What is $U(1)$ symmetry?

I saw there are three intrinsic symmetries in physics,U(1),SU(2) and SU(3).What's the U(1) symmetry talking about?I would appreciate it if you can give me some explaination.

• Are you wondering what U(1) is, which is a mathematical term dealing with symmetry, or are you wondering how it applies to the standard model (which deals with electromagnitism)? – Cort Ammon May 7 at 13:22
• I want to know its formula and some invariances in physics . – KarryMa May 7 at 13:27

Let us first refer to symmetry generically. When we say a theory is symmetric under $$G$$ ($$G$$ some group) we mean that the elements of $$G$$ transform the states, and the operators of a theory, (in the context of the Standard Model (SM)), in such a way that the Lagrangian won't change in form. One could then speak about space-time symmetries, such as Lorentz transformations, or to intrinsic symmetries which transform the fields of your theory. These are the ones you find associated to the SM.

Let us now speak about $$U(1)$$ in particular. It corresponds to the unitary group of dimension $$1$$. That is you can think about this group as unitary matrices of dimension 1, which means essentially (complex) numbers of norm 1 or as they are usually parametrized $$e^{i\theta}$$ with $$\theta\in[0,2\pi)$$. Observe that $$\theta$$ is just a parameter that helps us label the elements of the group. You can intuitively see that $$U(1)$$ corresponds geometrically to a circle, and multiplication among elements is equivalent to adding the parameter $$\theta$$, that is rotating with some angle around the circle.

Coming back to the physics, one has two cases.

Option 1, it might be the case that your theory doesn't change, is invariant, when you employ a transformation $$e^{i\theta}$$ were $$\theta$$ is independent of the point in space-time but otherwise arbitrary. This is called a global symmetry and corresponds to conserved quantities through Noether's theorem.

Option 2, the theory can be invariant through a more complicated rule, $$e^{i\theta(x)}$$, which is a transformation that depends on the point of space-time. These sort are called gauge symmetries and are associated to gauge freedom and gauge bosons. They essentially tell you that there is some redundancy in the way you are writing your theory and certain states must be identified (they should be considered the same).

There are different interpretations of $$U(1)$$ depending on what it refers in a given context. Particle number conservation, QED and photons, Hypercharge in SM. I hope I answered your question, and mentioned enough keywords so you can read more on your own.

• One thing I might add, just for completeness: the reason we see the idea of "groups" pop up when we're looking at symmetry is because group theory is mathematics' discipline for modeling the effects of symmetries. Some of the wording can be demanding, but in the end it's nothing more than a very formal way of capturing concepts like "If I spin 360 in place, the world will look exactly like it would have if I didn't spin" or capturing how a kung fu master can spin their staff without twisting themselves up nor having to let go. – Cort Ammon May 7 at 13:53