Rotation group representation and pauli matrices

[01030312]

Kindly ignore if some +- signs are placed wrongly in the equations. Thank you.
Rotation in three dimensions can be represented using pauli matrices $\sigma^{i}$, by writing coordinates as
$X= x_{i}\sigma^{i}$, and applying the transform $X'= AXA^{-1}$. Here $A= I + n_{i}\sigma^{i}d\theta/2$.

The pauli matrices are closely related to two-dimensional representations of $SO(3)$ and $SU(2)$ groups (SO(3) and SU(2) groups are isomorphic). And they are used in representing rotation. Necessary, and sufficient, condition for the rotation to be represented is that the matrices satisfy- $\sigma^{i}\sigma^{j}+\sigma^{j}\sigma^{i}= 2\delta^{ij}$ and $\sigma^{i}\sigma^{j} - \sigma^{j}\sigma^{i}= 2i\epsilon^{ijk}\sigma^{k}$.
But it can be easily verified that only two dimensional representations of SO(3) satisfy this property. My question is if there is a more fundamental reason for this, rather than just a mathematical coincidence. Or is there a more general result which relates all representations of SO(3) to rotation in three dimensions (like the one two-dimensional representations have, that is more than just having the same commutation relation for the generator). Also can the three dimensional representation of SO3 be written in terms of usual rotation matrices (Like $X'= RX$) ?

 

[syberraith]

I was just reading about this and one of the exercises was to show $\sigma$_i$\sigma$_j + $\sigma$_j$\sigma$_i = 0 for the the three sigmas.

 

[Fredrik]

I don't know the answer to your main question, so I'll just point out that SO(3) is isomorphic to SU(2)/Z₂, not to SU(2). Also, your last question has a trivial answer: The identity map on SO(3), defined by I(R)=R for all R in SO(3), satisfies the definition of a representation.

 

[01030312]

syberraith-
That relation is true when i and j are not equal. But we can prove it once we know how pauli matrices look like. Otherwise, in a different representation, the look different. Actually pauli matrices are defined by that relation, which helps us to get its form.

Fredrik-
I guess I did not state it clearly. Given the three dimensional representation of SO(3), that is written in the basis - |1,1>,|1,0>,|1,-1> (which is obtained by the usual way to find the eigenvectors form the relation $[L_{i},L_{j}]= i\epsilon_{ijk}L_{k}$), can we change the basis to obtain the usual rotation matrix written in coordinate basis? Or are three dimensional representations of SO(3) and rotation matrices acting on vectors $[x,y,z]^{T}$ same? I guess they are since photon is a spin one
particle and its polarization is a vector (this might be a reverse-argument).

And thank you for the correction.

 

[dextercioby]

[...] Given the three dimensional representation of SO(3), that is written in the basis - |1,1>,|1,0>,|1,-1> (which is obtained by the usual way to find the eigenvectors form the relation $[L_{i},L_{j}]= i\epsilon_{ijk}L_{k}$), can we change the basis to obtain the usual rotation matrix written in coordinate basis? Or are three dimensional representations of SO(3) and rotation matrices acting on vectors $[x,y,z]^{T}$ same? I guess they are since photon is a spin one
particle and its polarization is a vector (this might be a reverse-argument).[...]

Yes to the first question, yes to the second. Please, note that the abstract bra/ket formulation can be <realized> by choosing a function space to represent the abstract linear space in which you build the representations. Specifically, the Wigner functions (matrix elements of representation operators in case of SU(2)) could use either of the known paramentrizations of rotations: complex numbers (Cayley-Klein parameters), axis-angle, or Euler angles. Each time you use such a parametrization, you are turning abstract things like kets into real things like special functions (Gegenbauer or Jacobi polynomials whose particular cases are spherical harmonics) which are then easy to work with.

 

[dextercioby]

[...]
But it can be easily verified that only two dimensional representations of SO(3) satisfy this property. My question is if there is a more fundamental reason for this, rather than just a mathematical coincidence. Or is there a more general result which relates all representations of SO(3) to rotation in three dimensions (like the one two-dimensional representations have, that is more than just having the same commutation relation for the generator). [...] ?

Well, the fundamental result of representation theory of su(2) is that its lowest dimensional irreducible representation is 2-dimensional, so rotation/angular momentum operators are represented through 2x2 matrices obeying the fundamental commutation relations. The Pauli matrices are then the only inequivalent representation of ang. mom. operators in a basis made up of 2 linearly independent vectors. They are unique up to a similarity transformation, just as it happens for the Dirac matrices as well.

As for <is there a more general result which relates all representations of SO(3) to rotation in three dimensions>, well SO(3) enters the picture through the symmetry of Euclidean/Newtonian/Galilei space-time and would normally lead only to the existence of orbital angular momentum. So rotations in 3 dimensions and the rotational symmetry of space-time should be linked 1-1 to representations of SO(3). But we know that this doesn't happen. The group is actually SU(2) and there's more to angular momentum than just the orbital one.

 

[01030312]

dextercioby-
Thanks a lot. But two final questions. As Fredrik said, SO(3) is SU(2)/Z2. Is it because SO(3) demands rotation by 360 degrees to be an identity operation and SU(2) allows "fermionic property"?

Well, the fundamental result of representation theory of su(2) is that its lowest dimensional irreducible representation is 2-dimensional, so rotation/angular momentum operators are represented through 2x2 matrices obeying the fundamental commutation relations. The Pauli matrices are then the only inequivalent representation of ang. mom. operators in a basis made up of 2 linearly independent vectors. They are unique up to a similarity transformation, just as it happens for the Dirac matrices as well.

Can higher dimensional representations of SU(2) be used to represent rotation?

 

[dextercioby]

These are indeed deep results in representation theory. Yes, SO(3) is double-connected and its topology becomes important when switching from a classical description to a quantum one. Actually, to make an addendum to the 2 posts above, SO(3) means classical physics (rotational symmetry of rigid bodies for example; I remember having seen a chapter in an edition of Goldstein), SU(2) means quantum physics. SU(2) is not a starting point in the quantum theory, rather an ending point. Galilei symmetry includes SO(3). Putting it into a QM formalism of rays will lead you to a group containing SU(2) and no longer SO(3).

As for the <fermionic property>, as far as I know, this comes from relativity, not from purely quantum mechanics.

As for higher representations, well, yes, essentially the dynamical symmetry (Galilei or Poincare, both having as particular symmetries the rotational one) leads to the theoretical possibility of higher than 1/2 values of spin: So 0 dimensions (trivial representation) is a scalar, 1/2 is the basic spinor, 1 is a vector, etc.