7 21 Physics Again for Some Reason Idk if It s Different

Olena Shmahalo/Quanta Magazine

Over the past century, quantum field theory has proved to be the single most sweeping and successful physical theory ever invented. It is an umbrella term that encompasses many specific quantum field theories — the fashion "shape" covers specific examples like the foursquare and the circle. The near prominent of these theories is known equally the Standard Model, and information technology is this framework of physics that has been so successful.

"It can explicate at a central level literally every unmarried experiment that we've ever done," said David Tong, a physicist at the University of Cambridge.

Just quantum field theory, or QFT, is indisputably incomplete. Neither physicists nor mathematicians know exactly what makes a breakthrough field theory a quantum field theory. They have glimpses of the full picture, merely they tin can't yet brand it out.

"There are various indications that in that location could be a better way of thinking about QFT," said Nathan Seiberg, a physicist at the Found for Avant-garde Written report. "It feels like it's an animal yous tin can touch from many places, but you don't quite see the whole animal."

Mathematics, which requires internal consistency and attention to every last detail, is the language that might make QFT whole. If mathematics can learn how to describe QFT with the aforementioned rigor with which it characterizes well-established mathematical objects, a more consummate picture of the physical earth volition likely come forth for the ride.

"If y'all really understood quantum field theory in a proper mathematical manner, this would requite us answers to many open physics issues, perhaps even including the quantization of gravity," said Robbert Dijkgraaf, director of the Constitute for Advanced Study (and a regular columnist for Quanta).

Nor is this a one-way street. For millennia, the concrete earth has been mathematics' greatest muse. The ancient Greeks invented trigonometry to study the move of the stars. Mathematics turned information technology into a subject field with definitions and rules that students now larn without any reference to the topic'southward celestial origins. Almost 2,000 years later, Isaac Newton wanted to empathise Kepler's laws of planetary move and attempted to find a rigorous way of thinking about infinitesimal change. This impulse (along with revelations from Gottfried Leibniz) birthed the field of calculus, which mathematics appropriated and improved — and today could hardly exist without.

Now mathematicians want to practice the same for QFT, taking the ideas, objects and techniques that physicists accept developed to written report fundamental particles and incorporating them into the main body of mathematics. This means defining the basic traits of QFT so that future mathematicians won't have to think about the physical context in which the theory showtime arose.

The rewards are likely to exist great: Mathematics grows when it finds new objects to explore and new structures that capture some of the most of import relationships — between numbers, equations and shapes. QFT offers both.

"Physics itself, as a structure, is extremely deep and often a better mode to call up about mathematical things we're already interested in. It's just a better way to organize them," said David Ben-Zvi, a mathematician at the University of Texas, Austin.

For 40 years at least, QFT has tempted mathematicians with ideas to pursue. In contempo years, they've finally begun to understand some of the bones objects in QFT itself — abstracting them from the world of particle physics and turning them into mathematical objects in their own right.

Yet it's still early on days in the effort.

"We won't know until we become there, just it's certainly my expectation that we're only seeing the tip of the iceberg," said Greg Moore, a physicist at Rutgers University. "If mathematicians really understood [QFT], that would lead to profound advances in mathematics."

Fields Forever

It's common to think of the universe as being built from fundamental particles: electrons, quarks, photons and the like. Just physics long ago moved beyond this view. Instead of particles, physicists now talk about things chosen "quantum fields" equally the real warp and woof of reality.

These fields stretch beyond 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 and so vanish back into them, like the fleeting crests of a wave.

"Particles are not objects that are there forever," said Tong. "Information technology'due south a dance of fields."

To understand breakthrough fields, it's easiest to kickoff with an ordinary, or classical, field. Imagine, for instance, 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 sally whenever you take 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 infinite-fourth dimension, like, what's the electric field here versus over at that place," said Davide Gaiotto, a physicist at the Perimeter Institute for Theoretical Physics in Waterloo, Canada.

Quantum fields come about when yous're observing quantum phenomena, similar the energy of an electron, at every bespeak in space and fourth dimension. 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 tin merely be described probabilistically, by assigning values to every signal 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.

"Almost things in physics aren't simply objects; they're something that lives in every betoken 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 bespeak in a field chronicle to — or correlate with — measurements taken at another point.

Each quantum field theory describes physics in a specific number of dimensions. Two-dimensional breakthrough field theories are often useful for describing the behavior of materials, like insulators; vi-dimensional quantum field theories are peculiarly relevant to string theory; and four-dimensional quantum field theories describe physics in our actual iv-dimensional universe. The Standard Model is one of these; it's the single nearly important quantum field theory because it'due south the i that all-time describes the universe.

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

Information technology might sound strange, but physicists realized in the 1930s that physics based on fields, rather than particles, resolved some of their well-nigh pressing inconsistencies, ranging from issues regarding causality to the fact that particles don't alive forever. It besides 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 every bit ane that's been traveling for 10 billion years. That deserves some caption." QFT provides it: All protons are only fluctuations in the same underlying proton field (or, if you could await more closely, the underlying quark fields).

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

"Quantum field theories are past far the virtually complicated objects in mathematics, to the betoken where mathematicians have no idea how to make sense of them," said Tong. "Breakthrough field theory is mathematics that has non yet been invented by mathematicians."

Too Much Infinity

What makes it so complicated for mathematicians? In a give-and-take, infinity.

When you measure a breakthrough field at a point, the result isn't a few numbers like coordinates and temperature. Instead, information technology'due south a matrix, which is an assortment of numbers. And not just any matrix — a large one, called an operator, with infinitely many columns and rows. This reflects how a quantum field envelops all the possibilities of a particle emerging from the field.

"There are infinitely many positions that a particle tin accept, and this leads to the fact that the matrix that describes the measurement of position, of momentum, also has to exist infinite-dimensional," said Kasia Rejzner of the University of York.

And when theories produce infinities, it calls their physical relevance into question, because infinity exists as a concept, non as annihilation experiments can e'er measure. It also makes the theories difficult to piece of work with mathematically.

"We don't like having a framework that spells out infinity. That's why you lot start realizing you need a better mathematical understanding of what's going on," said Alejandra Castro, a physicist at the Academy of Amsterdam.

The bug with infinity become worse when physicists commencement thinking about how two quantum fields interact, as they might, for instance, when particle collisions are modeled at the Big Hadron Collider outside Geneva. In classical mechanics this type of calculation is easy: To model what happens when two billiard balls collide, just utilize the numbers specifying the momentum of each brawl at the indicate of collision.

When ii breakthrough fields collaborate, yous'd similar to do a similar thing: multiply the infinite-dimensional operator for one field by the infinite-dimensional operator for the other at exactly the bespeak in space-time where they meet. Merely this calculation — multiplying two infinite-dimensional objects that are infinitely close together — is hard.

"This is where things get terribly wrong," said Rejzner.

Great Success

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

"Nosotros can exercise experiments and measure things to 13 decimal places and they agree to all 13 decimal places. It's the most astonishing affair in all of science," said Tong.

One workaround starts by imagining that yous take a breakthrough field in which nothing is happening. In this setting — called a "free" theory because it's free of interactions — you lot don't accept to worry about multiplying infinite-dimensional matrices because nothing's in motility and nothing e'er collides. Information technology's a situation that'south easy to describe in full mathematical detail, though that description isn't worth a whole lot.

"It's totally boring, because you've described a lonely field with cypher to interact with, and then it's a fleck of an academic exercise," said Rejzner.

Just you can make it more interesting. Physicists dial up the interactions, trying to maintain mathematical control of the picture as they make the interactions stronger.

This arroyo is called perturbative QFT, in the sense that you allow for small changes, or perturbations, in a free field. You can utilize the perturbative perspective to quantum field theories that are similar to a gratuitous theory. Information technology'southward likewise extremely useful for verifying experiments. "You get amazing accuracy, amazing experimental agreement," said Rejzner.

But if you keep making the interactions stronger, the perturbative approach eventually overheats. Instead of producing increasingly accurate calculations that arroyo the existent physical universe, information technology becomes less and less accurate. This suggests that while the perturbation method is a useful guide for experiments, ultimately it'south non the right fashion to try and describe the universe: It's practically useful, but theoretically shaky.

"Nosotros practice not know how to add everything upwardly and go something sensible," said Gaiotto.

Another approximation scheme tries to sneak up on a full-fledged quantum field theory by other means. In theory, a quantum field contains infinitely fine-grained information. To melt up these fields, physicists commencement with a grid, or lattice, and restrict measurements to places where the lines of the lattice cross each other. So instead of beingness able to measure the quantum field everywhere, at first yous can only mensurate it at select places a fixed distance apart.

From at that place, physicists heighten the resolution of the lattice, drawing the threads closer together to create a finer and finer weave. As it tightens, the number of points at which you can have measurements increases, approaching the idealized notion of a field where you can accept measurements everywhere.

"The distance between the points becomes very minor, and such a affair becomes a continuous field," said Seiberg. In mathematical terms, they say the continuum quantum field is the limit of the tightening lattice.

Mathematicians are accustomed to working with limits and know how to establish that certain ones really exist. For example, they've proved that the limit of the infinite sequence $latex \frac{1}{two}$ + $latex \frac{i}{iv}$ +$latex \frac{one}{eight}$ +$latex \frac{1}{16}$ … is 1. Physicists would like to prove that quantum fields are the limit of this lattice process. They simply don't know how.

"It's not so clear how to take that limit and what it means mathematically," said Moore.

Physicists don't doubt that the tightening lattice is moving toward the idealized notion of a quantum field. The close fit between the predictions of QFT and experimental results strongly suggests that's the case.

"At that place is no question that all these limits really exist, because the success of quantum field theory has been actually stunning," said Seiberg. Merely having strong prove that something is correct and proving conclusively that it is are two different things.

It's a degree of imprecision that'south out of step with the other slap-up physical theories that QFT aspires to replace. Isaac Newton's laws of motion, quantum mechanics, Albert Einstein's theories of special and general relativity — they're all but pieces of the bigger story QFT wants to tell, only unlike QFT, they can all be written down in exact mathematical terms.

"Breakthrough field theory emerged as an well-nigh 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 then confident this is describing the globe? That sharpens the whole event," said Dijkgraaf.

Outside Anarchist

Fifty-fifty in this incomplete state, QFT has prompted a number of important mathematical discoveries. The general blueprint of interaction has been that physicists using QFT stumble onto surprising calculations that mathematicians so try to explain.

"It's an idea-generating car," said Tong.

At a bones level, concrete phenomena accept a tight relationship with geometry. To have a simple example, if you lot set up a brawl in motion on a smooth surface, its trajectory volition illuminate the shortest path between whatsoever two points, a belongings known as a geodesic. In this fashion, physical phenomena can find geometric features of a shape.

At present replace the billiard brawl with an electron. The electron exists probabilistically everywhere on a surface. By studying the quantum field that captures those probabilities, you can larn something about the overall nature of that surface (or manifold, to apply the mathematicians' term), like how many holes it has. That's a cardinal question that mathematicians working in geometry, and the related field of topology, want to answer.

"I particle even sitting there, doing nothing, volition start to know nigh the topology of a manifold," said Tong.

In the late 1970s, physicists and mathematicians began applying this perspective to solve basic questions in geometry. By the early on 1990s, Seiberg and his collaborator Edward Witten figured out how to use it to create a new mathematical tool — now called the Seiberg-Witten invariants — that turns quantum phenomena into an alphabetize for purely mathematical traits of a shape: Count the number of times breakthrough particles behave in a sure way, and you've effectively counted the number of holes in a shape.

"Witten showed that quantum field theory gives completely unexpected but completely precise insights into geometrical questions, making intractable problems soluble," said Graeme Segal, a mathematician at the Academy of Oxford.

Another example of this commutation also occurred in the early 1990s, when physicists were doing calculations related to cord theory. They performed them in 2 different geometric spaces based on fundamentally different mathematical rules and kept producing long sets of numbers that matched each other exactly. Mathematicians picked upwardly the thread and elaborated it into a whole new field of enquiry, called mirror symmetry, that investigates the concurrence — and many others similar it.

"Physics would come up upwardly with these amazing predictions, and mathematicians would try to prove them past our own ways," said Ben-Zvi. "The predictions were strange and wonderful, and they turned out to exist pretty much ever correct."

But while QFT has been successful at generating leads for mathematics to follow, its core ideas still exist near entirely exterior of mathematics. Quantum field theories are not objects that mathematicians understand well enough to utilise the way they can apply polynomials, groups, manifolds and other pillars of the discipline (many of which also originated in physics).

For physicists, this afar relationship with math is a sign that there'southward a lot more than they need to understand about the theory they birthed. "Every other idea that'south been used in physics over the past centuries had its natural place in mathematics," said Seiberg. "This is conspicuously not the case with quantum field theory."

And for mathematicians, it seems as if the relationship between QFT and math should be deeper than the occasional interaction. That's considering breakthrough field theories contain many symmetries, or underlying structures, that dictate how points in different parts of a field chronicle to each other. These symmetries have a concrete significance — they embody how quantities like energy are conserved as quantum fields evolve over fourth dimension. Merely they're also mathematically interesting objects in their own right.

"A mathematician might care about a certain symmetry, and nosotros can put it in a physical context. Information technology creates this beautiful bridge between these two fields," said Castro.

Mathematicians already utilise symmetries and other aspects of geometry to investigate everything from solutions to different types of equations to the distribution of prime numbers. Often, geometry encodes answers to questions about numbers. QFT offers mathematicians a rich new blazon of geometric object to play with — if they can go their hands on it directly, at that place's no telling what they'll be able to do.

"We're to some extent playing with QFT," said Dan Freed, a mathematician at the Academy of Texas, Austin. "We've been using QFT as an outside stimulus, but it would be nice if it were an within stimulus."

Make Way for QFT

Mathematics does not admit new subjects lightly. Many basic concepts went through long trials before they settled into their proper, canonical places in the field.

Take the real numbers — all the infinitely many tick marks on the number line. Information technology took math nearly two,000 years of practice to agree on a way of defining them. Finally, in the 1850s, mathematicians settled on a precise three-word statement describing the real numbers equally a "consummate ordered field." They're consummate because they contain no gaps, they're ordered because in that location's always a way of determining whether one real number is greater or less than some other, and they course a "field," which to mathematicians means they follow the rules of arithmetic.

"Those three words are historically difficult fought," said Freed.

In order to plough QFT into an within stimulus — a tool they can apply for their own purposes — mathematicians would like to give the same handling to QFT they gave to the real numbers: a precipitous listing of characteristics that any specific quantum field theory needs to satisfy.

A lot of the work of translating parts of QFT into mathematics has come from a mathematician named Kevin Costello at the Perimeter Institute. In 2016 he coauthored a textbook that puts perturbative QFT on firm mathematical footing, including formalizing how to piece of work with the space quantities that crop up as you increase the number of interactions. The work follows an earlier effort from the 2000s called algebraic quantum field theory that sought like ends, and which Rejzner reviewed in a 2016 volume. So now, while perturbative QFT withal doesn't really depict the universe, mathematicians know how to deal with the physically not-sensical infinities information technology produces.

"His contributions are extremely ingenious and insightful. He put [perturbative] theory in a nice new framework that is suitable for rigorous mathematics," said Moore.

Costello explains he wrote the book out of a desire to brand perturbative quantum field theory more coherent. "I simply found certain physicists' methods unmotivated and ad hoc. I wanted something more self-contained that a mathematician could go work with," he said.

By specifying exactly how perturbation theory works, Costello has created a basis upon which physicists and mathematicians tin construct novel breakthrough field theories that satisfy the dictates of his perturbation approach. It'southward been quickly embraced by others in the field.

"He certainly has a lot of young people working in that framework. [His book] has had its influence," said Freed.

Costello has too been working on defining merely what a quantum field theory is. In stripped-down form, a quantum field theory requires a geometric infinite in which you can make observations at every point, combined with correlation functions that express how observations at different points chronicle to each other. Costello's work describes the backdrop a collection of correlation functions needs to accept in order to serve as a workable basis for a breakthrough field theory.

The most familiar quantum field theories, similar the Standard Model, comprise additional features that may not be present in all breakthrough field theories. Breakthrough field theories that lack these features likely describe other, nevertheless undiscovered properties that could help physicists explain physical phenomena the Standard Model tin't account for. If your idea of a quantum field theory is fixed too closely to the versions nosotros already know about, y'all'll take a hard time even envisioning the other, necessary possibilities.

"At that place is a big lamppost under which yous can find theories of fields [like the Standard Model], and around it is a large darkness of [quantum field theories] we don't know how to define, merely nosotros know they're there," said Gaiotto.

Costello has illuminated some of that dark space with his definitions of quantum fields. From these definitions, he'south discovered ii surprising new quantum field theories. Neither describes our four-dimensional universe, simply they do satisfy the core demands of a geometric space equipped with correlation functions. Their discovery through pure thought is similar to how the kickoff shapes you lot might discover are ones present in the physical earth, just once yous have a general definition of a shape, you can think your way to examples with no physical relevance at all.

And if mathematics tin determine the full space of possibilities for quantum field theories — all the many different possibilities for satisfying a general definition involving correlation functions — physicists tin use that to find their mode to the specific theories that explain the important physical questions they care about about.

"I want to know the infinite of all QFTs because I want to know what quantum gravity is," said Castro.

A Multi-Generational Claiming

There'south a long style to go. And then far, all of the quantum field theories that have been described in full mathematical terms rely on various simplifications, which brand them easier to piece of work with mathematically.

One manner to simplify the trouble, going back decades, is to written report 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, only that make them more tractable from a mathematical perspective. These include "supersymmetric" and "topological" QFTs.

The side by side, and much more difficult, step volition exist to remove the crutches and provide a mathematical description of a quantum field theory that improve suits the physical world physicists near want to depict: the iv-dimensional, continuous universe in which all interactions are possible at once.

"This is [a] very embarrassing thing that nosotros don't have a single quantum field theory we can describe in four dimensions, nonperturbatively," said Rejzner. "It's a hard problem, and evidently it needs more than i or 2 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 blazon of object as they could promise for. Defining the characteristic properties shared by all breakthrough field theories will about 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 trouble just in math itself, because it combines ii great ideas," said Dijkgraaf.

If mathematicians can understand QFT, there's no telling what mathematical discoveries look in its unlocking. Mathematicians defined the characteristic properties of other objects, similar manifolds and groups, long ago, and those objects now permeate virtually every corner of mathematics. When they were first defined, it would have been impossible to anticipate all their mathematical ramifications. QFT holds at to the lowest degree as much promise for math.

"I like to say the physicists don't necessarily know everything, but the physics does," said Ben-Zvi. "If you inquire information technology the right questions, it already has the phenomena mathematicians are looking for."

And for physicists, a complete mathematical description of QFT is the flip side of their field's overriding goal: a complete description of physical reality.

"I experience there is 1 intellectual structure that covers all of it, and maybe it will encompass all of physics," said Seiberg.

At present mathematicians just have to uncover it.

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Source: https://www.quantamagazine.org/the-mystery-at-the-heart-of-physics-that-only-math-can-solve-20210610/