What Math Can Teach Physics

Question: In what new ways could math be applied to solve the problems of physics?

Peter Woit: The thing that most fascinates me about this whole subject is that something that probably quickly get very technical, but there's an area of mathematics which is known as representation theory. One way of thinking about it is in terms of what physicists often call symmetries. So, for instance, one of the basic facts about the laws of nature is that there are symmetric laws of nature are the same at - if you move in any direction, or you move in time back and forth, the laws of physics don't change. If you rotate things around in three dimensions, the laws of physics don't change. And so, these so-called symmetries have very important physical implications. The fact that the laws of physics don't change as if you move in time has physical implications that there's this thing called energy and energy is conserved, and the same thing - and the fact that the laws of physics don't change of you move back and forth in different directions in space implies that there is something called momentum and momentum is conserve and doesn't change as you evolve in time.

So, the very, very fundamental facts about physics are kind of deeply grounded in the symmetries of nature. And so to a mathematician this question is a question about what we call groups and representation of groups. So, if you go into any math department and look at what they're doing, you'll see that a lot of people in different kinds of mathematics are studying different structures which are also called groups and they're often studying what is called representations of the groups. So, even people studying number theory, abstract things about prime numbers or something, they're also studying groups and certain representations of these groups. So, there's kind of a, to the extent that mathematics has a kind of unifying theme and a unifying principle which shows up in different areas of mathematics, it's about these representations, or representation theory. And the thing that most strikes me about physics, and what fascinates me about it is, if you look at quantum mechanics, quantum mechanics initially looks like a very odd structure. It's not something were we have any kind of intuitive understanding of it. It doesn't look like the way we're used to thinking about physics, based on every day experience. But if you look at the mathematical structure and the basic structure of quantum mechanics, they're exactly the structures that show up in this theory of representations. So, there's a kind of a deep relation between math and physics which is surrounding this whole notion of symmetries and representation of symmetries.

So, that's one thing that's always fascinated me, and my own research and my own interests is in developing - taking a lot that has been learned in mathematics. There's a lot that's been learned in mathematics over the years about how to think about representations and how to construct them and how to work with them. Some of it has made its way into physics and have been used in physics and was used in physics since the early days of quantum mechanics. So, for instance, one of the great ****, there's a kind of hero of this book I wrote about this, it's The Mathematician called Herman Vial, who was one of the first people to understand how quantum mechanics worked and to understand the relation to representations.

But anyway, I think there's still a lot to be learned in that way and very specifically the so-called standard model has this group of symmetries which is called the gage symmetry and it's an infinitial group and the standard kind of physics understanding of the representations of this group is that that should not be an interesting question. There should only be a trivial representation of this group. But anyway, my conjecture is that there is actually a more interesting question there and in pursuing this question of how do you deal with the gauged symmetry of this theory in terms of using ideas from representation theory that are more well-known in mathematics that hadn't been used in physics before that you can actually get somewhere. Well that's maybe too technical, but that's as good as I can do with this.

Recorded on December 16, 2009
Interviewed by Austin Allen

Peter Woit explains the "deep relation" between the two disciplines and the most mind-bending new ways in which that relation is being explored.

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