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FQXI ARTICLE

August 23, 2019

The Art of Math

A pictorial branch of mathematics could help physicists draw new conclusions about quantum gravity and the nature of time.

July 31, 2013

John Baez

Exploring Shanghai

Credit: L. Raphals

Baez, who is based at the Center for Quantum Technologies in Singapore, champions rewriting—or rather redrawing—quantum physics using a pictorial branch of math called "category theory." It’s a system that mathematicians know well, but which physicists rarely exploit. "This math goes back to some very basic ideas and uses them to do some very new things," he says.

This yen for un-cluttering physics and simplifying it in an approachable way developed during Baez’s childhood, when his uncle, Albert Baez—father of well-known folksinger Joan Baez—visited the house. Albert had set up physics departments in Algeria and Iraq and so he was constantly devising cheap ways to demonstrate physics on the bench-top. He also wrote a physics textbook for college students which he gave to his eight-year-old nephew. "That made me decide early on that I wanted to do physics," says Baez.

Category theory is a very visual form of math that allows you to keep track of individual particles by linking objects in diagrams using arrows. If you ever studied set theory at school, you may remember grouping items that had something in common (objects you own that are items of clothing, say) into a large set, and then splitting them into subsets (socks, gloves, hats) and drawing Venn diagrams of intersecting circles to represent how some of the objects shared qualities (the color blue, say), while others did not.

For a more mathematical example, you could draw a set of all the positive integers (1, 2, 3, 4...) and another of all the positive even integers (2, 4, 6, 8...) and draw an arrow representing the function linking the elements in the two sets together (in this case, the function would be "multiply by two.") Category theory does much the same thing, but on a more abstract level, depicting connections between more mathematically sophisticated objects. If Venn diagrams can be thought of as the equivalent of kids’ crayon drawings, then category theory is more akin to the works of Picasso.

Multi-dimensional Lego

"There are papers on category theory where the diagrams are so large they go over several pages!" says another fan, John Barrett, a mathematical physicist at Nottingham University, UK. Barrett describes using category theory to click algebra together in space-time "like multi-dimensional lego."

Categorification

Category theory describes how to warp surfaces in 4-dimensional spacetime.

Credit: J. Scott Carter

"Categorification helps you understand conceptually what’s going on. You can stack up the algebra in spatial directions as well as time directions and ensure all the equations come out correctly," says Barrett.

This also means that if the physics that you are trying to understand is too tough to figure out in three-dimensional space, you can work out a simplified version in two or one dimensions, and then categorify it. Hey presto! You have your 3-D physics theory.

The Flow of Time

Category theory could also help illuminate an ancient saying about the nature of time. The Greek philosopher Heraclitus famously mused about the notion of change and the passage of time, stating that you cannot step into the same river twice. Why not? "Well, people usually say it’s because the river would be different later: all the water molecules in it will be different," says Baez. "But the writer Borges pointed out that

I expect that one day there

will be a good category

theory explanation of

fundamental physics.

will be a good category

theory explanation of

fundamental physics.

- John Barrett

Specifically, category theory replaces equations with

Thank you Mr. Feynman

Category theory may help our understanding of ancient Greek philosophy. But any sceptical physicist who does not want to change her mathematical ways will ask: Has categorification ever actually been shown to be useful in physics?

The answer is a resounding "yes." Most people don’t realise that Feynman diagrams—the squiggly representations that physicists routinely use to analyse what will be produced when subatomic particles collide—are a form of category theory.

Albert Baez

A physics inspiration

String theory’s loss was category theory’s gain. Crane told Baez that this alternative mathematical framework could be useful to the hunt for a theory to unite quantum mechanics with general relativity. Baez "took off on that and has been hooked on it ever since," says Crane. Today, Baez and his students, Chris Rogers, John Huerta and Christopher Walker, are busy categorifying quantum theory and classical mechanics, with the help of a $131,865 grant from FQXi.

Baez has also popularized category theory with physicists through his blog ,

Most importantly, Baez is a bridge builder. "John can write maths that other mathematicians understand, and can talk to physicists and try to understand what they are doing in a mathematical way," says Crane.

Barrett is optimistic about the strategy. "I expect that one day there will be a good category theory explanation of fundamental physics," he says. "It’s definitely a useful avenue to explore."

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