1. Introduction

Traditional technologies of transverse electromagnetic field (TEMF) for treating cardiovascular and cerebrovascular diseases primarily focus on the regulatory effects of physical therapies or bioactive substances (e.g., endothelin-1, nitric oxide) on vascular endothelial cells and blood circulation, with underlying mechanisms encompassing techniques such as TEMF stimulation, biofeedback, and microcurrent modulation ^{[1, 2, 3, 4, 5]}. EMF technology can act on vascular endothelial cells to promote nitric oxide release, thereby facilitating peripheral vasodilation and reducing blood flow resistance. The biofeedback module enables real-time monitoring of users’ heart rate variability, balancing sympathetic and parasympathetic neural activities by modulating autonomic nervous function. Microcurrent stimulation can influence local tissue metabolism, accelerate lactic acid clearance, and enhance cell membrane stability.

Typically, the amplitude of electromagnetic longitudinal waves (scalar waves, SWs) is considerably smaller than that of the source transverse electromagnetic waves (TEM waves) generating them ^[6], thus they are often neglected. However, in certain scenarios, such as the near-field region, existing Maxwell’s equations fail to fully characterize their electromagnetic properties, necessitating the consideration of SWs ^{[7, 8]}. Substantial SWs can be generated when two coherent, out-of-phase TEM waves interfere ^{[9, 10]}. The vortex electromagnetic field of SWs gives rise to a unique force field known as the torsion field ^{[11, 12, 13, 14, 15, 16]}. While the static torsion field exhibits extremely weak intensity (approximately ${10}^{27}$ times weaker than gravitational force), its dynamic intensity is substantially enhanced ^[17]. Torsion fields possess properties including superluminal propagation, non-shielding, zero-point vacuum energy extraction, and memory effects ^[18], and their potential and practical applications in medicine have gained widespread recognition and attention ^{[19, 20, 21, 22, 23]}.

Nevertheless, due to the absence of rigorous wave equations and particle propagation model ^[24], applied researches of torsion fields generated by SWs on energy medicine has encountered technical bottlenecks, such as optimizing energy intensity and therapeutic frequencies. Consequently, many scholars have questioned the efficacy and reliability of scalar wave and torsion field therapy ^[25]. Additionally, conventional perspectives maintain that generating pure torsion fields generated by SWs is highly challenging, requiring precise control of the phase difference between two source waves to the picosecond level, stable interference point spatial conditions, and mitigating environmental noise, metal reflections, and geomagnetic disturbances. Even minimal phase drift can cause SWs to collapse into ordinary TEM waves, resulting in the loss of all unique properties.

As a result, current researches on torsion fields generated by SWs remain within the realm of fringe science, and no clear consensus has been established in authoritative medical literature regarding their medical functions and physicochemical mechanisms in treating cardiovascular and cerebrovascular diseases. By constructing accurate models of the vacuum scalar wave equation and particle propagation, we aim to derive a theoretical framework for torsion fields generated by SWs and elucidate their intrinsic connections to mass and gravitational field. The development of these novel theoretical models may provide new tools and methodological approaches for analyzing the medical efficacy and physicochemical mechanisms of torsion fields generated by SWs in cardiovascular and cerebrovascular disease treatment.

2. The essence of electromagnetic fields

Both torsion field and SW wave are associated with spin effects. The torsion field stems from spacetime distortion ^[11], whereas SW waves originate from vortex electromagnetic fields ^[6]. Thus, torsion fields are closely related to SW waves, and investigating torsion fields necessitates a prior clarification of the essence of electromagnetic fields.

From the law of total current ^[26], the electromagnetic field $\overrightarrow{E}$/$\overrightarrow{B}$ in a conductor can be expressed as

Thus, it follows from Eq. (1) that

By synthesizing Eqs. (1) and (2), it is evident that the law of total current embodies a fundamental implication of the electromagnetic field: the relativistic effect of the propagation velocity of the electric field through positive charges (holes) in the conductor (specifically, the moving velocity of the electric field, $-{\overrightarrow{V}}_e$) is converted into the magnetic field within the conductor, which aligns with the principle that a moving electric field generates a magnetic field ^[27].

Employing the same analytical approach, Faraday’s Law of Electromagnetic Induction ^[26] yields that

Then, according to Eq. (3), we have

By synthesizing Eqs. (3) and (4), it is evident that Faraday’s law of electromagnetic induction embodies another fundamental implication of the electromagnetic field: the magnetic field within the conductor is converted into an electric field in the conductor via the relativistic effect of the magnetic field’s velocity vector ${\overrightarrow{V}}_B$, specifically, a moving magnetic field generates an electric field ^[28].

In Eq. (4), ${\overrightarrow{V}}_B$ serves as both the wave velocity of electromagnetic transverse waves ^[26] and the velocity of magnetic field motion ^[28]. According to the principle of duality, the component $-V_{ez}$ of $-{\overrightarrow{V}}_e$ along the electromagnetic field propagation direction $z$ is not only the velocity at which positive charges propagate the electric field (or the velocity of the electric field itself) ^[10], but should also correspond to the wave velocity of some form of wave. It is hypothesized that while the TEM waves ${\overrightarrow{E}}_t$/${\overrightarrow{B}}_t$ propagate along the $z$-direction within the conductor, there exists a type of electromagnetic longitudinal waves, i.e. SWs ${\overrightarrow{E}}_z$/${\overrightarrow{B}}_{\theta }$ that propagates along the same $z$-direction with frequency ${\omega }_P$ and wave velocity $V_{ez}$ ^[29].

Eq. (4), which is associated with Faraday’s law of electromagnetic induction for the TEM waves ${\overrightarrow{E}}_t$/${\overrightarrow{B}}_t$, can be expressed in terms of the electromagnetic scalar potential $\phi$ and magnetic vector potential $\overrightarrow{A}$ as follows ^[26]

which gives that

In accordance with the Lorenz gauge condition applicable to metallic conductors ^[26], we can obtain

Thus, it follows that

Eq. (6) reveals a constraint relation between $-{\overrightarrow{V}}_{ez}$ (the propagation velocity of the electric field of TEM waves in the conductor medium, i.e., the wave speed of SW waves) and ${\overrightarrow{V}}_B$ (the velocity of magnetic field motion, i.e., the wave speed of TEM waves): the product of the two quantities does not exceed the square of the speed of light in the conductor. For typical metallic conductors ^[30], we can obtain that

which implies that the wave velocity $V_{ez}$ of the SW propagating within the conductor could be superluminal.

3. The particle propagation mode for SWs

When two coherent light waves (termed "source light waves") with identical frequency, equal amplitude, opposite propagation directions, and a phase difference of $\pi$ are superposed in a vacuum, a pair of vacuum scalar waves ${\overrightarrow{B}}_z/{\overrightarrow{E}}_{\theta }$ are generated ^[6]. Upon superposition, the resultant TEM fields ${\overrightarrow{B}}_t/{\overrightarrow{E}}_t$ vanish, creating the illusion of energy disappearance. However, according to the Aharonov-Bohm effect ^[31], this process merely corresponds to the extinction of the TEM waves; the magnetic vector potential ${\overrightarrow{A}}_{\theta }$ and electromagnetic scalar potential ${\phi }_{\theta }$ in the vacuum do not cease to exist, but instead transform into the SWs ^[6].

By analogy with the electromagnetic fields in conductive media, the electromagnetic fields of vacuum SWs ${\overrightarrow{B}}_z/{\overrightarrow{E}}_{\theta }$ should also satisfy electromagnetic conversion formulas shown as Eqs. (2) and (4). We thus hypothesize

Here, $V$ denotes the propagation velocity of the electromagnetic field associated with the vacuum SW, and should also correspond to the velocity at which a charge or particle propagates the electromagnetic fields ^[10]. Given the absence of charges in a vacuum, $V$ must therefore represent the magnitude of the absolute velocity of the propagation particle of the SWs ${\overrightarrow{B}}_z/{\overrightarrow{E}}_{\theta }$ relative to the vacuum medium.

The particle propagation mode for the vacuum SWs ${\overrightarrow{B}}_z/{\overrightarrow{E}}_{\theta }$ is illustrated in Fig. 1 ^[18]. To uphold the conservation of angular momentum within this model, it is hypothesized that the propagating particles of the SWs consist of two left-handed neutrinos undergoing counter-rotating large-helical revolution, with a phase difference of $\pi$. The spin velocity of the left-handed neutrinos equals the speed of light, and both follow left-handed large-helical trajectories along the forward axis, with their axial velocity $V_z$ also matching the speed of light. Since neutrinos possess mass, their tangential rotational velocity $V_{\theta }$ cannot exceed the speed of light ^[32]. By assuming that the mechanism of neutrinos rotating around the SWs’ core is analogous to that of a ground-state electron orbiting the hydrogen nucleus ^[33], we can derive

From Ref. ^[10], the frequency of the helical motion of scalar wave neutrinos, denoted as ${\omega }_P$, corresponds to the frequency of the vacuum SWs ${\overrightarrow{B}}_z/{\overrightarrow{E}}_{\theta }$ and can be expressed as

Accordingly, the step length of the helical motion of their propagation particle (neutrino) can be expressed as

Herein, ${\tau }_2$ denotes the vortex decay period of the vortex electric field ${\overrightarrow{E}}_{\theta }$, $\omega$ represents the frequency of the source light waves that generate the SWs, and ${\sigma }_0$ and ${\epsilon }_0$ are the vacuum conductivity and vacuum permittivity, respectively.

Consequently, the energy ratio of ${\overrightarrow{B}}_z/{\overrightarrow{E}}_{\theta }$ to the source light waves can be expressed as

which implies that, in addition to the energy contribution $E_T$, the SWs ${\overrightarrow{B}}_z/{\overrightarrow{E}}_{\theta }$ must acquire a substantial amount of additional free energy from the zero-point vacuum field ^{[10, 16]}. The free energy acquisition mechanism of ${\overrightarrow{B}}_z/{\overrightarrow{E}}_{\theta }$ is associated with the local spatial vortex motions of the SW-propagating particles (neutrinos), where the vortex motion acts as the fundamental unit of energy and a bridge linking visible matter to invisible energy ^[18]. This vortex motion enables the extraction of zero-point vacuum energy via the coherent resonance effect between the torsion field they induce and the zero-point vacuum energy field ^{[13, 14]}.

Given that SWs ${\overrightarrow{B}}_z/{\overrightarrow{E}}_{\theta }$ absorb a significant quantity of vacuum free energy, this energy, analogous to the local spatial curvature effect induced by massive objects and the gravitational field generated by spatial curvature, gives rise to a distinct spatial distortion effect within the helical interior of the SWs. This distortion manifests as a tangential distortion motion $V_{\theta 1}$ and a $z$-directional bending motion $V_{z1}$ in the local space (see Fig. 2).

To simplify the model, we assume the bending effect of the Earth in this local space corresponds to a $z$-directional bending motion $V_{z2}$, while neglecting the influence of macroscopic $z$-directional spatial expansion. Accordingly, the absolute equation of motion for the vacuum SW-propagating particle (neutrino) relative to the vacuum is expressed as

Herein, $C{\overrightarrow{e}}_z$ denotes the linear motion of the SW neutrino at the speed of light along the $z$-direction, while $\frac{C{\overrightarrow{e}}_{\theta }}{137}$ corresponds to its tangential rotational motion. $V_{\theta 1}{\overrightarrow{e}}_{\theta }$ refers to the local spatial distortion motion induced by the scalar wave after it has absorbed a substantial amount of vacuum free energy, and $V_{z2}{\overrightarrow{e}}_z$ represents the $z$-direction bending motion of the Earth within this local space. Notably, $C{\overrightarrow{e}}_z$ and $\frac{C{\overrightarrow{e}}_{\theta }}{137}$ are the active motions of the scalar-wave neutrino, whereas $V_{z2}{\overrightarrow{e}}_z$ and $V_{\theta 1}{\overrightarrow{e}}_{\theta }$ are the passive motions of the SW neutrino that arise as a consequence of its following the spatial deformation.

4. The wave equations for SWs

As illustrated in Fig. 3, within the vacuum space, we establish a region $\mathbf{V}$ of the field point P centered at point O on the Earth’s surface. In the source region ${\mathbf{V}}^{\prime }$ (centered at the fixed source point O’ adjacent to field region $\mathbf{V}$), there exist certain continuous, non-moving, and time-varying charges $\rho ({\overrightarrow{r}}^{\prime },t)$ and a time-varying vortex current source ${\overrightarrow{J}}_{\theta }({\overrightarrow{r}}^{\prime },t)$ that is excited by external electromagnetic fields. These sources generate the SWs ${\overrightarrow{B}}_z/{\overrightarrow{E}}_{\theta }$ within the region $\mathbf{V}$.

Furthermore, a cylindrical coordinate system $(R_1,\theta ,z)$ is established with an observer $O_B$ as the origin on the Earth’s surface. This coordinate system is employed to describe the propagation of the neutrino motion of ${\overrightarrow{B}}_z/{\overrightarrow{E}}_{\theta }$. When ${\mathbf{V}}^{\prime }$ approaches zero and $O_B$ coincides with O’, the position vector $\overrightarrow{r}$ satisfies

and

Subsequently, in accordance with the relational formula between the scalar waves ${\overrightarrow{B}}_z/{\overrightarrow{E}}_{\theta }$ and ${\overrightarrow{A}}_{\theta }/{\phi }_{\theta }$ ^[6], within region P, when the radial dimension $r=R={10}^{-3}\lambda \sim {10}^{-28}\mathrm{m}\ll \lambda =C{\tau }_2\sim {10}^{-25}\mathrm{m}$ and $V_{\theta 1}\ll C$: