In the first part [1] of our study on the quantum field oscillator, we formulated a new representation of it in terms of sub-oscillators traversed by semi-quanta. The field oscillator expressed in this form has been denoted as IQuO and we have associated with it the IQuO-representation, see also ref. [2] - [4] . Consider two particles expressed by their respective scalar fields (₁, ₂), given by [5] :

$\left\{\begin{array}{c}{\Phi }_1={\displaystyle \underset{k}{\sum }{\varpi }_k}\left[\left(a_{1k}\left({\text{e}}^{-i\hat{r}{\omega }_{k}t+\alpha }\right)+a_{1\left(-k\right)}^{+}\left({\text{e}}^{i\hat{r}{\omega }_{k}t+\alpha }\right)\right)\left({\text{e}}^{ikx}\right)\right]\\ {\Phi }_2={\displaystyle \underset{k}{\sum }{\varpi }_k\left[\left(a_{2k}\left({\text{e}}^{-i\hat{r}{\omega }_{k}t+\beta }\right)+a_{2\left(-k\right)}^{+}\left({\text{e}}^{i\hat{r}{\omega }_{k}t+\beta }\right)\right)\left({\text{e}}^{ikx}\right)\right]}\end{array}\right\}$ (1)

where it is:

$\begin{array}{l}\Phi \left(x_i,t\right)=\left(\frac{1}{\sqrt{V}}\right){\displaystyle \underset{k}{\sum }{q}_k\left(t\right){\text{e}}^{i\left(k\cdot x\right)}}=\left(\frac{1}{\sqrt{V}}\right){\displaystyle \underset{k}{\sum }\left[\left(\sqrt{\frac{\hslash }{2{\omega }_k}}\right)\left(a_k+a_{-k}^{+}\right)\right]{\text{e}}^{i\left(k\cdot x\right)}}\\ {\epsilon }_{\left(n_1,n_2,\cdots ,n_k\right)}={\displaystyle \underset{k}{\sum }\left[\hslash {\omega }_k\left(n_k+\frac{1}{2}\right)\right]},P_{\left(n_1,n_2,\cdots ,n_k\right)}={\displaystyle \underset{k}{\sum }\left[\hslash k_k\left(n_k\right)\right]}\end{array}$ (2)

where we denote by (ϕ_k = k⋅x) the instantaneous phase of the oscillation relative to the k-mode associated with the wave function Φ. We will show that in the IQuO representation the interaction between two particle-fields (₁, ₂) takes place in two stages: the first stage is the “non-dynamic” process of wave function reduction while the second stage is the “dynamic” process of interaction. As is well known, the interaction between two particles occurs when there is an exchange of energy between them. If the particles are represented by extended fields, the coupling between the two particle-fields (₁, ₂) is obtained in space through a “local” elastic coupling between any two IQuO (I₁, I₂) of these two fields. This is because the exchange of energy between two particles cannot occur at all points of their respective representative fields. This condition leads us to admit, once again, the process of reduction of the wave function in quantum field theory (collapse of the wave function). In this way, if the interaction between particles implies an exchange of “integer” quanta of energy governed by the Hamiltonian H of the interacting system, it is intuitive to admit that the reduction process “precedes” the interaction and therefore cannot be described by the Hamiltonian of the system. This consideration implies a profound revision of the famous problem of the collapse of the wave function present in Quantum Mechanics (QM). If, for the QM, observing for an observer means “interacting”, it becomes mandatory to introduce the observer, with its instruments of observation and measure, into the Hamiltonian of the system as well. This gives rise to the problem of the boundary between the macroscopic and microscopic realities of physics. However, we must not forget the duplicity of behavior of a particle (wave-corpuscular), which leads us to represent it as a “field-quantum” so if we admit a “point interaction” between two particles, with an exchange of energy, we must admit that the respective fields must “reduce” to two local oscillators in mutual “elastic coupling” with each other. Even today, a definitive solution to these two aspects (the measurement problem or observation and the reduction problem) there is not in literature of QM. However, if we demonstrate in the IQuO representation, even thanks to the theory of oscillations and waves, that the interaction between particles, including also the interaction present in measuring instruments, is “subsequent” (t_R < t_int) to a reduction process from a non-local to a local state of the oscillator fields, then the two problems mentioned above are on the way to a probable and definitive solution. This would mean that the reduction process is separate from the interaction process and that it takes place in a different way. This is possible if one assumes that:

This last assert defines the reduction process as a “non-dynamic action”: the exchange (see refer [4] ) of a pair (•,o) between two IQuO (I₁, I₂) does not imply the presence of interaction (!) between the two particles. Therefore, it is the “initial” phase shifts produced by the exchanges of sq(•,o) between two IQuO (I₁, I₂) that “reduces” the physical system represented by the two respective fields (₁, ₂).

We note that at the end of the reduction process the coupling between the two IQuO (I₁, I₂) can result, see the one part [1] and ref. [4] , in two configurations (A, B), see Figure 1 :

These configurations [(Φ₁ ⊕ Φ₂)_a, (Φ₁ ⊕ Φ₂)]_b are given by adaptation phase shifts caused by sq switching from a sub-oscillator of I₁ to that of I₂ and vice versa. Energy exchange with phase shift between sq occurs between full sq (•) and empty sq (o) that is (• ⌠ o). When these configurations are reached, simultaneous exchange of two pairs (• ⌠ o) and thus an integer quantum (•, •) is possible, thus opening the “dynamic” interaction phase. In Figure 1(a) , the I₁-IQuO overlaps on the I₂-IQuO: the exchange of energy between full and empty sq (• ⌠ o) results in the possibility of originating a pair of IQuO single-moving in two opposite directions (r) of phase rotation (that is the operators (a, a⁺) of each IQuO all have the same direction of rotation (r)). In Figure 1(b) , a sub-oscillator of I₁ overlaps on the sub-oscillator of I₂: in this case, initiates a dynamic interaction phase of elastic and inelastic type between the particles.

The nondynamic process of reduction can be associated with an O_R operator [6] [7] that acts in two different ways, O_R ≡ (O_x, O_k):

1) Reduction to an eigenstate |k_i〉 relative of the moment p_i (with p_i = hk_i) to a k_i-mode of oscillation: O_R ≡ O_k

2) Reduction to a local eigenstate |x_i〉 of the position x_i: O_R ≡ O_x

First case O_R ≡ O_k

In the first case, the action of a reduction operator O_k will then only be focused on the operator q_k of in Equation (2), acting by phase shift [exp(iϕ_s)] on the q_ki oscillation mode, where s∈ (k₁… k_n). So, the O_k operator reduces the state vector to an eigenstate Φ_s: [O_k Φ = Φ_s]_{(k = s)}, where O_k is a projection operator (|k_i〉 〈k_i| with k_{(k = s)}= (|s〉〈s|). A due note: the physics presented here can be verified in any physics laboratory with teaching aids such as a “slinky” [8] . Recall that to induce a mode of oscillation an external force must act, see the experiments by a slinky:

1) with the same frequency as that of the k-mode: ω_est = ω_k

2) in an impulsive manner (see the swing example) at a given point (x_i) but in ϕ-phase (phase concordance) with the oscillation at that point of the k-mode: ϕ_est (x_i) = ϕ_k (x_i).

In this last case (concordant phase shift) the chains (C₁, C₂) can assume the mode characterized by the wavelength λ(k_i). We remember that along a chain of elastically coupled oscillators (I_k), a “constructive” superposition of oscillation modes can arise generating the so-called “wave packet”. Let us consider the state vector Φ describing a wave packet. For simplicity, we set [α = β = 0, r = +1] in Equation (2); it follows [5] :

$\left\{\begin{array}{c}\Phi \left(x,t\right)={\displaystyle \underset{k}{\sum }\left[A_k\left(t\right)\left({\text{e}}^{ikx}\right)\right]}\\ A_k\left(t\right)={\displaystyle \underset{k}{\sum }{\varpi }_k\left[\left(a_k\left({\text{e}}^{-i{\omega }_{k}t}\right)+a_{\left(-k\right)}^{+}\left({\text{e}}^{i{\omega }_{k}t}\right)\right)\right]}\end{array}\right\}$ (3)

Which we can represent in the following form:

$\begin{array}{c}\Phi \left(x,t\right)={\displaystyle \underset{k}{\sum }\left[A_k\left(t\right)\left({\text{e}}^{ikx}\right)\right]}={\displaystyle \underset{k}{\sum }\left[\left(a_k\left({\text{e}}^{-i{\omega }_{k}t}\right)+a_{(-k)}^{+}\left({\text{e}}^{i{\omega }_{k}t}\right)\right)\left({\text{e}}^{ikx}\right)\right]}\\ ={\displaystyle \underset{k}{\sum }\left[\left(a_k\left({\text{e}}^{-i{\omega }_{k}t}\right)\left({\text{e}}^{ikx}\right)+a_k^{+}\left({\text{e}}^{i{\omega }_{k}t}\right)\left({\text{e}}^{-ikx}\right)\right)\right]}\\ ={\displaystyle \underset{k}{\sum }\left[\left(a_k\left({\text{e}}^{\left[i\left(kx-i{\omega }_{k}t\right)\right]}\right)+a_k^{+}\left({\text{e}}^{\left[-i\left(kx-i{\omega }_{k}t\right)\right]}\right)\right)\right]}\\ ={\displaystyle \underset{k}{\sum }\left[\left(a_k\left({\text{e}}^{i\left(p_k\cdot x_k\right)/\hslash }\right)+a_k^{+}\left({\text{e}}^{-i\left(p_k\cdot x_k\right)/\hslash }\right)\right)\right]}\end{array}$ (4)

where (p_k ⋅ x_k)/h = (kx − ω_kt). We express by means of a column matrix the non-local state in k (with k = 2π/λ_k), that is Φ(k) for [h]:

$\Phi \left(k\right)=\left(\begin{array}{c}{A}_1\left(k_1\right)\\ ⋮\\ A_n\left(k_n\right)\end{array}\right)=\left(\begin{array}{c}\left(a_{k_1}\left({\text{e}}^{i{p}_{k_1}\cdot x_{k_1}}\right)+a_{k_1}^{+}\left({\text{e}}^{-i{p}_{k_1}\cdot x_{k_1}}\right)\right)\\ ⋮\\ \left(a_{k_1}\left({\text{e}}^{i{p}_{k_1}\cdot x_{k_1}}\right)+a_{k_1}^{+}\left({\text{e}}^{-i{p}_{k_1}\cdot x_{k_1}}\right)\right)\end{array}\right)\equiv \left(\begin{array}{c}{A}_{k_1}\left({\text{e}}^{i{\phi }_{k_1}}\right)\\ ⋮\\ A_{k_n}\left({\text{e}}^{i{\phi }_{k_n}}\right)\end{array}\right)$ (5)

The action of a “reduction operator” O_k will then only be focused on the operator A_ki of Equation (3). This operator can be regarded as a “projector” |k〉〈k| and will correspond to the product of three matrices A × B × C. The first matrix (A) correspond to the phase shift matrix {φ_sk = exp[i(ϕ_s − ϕ_k)]}, with [k = (k₁,….k_n), s = (s₁,….s_n)]_(k≡s), ϕ_k are the characteristic phases of k-modes and exp(ϕ_s) is the phase shift produced by I₁ on the I₂. We demonstrate, see the Equation (5), that {φ_sk ⋅ Φ(k) = λΦ(k)}, that is Φ(k_i) are eigenstates of the phase shifts matrix φ_sk. The second (B) matrix is a diagonal matrix (δ_ik), therefore we will have: an O_ik matrix-operator consisting of

$O_k\equiv \left(\left(O_{ik}\right)\right)=\left(\left({\varphi }_{sk}\right)\right)\times \left(\left({\delta }_{ik}\right)\right)=\left(\left(\exp \left[i\left({\phi }_s-{\phi }_k\right)\right]\right)\right)\times \left(\left({\delta }_{ik}\right)\right)\equiv \left(\left(O_{\phi }\right)\right)$

where the unique value s∈ (k₁,..., k_n). In matrix form it is:

$O_{\phi }=\left(\begin{array}{ccc}\exp \left[i\left({\phi }_s-{\phi }_{k_1}\right)\right]& 0& 0\\ 0& \ddots & 0\\ 0& 0& \exp \left[i\left({\phi }_s-{\phi }_{k_n}\right)\right]\end{array}\right)$ (6)

The third matrix (C) is another diagonal matrix δ_sk(A_k); therefore, we will have:

$\left(\left(O_R\right)\right)\equiv \left(\left(O_k\right)\right)=\left(\left(O_{ik}\left({\phi }_{\left(k=s\right)}\right)\right)\right)=\left(\left(\exp \left[i\left({\phi }_s-{\phi }_k\right)\right]\right)\right)\times \left(\left({\delta }_{ik}\right)\right)\times {\delta }_{sk}$

The O_R ≡ O_k action on the Φ(k) is:

$\begin{array}{c}{O}_k\Phi \left(k\right)=\left(\begin{array}{ccc}{\delta }_{sk}{\text{e}}^{\left[i\left({\varphi }_s-{\varphi }_{k_1}\right)\right]}& 0& 0\\ 0& \ddots & 0\\ 0& 0& {\delta }_{sk}{\text{e}}^{\left[i\left({\varphi }_s-{\varphi }_{k_n}\right)\right]}\end{array}\right)\left(\begin{array}{c}{A}_{k_1}\left({\text{e}}^{i{\varphi }_{k_1}}\right)\\ ⋮\\ A_{k_n}\left({\text{e}}^{i{\varphi }_{k_n}}\right)\end{array}\right)\\ =\left(\begin{array}{c}{\delta }_{sk}{A}_{k_1}{\text{e}}^{\left[i\left({\varphi }_s-{\varphi }_{k_1}\right)\right]}\left({\text{e}}^{i{\varphi }_{k_1}}\right)\\ ⋮\\ {\delta }_{sk}{A}_{k_n}{\text{e}}^{\left[i\left({\varphi }_s-{\varphi }_{k_n}\right)\right]}\left({\text{e}}^{i{\varphi }_{k_n}}\right)\end{array}\right)=\left(\begin{array}{c}{\delta }_{sk}{A}_{k_1}\left({\text{e}}^{i{\varphi }_{k_s}}\right)\\ ⋮\\ {\delta }_{sk}{A}_{k_n}\left({\text{e}}^{i{\varphi }_{k_s}}\right)\end{array}\right)=A_{k_s}\left({\text{e}}^{i{\varphi }_{k_s}}\right)\equiv \Phi \left(k_s\right)\end{array}$ (7)

thus obtaining, after reduction, a single eigenstate Φ(k_s = s) or an oscillation mode with λ_s.

Second case O_R ≡ O_x

Let us consider the reduction of the field to a local eigenstate of position |x_i〉. If the phase shifts are different from [exp(-iϕ_s)], that is “discordant”, then no oscillation mode is formed, and the two fields are reduced to the two oscillators (I₁, I₂) that are coupling. The “discordant” phase shift propagates along all the chains of the field at superluminal phase velocity (that is for spatial intervals) breaking the phase concordance between the contiguous oscillators and thus “destroying” the various monochromatic modes of oscillation of the chains (eigenstates Φ_k) and those of even a possible wave packet propagating along a chain. Discordant phase shifts induce along the chain an “indeterminate” (or “uncertainty”) state in the k-modes, consistent with the axioms of the quantum mechanics. This step can be described by the action of an “inverse” operator to the reduction operator (O_k) defined as [O_R ≡ (O_k)⁻¹]. The x position of the two IQuO (I₁, I₂)_x is now well determined; in this case the fields are reduced into a limited spatial range Δx in which the two IQuO with their oscillation amplitudes are located. Therefore, to a discordant phase shift action, we associate the position operator O_x, that is (|x_i〉 〈x_i|).

The Heisenberg uncertainty relations

In the case of discordant phase shifts, we can thus associate two operators with the O_R reduction process: the operator O_x (that is (|x_i〉 〈x_i|)), simultaneously with the operator (O_k)⁻¹. We will then say that the two operators are associated O_R = [(O_k)⁻¹⋅ O_x]. We associate an “uncertainty” of k with operator (O_k)⁻¹, that is Δk. By symmetry, just as there is a relation between O_x and (O_k)⁻¹, then there will also be a relation O_R = [O_k ⋅ (O_x)⁻¹], with an uncertainty Δx. Since in a position and pulse measurement we have a reduction process, the measurement errors (Δx, Δk) could be indeterminacy values of the measured physical quantity, that is: [(Δx =Δx), (Δk = Δk)]. Then, we can consider the products ${\left[\Delta x\cdot \underset{\_}{\Delta }k\right]}_{\left({\text{O}}_x,{\left({\text{O}}_k\right)}^{-1}\right)}$, ${\left[\underset{\_}{\Delta }x\cdot \Delta k\right]}_{\left({\text{O}}_k,{\left({\text{O}}_x\right)}^{-1}\right)}$; we suspect that it is:

${\left[\underset{\_}{\Delta }x\cdot \underset{\_}{\Delta }k\right]}_{\left({\text{O}}_x,{\left({\text{O}}_k\right)}^{-1}\right)}={\left[\underset{\_}{\Delta }x\cdot \underset{\_}{\Delta }k\right]}_{\left({\text{O}}_k,{\left({\text{O}}_x\right)}^{-1}\right)}={\left[\underset{\_}{\Delta }x\cdot \underset{\_}{\Delta }k\right]}_{\left({\left({\text{O}}_k\right)}^{-1},{\left({\text{O}}_x\right)}^{-1}\right)}$ (8)

where [((O_k)⁻¹, (O_x)⁻¹)⌠ Ψ] indicates a non-local state, as in a wave packet. Recall that in a wave packet there is [(Δν⋅Δt ≥ 1) ⌠(Δx/Δλ ≥ 1)] relations apply, which we will write as: {[(Δν⋅Δt ≥ 1) ⌠(Δx/Δλ ≥ 1)]⌠ Δx⋅Δk ≥ 2π}

Let us recall De Broglie’s relation p = hk; it is easily shown that if we go to indeterminacy (Δ) it would be Δp = ηΔk. Therefore, we have:

$\left[\underset{\_}{\Delta }x\cdot \underset{\_}{\Delta }k\ge 2\pi \right]\to \left[\underset{\_}{\Delta }x\times \left(\underset{\_}{\Delta }p/\hslash \right)\ge 2\pi \right]\to \left[\underset{\_}{\Delta }x\cdot \underset{\_}{\Delta }p\ge 2\pi \hslash \right]$ (9)

These are the well-known Heisenberg uncertainty relations [9] .

In whatever type of reduction, at the end of this process there is an “adaptation phase” with a Δt_ad, in which there is an instantaneous reciprocal induction of phase change (phase-shifting action) between two IQuO (I₁, I₂); let us conjecture that it is:

$\Delta {\phi }_1=-\Delta {\phi }_2$ (10)

and the phase shift will be expressed by the operator O_ϕ (phase shift action operator), with [O_ϕ = [exp(-iϕ)]. To the phase shifts given by exchanges of semi-quanta sq(•,o), not integers, occurring between the two IQuO (I₁, I₂), we will associate the operator O_ε (operator of semi-quanta exchange) that it exchange the sq(•,o) between two oscillators (I₁, I₂): [sq₁(•) ⌠ sq₂(o), sq₂(•) ⌠ sq₁(o)]. Therefore, we will associate a reduction operator O_R given by [O_R = O_ϕ O_ε]. Note that the exchange of a pair sq(•,•) can be given by the combination of two pair-exchange operators O_ε (sq(•,o)), that is: O_ε (sq(•,•)) = O_ε (sq(•,o)) ⋅ O_ε (sq(•,o)). In this case, the operator O_ε determines the exchange of a quantum (•,•) and is thus present in the dynamic phase of the interaction. Having finished the initial phase of reduction, the coupling of the two terminals becomes “dynamic” and the interaction phase begins with exchange of integer quanta (sq(•,•) pairs) of energy between the two terminals of the respective chains, now these in a given mode of oscillation. This reminds us of the interaction of particles [10] with given initial momentum (p_i1, p_j2). At this stage the system will be described by the Hamiltonian of the complete global system, from whose symmetries the conservations of total energy, momentum and total angular momentum will arise. In addition, the phase shifts produced in both the “nondynamic” and “dynamic” phases do not alter the signs of the electric charge, since the latter is given by invariance by gauge transformations involving just the phase [11] [12] . In the IQuO model [13] this corresponds to an invariance of the direction of rotation of the phase in the case of phase shifts produced in the reducing phase.

We denote by (_{1 2}) the coupling of two fields where the sign is the reciprocal action operation with components ≡ (O_R, ). O_R describes the nondynamic initial reduction phase while the sign indicates, after the reduction process, the dynamic coupling by exchanges of sq(•,•) pairs between two IQuO (I₁, I₂) of two fields (₁, ₂). As we well know, in inelastic processes other particles can be created than the initial ones, see the creation of pairs, where the final fields (Ψ₁, Ψ₂) are different from the initial ones (₁, ₂). To deal with such a process, let us consider for simplicity’s sake that we have already two reduced fields in a local k-moment state, that is (_1k, _2k), consequence of the previous reduction process with an operator O_R = O_k, see the Equation (7). Suppose that by chance two IQuO (I₁, I₂) of the two respective fields (_1k, _2k) couple together in the same x point (let us not forget that a quantum oscillator has a well-defined spatial dimension (oscillation amplitude), that is Δx). Initially, a transition phase there is (into a time Δt_tr) with a mutual adjustment between respective oscillation states of the IQuO (I₁, I₂) of two respective fields. During this time interval Δt_tr the reciprocal phase shifts (see Equation (10), that is |₁| = |−₂| = ) in discordance reduce the two fields to two (I₁, I₂) oscillators. This reduction process is operated by the operator O_R = O_x to which a phase shift operator (O_ϕ)_x is corresponding

${\left({\text{O}}_{\phi }\right)}_x={\left(\begin{array}{cc}{\text{e}}^{i{\phi }_1}& 0\\ 0& {\text{e}}^{i{\phi }_2}\end{array}\right)}_x={\left(\begin{array}{cc}{\text{e}}^{i\phi }& 0\\ 0& {\text{e}}^{-i\phi }\end{array}\right)}_x$ (11)

The operator (O_ϕ)_x, associated with the phase shifts (₁ = −₂), acts on the fields (_1k, _2k); since the matrix O_ϕ is a (2 × 2) matrix, the fields must be expressed by column matrices (2 × 1). The “unique” system will be represented by a state with two matrices:

$\Phi \equiv \left\{\left(\begin{array}{c}{\Phi }_{1k}\\ 1\end{array}\right),\left(\begin{array}{c}1\\ {\Phi }_{2k}\end{array}\right)\right\}$

Note that the operator (O_ϕ)_x acts both on the _1k by phase shifts (₁ = −₂) than on the _2k. We then will admit an operation:

$\left({\Phi }_1\underset{\_}{\oplus }{\Phi }_2\right)=\left\{{\oplus }_{{\text{O}}_{\phi }}\left({\Phi }_{1k},{\Phi }_{2k}\right)\right\}=\left\{{\left[{\Phi }_{1k}\times \left({\text{O}}_{\phi }\times {\Phi }_{2k}\right)\right]}_{\oplus }\right\}$ (12)

with O operating to right than left. Remember that ≡ (O_R, ) where is the dynamic operation and here O_R ≡ O_ϕ. Besides, we will associate the operator O_ε to the phase shifts (₁ = −₂) given by exchanges of semi-quants sq(•,o), not integers, occurring between the two IQuO (I₁, I₂). Therefore, in the coupling between the two fields (_1k, _2k), the operator O_R will correspond the operators O_R ≡ (O_ε, (O_ϕ)_x). Then, we will have:

$\begin{array}{l}{\Phi }_{1k}\underset{\_}{\oplus }{\Phi }_{2k}\\ ={\left\{{\text{O}}_{\epsilon }\cdot \left[{\Phi }_{1k}\cdot \left({\left({\text{O}}_{\phi }\right)}_x\cdot {\Phi }_{2k}\right)\right]\right\}}_{\oplus }={\left\{{\text{O}}_{\epsilon }\cdot \left[\left(\begin{array}{c}{\Phi }_{1k}\\ 1\end{array}\right)\cdot \left[{\left(\begin{array}{cc}{\text{e}}^{i\phi }& 0\\ 0& {\text{e}}^{-i\phi }\end{array}\right)}_x\cdot \left(\begin{array}{c}1\\ {\Phi }_{2k}\end{array}\right)\right]\right]\right\}}_{\oplus }\\ ={\left\{{\text{O}}_{\epsilon }\cdot \left[\left(\begin{array}{c}{\Phi }_{1k}\\ 1\end{array}\right)\cdot {\left[\left(\begin{array}{c}{\text{e}}^{i\phi }\\ {\text{e}}^{-i\phi }{\Phi }_{2k}\end{array}\right)\right]}_x\right]\right\}}_{\oplus }={\left\{{\text{O}}_{\epsilon }\cdot \left[{\Phi }_{1k}{\text{e}}^{i\phi }+{\text{e}}^{-i\phi }{\Phi }_{2k}\right]\right\}}_{\oplus }\\ ={\left[{\Phi }_{1k}{\text{e}}^{i\phi }+{\text{e}}^{-i\phi }{\Phi }_{2k}\right]}_{{\text{O}}_{\epsilon }\left(•\to \text{o}\right)}\end{array}$ (13)

The action of the second exchange operator O_ε(sq₂(•,o)) can induce a process of reduction but if we have a sequence of actions of O_ε(sq_i(•,o)), they will follow a sequence of phase shifts as long as at the end of the global process we have that [Δ₁ = −Δ₂] to which corresponds [Δk₁ = −Δk₂] with the final eigenstates ( ${k^{\prime }}_1$, ${k^{\prime }}_2$). There is this possibility when a coupling of type of Figure 1(b) happens.

Now we consider a coupling between two fields-particles as in Figure 1(a) , where the I₁-IQuO overlaps on the I₂-IQuO. Firstly, we need to precise that if we understand the particle always as a field, where all quantum oscillators of field are coupled, then any interaction between fields-particles must not alter the direction of phase rotation r of each field. This mean that when a field originates, all coupled field oscillators are in the same eigenstate |r’〉, and the interactions are of “gauge” type [14] , that is the interactions produce phase shifts but without altering the direction of phase rotation (r’). The eigenstates’ scheme of the r operator, relative to the physical system of the two coupled IQuO, is:

$\boxed{{|\Phi \rangle }_{\left(\text{IQuO}\text{1 + IQuO}\text{2}\right)}={|\Phi \rangle }_{\left(1\right)}\underset{\_}{\oplus }{|\Phi \rangle }_{\left(2\right)}\Rightarrow \{\begin{array}{l}{|\Phi \left(r^{\prime }=\pm 1\right)\rangle }_{\left(1\right)}\underset{\_}{\oplus }{|\Phi \left(r^{\prime }=\pm 1\right)\rangle }_{\left(2\right)}\\ {|\Phi \left(r^{\prime }=\pm 1\right)\rangle }_{\left(1\right)}\underset{\_}{\oplus }{|\Phi \left(r^{\prime }=\mp 1\right)\rangle }_{\left(2\right)}\end{array}}$ (14)

Let us first examine the two possible combinations of the “zero eigenvalue” of (r):

(|Φ₁(r’ = ±1)〉) ⊕ (|Φ₂(r’ = μ1)〉).

In Figure 1(a) , the coupling between two fields can occur at a given point (x) of the space between the two corresponding oscillators (I₁, I₂); the initial reduction phase begins with the action of the operator O_x. This operator, in addition to locally reducing a field, see the projector (|x_i〉 〈x_i|), makes k “indeterminate”, as if an inverse operator of O_k had been applied, that is $O_k^{-1}$. However, the local coupling of the two oscillators, after a preliminary adaptation phase, can determine a state in which the two IQuO constitute a single IQuO (see Figure 1(a) ), like a single field to which a single moment k₀ is associated. The pair creation process passes through this physical state of “uniqueness”. This means that the coupling (I₁, I₂) can be described by initially considering a system of two fields (₁, ₂) with a same single moment k₀. Recall in a pair creation the two exiting particles have the same energy and impulse |k| and the same mass (m₀ = ηω_0k/c²). So, to deal with such a process, we could consider for simplicity’s sake two reduced fields already in a local k-moment state, that is ≡ (_1k, _2k) on which an operator O_x acts. We’ll have (for simplicity one sets = in Equation (1)) from the Equation (1) that:

${\left\{\begin{array}{c}{\Phi }_{1x}\left(r^{\prime }=-1\right)={\varpi }_k\left(a_k{\text{e}}^{i{\omega }_{k}t}+a_k^{+}{\text{e}}^{-i{\omega }_{k}t}\right)\left({\text{e}}^{ikx}\right)\\ {\Phi }_{2x}\left(r^{\prime }=+1\right)={\varpi }_k\left(a_k{\text{e}}^{-i{\omega }_{k}t}+a_k^{+}{\text{e}}^{i{\omega }_{k}t}\right)\left({\text{e}}^{ikx}\right)\end{array}\right\}}_{\left(k=k_0\right)}$ (15)

By of Equation (12), applying the Equation (13), we will then have:

$\boxed{\begin{array}{l}{\Phi }_{1x}{\oplus }_{\left(\epsilon ,\phi \right)}{\Phi }_{2x}\\ =\left\{\left[{\varpi }_k\left(a_k{\text{e}}^{i{\omega }_{k}t}+a_k^{+}{\text{e}}^{-i{\omega }_{k}t}\right)\left({\text{e}}^{ikx}\right)\right]{\oplus }_{\left(\epsilon ,\phi \right)}\left[{\varpi }_k\left(a_k{\text{e}}^{-i{\omega }_{k}t}+a_k^{+}{\text{e}}^{i{\omega }_{k}t}\right)\left({\text{e}}^{ikx}\right)\right]\right\}\\ ={\left\{\left[{\varpi }_k\left(a_k{\text{e}}^{i{\omega }_{k}t}+a_k^{+}{\text{e}}^{-i{\omega }_{k}t}\right)\left({\text{e}}^{ikx}\right)\right]{\text{e}}^{i\phi }+{\text{e}}^{-i\phi }\left[{\varpi }_k\left(a_k{\text{e}}^{-i{\omega }_{k}t}+a_k^{+}{\text{e}}^{i{\omega }_{k}t}\right)\left({\text{e}}^{ikx}\right)\right]\right\}}_{{\oplus }_{\epsilon }}\\ ={\left\{\left[{\varpi }_k\left(a_k\left({\text{e}}^{i{\omega }_{k}t+i\phi }\right)+a_k^{+}\left({\text{e}}^{-i{\omega }_{k}t+i\phi }\right)\right)\left({\text{e}}^{ikx}\right)\right]+\left[{\varpi }_k\left(a_k\left({\text{e}}^{-i{\omega }_{k}t-i\phi }\right)+a_k^{+}\left({\text{e}}^{i{\omega }_{k}t-i\phi }\right)\right)\left({\text{e}}^{ikx}\right)\right]\right\}}_{{\oplus }_{\epsilon }}\text{}\text{}\text{}\text{}\end{array}}$(16)

In the first part of this study on the IQuO the calculation of the sum (_{1 2}) leads to two new fields (Ψ(), Ψ()) which are respectively expressed by IQuO with annihilation and creation operators (a, a⁺) having the same direction r of the phase rotation but with r’(Ψ) opposite to r’(Ψ_β):

$\boxed{\begin{array}{l}{\Phi }_{1x}\oplus {\Phi }_{2x}=\left({\text{e}}^{i\pi /4}\right)\left({\text{e}}^{ikx}\right)\left\{\left[\left(a_k^{+}+a_k\left({\text{e}}^{-i\pi /2}\right)\right)\left({\text{e}}^{-i{\omega }_{k}t}\right)\right]+\left[\left(a_k+a_k^{+}\left({\text{e}}^{-i\pi /2}\right)\right)\left({\text{e}}^{i{\omega }_{k}t}\right)\right]\right\}\\ =\left({\text{e}}^{i\pi /4}\right)\left({\text{e}}^{ikx}\right)\left\{\left[\left(a_k^{+}-i{a}_k\right)\left({\text{e}}^{-i{\omega }_{k}t}\right)\right]+\left[\left(a_k-i{a}_k^{+}\right)\left({\text{e}}^{i{\omega }_{k}t}\right)\right]\right\}\\ =\left({\text{e}}^{-i\pi /4}\right)\left({\text{e}}^{ikx}\right)\left\{\left[\left({\alpha }_k\right)\left({\text{e}}^{-i{\omega }_{k}t}\right)\right]+\left[\left(\beta \right)\left({\text{e}}^{i{\omega }_{k}t}\right)\right]\right\}\\ ={\Psi }_{\text{1}}\left(\alpha \left(t\right)\right)+{\Psi }_{\text{2}}\left(\beta \left(t\right)\right)\end{array}}$ (17)

The Hamiltonian operator H of the physical system is established, as usual, by the field operator and its conjugate (setting = ⁺), thus obtaining [11] :

$\boxed{H={\displaystyle \underset{-\infty }{\overset{+\infty }{\sum }}\hslash {\omega }_k\left({\alpha }_k^{+}{\alpha }_k+{\gamma }_k^{+}{\gamma }_k+1\right)}}$ (18)

As in the first reduction process, a second $s{q^{\prime }}_2$ (•, o) pair, initiates a reduction process by switching from the IQuO pair (I₁, I₂)_x to the field pair (Ψ(), Ψ()) with eigenstates ( ${k^{\prime }}_1$, ${k^{\prime }}_2$). If we have |ϕ₁| = |ϕ₂| = ϕ, it follows that the induced eigenstates by the reciprocal displacement action between the two IQuO are the same for both chains k₁ = k₂ = k_s. Recall in a pair creation the two particles have the same energy and impulse k_s (!) and the same mass (m₀ = ηω_0k/c²). In conclusion, we have assumed a coupling with an initial global reduction process (O_R) given by a sequence of operators:

${\text{<munder accentunder="true"> O <mo> \_ </mo> </munder>}}_R={\text{O}}_{k^{\prime }}\cdot {\text{O}}_k^{-1}\cdot {\text{O}}_x\cdot {\text{O}}_x^{-1}\cdot {\text{O}}_k={\text{O}}_{k^{\prime }}$ (19)

where we have considered [( ${\text{O}}_x\cdot {\text{O}}_x^{-1}=\text{I}$), ( ${\text{O}}_k^{-1}\cdot {\text{O}}_k=\text{I}$)] with I Identity Operator. Recall associating to operators (O_x, O_k) the operators (O_ε)_i. The action of the operation ⊕ given by Equation (12) determines that the fields (Ψ(), Ψ()) are invariants for not local gauge transformations on the wave function: in fact, an identical phase shift on all the oscillators in the chain does not change the |k〉 mode, see the Equation (15). For complex fields, see Equation (17), the literature assigns an electric charge to this type of particle-field. The combination (_{1 2}) determines two IQuO with representation:

$\left\{\begin{array}{c}\Psi {\left(\alpha \right)}_{cl}={\left\{\left[{\left({•}_{el}^{+}\right)}_{cl}^{}+{\left({\text{o}}_{in}^{+}\right)}_{cl}^{}{\text{e}}^{i\left(-\pi /2\right)}\right]\left({\text{e}}^{-i\omega t}\right),\left[{\left({\text{o}}_{el}^{}\right)}_{cl}^{}{\text{e}}^{i\left(-\pi /2\right)}+{\left({•}_{in}^{}\right)}_{cl}^{}\right]\left({\text{e}}^{-i\omega t}\right)\right\}}_{cl}\\ \Psi {\left(\beta \right)}_{\underset{\_}{cl}}={\left\{\left[{\left({•}_{el}^{+}\right)}_{\underset{\_}{cl}}^{}{\text{e}}^{i\left(-\pi /2\right)}+{\left({\text{o}}_{in}^{+}\right)}_{\underset{\_}{cl}}^{}{\text{e}}^{i\left(\pi \right)}\right]\left({\text{e}}^{i\omega t}\right),\left[{\left({\text{o}}_{el}^{}\right)}_{\underset{\_}{cl}}^{}+{\left({•}_{in}^{}\right)}_{\underset{\_}{cl}}^{}{\text{e}}^{i\left(+\pi /2\right)}\right]\left({\text{e}}^{i\omega t}\right)\right\}}_{\underset{\_}{cl}}^{}\end{array}\right\}$ (20)

where the cl-index indicates clockwise direction, and the cl-index indicates anti-clockwise direction. The graphic representation is, see Figure 2 :

We note that the two new IQuO have sq operators all with the same direction of phase rotation (“mono-verse” IQuO). One of the two fields ( Figure 2(a) ) is not as regular in shape as the other, so it requires the intervention of a field Φ_ϕ that with a phase shift makes it regular in shape. As we mentioned in the first part of the study, this indicates to us that around a mono-verse IQuO in the direction of phase rotation there are fields acting with corrective phase shifts called “gauge” fields. As it is well known, the Φ_ϕ field is a gauge field and the transformation induced by it is called “gauge transformation” [11] [14] . We also know the fields Φ_ϕ act on electrically charged particles-fields. Therefore, we can consider the hypothesis of associating the monoverse fields (Ψ_cl, Ψ_cl) with a sign (± q) of electric charge q connected to two directions of phase rotations: (Ψ_cl, Ψ_cl) ⌠(+q, -q). The existence of the Φ_ϕ corrector field indicates that a “mono-verse” IQuO cannot exist in a way “isolated”, see the QED. This, as we know from the literature, is analogous to the description in QED where every electric charge is “dressed” by virtual photon fields [11] [15] . This, as we know from the literature, occurs locally because the field in question operates local gauge transformations to make the interacting fields locally symmetrical. The correcting field will be the photon. The coupling mechanism (_{1 2}) of two B-IQuO [1] [2] [3] can have a particular situation in which the exchange of the sq to form two F-IQuO is direct and with a matrix of phase shift given by (|ϕ₁| = |ϕ₂| = 0). This happens with a particular configuration given by, see Figure 3 :

With matrices:

$\left\{\begin{array}{cc}\left(\left({\Phi }_1^{}\right)\right)\equiv \left(\begin{array}{c}\left({•}_{1el}^{+}{\text{e}}^{-i\left(5/4\right)\pi }+o_{1in}^{+}{\text{e}}^{-i\left(7/4\right)\pi }\right)\\ \left(o_{1el}^{}{\text{e}}^{i\left(5/4\right)\pi }+{•}_{1in}^{}{\text{e}}^{i\left(7/4\right)\pi }\right)\end{array}\right)& \left(\left({\Phi }_2^{}\right)\right)\equiv \left(\begin{array}{c}\left({•}_{2el}^{+}{\text{e}}^{i\left(3/4\right)\pi }+o_{2in}^{+}{\text{e}}^{i\left(1/4\right)\pi }\right)\\ \left(o_{2el}^{}{\text{e}}^{-i\left(3/4\right)\pi }+{•}_{2in}^{}{\text{e}}^{-i\left(1/4\right)\pi }\right)\end{array}\right)\end{array}\right\}$(21)

With indices (cl, cl). By Equation (11), in an “ideal” way (|ϕ₁| = |ϕ₂| = 0), we obtain:

$\begin{array}{c}{\Phi }_1\underset{\_}{\otimes }{\Phi }_2=\left(\begin{array}{c}{\Phi }_1\\ 1\end{array}\right)\cdot {\left[{\left(\begin{array}{cc}{\text{e}}^{i\phi }& 0\\ 0& {\text{e}}^{-i\phi }\end{array}\right)}_{{\oplus }_{\phi }}\cdot \left(\begin{array}{c}1\\ {\Phi }_2\end{array}\right)\right]}_{{\oplus }_{\epsilon }}\\ ={\left[\left(\begin{array}{c}{\Phi }_1\\ 1\end{array}\right){\oplus }_{\phi }\left(\begin{array}{c}1\\ {\Phi }_2\end{array}\right)\right]}_{{\oplus }_{\epsilon }}={\Psi }_{\alpha }+{\Psi }_{\beta }\end{array}$ (22)

That is:

$\begin{array}{l}\left({\Phi }_1\underset{\_}{\oplus }{\Phi }_2\right)=\left\{\left(\begin{array}{c}{\left({•}_{1el}^{+}{\text{e}}^{-i\left(5/4\right)\pi }+o_{1in}^{+}{\text{e}}^{-i\left(7/4\right)\pi }\right)}_{cl}\\ {\left(o_{1el}^{}{\text{e}}^{i\left(5/4\right)\pi }+{•}_{1in}^{}{\text{e}}^{i\left(7/4\right)\pi }\right)}_{\underset{\_}{cl}}\end{array}\right){\oplus }_{\varphi }{\left(\begin{array}{c}{\left({•}_{2el}^{+}{\text{e}}^{i\left(3/4\right)\pi }+o_{2in}^{+}{\text{e}}^{i\left(1/4\right)\pi }\right)}_{\underset{\_}{cl}}\\ {\left(o_{2el}^{}{\text{e}}^{-i\left(3/4\right)\pi }+{•}_{2in}^{}{\text{e}}^{-i\left(1/4\right)\pi }\right)}_{cl}\end{array}\right)}_{{\oplus }_{\epsilon }}\right\}\\ ={\left(\begin{array}{c}\left({•}_{1el}^{+}{\text{e}}^{-i\left(5/4\right)\pi }+{•}_{1in}^{+}{\text{e}}^{-i\left(7/4\right)\pi }\right)\\ \left(o_{2el}^{}{\text{e}}^{-i\left(3/4\right)\pi }+o_{2in}^{}{\text{e}}^{-i\left(1/4\right)\pi }\right)\end{array}\right)}_{cl}+{\left(\begin{array}{c}\left({•}_{2el}^{+}{\text{e}}^{i\left(3/4\right)\pi }+{•}_{2in}^{+}{\text{e}}^{i\left(1/4\right)\pi }\right)\\ \left(o_{1el}^{}{\text{e}}^{i\left(5/4\right)\pi }+o_{1in}^{}{\text{e}}^{i\left(7/4\right)\pi }\right)\end{array}\right)}_{\underset{\_}{cl}}\end{array}$ (23)

Graphically it is, see Figure 4 :

We proved that a Ψ field with a mono-verse IQuO has an electric charge Q with eigenvalue (q = +1, q = −1). Then, we calculated the electric charge of the (Ψ_cl) field in the semi-quanta representation with matrices:

${\Psi }_{cl}^{}\equiv \left(\begin{array}{c}{\left(o_{2el}^{}{\text{e}}^{-i\pi /2}+o_{2in}^{}\right)}_{cl}\\ {\left({•}_{1el}^{+}{\text{e}}^{-i\pi }+{•}_{1in}^{+}{\text{e}}^{-i3\pi /2}\right)}_{cl}\end{array}\right)$ (24)

${\left({\Psi }_{cl}\right)}^{+}\equiv \left(\begin{array}{c}{\left({•}_{el}^{+}{\text{e}}^{i\pi /2}+{•}_{in}^{+}\right)}_{\underset{\_}{cl}}\\ {\left(o_{el}^{}{\text{e}}^{i\pi }+o_{in}^{i3\pi /2}\right)}_{\underset{\_}{cl}}\end{array}\right)$ (25)

Using the well-known definition of the electric charge [2] and the commutation relations with semi-quanta (see appendix in ref. [5] ) it then follows that:

$\boxed{\begin{array}{l}{Q}_{cl}={\displaystyle \int \left[\left(\left({\Psi }_{\left(cl\right)}^{+}\right)\right){\sigma }_3\left(\left({\Psi }_{\left(cl\right)}^{}\right)\right)\right]\text{d}V}\\ ={\displaystyle \int \left(\begin{array}{c}\left({•}_{el}^{+}{\text{e}}^{i\pi /2}+{•}_{in}^{+}\right)\\ \left(o_{el}^{}{\text{e}}^{i\pi }+o_{in}^{i3\pi /2}\right)\end{array}\right)\left(\begin{array}{c}\left(o_{el}^{}{\text{e}}^{-i\pi /2}+o_{in}^{}\right)\\ -\left({•}_{el}^{+}{\text{e}}^{-i\pi }+{•}_{in}^{+}{e}^{-i3\pi /2}\right)\end{array}\right)\text{d}V}\end{array}}$ (26)

where σ₃ is a Pauli’s matrix. Calculating:

$\boxed{\begin{array}{c}{Q^{\prime }}_{cl}=\left[\left(\left({•}_{el}^{+}{\text{e}}^{i\pi /2}\right)+{•}_{in}^{+}\right)\left(o_{el}^{}{\text{e}}^{-i\pi /2}+o_{in}^{}\right)-\left(o_{el}^{}{\text{e}}^{i\pi }+o_{in}^{i3\pi /2}\right)\left({•}_{el}^{+}{\text{e}}^{-i\pi }+{•}_{in}^{+}{\text{e}}^{-i3\pi /2}\right)\right]\\ =\left(i{•}_{el}^{+}{o}_{in}^{}-i{•}_{el}^{+}{\text{e}}^{-i\pi }{o}_{in}^{i3\pi /2}\right)+\left({•}_{in}^{+}{o}_{el}^{}{\text{e}}^{-i\pi /2}-{•}_{in}^{+}{\text{e}}^{-i3\pi /2}{o}_{el}^{}{\text{e}}^{i\pi }\right)\\ =\left[{•}_{el}^{+},o_{el}^{}\right]+\left[{•}_{in}^{+},o_{in}^{}\right]+i\left[{•}_{el}^{+},o_{in}^{}\right]+i\left[o_{el}^{},{•}_{in}^{+}\right]=-1\end{array}}$ (27)

We can state that an IQuO with a clockwise direction of phase rotation has a negative value of the physical quantity Q of the Ψ_cl field. One also demonstrates that with the following matrix:

${\Psi }_{\underset{\_}{cl}}=\left(\begin{array}{c}{\left(o_{el}{\text{e}}^{-i\pi /2}+o_{in}\right)}_{\underset{\_}{cl}}\\ {\left({•}_{el}^{+}{\text{e}}^{i\pi }+{•}_{in}^{+}{\text{e}}^{i\pi /2}\right)}_{\underset{\_}{cl}}\end{array}\right)$ (28)

We find:

$\boxed{\begin{array}{c}{Q}_{\underset{\_}{cl}}={\displaystyle \int \left[\left(\left({\Psi }_{(+)}^{+}\right)\right){\sigma }_3\left(\left({\Psi }_{(+)}\right)\right)\right]\text{d}V}\\ ={\displaystyle \int \left(\begin{array}{c}\left(o_{el}^{}{\text{e}}^{-i\pi }+o_{in}^{}{\text{e}}^{-i\pi /2}\right)\\ \left({•}_{el}^{+}{\text{e}}^{i\pi /2}+{•}_{in}^{+}\right)\end{array}\right)\left(\begin{array}{c}\left({•}_{el}^{+}{\text{e}}^{i\pi }+{•}_{in}^{+}{\text{e}}^{i\pi /2}\right)\\ -\left(o_{el}{\text{e}}^{-i\pi /2}+o_{in}\right)\end{array}\right)\text{d}V}\\ =\left[\left(o_{el}^{}{\text{e}}^{-i\pi }+o_{in}^{}{\text{e}}^{-i\pi /2}\right)\left({•}_{el}^{+}{\text{e}}^{i\pi }+{•}_{in}^{+}{\text{e}}^{i\pi /2}\right)-\left({•}_{el}^{+}{\text{e}}^{i\pi /2}+{•}_{in}^{+}\right)\left(o_{el}^{}{\text{e}}^{-i\pi /2}+o_{in}^{}\right)\right]\\ =o_{el}^{}{•}_{el}^{+}+o_{el}^{}{•}_{in}^{+}{\text{e}}^{-i\pi /2}+o_{in}^{}{•}_{el}^{+}{\text{e}}^{i\pi /2}+o_{in}^{}{•}_{in}^{+}\\ -{•}_{el}^{+}{o}_{el}^{}-{•}_{el}^{+}{o}_{in}^{}{\text{e}}^{i\pi /2}-{•}_{in}^{+}{o}_{el}^{}{e}^{-i\pi /2}-{•}_{in}^{+}{o}_{in}^{}\\ =\left[o_{el}^{},{•}_{el}^{+}\right]+\left[o_{in}^{},{•}_{in}^{+}\right]+i\left[{•}_{in}^{+},o_{el}^{}\right]+i\left[o_{in}^{},{•}_{el}^{+}\right]=+1\end{array}}$ (29)

In conclusion, we can state that the sign of the electric charge is determined by the direction of rotation of the phase and the structure of the matrix representing the IQuO-particle. In conclusion, we can assert that: the (Φ₁ ⊕ Φ₂) coupling of two IQuO has generated two particles-IQuO with opposite electric charge (q): (Ψ_cl)₊₁, (Ψ_cl)₋₁. We have thus demonstrated that the (±) sign of the electric charge is related to the direction of the phase rotation of an oscillation IQuO. We denote the IQuO of type Φ_{(cl, cl)} as Boson-IQuO, having both directions of phase rotation, and the IQuO of type [(Ψ_cl)₊₁, (Ψ_cl)₋₁] as Fermion-IQuO. Note that an IQuO-chain is correspondent to a Field line.

Note that in Figure 3 , the two pairs of sq $\left[\left({\underset{\_}{\text{o}}}_{el},{\underset{\_}{•}}_{in}\right),\left({\text{o}}_{in}^{+},{•}_{el}^{+}\right)\right]$ in the two images Figure 3(a) and Figure 3(b) the two pairs of sq operators $\left[\left({\text{o}}_{el},{•}_{in}\right),\left({\underset{\_}{\text{o}}}_{in}^{+},{\underset{\_}{•}}_{el}^{+}\right)\right]$, act in “agreement”, in the same direction of propagation along the X axis (see the projection along X-axis): the operator ${\left({•}_{el}^{+}\right)}_{+\text{X}}$ creates a full sq (•) of elastic energy at the point x in the +X direction also (o_el)_+X annihilating an empty sq (o) of elastic energy in the point x in the +X direction, actually creates an sq full (•) of elastic energy, in the same +X direction, i.e. ${\left({•}_{el}^{+}\right)}_{-\text{X}}\equiv {\left(o_{el}\right)}_{+\text{X}}$. This is a characteristic of IQuO of boson type (B-IQuO).

Instead in Figure 4 , the sq-operators are “discordant”, act in opposite directions along the X axis of propagation, and are not coherent, that is ${\left({•}_{el}^{+}\right)}_{-\text{X}}\ne {\left(o_{el}\right)}_{+\text{X}}$. This makes “antagonists” the sq-operators of Figure 4 . We noticed immediately that the antagonism between operators (one creates energy in the +X direction while the other in the -X direction) makes “contradictory” the representation of a F-IQuO. It follows that the mono verso IQuO certainly represent “charged” IQuO (q = ±1) but, by themselves, they cannot represent particles. Since there are no massless charged particles, we could assume that monoverse IQuO are IQuO representative of massive coupling [2] [3] [16] , that is of massive particles. A first possibility to have F-IQuO with massive couplings, no-discordant sq operators and mono-verse could be a phase shift between the plane of oscillation of the pair ( $a_{el}^{+}$, $a_{in}^{+}$) and that of the pair (a_el, a_in). So, we proposed a form of F-IQuO in which the pair of the operators ( $a_{el}^{+}$, $a_{in}^{+}$) of each F-IQuO oscillates on a plan phase-shifted respect to that of the pair (a_el, a_in), see Figure 5 :

Where s is the spin vector associated to succession of planes which, as one can well see in Figure 5 , can exploit in two directions of rotation. The phase shift of the planes of oscillation implies, as we have already seen in ref. [16] , the existence of a semi-integer spin, given that the wave function results with a period of 4π, see just the Figure 5 , as in Fermions. Instead, the IQuO that do not have phase-shifted oscillation planes between a-operator and a⁺, then we will indicate them as B-IQuO and, by exclusion, they will describe particles of the Boson type.

The processes of pair creation and annihilation are found, in fundamental terms, in the operations: (Φ₁ ⊕ Φ₂) → [(Ψ_cl)_+e + (Ψ_cl)_-e], [(Ψ_cl)_+e ⊕ (Ψ_cl)_-e] → (Φ₁ + Φ₂)

Using the representations in Figure 3 , we can represent the creation and annihilation processes more simply with the following figures, Figure 6(a) , Figure 6(b) :

Note, in the pair creation of Figure 6(a) , that the two B-IQuO simultaneously perform the following sq exchanges:

$\begin{array}{l}\left\{{\left[e{l}^{+}{(•)}_{cl}\oplus El{\left(o\right)}_{cl}\right]}_{\left(•\to o\right)},{\left[i{n}^{+}{\left(o\right)}_{cl}\oplus In{(•)}_{cl}\right]}_{\left(o\leftarrow •\right)}\right\}\\ \equiv {\left\{\left[e{l}^{+}{\left(o\right)}_{cl},i{n}^{+}{(•)}_{cl}\right],\left[El{(•)}_{cl},In{\left(o\right)}_{cl}\right]\right\}}_{F^{-}}\end{array}$

$\begin{array}{l}\left\{{\left[E{l}^{+}{(•)}_{\underset{\_}{cl}}\oplus el{\left(o\right)}_{\underset{\_}{cl}}\right]}_{\left(•\to o\right)},{\left[I{n}^{+}{\left(o\right)}_{\underset{\_}{cl}}\oplus in{(•)}_{\underset{\_}{cl}}\right]}_{\left(o\leftarrow •\right)}\right\}\\ \equiv {\left\{\left[E{l}^{+}{\left(o\right)}_{\underset{\_}{cl}},I{n}^{+}{(•)}_{\underset{\_}{cl}}\right],\left[el{(•)}_{\underset{\_}{cl}},in{\left(o\right)}_{\underset{\_}{cl}}\right]\right\}}_{F^{+}}\end{array}$

Vice versa in the pair annihilation process ( Figure 6(b) ) we will have:

$\begin{array}{l}\left\{{\left[e{l}^{+}{\left(o\right)}_{cl}\oplus El{(•)}_{cl}\right]}_{\left(o\leftarrow •\right)},{\left[i{n}^{+}{(•)}_{cl}\oplus In{\left(o\right)}_{cl}\right]}_{\left(•\to o\right)}\right\}\\ \equiv {\left\{\left[e{l}^{+}{(•)}_{cl},i{n}^{+}{\left(o\right)}_{cl}\right],\left[El{\left(o\right)}_{cl},In{(•)}_{cl}\right]\right\}}_{B1}\end{array}$

$\begin{array}{l}\left\{{\left[E{l}^{+}{\left(o\right)}_{\underset{\_}{cl}}\oplus el{(•)}_{\underset{\_}{cl}}\right]}_{\left(o\leftarrow •\right)},{\left[I{n}^{+}{(•)}_{\underset{\_}{cl}}\oplus in{\left(o\right)}_{\underset{\_}{cl}}\right]}_{\left(•\to o\right)}\right\}\\ \equiv {\left\{\left[E{l}^{+}{(•)}_{\underset{\_}{cl}},I{n}^{+}{\left(o\right)}_{\underset{\_}{cl}}\right],\left[el{\left(o\right)}_{\underset{\_}{cl}},in{(•)}_{\underset{\_}{cl}}\right]\right\}}_{B2}\end{array}$

In the coupling of two B-IQuO generating a pair of F-IQuO, only one exchange of a full sq (•) could occur for each pair of operators (a, a⁺), that is, see Figure 7 :

We will have that:

$\begin{array}{l}\left\{\left[e{l}^{+}{(•)}_{cl}\oplus El{\left(o\right)}_{cl}\right],{\left[i{n}^{+}{\left(o\right)}_{cl}\oplus In{(•)}_{cl}\right]}_{\left(o\leftarrow •\right)}\right\}\\ \equiv {\left[\left[e{l}^{+}{(•)}_{cl},i{n}^{+}{(•)}_{cl}\right],\left[El{\left(o\right)}_{cl},In{\left(o\right)}_{cl}\right]\right]}_{F^{-}}\end{array}$