SU(2)

The following is modified from w:SU(2).

In mathematics, the special unitary group of degree n, denoted SU(n), is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication. The special unitary group is a subgroup of the unitary group U(n), consisting of all n×n unitary matrices, which is itself a subgroup of the general linear group GL(n, C).

The SU(n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in QCD.

The simplest case, SU(1), is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of absolute value 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), we have a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is $\{+I,-I\}$.

Properties

The special unitary group SU(n) is a real matrix Lie group of dimension n²− 1. Topologically, it is compact and simply connected. Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below). The center of SU(n) is isomorphic to the cyclic group Z_n. Its outer automorphism group, for n ≥ 3, is Z₂, while the outer automorphism group of SU(2) is the trivial group.

The SU(n) algebra is generated by n² operators, which satisfy the commutator relationship (for i,j,k,l = 1, 2, ..., n)

$\left[{\hat {O}}_{ij},{\hat {O}}_{kl}\right]=\delta _{jk}{\hat {O}}_{il}-\delta _{il}{\hat {O}}_{kj}.$

Additionally, the operator

${\hat {N}}=\sum _{i=1}^{n}{\hat {O}}_{ii}$

satisfies

$\left[{\hat {N}},{\hat {O}}_{ij}\right]=0$

which implies that the number of independent generators of SU(n) is n²-1.

Generators

In general the infinitesimal generators of SU(n), T, are represented as traceless hermitian matrices. I.e:

$\operatorname {tr} (T_{a})=0\,$

and

$T_{a}=T_{a}^{\dagger }.\,$

Fundamental representation

In the defining or fundamental representation the generators are represented by n×n matrices where:

$T_{a}T_{b}={\frac {1}{2n}}\delta _{ab}I_{n}+{\frac {1}{2}}\sum _{c=1}^{n^{2}-1}{(if_{abc}+d_{abc})T_{c}}\,$

where the f are the structure constants and are antisymmetric in all indices, whilst the d are symmetric in all indices.

As a consequence:

$\left[T_{a},T_{b}\right]_{+}={\frac {1}{n}}\delta _{ab}I_{n}+\sum _{c=1}^{n^{2}-1}{d_{abc}T_{c}}\,$

$\left[T_{a},T_{b}\right]_{-}=i\sum _{c=1}^{n^{2}-1}{f_{abc}T_{c}}.\,$

We also have

$\sum _{c,e=1}^{n^{2}-1}d_{ace}d_{bce}={\frac {n^{2}-4}{n}}\delta _{ab}\,$

as a normalization convention.

Adjoint representation

In the adjoint representation the generators are represented by $(n^{2}-1)$ $n$×$n$ matrices whose elements are defined by the structure constants:

$(T_{a})_{jk}=-if_{ajk}.\,$

SU(2)

$\operatorname {SU} _{2}(\mathbb {C} )$ and ${\mathfrak {su}}_{2}(\mathbb {C} )$

A general matrix element of $\operatorname {SU} _{2}(\mathbb {C} )$ takes the form

$U={\begin{pmatrix}\alpha &-{\overline {\beta }}\\\beta &{\overline {\alpha }}\end{pmatrix}}$

where $\alpha ,\beta \in \mathbb {C}$ such that $|\alpha |^{2}+|\beta |^{2}=1$. We can consider the following map $\varphi :\mathbb {C} ^{2}\to \operatorname {M} (2,\mathbb {C} )$, (where $\operatorname {M} (2,\mathbb {C} )$ denotes the set of 2 by 2 complex matrices), defined in the obvious way by

$\varphi (\alpha ,\beta )={\begin{pmatrix}\alpha &-{\overline {\beta }}\\\beta &{\overline {\alpha }}\end{pmatrix}}.$

By considering $\mathbb {C} ^{2}$ diffeomorphic to $\mathbb {R} ^{4}$ and $\operatorname {M} (2,\mathbb {C} )$ diffeomorphic to $\mathbb {R} ^{8}$ we can see that $\varphi$ is an injective real linear map and hence an embedding. Now considering the restriction of $\varphi$ to the 3-sphere, denoted $S^{3}$, we can see that this is an embedding of the 3-sphere onto a compact submanifold of $\operatorname {M} (2,\mathbb {C} )$. However it is also clear that $\varphi (S^{3})=\operatorname {SU} _{2}(\mathbb {C} )$. Therefore as a manifold we have $S^{3}$ is diffeomorphic to $\operatorname {SU} _{2}(\mathbb {C} )$ and so $\operatorname {SU} _{2}(\mathbb {C} )$ is a compact, connected Lie group.

Now considering the Lie algebra ${\mathfrak {su}}_{2}(\mathbb {C} )$, a general element takes the form

$U'={\begin{pmatrix}ix&-{\overline {\beta }}\\\beta &-ix\end{pmatrix}}$

where $x\in \mathbb {R}$ and $\beta \in \mathbb {C}$. It is easily verified that matrices of this form have trace zero and are antihermitian. The Lie algebra is then generated by the following matrices

$u_{1}={\begin{pmatrix}0&i\\i&0\end{pmatrix}}\qquad u_{2}={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}\qquad u_{3}={\begin{pmatrix}i&0\\0&-i\end{pmatrix}}$

which are easily seen to have the form of the general element specified above. These satisfy the relations $u_{3}u_{2}=-u_{2}u_{3}=-u_{1}$ and $u_{2}u_{1}=-u_{1}u_{2}=-u_{3}$. The commutator bracket is therefore specified by

$[u_{3},u_{1}]=2u_{2},\qquad [u_{1},u_{2}]=2u_{3},\qquad [u_{2},u_{3}]=2u_{1}.$

The above generators are related to the Pauli matrices by $u_{1}=i\sigma _{1}$, $u_{2}=-i\sigma _{2}$ and $u_{3}=i\sigma _{3}$.

SU(3)

The generators of ${\mathfrak {su}}$(3), T, in the defining representation, are:

$T_{a}={\frac {\lambda _{a}}{2}}.\,$

where $\lambda \,$, the Gell-Mann matrices, are the SU(3) analog of the Pauli matrices for SU(2):

$\lambda _{1}={\begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix}}$	$\lambda _{2}={\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}}$	$\lambda _{3}={\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}}$
$\lambda _{4}={\begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix}}$	$\lambda _{5}={\begin{pmatrix}0&0&-i\\0&0&0\\i&0&0\end{pmatrix}}$	$\lambda _{6}={\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}}$
$\lambda _{7}={\begin{pmatrix}0&0&0\\0&0&-i\\0&i&0\end{pmatrix}}$	$\lambda _{8}={\frac {1}{\sqrt {3}}}{\begin{pmatrix}1&0&0\\0&1&0\\0&0&-2\end{pmatrix}}.$

Note that they are all traceless Hermitian matrices as required.

These obey the relations

$\left[T_{a},T_{b}\right]=i\sum _{c=1}^{8}{f_{abc}T_{c}}\,$

where the f are the structure constants, as previously defined, and have values given by

$f_{123}=1\,$

$f_{147}=-f_{156}=f_{246}=f_{257}=f_{345}=-f_{367}={\frac {1}{2}}\,$

$f_{458}=f_{678}={\frac {\sqrt {3}}{2}},\,$

and all other $f_{abc}$ not related to these by permutation are zero.

The d take the values:

$d_{118}=d_{228}=d_{338}=-d_{888}={\frac {1}{\sqrt {3}}}\,$

$d_{448}=d_{558}=d_{668}=d_{778}=-{\frac {1}{2{\sqrt {3}}}}\,$

$d_{146}=d_{157}=-d_{247}=d_{256}=d_{344}=d_{355}=-d_{366}=-d_{377}={\frac {1}{2}}.\,$

Lie algebra

The Lie algebra corresponding to SU(n) is denoted by ${\mathfrak {su}}(n)$. Its standard mathematical representation consists of the traceless antihermitian $n\times n$ complex matrices, with the regular commutator as Lie bracket. A factor $i$ is often inserted by particle physicists, so that all matrices become Hermitian. This is simply a different, more convenient, representation of the same real Lie algebra. Note that ${\mathfrak {su}}(n)$ is a Lie algebra over $\mathbb {R}$.

For example, the following antihermitian matrices used in quantum mechanics form a basis for ${\mathfrak {su}}(2)$ over $\mathbb {R}$:

$i\sigma _{x}={\begin{bmatrix}0&i\\i&0\end{bmatrix}}$

$i\sigma _{y}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}$

$i\sigma _{z}={\begin{bmatrix}i&0\\0&-i\end{bmatrix}}$

(where $i$ is the imaginary unit.)

This representation is often used in quantum mechanics (see Pauli matrices and Gell-Mann matrices), to represent the spin of fundamental particles such as electrons. They also serve as unit vectors for the description of our 3 spatial dimensions in quantum relativity.

Note that the product of any two different generators is another generator, and that the generators anticommute. Together with the identity matrix (times $i$),

$iI_{2}={\begin{bmatrix}i&0\\0&i\end{bmatrix}}$

these are also generators of the Lie algebra ${\mathfrak {u}}(2)$.

Here it depends of course on the problem whether one works finally, as in non-relativistic quantum mechanics, with 2-spinors; or, as in the relativistic Dirac theory, one needs an extension to 4-spinors; or in mathematics even to Clifford algebras.

Note: make clearer the fact that under matrix multiplication (which is anticommutative in this case), we generate the Clifford algebra $\mathrm {Cl} _{3}$, whereas you generate the Lie algebra ${\mathfrak {su}}(2)$ with commutator brackets instead.

Back to general $\mathrm {SU} (n)$:

If we choose an (arbitrary) particular basis, then the subspace of traceless diagonal $n\times n$ matrices with imaginary entries forms an $n-1$ dimensional Cartan subalgebra.

Complexify the Lie algebra, so that any traceless $n\times n$ matrix is now allowed. The weight eigenvectors are the Cartan subalgebra itself and the matrices with only one nonzero entry which is off diagonal. Even though the Cartan subalgebra $\mathrm {h}$ is only $n-1$ dimensional, to simplify calculations, it is often convenient to introduce an auxiliary element, the unit matrix which commutes with everything else (which should not be thought of as an element of the Lie algebra!) for the purpose of computing weights and that only. So, we have a basis where the $i$th basis vector is the matrix with $1$ on the $i$th diagonal entry and zero elsewhere. Weights would then be given by $n$ coordinates and the sum over all $n$ coordinates has to be zero (because the unit matrix is only auxiliary).

So, $\mathrm {SU} (n)$ has a rank of $n-1$ and its Dynkin diagram is given by $A_{n-1}$, a chain of $n-1$ vertices.

Its root system consists of $n(n-1)$ roots spanning a $n-1$ Euclidean space. Here, we use $n$ redundant coordinates instead of $n-1$ to emphasize the symmetries of the root system (the $n$ coordinates have to add up to zero). In other words, we are embedding this $n-1$ dimensional vector space in an $n$-dimensional one. Then, the roots consists of all the $n(n-1)$ permutations of $(1,-1,0,\dots ,0)$. The construction given two paragraphs ago explains why. A choice of simple roots is

$(1,-1,0,\dots ,0),\$

$(0,1,-1,\dots ,0),\$

…,

$(0,0,0,\dots ,1,-1).\$

Its Cartan matrix is

${\begin{pmatrix}2&-1&0&\dots &0\\-1&2&-1&\dots &0\\0&-1&2&\dots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\dots &2\end{pmatrix}}.\$

Its Weyl group or Coxeter group is the symmetric group $S_{n}$, the symmetry group of the $(n-1)$-simplex.

Generalized special unitary group

For a field F, the generalized special unitary group over F, SU(p,q;F), is the group of all linear transformations of determinant 1 of a vector space of rank n = p + q over F which leave invariant a nondegenerate, Hermitian form of signature (p, q). This group is often referred to as the special unitary group of signature p q over F. The field F can be replaced by a commutative ring, in which case the vector space is replaced by a free module.

Specifically, fix a Hermitian matrix A of signature p q in GL(n,R), then all

$M\in SU(p,q,R)$

satisfy

$M^{*}AM=A\,$

$\det M=1.\,$

Often one will see the notation $SU_{p,q}$ without reference to a ring or field, in this case the ring or field being referred to is C and this gives one of the classical Lie groups. The standard choice for A when F = C is

$A={\begin{bmatrix}0&0&i\\0&I_{n-2}&0\\-i&0&0\end{bmatrix}}.$

However there may be better choices for A for certain dimensions which exhibit more behaviour under restriction to subrings of C.

Example

A very important example of this type of group is the Picard modular group SU(2,1;Z[i]) which acts (projectively) on complex hyperbolic space of degree two, in the same way that SL(2,Z) acts (projectively) on real hyperbolic space of dimension two. In 2003 Gábor Francsics and Peter Lax computed a fundamental domain for the action of this group on $HC^{2}$, see [1]. Another example is SU(1,1;C) which is isomorphic to SL(2,R).

Important subgroups

In physics the special unitary group is used to represent bosonic symmetries. In theories of symmetry breaking it is important to be able to find the subgroups of the special unitary group. Subgroups of SU(n) that are important in GUT physics are, for p>1, n-p>1:

$SU(n)\supset SU(p)\times SU(n-p)\times U(1).$

For completeness there are also the orthogonal and symplectic subgroups:

$SU(n)\supset O(n)$

$SU(2n)\supset USp(2n).$

Since the rank of SU(n) is n-1 and of U(1) is 1, a useful check is that the sum of the ranks of the subgroups is less than or equal to the rank of the original group. SU(n) is a subgroup of various other lie groups:

$SO(2n)\supset SU(n)$

$USp(2n)\supset SU(n)$

$Spin(4)=SU(2)\times SU(2)$ (see Spin group)

$E_{6}\supset SU(6)$

$E_{7}\supset SU(8)$

$G_{2}\supset SU(3)$ (see Simple Lie groups for E₆, E₇, and G₂).

There are also the identities SU(4)=Spin(6), SU(2)=Spin(3)=USp(2) and U(1)=Spin(2)=SO(2) .

One should finally mention that SU(2) is the double covering group of SO(3), a relation that plays an important role in the theory of rotations of 2-spinors in non-relativistic quantum mechanics.

See also

Mathematics portal

• Group theory
• Representation theory

References

• Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2. 
• Maximal Subgroups of Compact Lie Groups

External links

• Physics 558 - Lecture 1, Winter 2003