2. Pauli Matrices & Spinors
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Mathematical framework for spin-1/2: Pauli matrices form the basis of spinor algebra.
Definition
$$\sigma_1 = \sigma_x = \left(\begin{array}{cc}0&1\\1&0\end{array}\right)$$
$$\sigma_2 = \sigma_y = \left(\begin{array}{cc}0&-i\\i&0\end{array}\right)$$
$$\sigma_3 = \sigma_z = \left(\begin{array}{cc}1&0\\0&-1\end{array}\right)$$
Fundamental Properties
Hermitian: $\sigma_i^\dagger = \sigma_i$
Traceless: $\text{Tr}(\sigma_i) = 0$
Square to identity:
$$\sigma_i^2 = I \quad \text{for } i = 1,2,3$$
Determinant: $\det(\sigma_i) = -1$
Commutation Relations
$$[\sigma_i, \sigma_j] = 2i\epsilon_{ijk}\sigma_k$$
Explicitly:
$$[\sigma_x, \sigma_y] = 2i\sigma_z, \quad [\sigma_y, \sigma_z] = 2i\sigma_x, \quad [\sigma_z, \sigma_x] = 2i\sigma_y$$
Anticommutation Relations
$$\{\sigma_i, \sigma_j\} = \sigma_i\sigma_j + \sigma_j\sigma_i = 2\delta_{ij}I$$
Distinct Pauli matrices anticommute
Products of Pauli Matrices
$$\sigma_x\sigma_y = i\sigma_z, \quad \sigma_y\sigma_z = i\sigma_x, \quad \sigma_z\sigma_x = i\sigma_y$$
$$\sigma_y\sigma_x = -i\sigma_z, \quad \sigma_z\sigma_y = -i\sigma_x, \quad \sigma_x\sigma_z = -i\sigma_y$$
Pauli Vector
$$\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$$
Useful identity:
$$(\vec{\sigma}\cdot\vec{a})(\vec{\sigma}\cdot\vec{b}) = \vec{a}\cdot\vec{b}\,I + i\vec{\sigma}\cdot(\vec{a}\times\vec{b})$$
Exponentials of Pauli Matrices
For unit vector $\hat{n}$ and angle $\theta$:
$$e^{i\theta\vec{\sigma}\cdot\hat{n}/2} = \cos(\theta/2)I + i\sin(\theta/2)\vec{\sigma}\cdot\hat{n}$$
Rotation operator for spinors
Spinor Representation
Spinor: two-component complex column vector
$$\chi = \left(\begin{array}{c}\alpha\\\beta\end{array}\right), \quad |\alpha|^2 + |\beta|^2 = 1$$
Pauli matrices act on spinors:
$$\sigma_x\left(\begin{array}{c}\alpha\\\beta\end{array}\right) = \left(\begin{array}{c}\beta\\\alpha\end{array}\right), \quad \sigma_y\left(\begin{array}{c}\alpha\\\beta\end{array}\right) = \left(\begin{array}{c}-i\beta\\i\alpha\end{array}\right), \quad \sigma_z\left(\begin{array}{c}\alpha\\\beta\end{array}\right) = \left(\begin{array}{c}\alpha\\-\beta\end{array}\right)$$
Eigenvalues and Eigenvectors
All Pauli matrices have eigenvalues $\pm 1$
$\sigma_z$:
$$|+\rangle = \left(\begin{array}{c}1\\0\end{array}\right), \quad |-\rangle = \left(\begin{array}{c}0\\1\end{array}\right)$$
$\sigma_x$:
$$|+\rangle_x = \frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\1\end{array}\right), \quad |-\rangle_x = \frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\-1\end{array}\right)$$
Completeness
Any 2×2 matrix can be expanded in $\{I, \sigma_x, \sigma_y, \sigma_z\}$:
$$M = a_0 I + a_1\sigma_x + a_2\sigma_y + a_3\sigma_z = a_0 I + \vec{a}\cdot\vec{\sigma}$$
Applications
- Quantum computing: Single-qubit gates (X, Y, Z gates)
- NMR: Spin dynamics
- Condensed matter: Spin models, topological insulators
- Particle physics: Dirac equation, weak interactions