QUANTUM MECHANICS

0 downloads 0 Views 441KB Size Report
wave/particle proposal, rapidly led to a second generation quantum theory, usually ...... For all his controversial hypotheses, David Bohm (1917-1992) made a ...
UET7H

QUANTUM MECHANICS: A Contra-Revisionist History H. J. Spencer *

VIIH

ABSTRACT This is an essay to correct the false impression of the revisionist historical development of Quantum Theory that has unfortunately become accepted as orthodoxy. The thesis here is that the 100 years, beginning around 1840, has been “The Century of the Mathematicians”, who now dominate theoretical physics. The 2500 year old domination of western thinkers’ commitment to Geometry, Analysis (with its infinitesimals) and their associated metaphysical assumption of infinity and the Continuum, was threatened around 1850 by the inventions of discrete mathematics and non-Euclidean geometries. This threat became serious with the discrete discoveries around 1900 by experimental physicists: firstly of the electron, then the physical atom and their discrete electromagnetic (EM) spectra, compounded by Bohr’s radical model of the hydrogen atom with its very good agreement with its measured spectrum. Hilbert and others headed the successful mathematical Counter-Reformation with the invention of infinite Euclidean spaces and Analytic Function theory (known as Hilbert Spaces). These new mathematical tools, along with the misreading of de Broglie’s dramatic wave/particle proposal, rapidly led to a second generation quantum theory, usually called Wave Mechanics (WM) that completely overshadowed (by design) the “older” Quantum Mechanics (QM) of Bohr and its brilliant elliptical extension by Arnold Sommerfeld. Most new textbooks on quantum theory (usually written by mathematical physicists) begin with WM or its continuum vector space equivalence. Even some of the histories of the Quantum make the same omission (perhaps through ignorance, as the orthodox view is now so widely accepted). * Surrey, B.C. Canada (604) 542-2299 [email protected] © H. J. Spencer Version 1.011 31-10-2017 Begun 17-10-2017

UET7H

1. INTRODUCTION 1.1 SUMMARY This is an essay to correct the false impression of the revisionist historical development of Quantum Theory that has unfortunately become accepted as orthodoxy. The thesis here is that the 100 years beginning around 1840 has been “The Century of the Mathematicians”, who now dominate theoretical physics. The invention of discrete mathematics by William Hamilton, with his quaternions and their extension by Arthur Cayley (matrices) threatened the 2500 year old domination of geometry and its associated metaphysical assumption of the Continuum (Analysis). These techniques, along with the common-sense mapping between normal arithmetic and letters (algebra), had formed the basis for all the physics developed by mathematicians for making physics a metrical science for 300 years. This threat was removed around 1900 by the work of David Hilbert and others with the invention of Hilbert Spaces and Analytic Function Theory. A further threat arose in 1913, with Niels Bohr’s revolutionary circular model of the hydrogen atom, soon extended by Arnold Sommerfeld with his elliptical orbit theory. Both 2D theories produced very accurate agreements with observed results while maintaining the Newtonian model of particle mechanics; a new version called Quantum Mechanics (QM). While complying with Newton’s Classical Mechanics (CM), both theories ignored their non-compliance with Maxwell’s theory of the electromagnetic (EM) field: but this had already been sanctified by the mathematicians for its use of traditional calculus (continuous) differential equations (and incidentally, ending a bitter metaphysical war between the Newtonian supporters of discrete models of light and the powerful Cambridge mathematical supporters of wave models.) However, QM did fail to give a suitable explanation of the new discoveries of electron diffraction and scattering, which were readily ‘explained’ by de Broglie’s wave-particle hypothesis. This soon resulted in the famous 1926 Schrödinger Equation with its 3D spherical model that added a small correction to the earlier Hydrogen spectrum models. There was a brief appearance of ‘discrete’ mathematics with Heisenberg’s invention of Matrix Mechanics (MM), which is sometimes credited with the ‘discovery’ of quantum theory but it was soon dropped as far too opaque and difficult. This new form of atomic physics, quickly re-christened “Wave Mechanics” (WM), has dominated physics ever since and became the foundation for Quantum Field Theory (QFT) and all later models of micro-reality. The major problems with WM are reviewed below as they are usually omitted in triumphant retellings of the quantum mythology. New research (by the author) indicates that the early developments (up to Sommerfeld’s model) have been too rapidly dismissed, more for mathematical motives than for reasons of Natural Philosophy (physics). The early techniques of finite mathematics, when combined with the dismissal of the Continuum Hypothesis, can produce a full QM, with results comparable to WM, while providing a realistic explanation for all discrete quantum phenomena, including electron diffraction, without ever invoking continuum or wave-like concepts. This paper complements the proposal for a new method of atomic measurement and a broad critique of standard QM from a Natural Philosophy perspective. A major purpose here is to reacquaint readers with the real history of QM because the author now knows that the early pioneers were far more on the right track than their critics thought. The tragedy of QM is that the electron is the smallest level of material reality that human beings deal with on a regular basis but physics has failed to develop its full implications and has moved on too quickly to even more remote problems, such as nuclear physics and deep space cosmology. All versions of quantum theory focused on the atomic equivalence of the planetary gravity model, as this is the one example that is solvable exactly, in all cases. The central mistake is to assume that two electrons interact with each other, just like gigantic collections of them that we are familiar with (as was Maxwell).

1

UET7H A new revisionist view of the history of QM has developed since 1945, whereby all the quantum issues were considered to have been resolved in the Fifth Solvay Conference of 1927, when Heisenberg and the ‘Copenhagen School’ defeated Einstein, Bohr and the ‘Old School’. The opposite view has been documented recently in a new book by science historian Sheilla Jones. The present paper will directly pick up this challenge even though this new view will inevitably generate a huge resistance by professionals, who are heavily invested both in their math tools and the widely accepted orthodoxy.

1.2 OBJECTIVES

One of the global objectives of this research programme[1] is to refute the modern view that the goal of theoretical physics is to produce a set of equations that can be used to make predictions that numerically agree with the numbers obtained from the corresponding experiments. While this is a worthy goal, it is not seen here as sufficient. Most ordinary people in the last 300 years have expected physics to provide a comprehensible model of the material world, which it did with Classical Physics until 1900. After that year, as we show here, physics continued to produce mathematical theories but these became terribly difficult to interpret. Worse, experimental physicists accepted this difficulty with the defeatist view that the atomic world is too remote from normal experience, so that it operates in mysterious ways that can only be represented by mathematics (a position still very popular with anyone deeply committed to advanced mathematics). This approach was so deficient that even when conceptual contradictions arose between alternate mathematical images (such as waves and particles), these problems were swept away with the rhetorical flourish of simply naming them “paradoxes”. This programme refuses to go along with this professional consensus and has attempted to create a new, unified ontological model of the electron that can not only explain its apparent “weird” behavior but is readily visualizable by most nonmathematicians. This programme is committed to the view that physics is ‘Natural Philosophy’ and explicitly includes metaphysics. It was not the use of mathematics alone that made Newton’s worldview acceptable to educated Europeans for 300 years.

1.3 OVERVIEW 1.3.1 APPROACH The author’s research programme is founded on the dual pillars of history and philosophy, believing that both are required to make fundamental progress in understanding nature. Contrary to the modern view, (which sees its “discoveries” of the Truth as eliminating any need for context or background) knowledge we see the history of science is needed to understand how contemporary physics has reached its present stalled situation, especially when it finds itself in an impasse. A good rule in life, is that mistakes are due to bad assumptions and most assumptions in science remain unknown and are rarely examined. It is also believed here that metaphysics (all intuitions beyond science) is a necessary component of any theory of reality. What is rejected throughout this research programme is the present opinion that mathematical equations form a sufficient explanation of the world. This imperialistic mathematical perspective has now come to dominate theoretical physics (like much of modern life) and it is assumed that this approach is creating an asymptotic view of the Truth. Indeed, since Plato, mathematicians have believed that mathematics is not man-made but both true and eternal. Such arrogance (and its comeuppance) has arisen several times in the history of humanity. 1.3.1.1 History Historically, curious intellectuals have looked for patterns in nature. In the western tradition, this started to become formalized with the Ancient Greek philosophers. Over time, those who specialized in this aspect of investigating reality were referred to as “Natural Philosophers”. Much of this activity was purely mental speculation, which resulted in endless argumentation. The rise of modern science began when many of the Natural Philosophers agreed that Nature must resolve these disagreements[2]. 2

UET7H This was the vital empirical step were actual manipulations of material reality, also better known as experiments, began to play an increasing role in resolving speculations. Until the end of the Nineteenth Century, the study of matter[3] was dominated by astute experimenters, who used new technologies (such as the vacuum pump) or even invented new technologies, like electromagnetism, to discover new phenomena and new properties of matter in its various forms. This style of manipulating the material world was contrasted with abstract investigations that had begun with Pythagoras and actively promoted by Plato[4], as the only true form of knowledge: these conceptual investigations of timeless relationships became known as pure mathematics. Since mathematics could be readily taught (and examined), this style of human activity soon dominated the education of the social elites across western societies. It was not long before this scholastic approach began to encroach on the realistic model of what soon became known as science. The great pioneer here was the polymath, Isaac Newton, who combined admirable, experimental skills with a rare, imaginative talent for mathematical innovation. His most dramatic contribution was his explanation of planetary orbits using his conceptual model of inertial motion and his radical proposal for action-at-a-distance, attractive forces between remote masses (just called gravity) and its associated mathematical summarization as a continuous reciprocal force, whose strength varied inversely with their spatial separation. These ideas unified the ‘stuff’ of the Heavens with the common-or-garden matter, familiar to us all here on Earth. This was the foundation of the science of classical mechanics and the subsequent introduction of continuum calculus as the preferred mathematical description of nature. 1.3.1.2 PHILOSOPHY Since many intellectuals in the western tradition have raised mathematics to its role as “the Queen of the Sciences”, they see no need for any conceptual explanations beyond the symbology in their equations. In effect, they have hijacked physics, pushing empirical science into the background, doing no more than generating numbers that can ‘validate’ their theories. In many cases today, it is deemed sufficient to conduct experiments in their own heads: so-called “thought experiments”. This mind-before-matter rhetorical approach is fully rejected here, where conceptual real hypotheses are considered to be much more fruitful than inventing new equations. Recently, world-class specialists in mathematical physics, frustrated that their latest theories (e.g. “Strings”) cannot expect to be examined experimentally have suggested that rationalism alone can be relied on to advance knowledge of nature. This would remove science from the grounded world of physics before 1900, where new experiments dictated the new directions of physics and mathematics was viewed only as a tool to produce numbers, not explanations. 1.3.1.3 NATURAL LANGUAGE The idea of simply doing theoretical physics as an exercise in applied mathematics has been a demonstrated failure with no new concepts arising or even reaching the level of a useful technology, as happened to much of earlier physics that provided visualizable models for both professional physicists, engineers and common folk. Bohr and Heisenberg, the originators of the so-called ‘QM Copenhagen Interpretation’, believed that they could retain natural languages while forcing everyone to accept contradictory concepts (such as, particles and waves) as a new ‘mysterious’ property of the micro-world by simply invoking a new ‘scientific’ principle, which Bohr called ‘Complementarity’. Heisenberg expected the rest of the humanity to give up the “illusion of the world” that we experience on a daily basis when we come to think of atomic systems but still insists that the rest of the descriptions of such experiments can continue with the rest of our standard vocabulary and concepts[5]. The author’s new theory provides an explanation of all these atomic scale experiments using natural language and a model of particle physics that extends Newton’s original views with a few, reasonable hypotheses of electron interactions at this tiny scale of reality. These are summarized later herein.

3

UET7H

2. THREATENED MATHEMATICS 2.1 GEOMETRY

In one of his several recent books, (“God created the Integers” - 2005)[6], Stephen Hawking discusses the mathematical breakthroughs that changed history, claiming that “these have helped the human race achieve great insight into nature.” This is typical of the self-praise that mathematicians bestow on each other. Two of the five authors selected here contributed to the “science” of geometry. The Egyptians are dismissed by Hawking as they were mere builders, typical of the arrogance of the Greek (and later) intellectuals for their preference for theory and eternal results, such as the fact that the circular ratio π can be proved to not be the ratio of two ‘positive integers’ (see below).

2.1.1 EUCLID

The first work presented by Hawking is appropriately “The Elements” by Euclid (c. 325 BC), where years of work by many thinkers on geometry had been collected and organized into “models of proof”, which have set the standard for western thinking ever since. Since this provided the central educational subject for most western educated men over one thousand years, it is no surprise to realize that it the second-best selling book ever, after the Bible. The key to this was that Euclid started with plausible definitions, like a line is “a length without thickness”. He then provided rules (or postulates) for drawing common simple shapes (such as equal-sided triangles etc.) using only a straight-edged ruler and a device called a ‘compass’ for making circles. The constructions proved they were “real” but Plato then went beyond this demonstration to unjustly claim that the perfect examples (his ‘Forms’) must also “exist”.

2.2 NATURAL NUMBERS 2.2.1 ARITHMETIC Several ancient civilizations discovered the usefulness of the technique we call ‘counting’, particularly for merchants. When the objects being counted are distinct and stable then collections of such objects could be compared and matched on a one-for-one basis and assigned a standard name, we call a number. Actually, when such collections are finite, these are called ‘natural numbers’ (or “cardinals”). These collections could be combined, first as two sets, then larger, which could then be counted; this act of combining led to the idea of addition and its opposite action, called subtraction. These became the first ‘laws’ of arithmetic. What few noticed was that these rules only ‘work’ when the objects continue to exist; this is because our senses confirm existence and its determination maps to the binary basis: one and zero. Challenge people to count ice-cubes on a hot stove-top or indistinct ‘objects’ like clouds.

2.2.2 MULTIPLICATION

A few clever people noticed that objects in only some collections could be arranged into several rows of equal size; this led to the technique called multiplication (which generations of children memorized as ‘songs’). The smallest pattern involved two columns, and some collections failed to have two objects in every row; these we call ‘odd’ numbers, while the matched sets were called ‘even’ numbers. A few Greeks studied these patterns and also noticed that some collections were special with an equal number of objects in both the rows and the columns; these were called ‘square numbers’. Finally, the rules for manipulating the marks, representing these collections (number symbols) were formalized as arithmetic. Embarrassingly, Pythagoras used his own famous formula to prove that some lengths could not be given a unit ‘size’, while if the two lengths were equal (and defined as the ‘unit length’) then he proved (based, at least, on the myth) that there was no fraction that could be found that when ‘squared’ equals the small number 2; in other words there are some ‘numbers’ (like the square root of two = √2) that had to become known as “irrationals”. This demonstrated that lengths could not be represented by numbers, as there is no natural unit for measuring spatial separations. This was a deep insult to the ‘Masters of Rationality’.

4

UET7H

2.2.2 ALGEBRA Hawking again picks a famous Greek mathematician, Diophantus (c. 250 BC) for initiating a whole branch of mathematics; one that many school children will not thank him for: algebra, the substitution of letters (like X) for any number that remained unknown until an ‘Equation’ was ‘solved’. Ironically, he called his 13 books “Arithmetica”. Diophantus was the first Greek mathematician who recognized that most collections could be split into two parts, resulting in the concept of fractions as another type of number (together with ‘counting numbers’ known as ‘rational numbers’). Most knowledge of Greek mathematics (and Aristotle’s books) were lost in the West during the Dark Ages and only survived in Greek Byzantium and then saved by Arabic scholars who brought it to western scholars around 1300.

2.2.3 COMPLEX NUMBERS The Italian polymath, Cardano (1501-1576) was the first person to realize he needed a new idea to solve cubic equations; he constructed the concept of the square root of minus one , symbolized by the letter ‘i’ (for imaginary) with the defining property: i2 = –1. This cannot be imagined but it may be seen as a vertical line, drawn at right-angles to the real number line. Since this radical idea extended the concept of numbers to what we now call negative numbers, this is seen as one of the most imaginative steps in abstract thinking. When combined with a pair of regular numbers (like x and y) we can produce a new concept (or doublet) called complex numbers, such as : z = x + i y.

2.2.4 PROBABILITY Probability is a mathematical technique for estimating the likelihood that a possible event will occur. The trick is to calculate a number, whose value ranges between zero and one that are interpreted as never likely to happen (zero) or certainly likely (one). Numbers close to one are more likely to happen while rare situations are represented by a small number, close to zero. These methods are used when complex situations are too difficult for rational methods, so reliance on history is common; the assumption here is that the future will resemble the past. Probability now plays a central role in quantum theory where even the study of the simplest atom (hydrogen) is beyond our powers of sensory examination. It is no surprise to discover that much of this probability theory was developed by mathematicians with a propensity for gambling.

2.3 ANALYTIC GEOMETRY 2.3.1 DESCARTES Although René Descartes (1596-1650) is viewed more today as a philosopher, he really made his best contributions as a mathematician[6], in the latter half of his life. He formalized (an idea of Stevinus): the sums of inverse powers of ten as an infinite set of digits, he called rhetorically a ‘real’ number, like 0.123456789012... With this technique, he was able to link algebra to the geometry on plane surfaces, a blend now called Analytic Geometry. This was key step in using mathematics for physics.

2.4 ANALYSIS Descartes invention solved the problem of square roots and allowed continuous curves to be constructed. This provided a key step in evolving the old Greek idea of Analysis (or breaking anything into smaller pieces). The Greek thinker Zeno of Elea created a set of paradoxes by pushing these ideas to extremes, without limit, where it becomes continuous. Since Descartes, the analytic method has spread into philosophy, science and forms the basis for huge amounts of mathematics. Historically, the roots of this idea link to theological ideas on the infinitude of God and his creation. The ultimate goal of analysis is the inverse of infinity – dividing the target into an infinity of smaller parts: called infinitesimals.

5

UET7H

2.4.1 INFINITESIMALS In common speech, an infinitesimal thing is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, “infinitesimal” simply means “extremely small”. These ideas were politically contentious in the 16th century [see my essay: “Infinitesimals”[7]] but did become acceptable to imaginative intellectuals, like Gottfried Leibniz (1646-1716), who competed with Newton (see below) in developing the infinitesimal calculus. Leibniz’s use of infinitesimals relied upon heuristic principles (plausible analogies), such as: “what succeeds for the finite numbers succeeds also for the infinite and infinitesimal numbers and vice versa”.

2.4.2 NEWTON (Calculus) Isaac Newton (1642-1727) is widely viewed as the most creative scientist of the last 2000 years. Born under unusual circumstances that blighted his personal life[8], he eventually was knighted (‘for political services’) and became the president of the first scientific body called ‘The Royal Society’. He built on the innovations of Descartes and the esoteric discussions on infinitesimals, by creating the first version of the differential (infinitesimal) calculus and to invent the idea of instantaneous velocity. Newton also expanded on Galileo’s physics and his idea of a body’s “quantity of matter” (or ‘mass’) and invented the key concept of a particle’s momentum as the algebraic product of mass and velocity. He formalized the instantaneous change in momentum as being produced by a continuous interaction called force.

2.4.3 FOURIER SERIES One of the least appreciated innovators was the French mathematician, Joseph Fourier (1768-1830). He invented the technique of describing any variation across a finite space, in terms of an infinite sum of harmonic functions (the well-known trigonometric sines and cosines). He invented this method to create an investigation of heat diffusing through a solid body but it has now spread to become a central method used in many areas of physics, including wave mechanics.

6

UET7H

3. DISCRETE MATHEMATICS 3.1 ALGEBRA 3.1.1 HAMILTON William Rowan Hamilton[1] was born in Dublin in 1805; he was a child prodigy who could read English by three, Hebrew, Latin and Greek by the age of five and most of the major European languages (plus Arabic, Persian, Sanskrit and Bengali) by ten. He had mastered algebra by 13 and graduated from the leading university (Trinity College, Dublin) with the highest honors, winning gold medals in science and classics. At age 22 he was appointed Astronomer Royal for Ireland, for the rest of his life. He invented a new way of writing Newton’s Equations of Motion that are still being used in modern quantum theory. Hamilton was obsessed with generalizing the pairwise idea of complex numbers (see above) into three dimensions; he spent 13 years on this search until it dawned on him, while out walking, that he needed four parts (not three); so he names these new number combinations: “quaternions”. They were built around 3 unit imaginaries{i, j, k} that each acted like the infamous i (i.e. k2 = –1) but failed to obey one of the oldest rules of algebra: commutative multiplication (e.g. 2 x 3 = 3 x 2 = 6), so that: i j = – j i. This shocking equation lead to an explosion of interest in discrete mathematics, like arrays of numbers, later called matrices, where this non-commutative multiplication was quite common; so much so, that this is one of the ways of introducing quantum mechanics.

3.1.2 CAYLEY Sir Arthur Cayley (1821-1895) was a founder of British Pure Mathematics. He went to Trinity College, Cambridge at age 17 and by the end of his undergraduate years he had won the top two prizes in mathematics. After a spell as a successful lawyer, he returned, when 42, to Trinity as the first Sadleirian professor of pure mathematics (algebra). Cayley wrote a major treatise in 1858 on the algebra of finite arrays [square matrices]; this included definitions of addition, subtraction, multiplication and division. He was the first to show that the result of multiplying two matrices depended on the order involved – a result, known as the non-commutative property, that Hamilton had first shown for his quaternions.

3.2 FINITE DIFFERENCES Unfortunately, most mathematicians see Finite Differences as just an approximation (often numerical) of continuum mathematics (the theory of continuous – smooth-functions). The technique is actually more general and applies to any ordered set of objects {Xj}, which are identified and sequenced by a common subscript (j) e.g. {Xj = X1, X2, X3, ... }. The first (adjacent) finite difference (Δ - the Greek delta) is defined as simply the difference between two consecutive entries in the series; thus: ΔX2 = X3 – X2 or in more general terms: ΔXj = Xj+1 – Xj . This mathematical technique allows for examining discrete change with ‘quantum’ jumps, as in atoms.

3.3 NON-EUCLIDEAN GEOMETRIES In defining his traditional geometry, Euclid had begun with five plausible assumptions (usually called axioms). His fifth Axiom, called the ‘Parallels’ axiom, seemed reasonable; namely that two lines that remain at a constant nearest separation will never intersect (this assumes an infinite space). Over the centuries, several mathematicians puzzled whether this was necessary or what might happen if it were dropped, as it was much more complicated than the other four axioms. This remained a puzzle until the 19th century when a flurry of new interest made some major progress. Most of us are so saturated with studying flat spaces (books, blackboards, etc.) that we just assumed this is the only natural type of space.

7

UET7H

3.3.1 GAUSS Although the brilliant mathematician, Karl Friedrich Gauss (1777-1855) had solved this problem he did not publish his intuitions for fear of controversy. So, it was not until 1830, that two other professors of mathematics independently published treatises on hyperbolic geometry; they were the Hungarian Janos Bolyai (1802-1860) and the Russian Nikolai Lobachevsky (1792-1856). These theories depended on constructing lines on unusual 3D surfaces in normal space. Interestingly, the conclusion these theorists arrived at was that the geometry of the physical universe is either Euclidean (‘flat’) or not could not be decided by mathematics alone but would need large-scale measurements; some thinking that eventually resulted in Einstein’s General Theory of Relativity. The discovery of the non-Euclidean geometries had a ripple effect which went far beyond the boundaries of mathematics and science. The famous postNewtonian philosopher Immanuel Kant (1724-1804) relied on our awareness of space and geometry to illuminate the nature of all human knowledge. It was his prime example of synthetic a priori knowledge; not derived from the senses nor deduced through logic — our knowledge of space was a truth that we were apparently born with. Unfortunately for Kant, his implicit concept of this unalterably true geometry was Euclidean. Theology was also affected by the change from absolute truth to relative truth in the way that mathematics is related to the world around it, which was a result of this paradigm shift.

3.3.2 TOPOLOGY One of Gauss’s most imaginative students was Bernhard Riemann (1826-1866). He built on Gauss’s new ideas about geometry by investigating the general nature of curved surfaces and invented his very powerful idea of manifolds by his expanded concept of curvature. By formulating the geometry in terms of an advanced continuum, algebraic concept (multi-dimensional arrays) he could define a curvature tensor, Riemann expanded non-Euclidean geometries to (imaginary) higher dimensions.

3.3.3 THE INTUITIVE REBELS Although most mathematicians were educated to accept the assumptions of continuity as the foundation for geometry, calculus and function theory, a small band of rebels emerged in the 19th century. One of first was the German Leopold Kronecker (1821-1891), who upset his colleagues with the claim that: “God created the integers but man invented all the other numbers”. He was followed by L.E.J. Brouwer (1881-1966), the Dutch founder of ‘Intuitionism’ – as a foundational, philosophical opponent to the then-prevailing symbol-games or formalism of David Hilbert (see later) and his collaborators; Brouwer insisted that all concepts should be grounded in sense intuitions (a position I very much favor). He also challenged the beliefs that the rules of classical logic (which have come down to us essentially unchanged from Aristotle) have an absolute validity, independent of the subject matter to which they are applied. These critiques touched very sensitive, deep beliefs of academics, who revere the Old Ways.

8

UET7H

4. THE PHYSICS REVOLUTION 4.1 FAILURE OF CLASSICAL PHYSICS Here we remind readers of the areas where classical physics started to break down after 1900. We show how the continuous assumptions underlying Maxwell’s electromagnetic theory[9] could not account for the discrete behavior of EM interactions, especially when the oscillations moved into very high frequency ranges, such as X-rays. This story follows the historical sequence of major experimental observations, such as the energy spectrum of heated matter, the way in which crystals reacted anomalously to heat (specific heats), the bizarre results, now referred to as the photo-electric effect, and the problems of explaining the stability of atoms and nuclei, particularly the mystery of discrete atomic spectra. More problems arose when the continuous wave picture could not explain the discrete results found in several scattering experiments, again involving X-rays, such as Bragg scattering and Compton’s scattering off electrons. Further mysteries arose in the case of low-speed electrons, after de Broglie’s hypothesis prompted searches for evidence of electron diffraction and interference effects. In a later section, each of these experiments will be re-interpreted in terms of a new theory discussed here.

4.1.1 PLANCK It was the heat studies of Gustav Kirchhoff (1824-1887), which finally resulted in the necessity for introducing the quantum hypothesis. Kirchhoff was studying the heat characteristics of solids that had reached a stable temperature everywhere (a condition known as “thermal equilibrium”). He studied the heat emerging from a hole into a cavity carved out of such a hot body, which he called a blackbody, as all the radiation entering such a hole is absorbed 100% into the cavity. It is a pity that in 1859, he called the resulting radiation “blackbody radiation” instead of cavity-wall radiation, as this diverted later attention to the empty space surrounded by the actual hot body; however, since this was the æther era, it is understandable. In 1896, Wilhelm Wien measured the complete energy spectrum of a blackbody, as a function only of the frequency of the radiation produced. He found that the frequency at which the maximum energy is radiated increases as the temperature increases; this spectral distribution was independent of the type of material forming the hot body. Using classical methods, both Lord Rayleigh and James Jeans independently created mathematical models that derived Wien’s findings but only in the low frequency spectral range. Their theories predicted that the heat emitted at high frequencies (in the ultra-violet or UV) would become infinite: a result known as the “UV Catastrophe”. In 1900, Max Planck (1858-1947) created his own theory that fitted the experimental results at all measured frequencies, thus avoiding the UV Catastrophe. However, Planck was forced to introduce a “mathematical fiction” (later called the quantum of action) to achieve this result. Planck's radical hypothesis was the critical step that introduced the quantum era into modern physics, much to Planck's later chagrin (as he was a devoted believer in classical physics, particularly Maxwell's theory of EM radiation). Planck's hypothesis was driven only by mathematical necessity.

4.1.2 EINSTEIN 4.1.2.1 SPECIFIC HEATS In 1907, Einstein was the first to extend Planck’s blackbody radiation approach to new areas[10]. He invented a very simplified model of a crystalline solid by replacing each atom with 3 harmonic oscillators (‘elastic springs’ - one for each direction) and also assumed that these atoms (or oscillators) were only 'lightly' interconnected, i.e. effectively independent. He also assumed all the oscillators could only vibrate at the same single frequency f. He followed Planck, by assuming that the energy of each oscillator (E) could only take on discrete values; i.e. E = n h f; this fitted the experiments quite closely.

9

UET7H 4.1.2.2 PHOTO-ELECTRIC EFFECT In 1887, Heinrich Hertz (1857-1895) discovered the revolutionary photoelectric effect when he shone UV light on metallic electrodes finding that the voltage needed to induce sparking was lowered[11]. Hertz's student, Philipp Lenard soon replicated this effect using the recently discovered electrons; but in 1899, he proved that the electrodes themselves were emitting extra electrons, when he again focused UV light on metal foils. By 1902, he had discovered that the energies of the ejected electron where completely independent of the incoming light intensity but depended (linearly) on the frequency of this light. In fact, Lenard found that there was a minimum frequency, below which no electrons were ejected for a given type of metal foil. However, increasing the intensity of the light (of a given frequency - color) did increase the number of ejected electrons but only when the frequency exceeded the minimum. In 1905, Einstein proposed that the UV light was behaving like the concentrated electrons and was transferring energy to the foil's electrons in discrete “packets of energy” or light quanta (later called ‘photons’).

4.2 THE ELECTRON 4.2.1 FINITE PARTICLE

It cannot be repeated enough[12] that it was the ‘discovery’ of the electron by J. J Thomson (1856-1940) and published in 1897 that truly launched modern physics (and modern electrical technology) with his own accurate measurements of both the discrete electrical charge (e = – 4.8 x 10 –10 esu) and discrete mass (m = 9.1 x 10 –34 gm). Prior to these investigations, even the reality of the atom was still in question, while all quantum investigations can be shown to resolve back to the electron; this has been the central thrust of the author’s own research programme for many years. Even modern quantum physics has not properly absorbed this discovery; taking the electron for granted is now the norm. It was the technological advance of vacuum technology around 1870 that allowed William Crookes to develop the first cathode ray tube (CRT). This tube allowed a large external voltage to be applied between the two enclosed electrical terminals: the negative cathode and the positive anode. The “Crookes tube” permitted him to investigate the conduction of electricity in low-pressure gases. He discovered that as the pressure was lowered, the cathode appeared to emit rays, the so-called “cathode rays”, showing that they travel in straight lines, cause fluorescence in some materials upon which they impinge and by their impact produce great heat. He believed that the rays consisted of ordinary matter (atoms or molecules). In 1897, it was J. J. Thomson's investigations that made him suggest that the particles in these rays were over 1000 times smaller than a hydrogen atom and the mass of these particles were independent of the emitting cathode material. He readily concluded that these rays were composed of very light, negatively charged particles, which were a universal building block of atoms. He called the particles “corpuscles” but later scientists preferred the name electron, which had actually first been suggested by G. J. Stoney in 1891, prior to Thomson's actual research. The electron’s charge was more carefully measured later by the American physicist Robert Millikan (1868-1953) in his famous oil-drop experiment of 1909, the results of which were published in 1911.

4.2.2 ELECTRON DIFFRACTION (WAVE?)

De Broglie's radical hypothesis (see later) that matter particles should behave as waves inspired two independent experiments in 1927 that involved electron diffraction off crystalline structures. In Scotland, G. P. Thomson observed the circular interference patterns created by a thin gold film while at Bell Labs (New Jersey), C. J. Davisson and collaborators used a nickel crystalline grid with slow electrons (10 ~ 500 kV) to measure the back-scattered electrons off the crystal surface. Ten years later, Thomson and Davisson shared the Nobel Prize for Physics for their experiments.

10

UET7H

4.3 REAL ATOMS 4.3.1 RUTHERFORD Inspired by the discovery of the electron, the great experimentalist Ernest Rutherford (1871-1937) suggested a planetary-like model of the atom in 1911. Rutherford's team had been bombarding gold foil with the newly discovered alpha particles emitted by certain radioactive materials. He calculated that most of the backwards-scattering measurements could only be explained by assuming almost all the gold atom's mass and electrical charge was concentrated in a billionth of the atom, subsequently called the nucleus. Electrically neutral atoms would have to include a cloud of electrons. His famous paper mentioned the atomic model of Nagaoka, in which the electrons are arranged in one or more rings, with the specific analogical structure of the stable Rings of Saturn.

4.3.2 BOHR

Following his 1911 PhD on Lorentz Electron theory, Niels Bohr (1885-1962) left his native Copenhagen and traveled through England, including some time with Ernest Rutherford. This exposed him to Rutherford's model of the atom and Nagaoka's planetary suggestion for the orbits of the electrons. He also became aware of Nicholson's recent proposal for quantizing angular momentum. Bohr combined these ideas to construct a mathematical model of the atom that had a major impact on physics and the popular imagination[13]. He ignored Maxwell's electro-dynamical theory that predicted that circulating electrons should continuously radiate away their energy and fall into the nucleus but he did use Coulomb's model of static electrical attraction to repeat the mathematical treatment that Newton had used for instantaneous gravitational attraction between the Earth and its smaller moon. Bohr dismissed the relevance at the atomic level of Maxwell's EM theory, literally with the wave of his hand: he just assumed this was so; or in the words of theoretical physics: “he postulated it”. Like Planck's proposal, no mechanism was offered; it just had to be so. This model's major success was in explaining the empirical Balmer formula for the spectral emission lines of atomic hydrogen. Most importantly, Bohr assumed there was only one electron in the hydrogen atom. The Bohr model gives almost exact results but only for a system where two charged points orbit each other at speeds much less than that of light. This not only includes one-electron systems such as the hydrogen atom, singly ionized helium and doubly ionized lithium: all with a singly positively charged nucleus.

4.3.3 SOMMERFELD Although the Bohr atomic model was quite successful for predicting the spectrum of the hydrogen atom, it failed to include the fine structure found[13]. Arnold Sommerfeld (1868-1951) extended Bohr’s simple circular orbits to include elliptical orbits that Newton found were needed for Kepler’s planetary orbits in the Solar system[14]. In addition, he added Planck’s relativistic mass correction to allow for the faster speeds that might be present in extremely elliptical orbits. The relativistic mass effect produced an orbit that took on the shape of a precessing rosette[15].

4.4 MODERN QM 4.4.1 HEISENBERG The puzzling duality of the nature of light was compounded by de Broglie’s dramatic hypothesis that all material particles would also exhibit this duality between waves and particles (see later). It is not clear what motivated the second generation of quantum pioneers to dismiss the Bohr-Sommerfeld model of the atom so readily. Many of the deficiencies of this early approach persisted in the later, mathematical formulations although some improvements in the interpretation of some of the finer details of the hydrogen emission spectrum were achieved.

11

UET7H Heisenberg never considered that the electron was not pursuing a circular or even elliptical orbit around the hydrogen atom but some other more complicated motion that was consistent with the observed results; he simply dismissed all attempts to imagine a more appropriate trajectory as a “limitation of the concept of the electronic orbit.” He was too eager to push the contradictions in the classical physics approach, so that he could introduce his own revolutionary mathematics. In this regard, he was following the Zeitgeist at Göttingen, where he was assistant to Max Born. Here, they together applied an adaptation of the classical perturbation methods of the astronomers to atomic systems, as both problems were examples of the infamous Three-Body problem that had sunk Newton’s attempts to go beyond the 2-body simplification. As Born described later: “these results did not agree with the spectroscopic results for the helium atom”. As a result, the whole team became “more and more convinced that a radical change of the foundations of physics was necessary.” It became clear that a powerful clue was hiding in Bohr’s need to focus on the difference between two stationary states, not on one orbit alone, as in classical mechanics: this direction emphasized ‘transition quantities’[17] Born was the first to actually suggest that these transition ‘amplitudes’ might be handled by some kind of symbolic (matrix) multiplication: a key insight for later developments.

4.4.2 DE BROGLIE

Louis Victor, Duc de Broglie (1892-1987), became the Prince de Broglie in 1960 after his brother died, is famous for making the most radical proposal in 20th century science. As a man fascinated by music, he viewed the atom as “humming with vibrations” so that he saw the electron in Bohr’s model of the hydrogen atom as a mysterious wave spread out along the orbit, so that only full waves (“standingwaves”) were stable and incapable of emitting EM radiation (why?) until they jumped to another, standing-wave orbit. The mathematical equivalence of this hypothesis was that the wavelength (of the wave) was inversely proportional to the momentum of the particle, with Planck’s Quantum Constant playing the key role. De Broglie then made the revolutionary proposal that all material particles in motion are accompanied (“ill-defined”) by such a mysterious “matter-wave”. This was too radical a suggestion for his PhD committee, submitting it to Einstein, who had made a similar proposal for dual characteristics of light with his famous photon proposal. Einstein thought this was a magnificent hypothesis; indeed, it was so important that it gained de Broglie the Nobel Prize in Physics only five years later. In his 1924 thesis[18], de Broglie also explained this periodicity by conjecturing that every electron had its own internal clock - a view accepted in this research. This matter-wave proposal implied that electrons could undergo diffraction when impinging on a periodic structure such as a crystal with inter-atomic separations of about a billionth of a centimeter. This was immediately found to be the case, although no one understood what these waves consisted of (what medium they existed in), or how they could ‘bend’ around an orbital. Later, Born interpreted the (complex) square of these imaginary waves with the probability of being found in a small region of space.

4.4.3 SCHRÖDINGER Although the “Old Quantum Theory” (Bohr through Sommerfeld) actually provided a very good quantitative account of the energy levels in simple atoms, its quantum ‘postulates’ and its introduction of various quantum numbers seemed too arbitrary for the ‘rigorous’ mathematicians. This is why Heisenberg’s new ideas were initially found acceptable (especially to his mathematical colleagues in Göttingen). However, even much of Heisenberg’s approach could be subject to similar criticisms. It was de Broglie’s radical proposal of universal physical duality that really launched ‘modern’ quantum theory, supported by Einstein’s duality view of light. Following up on de Broglie’s ideas, Erwin Schrödinger (1887-1961) decided[19] to find a ‘proper’ three-dimensional wave equation for the electron.

12

UET7H

He was guided by William Rowan Hamilton’s analogy between classical mechanics and optics, summarized in the mathematical insight that the zero-wavelength limit of optics resembles a mechanical system: the trajectories of light rays become sharp tracks that obey Fermat’s optical principle of least time (an analog of Maupertuis’ principle of least action; a rule capable of deriving Newton’s Laws.) At first, Schrödinger was inspired to think of a particle as a group or ‘packet’ of waves that traveled together but all real examples of waves rapidly spread apart over time, while the electron’s charge and mass always remain together. The final form of Schrödinger’s equation resembled the Wave Equation of classical physics (hence “Wave Mechanics”): it is also in the form of the eigenvalue equation of classical physics that received a lot of attention in late 19th Century vibrational physics, so older solutions were readily available. The equation is called the time-independent Schrödinger Equation to acknowledge his key role in its development. It can now be used in several problems in classical physics where the energy of the system is conserved (independent of time). The wave (or psi functions), ψn[x,t] lead to standing waves, called stationary states (or orbitals in atomic chemistry). This wave corresponds to a quantum state of the whole system with a unique total energy; as such, it is sometimes called the energy eigenstate. Schrödinger wanted to impose the condition of linearity, as he wanted to be able to combine its parts (like Huygens) to produce the experimental interference effects and create wave packets. This implies that the most general solution is a linear combination (superposition) of plane waves; this is similar to the mathematical discovery by Joseph Fourier that any finite continuous curve may be represented by an infinite sum of suitably weighted harmonic functions (i.e. plane waves). The superposition hypothesis has the mysterious consequence that allows every particle to exist in two or more states with different classical properties, at the same time. For example, a particle can have several different energies at the same time, and can be in several different locations at the same time. This superposition is still viewed as a single quantum state, as shown by the interference effects, even though this idea conflicts with classical intuition. One problem was solved mathematically by creating a very much deeper philosophical one of meaning.

4.5 QUANTUM DIFFICULTIES Here we will briefly summarize several outstanding issues with the ‘new’ QM; each deserves at least a page of discussion but space is at a premium: suffice is to say, that they have only been raised by a few other unorthodox critics of QM, who reject orthodox QM as ‘proven truths’. 1. QM is not a physics theory but is formulated only as an abstract, mathematical scheme, with quite arbitrary and mostly un-interpretable symbols. 2. Contradictory concepts are offered as “complementary” descriptions and others (e.g. the ‘state’ idea) are often undefined. 3. The role of the macroscopic observer of atomic processes is unclear in fixing various outcomes. 4. Locally formulated theories result in unexpected, macroscopic correlations between remote objects. 5. QM, as a system’s size increases, generates rapidly rising complexity, far exceeding humans’ present calculational capacity. 6. As a result, QM fails to predict the energy levels of multi-electron (even 2) atoms, so its so-called explanation of the chemical Periodic Table is invalid.

13

UET7H

5. QUANTUM ORTHODOXY Even by 1920, theoretical physicists (aka mathematicians) in leading academic institutions were very unhappy with the new quantum theories of Bohr and Sommerfeld, which was soon dismissed as: ‘Old Quantum Theory’. They obsessed on small deficiencies, such as being unable to be extended to multielectron atoms and molecules (in which WM also later failed) and being limited to periodic systems (like atoms) instead of ‘open’ situations, such as ‘scattering’. Implicit in much of this criticism is that these initial theories began with arbitrary, “ad hoc” physical assumptions and contradicted the already sacrosanct Maxwellian EM theory[9] with its elegant mathematics, which they had invested much time in mastering. However, the real psychological force was that these earlier theories lacked the mathematical elegance they had come to expect from “solid” mathematics. So a series of papers were written by these specialists from Cambridge and Göttingen, offering a new approach that blended the particle concepts of CM with the wave-like mathematics of Maxwell’s ‘field’ theory. Within four years, a whole new theory was developed that was named THE Quantum Theory and students were weaned away from the older theories. As with much academic thought, the revolution was enforced by textbooks that became the only source of information for over-worked physics students. The best are summarized next.

5.1 QUANTUM TEXTS 5.1.1 DIRAC Probably the most influential text on the new approach was “The Principles of Quantum Mechanics”[20] by Paul Adrien Maurice Dirac (1902-1984) in 1930. This was published by the influential Oxford Clarendon Press as part of their definitive “International Series of Monographs on Physics”. It is still in print today (4th edition). Its title was deliberately chosen to allude to the world-famous text by Dirac’s predecessor as Lucasian professor of mathematics at Cambridge, i.e. Isaac Newton[21]. Dirac himself was “The Strangest Man” (to quote a very readable biography[22] by Graham Farmelo of this very singular individual), who always did his original thinking alone, was notoriously taciturn. This text influenced thousands, including the present author, who read it privately as an undergraduate. The book introduced Hilbert Space mathematics to many physicists, along with Dirac’s own notation. There is minimal historical context in the book which begins with two significant admissions: 1) the inability of CM to explain the new experimental discoveries; 2) the inadequacies of the EM forces to explain the stability of atoms (the unadmitted starting point of Bohr’s theory). The very first three sections try to preserve as much of Maxwell’s theory as possible; while by page 10, Dirac is defending his lack of imagery that will be developed by stating the mathematicians’ basic loyalty to equations. The heart of Dirac’s thinking here is his confusion on the meaning of the ‘state’ of an atomic system: implying a close analogy with CM’s mathematical view of a single system. Indeed, his whole theory was developed around a mathematical similarity he noticed between differences of derivatives of pairs of particle variables (location and momentum), called Poisson Brackets, and differences of quantum variables (actually matrices or differential operators), he called quantum commutators. So, Dirac simply postulates a relationship between the two sets introducing Planck’s Action Quantum. He calls this vital step, the method of Classical Analogy, recognizing this introduces the failure of the commutative law of multiplication exhibited by ordinary numbers and standard algebra. He even fails to acknowledge this key insight came from his long-term friend, Heisenberg but later, he does write that “the first form of QM was discovered by Heisenberg in 1925,” (which actually set Dirac off on his own lifelong study of QM). So, like the out-maneuvered Bohr, the Nobelist Dirac was unchallenged in his introduction of arbitrary mathematical ‘postulates’, such as the canonical “quantizing commutation relation”.

14

UET7H With his usual admirable honesty, Dirac concludes his masterpiece with the admission that: “The difficulties, being of a profound character, can be removed only by some drastic change in the foundations of the theory, probably a change as drastic as the passage from Bohr’s orbit theory to the present quantum mechanics.” Dirac spent the next 50 years of his dedicated life unsuccessfully pursuing this vision; it has

also inspired the present author on a similar quest. As can be seen from this summary, Dirac’s work provides the mathematical details needed to master QM but very little history. It is not recommended for the general reader but essential for many professional physicists. Einstein has written that Dirac’s book was “the most logically perfect presentation of QM”.

5.1.2 ISHAM Christopher J. Isham (b. 1944) has taught physics at Imperial College (my own Alma Mater). In 1995, he wrote an excellent text (“Lectures on Quantum Theory”)[23] based on lectures to final year undergraduates. It is strongly centered on the math of Hilbert (vector) Spaces but gives little historical context; the works of Bohr and Sommerfeld are not even mentioned while Heisenberg is credited with “discovering QM”; while Schrödinger’s Equation does not surface until page 6, where it exemplifies WM. The book clearly emphasizes the mathematical foundations of QM (Dirac’s notation) and (welcome, rare surprise!) includes QM’s philosophical underpinnings, such as a rare discussion of the key concept of things and their properties. He admits that QM presents a “striking gap between its mathematical structure and the physical objects it seeks to represent.” Like most authors, he introduces the quantizing commutation relation (like a magician pulls a rabbit out of the hat) and omits the time dependency. He does emphasize the centrality of measurement and probability but still implies that the wave function represents the ‘quantum state’ of a single point particle, with the traditional view that the ‘t’ parameter represents the physical (human) time being experienced by the electron, so he offers no new understanding of the mystery of the ‘collapse of the wave function’ upon measurement. The book is well written, logically organized and typical of the ahistorical approach to QM; it bears a strong resemblance to the more difficult text by Dirac, where I suspect it originated, but it is easier to read.

5.1.3 BOHM For all his controversial hypotheses, David Bohm (1917-1992) made a permanent contribution to physics with his massive text on “Quantum Theory”[24]. In contrast to most texts that have addressed this subject (see above), Bohm made a major effort to place wave mechanics in its historical context and made a valiant attempt to present the main ideas in non-mathematical terms; indeed, he explicitly does not introduce the mathematical formulation until his second part, after 170 pages. Bohm had a lifelong interest in history and philosophy and a deep admiration for Einstein and his work. Thus, Bohm reminds us (unlike most accounts) that Planck’s original idea in 1900 was not to quantize the radiation oscillators. Instead, Planck assumed that the EM cavity radiation was in equilibrium with material oscillators in the walls around the cavity, so that these material oscillators could give up or absorb radiant energy only in quantized exchanges of energy: ΔE = n h f. It is a pity that at this point, Bohm falls into the common assumption that there must be (aetherial) quantized radiation oscillators to explain the fact that the blackbody spectrum is independent of the materials forming the cavity walls. As the present theory prefers an explanation that acknowledges that all material atoms involve electrons, which are here treated as the sources and sinks of all the remote interactions with other electrons. This view avoids any need for introducing a new class of fundamental entity called “light” (or EM radiation) that travels from sources to sinks (an almost universal assumption made by most physicists). Like all other QM authors, Bohm invokes de Broglie’s wavelength equation and “derives” Schrodinger’s Equation. Here, he casually comments that: “Practically, the entire quantum theory is contained in this equation; once we know how to interpret the psi function.” [Bohm’s own emphasis].

15

UET7H Recognizing that this interpretation is the central philosophical challenge of QM, Bohm makes a massive effort to introduce some coherence into Bohr and Heisenberg’s confused metaphysical ramblings. Bohm builds his entire case around de Broglie’s wave-particle hypothesis (“matter is somehow ‘associated’ with oscillatory phenomena”) without any more justification than that this “explains” the scattering effects from crystals. Since he views light and electrons as two distinct types of natural entities, he is pleased to see this “great unification” of two mysterious quantum objects that both sometimes behave like waves and sometimes like particles. Unfortunately, Bohm never reconciles the ontological view (existence) with the conventional epistemological view (knowledge). He rejects the epistemological view of the “collapse of the wave function” when he dismisses the sometimes-used analogy of a life-insurance company suddenly discovering the real age of a client and recalculating the client’s new life expectancy. Bohm tries to contrast this statistical ‘knowledge’ interpretation with the deterministic view of classical mechanics, where he claims we could predict, in principle, a person’s lifespan in terms of the motions of all his atoms and molecules. A massive impossibility in all reality.

5.2 QUANTUM HISTORY

There are several academic books that claim to provide the “History of the Quantum” but few really tell the full story covered here. I have selected one excellent book written for the general reader.

5.2.1 JONES In her immensely readable book[25], Canadian science historian Jones retells the quantum tale very well, showing how a handful of men ‘fired by ambition, philosophical conflicts and personal agendas’ created the quantum revolution. She clearly shows that there was never a consensus, so that by the Fifth Solvay Conference in 1927 ‘there was such ill will that most were barely on speaking terms’ as they presented their three competing versions. The quantum revolution failed to produce a single philosophical worldview, which many leading academics demanded. The immediate result of this impasse was for physicists to abandon the need for a philosophical theory of quantum physics. Indeed, by 1940, philosophers of science were expelled from their long-time home (the Temple of Natural Philosophy). Ironically, the three competing versions were soon shown to be mathematically equivalent but the interpretation puzzle was still left bitterly unresolved. She concludes that: ‘the subsequent confusion and uncertainty that has bedeviled quantum physics undermine the idea that it was all figured out a long time ago’. In her introduction, she dares to write that: ‘If after such a long time, all the smart men and women who work in physics have not been able to reconcile the two sets of rules for the universe {classical and quantum}, it’s natural to wonder if one – or both – of the sets might just be wrong. … This suggestion is tantamount to goring a sacred cow, as much as questioning relativity or quantum physics.’ Few professional physicists will agree with her in public, as they all know that this would prove to be a “career-limiting” move because of the power of the orthodoxy (‘Group Think’) in most academic disciplines.

16

UET7H

6. THE MATHEMATICS COUNTER-REVOLUTION 6.1 ALGEBRAIC GEOMETRY 6.1.1 HILBERT David Hilbert (1862-1943), a German mathematician, is now recognized as one of the most influential mathematicians in the early part of the 20th century. He contributed to the study of the foundations of geometry (by creating his own extended set of Euclidean-style axioms) and he formulated the core mathematics of QM, known as Hilbert Spaces and Functional Analysis. He spent much of his time at his native Königsberg (Prussia), where he obtained his Ph.D. at age 23 and his first job. He became professor of mathematics there in 1893 and then two years later at Göttingen: Germany’s leading center of mathematics. His text “Foundations of Geometry” was published by Hilbert in 1899 and translated to English in 1902; it proposed a formal set (called Hilbert's axioms) substituting for the traditional axioms of Euclid that were being challenged by the ‘New Geometers’. The Hilbert Space concept was based on a Euclidean space of infinite dimensions (not our physical three) because he wanted to use Infinite Series throughout his mathematics. His best friend and colleague, Hermann Minkowski (1864-1909) introduced Hilbert to mathematical physics as early as 1905 and Hilbert helped Einstein with the difficult mathematics of Einstein’s theory of gravity (General Theory of Relativity). As a formalist, he viewed much work by theoretical physicists as “sloppy”.

6.2 FOUNDATIONS 6.2.1 RUSSELL There was a growing concern in the second half of the 19th century, amongst academic mathematicians, that the foundations of both mathematics and logic were weaker than many had ever suspected. Classical logic (derived originally from Aristotle) was revisited by a few academics after 1800 with the algebraic reformulation of logic by the autodidact, George Boole (1815-1864) with his invention in 1847 of binary ‘Boolean Logic’, which has led to digital computers. Boole planted the seed of approaching logic from a mathematical perspective and influenced the focus on the development of Set-Theory, where a set is defined as collection of distinct objects that share some ‘membership’ property that justifies their collective treatment as a single collection or ‘Set’. Sets may be combined or split into subsets. The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as in standard arithmetic addition and multiplication are ‘associative’ (e.g. 2+3 = 3+2) and ‘commutative’ (e.g. 2*3 = 3*2), so are the operations of set union and set intersection. It is because the union action resembles standard arithmetic that mathematicians seized on this to investigate the logical foundations of their core subject. It also an observed fact that all small sets of any kind, share the common characteristic that their number of members (called their cardinality) can be mapped to the common set of natural numbers. Several basic concepts of Boole re-appeared independently in the work of (at the time) an unknown German high-school teacher called Gottlob Frege (1848-1925). He is now considered to be a major figure in the history of mathematics, as he is seen as responsible for the development of modern logic. The main goal of Frege was to show that mathematics grows out of logic, and in so doing, he devised techniques that took him far beyond the Aristotelian syllogistic and Stoic propositional logic that had come down through history as the logical tradition. This has led to a mathematical form of logic that is now a complex area in its own right and goes far beyond normal intuitive understanding of the meaning of logic. So much so, that some modern logicians have admitted that it is embarrassing that there is no widely acceptable definition of logic itself. An irony based on logic being the ‘science’ of definitions. 17

UET7H The concept of ‘logical form’ is central to the study of logic. Here, logical form acts like an algebra of the language since the logical form of a sentence (or set of sentences) is the form obtained by abstracting out its ‘content terms’ from the subject matter or by regarding the content terms as mere placeholders or blanks on a standard form or language prototype. In an ideal logical language, the logical form can be determined from its defining rules (or syntax) alone; computer programming languages are now viewed in this light. If a computer programmer fails to follow the defining rules then his resulting programme will always be rejected. Natural languages have evolved over time and often contain anomalies, which academics try to eliminate to create a well-formed, unambiguous version. The validity of an argument is determined by its logical form, not by its content. Statements (or propositions) are assumed to represent binary states of the world, described by the common-use words ‘true’ or ‘false’; so that a statement that is proved to be ‘Not-True’ is always ‘False’. Greater complexity in the world, where there are more than two choices, are not represented in these types of schemes. This is why the assumptions of mathematics (or logic) implicitly refer to existence that is always assumed to have only two possibilities. This is why language semantics is needed to convey meaning between people, which is often only then subjective. As Bertrand Russell (1872-1970) completed his (equivalent Master’s) thesis on the “Foundations of Geometry” [see my essay], he was then recruited by his Cambridge tutor, Alfred North Whitehead (1861-1947) to work together on a ‘short’ study on the foundations of mathematics. This effort was seriously under-estimated at 12 months when it actually took ten years, with its three volumes being published in 1910, 1912 and 1913 with a revised second edition appearing in 1925 through 1927. Its title was deliberately chosen (I suspect by the young, ambitious Russell, rather than the modest, quiet Whitehead) to be “Principia Mathematica”, alluding to Newton’s revolutionary text of 1687 on a New Physics[21]. This is viewed (by many academics) as one of the most important publications written in the 20th century but ironically (like its Newtonian predecessor) is more often referred to than being read – even by specialists in the same profession. One reason for this lack of readership is that it takes almost 360 pages of dense symbolic manipulation to prove the intuitive definition that: “1 + 1 = 2”. One of its main inspirations and motivations was the earlier work of Frege’s work on sets, which Russell realized allowed for the construction of contradictions (more politely called ‘paradoxes’). This was avoided by ruling out the unrestricted creation of arbitrary sets, especially the highly unreal concept of “the set of all sets”. This was achieved by replacing the notion of a general set with the notion of a hierarchy of sets of different ‘types’, a set of a certain type only allowed to contain sets of strictly lower types. The goal of this project was to invent a basic list of axioms and symbolic processing rules (operations) from which all mathematical truths could, in principle, be proven. This ambitious project was of great importance in the history of mathematics and philosophy; being one of the foremost products of the belief that such an undertaking must be achievable. However, in 1931, the Austrian logician/mathematician Kurt Gödel (1906-1978) published his two revolutionary Incompleteness Theorems (based on his PhD thesis) that proved research programs, like Russell’s and Whitehead’s, could never achieve their goal. Worse, he ‘proved’ that any attempt to pre-define basic arithmetic must be inconsistent or there could be some results that will never be deduced from their axioms. Ironically, Gödel needed to invent a new mathematical technique, called ‘Gödel-Numbering’, (based on counting numbers) to identify formal expressions to develop his ‘proofs’, so he was reversing the ancient abstracting method of defining logical forms. Amusingly (at least to this author) was Gödel’s demonstration criticizing core statements about infinite sets.

18

UET7H

7. THE PHYSICS COUNTER-REVOLUTION The Counter-Revolution against the first generation quantum theory (that fitted so well with the facts of atomic spectra) was the work of only a small number of theoretical physicists (actually mathematicians) from two leading academic institutions that had established world-class reputations in mathematics. Here are several brief biographies that might help illustrate their motivations.

7.1 GÖTTINGEN GANG Hilbert joined Felix Klein at Göttingen and together they built a world-class team of mathematicians. They had several post-graduate students who went on to to win Nobel Prizes for their work in QM.

7.1.1 HEISENBERG Werner Heisenberg (1901-1976) was one of the leading theoretical physicists in Germany over much of the 20th century[26]. As the son of a professor of Greek, he followed in the ‘family business’, by aiming for his own academic career. He studied physics and mathematics at Göttingen and Munich university with Arnold Sommerfeld, where he wrote his thesis on classical fluid flows. At Göttingen, he studied physics with Max Born and mathematics with David Hilbert. He worked briefly with Niels Bohr in Copenhagen, after winning a Rockefeller Scholarship, before returning to Göttingen, where he developed his matrix mechanics version of QM. A year later he returned to Copenhagen as a lecturer and continued his QM research. From 1927, he was ‘ordinary’ professor of physics at Leipzig where he influenced many post-graduate and doctoral students. He and Pauli contributed to the development of Quantum Field Theory (QFT). Heisenberg shared the physics Nobel Prize for QM in 1932 with his long-time theoretical rival, Schrödinger[27]. Although Heisenberg was not Jewish, he was repeatedly attacked by pro-Nazi German scientists, to the point where Himmler (head of the SS) had to settle the affair, after a personal appeal by Werner’s mother. None-the-less, Heisenberg was the scientist who headed up the German ‘Uranium Club’, the group investigating a possible atomic bomb and actively sought funding when the military lost interest. Heisenberg was imprisoned in England after the war ended, along with other leading scientists, where their conversations were secretly examined for evidence of Germany’s atomic weapons program; these recordings exposed that many of the prisoners, including Heisenberg, regretted they had failed to make a bomb but claimed they were glad Hitler lost (implausible for such staunch nationalists). In 1946, he returned to German academic life (at several institutions) until his death from kidney cancer.

7.1.2 PAULI

Wolfgang Pauli (1900-1958) was born in Vienna and played a significant role in developing QM. Only two months after graduating top from High-School in 1918, he published his first scientific paper on Einstein’s theory of General Relativity, only recently published in 1915. Like so many others, he studied with Arnold Sommerfeld at Munich university, gaining his PhD in three years for his thesis on the quantum theory of the ionized hydrogen molecule. Immediately, Sommerfeld asked Pauli to write a Relativity article for the German Encyclopedia of Mathematical Sciences; he completed the 230 page article in two months; it was praised by Einstein, later published as a monograph and it has remained a standard reference ever since. Pauli’s initial career tracked Heisenberg’s closely: he too spent a year at Göttingen as Born’s assistant, a year with Bohr in Copenhagen, 5 years as a lecturer and then appointed professor of physics at ETH Zurich (Einstein’s alma mater). His major physics contributions include his Exclusion Principle[28], his theory of spin and his proposal for the mystery particle: the neutrino. Pauli was quite a mystic, spending time with C. G. Jung[29] on esoterica and eventually died of pancreatic cancer. His intelligence and sharp mind led to him being described by many as ‘acerbic’. He delighted to have the ‘Pauli Effect’ named after him; the anecdotal ability to break experimental equipment simply by entering a room. 19

UET7H

8. A NEW MICRO-PHYSICS ALTERNATIVE 8.1 QUANTUM ELECTRON MECHANICS (QEM) 8.1.1 ELECTRON MODEL OF QUANTUM MECHANICS It was only around 1900, that physicists finally had the chance to understand the real atomic basis of nature. It was only then the new discovery of the electron provided us with the true foundation of electricity. Before that event, there had been bitter controversies on the nature of Nature; many scientists (such as Ernst Mach) still disputed the reality of atoms. Unfortunately, science wishes to cling to its theories even when major discoveries cast grave doubt on their foundations. This was the case with light, where Maxwell's Aether-based theory of electromagnetism (EM) had become almost universally accepted by physicists with Hertz's revolutionary experiments on remote induced induction that was interpreted as a proof of Maxwell's EM wave theory. This research programme is dedicated to the proposition that Newton’s metaphysical scheme (including his laws of motion) is a universal foundation for all of physics, at all scales of nature, while Maxwell's theory of electromagnetism is strictly limited to interactions of macroscopic collections of electrons. Indeed, the author has shown that Newton’s corpuscular model of material reality is sufficient to describe the interactions of microscopic collections of electrons, when close together; this provides a new, visualizable model of quantum mechanics that avoids all reference to fields as either mathematics or even existential entities. It is also shown that QM has been the rearguard defense of field theory that, (it is believed here) has been the stumbling block of understanding atomic systems. As historian Jones concludes: ‘There have been no new fundamental laws of nature discovered since the 1970s and there is no math-driven theory that can reconcile the classical and quantum worlds.’ In this author’s view, this is the inevitable result of the mathematics-only approach initiated by the quantum theorists in 1925. The obvious dead-end should have alerted physicists that they have been driving down the wrong road for a long time.

8.1.2 MATRIX QM Although Heisenberg invented his matrix mechanics[30] there are few expositions of it for two good reasons – firstly it involves infinite matrices: columns and rows with an infinite number of numbers in them and secondly, its calculational techniques are difficult and opaque[31]. However, the present author has found that a small text from his undergraduate days (“Matrix Theory for Physicists” by J. Heading – 1958) does include a useful, readable chapter on finite matrices in QM (at least for angular momentum). The author admits that many of the matrices used in QM are infinite and points out that the infinite sum of product terms must be meaningful and so it is really a branch of analysis, rather than discrete algebra, while he declines to provide any interpretation of the individual elements of the matrices, especially those used to replace the standard algebra of CM. Like so many British authors, Heading follows Dirac’s method of classical analogy and begins with the dynamical canonical equations of Hamilton for generalized co-ordinates (and using the explicit potential energy function); this means he must develop techniques for ‘differentiating’ one matrix with respect to another. Again, the magical (matrix) quantizing commutator relations “must be adopted”. The author also introduces the math trick of arranging the answers (the spectral frequencies, f of the atoms as diagonal elements in a ‘Master Matrix’, T: fnm = Tnn – Tmm . These are also used for an exponentially (harmonically) oscillating time factor in the nth row and mth column: exp(i2πfnm t). All matrices have ‘roots’ (or eigenvalues) that not coincidentally form a complete set. Furthermore, these matrices are also Hermitian with real numbers as roots. The chapter concludes with usefully solving the angular momentum problem and that old physics ‘war-horse’: the 1D simple harmonic oscillator.

20

UET7H

8.2 ALTERNATIVE INTERPRETATIONS This section presents brief explanations of QM arising from our independent research into QM, which has resulted in an extensive series of papers providing the details. These views here are very unorthodox.

8.2.1 MICRO/MACRO THEORY OF MEASUREMENT

This research programme takes the view that QM was the result of an illegitimate alliance between the ancient Platonic view of reality as mathematics and the obsession with anti-metaphysical (especially anti-religious) philosophies of certain modern thinkers, such as exemplified by the Positivists around 1900. Indeed, QM is analyzed here as only a theory of measurement17] and not as a theory of material reality (i.e. physics). Indeed, this paper was motivated (in part) by the belief that the quantum wave function is only an eigenfunction of measured situations and only represents the microscopic reality of interactions between electrons in an idealized, statistical manner, so that the continuum mathematics, used in physics since 1690, can still be retained. Just as the Earth (since Copernicus) is no longer viewed as the center of the universe, it can no longer be maintained that humans are at the center of existence – a position implied by several interpretations of QM, including the orthodox ‘Copenhagen Interpretation’.

8.2.2 EXPLANATION OF THE UNCERTAINTY PRINCIPLE Many discussions of QM still obsess on the centrality of Heisenberg’s Uncertainty Principle, as the limit at which human observations of the micro-world inevitably generate statistical results. The new view re-emphasizes that though this is a logical consequence of the operator mathematics of standard QM, there is a deeper level of reality at work here, which will always produce such statistical consequences when macroscopic attempts are made to exactly replicate “the same” experimental configurations. Indeed, this programme adopts a many-body view[32] of the world where all attempts to isolate an ‘ideal experiment’ are doomed to failure, as electrons are shown now to retain a small finite interaction at extreme separations. These attempts to simplify our mental models of the world are viewed here as lastditch attempts to save the Continuum model of reality and its associated continuum mathematics that have dominated mathematical physics for several hundred years. These confusions have persisted because philosophers and scientists have had different ontological commitments, partly because they maintain different philosophies of language. Einstein and Heisenberg viewed light and matter as a single entity expecting field theory to supply the fundamental ontology. Einstein pushed this perspective to where he called QM ‘incomplete’; requiring the theory to reconcile field physics, deterministic causality and the physical continuum in 4 dimensions. These different perspectives result from deep differences in metaphysical world-views that might even be termed ‘religious’ differences.

8.2.3 EXPLANATION OF THE EXCLUSION PRINCIPLE It should be noted by now that QM has continued the ancient tradition (going back to medieval times) of what is now being called “Principle Physics”. In this style of metaphysics, a universal rule (or principle) is proposed that applies universally to all matter, at all times and in all circumstances. This approach can trace its roots to Ancient Greek philosophy, as it was reconstituted by medieval theologians, such as Aquinas. This is the quintessential ‘rationalist’ approach, where rules-of-the-world appears first in someone’s mind and are then conceived to be universal. At the very least, this old religious approach lends itself well to their translation into mathematics, where First Principles have long been held to be enshrined since Euclid.

21

UET7H As a mathematician of extra-ordinary power, Wolfgang Pauli was adept at introducing new principles into quantum physics. One of his most successful was the so-called Exclusion Principle[28]. It was critical to the early success of QM in providing an ‘explanation’ for the Periodic Table of elements and even atomic stability. As is known to some students of QM, this was an ad hoc imposition to compel only one electron to ‘occupy’ one quantum state at a time, even though the key concept of state was never clearly resolved. This principle was given its mathematical form immediately by Pauli, who decreed that all quantum wave functions must be anti-symmetric, when the variables corresponding to two electrons are exchanged. This was a very useful constraint on quantum recipes but did not provide much insight; it only applied to so-called fermionic particles, like electrons. However, the present theory abolishes all non-fermionic particles (also known as bosons) as no more than the real interactions between electrons, in what ultimately will be shown to be a Universal Electron Theory (UET). In the new theory, the Exclusion Principle is shown to be the result that two electrons cannot share the same sub-atomic orbital for more than one instant of time if their interactions are to satisfy the constraint of the finite exchange of one quantum of action. As will be seen, the UET does not feel constrained to abolish the intrinsic concept of a trajectory when discussing the dynamics of electrons, even though Heisenberg tried to remove this ancient corpuscular idea, by decree. Indeed, this was at the heart of Einstein and Schrödinger’s objections to Heisenberg’s revolutionary use of matrix mechanics to calculate ‘observables’ in atomic systems. The idea that below a certain scale of reality, the concepts of space and time were to be abolished seemed utter nonsense to them (and us). Meanwhile, the new wave mechanics drew in its armies of supporters (skilled in differential equations), even though it retained classical particle ideas of Hamiltonians, angular momentum and energy, at its very heart.

8.2.4 EXPLANATION OF THE DE BROGLIE WAVELENGTH Many modern expositions of quantum mechanics begin by focusing on de Broglie’s hypothesis that a wave is ‘associated’ with an electron. Attempts are then made to merge the equation of motion of an isolated particle with the standard wave equation in a continuous medium as a rationalization for the plausibility of Schrödinger’s Equation, which is the most popular form for representing the QM of a non-relativistic electron. Such attempts zero in on de Broglie’s equation that relates the linear momentum (P) to the new wave’s wavelength (λ), namely: λ = h/P, where h is Planck’s quantum of action. The final step was to show that using eigenfunction mathematics can readily generate QM’s central equation by hypothesizing that the momentum variable P can be replaced by the corresponding linear operator form: Px = ih ∂/∂x. As is readily seen, this is a purely mathematical approach and gives no physical insights into why such a method should work for electrons. It is not surprising that such a symbolic approach should have generated extreme semantic argumentation on what all this ‘machinery’ means. In contrast, this research programme positions itself in the middle of the Natural Philosophy tradition that attempts to make physical hypotheses first (such as: all matter is universally attracted to itself) and only then try to transform these philosophical stories into mathematics. The evolution of this research programme has been a gradual exposition of such new properties of the electron; in particular, the focus has been on understanding the interactions between electrons. The latest chapter of this saga now focuses on how these interactions quantize the activity of pairs of electrons. The new view begins with restrictions on the quantity of (inter)action exchanged between two electrons which are involved with each other. This quantity (Planck’s Constant) is a fundamental property of the relationships that define the very nature of being an electron, as much as its electric charge (e) and its mass (m).

22

UET7H

9. SUMMARY AND CONCLUSIONS 9.1 SUMMARY 9.1.1 RESISTANCE TO CHANGE The thesis of this essay is that powerful members of the western intellectual community have deeply resisted the innovations that began to emerge from a few imaginative members of their own academies. At first, this was in the areas of mathematics and logic; this was appropriate, as this had been the earliest origins of the awakening of intellectual abstraction, initiated by the Ancient Greeks 2,500 years ago. This area retained its hold on western thinking through its grip on the education of each generation of young men through the teaching of Euclidean (flat-spaced) geometry. The grip on the western mind was reinforced in the Late Middle Ages by innovative intellectuals, like Galileo and Descartes, who retained their deep loyalty to the centrality of mathematics as the ‘Royal Road to Understanding Reality’[33]. The orthodox story of science celebrates the mathematical innovations of Isaac Newton, who formalized the revolutionary idea of infinitesimals into his mathematics of the calculus, a useful tool for re-examining the observed behavior of the nearby Heavens (the planets). A major aspect of this story that is rarely told is that the human preference for spatial ‘explanations’ (based on our principal sense of vision) was always present in the implicit role of geometry (explicit in Newton’s masterpiece that is rarely read by modern scientists). This expulsion of the difficult idea of time was formalized by the masters of continuum mathematics, such as LaGrange, who replaced Newton’s idea of time-based interactions (or ‘force’ in its continuum form) with the time-free concept of spatial energy, now known as Potential Energy. This tactic ensured that changeless spatial sciences, like geometry, would retain their grip on the educated minds of western civilization. The tragedy of modern science is that most experts have become super-specialists. One consequence of this trend is that few physicists are aware of the recent findings of Newton scholars. I. B. Cohen, the first professor of the History of Science at Harvard, has written that most earlier scholars missed Newton’s intent to attribute changes due to the interactions between objects as acting continuously, leading to the centrality of the ‘force’ concepts in modern views, which are reinforced by common human-based experience of muscle-based changes[34]. In contrast, Cohen has proposed that Newton viewed changes being induced by a series of brief pulses, or to use the technical term: ‘impulse’. Another Newton scholar, Edwin Burtt, also a professor of the History of Science at Stanford, reminds us that Newton was far more empirical than the mathematical fanatics (Galileo and Descartes), particularly as Newton was aware that his great invention (differential calculus) “whose operations could not be fully represented geometrically”[35]. Furthermore, he writes: “For Newton, there was absolutely no a priori certainty, such as Kepler, Galileo and especially Descartes, believed in that the world is thoroughly mathematical; still less that its secrets can be fully unlocked by the mathematical methods already perfected . The world is what it is; in so far exact mathematical laws might be discovered in it, well and good; so far as not, we must resign ourselves to some other less certain methods.” As Burtt summarizes these views of Newton: “there is a distinct difference between mathematical and physical truths. ... Mathematics must be always modeled on experience. Thus, Newton was the true heir of the modern revolution: the experimental and experimental as well as the deductive and mathematical.” It is with superb irony that Burtt recognizes that: “he (Newton) had far less confidence in deductive reasoning as applied to physical problems than the average modern scientist.” The reader here will have noticed the repeated attempts by mathematicians to still ‘sell’ their explanations of physical phenomena by offering only equations; an attempt that the present writer (and physicist) also finds “frustrating”.

23

UET7H

9.1.2 QUANTUM MECHANICS This resistance to change (and preference for continuum mathematics) has been exposed here in the false attempts by mathematical physicists to obscure the physics-based explanations pioneered by both Bohr and Sommerfeld; pretending that quantum mechanics only really entered the world in 1926 with both Heisenberg’s infinite matrices (or equivalent Hilbert vector spaces) or Schrödinger’s wave mechanics. It is important to see how the mathematics of the particle model of classical dynamics was merged with the wave mathematics to produce Bohr’s confusing concept of wave-particle duality under its philosophical mask known as Complementarity. It has been shown how central is the original Newtonian definition of instantaneous momentum to all formulations of QM; again, a mathematical conception with no empirical meaning. This is why a deliberate effort was made here to tell the real history of QM. Only a historical approach can show how their various ideas stimulated later evolution of more sophisticated theories. This timebased story shows how much of the final versions of QM owed so much to Bohr’s earliest ‘planetary’ model, even though this was later severely criticized by most of the second generation quantum physicists. In fact, the author’s own research keeps returning to Sommerfeld’s incredibly accurate model of the hydrogen atom. The most bitter battles in physics have always occurred over the interpretations of the various theories (perhaps, this is why modern physicists are loathe to re-admit philosophers into their professional deliberations). 19th Century science was riven between the proponents of the corpuscular theory of light and the academic mathematicians, who promoted their ‘elegant’ wave theory of light. A similar battle arose in the last quarter of that same century centered on the reality of atoms when the Positivists, following Ernst Mach, decided that only evidence available to human senses could define reality; this was vigorously opposed by ‘Realists’ such as Einstein, who believed there was a world beyond humans. The Austrian, Ludwig Boltzmann (1844-1906) was the first physicist to challenge the idea of the classical continuum with his statistical theory of thermodynamics to estimate the degree of chaos (entropy) at the molecular level. These molecules were far too small to be seen, never mind tracked, but this did not prevent him calculating macroscopic averages that could be compared with experiments. This assumption of atomic scale entities seriously offended Mach, who was academically influential at that time. Worse, Boltzmann’s approach relied on probabilities – or the ‘mathematics of gambling’, as it was often called by its opponents. Ironically, both features eventually contributed to the development of QM. Even Max Planck eventually used some of Boltzmann’s techniques when he tried to fit the energyfrequency curves of so-called ‘Black-Body’ radiation: the first step in launching the quantum revolution. Poor old Boltzmann soon committed suicide because he could not handle the constant antagonism from his more powerful and prestigious Viennese academic colleague (Mach). Science has adopted the ancient religious debates that long occurred between educated men; again the contentious issues are those relating to meaning (or interpretations), which we call metaphysical because all we have are vague intuitions that are difficult to even state in non-subjective language. Clever men have built a whole history around the symbolic system we call mathematics but it has severe limitations.

24

UET7H

X REFERENCES [ 1] Spencer HJ, An Algebraic Representation for Action-at-a-Distance, (2007) https://jamescook.academia.edu/HerbSpencer [ 2] Kuhn TS, The Structure of Scientific Revolutions, Chicago, IL: University of Chicago Press, 2nd edition (1970) [ 3] Toulmin S, Goodfield J, The Architecture of Matter, Chicago, IL: University of Chicago Press (1962) [ 4] Herman A, The Cave and the Light: Plato versus Aristotle, New York, NY: Random House (2013) [ 5] Heisenberg W, Physics and Philosophy, New York, NY: Harper Collins (1958) p. 144; reprinted NY: Prometheus Books (1999) [ 6] Hawking S, God Created the Integers, Philadelphia, PA: Running Press, (2005) [ 7] Alexander A, Infinitesimal, Sci Am Books, New York, NY: (2014). [ 8] Westfall RS, Never at Rest, Cambridge, UK: Cambridge University Press, p.65 (1980) [ 9] Maxwell JC, Treatise on Electricity and Magnetism, Oxford, UK: Clarendon Press (1873); 3rd Ed.; reprinted New York, NY: Dover Books, (1953) [10] Einstein A, Planck's Theory of Radiation & the Theory of Specific Heat, Ann. Phys. 2 180 (1907) [11] Buchwald JZ, The Creation of Scientific Effects – Hertz, Chicago, IL: University of Chicago Press (1994) [12] Navarro J, A History of the Electron, Cambridge, UK: Cambridge University Press (2012) [13] Bohr N, On the Constitution of Atoms and Molecules, Phil. Mag., 26 pp. 1, 476, 875 (1913) [14] Sommerfeld A, On the Quantum Theory of Spectral Lines, Ann. der Phys., 51 1 (1916) [15] Born M, Atomic Physics, London, UK: Blackie & Son, p. 365, seventh edition (1962) [16] Van der Waerden BL, Sources of Quantum Mechanics, (Ed.), Amsterdam: North-Holland Publishing (1967); reprinted New York, NY: Dover Books (1968) [17] Pauli W, On the Hydrogen Spectrum from the Viewpoint of the New Quantum Mechanics, Z. Phys. 36 336 (1926); translated into English in [16] p. 387. [18] de Broglie L, Research into the Quantum Theory, (PhD thesis), Paris: Masson (1963) [19] Schrödinger E, Quantization as an Eigenvalue Problem, Ann. der Phys. 81 109 (1926) [20] Dirac PAM, The Principles of Quantum Mechanics, Oxford, UK: Clarendon Press; 4th edition (1958) [21] Newton I, Philosophiae Naturalis Principia Mathematica, (1687); reprinted as The Principia, translated by A. Motte, Amherst, NY: Prometheus Books (1995) [22] Farmelo G, The Strangest Man, New York, NY: Perseus Books, (2009) [23] Isham C, Lectures on Quantum Theory, London: Imperial College Press, p. 140 (1995) [24] Bohm D, Quantum Theory, Englewood Cliffs, NJ: Prentice-Hall Inc. (1951); reprinted New York, NY: Dover Books p. 335 (1989) [25] Jones S, The Quantum Ten, Toronto, ON: Thomas Allen Pubs, (2008) [26] Powers T, Heisenberg’s War, New York, NY: A. A. Knopf, (1993) [27] Moore W, A Life of Erwin Schrödinger, Cambridge (UK): Cambridge Univ. Press (1994) [28] Pauli W, On the Connection between the Completion of Electron Groups in an Atom with the Complex Structure of Spectra, Z. Phys., 31 765 (1925) [29] Miller AI, 123: Jung, Pauli & the Pursuit of a Scientific Obsession, New York, NY: Norton & Co., p. 101 (2009) [30] Heisenberg W, The Physical Principles of the Quantum Theory, New York, NY: Harper Collins (1930) [31] MacKinnon E, Heisenberg, Models and the Rise of Matrix Mechanics, Hist. Studies in Phys. Sci., 8 137 (1977) [32] Spencer HJ, Quantum Optical Mechanics, (2012) https://jamescook.academia.edu/HerbSpencer [33] Penrose R, The Road to Reality, New York, NY: A. A. Knopf, (2004) [34] Cohen IB, Smith GE, Cambridge Companion to Newton, Cambridge, UK: Cambridge University Press, p.65 (2002) [35] Burtt EA, The Metaphysical Foundations of Modern Science, New York, NY: Doubleday Anchor Books (1932); 2nd Ed.; reprinted New York, NY: Dover Books, p. 210 (2003)

25

UET7H

TABLE OF CONTENTS ABSTRACT .................................................................................................................................................... 1 1. INTRODUCTION ....................................................................................................................................... 1 1.1 SUMMARY ......................................................................................................................................... 1 1.2 OBJECTIVES ........................................................................................................................................ 2 1.3 OVERVIEW ......................................................................................................................................... 2 1.3.1 APPROACH .................................................................................................................................. 2 1.3.1.1 History ................................................................................................................................. 2 1.3.1.2 PHILOSOPHY ........................................................................................................................ 3 1.3.1.3 NATURAL LANGUAGE .......................................................................................................... 3 2. THREATENED MATHEMATICS.................................................................................................................. 4 2.1 GEOMETRY ........................................................................................................................................ 4 2.1.1 EUCLID ........................................................................................................................................ 4 2.2 NATURAL NUMBERS .......................................................................................................................... 4 2.2.1 ARITHMETIC................................................................................................................................ 4 2.2.2 MULTIPLICATION ........................................................................................................................ 4 2.2.2 ALGEBRA ..................................................................................................................................... 5 2.2.3 COMPLEX NUMBERS .................................................................................................................. 5 2.2.4 PROBABILITY ............................................................................................................................... 5 2.3 ANALYTIC GEOMETRY ........................................................................................................................ 5 2.3.1 DESCARTES ................................................................................................................................. 5 2.4 ANALYSIS............................................................................................................................................ 5 2.4.1 INFINITESIMALS .......................................................................................................................... 6 2.4.2 NEWTON (Calculus) .................................................................................................................... 6 2.4.3 FOURIER SERIES .......................................................................................................................... 6 3. DISCRETE MATHEMATICS ........................................................................................................................ 7 3.1 ALGEBRA ............................................................................................................................................ 7 3.1.1 HAMILTON .................................................................................................................................. 7 3.1.2 CAYLEY ........................................................................................................................................ 7 3.2 FINITE DIFFERENCES .......................................................................................................................... 7 3.3 NON-EUCLIDEAN GEOMETRIES ......................................................................................................... 7 3.3.1 GAUSS ......................................................................................................................................... 8 3.3.2 TOPOLOGY .................................................................................................................................. 8 3.3.3 THE INTUITIVE REBELS ................................................................................................................ 8 4. THE PHYSICS REVOLUTION ...................................................................................................................... 9 4.1 FAILURE OF CLASSICAL PHYSICS ........................................................................................................ 9 4.1.1 PLANCK ....................................................................................................................................... 9 4.1.2 EINSTEIN ..................................................................................................................................... 9 4.1.2.1 SPECIFIC HEATS.................................................................................................................... 9 4.1.2.2 PHOTO-ELECTRIC EFFECT .................................................................................................. 10

26

UET7H 4.2 THE ELECTRON ................................................................................................................................. 10 4.2.1 FINITE PARTICLE ....................................................................................................................... 10 4.2.2 ELECTRON DIFFRACTION (WAVE?) ........................................................................................... 10 4.3 REAL ATOMS .................................................................................................................................... 11 4.3.1 RUTHERFORD............................................................................................................................ 11 4.3.2 BOHR......................................................................................................................................... 11 4.3.3 SOMMERFELD ........................................................................................................................... 11 4.4 MODERN QM ................................................................................................................................... 11 4.4.1 HEISENBERG ............................................................................................................................. 11 4.4.2 DE BROGLIE............................................................................................................................... 12 4.4.3 SCHRÖDINGER .......................................................................................................................... 12 4.5 QUANTUM DIFFICULTIES ................................................................................................................. 13 5. QUANTUM ORTHODOXY ....................................................................................................................... 14 5.1 QUANTUM TEXTS ............................................................................................................................ 14 5.1.1 DIRAC ........................................................................................................................................ 14 5.1.2 ISHAM ....................................................................................................................................... 15 5.1.3 BOHM ....................................................................................................................................... 15 5.2 QUANTUM HISTORY ........................................................................................................................ 16 5.2.1 JONES ........................................................................................................................................ 16 6. THE MATHEMATICS COUNTER-REVOLUTION ....................................................................................... 17 6.1 ALGEBRAIC GEOMETRY ................................................................................................................... 17 6.1.1 HILBERT..................................................................................................................................... 17 6.2 FOUNDATIONS................................................................................................................................. 17 6.2.1 RUSSELL .................................................................................................................................... 17 7. THE PHYSICS COUNTER-REVOLUTION ................................................................................................... 19 7.1 GÖTTINGEN GANG .......................................................................................................................... 19 7.1.1 HEISENBERG ............................................................................................................................. 19 7.1.2 PAULI ........................................................................................................................................ 19 8. A NEW MICRO-PHYSICS ALTERNATIVE .................................................................................................. 20 8.1 QUANTUM ELECTRON MECHANICS (QEM) ..................................................................................... 20 8.1.1 ELECTRON MODEL OF QUANTUM MECHANICS ....................................................................... 20 8.1.2 MATRIX QM .............................................................................................................................. 20 8.2 ALTERNATIVE INTERPRETATIONS .................................................................................................... 21 8.2.1 MICRO/MACRO THEORY OF MEASUREMENT .......................................................................... 21 8.2.2 EXPLANATION OF THE UNCERTAINTY PRINCIPLE..................................................................... 21 8.2.3 EXPLANATION OF THE EXCLUSION PRINCIPLE ......................................................................... 21 8.2.4 EXPLANATION OF THE DE BROGLIE WAVELENGTH.................................................................. 22 9. SUMMARY AND CONCLUSIONS............................................................................................................. 23 9.1 SUMMARY ....................................................................................................................................... 23 9.1.1 RESISTANCE TO CHANGE .......................................................................................................... 23 9.1.2 QUANTUM MECHANICS ........................................................................................................... 24 X REFERENCES ........................................................................................................................................... 25 TABLE OF CONTENTS ................................................................................................................................. 26

27