The Math That Takes Newton Into the Quantum World

Image Credit. NAUTILUS
In my 50s, too old to become a real expert, I have finally fallen in love with algebraic geometry. As the name suggests, this is the study of geometry using algebra. Around 1637, René Descartes laid the groundwork for this subject by taking a plane, mentally drawing a grid on it, as we now do with graph paper, and calling the coordinates x and y. We can write down an equation like x2+ y2 = 1, and there will be a curve consisting of points whose coordinates obey this equation. In this example, we get a circle!

By John Baez | NAUTILUS

It was a revolutionary idea at the time, because it let us systematically convert questions about geometry into questions about equations, which we can solve if we’re good enough at algebra. Some mathematicians spend their whole lives on this majestic subject. But I never really liked it much until recently—now that I’ve connected it to my interest in quantum physics.

If we can figure out how to reduce topology to algebra, it might help us formulate a theory of quantum gravity.

As a kid, I liked physics better than math. My uncle Albert Baez, father of the famous folk singer Joan Baez, worked for UNESCO, helping developing countries with physics education. My parents lived in Washington, D.C. Whenever my uncle came to town, he’d open his suitcase, pull out things like magnets or holograms, and use them to explain physics to me. This was fascinating. When I was 8, he gave me a copy of the college physics textbook he wrote. While I couldn’t understand it, I knew right away that I wanted to. I decided to become a physicist and my parents were a bit worried, because they knew physicists needed mathematics, and I didn’t seem very good at that. I found long division insufferably boring and refused to do my math homework, with its endless repetitive drills. But later, when I realized that by fiddling around with equations I could learn about the universe, I was hooked. The mysterious symbols seemed like magic spells. And in a way, they are. Science is the magic that actually works.

In college I majored in math, and became curious about theoretical physicist Eugene Wigner’s question about the “unreasonable effectiveness” of mathematics: Why should our universe be so readily governed by mathematical laws? As he put it, “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” As a youthful optimist, I felt these laws would give us a clue to the deeper puzzle: why the universe is governed by mathematical laws in the first place. I already knew that there was too much math to learn it all, so, in grad school, I tried to focus on what mattered to me. And one thing that did not matter to me was algebraic geometry.

How I could any mathematician not fall in love with algebraic geometry? Here’s why: In its classic form, this subject considers only polynomial equations—equations that describe not just curves, but also higher-dimensional shapes called “varieties.” So, x2+ y2 = 1 is fine, and so is x432xy2 = y7, but an equation with sines or cosines, or other functions, is out of bounds—unless we can figure out how to convert it into an equation with just polynomials. As a graduate student, this seemed like a terrible limitation. After all, physics problems involve plenty of functions that aren’t polynomials.

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