LOG#091. Group theory(XI).

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Today, we are going to talk about the Lie groups U(n) and SU(n), and their respective Lie algebras, generally denoted by u(n) and su(n) by the physics community. In addition to this, we will see some properties of the orthogonal groups in euclidena “signature” and general quadratic metrics due to their importance in special relativity or its higher dimensional analogues.

Let us remember what kind of groups are U(n) and U(n):

1) The unitary group is defined by:

U(n)\equiv\left\{ U\in M_{n\times n}(\mathbb{C})/UU^+=U^+U=I\right\}

2) The special unitary group is defined by:

SU(n)\equiv\left\{ U\in M_{n\times n}(\mathbb{C})/UU^+=U^+U=I,\det (U)=1\right\}

The group operation is the usual matrix multiplication. The respective algebras are denoted as we said above by u(n) and su(n). Moreover, if you pick an element U\in U(n), there exists an hermitian (antihermitian if you use the mathematician “approach” to Lie algebras/groups instead the convention used mostly by physicists) n\times n matrix H such that:

U=\exp (iH)

Some general properties of unitary and special unitary groups are:

1) U(n) and SU(n) are compact Lie groups. As a consequence, they have unitary, finite dimensional and irreducible representations. U(n) and SU(n) are subgroups of U(m) if m\geq n.

2) Generators or parameters of unitary and special unitary groups. As we have already seen, the unitary group has n^2 parameters (its “dimension”) and it has rank n-1 (its number of Casimir operators). The special unitary group has n^2-1 free parameters (its dimension) and it has rank n-1 (its number of Casimir operators).

3) Lie algebra generators. The unitary group has a Lie algebra generated by the space of n^2 dimensional complex matrices. The special unitary group has a Lie algebra generated by the n^2-1 dimensional space of hermitian n\times n traceless matrices.

4) Lie algebra structures. Given a basis of generators L_i for the Lie algebra, we define the constants C_{ijk}, f_{ijk}, d_{ijk} by the following equations:

\left[L_i,L_m\right]=C_{ijk}L_k=if_{ijk}L_k

L_iL_j+L_jL_i=\dfrac{1}{3}\delta_{ij}I+d_{ijk}L_k

These structure constants f_{ijk} are totally antisymmetric under the exchange of any two indices while the coefficients d_{ijk} are symmetric under those changes. Moreover, we also have:

d_{ijk}=2\mbox{Tr}(\left\{L_i,L_j\right\}L_k)

f_{ijk}=-2i\mbox{Tr}(\left[L_i,L_j\right]L_k)

Remark(I):   From U=e^{iH}, we get \det U=e^{i\mbox{Tr} (H)}, and from here we can prove the statement 3) above.

Remark(II): An arbitrary element of U(n) can be expressed as a product of an element of U(1) and an element of SU(n). That is, we can write U(n)\cong U(1)\cup SU(n), where the symbol \cong means “group isomorphism”.

Example 1. The SU(2) group.

In particular, for n=2, we get

SU(2)=\left\{U\in M_{2\times 2})(\mathbb{C})/UU^+=U^+U=I_{2\times 2},\det U=1\right\}

This is an important group in physics! It appears in many contexts: angular momentum (both classical and quantum), the rotation group, spinors, quantum information theory, spin networks and black holes, the Standard Model, and many other places. So it is important to know it at depth. The number of parameters of SU(2) is equal to 3 and its rank is equal to one (1). As generators of the Lie algebra associated to this Lie group, called su(2), we can choose for free 3 any independent traceless (trace equal to zero) matrices. As a convenient choice, it is usual to select the so called Pauli matrices \sigma_i:

\sigma_1=\begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}

\sigma_2=\begin{pmatrix}0 & -i\\ i & 0\end{pmatrix}

\sigma_3=\begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}

In general, these matrices satisfy an important number of mathematical relations. The most important are:

\left\{\sigma_i,\sigma_j\right\}=2\sigma_i\sigma_j+2i\varepsilon_{ijk}\sigma_k

and

\sigma_i\sigma_j=i\varepsilon_{ijk}\sigma_k

The commutators of Pauli matrices are given by:

\left[\sigma_i,\sigma_j\right]=2if_{ijk}\sigma_k

The structure constants read

f_{ijk}=\dfrac{1}{2}\varepsilon_{ijk} d_{ijk}=0

The Casimir operator/matrix related to the Pauli basis is:

C(\sigma_i)=\sigma_i^2=\sigma_1^2+\sigma_2^2+\sigma_3^2

This matrix, by Schur’s lemma, has to be a multiple of the identity matrix (it commutes with each one of the 3 generators of the Pauli algebra, as it can be easily proved). Please, note that using the previous Pauli representation of the Pauli algebra we get:

\displaystyle{C=\sum_i\sigma_i^2=3I}

Q.E.D.

A similar relation, with different overall prefactor, must be true for ANY other representation of the Lie group algebra su(2). In fact, it can be proved in Quantum Mechanics that this number is “four times” the j(j+1) quantum number associated to the angular momentum and it characterizes completely the representation. The general theory of the representation of the Lie group SU(2) and its algebra su(2) is known in physics as the general theory of the angular momentum!

Example 2. The SU(3) group.

If n=3, the theory of SU(3) is important for Quantum Chromodynamics (QCD) and the quark theory. It is also useful in Grand Unified Theories (GUTs) and flavor physics.

SU(3)=\left\{U\in M_{3\times 3})(\mathbb{C})/UU^+=U^+U=I_{3\times 3},\det U=1\right\}

The number of parameters of SU(3) is 8 (recall that there are 8 different massless gluons in QCD) and the rank of the Lie algebra is equal to two, so there are two Casimir operators.

The analogue generators of SU(3), compared with the Pauli matrices, are the so-called Gell-Mann matrices. They are 8 independent traceless matrices. There are some “different” elections in literature, but a standard choice are the following matrices:

\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=\dfrac{1}{\sqrt{3}}\begin{pmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 &-2\end{pmatrix}

Gell-Mann matrices above satisfy a normalization condition:

\mbox{Tr}(\lambda_i\lambda_j)=2\delta_{ij}

where \delta_{ij} is the Kronecker delta in two indices.

The two Casimir operators for Gell-Mann matrices are:

1) \displaystyle{C_1(\lambda_i)=\sum_{i=1}^8\lambda_i^2}

This operator is the natural generalization of the previously seen SU(2) Casimir operator.

2) \displaystyle{C_2(\lambda_i)=\sum_{ijk}d_{ijk}\lambda_i\lambda_j\lambda_k}

Here, the values of the structure constans f_{ijk} and d_{ijk} for the su(3) Lie algebra can be tabulated in rows as follows:

1) For ijk=123,147,156,246,257,345,367,458,678 we have f_{ijk}=1,\dfrac{1}{2},-\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{2},-\dfrac{1}{2},\dfrac{\sqrt{3}}{2},\dfrac{\sqrt{3}}{2}.

2) If

ijk=118,146,157,228,247,256,338,344,355,366,377,448,558,668,778,888

then have

d_{ijk}=\dfrac{1}{\sqrt{3}},\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{\sqrt{3}},-\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{\sqrt{3}},\dfrac{1}{2},\dfrac{1}{2},-\dfrac{1}{2},-\dfrac{1}{2},-\dfrac{1}{2\sqrt{3}},-\dfrac{1}{2\sqrt{3}},-\dfrac{1}{2\sqrt{3}},-\dfrac{1}{2\sqrt{3}},-\dfrac{1}{\sqrt{3}}

Example 3. Euclidean groups, orthogonal groups and the Lorentz group in 4D and general D=s+t dimensional analogues.

In our third example, let us remind usual galilean relativity. In a 3D world, physics is the same for every inertial observer (observers moving with constant speed). Moreover, the fundamental invariant of “motion” in 3D space is given by the length:

L^2=x^2+y^2+z^2=\delta_{ij}x^ix^j \forall i,j=1,2,3

In fact, with tensor notation, the above “euclidean” space can be generalized to any space dimension. For a ND space, the fundamental invariant reads:

\displaystyle{L_N^2=\sum_{i=1}^NX_i^2=x_1^2+x_2^2+\cdots+x_N^2}

Mathematically speaking, the group leaving the above metrics invariant are, respectively, SO(3) and SO(N). They are Lie groups with dimensions 3 and N(N-1)/2, respectively and their Lie algebra generators are antisymmetric traceless 3\times 3 and N\times N matrices. Those metrics are special cases of quadratic forms and it can easily proved that orthogonal transformations with metric \delta_{ij} (the euclidean metric given by a Kronecker delta tensor) are invariant in the following sense:

A^\mu_{\;\;\; i}\delta_{\mu\nu}A^\nu_{\;\;\; j}=\delta_{ij}

or equivalently

A\delta A^T=\delta

using matric notation. In special relativity, the (proper) Lorentz group L is composed by every real 4\times 4 matrix \Lambda^\mu_{\;\;\;\nu} connected to the identity through infinitesimal transformations, and the Lorentz group leaves invariant the Minkovski metric(we use natural units with c=1):

s^2=X^2+Y^2+Z^2-T^2 if you use the “mostly plus” 3+1 metric (\eta=\mbox{diag}(1,1,1,-1)) or, equivalentaly

s^2=T^2-X^2-Y^2-Z^2 if with a “mostly minus” 1+3 metric (\eta=\mbox{diag}(1,-1,-1,-1)).

These equations can be also genearlized to arbitrary signature. Suppose there are s-spacelike dimensions and t-time-like dimensions (D=s+t). The associated non-degenerated quadratic form is:

\displaystyle{s^2_D=\sum_{i=1}^sX_i^2-\sum_{j=1}^tX_j^2}

In matrix notation, the orthogonal rotations leaving the above quadratic metrics are said to belong to the group SO(3,1) (or SO(1,3) is you use the mostly minus convention) real orthogonal group over the corresponding quadratic form. The signature of the quadratic form is said to be S=2 or (3,1) (equivalently \Sigma=3-1=2 and (1,3) with the alternative convention). We are not considering “degenerated” quadratic forms for simplicity of this example. The Lorentzian or Minkovskian metric are invariant in the same sense that the euclidean example before:

L^\mu_{\;\;\;\alpha}\eta_{\alpha\beta}L^\mu_{\;\;\;\beta}=\eta_{\alpha\beta}

LGL^T=G

The group SO(s,t) has signature (s,t) or s-t or s+t in non-degenerated quadratic spaces. Obviously, the rotation group SO(3) is a subgroup of SO(3,1) and more generally SO(s) is a subgroup of SO(s,t). We are going to focus now a little bit on the common special relativity group SO(3,1). This group have 6 parameters or equivalently its group dimension is 6. The rank of this special relativity group is equal to 1. We can choose as parameters for the SO(3,1) group 3 spatial rotation angles \omega_i and three additional parameters, that we do know as rapidities \xi_i. These group generators have Lie algebra generators S_i and K_i or more precisely, if we define the Lorentz boosts as

\xi=\dfrac{\beta}{\parallel\beta\parallel}\tanh^{-1}\parallel \beta\parallel

In the case of SO(3,1), a possible basis for the Lie algebra generators are the next set of matrices:

iS_1=\begin{pmatrix}0 & 0 & 0& 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & -1 & 0\end{pmatrix}

iS_2=\begin{pmatrix}0 & 0 & 0& 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\end{pmatrix}

iS_3=\begin{pmatrix}0 & 0 & 0& 0\\ 0 & 0 & 1 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}

iK_1=\begin{pmatrix}0 & 1 & 0& 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}

iK_2=\begin{pmatrix}0 & 0 & 1& 0\\ 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}

iK_3=\begin{pmatrix}0 & 0 & 0& 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0\end{pmatrix}

And the commutation rules for these matrices are given by the basic relations:

\left[S_a,S_b\right]=i\varepsilon_{abc}S_c

\left[K_a,K_b\right]=-i\varepsilon_{abc}S_c

\left[S_a,K_b\right]=i\varepsilon_{abc}K_c

Final remark: SO(s,t) are sometimes called isometry groups since they define isometries over quadratic forms, that is, they define transformations leaving invariant the “spacetime length”.

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4 Comments on “LOG#091. Group theory(XI).”

  1. Hello,

    I think that in your formula for the second Casimir operator for the Gell-Mann matrice you have forgotten some terms (may be proportional to f_{ijk} because of what you say after?

    I mean the formula for c_2,

    \displaystyle{C_2(\lambda_i)=\sum_{ijk}d_{ijk}\lambda_i\lambda_j\lambda_k}

    If could you complete it it would be great! Thanks in advance,

    Ezequiel. (sequi76@gmail.com)

  2. amarashiki says:

    Please, see this too…The Casimir C(3) is OK as it is written here. Please, check the matrix elements and your numerical procedure. A summary here http://scipp.ucsc.edu/~haber/ph251/gellmann13.pdf


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