2026-09-17
Geometric model of the tunneling effect in a system of channels and ports
A particle approaches a potential barrier, yet its energy is insufficient to pass through it in the conventional manner. Classically, it ought to be reflected. Nevertheless, experiments sometimes detect the intact particle on the other side of the barrier. It does not acquire the missing energy, nor does it shatter into pieces or fly over the obstacle. How can the possibility of its appearance persist in a region not accessible via a classically allowed trajectory?
This paper proposes a geometric model of this process framed in terms of spatial channels and ports. The barrier blocks the direct port of the particle's spatial realization but does not destroy the complete state or exhaust all avenues for its continuation. The blocked port becomes the entry point for a subsequent splitting - and then another - creating a sequence of deeper residual ports. Beyond the barrier, spatial realization becomes permissible once more; thus, the preserved potential can manifest as an intact particle.
This approach makes it possible to derive the fundamental exponential law of tunneling without recourse to the Schrödinger equation. The derivation relies on an energy invariant, the spatial component of the action, sequential idempotent splitting, and a specific rule governing transitions between ports. We also clearly distinguish between propositions derived from earlier work and those introduced here as novel physical principles.
1. The Particle and Its Spatial Manifestation
In the unified concept of Wave Electricity [1], a particle is viewed not as a pre-defined material point, but as a stable, closed state of a spatially manifested wave. The particle's internal periodicity and the motion of its center belong to distinct levels of description. The center may alter its speed and direction, while the internal closure maintains the object's integrity.
Consequently, an external obstacle does not necessarily destroy the particle.
It can alter the way its full state manifests in the observable space. Let us designate this manifestation prior to the barrier as port \(P_{mathrm{in}}\): \[\tag{1} P_{mathrm{in}}\xrightarrow{\;\mathcal M\;} \text{observable spatial state of the particle}. \] Here, \(\mathcal M\) is the physical mapping of the port's potential onto a specific spatial state. The port and its realization are not identical: the port defines the potential for manifestation, while the mapping determines exactly what that potential transforms into.
2. Channels, ports, and the realization of potentials
We employ the terminology introduced in the paper "Splitting of Potentials: Channels and Ports of Space" [2]. A channel is an independent direction in which the initial state can potentially continue. A port is the endpoint of a channel at the level under consideration, representing a distinct potential for manifestation.
\[\tag{2} X\longrightarrow \begin{cases} C_+\longrightarrow P_+,\\ C_-\longrightarrow P_-. \end{cases} \] This splitting does not divide the particle into two material objects. Rather, it reveals two distinguishable ways for a single initial whole to continue. The idempotent basis for such a division takes the form
\[\tag{3} \ep^2=\ep,\qquad \em^2=\em,\qquad \ep\em=0,\qquad \ep+\em=1. \] The equality \(\ep\em=0\) expresses the independence of the channels, while \(\ep+\em=1\) signifies the preservation of the initial whole. However, the algebra itself merely establishes the structure of potentials. It does not select a port, specify a transition probability, or automatically transform the port into a physical space.
Hereafter, it is necessary to distinguish between four operations: splitting creates channels and ports; mapping assigns a physical meaning to a port; transition transfers the state from one port to another; realization transforms the port's potentiality into an observable event.
3. What the potential barrier forbids
Let \(\varepsilon\) be the particle's external energy after separating out the invariant rest energy, and let \(U(s)\) be the potential energy along the actual one-dimensional coordinate \(s\). From the relativistic energy invariant
\[\tag{4} E_{\mathrm{rel}}^2-p_s^2c^2=m^2c^4 \] it follows, in the first non-relativistic approximation, that
\[\tag{5} E_{\mathrm{rel}}\approx mc^2+\frac{p_s^2}{2m}. \] After introducing the external interaction and excluding the rest energy, the energy balance takes the form
\[\tag{6} \boxed{\varepsilon=U(s)+\frac{p_s^2}{2m}.} \] If \(\varepsilon\geq U(s)\), there exists a real momentum
\[\tag{7} p_s=\sqrt{2m\bigl(\varepsilon-U(s)\bigr)}, \] and ordinary spatial continuation is kinematically allowed. If, however,
\[\tag{8} U(s)>\varepsilon, \] no real
The value \(p_s\) cannot satisfy formula (6). This signifies not the disappearance of the particle, but the closure of the direct realization of its external motion within the given spatial port. \[\tag{9} \boxed{U>\varepsilon \quad\Longrightarrow\quad \text{the port of ordinary spatial motion is closed to realization}.} \] The port itself, as a possibility, is not destroyed. It can become the entry point for a new bifurcation, opening up other ways for the full state to continue.
4. The closed port as the entry point for the next bifurcation
A port is a terminal point only relative to the chosen level of description. If the possibility allows for further refinement, the port becomes the starting point for new channels. Let us designate the initial port before the barrier as \(R_0\). We can write its first bifurcation as follows:
\[\tag{10} R_0\longrightarrow \begin{cases} C_1^-\longrightarrow P_1^-,\\ C_1^+\longrightarrow R_1. \end{cases} \] The port \(P_1^-\) corresponds to the possibility of completing the bifurcation and obtaining a direct spatial manifestation at the first level. Inside the barrier, this realization is forbidden by the condition \(U>\varepsilon\). The second channel leads to a residual port \(R_1\), which has not yet achieved a complete spatial manifestation and can therefore undergo further bifurcation:
\[\tag{11} R_1\longrightarrow \begin{cases} C_2^-\longrightarrow P_2^-,\\ C_2^+\longrightarrow R_2. \end{cases} \] If the port \(P_2^-\) also cannot be realized within the barrier, the continuation is preserved via \(R_2\). Repeating the process creates a chain \[\tag{12} \boxed{R_0\longrightarrow R_1\longrightarrow R_2 \longrightarrow\cdots\longrightarrow R_N.} \]
Each \(R_n\) represents not a new particle, but a deeper possibility for the continuation of the same complete state.
5. Port addresses and the residual branch
The sequence of selected channels forms a port address. A port reached after \(N\) continuations along the unfinished \(+\) channel has the address
\[\tag{13} \boxed{R_N=P_{\underbrace{+\,+\,\ldots\,+}_{N}}.} \] A port address is not a coordinate in an additional physical dimension. A coordinate answers the question of where a point is located within already manifested space. An address answers a different question: through what sequence of splittings the possibility in question has been preserved.
Algebraically, the residual branch corresponds to the product of continuation projectors:
\[\tag{14} R_N=\prod_{k=1}^{N}p_k^+, \qquad p_k^+=\frac{1+\j_k}{2}. \] And a completed port at level \(n\) corresponds to
\[\tag{15} P_n^-= \left(\prod_{k=1}^{n-1}p_k^+\right)p_n^-, \qquad p_n^-=\frac{1-\j_n}{2}. \] Distinct completed ports are orthogonal because the product \(p_n^+p_n^-=0\) arises at the level where they diverge. A residual port retains a portion of the full structure that has not yet undergone final realization.
6. A simple geometric explanation of tunneling
Before the barrier, the particle is realized via a standard spatial port. Inside the barrier, this mode of realization is forbidden by the energy balance. However, the full state does not vanish: the closed port opens more deeply, and the potential for continuation is transferred to the next residual port. This operation repeats for the duration of the barrier.
Beyond the barrier, the potential energy once again falls to a level no higher than the particle's energy. Direct spatial realization becomes permissible again, and the residual port can map to the output state:
\[\tag{16} P_{\mathrm{in}} \longrightarrow R_1 \longrightarrow R_2 \longrightarrow\cdots \longrightarrow R_N \longrightarrow P_{\mathrm{out}}. \] Tunneling is a transition from spatial realization to a sequence of unrealized residual ports and back to spatial realization.
Deep ports should not be visualized as hidden levels of ordinary three-dimensional space; rather, they are orthogonal channels representing the potentiality of the full state. Consequently, one cannot attribute a standard classical trajectory to the particle while it is within the barrier. Only the input state, the reflection, and the potential detection of the intact particle beyond the barrier are observed.
7. Splitting is not equivalent to transition
The mere existence of ports does not automatically imply a transition between them. Idempotent algebra answers the question of what independent possibilities exist, but the dynamics must separately determine how the state amplitude is transferred between ports. \[\tag{17} \boxed{\text{existence of ports} \neq \text{transition between ports} \neq \text{realization of a port}.} \]
Let us denote the operator for the transition to the next level by\(\mathcal T_n\):
\[\tag{18} R_{n-1}\xrightarrow{\;\mathcal T_n\;}R_n. \] Let \(A_n\) be the amplitude of the total state preserved in the residual port \(R_n\). We define a single transition by the coefficient \(q_n\):
\[\tag{19} \boxed{A_n=q_nA_{n-1},\qquad 0 < q_ n< 1.} \] After \(N\) successive transitions, we obtain
\[\tag{20} A_N=\left(\prod_{n=1}^{N}q_n\right)A_0. \] Under uniform conditions where \(q_n=q\) and the input amplitude is unity:
\[\tag{21} A_N=q^N, \qquad T_N=|A_N|^2=q^{2N}. \] It is precisely this geometric progression that links tunneling to the self-similar structure of the work "Origin of the Lorentz Factor from Infinite Idempotent Splitting" [3]. However, the coefficient \(q\) here is not equal to the velocity \(\beta=v/c\). While the mathematical structure of sequential self-similarity is the same, its physical mapping differs.
8. Barrier Width and Address Length
The wider the barrier, the longer the conventional spatial realization remains closed. Let us divide the forbidden region into successive small segments:
\[\tag{22} L=\sum_{n=1}^{N}\Delta s_n. \] We associate a single transition of residual capability with each segment:
\[\tag{23} \Delta s_n:\qquad R_{n-1}\longrightarrow R_n. \] For a uniform partition where \(\Delta s_n=\Delta s\):
\[\tag{24} N=\frac{L}{\Delta s}. \] This does not imply the existence of a fundamental cell of length \(\Delta s\). The partition serves as an intermediate method for describing the continuous barrier and will disappear in the final integral. The following correspondence holds physical significance:
\[\tag{25} \boxed{\text{barrier width}\longleftrightarrow \text{residual port address length}.} \] 9. Barrier height and port closure measure
In the forbidden region, we introduce a positive energy deficit:
\[\tag{26} \Delta(s)=U(s)-\varepsilon>0. \] From the energy balance (6), the following formal equality follows:
\[\tag{27} p_s^2=-2m\Delta(s). \] There is no real momentum associated with ordinary motion here. Therefore, we introduce the positive quantity
\[\tag{28} \boxed{p_B(s)=\sqrt{2m\Delta(s)} =\sqrt{2m\bigl(U(s)-\varepsilon\bigr)}.} \] The quantity \(p_B\) is not an observable momentum of the particle within the barrier. It measures the degree of incompatibility between direct spatial motion and the energy balance. The larger \(p_B\), the more rapidly the possibility of state preservation within the sequence of residual ports must diminish.
The formal expression \(p_s=\pm i p_B\) is possible as an analytic continuation of formula (7), but it is not required for the subsequent derivation.
10. Barrier action as splitting depth
In the phase description of external motion, the spatial part of the action takes the form
\[\tag{29} d\mathcal S_s=p_s\,ds. \] Since real motion within the barrier is forbidden, we define a positive barrier action:
\[\tag{30} d\mathcal D=p_B(s)\,ds. \] A natural dimensionless measure of this action is
\[\tag{31} \boxed{d\delta=\frac{d\mathcal D}{\hbar} =\frac{p_B(s)}{\hbar}\,ds.} \] Within the framework of the proposed model, the quantity \(d\delta\) acquires a new physical meaning: it represents the depth of sequential splitting required to maintain the possibility of continuation over the segment \(ds\).
For the entire forbidden region between points \(s_1\) and \(s_2\):
\[\tag{32} \boxed{\delta_B=\frac{1}{\hbar} \int_{s_1}^{s_2}p_B(s)\,ds.} \] The correspondence stating that "barrier action, measured in units of \(\hbar\), equals the depth of sequential splitting" constitutes the central new physical principle of this work. It does not follow solely from idempotent algebra.
11. Why exponential decay arises
Let \(Q(\delta)\) be the amplitude retention coefficient in the residual chain after traversing a depth \(\delta\). Two consecutive barrier segments form a single combined sequence of ports. Therefore, the coefficients must be multiplied:
\[\tag{33} Q(\delta_1+\delta_2)=Q(\delta_1)Q(\delta_2). \] We also require
\[\tag{34} Q(0)=1, \qquad 0 < Q(\delta) < 1\quad\text{for}\quad\delta > 0. \] The continuous solution to the functional equation (33) has the exponential form
\[\tag{35} Q(\delta)=e^{-C\delta}, \qquad C>0. \] We fix the depth scale \(\delta\) by the condition \(C=1\). Then the transmission coefficient for a small segment is
\[\tag{36} q_n=e^{-\Delta\delta_n} =\exp\left[-\frac{p_B(s_n)\Delta s_n}{\hbar}\right]. \] The amplitude after the entire sequence is:
\[\tag{37} \begin{aligned} A_N &=\prod_{n=1}^{N}q_n\\ &=\exp\left[-\frac{1}{\hbar} \sum_{n=1}^{N}p_B(s_n)\Delta s_n\right]. \end{aligned} \] In the continuous limit, the sum becomes an integral:
\[\tag{38} \boxed{A_B= \exp\left[-\frac{1}{\hbar} \int_{s_1}^{s_2}p_B(s)\,ds\right] =e^{-\delta_B}.} \] Thus, the exponential arises not from the Schrödinger equation, but from two geometric properties: the additivity of the depth of successive segments and the multiplicativity of transitions between ports.
12. Tunneling probability
We adopt the probabilistic rule: the probability of a specific output port being realized is proportional to the square of the modulus of the corresponding amplitude. For internal barrier traversal, this yields
\[\tag{39} T_B=|A_B|^2=e^{-2\delta_B}. \] Substituting formulas (28) and (32), we obtain the main result:
\[\tag{40} \boxed{ T_B= \exp\left[-\frac{2}{\hbar} \int_{s_1}^{s_2} \sqrt{2m\bigl(U(s)-\varepsilon\bigr)}\,ds \right].} \] The formula has an intuitive meaning.
The mass \(m\) characterizes the inertia of the closed state; the difference \(U-\varepsilon\) indicates the degree to which the direct port is closed; the extent \(ds\) accumulates the depth of the barrier; the constant \(\hbar\) converts the action into a dimensionless splitting depth. 13. Rectangular barrier
Consider a barrier of constant height \(U_0\) and width \(L\):
\[\tag{41} U(s)= \begin{cases} U_0,&0\leq s\leq L,\\ 0,&s < 0\, \text{or}\, s > L, \end{cases} \qquad U_0>\varepsilon. \] Inside it, the quantity \(p_B\) is constant:
\[\tag{42} p_B=\sqrt{2m(U_0-\varepsilon)}. \] The barrier depth is equal to
\[\tag{43} \delta_B=\frac{L}{\hbar} \sqrt{2m(U_0-\varepsilon)}. \] The amplitude of the preservation of residual possibility:
\[\tag{44} A_B= \exp\left[-\frac{L}{\hbar} \sqrt{2m(U_0-\varepsilon)}\right]. \] The probability:
\[\tag{45} \boxed{ T_B= \exp\left[-\frac{2L}{\hbar} \sqrt{2m(U_0-\varepsilon)}\right].} \] The formula reproduces the necessary dependencies: increasing the barrier width or height reduces the probability, while increasing the particle energy raises it; heavy objects tunnel significantly less readily than light ones. In the limit \(U_0\to\varepsilon\), the effective depth of the forbidden region tends to zero; however, near the limit of applicability, the entry and exit transitions must be accounted for separately.
14. Entry into and exit from the barrier
Formula (40) describes amplitude conservation within the forbidden region but does not yet account for the efficiency of state reconfiguration at the two boundaries. At the entry point, the observed port must transition into the residual chain:
\[\tag{46} P_{\mathrm{in}} \xrightarrow{\;\mathcal T_{\mathrm{in}}\;} R_1. \] At the exit, the reverse mapping occurs:
\[\tag{47} R_N \xrightarrow{\;\mathcal T_{\mathrm{out}}\;} P_{\mathrm{out}} \xrightarrow{\;\mathcal M\;} \text{detected particle}. \] Therefore, the total transmission amplitude must have the structure
\[\tag{48} \boxed{A_T=C_{\mathrm{out}}\,e^{-\delta_B}\,C_{\mathrm{in}},} \] and the probability is
\[\tag{49} T=|C_{\mathrm{in}}|^2|C_{\mathrm{out}}|^2e^{-2\delta_B}. \] The coefficients \(C_{\mathrm{in}}\) and \(C_{\mathrm{out}}\) must be derived from the geometry of the port coupling at the boundaries. Until they are derived, formula (40) should be regarded as the principal exponential factor rather than the complete, exact transmission coefficient for a barrier of arbitrary shape.
15. Reflection and preservation of the whole
At the input boundary, there are at least two possibilities: the restoration of the spatial manifestation in front of the barrier and the preservation of the residual branch through to the exit side. Formally:
\[\tag{50} |\Psi_{\mathrm{in}}\rangle \longrightarrow A_R|\Psi_R\rangle+A_T|\Psi_T\rangle. \] If there are no other interaction channels, normalization requires
\[\tag{51} |A_R|^2+|A_T|^2=1. \] This probabilistic decomposition does not imply a physical fragmentation of the detected particle. In each individual experiment, a single, integral outcome is realized: the particle is detected either in front of the barrier or beyond it. It is the possibilities of realization that split, not the material content of the particle.
16. What happens inside the barrier
In the proposed description, one should not look for the conventional trajectory of the center within the barrier. The direct spatial gateway is closed; therefore, the coordinate of motion along the classically allowed path is not realized there....is preserved. Only the sequence of possibilities for the continuation of the complete state remains: \[\tag{52} R_0\rightarrow R_1\rightarrow R_2\rightarrow\cdots\rightarrow R_N. \]
Therefore, it is misleading to say that the particle temporarily acquires additional energy, moves at an imaginary velocity, or decays into components. It is more accurate to say: its direct spatial manifestation is forbidden, but the amplitude of the complete state is preserved within a deep chain of ports and diminishes with each successive transition.
If the residual possibility reaches the rear boundary and is successfully mapped to \(P_{\mathrm{out}}\), a whole particle is registered. If a reflected port is realized, the particle is detected in front of the barrier.
17. Physical examples
Tunneling manifests in the decay of atomic nuclei, in electron transport across thin insulating layers, in tunnel diodes, and in the scanning tunneling microscope. In all these cases, the probability is extremely sensitive to the width of the forbidden region. A small change in distance alters the accumulated depth \(\delta_B\), and the exponential function transforms this into a significant change in current or decay probability.
The proposed model offers a general geometric explanation for this: increasing the width lengthens the address of the residual port; increasing the energy deficit reduces the transition coefficient at each segment; and increasing the mass intensifies the degree to which direct spatial manifestation is suppressed.
18. Results obtained and additional assumptions
The following results from previous work on Wave Electricity are utilized:
1. A particle constitutes a stable, internally closed state.
2. External motion is separated from internal periodicity.
3. Idempotent splitting creates independent channels that preserve the original whole.
4. A port at one level can serve as the input for the subsequent splitting.
5. The sequential residual branch possesses a self-similar structure.
6. Energy and momentum are linked by a relativistic invariant.
7. The spatial component of the action takes the form \(p_sds\).
2. External motion is separated from internal periodicity.
3. Idempotent splitting creates independent channels that preserve the original whole.
4. A port at one level can serve as the input for the subsequent splitting.
5. The sequential residual branch possesses a self-similar structure.
6. Energy and momentum are linked by a relativistic invariant.
7. The spatial component of the action takes the form \(p_sds\).
In the present work, additional physical rules are introduced:
1. The condition \(U>\varepsilon\) closes the direct port of spatial realization but allows for continuation via residual ports.
2. The quantity \(d\delta=p_Bds/\hbar\) determines the depth of sequential splitting across the barrier segment.
3. Sequential transition coefficients are multiplied.
4. The probability of realizing the output port is determined by the square of the amplitude's modulus.
2. The quantity \(d\delta=p_Bds/\hbar\) determines the depth of sequential splitting across the barrier segment.
3. Sequential transition coefficients are multiplied.
4. The probability of realizing the output port is determined by the square of the amplitude's modulus.
Adopting these rules yields the following:
\[\tag{53} \boxed{ \begin{aligned} U>\varepsilon &\longrightarrow \text{closure of the direct port},\\ p_B=\sqrt{2m(U-\varepsilon)} &\longrightarrow d\delta=\frac{p_B}{\hbar}ds,\\ \text{sequence of ports} &\longrightarrow A_B=e^{-\delta_B},\\ \text{probabilistic mapping} &\longrightarrow T_B=e^{-2\delta_B}. \end{aligned}} \] A separate task remains: the derivation of the operators \(\mathcal T_{\mathrm{in}}\) and \(\mathcal T_{\mathrm{out}}\), the exact boundary coefficients, the phase effects associated with multiple barriers, and the temporal characteristics of the process. These results cannot be considered to follow automatically from the port structure alone.
Conclusion
A potential barrier does not block the particle's entire state structure, but rather a specific possibility for its direct spatial continuation. A closed port becomes the input for the next splitting event. If spatial realization is again forbidden, the state transitions to the next residual port. Thus, a sequence of unrealized yet persisting possibilities belonging to a single integral state emerges within the barrier.
The barrier width determines the extent of this sequence; the energy deficit determines the difficulty of each transition; and the action (in units of \(\hbar\)) determines the total depth of the splitting. Since the depths of successive segments are additive while transition coefficients are multiplicative, the amplitude decreases exponentially.
\[\tag{54} \boxed{ \text{spatial realization} \longrightarrow \text{residual ports} \longrightarrow \text{spatial realization}.} \] In this framework, the particle is not required to traverse the forbidden region in a classical sense. Instead, the possibility of the full state's continuation is preserved across it. Tunneling thus appears not as motion along an impossible trajectory, but as a way of bypassing the prohibition deep within the split system of channels and ports.