\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 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 object-field we start to define an elementary scalar field \(\Xi\), whose basic oscillators will be called Intrinsic Quantum Oscillator (IQuO) \cite{guido2014}. The elementary structure of \(\Xi\) is a set of 1-dimensional chains af IQuOs elastically coupled, where the quanta associated to this fundamental field can propagate themselves.
The 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.
Here we summarize some fundamental aspects of the IQuOs, that will be recalled in the present paper.
In \cite{guido2014} one proves that the elastic coupling between quantum oscillators, which builds a field, transforms even each individual oscillator in one described by two components: the inertial and the one elastic. So, the field of the quantum oscillator will no longer described by the pair of operators \(a\), \(a^+\) \[\begin{aligned}
\nonumber
a_t = a_0 \text{e}^{-i \omega t}\\
a_t^+ = a_0^+ \text{e}^{i \omega t} \nonumber\end{aligned}\] but by two pair of operators \((a_{el}, \, a_{el}^+)\), \((a_{in}, \, a_{in}^+)\), where the absorptive component overlaps the inertial component (see figure \ref{fig_iquo}). \[\begin{aligned}
\label{eq_new_qo_operators}
a_t = a(t)_{elastic} + a(t)_{inertial} = a_{el} \text{e}^{-i \omega t} + a_{in} \text{e}^{-i (\omega t - \pi/2)}\\
a_t^+ = a^+(t)_{elastic} + a^+(t)_{inertial} = a_{el}^+ \text{e}^{i \omega t} + a_{in}^+ \text{e}^{i (\omega t - \pi/2)} \nonumber\end{aligned}\] This double structure of the operators \(a\), \(a^+\) splits the energy quanta of the quantum oscillator, that becames \[H_n = U_n + K_n = \left ( U_n \right )_{el} + \left ( K_n \right )_{in} = (2n+1) \left ( \frac{1}{4} \hbar \omega \right )_{el} + (2n+1) \left ( \frac{1}{4} \hbar \omega \right )_{in}\] 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, see figure \ref{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 \(\circ\). \[\begin{aligned}
\hat{a}_{1, r'}^+ (t) = & \, \full^+_{el} \text{e}^{i r' \omega t} + \empt^+_{in} \text{e}^{i (r' \omega t - \frac{\pi}{2})}\\
\hat{a}_{1, r'} (t) = & \, \empt^+_{el} \text{e}^{-i r' \omega t} + \full^+_{in} \text{e}^{-i (r' \omega t - \frac{\pi}{2})} \nonumber
\end{aligned}\] So the egenstates \(\left | 0 \right >\), \(\left | 1 \right >\) will be:
\[\begin{aligned} \left | 0 \right > & = \binom{\circ_{el}}{\circ_{in}} \label{eq-eigenstate0}\\ \left | 1 \right > & = \binom{\circ_{el}}{\bullet_{in}} + \binom{\bullet^+_{el}}{\circ^+_{in}} \label{eq-eigenstate1} \end{aligned}\]
A field line (gone through by a quantum \(\bullet + \bullet\) and with different values of frequency) will be represented by the following scalar field: \[\hat{\Phi}_R = \sum_k \omega_k \left ( \hat{a}_k \text{e}^{-i \hat{r} \omega_k t + \alpha} + \hat{a}_{-k} \text{e}^{i \hat{r} \omega_k t + \alpha} \right ) \text{e}^{ikx}\] where (see also the figure \ref{fig_iquo_chain}) \[\begin{aligned}
\left ( \hat{a}_{-k}^+ (t) \right )_{r'=-1} = & \left ( \full_{el}^+ \right )_{-k} + \left ( \empt_{in}^+ \right )_{-k} \text{e}^{-i \frac{\pi}{2}} \quad \text{clockwise} \label{eq-iquochain01}\\
\left ( \hat{a}_k (t) \right )_{r'=-1} = & \left ( \empt_{el} \right )_k + \left ( \full_{in} \right )_k \text{e}^{-i \frac{\pi}{2}} \quad \text{anticlockwise} \nonumber
\end{aligned}\] 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 \(\varepsilon_0 = 2 ( \frac{1}{4} + \frac{1}{4} ) h\).
So, the creation and annihilation operators will be (see the figure \ref{fig_iquo_chain01}): \[\begin{aligned}
\left ( \hat{a}_{-k}^+ (t) \right )_{r'=+1} = & \left ( \empt_{el}^+ \right )_{-k} + \left ( \empt_{in}^+ \right )_{-k} \text{e}^{-i \frac{\pi}{2}} \label{eq-iquochain02}\\
\left ( \hat{a}_k (t) \right )_{r'=+1} = & \left ( \empt_{el} \right )_k + \left ( \empt_{in} \right )_k \text{e}^{-i \frac{\pi}{2}} \nonumber
\end{aligned}\] And the vacuum state of an isolated IQuO will be represented by the following operators (see also the figure \ref{fig_iquo_vacuum_state}): \[\begin{aligned}
\hat{a}_{1, r'}^+ (t) = & \, \empt^+_{el} \, \text{e}^{i r' \omega t}\\
\hat{a}_{1, r'} (t) = & \, \empt_{in} \, \text{e}^{-i \left ( r' \omega t - \frac{\pi}{2} \right )}
\end{aligned}\] However, there is another chance to construct the vacuum state, which is not described in the 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 the figure \ref{fig_theta}.
The set of uncoupled vacuum sub-oscillators determines a physical system \(\Theta\) equipped with energy but without the presence of a field, that we call quantum vacuum of no-field.
In the same way, the state \(\Phi\) of vacuum of no-field \(\Theta\), any sub-oscillator will be: \[\hat{\Phi}_\Theta = \binom{\empt^+(t)_{el}}{\empt(t)_{in}}_\Theta = \binom{\empt^+(t)_{in}}{\empt(t)_{el}}_\Theta\] The no-field \(\Theta\) will be represented by the following matrix: \[\hat{\Phi}(\Theta) = \left (
\begin{array}{ccc}
\binom{\empt^+_1 (\omega_1)_{el}}{\empt_1 (\omega_1)_{in}} & \cdots & \binom{\empt^+_1 (\omega_m)_{el}}{\empt_1 (\omega_m)_{in}}\\
\cdots & \cdots & \cdots\\
\binom{\empt^+_n (\omega_1)_{el}}{\empt_n (\omega_1)_{in}} & \cdots & \binom{\empt^+_n (\omega_m)_{el}}{\empt_n (\omega_m)_{in}}
\end{array} \right )\] Where any element of the matrix represent a particular sub-oscillator with \(\omega_i\) frequency.