2026-10-05
Dirac equation in splitting operators: construction and connection with J
Why does the Dirac equation require four complex components and several matrices to describe a single particle? Underlying these objects, one can discern a system of simpler operations: the isolation of state channels, the alteration of their phases, and the exchange between them. In this paper, the four components are organized through two independent binary splittings, and the matrices are replaced by operators acting directly on these channels.
The advantage of the proposed formulation is the ability to trace the equation's structure from elementary transformations to relativistic dynamics and spin. We will derive the necessary anticommutation relations, obtain the operator form of the Dirac equation, demonstrate spinor periodicity, and express time evolution in the form of \(J\). The resulting construction is mathematically equivalent to the standard one under the conditions adopted below; its interpretation as a physical mechanism of splitting remains a hypothesis.
This work builds upon the approach presented in the article "The Wave Function as a Representation of a Split Event." The initial notation and the distinction between the algebraic structure and its physical mapping are outlined in the Wave Electricity concept. We consider a free particle in a pre-existing flat spacetime; the origins of space, mass, and interactions are not derived here.
1. Binary splitting and the phase operator
Let two channels be distinguished by mutually complementary idempotents:
\[\tag{1} \ep^2=\ep,\qquad \em^2=\em,\qquad \ep\em=0,\qquad \ep+\em=1,\qquad \j=\ep-\em,\qquad \j^2=1. \] We represent the state of a single binary factor as an element of a complex-extended algebra:
\[\tag{2} Q=\ep u+\em v,\qquad u,v\in\mathbb C,\qquad \langle Q,Q'\rangle=u^*u'+v^*v',\qquad \|Q\|^2=|u|^2+|v|^2. \] In this notation, the idempotents also serve as basis elements of the state space. Multiplication by them acts as a projection. The metric (2) is defined separately: the algebraic product of \(J\) and its conjugate is not identical to the unit Hilbert norm in (2).
\[\tag{3} J(a,b)=\ep e^{i\pi b}+\em e^{i\pi a},\qquad J(a,b)Q=\ep e^{i\pi b}u+\em e^{i\pi a}v. \] Thus, multiplication by \(J\) preserves the state norm and alters the two phases. It does not change the amplitudes \(u\) and \(v\) associated with the channels.
In the initial concept, \(a\) is associated with the internal state and \(b\) with external motion; new phase parameters introduced below for other splittings will be denoted separately. 2. Channel exchange: definition of the operator \(S\)
Let us introduce a complex-linear transformation:
\[\tag{4} \boxed{S[\ep u+\em v]=\ep v+\em u.} \] \(S\) maps each amplitude to the other channel, preserving its magnitude and phase. Two exchanges restore the initial state. With metric (2), the operator is Hermitian and unitary:
\[\tag{5} S^2=I,\qquad S^\dagger=S,\qquad S^\dagger S=I,\qquad \|SQ\|=\|Q\|. \] For example, \(S\) maps a state residing entirely in the first channel to the second: \(S[\ep]=\em\). If the amplitudes are equal and have the same phase, the state remains unchanged. Its action on the phase element \(J\) takes a particularly simple form:
\[\tag{6} S[J(a,b)]=J(b,a). \] Here, we are referring to the application of \(S\) to a state element. If, however, \(J\) is understood as a multiplication operator, the corresponding equality takes the form \(SJS=J(b,a)\). A phase permutation does not transform an internal physical parameter into a velocity.
Any element of the original commutative algebra acts via multiplication separately in each channel. Therefore, \(S\) cannot be represented as multiplication by a fixed \(J\): the exchange is an additional operation. For multidimensional channels, such a definition requires a chosen norm-preserving correspondence between them. In the present work, both channels of a single factor are one-dimensional over the complex numbers.
Physical exchange must be distinguished from the relabeling of channels. In an active transformation, the channels and observables remain fixed, while the state changes. The law governing this change must be specified by the dynamics; it may relate either to the internal motion of the free state or to an external influence.
3. How anticommutation arises
Let us denote the operator of multiplication by \(\j\) using the same symbol. It distinguishes the channels by opposite signs:
\[\tag{7} \j Q=\ep u-\em v,\qquad S[\j Q]=-\ep v+\em u,\qquad \j S[Q]=\ep v-\em u. \] Consequently, for any \(Q\), the following holds:
\[\tag{8} \boxed{S\j=-\j S.}\] This is evident in the state of the first channel without calculating the general formula: \(Sj[\epsilon]=\epsilon m\), whereas \(jS[\epsilon]=-\epsilon m\). The result depends on the order of operations. Anticommutation is derived from the definition of the swap, not added to it independently.
Hereafter, the symbols for idempotents and hyperbolic units in operator products denote the corresponding multiplication operators, and \(I\) denotes the identity operator. The operator applied last acts on the left.
4. Continuous mixing and three basic operators
The full swap \(S\) can be used as a generator of a continuous transformation:
\[\tag{9} U_S(\theta)=e^{-i\theta S}=\frac{I+S}{2}e^{-i\theta}+\frac{I-S}{2}e^{i\theta}. \] When the parameter is zero, \(U\) equals \(I\), and when \(\theta=\pi/2\), it equals \(-iS\): a full swap with a global phase occurs. For example, the Hamiltonian \(gS\) with real energy \(g\) defines coupled amplitude equations:
\[\tag{10} i\hbar\dot u=gv,\qquad i\hbar\dot v=gu,\qquad \theta=gt/\hbar. \] This is an example of possible dynamics, not an independent derivation of the swap law. Now, let us construct three operators from the two operations:
\[\tag{11} q_1=S,\qquad q_2=iSj,\qquad q_3=j. \] It follows from (8) that \((iS_j)^2=-S_j S_j=I\). The remaining relations are verified similarly:
\[\tag{12} q_i^\dagger=q_i,\qquad \{q_i,q_k\}=2\delta_{ik}I,\qquad q_1q_2=iq_3,\qquad [q_i,q_k]=2i\varepsilon_{ik\ell}q_\ell. \] Here, curly brackets denote the sum of the two multiplication orders, while square brackets denote their difference. The Pauli algebra is thus obtained through the isolation and exchange of channels, without the use of matrix element tables.
5. Two independent binary factors
Let us consider the tensor product of two such spaces. These represent two independent binary factors, rather than two successive operations on the same pair of channels. We associate \(j_1, S_1\) with the first factor and \(j_2, S_2\) with the second:
\[\tag{13} j_r^2=S_r^2=I,\qquad S_r j_r=-j_r S_r,\qquad [j_1,j_2]=[S_1,S_2]=[S_1,j_2]=[S_2,j_1]=0. \] Let us denote the projectors of the first factor as \(e_+, e_-\) and those of the second as \(p_+, p_-\). The four joint channels are distinguished by the products \(e_+ p_+\), \(e_+ p_-\), \(e_- p_+\), and \(e_- p_-\). Let us choose an orthonormal basis \(\xi_{rs}\), where \(r,s\) take the values (+1\) and \(-1\):
\[\tag{14} \Psi=\sum_{r,s=\pm1}\psi_{rs}\xi_{rs},\qquad \j_1\xi_{rs}=r\xi_{rs},\qquad \j_2\xi_{rs}=s\xi_{rs},\qquad S_1\xi_{rs}=\xi_{-r,s},\qquad S_2\xi_{rs}=\xi_{r,-s}. \] For example, \(S_1\xi_{++}=\xi_{-+}\) and \(S_2\xi_{++}=\xi_{+-}\). The two indices specify four complex components. They describe an internal state space, not four spacetime coordinates.
6. Replacing Dirac matrices with channel operators
Let us define four constant operators:
\[\tag{15} \boxed{B=\j_1,\qquad A_1=S_1S_2,\qquad A_2=iS_1S_2\j_2,\qquad A_3=S_1\j_2.} \] The squares of all these operators are equal to \(I\). Each \(A\) contains \(S_1\) and therefore anticommutes with \(B\). For example, \(A_1A_2=i\j_2\) and \(A_2A_1=-i\j_2\); other pairs can be verified similarly. In summary:
\[\tag{16} B^\dagger=B,\qquad A_i^\dagger=A_i,\qquad B^2=I,\qquad \{A_i,B\}=0,\qquad \{A_i,A_k\}=2\delta_{ik}I. \] These are precisely the relations required for the Dirac Hamiltonian form. The correspondence is as follows:
| Standard notation | Channel operator |
|---|---|
| Matrix \(\beta_D\) | \(\j_1\) |
| Matrix \(\alpha_1\) | \(S_1S_2\) |
| Matrix \(\alpha_2\) | \(iS_1S_2\j_2\) |
| Matrix \(\alpha_3\) | \(S_1\j_2\) |
Here, \(\beta_D\) does not denote the velocity \(v/c\). If one chooses the standard coordinate basis, \(\j_r\) and \(S_r\) are again represented by matrices, and (15) reproduces the standard Dirac representation. The advantage of the operator notation lies in the explicit definition of actions on the channels; this does not imply computational superiority over matrices.
7. Construction of the free-particle equation
We seek a Hamiltonian that is linear in momentum and consistent with relativistic energy.
We adopt a constant mass \(m\), the scales \(c\) and \(\hbar\), and the form \[\tag{17} H=c\sum_{i=1}^3A_ip_i+mc^2B. \] This is an additional dynamical condition: the splitting algebra alone does not determine the Hamiltonian. For a free particle, the momentum components commute with each other and with the constant internal operators. When (17) is squared, the cross terms vanish:
\[\tag{18} H^2=c^2\sum_i p_i^2I+c^2\sum_{i<k}\{A_i,A_k\}p_ip_k+mc^3\sum_i\{A_i,B\}p_i+m^2c^4I. \] \[\tag{19} \boxed{H^2=(c^2\mathbf p^2+m^2c^4)I.} \] Thus, the constructed operators ensure the required relativistic dependence. Adopting the standard representation of momentum \(p_i=-i\hbar\partial_i\) and the law of time evolution, we obtain
\[\tag{20} \boxed{i\hbar\partial_t\Psi=\left(-i\hbar c\sum_iA_i\partial_i+mc^2B\right)\Psi.} \] Substitution of (15) yields the equation directly in terms of splittings and exchanges:
\[\tag{21} i\hbar\partial_t\Psi=\left[-i\hbar c\left(S_1S_2\partial_x+iS_1S_2\j_2\partial_y+S_1\j_2\partial_z\right)+mc^2\j_1\right]\Psi. \] Applying the time derivative again, we obtain a second-order verification:
\[\tag{22} \left(\frac{1}{c^2}\partial_t^2-\nabla^2+\frac{m^2c^2}{\hbar^2}\right)\Psi=0. \] This relation holds for each component of the solution, but it does not by itself replace the first-order equation that couples the components to one another.
8. Norm conservation and covariant form
The Hermiticity of the operators allows one to directly obtain a local balance equation:
\[\tag{23} \rho=\Psi^\dagger\Psi,\qquad \mathcal J_i=c\Psi^\dagger A_i\Psi,\qquad \partial_t\rho+\sum_i\partial_i\mathcal J_i=0. \] The integral of the density is conserved if there is no flux at the boundary—for instance, if the state decays sufficiently rapidly at infinity. Consequently, exchanges and spatial dynamics are consistent with the conservation of the total norm.
For a covariant formulation, we introduce \(x^0=ct\), the metric \(\eta=\operatorname{diag}(1,-1,-1,-1)\), and the operators
\[\tag{24} \Gamma^0=B,\qquad \Gamma^i=BA_i,\qquad \{\Gamma^\mu,\Gamma^\nu\}=2\eta^{\mu\nu}I. \] In particular, the square of the temporal generator is \(I\), while that of the spatial generators is -\(I\). Multiplying (20) on the left by \(B/c\) yields
\[\tag{25} \left(i\hbar\Gamma^\mu\partial_\mu-mc\right)\Psi=0. \] This is the covariant form of the Dirac equation. For consistency under the coordinate transformation \(x'=\Lambda x\), the state must transform via a spinor operator \(L\):
\[\tag{26} \Psi'(x')=L\Psi(x),\qquad L^{-1}\Gamma^\mu L=\Lambda^\mu{}_{\nu}\Gamma^\nu. \] Products of the generators (24) define such a representation of Lorentz transformations. Below, we explicitly show its rotational part and provide an example of a boost. The standard connection between Clifford algebra and spinor transformations is outlined in [3]; here, it is realized by channel operators.
9. Spinors from two-level splitting
Four components constitute a spinor by virtue of the transformation law, not merely because of their number. Let us construct the rotation generators:
\[\tag{27} \Sigma_1=-iA_2A_3=S_2,\qquad \Sigma_2=-iA_3A_1=iS_2\j_2,\qquad \Sigma_3=-iA_1A_2=\j_2. \] We define the intrinsic spin operators \(\widehat s_i=\hbar\Sigma_i/2\). From the algebra already verified, it follows that
\[\tag{28} [\widehat s_i,\widehat s_k]=i\hbar\varepsilon_{ik\ell}\widehat s_\ell,\qquad \widehat{\mathbf s}^{\,2}=\sum_i\widehat s_i^2=\frac34\hbar^2I. \] This value is \(s(s+1)\hbar^2\) for \(s=1/2\). Note that for a moving particle, the individual \(\widehat s_i\) operators in this representation need not commute with \(H\); we use them as the internal part of the rotation generators. A full rotation of the spatial wave function also involves a change in its coordinate argument.
Rotation about a unit direction \(\mathbf n\) is defined by the operator
\[\tag{29} R_{\mathbf n}(\theta)=\exp\left(-\frac{i\theta}{2}\mathbf n\cdot\boldsymbol\Sigma\right),\qquad \Psi'(\mathbf x)=R_{\mathbf n}(\theta)\Psi(\mathcal R^{-1}\mathbf x), \] where \(\mathcal R\) is the corresponding standard spatial rotation. The relation \([\Sigma_i,A_k]=2i\varepsilon_{ik\ell}A_\ell\) ensures the consistent rotation of the momentum operators and the internal components of the Hamiltonian. It is precisely the factor of \(1/2\) that leads to the spinor law.
\[\tag{30} P_{\mathbf n,\pm}=\frac{I\pm\mathbf n\cdot\boldsymbol\Sigma}{2},\qquad R_{\mathbf n}(\theta)=P_{\mathbf n,+}e^{-i\theta/2}+P_{\mathbf n,-}e^{i\theta/2}. \] \[\tag{31} \boxed{R_{\mathbf n}(2\pi)=-I,\qquad R_{\mathbf n}(4\pi)=I.} \] The half-angle here arises within the framework of the chosen rotation law, which is consistent with the constructed Hamiltonian. The connection to the model's physical double-sheeted contour requires a separate mapping: identical periodicity is insufficient to identify the mechanisms.
Example 1. Spin along the \(z\)-axis. For any first ind index \(r\):
\[\tag{32} \widehat s_z\xi_{r,+}=\frac{\hbar}{2}\xi_{r,+},\qquad \widehat s_z\xi_{r,-}=-\frac{\hbar}{2}\xi_{r,-}. \] In the chosen representation, the second factor distinguishes two spin projections. At non-zero momentum, the first index does not automatically indicate the sign of the energy.
Example 2. Spin along the \(x\)-axis. Let us take the normalized combinations:
\[\tag{33} \chi_{x,\pm}=\frac{\xi_{r,+}\pm\xi_{r,-}}{\sqrt2},\qquad S_2\chi_{x,\pm}=\pm\chi_{x,\pm}. \] These states have a definite spin projection along the \(x\)-axis. When measured along the \(z\)-axis, both outcomes have a probability of \(1/2\) if the Born rule is applied. Consequently, a definite spin along one axis can be a combination of the two channels distinguished by another axis.
Example 3. Full rotation. Rotation around the \(z\)-axis yields \(R_z(\theta)\xi_{r,+}=e^{-i\theta/2}\xi_{r,+}\). At \(2\pi\), the amplitude changes sign; at \(4\pi\), it is restored. A global sign does not change the probability density in itself, but the relative sign between the transformed branch and the coherent reference branch can alter the interference.
Rotation also allows for a \(J\)-type representation:
\[\tag{34} J_{\mathbf n}(a_n,b_n)=P_{\mathbf n,+}e^{i\pi b_n}+P_{\mathbf n,-}e^{i\pi a_n},\qquad R_{\mathbf n}(\theta)=J_{\mathbf n}\left(\frac{\theta}{2\pi},-\frac{\theta}{2\pi}\right). \] 10. Example of a Lorentz boost
The full Dirac spinor must also transform under a change of inertial reference frame. For a boost along \(x\), we choose the convention \(x'^0=\cosh\eta\,x^0-\sinh\eta\,x^1\) and \(x'^1=-\sinh\eta\,x^0+\cosh\eta\,x^1\), where \(\tanh\eta=v/c\). The corresponding spinor operator is:
\[\tag{35} L_x(\eta)=e^{-\eta A_1/2}=\frac{I+A_1}{2}e^{-\eta/2}+\frac{I-A_1}{2}e^{\eta/2}. \] Using anticommutation relations, we verify (26):
\[\tag{36} L_x^{-1}\Gamma^0L_x=\Gamma^0\cosh\eta-\Gamma^1\sinh\eta,\qquad L_x^{-1}\Gamma^1L_x=-\Gamma^0\sinh\eta+\Gamma^1\cosh\eta. \] Thus, the construction defines not only rotations but also boosts. Unlike a rotation, \(L\) is not unitary with respect to the finite-dimensional positive metric of the components: a boost also alters the coordinates and the density. The preservation of physical normalization is expressed in terms of the current and consistent integration over a spatial hypersurface. The real exponentials in (35) should not be identified with the phase factor \(J\) involving real arguments.
11. Simple examples of dynamics
Particle at rest. At zero momentum, \(H=mc^2\j_1\). For \(m>0\), the two channels of the first factor possess opposite energies:
\[\tag{37} \Psi(t)=\sum_{s=\pm1}\left(c_{+,s}e^{-imc^2t/\hbar}\xi_{+,s}+c_{-,s}e^{imc^2t/\hbar}\xi_{-,s}\right). \] Each energy branch contains two spin components. Interpreting the negative energy branch as antiparticles requires field quantization; A mere permutation of indices is insufficient for this.
Motion along \(x\). Let the state have a definite eigenvalue \(S_2\), for example, \(+1\). We define \(\zeta_r=(\xi_{r,+}+\xi_{r,-})/\sqrt2\) and \(\Psi=f_+\zeta_++f_-\zeta_-\). Equation (21) reduces to two coupled equations:
\[\tag{38} i\hbar\partial_t f_+=mc^2f_+-i\hbar c\partial_xf_-,\qquad i\hbar\partial_t f_-=-mc^2f_--i\hbar c\partial_xf_+. \] For a momentum eigencomponent \(p\) with energy \(\varepsilon\), these equations yield
\[\tag{39} (\varepsilon-mc^2)f_+=cp f_-,\qquad (\varepsilon+mc^2)f_-=cp f_+,\qquad \varepsilon^2=m^2c^4+c^2p^2. \] For example, for positive energy \(E\) and \(p\ne0\), the ratio of amplitudes is \(cp/(E+mc^2)\). At low momentum, the second-channel component is small, but it becomes necessary during motion: the energy state is a combination of the channels of the first factor.
Massless limit. For \(m=0\) along \(x\), we have \(H=cA_1p\). For a fixed non-zero \(p\), the energies are \(\pm c|p|\). In coordinate form, the projections \(\Psi_\pm=(I\pm A_1)\Psi/2\) satisfy
\[\tag{40} (\partial_t+c\partial_x)\Psi_+=0,\qquad (\partial_t-c\partial_x)\Psi_-=0. \] Their profiles move in opposite directions at speed \(c\). This is massless spinor dynamics, not the photon equation: the disappearance of mass does not automatically change spin 1/2 to spin 1.
12. Time evolution in the form of the operator \(J\)
For a fixed momentum, let us denote the positive energy scale by \(E_p\). From (19), one can construct a new hyperbolic unit and its projectors:
\[\tag{41} E_p=\sqrt{c^2\mathbf p^2+m^2c^4}>0,\qquad \j_D=\frac{H(\mathbf p)}{E_p},\qquad \j_D^2=I,\qquad \ep_D=\frac{I+\j_D}{2},\qquad \em_D=\frac{I-\j_D}{2}. \] This represents a splitting into energy branches, with two components in each. Let us define
\[\tag{42} J_D(a_D,b_D)=\ep_D e^{i\pi b_D}+\em_D e^{i\pi a_D}. \] Then the exact evolution takes the form
\[\tag{43} \boxed{\Psi_{\mathbf p}(t)=J_D\left(\frac{E_pt}{\pi\hbar},-\frac{E_pt}{\pi\hbar}\right)\Psi_{\mathbf p}(0).} \] Verification is performed directly:
\[\tag{44} U_D(t)=\ep_De^{-iE_pt/\hbar}+\em_De^{iE_pt/\hbar},\qquad i\hbar\dot U_D=E_p(\ep_D-\em_D)U_D=HU_D. \] Thus, the form \(J\) describes the phase evolution of the energy channels, while form (34) describes the rotation of the spin channels. These are distinct projectors and distinct physical parameters. \(J_D\) depends on momentum and already incorporates the structure of the Hamiltonian; expression (43) neither replaces the construction of the Hamiltonian nor derives it from the original \(J\).
For a localized state, (43) is applied to each momentum component, after which the spatial wave function is reconstructed. If \(m=0\) and \(p=0\), then \(E_p=0\) and the division in (41) is impossible; for this component, the free Hamiltonian \(H\) is zero, and the evolution is the identity transformation.
13. Deeper splittings: a hypothesis
Two binary factors suffice for the constructed four-component representation. One might hypothesize that this is merely one accessible level of a more general state. Adding a third factor creates eight complex components but does not, in itself, generate new physics: the Hamiltonian \(H_D\otimes I_2\) describes two independent copies of the previous dynamics.
A new result may emerge if the additional factor is coupled to the original ones via a Hermitian operator \(V\):
\[\tag{45} \mathcal H_{\mathrm{ext}}=\mathcal H_D\otimes\mathbb C^2,\qquad H_{\mathrm{ext}}=H_D\otimes I_2+V,\qquad V^\dagger=V. \] For instance, the formal term \(V=gB\otimes\j_3\) splits the mass coefficient into two branches: \(mc^2+g\) and \(mc^2-g\). Setting \(g=0\) recovers the original equation. This serves as an illustration of an extension rather than a prediction of new particles or a derivation of the nature of mass. For a physical application, it would be necessary to define the meaning of \(g\), the observable states, and the admissible symmetries.
Hypothesis: known equations of motion may serve as an effective description of the accessible level of splitting of a more general state. Additional channels and their couplings could lead to more general equations, from which the familiar laws are recovered by restricting the scope to observable degrees of freedom or by eliminating inaccessible components.
In the general case, the elimination of additional channels may give rise to an effective dynamics that depends on energy and history, rather than merely introducing a new constant into an old formula. Therefore, the transition to a deeper level requires its own coupling law. Even an unlimited depth of algebraic splitting does not prove the existence of an infinite number of physical properties. Each extension must possess a consistent dynamics, conservation laws, and testable consequences.
14. Results and limitations of the construction
The advantage of the proposed formulation is confirmed by the sequence of results obtained. Anticommutation was derived from the isolation and exchange of channels; two binary factors made it possible to construct operators that replace the Dirac matrices. Their algebra ensures the relativistic energy-momentum relation, norm conservation, spin-1/2, and the spinor law of rotations. Energy evolution and rotations are represented by phase constructions of the \(J\)-type.
Thus, the matrix elements have been explicitly described in terms of operations on state channels. This structural advantage of the equivalent formulation allows for an investigation of its connection to the splitting hypothesis. The construction employs additional conditions: a tensor organization of factors, an \(S\)-exchange, a prescribed metric, a linear Hamiltonian, a spatial representation of momentum, and a state transformation law. The Dirac equation is not derived solely from norm conservation or a single phase \(J\).
The next step is to establish a physically grounded connection involving additional splitting and to demonstrate how, under specific conditions, the Dirac equation is recovered from the extended dynamics. If measurable corrections emerge in the process, the hypothesis acquires experimental content. The depth of the splitting characterizes the internal structure of the state and does not necessarily imply additional spatial dimensions or a smaller geometric scale.