A few trivia about the fine structure constant

Clifford Pickover

In a post I published in this blog over three years ago, I talked about the fine structure constant and some of its physical peculiarities. In this post I am going to talk about some of its mathematical peculiarities. I have taken them from a book I have mentioned on other occasions: A Passion for Mathematics, by Clifford A. Pickover.

As we know, the most exact value of this constant is this:

α = 1/137.035999206

Pickover points out that Eric W. Weisstein, in his World of Physics website, offers two mathematical approximations incredibly close to this value:

Mathematical trivia and quotes from mathematicians

I have taken the following trivia and quotes about mathematics from the book A Passion for Mathematics, by Clifford A. Pickover, which I have mentioned in another post in this blog. These are the trivia:

• Let's see four amazing properties of number 5: a) It is the hypotenuse of the smallest Pythagorean triangle. b) There are five Platonic solids. c) It is the smallest automorphic number. Automorphic numbers are those whose square ends in the number. d) It is probably the only odd untouchable number. Untouchable numbers are those that are not equal to the sum of the proper divisors of any other number.

Is science opposed to faith?

Charles Darwin

The opposition between science and faith is a nineteenth-century invention. And it was not scientists who invented it, since most of them were believers. Those responsible were atheist philosophers such as Marx, Feuerbach, Schopenhauer or Nietzsche. I count Marx among philosophers, even though he abhorred philosophy, which he considered dead (he said in the Manuscripts), just as Stephen Hawking did a century and a half later, as I commented in this post. I once said that Marx would have been horrified to know that he is studied today in the history of philosophy, for he did not consider himself a philosopher, but an economist.

The nature of the physical world

Arthur Eddington

The Nature of the Physical World is the title of a landmark work in the history of popular science. Published in 1928, it compiles the Gifford lectures given in Edinburgh by its author, Arthur Eddington, in 1927. Eddington was then famous, having been the scientist who, in 1919, on the occasion of a solar eclipse, organized the expedition that proved one of the predictions from Einstein's theory of General Relativity: the deflection of light when passing near a star. It was said of him that he was one of only three people in the entire world who understood General Relativity. In addition to this, Eddington was a pioneer researching on the origin of the energy of stars, for he was the first to propose that it came from the fusion of hydrogen to form helium.

The problem of the cosmological constant

Albert Einstein

The value of the cosmological constant Λ in Einstein's equation has gone through many vicissitudes and alternatives:

The fine-structure constant

Arnold Sommerfeld

One of the latest scientific advances of 2020 was a new, more precise measurement of the fine-structure constant. The last value officially accepted in 2018 by the Committee for Data on Science and Technology (CODATA) is equal to 1/137.035999084. The value obtained in December 2020 is equal to 1/137.035999206. It will be seen that the difference with the previous value is very small and affects only the seventh decimal place.

But what is the fine-structure constant? It's a dimensionless constant, defined by the following formula:

Travelling to the past?

S.Augustin, by Louis Comfort Tiffany
Lightner Museum

In his Confessions (Book XI, chapter 14), St. Augustine wrote these words, still valid today:

What then is time? If no one asks me, I know what it is. If I wish to explain it to him who asks, I do not know.

In the current situation of our scientific and philosophical knowledge, we still don’t know what time is.

·         For classical philosophy and Newton’s science, time is a property of the universe. Therefore, time would be absolute.

·         For Kant, time is an a priori form of human sensibility (i.e. a kind of mental container to which our sensory experiences adapt).

·         For Einstein, time is relative to the state of repose or movement of each physical object. There is, therefore, no absolute time.

·         For the standard cosmological theory, there is the possibility to define an absolute cosmic time for every physical object, measuring the time distance since the Big Bang to the present.

·         For the A theory of time (using J. McTaggart’s terminology) the flow of time is part of reality. The past no longer exists. The future does not yet exist. There is only the present. If the A theory is correct, travel to the past is impossible, because you cannot travel to what does not exist.

·         For the B theory of time, the flow of time is an illusion. Past, present and future exist simultaneously, but for each of us the past is no longer directly accessible, and the future is not yet accessible. Einstein adopted the B philosophy of time. In a condolence letter written to someone who had lost a beloved person, he wrote the following:

The distinction between past, present and future is only a stubbornly persistent illusion.

Irreversible processes

Those physicists who consider the arrow of time as an illusion have a problem: not all physics is compatible with a reversible time, as the equations and theories mentioned in an earlier article of this blog seem to indicate. The second principle of thermodynamics is known since the mid-nineteenth century (1850), when Clausius introduced the concept of entropy and it was proved that the value of this physical magnitude always increases, if it is measured in an isolated system that does not exchange matter or energy with its outside. Since the universe is an isolated system, we have at least one physical quantity that makes it possible to unequivocally signal the direction of time flow.

Aware of this problem, physicists in favor of the reversibility of time have answered in different ways: it has been said that the second principle of thermodynamics is a fictitious, subjective law that does not conform to reality; a mental illusion; an approximation; a consequence of the initial conditions of the universe. It has been hypothesized that, if the universe were cyclic, the arrow of time would be reversed during the contraction stage. (This theory has been abandoned). To escape the problem, Stephen Hawking proposed a universe without initial conditions in his book A Brief History of Time. It is curious, this desire to defend at any price the reversibility of time, when it was precisely Hawking who proposed the existence of an arrow of time in black holes, which rather than being permanent, would disintegrate.

In 1928, a year after inventing the term the arrow of time, Arthur Eddington challenged the physicists who defend the reversibility of time with the following devastating words: If your theory is found to be against the second law of Thermodynamics... there is nothing for it but to collapse in deepest humiliation (The Nature of the Physcal World, 1928).