2026-10-09
Total electric field: splitting into electric and magnetic branches
Why is a stationary charge detected through an electric field, while the motion of charges in a straight wire creates magnetic field lines that encircle the wire? One may hypothesize that motion opens additional channels of a single general electric state. The electric and magnetic components would then share a common origin, with their difference determined by which part of the state remains in the initial channel and which part is transferred into its orthogonal continuation.
This paper proposes a hypothesis of such splitting. Part I constructs a normalized general field in a multilevel idempotent basis, identifies electric and magnetic branches, and introduces a force state by multiplying the field by charge. Part II defines a mapping of these states into observable 3D space: the electric field direction, the circulating geometry of the magnetic field, and the conditions for recovering the Lorentz force. The possibility of additional field and force components at deeper levels of splitting is also considered.
The mathematical foundation is the paper “The Origin of the Lorentz Factor from Infinite Idempotent Splitting.” The construction is consistent with the distinction between an internal state and its spatial manifestation adopted in the Wave Electricity concept. Throughout this paper, the description in the multilevel idempotent basis refers to the internal state, rather than to a separate inertial reference frame.
Part I
Splitting the General Electric Field in a Multilevel Idempotent Basis
Splitting the General Electric Field in a Multilevel Idempotent Basis
1. The Source, Motion, and the Initial Field Scale
Consider a source with an initial signed field amplitude \(E_0\). Here, \(E_0\) has the dimensions of electric field strength and does not denote energy. The amplitude may depend on position and on the state of the source; the construction is initially carried out locally, at a selected point. The sign of the source is retained in \(E_0\), while the spatial direction will be specified by the mapping.
We associate the source's motion with the parameter \(\beta\) and the Lorentz factor:
\[\tag{1} \beta=\frac{v_s}{c},\qquad |\beta|<1,\qquad \gamma=\frac{1}{\sqrt{1-\beta^2}}. \] The quantity \(v_s\) is the signed velocity of the source along the selected axis. It must be distinguished from the velocity \(\mathbf u\) of the test charge. In a general charge distribution, each source may have its own motion parameters; we first consider a single source or a group sharing the same motion.
The central physical assumption is as follows: the source's electric state is transferred through the same motion channels and with the same splitting coefficients as the source's own state of motion. This assumption connects the algebraic construction to the field; it does not follow from idempotency alone.
2. Successive Splitting Channels
We briefly reproduce the required construction from the paper on the Lorentz factor. For independent commuting hyperbolic units, the projectors are defined as:
\[\tag{2} \j_k^2=1,\qquad \pmp_k=\frac{1+\j_k}{2},\qquad \pme_k=\frac{1-\j_k}{2}, \qquad \pmp_k^2=\pmp_k,\quad \pme_k^2=\pme_k,\quad \pmp_k\pme_k=0,\quad \pmp_k+\pme_k=1. \] At each level, the projector \(\pme_k\) selects the branch that is fixed at that level, while \(\pmp_k\) selects the remainder that continues to split. We denote the resulting channels by indexed idempotents:
\[\tag{3} \ep_n=\left(\prod_{k=1}^{n}\pmp_k\right)\pme_{n+1}, \qquad \ep_n\ep_m=\delta_{nm}\ep_n. \] This paper uses a multilevel extension of the original four-component idempotent basis. For the sequence of channels (3), it takes the form \(\mathcal I_\infty=\{\ep_0,i\ep_0,\ep_1,i\ep_1,\ldots\}\). The coefficients of the state under consideration are real, so the directions \(\ep_n\) are used; the directions \(i\ep_n\) allow phase components of the channels to be described when needed.
For \(n=0\), the product is absent. The indexed \(\ep_n\) denote channels of the deep splitting chain; they should not automatically be identified with the two original idempotents of the finite operator \(J\). For any finite number of levels, a residual branch remains:
\[\tag{4} 1=\sum_{n=0}^{N}\ep_n+R_{N+1}, \qquad R_{N+1}=\prod_{k=1}^{N+1}\pmp_k. \] To calculate norms, we specify a separate orthonormal metric: each channel corresponds to a unit direction \(\boldsymbol\xi_n\), with \(\boldsymbol\xi_n\cdot\boldsymbol\xi_m=\delta_{nm}\). Thus, for a state with coefficients \(a_n\), we adopt
\[\tag{5} \left\|\sum_{n=0}^{\infty}a_n\ep_n\right\|^2 =\sum_{n=0}^{\infty}|a_n|^2. \] This is the metric norm of the channel state. It is not obtained by simply squaring the idempotent sum algebraically.
3. Self-Similarity and Overall Normalization
Let the linear transfer \(S\) move a state to the next level while preserving its metric norm:
\[\tag{6} S\ep_n=\ep_{n+1},\qquad \|SX\|=\|X\|. \] The transfer acts on channel states; it is not assumed to be an automorphism of the entire idempotent algebra. At each transition, the amplitude is multiplied by \(\beta\). Repeating the same rule generates the coefficients \(1,\beta,\beta^2,\ldots\). The unnormalized sum has norm \(\gamma\), so the total electric state is written as:
\[\tag{7} \boxed{ \mathcal E_0= \frac{E_0}{\gamma} \sum_{n=0}^{\infty}\beta^n\ep_n = \frac{E_0}{\gamma} \left(\ep_0+\beta\ep_1+\beta^2\ep_2+\cdots\right). } \] The subscript zero in \(\mathcal E_0\) indicates its origin in the general electric state; the state itself depends on \(\beta\). Normalizing the entire series gives:
\[\tag{8} \|\mathcal E_0\|^2 =\frac{|E_0|^2}{\gamma^2} \sum_{n=0}^{\infty}\beta^{2n} =\frac{|E_0|^2}{\gamma^2(1-\beta^2)} =|E_0|^2. \] Motion redistributes amplitudes among the channels while preserving the total scale \(|E_0|\). This is conservation of the internal norm; without an additional physical mapping law, it cannot be identified with conservation of the energy density of the observable field.
At \(\beta=0\), only the initial branch remains. At low speeds, subsequent terms decay rapidly. The series converges for every \(|\beta|<1\), but as \(|\beta|\to1\), the number of significantly contributing levels increases. Substituting \(\beta=1\) does not define a normalized state in this space: a finite photon regime would require a separate limiting construction.
4. Electric and Magnetic Branches
The subscript \(J\) in the notation for field and force branches indicates the internal description of the source's state, represented in the multilevel idempotent basis. The operator \(J(a,b)\) itself retains its role as an operator of state and motion.
We isolate the initial channel and its entire orthogonal complement. Series (7) admits an exact recursive form:
\[\tag{9} \boxed{ \mathcal E_0= \underbrace{\frac{E_0}{\gamma}\ep_0}_{\mathcal E_J} + \underbrace{\beta S\mathcal E_0}_{c\mathcal B_J}. } \] In the proposed hypothesis, \(\mathcal E_J\) defines the electric branch in the multilevel idempotent basis. The orthogonal continuation excited by motion defines the magnetic branch \(\mathcal B_J\):
\[\tag{10} \mathcal E_J=\frac{E_0}{\gamma}\ep_0,\qquad \mathcal B_J= \frac{E_0}{c\gamma} \sum_{n=1}^{\infty}\beta^n\ep_n. \] The magnetic component thus arises from transferring the general electric state into the motion channels. Its existence within the model is determined by splitting, while its circulating spatial manifestation will be specified in Part II. Assigning a magnetic nature to this branch is a physical interpretation within the hypothesis.
The magnetic branch comprises the entire remainder, rather than only the individual term containing \(\ep_1\). Its norm is therefore:
\[\tag{11} \|\mathcal E_J\|=\frac{|E_0|}{\gamma}, \qquad c^2\|\mathcal B_J\|^2 =\frac{|E_0|^2}{\gamma^2}\sum_{n=1}^{\infty}\beta^{2n} =\beta^2|E_0|^2. \] \[\tag{12} \boxed{ \|\mathcal E_J\|^2+c^2\|\mathcal B_J\|^2=|E_0|^2. } \] The sign of \(E_0\) is retained throughout the state, and the sign of \(\beta\) is retained in the coefficients of the odd-numbered levels. The norm does not retain these signs, so orientation must be extracted from the state before taking its magnitude.
5. Connection with the Finite Operator J
In the model's concept, the finite operator has the form \(J(a,b)=\j^a(-\j)^b\), where \(a\) characterizes the internal state and \(b\) characterizes external motion. Its external phase branch gives:
\[\tag{13} b=\frac{\arcsin\beta}{\pi},\qquad \ep J(0,b)=\ep e^{i\pi b} =\ep\left(\frac1\gamma+i\beta\right). \] This is a compressed representation of the initial branch amplitude and the signed amplitude of the entire remainder. It does not reconstruct the individual deep channels. Nor are the real and imaginary parts themselves two idempotents. The connection between the electric and magnetic branches and the deep state is discussed in the paper on the geometric origin of fields; here, it is used as the basis for the hypothesis of a general electric state.
6. The Force State and Charge
We adopt a local law coupling the test charge \(q\) to the general field: multiplication by charge defines the total force state in the multilevel idempotent basis. Then
\[\tag{14} \boxed{ \mathcal F_J=q\mathcal E_0 =\frac{qE_0}{\gamma}\sum_{n=0}^{\infty}\beta^n\ep_n =q\mathcal E_J+qc\mathcal B_J. } \] Both terms have the dimensions of force. The first defines the electric branch of the force state, and the second defines the magnetic branch. With the adopted metric,
\[\tag{15} \|\mathcal F_J\|^2 =\|q\mathcal E_J\|^2+\|qc\mathcal B_J\|^2 =q^2|E_0|^2. \] Once law (14) is adopted, the decomposition and normalization of the force state follow automatically. However, the mechanical force on a test charge also depends on its motion. Equation (14) therefore represents an internal precursor of the electromagnetic force response; its correspondence to the observable Lorentz force is established by a separate mapping rule in Part II.
7. Possible Additional Fields and Forces
This paper identifies only the electric field and the magnetic field with a circulating spatial mapping. However, motion opens many channels, so the hypothesis allows for more than two field components to be identified within the general electric state in a more complete physical description. The number of channels alone does not prove the existence of independent new fields: each such field requires a distinct observable property, an interaction law, and a method of verification.
In the two-branch description, all levels \(n\ge1\) are combined into \(\mathcal B_J\). If part of the remainder is subsequently given an independent physical interpretation, the residual branch must be divided into nonoverlapping groups of channels. A contribution cannot simultaneously be included in the full magnetic branch and added again as a new field.
A generalized decomposition can be expressed using orthogonal projectors \(\pmp_E\), \(\pmp_B\), and \(\pmp_a\) onto groups of channels. Here, the subscripts denote groups rather than levels of binary splitting:
\[\tag{16} \pmp_E+\pmp_B+\sum_{a}\pmp_a=I,\qquad \pmp_r\pmp_t=0\quad(r\ne t), \] \[\tag{17} \mathcal E_0=\pmp_E\mathcal E_0+\pmp_B\mathcal E_0+\sum_a\pmp_a\mathcal E_0. \] In the extended description, \(\pmp_B\) selects only the part of the remainder that retains its magnetic interpretation. The other \(\pmp_a\) are candidates for additional components; their coefficients are provisionally treated as having the common dimensions of electric field strength.
The same applies to the force state:
\[\tag{18} \mathcal F_J=q\pmp_E\mathcal E_0+q\pmp_B\mathcal E_0+\sum_aq\pmp_a\mathcal E_0. \] The force state may have more than two internal components. In particular, the magnetic branch already contains many channels. If some of them have independent observable effects, the model's total force may include additional contributions. The classical Lorentz force, however, remains the sum of electric and magnetic terms; new independent observable terms would constitute a generalization of that law and would require experimental confirmation.
Part II
Mapping Fields and Forces into 3D Space
Mapping Fields and Forces into 3D Space
8. The Signed Amplitude of the Magnetic Branch
Before assigning a spatial direction, the magnetic branch must be compressed to a signed amplitude. Its norm alone is insufficient because the norm discards signs. We introduce a unit state for the transferred remainder:
\[\tag{19} N_\beta=\frac1\gamma\sum_{n=0}^{\infty}\beta^n\ep_n, \qquad U_\beta=SN_\beta,\qquad \|U_\beta\|=1. \] Then \(\mathcal E_0=E_0N_\beta\) and \(c\mathcal B_J=\beta E_0U_\beta\). In the adopted metric, the signed amplitude is obtained from the inner product:
\[\tag{20} e_J=\frac{E_0}{\gamma},\qquad b_J=\langle U_\beta,\mathcal B_J\rangle=\frac{\beta E_0}{c}. \] This compression is defined for the family of self-similar states under consideration. It does not imply that an arbitrary deep state can be replaced by a single number without loss of information.
9. Spatial Directions and Metric Factors
Let \(\mathbf n_s\) be a fixed unit direction along the axis of motion, \(\mathbf v_s=c\beta\mathbf n_s\), and let \(\mathbf n_E\) specify the direction of the initial electric state. We decompose this direction relative to the motion:
\[\tag{21} \mathbf n_{E\parallel}=(\mathbf n_s\cdot\mathbf n_E)\mathbf n_s,\qquad \mathbf n_{E\perp}=\mathbf n_E-\mathbf n_{E\parallel}. \] The initial spatial orientation of the electric state is specified by the vector \(\mathbf n_E\). The direction of the observable electric field is determined after separately mapping its longitudinal and transverse components. The magnetic direction is specified by the normal \(\mathbf n_s\times\mathbf n_E\). Unlike a unit normal, this cross product retains the factor \(\sin\vartheta\), where \(\vartheta\) is the angle between the directions. It vanishes when the directions are parallel.
Assigning directions alone does not determine the relativistic amplitudes. For a source with a purely electric field in its rest frame, we introduce a mapping rule consistent with the standard field transformations [1]:
\[\tag{22} \mathbf E_{\mathrm{obs}} =\gamma e_J\mathbf n_{E\parallel} +\gamma^2e_J\mathbf n_{E\perp}, \qquad \mathbf B_{\mathrm{obs}} =\gamma b_J(\mathbf n_s\times\mathbf n_E). \] Here, \(E_0\mathbf n_E\) corresponds to the source's rest-frame field evaluated at the event that transforms into the selected observation event. The coordinates must be transformed along with the field components; comparing values at the same numerical distance in two frames is not a substitute for this transformation.
Substituting (20) gives:
\[\tag{23} \boxed{ \mathbf E_{\mathrm{obs}} =E_0\mathbf n_{E\parallel}+\gamma E_0\mathbf n_{E\perp}, \qquad \mathbf B_{\mathrm{obs}} =\frac{\gamma\beta E_0}{c}(\mathbf n_s\times\mathbf n_E) =\frac{\mathbf v_s\times\mathbf E_{\mathrm{obs}}}{c^2}. } \] The factors in (22) are introduced to ensure relativistic consistency. They have not yet been derived from the channel recursion alone. The normalization of the internal state and the observable field strength refer to different objects, so the mapping need not preserve the Euclidean sum of the squares of the observable fields. Rule (23) concerns the transformation of a purely electric field and does not cover an arbitrary source with its own magnetic field or radiation due to acceleration.
10. Circular Magnetic Field Lines
Consider uniform motion along the \(z\) axis and an axisymmetric initial electric field. In this subsection, the coordinate \(z\) is measured relative to the current position of the source or the center of the distribution; it is a geometric coordinate used to describe field lines, rather than a transformed rest-frame coordinate. In a fixed laboratory frame, the field of a moving localized source generally depends on time. In a cylindrical basis, the initial direction lies in the meridional plane:
\[\tag{24} \mathbf n_s=\mathbf n_z,\qquad \mathbf n_E=a(r,z)\mathbf n_r+d(r,z)\mathbf n_z. \] Since \(\mathbf n_E\) is a unit vector, the coefficients satisfy \(a^2(r,z)+d^2(r,z)=1\).
Mapping the magnetic branch along the normal produces an azimuthal direction:
\[\tag{25} \mathbf n_z\times\mathbf n_E=a(r,z)\mathbf n_\varphi, \qquad \mathbf B_{\mathrm{obs}}=B_\varphi(r,z)\mathbf n_\varphi. \] The tangent to a field line is directed along \(\mathbf B\). Therefore, the radial and longitudinal coordinates remain constant along every nonzero field line:
\[\tag{26} dr=0,\qquad dz=0,\qquad x=r\cos\varphi,\quad y=r\sin\varphi. \] The resulting lines are circles around the axis of motion. Thus, the circulating geometry arises through the spatial mapping of the orthogonal branch. This does not imply that an individual charge necessarily moves along a magnetic field line. The circular shape applies to axisymmetric geometry; it is not universal for arbitrary currents. The geometry of the tangents also does not replace the full equation for the curl of the magnetic field.
11. Force Mapping and the Observable Lorentz Force
The internal force state has already been obtained by multiplying the field by charge. Its observable action must additionally account for the test charge velocity \(\mathbf u\), which generally differs from the source velocity:
\[\tag{27} \boldsymbol\beta_u=\frac{\mathbf u}{c},\qquad \mathcal F_J=q\mathcal E_J+qc\mathcal B_J. \] We propose the following force mapping rule. The electric branch gives \(q\mathbf E_{\mathrm{obs}}\), while the magnetic branch, after the field mapping, couples to the test charge's motion through the antisymmetric spatial cross product:
\[\tag{28} q\mathcal E_J\longmapsto q\mathbf E_{\mathrm{obs}},\qquad qc\mathcal B_J\longmapsto q\boldsymbol\beta_u\times(c\mathbf B_{\mathrm{obs}}). \] \[\tag{29} \boxed{ \mathbf F=q\left(\mathbf E_{\mathrm{obs}}+\mathbf u\times\mathbf B_{\mathrm{obs}}\right). } \] This rule recovers the standard Lorentz force [2]. At \(\mathbf u=0\), the magnetic force vanishes, even though the source's magnetic branch may be nonzero. For motion along \(\mathbf B\), the magnetic response also vanishes. The magnetic component is perpendicular to the velocity and does no work directly:
\[\tag{30} \mathbf u\cdot\mathbf F_B=0,\qquad \mathbf F\cdot\mathbf u=q\mathbf E_{\mathrm{obs}}\cdot\mathbf u. \] In the regime under consideration, substituting (23) explicitly reveals the contributions of the two motions:
\[\tag{31} \mathbf F=q\left[ \mathbf E_{\mathrm{obs}}+ \frac{\mathbf u\times(\mathbf v_s\times\mathbf E_{\mathrm{obs}})}{c^2} \right]. \] The relativistic mechanical force is defined as \(\mathbf F=d(\gamma_um\mathbf u)/dt\), where \(\gamma_u=(1-u^2/c^2)^{-1/2}\); it should not generally be replaced by \(m\,d\mathbf u/dt\).
The recovery of (29) is conditional: it follows after adopting rule (28). Splitting and multiplication by charge determine the internal force state, but do not by themselves derive the antisymmetric coupling to the test charge's motion. A rigorous derivation of the Lorentz force from the splitting model represented in the idempotent basis requires justification of this rule. A separate symbol for an interaction operator is not needed here, but the physical content of the interaction remains an independent assumption of the hypothesis.
12. Consistency Check: The Magnetic Field of a Straight Wire
In vacuum, for a steady current and low carrier speeds, mapping (23) takes the form \(d\mathbf B=\mathbf v_s\times d\mathbf E_0/c^2\). The Coulomb field scale of a charge element \(dq\) gives:
\[\tag{32} d\mathbf E_0=\frac{dq}{4\pi\varepsilon_0}\frac{\mathbf R}{R^3},\qquad d\mathbf B=\frac{\mu_0}{4\pi}\frac{dq\,\mathbf v_s\times\mathbf R}{R^3}, \qquad \mu_0\varepsilon_0c^2=1. \] For a straight wire, \(dq=\lambda\,dz\) and \(I=\lambda v_s\). At a distance \(r\) from the axis, with \(\mathbf R=r\mathbf n_r-z\mathbf n_z\), all magnetic contributions point along \(\mathbf n_\varphi\). Summation along an infinitely long wire gives:
\[\tag{33} \mathbf B(r)=\frac{\mu_0I}{4\pi}\mathbf n_\varphi \int_{-\infty}^{+\infty}\frac{r\,dz}{(r^2+z^2)^{3/2}} =\frac{\mu_0I}{2\pi r}\mathbf n_\varphi. \] This yields circular field lines and the dependence \(B\propto I/r\). For a current of 1 A and a distance of 1 cm, the magnitude is approximately 20 μT. This example checks the consistency of the adopted mapping in the magnetostatic regime; it is not independent experimental confirmation of the hidden channel structure.
In a neutral wire, the electric contributions of different carriers may cancel, while the magnetic contributions persist because their motions differ. The states and mappings of the individual sources must therefore be constructed first, and their observable fields then summed. A single common parameter \(\beta\) is insufficient for an arbitrary mixture of sources.
13. The Content of the Hypothesis and Further Work
The proposed hypothesis combines four assumptions. The general electric state participates in the splitting of the source's motion. Normalization through \(\gamma\) preserves its total metric norm. The initial channel is given an electric interpretation, and the orthogonal remainder a magnetic interpretation. Mapping the remainder into space along the normal to the plane defined by the motion and the electric direction produces a circulating magnetic geometry and, under axial symmetry, circular field lines.
Multiplication by charge defines the total force state. Mapping this state while accounting for the test charge velocity recovers the classical Lorentz force under the adopted interaction rule. The many internal channels allow a more detailed division into field and force components, but establishing new independent observable fields will require additional laws and experimental signatures.
The channel series, normalization, and norms of the two branches follow from the mathematical construction. The joint transfer of the field and source, the magnetic interpretation of the remainder, the metric factors of the observable mapping, and the force response law remain physical assumptions. The next tasks are to derive these mappings from a more general interaction rule for states represented in the idempotent basis, describe changes in motion over time, establish a connection with Maxwell's equations, and determine which deep components may have independent physical manifestations.