In Relativity and Quantum Mechanics (QM), one has for a free massive particle:

where E(energy) ≡ H and H is the Hamiltonian [4]. Applying to wave function Ψ, one obtains the Klein-Gordon equation (Σ_KG):

The covariant form of Σ_KG is:

With ε the energy of a particle. Besides, it is:

In the QM theory [4], it is possible doubles the solution Ψ of Equation (2) in two components: Ψ ≡ (ϕ, χ). A wave function with two components is called Spinor. Exactly, one sets:

By Equation (5), the Σ_KG can be written as:

The (2 × 2) matrices operating in Ψ-spinor are combinations of the Pauli matrices:

By making H_f explicit, Equation (6), we obtain the “dualequation” of Σ_KG by of Pauli matrices:

With solution of plate wave given by:

From continuity equation, Equation (4), derives the normalization of function:

Here, there is the first problem of Σ_KG given by ρdensitywith negative values; to avoid negative values, we can introduce a λ_ρ -parameter in ρ,which becomes ρ_λ = λ_ρρ such that is:

Note that the ρ_λ values are always positive. In literature, the λ_ρ parameter is given as electric charge e and ρ_e = λ_ρρ becomes the “electric charge density”. This solution (λ ⌠ ±e) is only phenomenology because does not investigate the origin of the electric charge (e) and its signs (±).The second problem of Σ_KG is given by solutions of the Equation (8) with energy negative, see the matrix σ₃, which has eigenvalues (±1) and, thus, it is: [±mc² → σ₃(mc²)]. Replacing the solution (9) in Equation (8), we obtain a system of two equations:

where (ε = ±E_p); the solutions are:

where the subscript (+) points out the solution with positive energies (ε = +E_p), while the subscript (−) points out the solution with negative energies (ε = −E_p). One obtains two conditions of normalization:

Here, we find again the negative value of the ρ density. Therefore, the problems of the presence of negative energy and negative density are correlated. The solutions of the dual equation, Equation (8), are:

where (Ψ₋) are solutions of Σ_KG associated to the negative values of energy. If we point out by Σ KG ∗ the conjugated equation of Σ_KG, to the conjugated solution Ψ^* of Σ KG ∗ , one can associate the “conjugated” wave functions Ψ_c = (σ₁Ψ^*), which could describe the “π_cconjugated particles” to those of Σ_KG. Recall the π-pion where π⁻ is the conjugated particle of π⁺. We consider a solution π_c with positive energy; by Equation (7), Equation (12) and Equation (14) one obtains:

where we have omitted the normalization volume V and applied the transformation (p→−p). Note the correlation (Ψ₊)_c = (Ψ₋)_(p→−p). It follows that the negative solution (Ψ₋) of Σ_KG can describe a conjugate particle π_c with positive energy (Ψ₊)_c and moment p (opposite to that −p of π particle of Σ_KG) only if we give physical meaning to the conjugate equation Σ KG ∗ . Remember the existence of two pions (π⁺, π⁻) in which one is conjugate to the other. In this way, it is also: (Ψ₋) = [(Ψ₊)_c]_(p→−p). In fact, we can have:

So, we highlight that the negative solution Ψ₍₋₎ with moment (p) are coincident with Ψ_+(c) (with positive energy) of (Σ_c)_KG but with moment (−p). We also have that: (Ψ₋)_c = (Ψ₊)_(p→−p). Therefore, the negative solutions Ψ₍₋₎ disappear only if one considers as “complete” the system of the two equations (Σ_KG, (Σ_c)_KG) and not one of the two single equations. We could expect that the two equations ((Σ_c)_KG, (Σ)_KG) describe two identical particles except for a “physical aspect” linked to the conjugation operation and transformation (p→−p). This physical aspect must be a physical quantity or descriptive physical characteristic. Remember the operation of conjugation changes the sign (±) of the exponential part of a wave function, and thus of E_p in Equation (12), therefore, the conjugate particles are different from the particles “ordinary” (omitting the moving in direction opposed) in a particular physical characteristic λ_ε that changes of sign (±) in the conjugate solutions. Recall the parameter λ_ρ associated with density ρ, see the Equation (11), which has two “eigenvalues” [ ( λ _ ′ ρ = ± 1 ) ] : we suspect λ_ρ ≡ λ_ε because the energy and density are correlated, see the Equation (13). We point out by [ ( λ _ ′ ε = ± 1 ) ] the sign (±) of two values of λ_ε; then, we admit the following correlations:

In this way, see the Equation (17), a solution of the Equation (8), describing a particle at “apparent” negative energy (ε = −E_p), is detected as a particle with positive energy [ ( λ _ ′ ε ) ε − = + E p > 0 ] but with the sign (−) of the parameter λ_ε, that is ( λ _ ′ ε = − 1 ) . To have this possibility, and so to eliminate the negative energies, it would need to redefine the energy in such a way that a possible negative value of this is attributable to the negative value of the parameter λ_ε, that is ( λ _ ′ ε = − 1 ) . This aspect can push us toward an alternative road to that travelled by Dirac to resolve the presence of the negative solutions of his equation. Then, we will state that the two equations (Σ_KG, Σ KG ∗ ) in “not separate” form constitute a complete physical system if the negative solutions Ψ₍₋₎ of Σ_KG (electrons with negative energy) coincide (that is physically are not distinguishable) with the positive solutions Ψ ( + ) ∗ of Σ^* (positrons with positive energy), while those negative Ψ ( − ) ∗ of Σ^* (that is positrons with negative energy) coincide with the positive solutions Ψ₍₊₎ of Σ (that is electrons with positive energy). This would seem a “contradiction” because there would be particles with negative energy detected as particles with positive energy: this contradiction can be eliminated by the parameter λ_ε that helps us to distinguish the negative solution from that positive and to have apparent negative solutions “without negative energy” thanks to the conjugation of the system of equations (Σ_KG, Σ KG ∗ ). But this is possible “only and only if” the negative value of energy is attributable to the negative value of the parameter λ_ε, ( λ _ ′ ε = − 1 ) , see the Equation (17). We ask then us how this parameter transforms the energy solutions negative in energy solutions positive.

Also, in the Dirac equation (Σ_D) is present the α_β matrix with the same form of the Pauli matrix σ₃, which is responsible of presence of the negative solutions [5]:

where (α_i, α_β) are matrices (4x4) built by the Pauli’s matrices (σ, 0, 1):

In fact, Dirac wrote “… our wave equation has a double number of solutions(Ψ₋,Ψ₊)_ε … in correspondence with them there are half of states with negative energy… these cannot be denied but cannot also be accepted”.After he adds: “Thesolutions corresponding tonegative values ofp₀don't correspond to the observable motions of the electron”. This would mean that the research of a solution of the negative energy problem is not in the electron motion but in an intrinsic characteristic of it, which is connected to the mathematical conjugation of two solutions (Ψ₋, Ψ₊) of Σ_D, as it happens in two equations (Σ_KG, Σ KG ∗ ), see in sect. 2.1. The term conjugated refers to the conjugate wave functions, solutions of the Dirac equation (Σ_D) and its conjugate ( Σ D ∗ ). To accept the solutions (Ψ_D, Ψ D ∗ ) as “reals”, an external parameter is admitted, historically interpreted as the electric charge e into the physical system (e, A), with (A) the 4-vector of the Electromagnetic Field (EMF). In fact, Dirac wrote “…the solution with negative energy(ofΣ(e,A)_D) is the conjugate to a solution with positive energy ofΣ^*(e,A)_Dcorrespondentto an electron with charge+e”. For symmetry is also true that “The conjugate images of the negative solutions ofΣ^*(e,A)_Dare in correspondence to particles with positive energy and electric charge(−e), that is electrons”. Here, the terms [Σ(e,A)_D, Σ^*(e,A)_D] are the two Dirac equations of an electron submitted to the action of an EMF (A):

Note that if Σ_D has physical meaning, then also Σ D ∗ must have a physical meaning; it follows that the Ψ^* describes a real particle with positive energy and electric charge (+e), that is the positron. We note that in Dirac, unlike the KG equation, the EMF appears correlated to solutions (Ψ₋, Ψ₊) and electric charge (±e). Recall the parameter λ in sect. 2.1 to the Equation (17), which allows us to treat the negative solution Ψ₍₋₎ connected to conjugate Ψ_+(c),see the Equation (16), as a particle with positive energy. Therefore, we need to have two equations (Σ_D, Σ D ∗ ) for describe the physics symmetry matter-antimatter: an individual Dirac equation represents a physical system, which is incomplete.This aspect elicited perplexity in Dirac: “We cannot affirm that solutions(Ψ_D)₋with negative energy represent positrons, because this would make the dynamic relations wrong(in presence of an EMF A_μ)…therefore, it is not true that a positron has a negative kinetic energy”.This perplexity pushed Dirac to introduce a background into an universe (the “Sea of Dirac”) where “almost all the states to negative energy are everyone busies by electrons (the negative solutions of Σ_D), while the no busies remainders ((Ψ_D)₋)_empties, (said gaps), will appear of positive energy…because to fill them one needs to add an electron (to negative energy (Ψ_D)₋)… we will suppose that(the gaps)are positrons…”.

Even if this road leads to the hypothesis of the quantum vacuum with its virtual processes of pair creation and annihilation, the interpretation of the positron as an empty negative gap leaves us perplexed and would not result, to my knowledge, a decisive road of the problem of having a negative energy in the theory of particles. We notice that the negative energy inevitably conducts to “paradoxes” (if not contradictions) what “negative mass” [6], “antigravity” and other [7], until reaching the procedure of “rinormalizzation” of the mass [8][9], postulating a negative inertial mass to infinite value that would balance an infinite electromagnetic mass (positive), to give a finite physical mass, (m_in(−∞) + m_em(+∞) = m_fis): clearly, this aspect is not correct mathematically. Note in ref. [10][11], instead, one shows “the mass asadditional couplingbetween oscillators of a massless scalar field”: this aspect leads us to argue that “mass can only be an observable with always positive and nevernegative values, because the“proper”mass(m₀)is the expression of the proper frequencyω₀of oscillations(ω₀ > 0 always) connected to the additional coupling:(m₀=hω₀/c²). If we want to avoid of follows the path of the “Dirac sea”, it is necessary to reconsider the system of the two equations Σ ≡ ( Σ D , Σ D ∗ ) , attempting to complete the Dirac’s theory through a representation in which the solutions, positive (Ψ₊) and “apparently” negative (Ψ₋), are different aspects of an unique physical system that don’t admits negative energy. This means that the particles of Σ_D and the particles conjugated of Σ D ∗ have to be two different aspects of a same particle with positive energy, individualized by two different “values” of a “label” physical greatness, in which one of these two values makes as “apparent” the negative energy of the solutions (Ψ₋) and turns it instead into a “positive” solution. We will show that the parameter λ_ε is just the electric charge (e), but this, in turn, for it not to be an external label, must be connected to an internal characteristic (r) to a “particle-system”, which allows of phagocyte the negative solutions, that is: (λ ⌠ e⌠ r). The completeness is reached thanks to the parameter (λ ⌠ r), which turns the negative solutions into positive solutions.

Remember, in sect. 2.1 and in Equation (17), that the negative value of energy is attributable to the negative value of the parameter λ_ε, ( λ _ ′ ε = − 1 ) . We can add that the passage to two “eigenvalues” [ ( λ _ ′ ε = − 1 ) , ( λ _ ″ ε = + 1 ) ] is connected to the conjugation operation of a wave function, see the relation between the two solutions (Ψ₊, Ψ₋) and ((Ψ₊)_c, (Ψ₋)_c) in the Equation (15) and Equation (16); it follows then that the parameterλ_εmathematically concerns the phaseϕof the exponential function related to the wave function and exactly the two signs(±)of the phaseϕ, that is [(±ϕ) ⌠(±λ)]. Note that the sign (±) can be correlated to the directions, clockwise and counterclockwise, of phase rotation. Then, we can represent these two directions by a parameter with two values, which could coincide with the parameter λ_ε. Recall in QM the following equations:

And the relativistic relation of energy:

The plain wave solution of KG equation is:

where (ε = ±E_p); here, one connects the energy ±E_p to the angular frequency ω, see the Equation (21). Then, one can introduce a parameter λ in the exponent of wave solutions of KG equation, exactly in the wave phase [4]:

By ε = λE_p, we will have:

Note if λ = −1 one has negative energy but also negative frequency. This parameter is in relation to (λ_ε ⌠ λ_ρ); recalling Equation (11) and Equation (17), with [(λ_ε ≡ λ_ρ) = λ = λe)], then we’ll have:

We ask ourselves because the parameter e is just the electric charge, as in literature. Let’s not forget that the phase transformations (Δϕ) on the field oscillations are connected to the gauge invariance of interactions. Let us recall, in fact, the generator g of the transformation U(Δϕ) = exp(igΔϕ) to which a constant variable Q is connected. As is well known, Q is the electric charge (±e), associated with the transformations on the phase ϕ. By relation (ε_k =hω_k), we could connect λ to the sign ± of the ω frequency as well as the negative values of energy; it follows that we cannot speak anymore of negative values of energy but of negative values of frequency. So, the problem of negative solutions moves toward the presence of negative values of the frequency ω associated with the phase ϕ of the oscillation, where it is (ϕ =ωt). As seen in Equation (26), if one detects the sign ± of the electric charge (±e), it is as if one detects the sign (±) of the parameter λ associated with the conjugation of the wave function, that is, the sign (±) associated with the phase ϕ of oscillation of the wave. In an undulatory way, a physical observer (O_γ) which “observes” a particle would read negative values of the energy (negative solutions) as negative values of the ω angular frequency and, thus, a negative value of the parameter λ, that is λ = −e. It also appears clear that the solutions with positive energy (ε > 0), that is, with positive frequency (ω > 0), will be described by a parameter λ = +e. In conclusion, the wave equations of KG and Dirac would describe particles characterized by the two values ((±e)) of the parameter λ, associated with two values of the frequency (±ω) and phase (±ϕ). Our physical observer (O_γ) it so must have the ability to detect the negative values and positive values of the angular frequency (that is, the energy) and to distinguish the positive signs (ω > 0) from those negative (ω < 0). From the relation ϕ(t) = ϕ(t₀) ± ωt, one infers that a negative frequency (ω < 0) corresponds to a phase rotation opposite to that associated with the positive frequency (ω > 0). For mathematical convention, we can associate the negative frequency with a clockwise rotation (cl) of the phase, while the positive frequency with an anticlockwise rotation (cl) of the phase. Therefore, an observerO_γ could see the negative values of energy or negative frequencies of oscillations as a clockwise rotation of phase. Having set (λ = λe), the physical parameter λ would then have two “eigenvalues”:

[(λ' = +e,with (λ = +1), counterclockwise rotation direction (cl)]

[(λ' = −e,with (λ = −1), clockwise rotation direction (cl)]

In this context, recalling the phenomenology of particles, we could then say, regarding the Matter-Antimatter theme, that it is described, in a complete manner, by the Dirac and KG equations (Σ) and by the conjugate equations (Σ^*), where the positive-energy solutions of Σ, that is (Ψ_+ε)_Σ,, describe electrons with electric charge (−e) and clockwise rotation (cl) of the ϕ, phase, while the negative-energy solutions (Ψ_−ε)_Σ correspond to positive-energy solutions (Ψ_+ε)_Σ* of the conjugate equations (Σ^*) but with electric charge (+e) and counterclockwise rotation (cl) of the ϕ phase. Symmetrically, the negative-energy solutions (Ψ_−ε)_Σ* of the conjugate equations (Σ*) are nothing other than electrons (!). Therefore, we have:

This scheme solves the problem of having negative energies (negative frequencies). We could observe that if there were a physical object that read the direction of phase rotation (value of the parameter λ) of the oscillation associated with a particle, it could be asserted that the negative frequency would be seen as the electric charge of a particle with positive energy: the solutions at negative energies would consequently disappear. This is coherent to the definition given of the proper massm₀, see the sect. 2.2, connected to proper frequency ω₀, of the additional coupling. It remains only to understand how the photon (O_γ) can read the direction of phase rotation and determine the phenomenology of electrical interaction where electric charges of the same sign repel each other while those of opposite sign attract each other [3].

If in the oscillations of a pendulum it is not possible to distinguish a direction of the phase rotation, instead, one could think that in a quantum oscillator this is possible. In a one-dimensional oscillator with its known descriptive degrees of freedom (amplitude and frequency), however, it is not possible to know the direction of phase rotation unless anadditional internal degree of freedom(fd)related to the phaseϕis added.The consequences of an additional degree of freedom internal to a quantum oscillator are significant. The new degree of freedom (fd) might be be related to the parameter λ and therefore to the two signs (±) of the electric charge e: (fd ⌠λ⌠e). Since the eigenvalues of λ are given by (±e) and are also connected to the direction of rotation of the phase (cl,cl), then it follows that the new fd could coincide with the direction (cl,cl) of the phase rotation. This last aspect allows us to represent a quantum oscillator in a two-dimensional way [1]. Therefore, the hypothesis of correlating the sign (±e) to the direction of phase rotation (cl, cl) of a wave can be proven true only if we provide a “2-dimensional” representation of the quantum oscillator where it is possible to identify the direction of the phase rotation. Nevertheless, to avoid falling into a vicious circle (the fd connected to the direction of phase rotation makes the oscillator 2-dimensional, or if an oscillator is 2-dimensional, then it has an additional fd given by the direction of phase rotation), we need to find another reason to induce us to represent a quantum oscillator in 2-dimensional terms. This reason exists. If we study an isolated oscillator, its 1-dim. character would be obvious, but if we consider a field oscillator instead, things change, because the latter is a forced oscillator: a quantum oscillator O_i along a field line, coupled to two lateral field oscillators (O_i-1, O_i+1), is “forced” by the adjacent oscillators when it exchanges energy with them. A forced oscillator is described in its oscillations by “Absorptive” and “Elastic” components, as prescribed by the theory of forced oscillations [12]. In the representation of operators of a quantum oscillator (a, a⁺), creation and annihilation operators, it is [1]:

Here, the subscript in indicates the absorptive or “inertial” component to distinguish it from the elastic component with the subscript el. The theory of forced oscillations tells us that the absorptive and elastic components are out of phase by π/2; we have:

Here, we can highlight two different directions of rotation of the phase of the two vectors (Oa, Oa⁺) representing the operators (a, a⁺), to which corresponds to the following geometric representation in the plane (q,ip), see Figure 1:

Figure 1. Representation of operators-vectors (a, a⁺) with two components in phase plain.

The phase of ωt = (3/4)π is reported to the elastic component with counterclockwise direction of rotation of adjoint operator a e l + . Note the “intrinsic2-dimensionalrepresentation”.A field quantum oscillator could thus be expressed by its intrinsic internal 2-dimensionality. At this point, the two directions of phase rotation (clockwise and counterclockwise) can be represented within a field quantum oscillator, see the Equation (28) and Figure 1.

We call IQuO the quantum oscillator of field: IQuO is the acronym for Intrinsic Quantum Oscillator (for mathematical proofs of IQuO, see ref. [1]-[3]). Recall that in quantum field theory, a field can be represented as a set of “lines” of mutually coupled oscillators along which oscillations (waves) propagate; see Huygens' theorem applied to a particle-field [12]. Without going into too much detail (for further mathematical and physical details, see ref. [1]), it is important to consider that an IQuO oscillator I_k couples laterally to the right and left with the other two adjacent oscillators (I₁,I₂); for this reason, it can be represented as having at least two parts (left and right parts), which we will refer to as two “sub-oscillators”. If an oscillator is composed of two parts (or two sub-oscillators), it is intuitive to assume that it must have its own spatial extension. Recall that a quantum oscillator [4][13] has a relationship that defines its oscillation amplitude A_n along an X axis in the oscillation eigenstate |n〉: A_n = (l₀/2)(2n+1)^1/2. Here, l₀ would be the elongation in the “vacuum” state, that is (n= 0). Therefore, we are led to assume that an IQuO has its own spatial extension and, thus, is “not point-like”. Note, these two sub-oscillators allow the oscillator to couple to the others sub-oscillators. We note that a coupling of two oscillators, to occur, must implement an exchange of energy between them and therefore between their respective sub-oscillators. It is therefore clear that it is not possible to have a coupling of I_k with the two lateral oscillators if we consider the energy of an oscillator expressed by a single and whole quantum. Therefore, in this case, it is necessary to distribute the quantum of energy (in the case n= 1) in the two component sub-oscillators, that is, it is necessary to split the quantum of energy into two “half-quanta”. The field IQuO, therefore, will turn out to be a 2-dim. quantum oscillator with sub-oscillators with half-quanta. This split is suggested by the representation of the probability function P(x) of a quantum oscillator in the Schrödinger treatment, P(x) = |Ψ(x)|², see Figure 2 [1]:

Figure 2. Probability function of quantum oscillator to varying of n quantum number.

The graph shows for (n= 1) two curves symmetrical with respect to the vertical axis (Y). If we assume that the graph is timeless and that each curve represents the probability zone where a quantum (1h) can be found, then it can be rationally thought that there are two oscillation zones in a quantum oscillator in the state (n= 1). If we identify these two oscillation zones with two sub-oscillators into an IQuO, always present, then we can unequivocally mean that the quantum (1h) must be divided into two half-quanta (1/2h). However, the graph of P(x) speaks clearly: in the two parts, the “quantum” must be revealed in its entirety in 50% of the observation cases and therefore, even admitting the existence of half-quanta (semi-quanta), we must have this possibility [1]. This possibility happens in an IQuO because the coupling between two adjacent IQuO of the field implies the overlapping of two their respective lateral sub-oscillators: in a superposition zone, two semi-quanta are equal to one quantum. In the remaining 50% of the cases, we do not have a quantum (1h) in each sub-oscillator but at least a half-quantum or “semi-quantum” (1/2h) which, obviously, cannot be revealed. We must admit this possibility. Note that a sub-oscillator without energy (kinetic energy and elastic potential energy) or without semi-quanta has no physical meaning. This implies that in the two sub-oscillators, the semi-quanta come and go if the following relationship is always valid in an IQuO:

where nis the number of integer quanta (1h). We should point out that the graphs in Figure 2 concern a single “isolated” quantum oscillator. The vacuum state of an isolated oscillator in (n= 0) is represented by a single curve centred on the Y-axis, see Figure 2(a), with a total energy of (1/2h). However, in a field line, the quantum oscillators (IQuO) are all “elastically” coupled to each other and, therefore, are not isolated oscillators and, thus, always consist of two sub-oscillators even in a field in a vacuum state. In an IQuO of a field in the vacuum state (n= 0) with two sub-oscillators and with a total energy of (1/2h), we must admit that in each sub-oscillator there is a half semi-quantum (1/4h), defined as “empty semi-quantum”. Let us therefore admit the existence into an IQuO of an empty semi-quantum, denoted by sq(o), with energy ε = (1/4h)ω, distinct from the “full semi-quantum”, sq(•), with energy ε = 1/2h)ω. The empty semi-quanta allow us to have an IQuO in state (n = 1) with a sub-oscillator having a semi-quantum at the 50% of probability. Returning to Figure 1, in which there are the representations of the operators (a, a⁺) with their double components, see [ ( a e l , a e l + ) , ( a i n , a i n + ) ] , and taking into account that we have full semi-quanta sq(•) and empty sq(o), we note that each of two sub-oscillators of the IQuO_(n=1) contains a pair of sq(o, •) which move following the oscillation vectors (Oa,Oa⁺) and the corresponding components [ ( a e l , a e l + ) , ( a i n , a i n + ) ] . Therefore, we could assign to each component of the operators (a, a⁺) a “semi-quantum operator” sq(o) or sq(•) with their respective elastic and inertial characteristics:

We will obtain, see Figure 2(b), the probabilistic representation of the IQuO at a given time, see Figure 3:

Figure 3. The probabilistic representation of the semi-quantum oscillator.

Precisely, it is:

1) ( • i n + , • e l + ) are creation operators of sq(•) (that is each creates an amount of energy (ε = 1/2hν ) or fill an sq(o) with energy (ε = 1/4hν)), that is the full sq(•)_x moves toward + X-axis.

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

3) ( o i n + , o e l + ) are creation operators of empty sq(o) (that is each annihilates an energy quantity of ε).

4) ( o _el, o _in) should be annihilation operators of sq(o).

Note in Figure 1 that the projections of the operators [ ( a e l , a e l + ) , ( a i n , a i n + ) ] along the X-axis give us the position of the semi-quanta (o, •) along this axis. In summary, we can represent an IQuO, see Figure 4:

Figure 4. The oscillation of a “quantum” along X-axis.

In this figure, the sq-operators have rotations in the opposite direction to those of Figure 1. Here, we note the “spatial extension” of the IQuO along the X-axis, and that in the apparently empty area, there are instead “semi-quanta” of energy available for couplings, that is, semi-quanta belonging to the adjacent oscillator along the same field line, as we will see later. Then the distribution of the semi-quanta in the various sub-oscillators of an only IQuO at level nwill be, see relation Equation (29):

The subscript numbers indicate the sub-oscillators; always see Figure 1. The various Ψ' are excited states. From the configurations of Equation (30), it can be deduced that to achieve internal coupling between field oscillators, it is sufficient for the extreme sub-oscillators of the two contiguous IQuO to exchange at least one full sq(•). It should be noted, however, that when two oscillators, each belonging to a different field, couple to describe the interaction between particles (dynamic phase), then they can only exchange one full quantum [(•) ≡ (•), (•)], that is, two full sq(•).

Looking the Figure 1 and Figure 4, we could use the representation of Equation (30) about an IQuO with eigenvalue of direction of phase rotation (λ' = −1 ⌠ cl) and phase angle of (5π/4):

Figure 5. IQuO with eigenvalue r' = −1.

Here, we can note that the clockwise direction of operator pair [ ( • e l + ) c l , ( o i n + ) c l ] + k indicates the movement of sq (•,o) toward the +X-axis; the same indication one has in the pair [ ( o e l ) c _ l _ , ( • i n ) c _ l _ ] + k ; from the scheme of pag. 14 (Equation (30)), one can see that the two operators [ ( • e l + ) c l ] + k ⌠ [ ( o e l ) c _ l _ ] + k are coherent and in action concordance. The same can be said about pairs [ ( o e l + ) c l ] − k and [ ( • i n ) c l ] − k .

The two movements in opposite directions (+k, −k) of two pairs are coherent with the characteristic of a forced oscillator of an IQuO, where the index (in) points out the absorbitive component of the forced oscillator (see Equation (27)), while the index (el) points out the elastic component. Recall the antagonistic behavior of two components elastic and absorbitive. In an no-isolated IQuO that is in a field line of coupling IQuOs for having the movement of quantum (•,•) along the line in the +X direction (+k) we must have another configuration:

In matrix form [1][3], for an “isolated” IQuO, see Figure 5, and for an IQuO of field, we will have:

Here, we have omitted the indices (cl,cl) because we have set the counter-clockwise operators as • c ̱ l ̱ , o c ̱ l ̱ • ̱ , o ̱ . Note, in Figure 5, the semi-quanta are both empty and full; then we have the following possibility for a field IQuO:

{ [ ( o e l ) c l , ( • _ e l + ) c _ l _ ] 1 , [ ( o i n ) c l , ( • _ i n + ) c _ l _ ] 2 } Field IQuO

o ̱ e l c ̱ l ̱ , • e l + c l 1 , o ̱ i n c ̱ l ̱ , • i n + c l 2 Field IQuO

where the subscript cl indicates the clockwise direction of the phase rotation, while the subscript cl indicates the counterclockwise direction. Here, we have a field IQuO with both directions of phase rotation in operators aand a⁺. If we have connected a sign of the electric charge to the direction of the phase rotation, then this type of IQuO is electrically “neutral”. In the IQuO representation, a photon appears as a quantum (•, •) that propagates along a field line (in this case, an electromagnetic field) made up entirely of “neutral” IQuO in the direction of phase rotation. An IQuO with both directions of phase rotation, and “neutral” in the direction of phase rotation, will be called type B. If we consider electrically charged particles, such as an electron or a positron, we will instead have all operators of sq with the same direction of the phase rotation (mono-verse IQuO), that is an IQuO type F.The positron has an electric charge (+e), so all the sq must rotate in the same direction, namely, counterclockwise (cl). The electron has an electric charge (−e), so all the sq must rotate in the same direction, namely, clockwise (cl). The following configurations are thus found, see Figure 6:

(a) (b)

Figure 6. Two “mono-verse” IQuO in regular shape with two opposite rotations of the phase.

Also, we can consider, as for B-IQuO, a sq-structure given by:

o ̱ e l c ̱ l ̱ , • ̱ e l + c ̱ l ̱ 1 , o ̱ i n c ̱ l ̱ , • ̱ i n + c ̱ l ̱ 2 Positron

{ [ ( o e l ) c l , ( • e l + ) c l ] 1 , [ ( o i n ) c l , ( • i n + ) c l ] 2 } Electron

The representation of the two types of F-IQuO in matrix form will be:

(33)

Here, in Figure 6(a), we can note that the clockwise direction of operators pair [ ( • e l + ) c l , ( o i n + ) c l ] + k indicates the movement of sq (•, o) toward the +X-axis; instead, the pair [ ( o e l ) c l , ( • i n ) c l ] − k indicates the movement of sq (•, o) toward the −X-axis; from scheme of Equation (30), one can see that the two operators [ ( • e l + ) c l ] + k ⌠ [ ( o e l ) c l ] − k are not coherent and, thus, are in action discordance; the same happens in the pair [ ( o i n + ) c l ] − k ⌠ [ ( • i n ) c l ] − k . This means that the following couplings are not compatible:

[ o p ( • e l + ) − k , o p ( • i n + ) − k ] c l ⌠ [ o p ( o e l ) + k , o p ( o i n ) + k ] c l

[ o p ( • e l + ) − k , o p ( • i n + ) − k ] c _ l _ ⌠ [ o p ( o e l ) + k , o p ( o i n ) + k ] c _ l _

where the sign (⌠) points out the pairs not compatible. If we admit the possibility given by Figure 6, then there must be a “possibility” of coupling the pairs without running into discordance. Since the oscillations of non-scalar fields such as vector and tensor fields, and hence the oscillations of the corresponding IQuOs, take place on multiple oscillation planes, the planes could be combined in such a way as to avoid incompatible couplings. This implies that the oscillation plane of the pair [ o p ( • e l + ) − k , o p ( • i n + ) − k ] z with the oscillation along the Z-axis is different from that of the pair [ o p ( o e l ) + k , o p ( o i n ) + k ] Y which oscillates along the Y-axis. We can have: { [ o p ( • e l + ) − k , o p ( • i n + ) − k ] z , [ o p ( o e l ) + k , o p ( o i n ) + k ] y } . That is:

where the moment k is along the X-axis, that is the propagation axis. However, we must admit that it is possible that the two pairs [ ( a e l + , a i n + ) , ( a e l , a i n ) ] of the Equation (31) can couple to create an IQuO with the pairs of annihilation and creation operators with the same phase rotation, see Equation (33). Then the coupling occurs following an exchange of sq(•), which can occur in overlapping or interpenetrating sub-oscillators.

Another relevant note is due: we point out that the phase is not an “observable”, while the direction of phase rotation may be observable, because its two eigenvalues λ' can be detected as signs of the that we point out as electric charge (±e).

The ability to detect the two directions of phase rotation in charged particles is possible through experimental observation using a probe photon. Therefore, we cannot associate an operator with the phase: in ref. [4] it is shown that only the exponential form of phase can be assumed in the role of a “functional operator” (Φ ≡ e^iϕ) [4]. It follows then that we can only associate two eigenstates |λ'〉, relating to the two eigenvalues of parameter λ' inserted in the functional (Φ ≡ e^iλϕ); however, this is possible only for fields with IQuO of F-type, see the pair creation where we can observe these two eigenvalues.

In a non-isolated IQuO, that is in a field line of coupling IQuOs, for having the movement of quantum (•, •) along the line in the +X direction (+k), one must admit the following configuration: { [ ( • e l + ) c l , ( • i n + ) c l ] + k , [ ( o e l ) c _ l _ , ( o i n ) c _ l _ ] + k } .

Recall that the quantum field Ψ is a set of coupled quantum oscillators expressed by operators (a, a⁺). The unique possibility to insert the 2-dimensional geometric aspect of an IQuO is in the component no time ( a 0 , a 0 + ) , which will be expressed by a matrix Ψ(t₀) ≡ Ψ₀ with elements describing the structure of a B-IQuO, see the Equation (32):