Gauge Symmetry for Dummies

By GrabBag | CrackpotPhysicist | 5 Oct 2020

Many concepts in physics are very difficult for a non-mathematician to understand. But with common language and analogy, the layperson can gain some glimpse of understanding. This is how I hope to explain Gauge Symmetry.

But first, let's review what a symmetry is. Here's a definition for symmetry that you may not have heard before: A symmetry is a property of some object or idea to change in one way but not another. For example, if you rotate a square by 90 degrees, it will look exactly the same as how it started. So, a square can change in one way (its rotation) and not change in another (how it looks).

Symmetries don't just apply to rotations of geometric objects. Take electrons for example - we are told that electrons are negatively charged, but the name of its charge is arbitrary. You could switch the terms "negative" and "positive" in electromagnetism without breaking anything. There's no mathematical or scientific basis for the naming. As long as pole A is attracted to pole B and repels pole A, it doesn't matter what you call it. This is also a symmetry - the two magnetic poles are separate but identical to each other. A car and a house don't have the same symmetry: I can't say a house is a car. So this isn't just some silly word game.

So let's talk about money. If money was symmetric over time, then the value of a dollar bill would be the same today as it was ten years ago. If we define the value of a dollar by its spending power, it can't be symmetric because milk was like 25¢ "back in the day". So we could conclude that the value of money is not symmetric over time, but actually we'd be wrong. Introducing: gauge symmetry!

We often see historical dollar amounts adjusted for inflation. This is great because it allows us to put historical events into a monetary context that is easy for us to understand. In other words, adjusting for inflation effectively makes the value of money symmetric over time. Now, here's the important bit: the function that adjusts for inflation defines a gauge symmetry of money. What makes something a "gauge" rather than a normal symmetry is the use of a specific mathematical device (function) to translate values between contexts in a mathematically rigorous way.

When a financial adviser looks at a stock portfolio, he can see the value of everything in terms of his country's currency. This is a very convenient and powerful feature. This is why symmetries are so important: they give a mathematician power and flexibility. If you can add more symmetry to a problem, you make it easier to solve. And if you can add symmetry in a mathematically rigorous way, it's even more powerful. That's what gauge symmetries do.

So now (hopefully) you know what a gauge symmetry is. That's all fine and interesting. So then why do mathematicians make such a big deal about them? Well, let's dive a little deeper.

In the examples above, I describe how mathematicians can create functions to add symmetry to a system. This seems like a human-invented contrivance, but it's not. It turns out that the function for an electromagnetic field is a gauge symmetry. It's not like a mathematician came up with some formula to give the electromagnetic field more symmetry, no, no. The field itself is a gauge symmetry, indeed all the forces of nature have this characteristic. It's symmetry within symmetry. It's a kind of beautiful thing that mathematicians appreciate. And maybe we can learn to appreciate it too.

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