- #1

- 133

- 2

## Main Question or Discussion Point

Hi,

I had a question about irreducible representations and elementary particles....

Namely, I've been told by teachers and read in a few texts that particles ARE irreducible representations, and I have never been able to wrap my mind around what that means.

Please keep in mind that I am no theorist or mathematician, so I may be a little loose in how I use my math jargon...

Correct me if I am wrong, but I thought particles were represented by a state vector in Hilbert space, i.e. in some particular basis an electron with a certain Hamiltonian is given by the state vector psi=(c1,c2,c3,.....) where c1, c2, c3, .... are complex.

Now, I thought that the representations of groups over some vector space were actual matrices. So if I take the group SU(2), then representations of that group would be some linear combination of the pauli matrices, since they form an irreducible representation of the group (kinda like a basis for a vector space, but instead we have matrices). I know that fermions are spin 1/2, and the spin operators in each direction (essentially the pauli matrices) are basically defined by the fact that the eigenvalues must be to +1/2, or -1/2.

This is where I get confused. I never thought of the matrices (or operators) as corresponding to particles, but instead correspond to the properties of the particles you can measure, such as the spin, energy, etc., and what I thought of the particles were the the state vectors or at least represented by them, NOT the matrices.

What do the matrices represent? There are 3 irreducible representations for SU(2), but not all three correspond to a different particle, since you can only have two, lets say an up electron or a down electron..... so does each matrix correspond to a different particle? Or is it the entire group that represents one spin half particle? Please, any clarity on the subject would be great. Thanks.

Christopher Betancourt

I had a question about irreducible representations and elementary particles....

Namely, I've been told by teachers and read in a few texts that particles ARE irreducible representations, and I have never been able to wrap my mind around what that means.

Please keep in mind that I am no theorist or mathematician, so I may be a little loose in how I use my math jargon...

Correct me if I am wrong, but I thought particles were represented by a state vector in Hilbert space, i.e. in some particular basis an electron with a certain Hamiltonian is given by the state vector psi=(c1,c2,c3,.....) where c1, c2, c3, .... are complex.

Now, I thought that the representations of groups over some vector space were actual matrices. So if I take the group SU(2), then representations of that group would be some linear combination of the pauli matrices, since they form an irreducible representation of the group (kinda like a basis for a vector space, but instead we have matrices). I know that fermions are spin 1/2, and the spin operators in each direction (essentially the pauli matrices) are basically defined by the fact that the eigenvalues must be to +1/2, or -1/2.

This is where I get confused. I never thought of the matrices (or operators) as corresponding to particles, but instead correspond to the properties of the particles you can measure, such as the spin, energy, etc., and what I thought of the particles were the the state vectors or at least represented by them, NOT the matrices.

What do the matrices represent? There are 3 irreducible representations for SU(2), but not all three correspond to a different particle, since you can only have two, lets say an up electron or a down electron..... so does each matrix correspond to a different particle? Or is it the entire group that represents one spin half particle? Please, any clarity on the subject would be great. Thanks.

Christopher Betancourt