Special Unitary Groups In Physics
Date: 11/09/98 at 21:20:19 From: David Winsemius Subject: Group theory in physics As a student, I left physics before I could see the applications of abstract algebra. Before I left physics, I did pick up a bit of group theory, but not enough to know what SU(2) and SU(3) groups are. These abstract objects are central to modern descriptions of quantum chromodynamics, so I was wondering if you could perhaps describe these entities, and what areas of study I should pursue? Thank you, David Winsemius, MD
Date: 11/09/98 at 22:32:03 From: Doctor Tom Subject: Re: Group theory in physics Hi David, SU(2) and SU(3) are the special unitary groups of dimensions 2 and 3. SU(2) is the set of all 2x2 matrices that are unitary and have determinant 1. SU(3) is the same thing, but with 3x3 matrices. A unitary matrix is a matrix whose conjugate transpose is its own inverse. The transpose of a matrix is gotten by flipping everything across the main diagonal. That is, you make the rows into the columns and the columns into the rows. So for example, the transpose of: [ a b c ] T [ a d g ] A = [ d e f ] is A = [ b e h ] [ g h i ] [ c f i ] The conjugate is gotten by replacing every element by its complex conjugate. This set of matrices forms a group under normal matrix multiplication. For example, the matrix: [ 0 i ] [ i 0 ] is unitary, since its transpose is itself, and the complex conjugate of the transpose is: [ 0 -i ] [ -i 0 ] and when you multiply that matrix by the one above, you get the identity. Unitary matrices are like (and include) the standard rotation matrices of real numbers. For example, the following real matrix is also unitary: [ cos(x) -sin(x) ] [ sin(x) cos(x) ] The transpose is: [ cos(x) sin(x) ] [ -sin(x) cos(x) ] and the conjugate leaves it unchanged, since all the entries are real. It's easy to check that the product of these two matrices is also the identity. In 3 dimensions, of course, it's much more complicated. - Doctor Tom, The Math Forum http://mathforum.org/dr.math/
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