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Pauli Basis Matrices, Pauli Vectors and Pauli Spinors

1. The Pauli Basis Matrices denoted \(\sigma_x\), \(\sigma_y\) and \(\sigma_z\) (or \(\sigma_1\), \(\sigma_2\) and \(\sigma_3\)) are a Set of 3 Unitary Traceless, Hermitian \(2\times 2\) Matrices given as follows
   
   \(\sigma_x=\sigma_1=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\),   \(\sigma_y=\sigma_2=\begin{bmatrix}0 & -i \\ i & 0\end{bmatrix}\),   \(\sigma_z=\sigma_3=\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}\)
2. Following are Properties of Pauli Basis Matrices
   1. They are Hermitian
   2. They are Unitary
   3. They are Involutary
   4. Their Trace is 0
   5. They have Determinant Value of -1
   6. Product of Any 2 Pauli Basis Matrices is given as
      
      \(\sigma_j\sigma_k = i\hspace{1mm}\epsilon_{jkl}\hspace{1mm}\sigma_l\)
      
      This means
      
      \(\sigma_x\sigma_y=i\sigma_z\),   \(\sigma_y\sigma_z=i\sigma_x\),   \(\sigma_z\sigma_x=i\sigma_y\)
      
      \(\sigma_y\sigma_x=-i\sigma_z\),   \(\sigma_z\sigma_y=-i\sigma_x\),   \(\sigma_x\sigma_z=-i\sigma_y\)
      
      This means that the Product of Any 2 Pauli Basis Matrices is Anti-Commutative. That is
      
      \(\sigma_{i}\sigma_{j}=-\sigma_{j}\sigma_{i}\)
   7. The Commutator of Any 2 Pauli Basis Matrices is given as
      
      \([\sigma_j,\sigma_k] = \sigma_j\sigma_k-\sigma_k\sigma_j = 2i\hspace{1mm}\epsilon_{jkl}\hspace{1mm}\sigma_l\)
   8. The Anti Commutator of Any 2 Pauli Basis Matrices is given as
      
      \(\{\sigma_j,\sigma_k\} = \sigma_j\sigma_k+\sigma_k\sigma_j = 2\hspace{1mm}\delta_{jk}\hspace{1mm}I\)
   9. Product of 3 Pauli Basis Matrices is given as
      
      \(\sigma_j\sigma_k\sigma_l=i\hspace{1mm}\epsilon_{jkl}\hspace{1mm}I\)
3. Using the Pauli Basis Matrices, any 3 Dimensional Real Vector can be represented as a Traceless, Hermitian, \(2\times 2\) Complex Matrix. Any 3 Dimensional Real Vector represented as such is called a Pauli Vector. For example, the 3D Vector \(V=\begin{bmatrix}x,y,z\end{bmatrix}^T\) can be represented as a Pauli Vector \(P\) as follows
   
   \(V=\begin{bmatrix}x\\y\\z\end{bmatrix}=x\mathbf{\hat{i}}+y\mathbf{\hat{j}}+z\mathbf{\hat{k}}=x\mathbf{\sigma_x}+y\mathbf{\sigma_y}+z\mathbf{\sigma_z}= x\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix} + y\begin{bmatrix}0 & -i \\ i & 0\end{bmatrix} + z\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}= \begin{bmatrix}z & x-yi \\ x+yi & -z\end{bmatrix}=V\cdot\sigma=P\)
   
   Similarly, Any Pauli Vector \(P\) can be converted to a 3D Vector as follows
   
   \(P=\begin{bmatrix}z & x-yi \\ x+yi & -z\end{bmatrix}=\begin{bmatrix}a_{00} & a_{01} \\ a_{10} & a_{11}\end{bmatrix}=\begin{bmatrix}{\Large\frac{a_{10}+a_{01}}{2}}\\{\Large\frac{a_{10}-a_{01}}{2i}}\\a_{00}\end{bmatrix}=\begin{bmatrix}x\\y\\z\end{bmatrix}=V\)
4. Any Property/Operation of/on a 3D Vector has a corresponding analog for it's Pauli Vector representation as follows
   1. The Square of a Pauli Vector is the Identity Matrix Scaled by the Square of the Magnitude of the Corresponding 3D Vector. For example, if \(|V|\) is the Magnitude of 3D Vector \(V\), then the Square of a Corresponding Pauli Vector \(P\), \(P^2=|V|^2I\).
   2. The Determinant of a Pauli Vector is the Negative of Square of the Magnitude of the Corresponding 3D Vector. For example, if \(|V|\) is the Magnitude of 3D Vector \(V\), then the Determinant of Corresponding Pauli Vector \(P\), \(|P|=-|V|^2\).
   3. The Anti-Commutator of any 2 Pauli Vectors is the Identity Matrix Scaled by Twice the Value of Dot Product of the Corresponding 3D Vectors. For example, if \(V_1\cdot V_2\) is the Dot Product of 2 3D Vectors \(V_1\) and \(V_2\), then the Anti-Commutator of their Corresponding Pauli Vectors \(P_1\) and \(P_2\), \(\{P_1,P_2\} = P_1P_2+P_2P_1 = 2(V_1\cdot V_2)I\).
   4. The Commutator of any 2 Pauli Vectors is the Pauli Vector Corresponding to Cross Product of the Corresponding 3D Vectors Scaled by \(2i\). For example, if \(V_1\times V_2\) is the Cross Product of 2 3D Vectors \(V_1\) and \(V_2\), then the Commutator of their Corresponding Pauli Vectors \(P_1\) and \(P_2\), \([P_1,P_2] = P_1P_2-P_2P_1 = 2iP_R\), where \(P_R\) is the Pauli Vector Corresponding to the Vector \(V_1\times V_2\).
5. The Product of 2 Pauli Vectors corresponding to 2 Parallel 3D Vectors is Commutative. For example if \(P_1\) and \(P_2\) are 2 Pauli Vectors corresponding to 2 3D Parallel Vectors \(V_1\) and \(V_2\), then \(P_1P_2 = P_2P_1\), that is, \([P_1,P_2] = P_1P_2-P_2P_1 = 0\).
6. The Product of 2 Pauli Vectors corresponding to 2 Perpendicular 3D Vectors is Anti-Commutative. For example if \(P_1\) and \(P_2\) are 2 Pauli Vectors corresponding to 2 Perpendicular 3D Vectors \(V_1\) and \(V_2\), then \(P_1P_2 = -P_2P_1\).
7. Any Pauli Vector \(P\) can be rotated by using \(SU(2)\) Rotation Matrix \(R\) as follows
   
   \(P_R = RPR^\dagger = (-R)P(-R)^\dagger\)
   
   where
   
   \(P =\) Given Pauli Vector
   
   \(P_R =\) Rotated Pauli Vector
   
   \(R,-R =\) \(SU(2)\) Rotation Matrices
   
   \(R^\dagger, -R^\dagger =\) Conjugate Transpose of Matrices \(R\) and \(-R\) respectively
   
   \(SU(2)\) Rotation Matrix \(R\) for an Axis of Rotation Vector \(A\) and Rotation Angle \(\theta\) Radians is calculated as follows
   
   \(R=\cos({\Large\frac{\theta}{2}})I - i\sin({\Large\frac{\theta}{2}}) (A\cdot\sigma)\)
   
   \(SU(2)\) Rotation Matrix \(-R\) can be calculated either by negating the above equation or as follows
   
   \(-R=\cos({\Large\frac{2\pi-\theta}{2}})I - i\sin({\Large\frac{2\pi-\theta}{2}}) (-A\cdot\sigma)\)
   
   Please note that Both \(SU(2)\) Rotation Matrices \(R\) and \(-R\) can be resolved into Counter Clockwise Rotation Angle \(\theta\) with respect to Axis \(A\) and Counter Clockwise Rotation Angle \(2\pi-\theta\) with respect to Axis \(-A\).
8. Any Pauli Vector \(P\) can be Reflected Across a Plane whose Unit Normal is given by a Pauli Vector \(A\) as follows
   
   \(P_{Ref}=-APA\)
   
   where
   
   \(P =\) Given Pauli Vector
   
   \(P_{Ref} =\) Reflected Pauli Vector
   
   \(A =\) Pauli Vector Corresponding to Unit Vector Normal to a Plane
9. Any Pauli Vector Matrix \(P\) having Magnitude \(M\) has Eigen Values \(\pm M\). The Unit Eigen Vectors corresponding to Eigen Values \(\pm M\) when Scaled by the their respective Eigen Values give rise to 2D Complex Vectors known as Pauli Spinors.
10. Any Pauli Spinor \(S\) can be converted to it's corresponding 3D Vector \(V\) as follows
    
    \(S=\begin{bmatrix}\psi_1 \\ \psi_2\end{bmatrix}=\begin{bmatrix}a+bi \\ c+di\end{bmatrix}\)  \(S^\dagger=\begin{bmatrix}\psi_1^* & \psi_2^*\end{bmatrix}=\begin{bmatrix}a-bi & c-di\end{bmatrix}\)
    
    Vector Via Direct Calculation \(V = \begin{bmatrix}\psi_1^*\psi_2 + \psi_1\psi_2^* \\ -i (\psi_1^*\psi_2 - \psi_1\psi_2^*) \\ \psi_1\psi_1^* - \psi_2\psi_2^*\end{bmatrix} =\begin{bmatrix}2Re(\psi_1^*\psi_2) \\ 2Im(\psi_1^*\psi_2) \\ {\psi_1}^2 - {\psi_2}^2\end{bmatrix}=\begin{bmatrix}2 (ac+bd) \\ 2 (ad-bc) \\ (a^2+b^2) - (c^2+d^2)\end{bmatrix}\)
    
    Vector Via Pauli Basis Matrix Multiplication \(V =\begin{bmatrix}S^\dagger \sigma_1 S \\ S^\dagger \sigma_2 S \\ S^\dagger \sigma_3 S\end{bmatrix}\)