Single-particle interference: SMM/DTT and Bohmian mechanics

01The question behind this comparison

Single-particle interference is interference between alternative amplitudes contributing to one detection event. The particle number can be one while the quantum state has support on several paths. An interference pattern accumulates over many trials because the relative phase changes the probability of each individual detection; it does not require two particles to collide at the detector.

The SMM/DTT idea of sequential inner production offers a possible physical narrative for how alternative contributions belong to one completed event. Its useful mathematical content would be a coherent rule for combining those contributions before registration. A classical source that randomly chooses one path and retains only its label produces a mixture, not an interference pattern. Phase memory, a recombination operation and the measurement context therefore carry the decisive explanatory work. ^{[3] [4]}

02The SMM/DTT starting point

The Single Monad Model begins with one nonspatial generative source. The Duality of Time Theory distinguishes the source’s inner production from the outer order in which completed records become accessible. Its six inner levels are nested functional roles of one process: a higher role organizes lower-level work. The seventh is the completion or observation interface. They are neither six independent clocks nor six additional time coordinates.

This distinction supplies a useful research question: which features of an observed world can follow from the way a source produces, retains and releases information? A concrete answer needs a state space, an autonomous update, retained memory and a rule connecting completion to an accessible measurement. In the constructions discussed here, those mathematical ingredients are stated explicitly. A successful finite example establishes what its specified mechanism can do; further arguments connect it to physical particles, spacetime or universal laws. ^[1]

03How the comparator works

Bohmian mechanics supplements a wavefunction with an actual configuration, ordinarily particle positions. The Schrödinger equation guides the wavefunction, and a velocity law guides the configuration. When configurations follow the quantum-equilibrium distribution, the theory reproduces the usual quantum statistics in its applicable domain. Apparatus outcomes are ultimately configurations too, so measurement is treated as a physical interaction rather than a separate primitive collapse.

This is an instructive comparison for SMM because it provides an explicit account of definite observed events while retaining an additional, nonclassical level of description. But a Bohmian particle follows a trajectory in a configuration space whose geometry is supplied. The Single Monad proposal instead seeks to explain how spatial and particle records arise at all. The challenge is therefore not to rename a guiding wave as inner time. It is to show what variables replace the configuration, how records obey the equilibrium statistics, and whether the replacement adds an observable consequence. ^[2]

04The decisive comparison

Bohmian interference involves a particle with an actual path guided by a wave that traverses the interferometer’s available alternatives. The empty branch of the wave can affect the eventual trajectory when the branches meet. DTT may also give one realized detection sequence while maintaining generative contributions that do not become separate particle records. This is a meaningful resemblance: an observed particle count of one need not mean that only one alternative participates in the dynamics.

The difference is the guiding structure. Bohmian mechanics specifies a configuration-space wavefunction and its velocity law. A source model cannot substitute the assertion that the Monad visits both paths unless its retained phase and readout reproduce the same guidance statistics. The benchmark must include changing the phase in an apparently unoccupied arm, because that operation changes the output distribution. A DTT explanation that uses only the finally realized path misses this dependence. Its distinctive ambition would be to generate both the effective wave and the path geometry from the source, rather than inherit them. That additional achievement requires an explicit dynamics beyond the interference formula itself. ^{[2] [3] [4]}

05Worked benchmark

Use an ideal balanced Mach–Zehnder interferometer. After the first beam splitter the path state is (|0⟩+|1⟩)/√2. A phase plate multiplies the second component by e^{iφ}. With a second balanced recombiner in the corresponding phase convention, the first output amplitude becomes (1+e^{iφ})/2. Its probability is cos²(φ/2), and the other output has probability sin²(φ/2).

For φ=0 every ideal trial reaches the first output; for φ=π none does; for φ=π/2 the outputs are equiprobable. If an environment records the paths in states with overlap γ, the interference term becomes Re(γe^{iφ}) and the visibility is |γ|. This calculation explains precisely what an inner-time mechanism must preserve: the relative phase and the coherence between alternatives. Changing the duration of private work while keeping these quantities and the registered phase fixed produces no new interference prediction. ^{[3] [4]}

06What follows for the source programme

The comparison rewards a model that identifies why a completion event combines amplitudes coherently, and why which-path information changes that combination. A finite-state simulation can exhibit the mechanism explicitly. Its result is an implementation of the interference law until independent restrictions on phase generation, memory capacity or recombination create a new testable relationship. A claimed fringe shift needs a formula tied to measured settings rather than an adjustable inner duration. ^{[3] [4]}

THE SAME QUESTION, DIFFERENT STARTING POINTS

The comparison at a glance

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