The longitudinal electromagnetic (LEM) wave, also named as scalar wave (SW), was first identified by Maxwell, who combined electricity and magnetism and developed the Maxwell’s equations [1] . In 1890, Tesla observed the LEM wave while refining Hertz’s experiment and introduced the concept of scalar field [2] . Aharonov and Bohm [3] experimentally demonstrated the presence of magnetic vector potential and electromagnetic scalar potential in a zero electromagnetic field in 1959, confirming the existence of LEM waves. Meyl [4] [5] conducted extensive research on LEM waves and proposed that Maxwell’s equations have solutions involving longitudinal waves. Zohuri [6] [7] presented a quantum electrodynamic theoretical framework for LEM waves, Monstein [8] [9] provided their existence from an astrophysical perspective, while Oschman [10] suggested that two anti-phase coherent light waves would generate a LEM wave. When studying about the reflection of incidence waves in initially stressed dissipative half space, Ivanovs [11] found two complex quasi-P-wave and quasi-TEM waves. Drawing upon the principles of momentum conservation and energy conservation, Shekhar and Parvez [12] introduced a novel approach for computing the amplitudes (coefficients) of reflected and transmitted longitudinal waves.

The exceptional properties of LEM waves have attracted widespread attentions [13] - [18] for their potential applications in medicine and energy areas due to their harmlessly free penetration of the human body, potentially superluminal speed, absorption of free energy, and lossless energy transmission. However, there are some skepticism [19] regarding their existence and effectiveness due to the lack of rigorous theoretical models for LEM waves. Therefore, establishing robust propagation mode and wave equation for LEM waves is essential for advancing the application of LEM waves. However, current Maxwell’s equations neglect the LEM wave item, rendering them incompatible with such waves. It is imperative to reintroduce the longitudinal wave item into Maxwell’s equations and develop a rigorous mathematical model for LEM waves, integrating it into the theoretical framework of Maxwell’s equations. Meanwhile, exploring the essence of LEM wave is helpful to unveil the underlying mechanisms of its absorption free energy and the applications in medical and energetic areas.

As shown in Figure 1 , $B/{\tau }_2=\left(\nabla \cdot B\right)V_B$, so $B/{\tau }_2$ has only the z-axis component. Therefore, Equation (10) needs to be rewritten as

$\begin{array}{c}-\frac{1}{\mu \epsilon }\nabla \times \left(\nabla \times E\right)=-\frac{1}{\mu \epsilon }\nabla \left(\nabla \cdot E\right)+\frac{1}{\mu \epsilon }{\nabla }^{2}E\\ =\frac{{\partial }^{2}E}{\partial t^2}+\frac{1}{{\tau }_1}\frac{\partial E}{\partial t}+\frac{1}{{\tau }_2}\frac{\partial E_z}{\partial t}+\frac{E_z}{{\tau }_2{\tau }_1}.\end{array}$ (14)

For TEM waves, since $\nabla \cdot E=0$, the projection of Equation (14) along the transverse direction of electromagnetic wave propagation in Figure 1 gives that

${\nabla }^2{E}_t-\mu \epsilon \frac{{\partial }^2{E}_t}{\partial t^2}-\mu \sigma \frac{\partial E_t}{\partial t}=0$, (15)

which is the wave equation for TEM wave in a good conductor.

For LEM wave in a good conductor, $\nabla \cdot E=\nabla \cdot E_z\ne 0$, when projected onto the axial direction of electromagnetic wave propagation (z-axis in Figure 1 ), Equation (14) yields

$\begin{array}{c}-\frac{1}{\mu \epsilon }\nabla \times \left(\nabla \times E_z\right)=-\frac{1}{\mu \epsilon }\nabla \left(\nabla \cdot E_z\right)+\frac{1}{\mu \epsilon }{\nabla }^2{E}_z=0\\ =\frac{{\partial }^2{E}_z}{\partial t^2}+\frac{1}{{\tau }_1}\frac{\partial E_z}{\partial t}+\frac{1}{{\tau }_2}\frac{\partial E_z}{\partial t}+\frac{E_z}{{\tau }_2{\tau }_1}.\end{array}$ (16)

Equation (16) is the wave equation for LEM wave in a good conductor. Based on it, we can obtain that

$\nabla \left(\nabla \cdot E_z\right)={\nabla }^2{E}_z.$ (17)

Combining Equation (16.1) and $E_z=\left(0,0,E_z\right)$ gives that $E_z=E_z\left(z,t\right)$. Similarly, for LEM wave in a vacuum, we can obtain $B_z=B_z\left(z,t\right)$.

Kozyrev introduced a significant concept [24] known as torsion field (TF). He posited that a fixed spin should have relation to a spin field-TF like a fixed particle mass to a gravitational field and a fixed charge to an electromagnetic field. Based on the assumption that the positive and negative photon components undergo double helix motion [22] , the TF model should consist of two neutrinos with reversely orbital rotation to sustain momentum balance. Figure 6 (left) represents the left-handed TF model composed of two neutrinos with left-handed spin. The neutrino spin velocity is the speed of light, and both of them move in a left-handed large spiral orbit along the forward axis, and the axial velocity is also the speed of light. The phase difference between the two neutrinos is π, and the total left-handed TF in a vacuum is equivalent to the left-handed magnetic P-wave B_L, representing a high energy state. The right-handed TF model is shown in Figure 6 (right), which is also composed of two neutrinos with left-handed spin, but moving in a right-handed large spiral orbit around the forward axis. Their phase difference is also π, and the total right-handed TF in a vacuum amounts to right-handed magnetic P-wave B_R, representing a low energy state. According to the law of electromagnetic induction, left-handed B_L can produce electric vortex potential E_L, whose vortex motion interacts with zero-point vacuum energy field to extracts positive free energy and generate high-energy rays, neutrons, and high-energy particles at the vortex center, accompanied by highly directed cold fusion [25] [26] . This process enables B_L to acquire significantly positive free energy. While right-handed B_R generated by right-handed TF in Figure 6 (right) produces electric vortex potential E_R, which extracts negative free energy from zero-point vacuum energy field and does harm to organism.

In general, LEM waves are usually allowed to be neglected due to their significantly smaller amplitudes compared with that of TEM waves. But in certain scenarios such as the near fields [13] , current Maxwell’s equations fail to elucidate the mechanism of them. Hence, consideration must be given to LEM waves. The generation of substantial LEM wave typically occurs when two coherent light waves with identical frequency, amplitude, and propagation direction, but a phase difference of π (referred to as source waves), are superimposed within a vacuum medium, they will generate a magnetic P-wave $B_z$ [10] . The electric field components of the source waves are represented by $E_1$ and $E_2$, while their corresponding magnetic field components are denoted as $B_1$ and $B_2$. Upon superposition of them, the resultant total electric fields are represented by $E_t$ and $B_t$, where

$\begin{array}{c}{E}_t=E_1+E_2=E_0\exp \left[\text{j}\left(k\cdot r-\omega t\right)\right]+E_0\exp \left[\text{j}\left(k\cdot r-\omega t+\pi \right)\right]\\ =E_0\exp \left[\text{j}\left(k\cdot r-\omega t\right)\right]\left[1+\text{exp}\left(\text{j}\pi \right)\right]=0,\end{array}$ $B_t=B_1+B_2=B_0\exp \left[\text{j}\left(k\cdot r-\omega t\right)\right]$

$\left[1+\text{exp}\left(\text{j}\pi \right)\right]=0.$ (18)

When two source waves are superimposed, the total electromagnetic fields seemly both vanish. However, in accordance with the principle of energy conservation, all energy of source waves is actually transferred to $B_z$. This energy transfer can be explained by definitions of magnetic vector potential $A$ and scalar potential $\phi$ [20] shown as

$B_t=B_1+B_2=\nabla \times \left(A_1+A_2\right)=0$, i.e., $A=A_1+A_2=\nabla \psi \ne 0$, (18.1)

$E_t=E_1+E_2=-\nabla \phi -\frac{\partial A}{\partial t}=0$, and $\nabla \phi =-\frac{\partial A}{\partial t}=-\nabla \frac{\partial \psi }{\partial t}\ne 0$, that is,

$\phi =-\frac{\partial \psi }{\partial t}\ne 0$. (18.2)

Here, $\psi$ represents a scalar field known as the “zero-point vacuum energy field” [25] , “torsion field” [26] . This phenomenon can be mutually corroborated by the Aharov-Bohm effect [3] , which highlights the significance of $A$ and $\phi$ in comparison to $E_t$ and $B_t$.

Our photonic structure model can effectively elucidate the essence of this phenomenon. The left-handed photon model depicted in the left of Figure 7 illustrates left-handed photon composed of positron and electron with equal charge. The positron, spinning to the right, follows a right-handed circular orbit around the forward axis, generating a left-handed magnetic P-wave $B_{z+}$ [27] . Similarly, the electron, spinning to the left, orbits the forward axis in a left-handed circular path, producing a left-handed magnetic P-wave $B_{z-}$. The combined effect of these motions results in a total left-handed magnetic P-wave given by $B_z=B_{z-}+B_{z+}$. As per reference [20] , $B_t/{\mu }_0{\epsilon }_0=C\times E_t$. Consequently, at point A where positron and electron intersect without colliding, the overall neutrality of the photon is observed along with zero electric field intensity $E_t$ and magnetic induction intensity $B_t$. At points B and H where positron and electron are at their maximum distance from each other; $E_t$ and $B_t$ exhibit negative and positive maximum values respectively. Figure 8 presents an illustration depicting the TEM waves $E_t$ and $B_t$ produced by left-handed photon alongside magnetic P-wave $B_L$. The right-handed photon model is depicted in the middle of Figure 7 , where the positron, spinning to the right, follows a left-handed circular orbit around the forward axis; the electron, spinning to the left, orbits the forward axis in a right-handed circular path. Their resultant motions leads to a total right-handed magnetic P-wave given as $B_R$.

The superposition of two source light waves is equivalent to a left-handed photon ${\gamma }_L$ superposing with a right-handed photon ${\gamma }_R$. As depicted in Figure 7 , at point E, H, positrons from ${\gamma }_L$ encounter with electrons from ${\gamma }_R$, while at point G, F, electrons from ${\gamma }_L$ meet with positrons from ${\gamma }_R$. Their superposition leads to the annihilations of positrons/electrons [28] , resulting in the generation of a substantially free energy (zero-point vacuum energy) and a magnetic P-wave $B_z$. This reaction can be mathematically expressed as follows

${\gamma }_R+{\gamma }_L\to B_z+\text{zero}\text{point}\text{vacuum}\text{energy}$. (18.3)

On this occasion, after superposition, although resultant total electric fields vanish in space, both magnetic vector potential $A$ and scalar potential $\phi$ still persist and consist of $B_z$, where $A=\nabla \psi$, $\phi =-\partial \psi /\partial t$ and $\psi$ is a scalar field related to $B_z$.

A certain region dΩ in a medium where light waves $E$ and $B$ propagate has a free charge density $\rho$ with charge number q. At any given moment, the Lorentz force acting on charge q in region dΩ can be expressed as $F=q\left(E+V\times E\right)$, where $V$ represents the motion speed of free charge. Consequently, after dt time, the work done by $F$ on q in dΩ is $\text{d}w=q\left(E+V\times E\right)\cdot V\text{d}t=\rho \text{d}\Omega \left(E+V\times E\right)\cdot V\text{d}t=\text{d}\Omega E\cdot \left(\rho V\right)\text{d}t=\text{d}\Omega E\cdot J\text{d}t$. By applying superposition principle, work done by Lorentz force $F$ on all free charges in entire region Ω per unit time can be expressed as

$\text{d}W/\text{d}t={\displaystyle {\int }_{\Omega }E\cdot J\text{d}\Omega },$ (18.4)

where $J$ denotes electric current density within dΩ. From Equation (6.1), we can obtain

$E\cdot J=E\cdot \frac{1}{\mu }\left(\nabla \times B\right)-\frac{\epsilon }{2}\frac{\partial E^2}{\partial t}=\frac{1}{\mu }B\cdot \left(\nabla \times E\right)-\frac{1}{\mu }\nabla \cdot \left(E\times B\right)-\frac{\epsilon }{2}\frac{\partial E^2}{\partial t}.$ (18.5)

Equation (8.1) is substituted into Equation (18.5), it is obtained that

$E\cdot J=-\frac{1}{\mu }B\cdot \left[\partial B/\partial t+\left(\nabla \cdot B_z\right)C\right]-\frac{1}{\mu }\nabla \cdot \left(E\times B\right)-\frac{\epsilon }{2}\frac{\partial E^2}{\partial t}.$ (18.6)

Substituting Equation (18.6) back to Equation (18.4) gives

$\begin{array}{c}{E}_T={\displaystyle {\int }_{\Omega }-\frac{1}{2\mu }\frac{\partial B^2}{\partial t}-\frac{\epsilon }{2}\frac{\partial E^2}{\partial t}\text{d}\Omega }=\text{d}W/\text{d}t+E_P+{\displaystyle {\int }_{\Omega }\frac{1}{\mu }\nabla \cdot \left(E\times B\right)\text{d}\Omega }\\ =\text{d}W/\text{d}t+{\displaystyle {\int }_{\Omega }\frac{1}{\mu }\left(\nabla \cdot B\right)\left(B\cdot C\right)\text{d}\Omega }+{\displaystyle {\int }_{\Omega }\frac{1}{\mu }\nabla \cdot \left(E\times B\right)\text{d}\Omega }\\ =\text{d}W/\text{d}t+{\displaystyle {\int }_{\Omega }\frac{1}{\mu }\left(\nabla B_z\right)\left(B_z\cdot C\right)\text{d}\Omega }+{\displaystyle {\int }_{\Omega }\frac{1}{\mu }\nabla \cdot \left(E\times B\right)\text{d}\Omega }.\end{array}$ (18.7)

In Equation (18.7), $E_T$ denotes the total light energy within Ω; the first term on the right-hand side represents the work done by the electromagnetic force on free charges, i.e., energy converted into joule heat; the second term signifies the energy $E_P$ transformed into LEM wave; and final term accounts for pure incoming and outgoing energy at Ω’s boundary surface. Equation (18.7) presents a modified Poynting theorem incorporating consideration of LEM wave terms. After superimposing two source lightwaves in a vacuum, as per Equations (6.3) and (8.3), $B_z$ gains non-zero value, while $E_z$ is equal to zero if electrical conductivity σ equals 0 for a vacuum, i.e., ${\displaystyle {\int }_{\Omega }\frac{1}{\mu }\nabla \cdot \left(E\times B\right)\text{d}\Omega }=\frac{1}{\mu }{\displaystyle {\oint }_s\left(0\times B_z\right)\text{d}s}=0$. Moreover, for a vacuum medium with no electric charge, $\text{d}W/\text{d}t=0$. And then $E_T$ is equal to $E_P$, that is,

$\begin{array}{c}{E}_P={\displaystyle {\int }_{\Omega }-\frac{1}{2\mu }\frac{\partial B_z^2}{\partial t}\text{d}\Omega }={\displaystyle {\int }_{\Omega }\text{j}\omega B_z^2\frac{1}{\mu }\text{d}\Omega }={\displaystyle {\int }_{\Omega }\frac{1}{\mu }\left(\nabla B_z\right)\left(B_z\cdot C\right)\text{d}\Omega }\\ =E_T={\displaystyle {\int }_{\Omega }-\frac{1}{\mu }\frac{\partial B_1^2}{\partial t}-\frac{1}{\mu }\frac{\partial B_2^2}{\partial t}\text{d}\Omega }={\displaystyle {\int }_{\Omega }2\text{j}\omega \frac{1}{\mu }\left(B_1^2+B_2^2\right)\text{d}\Omega },\end{array}$ (19)

From Equation (19), we can derive the wave equation for $B_z$, given by

$\frac{{\partial }^2{B}_z}{\partial t^2}-C^2\frac{{\partial }^2{B}_z}{\partial z^2}=0$, (20)

$\left|B_z\right|=B_{z0}=\sqrt{2\left|B_1^2+B_2^2\right|}=\sqrt{2+2{\exp }^2\left(\text{jπ}\right)}{B}_0=2B_0.$ (21)

$B_z=B_{z0}\exp \left[\text{j}\left(\frac{{\omega }_{\begin{array}{c}P\end{array}}}{C}z-{\omega }_{\begin{array}{c}P\end{array}}t\right)\right]$. (21.1)

Referring to the fundamental equations for a TEM wave in a vacuum $\nabla \times E_t=-\partial B_t/\partial t$ and $\nabla \times B_t/{\mu }_0{\epsilon }_0=\partial E_t/\partial t$, from Equation (4) and Table 2 , it can be inferred that within a vacuum, the magnetic P-wave $B_z$ and its electric counterpart $E_{\theta }$ can be converted into each other through their specific relationship formulas shown as

$\nabla \times E_{\theta }=-\frac{B_z}{{\tau }_2}=-\frac{1}{{\tau }_2}{B}_{z0}{e}_z\exp \left[\text{j}\left(\frac{{\omega }_{\begin{array}{c}P\end{array}}}{C}z-{\omega }_{\begin{array}{c}P\end{array}}t\right)\right]$. (21.2)

$E_{\theta }=-C\times B_z$, (21.3)

which gives that

$\nabla \times E_{\theta }=\text{j}k\times E=\text{j}k{E}_{\theta }{e}_z=-\frac{1}{{\tau }_2}{B}_{z0}{e}_z\exp \left[\text{j}\left(\frac{{\omega }_{\begin{array}{c}P\end{array}}}{C}z-{\omega }_{\begin{array}{c}P\end{array}}t\right)\right].$ (21.4)

Equation (13) is substituted into Equation (21.4), it is obtained that

$E_{\theta }=\frac{C{B}_{z0}}{{\omega }_{\begin{array}{c}P\end{array}}{\tau }_2}\exp \left[\text{j}\left(\frac{\omega }{C}z-\omega t+\frac{\pi }{2}\right)\right]e_{\theta }=C{B}_{z0}\exp \left[\text{j}\left(\frac{\omega }{C}z-\omega t+\frac{\pi }{2}\right)\right]e_{\theta }$. (21.5)

Equations (21.1) and (21.5) represent solutions to the wave equations governing the LEM waves $E_{\theta }$ and $B_z$. The propagation model for $E_{\theta }$ and $B_z$ is depicted in Figure 9 , illustrating that the phase angle of $E_{\theta }$ leads $B_z$ by 90˚. The conversion process from vortex electric field $E_{\theta }$ to scalar magnetic field $B_z$ is illustrated in Figure 9 as O to A and B to C, while the transformation from $B_z$ into $E_{\theta }$ is shown as A to B and C to D. This cyclic process involves alternating induction and conversion between $E_{\theta }$ and $B_z$, propagating along the positive z-axis at a phase velocity C.

If there is no signal interference, $E_{\theta }$ and $B_z$ can be transformed into each other and propagate in a straight line to infinity, with a vector tail of $B_z$ never connecting to its vector head. Consequently, $B_z$ can be considered as a magnetic monopole. Thus, we derive the condition for the generation of magnetic monopole as follows: two coherent light waves with identical frequency, amplitude, and propagation direction, but a phase difference of π are superimposed within a vacuum medium.

Under this circumstance, magnetic P-wave $\left|B_z\right|=2B_0\ne 0$, which cannot be disregarded; that is, $\left(\nabla \cdot B\right)C=B/{\tau }_2=-\text{j}k{B}_{z}C\ne 0$ in Equation (8). Otherwise, it would lead to a violation of energy conservation in Equation (19). It is deduced from Equations (19) and (21.5) that

$\begin{array}{c}{E}_P={\displaystyle {\int }_{\Omega }\left(-\frac{1}{2{\mu }_0}\frac{\partial B_z^2}{\partial t}-\frac{{\epsilon }_0}{2}\frac{\partial E_z^2}{\partial t}\right)\text{d}\Omega }={\displaystyle {\int }_{\Omega }\left(-\frac{1}{2{\mu }_0}\frac{\partial B_z^2}{\partial t}\right)\text{d}\Omega }\\ ={\displaystyle {\int }_{\Omega }\left(-\frac{{\epsilon }_0}{2}\frac{\partial E_{\theta }^2}{\partial t}\right)\text{d}\Omega }={\displaystyle {\int }_{\Omega }\frac{1}{{\mu }_0}\left(\nabla B_z\right)\left(B_z\cdot C\right)\text{d}\Omega }\\ ={\displaystyle {\int }_{\Omega }\nabla }\left(\frac{1}{2{\mu }_0}C{B}_z^2\right)\text{d}\Omega ={\displaystyle {\int }_{\Omega }\nabla }\left(\frac{{\epsilon }_0}{2C}{E}_{\theta }^2\right)\text{d}\Omega ,\end{array}$ (22)

which is energy equation for magnetic P-wave $B_z$.

1) Wave velocity

According to Equation (20), the wave speed of $B_z$ is equal to the speed of light C.

2) Frequency

According to Equation (13), the frequency ${\omega }_{\begin{array}{c}P\end{array}}$ of $B_z$ can be expressed as

$\left|{\omega }_{\begin{array}{c}P\end{array}}\right|=\left|-\text{j}\frac{1}{{\tau }_2}\right|=\frac{{\omega }^2{\epsilon }_0}{{\sigma }_0}={10}^{31}\text{Hz}$, (23)

where ${\epsilon }_0~{10}^{-11}\text{F}/\text{m}$, is the vacuum dielectric constant [20] , ${\sigma }_0~{10}^{-14}\text{S}/\text{m}$ is the atmospheric electrical conductivity [29] , which is approximately the vacuum electrical conductivity, $\omega$ taken as 10¹⁴ Hz (the frequency of light waves) is the source TEM waves frequency.

3) Wavelength

The typical wavelength of $B_z$ is calculated as ${\lambda }_z=\frac{C}{{\omega }_P}=\frac{{10}^8}{{10}^{31}}={10}^{-25}\text{m}$.

4) Exceptionally harmless penetration for conductive material

Owing to the skin effect, TEM waves are unable to penetrate good conductive medium and can only propagate within a thin layer $\delta$ on its surface [21] . The frequency ${\omega }_P$ of $B_z$ is approximately 10³¹ Hz, while the typical metal’s plasma resonance frequency ${\omega }_z$ is on the order of 1 × 10¹⁵ Hz, and damping coefficient ${\gamma }_s$ of metallic bound electrons is around 1 × 10¹³ Hz. Substituting these parameters [30] into the classical Drude equation [31] , yields a dielectric constant for metal corresponding to frequency 10³¹ Hz as ${\epsilon }_r=1-\frac{{\omega }_z^2}{{\gamma }_s^2+{\omega }_P^2}+\text{j}\frac{{\omega }_z^2{\gamma }_s}{{\omega }_P\left({\gamma }_s^2+{\omega }_P^2\right)}\approx 1$. Under this circumstance, it becomes evident that a conductive medium cannot shield against $B_z$; thus, it exhibits “high-frequency transparency” [32] [33] and has unhindered ability to permeate through human tissue. This observation aligns with Meyl K.’s assertion regarding LEM wave characteristic that “the Faraday cage cannot shield against LEM waves” [5] . What’s more, neutrinos are assumed as the propagating particles of $B_z$. The extremely small cross-section of 10⁻⁴³ cm² for neutrino interaction with atomic nuclei results in a minuscule probability of its capture by an atomic nucleus within a square centimeter area, thus demonstrating a harmlessly penetrative capability of $B_z$ to organism cells [34] .

5) The ability to absorb free energy

The total energy associated with two source light waves can be represented as

$\begin{array}{c}{E}_T={\displaystyle {\int }_{\Omega }-{\epsilon }_0\frac{\partial E_1^2}{\partial t}-{\epsilon }_0\frac{\partial E_2^2}{\partial t}\text{d}\Omega }\\ ={\displaystyle {\int }_{\Omega }2\text{j}\omega {\epsilon }_0\left(E_1^2+E_2^2\right)\text{d}\Omega }\\ ={\displaystyle {\int }_{\Omega }2\text{j}\omega {\epsilon }_0{E}_0^2\left\{{\exp }^2\left[\text{j}\left(k\cdot r-\omega t\right)\right]+{\text{exp}}^2\left[\text{j}\left(k\cdot r-\omega t+\pi \right)\right]\right\}\text{d}\Omega }\\ ={\displaystyle {\int }_{\Omega }4\omega {\epsilon }_0{E}_0^2\sin 2\left(\frac{\omega }{C}z-\omega t\right)\text{d}\Omega }\\ =4\Omega \omega {\epsilon }_0{E}_0^2=4\Omega \omega \frac{1}{{\mu }_0}{B}_0^2.\end{array}$ (24)

From Equation (19), after the superposition the total energy of $E_P$ can be expressed as

$\begin{array}{c}{E}_P={\displaystyle {\int }_{\Omega }-\frac{{\epsilon }_0}{2}\frac{\partial E_{\theta }^2}{\partial t}\text{d}\Omega }={\displaystyle {\int }_{\Omega }\text{j}{\omega }_{\begin{array}{c}P\end{array}}{\epsilon }_0{E}_{\theta }^2\text{d}\Omega }\\ =\text{j}{\omega }_{\begin{array}{c}P\end{array}}{\epsilon }_0{E}_{\theta 0}^2{\displaystyle {\int }_{\Omega }{\exp }^2\left[\text{j}\left(\frac{{\omega }_{\begin{array}{c}P\end{array}}}{C}z-{\omega }_{\begin{array}{c}P\end{array}}t\right)\right]\text{d}\Omega }\\ =\Omega {\omega }_{\begin{array}{c}P\end{array}}{\epsilon }_0{E}_{\theta 0}^2=\Omega {\omega }_{\begin{array}{c}P\end{array}}\frac{1}{{\mu }_0}{B}_{z0}^2=4\Omega {\omega }_{\begin{array}{c}P\end{array}}\frac{1}{{\mu }_0}{B}_0^2.\end{array}$ (25)

The ratio of the energy of LEM wave to TEM waves can be expressed as

$\frac{E_P}{E_T}=\frac{{\omega }_{\begin{array}{c}P\end{array}}}{\omega }=\frac{{10}^{31}\text{Hz}}{{10}^{14}\text{Hz}}={10}^{17}$. (26)

In accordance with Equation (26), the energy of the magnetic P-wave is 10¹⁷ times that of the source TEM waves. The mechanism for the additional energy gained by magnetic P-wave $B_z$ is likely associated with its induced vortex-magnetic field $E_{\theta }$, which has the capability to extract zero-point vacuum energy through the entangled interaction between its produced torsion field and zero-point vacuum energy field. As per Equation (19), the total energy of $B_z$ can be represented as

$E_P=-\text{d}W/\text{d}t+E_T$ (27)

Here, $-\text{d}W/\text{d}t$ denotes work performed by the zero-point vacuum energy field on $B_z$. As shown in Figure 10 , because $E_P\gg E_T$, it follows that $E_P\approx -\text{d}W/\text{d}t$; thus, indicating that most of $B_z$’s energy originates from its interaction with “zero-point vacuum energy field”. Although mysteries surrounding “zero-point vacuum energy” remain unresolved, it can be inferred that apart from $E_T$, $B_z$ must acquire additional free energy from zero-point vacuum fields to uphold genuine energy conservation. This aligns with Meyl K.’s assertion regarding characteristic of LEM waves having “the ability to absorb free energies” [5] .

In the main processes of life, such as photosynthesis and respiration, the essence of these chemical reactions is electron transfer. In these processes, electron transport is realized by protease which is actually a long chain connected by amino acid molecules with a left-handed chirality. In 2021, Ohio State University announced the identification of the protein deoxyuridine triphosphate nucleotidohydrolase (dUTPase) as a key modulator of the immune response that contributes to the immunological and neuro­logical abnormalities in individuals [35] . It suggests that dUTPase could be used as a biomarker of chronic fatigue syndrome. DUTPase, as a hydrolase, primarily utilizes pyrophosphate as its substrate and is involved in the hydrolysis of inorganic pyrophosphate (PPi, ${\text{P}}_2{\text{O}}_7^{4-}$) generated in the uridylation pathway of protein meTableolisminto orthophosphate ions (Pi, ${\text{HPO}}_4^{2-}$), which promotes thermodynamic equilibrium towards mitochondrial ATP (adenosine triphosphate) synthesis. ATP in mitochondria is a crucial cellular energy source. When it is hydrolyzed back to ADP (adenosine diphosphate) and Pi, energy is released to drive many biochemical processes within the cell, thereby enhancing the whole energy of the biological field of life and restoring the body’s health.

After organism receives the left-handed magnetic P-wave $B_L$ with drug or nutrition biochemical information and free energy emitted by the therapeutic apparatus by utilizing the tunnel nanotubes as biological signal waveguides in the cell membrane [36] [37] , $B_L$ induces the left-handed electric vortex potential $E_v$ on the long chain of dUTPase (see Figure 11 ). $E_v$ attracts the left-handed free electrons in the cell fluid to do left-handed helix movement along the long chain of dUTPase, which increases the ionization speed of the cell fluid, increases the concentration and the electrical potential energy of hydrogen ions (protons), breaks the concentration balance of ATP/ADP in mitochondria, then accelerating the chemical reaction speed of ATP synthesis. Moreover, the enrichment of free electrons in cell membrane also stimulates the efficiency of dUTPase, which increases the ability to catalyze the conversion of PPi into Pi, thus accelerating the speed of ATP synthesis and increasing the concentration ratio of ATP/ADP in the mitochondria, and cause organism in a high energy state (see Figure 11 ). While right-handed magnetic P-wave $B_R$ produces an opposite effect, and then reducing the concentration ratio of ATP/ADP in mitochondria and causing living body in a low energy state.

Schnabl and Meyl [15] [38] conducted a study using a Tesla magnetic P-wave generator (10 mW, 6.7 MHz) to investigate the effects of magnetic P-wave on plant growth. Their research revealed that the left-handed magnetic P-wave signal led to a 40% increase in ATP content in the mitochondria of Ephorbia pulcherrima (10 mW, 90 seconds, red columns in contrast to blue controls, see Figure 12 ). According to our magnetic P-wave model, left-handed magnetic P-wave can induce a left-handed electric vortex potential $E_v$ on the long chain of amino acid molecules within organisms. $E_v$ attracts more free electrons in

cell fluid to do left-handed helix movement into cell membrane, which results in a transition of plant molecules from the phenolic to the chinoidal stage (see Figure 13 ), thus enhancing mitochondrial efficiency and antioxidant effects while improving overall energy levels in plants.

Referring to the time attenuation constant of magnetic vortex potential in total current law for, we recover the longitudinal wave term $B/{\tau }_2$ in Faraday’s law, where ${\tau }_2$ represents the attenuation time constant of electric vortex potential. Based on reevaluated total current law and modified Faraday’s law, we develop the wave equation, energy equation, and propagation model for magnetic P-wave that involve inter-induction/conversion between scalar magnetic field and vortex electric field. These findings support the significant presence of magnetic P-wave reflecting profound electromagnetic essence. Rigorous wave and energy models for magnetic P-wave generated by the superposition of a left-handed photo and a right-handed photon can provide theoretical support and optimization strategies for its medical and energy applications while elucidating the underlying mechanisms for its absorption to free energy: left-handed magnetic P-wave can induce vortex electric field to generate high-energy rays, neutrons, and high-energy particles at its vortex center accompanied by highly directed cold fusion. This leads to substantial free energy absorption through interaction between zero-point vacuum energy field and the produced torsion field by vortex electric field. These models describing the ability of magnetic P-wave to absorb free energy are validated by the interaction between positive magnetic P-wave and cell fluid/long chains of amino acids in a plant organism.

I would like to thank my wife, Ms. Xue Jingwen, and my sister, Jiang Minye for their hard family work and support for my creation of this thesis.

Conceptualization: JJZ (JIANGJian-zhong)

Methodology: JJZ

Investigation: JJZ, WYF (WANGYu-feng)

Visualization: JJZ

Funding acquisition: WYF

Project administration: WYF

Supervision: WYF

Writing-original draft: JJZ

Writing-review & editing: JJZ

All data are available in the main text or the supplementary materials.