# Could a charge with more than 3 types be self consistent?

I know color charge has three types: red, green, and blue. There is also just one type of electric charge, with there being negative and positive electric charges.

I was thinking of a universe, in which there is a charge with a number of types that is greater than 3, known as A Charge. Equal amounts of all types of A Charge cancel each other out, to produce a system with 0 A Charge. Also all the types of A Charge are equivalent. In this universe relativity Special Relativity, and General Relativity apply. Also the Uncertainty Principle applies.

Could A Charge be self consistent?

• I am not sure I understand what you are trying to say: you state there is just one type of electric charge and then you list two, and then what is a color charge?
– L.Dutch
Jun 9, 2020 at 4:16
• Color charge is a real concept in physics and relates to quarks, which are elementary particles. I added a link to Wikipedia's page on the subject. Note that quarks have both an electric charge and a color "charge". Jun 9, 2020 at 4:23
• I think you missed out anti-red, anti-green and anti-blue... Jun 9, 2020 at 4:28
• I enjoyed this question and the HDEs answer but I don't think it should be on Worldbuilding SE. Physics SE would make more sense. I just don't see any context (book, RPG, computer game) in which it would ever be remotely important for your setting whether or not the forces and physical laws of your universe would make sense to some people in a different universe (ours) with different rules. You are the worldbuilder - it works however you say. Real life can only effect how plausible people find it, and stuff people mostly don't know (symmetry groups) won't impact what they find plausible.
– Dast
Jun 9, 2020 at 17:33
• @Dast 100% of the questions on worldbuilding can be answered by "you're the worldbuilder-- it works however you say." But that's a boring answer, so generally people like to furnish fun and interesting explanations instead-- yet for whatever reason, hard sci fi questions like this almost always attract a bunch of responses like yours. I agree that if you want to engage a wider audience, this is way too specific. But this isn't writing stack exchange, or RPG stack exchange. If a person wants to construct a world for just themselves to ponder, that seems like a perfectly valid use of this site. Jun 10, 2020 at 4:02

Yes, it could be consistent.

For each of the fundamental forces, we have a certain conserved quantity, which we refer to as a charge. The converse of a result called Noether's theorem tells us that in most cases, a conservation law leads to something called a symmetry$$^{\dagger}$$, and each symmetry is associated with a mathematical structure called a symmetry group. If we were given a charge, then by studying the symmetry group associated with it, we could learn about the quantum field theoretic interactions that arise from it, and vice versa.

The fundamental forces have the following symmetry groups, respectively:

• Electromagnetism: The very simple unitary group $$\text{U}(1)$$
• Strong nuclear force: The more complicated special unitary group $$\text{SU}(3)$$
• Weak nuclear force: The special unitary group $$\text{SU}(2)$$

We can then learn something about the charges associated with the force and the bosons mediating its interactions. The number of distinct fundamental charges is given by the dimensions of the irreducible representations of the symmetry group, and the number of bosons is given by the number of generators of the group.$$^{\ddagger}$$ $$\text{SU}(2)$$ has three generators, and so we have three gauge bosons associated with the weak nuclear force: the $$W^+$$, $$W^-$$ and $$Z$$ bosons. Its representations are two-dimensional, and there are two charges associated with the weak force.

You've simply given us a new conservation law (which at a glance looks "nice" enough for us to be able to apply the converse of Noether's theorem), and therefore a new symmetry group. There's nothing prohibiting us from considering higher-dimensional groups (and as pregunton mentioned, we can go beyond unitary and special unitary groups) that would in turn be associated with new types of charge, giving us new bosons to play around with.

$$^{\dagger}$$ Noether's theorem itself says that any continuous symmetry has a corresponding conservation law.
$$^{\ddagger}$$ In particular, the group $$\text{SU}(n)$$ has $$n^2-1$$ generators and therefore $$n^2-1$$ bosons. The group $$\text{U}(n)$$ has $$n^2$$ generators and $$n^2$$ bosons.

• Since Electromagnetism is $\text{U}(1)$ would a $\text{U}(n)$ force work, and if so how would an $\text{U}(n)$ force be different from a $\text{SU}(n)$ force? Also seeing as $1^2-1=0$ would this mean that a $\text{SU}(1)$ charge wouldn't work? Would a charge that is neither $\text{U}(n)$ nor $\text{SU}(n)$ work? Jun 9, 2020 at 8:04
• @AndersGustafson $\text{U}(n)$ is the same as $\text{SU}(n)\times\text{U}(1)$ (up to a finite factor), so the number of bosons would be $n^2$ (the $n^2-1$ from $\text{SU}(n)$ and an extra "photon"). The group $\text{SU}(1)$ is trivial, so there is no $\text{SU}(1)$-charge. There are many other groups you can work with, the complete list is roughly $\text{SU}(n), \text{SO}(n), \text{Sp}(n), U(1), G_2, F_4, E_6, E_7, E_8$ and products of these (see here). Jun 9, 2020 at 8:39
• @pregunton Where I see the dimension Column, is that what indicates the number of force carrying bosons there would be? Jun 9, 2020 at 10:11
• @AndersGustafson Yep, that's correct! The group's dimension corresponds to the number of bosons there would be. The number of charges is more difficult to determine, you have to look at the dimensions of something called irreducible representations. For example $G_2$ is itself 14-dimensional and has a 7-dimensional smallest irreducible representation, so there would be 7 fundamental charges and 14 force carriers (in this particular case charges still combine in groups of three however, due to complicated reasons). Jun 9, 2020 at 10:22
• @AndersGustafson As I said you have to look for the dimensions of irreducible representations (there should be tables online). And no, the only groups with uncharged carriers (also called Abelian groups) are essentially products $U(1)\times U(1)\times\ldots\times U(1)$. Anyways, if you are interested about all this you should learn about representation theory. There are whole books about this and it would take too long to explain everything in a comment thread. Jun 15, 2020 at 8:39