In Dirac, just speaking of fermions, points out that, Ref. [1] : “… if we consider spin operators σ (operators indicating rotations) and combine them with an operator ζ that anti-commutes, then we can obtain quantum oscillators of (fermion) field described by operators (η,η⁺) that anti-commute”. After, Dirac posed the following hypothesis:

$\eta =\frac{1}{2}\varsigma \left({\sigma }_x-i{\sigma }_y\right),{\eta }^{+}=\frac{1}{2}\varsigma \left({\sigma }_x+i{\sigma }_y\right)$ (1)

With ζ operator which anti commute ([(ζ⁺,ζ)]₍₊₎ = ih), but also commutes with σ operators and it is (ζ)² = 1. Then Dirac proved:

$\begin{array}{l}\left\{{\eta }^2=0,{\eta }^{+}{}^2=0,{\left(\eta {\eta }^{+}\right)}^2=\eta {\eta }^{+},\left(\eta {\eta }^{+}\right)=n\right\}\\ {\left[{\eta }^{+}\eta +\eta {\eta }^{+}\right]}_{(+)}=1\end{array}$ (2)

The Dirac showed (ηη⁺)' = n' = (0,1), that is (η, η⁺) are operators describing quantum oscillators associated with fermion particles. Therefore, the anti-commutation property of the operator ζ combined with the properties of rotations result in particles with spin (1/2). Recall that in the quantum (harmonic) oscillator the operators (a, a⁺) can be placed in the form [2] :

$a_t=a_0{\text{e}}^{i\omega t},a_t^{+}=a_0^{+}{\text{e}}^{-i\omega t}$ (3)

These relations represent the time evolution of the operators (a, a⁺) where (ωt) is the phase φ of the oscillation. Since (η,η⁺) also represent quantum oscillators (Fermion field oscillators), then we can write:

${\eta }_t={\eta }_0{\text{e}}^{i\omega t},{\eta }_t^{+}={\eta }_0^{+}{\text{e}}^{-i\omega t}$ (4)

Where η_t is the operator η at time t and η₀ the operator at the initial time t = 0. From the Equations (4) and (2d), it follows that:

$\left\{\begin{array}{c}\eta {\eta }^{+}=n\\ \eta {\eta }^{+}={\eta }_0{\eta }_0^{+}\end{array}\right\}\Rightarrow {\eta }_0{\eta }_0^{+}=n$ (5)

where n is self-adjoint, n = n⁺. For an oscillator (n_t = n₀). Dirac posed that [3] :

${\eta }_t={\text{e}}^{i{\phi }_{op}}\sqrt{n},{\eta }_t^{+}=\sqrt{n}{\text{e}}^{-i{\phi }_{op}}$ (6)

Introducing a phase operator (φ_op). Using Equations (4)-(6), we obtain that:

$\begin{array}{l}\left\{\begin{array}{c}\left(\eta ={\eta }_0{\text{e}}^{i\omega t}\right),\left({\eta }^{+}={\eta }_0^{+}{\text{e}}^{-i\omega t}\right)\\ \left(\eta ={\text{e}}^{i{\left(\omega t\right)}_{op}}\sqrt{n}\right),\left({\eta }^{+}=\sqrt{n}{\text{e}}^{-i{\left(\omega t\right)}_{op}}\right)\end{array}\right\}\\ \Rightarrow \left\{\eta ={\eta }_0{\text{e}}^{i\omega t}={\text{e}}^{i{\left(\omega t\right)}_{op}}\sqrt{n},{\eta }^{+}={\eta }_0^{+}{\text{e}}^{-i\omega t}=\sqrt{n}{\text{e}}^{-i{\left(\omega t\right)}_{op}}\right\}\\ \left\{\begin{array}{c}\eta {\eta }^{+}=n\\ \eta {\eta }^{+}={\eta }_0{\text{e}}^{i\omega t}{\eta }_0^{+}{\text{e}}^{-i\omega t}={\eta }_0{\text{e}}^{i\omega t}{\text{e}}^{-i\omega t}{\eta }_0^{+}={\eta }_0{\eta }_0^{+}=n\\ \eta {\eta }^{+}={\text{e}}^{i{\left(\omega t\right)}_{op}}\sqrt{n}\sqrt{n}{\text{e}}^{-i{\left(\omega t\right)}_{op}}={\text{e}}^{i{\left(\omega t\right)}_{op}}n{\text{e}}^{-i{\left(\omega t\right)}_{op}}={\text{e}}^{i{\left(\omega t\right)}_{op}}{\text{e}}^{-i{\left(\omega t\right)}_{op}}\end{array}\right\}\end{array}$ (7)

The last equation comes from the relation (ηη⁺) = n which leads us to admit the commutation of [(exp(iωt)_op)] with n. This would mean that we can simultaneously observe the number of quanta n of an oscillator and its phase φ, that is, its position x. But this is not true because n commutes with the energy (E ≡ H), where H is Hamiltonian, while the x-position does not. There is something wrong with Dirac’s hypothesis, Equation (6). Instead, if the φ-phase (with φ = ωt) is a “number” and we believe that (exp(iωt)) is also a number, then we can have, by Equation (4), that η₀ = (n)^1/2. We recall, from the properties of the spin operator, that: σ_± = σ_x ± iσ_y. We can then express (η, η⁺), see Equation (1), such as:

$\eta =\frac{1}{2}\varsigma {\sigma }_{+},{\eta }^{+}=\frac{1}{2}\varsigma {\sigma }_{-}$ (8)

By means of the properties of matrices σ [(σ_x)² = (σ_y)² = 1], we find that

ηη⁺ = ζ² = n, that is ζ = (n)^1/2. Thus, we will find the Equation (6) if we place the operator σ in exponential form of Euler: σ→exp(iωt). Comparing Equation (3) with Equation (8) we obtain:

$\begin{array}{l}\eta ={\eta }_0{\text{e}}^{i\omega t}=\frac{1}{2}\varsigma {\sigma }_{+},{\eta }^{+}={\eta }_0^{+}{\text{e}}^{-i\omega t}=\frac{1}{2}\varsigma {\sigma }_{-}\\ \sigma \iff {\text{e}}^{i\omega t},\varsigma \iff \left({\eta }_0\equiv \sqrt{n}\right)\end{array}$ (9)

Since σ_i -spin are operators, we expect from Equation (8) that [(exp(iωt))_σ+, (exp(−iωt))_σ−] would also be operators (!): this tell us that even if the phase of an oscillation is a number (angle), instead, (exp(iωt)) should be an operator. Dirac’s hypothesis would be partially restored: the phase φ is not an operator but, instead [(exp(iωt))_σ+, (exp(−iωt))_σ−] must be. Let us note that in these relations we are linking phase to spin, that is to rotations. We ask ourselves to which rotations we could connect the phase. Besides, we wonder what could make the terms [(exp(iωt))_σ+, (exp(−iωt))_σ−] into operators. We again think that something is missing in the expression (exp(iωt)). Don't be in contradiction with the Equation (9) is necessary to conjecture that an operator needs to be added in (exp(iωt)). To understand well this hypothesis, it is necessary to return to the spin. In the case of a spin, we could indeed associate an internal rotation with the particle spin, as Dirac conjectured [1] , and to the phase of this rotation an intrinsic oscillation described by η-operators, see the Equations (9) and (1). Note that this aspect would be valid for quantum oscillators of fermion type, where the phase exp(iωt) is linked to the “sign” of the spin operator, see [(exp(iωt) ⌠σ₊), (exp(−iωt) ⌠σ₋)]. Remember that the spin eigenvalues are connected to two rotation directions. This makes us realise the direction of phase rotation in the rotations is an operator (r’) with eigenvalues coincident to that of spin. Now, recall, the connection [phase rotation ⌠ oscillation] would be a characteristic of any type of oscillation. Then, we can conjecture that in every oscillation the rotation of a certain phase (φ) can be related to a physics variable (r) with two eigenvalues (±r'). Note that a rotation has two rotation directions and since the two eigenvalues ±r' are observables in the “spin” then also the two rotation directions of phase (±r') of any oscillator, must be observables. Let us recall that Dirac thought [4] to a phase operator (φ) in a quantum oscillator and, therefore, also of field oscillators, but this last aspect, as before we have noted and as Davidov [3] pointed out referring to the works of Susskind and Glogower [5] , could lead to some “unclear” aspects in quantum oscillator theory, if not even to contradictions, see next section. This unclearness could stem from an incompleteness: something is missing in the description of the quantum oscillator. The first step to complete the quantum oscillator is proper to introduce an r-operator in the phase rotation. So, we suppose that the phase exp(iωt) can become an operator (or precisely functional operator) in any quantum oscillator only if we introduce an r-operator of the direction of phase rotation, that is [exp(iωt) → exp(irωt)], and if we admit (as is the case with spin) that the phase exp(iωt) is also connected to the sign of a specific operator q: [(exp(iωt) ⌠q₊), (exp(−iωt) ⌠q₋)]. Note that with the operator r [6] [7] , we associate an “internal degree of freedom” with any quantum oscillator. In support of our conjecture about r operator, we report what Fain demonstrated [3] : “The eigenstate of energy (eigenstate of n) is degenerate concerning to the sign relative to the direction of phase rotation (±φ); the sign (±) is associated with the operator I (we have called it r) with eigenvalues ±1, which commutes with the Hamiltonian operator H”. Therefore, to complete the theory of the quantum oscillator, we must associate an operator r to the phase variable φ in exp(iωt), whose eigenvalues are r' = ±1 relative to the two rotation directions of the phase. Recall the Hamiltonian operator H and n the operator number of the quanta: these operators endowed with simultaneously definable eigenstates, are commuting. In fact, the eigenvalues of the Hamiltonian operator are:

$E_n=\left(n+\frac{1}{2}\right)\hslash \omega$

where n (1, 2, …) are the eigenvalues of the n operator. As a novelty in QM, in any type of oscillation, we introduce an operator r expressing the direction of the phase rotation: this operator has the eigenvalues r' = ±1 and commutes with H and, accordingly, with n. In an isolated quantum oscillator, the energy can degenerate into eigenvalues of (r), as it is degenerated in the direction of the phase rotation. We have:

${\left[E_n=\left(n+\frac{1}{2}\right)\hslash \omega \right]}_{\left(r^{\prime }=\pm 1\right)}$

Therefore, we define by S(H, n)_{(r' = ±1)} a space of Ψ states, having as basis the eigenvectors of the (H, n)_{(r' = ±1)} operators, that is $\left[{\left(|\Psi \rangle \right)}_{\left(r^{\prime }=\pm 1\right)}\equiv {\left(|{\Psi }_H\rangle |{\Psi }_n\rangle \right)}_{\left(r^{\prime }=\pm 1\right)}\right]$. If we have diversified σ into (σ₊, σ₋), see the Equation (9), by of the eigenvalues (r' = ±1), we could also try to diversify η₀ into [(η₀)₊, (η₀)₋], see again Equation (9). In this way, we could suppose:

$\begin{array}{l}{\varsigma }_{\eta }\to \sqrt{n_{+}},{\varsigma }_{{\eta }^{+}}\to \sqrt{n_{-}}\\ {\sigma }_{+}\iff {\text{e}}^{ir\omega t}={\left({\text{e}}^{i\omega t}\right)}_{\left(r^{\prime }=+1\right)},{\sigma }_{-}\iff {\text{e}}^{ir\omega t}={\left({\text{e}}^{-i\omega t}\right)}_{\left(r^{\prime }=-1\right)}\end{array}$ (10)

And so, to have:

$\left\{\eta =\left(\sqrt{n_{+}}\right)\left({\text{e}}^{ir\omega t}\right)=\left(\sqrt{n_{+}}\right){\left({\text{e}}^{i\omega t}\right)}_{\left(r^{\prime }=+1\right)},{\eta }^{+}=\left(\sqrt{n_{-}}\right)\left({\text{e}}^{ir\omega t}\right)=\left(\sqrt{n_{-}}\right){\left({\text{e}}^{-i\omega t}\right)}_{\left(r^{\prime }=-1\right)}\right\}$(11)

We then wonder so if the definition of functional operator given to the term exp(irωt), with the eigenvalues of r-operator (r' = ±1), is just connected to the difference {[(η₀)₋ ≠ (η₀)₊], [(η)₋ ≠ (η)₊]}, see the Equation (11) where [(η₀)₊ = (n₊)^1/2, (η₀)₋ = (n₋)^1/2]. In order not to contradict Equations (6) and (8), we could assume that the ζ operator undergoes a splitting into two components (ζ₊,ζ₋), becoming ζ₊ or ζ₋. Obviously, ζ is intact when the oscillator is in an indeterminate state of the phase direction. The veracity of this “paradoxical” hypothesis is linked to the possibility that the direction r of the rotation phase can only be observed in a quantum oscillator but no into classic oscillator. To demonstrate this connection (see Equation (11)), we recall the theory of oscillations.

Be the classical oscillator in the (q, p) coordinates of the phase plane; its Hamiltonian is:

$H=\left(\frac{p^2}{2m}\right)+\left(\frac{m{\omega }_0^2{q}^2}{2}\right)$

With equations of motion

$\{\begin{array}{l}\frac{\text{d}p}{\text{d}t}=-m{\omega }_0^{2}q\\ \frac{\text{d}q}{\text{d}t}=\left(\frac{p}{m}\right)\end{array}$

Its general solutions are linear combinations of two independent oscillations [2] :

$\left\{\begin{array}{c}q\left(t\right)=q\left(0\right)\cos \left({\omega }_{0}t\right)+\left(\frac{p\left(0\right)}{m{\omega }_0}\right)\sin \left({\omega }_{0}t\right)\\ p\left(t\right)=p\left(0\right)\cos \left({\omega }_{0}t\right)-\left(m{\omega }_0\right)q\left(0\right)\sin \left({\omega }_{0}t\right)\end{array}\right\}$ (12)

Instead, in forced damped oscillator (Γ≠0), the general solution is [9] [10] :

$\left(q{\left(t\right)}_{damp}\right)=\left({\text{e}}^{-\left(1/2\right)\Gamma t}\right)\left[A_s\sin \left({\omega }_{s}t\right)+B_s\cos \left({\omega }_{s}t\right)\right]$

With ω_s = ω_est and ω_s ≠ ω₀, where q(t)_damp is any coordinate. Note that there are two terms. The energy of the oscillator is:

$E\left(t\right)=E_0\left({\text{e}}^{-\Gamma t}\right)=\left({\text{e}}^{-\Gamma t}\right)\left[\frac{1}{2}M\left({\omega }_s^2+{\omega }_0^2\right)\left(\frac{1}{2}{A}_s^2+\frac{1}{2}{B}_s^2\right)\right]$ (13)

If (Γ≈ 0), the exponential decay term is approximately constant during an oscillation; in this case we have:

$\left(q{\left(t\right)}_{damp}\right)\approx \left(\cos t\right)\left[A_s\sin \left({\omega }_{s}t\right)+B_s\cos \left({\omega }_{s}t\right)\right]$ (14)

The energy (with ω_s ≈ ω₀) is:

$E\left(t\right)\approx E_0=\left[\frac{1}{2}M\left(2{\omega }_s^2\right)\left(\frac{1}{2}{A}_s^2+\frac{1}{2}{B}_s^2\right)\right]=M{\omega }_s^2\left(\frac{1}{2}{A}_s^2+\frac{1}{2}{B}_s^2\right)$ (15)

We also note here two energy components. In the driven oscillator there is also a two-term solution (again with damping). Outside the initial phase, the oscillator adapts to the external force, oscillates with frequency ω_a = ω_ext ≠ ω₀ and admits a solution with two components: one absorptive amplitude (A_abs), and the other elastic (A_el). Thus, it is:

$\left(q{\left(t\right)}_{adaptive}\right)=\left[A_{abs}\sin \left({\omega }_{a}t\right)+B_{el}\cos \left({\omega }_{a}t\right)\right]$ (16)

with a mean energy (in a cycle) given by:

${\langle E\left(t\right)\rangle }_{T_a}=\left[\frac{1}{2}M\left({\omega }_a^2+{\omega }_0^2\right)\left(\frac{1}{2}{A}_{abs}^2+\frac{1}{2}{A}_{el}^2\right)\right]$ (17)

So, the energy is expressed through the two absorptive and elastic components. If Γ≈ 0 and (ω_a = ω_ext) ≈ ω₀, the oscillator has an adaptive oscillation (resonance) with two energy components:

${\langle E\left(t\right)\rangle }_{T_a}\approx \left[\frac{1}{2}M\left({\omega }_a^2\right)\left(A_{abs}^2+A_{el}^2\right)\right]$

where it is τ_a = 2π/ω_a. Now, let us we consider an oscillator belonging to a chain of elastically coupled oscillators (a field). The coupling between two oscillators can determine a reciprocal action of one on the other. Then, we think of each oscillator of the chain as a driven oscillator by the neighboring oscillators, with frequency set ω_ext ≠ ω₀ (where ω₀ is the frequency of the individual single “isolated” oscillator q). We’ll say that the system of coupled oscillators is well adapted to the external frequency (ω_ext = ω_k), where ω_k is now the frequency of the k-mode oscillation of the chain of a field [8] [9] . It follows that the single field oscillator is like an oscillator (with a damping coefficient Γ≈ 0) adapted to the field frequency ω_k. In conclusion, the field-oscillator is an adapted oscillator which can be described by two oscillating components, see the Equation (16). This has notable consequences in the field theory. To discover these consequences, we recall the classical oscillator expressed in (a, a⁺) complex coordinates; it is:

$a=\left[\left(\sqrt{\frac{m\omega }{2\hslash }}\right)\left(q+\frac{ip}{m\omega }\right)\right]\text{,}{a}^{+}=\left[\left(\sqrt{\frac{m\omega }{2\hslash }}\right)\left(q-\frac{ip}{m\omega }\right)\right]$ (18)

Note that the generalised coordinates (q, p) are expressed in the plane (q, ip), see the operators (a, a⁺). We could consider the plane (iq, p) for symmetry, see the operators (b, b⁺). We might also have:

$\left\{\left[b=\left(\sqrt{\frac{m\omega }{2\hslash }}\right)\left(\frac{p}{m\omega }+iq\right)\right]\text{,}{b}^{+}=\left[\left(\sqrt{\frac{m\omega }{2\hslash }}\right)\left(\frac{p}{m\omega }-iq\right)\right]\right\}$

$\left\{\left[\left(b=i{a}^{+}\right),\left(b^{+}=-ia\right)\right],\left(a=i{b}^{+}\right),\left(a^{+}=-ib\right)\right\}$

It is well known that the equations of motion of the oscillator are invariant by change of the coordinate system used; therefore, we can choose the coordinate plane (q, ip) with operators (a, a⁺) and equation of motion given by:

$\left\{\begin{array}{c}\left(\frac{\text{d}{a}_t}{\text{d}t}\right)=-i\omega a_t\\ \left(\frac{\text{d}{a}_t^{+}}{\text{d}t}\right)=i\omega a_t^{+}\end{array}\right\}$

and with solutions

$\left\{a_t=a_0\exp \left(-i\omega t\right)\text{,}{a}_t^{+}=a_0^{+}\exp \left(i\omega t\right)\right\}$ (19)

where $\left\{\left[a_0=a\left(0\right),a_0^{+}=a^{+}\left(0\right)\right]\right\}$. The solutions $\left(a_t,a_t^{+}\right)$ could be related to a pair of vectors (Oa, Oa⁺) with opposite phase rotation, see Figure 1 .

The two solutions may concern the two opposite rotations of the phase. In classical mechanics the two possibilities are hidden: an oscillating pendulum going backwards in time (clockwise rotation of the representative curve in Figure 1 is indistinguishable from one that goes forwards in time (counterclockwise rotation). Considering, instead, the eigenvalues of r (that is, one detects the two directions of rotation of the phase) as “physical observables”, the two rotating vectors (Oa, Oa⁺), associated with classical coordinates (a, a⁺), must be somehow different when we observe the direction of rotation of the phase in an oscillation, but they must not be different when we observe the oscillator in its “uniqueness”, for example we are observing its energy but we don't look at where the oscillator is. Recall that the coordinates (a, a⁺) always describe the same oscillator. One might think that a “new physics” could be devised, in which these two different possibilities (eigenstates) can be individually detected in some physical phenomena but can coexist in other phenomena of non-locality: we will talk about complementarity of two possibilities in local states. An intuition suggests, looking at Figure 1 , that the only possibility of having a difference between the two rotating vectors is in their norms or modules. This possibility was raised in the previous section when we discussed the norm of the two operators (η, η⁺). In this new original physics [1] , the two different possibilities of rotation would be expressed by a difference in the norm of the two vectors (Oa) and (Oa⁺) of the phase rotation, see Figure 2 .

The new physics could be the “Quantum Mechanics”, where the possible local states of a system can individually exist, but are experimentally complementary, or can also exist in a superposition state (non-locality). Let us recall that in analytical mechanics [8] to the two rotations of phase are associated two fluxes of phase ((g_cl(t), g_cl(t)) with their respective Poisson operators and relative Poisson brackets (cl is clockwise, cl is counterclockwise). Well, a difference between the two fluxes of phase (that is the difference (Oa ≠ Oa⁺)) would imply a non-commutation of the phase fluxes and therefore a non-zero value of their Poisson bracket. If we then associate a physical quantity with an operator because experimentally observing and measuring perturbs the observed system and we admit the possibility of a non-local state Ψ of individual possibilities (eigenstates) or measurements (Ψ_k), then physics “changes” and instead of classical Poisson brackets we have commutators between operators. So, in quantum mechanics, see the Heisenberg’s representation, the Poisson brackets correspond to the commutators. It follows that the difference (Oa ≠ Oa⁺) must correspond to a commutation bracket relating to two operators (a, a⁺), that is [a, a⁺] = (aa⁺ − a⁺a) ≠ 0. Recall that the difference (Oa ≠ Oa⁺) is connected to the two eigenvalues (±r'), now assumed to be observable. Therefore, the “new physics” admits the possibility of detecting the two rotation directions of the phase (classical physics does not) but this implies that the coordinates (a, a⁺) become non-commutative operators and that there is the possibility of a non-local state in which the two rotation eigenvalues coexist. It follows, as far as quantum oscillators are concerned, we consider that the difference (Oa ≠ Oa⁺) is related to the commutation bracket (non-zero value) of the corresponding Hermitian operators (a, a⁺), that is [a, a⁺] = 1, and vice versa. Note that “vice versa” would mean that the entire physics of quantum oscillators is built on the possibility of detecting the direction of phase rotation. In conclusion, if we admit that QM is “true” we must admit that the two different directions of rotation of the phase assigned to a quantum oscillator (note that the energy is degenerate in these two eigenstates), can instead be detected, under certain conditions, and observed. The solutions of the motion equations in the phase plane of the quantum Hermitian operators (q, p) are the same as Equation (12), but with operators (q_op, p_op). In an adaptive oscillator, the solutions (see Equation (16)) for a quantum oscillator, expressed by operators, will be:

$q\left(t\right)=q_{el}\left(0\right)\cos \left(\omega t\right)+q_{abs}\left(0\right)\sin \left(\omega t\right)$ (20)

where q_el is the elastic component in phase with the external force, while q_abs has a phase delay of π/2 with respect to the external force. We remember that any oscillator has two components: elastic and inertial; therefore, we admit the following equivalence: q_abs ≡ q_inertial. It follows:

$q\left(t\right)=q_{el}\left(0\right)\cos \left(\omega t\right)+q_{in}\left(0\right)\sin \left(\omega t\right)$ (21)

This equation can also describe, as we have already mentioned, an oscillator belonging to a chain of coupled oscillators (field). We state that a quantum field oscillator should always be represented with two components. The graphic representation [6] [7] of the quantum oscillator with two components is, Figure 3 :

Recall that the quantum field Φ-field [3] is as a set of coupled quantum oscillators express by operators (a, a⁺):

$\begin{array}{l}\Phi \left(x_i,t\right)=\left(\frac{1}{\sqrt{V}}\right){\displaystyle \underset{k}{\sum }{q}_k\left(t\right){\text{e}}^{i\left(k\cdot x\right)}}=\left(\frac{1}{\sqrt{V}}\right){\displaystyle \underset{k}{\sum }\left[\left(\sqrt{\frac{\hslash }{2{\omega }_k}}\right)\left(a_k+a_{-k}^{+}\right)\right]{\text{e}}^{i\left(k\cdot x\right)}}\\ {\epsilon }_{\left(n_1,n_2,\cdots ,n_k\right)}={\displaystyle \underset{k}{\sum }\left[\hslash {\omega }_k\left(n_k+\frac{1}{2}\right)\right]},P_{\left(n_1,n_2,\cdots ,n_k\right)}={\displaystyle \underset{k}{\sum }\left[\hslash k_k\left(n_k\right)\right]}\end{array}$(22)

where k_k = (2π/λ_k), ω_k = (v_φ/k_k). Considering the Equations (18) and (19), we have [q_t = q₀exp(±iωt), p_t = p₀exp(±iωt)] and resorting to the Euler’s relations, easily one finds that:

$\left\{\begin{array}{c}{a}_t=\left(\sqrt{\frac{m\omega }{2\hslash }}\right)\left[q_0{\text{e}}^{-i\omega t}+i\left(\frac{p_0}{m\omega }\right){\text{e}}^{-i\omega t}\right]=k_q{\text{e}}^{-i\omega t}+i{k}_p{\text{e}}^{-i\omega t}=k_q{\text{e}}^{-i\omega t}+k_p{\text{e}}^{-i\left(\omega t-\pi /2\right)}\\ a_t^{+}=\left(\sqrt{\frac{m\omega }{2\hslash }}\right)\left[q_0{\text{e}}^{i\omega t}-i\left(\frac{p_0}{m\omega }\right){\text{e}}^{i\omega t}\right]=k_q{\text{e}}^{i\omega t}-i{k}_p{\text{e}}^{i\omega t}=k_q{\text{e}}^{i\omega t}+k_p{\text{e}}^{i\left(\omega t-\pi /2\right)}\end{array}\right\}$(23)

where it’s [q₀ = q(0), p₀ = p(0)] and

$\left\{\begin{array}{c}\left(k_q=\left(\sqrt{\frac{m\omega }{2\hslash }}\right)q_0\text{,}{k}_p=\left(\sqrt{\frac{1}{2m\hslash \omega }}\right)p_0\right)\\ \left[q_0,p_0\right]=i\hslash ,\left[k_q{k}_p-k_p{k}_q\right]=\frac{i}{2},\left[a_0,a_0^{+}\right]=1\end{array}\right\}$ (24)

We thus note that the [(a, a⁺)] operators can be expressed by two components: elastic and inertial, see the Equation (21). We could then say that the operators [(a, a⁺)] will be expressed by the following equations, see Equation (23):

$\left\{\begin{array}{c}{a}_t=k_{\left(elastic\right)}{\text{e}}^{-i\omega t}+k_{\left(inertial\right)}{\text{e}}^{-i\left(\omega t-\pi /2\right)}\\ a_t^{+}=k_{\left(elastic\right)}{\text{e}}^{i\omega t}+k_{\left(inertial\right)}{\text{e}}^{i\left(\omega t-\pi /2\right)}\end{array}\right\}$ (25)

We could then say that the operators (a, a⁺) will be expressed by the following equation (see Equation (25)):

$\left\{\begin{array}{c}{a}_t=a{\left(t\right)}_{elastic}+a{\left(t\right)}_{inertial}\equiv Oa\\ a_t^{+}=a^{+}{\left(t\right)}_{elastic}+a^{+}{\left(t\right)}_{inertial}\equiv O{a}^{+}\end{array}\right\}\iff \left\{\begin{array}{c}{a}_t=a_{\left(el\right)}{\text{e}}^{-i\omega t}+a_{\left(in\right)}{\text{e}}^{-i\left(\omega t-\pi /2\right)}\\ a_t^{+}=a_{\left(el\right)}^{+}{\text{e}}^{i\omega t}+a_{\left(in\right)}^{+}{\text{e}}^{i\left(\omega t-\pi /2\right)}\end{array}\right\}$ (26)

This is the generalization of the Equation (25). Let’s duly observe that the elastic component has a phase delay of π/2, in comparison to the inertial one. Hereafter, we will resort to an oscillator with two components to describe a field oscillator associated with a particle, Figure 4 .

The two descriptive circles of the oscillation in Figure 4 and the doubling of the components of each quantization operator (a, a⁺) see $\left[\left(a_{el},a_{el}^{+}\right),\left(a_{in},a_{in}^{+}\right)\right]$, lead us to describe a quantum field oscillator as a 2-dim. oscillator.

We return to the Equations (3), (4) and (6), which can be extended to any type of

quantum oscillator. As we already said, Dirac [4] proposed the following relations for the (a, a⁺) operators of a quantum oscillator, see the Equation (6):

$\{\begin{array}{l}a=\left(\exp \left(i\phi \right)\right)\left(\sqrt{n}\right)\\ a^{+}=\left(\sqrt{n}\right)\left(\exp \left(-i\phi \right)\right)\end{array}$ (27)

For Dirac, φ is the φ-operator of the phase angle (remember that (a, a⁺) are operators). However, it’s evident that this operator (φ) does not commute with the operators (H), (n), and (r), see the Equation (6): the phase angle [θ = ωt] is uncertain, if, instead, n'-eigenvalue of (n) is certain. Note that, if the oscillator energy (degenerate in an eigenvalue of (r)) is certain, then the phase is uncertain. As already demonstrated by us and by Susskind and Glogower [5] , the Equation (27) leads to some contradictions, showing so that the phase must be seen as a physical variable but not as an operator. In Ref. [3] , however, the functional operator (Φ) is proposed:

$\Phi \left(\phi \right)=\exp \left(i\phi \right),{\Phi }^{+}\left(\phi \right)=\exp \left(-i\phi \right)$ (28)

Considering Fain [3] and our conjecture of the existence of an operator r, see Sects. 2.1 and 2.2, we can introduce the operator r into the function Φ, and have:

$\{\begin{array}{l}a=\left(\Phi \left(r,\phi \right)\right)\left(\sqrt{n}\right)\\ a^{+}=\left(\sqrt{n}\right)\left({\Phi }^{+}\left(r,\phi \right)\right)\end{array}$ (29)

where r commutes whit Hamiltonian operator H and the function operator Φ consists in:

$\Phi \left(r,\phi \right)=\exp \left(ir\phi \right),{\Phi }^{+}\left(r,\phi \right)=\exp \left(-ir\phi \right)$ (30)

From the Equations (11), (23) and (25) it is:

$\{\begin{array}{l}a=\left(\Phi \left(r,\phi \right)\right)\left(\sqrt{n}\right)=\left[\left({\text{e}}^{-ir\omega t}\right)\left(k_{el}+i{k}_{in}\right)\right]\\ a^{+}=\left(\sqrt{n}\right){\Phi }^{\ast }\left(r,\phi \right)=\left[\left(k_{el}-i{k}_{in}\right)\left({\text{e}}^{ir\omega t}\right)\right]\end{array}$ (31)

But

$\left\{\begin{array}{c}{a}^{+}a=n=\left(\sqrt{n}\right)\left(\sqrt{n}\right)\\ a^{+}a=n=\left[\left(k_{el}-i{k}_{in}\right)\left({\text{e}}^{ir\omega t}\right)\right]\left[\left({\text{e}}^{-ir\omega t}\right)\left(k_{el}+i{k}_{in}\right)\right]=\left(k_{el}-i{k}_{in}\right)\left(k_{el}+i{k}_{in}\right)\end{array}\right\}$ (32)

Note that the last equality (Equation 32(b)) implies a contradiction with the first one (Equation 32(a)) because it’s (k_el − ik_in) ≠ (k_el + ik_in). To avoid this contradiction, we recall the previous assumption, see Figure 2 : the difference (Oa ≠ Oa⁺) between vectors (Oa, Oa⁺) is implicitly contained in the commutator [a, a⁺] (non-zero value). Since the no commutation [a, a⁺] is a physical reality, then the difference (|Oa| ≠ |Oa⁺|) should be permissible. Vice versa if we physically admit the difference (|Oa| ≠ |Oa⁺|) as the existence of detect the two possible directions of phase rotation, then as consequence we have a commutator [a, a⁺] = 1. This means that we can distinguish the norms of these operators, or we can distinguish the (n)^1/2 operator contained in (a) from that contained in (a⁺). Then we can conjecture the following form:

$\left\{\begin{array}{c}a=\Phi \left(r,\phi \right)\left(\sqrt{n_{+}}\right)\\ {\hat{a}}^{+}=\left({\sqrt{n}}_{-}\right)\left({\Phi }^{+}\left(r,\phi \right)\right)\end{array}\right\}$ (33)

where it is

$\sqrt{n_{+}}=\left(k_{el}+i{k}_{in}\right)\text{;}\sqrt{n_{-}}=\left(k_{el}-i{k}_{in}\right)$ (34)

with

$\left\{\begin{array}{c}\sqrt{n_{+}}\ne \sqrt{n_{-}}\\ {\left(\sqrt{n_{+}}\right)}^{+}=\sqrt{n_{-}}\iff {\left(\sqrt{n_{-}}\right)}^{+}=\sqrt{n_{+}}\end{array}\right\}$ (35)

We’ll have, for k operator, see the Equation (24):

$\left(\sqrt{n_{+}}\right)\left(\sqrt{n_{-}}\right)=\left(k_{el}+i{k}_{in}\right)\left(k_{el}-i{k}_{in}\right)=\left(k_{el}^2+k_{in}^2\right)-\left(\frac{1}{2}\right)$ (36)

Thus, we find (by Equation (31) and Equation (33)) that:

$\{\begin{array}{l}{a}^{+}a=\left[\left(k_{el}\right)\left({\text{e}}^{ir\omega t}\right)-\left(i{k}_{in}\right)\left({\text{e}}^{ir\omega t}\right)\right]\left[\left({\text{e}}^{-ir\omega t}\right)k_{el}+\left(i{\text{e}}^{-ir\omega t}\right)k_{in}\right]=\left[\left(\sqrt{n_{+}}\right)\left(\sqrt{n_{-}}\right)\right]\\ a^{+}a=n\end{array}$(37)

confirming that

$n=\left[\left(\sqrt{n_{+}}\right)\left(\sqrt{n_{-}}\right)\right]$ (38)

In this way, Equation (33) becomes:

$\{\begin{array}{l}a=\left(\Phi \left(r,\phi \right)\right)\left(\sqrt{n_{+}}\right)=\left[\left({\text{e}}^{-ir\omega t}\right)\left(k_{el}+i{k}_{in}\right)\right]\\ a^{+}=\left(\sqrt{n_{-}}\right){\Phi }^{+}\left(r,\phi \right)=\left[\left(k_{el}-i{k}_{in}\right)\left({\text{e}}^{ir\omega t}\right)\right]\end{array}$ (39)

A new Dirac’s form of the (a, a⁺) operators will be:

$\{\begin{array}{l}a=\left(\Phi \left(r,\phi \right)\right)\left(\sqrt{n_{+}}\right)=\left[\left(\Phi \left(r,\phi \right)\right)\left(k_{el}\right)+\left(\Phi \left(r,\phi \right)\right)\left(i{k}_{in}\right)\right]=\left(a_{el}+a_{in}\right)\\ a^{+}=\left(\sqrt{n_{-}}\right)\left({\Phi }^{+}\left(r,\phi \right)\right)=\left[\left(k_{el}\right)\left({\Phi }^{+}\left(r,\phi \right)\right)-\left(i{k}_{in}\right)\left({\Phi }^{+}\left(r,\phi \right)\right)\right]=\left(a_{el}^{+}+a_{in}^{+}\right)\end{array}$ (40)

This form is valid also for fermionic operators (η,η⁺) in Sect. 2.1. We have thus shown that the differences {[(η₀) ≠ (η₀)₊], [(η)₋ ≠ (η)₊]} are given and are connected to the Φ via the r-operator: (exp(irωt). Note that the (a, a⁺) become complex operators, consisting of the operator Φ (with real norm ρ = 1) and the $\left(\sqrt{n_{\pm }}\right)$ complex operator. Here, we point out that new form of the operators (a, a⁺) contains one more piece (see the $\left(\sqrt{n_{\pm }}\right)$) than the old form, Equation (27).

Let us consider that a two-component quantum oscillator, see equations (17), (26) and (40), as also Figure 3 and Figure 4 , has the two components oscillating even in the vacuum state. Therefore, if the vacuum state is with energy ε = 1/2ηω, each component will have energy ε = 1/4ηω. We define “full semi-quantum (•)” as the oscillating quantum with energy ε = 1/2ηω, and “empty semi-quantum (o)” as the quantum with energy ε = 1/4ηω [6] [7] . In this case, we can have one quantum (•) in way that:

$\left\{(•)=\left[(•)+(•)\right]\equiv \left(1\hslash \omega \right)\right\}$ (41)

To better understand this last aspect, let’s consider the Schrödinger’s representation of a quantum oscillator, see Figure 5 [10] :

The probability function P_n(x) shows some peaks indicating where the energy quanta of the oscillation are most likely to be found at time t₀. Then, we could conjecture that the peaks of P_n(x) could represent the distinct parts (S_i) of the same quantum oscillator, each always with elastic characteristics and kinetic (inertial). The quantum oscillator is thus composed of (n + 1) not-separable oscillating parts or “sub-oscillators” S_i, see Figure 5 . We also believe that the presence of two or more oscillating components in an oscillator causes the “splitting” of its energy quanta into two energetic components in each sub-oscillators: this introduces the idea of “semi-quanta” (or singularly a “semi-quantum” (sq)). If (n = 0), there is only one sub-oscillator with energy (ε = 1/2ηω), then the two sq elastic and inertial must each have an energy of [(ε = 1/4ηω)]. These relations are consistent with the actions of the operators (a, a⁺) on the eigenvectors $|n\rangle$ of the operator (n); indeed, recall:

$\left\{\begin{array}{c}a|n\rangle =\left(\sqrt{n}\right)|n-1\rangle \\ a^{+}|n\rangle =\left(\sqrt{n+1}\right)|n+1\rangle \end{array}\right\}$ (42)

With the H-Hamiltonian operator:

$H=\left(a^{+}a+\frac{1}{2}\right)\hslash \omega =\left(n+\frac{1}{2}\right)\hslash \omega$ (43)

Recall that in any oscillator the energy ε is given as (ε = U + K), where U is the potential energy or elastic, while K is the kinetic energy. Then, through the inverse of the two relations in Equation (18), we have, see also ref. [2] :

$\left\{\begin{array}{c}U|n\rangle =\left(\frac{1}{2}m{\omega }^2{x}^2\right)|n\rangle ={\left(\frac{1}{4}\hslash \omega \right)}_U\left[a^2+{\left(a^{+}\right)}^2+a^{+}a+a{a}^{+}\right]|n\rangle \\ K|n\rangle =\left(\frac{p^2}{2m}\right)|n\rangle ={\left(\frac{1}{4}\hslash \omega \right)}_K\left[-a^2-{\left(a^{+}\right)}^2+a^{+}a+a{a}^{+}\right]|n\rangle \end{array}\right\}$ (44)

Note

$\left\{\begin{array}{c}a{a}^{+}|n\rangle =a\left(\sqrt{n+1}|n+1\rangle \right)=\sqrt{n+1}\left(\sqrt{n+1}\right)|n\rangle =\left(n+1\right)|n\rangle \\ a^{+}a|n\rangle =a^{+}\left(\sqrt{n}|n-1\rangle \right)=\sqrt{n}\left(\sqrt{n}\right)|n\rangle =\left(n\right)|n\rangle \end{array}\right\}$ (45)

That is:

$\left(a{a}^{+}+a^{+}a\right)|n\rangle =\left(n+1\right)|n\rangle +\left(n\right)|n\rangle =\left(2n+1\right)|n\rangle$ (46)

So, we can calculate, by Equation (46), the value of the potential and kinetics energy in the |0〉 eigenstate:

$\left\{\begin{array}{c}U|0\rangle =\left(\frac{1}{2}m{\omega }^2{x}^2\right)|0\rangle ={\left(\frac{1}{4}\hslash \omega \right)}_U\left[{\left(a^{+}\right)}^2+1\right]|0\rangle \\ K|0\rangle =\left(\frac{p^2}{2m}\right)|0\rangle ={\left(\frac{1}{4}\hslash \omega \right)}_K\left[-{\left(a^{+}\right)}^2+1\right]|0\rangle \end{array}\right\}$ (47)

Thus, the form of the Hamiltonian operator in |0〉 is found:

$\hat{H}|0\rangle =U|0\rangle +K|0\rangle ={\left(\frac{1}{4}\hslash \omega \right)}_U|0\rangle +{\left(\frac{1}{4}\hslash \omega \right)}_K|0\rangle$ (48)

In this way, in the empty state of quantum oscillator one demonstrates the existence of empty semi-quanta sq(o) with energy ε = 1/4ηω. Looking at the Equation (48), one obtains, Figure 6 .

We demonstrate by Equation (43) also that in (n ≠ 0) one has:

$H|n\rangle =U|n\rangle +K|n\rangle =\left(2n+1\right){\left(\frac{1}{4}\hslash \omega \right)}_U|n\rangle +\left(2n+1\right){\left(\frac{1}{4}\hslash \omega \right)}_K|n\rangle$ (49)

with eigenvalues in the eigenstate S(H,n) such that:

${\left[H_{\left(n\right)}\right]}_{r^{\prime }}={\left[U_{\left(n\right)}+K_{\left(n\right)}\right]}_{r^{\prime }}={\left[\left(2n+1\right){\left(\frac{1}{4}\hslash \omega \right)}_U+\left(2n+1\right){\left(\frac{1}{4}\hslash \omega \right)}_K\right]}_{r^{\prime }}$ (50)

$\begin{array}{c}{\left(H_{\left(n\right)}\right)}_{r\text{'}}={\left(U_{\left(n\right)}+K_{\left(n\right)}\right)}_{r^{\prime }}\\ ={\left[n\left({\left(\frac{1}{2}\hslash \omega \right)}_U\right)+{\left(\frac{1}{4}\hslash \omega \right)}_U\right]}_{r^{\prime }}+{\left[n\left({\left(\frac{1}{2}\hslash \omega \right)}_K\right)+{\left(\frac{1}{4}\hslash \omega \right)}_K\right]}_{r^{\prime }}\end{array}$

Here it is a further confirmation that in the quantum oscillator of field there are full semi-quanta and some empty semi-quanta. We denote by the acronym IQuO (Intrinsic Quantum Oscillator) an oscillator composed of sub-oscillators traversed by semi-quanta energy. We have, see Figure 7 :

Even if the maximum potential energy and the maximum kinetic energy always have a constant value over time, the semi-quantum goes back and forth from a sub-oscillator to another.

Now we can state the following correspondences:

$\boxed{\begin{array}{l}\left\{{\epsilon }_{|1\rangle }=\begin{array}{c}U\equiv \left(•+\text{o}\right)\\ K\equiv \left(•+\text{o}\right)\end{array}\right\}\iff \left\{{\Psi }_{osc}\equiv \left(\begin{array}{c}Oa=O{a}_{el}+O{a}_{in}\\ O{a}^{+}=O{a}_{el}^{+}+O{a}_{in}^{+}\end{array}\right)\right\}\\ \iff \left\{\left(\begin{array}{c}{a}_{el}\equiv U_{el}(•)\\ a_{el}^{+}\equiv U_{el}\left(\text{o}\right)\end{array}\begin{array}{c}{a}_{in}\equiv K_{in}\left(\text{o}\right)\\ a_{in}^{+}\equiv K_{in}(•)\end{array}\right)\right\}\end{array}}$ (51)

There is also the following equivalence (invariance for (o) ↔ (•) change) between matrices:

$\boxed{\left\{\left(\begin{array}{c}{a}_{el}\equiv U_{el}(•)\\ a_{el}^{+}\equiv U_{el}\left(\text{o}\right)\end{array}\begin{array}{c}{a}_{in}\equiv K_{in}\left(\text{o}\right)\\ a_{in}^{+}\equiv K_{in}(•)\end{array}\right)\right\}\iff \left\{\left(\begin{array}{c}{a}_{el}\equiv U_{el}\left(\text{o}\right)\\ a_{el}^{+}\equiv U_{el}(•)\end{array}\begin{array}{c}{a}_{in}\equiv K_{in}(•)\\ a_{in}^{+}\equiv K_{in}\left(\text{o}\right)\end{array}\right)\right\}}$ (52)

Recall that the two sub-oscillators of the IQuO_{(n = 1)} contain the sq(o, •) which move following the oscillation vectors (Oa, Oa⁺) and the corresponding components $\left[\left(a_{el},a_{el}^{+}\right),\left(a_{in},a_{in}^{+}\right)\right]$. We assign to each component of the operators (a, a⁺) the “semi-quantum operators” sq(o, •) with their respective elastic and inertial characteristics: $\left\{\left[\left(a_{el},a_{el}^{+}\right),\left(a_{in},a_{in}^{+}\right)\right]\iff \left[\left(o_{el},{•}_{el}^{+}\right),\left({•}_{in},o_{in}^{+}\right)\right]\right\}$.

The projections of the operators $\left[\left(a_{el},a_{el}^{+}\right),\left(a_{in},a_{in}^{+}\right)\right]$ along the X-axis give us the position of the semi-quanta (o, •) along this axis. As for the (a, a⁺) operators, by means of the Equation (53), we can set the following symbols:

$\boxed{\begin{array}{l}\left\{\left(\begin{array}{c}{a}_{el}\equiv {(•)}_{el}\\ a_{el}^{+}\equiv {\left({\text{o}}^{+}\right)}_{el}\end{array}\begin{array}{c}{a}_{in}\equiv {\left(\text{o}\right)}_{in}\\ a_{in}^{+}\equiv {\left({•}^{+}\right)}_{in}\end{array}\right)\right\}\iff \left\{\left(\begin{array}{c}{a}_{el}\equiv {\left(\text{o}\right)}_{el}\\ a_{el}^{+}\equiv {\left({\widehat{•}}^{+}\right)}_{el}\end{array}\begin{array}{c}{a}_{in}\equiv {(•)}_{in}\\ a_{in}^{+}\equiv {\left({\text{o}}^{+}\right)}_{in}\end{array}\right)\right\}\\ {\text{B}}_{\text{1}}{\text{-Matrix B}}_{\text{2}}\text{-Matrix}\end{array}}$ (53)

where ((o⁺)_el, (o⁺)_in) are the creation operators of empty semi-quanta (with ε = 1/4 ηω), while ((•⁺)_el, (•⁺)_in) are the creation operators of full semi-quanta (ε = 1/2ηω). Instead, ((o)_el, (o)_in) are annihilation operator of empty semi-quanta, while ((•)_el, (•)_in) are the annihilation operator of full semi-quanta. Precisely we have:

1) ( ${•}_{in}^{+}$, ${•}_{el}^{+}$) are creation operators of sq(•) (that is each creates an amount of energy (ε = 1/2hν) or fill an sq(o) with the amount of energy (ε = 1/4hν), (o → •))

2) (•_in, •_el) are annihilation operators of sq(•) (that is annihilate an energy quantity of (ε = 1/2hν) or empty sq(•) of the energy quantity (ε = 1/4hν) making it so empty, (• → o))

3) ( $o_{in}^{+}$, $o_{el}^{+}$) are creation operators of empty sq(o) (that is annihilate an energy quantity of (ε = 1/2hν) or empty a sq(•) of the energy quantity (ε = 1/4hν), (• → o))

4) (o_el, o_in) should be annihilation operators of sq(o)

How are these last two operators interpreted? We remind you that in QM the annihilation op. (a) does not act in eigenstates $|n\rangle$ with n = 0. The same it happens to operators (o_el, o_in). In an eigenstate of (r) and (n), it will be (see Equation (25) and Equation (26)):

$\{\begin{array}{l}O{a}_{n,r^{\prime }}^{+}\left(t\right)\equiv a_{n,r^{\prime }}^{+}\left(t\right)={\left({•}^{+}\right)}_{el}^{\left(n\right)}\left[\exp \left(i{r}^{\prime }\omega t\right)\right]+{\left({\text{o}}^{+}\right)}_{in}^{\left(n\right)}\left[\exp \left(i\left(r^{\prime }\omega t-\pi /2\right)\right)\right]\\ O{a}_{n,r^{\prime }}\left(t\right)\equiv a_{n,r^{\prime }}\left(t\right)={\left(\text{o}\right)}_{el}^{\left(n\right)}\left[\exp \left(-i{r}^{\prime }\omega t\right)\right]+{(•)}_{in}^{\left(n\right)}\left[\exp \left(-i\left(r^{\prime }\omega t-\pi /2\right)\right)\right]\end{array}$ (54)

or equivalently

$\{\begin{array}{l}O{a}_{n,r^{\prime }}^{+}\left(t\right)\equiv a_{n,r^{\prime }}^{+}\left(t\right)={\left({\text{o}}^{+}\right)}_{el}^{\left(n\right)}\left[\exp \left(i{r}^{\prime }\omega t\right)\right]+{\left({•}^{+}\right)}_{in}^{\left(n\right)}\left[\exp \left(i\left(r^{\prime }\omega t-\pi /2\right)\right)\right]\\ O{a}_{n,r^{\prime }}\left(t\right)\equiv a_{n,r^{\prime }}\left(t\right)={(•)}_{el}^{\left(n\right)}\left[\exp \left(-i{r}^{\prime }\omega t\right)\right]+{\left(\text{o}\right)}_{in}^{\left(n\right)}\left[\exp \left(-i\left(r^{\prime }\omega t-\pi /2\right)\right)\right]\end{array}$ (55)

These mathematical relations give to us the mathematical form of the field oscillator called “IQuO”. In an IQuO at first level (n = 1) it will be:

$\{\begin{array}{l}{a}_{1,r^{\prime }}^{+}\left(t\right)={\left({•}^{+}\right)}_{el}\left[\exp \left(i{r}^{\prime }\omega t\right)\right]+{\left({\text{o}}^{+}\right)}_{in}\left[\exp \left(i\left(r^{\prime }\omega t-\pi /2\right)\right)\right]\\ a_{1,r^{\prime }}\left(t\right)={\left(\text{o}\right)}_{el}\left[\exp \left(-i{r}^{\prime }\omega t\right)\right]+{(•)}_{in}\left[\exp \left(-i\left(r^{\prime }\omega t-\pi /2\right)\right)\right]\end{array}$ (56)

With

$\boxed{\begin{array}{l}{\left(r\right)}^{\prime }=+1\Rightarrow \left[\begin{array}{c}\left[a_{el}={\text{o}}_{el}\left(\text{clockwise direction}\right)\right]\\ \left[a_{in}={•}_{in}\left(\text{clockwise direction}\right)\right]\end{array}\right]\\ {\left(r\right)}^{\prime }=+1\Rightarrow \left[\begin{array}{c}\left[a_{el}^{+}={•}_{el}^{+}\left(\text{anti-clockwise direction}\right)\right]\\ \left[a_{in}^{+}={\text{o}}_{in}^{+}\left(\text{anti-clockwise direction}\right)\right]\end{array}\right]\end{array}}$ (57a)

$\boxed{\begin{array}{l}{\left(r\right)}^{\prime }=-1\Rightarrow \left[\begin{array}{c}\left[a_{el}={\text{o}}_{el}\left(\text{anti-clockwise direction}\right)\right]\\ \left[a_{in}={•}_{in}\left(\text{anti-clockwise direction}\right)\right]\end{array}\right]\\ {\left(r\right)}^{\prime }=-1\Rightarrow \left[\begin{array}{c}\left[a_{el}^{+}={•}_{el}^{+}\left(\text{clockwise direction}\right)\right]\\ \left[a_{in}^{+}={\text{o}}_{in}^{+}\left(\text{clockwise direction}\right)\right]\end{array}\right]\end{array}}$ (57b)

Considering Figure 4 , we could use the following representation (see Equation (56)) about an IQuO with (r' = −1) and phase angle of (5π/4), see also the Equation (57b), in Figure 8 :

Note that by convention the positive angles increasing in anti-clock direction. This is the graphic representation of an IQuO, where (o_el, •_in) is the pair with anti-clockwise rotation of the phase, while ( ${\text{o}}_{in}^{+}$, ${•}_{el}^{+}$) is the couple with clockwise rotation. This representation could be useful to represent a particle moving along the X axis, seeing this as a quantum moving along a line of coupled field-oscillators. Considering (r' = ±1), we will have Figure 9 .

This is the graphic representation of an IQuO, where we will denote by [(o_el, •_in), ( ${\underset{\_}{\text{o}}}_{in}^{+}$, ${\underset{\_}{•}}_{el}^{+}$)] the pairs with anti-clockwise rotation of the phase, and by [(o_el, •_in), ( ${\text{o}}_{in}^{+}$, ${•}_{el}^{+}$)] the pairs with clockwise rotation. Looking Figure 6 and Figure 9 , and the Equation (56) with eigenvalue r' = +1, we set up the following representation, see Figure 10 .

Note that the two-component oscillator (forced oscillator of field) admits the presence of an internal degree of freedom, that is the phase rotation. This aspect is very important in the gauge fields’ theory because this internal degree of freedom allows us of determine the origin of the sign of the electric charge [6] [7] and that of color [11] [12] by sub-oscillators, see the first image in Figure 10 , correspondent to gluons. It is evident that an IQuO could not spontaneously change the r' eigenvalue, from r' = +1) to (r' = -1). We show (as an example) the first three possible configurations after those of the second image in Figure 9 , of the IQuO during the phase change, with (r' = +1), see Figure 11 .

In analogy with the classical oscillator, we project on the X-axis the motion of the quanta which represent the oscillator of Figure 11 ; we’ll have, Figure 12 .

Where the position of a “quantum” [(•,•) ≡ (•)] follows the law of probabilistic function |Ψ₁(x)|². We underline that in the $|r^{\prime }\rangle$ eigenstate (local state) the phase is uncertain (that is the configuration is indefinite), see the Sect. 2.1. The non-local state of the phase involves that the transition instant of a semi-quantum from a sub-oscillator to other one is uncertain, even if the configurations during an unobserved oscillation follow each other in an orderly way, see Figure 11 . Note, from Figure 8 , that the projections of the op. of semi-quanta (o, •) along the X axis give us the position of the semi-quanta, [sq(o, •)], along this axis and therefore the position of the quantum (•,•), respect to two sub-oscillators, see Figure 12 . In this figure, the upper image represents the oscillation of one quantum (•) moving along the X-axis, between the two extreme points (x₁, x₂); the lower image represents an oscillator with its two sub-oscillators and the sq pairs (o, •) moving (see the arrow) along the X axis between the two points (x₁,x₂). Note the sq are matched to their respective operators. Let us return to Figure 9 ; the image reads as follows:

1) the sq (el⁺(•)) moves in the +X direction with clockwise phase rotation; then, the ${(•)}_{el}^{+}$ operator works by creating a sq(•_el) in the +X direction

2) the sq (el(o)) moves in the +X direction with counterclockwise phase rotation; then the (o)_el operator works by annihilating a sq (o_el) in the +X direction, but this is equivalent to creating a sq(•_el) in the +X direction

These two aspects give us the following equivalences:

${\left[{(•)}_{el}^{+}\equiv {\left(\underset{\_}{o}\right)}_{el}\right]}_{+\text{X}},{\left[{(•)}_{in}^{+}\equiv {\left(\underset{\_}{o}\right)}_{in}\right]}_{+\text{X}}$

This implies the equivalence of the matrices, see Equation (54), and, that is, during the oscillation the full sq( •) change with empty sq(o) and vice versa: [(•)⌠(o)].

We define the Φ(B)-matrix associated with each configuration in Figure 9 , see also the Equation (56):

$\boxed{{\left[\Phi \left(\text{B}\right)\right]}_{\left(r^{\prime }=+1\right)}\equiv \left\{\left(\begin{array}{cc}{\left(a_{el}\right)}_{\left(cl\right)}& {\left(a_{in}\right)}_{\left(cl\right)}\\ {\left(a_{el}^{+}\right)}_{\left(\underset{\_}{cl}\right)}& {\left(a_{in}^{+}\right)}_{\left(\underset{\_}{cl}\right)}\end{array}\right)\iff \left[\left(\begin{array}{cc}{\left({\text{o}}_{el}\right)}_{(cl)}& {\left({•}_{in}\right)}_{\left(cl\right)}\\ {\left({•}_{el}^{+}\right)}_{\left(\underset{\_}{cl}\right)}& {\left({\text{o}}_{in}^{+}\right)}_{\left(\underset{\_}{cl}\right)}\end{array}\right)\equiv \left(\begin{array}{cc}{\left({•}_{el}\right)}_{\left(cl\right)}& {\left({\text{o}}_{in}\right)}_{\left(cl\right)}\\ {\left({\text{o}}_{el}^{+}\right)}_{\left(\underset{\_}{cl}\right)}& {\left({•}_{in}^{+}\right)}_{\left(\underset{\_}{cl}\right)}\end{array}\right)\right]\right\}}$(58a)

$\boxed{{\left[\Phi \left(\text{B}\right)\right]}_{\left(r^{\prime }=-1\right)}\equiv \left\{\left(\begin{array}{cc}{\left(a_{el}\right)}_{\left(\underset{\_}{cl}\right)}& {\left(a_{in}\right)}_{\left(\underset{\_}{cl}\right)}\\ {\left(a_{el}^{+}\right)}_{\left(cl\right)}& {\left(a_{in}^{+}\right)}_{\left(cl\right)}\end{array}\right)\iff \left[\left(\begin{array}{cc}{\left({\text{o}}_{el}\right)}_{\left(\underset{\_}{cl}\right)}& {\left({•}_{in}\right)}_{\left(\underset{\_}{cl}\right)}\\ {\left({•}_{el}^{+}\right)}_{(cl)}& {\left({\text{o}}_{in}^{+}\right)}_{\left(cl\right)}\end{array}\right)\equiv \left(\begin{array}{cc}{\left({•}_{el}\right)}_{\left(\underset{\_}{cl}\right)}& {\left({\text{o}}_{in}\right)}_{\left(\underset{\_}{cl}\right)}\\ {\left({\text{o}}_{el}^{+}\right)}_{\left(cl\right)}& {\left({•}_{in}^{+}\right)}_{\left(cl\right)}\end{array}\right)\right]\right\}}$ (58b)