The Galois Group of Quarks: How is a Group Assigned to a Particle?

In summary, the conversation discusses the concept of assigning groups to particles in physics. Different examples of groups being assigned to shapes, polynomials, permutations, rotations, transformations, and point particles are mentioned. It is explained that groups in nature reflect symmetry and are not necessarily spacetime groups but can also be gauge groups. The SU(3)xSU(2)xU(1) group is used as an example to explain the symmetries of known particles, such as the U(1) phase change and the existence of massless fields. The conversation also touches on the GIM mechanism and the use of SU2 doublets to explain parity violation in weak interactions. Lastly, the idea of using matrix group versions of Feynman diagrams for
  • #1
Ray
25
0
Can anyone give an answer (or give a web reference) to the following question: How is a group assigned to a particle? I've seen groups assigned to shapes, polynomials, permutations, rotations and transformations. But how is a group assigned to a point particle?
 
Physics news on Phys.org
  • #2
Is your question why particles are chosen to be in certain representations of a group?

Or something else I'm not aware of?
 
  • #3
No, it's just that whenever I've seen groups in the past there has been a clear mapping between the objects in question and the group elements and operation. E.g., the vertices of an equilateral triangle and S3.
 
  • #4
PS the reference to Galois was meant to be ironic...
 
  • #5
Why do we assign groups to particles?
We don't assign groups to particles.
We assign groups in nature. Those groups reflect some symmetry there is, or some symmetry that there is not. Plus those groups are not spacetime groups, but gauge groups.
The SU(3)xSU(2)xU(1) group for example, explains the symmetries that the particles we know obey. For example the U(1) corresponds to some phase change of your field, and we know that if that's a global ohase (so you can't distinguish between points) nothing changes, while if the phase is given locally to each different spacetime point you get the existence of massless field similar to the EM field in order to keep it.
And so on...
I hope I was clear and correct.
 
Last edited:
  • #6
The galois reference confused me ;)

I agree with ChrisVer.
1) I either observe some particles, such as left handed neutrinos and massive charged leptons. I can then try and embed these in my theory. Which requires some understanding of the representation of different groups.

For example, the measurements of the ratio of hadrons to muons in e+e- suggest there are 3 colours. So, the choice of placing the quarks in a 3 of SU(3) in colour space followed.

There are lots of other things you can do, for example the approximate symmetries in QCD. The model of placing the u,d and s quarks in a 3 of SU(3) flavour space. as proposed by gell-man etc.

Sometimes knowing that you want to place particle in for example, a doublet of SU2 when you have only so far seen one of these particles (such as the strange in the strange/charm doublet) allows the prediction of undiscovered particles.
 
  • #7
Well, the reason of the strange-charm quark lies in GIM mechanism (which explained the lack of observation of flavor changing neutral currents), which in fact predicted the existence of the extra quark that was discovered later putting s and c in the same flavor SU2 doublet.
The other thing with SU2 doublets was the left-right movers which we introduced to explain the parity violation of the weak interactions. Someone before that would expect that both left/right handed movers would interact in the same way in every case. SU2 allowed us to distinguish between them two, putting the left movers in SU2 doublets and the right ones in SU2 singlets, thus under SU(2) they would be seen differently.

Why SU(3)xSU(2)xU(1)? Well this in fact, I don't know to answer. I don't think that people know... since we already know that it's just a broken subgroup in lower energies of higher symmetries (eg leptogenesis, baryogenesis, dark matter, massive neutrinos etc)
 
  • #8
Many thanks for the replies, still don't get it. Suppose we have a family of related particles which share in several properties, then anyone particle can be thought of as a state of a generic particle. Using column vectors of ones and zeros we can describe an interaction between two particular particles by giving their 'before' and 'after' reaction vectors. Interactions can then be modeled by a collection of linear transformations which would have group properties reflecting the internal symmetries. I suppose that what I'm asking for is a matrix group version of Feynman diagrams.
 
  • #9
Feynman diagrams are computational shorthand.

When you do a calculation in QCD, at each vertex you place a T-matrix, which are the generators of SU(3) color.

When you do a calculation in QED, at each vertex you place a number, and a single number is the U(1) equivalent of the T-matrix.

So they are already there.
 
  • #10

1. What is the Galois group of a quark?

The Galois group of a quark is a mathematical concept used to describe the symmetry of the internal structure of a quark. It is a subgroup of the larger group of symmetries known as the Standard Model group.

2. How is the Galois group of a quark related to its properties?

The Galois group of a quark is related to its properties through the principles of group theory. By understanding the symmetries of the Galois group, scientists can make predictions about the behavior and properties of quarks.

3. What role does the Galois group play in particle physics?

The Galois group is important in particle physics because it helps to explain the symmetries and interactions of fundamental particles, such as quarks. It also plays a crucial role in the development of mathematical models and theories in this field.

4. Can the Galois group of a quark change?

No, the Galois group of a quark is a fundamental property of the particle and does not change. However, our understanding and mathematical descriptions of the Galois group may evolve and change as we continue to learn more about quarks and their behavior.

5. Are there different Galois groups for different types of quarks?

Yes, there are different Galois groups for different types of quarks, as each type of quark has its own unique properties and symmetries. For example, there is a different Galois group for up quarks than there is for down quarks.

Similar threads

  • High Energy, Nuclear, Particle Physics
Replies
3
Views
1K
  • High Energy, Nuclear, Particle Physics
Replies
5
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
17
Views
1K
  • High Energy, Nuclear, Particle Physics
Replies
4
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
2
Views
1K
  • High Energy, Nuclear, Particle Physics
Replies
1
Views
1K
  • High Energy, Nuclear, Particle Physics
Replies
6
Views
1K
  • High Energy, Nuclear, Particle Physics
Replies
1
Views
1K
  • High Energy, Nuclear, Particle Physics
Replies
3
Views
2K
  • Linear and Abstract Algebra
Replies
3
Views
1K
Back
Top