SU(2), SU(3) and other symmetries

[shounakbhatta]

Hello,

I am trying to understand the concept of symmetries, SU(2), SU(3), unitary group, orthogonal group SO(1)...so on.

I don't know from where to start and what would be the first group to study and then move on step by step into the other.

Also, I need to have a basic (theoretical) understanding, what this mean and the significances in modern physics.

Can anybody please guide me?

Thanks

 

[ChrisVer]

In general the same ways are followed in every book which deals with theoretical particle physics or maybe group theory books. it goes from U(1) up to SU(3) [maybe SU(4) or SU(5) for BSM- although they almost are the same each step higher you get]. The SU(2) in general is studied in parallel with SO(3) or vice versa. I think the "easiest" is SU(2) and SO(3) because everyone has some basic understanding on spin physics or angular momenta.

As for their significance, it wouldn't be much to say that they are very important... from the fact that they are the Standard Model, as well as they are used in GUTs or sugra/strings etc... Some of them can also be applied in quantum physics or even classical mechanics. I don't understand the question "what this mean"

I'd recommend Georgi's book- Lie Algebras in Particle Physics. If you find you are not so accustomed with the context, you should try at first some introduction to group theory and Lie algebras.

PS. This could go to group theory or high energy physics?

 

[haushofer]

You might try Zee's treatment of group theory first. It's short and very instructive. Also, I liked 't Hooft his notes (Lie groups in physics).

 

[arivero]

I try to remember when I first did this same question, more than 20 years ago.

A first hint: uppercase SU(2) is group, lowercase su(2) is algebra.

All of them are matrix groups, so this is the first prerequiste: understand matrix multiplication.

 

[microsansfil]

All of them are matrix groups, so this is the first prerequiste: understand matrix multiplication.

Isn't it a concret representation of abstract algebraic structures ?

: http://en.wikipedia.org/wiki/Representation_theory

In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication

Patrcik

 

[haushofer]

No. The elements of matrix groups are matrices per definition, the elements form the fundamental representation.

 

[microsansfil]

No. The elements of matrix groups are matrices per definition, the elements form the fundamental representation.

Physics use the concept of "Irreducible representation". The Group elements can be represented by matrices. A representation of a group is a mapping from the group elements to the general linear group of matrices.

In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication.

More generally still, the general linear group of a vector space GL(V) is the abstract automorphism group, not necessarily written as matrices.

Example : Representation theory of SU(2)

Patrick

 

[VRT]

I think by "what this mean" OP asked about the link between math and physics. Why particles can be described by those math groups? Why "there are 8 but not 9 gluons" followed from "there are 8 generators of SU(3)"? Why math rules (8 generators) took over physics logic (3 gluons and 3 anti-gluons make 9 pairs)? Etc...

 

[ohwilleke]

Most of the jargon involved (e.g. "groups") is first taught to mathematicians in a class typically called "Abstract Algebra" which is essentially the study of mathematical operators that don't follow the same rules of analogous addition, subtraction, multiplication, division, exponent, and parentheses operators in ordinary algebra with real numbered variables. For example, b*a and a*b might have different values.

Algebra using matrixes obeys some of these modified rules of algebra and is usually taught to mathematicians in a class called "Linear Algebra". Moduli mathematics are also relevant and typically are first introduced in the concrete case of trigonometry since any angle +2*pi*N for any integer value of N has the same value as the original angle. Moduli mathematics are also important in the mathematics of imaginary numbers which are often first addressed really rigorously in a course called "complex analysis."

Another challenge is that almost all math with physics applications has both an algebric form and a geometric form. Math classes tend to emphasize the algebric form, but understanding the physics typically requires a good grasp of the concepts from a geometric perspective. My background is in mathematics, for example, and I understood how to calculate the cross product and dot product of two vectors for years before I learned their geometrical meaning in a physics class.

 

[arivero]

I think by "what this mean" OP asked about the link between math and physics. Why particles can be described by those math groups? Why "there are 8 but not 9 gluons" followed from "there are 8 generators of SU(3)"? Why math rules (8 generators) took over physics logic (3 gluons and 3 anti-gluons make 9 pairs)? Etc...

For this, it is better to start by understanding what U(1) "means".

 

[ChrisVer]

I'd like to see a geometrical meaning of U(1), U(2) or U(n) in general... same for SU(n)

 

[VRT]

Here're my favorite lectures on the subject - "Group Theory, Robert de Mello Koch" - very intense. I wish quality of the video would be better, though...