Majorana representation of higher spin states

fsh26 · February 5, 2022

In the article by E. Majorana "Oriented atoms in a variable magnetic field", in particular, it's considered (and solved) the problem of describing a state with spin J using 2J points on the Bloch sphere.

That is, if $\psi$ the general state of the spin system

$\psi=C_j\psi_j+C_{j-1}\psi_{j-1}+...+C_{-j}\psi_{-j}$ , (1)

then, according to the article, those 2J points in on the Bloch sphere are described by the following complex numbers ( $\zeta$ ):

$\zeta_j=\tg\frac{\theta_j}{2}e^{i\varphi_j}$ . (2)

Here $\theta$ - is the angle between the unit vector and the Z axis, $\varphi$ - is the angle between the projection of the given vector (on the XY plane) with the X axis (Bloch spheres), $\zeta_j$ - are the roots of the polynomial

$a_0\zeta^{2j}+a_1\zeta^{2j-1}+...+a_{2j}=0$ , (3)

where
$a_r=(-1)^r\frac{C_{j-r}}{\sqrt{(2j-r)!r!}}$ . (4)

In the case of J=1/2, is very simple

$\zeta=\tg\frac{\theta}{2}e^{i\varphi}=\frac{C_{-1/2}}{C_{1/2}}$ (5)

Here $C_{-1/2}$ ( $C_{1/2}$ ) can be considered as the probability of finding the end of the unit vector at the lower (upper) pole of the Bloch sphere, and $\zeta$ - is a point of the complex plane that is drawn through the center given sphere (stereographic projection of the end of the unit vector from the south pole onto the given plane).

But, for the cases J>1/2, I encountered difficulties in following the idea of deriving formulas (3) and (4). Of course, there are several papers where this representation of Majorana is used, but so far I have not found such a work where the derivation of formulas is discussed in detail. I will be grateful if you advise literature or sources that can help to clearly understand the derivation of formulas (3) and (4).

joigus · February 5, 2022

Here are some references:

https://chaos.if.uj.edu.pl/~karol/geometry.htm

https://arxiv.org/pdf/1601.07612.pdf

https://www.reed.edu/physics/faculty/wheeler/documents/Quantum Mechanics/Miscellaneous Essays/Angular Momentum, Spin/D2. Majorana.pdf

Notation differs a little from source to source, and approaches may differ a bit (the second one is based on coherent states).

If anything else fails, do as I do: Test it for low-dimensional cases. See that it works. Consider the general case. Make sure it makes sense. Try to build up a recurrence... Maybe take it on faith for a while until you get to a satisfactory proof.

Please, tell me if the references helped.

fsh26 · March 26, 2022

Many thanks Joigus, the literature you indicated helped to deal with the problem.

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