text:
\chapter{Masses and Coupling Constants}

In this chapter, the coupling constants and masses for the standard model
vector fields are given in a formulation adapted for an interpretation as group
normalization constants.

\section{Particle and Interaction Normalizations\\ in the Standard Model}
Obviously, coupling constants and masses play an important role in the
standard model interactions\cite{WEIN}. They allow
the experimental tests of the model by quantifying
 the transition from the interaction
oriented field language to the  ground state related
particle language.

The standard model ground state properties
are implemented by the complex $2$-dimensional
Higgs field $\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^{\alpha}  \def\be{\beta} \def\ga{\gamma=1,2}(x)$ with a mass $M$ as ground state value,
characterizing the minima of the potential $V(\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta)$
\begin{eqn}{l}
V(\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta)=\la_0(\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^\star \varphi}  \def\Ga{\Gamma}  \def\Th{\Theta-M^2)^2,~~\la_0>0\cr
\angle {\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta(x)}=\ul\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta={\scriptsize\pmatrix{
0\cr M\cr}},~~M\ne 0
\end{eqn}Therewith
a translation invariant  internal  reference system is fixed
for the hyperisospin field symmetry
group ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)={\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_2)\circ}\def\x{\times}\def\oo{\infty\SU(2)$
up to a
 remaining abelian  particle symmetry with respect to
 the electromagnetic ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)_+\cong{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)$ subgroup
\begin{eqn}{rll}
&\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta(x)\longmapsto u(\alpha}  \def\be{\beta} \def\ga{\gamma_0,\rvec\alpha}  \def\be{\beta} \def\ga{\gamma)\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta(x),~~
&u(\alpha}  \def\be{\beta} \def\ga{\gamma_0,\rvec\alpha}  \def\be{\beta} \def\ga{\gamma)=e^{i{\alpha}  \def\be{\beta} \def\ga{\gamma_0\bl 1_2+\rvec\alpha}  \def\be{\beta} \def\ga{\gamma\rvec\tau\over\sqrt 2}}\in{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)\cr
&\ul\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta\longmapsto
u(\alpha}  \def\be{\beta} \def\ga{\gamma_+)\ul\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta,~~&u(\alpha}  \def\be{\beta} \def\ga{\gamma_+)=e^{i\alpha}  \def\be{\beta} \def\ga{\gamma_+{\bl 1_2+\tau_3\over\sqrt2}}\in{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)_+\cr
\end{eqn}The electromagnetic ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)_+$ is spanned with the projector
${\bl 1_2+\tau_3\over2}$ connecting hypercharge ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_2)\cong{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)$ and
isospin $\SU(2)$.
With respect to the Killing form the  $\sqrt2$-normalization
is used in the ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$ Lie algebra basis
$\{i{\bl 1_2\over \sqrt2},i{\rvec\tau\over\sqrt2}\}$ with the
convenient 'double' trace
$\tr {\tau_a\over\sqrt2}{\tau_b\over\sqrt2}=\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}$.

With regard to the internal symmetries
the real $3$-dimensional Goldstone manifold ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)/{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)_+$
characterizes the transition from fields to  particles.
In contrast to the
${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$-symmetry for fields, particles can have only a nontrivial
electromagnetic ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)$-symmetry.
All particles come as $\SU(2)$-singlets - there is no mass degenerated nontrivial
particle isomultiplet.

The kinetic terms for the Higgs field
in connection with the vector gauge fields
$B^{j=0,1,2,3}(x)$
for hypercharge ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_2)$ symmetry  and $\rvec W^j(x)$ for isospin $\SU(2)$
symmetry are given in the
 Lagrangian
\begin{eqn}{rl}
\bl L(\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta,B,W)
=&|\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_j-i{B_j+\rvec W_j\rvec\tau\over\sqrt2}\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta|^2- V(\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta)\cr
&+F^{jk}{\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_jB_k-\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_kB_j\over2}+g_1^2{F^{jk}F_{jk}\over4}\cr
&+\rvec F^{jk}{\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_j\rvec W_k-\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_k\rvec W_j\over2}+
g_2^2{\rvec F^{jk}\rvec F_{jk}\over4}
-{(\rvec F^{jk}\x\rvec W_k)\rvec W_j\over2}\cr
\end{eqn}The gauge fixing terms are omitted. For the vector fields a 1st
order derivative formalism is used
with gauge fields and curvatures
as canonical pairs, $(B,F)$ and $(\rvec W,\rvec F)$,
connected in the derivative terms.
The curvatures arise by variation in the equations of motion
\begin{eqn}{l}
g_1^2F^{jk}=\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla^kB^j-\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla^jB^k,~~
g_2^2\rvec F^{jk}=\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla^k\rvec W^j-\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla^j\rvec W^k+\rvec W^k\x\rvec W^j\cr
\end{eqn}where the coupling constants show up
as curvature normalizations. $g_1^2$ and $g_2^2$ are called internal
normalizations for hypercharge and isospin interaction resp.

A more familiar notation uses the coupling constants in the current gauge field
coupling with the rescaling
\begin{eqn}{l}
\left.\begin{array}{ll}B= g_1\ul B,~~&F={1\over g_1}\ul F\cr
\rvec W= g_2\ul{\rvec  W},~~&\rvec F={1\over g_2}\ul{\rvec  F}
\end{array}\right\}
~\Rightarrow~}\def\sub{\subseteq}\def\subnoteq{\subset
 J^kB_k+\rvec J^k\rvec W_k=
g_1J^k\ul B_k+g_2\rvec J^k\ul{\rvec W}_k
\end{eqn}which shows that the coupling constants are
dynamically  relevant only for the case
of a nontrivial interaction. Only the squares $g^2$ occur,
therefore
$g=+\sqrt{g^2}$ can be defined as positive definite. Positive and negative
electromagnetic  charges arise by ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)_+$-representations with positive or
negative winding numbers.

The hyperisospin transformation on the gauge sector is given by
\begin{eqn}{rll}
&B_j\longmapsto B_j+\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_j\alpha}  \def\be{\beta} \def\ga{\gamma_0,~~&F_{jk}\longmapsto F_{jk}\cr
&\rvec W_j \longmapsto O(\rvec \alpha}  \def\be{\beta} \def\ga{\gamma)(\rvec W_j)+\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_j\rvec \alpha}  \def\be{\beta} \def\ga{\gamma,~~
&\rvec F_{jk}\longmapsto O(\rvec\alpha}  \def\be{\beta} \def\ga{\gamma)(\rvec F_{jk})\cr
&&O(\rvec\alpha}  \def\be{\beta} \def\ga{\gamma)\in\SO(3)\cong{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)/{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_2)\cr
\end{eqn}

The projection of the interaction fields to the ground state gives
the vector particle  mass terms
\begin{eqn}{rl}
\bl L(\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta,B,W)=&|{\scriptsize\pmatrix{
B_j-W^3_j\cr
W_j^1+iW_j^2\cr}}|^2{M^2\over2}+\dots\cr
&+F^{jk}{\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_jB_k-\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_kB_j\over2}+g_1^2{F^{jk}F_{jk}\over4}
+\rvec F^{jk}{\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_j\rvec W_k-\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_k\rvec W_j\over2}+
g_2^2{\rvec F^{jk}\rvec F_{jk}\over4}+\dots
\end{eqn}diagonalizable with the Weinberg rotation
$O(\theta}   \def\vth{\vartheta} \def\io{\iota)$ defining the massive vector field $Z^j(x)$ and the massless
electromagnetic one $ A^j(x)$
\begin{eqn}{l}
O(\theta}   \def\vth{\vartheta} \def\io{\iota)\!=\!{\scriptsize\pmatrix{
\cos\theta}   \def\vth{\vartheta} \def\io{\iota&\tan\theta}   \def\vth{\vartheta} \def\io{\iota\cr
-\sin\theta}   \def\vth{\vartheta} \def\io{\iota&\cos\theta}   \def\vth{\vartheta} \def\io{\iota\cr}}\in\SO(2):
\cases{
{\scriptsize\pmatrix{
\ul{F_A}\cr
\ul{F_ Z}\cr}}
\!=\!{\scriptsize\pmatrix{
e F_A\cr
g_ZF_Z\cr}}
\!=\!O(\theta}   \def\vth{\vartheta} \def\io{\iota)
{\scriptsize\pmatrix{
g_1 F\cr
g_2 F_3\cr}}
\!=\!O(\theta}   \def\vth{\vartheta} \def\io{\iota)
{\scriptsize\pmatrix{
\ul{  F}\cr
\ul{  F_3}\cr}}                   \cr
{\scriptsize\pmatrix{
\ul A\cr
\ul Z\cr}}\!=\!
{\scriptsize\pmatrix{
{1\over e} A\cr
{1\over g_Z} Z\cr}}
\!=\!O(\theta}   \def\vth{\vartheta} \def\io{\iota)
{\scriptsize\pmatrix{
{1\over g_1} B\cr
{1\over g_2}W_3\cr}}
\!=\!O(\theta}   \def\vth{\vartheta} \def\io{\iota)
{\scriptsize\pmatrix{
\ul{B}\cr
\ul{W_3} \cr}}\cr}
\end{eqn}The Weinberg angle $\theta}   \def\vth{\vartheta} \def\io{\iota$ and the
electromagnetic ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)_+$-normalization (coupling constant) $ e^2$ are related
to the ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_2)$ and $\SU(2)$ normalizations (coupling
constants) $g_{1,2}^2$  by  two conditions:
The massive field $Z$ has
to occur in the combination of the $M^2$-proportional term
in the Lagrangian and
the massless field $A$ has to be
normalized with respect to the remaining gauge invariance
\begin{eqn}{rl}
Z_j\sim B_j-W_j^3,~~
&~\Rightarrow~}\def\sub{\subseteq}\def\subnoteq{\subset\cases{
\tan\theta}   \def\vth{\vartheta} \def\io{\iota={g_1\over g_2}\cr
O(\theta}   \def\vth{\vartheta} \def\io{\iota)={1\over\sqrt{g_1^2+g_2^2}}{\scriptsize\pmatrix{
g_2&g_1\cr
-g_1&g_2\cr}}\cr}\cr
 A_j\longmapsto  A_j+\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_j\alpha}  \def\be{\beta} \def\ga{\gamma_+&~\Rightarrow~}\def\sub{\subseteq}\def\subnoteq{\subset
{1\over  e^2}={1\over g_1^2}+{1\over g_2^2}\cr
\end{eqn}These two relations for $\theta}   \def\vth{\vartheta} \def\io{\iota$ and $e^2$ define
the electroweak rectangular triangle\cite{S921} with the ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)_+$-hypotenusis
${1\over e}$ and  the ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_2)$ and $\SU(2)$ sides ${1\over g_1}$ and ${1\over g_2}$
resp. The $Z$-coupling constant ${1\over g_Z}$ is determined as the
height of this triangle
\begin{eqn}{l}
eg_Z=g_1g_2~\Rightarrow~}\def\sub{\subseteq}\def\subnoteq{\subset g_Z^2=g_1^2+g_2^2
\end{eqn}

The orthogonal Weinberg transformation $O(\theta}   \def\vth{\vartheta} \def\io{\iota)$ for the rescaled fields $\ul B,\ul F$ etc.
is a special linear transformation
$S(\theta}   \def\vth{\vartheta} \def\io{\iota)$ for the unscaled fields $B,F$ etc.,
contragredient (inverse transposed, denoted by $-1T$) to each other
for the canonical pairs
\begin{eqn}{l}
\begin{array}{l}
S(\theta}   \def\vth{\vartheta} \def\io{\iota)={\scriptsize\pmatrix{
1&1\cr
-\sin^2\theta}   \def\vth{\vartheta} \def\io{\iota&\cos^2\theta}   \def\vth{\vartheta} \def\io{\iota\cr}}\in\SL({\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}^2)\cr
\d S(\theta}   \def\vth{\vartheta} \def\io{\iota)=S(\theta}   \def\vth{\vartheta} \def\io{\iota)^{-1 T}={\scriptsize\pmatrix{
\cos^2\theta}   \def\vth{\vartheta} \def\io{\iota&\sin^2\theta}   \def\vth{\vartheta} \def\io{\iota\cr
-1&1\cr}}\end{array}:
\cases{
{\scriptsize\pmatrix{
F_A\cr
F_ Z\cr}}
=S(\theta}   \def\vth{\vartheta} \def\io{\iota)
{\scriptsize\pmatrix{
F\cr
 F_3\cr}}\cr
{\scriptsize\pmatrix{
A\cr
 Z\cr}}
=\d S(\theta}   \def\vth{\vartheta} \def\io{\iota)
{\scriptsize\pmatrix{
 B\cr
W_3\cr}}\cr}
\end{eqn}The  transformation
$S(\theta}   \def\vth{\vartheta} \def\io{\iota)$ shows the ground state induced
direction $\bl 1_2+\tau_3$ in the first line  as $(1,1)$,
i.e. $F+F_3$, the contragredient
transformation $\d S(\theta}   \def\vth{\vartheta} \def\io{\iota)$ involves the complementary 'broken'  direction
$\bl 1_2-\tau_3$ in the 2nd line $(-1,1)$, i.e. $-W_3+B$.

The normalizations (coupling constants) of the ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_2)$ and $\SU(2)$ interaction
fields on the one side and ground state projected  fields on the
other side are related to each other as follows
\begin{eqn}{l}
S(\theta}   \def\vth{\vartheta} \def\io{\iota){\scriptsize\pmatrix{
{1\over g_1^2}&0\cr 0&{1\over g_2^2}\cr}}S(\theta}   \def\vth{\vartheta} \def\io{\iota)^T
=
{\scriptsize\pmatrix{
{1\over e^2}&0\cr 0&{1\over g_Z^2}\cr}}\hbox{ for }\tan^2\theta}   \def\vth{\vartheta} \def\io{\iota={g_1^2\over g_2^2}
\end{eqn}

The Weinberg rotation from interaction fields to particle fields leads
to the kinetic terms for
the vector fields
\begin{eqn}{rl}
\bl L(B,W)=& F_ A^{jk}{\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_j A_k-\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_k A_j\over2}
+ e^2{F_ A^{jk}F_{ A jk}\over4}\cr
&+F_Z^{jk}{\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_jZ_k-\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_kZ_j\over2}+
g_Z^2{F_Z^{jk}F_{Zjk}\over4}
  +{M^2(g_1^2+g_2^2)\over g_Z^2}{Z_jZ^j\over2}\cr
&+ {\displaystyle \sum}\def\PROD{\displaystyle \prod_{a=1,2}}(F_a^{jk}{\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_jW_{ak}-\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_kW_{aj}\over2}+
g_2^2{F^{jk}_a F_{ajk}\over4}+M^2{W_{aj}W^j_a\over2})
\end{eqn}

The particle masses $\mu$ arise from the products of
the coefficients for the curvature ${F^2\over 4}$ and the vector field
${Z^2,W^2\over2}$ whereas the quotients of the coefficients
define  constants
$\ell^2$ and ${1\over\ell^2}$, called external normalizations related to the
particles
\begin{eqn}{rl}
\bl L(B,W)=&F_ A^{jk}{\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_j A_k-\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_k A_j\over2}
+ e^2{F_ A^{jk}F_{ A jk}\over4}\cr
&+F_Z^{jk}{\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_jZ_k-\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_kZ_j\over2}+
\mu_Z(\ell^2_Z{F_Z^{jk}F_{Zjk}\over4}
  +{1\over\ell^2_Z}{Z_jZ^j\over2})\cr
&+ {\displaystyle \sum}\def\PROD{\displaystyle \prod_{a=1,2}}(F_a^{jk}{\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_jW_{ak}-\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_kW_{aj}\over2}+
\mu_W(\ell^2_W{F^{jk}_a F_{ajk}\over4}+{1\over\ell^2_W}{W_{aj}W^j_a\over2}))\cr
\hbox{with }&\left\{
\begin{array}{ll}
\mu^2_Z=(g_1^2+g_2^2)M^2=g_Z^2M^2,~~
&\ell^4_Z={g_1^2+g_2^2\over M^2}={g_Z^2\over M^2}\cr
\mu^2_W=g_2^2M^2,~~&\ell^4_W={g_2^2\over M^2}\end{array}\right.\cr
\end{eqn}Planck's constant
$\hbar$ and the maximal action
velocity $c$ are used as natural units.

The currents $J^k(x)$ for hypercharge ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_2)$ and $\rvec J^k(x)$
for isospin $\SU(2)$ have to be transformed as the curvatures
\begin{eqn}{rll}
{\scriptsize\pmatrix{
J_A\cr
J_Z\cr}}
&=S(\theta}   \def\vth{\vartheta} \def\io{\iota)
{\scriptsize\pmatrix{
 J\cr
 J_3\cr}}&={\scriptsize\pmatrix{
J+J_3\cr
-\sin^2\theta}   \def\vth{\vartheta} \def\io{\iota J+\cos^2\theta}   \def\vth{\vartheta} \def\io{\iota J_3\cr
}}\cr
\end{eqn}to obtain the interaction in the asymptotic particle basis
\begin{eqn}{rl}
-\bl L(J,B,W)&=J^kB_k+\rvec J^k\rvec W_k=
J_ A^k  A_k+ J_Z^k Z_k+{\displaystyle \sum}\def\PROD{\displaystyle \prod_{a=1,2}}J_a^kW_{ak}\cr
&=(J^k+J^k_3)A_k+(-\sin^2\theta}   \def\vth{\vartheta} \def\io{\iota J^k+\cos^2\theta}   \def\vth{\vartheta} \def\io{\iota J_3^k)Z_k
+{\displaystyle \sum}\def\PROD{\displaystyle \prod_{a=1,2}}J_a^kW_{ak}\cr
\end{eqn}

The standard model cannot determine the  dimensionless values
${ e^2\over4\pi}\sim{1\over137}$, $\tan^2\theta}   \def\vth{\vartheta} \def\io{\iota\sim0.3$ and
the ratio of $M\sim 123 {\ro{GeV}\over c^2}$ to any reference mass.
 Those values  are taken from the experiments with the energy momentum
 dependent Weinberg angle at the $Z$-mass.
 Using dual
 rectangular triangles
 \begin{eqn}{l}
 \left.\begin{array}{rl}
\cl T=  (a_1,a_2;c|h):~~&a_1^2+a_2^2=c^2\cr
 \d{\cl T}= ({1\over a_2},{1\over a_1};{1\over h}|{1\over c}):~~
 &{1\over a_2^2}+{1\over a_1^2}={1\over h^2}\end{array}\right\}
 ,~~a_1a_2=ch,~~\tan\theta}   \def\vth{\vartheta} \def\io{\iota={a_1\over a_2}
 \end{eqn}one has for the coupling constants $\cl G$,  the
 masses $\cl M$ and the normalization constants $\cl L^2$
 \begin{eqn}{rl}
\hbox{couplings:}&
 \cl G=(g_1,g_2;g_Z|e)
 \sim({1\over 2.9},{1\over 1.6};{1\over 1.4}|{1\over 3.3})\cr
\hbox{masses:}&\cl M=(\mu_1,\mu_W;\mu_Z|\mu_e)
\sim(43.4,80.2;91.2|38.2){\ro{GeV}\over c^2}\cr
\hbox{normalizations:}&\cl L^2=(\ell_1^2,\ell_2^2;\ell_Z^2|\ell_ e^2) \cr
&\cl M=M\cl G,~\cl L^2={1\over M}\cl G\cr
\end{eqn}$\mu_e$ and $\mu_1$
are no particle masses
in contrast to the $W$- and $Z$-masses $\mu_W$ and $\mu_Z$ resp.

The {\it internal interaction
normalizations} are $\cl G^2$ and ${1\over \cl G^2}$,
the {\it external particle normalizations} $\cl L^2$ and ${1\over\cl L^2}$,
related to each other by the ground state mass value $M$.

The  Feynman
propagators for the particle vector fields
are characterized by the mass as momentum space pole  and the
residue as the coupling constant involved
\begin{eqn}{rll}
\angle{\cl T Z^kZ^j}(x)&=
-{i\over\pi}\int {d^4 q e^{iqx}\over(2\pi)^3}(\eta^{kj}-{q^kq^j\over \mu_Z^2})
{g_Z^2\over q^2+io-\mu_Z^2},&g_Z^2={\mu_Z^2\over M^2}=\mu_Z\ell_Z^2\cr
\angle{\cl T A^kA^j}(x)&=
-{i\over\pi}\int {d^4 qe^{iqx}\over(2\pi)^3}\eta^{kj}
{ e^2\over q^2+io}+\hbox{gauge dep.},& e^2={\mu_ e^2\over M^2}=\mu_e\ell_e^2\cr
\angle{\cl T W_a^kW_b^j}(x)&=
-\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}{i\over\pi}\int {d^4 qe^{iqx}\over(2\pi)^3}(\eta^{kj}-{q^kq^j\over \mu_W^2})
{g_2^2\over q^2+io-\mu_W^2},&g_2^2={\mu_W^2\over M^2}=\mu_W\ell_W^2\cr
\end{eqn}

\section{The Quantum Mechanical Oscillators}

The quantum mechanical oscillators can be used as a simple model
to illustrate the interplay
of masses and coupling constants.

The isotropic Bose oscillator in $n$ dimensions
has as Hamiltonian with mass $m$ and spring constant $k$
\begin{eqn}{l}
H_n={\rvec p^2\over 2m}+k{\rvec q^2\over 2},~~(\rvec p,\rvec q)=(p_a,q_a)_{a=1}^n
\end{eqn}to be compared with the field theoretical expression
$g^2{F^2\over4}+M^2{W^2\over2}$ e.g. for the massive $W_{1,2}$-fields.
The quantum mechanical analogue of the coupling constants $g^2$ for
relativistic fields and the ground state mass $M^2$ is the inverse inert
mass
${1\over m}$ for mass points and the spring constant $k$ resp.
The characteristic frequency $\mu$ and the intrinsic  length $\ell$
of the oscillator  are
given with the product and the quotient of inverse mass and spring constant
\begin{eqn}{l}
H_n=\mu{\ell^2\rvec p^2+{1\over\ell^2}\rvec q^2\over2},~~
\mu^2={k\over m},~~\ell^4={1\over mk}
\end{eqn}$\hbar$ is used as natural unit.
The mass of a  particle, e.g. the $Z$ or $W$ mass  $\mu_{Z,W}$, has its
analogue in the mass point frequency $\mu$, not in the inert mass $m$.

The time dependent quantization of the harmonic oscillator
with the shorthand notation
$\com ab(t-s)=\com{a(s)}{b(t)}$ is a ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)\cong\SO(2)$
representation
\begin{eqn}{l}
D_{ab}(t|\mu)={\scriptsize\pmatrix{
\com{i p_a}{q_b}&
\com{q_a}{q_b}\cr
\com{ p_a}{p_b}&
\com{q_a}{-ip_b}\cr}}(t)=
\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}D_1(t|\mu)\cr
~~\cr
D_1(t|\mu)={\scriptsize\pmatrix{
\cos t\mu& i\ell^2\sin t\mu\cr
{i\over\ell^2}\sin t\mu&\cos t\mu\cr}}=e^{it\mu N(\ell^2)},~~N(\ell^2)
={\scriptsize\pmatrix{
0& \ell^2\cr
{1\over\ell^2}&0\cr}}
\end{eqn}

The harmonic oscillator  mass arises in the residue ${1\over m}$ of the
integral representation for the time dependent quantization. One has
with the principial
value $(\mu^2)_P$ integration contour
in the energy plane
\begin{eqn}{l}
\com {q_a}{q_b}(t)=
\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}i\ell^2\sin t\mu=-\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}{i\ep(t)\over\pi}
\int dE e^{it E}{{1\over m}\over
E^2-(\mu^2)_P},~~~{1\over m}={\mu^2\over k}=\mu\ell^2
\end{eqn}

The characteristic length $\ell$ can be absorbed
in the 'free' oscillator by the rescaling
\begin{eqn}{l}
\rvec q=\ell\ul{\rvec q},~~\rvec p={1\over\ell}\ul{\rvec p}
\end{eqn}i.e. it has no physical relevance.
If, however, a nontrivial
rotation $\O(n)$-invariant $\rvec q^2$-dependent potential $V$ is added,
e.g.  $(\rvec q^2)^2$  for an  anharmonic oscillator or
$-k{\rvec q^2\over2}-{1\over\sqrt{\rvec q^2}} $ for the Kepler dynamics,
$\ell^2$ is relevant for the coupling constant
\begin{eqn}{rl}
H_n+V&=
\mu{\ell^2\rvec p^2+{1\over\ell^2}\rvec q^2\over2}+V(\rvec q^2)=
\mu{\ul{\rvec p}^2+\ul{\rvec q}^2\over2}+V(\ell^2\ul{\rvec q}^2)\cr
\end{eqn}

The oscillator length $\ell$  normalizes the 'real' and 'imaginary'
combination of the creation operator $\ro u$ and the annihilation operator
$\ro u^\star$
\begin{eqn}{l}
\rvec q=\ell{\rvec{\ro u}+\rvec {\ro u}^\star\over\sqrt2},~~
\rvec p={i\over\ell}{\rvec {\ro u}-\rvec {\ro u}^\star\over\sqrt2}\cr
\end{eqn}In the complex creation-annihilation formulation the
characteristic length does not show up in the ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(n)$-symmetric Hamiltonian
\begin{eqn}{l}
H_n=\mu{\acom{\rvec \ro u}{\rvec {\ro u}^\star}\over2},~~
\com{\ro u^\star_ a}{\ro u_b}(t)
=\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}e^{it\mu}
\end{eqn}

The quantum mechanical meaning of the
particle normalization $\ell^2$
is given with the Fock ground state of the harmonic oscillator.
For the Fock space with the ground state values
\begin{eqn}{l}
n=1:~~\angle{(\ro u^\star \ro u)^N}=1,~~N=0,1,\dots\cr
n\ge1:~~\angle{
(\ro u^\star_{a_1}\ro u_{b_1})
\cdots( \ro u^\star_{a_N}\ro u_{b_N})}=\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{a_1b_1}\cdots\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{a_Nb_N}
\end{eqn}the time dependent Fock values for position and momenta
with the shorthand  notation
$\angle{\acom ab}(t-s)=\angle{\acom{a(s)}{b(t)}}$
are the product of the particle normalization matrix $N(\ell^2)$ with a time
representation
\begin{eqn}{l}
{\scriptsize\pmatrix{
\angle{\acom{i p_a}{q_b}}&\angle {\acom{q_a}{q_b}}\cr
\angle{\acom{ p_a}{p_b}}&\angle {\acom{q_a}{-ip_b}}\cr}}(t)=
\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}{\scriptsize\pmatrix{
i\sin t\mu& \ell^2\cos t\mu\cr
{1\over\ell^2}\cos t\mu &i\sin t\mu\cr}}=\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}N(\ell^2) D_1(t|\mu)\cr
{\scriptsize\pmatrix{
\angle{\acom{i p_a}{q_b}}&\angle {\acom{q_a}{q_b}}\cr
\angle{\acom{ p_a}{p_b}}&\angle {\acom{q_a}{-ip_b}}\cr}}(0)=
\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}N(\ell^2)=\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}{\scriptsize\pmatrix{
0& \ell^2\cr
{1\over\ell^2} &0\cr}}
\end{eqn}$N(\ell^2)$ contains the dual normalizations of positions
and momenta, i.e. $\ell^2$ and  ${1\over\ell^2}$ resp.

The sum of
time ordered  quantization and Fock values is the
quantum mechanical analogue of the Feynman propagator, given
for the position with
the oscillator frequency as energy pole and the inverse mass as residue
\begin{eqn}{rl}
\angle{\cl T qq}(t)&=\angle{\acom qq}(t)-\ep(t)\com qq(t)
=\ell^2(\cos t\mu-i\ep(t)\sin t\mu)\cr
&=\ell^2 e^{-i|t|\mu}=-{i\over\pi}\int dE e^{itE}{{1\over m}\over E^2+io-\mu^2}
,~~{1\over m}={\mu^2\over k}=\mu\ell^2\end{eqn}

\section{Fock Normalization for Particles}

The particle expansion of the vector
fields in the standard model\cite{SBH}
\begin{eqn}{rl}
Z^j(x)&={\displaystyle \sum}\def\PROD{\displaystyle \prod_{n=1,2,3}}\int{d^3q\over(2\pi)^3}
\sqrt{{\mu_Z\over q_0}}\La({q\over\mu_Z})^j_n
~\ell_Z{e^{ixq}\ro u^n(\rvec q)+e^{ -ixq}\ro u^{\star n}(\rvec q)\over\sqrt2}\cr
W_a^j(x)&={\displaystyle \sum}\def\PROD{\displaystyle \prod_{n=1,2,3}}\int{d^3q\over(2\pi)^3}
\sqrt{{\mu_W\over q_0}}\La({q\over\mu_W})^j_n
~\ell_W{e^{ixq}\ro u_a^n(\rvec q)+e^{ -ixq}\ro u_a^{\star n}(\rvec q)\over\sqrt2}
,~~a=1,2\cr
A^j(x)&={\displaystyle \sum}\def\PROD{\displaystyle \prod_{n=0,1,2,3}}\int{d^3q\over(2\pi)^3}
\sqrt{{ \mu_e\over q_0}}H({\rvec q\over q_0})^j_n
~\ell_e {\scriptsize\pmatrix{
\dots\dots\cr
~\cr
{e^{ixq}\ro U^1(\rvec q)+e^{ -ixq}\ro U^{\star 1}(\rvec q)\over\sqrt2}\cr
{e^{ixq}\ro U^2(\rvec q)+e^{ -ixq}\ro U^{\star 2}(\rvec q)\over\sqrt2}\cr
~\cr
\dots\dots\cr}}\cr
\end{eqn}uses the energy $q_0=\sqrt{\mu^2+\rvec q^2}$ for
$\mu=\mu_Z,\mu_W,0$ and the boost $\La({q\over\mu})$
with positive mass $\mu>0$ for $\SO^+(1,3)/\SO(3)$
to transform to a rest system and
$H({\rvec q\over q_0})$,  $q_0=|\rvec q|$, i.e. with vanishing mass $\mu=0$,
for $\SO^+(1,3)/\SO(2)$ to transform to a rest system with a polarization
direction.
The two nonparticle components $(\dots)$
(Coulomb and gauge degree of freedom for $n=0,3$)
for the massless electromagnetic
field are not given explicitly.

The quantization and Fock values for the creation and annihilation operators
\begin{eqn}{l}
\com{\ro u^\star(\rvec p)}{\ro u(\rvec q)}=\delta}  \def\ep{\epsilon}  \def\ze{\zeta(\rvec q-\rvec p)=
\angle {\acom{\ro u^\star(\rvec p)}{\ro u(\rvec q)}}
\end{eqn}exhibit  the external normalizations
$\ell_Z^2,\ell_W^2$ for the massive particles involved
\begin{eqn}{rl}
\angle{\acom{Z^k}{Z^j}}(x)&=
{\displaystyle \sum}\def\PROD{\displaystyle \prod_{n,m=1,2,3}}
\int{d^3qe^{-i\rvec x\rvec q}\over(2\pi)^3}
{\mu_Z\over q_0}
\La({q\over\mu_Z})^k_m
~\ell_Z^2\delta}  \def\ep{\epsilon}  \def\ze{\zeta^{mn}\cos x_0q_0~
\La({q\over\mu_Z})^j_n\cr
\angle{\acom{W_a^k}{W_b^j}}(x)&=
{\displaystyle \sum}\def\PROD{\displaystyle \prod_{n,m=1,2,3}}
\int{d^3qe^{-i\rvec x\rvec q}\over(2\pi)^3}
~{\mu_W\over q_0}
\La({q\over\mu_W})^k_m
~\ell_W^2\delta}  \def\ep{\epsilon}  \def\ze{\zeta^{mn}\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}\cos x_0q_0~
\La({q\over\mu_W})^j_n\cr
\end{eqn}The Fock values for the two massless particle degrees of freedom
(photons) $m,n=1,2$ in the  electromagnetic  field
\begin{eqn}{l}
\angle{\acom{A^k}{A^j}}(x)=
{\displaystyle \sum}\def\PROD{\displaystyle \prod_{n,m=0,1,2,3}}
\int{d^3qe^{-i\rvec x\rvec q}\over(2\pi)^3}
{\mu_e\over q_0}H({\rvec q\over q_0})^k_m
~\ell_e^2{\scriptsize\pmatrix{
\cdots & 0&\cdots\cr
0&\cos x_0q_0&0\cr
\cdots&0&\dots\cr}}^{mn}
H({\rvec q\over q_0})^j_n\cr
\end{eqn}have as external particle normalization $\ell_ e^2={ e^2\over \mu_e}$.

\chapter{Normalization of Symmetries}
In this chapter, the mathematical structure for  symmetry normalizations
(coupling constants and particle masses) is given together with its application to the standard
model vector fields.

\section{Dual Normalizations}

A nondegenerate  quadratic form $\ze$, i.e. a  symmetric  real bilinear  or
complex sesquilinear form of a finite dimensional vector space $V$
with scalars $\K={\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}$ and $\K=\C$ resp.
\begin{eqn}{l}
\ze(~,~):V\x V\longrightarrow \K,~~\cases{
\ze(v,w)=\ze(w,v)\hbox{, bilinear}\cr
\ze(v,w)=\ol{\ze(w,v)}\hbox{, sesquilinear}\cr }
\end{eqn}induces
an (anti)linear isomorphism to the dual space $V^T$,
containing  the linear $V$-forms
\begin{eqn}{l}
\ze:V\longrightarrow V^T, v\longmapsto \ze(v)=\ze(v,~),~~\cases{
\ze(v+w)=\ze(v)+\ze(w)\cr
\ze(\alpha}  \def\be{\beta} \def\ga{\gamma v)=\alpha}  \def\be{\beta} \def\ga{\gamma\ze(v),~\alpha}  \def\be{\beta} \def\ga{\gamma\in{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}\hbox{, bilinear}\cr
\ze(\alpha}  \def\be{\beta} \def\ga{\gamma v)=\ol\alpha}  \def\be{\beta} \def\ga{\gamma\ze(v),~\alpha}  \def\be{\beta} \def\ga{\gamma\in\C\hbox{, sesquilinear}\cr}
\end{eqn}and therewith the inverse quadratic form $\d\ze$ for
the linear forms
\begin{eqn}{l}
\d\ze(~,~):V^T\x
V^T\longrightarrow\K,~~\d\ze(\om,\theta}   \def\vth{\vartheta} \def\io{\iota)=\ze(\ze^{-1}(\om),\ze^{-1}(\theta}   \def\vth{\vartheta} \def\io{\iota))
\end{eqn}

With the isomorphism $\ze$, the quadratic form
can be expressed by the dual product
of vectors $V$ and linear forms $V^T$
\begin{eqn}{rll}
\hbox{dual product}:&V^T\x V\longrightarrow \K,~~&(\om,v)\longmapsto\om(v)=\dprod\om v\cr
\ze(~,~):&V\x V\longrightarrow\K,~~&\ze(v,w)=\dprod{\ze(v)}w\cr
\d\ze(~,~):&V^T\x V^T\longrightarrow\K,~~&\d\ze(\om,\theta}   \def\vth{\vartheta} \def\io{\iota)=\dprod\om{\ze^{-1}(\theta}   \def\vth{\vartheta} \def\io{\iota)}\cr
\end{eqn}

Nondegenerate quadratic forms are
characterized by a signature $(n_+,n_-)$,
i.e. by their real orthogonal
invariance groups $\O(n_+,n_-)$ for real vector spaces
$V\cong{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}^n$, $n=n_++n_-$, or unitary ones ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(n_+,n_-)$
for complex vector spaces $V\cong\C^n$.
If $\{e^a\}_{a=1}^n$ and
$\{\d e_a\}_{a=1}^n$ are dual bases of $V$ and $V^T$, i.e.
with the dual product $\dprod{\d e_a}{e^b}=\delta}  \def\ep{\epsilon}  \def\ze{\zeta_a^b$, their squares
build the matrices
 $\ze(e^a,e^b)=\ze^{ab}$ and $\d\ze(\d e_a,\d e_b)=\ze_{ab}$.

A quadratic form of a complex vector space defines a
${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(n_+,n_-)$-conjugation
\begin{eqn}{l}
\bl U(n_+,n_-):~~\ze(v)=v^*=\ze(v,~),~~\ze^{-1}(\om)=\om^*=\d\ze(\om,~)
\end{eqn}

 Only  real, not complex quadratic forms are expressable by
power two tensors, written in, but independent of bases
\begin{eqn}{l}
\bl O(n_+,n_-):~~\left\{\begin{array}{rl}
\d\ze&=\ze_{ab} e^a\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes e^b\in V\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes V\cr
\ze&=\ze^{ab} \d e_a\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes \d e_b\in V^T\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes V^T\end{array}
\right.
\end{eqn}

Orthogonal  bases of vector spaces with definite bilinear forms, i.e.
of scalar products with $n_-=0$, define
dual normalizations $\ell^2$ and ${1\over\ell^2}$ for the
compact symmetries acting on  the vector spaces $V$ and $V^T$
\begin{eqn}{rll}
&\hbox{for }\O(n),{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(n):~~
&\left\{\begin{array}{rl}
\ze(e^a,e^b)&= \ell^2\delta}  \def\ep{\epsilon}  \def\ze{\zeta^{ab} \cr
\d\ze(\d e_a,\d e_b)&={1\over\ell^2}\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}\cr
\end{array}\right.,~~\ell^2>0\cr
&\hbox{for }\O(n):~~&\left\{\begin{array}{rl}
\d\ze&={1\over\ell^2}\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab} e^a\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes e^b\in V\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes V\cr
\ze&=\ell^2 \delta}  \def\ep{\epsilon}  \def\ze{\zeta^{ab} \d e_a\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes \d e_b\in V^T\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes V^T\end{array}
\right.
\end{eqn}

\section{Dual Normalizations for Oscillators}

The harmonic Bose oscillator (section 1.2) has
an $\O(n)$-invariant scalar product for the dual spaces
 of positions $q\in\S\cong{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}^n$  and momenta $p\in\S^T\cong{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}^n$
\begin{eqn}{rll}
&\si(~,~):\S\x\S\longrightarrow {\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}},~~&\cases{
\si(q,q')=\si(q',q)\cr
\si(q,q)>0\iff q\ne0\cr}\cr
&\d\si(~,~):\S^T\x \S^T\longrightarrow {\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}},~~&\cases{\d\si(p,p')=\d\si(p',p)\cr
\d\si(p,p)>0\iff p\ne0\cr}\cr
\end{eqn}Using dual and orthonormal bases $\{q_a,p_a\}_{a=1}^n$
one has the invariant tensors with the intrinsic oscillator lenght $\ell$
\begin{eqn}{l}
\dprod{p_a}{q_b}=\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab},~~\left\{\begin{array}{rll}
&\si(q_a,q_b)=\ell^2\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}&~\Rightarrow~}\def\sub{\subseteq}\def\subnoteq{\subset
 \si=\ell^2 p_a\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes p_a\cr
& \d\si(p_a,p_b)={1\over\ell^2}\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}&~\Rightarrow~}\def\sub{\subseteq}\def\subnoteq{\subset
\d \si={1\over\ell^2} q_a\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes q_a\cr
 \end{array}\right.
\end{eqn}The sum of the dual scalar products, multiplied
 with a constant $\mu$ leads to the quantum
mechanical   Hamiltonian
\begin{eqn} {rl}
\dprod{p_a}{q_b}=\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}&~\Rightarrow~}\def\sub{\subseteq}\def\subnoteq{\subset
\com{ip_a}{q_b}=\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}\cr
\mu{\d\si+\si\over2}=
\mu{{1\over \ell^2}p_a\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes p_a+\ell^2 q_a\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes q_a\over 2}&~\Rightarrow~}\def\sub{\subseteq}\def\subnoteq{\subset H_n
=\mu{{1\over \ell^2}p_a p_a+\ell^2 q_aq_a\over 2}\cr
\end{eqn}$\ell^2$ and ${1\over\ell^2}$ are the dual
particle normalizations of $\O(n)$
acting on positions and momenta.

\section{Interaction Normalizations\\for the Standard  Gauge Fields}

With respect to the
internal degrees of freedom
the gauge field sector in the standard model is built with dual
real 4-dimensional vector spaces for gauge vector fields and
curvatures
which are the direct sums of 1- and 3-dimensional subspaces
\begin{eqn}{rl}
\hbox{curvatures: } G= G_1\oplus}\def\Pl{\bigoplus}\def\PL{\displaystyle \bigoplus  G_3\cong{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}\oplus}\def\Pl{\bigoplus}\def\PL{\displaystyle \bigoplus{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}^3&\hbox{with basis }\{F,F_a\}\cr
\hbox{gauge vectors: } G^T= G_1^T\oplus}\def\Pl{\bigoplus}\def\PL{\displaystyle \bigoplus  G_3^T\cong{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}\oplus}\def\Pl{\bigoplus}\def\PL{\displaystyle \bigoplus{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}^3&\hbox{with basis }\{B,W_a\}\cr
\hbox{dual bases: }\dprod B F=1,&~~\dprod {W_a}{F_b}=\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}
\end{eqn}The hyperisospin group ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)={\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_2)\circ}\def\x{\times}\def\oo{\infty\SU(2)$ acts nontrivially
only on the
isospin subspaces $ G _3, G _3^T$
as ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)/{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_2)\cong\SO(3)$.

The squares of the coupling constants reflect a ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$-invariant
scalar product with dual normalizations
$g^2$ and ${1\over g^2}$ of the internal groups
acting on gauge vectors and curvatures
\begin{eqn}{rll}
 g(~,~): G \x  G \longrightarrow {\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}},& g(F,F)={1\over g_1^2}
,&g(F_a,F_b)=\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}{1\over g_2^2}\cr
\d g(~,~): G ^T\x  G ^T\longrightarrow {\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}},&\d g(B,B)=g_1^2,&\d g(W_a,W_b)=\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}g_2^2\cr
\d g=g_1^2 F^2+g_2^2\rvec F^2,&
 g={1\over g_1^2}B^2+{1\over g_2^2}\rvec W^2
 \end{eqn}The Lagrangian
 uses the gauge invariant metrical tensor $\d g$ for the curvatures.

The distinction of the electromagnetic group ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)_+$-direction $\bl 1_2+\tau_3$
in the  ground state defines
the basis transformation  $S(\theta}   \def\vth{\vartheta} \def\io{\iota)\in\SL({\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}^2)$ in the 2-dimensional
curvature subspace spanned by $\{F,F_3\}$
and to the contragredient transformation
in the vector field space spanned by $\{B,W_3\}$ (section 1.1)
\begin{eqn}{lll}
\Th: G \longrightarrow G ,&
{\scriptsize\pmatrix{
F_A\cr F_ Z\cr F_{1,2}\cr}}=\Th
{\scriptsize\pmatrix{F\cr F_3\cr F_{1,2}\cr}},&
\Th={\scriptsize\pmatrix{
1&1&0\cr
-\sin^2\theta}   \def\vth{\vartheta} \def\io{\iota&\cos^2\theta}   \def\vth{\vartheta} \def\io{\iota&0\cr 0&0&\bl 1_2\cr}}\in\SL({\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}^4)\cr
\d \Th: G ^T\longrightarrow G ^T,&{\scriptsize\pmatrix{
A\cr  Z\cr W_{1,2}\cr}}=
\d \Th{\scriptsize\pmatrix{B\cr W_3\cr W_{1,2}\cr}},&
\d \Th={\scriptsize\pmatrix{
\cos^2\theta}   \def\vth{\vartheta} \def\io{\iota&\sin^2\theta}   \def\vth{\vartheta} \def\io{\iota&0\cr
-1&1&0\cr 0&0&\bl 1_2\cr}}\in\SL({\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}^4)
\end{eqn}The internal normalizations read in the new basis
\begin{eqn}{rl}
\Th{\scriptsize\pmatrix{
{1\over g_1^2}&0\cr
0&{1\over g_2^2}\bl 1_3\cr}}\Th^T&=
{\scriptsize\pmatrix{
{1\over e^2}&0&0\cr
0&{1\over g_Z^2}&0\cr
0&0&{1\over g_2^2}\bl 1_2\cr}}\cr
\d g=g_1^2 F^2+g_2^2\rvec F^2&=e^2 F_A^2+g_Z^2F_Z^2+g_2^2{\displaystyle \sum}\def\PROD{\displaystyle \prod_{a=1,2}}F_a^2
\end{eqn}

\section{Higgs Mass as Unit\\for  the Goldstone Manifold}

The algebraic framework for the
ground state properties implementing complex Higgs vectors
$\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta$ (section 1.1)
uses  a ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$-invariant
scalar product of the complex 2-dimensional Higgs vector space $\H\cong\C^2$
\begin{eqn}{l}
\sprod{~}{~}:\H\x \H\longrightarrow \C,~~\cases{
\sprod{\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta}{\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta'}=
\ol{\sprod{\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta'}{\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta} }\cr
\sprod{\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta}{\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta}>0\iff\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta\ne 0\cr
\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^\star=\sprod\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta{~}\in\H^T\cr}
\end{eqn}

In the  real $4$-dimensional manifold $\GL(\C^2)_{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}/{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$  of the
 classes of
 ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$-equivalent bases with $\GL(\C^2)_{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}$ taken as real $8$-dimensional
Lie group, the orthogonal bases
$\{\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^\alpha}  \def\be{\beta} \def\ga{\gamma\}_{\alpha}  \def\be{\beta} \def\ga{\gamma=1,2}$
with the diagonal matrix
\begin{eqn}{l}
\sprod{\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^\alpha}  \def\be{\beta} \def\ga{\gamma}{\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^\be}=M^2\delta}  \def\ep{\epsilon}  \def\ze{\zeta^{\alpha}  \def\be{\beta} \def\ga{\gamma\be},~~
\sprod{~}{~}\cong M^2{\scriptsize\pmatrix{1&0\cr 0&1\cr}}
\end{eqn}define the normalization $M^2$  of the
hyperisospin group ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$ acting on the Higgs space.

Any nontrivial vector $\ul\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta\in\H$ fixes an orthogonal basis up to
the electromagnetic ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)_+$-direction. Therefore $M$ can also be called
the unit of the Goldstone manifold ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)/{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)_+$ which relates fields and
particles with repect to the internal degrees of freedom.

\section{Induced Normalizations \\for Lie Algebras}

If a real Lie algebra $L$ acts
on  a real or complex vector space $V$   by linear endomorphisms
\begin{eqn}{l}
L\x V\longrightarrow V,~~ l\bullet}\def\tria{\triangleleft v=\cl D(l)(v)\cr
\end{eqn}each vector $v\in V$ induces via a
quadratic form $\ze$ of $V$  a
symmetric bilinear form on the Lie algebra,
definite for a scalar product
\begin{eqn}{l}
\ze_v(~,~):L\x L\longrightarrow {\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}},~\cases{
\ze_v(l,l')=\ze(l\bullet}\def\tria{\triangleleft v,l'\bullet}\def\tria{\triangleleft v)
=\ze_v(l',l)\cr
\hbox{if }\ze(v,v)> 0\hbox{ for }v\ne0~\Rightarrow~}\def\sub{\subseteq}\def\subnoteq{\subset \ze_v(l,l)\ge 0\cr}
\end{eqn}The induced  bilinear Lie algebra form is trivial for the
Lie subalgebra which acts trivially on the vector $v$
\begin{eqn}{l}
\ze_v(l,L)=\{0\}~~\hbox{ for }~~l\in\,\hbox{FIX}}\def\STAB{\,\hbox{STAB}_vL
=\{l\in L\big\bracevert}\def\Cup{\bigcup}\def\Cap{\bigcap l\bullet}\def\tria{\triangleleft v=0\}\cr
\end{eqn}$\ze_v$ is nondegenerate for the quotient vector space
$L/\,\hbox{FIX}}\def\STAB{\,\hbox{STAB}_v L$ and a scalar product thereon  for a scalar product $\ze$.

In the standard  model (section 1.1)
the vector fields obtain their masses in the
mechanism just described:
The Lie algebras for
hypercharge and for isospin
$\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)=\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_2)\oplus}\def\Pl{\bigoplus}\def\PL{\displaystyle \bigoplus\log\SU(2)$ act on the
Higgs vectors $\H\cong\C^2$
\begin{eqn}{rll}
&l^0\bullet}\def\tria{\triangleleft\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^\alpha}  \def\be{\beta} \def\ga{\gamma={i\over\sqrt2}\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^\alpha}  \def\be{\beta} \def\ga{\gamma,~~&l^0\in\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)\cong{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}\cr
&l^a\bullet}\def\tria{\triangleleft\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^\alpha}  \def\be{\beta} \def\ga{\gamma={i\over\sqrt 2}(\tau^a)^\alpha}  \def\be{\beta} \def\ga{\gamma_\be\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^\be,~~
&l^a\in\log\SU(2)\cong{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}^3\cr
\end{eqn}The electromagnetic Lie algebra
$\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)_+$ is the Lie algebra acting trivially on each
nontrivial Higgs vector
\begin{eqn}{l}
\H\ni\ul\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta\cong{\scriptsize\pmatrix{0\cr M\cr}}\ne 0
,~~\sprod{\ul\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta}{\ul\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta}=M^2\cr
\,\hbox{FIX}}\def\STAB{\,\hbox{STAB}_{\ul\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta}\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)=\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)_+
\cong i{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}{\bl 1_2+\tau_3\over\sqrt2}\cr
\end{eqn}The  scalar product
induced on the Lie algebra $\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$  is nontrivial on the
vector space $\log {\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)/{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)\cong{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}^3$ for the massive $Z$- and
$W_{1,2}$-particle fields
\begin{eqn}{rll}
\sprod{l^0-l^3}{l^0-l^3}_{\ul\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta}&=
\sprod{i{\bl 1_2-\tau^3\over\sqrt2}}{i{\bl 1_2-\tau^3\over\sqrt2}}_{\ul\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta}
&={M^2\over2}\cr
\sprod{l^a}{l^b}_{\ul\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta}&=
 \sprod{i{\tau^a\over\sqrt2}}{i{\tau^b\over\sqrt2}}
 _{\ul\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta}&=\delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}{M^2\over2}\hbox{ for }
a,b=1,2\cr
\end{eqn}

\section{Particle Normalization\\for the Standard Gauge Fields}

The
normalization of the Lorentz group $\SO^+(1,3)$-action shows up in the
particle normalizations $\ell^2$ (section 1.1).
The distinction of the definite subgroups of the Lorentz group
goes parallel with the Wigner classification\cite{WIG,SBH,S961}
 of the representations of
the Poincar\'e group: Via the massive particles  the definite spin group
$\SO(3)\cong\SU(2)/{\,{\rm I\kern-.2emI}}}\def\K{{\,{\rm I\kern-.2emK}}_2$ is normalized, via
the massless ones the definite cirularity
(polarization) group $\SO(2)\cong{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)$ inside the Lorentz group
$\SO^+(1,3)\cong\SL(\C^2)_{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}/{\,{\rm I\kern-.2emI}}}\def\K{{\,{\rm I\kern-.2emK}}_2$.

Similar to the internal symmetries with the
Goldstone manifold ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)/{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)_+$
characterizing the field particle transition, there arise
for the external symmetries the Sylvester and
Witt
manifolds $\SO^+(1,3)/\SO(3)$ and $\SO^+(1,3)/\SO(2)$ resp.
relevant for the harmonic particle analysis of the relativistic fields.
In contrast to the fields, particles have no full Lorentz symmetry, massive particles keep the
spin $\SO(3)$ symmetry, massless ones only the circularity
(polarization) $\SO(2)$ symmetry.

To be more explicit:
The  Lagrangian for a massive  vector field
\begin{eqn}{l}
\bl L(Z,F_Z)={1\over2}F_Z^{jk}\ep^{lm}_{jk}\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_l Z_m
+\mu_Z(\ell_Z^2{F_Z^{jk}F_{Zjk}\over 4}+{Z^jZ_j\over2\ell_Z^2})
\end{eqn}with the Clebsch-Gordan coefficients
$\ep^{lm}_{jk}=\delta}  \def\ep{\epsilon}  \def\ze{\zeta^l_m\delta}  \def\ep{\epsilon}  \def\ze{\zeta^m_k-\delta}  \def\ep{\epsilon}  \def\ze{\zeta^l_k\delta}  \def\ep{\epsilon}  \def\ze{\zeta^m_j$  and
dual normalization constant $\ell_Z^2$ and ${1\over\ell_Z^2}$
gives rise to the commutators of the canonical pair $(Z,F_Z)$
\begin{eqn}{rl}
&{\scriptsize \pmatrix{
\com{iF_Z^{kl}}{Z^j}&\com {Z^k}{Z^j}\cr
\com{F_Z^{kl}}{F_Z^{jm}}&\com {Z^k}{-iF_Z^{jm}}\cr}}(x)\cr
&~~\cr
&={\scriptsize \pmatrix{
-i\ep_{ut}^{lk}\delta}  \def\ep{\epsilon}  \def\ze{\zeta^j_s{\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla^u\over \mu_Z} &\ell_Z^2\delta}  \def\ep{\epsilon}  \def\ze{\zeta_t^k\delta}  \def\ep{\epsilon}  \def\ze{\zeta_s^j\cr
-{1\over\ell_Z^2}\ep_{ut}^{lk}\ep_{rs}^{mj}{\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla^r\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla^u\over \mu_Z^2}
&-i\delta}  \def\ep{\epsilon}  \def\ze{\zeta^k_t\ep_{rs}^{mj}{\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla^r\over \mu_Z}\cr}}
\int {d^4 qe^{ixq}\over(2\pi)^3}\ep(q^0)\mu_Z\delta}  \def\ep{\epsilon}  \def\ze{\zeta(q^2-\mu_Z^2)
(-\eta^{kj}+{q^kq^j\over\mu_Z^2})\cr
\end{eqn}
The $Z$-commutators
\begin{eqn}{l}
\com {Z^k}{Z^j}(x)
= \int {d^3 q e^{-i\rvec x\rvec q}\over(2\pi)^3}{\mu_Z\over q_0}~
\La({q\over\mu_Z})_m^k\brack{ZZ}^{mn}(x_0)
\La({q\over \mu_Z})^j_n\cr
\end{eqn}involve the $\SO(3)$ normalization
$\ell_Z^2$
\begin{eqn}{l}
\brack{ZZ}(x_0)=
\ell_Z^2~{\scriptsize \pmatrix{0&0\cr 0&\bl 1_3\cr}}i\sin x_0q_0
,~~\ell_Z^2={g_Z^2\over\mu_Z}={\mu_Z\over M^2} \cr
\end{eqn}and the boost $\La({q\over\mu_Z})$ for the
transformation to a Sylvester basis
\begin{eqn}{l}
\La({q\over\mu_Z})\cong{1\over\mu_Z}{\scriptsize\pmatrix{
q_0&\rvec q\cr \rvec q& \delta}  \def\ep{\epsilon}  \def\ze{\zeta_{ab}\mu+{q_aq_b\over q_0+\mu_Z}\cr}},~~q^2=\mu_Z^2
\end{eqn}The boosts represent $\SO(3)$-classes of the
Sylvester manifold $\SO^+(1,3)/\SO(3)$.
The Fock values of the anticommutators use the analogue
structures with
\begin{eqn}{l}
\angle{\acom {Z^k}{Z^j}}(x)
= \int {d^3 qe^{-i\rvec x\rvec q}\over(2\pi)^3}{m\over q_0}~
\La({q\over\mu_Z})_m^k
\angle{\brace{ZZ}^{mn}}(x_0)
\La({q\over \mu_Z})_n^j\cr
\angle{\brace{ZZ}}(x_0)=
\ell_Z^2~{\scriptsize \pmatrix{0&0\cr 0&\bl 1_3\cr}}\cos x_0q_0\cr
\end{eqn}For  massive vector particles a rest system with a time direction
has to be fixed up to space rotations $\SO(3)$.

For the massless electromagnetic vector fields the Lagrangian contains in addition to
the coupling constant $ e^2$ also the gauge fixing terms with a scalar field
$L(x)$ and gauge fixing parameter $\kappa}   \def\la{\lambda}   \def\si{\sigma$
\begin{eqn}{l}
\bl L( A, F_A, L)=
 {1\over2}F_A^{jk}\ep^{lm}_{jk}\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_l A_m+L\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla^jA_j
+ e^2  { F_A^{jk} F_{Ajk}\over4}+ \kappa}   \def\la{\lambda}   \def\si{\sigma { L^2\over2}
\end{eqn}The curvature $F_A$ and the gauge fixing field $L$
are the canonical partners for the gauge field $A$.
The nontrivial commutators are given by
\begin{eqn}{rl}
&{\scriptsize\pmatrix{
\com{i F_A^{kl}}{ A^j}&\com { A^k}{ A^j}\cr
\com{ F_A^{kl}}{ F_A^{jm}}&\com { A^k}{i L}\cr}}(x)
={\scriptsize\pmatrix{
-i\ep_{ut}^{lk}\delta}  \def\ep{\epsilon}  \def\ze{\zeta^j_s{\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla^u\over e^2   } &\delta}  \def\ep{\epsilon}  \def\ze{\zeta_t^k\delta}  \def\ep{\epsilon}  \def\ze{\zeta_s^j\cr
-\ep_{ut}^{lk}\ep_{rs}^{jm}{\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla^r\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla^u\over e^4  }
&-i\delta}  \def\ep{\epsilon}  \def\ze{\zeta^k_t{\partial}\def\pd#1#2{{\partial#1\over{\partial #2}}}\def\cd{\nabla_s\over \kappa}   \def\la{\lambda}   \def\si{\sigma  }\cr}}   \com{ A^t}{ A^s}(x)\cr
&~~\cr
&=
\int{d^4qe^{ixq}\over(2\pi)^3}\ep(q_0) \mu_e
{\scriptsize\pmatrix{
{q^u\over\mu_e }\ep^{kl}_{ut}\eta^{tj}\delta}  \def\ep{\epsilon}  \def\ze{\zeta(q^2)&
\ell_e^2[-\eta^{kj}\delta}  \def\ep{\epsilon}  \def\ze{\zeta(q^2)-{ e^2+\kappa}   \def\la{\lambda}   \def\si{\sigma \over e^2}q^kq^j\delta}  \def\ep{\epsilon}  \def\ze{\zeta'(q^2)]\cr
{1\over\ell^2_e}\ep^{kl}_{ut}\ep^{jm}_{rs}\eta^{ts}{q^rq^u\over\mu_e^2}\delta}  \def\ep{\epsilon}  \def\ze{\zeta(q^2)
&{q^k\over \mu_e }\delta}  \def\ep{\epsilon}  \def\ze{\zeta(q^2)\cr}} \cr
\end{eqn}In the space-time translations analysis
the vector field components are expressed in a Witt basis
\begin{eqn}{l}
\com{ A^k}{ A^j}(x)
=\int {d^3 q e^{-i\rvec x\rvec q}\over(2\pi)^3}{\mu_e\over q_0}
H({\rvec q\over q_0})_{\ul k}^k
~\brack{ A A}^{\ul k\ul j}(x_0) ~
H({\rvec q\over q_0})_{\ul j}^j\cr
\end{eqn}The transversal components (photons) have the $\SO(2)$-normalization
$\ell^2_e$,
the nondiagonal light fixing parameter $\alpha}  \def\be{\beta} \def\ga{\gamma$ is combined from $ e^2$ and the gauge fixing
parameter $\kappa}   \def\la{\lambda}   \def\si{\sigma $
\begin{eqn}{l}
\brack{ A A}(x_0)=\ell_e^2\Bigl\{
{\scriptsize\pmatrix{
0&0&-\alpha}  \def\be{\beta} \def\ga{\gamma\cr 0&\bl 1_2&0\cr- \alpha}  \def\be{\beta} \def\ga{\gamma&0&0\cr}}i\sin x_0q_0
+ix_0q_0\be{\scriptsize\pmatrix{
e^{-ix_0q_0}&0&0\cr 0&0&0\cr 0&0& e^{ix_0q_0}\cr}}   \} \cr
\ell^2_e={ e^2\over\mu_e}={\mu_e\over M^2},~~
\alpha}  \def\be{\beta} \def\ga{\gamma=-{3  e^2+\kappa}   \def\la{\lambda}   \def\si{\sigma \over 2 e^2},~~\be=-{ e^2+\kappa}   \def\la{\lambda}   \def\si{\sigma \over 2 e^2}\cr
\end{eqn}

The transmutation to a Witt basis
is performed by $H({\rvec q\over q_0})$
(Sylvester indices $k,j,\dots=0,1,2,3$,
Witt indices $\ul k,\ul j,\dots=0,1,2,3$)
\begin{eqn}{rl}
&H({\rvec q\over q_0})^k_{\ul k} =O({\rvec q\over q_0})_j^k\circ}\def\x{\times}\def\oo{\infty w^j_{\ul k}
\cong{\scriptsize {1\over q^0}}{\scriptsize\pmatrix{
{q^0\over\sqrt2}&0&0&{q^0\over\sqrt2}\cr
{q^1\over\sqrt2}
&q^0-{(q^1)^2\over q^0+q^3}&{q^1q^2\over q^0+q^3}&-{q^1\over\sqrt2}\cr
{q^2\over\sqrt2}
&-{q^1q^2\over q^0+q^3}&q^0-{(q^2)^2\over q^0+q^3}&-{q^2\over\sqrt2}\cr
{q^3\over\sqrt2}&- q^1&- q^2&-{q^3\over\sqrt2}\cr}},~q^2=0  \cr
&H({\rvec q\over q_0})
{\scriptsize\pmatrix{0&0&-1\cr 0&\bl 1_2&0\cr- 1&0&0\cr}}
H({\rvec q\over q_0})=
{\scriptsize\pmatrix{-1&0\cr 0&\bl 1_3\cr}}\cr
\end{eqn}For the massless photons a rest system with
one space axis, e.g. a 3rd direction, has to be fixed up to  space plane
rotations $\SO(2)$.

The 1st and 2nd component have a particle structure with Fock value
\begin{eqn}{l}
\ul k,\ul j\in\{1,2\}:
\angle{\brace{ A A}}(x_0)\cong\ell_e^2 \bl1_2\cos x_0q_0
\end{eqn}

\chapter{Determination of\\ the Weinberg Angle}

In this chapter a symmetry normalization related framework is proposed
and conditions are given therein leading to a numerical
determination of the Weinberg angle.

To recapitulate:
The four coupling constants $g$
or masses $\mu$ in the electroweak triangle for the standard model
\begin{eqn}{rl}
\cl G=(g_1,g_2;g_Z|e)&
={1\over M}(\mu_1,\mu_W;\mu_Z|\mu_e)
=M(\ell_1^2,\ell_2^2;\ell_Z^2|\ell_e^2)\cr
& \sim({1\over 2.9},{1\over 1.6};{1\over 1.4}|{1\over 3.3})\cr
\end{eqn}contain the
interaction symmetry normalizations $g_1^2$ for hypercharge ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_2)$ and
$g_2^2$ for isospin $\SU(2)$. The particle normalizations $\ell_Z^2$
and $\ell_e^2$ are related to the stability groups $\SO(3)$ and $\SO(2)$ resp.
of the neutral massive and massless particles
resp. in the Lorentz group $\SO^+(1,3)$. $M^2$ normalizes the internal group
${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$.

To compute the interaction and particle normalizations
(for a first  purely algebraic
orientation without the interaction related running of the coupling
constants), one can proceed in three steps:
First,
the ratio of the ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_2)$ and $\SU(2)$ normalizations has to be understood,
i.e. the Weinberg angle $\tan ^2\theta}   \def\vth{\vartheta} \def\io{\iota={g_1^2\over g^2_2}$.
An attempt for this first step is given in this chapter.
Therewith the
form of the electroweak triangle and the ratios of electroweak masses,
coupling constants and normalizations are determined. As a second step,
the absolute value of one internal coupling constant, e.g.
the fine structure constant ${e^2\over 4\pi}$, has to be given.
Finally, the  ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$ unit $M^2$
has to be related to other masses,
e.g. to the Planck mass or to the proton mass.

\section{Invariant Lie Algebra Forms}

Representations of Lie symmetries have their intrinsic metrical structure:
Any representation
of a Lie algebra $L$ on a vector space $V$, both finite dimensional, in
the endomorphisms algebra with its natural Lie algebra
structure
\begin{eqn}{l}
L\x V\longrightarrow V,~~l\bullet}\def\tria{\triangleleft v=\cl D(l)(v)\cr
\end{eqn}gives rise to the
$V$-associated $L$-invariant multilinear trace forms
\begin{eqn}{l}
t_V^k:\underbrace{L\x\cdots\x L}_{k-\rm{times}}\longrightarrow\K,~~
t _V^k(l_1,\dots,l_k)=\tr\cl D(l_1)\circ}\def\x{\times}\def\oo{\infty\cdots\circ}\def\x{\times}\def\oo{\infty \cl D(l_k)\cr
l\bullet}\def\tria{\triangleleft t^k_V(l_1,\dots,l_k)=
t^k_V(\com l{l_1},\dots,l_k)+\dots+
t^k_V(l_1,\dots,\com l{l_k})=0\cr
\end{eqn}

Any  irreducible representation of a semisimple
Lie algebra $L$
on a complex vector space $V$ gives an  invariant bilinear Lie algebra form,
unique up to a  normalization. These associated bilinear forms
are multiples $\alpha}  \def\be{\beta} \def\ga{\gamma t^2_V$, $\alpha}  \def\be{\beta} \def\ga{\gamma\in\K$, of the 'double' trace
\begin{eqn}{l}
 t_V^2(~,~):L\x L\longrightarrow\K,~~\cases{
 t_V^2(l,m)=\tr\cl D(l)\circ}\def\x{\times}\def\oo{\infty\cl D(m)= t_V^2(m,l)\cr
 t_V^2(\com kl,m)+ t_V^2(l,\com km)=0\cr}
\end{eqn}The bilinear form  associated  to the adjoint representation is the
Killing form.

E.g. the real Lie algebras $\log\SU(n)$,
especially the isospin Lie algebra $\log\SU(2)$,
can be spanned with the $(n^2-1)$ generalized
Pauli matrices
as basis
\begin{eqn}{l}
\hbox{basis of $\log\SU(n)$: }\{l^a={i\over\sqrt2}\tau(n)^a\big\bracevert}\def\Cup{\bigcup}\def\Cap{\bigcap a=1,\dots,n^2-1\}\cr
\hbox{for }\SU(2):~~\{\tau(2)^a=\tau^a\big\bracevert}\def\Cup{\bigcup}\def\Cap{\bigcap\hbox{Pauli matrices}\}\cr
\hbox{for }\SU(3):~~\{\tau(3)^a=\la^a\big\bracevert}\def\Cup{\bigcup}\def\Cap{\bigcap\hbox{Gell-Mann matrices}\}\cr
\end{eqn}The generalized Pauli matrices  have the structure constants
\begin{eqn}{ll}
\tau(n)^a\tau(n)^b&={2\over n}\delta}  \def\ep{\epsilon}  \def\ze{\zeta^{ab}\bl 1_n+
(\delta}  \def\ep{\epsilon}  \def\ze{\zeta^{abc}+i\alpha}  \def\be{\beta} \def\ga{\gamma^{abc})\tau(n)^c\cr
&\hbox{totally symmetric }\delta}  \def\ep{\epsilon}  \def\ze{\zeta^{abc}\in{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}\cr
&\hbox{totally antisymmetric }\alpha}  \def\be{\beta} \def\ga{\gamma^{abc}\in{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}\cr
\end{eqn}The defining representation of $\log\SU(n)$ on $V\cong\C^n$
has as associated bilinear forms
\begin{eqn}{rl}
t^2_n(~,~):&\log\SU(n)\x\log\SU(n)\longrightarrow{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}\cr
&t^2_n(l^a,l^b)=\alpha}  \def\be{\beta} \def\ga{\gamma_n\tr{i\over\sqrt2}\tau(n)^a{i\over\sqrt2}\tau(n)^b=
-\alpha}  \def\be{\beta} \def\ga{\gamma_n\delta}  \def\ep{\epsilon}  \def\ze{\zeta^{ab},~~\alpha}  \def\be{\beta} \def\ga{\gamma_n\in{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}
\end{eqn}$\SU(n)$ has no  structure which distinguishes any
normalization $\alpha}  \def\be{\beta} \def\ga{\gamma_n$.

Since for a semisimple Lie algebra the derived Lie algebra $\com LL$
coincides with $L$,
all nontrivial $L$-representations have to be traceless, i.e.
there do not exist nontrivial invariant linear forms for
semisimple Lie algebras.

This is different for abelian Lie algebras where the 'single' trace is
a basis for the invariant linear forms
\begin{eqn}{l}
L\x V\longrightarrow V,~~l\bullet}\def\tria{\triangleleft v=\cl D(l)(v)\cr
t^1_V: L\longrightarrow \K,~~\cases{t^1_V(l)=\tr\cl D(l)\cr t^1_V(\com kl)=0\cr}
\end{eqn}Therewith one obtains the invariant bilinear forms
of abelian Lie algebras from the squared 'single' trace
\begin{eqn}{l}
t^{1,1}_V(~,~):L\x L\longrightarrow\K,~~
t^{1,1}_V(l,m)=\tr\cl D(l)\tr \cl D(m)\cr
\end{eqn}

E.g. the real Lie algebras $\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_n)$ with basis
$\{l^0={i\over\sqrt2}\bl 1_n\}$,
especially the hypercharge Lie algebra $\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_2)$,
have as invariant bilinear forms in the defining complex $n$-dimensional
representation
\begin{eqn}{rl}
t^{1,1}_n(~,~):&\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_n)\x\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_n)\longrightarrow{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}\cr
&t^{1,1}_n(l^0,l^0)=
\be_n(\tr {i\over\sqrt2}\bl 1_n)^2=-{n^2\over2}\be_n,~~\be_n\in{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}
\end{eqn}Again, ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_n)$ has no structure which distinguishes any
normalization $\be_n$.

\section{Relative Normalization\\of
Hypercharge and Isospin}

In the electroweak sector of the standard model
the internal group
comes as product group ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)={\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_2)\circ}\def\x{\times}\def\oo{\infty\SU(2)$, but apparently
not as the direct product group ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1)\x \SU(2)$.
The correlation of hypercharge and isopin via the
common subgroup ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_2)\cap\SU(2)=\{\pm\bl 1_2\}$
can be  seen  on the colourless sector of the standard model\cite{RAIF,S921}
where fields with  integer and half integer isospin $T$
have always integer and
half integer hypercharge $Y$ resp., e.g. the left handed electron
fields $(T,Y)=({1\over2},-{1\over2})$, the right handed electron fields $(0,-1)$,
the $\SU(2)$ gauge fields $(1,0)$ etc.

The situation is more complicated for the colour nontrivial sector,
e.g. for the quark fields, where
in addition to the  $\SU(2)$-center $\{\sqrt{\bl 1_2}=\pm\bl 1_2\}$
(two-ality) also
the $\SU(3)$-center
$\{\sqrt[3]{\bl 1_3}\}$ (tri-ality) has to be taken into account.
Triality is correlated to  third  integer hypercharges:
E.g., the left handed quarks as
isospin
doublets $d_T=2$ and colour triplets $d_C=3$ have hypercharge
$Y={1\over d_Td_C}$.
This relation has been discussed to some extent in\cite{S926,S941}.
I will consider
in this paper only the colour trivial electroweak sector.

The invariant bilinear forms for the defining representation of
the real $n^2$-dimensional Lie algebra $\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(n)$
allow two real normalization constants $\be_n$ and $\alpha}  \def\be{\beta} \def\ga{\gamma_n$ for
the Lie subalgebras $\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_n)$ (abelian) and $\log\SU(n)$ (simple) resp.
\begin{eqn}{l}
g_n(~,~):\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(n)\x\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(n)\longrightarrow{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}},~~g_n(l,m)\cases{
=t^{1,1}_n(l,m)+t^2_n(l,m)\cr
=\be_n\tr l\tr m+\alpha}  \def\be{\beta} \def\ga{\gamma_n\tr l\circ}\def\x{\times}\def\oo{\infty m\cr}
\end{eqn}These bilinear forms read in the basis with Pauli matrices
\begin{eqn}{l}
\hbox{basis of $\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(n)$: }\{l^0,l^a\}=
\{{i\over\sqrt2}\bl 1_n,{i\over\sqrt2}\tau(n)^a\}\cr
g_n(~,~)\cong -{\scriptsize\pmatrix{
{\be_n n^2+\alpha}  \def\be{\beta} \def\ga{\gamma_n n\over2}&0\cr0&\alpha}  \def\be{\beta} \def\ga{\gamma_n\bl 1_{n^2-1}\cr}}
\end{eqn}

The real 2-dimensional space of the invariant bilinear forms of $\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(n)$
(coefficients $\alpha}  \def\be{\beta} \def\ga{\gamma_n$, $\be_n$)
can be decomposed into the bilinear
forms with symmetric and antisymmetric 'double' trace,
called Fierz symmetrical and antisymmetrical forms with
normalizations $\alpha}  \def\be{\beta} \def\ga{\gamma_n^\pm$
\begin{eqn}{l}
g^\pm_n(~,~):\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(n)\x\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(n)\longrightarrow{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}},~\cases{
g^\pm_n(l,m)=\alpha}  \def\be{\beta} \def\ga{\gamma_n^\pm(\tr l\tr m\pm\tr l\circ}\def\x{\times}\def\oo{\infty m)\cr
\alpha}  \def\be{\beta} \def\ga{\gamma_n^\pm\in{\,{\rm I\kern-.2emR}}}\def\S{{\,{\rm I\kern-.42emS}}\cr }\cr
g^+_n(~,~)\cong -\alpha}  \def\be{\beta} \def\ga{\gamma_n^+{\scriptsize\pmatrix{
{n+1\choose2}&0\cr0&\bl 1_{n^2-1}\cr}},~~
g^-_n(~,~)\cong -\alpha}  \def\be{\beta} \def\ga{\gamma_n^-{\scriptsize\pmatrix{
{n-1\choose2}&0\cr0&-\bl 1_{n^2-1}\cr}}\cr
\end{eqn}Only the Fierz symmetrical forms $g_n^+$ are definite.
The indefinite Fierz antisymmetrical forms have signature $(1,n^2-1)$.
The invariance groups of the quadratic
forms $g_n^+$ and $g_n^-$ are
the compact group $\O(n^2)$ and the noncompact group
$\O(1,n^2-1)$ with $\SO(n^2-1)$
 embedding ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(n)/{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_n)$.

For the hyperisospin group ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$ one obtains
as Fierz (anti)symmetrical quadratic forms
\begin{eqn}{l}
g^+_2(~,~)\cong -\alpha}  \def\be{\beta} \def\ga{\gamma_2^+{\scriptsize\pmatrix{
3&0\cr0&\bl 1_3\cr}},~~
g^-_2(~,~)\cong -\alpha}  \def\be{\beta} \def\ga{\gamma_2^-{\scriptsize\pmatrix{
1&0\cr0&-\bl 1_3\cr}}\cr
\end{eqn}If one can give an argument in favour of the
definite Fierz symmetrical form $g_2^+$
as the physically relevant hyperisospin form (next section),
the ratio of the normalizations
of abelian ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_n)$-symmetry and simple $\SU(n)$-symmetry is fixed
\begin{eqn}{l}
{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2):g_2^+(~,~)\cong{\scriptsize \pmatrix{
{1\over g_1^2}&0\cr
0&{1\over g_2^2}\bl 1_3\cr}}
~\Rightarrow~}\def\sub{\subseteq}\def\subnoteq{\subset \tan^2\theta}   \def\vth{\vartheta} \def\io{\iota={g_1^2\over g_2^2}={1\over 3}\cr
\end{eqn}and in the general case
\begin{eqn}{l}
{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(n):g_n^+(~,~)\cong{\scriptsize\pmatrix{
 {1\over g_1^2}&0\cr
0&{1\over g_n^2}\bl 1_{n^2-1}\cr}}
~\Rightarrow~}\def\sub{\subseteq}\def\subnoteq{\subset \tan^2\theta}   \def\vth{\vartheta} \def\io{\iota={g_1^2\over g_n^2}={1\over{n+1\choose 2}}\cr
\end{eqn}

\section{Fierz (Anti)Symmetrical Forms}

Statistical structures can determine the Fierz (anti)symmetry of
the bilinear forms for the ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$-normalization: The endomorphisms
of a finite dimensional vector space $V$
are isomorphic to the tensor product $V\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes V^T$ of space and dual space,
the invariant linear trace form  is the dual product
\begin{eqn}{l}
\tr: V\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes V^T\longrightarrow\K,~~\cases{
\tr v\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes\om=\dprod\om v\cr
\tr\com fg=0\cr}\cr
\hbox{dual bases }\tr e^a\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes\d e_b=\dprod{\d e_b}{e^a}=\delta}  \def\ep{\epsilon}  \def\ze{\zeta_b^a\cr
\end{eqn}

The extension of the trace from $V\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes V^T$ as 'double trace' for
invariant linear forms for the power two tensors $(V\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes V^T)^2$
can be done in two ways,  symmetrically  or antisymmetrically
\begin{eqn}{l}
\tr_\pm^2:(V\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes V)\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes (V\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes V)^T\longrightarrow\K\cr
\tr_\pm^2(v_1\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes v_2)\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes (\om_1\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes\om_2)=
\dprod{\om_1}{v_1}\dprod{\om_2}{v_2}
\pm \dprod{\om_1}{v_2}\dprod{\om_2}{v_1}
\end{eqn}These two possibilities are
the appropriate nontrivial forms for the symmetrical
$\vee}\def\Od{\bigvee}\def\OD{\displaystyle\bigvee $ or
antisymmetrical $\wedge}\def\And{\bigwedge}\def\AND{\displaystyle\bigwedge$ 'square' of the vector space $V$
\begin{eqn}{l}
\tr_+^2:(V\vee}\def\Od{\bigvee}\def\OD{\displaystyle\bigvee V)\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes (V\vee}\def\Od{\bigvee}\def\OD{\displaystyle\bigvee V)^T\longrightarrow\K\cr
\tr_-^2:(V\wedge}\def\And{\bigwedge}\def\AND{\displaystyle\bigwedge V)\otimes}\def\Ox{\bigotimes}\def\OX{\displaystyle\bigotimes (V\wedge}\def\And{\bigwedge}\def\AND{\displaystyle\bigwedge V)^T\longrightarrow\K\cr
\end{eqn}

The gauge interactions of the standard model act on vector spaces   $V$
with ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$ representations. Those vector spaces carry in the quantum
structure either  Fermi or  Bose statistics, characterized by
$\com vu=0$ or $\acom vu=0$ resp. for $v,u\in V$. Therefore only
the symmetrical product $V\vee}\def\Od{\bigvee}\def\OD{\displaystyle\bigvee V$ or
the antisymmetrical product  $V\wedge}\def\And{\bigwedge}\def\AND{\displaystyle\bigwedge V$  is nontrivial,
e.g. for the Higgs vectors $\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta\in\H\cong\C^2$ with Bose statistic
\begin{eqn}{l}
2\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^A\wedge}\def\And{\bigwedge}\def\AND{\displaystyle\bigwedge \varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^B=\com{\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^A}{\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^B}=0,~~
\hbox{nontrivial }2\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^A\vee}\def\Od{\bigvee}\def\OD{\displaystyle\bigvee\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^B=\acom{\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^A}{\varphi}  \def\Ga{\Gamma}  \def\Th{\Theta^B}
\end{eqn}

In a quantum structure where the hyperisospin ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$-interactions get
their normalization via a vector space
acted on only with the defining ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$ representation,
the associated bilinear form
for the Lie algebra of ${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$ has to be either Fierz symmetrical or
antisymmetrical. The definite forms $g_2^+$
require  symmetrical combinations of the underlying 2-dimensional
${\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$ representations.

Obviously only the normalization ratio
(mixing angle) for
$\log{\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(1_2)$ (hypercharge)  and  $\log \SU(2)$
as Lie subalgebras of $\log {\bf U}} \def\O{{\bf O}}   \def\SU{{\bf SU}} \def\SO{{\bf SO}(2)$ is fixed
by such a statistical argument, not
the absolute normalizations.

redpajama_set_name: RedPajamaArXiv


A very big number of cars in America are under a lease agreement. Trying a car with this method is really useful if you want to drive a newer one without having to cash out a huge amount of money that might cause some economical imbalance in your life.
Although this is a very preferred method and about one-quarter of the new cars in America are currently on lease, there are some things that you need to take in consideration before getting one. Here's what you need to know.
Which it really is. A lease is basically a pay to use contract that allows you to "own" the car for a certain time period, and then you pay for it accordingly. You can try a car that you like, drive it for some time and then you are free to walk away after paying a much smaller amount than you would've had if you were to purchase it instead. A lot of people found out that the car they thought is going to be their best ride, is really not that great, and they would've made a huge mistake if they bought it instead. That's why getting a car on a lease is such a popular method. You should be careful though, if you really want to lease the same car over and over again for a long time, you will end up wasting more money than you were if you fully purchased it instead.
It is not very common, but it is definitely possible to negotiate the price for a lease. Unlike car loans, leases are coming from car manufacturers, not the banks. You are able to reduce the price by some percentage if you try and discuss some of the details with the manufacturers. Negotiations will sometimes help you end up with a much lower price than you expected, so why not try?
This is the way Canadians lease their cars. The service provides you with the ability to instantly get a detailed list which has all the interest rates, discounts and price changes for all car models that are available in the country. Every single month, more than one hundred thousand Canadians are using this service in order to find the best deals and never make a mistake when it comes to getting a car on lease.
Is leasing the right thing for me?
There are a few things that you need to consider if you're thinking about getting yourself a car on leasing. Also, if you're looking to find the most inexpensive way of driving then you should probably avoid leasing. If you don't care about how your vehicle looks and you're just looking for the basic functionality, leasing might not be really suitable for you.
However, if you're a car lover that really wants to enjoy the comfort and luxury of new rides, while also not minding the monthly payments that will occur, then leasing can be a great option for you.
Thanks for sharing detail about leasing a car!! Great Post!!!


Rising superstar Sage Northcutt pulled off another victory in just his second Octagon appearance at Fight Night Las Vegas. The Cady, Tx. product spoke to UFC correspondent Megan Olivi backstage about his performance.


text:
Tragedy Assistance
Warfighter Support
Visible & Invisible Wounds Support
Meet TOMLF
Air Force Marathon
They give their lives to save others.
We're here to help them when
they've given their all.
We support the families and children of U.S. Air Force Rescue Heroes who are killed or severely wounded in operational or training missions with scholarships, immediate tragedy assistance, and other critical support.
We provide an all-around approach to supporting our rescue warriors when it's needed most.
Our events raise much-needed funds and awareness for rescue families in need.
Support OUR NEXT EVENT
Our rescue families need your support during times of tragedy.
WE CAN HELP YOU GET INVOLVED WITH THE RESCUE COMMUNITY
Want to know about our next event or new ways to help us support rescue families in need?
SUPPORT A HERO
There are many ways you can help us to reach as many rescue families as possible by making a tax-deductible donation to TOMLF, volunteering, or shopping at our store.
I consent to the Privacy Policy
You Can Help Change Lives
The Foundation is a Guidestar Exchange Platinum Level Participant and "Best in America" 501(c)3 non-profit charitable organization established in 2002.
info@thatothersmaylive.org
871 Coronado Center Drive, Suite 200
That Others May Live Foundation is a non-profit organization. It is not endorsed by the USAF and receives no federal funding.
That Others May Live Foundation is a qualified 501 (c)(3) and donations are tax-deductible to the full extent of the law.
All Rights Reserved | (C) 2020 That Others May Live Foundation | Site by: Creatief Marketing Group

redpajama_set_name: RedPajamaCommonCrawl


Does this mean that acceptance is a bad thing? No, it doesn't. But it doesn't mean that it is something that you have to be careful about when approaching.
Taking the positives and negatives from the concepts of awareness and acceptance, I think that there is a good ground to be reached somewhere in the middle. Most people are awareof autism to some degree – even if it is just through Rainman. Some people are acceptingof it without really knowing what it is. For me the real key is understanding; you can be aware of autism, but not really know what it is. Often this will lead to pity, people being patronising, or a belief that people with autism are either dangerous, or completely pointless trying to communicate with. A lot of negative stereo-types pervade the public's perception of autism; they are accepting yes, but quite often they are not accepting of real autistic people. They might be aware, but in reality they are not aware of real autism, they are aware of the Medias` portrayal of autism. That is why understanding is so important: you are aware of autism – good. You are accepting of autism – good. But do you understand autism?


text:
Mukesh Ambani's earnings during Covid 'can lift' 40% informal workers out of poverty
By Dr Gian Singh*
The Inequality Virus Report released by Oxfam, a non-profit organization, on January 25, 2021 on the growing inequalities in different parts of the world, sheds light on the growing economic, educational, healthcare and gender inequalities in India. The report has revealed that the wealth of billionaires has increased by 35 per cent during the lockdown period in the country.
The Indian economy ranks sixth in the world after the United States, China, Germany, Russian Federation and France. With increased wealth of 100 billionaires of the country during the lockdown period, a check of Rs. 94045 could be given to every poor man. The wealth raised by just 11 billionaires in the country during the Coronavirus Coronavirus Pandemic could cover the expenses of MGNREGA or the Ministry of Health in the next 10 years. While billionaires continued to amass wealth by escaping the deadly effects of the Coronavirus pandemic, poor workers continued to suffer from unemployment, starvation, and death.
The report reveals that Mukesh Ambani, Gautam Adani, Shiv Nader, Cyrus Poonawala, Uday Kotak, Azim Premji, Sunil Mittal, Radhakrishnan Damani, Kumar Mangalam Birla and Lakshmi Mittal were among the richest billionaires who have amassed huge wealth after the March 2020 lockdown. In April 2020, 170,000 working people lost their jobs every hour.
Mukesh Ambani, who has become the richest man in India and Asia, earned Rs 90 crore per hour during the Coronavirus pandemic, while 24 per cent of the country's population earned less than Rs 3,000 per month. With the increase in his wealth, 40 per cent of the informally employed workers can be lifted out of poverty for 5 months. The report has also revealed that it would take 10,000 years for an ordinary worker to earn as much as Mukesh Ambani earned in an hour during the pandemic.
In terms of health services, 93.4 per cent of the country's top 20 per cent have their own sanitation facilities, while only 6 per cent of the poorest 20 per cent have such facilities. As many as 59.6 per cent of the country's population lives in one room or less, as a result of which they are not even in a position to maintain the required physical distance to survive the Coronavirus pandemic.
As many as 32 crore students were affected by the closure of schools till October 2020 out of which 84 per cent were villagers and 70 per cent were studying in government schools. Forty per cent of government school teachers in the country's five states fear that one-third of their children will not be able to attend school when the schools reopen. Dalits, Adivasis and Muslims will outnumber other school dropouts. The female students would be hit hard the most because of their early forced marriages, abortions and different types of violence.
The pre-lockdown unemployment rate for women workers, which was 15 per cent, has risen to 18 per cent during this period, which could lead to a decline in India's GDP by about 8 per cent. Women who were employed before the lockdown period may have a 23.5 per cent lower chance of regaining employment after the lockdown period than male workers.
Inequalities in India have not arisen spontaneously, but are the result of policies formulated and implemented by our rulers. The people who fought and sacrificed a lot to liberate the country from the British rule had a dream that an independent India would move forward by overcoming all forms of inequalities and making the lives of the people here comfortable and dignified.
To fulfill these dreams after independence, the Planning Commission was set up in 1950 and Five-Year Plans were introduced from 1951. The implementation of these plans led to the emergence of a mixed economy in which the public sector was created, expanded and developed, and the functioning of the private sector was monitored and regulated.
Although there were some shortcomings in the functioning of the public sector, but they were misrepresented by the capitalist world and their courtiers, there was a commendable increase in employment opportunities and standards for the working people during the planning period (1951-80). As a result, free services were provided to the general public which reduced economic inequalities in the country.
Planning was put in the reverse gear by the countries' rulers since 1980, and the NDA government by abolishing the Planning Commission established a pro-capitalist NITI Aayog. The 'new economic policies' of liberalization, privatization and globalization introduced in the country since 1991 have hit the public sector enterprises hard and started selling them to the capitalist/ corporate world for pennies which has continued unabated.
As a result of these policies, numerous concessions/favours are being given to the capitalist/corporate world. The opportunities and quality of employment are being reduced which are resulting in growing numerous inequalities in the country and the ordinary working people struggle to control their appetite by eating only two bites of bread, what to talk fulfilling their basic needs of food, clothing, housing, education, healthcare, clean environment and social security.
It would take 10,000 years for an ordinary worker to earn as much as Mukesh Ambani earned in an hour during the pandemic
About 50 per cent of the country's population depends on agriculture for their livelihood, which is being given about 16 per cent of the national income. In 1951, 82 per cent of the country's population depended on agriculture for their livelihood, was being given 55 per cent of the national income. The plight of peasants, farm labourers and rural artisans out of the country's agriculturally dependent population has made it extremely unsatisfactory.
Landless farm labourers and rural artisans have no other means of production but to sell their manual labour. The use of machinery and herbicides in the package of New Agricultural Technology adopted to meet the food needs of the country has reduced the employment opportunities and quality of employment of these workers in the agricultural sector.
Research studies have revealed that due to the economic and agricultural policies formulated and adopted by the rulers, almost all peasants, farm labourers and small artisans are born in poverty and debt, live their hard life in poverty and debt and after leaving huge debt and extreme poverty for the coming generations either they die a death of deprivation or when all their hopes of life are dashed, they start committing suicides.
Leaving the mountain of debt and abject poverty for their future generations, they either die a miserable death or even commit suicide when all hopes for their lives are dashed. The Union government has enacted three laws relating to agriculture paving the way for the eviction of farmers, farm labourers, rural small artisans and other sections dependent on the agricultural sector. When these laws are fully implemented these would further increase the already sharply growing inequalities.
The 66th round of the National Sample Survey has revelated that 92.8 per cent of the country's workers were in informal employment during 2009-10. In the last 10 years, not only has the number of formally employed workers been reduced, but the quality of employment has been brought down to a very low level. The latest example is the lockdown introduced during the Coronavirus pandemic.
During this time the informally employed workers were evicted without any compensation or financial assistance due to which these workers had to travel long distances to reach their homes and faced many hardships and many families have lost their bread earners. The way in which the public sector is being axed and public institutions are being sold at very low prices to the capitalist / corporate world. The problems faced by the employed workers will increase.
In the development of the industrial sector in the country, all possible help is being given to the large industrial units, as a result of which their dominance in increasing their wealth and making and implementing government policies in their favour is constantly increasing. The increasing use of machinery and automated machinery in large industrial units is seriously eroding the employment and quality of employment.
Small scale industrial units, which mostly boost employment, are being discouraged by the government in various ways. The services sector has made a lot of gains, but the employment opportunities in this sector are also very limited and they are also generally for those who speak and write English and have knowledge of computer work. Most of the workers employed in this sector have such a meager income that they make a living with great difficulty. The services rendered by this sector have become so expensive that the working class cannot even dream of taking advantage of them.
An increase in the economic growth rate of the country can be significant only if the economic development caused by it raises the living standards of the working class. In order to overcome the growing inequalities in the country, it is necessary to adopt a people-friendly and nature-friendly economic development model which ensures expansion and development of the public sector and monitoring and control over the private sector.
According to the Oxfam report and the suggestions made by Nobel Prize-winning economist Stiglitz and other economists, taxes on the rich should be increased, markets should be monitored and regulated, and quality education and healthcare services should be provided free of charge to everyone. The government should ensure quality employment to all the working people.
*Former professor, department of economics, Punjabi University, Patiala
Business Development Economy Governance People Society
indian_patriot said…
This was an Indian Flag rally which the photos you have posted on top of the article, not a Hindu rally. That itself shows your agenda against Hindus. The rally was diverted by police because of the accident and because of the risk of the Khalistani mob attack. You should be ashamed of saying 'Hindu Extremism. There is/was never Hindu extremism, if there was, Pakistan would've been destroyed for destroying Kashmiri Hindu lives, for making them refugees in their own country, for raping, killed and looting them, for kicking them out of their own houses. Hindus can let go of the insults made on their faith but they can never forgive those who insult their country, who go against their country. Did you ever ask how did farmer protests have become a project against Hindus? Why only Sikhs are protesting against the laws? You all have a problem with the Hindu-centric Modi government and the Hindus. You are all afraid of the ever-growing confidence of the Hindus. You are all afraid that you won't be able to further your agendas anymore. Islamic extremism is the major source of terrorism around the world and you are accusing Hindus of extremism and saying the Muslims and Christians are not safe in India? You probably have noticed how fast so-called minorities are growing their population. Some by converting, some by breeding, if there was ever Hindu extremism, there wouldn't have been any Muslims and Christians in India or they grow so fast. Stop these biased articles against Hindus. If you have a problem with the Indian government, take it up with them. Don't equate government with Hindus.
Tarunjot Singh said…
Your so called indian rally were yelling jai shri ram and attempting to attack a sikh gurudwara and raise rssor BJP flag but were stopped by police. If it had been in india police qould have let them go and attack gurudwara and kill sikhs, but since australia is real democracy and secular that is not allowed. Why is sikh youths attacked for no reason by hindutavis in sydeny? I wonder how bad they are treated in india.... were hindutavis outnumber them by many times and have whole police and hindu population amd govt to help them terrorise minority....
Were did u get khalistani from, there was 0 khalistani mentioned except you made up ur own lies. Australian media even said they were trying to attack gurudwara
Vendetta politics of govt: Over 600 activists on F...
'Bird, take me flying with you too!' Being Devanga...
Young Dalit labour leaders were 'mercilessly tortu...
Surface irrigation won't improve 'dramatic' ground...
IMPRI survey: 72% Bihar casual workers reported th...
Union budget: Except metro rail projects, no 'appr...
In a city and an industry obsessed with speed, Viv...
Withdraw prosecution against Disha Ravi, Nikita Ja...
Revealing the real ways of Tibetan govt in exile t...
Rally asks Punjab govt to repeal APMC amendment, c...
Odisha's Kulda mines 'heavily polluted', yet Centr...
India needs accountability, should clean up politi...
UP STF raid at Delhi state office part of 'ongoing...
Kashmiri diaspora group sheds equidistance, ignore...
Chauri Chaura: Why Modi govt decided to 'question'...
Sreedharan's keenness to be part of saffron agenda...
Aged 88, facing detractors, will Sreedharan's expe...
Golwalkar's views on tricolour, martyrs, minoritie...
Favouring industrial lobby? 12,635 acres land acqu...
Budget 2021-22 is 'uninspiring' for Dalits: Whithe...
Intimidating atmosphere, as 'ramshackled' judiciar...
As 'self-centred' politicians lose credibility, ar...
Land acquisition for coalmining: Outdated law bein...
Digital fascism? Incriminating, 'fabricated' evide...
Disha Ravi case: Conspiracy theory is being floate...
Catholic Union: Hounding of young climate change a...
Attack on young girls: Madhya Pradesh tribal women...
Dholi Ganga disaster: People just don't want dam, ...
West Bengal tea garden workers protest 'unjust' ta...
Clampdown on media: Why even well-meaning people f...
Our members arrested in UP citing 'fake' terrorism...
Overcoming trade imbalance in South Asia and IMF's...
Aadhaar a 'major hurdle' in disbursement of social...
Sinful forms of escapism? 'Buzzwords' in Church of...
Ahmedabad slum dwellers lathicharged: Demand for b...
Odisha NGOs launch #Mission1lakh to 'bridge' educa...
Attack on Newsclick: Criminalising free speech as ...
20% of FIRs against journalists in 2020 alone, tar...
TINA factor? Why so many Indians still think: Modi...
Young environmentalist's arrest 'sinister', even p...
Tikait's emotional outburst: Is emphasis shifting ...
Govt of India 'treating' its own citizens as enemi...
A peep into Australia's South Asians obsessed with...
Nothing new about Gujarat's solar misadventures. T...
US forensic revelation enough evidence to release ...
'Andolanjeevi' vs 'hum do hamare do': Will non-BJP...
Reasons I decided to install and uninstall the nat...
Easing eco-norms? Uttarakhand havoc result of Govt...
Union budget ignores jobless urban labour, 'reduce...
Hitler, Mussolini, Churchill, Thatcher have used t...
Tamil Nadu's Salem sanitation workers 'not paid' f...
Video reveals caste is still powerful means to sup...
Mukesh Ambani's earnings during Covid 'can lift' 4...
Will UP govt cease 'attempts' to muzzle protests, ...
Elgar Parishad event: 'BJP inspired' FIRs against ...
Uttarakhand tragedy: How govt 'ignored' dam report...
Kutch communal violence 'provoked' to cause physic...
Why farmers' movement should seriously address #Re...
Andolanjivi: Modi's new term meant as 'fodder' to ...
Sovereignty: Ignoring anti-colonial concept, celeb...
India's business executives regret further hardeni...
Time to ponder: Why activists, politicians target ...
How Australian cops protected protesters favouring...
Disaster in Uttarakhand: Hindutva project? Religio...
NAPM extends support to Indian, Aussie citizen g...
BJP rulers' effort to 'undermine' Frontier Gandhi'...
Villagers' bovine march 'forces' Yogi to hold govt...
Reduced concern for rescued child bonded labour i...
Despite pandemic, allocation for education down fr...
Whither right to food? Social security scheme allo...
Protests in support of farmers' Chakka Jam call: A...
Aftermath of Tikait's tears: Political scenario in...
Economic loss of Jan 26 internet shutdown in Delhi...
Global NGO objects to 'fortification' of Delhi wit...
Farmers' protest 'highlights' sprawling, spoilt ur...
Ex-civil servants support agitating farmers, seek ...
Unexpected? Western regional meet objects to Gujar...
Gujarat electricity plant in Bhavnagar operates wi...
Majoritarianism is not a crisis of democracy but o...
Assam meet protests 'continuing' eviction of adiva...
Modi's budget 'reflects' minimum govt for people, ...
Journalists' arrest: Govt's 'scare tactics' won't ...
Why Modi's Mann ki Baat was silent on who 'insulte...
Union budget fails to address 'urgent' spending ne...
Spate of anti-conversion laws leading to state 'co...
'Viability' of agricultural cooperatives vs govt p...
Total education budget down by 6% amidst 'fillip' ...
New seed of hope for peace, unity? Solidarity prot...
Farmers' protest: Excessive democracy allows expor...
Punia arrest, criminal cases on journalists meant ...
Doublespeak? Why India's authorities are silent on...
Gandhi 'touched' soul of India's masses as Communi...


Our first tryst with Asif Khan Dehlvi of Delhi Karavan, where we had explored the rich heritage of Mehrauli Archaeological Park, had turned out to be a huge success and had instilled in us a craving to explore Delhi's gems furthermore. Therefore, on the bright morning of 13th February, when the world was caught up in celebrating their love for each other, the highly spirited POSist Team decided to celebrate our love for nature instead!
Super excited for the trip, we all assembled at the Qutub Minar once again, eagerly waiting for the walk to begin. This time, we'd decided to trek our way through the famous Sanjay Van to Qila Lal Kot which is the ruins of the once formidable palace of Tomar Rajputs. The 10 km forest is said to be haunted, and the sprawling trees seemed to whisper old folklores and tales that they'd witnessed throughout the years. Asif further set the mood by telling us little anecdotes of the forest and some popular spooky tales as well.
As we moved forward the jungle trail, taking in the quiet beauty of the forest, we spied some Neelgai, basking in the winter sun, a majestic peacock strutting, looking a tad surprised to see some eager visitors early in the morning, and exotic birds serenading. We also picked up a companion on the way, a stray canine whom we fondly named Honey, who accompanied us throughout the walk.
The jungle seemed to be alive with the chirping of the birds, the quiet rustle of trees, and our gay chatter. With the cool breeze kissing our faces, and the sun peeking from behind the clouds just enough to keep us warm and energetic, we had a great time chatting amongst ourselves, taking pictures, and also taking in the natural beauty. The awesome hoodies done for our team by Champu.in kept us warm and snug in the cool morning hours.
We sought a trail that took us to a rocky stairway, which led us to the tomb of Haji Roz Bih. We listened with rapt attention as Asif told us about the Slave Dynasty, and shared some beautiful anecdotes of the time.
We further learned about the seven cities of Delhi, Qila Lal Kot, Mehrauli, Siri, Tughlakabad, Firozabad, Shergarh, and Shahjahanabad. Asif carved a map into the dust with a stick, and we were transported back to the times of medieval Delhi while listening to the historical escapades of rulers of the time.
Moving on, we trekked a little upwards and reached the rocky wall of the Qila Lal Kot fortress, wide enough to accommodate the horses racing on them, where we paused to take in the breathtaking view. High above the ground, with the cool breeze and light drizzle, we had a great time capturing those unforgettable moments in pictures.
A little ahead we reached a rocky enclosure where we decided to sit and relax, as we listened to the mesmerizing storytelling of Asif. His beautiful narration of the story of Raziya Sultan, and how she came into power is sure to stay with us forever.
As the sun started to shine on our heads and the clock ticked off, with a heavy heart we began our descent. Once back on the planes, Asif addressed us for the last time, but we, of course, would not let him go without a sher(couplet), and he obliged us with beautiful lines of Ghalib that seemed perfectly apt for the day.
I am in wilderness and spring has arrived at my house".
Qila Lal Kot one of the lesser known trekking and tourist destinations of our city Delhi, is the abode of the kind of peace and serenity that we city dwellers often seek in foreign mountains. The beautiful couplet says that we seek happiness and prosperity outside in the world, which is however present instead in our homes and hearts.


Marc Bakker and Marjolijn Botje have been the proud owners of African Queen Guesthouse in Noordhoek since February 2014. Together with our daughters Floor and Kiki, we are operating the guesthouse in a similar way as our predecessors, Bram and Margaux.
We are living on the premises and our guests can expect a high level of hands-on service and hospitality. Our aim is to offer our guests the most enjoyable stay that make them want to come back to our guesthouse time and time again.
Marc has a long history with South Africa where he first came to the country in 1983 as a Rotary Exchange Student. He spent a year at Dale College in King William's Town. In 2002 we bought a house in Kommetjie where we spent many happy holidays. We have been living permanently in the Southern Peninsula since 2006 and have an excellent knowledge of Cape Town and the surrounding areas.
Marjolijn has lived an international life as a daughter of 'Shell expats'. She has real African roots as she was born in Nigeria and as such our daughters are, interestingly, eligible for Nigerian citizenship. She has a degree from Amsterdam University in European studies and has had a previous career in telecommunications. Marc has a law degree from Leiden University but went on to work in sports – and lifestyle management.
Through Marc's brother, Dave Bakker, we got to know the hospitality industry as he and his wife Anouk are managers/owners of Wedgeview Country House & Spa in Stellenbosch and WhaleRock Luxury Lodge in Hermanus. Both Marc and Dave have been active cricket players and played representative cricket for The Netherlands. Although a double knee operation has 'slowed' down Marc considerably he still enjoys playing golf (hc. 4) and he and Marjolijn are members at the Clovelly Golf Club.
We have a passion for good food & wine and we love the outdoors; we feel privileged we live in one of the most beautiful places on the planet. Also we are strong on sustainability; we recycle & reuse and we don't use pesticides in our organic vegetable garden. Our 2 pools and showers are heated through solar power and eggs for breakfast come mostly from our own chickens.
In the last year we added a new salt water solar heated swimming pool (10 x 3,5m), 2 new rooms (Ocean and Sunset), a deck for the Honeymoon Suite and a sunset lapa. Also a sauna is now in operation; the little 'shack' where the sauna is housed in has been built with mainly recycled materials.
War bei uns seit Anfang an dabei. Sie lebt mit ihrem Mann Wellington in Masimphumele. Ihre drei Kinder Lindile, Sandile und Odwa wohnen mit ihrer Familie in der Provinz Eastern Cape, wo sie zur Schule gehen. Sie hat immer ein Lächeln auf ihrem Gesicht und wir sind wirklich beneidenswert, sie bei uns zu haben.
Nach der Eröffnung des Gästehauses im September 2008 haben wir schnell gemerkt, dass wir mehr Hilfe brauchten. Siphokazi war die Antwort auf unser Problem! Sie lebt in Masimphumele mit ihrem Mann Thomas und ihrem Sohn Banele. Ihr anderer Sohn Anovuyo wohnt mit ihrer Familie in der Provinz Eastern Cape. Ein wenig schüchtern, aber sehr nett, ist sie immer da, wenn wir sie brauchen.
Lameck wirkt wunder in unserem Garten. Ursprünglich ist er aus Malawi, und kam im Jahr 2003 auf der Suche nach einem besseren Leben nach Kapstadt . Er lebt jetzt außerhalb Masi mit seinem Sohn Goli, während seine Frau Eunice mit dem Rest der Familie in Malawi wohnt. Seine Fähigkeiten sind so vielseitig, dass er uns von einem Pferdestall bis zu einem Nistkasten für Eulen alles bauen kann.


Video provides a powerful way to help you prove your point. When you click Online Video, you can paste in the embed code for the video you want to add. You can also type a keyword to search online for the video that best fits your document. code for the video you want to add. You can also type a keyword to search online for the video that best fits your document.code for the video you want to add.