In mathematics, the spinor concept as specialised to three dimensions can be treated by means of the traditional notions of dot product and cross product. This is part of the detailed algebraic discussion of the rotation group SO(3).
Formulation The association of a spinor with a 2×2 complex traceless Hermitian matrix was formulated by Élie Cartan. In detail, given a vector x = (x1, x2, x3) of real (or complex) numbers, one can associate the complex matrix
x → → X = ( x 3 x 1 − i x 2 x 1 + i x 2 − x 3 ) . {\displaystyle {\vec {x}}\rightarrow X\ =\left({\begin{matrix}x_{3}&x_{1}-ix_{2}\\x_{1}+ix_{2}&-x_{3}\end{matrix}}\right).}
In physics, this is often written as a dot product X ≡ σ → ⋅ x → {\displaystyle X\equiv {\vec {\sigma }}\cdot {\vec {x}}} , where σ → ≡ ( σ 1 , σ 2 , σ 3 ) {\displaystyle {\vec {\sigma }}\equiv (\sigma _{1},\sigma _{2},\sigma _{3})} is the vector form of Pauli matrices. Matrices of this form have the following properties, which relate them intrinsically to the geometry of 3-space:
det X = − | x → | 2 {\displaystyle \det X=-|{\vec {x}}|^{2}} , where det {\displaystyle \det } denotes the determinant.
X 2 = | x → | 2 I {\displaystyle X^{2}=|{\vec {x}}|^{2}I} , where I is the identity matrix.
1 2 ( X Y + Y X ) = ( x → ⋅ y → ) I {\displaystyle {\frac {1}{2}}(XY+YX)=({\vec {x}}\cdot {\vec {y}})I}
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