Dear readers, after a long pause, due mainly to my work commitments and to lack of important news, waiting for the demo, announced by Andrea Rossi and scheduled in November, I decided to write this post taking as a starting point the preprint of two works that will be published in Volume 25 of the JCMNS (Journal of Condensed Matter Nuclear Science), having an unusual title: “Maxwell’s Equations and Occam’s Razor” and “The Electron and Occam’s Razor”. Some of the arguments presented in these two papers were briefly introduced by the same authors at the ICCF20 Conference (held in Sendai, Japan), in a poster titled “The Zitterbewegung interpretation of quantum mechanics as theoretical framework for Ultra Dense Deuterium and Low Energy Nuclear Reactions”. Not having the scientific knowledge needed to make a judgment on these theoretical physics works, I decided to ask for the collaboration of a physicist and science writer, Mario Menichella, during his summer holidays in the city where I live, Viareggio. Following his suggestion, we publish, with the help of the authors, some intriguing hypotheses presented in the above cited publications. We put it in the form of an interview, to make all more interesting and easy. I hope that what you will read could be stimulating for those who want to deepen these fascinating themes with an open mind.

Enjoy the reading!

Vessela Nikolova

“It is now easier to smash an atom than to break a prejudice”

Ronald Lippitt


A Zitterbewegung Model for Ultra-Dense Hydrogen and Low Energy Nuclear Reactions

In the two papers, the Authors suggest a “purely electromagnetic model of the electron”. What is the fundamental feature of the proposed model?

It’s the attempt to respect, as much as possible, Occam’s razor, a principle proposed by the English philosopher William of Occam, which suggests to not introduce information and concepts that are not strictly necessary in solving problems. This principle can be considered as an excellent epistemological tool for the creation and evaluation of models. If we want to formalize the concept, we can say that the quality of a model is defined by two fundamental parameters: the first one is related to the achievement of the desired goals, such as the adherence of the model predictions to the data and concepts that we want to encode or interpret, while the other one is the simplicity of the model, a parameter that is inversely proportional to the number of informations, concepts, exceptions, postulates and parameters needed by the model itself.

Which mathematical formalism was used?

Scientific knowledge is based on mathematical language, but the importance of choosing the right tools is often underestimated, as the Authors point out. The formalism used is based on space-time algebra, one of the Clifford algebras introduced by the mathematician William K. Clifford in 1878. The advantages of such formalism in physics have been described in detail by prof. David Orlin Hestenes in the work “Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics”. Space-time algebra respects the Occam’s razor criteria in terms of simplicity and universality, and allows a precise geometric interpretation of concepts often hidden by the formalism of complex matrix algebra traditionally used in modern physics.

Prof. David Orlin Hestenes (on the left)

Can you briefly describe the currently widespread and accepted model of the electron and the differences with the model proposed by the Authors?

Simplifying, we could say that in Quantum Mechanics the electron is a point-like particle having an “intrinsic” mass, a charge, a magnetic moment, an angular momentum and “spin”. The particle behavior is described by a complex function of space and time. The “square” of this function represents the “probability density” of finding the particle in a particular point of space-time. According to classical physics, the point-like particle concept is incompatible with the observed electron properties. In order to justify such incompatibility, some exceptions are introduced, thus seriously violating the Occam razor’s principle. According to the laws of mechanics and electromagnetism, a point-like particle cannot have an “intrinsic angular momentum”, and a magnetic moment must necessarily be generated by a current, that cannot exist in a point-like particle. Moreover, the electric field generated by a point-like charged particle should have an infinite energy! Moreover, Quantum Mechanics does not even try to derive the concepts of charge and mass, which are simply considered as “intrinsic properties” of the particle. Simplifying, the model proposed by the Authors consists in a current ring generated by a massless charge that rotates at speed of light along a circumference whose length is equal to the Compton wavelength of the electron: about 2.4 \cdot 10^{-12} meters. The charge is not point-like but distributed on the surface of a sphere, whose radius is equal to the classic radius of the electron: about 2.8 \cdot 10^{-15} meters. Similar models, based on the concept of “current ring”, have been proposed by many authors but have often been ignored for their incompatibility with the most widely accepted interpretations of Quantum Mechanics. It is interesting to note how, already in his “Nobel lecture” of 1933, P.A.M. Dirac made reference to a high-frequency internal oscillation of the electron: “It is found that an electron which seems to us to be moving slowly, must actually have a very high frequency oscillatory motion of small amplitude superposed on the regular motion which appears to us. As result of this oscillatory the velocity of the electron at any time equals the velocity of light”.

In scientific literature the German term Zitterbewegung is often used to indicate this very fast swing / rotation.

Zitterbewegung trajectory radius rE ~ 0.386e-12 m [1.957e-6 1/eV]. Charge [in red] radius ~ rE/137.04.

How can you reconcile the concept of massless charge with the experimental value of 511 keV for the electron mass ?

In this model the mass is not an “intrinsic property” of the particle, but it is a value that can be derived from other parameters of the model itself. A key point of the model consist in assigning to the rotating charge a purely electromagnetic momentum qA = mc, whose value is equal to the product between the charge q and the vector potential A associated with the current generated by the charge itself. In natural units both the light speed and the Planck constant are adimensional scalars with value c = \hbar = 1 and the physical quantities dimensions are integer (positive, negative, or zero) powers of one electron-volt (eV). The rotating charge momentum has – when expressed in natural units – the dimension of an energy and a value equal to electron mass. Using natural units, the electron mass is also equal to the angular frequency of the charge rotation, and is equal to the inverse of the ring radius. But the mass value can also be obtained by integrating the square of the electric and magnetic field or by integrating the product of the current density for the vector potential. Starting from such model, it is thus possible to obtain the electron rest mass in six different ways.

There is a point which needs to be cleared up: how is it possible that a charge having a momentum is subject to a circular motion without a positive charge at the center of the orbit, as in Bohr atomic model ?

A magnetic flux equal to the ratio h/q –where h is the Planck constant and q is the elementary charge- is associated to the current ring: \Phi_M = h/q. The centripetal force, at the origin of the rotating movement, is the Lorentz Force due to the magnetic field.

I observe that, multiplying the amount of momentum of the rotating charge by the radius, we obtain a value of the angular momentum of the free electron equal to a single quantum of action \hbar. I would expect a value equal to half of this value, the commonly accepted value of the electron spin…

The hypothesis proposed by the Authors distinguishes spin and “intrinsic” angular momentum. Spin is interpreted as the component of the intrinsic angular momentum vector along the direction of an external magnetic field. This component can have only the two values \pm \hbar/2 when the electron is subject to the well-known Larmor precession.

Spin as a component of intrinsic angular momentum

Interesting hypothesis, but the concept of light-speed moving charges, fundamental for the proposed model, does not seem widespread discussed and studied in the mainstream scientific literature. What theoretical foundations suggest the existence of such charges ?

The conceptual foundation is the application of Occam’s razor to Maxwell’s equations. In the mainstream scientific literature, the so-called “Lorenz gauge” is often applied. It is a particular operation consisting essentially of zeroing an expression that appears in Maxwell’s equations. This expression represents a “scalar field”, a function that associates a single real value with the space-time coordinates. In the paper Maxwell’s equations and Occam’s razor, Lorenz’s gauge is considered as a normal “boundary condition”, that cannot be always applied. The hypothesis of the existence of a scalar field is not new: many authors, often quoting Nicola Tesla’s works, dealt with this subject. In particular, it is interesting to mention a recent project of the Oak Ridge National Laboratories (ORNL) entitled “Electrodynamic Scalar Wave Transmission and Reception” evaluating the possibility of an innovative communication system based on the concept of transmission and reception of scalar waves. The acceptance of the existence of a scalar field allows an interesting interpretation of the concept of “charge density” as the time derivative of the scalar field, as suggested by Giuliano Bettini in the work “Manuscripts of the late century”, published in the viXra archives. In this case, Maxwell’s equations describe only light-speed moving charges. It is interesting to note that Richard Feynman’s intriguing hypothesis that positron can be interpreted as an “electron traveling back in time” emerges immediately from this particular charge definition. Positron, differs from the electron only for the charge sign. Obviously, if you consider the charge density as the time derivative of the scalar field, the sign change of the time variable automatically flip the sign of the charge.

Nicola Tesla

In natural units, the electron rest mass-energy is equal to the Zitterbewegung angular frequency and is equal to the momentum of the rotating charge. Starting from these observations, is it possible to formulate a purely electromagnetic interpretation of both Newton’s laws and Special Relativity?

Yes. An example particularly easy to study describes an electron moving at constant speed vz along a direction orthogonal to the xy plane of charge rotation, which, consequently, will follow an helicoidal path at light speed. In this case the charge momentum vector qA = mc will have a component along the z axis. Calling m_e the rest mass of the electron we observe that the angular frequency \omega_e and the module of the component of momentum qA_\perp = m_ec = \hbar \omega_e/c  in the plane xy is a value that does not depend from v_z. It is thus possible to derive directly the value of the electron relativistic mass m by applying the Pythagorean theorem, considering that the component of the momentum m_ec  is orthogonal to the component mv_z = qA_z:

m_e^2c^2 + m^2v_z^2 = m^2c^2

A variation in speed will therefore result in an electric force f_z

f_z = \frac{d(mv_z)}{dt} = \frac{qdA_z}{dt} = qE_z

or, for non-relativistic speeds:

f_z =\frac{qdA_z}{dt} = qE_z \simeq m_e\frac{dv_z}{dt}  = m_ea

Zitterbewegung trajectories for different electron speeds: v/c = 0, 0.43, 0.86, 0.98

Can you describe the proposed relativistic model using a simple, easy to understand metaphor?

Considering the invariance of light speed c of the electric charge, it is possible to visualize the charge helicoidal trajectory of an electron moving at velocity v_z within a fixed time interval \Delta t, as a spring of length v_z \Delta t  formed by a thin elastic wire of constant length c\Delta t. The electron mass m = \hbar/rc is exactly equal to the inverse of the radius r of the spring when expressed in natural units: m = 1/r. An increase of v_z, will be represented by a spring elongation and a spring radius reduction. The radius reduction is inversely proportional to the relativistic mass increase. If we call r_e the spring rest radius \left(v_z = 0\right), it is possible to write the value of the radius r and mass m as a function of v_z:

r = r_e \sqrt{1-\frac{v_z^2}{c^2}}

m = \frac {m_e} {\sqrt{1-\frac{v_z^2}{c^2}}}

Of course, if the electron is observed at a spatial scale far greater than its Compton wavelength and at a timescale far greater than the very short period (\approx 8.1 \cdot 10^{-21} sec ) of the Zitterbewegung rotation, the electron can be approximated by a point-like particle that moves along the helix axis.

How can we describe shortly, by using simple concepts, the relation between Maxwell’s equations and the proposed model?

Space-time algebra uses an orthogonal basis of four unit vectors that obey the following simple rules:

\gamma_x^2 = \gamma_y^2 = \gamma_z^2 = -\gamma_t^2 = 1

\gamma_i\gamma_j = - \gamma_j\gamma_i \quad \forall \:\{i, j\}\: \in\: \{x, y, z, t\}\quad and \quad i \neq j

The algebra thus defined is isomorphic to the algebra of Majorana matrices. Now we define a generic function

A = A\left(x,y,z,t\right)=\left(\gamma_xA_x, \gamma_yA_y, \gamma_zA_z, \gamma_tA_t \right)

that associates each point of space-time with a four values vector. We define also a special vector \partial

\partial = \left(\gamma_x\partial_x, \gamma_y\partial_y, \gamma_z\partial_z, \gamma_t\partial_t\right)

whose components represent the derivative operators along the four directions of space-time.

Applying the derivative operators of \partial to the vector field A, a “spinorial field” is obtained, a function that associates a spinor to each point of space-time. The spinor, in space-time algebra, is a particular mathematical structure identified by seven values: a scalar field S, characterized by a single value and a “bivectorial” field F with six distinct values. The number six corresponds to the number of possible orthogonal planes (bivectors) of space-time: xy, xz, yz, xt, yt, zt.

\partial A = \gimel = S + F

If A is the electromagnetic four potential, the six F values are related to the three values (Ex, Ey, Ez) of the electric field and to the three magnetic field values (Bx, By, Bz). The S field, defined by a single value, is generally ignored in mainstream literature, where the “Lorenz gauge” is often applied, an operation that, as already mentioned, consists in assuming S = 0.

Applying the operator \partial to the field \gimel and setting the result to zero,

\partial\gimel = \partial^2A = \partial S + \partial F = 0

we obtain the Maxwell equations rewritten in a compact form, if we identify the four partial derivatives \partial S of the scalar field S, along the four space-time coordinates, as the electromagnetic field sources, i.e. the three values of the current density J and the charge density ρ. According to Occam’s razor principle, therefore, charge and current concepts are not introduced ad hoc in the model, but are derived from a single core entity, the electromagnetic four potential. The equation \partial \gimel = 0, if expanded, leads to a system of eight equations that link together the six values of the electromagnetic field F and the sources.

Beyond the formalism, what are the substantial consequences of this particular interpretation of Maxwell’s equations?

This rewriting of Maxwell equations implies the existence of scalar waves and the existence of light speed moving charges. Their equations are very simple:

\partial^2 S = 0

\partial^2 \rho = 0

How this proposed model relates with Dirac’s equation ?

For m = 0, the Dirac equation

\left(i\partial - m\right)\psi = 0

becomes the Weyl equation

\partial\psi = 0

an expression similar to equation \partial \gimel = 0, if rewritten using the formalism of space-time algebra.

The solution of these equations is a field of spinors. A spinor is a mathematical structure that has some analogies with complex numbers. As it is well known, a complex number z=\exp(i\theta) with module 1 and argument \theta encodes a generic rotation of \theta radians. In space-time algebra, the product \gamma_x\gamma_y has, like imaginary unit i, a negative square:

\left(\gamma_x\gamma_y\right)^2 = \gamma_x\gamma_y\gamma_x\gamma_y = -\gamma_x\gamma_y\gamma_y\gamma_x = -1

and the expression R_{xy} = \exp(\gamma_x\gamma_y\theta) represents a spinor that encodes a rotation in the xy plane. The product γzγt has a positive square (we remember that \gamma_t^2 = -1):

\left(\gamma_z\gamma_t\right)^2 = \gamma_z\gamma_t\gamma_z\gamma_t = -\gamma_z\gamma_t\gamma_t\gamma_z = 1

in this case the spinor R_{zt} = \exp(\gamma_z\gamma_t\phi) implements a hyperbolic rotation in the zt plane. Simplifying, the (non-commutative) product of the two spinors encodes the helicoidal trajectory of the electron charge if we set \theta = \omega_et and \phi = \tanh^{-1}(v_z/c).

What are the main differences with Hestenes model?

In the Hestenes model the charge is point-like shaped. In his more recent works, the Zitterbewegung radius is equal to half the value of the reduced electron Compton wavelength. Moreover, in the Hestenes model, the Zitterbewegung angular speed decreases as a result of the relativistic time dilation when the electron is accelerated. This point, in particular, is not compatible with the model proposed in the two papers of Vol. 25 of the JCMNS, where the value of the mass, the ZBW radius, the Zitterbewegung angular frequency, the current and the vector potential associated with the charge motion are strictly interdependent parameters. The correlation between these parameters demands a relativistic contraction of the radius, an increase in the instantaneous angular speed \omega=c/r and the invariance of angular frequency \omega_e in the xy plane orthogonal to the direction of motion.

Are there any experimental results that could be interpreted using this particular electron model?

A series of experiments conducted over the last ten years by prof. Leif Holmlid (University of Gothenburg) have proved the existence of a very compact form of deuterium. Starting from the kinetic energy value (about 630 eV) of the nuclei emitted in some experiments where this particular form of ultra-dense deuterium is irradiated by a small laser, he calculates a distance between deuterium nuclei of about 2.3 \cdot 10^{-12} m, a much smaller value than the distance of about 74 \cdot 10^{-12} between the nuclei of a normal deuterium molecule. A preliminary hypothesis about the structure of the ultra-dense hydrogen (or deuterium) structure can be proposed starting from the electron and proton Zitterbewegung models. The proton can be seen as a current ring generated by a positive elementary charge that moves at the velocity of light along a circumference whose length is equal to the Compton proton wavelength \left(\lambda_p \approx 1.3 \cdot 10^{-15} m\right). According to this hypothesis, the proton would be much smaller than the electron, being the ratio between the radii of the two current rings equal to the inverse of masses ratio: r_e/r_p = m_p/m_e \approx 1836. An hypothetical structure (Z-Hydrino or Zitterbewegung Hydrino) formed by an electron with a proton (or a deuteron) in its center would have a potential energy of -q^2/r_e \approx -3.7 keV, a value equal to the energy of an X-ray photon with a wavelength of about 3.3 \cdot 10^{-10} m. The distance between the deuterium nuclei in the Holmlid experiment could be explained by an aggregate of these structures. In these hypothetical aggregates, the Zitterbewegung phase difference of two neighboring electrons is \pi radians and the distance d_c between the charges of the two electrons is equal to electron Compton wavelength d_c = \lambda_c \approx 2.42 \cdot 10^{-12} m. In this case the distance between the nuclei can be obtained by applying the Pythagorean theorem:

d_i = \sqrt{\lambda_c^2 - \lambda_c^2/\pi^2} \approx 2.3 \cdot 10^{-12} m

Ultra Dense Hydrogen model. Proton distance ~ 2.3e-12 m [1.16e-5 1/eV]

at this point, it is important to briefly mention the interesting work of Jan Naudts, “On the hydrino state of the relativistic hydrogen atom“, where the author, applying the Klein-Gordon equation to hydrogen atom, finds an energy level E_0 \approx m_ec^2\alpha \approx 3.7 keV.

Prof. Leif Holmlid

In the Iwamura experiment, a low energy nuclear transmutation of deposited elements was observed on a system consisting of alternate thin layers of palladium (Pd) and calcium oxide (CaO). A transmutation occurs when the system is crossed by a deuterium flow. The CaO layer, essential for the transmutation, is hundreds of atomic layers away from the area, near the surface, where the atoms to be transmuted have been implanted. It is therefore interesting to find an hypothesis that explains the action at distance and the role of CaO and the deuterium nuclei overcoming of the coulomb barrier.

Prof. Yasuhiro Iwamura

A possible hypothesis may arise from considering essential the ultra-dense deuterium (UDD) formation at interface between calcium oxide and palladium, an area where the high work function difference between Pd and CaO favors the formation of a layer with an High Electron Density (SEL, Swimming Electron Layer). The ultra-dense deuterium could later migrate to the area where the atoms to be transmuted have been implanted. This hypothesis seems more realistic than the hypothesis of di-neutrons (couples of neutrons) formations, consequence of an hypothetical nuclear capture of the electron, considering the very high energy required to balance this process (~ 0.78 MeV). A more realistic hypothesis sees the Ultra-dense deuterium aggregates, having no charge, as the be probable cause of the transmutation of Cs into Pr and Sr into Mo. Using the Holmlid’s notation “D(0)” to indicate the “mini-atoms” of ultra-dense deuterium, the hypothesized reaction for Cesium transmutation into Praseodymium in the Iwamura experiments would be very simple:

^{133}_{55}Cs + 4D(0)\:\rightarrow \:^{141}_{59}Pr + 4e

In this context, the electrons would have the role of carrier of deuterons towards the nuclei to be transmuted.

Andrea Rossi in his Miami office with one of his favorite paintings

Seems that a possible role of electrons in low-energy nuclear reactions has also been proposed by Gullström and Rossi in the their last theoretical work “Nucleon polarizability and long range strong force from σI = 2 meson exchange potential”:

A less probable alternative to the long range potential is if the e-N coupling in the special EM field environment would create a strong enough binding to compare an electron with a full nuclide. In this hypothesis, no constraints on the target nuclide are set, and nucleon transition to excited states in the target nuclide should be possible. In other words these two views deals with the electrons role, one is as a carrier of the nucleon and the other is as a trigger for a long range potential of the nucleon”.

Ultra-dense hydrogen or deuterium aggregates could be the cause of “many-body” nuclear reactions ? These reactions are currently considered impossible or highly unlikely.

We do not know, but if confirmed it would be difficult to find an alternative explanations for such reactions. Interestingly to note that, already in the 1990s, Brian Ahern’s patent US5411654 refers to “many-body” nuclear reactions:

“Condensed matter systems in which the deuteron nuclei motions are synchronized to such a high degree are expected to generally tend toward conditions that favor 3- and 4-body strong force interactions. Such many-bodied, cooperative oscillations permit 3 nuclei to be confined in, or close, to, the strong force envelope simultaneously, providing a corresponding increase in interaction potential. Prediction of reaction by-products of 3- and 4-body strong force interactions are beyond current understanding. High energy scattering experiments are of no predictive use, owing to the immeasurably low probability of even a 3-body interaction.”.

Incidentally, the cited patent also addresses other key issues such as, for example, the phenomena of energy localization in nano-structured materials.

What are the main differences with Randell L. Mills’ hydrino?
In Mills’ theory, the charge density equation ρ involves the existence of charges moving at speed v < c:

\left(\partial_x^2 + \partial_y^2 + \partial_z^2 - \frac{1}{v^2}\partial_t^2\right)\rho = 0

while the equation of the model proposed by the Authors

\partial^2\rho =\left(\partial_x^2 + \partial_y^2 + \partial_z^2 - \frac{1}{c^2}\partial_t^2\right)\rho = 0

demands, as already mentioned, the existence of light-speed moving charges. Moreover Mills’ theory contemplates the possibility of many energy levels for the hydrogen atom below the commonly accepted fundamental level.

Other experimental results that suggest the hypothesis of compact forms of hydrogen?

In the 1960s, in an attempt to demonstrate the hypothesis that a neutron is a compressed form of hydrogen, Don Carlo Borghi has conducted an experiment in which partially ionized hydrogen was crossed by a 10 GHz microwaves beam generated by a Klystron. The experiment aimed to test the possibility of synthesizing neutrons from protons and electrons. The neutron synthesis should not be a considered a realistic hypothesis, taking into account the energy necessary to balance the nuclear electron capture p + e \rightarrow n. More likely, though not yet proven, is the possibility of ultra-dense hydrogen formation: p + e \rightarrow H (0).


In conclusion, I would like to point out the possibility of asking questions and commenting on my blog for those who want to deepen the topics discussed or want clarifications about them. A sincere thank to Mario Menichella and to those who have collaborated for the realization of this post.