SU (2)

In mathematics , the special unitary group is of order 2, i.e. H. the linear group of unitary - matrices with determinant  1. It is (along with the rotation group , the two-time overlay is) the smallest non-Abelian Lie group . ${\displaystyle \operatorname {SU} (2)}$ ${\displaystyle (2\times 2)}$ ${\displaystyle \operatorname {SO} (3)}$

The group plays an important role in physics , including in the standard model of elementary particle physics and in quantum mechanics , where it is also referred to as the complex rotation group (group of “complex rotations” in two-dimensional complex space ) or spin group . Bundles with structure groups are used in the theory of the 4-manifolds to define the Donaldson invariants and in the theory of the 3-manifolds to define the Casson invariant and the Instanton- Floer homology . ${\displaystyle \operatorname {SU} (2)}$ ${\displaystyle \mathbb {C} ^{2}}$ ${\displaystyle \operatorname {SU} (2)}$

definition

This is the group of unitary matrices with determinant  1: ${\displaystyle \operatorname {SU} (2)}$ ${\displaystyle (2\times 2)}$

${\displaystyle \operatorname {SU} (2)={\bigl \{}A\in {\text{Mat}}(2,\mathbb {C} ):A{\bar {A}}^{T}={\bar {A}}^{T}A=1,\det(A)=1{\bigr \}}\subset {\text{Mat}}(2,\mathbb {C} )}$.

All matrices are made out of shape ${\displaystyle \operatorname {SU} (2)}$

${\displaystyle {\begin{pmatrix}z_{1}&z_{2}\\-{\overline {z}}_{2}&{\overline {z}}_{1}\end{pmatrix}}}$with . ${\displaystyle \vert z_{1}\vert ^{2}+\vert z_{2}\vert ^{2}=1}$

${\displaystyle \operatorname {SU} (2)}$is a Lie group. It is the simplest non-Abelian Lie group.

The Lie algebra of the Lie group is the Lie algebra of the lopsided Hermitian matrices ${\displaystyle {\mathfrak {su}}(2)}$ ${\displaystyle \operatorname {SU} (2)}$ ${\displaystyle (2\times 2)}$

${\displaystyle {\mathfrak {su}}(2)={\bigl \{}A\in {\text{Mat}}(2,\mathbb {C} ):A+{\bar {A}}^{T}=0,\operatorname {tr} (A)=0{\bigr \}}\subset {\text{Mat}}(2,\mathbb {C} )}$.

All matrices are made out of shape ${\displaystyle {\mathfrak {su}}(2)}$

${\displaystyle {\begin{pmatrix}ia&-{\bar {z}}\\z&-ia\end{pmatrix}}}$with . ${\displaystyle a\in \mathbb {R} ,z\in \mathbb {C} }$

topology

• The Lie group is a compact Lie group . ${\displaystyle \operatorname {SU} (2)}$

• It is connected and simply connected .

• Like every Lie group, it can be parallelized .

• It is diffeomorphic to the 3-sphere , the diffeomorphism is given by ${\displaystyle S^{3}}$

${\displaystyle {\begin{pmatrix}z_{1}&z_{2}\\-{\bar {z}}_{2}&{\bar {z}}_{1}\end{pmatrix}}\mapsto (z_{1},z_{2})\in S^{3}\subset \mathbb {C} ^{2}}$.

SU (2) as a spin group

${\displaystyle \operatorname {SU} (2)}$is a 2-fold superposition of the rotating group , so it realizes the spin group . The natural operation from on is a so-called spinor representation . ${\displaystyle \operatorname {SO} (3)}$ ${\displaystyle \operatorname {Spin} (3)}$ ${\displaystyle \operatorname {SU} (2)}$ ${\displaystyle \mathbb {C} ^{2}}$

The superposition is explicitly given by the adjoint representation of on its 3-dimensional Lie algebra . This leaves the killing form and thus also invariant. Because positive is definite, the group of linear maps obtained is isomorphic to . It can be shown that the mapping defined in this way defines a double overlay . ${\displaystyle \operatorname {SU} (2)}$ ${\displaystyle {\mathfrak {su}}(2)}$ ${\displaystyle B}$ ${\displaystyle -B}$ ${\displaystyle -B}$ ${\displaystyle -B}$ ${\displaystyle O(3)}$ ${\displaystyle \operatorname {SU} (2)\to \operatorname {SO} (3)}$

Pauli Matrices and Complex Rotations

The Pauli matrices are

${\displaystyle \sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{2}={\begin{pmatrix}0&-\mathrm {i} \\\mathrm {i} &0\end{pmatrix}},\quad \sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.}$

The imaginary multiples are elements of Lie algebra . It applies ${\displaystyle \mathrm {i} \sigma _{j},j=1,2,3}$ ${\displaystyle {\mathfrak {su}}(2)}$

${\displaystyle \operatorname {SU} (2)=\left\{\left.\exp \left(-{\tfrac {\mathrm {i} }{2}}{\vec {\alpha }}\cdot {\vec {\sigma }}\right)\right|{\vec {\alpha }}\in \mathbb {R} ^{3}\right\}}$

with real vector components and , the "angles of rotation" ( runs through the interval, for example ), and with the basic elements of the quaternions converted into the three Pauli matrices, i.e. the formal three-vector formed from the three 2x2 Pauli matrices  (in the language of Physics: "the double (!) Spin angular momentum operator"). The point is the formal scalar product . The apparently only physically motivated factor 1/2 has mathematically u. a. As a result, in contrast to vectors , the spinors do not   reproduce with rotations of 2 π  (= 360  ^o ), but only at twice the value. On the other hand, the usual rotation group in three-dimensional real space, the SO (3), is obtained by replacing it with the spatial angular momentum operator (expressed by differential quotients, e.g. ). It was , the reduced Planck's constant , as usual replaced by one, and is the azimuthal angle (rotation around the z-axis). Now the rotation by 360  ^{o is} sufficient to reproduce an ordinary function - instead of a spinor. ${\displaystyle \alpha _{1},\,\alpha _{2}}$ ${\displaystyle \alpha _{3}}$ ${\displaystyle \alpha _{3}}$ ${\displaystyle [-2\pi ,+2\pi ]}$ ${\displaystyle {\vec {\sigma }}}$ ${\displaystyle {\vec {\alpha }}\cdot {\vec {\sigma }}=\alpha _{1}\sigma _{1}+\alpha _{2}\sigma _{2}+\alpha _{3}\sigma _{3}}$ ${\displaystyle {\vec {\sigma }}/2}$ ${\displaystyle {\vec {\mathcal {L}}}}$ ${\displaystyle {\mathcal {L}}_{3}={\tfrac {\partial }{\mathrm {i} \,\partial \varphi }}}$ ${\displaystyle \hbar }$ ${\displaystyle \varphi }$

In this sense, the group of complex rotations is “generated” by the Pauli matrices, which is used in quantum mechanics, especially in the theory of the spin angular momentum . ${\displaystyle \operatorname {SU} (2)}$

SU (2) as a group of the unit quaternions

→ Main article : “Complex matrices” in the article Quaternion

Each quaternion can be clearly identified in the form ${\displaystyle x\in \mathbb {H} }$

${\displaystyle x_{0}+x_{1}\mathrm {i} _{\mathbb {H} }+x_{2}\mathrm {j} _{\mathbb {H} }+x_{3}\mathrm {k} _{\mathbb {H} }}$

with real numbers , , , Write. The amount of a quaternion is defined by ${\displaystyle x_{0}}$ ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle x_{3}}$

${\displaystyle |x|={\sqrt {x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}}}$.

The group of unit quaternions

${\displaystyle \left\{x\in \mathbb {H} :|x|=1\right\}}$

is isomorphic to , under isomorphism correspond to each other ${\displaystyle \operatorname {SU} (2)}$

${\displaystyle 1\mapsto {\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad \mathrm {i} _{\mathbb {H} }\mapsto {\begin{pmatrix}\mathrm {i} &0\\0&-\mathrm {i} \end{pmatrix}},\quad \mathrm {j} _{\mathbb {H} }\mapsto {\begin{pmatrix}0&1\\-1&0\end{pmatrix}},\quad \mathrm {k} _{\mathbb {H} }\mapsto {\begin{pmatrix}0&\mathrm {i} \\\mathrm {i} &0\end{pmatrix}}}$.

Finite subgroups of the SU (2)

The finite subgroups were classified by Felix Klein .

Every finite subgroup is isomorphic to one of the following subgroups of : ${\displaystyle \operatorname {SU} (2)}$

• of the cyclic group generated by the diagonal matrix

${\displaystyle {\rm {{diag}(\exp(2\pi {\rm {i/{\it {n{\rm {),\exp(-2\pi {\rm {i/{\it {n{\rm {))}}}}}}}}}}}}}}}$,

• of the dihedral group generated by

${\displaystyle {\rm {{diag}(\exp(\pi {\rm {i/{\it {n{\rm {),\exp(-\pi {\rm {i/{\it {n{\rm {))}}}}}}}}}}}}}}}$and , ${\displaystyle \left({\begin{array}{cc}0&{\rm {i}}\\{\rm {i}}&0\end{array}}\right)}$

• the archetype of the symmetry group of one of the regular Platonic solids (i.e. apart from the duality of either the regular tetrahedron , octahedron or icosahedron ) under the overlay . ${\displaystyle \operatorname {SU} (2)\to \operatorname {SO} (3)}$

These subgroups correspond to the Dynkin diagrams . See also Quaternion # The finite subgroups . ${\displaystyle A_{n-1},D_{n+2},E_{6},E_{7},E_{8}}$

Differential geometry

The negative of the Killing form defines a bi-invariant Riemannian metric on their sectional curvature is constant . The is therefore isometric to three-dimensional unit sphere. ${\displaystyle \operatorname {SU} (2)}$ ${\displaystyle 1}$ ${\displaystyle \operatorname {SU} (2)}$

Representation theory

The Lie algebra is a real form of the Lie algebra ; H. is the complexification of . All representations of are thus obtained by restricting representations of . In particular, it follows from the classification of the representations of that for every natural number there is a -dimensional, irreducible representation of which is unique except for isomorphism . ${\displaystyle {\mathfrak {su}}(2)}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {su}}(2)}$ ${\displaystyle {\mathfrak {su}}(2)}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle m}$ ${\displaystyle (m+1)}$ ${\displaystyle {\mathfrak {su}}(2)}$

According to Lie's second theorem , the Lie algebra representations of exactly correspond to the Lie group representations of . So for every natural number there is a -dimensional, irreducible representation of which is unique except for isomorphism . In physics, this is called the spin representation. ${\displaystyle {\mathfrak {su}}(2)}$ ${\displaystyle \operatorname {SU} (2)}$ ${\displaystyle m}$ ${\displaystyle (m+1)}$ ${\displaystyle \operatorname {SU} (2)}$ ${\displaystyle {\tfrac {m}{2}}}$

An explicit realization of the -dimensional representation of goes as follows. Let it be the vector space of the complex-valued homogeneous polynomials of degree in two variables, i.e. the complex vector space spanned by. Then acts on through . ${\displaystyle (m+1)}$ ${\displaystyle \operatorname {SU} (2)}$ ${\displaystyle V_{m}}$ ${\displaystyle m}$ ${\displaystyle x^{m},x^{m-1}y,\ldots ,xy^{m-1},y^{m}}$ ${\displaystyle A\in \operatorname {SU} (2)}$ ${\displaystyle V_{m}}$ ${\displaystyle (AP)(x,y):=P(A^{-1}(x,y))}$

physics

The angular momentum algebra is isomorphic to the complexification of the Lie algebra of . Many physical situations are rotation-invariant and can therefore be described as representations of those, which are usually infinite-dimensional but can be broken down into finite-dimensional irreducible representations. In the case of the hydrogen atom, the numbers of states of equal energy correspond precisely to the dimensions of these irreducible representations. However, certain effects can only be explained if the dimensions are doubled, i.e. instead of the SO (3) representations, the SU (2) representations resulting from tensing with the standard representation are considered. ${\displaystyle \operatorname {SU} (2)}$ ${\displaystyle \operatorname {SO} (3)}$ ${\displaystyle \mathbb {C} ^{2}}$

The strong interaction and thus the standard model of elementary particle physics is invariant. ${\displaystyle \operatorname {SU} (2)}$

See also

• Special unitary group

literature

• Theodor Bröcker , Tammo tom Dieck : Representations of Compact Lie Groups (= Graduate Text in Mathematics. Vol. 98). Springer, New York NY a. a. 1985, ISBN 3-540-13678-9 .
• Walter Pfeifer: The Lie Algebras see below (N). An Introduction. Birkhäuser, Basel a. a. 2003, ISBN 3-7643-2418-X .
• Jean-Marie Normand: A Lie group. Rotations in quantum mechanics. North-Holland Publishing Co., Amsterdam u. a. 1980, ISBN 0-444-86125-4 .
• Max Wagner: Group theoretical methods in physics. A teaching and reference work. Friedr. Vieweg & Sohn, Braunschweig 1998, ISBN 3-528-06943-0 .

Web links

• Stephen Haywood: SU (2) (emphasizes physical applications)
• Julien Marché: Geometry of representation spaces in SU (2)

Individual evidence

1. ^ Simon K. Donaldson : Polynomial invariants for smooth four-manifolds. In: Topology. Vol. 29, No. 3, 1990, pp. 257-315, doi : 10.1016 / 0040-9383 (90) 90001-Z
2. ↑ Andreas Floer : An instanton-invariant for 3-manifolds. In: Communications in Mathematical Physics. Vol. 118, No. 2, 1988, pp. 775-813, doi : 10.1007 / BF01218578
3. ^ Clifford Henry Taubes : Casson's invariant and gauge theory. In: Journal of differential geometry. Vol. 31, No. 2, 1990, pp. 547-599, online .
4. ↑ That is not , but ${\displaystyle {\vec {\sigma }}}$ ${\displaystyle {\vec {\sigma }}/2}$ the spin angular momentum operator, results u. a. from the associated Lie algebra, the angular momentum algebra.