Why quantum field theory

The field emerging from a magnet is shown on the right. If you look closely enough at electromagnetic waves, you'll find that they are made out of particles called photons. The ripples of the electric and magnetic fields get turned into particles when we include the effects of quantum mechanics. But this same process is at play for all other particles that we know of. There exists, spread thinly throughout space, something called an electron field.

Ripples of the electron field get tied up into a bundle of energy by quantum mechanics. And this bundle of energy is what we call an electron. Similarly, there is a quark field, and a gluon field, and Higgs boson field. Every particle your body indeed, every particle in the Universe is a tiny ripple of the underlying field, moulded into a particle by the machinery of quantum mechanics.

In part this is because it contains all of physics: the field can describe vast numbers of particles, interacting in a myriad of different ways. Yet, even before we get to these difficulties, there is another reason that quantum field theory is hard.

From there, physicists enhance 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 take measurements increases, approaching the idealized notion of a field where you can take measurements everywhere. 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. Physicists would like to prove that quantum fields are the limit of this lattice procedure. But having strong evidence that something is correct and proving conclusively that it is are two different things. Even in this incomplete state, QFT has prompted a number of important mathematical discoveries. The general pattern of interaction has been that physicists using QFT stumble onto surprising calculations that mathematicians then try to explain.

At a basic level, physical phenomena have a tight relationship with geometry. To take a simple example, if you set a ball in motion on a smooth surface, its trajectory will illuminate the shortest path between any two points, a property known as a geodesic. In this way, physical phenomena can detect geometric features of a shape. Now replace the billiard ball with an electron. The electron exists probabilistically everywhere on a surface.

In the late s, physicists and mathematicians began applying this perspective to solve basic questions in geometry. Another example of this exchange also occurred in the early s, when physicists were doing calculations related to string theory. They performed them in two different geometric spaces based on fundamentally different mathematical rules and kept producing long sets of numbers that matched each other exactly.

Mathematicians picked up the thread and elaborated it into a whole new field of inquiry, called mirror symmetry , that investigates the concurrence — and many others like it. But while QFT has been successful at generating leads for mathematics to follow, its core ideas still exist almost entirely outside of mathematics.

Quantum field theories are not objects that mathematicians understand well enough to use the way they can use polynomials, groups, manifolds and other pillars of the discipline many of which also originated in physics. And for mathematicians, it seems as if the relationship between QFT and math should be deeper than the occasional interaction.

These symmetries have a physical significance — they embody how quantities like energy are conserved as quantum fields evolve over time. Mathematicians already use 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. 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. It took math nearly 2, years of practice to agree on a way of defining them.

In order to turn QFT into an inside stimulus — a tool they can use for their own purposes — mathematicians would like to give the same treatment to QFT they gave to the real numbers: a sharp list 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 he coauthored a textbook that puts perturbative QFT on firm mathematical footing, including formalizing how to work with the infinite quantities that crop up as you increase the number of interactions.

The work follows an earlier effort from the s called algebraic quantum field theory that sought similar ends, and which Rejzner reviewed in a book. Costello explains he wrote the book out of a desire to make perturbative quantum field theory more coherent. By specifying exactly how perturbation theory works, Costello has created a basis upon which physicists and mathematicians can construct novel quantum field theories that satisfy the dictates of his perturbation approach.

Costello has also been working on defining just what a quantum field theory is. In stripped-down form, a quantum field theory requires a geometric space in which you can make observations at every point, combined with correlation functions that express how observations at different points relate to each other. The most familiar quantum field theories, like the Standard Model, contain additional features that may not be present in all quantum field theories.

Costello has illuminated some of that dark space with his definitions of quantum fields. It was the first of a whole new menagerie of particles that theorists proposed as quantum field theories evolved — and that later popped up in reality.

Two quantum field theories lie at the heart of the standard model of particle physics. The product of many decades of theoretical work, meticulously confirmed by experiment, this model covers the workings of three of the four forces of nature through interactions of force-carrying boson particles with matter-making fermions. Quantum chromodynamics QCD , meanwhile, is the theory of the strong nuclear force.

Transmitted by bosons called gluons, this strong, very short-range force binds quarks together to make particles such as protons and neutrons.