The Quantum Bicycle Society is pleased to share open access to the following selection of articles. Below you will find:

**1. The Quantum Bicycle Journal**: papers published exclusively by us.

**2. Papers Published Elsewhere** by Quicycle members and contributors.

**3. Classic Papers** we felt it worthwhile to include.

## 1. The uantum Bicycle Journal

**Authors Listed:**Williamson, J.G., Benn, A., Mercury, M. (2022)

Robinson, V.N.E. (2020)

Ord, G.N. (2020)

Van Der Mark, M.B., Williamson, J.G. (2016)

**Quantum Spin Coherence In Four Derived 3-Spaces**

WILLIAMSON, J.G., Benn, A., Mercury, M. (2022)

**Abstract**:

A new chemical theory is proposed that integrates sub-quantum mechanics — a development of relativistic quantum mechanics and electromagnetism — in a way that allows new insights into the modeling of electrons in atomic and molecular orbitals. Central to this approach is the mathematical treatment of each aspect of a system’s energy flow in its appropriate space-time components, as well as the fact that the internal wave nature of energy and matter results in quantum systems with a coherent, resonant, harmonic nature. It is this wave nature that lies at the root of quantisation itself. This paper presents an approach to the modeling of a quantum system in a more complete way. It is proposed that derivatives in space-time lead to four, 3-dimensional ‘dynamical spaces.’ This proves significant because these derivative spaces describe physical observables such as spin, electric field, and magnetic field. These have different properties, and this approach allows each to be considered either separately or together. The four 3-component derivative spaces are: energy (or frequency) space, electric field space, magnetic field space, and spin (or angular momentum) space. At the root of each of these are the properties of the *one* underlying 4-dimensional space-time. In particular, emerging from this new theory is a new way to model quantum spin and its contributions to energy, harmonic stability, and quantisation. This approach may also account for several phenomena that relate to elemental magnetic properties and solid state crystalline properties.Keywords:

*quantum spin, harmonic resonance, field cancellation, relativistic quantum mechanics, quantum chemistry*

**Moving Clocks And Special Relativity Rest Reference Frames**

ROBINSON, V.N.E. (2020)

**Abstract**:

Einstein’s special relativity theory predicts that time will go slower on moving clocks than stationary clocks. Without an absolute rest reference frame, how do we know which clock is moving and which is stationary? Although there is no absolute rest reference frame there are some practical rest reference frames. When the appropriate rest reference frame is chosen, clocks moving with respect to that rest reference pass time slower than those at rest in that reference frame.

*Below, the second version was released following a video peer review session, viewable HERE .*Keywords:

*Relativity, Special Relativity, Special Relativity corrections, General Relativity, inertial reference frame*

**Perspective, Images, And Quantum Mechanics As A Relativistic Effect**

ORD, G.N. (2020)

**Abstract:**

Much of human cognition is devoted to image analysis. From infancy we learn to process the two dimensional images we receive on our retinas, decoding what we believe to be a representation of a world of objects located in a three dimensional space, viewed from a particular perspective. The consistency of that decoding is very strong: ‘seeing is believing.’ However, it is not infallible, and visual clues can give rise to mistaken perspectives. There is an analog of mistaken perspective in a ‘received view’ on the relation between quantum mechanics and special relativity. We explore a counter-argument, that quantum mechanics is in fact a relativistic effect, primarily through the use of images.

Keywords:*special relativity, light cone, quantum mechanics, perspective*, *Twins Paradox*

**Inversion And General Invariance In Spacetime**

VAN DER MARK, M.B., WILLIAMSON, J.G. (2016)

**Abstract**:

A general formula for division in a relativistic Clifford-Dirac algebra is derived. Where division is undefined turns out, in many cases, to correspond to areas of physical interest, such as the light cone, invariant quantities in electromagnetism, and the basis set of quantities in the Dirac equation. Apart from such areas, where there has already been significant development in science, new sets of inter-related quantities, involving the spin and the total energy for example, are suggested as possibilities for further investigation and development.

Keywords:*non-division algebra, invariants, inverses, scaling, Dirac algebra, Clifford algebra, geometric algebra, special relativity, generalized covariance, rest mass, light cone, diamond conjugate*

## 2. Papers Published Elsewhere

**Authors Listed:**Van Der Mark, M.B., Williamson, J.G. (2021)

Robinson, V.N.E. (2021)

Williamson, J.G. (2019), (2015), (2014), (2012)

Van Der Mark, M.B. (2019), (2015), (2015)

Williamson, J.G., Leary, S.J. (2015)

Hagen, W.F. (2015)

Butler, P.H., Gresnigt, N.G., Van Der Mark, M.B., Renaud, P.F. (2012)

Van Der Mark, M.B., ‘t Hooft, G.W. (2000)

Williamson, J.G., Van Der Mark, M.B. (1997)

Williamson, J.G., Timmering, C.E. (1990)

**RELATIVISTIC INVERSION, INVARIANCE AND INTER-ACTION**

VAN DER MARK, M.B., WILLIAMSON, J.G. (2021)

**Abstract:**A general formula for inversion in a relativistic Clifford-Dirac algebra has been derived. Identifying the base elements of the algebra as those of space and time, the first order differential equations over all quantities proves to encompass the Maxwell equations, and leads to a natural extension incorporating rest mass and spin. Despite the fact that this algebra is not a division algebra it seems to parallel reality well: where division is undefined turns out to correspond to physical limits, such as that of the light cone. The divisor corresponds to invariants of dynamical significance, such as the invariant interval, the general invariant quantities in electromagnetism, and the basis set of quantities in the Dirac equation. The study suggests other Lorentz invariants that may prove of interest, including one relating the spin and total energy. It is speculated that the apparent 3-dimensionality of nature arises from a beautiful symmetry between the three-vector algebra and each of four sets of three derived spaces in the full 4-dimensional algebra. It is conjectured that elements of inversion may play a role in the interaction of fields and matter.

**Keywords**: invariants ; inversion; division; non-division algebra ; Dirac algebra ; Clifford algebra ; geometric algebra ; special relativity, photon interaction

**PHYSICAL EXPLANATIONS OF EINSTEIN’S GRAVITY**

ROBINSON, V.N.E. (2021)

**Abstract:**Einstein’s gravitational field equations from his general theory of relativity have formed the foundations of gravitational studies since their publication. His work is widely acknowledged as an example of a theoretical study that made a great contribution to our understanding of gravity. Einstein’s mathematical approach has made the topic complex and open to misinterpretation. This study evaluates the physics upon which his mathematics operates. It shows that mass distorts space–time by the redshift of photons. It derives two alternative metrics to the Schwarzschild metric. One was derived directly from Einstein’s early work on gravity. The other was derived from Einstein’s field equations by removing the approximations introduced in the Schwarzschild metric derivation. Both match observation better than the Schwarzschild metric and show why Einstein did not believe in black holes. The metric derived directly from Einstein’s early gravitational study, predicts the torus shape shown in the Event Horizon Telescope collaboration image. In showing the physics involved, this study suggests it is easier to understand the complexities of his work.

**A NEW LINEAR THEORY OF LIGHT AND MATTER**

WILLIAMSON, J.G. (2019)

**Abstract:**A new theory is proposed with similar form and mechanics to the Dirac equation, but where the rest-mass is introduced in a more fundamental way. The new theory proves to encompass and extend classical electromagnetism, and reduce to it exactly in the field only case. A sharpening of the principle of relativity allows fully relativistic field-only solutions, which take on some of the properties more usually associated with quantised states. The extended theory has four new solutions which exhibit a non-zero electric field divergence and a double-covering symmetry. These are necessarily charged, essentially fermionic states, two with positive and two with negative charge, differing by an inner spin symmetry. The new configurations are identified with the spin “up” and spin “down” electron and positron.

**QUANTUM PARTICLE, LIGHT CLOCK OR HEAVY BEAT BOX?**

VAN DER MARK, M.B. (2019)

**Abstract:**

A standing wave of light inside a cavity, while observed from a moving frame, is seen to have a spatial beat of precisely the same nature as that of the de Broglie wave of a quantum particle. The structural forces that hold together the cavity against the internal radiation pressure are compared to the so-called Poincaré stresses inside an elementary particle, as are required for the particle’s stability. This seems to work best if the cavity is non-rigid and hence is itself oscillating, coupled to the radiation inside.

**ON THE NATURE OF “STUFF” AND THE HIERARCHY OF FORCES**

VAN DER MARK, M.B. (2015)

**Abstract:**

From super clusters of galaxies down to the quarks in the proton, at all length scales the structure of matter is the result of a balance of forces. In this paper it is argued that with decreasing size there must necessarily be an increase of the fraction of kinetic and binding energy with respect to the total energy. Smaller sizes require stronger forces which represent more of the energy available. The smallest possible size of granularity is found to be where the internal kinetic energy and total energy become comparable, which occurs at the size of the proton. We infer that the proton is the smallest stable particle, being a light speed circulation of energy.

Keywords: primordial stuff, hierarchy of forces, rest mass, energy balance, proton structure, light speed knot of energy.

**ON THE NATURE OF THE PHOTON AND THE ELECTRON**

WILLIAMSON, J.G. (2015)

**Abstract:**A new theory, describing both light and material particles, is proposed. The experimentally-observed nature of space and time are brought into the theory at the most fundamental level. An equation encompassing the usual free-space Maxwell equations but similar in form to the Dirac equation is proposed. This equation has new kinds of solutions. Propagating, pure-field solutions may have any energy, but the energy transferred must be proportional to the frequency. These are identified with the physical photon. Solutions with a rest-mass term allow any incoming propagating field to merge into re-circulating vortex-like solutions. The minimum energy configuration “rectifies” the oscillating electric field of light into a uni-directional, radial (inward or outward directed) configuration. The resulting apparent external charge may be readily estimated and is found to be of the order of the elementary charge. The spin may, likewise, be calculated, and is found to be half integral, exhibiting a double-covering internal symmetry. Charge is then not a fundamental quantity in the theory – but is a result of the way field folds from a rest-massless bosonic to a rest-massive fermionic configuration. The simplest such charged, fermionic particles are identified with the electron and positron.

Keywords: Electromagnetic electron positron photon.

**ABSOLUTE RELATIVITY IN CLASSICAL ELECTROMAGNETISM: THE QUANTISATION OF LIGHT**

WILLIAMSON, J.G., LEARY, S.J. (2015)

**Abstract:**A rigorous introduction of the underlying nature of space and time forces qualitatively new kinds of solutions in the classical theory of electromagnetism. A class of relativistic wave-functions are derived which are solutions to the first-order, free-space Maxwell equations. These describe all photons from radio to gamma waves and are governed by a single parameter: the exchange frequency. Though the theory remains that of classical, continuous electromagnetism, allowed traveling-wave solutions are quantised in that they come in “lumps” and their characteristic energy is proportional to frequency.

Keywords: light quantisation.

**MATTER IN THE FORM OF TOROIDAL ELECTROMAGNETIC VORTICES**

HAGEN, W.F. (2015)

**Abstract:**The creation of charged elementary particles from neutral photons is explained as a conversion process of electromagnetic (EM) energy from linear to circular motion at the speed of light into two localized, toroidal shaped vortices of trapped EM energy that resist change of motion, perceptible as particles with inertia and hence mass. The photon can be represented as a superposition of left and right circular polarized transverse electric fields of opposite polarity originating from a common zero potential axis, the optical axis of the photon. If these components are separated by interaction with a strong field (nucleon) they would curl up into two electromagnetic vortices (EMV) due to longitudinal magnetic field components forming toroids. These vortices are perceptible as opposite charged elementary particles. These spinning toroids generate extended oscillating fields that interact with stationary field oscillations. The velocity-dependent frequency differences cause beat signals equivalent to matter waves, leading to interference. The extended fields entangled with every particle explain wave particle duality issues. Spin and magnetic moment are the natural outcome of these gyrating particles. As the energy and hence mass of the electron increases with acceleration so does its size shrink proportional to its reduced wavelength. The artificial weak and strong nuclear forces can be easily explained as different manifestations of the intermediate EM forces. The unstable neutron consists of a proton surrounded by a contracted and captured electron. The associated radial EM forces represent the weak nuclear force. The deuteron consists of two axially separated protons held together by a centrally captured electron. The axial EM forces represent the strong nuclear force, providing stability for “neutrons” only within nucleons.The same principles were applied to determine the geometries of force-balanced nuclei. The alpha-particle emerges as a very compact symmetric cuboid that provides a unique building block to assemble the isotopic chart. Exotic neutron-4 appears viable which may explain dark matter. The recognition that all heavy particles, including the protons, are related to electrons via muons and pions explains the identity of all charges to within 10

^{–36}. Greater deviations would overpower gravitation. Gravitation can be traced to EM vacuum fluctuations generated by standing EM waves between interacting particles. On that basis, gravity can be correlated via microscopic quantities to the age of the universe of 13.5 billion years. All forces and particles and potentially dark matter and dark energy are different manifestations of EM energy.

Keywords: photon, pair-creation, electron, waves, electromagnetic vortices, matter, elementary particles, neutron, deuteron, gravitation.

**QUANTUM MECHANICAL PROBABILITY CURRENT AS ELECTROMAGNETIC 4-CURRENT FROM TOPOLOGICAL EM FIELDS**

VAN DER MARK, M.B. (2015)

**Abstract:**

Starting from a complex 4-potential A=αdβ we show that the 4-current density in electromagnetism and the probability current density in relativistic quantum mechanics are of identical form. With the Dirac-Clifford algebra Cl_{1,3} as mathematical basis, the given 4-potential allows topological solutions of the fields, quite similar to Bateman’s construction, but with a double field solution that was overlooked previously. A more general null-vector condition is found and wave-functions of charged and neutral particles appear as topological configurations of the electromagnetic fields.

Keywords: topological fields, electromagnetism, Clifford algebra, Dirac equation, 4-current, knots.

**A NEW THEORY OF LIGHT AND MATTER**

WILLIAMSON, J.G. (2014)

**Abstract:**Within a new mathematical framework, an extension to the free-space Maxwell equations is pro-posed with four further linear coupled differential equations. Within the formalism, a sharpening of the principle of relativity leads to new solutions of the Maxwell equations corresponding to uncharged rest-massless propagating states identified with the photon. The simplest extension, including a rest-mass density, allows for a self-confinement of the resulting system. The resulting new solutions are necessarily charged and essentially fermionic. These are identified with the electron and positron.

**FERMIONS FROM BOSONS AND THE ORIGIN OF THE EXCLUSION PRINCIPLE**

WILLIAMSON, J.G. (2012)

**Abstract:**Based on a model of the electron as a localised photon, a possible physical basis for the Pauli exclusion principle is proposed. It arises from the harmonic interference of the underlying fields. One possible theory leading to the self-confinement of electromagnetism is explored: an extension of the Maxwell equations.

Keywords: Electron, electromagnetism, exclusion principle, electromagnetic vortex.

**A FIELDS ONLY VERSION OF LORENTZ FORCE LAW: PARTICLES REPLACED BY THEIR FIELDS**

BUTLER, P.H., GRESNIGT, N.G., VAN DER MARK, M.B., RENAUD, P.F. (2012)

**Abstract:**

We show that the Lorentz force law, F^{L}_{1}=q_{1}(E+v_{1}×B) being the charge on particle 1interacting with the electromagnetic fields due to all other particles, can be written in a pure field form F^{L}_{1}=−∇_{1}U^{EM}. In this expression U^{EM} is the total electromagnetic energy of the system of particle1 and all other particles. In deriving this result we review the old but not widely known results thatMaxwell’s equations follow uniquely from Special Relativity, and that the Lorentz force law follows from applying Hamilton’s variational principle to this result.For a two particle system, the standard view is that the electromagnetic force on particle 1 is the result of the charge of particle 1 interacting locally with the field of particle 2, and conversely. Both charges 1 and 2, and fields 1 and 2 are needed. In our approach, the fields of all particles contribute to the electromagnetic interaction everywhere, over all of space. The charges of the particles do not enter the theory except incidentally, via Gauss’s law. This has novel interpretational consequences. In particular, it allows a charged particle to be replaced by its electric and magnetic fields, much as a particle in quantum mechanics is replaced by its complex valued wave function.

Keywords: Lorentz force, Maxwell’s equations, Coulomb force. Hamilton’s principle, energy density.

**LIGHT IS HEAVY**

VAN DER MARK, M.B., ‘t HOOFT, G.W. (2000)

**Abstract:**

Einstein’s relativity theory appears to be very accurate, but at times equally puzzling.On the one hand, electromagnetic radiation must have zero rest mass in order to propagate at the speed of light, but on the other hand, since it definitely carries momentum and energy, it has non-zero inertial mass. Hence, by the principle of equivalence, it must have non-zero gravitational mass, and so, light must be heavy.In this paper, no new results will be derived, but a possibly surprising perspective on the above paradox is given.

**IS THE ELECTRON A PHOTON WITH TOROIDAL TOPOLOGY?**

WILLIAMSON, J.G., VAN DER MARK, M.B. (1997)

**Abstract:**We study the properties of a simple semi-classical model of a photon confined in periodic boundary conditions of one wavelength. the topology of this object, together with the photon electric field, give rise to a charge of the order of 10

^{-19}Coulombs and a half-integral spin, independent of its size. The ratio of the electromagnetic energy inside and outside the object leads to an anomalous spin

*g*factor which is close to that of the electron. Although a finite size of order 10

^{-12}meter arises in a natural way, the apparent size of the object will be much smaller iin energetic scattering events.

**QUANTUM POINT CONTACT AS A LOCAL PROBE OF THE ELECTROSTATIC POTENTIAL CONTOURS**

WILLIAMSON, J.G., TIMMERING, C.E. (1990)

**Abstract:**Quantum point contacts, shifted laterally by +-50nm have been used to study the local electrostatic potential in the two-dimensional electron gas of a GaAs-Al

_{x}Ga

_{1-x}As heterostructure. The resistance of the quantised plateaux is observed to vary with this lateral position. In one device, an alternate suppression of the quantised resistance plateaux is observed, which we interpret as being due to an overlap of strong local potential fluctuations with the transverse profile of the wave function in the quantum point contact.

## 3. Classic Papers

**Authors Listed:**

De Broglie, L. (1924)

Clifford, W. (1878)

**RECHERCHES SUR LA THéORIE DES QUANTA**

DE BROGLIE, L. (1924)

**APPLICATIONS OF GRASSMANN’S EXTENSIVE ALGEBRA**

CLIFFORD, W. (1878)