RECENT ARTICLES

New photon experiment gives new meta-framework, 'constructor theory,' a boost.

Experiments seek to use quantum observations as fuel to power mini motors.

Sally Shrapnel, a quantum physicist and medical practitioner, on her experiments into cause-and-effect that could help us understand time’s arrow—and build better healthcare algorithms.

'Artificial atoms' allow physicists to manipulate individual electrons—and could help to reduce energy wastage in electronic devices.

Two teams are developing ways to detect quantum-gravitational effects in the lab.

FQXI ARTICLE

September 26, 2021

The Art of Math

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

October 20, 2010

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."

Comment on this Article

Please read the important Introduction that governs your participation in this community. Inappropriate language will not be tolerated and posts containing such language will be deleted. Otherwise, this is a free speech Forum and all are welcome!

function ValidatePostText_main () {
form = document.addPostForm_main;
var recaptcha = $("#g-recaptcha-response").val();
if (recaptcha === "") {
event.preventDefault();
alert("The reCaptcha Box below must be checked before you submit the form");
}
else if (form.postText_main.value == '') {
alert ("The post contains no text");
return false;
}
else {
return true;
}
}

**Your name:**
(optional)

Recent Comments

read all article comments

Please read the important Introduction that governs your participation in this community. Inappropriate language will not be tolerated and posts containing such language will be deleted. Otherwise, this is a free speech Forum and all are welcome!

Please enter the text of your post, then click the "Submit New Post" button below. You may also optionally add file attachments below before submitting your edits.

HTML tags are not permitted in posts, and will automatically be stripped out. Links to other web sites are permitted. For instructions on how to add links, please read the link help page.

You may use superscript (10

^{100}) and subscript (A_{2}) using [sup]...[/sup] and [sub]...[/sub] tags.You may use bold (

**important**) and italics (*emphasize*) using [b]...[/b] and [i]...[/i] tags.You may also include LateX equations into your post.

Insert LaTeX Equation
[hide]

LaTeX equations may be displayed in FQXi Forum posts by including them within [equation]...[/equation] tags. You may type your equation directly into your post, or use the LaTeX Equation Preview feature below to see how your equation will render (this is recommended).

For more help on LaTeX, please see the LaTeX Project Home Page.

LaTeX Equation Preview

preview equation

clear equation

insert equation into post at cursor

LaTeX equations may be displayed in FQXi Forum posts by including them within [equation]...[/equation] tags. You may type your equation directly into your post, or use the LaTeX Equation Preview feature below to see how your equation will render (this is recommended).

For more help on LaTeX, please see the LaTeX Project Home Page.

LaTeX Equation Preview

preview equation

clear equation

insert equation into post at cursor

Attachments
[hide]

You may optionally attach up to two documents to your post. To add an attachment, use the following feature to browse your computer and select the file to attach. The maximum file size for attachments is 1MB.

Once you're done adding file attachments, click the "Submit New Post" button to add your post.

You may optionally attach up to two documents to your post. To add an attachment, use the following feature to browse your computer and select the file to attach. The maximum file size for attachments is 1MB.

Once you're done adding file attachments, click the "Submit New Post" button to add your post.

JOHN R. COX wrote on August 7, 2017

Nicely written introductory piece. Even makes me curious. Topology has become something of a unification of mathematics with its role in computer sciences, and I have just read that the techies are now looking at 'the 2nd Billion' smart-phone users in emerging countries with high rates of poverty and illiteracy and the developing of 4Gig cheap phones using non-alphabetic symbology and/or voice commands. The n-category café sounds like the route this interface might take, and evokes a...

Nicely written introductory piece. Even makes me curious. Topology has become something of a unification of mathematics with its role in computer sciences, and I have just read that the techies are now looking at 'the 2nd Billion' smart-phone users in emerging countries with high rates of poverty and illiteracy and the developing of 4Gig cheap phones using non-alphabetic symbology and/or voice commands. The n-category café sounds like the route this interface might take, and evokes a...

ANONYMOUS wrote on August 7, 2017

Comment on this Article

Please read the important Introduction that governs your participation in this community. Inappropriate language will not be tolerated and posts containing such language will be deleted. Otherwise, this is a free speech Forum and all are welcome!

Please enter the text of your post, then click the "Submit New Post" button below. You may also optionally add file attachments below before submitting your edits.

HTML tags are not permitted in...

Comment on this Article

Please read the important Introduction that governs your participation in this community. Inappropriate language will not be tolerated and posts containing such language will be deleted. Otherwise, this is a free speech Forum and all are welcome!

Please enter the text of your post, then click the "Submit New Post" button below. You may also optionally add file attachments below before submitting your edits.

HTML tags are not permitted in...

STEVE A JEFFREY wrote on September 23, 2011

The Grand design states that in 10 dimensions only circular orbit can exist and these are unstable..

And this applies to atoms too atoms would be unstable in higher dimensions and the electrons would either spiral into the nucleus or fly off into space.

If atoms cuold not exist in 10^500 other universes each with it's own set of rules.

Then life cannot exist in another universe.

Does that mean life was intelligently designed. What do you think.?

The Grand design states that in 10 dimensions only circular orbit can exist and these are unstable..

And this applies to atoms too atoms would be unstable in higher dimensions and the electrons would either spiral into the nucleus or fly off into space.

If atoms cuold not exist in 10^500 other universes each with it's own set of rules.

Then life cannot exist in another universe.

Does that mean life was intelligently designed. What do you think.?

read all article comments