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arxiv: 2604.26716 · v1 · submitted 2026-04-29 · 🪐 quant-ph

Delayed Choice Phenomena in the Projection Evolution Model

Marek G\'o\'zd\'z , Andrzej G\'o\'zd\'z , Krzysztof Lider This is my paper

Pith reviewed 2026-05-07 10:40 UTC · model grok-4.3

classification 🪐 quant-ph
keywords delayed choiceprojection evolutionMach-Zehnder interferometertime operatortemporal overlapphoton wave functionfour-dimensional spacetime
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The pith

Delayed-choice interference arises from temporal overlap between a photon's wave function and interferometer devices in the projection evolution model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that treating time as a quantum observable allows the wave function to be defined across four-dimensional spacetime. In this projection evolution framework the delayed-choice behavior of photons in a Mach-Zehnder interferometer follows directly from the temporal overlap between the photon's wave packet and the physical devices that may be inserted or removed after the photon has entered the setup. A reader would care because the account reproduces standard quantum interference patterns while keeping time internal to the dynamics rather than an external parameter. This removes the need to invoke retrocausality or measurement-induced collapse at a specific instant.

Core claim

In the projection evolution model the wave function is defined in both space and time, permitting construction of a time operator. For a photon traveling through a Mach-Zehnder interferometer the delayed-choice experiments are explained by the temporal overlap of the photon and the devices in the interferometer.

What carries the argument

Temporal overlap of the photon's four-dimensional wave function with the time-dependent configuration of the interferometer components.

If this is right

  • Delayed-choice results do not require backward-in-time influences.
  • Interference visibility is determined by an overlap integral that includes both spatial and temporal coordinates.
  • Quantum evolution proceeds consistently inside four-dimensional spacetime without an external time parameter.
  • Standard predictions of quantum optics are recovered whenever the temporal profiles overlap as in conventional setups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Ultrafast control of optical elements synchronized to photon wave-packet duration could test whether overlap alone sets the observed fringe visibility.
  • The same spacetime-overlap mechanism may apply to other time-dependent quantum effects such as the quantum Zeno effect or time-bin entanglement.
  • If the model is correct, precise timing measurements could distinguish it from interpretations that treat time as strictly classical.

Load-bearing premise

The projection evolution model is a valid description of quantum processes and temporal overlap by itself reproduces all observed delayed-choice interference patterns.

What would settle it

An experiment that eliminates temporal overlap between the photon wave packet and the active interferometer devices yet still produces the interference or which-path statistics predicted by standard quantum mechanics.

Figures

Figures reproduced from arXiv: 2604.26716 by Andrzej G\'o\'zd\'z, Krzysztof Lider, Marek G\'o\'zd\'z.

Figure 1
Figure 1. Figure 1: The Mach-Zehnder interferometer The state space of the interferometer consists of two quantum channels based on functions dependent on spacetime coordinates. Assuming channel separation the simplified description contains two orthogonal channels in a two-dimensional spacetime, as indicated view at source ↗
Figure 2
Figure 2. Figure 2: The symmetric photon time profile. The detection probabilities for none or both beamsplit￾ters present in the setup (upper figure); for only one of the beamsplitters present in the setup (lower figure). will know about all the manipulations of the beamsplitters before reaching their spacetime localization. Contrary, the photon with a time tail directed forward in time will react to the changes made after h… view at source ↗
Figure 3
Figure 3. Figure 3: The asymmetric photon time profile. The detection probabilities for none or both beamsplit￾ters present in the setup (upper figure); for only one of the beamsplitters present in the setup (lower figure). 4.3. Other examples We will present below few special cases in which the beamsplitters are inserted in the interferometer before or after photon has reached their spatial location. In all of the cases the … view at source ↗
Figure 4
Figure 4. Figure 4: The asymmetric photon time profile. The detection probabilities for none or both beamsplit￾ters present in the setup (upper figure); for only one of the beamsplitters present in the setup (lower figure). Scenario 1: In the first scenario BS1 is absent and BS2 is present for 18 ≤ t ≤ 21. The temporal part of the photon is a Gaussian. In this case a small detection probability appears in the second detector … view at source ↗
Figure 5
Figure 5. Figure 5: The detection probabilities in Scenario 1. Scenario 2: The first beamsplitter is present before the photon can reach its spatial position, 1.5 ≤ t ≤ 4.5. The second beamsplitter appears later, 16.5 ≤ t ≤ 19.5. The photon has a temporal tail directed forwards. In this case the second detector gets a small probability of detection (see view at source ↗
Figure 6
Figure 6. Figure 6: The detection probabilities in Scenario 2 for a forward directed temporal profile (upper figure) and a Gaussian (lower figure). 5 10 15 20 25 t 0.1 0.2 0.3 0.4 0.5 Prob(D1) 5 10 15 20 25 t 0.1 0.2 0.3 0.4 0.5 Prob(D2) view at source ↗
Figure 7
Figure 7. Figure 7: The detection probabilities in Scenario 3. about causality only between points in which the object was localized on the time axis. Between them the object is smeared over some time interval, exactly like in the case of spatial localization. In the presented here example of the interferometer the photon was not localized until detected, having the possibility to react to the changes in the setup. The free p… view at source ↗
read the original abstract

In the Schr\"odinger evolution of a quantum state time enters as a real parameter representing the coordinate. In a more consistent approach time should be defined as a quantum observable, with the evolution taking place in a four-dimensional spacetime. This is possible in the projection evolution model in which the wave function is defined in both space and time. This allows to construct the time operator and discuss the temporal structure of quantum processes. In this paper we discuss a photon travelling through a Mach-Zehnder interferometer, focusing the description on the temporal profile of the wave function. We show that in this approach the delayed-choice experiments can be explained by the temporal overlap of the photon and the devices in the interferometer.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes that delayed-choice experiments in a Mach-Zehnder interferometer can be accounted for in the projection evolution model by treating time as a quantum observable. The wave function is defined over four-dimensional spacetime, and the observed interference patterns are attributed to the temporal overlap between the photon's wave packet and the time-dependent configurations of the interferometer devices (such as beam splitters), rather than retrocausality.

Significance. If the temporal-overlap mechanism is shown to reproduce the exact output-port probabilities and visibility limits of standard quantum mechanics for delayed-choice setups, the work would offer a spacetime-based alternative interpretation that embeds time explicitly in the dynamics. This could inform discussions of the role of time in quantum theory. However, its broader significance is constrained by dependence on a non-standard framework whose equivalence to conventional QM predictions is not demonstrated quantitatively in the provided description.

major comments (2)
  1. [Abstract] Abstract: The claim that delayed-choice phenomena are explained by temporal overlap requires an explicit derivation of the detection probabilities using the 4D photon wave function and time-projection operators. No overlap integrals, time-dependent device operators, or computed visibilities (e.g., confirming 0 % / 100 % limits) are referenced, leaving open whether the model matches standard QM or deviates.
  2. [Main text] Main text (Mach-Zehnder description): The explanation hinges on the temporal overlap determining which-path information, yet the manuscript does not supply the concrete form of the spacetime wave function or the projection operators that would allow verification that the output-port statistics coincide with ordinary quantum mechanics for time-dependent beam-splitter insertion/removal.
minor comments (1)
  1. [Abstract] Abstract: The phrase 'temporal profile of the wave function' is used without specifying the coordinate representation or normalization conventions in the four-dimensional spacetime.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The points raised correctly identify that the manuscript emphasizes the conceptual role of temporal overlap in the projection evolution model but does not yet include the full quantitative derivations needed to verify exact agreement with standard quantum mechanics. We will incorporate these explicit calculations in the revised version.

read point-by-point responses

  1. Referee: [Abstract] The claim that delayed-choice phenomena are explained by temporal overlap requires an explicit derivation of the detection probabilities using the 4D photon wave function and time-projection operators. No overlap integrals, time-dependent device operators, or computed visibilities (e.g., confirming 0 % / 100 % limits) are referenced, leaving open whether the model matches standard QM or deviates.

    Authors: We agree that the abstract and current text present the mechanism descriptively. In the revision we will add the explicit 4D Gaussian wave-packet form, the time-dependent projection operators for the beam splitters, and the resulting overlap integrals. These will be shown to reproduce the standard 0 % / 100 % visibility limits and output-port probabilities for the relevant temporal configurations. revision: yes

  2. Referee: [Main text] The explanation hinges on the temporal overlap determining which-path information, yet the manuscript does not supply the concrete form of the spacetime wave function or the projection operators that would allow verification that the output-port statistics coincide with ordinary quantum mechanics for time-dependent beam-splitter insertion/removal.

    Authors: The manuscript currently focuses on the conceptual picture. We will supply the concrete spacetime wave function and the explicit time-projection operators in a new subsection, together with sample calculations demonstrating that the output-port statistics match those of ordinary quantum mechanics for time-dependent insertion and removal of the beam splitters. revision: yes

Circularity Check

0 steps flagged

No circularity: explanation follows from independent model framework

full rationale

The paper defines its framework (projection evolution with time as observable and 4D wave functions) as an alternative to standard Schrödinger evolution, then applies it to Mach-Zehnder delayed-choice setups via temporal overlap of the photon wave packet with time-dependent devices. No step reduces a claimed prediction to a fitted parameter, self-referential definition, or load-bearing self-citation that itself assumes the target result. The abstract and described derivation treat the model as given input whose consequences are then inspected; the output (reproduction of interference patterns via overlap) is not shown to be identical to the input by construction. External benchmarks (standard QM visibility limits) are referenced as consistency checks rather than smuggled in via ansatz or renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the projection evolution model being a consistent quantum framework and on temporal overlap being the mechanism that reproduces delayed-choice results.

axioms (1)
  • domain assumption Time can be defined as a quantum observable with the wave function defined in four-dimensional spacetime.
    Invoked in the abstract as the basis for the projection evolution model.
invented entities (1)
  • Temporal overlap between photon and devices no independent evidence
    purpose: To explain delayed-choice interference without retrocausality.
    Postulated as the explanatory mechanism; no independent falsifiable evidence given in abstract.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Description and error analysis of quantum alghorithms in the projection evolution model -- the Deutsch algorithm case

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    The Deutsch algorithm is modeled using a two-level harmonic oscillator in second quantization formalism, with derived evolution operators and a projection evolution model that describes quantum gate transformations in...

Reference graph

Works this paper leans on

22 extracted references · 1 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Delayed Choice Phenomena in the Projection Evolution Model

    Introduction In the standard approach to non-relativistic quantum mechanics the time evolution of a quantum stateψis given by the equation ψ(t,x) =U(t)ψ(x, 0), (1) where U(t) is a unitary evolution operator generated by a Hamiltonian H and t denotes time. In the case of time independent Hamiltonian U(t) =exp(−iHt) , where here and in the following we set ...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.3390/e0000000arxiv:2604.26716v1 1978

  2. [2]

    We start from a four-dimensional formulation in which time is a quantum observable similar to the three spatial coordinates

    The projection evolution We describe the time evolution of a quantum state using the projection evolution (PEv) formalism [ 13]. We start from a four-dimensional formulation in which time is a quantum observable similar to the three spatial coordinates. It follows that the wave function ψ=ψ(x) depends on all four coordinates x= (x 0, x1, x2, x3) and the s...

    2026

  3. [3]

    one dimensional photon

    The Mach-Zehnder interferometer The Mach-Zehnder interferometer consists of two beamsplitters, two mirrors, and two detectors, as shown in Fig. 1. k= 1 k= 1 k= 1 k= 1 t x( , ) = (0,0) BS 2 1BS mirror mirror D1 D2 k= k= k= 2 2 2 source Figure 1.The Mach-Zehnder interferometer The state space of the interferometer consists of two quantum channels based on f...

    2026

  4. [4]

    In the following we assumeκ 1 =κ 2 =π

    Discussion We assume that the photon enters the interferometer at (t, x) = ( 0, 0), and that the first beamsplitter, the mirrors, the second beamsplitter, and the detectors are all 5 units of space apart, ie., BS1 can be reached at x= 5, the mirrors at x= 10, BS2 at x= 15, and the detectors atx=20. In the following we assumeκ 1 =κ 2 =π. If there is only o...

    2026

  5. [5]

    In this paper we have shown that the delayed-choice experiment on the example of a Mach- Zehnder interferometer can be successfully described by the temporal interaction

    Closing remarks The temporal part of the wave function provides a natural way to describe the time structure of quantum events like the temporal interference, time-of-arrival, nuclear fission and fusion processes, time structure of elementary particles interactions and many more. In this paper we have shown that the delayed-choice experiment on the exampl...

    2026

  6. [6]

    Wheeler.Mathematical Foundations of Quantum Theory; Academic Press, 1978

    J.A. Wheeler.Mathematical Foundations of Quantum Theory; Academic Press, 1978

    1978

  7. [7]

    Walborn, M.O

    S.P . Walborn, M.O. Terra Cunha, S. Pádua, C.H. Monken. A double-slit quantum eraser.Phys. Rev. A2002,65, 033818

  8. [8]

    Jacques, E

    V . Jacques, E. Wu, F. Grosshans, F. Treussart, P . Grangier, A. Aspect, J-F. Roch. Experimental Realization of Wheeler’s Delayed-Choice Gedanken Experiment.Science2007,315, 966–968

  9. [9]

    Houser, W

    U. Houser, W. Neuwirth, N. Thesen. Time-dependent modulation of the probability amplitude of single photons.Phys. Lett. A1974,49, 57–58

  10. [10]

    Lindner, M

    F. Lindner, M. Schätzel, H. Walther, A. Baltuška, E. Goulielmakis, F. Krausz, D. Miloševi´ c, D. Bauer, W. Becker, G. Paulus. Attosecond Double-Slit Experiment.Phys. Rev. Lett.2005, 95, 040401

    2005

  11. [11]

    Tirole, S

    R. Tirole, S. Vezzoli, E. Galiffi, I. Robertson, D. Maurice, B. Tilmann, S.A. Maier, J.B. Pendry, R. Sapienza. Double-slit time diffraction at optical frequencies.Nature Physics2023,19, 999–1002

  12. [12]

    Aharonov, D

    Y. Aharonov, D. Bohm. Time in the Quantum Theory and the Uncertainty Relation for Time and Energy.Phys. Rev.1961,122, 1649

    1961

  13. [13]

    Rosenbaum

    D.M. Rosenbaum. Super Hilbert Space and the Quantum-Mechanical Time Operators.J. Math. Phys.1969,10, 1127–1144. Version April 30, 2026 submitted toEntropy 12 of 12

    1969

  14. [14]

    N.N. Grot, C. Rovelli, R.S. Tate. Time of arrival in quantum mechanics.Phys. Rev. A1996, 54, 4676

  15. [15]

    Olkhovsky, E

    V .S. Olkhovsky, E. Recami, A.J. Gerasimchuk. Time operator in quantum mechanics.Il Nuovo Cimento A1974,22, 263–278

  16. [16]

    Kijowski

    J. Kijowski. On the time operator in quantum mechanics and the heisenberg uncertainty relation for energy and time.Rep. Math. Phys.1974,6, 361–386

    1974

  17. [17]

    E. Galapon. Pauli’s theorem and quantum canonical pairs: the consistency of a bounded, self–adjoint time operator canonically conjugate to a Hamiltonian with non–empty point spec- trum.Proc. R. Soc. Lond. A2002,458, 451–472

  18. [18]

    Gó´ zd´ z, M

    A. Gó´ zd´ z, M. Gó´ zd´ z, A. P˛ edrak. Quantum Time and Quantum Evolution.Universe2023,9, 256

  19. [19]

    M. Choi. Completely positive linear maps on complex matrices.Lin. Alg. App.1975,10, 285–290

    1975

  20. [20]

    Krauss.States, Effects and Operations: Fundamental Notions of Quantum Theory; Springer: Berlin/Heidelberg, Germany, 1983

    K. Krauss.States, Effects and Operations: Fundamental Notions of Quantum Theory; Springer: Berlin/Heidelberg, Germany, 1983

    1983

  21. [21]

    Gó´ zd´ z, M

    A. Gó´ zd´ z, M. Gó´ zd´ z. Quantum Clock in the Projection Evolution Formalism.Universe2024, 10, 116

  22. [22]

    Szpikowski.Podstawy mechaniki kwantowej; Wydawnictwo UMCS, Poland, 2011

    S. Szpikowski.Podstawy mechaniki kwantowej; Wydawnictwo UMCS, Poland, 2011. (in Polish)

    2011