‘Reverse physics’ could help bridge the classical and quantum worlds

There’s a bit of an open secret in physics. Even when our theories work well – and some have been rigorously tested for centuries – we don’t always fully understand them. We start with experimental evidence, then build mathematical models and theories, test and update them with more experiments and hope that it all adds up. Often, it does. But were a mathematician to ask what a theory we built this way is – that is, what type of well-defined mathematical objects comprise it – that isn’t a question that researchers can always fully answer.

“The sense from a lot of physicists is these are maths details and that is not really physics, so it’s not their job to set them straight,” says Gabriele Carcassi at the University of Michigan. But if we are to tackle the biggest problems in physics, such as marrying quantum mechanics with general relativity to produce a theory of quantum gravity, we may have to take a different tack.

That’s where the “reverse physics” programme comes in. It is Carcassi’s passion project: he spends his days trying to put physics on accurate, rigorous and meaningful mathematical foundations. He is mapping out the bridge you must be able to walk to fully connect mathematics and physics. Carcassi puts it like this: “When the maths is derived from the right physical assumptions, the physical assumptions can be derived from the maths.”

Like most physicists, I didn’t learn to think this way. All the concepts that were taught to me came from physics, such as fields or forces or even slices of space-time, then I’d learn just enough about what they are mathematically to know which equations predict their behaviour. If you asked me to define a mathematical object called a Lorentzian manifold, I am really not sure that I could, yet I have a fairly good mental image of the physical thing it is representing: space-time as the fabric of reality. Reverse-engineering a theory of a changing space-time, essentially the theory of general relativity, from ideas like Lorentzian manifolds and other mathematical ingredients certainly hadn’t been put on my radar throughout my education.

But Carcassi and his colleagues at the Assumptions of Physics project want to establish these equivalent, two-way definitions for nothing less than all of physics. It’s an enormous undertaking, to say the least.

Trained as an engineer, Carcassi fell into this line of work because he married a physicist, Christine Aidala at the University of Michigan, who had a lot of physics books lying around. Reading them puzzled him, particularly those on quantum mechanics. Whereas most people get hung up on the matter of which interpretation of quantum mechanics is correct – if and how a theory captures the objective physical reality of our world, or if there even is an objective reality – Carcassi found that every idea was missing something.

“To me, all these [theories, such as quantum mechanics] are models, and I need to understand when they’re applicable and when not, essentially what they’re supposed to describe and under what assumptions,” he says. While he was grappling with this, the situation got worse. “I realised that we don’t really understand classical mechanics,” he says. This isn’t to say that classical mechanics doesn’t work, but that some of its theoretical ingredients are ill-defined, in Carcassi’s view.

As we talked, I was reminded of my advanced mechanics courses where I learned about objects called the “action” and “Hamiltonian”. They are incredibly useful, and I can define them loosely in English, saying something about an object’s preferred path through space or about its energy, but I’d be offering intuition that I got through homework problems more than mathematical rigor.

Carcassi says his experience was similar. “You know how to use them. You know their properties, but what exactly are they describing? You don’t know. I basically started reverse engineering a lot of these things.” This is just one example of the kind of definitional work reverse physics could offer.

“The tagline is to find a minimal set of physical assumptions from which we can rigorously re-derive the laws,” he says. Ideally, this would resolve many existing physics paradoxes and puzzles, such as the role of measurements in quantum physics where objects seem to acquire a fixed reality only after being observed. If we understood the maths underpinning those scenarios better, we could ascertain whether the world is fundamentally fuzzy or if a solid reality exists after all.

In the past 11 years, the Assumptions of Physics team, primarily consisting of Carcassi and Aidala, has broken down essentially all of classical physics into a set of mathematical equivalencies that allow the researchers to run over that bridge between maths and physics in either direction. Notably, entropy – which quantifies whether a physical system must assume one very precisely ordered state, or can find itself in many more equally likely states – has emerged as a key player in their effort.

For instance, they have shown that restricting any change in the entropy of a system is equivalent to previously known rules for how such a system can evolve over time. Reverse physics has already illuminated connections between quantum and classical physics – according to the team’s list of mathematical concepts and assumptions that undergird all of physics, it seems that classical physics is akin to quantum physics, but with entropy made high.

Carcassi is currently working on a slew of quantum problems, and candidly notes that it will be difficult to tackle various features of quantum field theory (QFT), where quantum fields extend through all of space and infinitely large numbers are known to crop up. One of the central objects that physicists use for QFT computations – for instance, to predict outcomes of particle accelerator experiments – is the “path integral”, which even professors who teach it often admit is an poorly defined headache for mathematicians. I’ve personally been fascinated by this enough to have devoted to a chapter to the path integral in my book.

But Carcassi’s quantum hopes soar higher: he thinks reverse physics could help researchers marry quantum physics and Albert Einstein’s theory of gravity by providing sufficient clarity on how both can be built from a shared mathematical base. “How can we even proceed and try to get new theories of physics, if we don’t even understand exactly what we have in the current one,” says Carcassi. The quest for a theory of quantum gravity has been one of physics’ biggest and least fruitful efforts for decades, which makes any new approach welcome.

Intriguingly, the way reverse physics could interweave these two pillars is indifferent to many of the common quantum fights among physicists. Carcassi and his collaborators aren’t interested in whether the equations of quantum theory imply that we live in the multiverse or that, for example, everything in the cosmos is inextricably connected with everything else. They are interested in the commonalities between every possible mathematical framework for quantum physics, and whether any resultant theory that encompasses them all makes sense with how we can measure and observe the universe.

Trying to redo all of physics from the ground up in this manner isn’t for the faint of heart. But what intimidates Carcassi is mostly the lack of funding and broad interest into this type of nitty-gritty work in physics’ very foundations. With appropriate funding and a big enough team, the reverse physics programme could be completed within a decade, he says. It brings to mind a story from my college years.

Once, during office hours, I told a physics professor about all the maths courses I was taking in the hope that they’d make me a better physicist. “I just know there will be a moment when all the maths and physics come together, and I finally get it all,” I said. The professor chuckled. He said he’d been waiting for that moment for his whole career. I hope the wait will soon be over.