2026-10-15
A geometric explanation of the Zeeman effect via splitting in an i-basis
A single spectral line can split into several components when a light source is placed in a magnetic field. This Zeeman effect shows that states with initially equal energies become energetically distinguishable. In the geometry of Wave Electricity, a natural question arises: can these states be represented as splitting channels in the idempotent i-basis, with the energy separation related to the oriented geometry of an internal charge-carrying loop? We develop such a description below, starting with two idempotent channels and a spin doublet, then extending the discussion to atomic sublevels and spectral lines.
This work develops the Concept of Wave Electricity, uses the result of the article "Internal Motion of the Electron and the Origin of Its Magnetic Moment," and employs the framework of channel selection and exchange operators. The magnetic-moment scale is not fitted anew. The main aim is to show how it enters the energy geometry of the split state.
The derivation is conditional on explicitly specified physical correspondences: the geometry of the charge-carrying loop, its electromagnetic coupling law, and the spin mapping. The projector algebra alone does not determine the magnetic energy. All quantities are expressed in SI units; the main calculation concerns spin splitting linear in the field, without the anomalous correction.
1. One Energy Level and Several State Channels
We first distinguish state splitting from energy splitting. The former means that independent components exist. The latter occurs when the generator of time evolution assigns different eigenvalues to them. Thus, two orthogonal states may correspond to the same energy level.
Let a pair of channels be selected by the projectors \(\pmp\) and \(\pme\):
\[\tag{1} \pmp^2=\pmp,\qquad \pme^{\,2}=\pme,\qquad \pmp\pme=0,\qquad \pmp+\pme=I. \] Here \(I\) is the identity operator in the state space under consideration. If both channels have energy \(E_0\), then
\[\tag{2} H_0=E_0\pmp+E_0\pme=E_0I. \] The energies coincide, although other observables can still distinguish the states. This is energy degeneracy. When the field is switched on, we need an operator that acts on the orientation channels with opposite signs.
2. The Idempotent i-Basis and the Geometry of Orientation Channels
In the original phase construction, the state operator has the form
\[\tag{3} J(a,b)=\j^a(-\j)^b=\ep e^{i\pi b}+\em e^{i\pi a},\qquad \j=\ep-\em,\qquad \j^2=1. \] The parameter \(a\) describes the internal state, while \(b\) describes the external motion of the center. In what follows, \(\pmp\) and \(\pme\) denote projectors for a separately identified binary orientation factor; they are not automatically identified with the primary phase planes \(\ep\), \(\em\). Passing from the phase algebra to physical orientation requires an amplitude space and a rule for physical rotations.
On this binary factor, define the channel-distinguishing operator:
\[\tag{4} \j_{\mathrm s}=\pmp-\pme,\qquad \j_{\mathrm s}^{\,2}=I,\qquad \pmp=\frac{I+\j_{\mathrm s}}2,\quad \pme=\frac{I-\j_{\mathrm s}}2. \] Idempotents can be visualized as independent directions in state space. Their algebraic orthogonality does not mean that the physical loops lie in mutually perpendicular spatial planes. These are different levels of geometric description.
The two sheets traversed sequentially by a single internal wave and the two alternative values of the spin projection are different structures. Each spin-projection state includes the entire internal loop. In this article, energy splitting concerns orientation states.
3. Channel Selection and Exchange: Constructing the Measurement Axis
Changing the orientation basis requires exchange operations. We use the construction from the work on the Dirac equation in splitting operators. For a state \(\Psi=\pmp u+\pme v\) with complex amplitudes, define the exchange:
\[\tag{5} \mathsf S[\pmp u+\pme v]=\pmp v+\pme u,\qquad \mathsf S^2=I,\qquad \mathsf S\j_{\mathrm s}=-\j_{\mathrm s}\mathsf S. \] In amplitude space, adopt the norm \(\|\Psi\|^2=|u|^2+|v|^2\). The distinguishing and exchange operators generate three Hermitian operators:
\[\tag{6} Q_x=\mathsf S,\qquad Q_y=i\mathsf S\j_{\mathrm s},\qquad Q_z=\j_{\mathrm s}, \qquad Q_iQ_j=\delta_{ij}I+i\varepsilon_{ijk}Q_k. \] This is the Pauli algebra obtained through operations on the channels. Adopt its physical spin mapping and rotation law:
\[\tag{7} \widehat{\mathbf S}=\frac{\hbar}{2}\mathbf Q,\qquad U_{\mathbf n}(\Theta)=\exp\!\left(-\frac{i\Theta}{2}\mathbf Q\cdot\mathbf n\right). \] For a nonzero field, introduce the unit axis \(\mathbf n_B=\mathbf B/B\), where \(B=|\mathbf B|\), giving
\[\tag{8} \j_B=\mathbf Q\cdot\mathbf n_B,\qquad \j_B^2=I,\qquad \pmp_B=\frac{I+\j_B}{2},\qquad \pme_B=\frac{I-\j_B}{2}. \] The identity \(\j_B^2=I\) follows from (6): the antisymmetric part of the product vanishes when contracted with \(n_{B,i}n_{B,j}\), while the symmetric part gives \(|\mathbf n_B|^2I=I\). Thus, the possible eigenvalues of \(\j_B\) are \(+1\) and \(-1\). They correspond to spin projections \(+\hbar/2\) and \(-\hbar/2\). At \(B=0\), the axis can be chosen arbitrarily.
The field therefore selects a physically relevant axis from the possible orientation bases. The new pair of projectors describes the same two-dimensional spin factor; introducing it does not imply an additional level of internal structure.
Fig. 1. Orthogonal orientation channels have equal energies without a field and opposite energy shifts in a field.
4. The Geometric Origin of the Magnetic-Moment Scale
For the coupling strength, use the result of the previous magnetic-moment calculation. To zeroth order, assume two circular traversals with the same orientation and mean radius \(R\), steady transport of a single charge \(q=-e\) at speed \(c\), and negligible connecting segments:
\[\tag{9} R=\frac{\hbar}{m_ec},\qquad L_\Gamma=4\pi R,\qquad T_\Gamma=\frac{4\pi R}{c},\qquad I_q=\frac{q}{T_\Gamma}. \] The oriented area of the complete path is defined by the integral
\[\tag{10} \mathbf A_\Gamma=\frac12\oint_\Gamma\mathbf r\times d\mathbf r =2\pi R^2\mathbf n_\Gamma. \] The magnitude of the magnetic moment of the thin loop is
\[\tag{11} \mu_0=\frac{e}{T_\Gamma}|\mathbf A_\Gamma| =\frac{ecR}{2}=\frac{e\hbar}{2m_e}=\mu_B. \] The period and area double together, so the two-sheet structure alone does not enhance the moment. Comparing (11) with the spin projection \(\hbar/2\) yields the previously obtained result \(g_0=2\). The radius, the charge-transport law, and the correspondence between the loop normal and spin polarization remain physical assumptions of this mapping.
5. The Field Distinguishes Orientations Through Magnetic Flux
Consider an external static field that is effectively uniform on the scale of the internal loop. Introduce the standard electromagnetic coupling of the current to the vector potential:
\[\tag{12} L_{\mathrm{int}}=\int\mathbf j_q\cdot\mathbf A\,d^3r,\qquad \mathbf B=\nabla\times\mathbf A. \] This is an additional dynamical law corresponding to the term \(q\mathbf v\cdot\mathbf A\) in the Lagrangian of a charge. For a thin, steady-current loop:
\[\tag{13} L_{\mathrm{int}}=I_q\oint_\Gamma\mathbf A\cdot d\mathbf r =I_q\,\mathbf B\cdot\mathbf A_\Gamma. \] The last equality follows from Stokes' theorem. For two traversals, the flux is summed with their orientation and multiplicity taken into account. Under a gauge transformation \(\mathbf A\mapsto\mathbf A+\nabla\chi\), the circulation around the closed path is unchanged for a single-valued regular function \(\chi\).
Substituting the geometry (9)-(10) gives
\[\tag{14} L_{\mathrm{int}}=-\frac{ecR}{2}\,\mathbf n_\Gamma\cdot\mathbf B. \] This is the geometric mechanism of orientation dependence: the field couples to the projection of the loop's oriented area onto its axis. Reversing the normal reverses the sign of the flux and of the linear coupling. However, a classical normal admits a continuous range of directions; the existence of exactly two eigenvalues is supplied by the separately adopted spin mapping (7).
Fig. 2. The field couples to the oriented area of the complete loop. Both traversals belong to a single normalized charge.
6. Interaction Energy: Checking the Sign
Because the electromagnetic term depends on velocities, energy cannot be obtained by simply identifying it with the Lagrangian. Assume that the internal motion admits a consistent Hamiltonian description. Introduce a field-strength parameter \(\lambda\):
\[\tag{15} L(\xi,\dot\xi;\lambda)=L_0(\xi,\dot\xi)+\lambda L_1(\xi,\dot\xi),\qquad p_\alpha=\frac{\partial L}{\partial\dot\xi_\alpha},\qquad H=\sum_\alpha p_\alpha\dot\xi_\alpha-L. \] Here \(\xi_\alpha\) are internal generalized coordinates. When differentiating with respect to \(\lambda\) at fixed \(\xi,p\), the terms containing \(\partial\dot\xi_\alpha/\partial\lambda\) cancel:
\[\tag{16} \left.\frac{\partial H}{\partial\lambda}\right|_{\xi,p} =-\left.\frac{\partial L}{\partial\lambda}\right|_{\xi,\dot\xi}. \] Consequently, the linear correction to the Hamiltonian is minus the interaction evaluated on the unperturbed motion. For a fast periodic regime, use its average over a period:
\[\tag{17} H_B^{(1)}=-\langle L_{\mathrm{int}}\rangle_0 =\frac{ecR}{2}\mathbf n_\Gamma\cdot\mathbf B =-\boldsymbol\mu_0\cdot\mathbf B,\qquad \boldsymbol\mu_0=-\mu_B\mathbf n_\Gamma. \] In the steady-current mapping, (13) already expresses this average. Applying (16) requires regular dynamics or a consistent reduction of constraints; a prescribed trajectory at speed \(c\) alone does not construct an internal Lagrangian. Thus, (17) follows from minimal coupling provided that consistent loop dynamics exists. The external field is treated as prescribed, while the energy of its source is included in the full energy balance of the system.
7. Energy Splitting in the Idempotent i-Basis
Under the adopted spin mapping, the electron's magnetic moment is a vector operator
\[\tag{18} \widehat{\boldsymbol\mu}_0=-\mu_B\mathbf Q =-\frac{e}{m_e}\widehat{\mathbf S}. \] This uses the correspondence between loop orientation and spin polarization, together with rotational covariance in the two-dimensional spin space. Equation (18) does not follow from the existence of an area alone: it specifies the passage from current-loop geometry to a spin observable.
Then (17) gives
\[\tag{19} H_B^{(1)}=-\widehat{\boldsymbol\mu}_0\cdot\mathbf B =\mu_BB\j_B =\mu_BB(\pmp_B-\pme_B). \] For an isolated initial doublet, \(H_0=E_0I\). In the linear approximation, the full effective Hamiltonian has the spectral decomposition
\[\tag{20} \boxed{H=(E_0+\mu_BB)\pmp_B+(E_0-\mu_BB)\pme_B.} \] Acting on each projector selects its eigenenergy: \(H\pmp_B=(E_0+\mu_BB)\pmp_B\), \(H\pme_B=(E_0-\mu_BB)\pme_B\). Therefore,
\[\tag{21} \boxed{E_\pm=E_0\pm\mu_BB,\qquad \Delta E=E_+-E_-=2\mu_BB=\frac{e\hbar}{m_e}B.} \] At \(B=0\), the coefficients of the two projectors coincide: an energy measurement does not distinguish the channels. At \(B\ne0\), the coefficients differ. The geometric content of spin splitting is that the field selects an axis and associates opposite orientation projections with opposite energy shifts.
To discuss field reversal, it is convenient to retain a fixed \(z\) axis and write \(\mathbf B=B_z\mathbf e_z\), where \(B_z\) may be negative. Then \(E_\pm=E_0\pm\mu_BB_z\), and the energy ordering of the fixed channels reverses. If the axis is instead always directed along the field, the projectors \(\pmp_B\) and \(\pme_B\) themselves interchange when the field reverses.
Equations (20)-(21) isolate the spin contribution. A free charged electron in a uniform field also has orbital dynamics and Landau quantization; a bound electron has orbital and other atomic contributions. The doublet formulas are not the complete electron spectrum in an arbitrary magnetic field.
8. Frequency Splitting and Relative Phase
Use the correspondence between energy and angular frequency, \(E=\hbar\omega\). Then
\[\tag{22} \omega_\pm=\omega_0\pm\Omega_B,\qquad \omega_0=\frac{E_0}{\hbar},\qquad \Omega_B=\frac{\mu_BB}{\hbar}=\frac{eB}{2m_e}. \] With the convention \(i\hbar\partial_t\Psi=H\Psi\), the evolution is written directly in terms of idempotents:
\[\tag{23} U(t)=e^{-i\omega_0t} \left(\pmp_Be^{-i\Omega_Bt}+\pme_Be^{+i\Omega_Bt}\right). \] For a superposition of channels, the amplitude magnitudes remain constant, but their relative phase changes at the rate \(2\Omega_B\). The transition frequency between the branches is therefore
\[\tag{24} \boxed{\omega_{\mathrm{res}}=\frac{\Delta E}{\hbar}=2\Omega_B =\frac{eB}{m_e},\qquad \nu_{\mathrm{res}}=\frac{\Delta E}{h}.} \] The quantity \(\Omega_B\) is the magnitude of the phase-frequency shift of one branch. The frequency \(2\Omega_B\) determines the relative phase and the precession of spin polarization. The unit of \(\omega\) is rad/s, while that of \(\nu\) is Hz. A static diagonal Hamiltonian does not transfer population between its channels. Exciting a transition requires a time-dependent perturbation with a nonzero interchannel matrix element, such as a transverse magnetic field at the appropriate frequency.
This gives electron spin resonance its practical meaning: the position of the resonant response measures the energy difference between orientation channels. For real systems, the coefficient is refined to account for the anomaly, binding, and the environment.
9. From a Binary Doublet to Atomic Sublevels
An atom may have a multichannel orientation space. In a field weak relative to the fine-structure splitting, with hyperfine structure neglected, a level with a definite total electronic angular momentum \(J\) has projections \(m=-J,-J+1,\ldots,J\). Here \(J\) is an angular-momentum quantum number, rather than the operator \(J(a,b)\) in (3).
Denote the projectors onto the magnetic sublevels by \(\Pi_m\). Their spectral organization is
\[\tag{25} \Pi_m\Pi_{m'}=\delta_{mm'}\Pi_m,\qquad \sum_{m=-J}^{J}\Pi_m=I_J,\qquad \frac{\widehat J_z}{\hbar}=\sum_{m=-J}^{J}m\Pi_m. \] After adopting the atomic magnetic mapping with factor \(g_J\), the standard weak-field splitting law can be written in the language of channels:
\[\tag{26} H_Z=g_J\mu_BB\sum_{m=-J}^{J}m\Pi_m,\qquad E_{Jm}=E_J+g_J\mu_BBm. \] This law and the factors for atomic levels are given in the NIST reference on the Zeeman effect. Geometrically, (26) represents the field as assigning different energy weights to the orientation channels. The number of sublevels, \(2J+1\), is determined by the representation of physical rotations. It is not obtained by simply identifying them with all \(2^N\) channels of a binary tree.
For a single electron spin with \(J=1/2\), \(m=\pm1/2\), and \(g_J=2\), equation (26) reproduces (21). In a multielectron atom, the factor \(g_J\) depends on the coupling of orbital and spin angular momenta. When hyperfine structure is resolved, weak-field states are classified by the total angular momentum \(F\), including the nucleus, and the corresponding magnetic mapping is required. These are separate extensions of the simplest doublet.
10. Why the Number of Spectral Lines Differs from the Number of Sublevels
Radiation connects the upper and lower levels. For an allowed transition \((u,m_u)\to(l,m_l)\), the angular frequency is determined by their energy difference:
\[\tag{27} \omega_{ul}=\frac{E_u-E_l}{\hbar} +\frac{\mu_BB}{\hbar}(g_um_u-g_lm_l). \] The spectrum therefore depends on the splitting of both levels, the transition operator, and coincidences between the frequencies of different transitions. For electric dipole radiation, the adopted selection rules for the magnetic quantum number are \(\Delta m=0,\pm1\); additional rules apply to \(J\), parity, and other quantum numbers. These rules require a dipole operator, its rotational properties, and a physical mapping to radiation. The idempotent algebra alone does not establish them.
In the normal Zeeman effect, for a suitable transition between states of zero total spin, both nonzero angular momenta have the orbital factor \(g=1\). Allowed transitions are grouped by \(\Delta m\), producing three possible shifts:
\[\tag{28} \delta\omega=0,\qquad \delta\omega=+\frac{\mu_BB}{\hbar},\qquad \delta\omega=-\frac{\mu_BB}{\hbar}. \] For example, in the dipole transition \(J_u=1\to J_l=0\), the three upper sublevels can decay to a single lower sublevel. This produces a central \(\pi\) component and two \(\sigma\) components. Their polarizations differ when viewed perpendicular to the field; along the field, the \(\pi\) component is absent, while the two \(\sigma\) components have opposite circular polarizations. Thus, the observed pattern also depends on the viewing direction.
More generally, different \(g_u,g_l\) produce a more complex set of frequencies. The term "anomalous Zeeman effect" refers to the spin-dependent structure of the atomic spectrum and is not identical to the electron's anomalous magnetic moment: a complex pattern can arise even at \(g_s=2\).
Fig. 3. Example of an allowed dipole transition J = 1 → J = 0 with normal Zeeman splitting.
11. What Can Be Tested Experimentally
For an isolated spin doublet, the predicted dependence is linear: \(\Delta E=2\mu_BB\). It is tested by measuring the resonance position. For atomic spectra, the shifts of allowed lines and their polarizations are measured; comparison requires knowledge of both participating levels and their magnetic factors. Such observations test the quantitative consistency of the construction.
Agreement with a known law must be distinguished from a test of a new mechanism. The formulas obtained here reproduce the standard weak-field description under the adopted physical correspondences. For an experiment to distinguish the geometric model from conventional quantum theory, an additional consequence must be derived in advance: for example, a definite correction to the coupling coefficient or to interchannel dynamics. Merely introducing new projector notation creates no such distinction.
The calculation does not include the anomalous magnetic moment, binding corrections, deformation of the internal state, strong-field mixing, or quadratic shifts. In fields comparable to internal atomic splittings, classification by a fixed \(J\) may cease to apply. The contribution of the field source and possible losses also require a dynamical description.
12. The Geometric Content of the Result
Idempotent splitting selects independent orientation channels. Their energies may coincide without a field. The magnetic field selects an axis, and electromagnetic coupling to the oriented area of the internal loop assigns different energy coefficients to the channels. For the spin factor, this is expressed by
\[\tag{29} \boxed{\begin{aligned} H_0&=E_0(\pmp_B+\pme_B),\\ H_B^{(1)}&=\mu_BB(\pmp_B-\pme_B),\\ \Delta E&=2\mu_BB. \end{aligned}} \] We have obtained a consistent geometric interpretation of spin splitting and its operator generalization to atomic magnetic sublevels. Channel geometry explains the structure of the spectral decomposition; the adopted current and spin mappings, together with minimal coupling, determine its physical scale. Deriving the internal loop dynamics, justifying the mappings, and calculating new observable corrections remain separate tasks.
References
- David Tong. Classical Dynamics. The Hamiltonian Formalism: electromagnetic coupling and the Legendre transformation.
- NIST. Atomic Spectroscopy — Zeeman Effect: magnetic sublevels and g-factors.
- David Tong. Applications of Quantum Mechanics: atomic interactions with the electromagnetic field.