\begin{align*} \sigma_1 = \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \\ \sigma_2 = \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \\ \sigma_3 = \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \\ \end{align*}

• Sometimes the identity matrix is included as \(\sigma_0\).

• Hermitian, Involutory, Unitary.
• They, together with the identity matrix \(I = \sigma_0\), form the basis for the real vector space of \(2\times 2\) Hermitian matrices.
• Anti-Commutativity under Multiplication.
• It is the linear map (order 3 tensor) from the three dimensional vector space \(V\) to the spinor-pair space \(S\otimes S^*\):
  • \[ \sigma_i\vphantom{\sigma}^{a}\vphantom{\sigma}_b \]
  • where \(a\) is the index for the spinor space \(S\) and \(b\) is the index for the dual spinor space \(S^*\).
• It is with respect to the standard spinor basis.

• It represents the interaction between the spin of a particle with an external electromagnetic field, which is best represented in the
  • \(\sigma s = s'\)
• It also represents the observables which are the Hermitian matrices.
  • \(\langle s | \sigma | s \rangle\)
• It is equivalent to the three basis vector of the Clifford Algebra \(\mathrm{Cl}_{3,0}(\mathbb{R})\).

\[ \sigma_\mu^{a\dot{b}} \]

• Generalization of the Pauli matrices to the four dimensional spacetime.
• Using the 5.3.1.

• Conjugation rotates \(\pi\) radians around the axis:

  \begin{align*} \sigma_x &\rightsquigarrow \sigma_z\sigma_x\sigma_z^{-1} = \sigma_z\sigma_x\sigma_z = -\sigma_z\sigma_z\sigma_x = -\sigma_x \\ \sigma_y &\rightsquigarrow \sigma_z\sigma_y\sigma_z^{-1} = \sigma_z\sigma_y\sigma_z = -\sigma_z\sigma_z\sigma_y = -\sigma_y \\ \sigma_z &\rightsquigarrow \sigma_z\sigma_z\sigma_z^{-1} = \sigma_z\sigma_z\sigma_z = +\sigma_z \end{align*}
• And negative conjugation by \(\sigma_z\) reflects along the \(z\) axis.

• Two reflections is a rotation.
• Pauli vectors are reflected by the Pauli matrices by the conjugation.
• And rotated by the two pairs of unit Pauli vectors, which are the members of the \(\mathrm{SU}(2)\) group.
• Using some abstract algebra, the \(\mathrm{SU}(2)\) matrices are all the matrix the rotates a Pauli vector, (up to a complex multiple, which does not matter what that is so the determinant is just taken to be 1).

• Traceless Hermitian Matrix
• It squares to the square magnitude of the vector.
• The determinant is the negative of the square magnitude.
• The sum of pairs of sigma matrices are equivalent to the quaternion.
• Both the group of unit quaternions and \(\mathrm{SU}(2)\) are isomorphic to the \(\mathrm{Spin}(3)\) Group, describing the spin in three dimensional space, which is the double cover of rotation group \(\mathrm{SO}(3)\).

• It is rotated by a single \(\mathrm{SU}(2)\) matrix, which is half of the rotation of the Pauli vector, that does not change the length of the spinor: \[ \lVert U\xi \rVert^2 = (U\xi)^\dagger (U\xi) = \xi^\dagger U^\dagger U\xi = \xi^\dagger \xi = \lVert \xi \rVert^2. \]
• The spinors on the Poincaré sphere and the Bloch sphere is also rotated in the same way.

• Hermitian matrix
• The determinant is the spacetime interval.
• It transforms with \(\mathrm{SL}(2, \mathbb{C})\) matrices.
  • The transformation matrix is special and Hermitian
  • The boosts uses the hyperbolic functions

• \(\psi^\dagger \epsilon \phi\) is never conserved.
• So \(\psi^\top \epsilon \phi\) is chosen as the inner product with
  • \[ \epsilon = \begin{bmatrix} 0 & 1 \\ -1 & 0\end{bmatrix}. \]
• It is alternating.
• \(\epsilon\) is called a

• From the inner product

\[ \psi^\top\epsilon \]

• The complex conjugate of a left spinor \(\psi\) becomes the right dual spinor \(\psi^*\).
• The right spinor would be \(\psi^\dagger \epsilon\).

• It has left and right chirality, which is notated by \(\psi^a, \psi_a\) and \(\psi^{\dot{a}}, \psi_{\dot{a}}\).
• This happens because the \(\mathrm{SO}(2,\mathbb{C})\) has two different representation, left and right. The two are connected by complex conjugation.
  • Unlike \(\mathrm{SU}(2)\), there is no invertible matrix \(P\), such that \(L^* = P^{-1}LP\).