To ascertain a mathematical model capable of generating m, the introduction of a function termed the Mass Generator (represented as Gm) becomes imperative. The primary objective of this function is to accept the parameters inherent to m, effect modifications upon them, and subsequently yield the resultant m. A schematic depiction of this process is presented below:

$\left(x_i\right)\to Gm\left(x_i\right)\to m\left(x_i\right)$ (1)

where $x_i$ is a quantity that is used as input to the Gm operator and $m\left(x_i\right)$ is the output of the Gm, i.e. the m that is produced and is related to the initial quantity $x_i$.

Equation (1) points out that mass is a product quantity, thus other quantities of physics must be assumed as fundamental in order to be inserted in the Gm and interact for the formation of mass. As a guidance, it is observed that mass is linked to energy, e.g. when mass moves, it is assigned kinetic energy. So, there must be a relation between these quantities, i.e. $m=c_m\cdot \epsilon$. Further, two basic definitions of physics are used so that the quantities can be assigned through their measuring units. These definitions are the definition of Joule and that of Newton.

“1 Joule is the quantity of energy produced if a force of 1 Newton displaces its center for 1 meter”.

While the definition of Newton (Nt) is:

“1 Newton is the force that is produced of a mass of 1 kg is accelerated by 1 meter by square second”.

Based on the analysis so far, the concepts that most match with the observation mentioned here above, basis their units, and can be assumed as the fundamental quantities that can be used as inputs to form the mass, are the time (t) the energy (ε) and the space ( $\stackrel{\rightharpoonup }{s}$). Then, the introduction of the fundamental concepts in the Mass Generator presents a schematic approach as follows:

$\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right)\to Gm\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right)\to m\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right)$ (2)

Considering that the concepts interact, probably within an entity {O}, the Mass Generator is:

$Gm\left(e,\stackrel{\rightharpoonup }{s},t\right)=g\left(m\right)$x $g\left(t\right)$x $g\left(\epsilon \right)$ (3)

In which relation, the functions $g(\ast )$ are referred to as the concepts generators and the symbol (x) denotes the operator of interaction of concepts. The generators are the functions that describe the manner in which each concept completes and interacts with other concepts for the formation of m within the entity {O}.

The examination of m is conducted within the domain of complex numbers, with particular emphasis on the exponential expression. This choice is predicated upon the expansive nature of the complex number set, encompassing a vast array of solutions. Utilising this comprehensive set ensures the inclusion of all potential outcomes. Conversely, restricting the analysis to a smaller field, such as the real numbers, may result in insufficient information and an incomplete depiction of the phenomenon under investigation. Besides, even if the wider field, i.e. $ℂ$, initially contains more terms than necessary, the mathematical procedure will guide towards removing the unnecessary ones.

The exponential expression of a complex number is expressed as $z\left(\vartheta \right)$ and is expressed as $z\left(\vartheta \right)=z_0{\text{e}}^{i\theta }$ [1] - [3] , in which $z_0\ge 0,\in ℝ$ and $\vartheta \in ℝ$ is described in (rad). For the purposes of studying the generator, let us use the exponential expression:

$g\left(\stackrel{\rightharpoonup }{y}\right)=g_0{\text{e}}^{i\vartheta \left(\stackrel{\rightharpoonup }{y}\right)}$, (4)

so that $g\left(\stackrel{\rightharpoonup }{y}\right):{ℝ}^n\to ℂ,n\in \mathbb{N}$ wherein $g_0$ is the magnitude of the generator and $\vartheta \left(\stackrel{\rightharpoonup }{y}\right)$ denotes the argument that is affected by the concept $\stackrel{\rightharpoonup }{y}$. The concept $\stackrel{\rightharpoonup }{y}$ is measured in a certain unit, i.e. (unit), so, let us say $\stackrel{\rightharpoonup }{c}$, must be used in order to be converted into (rad) a factor, so that:

$\vartheta \left(\stackrel{\rightharpoonup }{y}\right)=\stackrel{\rightharpoonup }{c}·\stackrel{\rightharpoonup }{y}$, (5)

in which $\stackrel{\rightharpoonup }{c}$ is measured in $\frac{\text{rad}}{\text{unit}}$.

Special attention is accorded to the Euler’s function:

$c\left(\stackrel{\rightharpoonup }{y}\right)={\text{e}}^{i\vartheta \left(\stackrel{\rightharpoonup }{y}\right)}$, (6)

which, in this instance, is referred to as the contributor of concept $\stackrel{\rightharpoonup }{y}$. The function in question, denoted as $c\left(\stackrel{\rightharpoonup }{y}\right)$ is a non-dimensional complex function. It serves as the foundation for the conversion of the conceptual entity represented as {O} into a constituent that actively participates in the formation of m. Moreover, it stands as the central function upon which the creation of other functions is predicated.

The contributor can apply both to scalar and vector concepts as long as the term reflecting the argument $\vartheta \left(\stackrel{\rightharpoonup }{y}\right)$ is formed, as indicated in relation (5). Generally, the contributors can be elucidated as $c\left(\stackrel{\rightharpoonup }{y}\right)$, so that $c\left(\stackrel{\rightharpoonup }{y}\right):{ℝ}^n\to ℂ,n\in \mathbb{N}$ It is understood that when n = 1, it corresponds to a scalar concept. In this case, the contributor is expressed as:

$c\left(y\right)={\text{e}}^{icy}$, (7)

in which c signifies a constant and y represents a scalar concept set in the domain $A=\left[y_1,y_m\right]\subseteq ℝ$ and $m\in \mathbb{N}$. In this respect, the contributor is set $c\left(y\right):D\subseteq ℝ\to B\subseteq ℝ$. Implementing (5) to (6), for the case wherein the concept $\stackrel{\rightharpoonup }{y}$ is a vector, the contributor can be expressed as:

$c\left(\stackrel{\rightharpoonup }{y}\right)={\text{e}}^{i\stackrel{\rightharpoonup }{c}·\stackrel{\rightharpoonup }{y}}$ [4] , (8)

in which $\stackrel{\rightharpoonup }{c}$ is a vector with constant values, i.e. $\stackrel{\rightharpoonup }{c}=\left(c_1,c_2,\cdots ,c_n\right)\in {ℝ}^n$ and $\stackrel{\rightharpoonup }{y}$ is set $\stackrel{\rightharpoonup }{y}=\left(y_1,y_2,\cdots ,y_n\right)\in {ℝ}^n$. Then, the contributor is set $c\left(\stackrel{\rightharpoonup }{y}\right):D\subseteq {ℝ}^n\to B\subseteq ℂ$.

The Mass Generator can be expressed as follows:

$Gm\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right)=g_t\times g_{\epsilon }\times g_{\stackrel{\rightharpoonup }{s}}\times {\text{e}}^{i\left[\left(\omega t+\alpha \epsilon +\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)+\left(\phi \left(t\right)+\phi \left(\epsilon \right)+\phi \left(\stackrel{\rightharpoonup }{s}\right)\right)\right]}$ (9)

The result is that the function that describes m is:

$m\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right)=g_t\times g_{\epsilon }\times g_{\stackrel{\rightharpoonup }{s}}\times {\text{e}}^{i\left[\left(\omega t+\alpha \epsilon +\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)+\left(\phi \left(t\right)+\phi \left(\epsilon \right)+\phi \left(\stackrel{\rightharpoonup }{s}\right)\right)\right]}$ (10)

considering:

$g_t\to \frac{1}{{\omega }^2}:\left({\text{s}}^{\text{2}}\right)$

$g_{\epsilon }\to \frac{1}{\alpha }:\left(\text{J}\right)$

$g_{\stackrel{\rightharpoonup }{s}}\to k^2:\left(\frac{\text{1}}{{\text{m}}^{\text{2}}}\right)$

In order to describe the generic model, let us analyse the proposed way to insert the probability by considering a function $f\left(x\right)$, so that $x\in D=\left[x_0,x_n\right]\subseteq ℝ$ and $f\left(x\right):D\to B$, so that $B=\left[f\left(x_0\right),f\left(x_n\right)\right]\subseteq ℂ$. If the factor $b\in ℝ$ is introduced in $f\left(x\right)$, the latter will be transformed to function $f\left(bx\right)$. The result will change in this case, but the type of function remains unchanged, and the function operates in the same pattern for the entire domain.

Probability is introduced via the parameter b, which is regarded as a random variable; b receives values from a sample space denoted as ${\Omega }_b$ and possesses the attributes of one. In this case, the domain D is separated in sub-domains, i.e. $D=d_0\cup d_1\cup \cdots \cup d_n$, so that $d_{i-1}=\left[x_{i-1},x_i\right]$. In each sub-domain, the value of b changes. When $f\left(bx\right)$ begins to run the domain $d_{i-1}$, an experiment of luck takes place and b receives a value from ${\Omega }_b$, which for this case is $b_{k_{i-1}}$. It receives the value $b_{k_i}$ in the next sub-domain ( $d_i=\left[x_i,x_{i+1}\right]$). Given that the domains are closed, during each field a different factor b applies during each domain. During $d_{i-1}$ the factor is $b_{k-1}$ and applies until the latest point, i.e. $x_i$ and the value of the function is $f\left(\left(b_{k-1}\right)x_i\right)$. Meanwhile, in the domain d_i, the factor that applies through the entire domain is $b_{k_i}$ different $b_{k_{i-1}}$ ( $b_{k_i}\ne b_{k_{i+1}}$) and applies from the first possible value, i.e. $x_i$. So, the value at the first point of domain $d_i$ is $f\left(\left(b_k\right)x_i\right)$. However, $f\left(\left(b_{k-1}\right)x_i\right)\ne f\left(\left(b_k\right)x_i\right)$ which means that there is a discontinuity at that point. Considering the use of open domain of the form $\left(x_i,x_{i+1}\right]$ is one way of avoiding this. In this scenario, the function $f\left(\left(b_{i-1}\right)x\right)$ can receive the value $f\left(\left(b_{i-1}\right)x_i\right)$ at point $x_i$, soit ends at $f\left(\left(b_{i-1}\right)x_i\right)$ (since $x_i\in \left(x_{i-1},x_i\right]$). On the other hand, the function $f\left(\left(b_i\right)x\right)$ starts from $f\left(\left(b_{i-1}\right)x_i\right)$. Continuity at this single point can be ensured because $f\left(bx\right)$ is set to the entire domain $D=\left[\left[x_0,x_1\right]\cup \left(x_1,x_2\right]\cup \cdots \cup \left(x_{n-1},x_n\right]\right]$, which can be separated as $D=\left[\left[x_0,x_1\right]\cup \left(x_1,x_2\right]\cup \cdots \cup \left(x_{n-1},x_n\right]\right]$.

The combination of the previous paragraphs allows for the elucidation of m as a probabilistic function. Each generator is described in the following manner:

Time generator:

Space generator:

Energy generator:

Let us consider a function $g\left(\theta \right)=g_0{\text{e}}^{i\theta }$. Following the steps so far, it can be inferred that $\theta =\omega \cdot t+\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}+\alpha \cdot \epsilon +{\phi }_i$, and $g_0=g_t\times g_{\stackrel{\rightharpoonup }{s}}\times g_{\epsilon }$, and the complex function is expressed as follows:

$g\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right)=g_t\times g_{\stackrel{\rightharpoonup }{s}}\times g_{\epsilon }\times {\text{e}}^{i\left(\omega \cdot t+\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}+\alpha \cdot \epsilon +{\phi }_i\right)}$ (11)

so that $g\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right):\left[{\theta }_0,{\theta }_N\right]\subset ℝ\to B\subset ℂ$, considering that: $t\in \left[t_0,t_k\right]\subset ℝ,t_0\ge 0$ and $\epsilon \in \left[{\epsilon }_0,{\epsilon }_k\right]\subset ℝ$ and $\stackrel{\rightharpoonup }{s}\in \left[x_{1,0},x_{1,k}\right]\times \left[x_{2,0},x_{2,k}\right]\times \cdots \times \left[x_{n,0},x_{n,k}\right]\subseteq {ℝ}^n$.

The domain to be examined is $\Theta =\left[{\theta }_0,{\theta }_n\right]$. When $g\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right)$ starts running in Θ, an experiment of luck takes place at the beginning of the domain and the parameters $g_t,g_{\stackrel{\rightharpoonup }{s}},g_{\epsilon },\omega ,\stackrel{\rightharpoonup }{k},\alpha$ and ${\phi }_i$ take an initial value, i.e. $g_{t0},g_{\stackrel{\rightharpoonup }{s}0},g_{\epsilon 0},{\omega }_0,{\stackrel{\rightharpoonup }{k}}_0,{\alpha }_0$. Then, the domain’s initial value is calculated as ${\theta }_0={\omega }_0\cdot t_0+{\stackrel{\rightharpoonup }{k}}_0·{\stackrel{\rightharpoonup }{s}}_0+{\alpha }_0\cdot {\epsilon }_0+{\phi }_i$, having received the values $t_0,{\stackrel{\rightharpoonup }{s}}_0$ and ${\epsilon }_0$, from T, S and $ℰ$. The initial value of $g\left(\theta \right)$ is:

$g\left({\theta }_0\right)=g_{t0}\times g_{\stackrel{\rightharpoonup }{s}0}\times g_{\epsilon 0}\times {\text{e}}^{i\left({\omega }_0\cdot t_0+{\stackrel{\rightharpoonup }{k}}_0·{\stackrel{\rightharpoonup }{s}}_0+{\alpha }_0\cdot {\epsilon }_0+{\phi }_i\right)}$ (12)

As $g\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right)$ continues running in Θ, it also runs the domains T, S, and $ℰ$, so that each parameter processes in this domain. During this process, experiments of luck are performed in the subdomains and the factors $g_t,g_{\stackrel{\rightharpoonup }{s}},g_{\epsilon },\omega ,\stackrel{\rightharpoonup }{k}$ and α receive varying values, as mentioned in Paragraph 2.3. In this context, the function $g\left(\theta \right)$ receives different values and takes its probabilistic form.

The integration of the contributor provides the behaviour function $\stackrel{\rightharpoonup }{b}\left(\stackrel{\rightharpoonup }{y}\right)$, so that $\stackrel{\rightharpoonup }{b}\left(\stackrel{\rightharpoonup }{y}\right):{ℝ}^n\to {ℂ}^n,n\in \mathbb{N}$, which explicates how the concept acts within the place {O}. In the case that n = 1, the behaviour refers to scalar functions, which are calculated as follows for time and energy:

$b\left(t\right)=-\frac{i}{\omega }\left({\text{e}}^{i\omega t}+{\text{e}}^{i\omega t_1}\right),t\ge 0,\omega \in ℝ$ and $\omega \ne 0$ and $t_1\in ℝ$ for time (13)

and

$b\left(\epsilon \right)=-\frac{i}{\alpha }\left({\text{e}}^{i\alpha \epsilon }+{\text{e}}^{i\alpha {\epsilon }_1}\right),\epsilon \in ℝ,\alpha \in ℝ$ and $\alpha \ne 0$ and ${\epsilon }_1\in ℝ$ for energy (14)

considering that the time contributor is $c\left(t\right)={\text{e}}^{i\omega t}$ and the energy contributor is $c\left(\epsilon \right)={\text{e}}^{i\alpha \epsilon }$.

The space behaviour is calculated after indefinite integration is implemented over the contributor on tis vector format, i.e.:

$\stackrel{\rightharpoonup }{b}\left(\stackrel{\rightharpoonup }{s}\right)=\left(-i\frac{c_1}{k_1}{\text{e}}^{i\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}}+{\gamma }_1,-i\frac{c_2}{k_2}{\text{e}}^{i\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}}+{\gamma }_2,-i\frac{c_3}{k_3}{\text{e}}^{i\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}}+{\gamma }_3\right)$ (15)

where $c_1,c_2$ and $c_3$ are constants so that $c_1,c_2$ and $c_3\in ℝ$

The considerations outlined thus far provide insights into the direction that warrants attention for crafting a more intricate model of m. By employing relation (9) as a guiding principle and employing mathematical techniques like integration, the generation of each concept’s generator can be computed based on its contributing factors.

Upon observing m, it becomes apparent that each individual unit possesses a distinct quantity. This observation prompts the consideration that a specific amount of energy is encapsulated within each m unit. To determine the total energy quantity, integration can be employed as a method of calculation.

Let us assume that the energy captured in a portion of m which is necessary for retaining this quantity of m lies between ${\epsilon }_1$ and ${\epsilon }_2$, thus creating a domain ${\rm E}=\left[{\epsilon }_1,{\epsilon }_2\right]\subseteq ℝ$. Therefore, the total energy is calculated as:

$g\left({\epsilon }_c\right)={\displaystyle \underset{{\epsilon }_1}{\overset{{\epsilon }_2}{\int }}{\text{e}}^{i\alpha \epsilon }\text{d}\epsilon }=\frac{i}{\alpha }\left({\text{e}}^{i\alpha {\epsilon }_1}-{\text{e}}^{i\alpha {\epsilon }_2}\right),\alpha \in ℝ\backslash \left\{0\right\}$ (16)

whereas a compact format is:

$g\left({\epsilon }_c\right)=g_{\epsilon ,c}{\text{e}}^{i\theta \left({\epsilon }_c\right)}$ (17)

considering:

$g_{\epsilon ,c}=\frac{2}{\alpha }\sin \left(\frac{\alpha }{2}\left({\epsilon }_2-{\epsilon }_1\right)\right),\alpha \in ℝ\backslash \left\{0\right\}$ (18)

and

$\theta \left({\epsilon }_c\right)=\frac{\alpha }{2}\left({\epsilon }_2-{\epsilon }_1\right)$ (19)

Additionally, considering a domain $\left[{\epsilon }_1,\epsilon \right]\subseteq {\rm E}$, the generic model describing the movements of energy within E is:

$g\left(\epsilon \right)=g_{\epsilon }{\text{e}}^{i\theta \left(\epsilon \right)}$ (20)

considering:

$g_{\epsilon }=\frac{2}{\alpha }\sin \left(\frac{\alpha }{2}\left(\epsilon -{\epsilon }_1\right)\right),\alpha \in ℝ\backslash \left\{0\right\}$ (21)

and

$\theta \left(\epsilon \right)=\frac{\alpha }{2}\left(\epsilon +{\epsilon }_1\right)$ (22)

Some important findings emerge from both these relations:

When observing m, it becomes evident that the influence of time extends beyond a confined time domain; rather, it manifests as a continuous phenomenon. Furthermore, there are instances where time exerts diminishing effects on certain types of m, a characteristic contingent upon the specific nature of the m in question. Given these observations, the deployment of the time generator is examined in a continuous manner, thereby precluding the application of definite integration.

The description provided in Equation (13) inadequately characterises the time generator, as the magnitude of $g_t$ should be (s²). Consequently, the time generator is computed through infinite integration over the behaviour of time, as follows:

$g\left(t\right)=-\frac{1}{{\omega }^2}\left({\text{e}}^{i\omega t}-{\omega }^2{C^{\prime }}_1{\text{e}}^{i\omega t_1}t-{\omega }^2{C^{\prime }}_2{\text{e}}^{i\omega t_1}\right),t\ge 0$ (23)

$C_1=-{\omega }^2{C^{\prime }}_1{\text{e}}^{i\omega t_1}t,{C^{\prime }}_1\in ℝ$, so that ${C^{\prime }}_1$ is measured in units of (s) (24)

$C_2=-{\omega }^2{C^{\prime }}_2{\text{e}}^{i\omega t_1},{C^{\prime }}_2\in ℝ$ and ${C^{\prime }}_2\ge 0$, so that ${C^{\prime }}_2$ is measured in units of (s²) (25)

Whereas a more compact format is:

$g\left(t\right)=g_t{\text{e}}^{i\theta \left(t\right)}$ (26)

so that:

$g_t=\left|g\left(t\right)\right|=\sqrt{\frac{1}{{\omega }^4}+{\left({C^{\prime }}_{1}t+C_2\right)}^2-\frac{2}{{\omega }^2}\left({C^{\prime }}_{1}t+{C^{\prime }}_2\right)\cos \left(\omega \left(t-t_1\right)\right)},\left|g\left(t\right)\right|\ge 0$ (27)

and

$\tan \left(\theta \left(t\right)\right)=\frac{\sin \left(\omega t\right)-{\omega }^2\left({C^{\prime }}_{1}t+{C^{\prime }}_2\right)\sin \left(\omega t_1\right)}{\cos (\omega t)-{\omega }^2\left({C^{\prime }}_{1}t+{C^{\prime }}_2\right)\cos \left(\omega t_1\right)}$ (28)

Mass is a scalar concept, which implies that the space generator must also be a scalar function. In addition, the correct unit must be applied so that m can be described properly. The behaviour of space is given in relation (15) and it is:

$\stackrel{\rightharpoonup }{b}\left(\stackrel{\rightharpoonup }{s}\right)=\left(-i\frac{c_1}{k_1}{\text{e}}^{i\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}}+{\gamma }_1,-i\frac{c_2}{k_2}{\text{e}}^{i\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}}+{\gamma }_2,-i\frac{c_3}{k_3}{\text{e}}^{i\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}}+{\gamma }_3\right)$ (29)

Put succinctly, to calculate the space generator, it is considered that the vector $\stackrel{\rightharpoonup }{b}\left(\stackrel{\rightharpoonup }{s}\right)$, can derive form a scalar function considering [8] - [10] :

$\stackrel{\rightharpoonup }{b}\left(\stackrel{\rightharpoonup }{s}\right)=\nabla G\left(\stackrel{\rightharpoonup }{s}\right)$ (30)

in which $G\left(\stackrel{\rightharpoonup }{s}\right)$ is a scalar function so that $G\left(\stackrel{\rightharpoonup }{s}\right):{ℝ}^3\to ℂ$.

The space generator is calculated as a function extracted after indefinite integration over the behaviour $\stackrel{\rightharpoonup }{b}\left(\stackrel{\rightharpoonup }{s}\right)$, i.e.:

$g\left(\stackrel{\rightharpoonup }{s}\right)=-{\left|\stackrel{\rightharpoonup }{\kappa }\right|}^2{\text{e}}^{i\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}}$ (31)

where:

${\left|\stackrel{\rightharpoonup }{\kappa }\right|}^2=\left(\frac{k_1^2{k}_2^2{k}_3^2}{k_1^2{k}_2^2+k_2^2{k}_3^2+k_3^2{k}_1^2}\right)$ (32)

whereas the generic format is:

$g^{\prime }\left(\stackrel{\rightharpoonup }{s}\right)=-g_{\stackrel{\rightharpoonup }{s}}{}^{\prime }{\text{e}}^{i\theta \left(\stackrel{\rightharpoonup }{s}\right)}$ (33)

so that:

$g_{\stackrel{\rightharpoonup }{s}}{}^{\prime }=\left|g\left(s\right)\right|=\frac{1}{\sqrt{c_{\kappa }^2+{\left(\lambda \stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}+{\lambda }_{\nu }\right)}^2-2c_{\kappa }\lambda \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}+c_{\lambda }\right)\cos \left(\stackrel{\rightharpoonup }{k}·\left(\stackrel{\rightharpoonup }{s}-{\stackrel{\rightharpoonup }{s}}_0\right)\right)}}$ (34)

and

$\tan \theta \left(\stackrel{\rightharpoonup }{s}\right)=\frac{-c_{\kappa }\sin \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)+\lambda \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}+c_{\lambda }\right)\sin \left(\stackrel{\rightharpoonup }{k}·{\stackrel{\rightharpoonup }{s}}_0\right)}{-c_{\kappa }\cos \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)+\lambda \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}+c_{\lambda }\right)\cos \left(\stackrel{\rightharpoonup }{k}·{\stackrel{\rightharpoonup }{s}}_0\right)}$ (35)

A function $G\left(\stackrel{\rightharpoonup }{s}\right)$ that that satisfies (34) is calculated as follows:

$G\left(s\right)=-\left(\frac{k_1^2{k}_2^2+k_2^2{k}_3^2+k_3^2{k}_1^2}{k_1^2{k}_2^2{k}_3^2}\right){\text{e}}^{i\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}}$ (36)

Whereas the generic format is:

$G^{\prime }\left(\stackrel{\rightharpoonup }{s}\right)=\left|G^{\prime }\left(\stackrel{\rightharpoonup }{s}\right)\right|{\text{e}}^{i{\theta }_{\stackrel{\rightharpoonup }{s}}}$ (37)

In which:

$\left|G^{\prime }\left(\stackrel{\rightharpoonup }{s}\right)\right|=\sqrt{{\left(c_{\kappa }\right)}^2+{\left(\lambda \stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}+{\lambda }_{\nu }\right)}^2-2c_{\kappa }\lambda \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}+c_{\lambda }\right)\cos \left(\stackrel{\rightharpoonup }{k}·\left(\stackrel{\rightharpoonup }{s}-{\stackrel{\rightharpoonup }{s}}_0\right)\right)}\ge 0$ (38)

and

$\tan \theta \left(\stackrel{\rightharpoonup }{s}\right)=\frac{-c_{\kappa }\sin \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)+\lambda \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}+c_{\lambda }\right)\sin \left(\stackrel{\rightharpoonup }{k}·{\stackrel{\rightharpoonup }{s}}_0\right)}{-c_{\kappa }\cos \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)+\lambda \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}+c_{\lambda }\right)\cos \left(\stackrel{\rightharpoonup }{k}·{\stackrel{\rightharpoonup }{s}}_0\right)}$ (39)

and

$c_{\kappa }=\left(\frac{k_1^2{k}_2^2+k_2^2{k}_3^2+k_3^2{k}_1^2}{k_1^2{k}_2^2{k}_3^2}\right)$ (40)

Finally, another function that must be included is $H\left(\stackrel{\rightharpoonup }{s}\right)$, which is related to the inverted space contributor ɔ $\left(s\right)={\text{e}}^{-i\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}}$ (41), ɔ: $A\subseteq {ℝ}^n\to D\subseteq ℂ$. It is then:

$H\left(s\right)=\left(\frac{k_1^2{k}_2^2+k_2^2{k}_3^2+k_3^2{k}_1^2}{k_1^2{k}_2^2{k}_3^2}\right){\text{e}}^{-i\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}}$ (42)

so that [8] - [10] :

${\nabla }^{2}H\left(s\right)={\text{e}}^{-i\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}}$ (43)

While the generic format is as follows:

$H^{\prime }\left(\stackrel{\rightharpoonup }{s}\right)=h\left(\stackrel{\rightharpoonup }{s}\right){\text{e}}^{-i\theta \left(\stackrel{\rightharpoonup }{s}\right)}$ (44)

so that $h\left(\stackrel{\rightharpoonup }{s}\right)$ is a scalar function assigned ${ℝ}^3\to ℂ$.

In this case, it is:

${\nabla }^2{H}^{\prime }\left(\stackrel{\rightharpoonup }{s}\right)={\text{e}}^{-i\theta \left(\stackrel{\rightharpoonup }{s}\right)}{\displaystyle \underset{a=1}{\overset{3}{\sum }}{f}_a}$ (45)

considering:

$f_a={\text{e}}^{-i\theta \left(\stackrel{\rightharpoonup }{s}\right)}\left(\frac{{\partial }^2}{\partial x_a^2}h\left(\stackrel{\rightharpoonup }{s}\right)+\frac{\partial }{\partial x_a}h\left(\stackrel{\rightharpoonup }{s}\right)\frac{\partial }{\partial x_a}\left(-i\theta \left(s\right)\right)+h\left(\stackrel{\rightharpoonup }{s}\right){\left(\frac{\partial }{\partial x_a}\left(-i\theta \left(\stackrel{\rightharpoonup }{s}\right)\right)\right)}^2+h\left(\stackrel{\rightharpoonup }{s}\right)\frac{{\partial }^2}{\partial x_a^2}\left(-i\theta \left(\stackrel{\rightharpoonup }{s}\right)\right)\right)$ (46)

so that a = 1, 2, 3 in a 3-dimensional Euclidian space.

Finally:

${\text{e}}^{-i\theta \left(\stackrel{\rightharpoonup }{s}\right)}=\frac{1}{f\left(\stackrel{\rightharpoonup }{s}\right)}{\nabla }^2{H}^{\prime }\left(\stackrel{\rightharpoonup }{s}\right),f\left(\stackrel{\rightharpoonup }{s}\right)\ne 0$

so that:

$f\left(\stackrel{\rightharpoonup }{s}\right)={\displaystyle \underset{a=1}{\overset{n}{\sum }}{f}_a}$ (47)

considering a n-dimensional space.

The objective of this section is to formulate the model of m. Following the elucidation of the methodology and the computation of the requisite factors and generators necessary for composing the model of m, this section amalgamates this information to construct the model. The section is divided into two parts: the first part considers the composition of m from its constituent components, while the second part explores the behaviour of m within specific constraints.

Let us consider the formation of a specific quantity of m. This m component must embody a certain quantum of energy influenced by temporal interaction within space. Although time contributes to m creation and exerts an ongoing influence, it is not bound by any limitations and continues to affect m thereafter.

In this case, the generators are combined to create m:

$m\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right)=g\left(t\right)\cdot g\left(\epsilon \right)\cdot g\left(\stackrel{\rightharpoonup }{s}\right)$ (48)

The outcome comprises the generation of several relationships delineating m, among which three are of paramount significance and are outlined below. The primary model, denoted as the general model, is expressed as:

$m\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right)=\left|{\mu }_A\right|{\text{e}}^{i{\phi }_{{\mu }_A}}-\left|{\mu }_B\right|{\text{e}}^{i{\phi }_{{\mu }_B}}$ (49)

considering:

$\left|{\mu }_A\left(\epsilon \right)\right|={\mu }_{0,\epsilon }=2\frac{{\left|\stackrel{\rightharpoonup }{\kappa }\right|}^2}{\alpha {\omega }^2}\sin \left(\frac{\alpha }{2}\left(\epsilon -{\epsilon }_1\right)\right)\ge 0$ (50)

and

${\phi }_{{\mu }_A}=\omega t-\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}+\frac{\alpha }{2}\left({\epsilon }_1+\epsilon \right)+\phi$ (51)

Also:

$\left|{\mu }_B\left(\epsilon \right)\right|=2\frac{{\left|\stackrel{\rightharpoonup }{\kappa }\right|}^2}{\alpha {\omega }^2}\sin \left(\frac{\alpha }{2}\left(\epsilon -{\epsilon }_1\right)\right){\omega }^2\left({C^{\prime }}_{1}t+{C^{\prime }}_2\right)\ge 0$ (52)

and

${\phi }_{{\mu }_B}=\omega t_1-\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}+\frac{\alpha }{2}\left({\epsilon }_1+\epsilon \right)+\phi$ (53)

This format attributes the dual nature of m, encompassing both its particulate and wave manifestations, with the second component denoting fluctuations in space and energy, influenced continuously by the passage of time.

The second model is expressed as:

$m\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right)=\frac{1}{{\left({\omega }_{\epsilon }-{\omega }_{s^2}\right)}^2}\frac{{\epsilon }_0}{s_0^2}\left({\text{e}}^{i\left({\omega }_{\epsilon }-{\omega }_{s^2}\right)t}-{\left({\omega }_{\epsilon }-{\omega }_{s^2}\right)}^2{\text{e}}^{i\left({\omega }_{\epsilon }-{\omega }_{s^2}\right)t_1}\left({C^{\prime }}_{1}t+{C^{\prime }}_2\right)\right){\text{e}}^{i\left(\phi -\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)}$ (54)

This relation comprises the factor $\frac{1}{{\left({\omega }_{\epsilon }-{\omega }_{s^2}\right)}^2}$, which indicates that it is possible to tune m.

The generic formats describing m and are similar to what was expected are:

$m\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right)={\mu }_{0,c}{\text{e}}^{i\left(\theta \left({\epsilon }_c\right)+\theta \left(t\right)-\theta \left(\stackrel{\rightharpoonup }{s}\right)+\phi \right)}$ (55)

so that:

${\mu }_{0,c}=g_{\epsilon }{g}_t{g^{\prime }}_{\stackrel{\rightharpoonup }{s}}\ge 0$ (56)

The utilisaiton of the generic format of energy generator provides:

$m\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right)={\mu }_{0,g}{\text{e}}^{i\left(\theta \left(\epsilon \right)+\theta \left(t\right)-\theta \left(\stackrel{\rightharpoonup }{s}\right)+\phi \right)}$ (57)

setting:

${\mu }_{0,g}=g_{\epsilon }{g}_t{g^{\prime }}_{\stackrel{\rightharpoonup }{s}}\ge 0$ (58)

The parameters that are introduced in the relations are:

$t\in \left[t_j,t_k\right]\subseteq ℝ$, where $t_j<t_k$ and $t_j\ge 0$.

$\stackrel{\rightharpoonup }{s}=\stackrel{\rightharpoonup }{s}\left(x_1,\cdots ,x_n\right)$ set to $D=\left[x_{1,j},x_{1,k}\right]\times \left[x_{2,j},x_{2,k}\right]\times \cdots \times \left[x_{n,j},x_{n,k}\right]\subseteq {ℝ}^n$.

$\epsilon \in \left[{\epsilon }_j,{\epsilon }_k\right]\subseteq ℝ$, where ${\epsilon }_j<{\epsilon }_k$.

φ is a phase difference.

${\mu }_0$ is a random variable composed from the other random variables $\left(\omega ,\stackrel{\rightharpoonup }{k},\alpha ,\epsilon \right)$ with sample space ${\Omega }_{{\mu }_{0,\epsilon }}:{\Omega }_{{\mu }_{0,\epsilon }}\to \left[{\mu }_{0,\epsilon ,\min },{\mu }_{0,\epsilon ,\max }\right]\subseteq {ℝ}^{+}$, where $ℝ$ the set of real numbers, distribution $f\left({\mu }_{0,\epsilon }\right)$, mean value ${\mu }_{{\mu }_{0,\epsilon }}$, dispersion ${\sigma }_{{\mu }_{0,\epsilon }}^2$, measured in (kg) and defined as ${\mu }_{0,\epsilon }=2\frac{{\left|k\right|}^2}{\alpha {\omega }^2}\sin \left(\frac{\alpha }{2}\left({\epsilon }_1-\epsilon \right)\right)$ and a summary of the probabilistic components is in the following manner:

ω represents a random variable with a sample space ${\Omega }_{\omega }:{\Omega }_{\omega }\to \left[{\omega }_{\min },{\omega }_{\max }\right]\subseteq {ℝ}^{\ast }$, where $ℝ$ denotes the set of real numbers, distribution $f\left(\omega \right)$, mean value ${\mu }_{\omega }$ and dispersion ${\sigma }_{\omega }^2$ and measured in rad/s [5] - [7] .

$\stackrel{\rightharpoonup }{k}$ denotes a random variable with a sample space ${\Omega }_{\stackrel{\rightharpoonup }{k}}:{\Omega }_{\stackrel{\rightharpoonup }{k}}\to \left[{\stackrel{\rightharpoonup }{k}}_{\min },{\stackrel{\rightharpoonup }{k}}_{\max }\right]\subseteq {ℝ}^n$, where n signifies the number of dimensions that the vector is set, distribution $f\left(\stackrel{\rightharpoonup }{k}\right)$, mean value ${\mu }_{\stackrel{\rightharpoonup }{k}}$ and dispersion ${\sigma }_{\stackrel{\rightharpoonup }{k}}^2$ and measured in rad/m [5] - [7] .

α is a random variable with a sample space ${\Omega }_{\alpha }:{\Omega }_{\alpha }\to \left[a_{\min },{\alpha }_{\max }\right]\subseteq {ℝ}^{\ast }$, where $ℝ$ is the set of real numbers, distribution $f\left(\alpha \right)$, mean value ${\mu }_{\alpha }$ and dispersion ${\sigma }_{\alpha }^2$ and measured in rad/J [5] - [7] .

φ is a random variable with a sample space ${\Omega }_{\phi }:{\Omega }_{\phi }\to \left[{\phi }_{\min },{\phi }_{\max }\right]\subseteq ℝ$, where $ℝ$ signifies the set of real numbers, distribution $f\left(\phi \right)$, mean value ${\mu }_{\phi }$ and dispersion ${\sigma }_{\phi }^2$ and measured in (rad) [5] - [7] .

${C^{\prime }}_1\in ℝ$ and ${C^{\prime }}_2\ge 0$ and ${C^{\prime }}_2\in ℝ$.

The methodology employed thus far, involving the incorporation of complex numbers in the probabilistic m generation process, demonstrates that m can indeed arise from the interplay between time, space, and energy. The introduction of contributors has facilitated the integration of concepts into the m formation process, each contributing in a distinct manner. The energy contributor has been utilised to attribute the rate of change of the energy generator, leading to the establishment of a specific energy quantity within m through definite integration. The space generator functions as a scalar function, indicating the manner in which the spatial vector can be configured to participate in m formation. Lastly, the time contributor emerges as the second derivative of the time generator, essentially representing the acceleration of time, which consequently impacts the form and magnitude of m. m is depicted across various models, stemming from the initial interaction of generators and culminating in the most generic model akin to that considered in the methodology. This serves as validation of the initial premise.

At this infant level, m can be named as point energy mass (pem), in order to distinguish from the general concept of m.

Other conclusions drawn from the analysis thus far include:

The findings confirm that m is indeed produced through the interaction between energy, space, and time. Furthermore, it is demonstrated that it is feasible to manipulate the m such that it undergoes decomposition. However, it is imperative to note that certain conditions within the surrounding environment may be necessary to facilitate this decomposition, warranting further investigation.

Following the format provided in relation (49), the real and imaginary components of m can be expanded utilising the trigonometric format. The inclusion of the energy variable ε, in addition to the space and time variables, is the advantage of this form, since a value of energy ε randomly chosen from the domain $\left[{\epsilon }_1,\epsilon \right]:\left[{\epsilon }_1,\epsilon \right]\subseteq \left[{\epsilon }_1,{\epsilon }_2\right]$, is used. Based on this analysis, it becomes apparent that each part is composed of other parts.

The real part of (49) is:

$\begin{array}{l}\mathrm{Re}\left[m\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right)\right]\\ ={\mu }_{0,\epsilon }\left(\cos \left(\omega t-\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}+\frac{\alpha }{2}\epsilon +{\phi }^{\prime }\right)-{\omega }^2\left({C^{\prime }}_{1}t+{C^{\prime }}_2\right)\cos \left(\frac{\alpha }{2}\epsilon -\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}+{\phi }^{″}\right)\right)\end{array}$ (59)

which can be analysed as follows:

$\begin{array}{c}{\mathrm{Re}}_m={\mu }_{0\epsilon }\{\cos \left(\omega t\right)\cos \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)\cos \left(\frac{\alpha }{2}\epsilon \right)\cos \left({\phi }^{\prime }\right)-\cos \left(\omega t\right)\cos \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)\sin \left(\frac{\alpha }{2}\epsilon \right)\sin \left({\phi }^{\prime }\right)\\ +\cos \left(\omega t\right)\sin \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)\sin \left(\frac{\alpha }{2}\epsilon \right)\cos \left({\phi }^{\prime }\right)+\cos \left(\omega t\right)\sin \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)\cos \left(\frac{\alpha }{2}\epsilon \right)\sin \left({\phi }^{\prime }\right)\\ +\sin \left(\omega t\right)\sin \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)\cos \left(\frac{\alpha }{2}\epsilon \right)\cos \left({\phi }^{\prime }\right)-\sin \left(\omega t\right)\sin \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)\sin \left(\frac{\alpha }{2}\epsilon \right)\sin \left({\phi }^{\prime }\right)\end{array}$

$\begin{array}{c}-\sin \left(\omega t\right)\cos \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)\sin \left(\frac{\alpha }{2}\epsilon \right)\cos \left({\phi }^{\prime }\right)-\sin \left(\omega t\right)\cos \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)\cos \left(\frac{\alpha }{2}\epsilon \right)\sin \left({\phi }^{\prime }\right)\\ -{\omega }^2\left({C^{\prime }}_{1}t+{C^{\prime }}_2\right)[\cos \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)\cos \left(\frac{\alpha }{2}\epsilon \right)\cos \left({\phi }^{″}\right)+\cos \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)\sin \left(\frac{\alpha }{2}\epsilon \right)\sin \left({\phi }^{″}\right)\\ +\sin \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)\sin \left(\frac{\alpha }{2}\epsilon \right)\cos \left({\phi }^{″}\right)-\sin \left(\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}\right)\cos \left(\frac{\alpha }{2}\epsilon \right)\sin \left({\phi }^{″}\right)]\}\end{array}$ (60)

Due to this dynamic and probabilistic approach, the m cannot be perceived as isolated from its environment; it must engage with it. This observation is evident from the movement of each component. This oscillatory behaviour indicates that the m not only transfers energy to the environment but also reacts to it. This interaction of the m with its environment can be conceptualised as an area surrounding the m within which energy is exchanged between the m and its surroundings.

Considering that the field emanates from the m as a consequence of its energy interaction with the environment, it becomes imperative to ascertain the characteristics of the field and identify potential correlations with the m.

The m is proven to be the source of an energy field () since it can be expressed as:

$m\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right)={\epsilon }_0{c}^{\prime }\left(t,\stackrel{\rightharpoonup }{s}\right){\nabla }^2\left(H^{\prime }\left(\stackrel{\rightharpoonup }{s}\right)\right)$ (61)

given:

${\epsilon }_0=\frac{2}{\alpha }\sin \left(\frac{\alpha }{2}\left({\epsilon }_2-{\epsilon }_1\right)\right){\text{e}}^{i\left(\frac{\alpha }{2}\left({\epsilon }_1+{\epsilon }_2\right)\right)}$ (62)

and

$c^{\prime }\left(t,\stackrel{\rightharpoonup }{s}\right)=\frac{1}{{\omega }^2}\frac{1}{f\left(\stackrel{\rightharpoonup }{s}\right)}{g^{\prime }}_{\stackrel{\rightharpoonup }{s}}\left({\text{e}}^{i\omega t}-{\omega }^2\left({C^{\prime }}_{1}t+{C^{\prime }}_2\right){\text{e}}^{i\omega t_1}\right){\text{e}}^{i\phi }$ (63)

In this context, let us contemplate a scenario where, at a specific distance from the m and at a particular moment in time, the energy assumes a defined value. The most suitable function to articulate this relationship is the wave function, enabling the expression of energy in this region as a complex number utilising the exponential format, i.e.:

$E\left(\stackrel{\rightharpoonup }{s},t\right)=E_0{\text{e}}^{i\left({\omega }_{E}t+{\stackrel{\rightharpoonup }{k}}_E·{\stackrel{\rightharpoonup }{s}}_E+{\phi }_E\right)}$ (64)

Let us assume a point $\stackrel{\rightharpoonup }{s}=\left(x_1,x_2,x_2\right)$ in a Euclidian space $0x_1{x}_2{x}_3$, as described in Figure 1 . The way in which the energy is transferred from the m to that point is through a field.

In a more generalised format, (64) can be expressed as:

$E\left(\stackrel{\rightharpoonup }{s},t\right)=E_0{\text{e}}^{i\left({\theta }_E\left(t\right)+{\theta }_E\left(\stackrel{\rightharpoonup }{s}\right)+{\phi }_E\right)}$ (65)

To devise a mathematical framework for delineating the interaction between the m and its environment, given that the m has been established as the origin of a field, it becomes imperative to introduce a novel parameter, called the “energy field” (or simply “field”) which is defined by the symbol . This concept elucidates the mechanism through which energy is generated by the m, dispersed into the surrounding space, and engages with the energy of the surrounding environment. This parameter takes the form of a vector, with measurement units denoting energy per unit length, that is, [] $=\frac{E}{\stackrel{\rightharpoonup }{s}}=\frac{\left(\text{J}\right)}{\left(\text{m}\right)}=\frac{\left(\text{kg}\right)\cdot \left(\text{m}\right)}{\left({\text{s}}^2\right)}$, based on the S.I. system. It would not be unreasonable to infer that the field is the gradient of energy [11] , that is:

$=\nabla E$ (66)

The description of this field is premised on its characteristics, i.e. divergence, cult, time change, acceleration, and wave form. These characteristics have been summarised in Table 1 and Table 2 .

The factor $c_u$ is measured in units of m/s. It is considered that $\stackrel{\rightharpoonup }{p}\left(\stackrel{\rightharpoonup }{s}\right)=\nabla \left({\theta }_E\left(\stackrel{\rightharpoonup }{s}\right)\right)$ and since the argument ${\theta }_E\left(\stackrel{\rightharpoonup }{s}\right)$ is related to space it is equal to ${\theta }_E\left(\stackrel{\rightharpoonup }{s}\right)={\theta }_E\left(x_1\right)+{\theta }_E\left(x_2\right)+{\theta }_E\left(x_3\right)$ or ${\theta }_E\left(s\right)={\theta }_E\left(x_1\right)+\cdots +{\theta }_E\left(x_n\right)$ for a space of more than n dimensions.

It is possible to relate the field with m and their relation is as follows:

$=-i\stackrel{\rightharpoonup }{v}{\omega }^2{m}_A\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right)$(67)

considering that:

$m_A=m_0\left({\text{e}}^{ia{\epsilon }_1}-{\text{e}}^{i\alpha {\epsilon }_2}\right){\text{e}}^{i\left(\omega t-\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{s}+\phi \right)}$ (68)

Whereas utilising the generic generators of space and time:

$={\stackrel{\rightharpoonup }{v}}_g\frac{1}{g_t}m\left(\epsilon ,\stackrel{\rightharpoonup }{s},t\right),g_t\ne 0$(69)

Using the trigonometric representation of complex numbers facilitates the dissection of the field into its constituent elements, yielding a total of 16 components. Each of these components attributes a distinct manner in which the m interacts with the environment.

${\stackrel{\rightharpoonup }{C}}_{{\stackrel{\rightharpoonup }{f}}_1}={\stackrel{\rightharpoonup }{f}}_0\cos \left(k_1{x}_1\right)\cos \left(k_2{x}_2\right)\cos \left(k_3{x}_3\right)\cos \left(\theta \right){\text{e}}^{i\omega t}$ (70)

${\stackrel{\rightharpoonup }{C}}_{{\stackrel{\rightharpoonup }{f}}_2}={\stackrel{\rightharpoonup }{f}}_0\cos \left(k_1{x}_1\right)\cos \left(k_2{x}_2\right)\cos \left(k_3{x}_3\right)\sin \left(\theta \right){\text{e}}^{i\left(\omega t+\frac{\pi }{2}\right)}$ (71)

${\stackrel{\rightharpoonup }{C}}_{{\stackrel{\rightharpoonup }{f}}_3}=-{\stackrel{\rightharpoonup }{f}}_0\sin \left(k_1{x}_1\right)\sin \left(k_2{x}_2\right)\cos \left(k_3{x}_3\right)\cos \left(\theta \right){\text{e}}^{i\omega t}$ (72)

${\stackrel{\rightharpoonup }{C}}_{{\stackrel{\rightharpoonup }{f}}_4}=-{\stackrel{\rightharpoonup }{f}}_0\sin \left(k_1{x}_1\right)\sin \left(k_2{x}_2\right)\cos \left(k_3{x}_3\right)\sin \left(\theta \right){\text{e}}^{i\left(\omega t+\frac{\pi }{2}\right)}$ (73)

${\stackrel{\rightharpoonup }{C}}_{{\stackrel{\rightharpoonup }{f}}_5}=-{\stackrel{\rightharpoonup }{f}}_0\sin \left(k_1{x}_1\right)\cos \left(k_2{x}_2\right)\sin \left(k_3{x}_3\right)\cos \left(\theta \right){\text{e}}^{i\omega t}$ (74)

${\stackrel{\rightharpoonup }{C}}_{{\stackrel{\rightharpoonup }{f}}_6}=-{\stackrel{\rightharpoonup }{f}}_0\sin \left(k_1{x}_1\right)\cos \left(k_2{x}_2\right)\sin \left(k_3{x}_3\right)\sin \left(\theta \right){\text{e}}^{i\left(\omega t+\frac{\pi }{2}\right)}$ (75)

${\stackrel{\rightharpoonup }{C}}_{{\stackrel{\rightharpoonup }{f}}_7}=-f_0\cos \left(k_1{x}_1\right)\sin \left(k_2{x}_2\right)\sin \left(k_3{x}_3\right)\cos \left(\theta \right){\text{e}}^{i\omega t}$ (76)

${\stackrel{\rightharpoonup }{C}}_{{\stackrel{\rightharpoonup }{f}}_8}=-{\stackrel{\rightharpoonup }{f}}_0\cos \left(k_1{x}_1\right)\sin \left(k_2{x}_2\right)\sin \left(k_3{x}_3\right)\sin \left(\theta \right){\text{e}}^{i\left(\omega t+\frac{\pi }{2}\right)}$ (77)

${\stackrel{\rightharpoonup }{C}}_{{\stackrel{\rightharpoonup }{f}}_9}=f_0\cos \left(k_1{x}_1\right)\cos \left(k_2{x}_2\right)\sin \left(k_3{x}_3\right)\cos \left(\theta \right){\text{e}}^{i\left(\frac{\pi }{2}-\omega t\right)}$ (78)