31 lines
1.3 KiB
Markdown
31 lines
1.3 KiB
Markdown
# Math 401, Fall 2025: Thesis notes, S2, Majorana stellar representation of quantum states.
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## Majorana representation of quantum states
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> [!TIP]
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>
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> A helpful resource is [Geometry of Quantum states](https://www.cambridge.org/core/books/geometry-of-quantum-states/46B62FE3F9DA6E0B4EDDAE653F61ED8C) Section 4.4 and Chapter 7.
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Vectors in $\mathbb{C}^{n+1}$ can be represented by a set of $n$ degree polynomials.
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$$
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\vec{Z}=(Z_1,\cdots,Z_n)\sim w(z)=Z_0+Z_1z+\cdots+Z_nz^n
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$$
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If $Z_0\neq 0$, then we can rescale the polynomial to make $Z_0=1$.
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Therefore, points in $\mathbb{C}P^{n}$ will be one-to-one corresponding to the set of $n$ degree polynomials with $n$ complex roots.
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$$
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Z_0+Z_1z+\cdots+Z_nz^n=0=Z_0(z-z_1)(z-z_2)\cdots(z-z_n)
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$$
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If $Z_0=0$, then count $\infty$ as root.
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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.
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> [!NOTE]
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>
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> TODO: Check more definition from different area of mathematics (algebraic geometry, complex analysis, etc.) of the Majorana representation of quantum states.
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>
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> Read Chapter 5 and 6 of [Geometry of Quantum states](https://www.cambridge.org/core/books/geometry-of-quantum-states/46B62FE3F9DA6E0B4EDDAE653F61ED8C) for more details. |