Editing Special unitary group (section)

== The group SU(2) ==
{{see also|Versor|Pauli matrices|3D rotation group#A note on Lie algebras|Representation theory of SU(2)}}

Using [[matrix multiplication]] for the binary operation, {{math|SU(2)}} forms a group,<ref>{{harvnb|Hall|2015}} Exercise 1.5</ref>

<math display="block">\operatorname{SU}(2) = \left\{ \begin{pmatrix} \alpha & -\overline{\beta} \\ \beta & \overline{\alpha} \end{pmatrix}: \ \ \alpha,\beta \in \mathbb{C}, |\alpha|^2 + |\beta|^2 = 1 \right\}~,</math>

where the overline denotes [[complex conjugate|complex conjugation]].

=== Diffeomorphism with the 3-sphere ''S''<sup>3</sup> ===
If we consider <math>\alpha,\beta</math> as a pair in <math>\mathbb{C}^2</math> where <math>\alpha=a+bi</math> and <math>\beta=c+di</math>, then the equation <math>|\alpha|^2 + |\beta|^2 = 1</math> becomes

<math display="block"> a^2 + b^2 + c^2 + d^2 = 1 </math>

This is the equation of the [[3-sphere|3-sphere S<sup>3</sup>]]. This can also be seen using an embedding: the map

<math display="block">\begin{align}
  \varphi \colon \mathbb{C}^2 \to{} &\operatorname{M}(2, \mathbb{C}) \\[5pt]
         \varphi(\alpha, \beta) ={} &\begin{pmatrix} \alpha & -\overline{\beta}\\ \beta & \overline{\alpha}\end{pmatrix},
\end{align}</math>

where <math>\operatorname{M}(2,\mathbb{C})</math> denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering <math>\mathbb{C}^2</math> [[diffeomorphism|diffeomorphic]] to <math>\mathbb{R}^4</math> and <math>\operatorname{M}(2,\mathbb{C})</math> diffeomorphic to <math>\mathbb{R}^8</math>). Hence, the [[restriction (mathematics)|restriction]] of {{math|''φ''}} to the [[3-sphere]] (since modulus is 1), denoted {{math|''S''<sup>3</sup>}}, is an embedding of the 3-sphere onto a compact submanifold of <math>\operatorname{M}(2,\mathbb{C})</math>, namely {{math|1=''φ''(''S''<sup>3</sup>) = SU(2)}}.

Therefore, as a manifold, {{math|''S''<sup>3</sup>}} is diffeomorphic to {{math|SU(2)}}, which shows that  {{math|SU(2)}} is [[simply connected space|simply connected]] and that {{math|''S''<sup>3</sup>}} can be endowed with the structure of a compact, connected [[Lie group]].

=== Isomorphism with group of versors ===
[[Quaternion]]s of norm 1 are called [[versor]]s since they generate the [[Rotation group SO(3)#Using quaternions of unit norm|rotation group SO(3)]]:
The {{math|SU(2)}} matrix:

<math display="block"> \begin{pmatrix} 
    a + bi & c + di \\ 
    -c + di &  a - bi
  \end{pmatrix} \quad
  (a, b, c, d \in \mathbb{R})
</math>

can be mapped to the quaternion

<math display="block">a\,\hat{1} + b\,\hat{i} + c\,\hat{j} + d\,\hat{k}</math>

This map is in fact a [[group isomorphism]]. Additionally, the determinant of the matrix is the squared norm of the corresponding quaternion. Clearly any matrix in {{math|SU(2)}} is of this form and, since it has determinant&nbsp;{{math|1}}, the corresponding quaternion has norm {{math|1}}. Thus {{math|SU(2)}} is isomorphic to the group of versors.<ref>{{cite web |url=http://alistairsavage.ca/mat4144/notes/MAT4144-5158-LieGroups.pdf |series=MATH 4144 notes |title=LieGroups |last=Savage |first=Alistair}}</ref>

=== Relation to spatial rotations ===
{{Main|3D rotation group#Connection between SO(3) and SU(2)|Quaternions and spatial rotation}}
Every versor is naturally associated to a spatial rotation in 3 dimensions, and the product of versors is associated to the composition of the associated rotations. Furthermore, every rotation arises from exactly two versors in this fashion. In short: there is a 2:1 surjective homomorphism from {{math|SU(2)}} to [[3D rotation group|{{math|SO(3)}}]]; consequently {{math|SO(3)}} is isomorphic to the [[quotient group]] {{math|SU(2)/{{mset|±I}}}}, the manifold underlying {{math|SO(3)}} is obtained by identifying antipodal points of the 3-sphere {{math|''S''<sup>3</sup>}}, and {{math|SU(2)}} is the [[Covering group#Universal covering group|universal cover]] of {{math|SO(3)}}.

=== Lie algebra <span class="anchor" id="Lie algebra basis"></span> ===
The [[Lie algebra]] of {{math|SU(2)}} consists of {{math|2 × 2}} [[skew Hermitian matrix|skew-Hermitian]] matrices with trace zero.<ref>{{harvnb|Hall|2015}} Proposition 3.24</ref> Explicitly, this means

<math display="block">\mathfrak{su}(2) = \left\{ \begin{pmatrix} i\ a & -\overline{z} \\ z & -i\ a \end{pmatrix}:\ a \in \mathbb{R}, z \in \mathbb{C} \right\}~.</math>

The Lie algebra is then generated by the following matrices,

<math display="block">u_1 = \begin{pmatrix}
    0 & i \\
    i & 0
  \end{pmatrix}, \quad
  u_2 = \begin{pmatrix}
    0 & -1 \\
    1 &  0
  \end{pmatrix}, \quad
  u_3 = \begin{pmatrix}
    i &  0 \\
    0 & -i
  \end{pmatrix}~,
</math>

which have the form of the general element specified above.

This can also be written as <math>\mathfrak{s u}(2)=\operatorname{span}\left\{i \sigma_{1}, i \sigma_{2}, i \sigma_{3}\right\}</math> using the [[Pauli matrices#SU(2)|Pauli matrices]].

These satisfy the [[quaternion]] relationships  <math>u_2\ u_3 = -u_3\ u_2 = u_1~,</math>  <math>u_3\ u_1 = -u_1\ u_3 = u_2~,</math>  and  <math>u_1 u_2 = -u_2\ u_1 = u_3~.</math> The [[commutator bracket]] is therefore specified by

<math display="block">\left[u_3, u_1\right] = 2\ u_2, \quad \left[u_1, u_2\right] = 2\ u_3, \quad \left[u_2, u_3\right] = 2\ u_1~.</math>

The above generators are related to the [[Pauli matrices]] by <math>u_1 = i\ \sigma_1~, \, u_2 = -i\ \sigma_2</math>  and  <math>u_3 = +i\ \sigma_3~.</math> This representation is routinely used in [[quantum mechanics]] to represent the [[Spin (physics)|spin]] of [[fundamental particle]]s such as [[electron]]s. They also serve as [[unit vector]]s for the description of our 3&nbsp;spatial dimensions in [[loop quantum gravity]]. They also correspond to the [[quantum logic gate#Pauli gates (X,Y,Z)|Pauli X, Y, and Z gates]], which are standard generators for the single qubit gates, corresponding to 3d rotations about the axes of the [[Bloch sphere]].

The Lie algebra serves to work out the [[representation theory of SU(2)|representations of {{math|SU(2)}}]].