Edit Porkson Edit Porkson has been a reporter on the news desk since Related Post. Nov 13, Edit Porkson. Leave a Reply Cancel reply Your email address will not be published. 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. Neither describes our four-dimensional universe, but they do satisfy the core demands of a geometric space equipped with correlation functions. Their discovery through pure thought is similar to how the first shapes you might discover are ones present in the physical world, but once you have a general definition of a shape, you can think your way to examples with no physical relevance at all.
And if mathematics can determine the full space of possibilities for quantum field theories — all the many different possibilities for satisfying a general definition involving correlation functions — physicists can use that to find their way to the specific theories that explain the important physical questions they care most about.
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. 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.
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.
Mathematicians defined the characteristic properties of other objects, like 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 least as much promise for math. Get highlights of the most important news delivered to your email inbox.
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