The mystery at the heart of physics that only math can solve

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EXCERPTS: . . . It’s common to think of the universe as being built from fundamental particles: electrons, quarks, photons and the like. But physics long ago moved beyond this view. Instead of particles, physicists now talk about things called “quantum fields” as the real warp and woof of reality.

These fields stretch across the space-time of the universe. They come in many varieties and fluctuate like a rolling ocean. As the fields ripple and interact with each other, particles emerge out of them and then vanish back into them, like the fleeting crests of a wave.

“Particles are not objects that are there forever,” said Tong. “It’s a dance of fields.”

To understand quantum fields, it’s easiest to start with an ordinary, or classical, field. Imagine, for example, measuring the temperature at every point on Earth’s surface. Combining the infinitely many points at which you can make these measurements forms a geometric object, called a field, that packages together all this temperature information.

In general, fields emerge whenever you have some quantity that can be measured uniquely at infinitely fine resolution across a space. “You’re sort of able to ask independent questions about each point of space-time, like, what’s the electric field here versus over there,” said Davide Gaiotto, a physicist at the Perimeter Institute for Theoretical Physics in Waterloo, Canada.

Quantum fields come about when you’re observing quantum phenomena, like the energy of an electron, at every point in space and time. But quantum fields are fundamentally different from classical ones.

While the temperature at a point on Earth is what it is, regardless of whether you measure it, electrons have no definite position until the moment you observe them. Prior to that, their positions can only be described probabilistically, by assigning values to every point in a quantum field that captures the likelihood you’ll find an electron there versus somewhere else. Prior to observation, electrons essentially exist nowhere — and everywhere.

“Most things in physics aren’t just objects; they’re something that lives in every point in space and time,” said Dijkgraaf.

A quantum field theory comes with a set of rules called correlation functions that explain how measurements at one point in a field relate to — or correlate with — measurements taken at another point.

Each quantum field theory describes physics in a specific number of dimensions. Two-dimensional quantum field theories are often useful for describing the behavior of materials, like insulators; six-dimensional quantum field theories are especially relevant to string theory; and four-dimensional quantum field theories describe physics in our actual four-dimensional universe. The Standard Model is one of these; it’s the single most important quantum field theory because it’s the one that best describes the universe.

There are 12 known fundamental particles that make up the universe. Each has its own unique quantum field. To these 12 particle fields the Standard Model adds four force fields, representing the four fundamental forces: gravity, electromagnetism, the strong nuclear force and the weak nuclear force. It combines these 16 fields in a single equation that describes how they interact with each other. Through these interactions, fundamental particles are understood as fluctuations of their respective quantum fields, and the physical world emerges before our eyes.

It might sound strange, but physicists realized in the 1930s that physics based on fields, rather than particles, resolved some of their most pressing inconsistencies, ranging from issues regarding causality to the fact that particles don’t live forever. It also explained what otherwise appeared to be an improbable consistency in the physical world.

“All particles of the same type everywhere in the universe are the same,” said Tong. “If we go to the Large Hadron Collider and make a freshly minted proton, it’s exactly the same as one that’s been traveling for 10 billion years. That deserves some explanation.” QFT provides it: All protons are just fluctuations in the same underlying proton field (or, if you could look more closely, the underlying quark fields).

But the explanatory power of QFT comes at a high mathematical cost.

“Quantum field theories are by far the most complicated objects in mathematics, to the point where mathematicians have no idea how to make sense of them,” said Tong. “Quantum field theory is mathematics that has not yet been invented by mathematicians.”
Too Much Infinity

What makes it so complicated for mathematicians? In a word, infinity [...] And when theories produce infinities, it calls their physical relevance into question, because infinity exists as a concept, not as anything experiments can ever measure. It also makes the theories hard to work with mathematically. “We don’t like having a framework that spells out infinity. That’s why you start realizing you need a better mathematical understanding of what’s going on,” said Alejandra Castro, a physicist at the University of Amsterdam.

[...] Physicists and mathematicians can’t calculate using infinities, but they have developed workarounds — ways of approximating quantities that dodge the problem. These workarounds yield approximate predictions, which are good enough, because experiments aren’t infinitely precise either.

[...] “Quantum field theory emerged as an almost universal language of physical phenomena, but it’s in bad math shape,” said Dijkgraaf. And for some physicists, that’s a reason for pause. “If the full house is resting on this core concept that itself isn’t understood in a mathematical way, why are you so confident this is describing the world? That sharpens the whole issue,” said Dijkgraaf.

[...] There’s a long way to go. So far, all of the quantum field theories that have been described in full mathematical terms rely on various simplifications, which make them easier to work with mathematically.

One way to simplify the problem, going back decades, is to study simpler two-dimensional QFTs rather than four-dimensional ones. A team in France recently nailed down all the mathematical details of a prominent two-dimensional QFT.

Other simplifications assume quantum fields are symmetrical in ways that don’t match physical reality, but that make them more tractable from a mathematical perspective. These include “supersymmetric” and “topological” QFTs.

The next, and much more difficult, step will be to remove the crutches and provide a mathematical description of a quantum field theory that better suits the physical world physicists most want to describe: the four-dimensional, continuous universe in which all interactions are possible at once.

“This is [a] very embarrassing thing that we don’t have a single quantum field theory we can describe in four dimensions, nonperturbatively,” said Rejzner. “It’s a hard problem, and apparently it needs more than one or two generations of mathematicians and physicists to solve it.”

But that doesn’t stop mathematicians and physicists from eyeing it greedily. For mathematicians, QFT is as rich a type of object as they could hope for. Defining the characteristic properties shared by all quantum field theories will almost certainly require merging two of the pillars of mathematics: analysis, which explains how to control infinities, and geometry, which provides a language for talking about symmetry.

“It’s a fascinating problem just in math itself, because it combines two great ideas,” said Dijkgraaf... (MORE - missing details)

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