# Surprises of spin

by Jesse Liu

Let’s start with two big ideas from two great physicists of the early 20th century. Hideki Yukawa first proposed in 1935 the picture that forces are due to the exchange of particles. Just four years later, Eugene Wigner worked out that every particle is labelled by just two numbers: its mass and spin. Remarkable in their own right, combining these two ideas reveals profound surprises about our universe. The mass and spin of the exchanging particle precisely determine the characteristics of the fundamental forces of nature.

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Left: Natalia Toro and Philip Schuster who discovered the first non-interacting example of a quantum field theory of continuous spin particles. Right: Steven Weinberg, who found that many of the essential properties of fundamental forces arise from using the principles relativity and quantum mechanics. Image source: Gabriela Secara.

Firstly, mass governs the range of the force. The lower the mass of the exchange particle, the farther the force is felt. Yukawa used this relation to predict a massive particle called the meson that binds protons and neutrons together with what we call the strong nuclear force. We do not feel the strong force in everyday life precisely because this force’s range is exponentially suppressed by the exchanging particle having a large mass.

Make that exchanging particle massless and the force suddenly becomes ‘long-range’. This means we can feel it not only at everyday human scales but stars and galaxies can be bound by forces reaching between its vast distances. Electromagnetism and gravity are precisely such long-range forces and they are mediated by massless photons and gravitons.

Things get much more interesting with spin. Forces are mediated by the exchange of integer spin particles called bosons. (Technically for massless particles, I should talk about the helicity, which is the spin-aligned along the direction of motion, but that’s unimportant for our narrative.) In 1964, Steven Weinberg derived some deep relations that tell us how massless particles of different spin behave. He did this by looking at the particles’ scattering amplitudes. These are the primitive mathematical objects quantifying how particles interact.

Weinberg found that if massless spin 1 particles are responsible for a long-range force, the theory must have a conserved charge. Indeed, that is precisely what we observe for the electric charge in our universe. He then derived that interacting massless spin 2 particles must give rise to the equivalence principle. Surprising as it seemed, the most salient features of electromagnetism and gravity were emerging. Could the entire theory be there?

And so it was. The following year in 1965, Weinberg went for the home run. He showed that the whole mathematical structure of Maxwell’s electromagnetic equations inevitably arise in theories with an interacting massless spin 1 particle. Similarly, Einstein’s General Relativity equations naturally appear in theories with an interacting massless spin 2 particle.

Why is this deep? Because all Weinberg had to assume was that scattering amplitudes, which dictate how particles interact, obeyed Lorentz symmetry (due to special relativity) and were unitary (due to quantum mechanics). It’s astonishing how constraining those two fundamental principles of nature are on interactions between particles. All we need to do next is input the labels of the mediating particle, mass, and spin, and out comes the form of the force.

Let me reiterate. The unique theories corresponding to interacting massless particles with spin 1 and 2 are Maxwell’s electromagnetism (technically its generalisation called Yang-Mills theory but that’s a subplot for another day) and Einstein’s General Relativity.

Weinberg’s conclusions were certainly a surprise for me while I was learning quantum field theory. History unfolded in the reverse order and so most of us physicists learn it so. But in fact, we never needed to know about Faraday’s experiments or falling elevators to discover electromagnetism or gravity in its modern form. All we needed were special relativity and quantum theory.

That is the power of physics and hindsight. Enthused, we naturally want to ask about spin 3 or heck, why not spin 300 particles? If spin governs the nature of a force, why don’t we see any other long-range forces in our universe? Worry not. Weinberg had us covered. He found was that massless spin 3 and higher particles cannot interact consistently to give a long-range force. More precisely, their scattering amplitudes are zero.  No interaction, no particle exchange, no force.

Perhaps there are vestiges of hope with spin 0? Well, the bottom line is that massless spin 0 particles very easily become massive, so can only mediate short-range forces. Theories governing photons (spin 1) and gravitons (spin 2) automatically have a technical feature called ‘gauge invariance’ that keeps these particles massless. The upshot: electromagnetism and gravity are the only long-range forces we can see in our universe ruled by special relativity and quantum theory. Profound it is but there’s a detail lurking in the fine print.

Our story now ventures into largely uncharted territories. In the mathematics that Wigner was exploring to label particles, he unearthed a strange surprise in the massless case. So far we only thought about particles labelled by a single spin. Photons only have the spin 1 label while gravitons only have spin 2.

However photons and gravitons are actually special cases of a more general massless particle. Peculiar as it may sound, Wigner found that this particle has an infinite list of integer spins. That’s right, this massless particle is labelled by spin 0, 1, 2, 3, all the way to infinity! For historical reasons, Wigner called these ‘continuous spin particles’ (CSPs). The spins are definitely discrete but we’re stuck with that name.

Could nature seriously exhibit a particle with an infinite set of spins? The common lore believed the contrary for over seven decades since Wigner first described the CSP. But a decisive ‘no’ remained elusive even at the beginning of the 2010s. It was a question Philip Schuster and Natalia Toro at the Perimeter Institute wanted to close once and for all by endowing firm theoretical foundation for the ‘no’ answer.

Weinberg earlier showed that massless particles with spin 3 or higher could not be responsible for long-range forces because their scattering amplitudes were zero. Schuster and Toro’s strategy was to pursue a similar argument for CSPs. To their surprise, they instead discovered that CSP scattering amplitudes were remarkably non-zero, just like the photon and graviton ones!

In other words, CSPs could potentially be responsible for long-range forces like electromagnetism and gravity. An immediate question springs to mind. If CSP scattering amplitudes are non-zero, meaning they could interact, why have we not seen them? More intriguingly, this begs sober speculation. Could our two familiar long-ranged forces actually be mediated by a CSP that looks a lot like the photon and graviton?

To answer these questions precisely, we need a full interacting quantum field theory describing CSPs. All the usual Standard Model particles with single spins are described by such theories. But sadly so little was known about CSPs that no such theory was ever written down for it. Schuster and Toro sought to fill this gap in the field theory market. In 2013, they succeeded in finding the first free field theory of a bosonic CSP, the massless particle with an infinite label of integer spins.

A free field theory has no interactions, which means no particle exchange or forces. But it nonetheless represents a significant step towards a fully fledged theory of CSPs we can use to make precise predictions. For over seven decades, theoretical progress struggled and few expected CSPs to come even this far. Suddenly, the surprising became the tantalising.

And this is where the story of CSPs meets an unsatisfying cliffhanger. The sequel to this narrative involves adding interactions to the quantum field theory of CSPs, which at the time of writing remains an open question. As well as those much desired experimental tests, it could moreover provide a new building block for theorists to whet their imagination in constructing new models of nature. The theoretical hints are intriguing and without further research, these musings will remain speculation rather than science.

Weinberg showed how the humble spin of a particle can surprise us with its profound insights into electromagnetism and gravity. Schuster and Toro recently demystified over seven decades of lore surrounding the most mysterious guise of particle spin: the continuous spin particle. As the largely ignored generalisation of photons and gravitons, a window has finally been open to shine light on the CSP. What more surprises could this story of spin have in store for us?

To their surprise, [Schuster and Toro] discovered that CSP scattering amplitudes were remarkably non-zero, just like the photon and graviton ones!

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Jesse Liu is a former PSI student. He is currently a Ph.D. student at the University of Oxford.