Tag Archives: Gauge fields

Math can be hard even for very smart people

50 McCosh Hall an autumn evening in 1956. The place was packed. Chen Ning Yang was speaking about parity violation. Most of the people there had little idea (including me) of what he did, but wanted to be eyewitnesses to history.. But we knew that what he did was important and likely to win him the Nobel (which happened the following year).

That’s not why Yang is remembered today (even though he’s apparently still alive at 98). Before that he and Robert Mills were trying to generalize Maxwell’s equations of electromagnetism so they would work in quantum mechanics and particle physics. Eventually this led Yang and Mills to develop the theory of nonAbelian gauge fields which pervade physics today.

Yang and James Simons (later the founder of Renaissance technologies and already a world class mathematician — Chern Simons theory) later wound up at Stony Brook. Simons, told him that gauge theory must be related to connections on fiber bundles and pointed him to Steenrod’s The Topology of Fibre Bundles. So he tried to read it and “learned nothing. The language of modern mathematics is too cold and abstract for a physicist.”

Another Yang quote “There are only two kinds of math books: Those you cannot read beyond the first sentence, and those you cannot read beyond the first page.”

So here we have a brilliant man who invented significant mathematics (gauge theory) along with Mills, unable to understand a math book written about the exact same subject (connections on fiber bundles).

Book recommendation

Tired of reading books about physics?  Want the real McCoy”?  Well written and informal?  Contains stuff whose names you know but don’t understand — Jones Polynomial, Loop Quantum Gravity, Quantum field theory, Gauge groups and transformations —  etc. etc.

Up to date?  Well no, it’s 25 years old but still very much worth a read, so very unlike molecular biology, chemistry, computer science etc. etc.

Probably you should know as much physics and math as a beginning chemistry grad student. If you studied electromagnetism through Maxwell’s equations it would be a plus.  I stopped at Coulomb’s Law, and picked up enough to understand NMR.

This will give you a sample of the way it is written

“Much odder is that we are saying the vector field v is the linear combination of . .  partial derivatives.  What we are doing might be regarded as rather sloppy, since we are identifying two different although related things: the vector  field and the operator v^i * d-/dx^i which takes a directional derivative in the direction of v.”

“Now let us define vector fields on a manifold M. .. . these will be entities whose sole ambition in life is to differentiate functions”

The book is “Gauge FIelds, Knots and Gravity” by John Baez and Javier P. Muniain.

The writing, although clear has a certain humility.  “Unfortunately understanding these new ideas depends on a through mastery of quantum field theory, general relativity, geometry, topology and algebra.  Indeed, it is almost certain that nobody is sufficiently prepared to understand these ideas fully.”

I’m going to take it with me to the amateur chamber music festival.  As usual, at least 2 math full professors will be there to help me out.  Buy it and enjoy