The following is an extract from our Lost in Space-Time newsletter. Each month, we hand over the keyboard to a physicist or mathematician to tell you about fascinating ideas from their corner of the universe. You can sign up for Lost in Space-Time here.
Before I entered university, I learned quantum physics is the most mysterious type of physics, full of particles that exist in two places at once, waves that don’t actually wave and objects whose behaviour can depend on things happening on the other side of our universe. I was in awe – until one dinner at my university’s physics department dissuaded me. Over pizza, I asked an expert in quantum optics about their work. “It’s all just linear algebra,” they said between greasy bites.
I was shocked by how swiftly they had taken the mystery out of it all. But it is true that when it comes to comparing the maths and what researchers see in experiments, quantum physics stacks up ludicrously well. In that sense, it is one of the most successful – if not the most successful – scientific theories. There is nothing inaccurate about boiling down all the counterintuitive, mind-bending quantum weirdness to something as simple as “just” a type of algebra. So why can’t it escape its reputation for being bizarre?
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The answer lies in the way quantum mechanics aligns – or doesn’t – with our experience of the world. Often, it seems as soon as anything quantum leaves the realm of mathematics, it becomes nearly inconceivable. So, for me, quantum physics’ true weirdness, and much of its appeal, lies in how it confronts us with the possibility that the tools we invented for understanding the world only go so far. It also makes me wonder whether I could ever truly develop a quantum intuition.
Each physics theory uses its own kind of maths. Take Isaac Newton’s laws, which can help predict how fast you must pedal a bicycle up a hill to avoid sliding downwards – they are just a set of differential equations. Another example is electromagnetism, the theory of which is expressed through three-dimensional calculus. Crucially, however, you don’t need expertise in taking derivatives of functions to stop yourself from rolling down a hill, nor do you need to know what a vector product is to stick a magnet to your fridge. Your experience and your body’s reflexes are all that is required. Learning the maths just explains it.
Quantum physics has also claimed its own corner of mathematics, but how we use that to make sense of the theory is much less straightforward. In some cases where our intuition says “impossible”, the maths proves otherwise.
Quantum theory was built upon work with matrices, wave equations and probability by some of its most well-known founders, such as Werner Heisenberg, Erwin Schrödinger and Max Born. Even they suspected the leap from the mathematics on the page to the physical world we experience would be unusually difficult. Intervening decades proved this difficulty to be so great as to bring into question whether the practice of physics can fully capture reality at all.
“Reality resists imitation through a model,” Schrödinger wrote in 1935, lamenting the gaps in physicists’ understanding of the era’s pioneering quantum experiments. Heisenberg was even more distraught about the relationship between reality and mathematics, deliberating in his 1958 book Physics and Philosophy: The Revolution in Modern Science about whether phenomena that cannot be described by mathematics could even arise.
Let’s consider one of the quantum behaviours often deemed the weirdest: particles can exist in a superposition of states, neither precisely here nor there. This is a key part of the famous double-slit experiment. Here, a quantum object, such as an electron or another subatomic particle, travels towards a barrier that has two narrow slits cut out of it. Behind the barrier is a screen where the particle will land and make a mark when it passes through a slit.
Suppose you shoot particles towards the barrier one at a time. After doing so a hundred times, you may expect to find a big mark on the screen right behind one of the slits, and another big mark right behind the other, but no marks between them where the particles’ paths were blocked. That’s a perfectly reasonable assumption – intuitive, even – but you’d be wrong.
Instead, you’d see a pattern spread across the screen, suggesting the particles travelled to many more places than immediately behind each slit.
Quantum theory has an explanation for how this happens that doesn’t feel like it should work – yet the maths are perfectly sound. It all hinges on a plus sign.
I learned this calculation in my first quantum physics course at university. First, you write down a function that means “particle goes through left slit” and a function that means “particle goes through right slit”. Then, with the flair of a minor magician, you put a plus sign between them. Next, you follow the perfectly good mathematical rules for how to deal with these types of functions, called wavefunctions, and end up predicting – counterintuitively, yet truthfully – the pattern that shows up on the screen.
At this point, you might reach for a textbook in the hope of understanding what exactly it is you just calculated. It will tell you that when you wrote down that plus sign, you created a special sum that is called a “superposition state” where it is impossible to tell whether the particle goes through the left slit or the right one. Sometimes, this claim is interpreted more radically, positing the plus sign means that the particle moved through both slits simultaneously. That’s an awful lot of power for one mathematical symbol, and so far, no one has been able to directly watch the particle execute that magical act.
It doesn’t get better if you consider intervening. In the double-slit experiment, if you add a detector right next to one of the slits to more directly see when a particle does or does not definitively go through it, the pattern on the screen changes.
Some quantum phenomena are stranger still. Entanglement, for one. The maths suggests – and we have seen evidence over the past several decades – that this phenomenon can be used to teleport the quantum properties of one particle to another across vast distances. A few experiments that have shown this works so well that there are efforts underway to use them as the foundation for a whole new system of communication. This is a triumph of quantum theory: even its most counterintuitive predictions have withstood experimental scrutiny and proven useful.
Yet, it’s an irksome kind of triumph, because there is still no consensus on the best way to interpret quantum theory and say with certainty exactly what happens to quantum objects at all times, in all situations. Will quantum theory ever be able to tell me what that plus sign actually means in a way that isn’t completely foreign to my intuition and experience? I lose sleep over this question less than I did as a student, but the tension between being able to do the maths and experiencing reality still lives in my head rent-free.
Heisenberg argued this gap stems from the way we conceptualise what it means to do science, an idealised view where we separate ourselves from the rest of the world and treat it as a perfectly understandable and independent object. You might think if we can break the world down into smaller and smaller bits, we can fully understand, well, everything. Quantum physics repeatedly encourages us to ask whether it can really be so simple.
This is what makes quantum physics so intoxicating. I have long given up being upset about the possibility of a particle being in two places at once. But no other theory or branch of science has ever made me worry so much about whether I – or anyone – will ever be able to understand physical reality beyond “just linear algebra” and fully know it, deep in my bones. If you’ve achieved this, please write in to tell me what linear algebra feels like.
