The number that fascinates physicists


“God is a pure mathematician!” declared British astronomer Sir James Jeans. The physical Universe does seem to be organised around elegant mathematical relationships. And one number above all others has exercised an enduring fascination for physicists: 137.03599913.

Let me explain. When scientists measure any quantity they must specify the units being used. The speed of light, for example, is either 186,000 or 300,000 depending on whether it is expressed as miles per second or kilometres per second. Likewise your weight might be 150 or 68 according to whether you are measuring in pounds or kilograms. Without knowing the units being used the number is meaningless – unless it is a pure number.

The best known example of a pure number comes from combining three of nature’s most fundamental quantities: the speed of light, the electric charge carried by a single electron and Planck’s constant of quantum mechanics. In symbols, that’s c, e and h. Put them together as follows, hc/2πe2, and the units of c, e and h cancel out to leave a pure number, 137.03599913. If c, e and h were measured by Vulcan scientists using Vulcan units, they would still get 137.03599913. This curious number is therefore a universal constant of nature – “God given” Jeans might have said.

In view of its importance, hc/2πe2 has acquired a name and a symbol all of its own. For historical reasons the inverse,
2πe2/hc = 1/137.03599913, is used. It is known as the fine-structure constant and is denoted by the Greek letter alpha (α).

What is α good for? It is used to measure how strongly charged particles such as electrons interact with electromagnetic fields. For example, it determines how quickly an excited atom emits a photon. If  were twice as big, atoms would decay twice as fast. Alpha also determines the trajectory of a charged particle moving through an electromagnetic field – this was the basis of old-fashioned TV projection.

Alpha also determines details in the pattern of light emitted by atoms. As electrons whirl around the atomic nucleus they create magnetic fields that in turn interact with other electrons in the same atom causing subtle shifts in their energy levels. These shifts show up in the particular wavelengths of light the atoms emit, producing a pattern of small ‘splits’ (missing wavelengths) known as fine structure. The size of the splitting is proportional to α – that is why it is called the fine structure constant. This fine structure is observed in the light from the Sun and stars, for example.

Alpha also raises some profound theoretical questions. If it is compared to another fundamental force, gravitation, the ratio is a huge number – about 1040 – which expresses the weakness of gravity compared to the electric and magnetic forces. Physicists and cosmologists have long wondered where these numbers, 1/137.03599913 and 1040, come from. Are they arbitrary, or do they flow from some deeper theory of the Universe?

There is a long history of attempts to derive α from physical theory or to concoct a mathematical formula that has this value. For a brief time in the 1920s, when it looked as if α might be exactly 1/137, astronomer Arthur Eddington searched for a theory that would throw up both the numbers 137 and 1040 naturally, but his ideas ultimately led nowhere. Then in 1969 a young Swiss mathematician, Armand Wyler, pointed out that (9/16π3)(π/5!)¼ comes close to 1/137.036, which matched the value of α to the precision known at the time. However, his formula was not accompanied by any credible theory and was regarded as little more than a numerical curiosity. Several other attempts at α numerology have been made since, none of which have gained traction in the physics community.
So what are we to make of 1/137.03599913? Is there a deep reason why α has to be precisely this number for the world to function as it does? Suppose by some magic we woke up tomorrow and α was 1/138 instead, would it make much difference?

A few years ago a new slant on this old problem emerged. John Webb and his colleagues at the University of New South Wales carried out an intensive study of the fine structure in the spectral lines of extremely distant astronomical objects. The data analysed by Webb provides a way to measure the value of α billions of light years away and, because of the finite speed of light, billions of years ago.

What they found threatens to turn the world of theoretical physics upside down. On the face of it, α has slightly different values in different parts of the Universe, implying that the fine structure constant is not a constant at all, but varies over cosmological distances and times.

While it is too soon to rewrite the textbooks, the prospect that one of nature’s critical pure numbers might turn out to be variable has created considerable controversy. If James Jeans were alive today, he might conclude that God may well be a pure mathematician, but one with a whimsical bent.

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