Special unitary group

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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. As a compact classical group, U(n) is the group that preserves the standard inner product on Cn. It is itself a subgroup of the general linear group, SU(n) ⊂ U(n) ⊂ 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.1

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 norm 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), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+I, −I}. SU(2) is also identical to one of symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations.

Properties

The special unitary group SU(n) is a real Lie group (though not a complex Lie group). Its dimension as a real manifold is n21. 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 Zn. Its outer automorphism group, for n3, is Z2, while the outer automorphism group of SU(2) is the trivial group.

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

\left [ \hat{O}_{ij} , \hat{O}_{k \ell} \right ] = \delta_{jk} \hat{O}_{i \ell} - \delta_{i \ell} \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 is n21 .2

Generators

In general the infinitesimal generators (elements of the Lie algebra) 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-coefficients 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 take

\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 (n21) × (n21) matrices, n21 of them, whose elements are defined by the structure constants themselves:

 (T_a)_{jk} = -if_{ajk}.

n = 2

SU(2) is the following group:

 \mathrm{SU}(2) = \left \{ \begin{pmatrix} \alpha&-\overline{\beta}\\ \beta & \overline{\alpha} \end{pmatrix}: \ \ \alpha,\beta\in\mathbf{C}, |\alpha|^2 + |\beta|^2 = 1\right \}  ~,

where the overline denotes complex conjugation. Now consider the following map:

 \begin{align} \varphi : \mathbf{C}^2 &\to \operatorname{M}(2,\mathbf{C}) \\ \varphi(\alpha,\beta) &= \begin{pmatrix} \alpha&-\overline{\beta}\\ \beta & \overline{\alpha}\end{pmatrix},\end{align}

where M(2, C) denotes the set of 2 by 2 complex matrices. By considering C2 diffeomorphic to R4 and M(2, C) diffeomorphic to R8 we can see that φ is an injective real linear map and hence an embedding. Now, considering the restriction of φ to the 3-sphere (since modulus is 1), denoted S3, we can see that this is an embedding of the 3-sphere onto a compact submanifold of M(2, C). However it is also clear that φ(S3) = SU(2). Therefore as a manifold S3 is diffeomorphic to SU(2) and so SU(2) is a compact, connected Lie group.

The Lie algebra of SU(2) is:

\mathfrak{su} (2) =  \left \{ \begin{pmatrix} ix & -\overline{\beta}\\ \beta & -ix \end{pmatrix}: \ x \in \mathbf{R}, \beta \in \mathbf{C} \right \}

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 u3u2 = −u2u3 = −u1 and u2u1 = −u1u2 = −u3. 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 u1 = i σ1,u2 = −i σ2 and u3 = i σ3. This representation is often used in quantum mechanics 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 loop quantum gravity.

The Lie algebra is used to work out the representations of SU(2).

n = 3

The generators of su(3), T, in the defining representation, are:

T_a = \frac{\lambda_a }{2}.\,

where λ 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 the \lambda_a span 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} \,
 \{T_a, T_b\} = \frac{1}{3}\delta_{ab} + \sum_{c=1}^8{d_{abc} T_c} \,,
(or, equivalently,  \{\lambda_a, \lambda_b\} = \frac{4}{3}\delta_{ab} + 2\sum_{c=1}^8{d_{abc} \lambda_c} ).

The f are the structure constants, 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}, \,

while 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 above representation bases generalize to n > 3. The Lie algebra corresponding to SU(n) is denoted by su(n). Its standard mathematical representation consists of the traceless antihermitian n×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 su(n) is a Lie algebra over R.

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

Complexify the Lie algebra, so that any traceless n×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 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, SU(n) is of rank n − 1 and its Dynkin diagram is given by An−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, ..., 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 Sn, the symmetry group of the (n1)-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^{*} A M = 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(2,9;Z) acts (projectively) on real hyperbolic space of dimension two. In 2005 Gábor Francsics and Peter Lax computed an explicit fundamental domain for the action of this group on HC2.3 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),

where × denotes the direct product and U(1), known as the circle group, is the multiplicative group of all complex numbers with absolute value 1.

For completeness there are also the orthogonal and symplectic subgroups:

 SU(n) \supset SO(n),
SU(2n) \supset Sp(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)
Sp(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 , 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

Notes

  1. ^ Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2. 
  2. ^ R.R. Puri, Mathematical Methods of Quantum Optics, Springer, 2001.
  3. ^ Francsics, Gabor; Lax, Peter D.. "An Explicit Fundamental Domain For The Picard Modular Group In Two Complex Dimensions". arXiv:math/0509708v1.pdf.

References








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