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\subsection{The intrinsic quantum oscillators}\label{iquo}
%
It is known that the most basic field structure (whatever it may be its nature) is the scalar field; then we will say that any observer in their frame reference in order to describe the Universe, in its \emph{most basic} form, uses a scalar field, which we specifically denoted by $\Xi$.\\
It's obvious that $\Xi$ may be represented in any reference system by the scalar field $\Xi (x, y, z, t)$, where $(x, y, z, t)$ are the coordinates defined in that reference system. Therefore to treat the Universe-(ST) as an \emph{object-field} we start to define an elementary scalar field $\Xi$, whose basic oscillators will be called Intrinsic Quantum Oscillator (IQuO)
\cite{Guido_2012}. \cite{guido2014}. The elementary structure of $\Xi$ is a set of 1-dimensional chains af IQuOs elastically coupled, where the \emph{quanta} associated to this
\emph{Fundamental Field} \emph{fundamental field} can propagate themselves.\\
The
following figure
\ref{fig_iquo_lattice} makes us better understand this definition: a scalar field with a system of beads that are mutually linked by
springs:\\
(figure spring layer)\\ springs.\\
%\label{fig_iquo_lattice}
Here we summarize some fundamental aspects of the IQuOs, that will be recalled in the present paper.\\
First of all we explain the origin of the idea behind In \cite{guido2014} one proves that the
IQuO. The elastic coupling between quantum oscillators, which builds a field, transforms
even each individual
\emph{isolated} quantum field oscillator in
a coupled oscillator. This makes each field oscillator a \emph{forced} oscillator, that, following the oscillations, is one described by two components: the
absorptive amplitude inertial and the
elastic amplitude (see ref. [3]).\\
In the phase plane $(q, \, p)$ we have:
\begin{equation}
q(t) = q_{el} (0) \cos (\omega t) + q_{abs} (0) \sin (\omega t)
\end{equation}
Using operators. it becames:
\begin{equation}
\hat{q} (t) = \hat{q}_{el} (0) \cos (\omega t) + \hat{q}_{abs} (0) \sin (\omega t)
\end{equation}
That we can draw in this way:\\
(figure)\\
And one elastic. So, the field
of the quantum oscillator will no longer described by the pair of operators $a$, $a^+$
\begin{align} \begin{align}\nonumber
a_t = a_0 \text{e}^{-i \omega t}\\
a_t^+ = a_0^+ \text{e}^{i \omega t} \nonumber
\end{align}
but by two pair of operators $(a_{el}, \, a_{el}^+)$, $(a_{in}, \, a_{in}^+)$, where the absorptive component overlaps the inertial
component.
\begin{align}\label{eq:pairelin} component (see figure \ref{fig_iquo}).
\begin{align}\label{eq_new_qo_operators}
a_t = a(t)_{elastic} + a(t)_{inertial} = a_{el} \text{e}^{-i \omega t} + a_{in} \text{e}^{-i (\omega t - \pi/2)}\\
a_t^+ = a^+(t)_{elastic} + a^+(t)_{inertial} = a_{el}^+ \text{e}^{i \omega t} + a_{in}^+ \text{e}^{i (\omega t - \pi/2)} \nonumber
\end{align}
(Graphical representation of the operators \ref{eq:pairelin}: the inertial pair $(a_{in}, \, a_{in}^+)$ is shifted of $\frac{\pi}{2}$ than the elastic pair $(a_{el}, \, a_{el}^+)$)\\ %\label{fig_iquo}
This double structure of the operators $a$, $a^+$ splits the energy quanta of the quantum oscillator,
given by:
\begin{equation}
\hat {H} = \left ( \hat{a}^+ \hat{a} + \frac{1}{2} \right ) \hbar \omega = \left ( \hat{n} + \frac{1}{2} \right ) \hbar \omega
\end{equation} that becames
\begin{equation}
H_n = U_n + K_n = \left ( U_n \right )_{el} + \left ( K_n \right )_{in} = (2n+1) \left ( \frac{1}{4} \hbar \omega \right )_{el} + (2n+1) \left ( \frac{1}{4} \hbar \omega \right )_{in}
\end{equation}
The splitting of the oscillator is shown also in the form of the wave function $\psi$ of the quantum oscillator: we observe the presence in $\psi$ of a pair of peaks in the probability of detecting the quantum of the oscillation that we will describe as well as composed of two sub-units of oscillation
(sub-oscillators).\\
(figure)\\ (sub-oscillators, see figure \ref{fig_suboscillators}).\\
% \label{fig_suboscillators}
An IQuO ($\varepsilon_n = \left ( n+\frac{1}{2} \right ) h \nu$) will be represented by empty half-quantum ($\varepsilon (\circ) = \frac{1}{4} h$), and full half-quantum ($\varepsilon (\bullet) = \frac{1}{2} h$); so $\varepsilon_n = \left ( n+\frac{1}{2} \right ) h \nu = ( n (\frac{1}{2} + \frac{1}{2}) + \frac{1}{4} + \frac{1}{4} ) h$.\\
We can represent the annihilation operator $a$, and the creation operator $a^+$ using the full half-quantum $\bullet$ and the empty half-quantum $\circ$.
\begin{equation} %\begin{equation}
% B_1 = \left (
% \begin{array}{cc}
% \hat{a}_{el} \equiv \full_{el} & \hat{a}_{in} \equiv \empt_{in}\\
% \hat{a}^+_{el} \equiv \empt^+_{el} & \hat{a}_{in} \equiv \full^+_{in}
% \end{array}
% \right ) \Leftrightarrow
% B_2 = \left (
% \begin{array}{cc}
% \hat{a}_{el} \equiv \empt_{el} & \hat{a}_{in} \equiv \full_{in}\\
% \hat{a}^+_{el} \equiv \full^+_{el} & \hat{a}_{in} \equiv \empt^+_{in}
% \end{array}
% \right )
\end{equation}
or, %\end{equation}
%or, in a more classical representation:
\begin{align}
\hat{a}_{1, r'}^+ (t) = & \, \full^+_{el} \text{e}^{i r' \omega t} + \empt^+_{in} \text{e}^{i (r' \omega t - \frac{\pi}{2})}\\
\hat{a}_{1, r'} (t) = & \, \empt^+_{el} \text{e}^{-i r' \omega t} + \full^+_{in} \text{e}^{-i (r' \omega t - \frac{\pi}{2})} \nonumber
\end{align}
(Bidimensional graphical representation of the IQuO $(a_{el}, \, a_{el}^+)$, $(a_{in}, a_{in}^+)$)\\
One % \label{fig_iquo01}
%One of the advantages in treating the field oscillators in terms of IQuOs is that the bidimensional representation
(fig. 4*) (figure \ref{fig_iquo01}) allows to distinguish the direction of rotation of the phase associated with oscillations. In
[art 1] \cite{guido2014} has showed that the two directions of rotation emerge in the physical world as the two signs associated with the electric charge. This important result marks a turning point in the understanding of the physics of the interactions.\\
So the egenstates $\left | 0 \right >$, $\left | 1 \right >$ will be:
\begin{subequations}
\begin{align}
\left | 0 \right > & =
\binom{\circ_{el}}{\circ_{in}}\\ \binom{\circ_{el}}{\circ_{in}} \label{eq-eigenstate0}\\
\left | 1 \right > & = \binom{\circ_{el}}{\bullet_{in}} + \binom{\bullet^+_{el}}{\circ^+_{in}}
\nonumber \label{eq-eigenstate1}
\end{align}
\end{subequations}
% \label{fig_eigenstates}
A field line (gone through by a quantum $\bullet + \bullet$ and with different values of frequency) will be represented by the following scalar field:
\begin{equation}
\hat{\Phi}_R = \sum_k \omega_k \left ( \hat{a}_k \text{e}^{-i \hat{r} \omega_k t + \alpha} + \hat{a}_{-k} \text{e}^{i \hat{r} \omega_k t + \alpha} \right ) \text{e}^{ikx}
\end{equation}
with where (see also the figure \ref{fig_iquo_chain})
\begin{align}
\left ( \hat{a}_{-k}^+ (t) \right )_{r'=-1} = & \left ( \full_{el}^+ \right )_{-k} + \left ( \empt_{in}^+ \right )_{-k} \text{e}^{-i \frac{\pi}{2}} \quad
\text{clockwise}\\ \text{clockwise} \label{eq-iquochain01}\\
\left ( \hat{a}_k (t) \right )_{r'=-1} = & \left ( \empt_{el} \right )_k + \left ( \full_{in} \right )_k \text{e}^{-i \frac{\pi}{2}} \quad \text{anticlockwise} \nonumber
\end{align}
% \label{fig_iquo_chain}
If the vacuum state of the field is represented by coupling empty oscillators with energy $\varepsilon_0 = \frac{1}{2} h \nu$, in the IQuO-representation the empty state of an IQuO's field will be $\varepsilon_0 = 2 ( \frac{1}{4} + \frac{1}{4} ) h$.\\
And So, the creation and annihilation
operators: operators will be (see the figure \ref{fig_iquo_chain01}):
\begin{align}
\left ( \hat{a}_{-k}^+ (t) \right )_{r'=+1} = & \left ( \empt_{el}^+ \right )_{-k} + \left ( \empt_{in}^+ \right )_{-k} \text{e}^{-i
\frac{\pi}{2}}\\ \frac{\pi}{2}} \label{eq-iquochain02}\\
\left ( \hat{a}_k (t) \right )_{r'=+1} = & \left ( \empt_{el} \right )_k + \left ( \empt_{in} \right )_k \text{e}^{-i \frac{\pi}{2}} \nonumber
\end{align}
% \label{fig_iquo_chain01}
And the vacuum state of an
“isolated” \emph{isolated} IQuO will be
graphically represented:\\
(figure)\\
With operators: represented by the following operators (see also the figure \ref{fig_iquo_vacuum_state}):
\begin{align}
\hat{a}_{1, r'}^+ (t) = & \, \empt^+_{el} \, \text{e}^{i r' \omega t}\\
\hat{a}_{1, r'} (t) = & \, \empt_{in} \, \text{e}^{-i \left ( r' \omega t - \frac{\pi}{2} \right )}
\end{align}
% \label{fig_iquo_vacuum_state}
However, there is another chance to construct the vacuum state, which is not described in the \emph{classical} theory of quantum fields: because a vacuum quantum oscillator is composed of a sub-oscillator, then we can assume the existence of a configuration of not coupled empty sub-oscillators, given
by\\
(figure)\\ by the figure \ref{fig_theta}.\\
% \label{fig_theta}
The set of
not coupled uncoupled vacuum sub-oscillators determines a physical system $\Theta$ equipped with energy but without the presence of a field, that we call \emph{quantum vacuum of no-field}.\\
In the same way, the state $\Phi$ of vacuum of no-field $\Theta$, any sub-oscillator will be:
\begin{equation}
\hat{\Phi}_\Theta = \binom{\empt^+(t)_{el}}{\empt(t)_{in}}_\Theta = \binom{\empt^+(t)_{in}}{\empt(t)_{el}}_\Theta
...
\binom{\empt^+_n (\omega_1)_{el}}{\empt_n (\omega_1)_{in}} & \cdots & \binom{\empt^+_n (\omega_m)_{el}}{\empt_n (\omega_m)_{in}}
\end{array} \right )
\end{equation}
Where any element of the matrix represent a particular sub-oscillator with
$\omega$ $\omega_i$ frequency.
