QED is regarded as the first and simplest physical gauge theory.

Gauge theories are a class of quantum theories that describe the fundamental forces of nature through the concepts of symmetry and the exchange of elementary particles.
Gauge theory is a field theory in which the equations of motion remain unchanged after a transformation. A local gauge is a coordinate system that can change from point to point in space-time, while a gauge transformation allows one to move from one location in the coordinate system to another. A gauge field theory is a special type of quantum field theory. QED is regarded as the first and simplest physical gauge theory. The set of gauge transformations of the entire configuration of a given gauge theory also forms a group, the gauge group of the theory.
Gauge theory, a class of quantum field theory, is a mathematical theory involving both quantum mechanics and Einstein’s special theory of relativity that is commonly used to describe subatomic particles and their associated wave fields.
If you’re like me, you’ve heard about gauge theory for “forever” but may not understand what it’s all about. It’s hard to find real enlightenment about.
Let’s take an example. Start with quantum theory. It’s all about these complex quantities, and from those complex quantities, you can calculate probabilities. Well, it turns out that if you go through all the complex quantities in your problem and add some arbitrary amount to the “phase” of every one of them (basically, what that means is that you’re rotating your complex plane), that has absolutely no effect on the ultimate probabilities.
You calculate. The probabilities depend only on the relative phases of all those quantities—shifting all of them by the same amount has no effect.
Okay, so far, so good, and that’s just simple quantum theory. But now, what if you imagine doing something a little different? Instead of adding the same phase to every quantity, you add different, entirely arbitrary phases? I say “entirely arbitrary,” but what I mean is that you imagine some smooth function that gives you the adjustment amount.
Well, this does break physics. Now you will get different probabilities, and it’s even possible that your probabilities won’t add up to 1.0 anymore. That’s a completely fatal thing—you no longer have a valid description of the world. So this is an unacceptable situation.
Turns out, though, that there is a way to fix that probability pathology. You can add a new field to your system—to the Lagrangian. That will affect the Schrodinger equation you derive from that Lagrangian, and if you choose that field properly, you can cause it to exactly cancel out the phase change you made in step one. Now your probabilities add up to one again, and you have a physically tenable situation.
However, those new items in the Lagrangian not only “fix” your original equations of motion—they also lead to altogether new equations of motion. And here is the truly remarkable and unexpected thing: our universe appears to work in such a way that these new equations of motion describe legitimate, valid physics.
The first time I saw this workout, I found it utterly and completely shocking. I saw no reason whatsoever to expect such a “magic trick” to work; it all feels so arbitrary. But—it does. We’ve used it over and over across the decades to extend our understanding of the world. Honestly, it feels like a “gift from Nature.” Here, as we’ve discussed, the new field you just made up and shoved into the problem turns out to be the vector magnetic potential, and the new physics you get is electromagnetic theory.
This isn’t how we discovered electromagnetism. Maxwell arrived at that “the hard way,” based on all kinds of weird models of the “aether,” pondering experimental results, and so on. But it turned out later to be our first and simplest gauge theory.
The general description of this procedure runs like this. Identify a global symmetry in your system (in the case we discussed, it’s the “global phase symmetry” of quantum mechanics). Promote that global symmetry to a local symmetry. Add a new field or fields to the Lagrangian to cancel out the adverse effects of forcing that symmetry to be local. Derive the new equations of motion that result. Interpret those new equations. That’s it. It’s a “recipe for new physics.”
This arbitrary process “works” and appears to be part of how nature operates. As far as I can tell, the gist of this is that any process you can imagine that is not forbidden by the laws of physics (e.g., processes that don’t conserve energy or momentum would be “forbidden”) can What are some specific examples of gauge theories beyond electromagnetism, and how do they apply to other fundamental forces like the weak and strong nuclear forces?
Double copy theory is a theory in theoretical physics, specifically in quantum gravity, that hypothesizes a perturbative duality between gauge theory and gravity.

What are the implications of these new equations of motion for future scientific discoveries and advancements in technology, particularly in fields like particle physics and cosmology? Happen in the world. It might not happen often—it might have a very low probability—but you can find it out there somewhere. Anything that can happen (is allowed) will happen, with some probability.

QED is regarded as the first and simplest physical gauge theory.

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