Meaning of representations?

[sanctus]

I'm following a Quantum Mechanics course ant there we talk a lot about group therory. To understand more I already posted about lie algebras and found some help (thanks Bo), now I'm reading a book called "Group theory and quantum mechanics" by Tinkham and post again a question.

I understand that a representation of a group can be seen as a homomorphisme between the abstract group elements and, for example, linear square matrices. What I didn't understand yet is what what this representations exactly mean? for example a representation of the su(2) algebra is the cinétique moment about the z-axis, mathematically I understand why, but what's the concept behind? what's the concept in general?

 

Thanks very much already now.

[Bo]

(the following is technical; as with Sanctus'previous question on group theory: group theory is difficult and not necessary for understanding the basic, 'popular' features of quantum mechanics. However it is essential for advanced quantum mechanics, field theory, string theory etc. Any 'normal' reader should ignore group theory i think :wink:)

 

the following is only about non-relativistic quantum mechanics. I suppose simular arguments hold for field theory, but that makes it all a lot more complicated i guess.

First of all, you might know that every symmetry, has an associated conserved quantity (Noether's theorem). For example: rotation symmetry leads to the conservation of angular momentum, translation symmetry to the conservation of momentum. And it is indeed the case that the generators of a certain group, are the quantum mechanical operators of the quantity, conserved by the symmetry of the group.

 

Unlike classical mechanics, where the symmetries just act on the equations of motion, in quantum mechanics, the system is characterized by a vector (the 'state') in something called the Hilbert space. If we want this state to have certain symmetries, we can write the Hilbert space (which is a complex vector space) as a unitary representation of the symmetry group. The operators are then the generators of the group and they act (naturally) on the associated Hilbert space, giving the expectation value of the associated conserved quantity.

 

 

So for an example, the Group SU(2) (symmetry of spin) in the 2 dimensional representation, (so "spin=1/2") has a hilbert space, which just has 2 basis vectors: spin up and spin down. any state of a quantum system can be represented by a linear superposition of these 2 vectors. The associated operators are the 2 dimensional generators of SU(2): the pauli matrices.

 

Hope this helps.

Bo