A billiards table
Start by imagining a billiard table with a ball travelling across it in a straight line

For centuries, electricity and magnetism had everyone baffled. In the 19th century, Michael Faraday not only discovered new phenomena but summarised all that was already known in his three-volume Experimental Researches in Electricity. It enabled the great physicist James Clerk Maxwell to distill everything down to 20 equations in 1861-1862. A couple of decades later, this was reduced to four by the physicist and mathematician Oliver Heaviside.

Maxwell’s equations are considered to be the high-point of 19th-century physics. They contain the seeds of the revolutions that took place in the 20th century in the fields of both relativity and quantum theory. However, what Maxwell did not realise was that the bewildering array of electrical and magnetic phenomena he had summarised was no more than a consequence of one remarkably simple principle: “local gauge invariance”.

Picture a billiard table and a billiard ball travelling across it in a straight line. Now imagine that the table is raised by a metre, or two metres, or by any height whatsoever. The ball will continue to follow the same trajectory. In physics, the height of the billiard table is an example of a “gauge” and the fact that it can be changed without altering the physics is a “gauge symmetry”.

In 1918, the German mathematician Emmy Noether discovered that an unavoidable consequence of every such “global” symmetry is a “conservation law”. So, for instance, the fact that the outcome of an experiment will be the same if it is carried out today or tomorrow – that is, that it obeys “time translational symmetry” – leads to the “conservation of energy”, one of the cornerstones of physics, which dictates that energy can neither be created nor destroyed, merely transformed from one type to another.

But it turns out that in a Universe in which the cosmic speed limit – set by Einstein – is the speed of light, it is impossible to lift all parts of a billiard table simultaneously. Imagine, for instance, the billiard table is ten light years across. The best that can be done is to raise one end and watch a hummock propagate across the table at the speed of light, reaching the far end ten years later. It would be reasonable to expect the laws of motion to remain the same everywhere in the billiard-table Universe and for the ball to continue following a straight-line path. However, this would be possible only if a force were introduced at every location to compensate for the undulation. This is the key thing: maintaining local gauge symmetry requires the existence of a force.

In the 20th century, physicists discovered that a particle like an electron is described by a “wave function”, which propagates according to the Schrödinger equation. The probability of finding the particle at any location is given by the square of the “amplitude” of the wave at that location. But, crucially, the probability does not depend on where the wave happens to be in its undulating pattern. This “phase” is the gauge, and changing it everywhere by the same amount merely moves the peaks and troughs of the wave along, but does not change anything observable.

So what happens if we insist that not only global but local gauge symmetry is maintained – that is, that the probability is not changed by altering the phase of the electron wave function by a different amount at each location in space and time? Remember, from the billiard table example, that this requires the existence of a force. So, what is the force that is needed? Remarkably, it is precisely the electromagnetic force described by Maxwell.

There are four fundamental forces that glue together the particles of matter and make the Universe possible. And, amazingly, three of those have been shown to exist only to maintain local gauge symmetry. In fact, 2023 marks 40 years since this was proved for nature’s “weak” force, which is responsible for radioactivity. The proof was the discovery in 1983 of the particles that “carry” the weak force – the W and Z bosons – for which Carlo Rubbia and Simon van der Meer won the Nobel Prize.

In all the chaos of the world, it is easy to lose sight of how amazingly far we humans have come. We are a species of puny apes that arose on an African plain a mere blink of an eye ago in geological time. Yet we have discovered the genetic secret of life, made footprints in the Moon’s Sea of Tranquility and discovered the principle that makes everything – the stars and galaxies and us – possible. Of course, we are left with the mystery of why nature is so obsessed by enforcing local gauge symmetry. What is the significance of the “one rule that binds them all”?

This piece is from the New Humanist spring 2023 edition. Subscribe here.