- Pauli Matrices and Rotation

1. Vector Space of Traceless Hermitian Matrices

(1) Basis: ${\sigma_x, \sigma_y, \sigma_z}$

  • $\mathbf{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$
  • $\mathbf{n} = (n_x, n_y, n_z)$
  • $\mathbf{n} \cdot \mathbf{\sigma} = n_x \sigma_x + n_y \sigma_y + n_z \sigma_z \in V$
  • Inner product: $X \cdot Y = \frac{1}{2}\text{tr}(XY)$

(2) Basis: ${\frac{1}{2}\sigma_x, \frac{1}{2}\sigma_y, \frac{1}{2}\sigma_z}$

  • $\tilde{\sigma}_x = \frac{1}{2}\sigma_x$, $\tilde{\sigma}_y = \frac{1}{2}\sigma_y$, $\tilde{\sigma}_z = \frac{1}{2}\sigma_z$
  • $\mathbf{\tilde{\sigma}} = (\tilde{\sigma}_x, \tilde{\sigma}_y, \tilde{\sigma}_z)$
  • $\mathbf{n} = (n_x, n_y, n_z)$
  • $\mathbf{n} \cdot \mathbf{\tilde{\sigma}} = n_x \tilde{\sigma}_x + n_y \tilde{\sigma}_y + n_z \tilde{\sigma}_z \in V$
  • Inner product: $X \cdot Y = 2 \text{tr}(XY)$

2. Rotation in Vector Space $S$ with Basis ${|+\rangle_s, |-\rangle_s}$

Rotation in Complex Plane and Coordinate Plane

  • $e^{i\theta}$: Rotation of complex numbers in the complex plane
  • $e^{-i\theta \mathbf{n} \cdot \mathbf{\sigma}}$: For a rotation axis $\mathbf{n}$ (corresponding to the y-axis) and a measurement axis $\mathbf{n’}(S)$(corresponding to the $z$-axis) that is perpendicular to the rotation axis, rotate a vector in the eigenvector space $S$ (a vector space with basis {|+⟩_s, |-⟩_s}) by an angle $\theta$.

Example: $\mathbf{n} = (0, 1, 0) \Rightarrow \mathbf{n} \cdot \mathbf{\sigma} = \sigma_y$

$$e^{-i\theta\sigma_y} = \cos\theta - i\sin\theta\sigma_y = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$

Example: $\mathbf{n} = (0, 1, 0) \Rightarrow \mathbf{n} \cdot \mathbf{\tilde{\sigma}} = \tilde{\sigma}_y$

$$\begin{align*} e^{i\theta\tilde{\sigma}_y} &= \cos\frac{\theta}{2} + i\sin\frac{\theta}{2}\sigma_y\\ &= \begin{pmatrix} \cos\frac{\theta}{2} & \sin\frac{\theta}{2} \\ -\sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix} \end{align*}$$

Note: The rotation of vectors in vector space $S$ is given by:

$$e^{-i\theta \mathbf{n} \cdot \mathbf{\sigma}}, e^{-i\theta \mathbf{n} \cdot \mathbf{\tilde{\sigma}}}$$

3. Rotation of Traceless Hermitian Matrices in Vector Space $V$

For $X \in V$ with basis ${\tilde{\sigma}_x, \tilde{\sigma}_y, \tilde{\sigma}_z}$:

  • $U(R(\theta)) = e^{-i\theta \mathbf{n} \cdot \mathbf{\tilde{\sigma}}}$
  • $R(\theta)[X] \leftrightarrow U(R)XU^{\dagger}(R)$
  • $\mathbf{n} = (0,0,1) \Rightarrow \mathbf{n} \cdot \mathbf{\tilde{\sigma}} = \tilde{\sigma}_z$
In this case, $[X] = \left[\begin{pmatrix} z & x-iy \\ x+iy & -z \end{pmatrix}\right ] = (x,y,z)$

Example: $\mathbf{n} \cdot \mathbf{\tilde{\sigma}} = \tilde{\sigma}_z$, $U(R) = e^{-i\theta\tilde{\sigma}_z}$, $X=\sigma_x$

$$X' = U(R)XU^{\dagger}(R) = \begin{pmatrix} 0 & e^{-i\theta} \\ e^{i\theta} & 0 \end{pmatrix}$$$$= \begin{pmatrix} 0 & \cos\theta - i\sin\theta \\ \cos\theta + i\sin\theta & 0 \end{pmatrix}$$

This means: $(1,0,0) \rightarrow (\cos\theta, \sin\theta, 0)$

Important Notes:

  1. In a vector space $V$, the coordinates of a vector are represented as $\mathbf{n}$, which is an expression in terms of axes, and it should be noted that the axis of rotation is expressed in the same format.
  2. This can be understood by setting the reduced Planck constant $\hbar = 1$.
← - Properties of Pauli Matrices
- Translation Operator →