Understanding Expressions like SU(2), O(3) and E(8)

[Jack]

Could some one please explain to me, in the simplest way possible and withought lots of mathematical jargon, what expressions such as SU(2), O(3), E(8) etc. mean?

I think they all have something to do with rotation (although I could be completely wrong). Thanks.

 

[Tyger]

Probably not.

These are basically mathematical concepts and they require that kind of explanation. I don't know of any math books that give a good explanation but Schiff's Quantum Mechanics, still one of the best ones around, gives about as clear a description, without talking down to you, as any I know of.

 

[chroot]

Well, a book on quantum mechanics will most likely not include a section on group theory.. lol.

These terms refer to mathematical formalisms called 'groups.' A group is a deceptively simple thing: it's a set of elements along with an operator that operates upon them. A group must be such that the result of an operation on any two elements of the set results in another element that is also a member of the set. There are, of course, more restrictions, but they're not terribly important -- you can find them for yourself.

An example of a group is addition defined over the integers. The intergers are a set of elements, and addition is an operation. The addition of any two integers is, of course, another integer.

The SU(3) and so on are groups of n x n (square) matrices. The number in parentheses is the number of dimensions, i.e., rows and columns. These groups of matrices are closed under matrix multiplication -- the multiplication of any two members of SU(3) results in another member of SU(3).

O(3) is referred to as the orthogonal group in 3-dimensions. Its members represent rotations in space about an abritrary axis. The matrix multiplication (concatenation) of any two rotation matrices is another rotation matrix.

U(3) is referred to as the unitary group in 3-dimensions. All of its members have determinant -1 or +1. The SU(3) group is referred to as the special unitary group in 3-dimensions. The modifier 'special' serves only to fix that all its members have determinant +1 (unlike U(3), whose members can also have determinant -1.)

These groups are important to physics because they provide a rigorous way to model interactions between particles.

- Warren

 

[Jack]

I still haven't got a clue what they are. That explanation was still a bit too complicated for me to understand chroot but thanks for trying.

 

[damgo]

One way to think of SO(n) is as the group of rotations in n-dimensional space. eg, the rotation of 45 degrees around the z-axis would be an element of SO(3).

O(n) contains all of SO(n) but also reflections.

Similarly SU(n) and U(n) are rotations and rotations+reflections, but in n-dimensional complex space.

Most groups don't, in fact, correspond to rotations in some space, but a number of the important physics ones do. They are very useful in dicussing symmetries, and in many other things besides...

 

[rutwig]

Originally posted by Jack
Could some one please explain to me, in the simplest way possible and withought lots of mathematical jargon, what expressions such as SU(2), O(3), E(8) etc. mean? I think they all have something to do with rotation (although I could be completely wrong). Thanks.

Any elementary book on Lie groups will give you precise definitions and properties of the ennounced groups, with the possible exception of E(8), which corresponds to the exceptional Lie group of rank 8, of great importance in HEP, specially string theory. For an eementary treatment (that is, plenty of examples and few, if any, proofs from the general representation theory), see the book by R E Gilmore, Lie groups, Lie algebras and some of their applications, Wiley 1974.

The covering SU(2)-->SO(3) is fundamental for the theory and the physical applications (see boson formalism of Schwinger), and its analysis contains implicitly the concept of Clifford algebra and spinors.

Just an addition, Lie groups are more than merely groups, they also carry the structure of differentiable (or analytic) manifold. This is the reason for their i,portance in the description of (continuous!) symmetries in physics.

 

[Parsons]

Originally posted by rutwig
Any elementary book on Lie groups will give you precise definitions and properties of the ennounced groups, with the possible exception of E(8), which corresponds to the exceptional Lie group of rank 8, of great importance in HEP, specially string theory.

1) What are Lie groups?
2) What does HEP stand for?

 

[chroot]

Originally posted by Parsons
1) What are Lie groups?
2) What does HEP stand for?

SO(3) and U(3) and so on are Lie groups.

HEP stands for high-energy physics -- i.e. particle physics.

- Warren

 

[Tyger]

I still think the Schiff book

will give a better explanation than most books on Lie algebras, it gives concrete examples and is clearly written for first year QM students. You may find something in Schuam Outlines, if so you can expect it to be easily understood and have examples. Then work with it a little with pencil and paper.