Handedness of Mobius transformations

[Assaf]

"Handedness" of Mobius transformations

Hello everyone,

I've been trying to derive the SU(2) (right-handed) rotation matrix by using a projection of the sphere and Mobius transformations, but I'm having some issues which I was hoping someone here could help me out with. My apologies for the long question, but I couldn't find any way to make it shorter. :)

What I'm trying to re-derive is the SU(2) 2x2 complex rotation matrix appearing in Sakurai's book "Modern Quantum Mechanics", on page 166 of the revised (red) edition. I unfortunately can't seem to get the Latex to work on my end, but the same matrix also appears, say, in these lecture notes (p. 10):

http://www.physics.rutgers.edu/grad/502/Lectures_Final/Lec04_SU2.pdf"

I've started by looking at the stereographic projection of points on the sphere onto the complex plane: the projection P' of P=(Px,Py,Pz) (P is on the sphere) is the intersection of the line NP with the xy-plane, where N is the sphere's north pole (N=(0,0,1)). This leads to the following coordinates for P':

P' = (Px/(1-Pz), Py/(1-Pz), 0)

Since I want to identify the xy-plane with the complex plane, I set the correspondence

P' <--> z = Px+i*Py = Pxy/(1-Pz)

where Pxy = Px+i*Py.

So far so good. Here comes the puzzling bit. To deduce the form of the 2x2 rotation matrix about z, I start by applying a z-rotation to a vector in xyz. Such a rotation is represented by the "regular" real 3x3 rotation matrices, as found here:

http://en.wikipedia.org/wiki/Rotation_matrix

(To the best of my knowledge my choice of handedness is consistent: the rotation matrices shown here, both real and complex, are right-handed. Please correct me if I'm wrong.)

So, under a z-rotation, the vector P=(Px,Py,Pz) transforms into

P ---> Prot = (Pxrot, Pyrot, Pzrot) = (Px*cos(phi) - Py*sin(phi), Px*sin(phi) + Py*cos(phi), Pz)

The corresponding complex number, obtained by stereographic projection, transforms as:

z = (Px + iPy)/(1-Pz) --> (Pxrot+i*Pyrot)/(1-Pzrot) = e^{i*phi}*z = exp(i*phi/2)*z / (exp(-i*phi/2))

So, this is a Mobius transformation of the form M(z) = (a*z+b)/(c*z+d) with ab-cd=1, a=exp(i*phi/2), d = exp(-i*phi/2), b=c=0. The corresponding matrix is then:

exp(i*phi/2) 0
0 exp(-i*phi/2)

But this is the exact opposite (in terms of handedness) of the matrices appearing in Sakurai
i.e., that's a left-handed rotation. Can the two views be brought into agreement? And if not, why?

 

[jvicens]

Thank you for your question regarding the "handedness" of Mobius transformations and how they relate to the SU(2) rotation matrix. I can understand the difficulty in trying to make sense of the seemingly contradictory results. After looking into the matter, I believe I have found the explanation for the discrepancy.

Firstly, it is important to note that the term "handedness" refers to the direction of rotation and is not inherently tied to left or right. In fact, the concept of handedness is often used in physics to distinguish between clockwise and counterclockwise rotations. With that in mind, let's take a closer look at the two different approaches you have described.

In the first approach, you are using the stereographic projection of points on a sphere onto the complex plane to derive the rotation matrix. This approach results in a left-handed rotation matrix, as you have correctly identified. However, in the second approach, you are using the "regular" real 3x3 rotation matrices, which are right-handed. This is where the discrepancy arises.

The key difference between these two approaches is the way in which they define the rotation axis. In the stereographic projection approach, the rotation axis is defined as the line connecting the point on the sphere and its projection onto the complex plane. This results in a left-handed rotation. However, in the "regular" approach, the rotation axis is defined as the direction of the rotation itself, which is right-handed.

To reconcile these two views, we need to define the rotation axis in terms of the direction of rotation, rather than the line connecting the point and its projection. This can be achieved by using the inverse stereographic projection, which maps points on the complex plane back to the sphere. In this case, the rotation axis will be defined as the line connecting the point and its inverse projection, resulting in a right-handed rotation.

To summarize, the discrepancy in handedness arises due to the different definitions of the rotation axis in the two approaches. By using the inverse stereographic projection, we can reconcile these two views and obtain the desired right-handed rotation matrix.

I hope this explanation helps to clarify the issue. If you have any further questions or concerns, please do not hesitate to reach out.