Special unitary group

The special unitary group consists of the unitary n × n matrices with complex entries, the determinant of which is  1. It is a compact , simple Lie group of the real dimension, in particular also a differentiable manifold . ${\displaystyle \mathrm {SU} (n)}$ ${\displaystyle n^{2}-1,}$

It is also a subgroup of the unitary group as well as the special linear group . ${\displaystyle \mathrm {U} (n)}$ ${\displaystyle \mathrm {SL} (n,\mathbb {C} )}$

Lie algebra

The corresponding Lie algebra corresponds to the tangent space on the unit element of the group. It consists of the space of all oblique Hermitian matrices with trace 0. The surjective mapping ${\displaystyle \mathrm {SU} (n)}$ ${\displaystyle {\mathfrak {su}}(n)}$

${\displaystyle f\colon {\mathfrak {su}}(n)\to \mathrm {SU} (n),\quad \quad g\mapsto \exp {g}}$

maps an element of Lie algebra to the group.

center

The center of consists of all the multiples of the identity matrix that lie in. There , these multiples must be roots of unity . Therefore the center is isomorphic to the remainder class group . ${\displaystyle \mathrm {SU} (n)}$ ${\displaystyle \xi E_{n}}$ ${\displaystyle E_{n}}$ ${\displaystyle \mathrm {SU} (n)}$ ${\displaystyle \det(\xi E_{n})=\xi ^{n}=1}$ ${\displaystyle n}$ ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$

Importance in physics

The special unitary group plays a special role in theoretical physics , since the current standard model of elementary particle physics has several symmetries. The internal symmetry group of the Standard Model is given by (the three factors referring to different degrees of freedom, namely color , flavor and electrical charge ). In addition, there is the approximately valid symmetry for the classification of hadrons , which consist of the “light” up, down and strange quarks (the masses of these quarks are neglected, the three “heavy” quarks are not described by this group ). ${\displaystyle \mathrm {SU} (n)}$ ${\displaystyle \mathrm {SU} (3)\times \mathrm {SU} (2)\times \mathrm {U} (1)}$ ${\displaystyle \mathrm {SU} (3)}$

Furthermore, the compact portion of the special orthochronous Lorentz group is isomorphic to . ${\displaystyle \mathrm {SU} (2)\times \mathrm {SU} (2)}$

The group is also the so-called double group of the ordinary rotating group in three-dimensional space: ${\displaystyle \mathrm {SU} (2)}$ ${\displaystyle \mathrm {SO} (3)}$

SU (2) as "superposition" of the rotation group SO (3)

The SU (2) , the group of “complex rotations” of two-dimensional complex space , with main applications in quantum mechanics (→  spin angular momentum ), is generated by the three Pauli matrices . It is the two-leaf superposition group of the SO (3), the rotating group of three-dimensional real space , which is generated by the spatial angular momentum . With the imaginary unit it applies : ${\displaystyle \mathbb {C} ^{2}}$ ${\displaystyle \sigma _{i}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathrm {i} }$

${\displaystyle \mathrm {SU} (2)=\left\{\left.\exp \left(-{\tfrac {\mathrm {i} }{2}}{\vec {\alpha }}\cdot {\vec {\sigma }}\right)\right|{\vec {\alpha }}\in \mathbb {R} ^{3}\right\}}$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 2 × 2 Pauli matrices  (in the The language of physics: "the double (!) Spin angular momentum operator"). The point means the formal scalar product, the apparently only physically motivated factor 1/2 has mathematical u. a. As a result, in contrast to vectors , the spinors do not   reproduce when they are rotated , but only when they are 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. ${\displaystyle \,\alpha _{1},\,\alpha _{2}}$ ${\displaystyle \,\alpha _{3}}$ ${\displaystyle \,\alpha _{3}}$ ${\displaystyle [-2\pi ,+2\pi ]}$ ${\displaystyle {\vec {\sigma }}}$ ${\displaystyle {\vec {\alpha }}\cdot {\vec {\sigma }}=\alpha _{1}\sigma _{1}+\alpha _{2}\sigma _{2}+\alpha _{3}\sigma _{3}\,.}$ ${\displaystyle 2\pi (=360^{\circ })}$ ${\displaystyle {\vec {\sigma }}/2}$ ${\displaystyle {\vec {\mathcal {L}}}}$ ${\displaystyle {\mathcal {L}}_{3}={\tfrac {\partial }{\mathrm {i} \,\partial \varphi }}}$ ${\displaystyle \hbar }$ ${\displaystyle \varphi }$

In an analogous way, the SU (3), the symmetry group of quantum chromodynamics , is generated from the eight Gell-Mann matrices . The rotation group im , the SO (4), in this case does not fit the SU (3) for dimensional reasons, but SO (4) = SU (2) × SU (2) applies (see again the article Quaternions ). ${\displaystyle \mathbb {R} ^{4}}$

literature

Textbooks

• Joachim Hilgert, Karl-Hermann Neeb: Lie groups and Lie algebras. Vieweg, Braunschweig a. a. 1999, ISBN 3-528-06432-3 .
• Nicolas Bourbaki : Lie Groups and Lie Algebras . Springer, Berlin a. a. 2002, ISBN 3-540-42650-7 , Chapters 4-6.
• Theodor Bröcker , Tammo tom Dieck : Representations of Compact Lie Groups (= Graduate Texts in Mathematics. Vol. 98). Corrected 2nd printing. Springer, New York NY a. a. 1995, 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 .

items

• Jonathan L. Rosner: An Introduction to Standard Model Physics . TASI 1987, Scanned version from KEK .
• Erhard Scholz: Introducing Groups into Quantum Theory (1926–1930) . arxiv : math.HO / 0409571

Web links

• Eric Weisstein: Special Unitary Group . In: MathWorld (English).

References and comments

1. ↑ 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.