates.org/docs/eprreviewiii.pdf.
DunningDavies, J. A present day perspective on Einstein–Podolsky–Rosen and its consequences. J. Mod. Phys. 12, 887–936 (2021).
BeghellaBartoli, S. & Santilli, R.M. (eds) Proceedings of the 2020 Teleconference on the Einstein–Podolsky–Rosen Argument That “Quantum Mechanics is not a Compete Theory” (Curran Associates, New York, NY 2021).
Aringazin, A. K., Jannussis, A., Lopez, F., Nishioka, M. & Veljanosky, B. Santilli’s LieIsotopic Generalization of Galilei and Einstein Relativities (Kostakaris Publishers, Athens, 1991).
Sourlas, D. S. & Tsagas, G. T. Mathematical Foundation of the LieSantilli Theory (Ukraine Academy of Sciences, Kyiv, 1993).
Lohmus, J., Paal, E. & Sorgsepp, L. Nonassociative Algebras in Physics (Hadronic Press, Palm Harbor, 1994).
Jiang, C.X. Foundations of Santilli Isonumber Theory (International Academic Press, 2001).
Ganfornina, R. M. F. & Valdes, J. N.: Fundamentos de la Isotopia de Santilli (International Academic Press, 2001). http://www.ibr.org/docs/spanish.pdf English Translation: Algebras, Groups and Geometries 32, p. 135–308 (2015). http://www.ibr.org/docs/Aversatranslation.pdf
Davvaz, B. & Vougiouklis, T. A Walk Through Weak Hyperstructures and\(H_v\)Structures (World Scientific, 2018).
Gandzha, I. & Kadeisvili, J. New Sciences for a New Era: Mathematical, Physical and Chemical Discoveries of Ruggero Maria Santilli (Sankata Printing Press, Nepal, 2011).
Georgiev, S. Foundations of IsoDifferential Calculus Nova Publishers, New York, Vol. 1: IsoDifferential and IsoIntegral Calculus for IsoFunctions in One Variable (2014), Vol. 2: IsoDifferential and IsoIntegral Calculus for IsoFunctions in Several Variables (2014), Vol. 3: IsoOrdinary IsoDifferential Equations (2014), Vol. 4: IsoDifference Equations (2015), Vol. 5: IsoStochastic IsoDifferential Equations (2015), Vol. 6: Theory of IsoMeasurable IsoFunctions (2016), New Edition of Vol. 1: IsoDifferential and IsoIntegral Calculus for IsoFunctions in One Variable (2022), IsoMathematics, Lambert Academic Publishing (2022).
Santilli, R. M. IsoRepresentation of the Deuteron Spin and Magnetic Moment via Bohm’s Hidden Variables. Prog. Phys. 18, 74–81 (2022) http://www.santillifoundation.org/docs/PiPpaper322.pdf.
Hestenes, D. & Sobczyk, G. Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics 2nd edn (Kluwer, 1992).
Doran, C. & Lasenby, A. Geometric Algebra for Physicists (Cambridge University Press, Cambridge, 2003).
Sobczyk, G. New Foundations in Mathematics: The Geometric Concept of numbers (Birkhauser, New York, 2013).
Sobczyk, G. Spacetime vector analysis. Phys. Lett. A 84, 45–49 (1981).
Sobczyk, G. Conformal mappings in geometric algebra. Notices AMS 59, 264–273 (2012).
Santilli, R. M. Isonumbers and genonumbers of dimensions 1, 2, 4, 8, their Isoduals and Pseudoduals, and “hidden numbers,’’ of dimension 3, 5, 6, 7. Algebras Groups Geom. 10, 273–295 (1993).
Santilli, R. M. A quantitative isotopic representation of the deuteron magnetic moment. in Proceedings of the International Symposium ’Dubna Deuteron93 (Joint Institute for Nuclear Research, Dubna, Russia, 1994). http://www.santillifoundation.org/docs/Santilli134.pdf
Santilli, R. M. The physics of new clean energies and fuels according to hadronic mechanics, in Spec. Issue J. New Energy (1998). http://www.santillifoundation.org/docs/Santilli114.pdf
Santilli, R. M. Nonlocal formulation of the BoseEinstein correlation within the context of hadronic mechanics. Hadron. J. 15, 1–50 and 15, 81–133 (1992). www.santillifoundation.org/docs/Santilli116.pdf.
Cardone, F. & Mignani, R. Nonlocal approach to the BoseEinstein correlation. Eur. Phys. J. C 4, 705 (1998).
Santilli, R. M. The novel intermediate controlled nuclear fusions, a report for its industrial realization. Hadron. J. 31, 15 (2008).
Cardone, F., Mignani, R. & Santilli, R. M.: On a possible energydependence of the Ko/s lifetime. J. Phys. G Part. Phys. 18, L61–L65 (1992) and 18, L141–L144 (1992).
Arestov, Yu., Santilli, R. M. & Solovianov, V. Experimental evidence on the isominkowskian character of the hadronic structure. Found. Phys. Lett. 11, 483–492 (1998) http://www.santillifoundation.org/docs/Santilli52.pdf.
Santilli, R. M. Representation of the anomalous magnetic moment of the muons via the Einstein–Podolsky–Rosen completion of quantum into hadronic mechanics. Prog. Phys. 17, 210–215 (2021).
Muktibodh, A. S. & Santilli, R. M. Studies of the regular and irregular isorepresentations of the Lie–Santilli isotheory. J. Gen. Lie Theor. 11, 1–7 (2007).
Bell, J. S. On the Einstein Podolsky Rosen paradox. Physics 1, 195–207 (1964).
Santilli, R. M. Foundations of Hadronic Chemistry, with Applications to New Clean Energies and Fuels (Kluwer Academic Publishers, 2001).
Santilli, R. M. & Shillady, D. D. A new isochemical model of the hydrogen molecule. Int. J. Hydrog. Energy 24, 943–956 (1999).
Santilli, R. M. & Shillady, D. D. A new isochemical model of the water molecule. Int. J. Hydrog. Energy 25, 173–183 (2000).
Santilli, R. M. Nonlocalintegral isotopies of differential calculus, mechanics and geometries. Rendiconti Circolo Matematico Palermo Suppl. 42, 7–82 (1996).
Santilli, R. M. Recent theoretical and experimental evidence on the synthesis of the neutron. in Communication of the JINR, Dubna, Russia, No. E493252 (1993), published in the Chin. J. Syst. Eng. Electr. 6, 177–194 (1995). http://www.santillifoundation.org/docs/Santilli18.pdf
Myung, H. C. & Santilli, R. M. Modularisotopic Hilbert space formulation of the exterior strong problem. Hadron. J. 5, 1277–1366 (1982) http://www.santillifoundation.org/docs/myungsantilli1982.pdf.
Bhalekar, A. A. & Santilli, R. M. Exact and invariant representation of nuclear magnetic moments and spins according to hadronic mechanics. Am. J. Mod. Phys. 5, 56–118 (2016).
da Rocha, R. & Vaz, J. Jr. Isotopic liftings of Clifford algebras and applications in elementary particle mass matrices. Int. J. Theor. Phys. 46, 2464–2487 (2007) arxiv.org/abs/0710.0832.
Acknowledgements
Thanks are due for penetrating critical comments received from the participants of the 2020 International Teleconference on the EPR argument, the 2021 International Conference on Applied Category Theory and GraphOperadLogic and the Seminars on Fundamental Problems in Physics. Additional thanks are due to various colleagues for technical controls and to Mrs. Sherri Stone for linguistic control of the manuscript. The author is solely responsible for the content of this paper.
Author information
Authors and Affiliations
Contributions
Both authors have contributed to the writing of the paper. Both Authors have reviewed the submitted manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher's note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Santilli, R.M., Sobczyk, G. Representation of nuclear magnetic moments via a Clifford algebra formulation of Bohm’s hidden variables. Sci Rep 12, 20674 (2022). https://doi.org/10.1038/s41598022249704
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41598022249704
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.