UET3 Continuous Two-Electron Theory of Electromagnetism H. J. Spencer * III ABSTRACT This research programme continues with its fundamental analysis of the electromagnetic interaction. In contrast to the continuous charge model of electricity that is today used as the foundation for presentations of classical electro- magnetism (CEM), this paper now analyzes the continuous interaction of pairs of charged point particles that better reflects the known basis of electricity – electrons. This analysis first demonstrates that all continuous theories of interaction between point particles that exhibit inertial resistance to changes in their motion are inconsistent with all asynchronous action-at-a-distance forms of interaction or equivalently, interactions limited to points on their ‘light- cone’. This research programme is an extension of the Newtonian scheme of classical mechanics that represents the locations of point particles, not by standard algebraic vectors but by a more powerful, non-commutative complex algebra, based on Hamilton’s quaternions, called here ‘Natural Vectors’. This NV representation is extended here from representing a single location (the ‘field point’) to representing the differences between pairs of point objects; this automatically advances the idea that even ‘classical’ electrons must be treated as ‘fermions’ (as this is an anti- symmetric algebraic representation). Adding the assumption of separability of the electromagnetic momentum to the previous single-time version now reproduces Planck’s 1907 infamous proposal (not Einstein’s) for defining relativistic forms of single particle momentum and energy but now in terms of EM electro-kinetic momentum between two particles, in contrast to Planck’s original but unphysical assumption of a constant, mechanical force on a single particle that required the Lorentz transformation. This paper also extends this new two-electron viewpoint to many-body situations involving myriads of pair-wise interactions by showing that classical electromagnetism is a consequence of the statistical effects of very many of these interactions arising from multiple, remote electrons moving within metallic conductors on one or many ‘target’ electrons. A new discrete, many-body approximation model (“Mesoscopic Electrodynamics”) is developed here that is shown to be a covering theory for the standard (continuum) model of CEM. The emphasis here is shifted back from empty space to the actual experiments involving electrical currents in metallic wires that were the real foundation for CEM’s integral and differential equations, which only summarized these effects mathematically but never provided any physical justification or insights. This theory now extends the rival, forgotten (‘continental’) approach to CEM to directly include radiation, as just a long-range induction effect, removing the only advantage previously associated with Maxwell’s field theory. It also links directly to Newtonian mechanics to provide a seamless unity to all of classical physics. These results now demonstrate that, contrary to the orthodox consensus, Maxwell’s Equations (as a field theory) are not a fundamental model for understanding the basic interaction between any types of elementary particles. This view challenges the last 150 years in theoretical physics that has been constructed only on the mathematics of continuous fields, leading to quantum field theories. This approach eliminates the force densities (electric and magnetic fields), the ‘instant’ Coulomb potential and the single-time (‘God-like’) view of nature that have dominated physics for 300 years, only because these ideas could use the simplified (but well-studied) mathematical representation of differential equations. * Surrey B.C. Canada (604) 542-2299 [email protected] © H. J. Spencer Version 2.180 29-04-2011 Version 1.0 21-10-2007 [123 pp;94Kw] 1 UET3 1. INTRODUCTION & OVERVIEW 1.1 INTRODUCTION A major contention of this research programme is that the theoretical investigations into the foundations of the phenomenon of electromagnetism ended prematurely due to the exciting discoveries of the electron and atomic physics in the early part of the 20th Century. As a result, physics has relied excessively on the mathematical formulation of electromagnetism, known as Maxwell’s Equations, without realizing that these equations have no foundations in the modern understanding of the basis of electricity, namely, the electron: or the view that electricity exists as point charges and not as continuous charge densities. The previous paper critically revisited Maxwell’s EM theory, beginning with his model of the æther and ending with the model of continuous electric charge density that is the basis for today’s view of classical EM. This paper moves through the era when the electronic model of electricity appeared around 1900 and then merged by Lorentz into a dualistic continuum-and-particle model of EM. Subsequent papers will focus exclusively on the electron as the basis of EM. The authors of a widely read graduate text [64] on CEM begin their penultimate chapter with the important statement that: “The fundamental problem of EM theory deals with the motion of charged particles interacting via the EM forces. The field concept is not necessary in developing equations for such motion; formulation in terms of action-at-a-distance is possible in a relativistically covariant way. … Separation of the problem into ‘radiation’ (production of a field by a charged system) and ‘motion of a charged particle in a field’ is in a certain sense artificial and introduces some difficulties.” This is a rather obtuse admission that Maxwell’s Equations are incompatible with the concept of a point particle. In fact, all field theories are incompatible with this fundamental point- particle model of matter and such theories have led to the spurious concept of ‘renormalization’. This should be acutely embarrassing for CEM, as the point-nature of electrons has been demonstrated repeatedly to the highest energy levels experimentally available, while CEM itself was built on Maxwell’s rejected model of the continuous æther. The present theory now returns to the ‘Continentalist’ tradition of CEM that was the principal rival to Maxwell’s field theory in the Nineteenth Century but is now largely forgotten. This ‘old’ theory was an action-at-a-distance, mathematical model based on the intuitively correct proposal that electricity occurred in the form of point particles (now known as electrons). The present theory adds a physical model of conduction in metallic wires, as these were central to all experimental studies in CEM. The present theory shows that a relativistically invariant form of the ‘mathematics of natural vectors’ is the optimum representation to construct a theory of EM that is comprehensive and consistent at all levels of interaction: from two electrons up to mesoscopic aggregations that are actually the reality underlying CEM. 1.1.1 RESEARCH PROGRAMME This paper is the third in the series documenting a new research programme that challenges some of the major assumptions of the last 150 years of theoretical physics. In addition to returning to the original approach to undertaking investigations in natural philosophy, emphasizing the value of philosophy and the history of science, this programme challenges the universal “Continuum Hypothesis”. This is the (now implicit) metaphysical view that reality is fundamentally continuous. However, since 1900 physicists have reluctantly come to the realization that electricity is particulate (electrons) and matter is atomic. This has meant that only space itself (the ‘æther’) is left to carry the burden of physical continuity. This research programme takes an aggressively discrete position: its introduction was announced with the description of a new form of algebra [01] suitable for representing asynchronous action-at-a-distance. The goal is to describe the discrete nature of reality using only non-continuous concepts and discrete forms of mathematics to represent these concepts. Continuum mathematics will be reserved only for aggregating from the discrete microscopic world-view to the apparently continuous view of the macro world: the approach pioneered by Newton himself. In 1912, J. Henri Poincaré (1854-1912), recognizing the potentially disruptive impact of Planck’s quantum of action, wrote that ever since Newton, the laws of physics were assumed to be inseparable from differential equations but the introduction of discontinuities associated with the new quantum theory now seemed to demand another form of mathematics [02]. The present research programme shares this viewpoint and this was one of the principal motivations for the creation of a new algebra (“Natural Vectors”) for the investigation of all forms of electromagnetism that can be readily formulated as either continuous or discrete interactions. 1.1.2 OBJECTIVES This paper is deeply indebted to the major efforts of one of the giants of science: James Clerk-Maxwell (1831-1879). In this light, an author can do no better than repeat one of Maxwell’s recent biographers [03] when he quoted (with approval) how Maxwell wrote about scientific innovation that should “jolt the reader into new patterns of thought … especially those readers who have become accustomed to mathematical forms of thought.” 1 UET3 1.1.2.1 Need for a New EM Theory The principal objective of this paper is to demonstrate that Maxwell’s theory is not a suitable model for the foundations of EM as electricity is