Hi all, my 13yro student dropped a bomb on me asking me what are Gauge Theories.. How do I explain it to him in a meaningful way?
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Gauge theories are about simmetries and “change of coordinates”.
For example, if you take the phenomenon of “uniform motion”, it doesn’t matter where exactly it’s taking place. The physical laws are exactly the same. It also doesn’t matter *when* it happens. Again, the physical laws are always the same. So uniform motion in 2021 New York is exactly the same as uniform motion in 1800 Tokyo. From a mathematical standpoint, if you “change coordinates” (in space-time) you’re simply moving a phenomenon in space and time but you’re not changing how that works from a physics standpoint.
Of course this is a very simple example, but gauge theories are essentially about simmetries and how you can “change coordinates” (not just space and time, it’s an abstract concept where you apply a function to any mathematical variable you want) and still keep the same physical laws, and what this implies.
Gauge theory allow you to extend a physical theory that require a nice global coordinate system to one where such global coordinate system is no longer possible, by introducing a gauge field that take into account the coordinate system. Additional physical laws can be put onto which gauge field.
For example, let’s say we start with Newtonian physics. Newtonian physics assume there is a Cartesian coordinate for the entire universe. And it seems to work pretty well in experiment. But the experiment only show that the physics at small scale, there is nothing it does that indicates such universal Cartesian coordinate exist. So if we suspect that there might not be an universal Cartesian coordinate, what we could do is find a physical theory that at small scale still act approximately like Newtonian physics, but at large scale no longer require a Cartesian coordinate.
So how can it be done? In the original physics, the specific choice of coordinate system doesn’t matter, we just assume we had one. But now it does, and we need to care about it explicitly. So what we will do is allow a varying coordinate system, different point use different coordinate system, but our coordinate system still allow the original physics to work. For example, let’s say along a line, we allow the coordinate system to rotate. Then a ball moving along that line will have its coordinate changing in a way that looks like it’s rotating, even though it’s moving straight. You might wonder what’s the point of having such a seemingly misleading coordinate system, but it’s really helpful for situation such as when you want to study the perspective of someone on a carousel: they will feel a centrifugal force, and this can be explained by the fact that their frame of reference is a rotating one, and this is when a rotating coordinate system can be helpful.
But we also require that even for varying coordinate system, it still allows the original physical law to work. In Newtonian physics, if you change your coordinate system by rotating it, then the original physical laws still work; but if you stretch an axis, then everything break because now one direction has a different number. So only coordinate system that are rotation of each other preserve the original physical laws. So we require that our varying system have to be rotating with respect to each other. That way if we don’t rotate our coordinate system, then all the original laws work. For example, if the universe really have a global Cartesian coordinate, then we can choose to never rotate our coordinate system. This ways we had set the stage for a extending our original laws.
Now we allow the coordinate to rotate. This force us to take into account the fact that the object might “move” in our coordinate purely due to this rotation of the coordinate system itself. To deal with this, we introduce a gauge field, a mathematical construct that describe how much we rotate the coordinate system in different point. Then we modify our old physical laws to be able to make use of this gauge field to account for these additional movement.
What we got in the end is now a theory that no longer assume a global coordinate system. The original theory is now a special case where the gauge field is 0. The new theory produce the same physics at small scale, but allow the possibilities that gauge field is not 0, and even better, this allows additional physical possibilities that the gauge field cannot be made to be 0 (no global coordinate system).
What’s next? The thing is, now that we have this gauge field, we can introduce additional physical laws to the gauge field itself. For example, original Newtonian physics could be considered to be a special case where the law said “gauge field is always 0”, and that tells us that the original Newtonian physics assume that there is a global coordinate. So far we had relaxed this stringent assumptions by making no assumptions whatsoever on what kind of coordinate system we can have, so we allow the possibilities that the universe is curved. But maybe we had gone too far? Perhaps the universe is curved, but not every possible way of curving is possible. This require an additional physical law that describe how the universe is curved, and this leads to limitation on the gauge field. This gives us additional laws on the gauge field.
The above explanation make use of Newtonian physics and curvature of the universe, because Newtonian physics is familiar to secondary school student and “universe is curved” is often seen in popsci. In actual physics, gauge theory is often used for quantum physics. However the upgrade from special relativity to general relativity can be seen as very similar to how gauge theory work as well: general relativity introduce a metric tensor to account for coordinate change in an arbitrary coordinate system when special relativity doesn’t allow arbitrary coordinate system, and then a new physical law about how the metric tensor should work and this law describe gravity. It is not a gauge theory – if it were a gauge theory instead of a metric tensor it will has a field of infinitesimal Lorentz transform instead – but the spirit is there.
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