Math 401, Fall 2025: Thesis notes, S2, Majorana stellar representation of quantum states

Majorana stellar representation of quantum states

Vectors in $\mathbb{C}^{n+1}$ can be represented by a set of $n$ degree polynomials.

If $Z_0\neq 0$, then we can rescale the polynomial to make $Z_0=1$.

Therefore, points in $\mathbb{C}P^{n}$ will be one-to-one corresponding to the set of $n$ degree polynomials with $n$ complex roots.

If $Z_0=0$, then count $\infty$ as root.

Using stereographic projection of each root we can get a unordered collection of $S^2$. Example: $\mathbb{C}P=S^2$, $\mathbb{C}p^2=S^2\times S^2\setminus S_2$ where $S_2$ is symmetric group.