\documentclass[12pt]{article} % Specifies the document class
% The preamble begins here.
%<-------------------------------------------Included Packages---------------------------------------------------->
%\usepackage[dvips]{epsfig}
% for displaying pictures
%\usepackage[b]{esvect}
\usepackage{amssymb}
\usepackage{cool}
%<-----------------------------------------End Included Packages-------------------------------------------------->
%<------------------------------------------Document Properties--------------------------------------------------->
\title{Don't Look Directly into the $SU(n)$} % Declares the document's title.
\author{N. Setzer} % Declares the author's name.
%\date{} % Declares the date. Aren't you glad you have that kind of power?
\setlength{\topmargin}{-0.8in}
\setlength{\topskip}{0.2in} % between header and text
\setlength{\textheight}{9.0in} % height of main text
\setlength{\textwidth}{7.3in} % width of text
\setlength{\oddsidemargin}{-0.4in} % odd page left margin
\setlength{\evensidemargin}{-0.4in} % even page left margin
%<----------------------------------------End Document Properties------------------------------------------------->
%<----------------------------------------Modified LaTeX Command Definitions--------------------------------------->
\newcommand{\var}[1]{}
\newenvironment{declaration}{\hide}{}
\newcommand{\hide}[1]{}
\newenvironment{derivation}{\begin{eqnarray*}}{\end{eqnarray*}}
\newenvironment{der}{\begin{eqnarray*}}{\end{eqnarray*}}
%<--------------------------------------End Modified LaTeX Command Definitions------------------------------------->
%<-------------------------------------------Command Definitions--------------------------------------------------->
%% Command Definitions Path
\newcommand{\cmdDefPath}{../command_def}
%% Quantum Notation
\input{\cmdDefPath/quantum_notation.cmdef}
%% Vector Notation
\input{\cmdDefPath/vectors_notation.cmdef}
%% Math
%\input{\cmdDefPath/math_probability.cmdef}
%\input{\cmdDefPath/math_calculus_derivatives.cmdef}
\input{\cmdDefPath/math_fractions.cmdef}
\input{\cmdDefPath/math_miscellaneous.cmdef}
%\input{\cmdDefPath/math_complex.cmdef}
%%%% add to complex eventually
%\newcommand{\real}[1]{\, \mbox{Re} \ifthenelse{\equal{#1}{^}}{^}{\, #1} }
%\newcommand{\imag}[1]{\, \mbox{Im} \ifthenelse{\equal{#1}{^}}{^}{\, #1} }
%% Paranthesis
% \input{\cmdDefPath/parentheses.cmdef}
%% Miscellaneous
\input{\cmdDefPath/format_sci_miscellaneous.cmdef}
\input{\cmdDefPath/format_text.cmdef} % for the superscript
%% Matrices
\input{\cmdDefPath/math_matrices_and_vectors.cmdef}
%% Matrix Operations
%\newcommand{\Tr}{\,\mbox{Tr}\,}
%% Equation Boxes
%\newcommand{\eqbox}[3]{\vspace{0.3cm} \noindent \framebox{\parbox[h][#1][c]{#2}{#3}} \vspace{0.3cm}}
%defines a command that boxes an equation \eqbox{ht box}{width box}{equation(s)}
%\newcommand{\eqdbx}[3]{\noindent \framebox{\frame{\parbox[h][#1][c]{#2}{#3}}}}
% defines a command that double boxes an equation \eqdbx{height box}{width box}{ equation(s) }
%<-----------------------------------------End Command Definitions------------------------------------------------->
%############################################Sectioning Templates###################################################
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%\subsection{}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %
%\subsubsection{}
%% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%%
%\subsubsubsection{}
%%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% | % | % | % | % | % | % | % | % | % | % | % | % | % | % |
%\appendix
% | % | % | % | % | % | % | % | % | % | % | % | % | % | % |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%##########################################End Sectioning Templates#################################################
\begin{document} % End of preamble and beginning of text.
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Normalization}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\label{Section:Normalization}
%
%% Declare the Variables for this section
\var{\Lambda^$, the $th matrix of a set that are potentially generators for the group}
%% alternative notation
\begin{declaration}
{
\Lambda^$ & The $th matrix of a set that are potentially generators for the group
}
\end{declaration}
%%
%
let $\Lambda^A$ be a set of matrices in the fundamental representation of a group that obey
%
$$
\commutate{\Lambda^A}{\Lambda^B} = \I g^{ABC} \Lambda^C
$$
%
Evaluate $\Tr\inp{\Lambda^A \Lambda^B}$ to find $S_\Lambda$, where
%
$$
\Tr\inp{\Lambda^A \Lambda^B} = S_\Lambda \KroneckerDelta{AB}
$$
%
Define the new set of matrices, $\lambda^A$, by
%
$$
\lambda^A \equiv \inp{\sqrt{\frac{2}{S_\Lambda}}} \Lambda^A
$$
%
With this definition, we then have
%
$$
\Tr\inp{\lambda^A \lambda^B} = 2 \KroneckerDelta{AB}
$$
%
which means that the $\lambda_i$'s are matrices like the Pauli Matrices of $SU(2)$ and the Gell-Mann lambda matrices of $SU(3)$.
The generators, $T^A$ for the fundamental representation of the group are then given by
%
$$
T^A = \frac{\lambda_i}{2}
$$
%
This then defines the structure constants of the group as
%
$$
f^{ABC} = \frac{1}{\sqrt{2 S_\Lambda}} g^{ABC}
$$
%
So, for any representation of the group with generators denoted by $T_R^A$, we have
%
$$
\commutate{T_R^A}{T_R^B} = \I f^{ABC} T_R^C
$$
%
and
%
$$
\Tr\inp{T_R^A T_R^B} = S_R \KroneckerDelta{AB}
$$
%
For the fundamental representation, $S_R \equiv S_0 = \half$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Completeness}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
\subsection{The Seemingly General Relation}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $U \in SU(n)$, so that we can express $U$ as
%
$$
U = e^{-i \theta^A T^A}
$$
%
where a sum on $A$ is implied ($A \in \{1,2, \ldots, n^2 - 1\}$) and $T^A$ are the generators of $SU(n)$, in any representation.
Taking the natural log gives
%
$$
\ln U = - \I \theta^A T^A
$$
%
Multiply this by an arbitrary generator $T^B$:
%
$$
T^B \ln U = - \I \theta^A T^B T^A
$$
%
and then take the trace
%
\begin{eqnarray*}
\Tr \inp{T^B \ln U}
& = & - \I \theta^A \Tr \inp{T^B T^A} \\
& = & - \I \theta^A S_R \delta^{AB}
\end{eqnarray*}
%
where the normalization condition has been used (see Section \ref{Section:Normalization}).
The above can be solved for $\theta^A$, yielding
%
$$
\theta^A = \frac{\I}{S_R} \Tr \inp{T^A \ln U}
$$
%
This is then substituted back into the expression for $\ln U$:
%
$$
\ln U = - \I \theta^A T^A = - \I \inp{ \frac{\I}{S_R} \Tr \inp{T^A \ln U} } T^A
$$
%
or
%
$$
\ln U = \frac{1}{S_R} T^A \Tr \inp{T^A \ln U}
$$
%
which, written out in component form, says
%
$$
\inp{\ln U}_{\mu \nu} = \frac{1}{S_R} T^A_{nm} \inp{\ln U}_{mn} T^A_{\mu \nu}
$$
%
Since this must be generally true for any $\ln U$ of the fundamental representation, we can conclude that
%
\begin{equation}
T^A_{nm} T^A_{\mu \nu} = S_0 \delta_{m \mu} \delta_{n \nu} + S_0 F_{\mu \nu} \delta_{mn}
\label{Eqn:Generator Completeness Incomplete}
\end{equation}
%
where the $F_{\mu \nu}$ are as of yet undetermined.
The necessity of the first term in Equation (\ref{Eqn:Generator Completeness Incomplete}) is obvious in that we need to get $\inp{\ln U}_{\mu \nu} = \inp{\ln U}_{\mu \nu}$ and that term does just that. The second term comes in because we have that
%
$$
\Tr{\ln U} = -\I \theta^A \Tr T^A = 0
$$
%
So, in Eqn (\ref{Eqn:Generator Completeness Incomplete}), the second term vanishes, and therefore must be included.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
\subsection{Fundamentally Valid and Nowhere Else}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let's now look at why the conclusion is only true in the fundamental representation. To do that, we need to investigate $\ln U$ in more detail. First, we find
\begin{der}
\inp{\ln U}^\dagger
& = & - \I \theta^A \inp{T^A}^\dagger
\\ % BEGIN
& = & - \I \theta^A T^A
\\ % RANK: 0; DESC: the generators are hermitian;
& = & - \inp{\ln U} % END
\end{der}
%
So, $\ln U$ is anti-hermitian.
An $(m \times m)$ anti-hermitian matrix $A$ has $m^2$ degrees of freedom: initially there are $2m^2$, but $A^\dagger = -A$ gives $m$ contraints from the diagonal and $2\pfrac{m(m-1)}{2}$ from the off-diagonal.
We also know that $\ln U$ is traceless -- an $(m \times m)$ traceless anti-hermitian matrix $\bar{A}$ has $m^2 - 1$ degrees of freedom.
Now, if $\ln U$ were taken in any representation $R$ as an $(N_R \times N_R)$ matrix, then if it were truly general, it would have $N_R^2 - 1$ degrees of freedom. However, we know that it only has $n^2 - 1$ degrees of freedom since it is a member of $SU(n)$. For $N_R > n$ there must therefore be more constraints on $\ln U$. Thus, the expression can only be generally true in the fundamental representation because it is only in that representation where the degrees of freedom match up on both sides for a general $(N_R \times N_R)$ matrix.
Perhaps an easier way to see it is to look at a general $(m \times m)$ complex matrix, $C$. $C$ has $2m^2$ degrees of freedom which can be split evenly among a hermitian part $H$ and an anti-hermitian part $A$
%
$$
H = \half \inp{C + C^\dagger}
$$
%
$$
A = \half \inp{C - C^\dagger}
$$
%
$C$ can be expressed as some complex coefficients times the identity and the generators of $SU(m)$ in the fundamental representation as follows:
%
$$
C = a_0 \IdentityMatrix + a_A T^A
$$
%
which can be verified by checking degrees of freedom: the right hand side has $2m^2$ free parameters, while the left hand side has $m^2 - 1$ for each generator, plus $1$ with the identity, and since the coefficients are complex they each have $2$ -- giving a grand total of $2m^2$.
%
If we write out the coefficients in terms of real parameters, we get
%
\begin{der}
C & = & \inp{\alpha_0 + \I \beta_0} \IdentityMatrix + \inp{ \alpha_A + \I \beta_A} T^A
\\ % BEGIN
& = & \inp{ \alpha_0 \IdentityMatrix + \alpha_A T^A }
+ \I \inp{ \beta_0 \IdentityMatrix + \beta_A T^A }
\\ % RANK: *; DESC: regroup elements into real and imaginary parts;
& = & H + A
% END
\end{der}
%
Thus individually, we can express these matrices as
%
$$
H = \inp{ \alpha_0 \IdentityMatrix + \alpha_A T^A }
$$
%
$$
A = \I \inp{ \beta_0 \IdentityMatrix + \beta_A T^A }
$$
%
Of course, $\ln U$ falls into the latter category and being traceless requires $\beta_0 = 0$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
\subsection{Completing Completeness}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We can now find the $F_{\mu \nu}$ (which we now know are only pertinent for the fundamental representation). To do this, we take the incomplete completeness expression, and let $m = \mu$, so we get
%
\begin{eqnarray*}
\inp{T^A T^A}_{\alpha \nu}
& = & S_0 \KroneckerDelta[d]{m m} \KroneckerDelta[d]{\alpha \nu} + S_0 F_{m \nu} \KroneckerDelta[d]{m \alpha}
\\
& = & S_0 n \KroneckerDelta[d]{\alpha \nu} + S_0 F_{\alpha \nu}
\end{eqnarray*}
%
or
%
\begin{eqnarray*}
F_{\mu \nu}
& = & \frac{1}{S_0} \inp{T^A T^A}_{\mu \nu} - n \KroneckerDelta[d]{\mu \nu}
\\
& = & \frac{1}{S_0} C_0 \KroneckerDelta[d]{\mu \nu} - n \KroneckerDelta[d]{\mu \nu}
\\
& = & \inp{ \frac{C_0}{S_0} - n } \KroneckerDelta[d]{\mu \nu}
\end{eqnarray*}
%
That's not helpful it you don't know $C_0$ (we do know $S_0$), so we either derive $C_0$ another way or find a differnt route for the completeness relation. Since we know about a general hermitian matrix and its relation to the generators, let's get the completeness another way. We have
%
$$
\alpha_0 = \frac{1}{n} \Tr{H}
$$
%
$$
\alpha_A = \frac{1}{S_0} \Tr\inp{H T^A}
$$
%
which says
%
$$
H = \frac{1}{n} \inp{ \Tr{H} } \IdentityMatrix + \frac{1}{S_0} \inb{\Tr\inp{H T^A}} T^A
$$
%
or
%
\begin{eqnarray*}
H_{\alpha \beta}
& = & \frac{1}{n} H_{\mu \nu} \KroneckerDelta[d]{\nu \mu} \KroneckerDelta[d]{\alpha \beta}
+ \frac{1}{S_0} H_{\mu \nu} T^A_{\nu \mu} T^A_{\alpha \beta}
\\
& = & H_{\mu \nu} \inb{
\frac{1}{n} \KroneckerDelta[d]{\nu \mu} \KroneckerDelta[d]{\alpha \beta}
+ \frac{1}{S_0} T^A_{\nu \mu} T^A_{\alpha \beta}
}
\end{eqnarray*}
%
and since $H$ is both arbitrary and general
%
$$
\frac{1}{n} \KroneckerDelta[d]{\nu \mu} \KroneckerDelta[d]{\alpha \beta} + \frac{1}{S_0} T^A_{\nu \mu} T^A_{\alpha \beta}
= \KroneckerDelta[d]{\mu \alpha} \KroneckerDelta[d]{\nu \beta}
$$
%
giving, finally:
%
$$
T^A_{\nu \mu} T^A_{\alpha \beta}
= S_0 \inb{ \KroneckerDelta[d]{\mu \alpha} \KroneckerDelta[d]{\nu \beta}
- \frac{1}{n} \KroneckerDelta[d]{\nu \mu} \KroneckerDelta[d]{\alpha \beta}
}
$$
%
Comparing this to our other expression gives
%
$$
S_0 F_{\mu \nu} = - \frac{S_0}{n} \KroneckerDelta[d]{\mu \nu}
$$
%
hence
$$
C_0 - n S_0 = - \frac{S_0}{n}
$$
%
$$
C_0 = n S_0 - \frac{S_0}{n}
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Representation Independent Relations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For any representation, the only thing that we know for sure is that
%
$$
\commutate{T^A}{T^B} = \I f^{ABC} T^C
$$
%
But this can be used to show quite a few things. For starters, let's find the structure constants in terms of the generators.
Multiply the above expression by $T^D$
%
$$
\commutate{T^A}{T^B} T^D = \I f^{ABC}T^C T^D
$$
%
and take the trace
%
\begin{der}
\Tr \inp{ \commutate{T^A}{T^B} T^D }
& = & \I f^{ABC} \Tr\inp{T^D T^C}
\\ % BEGIN
& = & \I f^{ABC} S_R \KroneckerDelta{CD}
\\ % RANK: 0; DESC: Use the Dynkin Index Relation;
& = & \I S_R f^{ABD}
% END
\end{der}
%
which implies
%
\newcommand{\fABCinTermsofTAAnyRep}{%
f^{ABC} = \frac{-\I}{S_R} \Tr \inp{\commutate{T^A}{T^B} T^C }%
}
\begin{equation}
\fABCinTermsofTAAnyRep
\end{equation}
%
We can also sums of structure constants: take the commutation relationship and multiply it by itself
%
$$
\inp{ \I f^{ABC} T^C }\inp{ \I f^{DEF} T^F } = \commutate{T^A}{T^B} \commutate{T^D}{T^E}
$$
%
and take the trace
%
$$
- f^{ABC} f^{DEF} \Tr\inp{T^C T^F} = \Tr \inp[0.75cm]{\commutate{T^A}{T^B} \commutate{T^D}{T^E}}
$$
%
Invoking the definition of the Dynkin Index again, we have
%
$$
- f^{ABC} f^{DEC} S_R = \Tr \inp[0.75cm]{\commutate{T^A}{T^B} \commutate{T^D}{T^E}}
$$
%
or
%
\newcommand{\fABCfDECAnyRep}{%
f^{ABC} f^{DEC} = - \frac{1}{S_R} \Tr \inp[0.75cm]{\commutate{T^A}{T^B} \commutate{T^D}{T^E}}%
}
\begin{equation}
\fABCfDECAnyRep
\end{equation}
%
We can then sum over another index as well:
%
$$
f^{ABC} f^{DBC} = - \frac{1}{S_R} \Tr \inp[0.75cm]{\commutate{T^A}{T^B} \commutate{T^D}{T^B}}
$$
%
Since it turns out that this trace appears frequently we'll evaluate it in a general manner
%
\begin{eqnarray*}
\Tr \inp[0.75cm]{ \inp{ T^A T^B \pm_L T^B T^A } \inp{T^D T^B \pm_R T^B T^D} }
& & \hspace{3in}
\end{eqnarray*}
\vspace{-1cm}
\begin{eqnarray*}
& = & \Tr \inp[0.75cm]{
T^A T^B T^D T^B
\pm_R T^A T^B T^B T^D
\pm_L T^B T^A T^D T^B
+ T^B T^A T^B T^D
}
\\
& = & \Tr \inp[0.75cm]{
2 T^A T^B T^D T^B
\pm_R T^D T^A T^B T^B
\pm_L T^A T^D T^B T^B
}
\\ %RANK: 0; DESC: Use cyclic property of the Trace;
& = & 2 \Tr \inp[0.75cm]{ T^A T^B T^D T^B }
+ \Tr \inp[0.75cm]{ \inp{\pm_L T^A T^D \pm_R T^D T^A} T^B T^B }
\end{eqnarray*}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Fundamental Representation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We have thus far shown the following for this representation
%
$$
T^A_{\alpha \beta} T^A_{\mu \nu}
= S_0 \inb{ \KroneckerDelta[d]{\nu \alpha} \KroneckerDelta[d]{\mu \beta}
- \frac{1}{n} \KroneckerDelta[d]{\mu \nu} \KroneckerDelta[d]{\alpha \beta}
}
$$
%
$$
C_0 = n S_0 - \frac{S_0}{n}
$$
%
and we have chosen
%
$$
S_0 = \half
$$
%
Let's see what else we can get for the fundamental representation. First, we notice that the fundamental representation is special because we can actually get the anticommutation relations to close -- that is, in addition to
%
$$
\commutate{T^A}{T^B} = \I f^{ABC} T^C
$$
%
we have
%
$$
\acommutate{T^A}{T^B} = \frac{2 S_0}{n}\KroneckerDelta{AB} + d^{ABC} T^C
$$
%
where that $\frac{1}{n}$ pops up because of the implicit identity matrix multiplying the kronecker delta and the $2 S_0$ is necessary because if we take the trace, then we get $2 S_0 \KroneckerDelta{AB}$ on the left-hand side.
So, we have even more constants in the fundamental representation -- yet for the time being, we'll just focus on the structure constants. We know
%
\begin{eqnarray*}
f^{ABC} f^{DBC}
& = & - \frac{1}{S_0}
\inb[1cm]{
2 \Tr \inp[0.75cm]{ T^A T^B T^D T^B }
+ \Tr \inp[0.75cm]{ \inp{- T^A T^D - T^D T^A} T^B T^B }
}
\\
%====
%====
& = & - \frac{1}{S_0}
\inb[1cm]{
2 \Tr \inp[0.75cm]{ T^A T^B T^D T^B }
- \Tr \inp[0.75cm]{ \acommutate{T^A}{T^D}T^B T^B }
}
\\
%====
%====
& = & - \frac{2}{S_0} \Tr \inp[0.75cm]{ T^A T^B T^D T^B }
+ \frac{1}{S_0} \Tr \inp[0.75cm]{ \acommutate{T^A}{T^D}T^B T^B }
\\
%====
%====
& = & - \frac{2}{S_0} T^A_{\nu \alpha} T^B_{\alpha \beta} T^D_{\beta \mu} T^B_{\mu \nu}
+ \frac{1}{S_0} \acommutate{T^A}{T^D}_{\nu \alpha} T^B_{\alpha \beta} T^B_{\beta \nu}
\\ % DESC: Write trace out in component form;
%====
%====
& = & - \frac{2}{S_0} T^A_{\nu \alpha} T^D_{\beta \mu}
\inp{ S_0 }
\inb{ \KroneckerDelta[d]{\nu \alpha} \KroneckerDelta[d]{\mu \beta}
- \frac{1}{n} \KroneckerDelta[d]{\mu \nu} \KroneckerDelta[d]{\alpha \beta}
}
\\
& &
+ \frac{1}{S_0} \acommutate{T^A}{T^D}_{\nu \alpha}
\inp{ S_0 }
\inb{ \KroneckerDelta[d]{\nu \alpha} \KroneckerDelta[d]{\beta \beta}
- \frac{1}{n} \KroneckerDelta[d]{\beta \nu} \KroneckerDelta[d]{\alpha \beta}
}
\\ % DESC: Use Completeness relation
%====
%====
& = & - 2 T^A_{\alpha \alpha} T^D_{\beta \beta}
+ \frac{2}{n} T^A_{\mu \alpha} T^D_{\alpha \mu}
\\
& &
+ n \acommutate{T^A}{T^D}_{\alpha \alpha}
- \frac{1}{n} \acommutate{T^A}{T^D}_{\alpha \alpha}
\\ % DESC: Simplify Kroneckers
%====
%====
& = & \frac{2}{n} \Tr \inp{T^A T^D}
+ n \Tr \inp{ T^A T^D }
- \frac{1}{n} \Tr\inp{T^A T^D}
\\ % DESC: Generators are traceless
%====
%====
& = & \frac{2}{n} S_0 \KroneckerDelta{AD}
+ 2 n S_0 \KroneckerDelta{AD}
- \frac{2}{n} \KroneckerDelta{AD}
\\ % DESC: Use Dynkin Index Definition
%====
%====
& = & 2 n S_0 \KroneckerDelta{AD}
\end{eqnarray*}
%
\newcommand{\eqnfABCfDBCAnyRep}{%
f^{ABC} f^{DBC} = 2 n S_0 \KroneckerDelta{AD}%
}
$$
\eqnfABCfDBCAnyRep
$$
%
Since this is a statement on the structure constants themselves, independent of any generators, this equation must actually be true for all representations\footnote{yes, there is an $S_0$ in that equation, however, since we get to choose this parameter, it's a reflection of the fact that there is an overall free normalization constant and not tying that expression to the fundamental representation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Adjoint Representation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent In any representation we have found
%
$$
f^{ABC} f^{DEC} = - \frac{1}{S_R} \Tr \inp{ \commutate{T^A}{T^B} \commutate{T^D}{T^E} }
$$
%
$$
f^{ABC} f^{DBC} = n \KroneckerDelta{AD}
$$
%
Now we know that in the adjoint representation ($\cal A$)
%
$$
T^A_{\alpha \beta} = f^{A\alpha \beta}
$$
%
Hence
%
\begin{eqnarray*}
T^A_{\alpha \beta} T^A_{\mu \nu}
& = & f^{A \alpha \beta} f^{A \mu \nu}
\\
& = & - f^{\alpha \beta A} f^{\mu A \nu}
\\
& = & - T^\alpha_{\beta A} T^\mu_{A \nu}
\\
& = & - \inp{ T^\alpha T^\mu }_{\beta \nu}
\end{eqnarray*}
%
So we can find $C_{\cal A}$:
%
\begin{eqnarray*}
f^{A \mu \beta} f^{A \mu \nu}
& = & - \inp{ T^\mu T^\mu }_{\beta \nu}
\\
& = & - C_{\cal A} \KroneckerDelta{\beta \nu}
\end{eqnarray*}
%
$$
\Rightarrow C_{\cal A} = - n
$$
%
We can also get $S_{\cal A}$
%
\begin{eqnarray*}
\Tr\inp{T^A T^D}
& = & T^A_{BC} T^D_{CB}
\\
& = & f^{ABC} f^{DCB}
\\
& = & - n \KroneckerDelta{AD}
\end{eqnarray*}
%
$$
\Rightarrow S_{\cal A} = - n
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Genius of Dragt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Extension of Dragt's Genius}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{eqnarray*}
T_\pm & = & F_1 \pm i F_2 \\
T_3 & = & F_3 \\
V_\pm & = & F_4 \pm i F_5 \\
U_\pm & = & F_6 \pm i F_7 \\
Y & = & \frac{2}{\sqrt3} F_8
\end{eqnarray*}
%
Yielding
%
\begin{eqnarray*}
F_1 & = & \half \inp{T_+ + T_-} \\
F_2 & = & \frac{i}{2} \inp{T_+ - T_-} \\
F_3 & = & T_3 \\
F_4 & = & \half \inp{V_+ + V_-} \\
F_5 & = & \frac{i}{2} \inp{V_+ - V_-} \\
F_6 & = & \half \inp{U_+ + U_-} \\
F_7 & = & \frac{i}{2} \inp{U_+ - U_-} \\
F_8 & = & \frac{\sqrt3}{2} Y
\end{eqnarray*}
%
So, we find
%
\begin{der}
\DotProduct{\vect{F}^{(1)}}{\vect{F}^{(2)}}
& = & F^{(1)}_1 F^{(2)}_1
+ F^{(1)}_2 F^{(2)}_2
+ F^{(1)}_3 F^{(2)}_3
+ F^{(1)}_4 F^{(2)}_4
+ F^{(1)}_5 F^{(2)}_5
+ F^{(1)}_6 F^{(2)}_6
+ F^{(1)}_7 F^{(2)}_7
+ F^{(1)}_8 F^{(2)}_8
\\ %BEGIN
& = & \fourth \inp{T^{(1)}_+ + T^{(1)}_-} \inp{T^{(2)}_+ + T^{(2)}_-}
- \fourth \inp{T^{(1)}_+ - T^{(1)}_-} \inp{T^{(2)}_+ - T^{(2)}_-}
+ T^{(1)}_3 T^{(2)}_3
\\
& & + \; \fourth \inp{V^{(1)}_+ + V^{(1)}_-} \inp{V^{(2)}_+ + V^{(2)}_-}
- \fourth \inp{V^{(1)}_+ - V^{(1)}_-} \inp{V^{(2)}_+ - V^{(2)}_-}
\\
& & + \; \fourth \inp{U^{(1)}_+ + U^{(1)}_-} \inp{U^{(2)}_+ + U^{(2)}_-}
- \fourth \inp{U^{(1)}_+ - U^{(1)}_-} \inp{U^{(2)}_+ - U^{(2)}_-}
+ \frac{3}{4} Y^{(1)} Y^{(2)}
\\ %RANK: 1; DESC: Substitute identities;
& = & \fourth \left[ \parbox[h][1cm]{0cm}{}
T^{(1)}_+ T^{(2)}_+
+ T^{(1)}_+ T^{(2)}_-
+ T^{(1)}_- T^{(2)}_+
+ T^{(1)}_- T^{(2)}_-
\right.
\\
& &
- T^{(1)}_+ T^{(2)}_+
+ T^{(1)}_+ T^{(2)}_-
+ T^{(1)}_- T^{(2)}_+
- T^{(1)}_- T^{(2)}_-
+ T^{(1)}_3 T^{(2)}_3
\\
& &
+ V^{(1)}_+ V^{(2)}_+
+ V^{(1)}_+ V^{(2)}_-
+ V^{(1)}_- V^{(2)}_+
+ V^{(1)}_- V^{(2)}_-
\\
& &
- V^{(1)}_+ V^{(2)}_+
+ V^{(2)}_- V^{(1)}_+
+ V^{(1)}_- V^{(2)}_+
- V^{(1)}_- V^{(2)}_-
\\
& &
+ U^{(1)}_+ U^{(2)}_+
+ U^{(1)}_+ U^{(2)}_-
+ U^{(1)}_- U^{(2)}_+
+ U^{(1)}_- U^{(2)}_-
\\
& & \left. \parbox[h][1cm]{0cm}{}
- U^{(1)}_+ U^{(2)}_+
+ U^{(2)}_- U^{(1)}_+
+ U^{(1)}_- U^{(2)}_+
- U^{(1)}_- U^{(2)}_-
+ 3 Y^{(1)} Y^{(2)}
\right]
\\ %RANK: 0; DESC: Expand the terms;
& = & \fourth \left[ \parbox[h][1cm]{0cm}{}
2 T^{(1)}_+ T^{(2)}_-
+ 2 T^{(1)}_- T^{(2)}_+
+ T^{(1)}_3 T^{(2)}_3
+ 2 V^{(1)}_+ V^{(2)}_-
\right.
\\
& & \left. \parbox[h][1cm]{0cm}{}
+ 2 V^{(1)}_- V^{(2)}_+
+ 2 U^{(2)}_- U^{(1)}_+
+ 2 U^{(1)}_- U^{(2)}_+
+ 3 Y^{(1)} Y^{(2)}
\right]
\\ %RANK: 1; DESC: Simplify;
& = & \half \left[ \parbox[h][1cm]{0cm}{}
T^{(1)}_+ T^{(2)}_-
+ T^{(1)}_- T^{(2)}_+
+ \half T^{(1)}_3 T^{(2)}_3
+ V^{(1)}_+ V^{(2)}_-
\right.
\\
& & \left. \parbox[h][1cm]{0cm}{}
+ V^{(1)}_- V^{(2)}_+
+ U^{(2)}_- U^{(1)}_+
+ U^{(1)}_- U^{(2)}_+
+ \frac{3}{2} Y^{(1)} Y^{(2)}
\right]
\\ %END; DESC: Distribute one-half;
\end{der}
%
We now consider
%
\begin{eqnarray*}
\bra{j_t m^{(1)}_t j_y m^{(1)}_y}\bra{j_t m^{(2)}_t j_y m^{(2)}_y}
\DotProduct{\vect{F}^{(1)}}{\vect{F}^{(2)}}
\ket{j_t n^{(1)}_t j_y n^{(1)}_y} \ket{j_t n^{(2)}_t j_y n^{(2)}_y}
\end{eqnarray*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Relationships Summary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
\subsection{Representation Independent}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$
\fABCfDECAnyRep
$$
%
$$
\fABCinTermsofTAAnyRep
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ REFERENCES ]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
%\begin{thebibliography}{hello}
%\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ END REFERENCES ]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document} % End of document