matrix operations


Registered Senior Member
hey all. I've got a great understanding of physics. a degree will do that for you. unfortunately, I have my reservations about the quality of my degree. there are bits and pieces that seem to be "common knowledge" among those in the field. such things as using special unitary matricies do describe the properties of particles and whatnot. I was never taught matrix operations past using determinants to solve differential equations. certainly it was never spoken of in my physics classes. what sorts of matrix operations or properties should I learn to get a better understanding of this all? what other mathamatical principles should I learn? I learned so much math but none of it was ever really tied in. all the proofs were skimmed over, and we were handed algebraic equations. BS if you ask me. I jsut hate skimming through the physical review and feeling totally lost sometimes.

so yeah. what do I need to learn to understand what "SUxSU(2)xSU(3)" means?

waht kinds of transforms and relationships should I study?

bah. I just wish they'd challenged me a litle more in school instead of pushing me through and slapping a degree on my head. I've only been out for 2 years and I swear I have learned more outside school than I ever did inside.

If you're talking about an undergrad degree, then, I've never heard of anyone learning more in school than after they graduated. It is 90% B.S. at best. If you have a master's or higher, then I don't understand how you would have gotten the degree without an understanding of matrices.

The most important types of matrices are the Hermitian and unitary matrices. First, you need to understand eigenvalues and eigenfuctions, before you appreciate the properties of these two types. Basically, Hermitian matrices represent stuff that's really there; unitary matrices are a generalization of a rotation.

I'm still new to group theory typr stuff, but the SUxSU(2)xSU(3) looks like some combination of three special unitary groups, one in 1-D, one in 2-D, and one in 3-D.
I understand...

I have a bachelor's in mathematics now and I'm still lost when I pick up certain magazines. It seems that a bachelor's is a good start...that's why I'm getting my master's degree now (and a secondary teaching license to fall back on, though I hope I won't need it - I'll still teach some even if I don't need to, though.). Anyway, don't feel bad, I'm sure you understand A LOT more about physics than you did before going to school, and that's important! I think it's great that you're not just sitting with your thumb up... well, many people do, but you're trying to learn and improve - keep it up! We need more ambitious people who take the initiative to learn in the sciences.
Specific to your question, I don't have a lot to add about matrices, they just seem to muddle things up when dealing with systems of eqn's, but they do have the valuable property that they are noncommutative, and they are efficient at handling LARGE systems of eqn's. Maybe a physicist will have a different take on it, I really don't know much about matrix mechanics.
errandir wrote, [
I'm still new to group theory typr stuff, but the SUxSU(2)xSU(3) looks like some combination of three special unitary groups, one in 1-D, one in 2-D, and one in 3-D.

Hey, it makes perfect sense to me. I think I even used to know what those groups were, though I don't remember now. We need some abstract algebra here.
thanks for the words, synergy. I'm an undergrad, and I wish I had the time for grad school. very badly. but I don;t :( . the big problem is that I never had a dedicated course in linear algebra. it was always touched on, but I never had the actual solid course. I'm trying very hard to continue my education by myself. it's certainly taking a lot longer than it would in school. I keep running up against walls, and I really need to start breaking them down. but even so, I'm learning quite a bit and I'm glad I kept it up.

that group is often used as what seems to be a "scope." people will reference the Standard Model and put ( SUxSU(2)xSU(3) ) in parenthesis after it. I think it represents teh properties of certain particles or something. if you've ever read through teh Physical Review you've prolly seen it a lot of times.

the gauge group of the standard model is SU(3)xSU(2)xU(1)

the SU(3) stands for special unitary matrices on a 3-dimensional vector space, which physically represents the symmetry between exchange of the 3 colors quarks can have. this it is sometimes called color SU(3) (to distinguish it from isospin SU(3))

the SU(2) stands for the symmetries present in the weak force, the quantum number being exchanged is called weak isospin.

the U(1) stands for hypercharge, another quantum number that particles can have.

a U(1) symmetry is the kind of symmetry that electromagnetism has, i.e. the electromagnetic field remains unchanged if you at a total derivative to the vector potential. the other gauge groups are similar, except they are nonabelian (i.e. given two matrices A and B, AB is not equal to BA unless they are 1x1 matrices, as in the case of U(1))

these are matrices of complex numbers. you can take the complex comjugate of each number in a matrix, and then exchange the rows with the columns. this new matrix is called the hermitian adjoint of the original. if a matrix is equal to its hermitian adjoint, then it is called a hermitian matrix. if the matrix is equal to the inverse of its hermitian adjoint, then it is a unitary matrix. if, in addition, the determinant of the matrix is 1, then it is called unimodular. the 'S' in SU(N) means unimodularity.