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Math 401, Fall 2025: Thesis notes, S2, Majorana stellar representation of quantum states
Majorana stellar representation of quantum states
Tip
A helpful resource is Geometry of Quantum states Section 4.4 and Chapter 7.
Vectors in \mathbb{C}^{n+1} can be represented by a set of n degree polynomials.
\vec{Z}=(Z_1,\cdots,Z_n)\sim w(z)=Z_0+Z_1z+\cdots+Z_nz^n
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.
Z_0+Z_1z+\cdots+Z_nz^n=0=Z_0(z-z_1)(z-z_2)\cdots(z-z_n)
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.
Note
TODO: Check more definition from different area of mathematics (algebraic geometry, complex analysis, etc.) of the Majorana stellar representation of quantum states.
Read Chapter 5 and 6 of Geometry of Quantum states for more details.