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arXiv:quant-ph/0703039

Implications for a spatially discrete transition amplitude in the twin-slit experiment

W. M. Stuckey

Published Mar 5, 2007• 1 citation
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Abstract

A discrete path integral formalism is used to obtain the transition amplitude between 'sources' (slits and detector) in the twin-slit experiment of quantum mechanics. This method explicates the normally tacit construct of dynamic entities with temporal duration. The resulting amplitude is compared to that of Schrodinger dynamics in order to relate 'source' dynamics and spatial separation. The implied metric embodies non-separability, in stark contrast to the metric of general relativity. Thus, this approach may have implications for quantum gravity.

DOI
10.48550/arxiv.quant-ph/0703039

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A discrete path integral formalism is used to obtain the transition amplitude between 'sources' (slits and detector) in the twin-slit experiment of quantum mechanics. This method explicates the normally tacit construct of dynamic entities with temporal duration. The resulting amplitude is compared to that of Schrodinger dynamics in order to relate 'source' dynamics and spatial separation. The implied metric embodies non-separability, in stark contrast to the metric of general relativity. Thus, this approach may have implications for quantum gravity.
We propose a discrete path integral formalism over graphs fundamental to quantum mechanics (QM) based on our interpretation of QM called Relational Blockworld (RBW). In our approach, the transition amplitude is not viewed as a sum over all field configurations, but is a mathematical machine for measuring the symmetry of the discrete differential operator and source vector of the discrete action. Therefore, we restrict the path integral to the row space of the discrete differential operator, which also contains the discrete source vector, in order to avoid singularities. In this fashion we obtain the two-source transition amplitude over a "ladder" graph with N vertices. We interpret this solution in the context of the twin-slit experiment.
This study proposes a conceptual framework in which time and space are derived from quantum state transitions rather than being treated as fundamental background entities. Temporal progression is interpreted as the ordered sequence of discrete quantum state changes, while spatial structure emerges from relational properties among quantum states.Within this framework, time is defined operationally as the measurable rate of quantum state evolution, and space arises from correlations and structural relations between quantum systems. The approach suggests that spacetime may be understood as an emergent construct grounded in quantum dynamics rather than as a primary ontological structure.The implications of this interpretation are discussed in relation to general relativity and quantum mechanics, with attention to cosmological considerations and foundational questions in theoretical physics.
The modern concept of spacetime usually emerges from the consideration of moving clocks on the assumption that world-lines are continuous. In this paper we start with the assumption that natural clocks are digital and that events are discrete. By taking different continuum limits we show that the phase of non-relativistic quantum mechanics and the odd metric of spacetime both emerge from the consideration of discrete clocks in relative motion. From this perspective, the continuum limit that manifests itself in 'spacetime' is an infinite mass limit. The continuum limit that gives rise to the Schrödinger equation retains a finite mass as a beat frequency superimposed on the 'Zitterbewegung' at the Compton frequency. We illustrate this in a simple model in which a Poisson process drives a relativistic clock that gives rise to a Feynman path integral, where the phase is a manifestation of the twin paradox. The example shows that the non-Euclidean character of spacetime and the wave-particle duality of quantum mechanics share a common origin. They both emerge from the necessity that clocks age at rates that are path dependent.
One of the important classic experiments that led to the fundamental difference between quantum and classical mechanics is the double slit experiment, which is interesting with respect to the path integral formalism since it leads to a conceptual motivation for introducing it. From Quantum Mechanics, the probability amplitude can also be determined from the Schrodinger equation as it describes the wave properties of the particles. From Quantum Mechanics, the probability amplitude (square of the probability function or wave function) can also be determined from the Schrodinger equation as it describes the wave properties of the particles. Here, the role of the trajectory of a point-like particle, formally is revived in a way compatible with the indetermination principle and is indeed the path integral approach to quantum mechanics.
The previous statistical formulation of Quantum Gravity Theory based on Inverted Rel-ativity (QGT:IR) established a constitutive description of the vacuum through a hierarchyof correlation functions, effective Green functions, and constitutive electromagnetic kernels.Those results demonstrated that the constitutive vacuum possesses the mathematical struc-tures required for correlated propagation and interference-like dynamics, while explicitlyavoiding any claim of deriving Quantum Mechanics.The present work investigates whether this constitutive statistical framework can beextended to reproduce the mathematical structure of two fundamental quantum phenomena:double-slit interference and nonlocal correlations. Starting exclusively from the constitutivescalar action and its statistical extension, we derive the effective electromagnetic responsegenerated by multiple constitutive sources, construct the corresponding intensity functionalmeasured by an ideal detector, and establish the conditions under which interference patternsemerge from constitutive vacuum correlations.The same formalism is then generalized to spatially separated constitutive regions, al-lowing the construction of normalized constitutive correlation functions analogous to thoseemployed in quantum correlation experiments. Their mathematical properties are analyzedtogether with the conditions required for long-range constitutive coherence.Throughout the article a strict distinction is maintained between exact mathematicalderivations, effective constitutive models, and working hypotheses. No claim is made thatthe Born rule, Hilbert-space structure, wavefunctions, quantum entanglement, or violationsof Bell inequalities are derived. Instead, the present work establishes a unified constitutiveframework from which experimentally testable interference and nonlocal correlation phenom-ena may emerge as effective macroscopic consequences of statistical vacuum dynamics.
This work presents a canonical General Relativity–Clock Shift (GR-CS) interpretation of the double-slit experiment that preserves all standard quantum and quantum-field-theoretic predictions while replacing ambiguous measurement language with a geometric criterion. In GR-CS, space is static and non-gravitating, and the only geometric degree of freedom is the scalar clock-rate field N(x). A single photon is emitted and absorbed once and never splits; interference arises from the coherent evolution of an unanchored quantum field amplitude constrained by slit boundary conditions, not from a particle traversing multiple paths. Prior to detection, the excitation carries no proper time and does not couple to the clock field (J_N = 0). Measurement corresponds to clock anchoring: a physical transition in which energy becomes bound into matter, acquires proper time, produces a localized classical record, and terminates coherent evolution (J_N \neq 0). Which-path detection suppresses interference because anchoring occurs upstream, not because of observer knowledge or information gain. This paper provides explicit definitions of anchored and unanchored sectors, a minimal detector model, a density-matrix mapping of coherence loss to anchoring, and an optional connection to path-integral and QFT formulations on a static clock-field background.