Culture & Ideas

The Mystery of the Number 1/137


The universe holds many mysteries. One of them is a number that nobody can really explain, though many have tried.

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Senior Futurist, Editor

Posted Sep 26, 2019 in Culture & Ideas Article from Scenario 04:2019

Mathematics is a nice science to work with, since it isn’t based on the measurement of anything in the real world. All mathematics takes its basis in a range of definitions and analyses based on these definitions. E.g. a circle is defined as a curve where all points are equidistant from a centre. The number π is defined as the ratio of a circle’s circumference to its diameter, and you can mathematically calculate the precise value of π – you don’t have to go out and measure any actual physical circles. Similarly, the number e (or Euler’s number), the basis for natural logarithms, is defined by the area beneath the curve 1/x from x =1 to x = e being equal to 1 (mathematically written as ln e = 1). π and e are both so-called irrational numbers that can’t be written as the ratio of any two whole numbers, and even though the numbers may seem a bit odd, they are beautifully mathematically related as expressed by Euler’s identity, eiπ + 1 = 0, where i is the imaginary unit, equal to the square root of minus 1. In this manner, everything in mathematics comes together in a larger whole.

In the real world, the world of physics, things don’t come together that well.

When I studied physics at Copenhagen University as a youth, I had a thought that many before me have probably also had: instead of using the metric system or the system of feet, pounds, and gallons, which are all based on our bodies or our planet, could we not develop a system of units where all the basic physical constants have the value 1? Where the speed of light, c, is 1; where the elementary charge, e, is 1; where Planck’s constant, h is 1, and so on? That would be a truly universal system of measurement, independent of special earthly or human conditions. It might not be the most practical system of measurement in our everyday lives; for instance, we would have to measure the speed of our cars in billionths of light speeds. But then, many everyday measurements are made with those sorts of skewed numbers anyway: millibar, megahertz, kilojoule, etc., so we could probably live with it. 

However, I quickly found that everything did not come together neatly. If a number of physical constants are set to be one, others are forced to have non-simple numerical values. This was my first encounter with the number usually known as the fine structure constant, even though it is uncertain whether the number in fact has anything to do with the fine structure of atoms.

The Fine Structure Constant

The fine structure constant has a value very close to 1/137; so close, that the physicist Arthur Eddington argued for many years that the value was precisely 1/137. Later, more precise measurements of basic physical constants showed that this wasn’t the case. The best value we have today is 1/137.035999084 (with some uncertainty about the last two digits). Many have over the years attempted to re-create this number as a combination of known mathematical numbers like π and Euler’s number, but it has proven impossible. The number remains a mystery. 

There are many ways to arrive at the fine structure constant, which is typically denoted by the Greek letter α. One is that α α = kee²/αhc, where ke is Coulomb’s constant, a basic unit of electrical power. Other ways involve more exotic physical constants like the vacuum permittivity, the vacuum permeability, or the vacuum impedance, but the result remains the same.

Since the fine structure constant was first calculated about a hundred years ago, physicists have philosophised over what the number actually means. Why does this number exist; why doesn’t everything simply come together neatly as with Euler’s identity? The first suggestion was that it might be the ratio between an electron’s velocity in the first circular orbit of a relativistic Bohr atom and the speed of light. It is this interpretation that has given the number its name. It could also be the ratio between the energy required to counteract the electrostatic repulsion between two electrons at distance d and the energy of a photon of wavelength 2πd. The inverse of the fine structure constant, 137, is (possibly) the highest atomic number for an atom that can have stable electron orbits. 

There are at least a dozen other interpretations of the meaning of this number, and most of them have to do with atoms, but they remain interpretations. Is the fine structure constant simply a relationship that derives naturally from the basic structure of atoms? Or is this structure shaped by a deeper, universal relationship represented by the fine structure constant? If so, what is this deeper relationship? The famous physicist Richard Feynman called the fine structure constant “a magic number that comes to us with no understanding by man.”

The Deeper Mystery

The mystery surrounding the fine structure constant is deepened by recent measurements, which suggest that the constant isn’t a constant at all – it has most likely changed over the lifetime of the universe. In 1999, studies of quasars – very energetic, cosmic objects – showed that the fine structure constant was lower 10-12 billion years ago. Not a lot lower; only about 6 parts in a million smaller than today, but still enough to make scientists sure that there has been a real change.

What does this mean? Since the fine structure constant is derived from other physical constants, it must mean that one or more of these are changing over time – what makes them do this? What role does this play in the evolution of our universe? And not least, since the fine structure constant is closely connected to atomic structure, does this mean that atoms were fundamentally different in the very early universe – and may be different in the future? 

To muddy everything even more, in 2010, Australian scientists found that the fine structure constant doesn’t just change across time, but also across space. Like the earlier studies, this study used quasars as points of measurement. The scientists found that the fine structure constant 10 billion years ago was one part in a hundred thousand higher if we look in the direction of the southern constellation Ara, and conversely about the same amount lower if we look in the opposite direction. This result was later confirmed, and it now looks as if the fine structure constant increases with distance in one direction and decreases with distance in the opposite direction.

This is a very remarkable result, since a basic assumption in astronomy is that the universe is the same in all directions and there is no centre as such in the universe – the so-called isotropic principle. If there really is an axis going through the universe, as the measurements show, astronomers will have to abandon this principle.

This also means that different parts of the universe can have different properties. A lower value of the fine structure constant, for instance, means that supernovae become more luminous, and vice versa. This is very important since supernovae are usually thought to have luminosities so similar that their apparent brightness can be used to calculate their distance with very high precision; an important tool to measure the universe’s size and age. If supernovae are in fact more luminous in a particular direction because of a lower value of the fine structure constant, they may be more distant than we thought,
and less distant in the opposite direction. Our entire image of the universe hence becomes skewed. 

There may be very distant regions in the universe where the fine structure constant is so different that life as we know it can’t exist, or where stars don’t shine because fusion of light elements isn’t possible.

It is still possible that the observed variations in the fine structure constant are due to systematic errors in measurements or conditions that haven’t been considered. Newer and better instruments will be ready over the next few years, and they may provide a clearer picture. It may be confirmed that the fine structure constant in fact varies across time and space, or it may be refuted. No matter what, one question will most likely remain: What is the meaning of the number 1/137?


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