, the particles exchange virtual quanta of momentum. \end{em}

An exception to this rule is the representation where the 
3-momenta $\mathbf{p}_1$ and $\mathbf{p}_2$, and therefore also $\mathbf{p}$,
are parallel. 
In this representation, there is no entanglement and hence no exchange of virtual 
quanta.

In conjunction with the conservation laws for total momentum and angular 
momentum, this rule has the character of a \begin{em}complementary 
dynamical law\end{em}, dictated by Poincar\'e symmetry.
It defines a mechanism that forces two particles to exchange quanta of momentum 
in a controlled way.
I call this rule `a law' (in the following: `the Dynamical Law') for two reasons:
Firstly, it derives directly from the commutation relations of the Poincar\'e group
and is, therefore, fundamental for relativistic quantum mechanics, as fundamental 
as the conservation laws of energy, momentum, and angular momentum.
Secondly, it has, just as the mentioned conservation laws, far-reaching implications
for the physics of elementary particles. 

Actually, the Standard Model uses a concept of momentum exchange to describe 
three of the fundamental forces between elementary particles.
Feynman, when formulating the perturbation theory of quantum electrodynamics 
\cite{rf1,rf2,rf3}, spoke of `virtual quanta'.
Based on the postulate of gauge invariance, the exchanged quanta became 
`gauge bosons'.
Integration over the exchanged momenta is an integral part of the Feynman rules 
in momentum space (cf., e.g., \cite{sss1}).
Together with the conservation of momentum at each vertex, this integration 
generates the kind of momentum entanglement that is demanded by the Dynamical Law.

As just shown, there is no need to explicitly postulate the presence of gauge 
bosons: they are there not for reasons of gauge invariance, but because the 
structure of the Poincar\'e group requires their existence.
However, in contrast to the common understanding of QED, these gauge boson are 
not the quanta of an independent field.
They rather stand for a field that describes the intrinsic structure of 
many-particle states as determined by Poincar\'e symmetry. 
Therefore, this field may be referred to as an {\it intrinsic gauge field}.


\section{Irreducible two-particle states}
\label{section_5}

The parameter space of an irreducible two-particle representation derives from
the two-particle mass-shell relation
\begin{equation}
(p_1 + p_2)^2 = M^2 ,   \label{5-1}
\end{equation}
which defines a seven-dimensional surface in an eight-dimensional space--time continuum.
After inserting the mass-shell relations for the individual particles, this parameter space
is described by a five-dimensional space $\Omega$ of independent momentum parameters.
$\Omega$ is embedded into the six-dimensional momentum space $\mathbb{R}^3\times\mathbb{R}^3$. 
It has spherical properties with respect to four dimensions and is unbounded in 
the radial direction.
Accordingly, $\Omega$ has a spherical infinitesimal volume element 
$d\omega(\mathbf{p}_1,\mathbf{p}_2)$. 
On the other hand, the embedding into $\mathbb{R}^3\times\mathbb{R}^3$
induces a Cartesian infinitesimal volume element $d^5\!p$ on $\Omega$.

Since the two-particle states of an irreducible representation are momentum entangled,
a two-particle state is represented by an integral
\begin{equation}
\left|\Phi\right> = \! \int\!d\omega(\mathbf{p_1},\mathbf{p_2})\; 
c_{\mathbf{p_1}\mathbf{p_2}}\,\left|\mathbf{p_1},\mathbf{p_2}\right>  
\label{5-2} 
\end{equation}
over pure product states $\left|\mathbf{p_1},\mathbf{p_2}\right>$, with state-specific
coeffients $c_{\mathbf{p_1}\mathbf{p_2}}$.
The range of integration is a finite (bounded) subspace of the full parameter space 
$\Omega$.
Since the product states are normalized to $\mathbb{R}^3\times\mathbb{R}^3$, a
normalization factor that maintains the normalization of the two-particle state is required.
It equals the square root of the volume of the integration area.
This area is essentially the spherical subspace of $\Omega$.
It makes sense to split the normalization factor into a state-independent factor $\omega$, 
directly related to the square root of the volume of the spherical subspace, and a 
state-specific factor.
In state (\ref{5-2}), the first factor is assumed to be included in the volume element 
$d\omega$, the second one in the coefficients $c_{\mathbf{p_1}\mathbf{p_2}}$.
The spherical volume element can then be replaced by the Cartesian volume element $d^5\!p$
\begin{equation}
d\omega(\mathbf{p}_1,\mathbf{p}_2) \;\;\Rightarrow\;\; \omega \, d^5\!p.  \label{5-3}
\end{equation}
Using (\ref{5-3}), a two-particle state can be written as
\begin{equation}
\left|\Phi\right> = \! \int_\Omega\! \omega \, d^5\!p\;
c_{\mathbf{p_1}\mathbf{p_2}}\,\left|\mathbf{p_1},\mathbf{p_2}\right>.
\label{5-4} 
\end{equation}


\section{Scattering experiments}
\label{section_6}

The experimental setup of a typical scattering experiment (Fig.\,1) will help to reveal 
the physical meaning of $\omega$.
An incoming plane wave (lower left) with momentum $\mathbf{p}$ passes an aperture 
and hits a target (centre).
Parts of the scattered wave pass the second aperture and are registered by 
a detector (right).
Between the apertures, an incoming particle together with a particle of the target 
form an intermediate, correlated two-particle system.
By positioning the first aperture at a distance $\mathbf{d}$ from the 
target, an orbital angular momentum $\mathbf{d} \times \mathbf{p}$ is selected.
The observed transition is represented by a hyperbolic trajectory.

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