Electrons and Group Theory

A colleague and friend of mine has two degrees, one in Mathematics and a second in Engineering. He now teaches at Birkbeck College in London. He recommended a book (or series of books) to me: The World of Mathematics, edited by James Newman.

James Newman was an interesting man. He was a practicing lawyer in New York, then Chief Intelligence Office at the US embassy in London during the war. In the meantime he was an expert mathematician. In 1940, together with Edward Kasner, he adopted the term “googol” for a very large number, which term was used by the founders of Google, to signify very large amounts of data. In 1945 Newman became a member of the board of editors for Scientific American. In 1956 he published a four-volume series of books, The World of Mathematics, representing 15 years of work to collect the best essays on mathematical topics, together with his own introduction on each topic.

One of the topics (Volume 3, Part IX The Supreme Art of Abstraction: Group Theory) has an essay by Sir Arthur Stanley Eddington, on The Theory of Groups. Eddington was a very eminent mathematician, physicist and astronomer from Cambridge in the first half of the 20th century. In the essay, Eddington uses as an example something that has always puzzled me. What exactly is an electron?

My own puzzlement is simply from ignorance. I am sure a physicist could tell me what he thinks an electron is. What puzzles me is that a particle is described as a physical entity; and yet its scale is so small it must be beyond the ability of any device to measure it. I understand that we could observe sub-atomic behaviour. But that means we have a set of observed behaviours, not necessarily a particle.

Eddington uses an electron as an example of the application of Group Theory. He starts with a quotation from Bernard Russell, about the relationship between our observations and the objective reality that gives rise to them. The quotation is from Russell’s Introduction to Mathematical Philosophy: “In short, every proposition having a communicable significance must be true of both worlds [observation and reality] or of neither”.

Eddington says that using words like “orbit” and “jump” in relation to an electron causes us to create in our mind a misleading impression of reality, and to draw false conclusions from it. Instead what matters is the mathematical statement of the operations that define the observed behaviour.

“In describing the behaviour of an atom reference is often made to the jump of an electron from one orbit to another. We have pictured the atom as consisting of a heavy central nucleus together with a number of light and nimble electrons circulating round it like the planets round the sun. In the solar system any change of the orbit of a planet takes place gradually, but in the atom the electron can only change its orbit by a jump. Such jumps from one orbit to an entirely new orbit occur when an atom absorbs or emits a quantum of radiation.”

“You must not take this picture too literally. The orbits can scarcely refer to an actual motion in space, for it is generally admitted that the ordinary conception of space breaks down in the interior of an atom; nor is there any desire nowadays to stress the suddenness or discontinuity conveyed by the word “jump”. It is found also that the electron cannot be localised in the way implied by the picture. In short, the physicist draws up an elaborate plan of the atom, and then proceeds critically to erase every detail in turn. What is left is the atom of modern physics!”

[And later] “In the early days of atomic theory, the atom was defined as an indivisible particle of matter. Nowadays dividing the atom seems to be the main occupation of physicists. The definition contained an essential truth; only it was wrongly expressed. What was really meant was a property typically manifested by indivisible particles but not necessarily confined to indivisible particles. That is the way with all models and pictures and familiar descriptions; they show the property that we are interested in, but they connect it with irrelevant properties which may be erroneous and for which at any rate we have no warrant. You will see that the mathematical method here discussed is much more economical of hypothesis. It says no more about the system than that which it is actually going to embody in the formulae which yield the comparison of theoretical physics with observation. And, is so far as it can surmount the difficulties of investigation, its assertions about the physical universe are the exact systematised equivalent of the observational results on which they are based. I think it may be said that hypotheses in the older sense are banished from those parts of physical science to which the group method has been extended. The modern physicist makes mistakes, but he does not make hypotheses.”

This is the gist of the essay:

  1. A Group is a set of operators, such that the product of any two of them always gives an operator belonging to the set.
  2. Eddington describes a construction of operators that starts simple and gets a bit more complex by combining the operators in various ways.
  3. He describes how he first worked out this more complex set of operators (but still made up of simple operations performed in various sequences) in connection with Dirac’s theory of the electron. He then learnt that it was the same set of operators contained in Kummer’s quartic surface.
  4. Taking the base operators to construct more complex operators; and then taking a combination of operators; and then squaring the result; gives a result analogous to the three dimension in space.
  5. If you add one more operator and square the result, it breaks down (just as there is no fourth dimension in space). But if you takes the inverse of the additional operator, then you obtain a result that is analogous to the dimension of time, where time is distance at the speed of light.
  6. So we move from a simple set of operators; to a more complex set that describes the wave mechanics of an electron; to a further set that describes relativity.

“Thus the distinction between space and time is already foretold in the structure of the set of E-operators [the more complex operators I mentioned above]. Space can have only three dimensions, because no more than three operators fulfil the necessary relationship pf perpendicular displacement. A fourth displacement can be added, but it has a character essentially different from a space displacement. Calling it a time displacement, the properties of its associated operator secure that the relation of a time displacement to a space displacement shall be precisely that postulated in the theory of relativity.”

The World of Mathematics, edited by James Newman, published by Simon and Schuster, New York, 1956 ISBN 978-0-486-41153-8 / 41150-7 / 41151-4 / 41152-1 reprinted unabridged by Dover Publications.

Axioms

Isaac Newton, in his Mathematical Principles of Natural Philosophy of 1687, used the heading “Axioms (or laws)” for what we know as the three laws of motion. The term “axiom” puzzles me, because an axiom should be a given, requiring no further proof, and used as a premise when reaching a conclusion. But in this case the axiom (or law) seems to be something that Isaac Newton is setting out to prove applies universally, to celestial bodies as well as objects on Earth.

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Exoplanets

The study of planets outside our solar system.

planet_hunters
Image: Planet Hunters* transit mapping tool.

An exoplanet is a planet orbiting a star other than the Sun. The first confirmed detection was only in 1992, orbiting a pulsar, and in 1995 around a normal star. Early detections were by terrestrial telescope. The large number of recent detections has been made by the NASA Kepler telescope, launched in 2009. Apart from being fascinating in itself, the significance of exoplanets is that they form in the early years of a star’s lifecycle. Observing large numbers, at different stages of a star’s life, tells us about planet formation. The light from an exoplanet can be used to infer the chemical composition of the planet and atmosphere.

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Hydrodynamic Quantum Analogs: Prof. John W.M. Bush, MIT

Hydrodynamic Quantum Analogs

Professor John W.M. Bush, Professor of Applied Mathematics, MIT

orbiter-JFM2

The wave field generated by a droplet (black dot) executing an inertial orbit (dashed circle).

“Yves Couder and Emmanuel Fort have recently discovered that droplets walking on a vibrating fluid bath exhibit features previously thought to be peculiar to the quantum realm, including single-particle diffraction, tunneling, quantized orbits and orbital level splitting. Much of my group’s recent research has been directed towards elucidating the subtle pilot-wave dynamics of these walking droplets, and rationalizing their quantum-like behaviour.”