Single-particle interference: SMM/DTT and standard Hilbert-space quantum 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. ^{[2] [3]}

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

Operational quantum mechanics represents a preparation by a positive density operator of unit trace. A measurement outcome corresponds to an effect, and an instrument specifies both its probability and the conditional state left behind. Closed evolution is unitary; an open system can evolve by a completely positive, trace-preserving map. These ingredients are sufficient to calculate laboratory statistics without deciding whether the wavefunction describes an ontic object, a relation or an agent’s expectations.

The instrument language is especially close to DTT’s distinction between production and accessible records. An unobserved interaction and a registered outcome are mathematically different objects. Yet an ordinary instrument does not itself contain a hidden generative time, nor does a sequence of instruments derive the space in which laboratories operate. It provides a precise interface against which a proposed inner process can be tested: if the process reproduces exactly the same instruments, the difference is explanatory or structural unless another accessible measurement distinguishes it. ^[2]

04The decisive comparison

Standard quantum mechanics produces interference by adding amplitudes before taking their squared modulus. A DTT implementation can produce the two contributions sequentially in inner work, provided it retains their phase coherently and recombines them before the relevant record is registered. Simultaneous support of a wavefunction on two paths and sequential processing of its amplitudes are compatible descriptions of the same calculation; private computation order is not an additional measured path.

The actual explanatory opportunity is to account for the coherent addition rule from the source’s composition law. An ordinary classical register that records which alternative was selected cannot produce the phase-sensitive cross term by simple frequency addition. It needs an additional complex-amplitude resource or another mechanism with the same operational effect. The benchmark above makes this requirement explicit at all phase settings. If the source rule derives a restricted visibility or a phase relation from independently fixed resources, it can become physically informative. Reproducing cos²(φ/2) after importing a conventional qubit and recombiner is a valid realization, but the quantum interference law remains an input. ^{[2] [3]}

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. ^{[2] [3]}

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. ^{[2] [3]}

THE SAME QUESTION, DIFFERENT STARTING POINTS

The comparison at a glance

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