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• {$SU(2)$} is the Lie group of {$2\times 2$} unitary matrices with determinant {$1$}.
  • An invertible complex square matrix {$U$} is unitary if its conjugate transpose {$U^*$} is also its inverse: {$U^*U = UU^* = I$}
    • The columns of {$U$} form an orthonormal basis of {$\mathbb {C} ^{n}$} with respect to the usual inner product. In other words, {$U^{*}U=I$}.
    • The rows of {$U$} form an orthonormal basis of {$\mathbb {C} ^{n}$} with respect to the usual inner product. In other words, {$ UU^{*}=I$}.
  • For real numbers, the analogue of a unitary matrix is an orthogonal matrix.
  • Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.
  • {$U$} has a decomposition of the form {$U = V D V^∗$}, where {$V$} is unitary, and {$D$} is diagonal and unitary.
  • {$U$} can be written as {$U = e^{iH}$} where {$H$} is a Hermitian matrix (a self-adjoint matrix, a matrix equal to its own conjugate transpose, so that {$a_{ij}=\bar{a}_{ji}$})

• The three Pauli matrices and the identity matrix together form a basis for the {$2\times 2$} Hermitian matrices.
• The three Pauli matrices have trace {$0$}.
• Multiplying the Pauli matrices by {$i$} makes them anti-Hermitian and a basis for the Lie algebra {$\frak{su}(2)$}.

• {$SU(2)$} is isomorphic to the unit quaternions (the versors)(the group of quaternions of norm 1).
• {$SU(2)$} is diffeomorphic to the 3-sphere.
• {$SU(2)$} is the spin group {$\textrm{Spin}(3)$}, which is the double cover of {$SO(3)$}.
  • There is a surjective homomorphism from {$SU(2)$} to the rotation group {$SO(3)$} whose kernel is {$\{+I, −I\}$}. This is because unit quaternions can be used to represent rotations in 3-dimensional space (up to sign).
  • The Lie algebra for the group {$SU(2)$} is isomorphic to the Lie algebra of the group of three dimensional rotations {$SO(3)$}.
  • {$SO(3)$} is {$\mathbb{R}P^3$}
  • The evolution of the qubit state is describable by rotations of the Bloch sphere.
  • Rotations of the Bloch sphere about the x axis are quaternion rotations, about the y axis are real rotations and about the z axis are complex rotations.

• {$SU(2)$} is also identical to one of the symmetry groups of spinors, {$\textrm{Spin}(3)$}, that enables a spinor presentation of rotations.
• {$SU(2)$} is important in quantum computing because it represents the possible quantum logic gate operations in a quantum circuit with 1 qubit and thus 2 basis states.
• {$M=H_Ae^{iH_B}$} A special linear matrix can be written as the product of a special unitary matrix (or special orthogonal matrix in the real case) and a positive definite hermitian matrix (or symmetric matrix in the real case) having determinant 1.
  • The topology of the group {$SL(n,\mathbb{C})$} is the product of the topology of {$SU(n)$} and the topology of the group of hermitian matrices of unit determinant with positive eigenvalues.
• The Möbius group is the projective linear group {$PGL(2, \mathbb{C})=PSL(2, \mathbb{C})=SL(2, \mathbb{C})/\mathbf{Z}(2,\mathbb{C})=SL(2, \mathbb{C})/\{I,-I\}$}.

Polar decomposition

• Intuitively, if a real {$n\times n$} matrix {$A$} is interpreted as a linear transformation of {$n$}-dimensional space {$\mathbb {R} ^{n}$}, the polar decomposition separates it into a rotation or reflection {$U$} of {$\mathbb {R} ^{n}$}, and a scaling of the space along a set of {$n$} orthogonal axes.
• Thus there is a division into a changing of axes (by way of rotation or reflection) and respecting the axes (simply rescaling them).

读物

维基百科

• https://en.wikipedia.org/wiki/Special_unitary_group
• https://en.wikipedia.org/wiki/Unitary_matrix
• https://en.wikipedia.org/wiki/Spin_group
• https://en.wikipedia.org/wiki/Hermitian_matrix
• https://en.wikipedia.org/wiki/Representation_theory_of_SU(2)
• https://en.wikipedia.org/wiki/Bloch_sphere
• https://en.wikipedia.org/wiki/M%C3%B6bius_transformation
• https://en.wikipedia.org/wiki/Polar_decomposition

书

• Feynman lectures Volume III

• Open and closed system. Three-cycle is a closed system. X-Y-H is an open system.

一般线性群 {$GL(2,\mathbb{C})$}

• {$GL(2,\mathbb{C})$} consists of the invertible {$2\times 2$} matrices of complex numbers. The condition for these matrices is that the determinant is not equal to {$0$}.
• The noninvertible matrices - the matrices for which the determinant is equal to {$0$} - is a three (complex) dimensional space.

{$\textrm{det}\begin{pmatrix} z_{11} & z_{12} \\ z_{21} & z_{22} \end{pmatrix} = 0$}, thus {$z_{21}=z_{22}\frac{z_{11}}{z_{12}}$} where {$z_{12}\neq 0$}. We can have any {$z_{11},z_{12},z_{22}$} where {$z_{12}\neq 0$}.

• So this means that the invertible matrices {$GL(2,\mathbb{C})$} is a four (complex) dimensional space. For subtracting a three (complex) dimensional space leaves a four (complex) dimensional space.
• 2x2 complex numbers is 8 real parameters. Of these, 1 parameter (the nullsome) is determined by normalization, and 1 parameter (slack) is the phase. Thus 8 = 1 + 6 + 1.
• {$GL(2,\mathbb{C})$} has 7 =8-1 dimensions thus it may model structures that have a sevensome-eightsome nature.

Pauli matrices

• Pauli matrices. Wikipedia. Minkowski space. Time is no rotation. Time is scalar times identity. Space is rotations.

{$e^{\theta u_1}=\begin{pmatrix} \cos\theta & i\sin\theta \\ i\sin\theta & \cos\theta \end{pmatrix} \;\;\; e^{\theta u_2}=\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix} \;\;\; e^{\theta u_3}=\begin{pmatrix} e^{i\theta} & 0 \\ 0 & e^{-i\theta} \end{pmatrix}$}

https://math.stackexchange.com/questions/1354627/why-is-it-so-that-a-unit-quaternion-t-can-be-written-as-t-cos-thetau-sin The quaterions have a subgroup of rotations related to any unit vector u, where that u plays the role of i, thus this divides up cos theta and i sin theta, and grounds the related 2x2 Pauli matrix.

https://math.stackexchange.com/questions/302465/half-sine-and-half-cosine-quaternions The quaternion represents a directed area, and you're rotating by that area.

{$𝑥⋅𝑦=\frac{1}{2}(𝑥𝑦+𝑦𝑥)$}

{$𝑥×𝑦=\frac{1}{2}(𝑥𝑦−𝑦𝑥)$}

{$𝑥𝑦=𝑥⋅𝑦+𝑥×𝑦$}

• real - spread {$s=\textrm{sin}^2\;\theta$} (projection onto y axis)
• complex - angle {$\theta$} (fraction of circumference {$2\pi$})
• quaternion - directed area {$\frac{\theta}{2}$} (fraction of area {$\pi$})

https://physics.stackexchange.com/questions/144294/what-do-the-pauli-matrices-mean

A rotation is given by i, and its opposite by j=-i. Pauli matrices give different ways of writing i as a 2x2 matrix. What other roles does i play with regard to rotations?

https://www.nature.com/articles/s41534-018-0106-y#:~:text=Surface%20codes%20are%20building%20blocks,models%20describing%20random%20Pauli%20errors

Quaternions, Dirac equation: Pauli matrices are the three-cycle for learning and they are extended by a fourth dimension of non-learning (what is absolutely true or false) for the foursome.

笔记

• SU(2) precursor for Lorentz group
• The sum x_1 + x_2 + x_3 + x_4 = 0 is the condition placed on the roots of su(n) that the trace is zero. And this is precisely the condition that lets us use quadrays.
• Double covering nature of SO(3) and SU(2) is the basis for the nature of spin. (alpha, beta) and (i alpha, i beta) have the same squares so give the same probability which yields the double cover.
• SU(2) is a three-sphere in four-dimensional space.