Representations of Symmetry Operators

[stone]

For spin 1/2 particles, I know how to write the representations of the symmetry operators
for instance $$T=i\sigma^{y}K$$ (time reversal operator)
$$C_{3}=exp(i(\pi/3)\sigma^{z})$$ (three fold rotation symmetry) etc.

My question is how do we generalize this to, let's say, a basis of four component spinor with spins localized on two sites $$a$$ and $$b$$
$$(|a, up>, |a, down>, |b, up>, |b, down>)^{T}$$

Is it a direct product $$i\sigma^{y}K \otimes i\sigma^{y}K$$
Or $$i\sigma^{y}K \otimes I_{2 \times 2}$$

Or is it something else?
It would be wonderful if you could point to any references.

 

[Bill_K]

stone, No generalization is necessary. Using tensor products would be appropriate if you were talking about a system with several particles, but my understanding is that you have just one.

The Pauli matrices you've written, σ_y and so forth, act on the particle's spin coordinate. They do not affect |a> and |b>. Furthermore the antilinear operator K may be defined so as to also leave the basis states |a> and |b> unchanged, and therefore its only effect will be to complex conjugate the coefficients. That is, if you have a state |ψ> = α|a> + β|b>, then K|ψ> = α*|a> + β*|b>.

 

[stone]

Thanks for the reply.
Yes the number of particles is still one, but the basis is now 4x4 instead of the usual 2x2, then we need to represent the symmetry operators in terms of 4x4 matrices.
I am still not sure how to go about doing this.

 

[stone]

Some more help would be appreciated.

 

[Bill_K]

Ok, I yield! If you want to represent your four states as a tensor product of two 2-spaces, S ⊗ T say, then an operator R that acts only on the spin part will be of the form R ⊗ I.

I have a couple of reasons for resisting this, one is (IMHO) it's a rather cumbersome way of stating a simple fact, namely that the rotation operator acts on just the spin states. For a more general example, in which instead of |a> and |b> you had states |l m> say, which were also affected by rotations, you'd have to write the action as (S ⊗ I) ⊕ (I ⊗ L).

 

[stone]

thanks for yielding!
I understand now.