An Event Driven First Passage Model from Quantum to Classical Transition (EDFPM)

An Event Driven First Passage Model from Quantum to Classical Transition Stefan-Alexandru Gheorghe [email protected] 15 September 2025 Abstract A comprehensive, falsifiable model in which classical definiteness, motion, and structure arise from a multi scale cascade of first passage events in invariant proper time. Building on the Theory of Emergent Motion (ToEM), in which directional motion emerges across a discrete temporal threshold T0, this aims to extend to event dependent thresholds, relativistic Continuous Spontaneous Localization (CSL) with quantized time (Bedingham and Pearle), and Loop Quantum Gravity (LQG) as a discrete spacetime arena. Each degree of freedom carries a hazard λhit (τ) for its first event at proper time T0along a world line or world tube. Before T0, evolution is Hamiltonian; after T0, CSL type de phasing applies with rate A. Higher scale events arise from OR/AND/ k-ofngates over lower scale first passage times, inducing either constant or ageing hazards without new postulates. Derived closed form gate calculus, ensemble visibility, and paired shot covariance, and validate with analytic checks and Monte Carlo simulations. Experimental implementations include ultracold 87/88Sr atom interferometry (paired shot T0protocol), as well as high energy analogues in gluon splitting. Key predictions are: a strict early time plateau in visibility, a positive covariance shoulder C(δ)up to delays ≲E[T0], delayed CSL heating, and variance scaling signatures that separate independent gates from clustered micro events. The model provides a unified, Lorentz covariant account of the quantum to classical transition using only proper time first passage plus CSL, without exotic fields or dualist assumptions. 1 1 Introduction The emergence of classical definiteness and motion from underlying quantum indeterminacy remains one of the central challenges in theoretical physics. Standard quantum mechanics prescribes unitary Hamiltonian evolution in Hilbert space, yet provides no mechanism for the actualization of classical trajectories. Collapse models such as Continuous Spontaneous Localization (CSL) introduce a stochastic, norm-preserving noise that tends to localize macroscopic degrees of freedom ( 1; 2; 3 ). Recent advances by Bedingham and Pearle cast CSL in a relativistic framework where time itself is treated as a quantum operator conjugate to energy, preserving Lorentz covariance (4). Independently, the Theory of Emergent Motion (ToEM) proposed that classical motion emerges only after a discrete temporal threshold T0has been exceeded (5). Below this scale, systems exist in a superposed, directionally unresolved state. Beyond it, trajectories resolve probabilistically into classically deterministic paths. Building upon these ideas, the present work integrates multiple strands: 1. The ToEM foundation, with its temporal threshold T0and switching function governing emergence of motion. 2. The hypothesis of event dependent T0, varying stochastically between experimental runs, producing observable run to run fluctuations in interferometric visibility. 3. A relativistic CSL model with quantized time (Bedingham and Pearle) providing a Lorentz covariant stochastic dynamics, linked to first-passage hazards in proper time. 4. A multi scale gate calculus ( OR/AND/k-ofn) that propagates microscopic first passage events into molecular, neural, and macroscopic scales, producing constant or ageing hazards without new postulates. 5. A concrete experimental program: ultracold strontium atom interferometry for detecting T0signatures, and high energy collider data for probing analogous waiting time distributions in jet fragmentation. This unified model yields sharp, falsifiable predictions: an early time plateau in ensemble visibility, a positive paired shot covariance shoulder, delayed CSL heating, and diagnostic variance scaling. The remainder of this paper develops these foundations, derives the core mathematics, and outlines the experimental roadmap. 2 2 Foundational Model: Theory of Emergent Motion (ToEM) The Theory of Emergent Motion (ToEM) proposes that classical motion is not fundamental but arises as a probabilistic resolution across a discrete temporal threshold T0(5). Below this scale, directional motion does not exist in coherent form; instead, particles remain in a regime of unresolved path uncertainty, consistent with quantum mechanical superposition. As the elapsed interval ∆t grows beyond T0, the probability of resolving into a classically deterministic trajectory increases. This transition is governed by the switching function: F(∆t) = 1 −e−∆t/T0(1) which satisfies: •F(0) = 0 : at zero elapsed time, directionality is undefined. •lim∆t→∞ F(∆t) = 1 : classical motion emerges asymptotically. •F(∆t)is strictly increasing and smooth for ∆t > 0. Directional uncertainty contracts over time, described by a spatial spread function σ(∆t) = σ0e−∆t/T0(2) with σ0the maximal uncertainty at ∆t= 0. For ∆t≪T0, the spread remains near maximal, while for ∆t≫T0it collapses to zero, recovering classical predictability. 2.1 Path Integral Reinterpretation Traditionally, the Feynman path integral expresses transition amplitudes as a sum over an uncountable infinity of paths: ⟨xf, tf|xi, ti⟩=ZP D[x(t)]ei ℏS[x(t)] (3) In ToEM, this is reinterpreted as a finite sum over paths on a discrete time lattice with minimum interval δt ≥T0: ⟨xf, tf|xi, ti⟩ ≈ M X j=1 Pj(∆t)ei ℏSj(4) where Pj(∆t)is a probability weight derived from the switching function, and Sjis the action along the j-th path. This construction removes divergences and introduces a bounded transition regime between quantum and classical domains. 3 2.2 Time Symmetry and the Arrow of Motion At ∆t= 0, the system is directionally unresolved, preserving time symmetry. As ∆tgrows, F(∆t)progressively introduces an arrow of time, consistent with observed macroscopic irreversibility. Unlike the thermodynamic arrow, this mechanism is intrinsic to motion emergence itself. 3 Event Dependent Thresholds The original ToEM formulation left T0unspecified. A natural extension is to treat T0as event dependent, varying stochastically between nominally identical experimental runs (6). This acknowledges that microscopic first passage events may differ across realizations, producing observable consequences. We formalize this with T(event ) 0=Ntp(5) where tp≈5.39 ×10−44 sis the Planck time and Nis a stochastic variable differing run by run. Tmin is conservatively located at Planck time, with the upper bound Tbeing ever expanding and governed by entropy. The quantum to classical crossover thus occurs at different effective rates depending on N. 3.1 Experimental Consequences Event dependent thresholds predict: •Run to run variations in interferometric visibility curves. •Microsecond scale covariance shoulders detectable with paired shot protocols. •Mass scaling and environmental dependence of effective T0. This hypothesis leads directly to falsifiable experiments using cold atom interferometers, as described later in Section 6. 4 4 Relativistic CSL and Discrete Spacetime 4.1 Relativistic CSL with Quantized Time Bedingham and Pearle recently formulated CSL in a relativistic setting where time is treated as a quantum operator ˆ tconjugate to energy ˆ Ewith [ˆ t, ˆ E] = iℏ. Collapses are generated by a Lorentz scalar operator ˆ A, and evolution is driven by a stochastic equation in an auxiliary parameter s(4). It is a known feature of this class of relativistic CSL models that the stochastic field can induce particle production from the vacuum, leading to a divergent energy increase.[1, 2] While the resolution of this issue is a subject of ongoing research and beyond the scope of the present phenomenological framework, we note that the core mechanism of a stochastically triggered onset of CSL dynamics can be formulated independently of this specific pathology. The Bedingham and Pearle model is employed here as a concrete, Lorentz covariant realization of such a trigger, with the understanding that a future, fully-realized theory would need to incorporate a mechanism to regulate this energy production. The master equation admits a quantum trajectory unravelling in which the first occurrence of a collapse defines a stochastic proper time threshold T0. Before T0, dynamics are unitary and Hamiltonian. After T0, CSL-type de phasing dominates with rate Λ. Formally, the survival probability for no event up to proper time τis S(τ) = exp −Zτ 0 λhit(u)du(6) with hazard λhit (τ)=κDˆ A2Eτ.(7) Thus, T0is exponential if λhit is constant, and more generally follows a nonhomogeneous Poisson process in proper time. 4.2 Granular Arena from Loop Quantum Gravity Loop Quantum Gravity (LQG) represents space via spin networks and their histories via spin foams. Geometric operators (area, volume) exhibit discrete spectra. We use this conservatively: spin-network sub graphs provide candidate microscopic sites for events, and spin foam coarse graining maps many micro-sites to mesoscopic hazards. No claim of universal ultraviolet finiteness or generic topology change is required. This connection is intended to be motivational, providing a plausible physical origin for discreteness without relying on the specifics of LQG’s unresolved dynamics.[3] 5 4.3 First Passage Law and Normalization The Lorentz covariant dynamics of the Bedingham and Pearle model provide a sophisticated starting point for a relativistic trigger mechanism.[1] However, the waiting time probability density for the first collapse event, as formulated in the original work, presents a normalization challenge that requires careful consideration. The proposed density is given by: P(∆τ) = λ 2πZ+S −S e−ip∆τdp, (8) where Sis an unobservable momentum cut off introduced to regulate the associated energy spread.[2] A direct evaluation of the equation reveals that its integral over all positive proper time, R∞ 0P(∆τ)d(∆τ), converges to unity only in the formal limit where S→ ∞. For any finite cut off S, the integral is less than one, meaning the distribution is not properly normalized.[2] While subsequent smoothing procedures can tame the divergence of the peak at ∆τ= 0, they do not resolve this underlying normalization issue. To construct a phenomenologically robust model built upon a consistent probabilistic foundation, we adopt an alternative first passage law based on a simple and well motivated Poisson switching process. In this picture, the probability that no collapse event has occurred by proper time τis given by a pure exponential survival function: P(T0> τ) = exp(−ατ),(9) where α≡λN is the constant hazard rate, with Nrepresenting the number of constituent particles or relevant degrees of freedom. This distribution is properly normalized for all τ≥0without requiring a cut off, and it yields a mean first passage time of ⟨T0⟩= 1/α. By identifying the event dependent threshold T0of our framework with the first passage time of this exponential process, we achieve two crucial goals. First, we ensure that the underlying waiting time law is mathematically sound. Second, T0is no longer an externally imposed parameter but becomes a derived, stochastic quantity whose statistics are governed by the microscopic hazard rate α.[2] This approach collapses the parameter sets and yields the distinct experimental signature of an early time visibility plateau, which persists until the first "hit" at T0, followed by a standard CSL type decay.[2] 6 5 First Passage and Multi scale Gate Calculus Each relevant degree of freedom carries a hazard λhit (τ)for a first event at proper time T0. Before T0, evolution is Hamiltonian; after T0, CSL type de phasing applies with rate Λ. 5.1 Minimal Axioms 1. Proper time gate. Each degree of freedom carries a hazard λhit (τ), with survival given by Eq. 6. 2. Two phases. Pre event (τ < T0): Hamiltonian evolution. Post-event ( τ≥T0): CSL type de phasing at rate Λ. 3. Gating hierarchy. Higher scale events are formed by OR/AND/ k-ofn gates over lower scale events. 4. Discrete arena. Spin network sub graphs supply micro-sites for events; spin foam coarse graining yields mesoscopic hazards. 5.2 Primitive Gates Let Tibe i.i.d. exponential with hazard λ. OR (any-ofn). TOR = min iTi, λOR = n X i=1 λi(10) Thus TOR ∼Exp(nλ), memoryless. AND (all-ofn). SAND(τ)=1−1−e−λτ n, λAND(τ) = nλe−λτ 1−e−λτ n−1 1−(1 −e−λτ )n(11) This produces an ageing hazard, increasing with τ.k-ofn(order statistics). Tk:n d = k−1 X j=0 Yj, Yj∼Exp((n−j)λ)(12) Expectation and variance: E[Tk:n] = 1 λ n X m=n−k+1 1 m,Var (Tk:n) = 1 λ2 n X m=n−k+1 1 m2.(13) 7 5.3 Observables 5.4 Run wise Law For elapsed proper time ∆τin one run: Vrun (∆τ) = (V0,∆τ < T0 V0e−Λ(∆τ−T0),∆τ≥T0 (14) 5.5 Ensemble Visibility Averaging over hazards yields ⟨V(∆τ)⟩ V0 =S(∆τ) + Z∆τ 0 λhit (u)S(u)e−Λ(∆τ−u)du (15) For constant hazard α: ⟨V⟩ V0 =Λe−α∆τ−αe−Λ∆τ Λ−α Λ→α −−−→ e−α∆τ(1+α∆τ)(16) This exhibits a strict early time plateau (zero slope). 5.6 Paired Shot Covariance For runs A and B of equal duration ∆, with B delayed by δ: C(δ)=⟨VAVB⟩−⟨V⟩2(17) For constant hazard αand decay β= Λ, closed forms are piecewise (see Appendix). In all cases C(δ)>0for δ≲E[T0]and decays to zero as δ→ ∞. 5.7 Variance Scaling The Fano factor F=Var (T0) E[T0]2(18) distinguishes aggregation mechanisms: F≈1for independent OR gates, F > 1 for clustered or AND-like gating. 8 6 Experiments and Analysis 6.1 Sr Interferometer: Paired Shot T0Protocol We propose a strontium atom Mach-Zehnder interferometer to detect two signatures: 1. An early zero slope plateau in ensemble visibility ⟨V(∆τ)⟩. 2. A positive paired shot covariance shoulder C(δ)up to delays ∼1/α. Species and Cooling. 88Sr (bosonic) or 87Sr (fermionic, clock heritage). Blue MOT ( 461 nm ) →Red MOT ( 689 nm ), ≤300nK; loaded into a shallow optical dipole trap. Interferometer. Light pulse Mach-Zehnder with Bragg π/2−π−π/2pulses, order n= 4 −10. Counter propagating beams via retro reflector on vibration isolated stage. Phase controlled by AOMs (phase locked RF; DDS step ≤100 ns ). Timings. Pulse separation T= 5 −15 ms(∆τ= 2T= 10 −30 ms). Runpair micro-offsets δ∈ {1,3,10,30,100}µsrandomized. Environment. UHV <10−10 mbar, magnetic shielding, stabilized beam powers, vibration isolation. Detection SNR ≥50. Analysis. •Visibility per run: Vrun = (N1−N2)/(N1+N2). •Ensemble curve: fit constant hazard vs ageing (Weibull); compare AIC/BIC. •Paired covariance: compute C(δ)with bootstrap CIs; expect positive shoulder. •Variance scaling (if sub clouds available): Fano factor, from Eq. 18, to fingerprint OR vs AND. 6.2 Systematic Effects and Controls The central experimental challenge is the robust detection of the paired shot covariance C(δ)in the presence of correlated technical noise, which can mimic the predicted signature. A convincing measurement requires a rigorous protocol to characterize, mitigate, and subtract these systematic effects. The primary confounders include: •Laser Noise: Slow drifts or 1/f noise in laser intensity and phase/frequency can induce strong correlations in visibility between two closely timed shots. Mitigation requires active power and frequency stabilization, supplemented by high bandwidth, out of loop monitoring of these parameters to be used in post correction analysis. 9 10 HEP Archival Analysis - Hazard & Covariance (Lab Ready Run Card) 10.1 Overview Test whether inter emission "waiting times" in gluon rich jets are memoryless (exponential) or ageing, and search for a paired sample covariance shoulder analogue using open LHC data + MC. 10.2 Datasets & Tools •ATLAS/CMS Open Data (13 TeV Run-2; start 2016). •MC: PYTHIA 8, HERWIG 7 (detector/particle level). •FastJet (anti-kT R=0.4/0.6), RIVET/ROOT/Python; Delphes if needed. 10.3 Repo Structure repo/ data/ # open data, MC, metadata configs/ # selections, tagger WPs, grooming src/ # code packages: selection/ declustering/ hazards/ covariance/ utils/ notebooks/ # 00_quickstart, 10_waiting_times, 20_covariance_pairs plots/ reports/ README.md env.yml 10.4 Setup conda env create -f env.yml conda activate t0-hep python -m pip install awkward uproot vector matplotlib numpy scipy scikitlearn iminuit 10.5 Selection (gluon enriched +g→Q¯ Qtags) •Dijet events; leading jet quark/gluon taggers (track multiplicity, width). •pT ∈[200,400]GeV,|η|<2.0; good PV; data-quality flags. •Heavy flavour splits: g→bb/g→cc via SV/soft-muon; validate with MC truth. 16 10.6 Waiting Time Observable Build a decluttering chain ( C/Aor kT ). Define inter emission intervals in a monotone "evolution" variable: e.g., ∆t _shower ∝1/k_T∧2step (formation time proxy), or ∆ ln k−T,∆ ln(1/θ)on the Lund plane. Extract sequence ( ∆t_i)per jet to groom cutoff. 10.7 Pairing (covariance analogue) Form kinematic twins: for each jet A, match jet Bin ( pT, η, pileup, run period), ensure independence (separate events). Define tiny pre emission offset δ_Lund via nearly identical upstream state. Compute binary "resolved by step k" flags; estimate C(δ_Lund ) = ⟨X_AX_B⟩ − ⟨X⟩2across δ_Lund bins. 10.8 Statistics •Hazard shape: fit exponential vs Weibull (ageing); compare AIC/BIC; KS/CvM tests; parameter CIs. •Covariance: bootstrap C (δ_Lund); expect small positive shoulder at small δ−Lund if shared pre event history exists. - Systematics: jet R , grooming, tagger WP, pileup; compare MC tunes (Monash, CP5). - Nulls: pair shuffle ( C→0), event time scrambling, MC truth toggles. 10.9 Plots & Reports •Waiting time PDFs/CDFs with exp vs Weibull fits; △AIC table. -C (δ_Lund )with CIs+ surrogate null overlay. - Stability grids vs grooming/algorithm/pileup. - One-pager summary: memoryless vs ageing decision; any shoulder seen/bounded. 10.10 Quick start Checklist [ ] Env built [ ] One dataset + PYTHIA pulled [ ] De clustering ∆t extraction validated on MC [ ] Pairing path + surrogates working [ ] First plots (waiting time + C ) produced [ ] Systematics sweep grid defined 17 Figure 1: Hazard vs. proper time. OR dominated (constant) hazards are memoryless; AND gates produce ageing hazards with a rising profile. Figure 2: Ensemble visibility for a constant hazard shows a strict early time plateau (zero slope) followed by decay. Ageing hazard curves (AND/ k-ofn) have the same plateau with distinct early curvature (see text). 18 11 Scale-setting for T0(proper time pull-back) Let xµ(τ)be a chosen world line (or an average over a world tube). Define a local Lorentz scalar intensity built from a collapse generator ˆ O(y)(e.g., the scalar used in relativistic CSL): λ(y) = κ⟨ˆ O(y)ˆ O(y)⟩, κ > 0.(31) With a space like smearing kernel gσof width σ, pull back to proper time: λhit (τ) = Zd4ygσ(x(τ)−y)2λ(y)(32) Then E[T0] = R∞ 0S(τ)dτ is system dependent (geometry, mass density, path separation), not fixed at a Planck scale. Planckian structure enters only via ultraviolet cutoffs in λ(y)or gσ. Empirical anchor. Fit λhit from the plateau width and Λfrom the decay tail using the constant hazard closed form (Eq. 16) when adequate, or the general hazard formula (Eq. 15) when curvature indicates ageing (Sec. 5). Cross check with independent CSL heating limits. 11.1 Counting statistic diagnostics for T0 Write T0as a random sum of i.i.d. exponentials Xi∼Exp (1/tp)and a random count Ndetermined by gate structure: T0= N X i=1 Xi,E[T0] = tpE[N],Var (T0)=t2 p(E[N] + Var(N)).(33) Define the Fano factor using Eq. 18 to fingerprint gates: Gate /N law E[N] Var(N)F FixedN0(Erlang; AND) N001/N0<1 Geometric (p)(OR cascade) 1/p (1 −p)/p21 NegBin (r, p)(clusters) r(1 −p)/p r(1 −p)/p2(p+ 1)/(r(1 −p)) >1 Variance scaling test. Estimate Ffrom T0histograms while changing the number of parallel sub units. Independence predicts F↓with more sub units (toward OR/memoryless); persistent F > 1signals clustering. 11.2 Unravelling robustness and field-theoretic lift Let the ensemble dynamics be ˙ρ=L[ρ]. Define the completely positive "noevent" map Φτgenerated by the effective non Hermitian evolution. The survival and hazard are S(τ) = Tr (Φτ[ρ0]) , λhit(τ)=−d dτln S(τ).(34) 19 Figure 3: Fano factor Fdistinguishes gates: under dispersed ( F < 1) for fixed NAND like, Poisson like ( F= 1 ) for OR/geometric, over dispersed ( F > 1) for clustered micro events. Any stochastic unravelling that realizes Lyields trajectories with the same S(τ); therefore the observables used in the multi scale paper ensemble visibility and paired shot covariance are unravelling robust (they depend only on Sand Λ). A field-theoretic form consistent with relativity is given by equations similar to Eq. 1 and Eq. 2 with a Lorentz scalar ˆ Oand space like smearing gσ. All waiting time statistics are then defined in invariant proper time. 11.3 General hazard visibility and covariance For the run wise law in the main paper, the ensemble visibility at elapsed proper time ∆τis given by Eq. 15. The paired-shot covariance C(δ)=⟨VAVB⟩ − ⟨V⟩2 admits a single integral form for any hazard by conditioning on T0(see Appendix C of the main paper). Numerically, both Eq. 7 and the covariance integral are stable and fast (one quadrature each). 20 11.4 General hazard covariance (integral form) Let S(τ)and λ(τ)denote the shared pre event survival and hazard. With run durations ∆and delay δ, conditioning the event time T0yields the following. ⟨VAVB⟩ V2 0 = Pr [T0>∆] + Zmax(0,∆−δ) 0 λ(u)S(u)e−Λ(2∆−δ−2u)du +Z∆ max(0,∆−δ) λ(u)S(u)e−Λ(∆−u)du +Z∆+δ ∆ λ(u)S(u)e−Λ(∆+δ−u)du (35) from which C(δ)follows by subtracting ⟨V⟩2with ⟨V⟩from an expression similar to Eq. 31. 11.5 Ageing Hazard - Minimal Monte Carlo Code calculates the variance, which is equivalent to the covariance at δ= 0. def paired_cov_variance(Npairs, Delta, lam_hit, Lam, mode="constant", n_and=3): if mode == "constant": T0_shared = sample_T0_constant(Npairs, lam_hit) else: T0_shared = sample_T0_AND(Npairs, lam_hit, n_and) VA = np.array() # Placeholder for actual visibility calculation # For a shared T0 and delta=0, VB is identical to VA VB = VA return np.mean(VA * VB) - np.mean(VA)**2 np.random.seed(0) lam_hit, Lam, Delta = 2e5, 5e5, 10e-6 print(f"Covariance (constant hazard): \ {paired_cov_variance(200000, Delta, lam_hit, Lam, ’constant’):.2e}") print(f"Covariance (aging hazard): \ {paired_cov_variance(200000, Delta, lam_hit, Lam, ’aging’, n_and=4):.2e}") 21 12 Limitations and open problems •Microsite correlations. The gate calculus assumes independent sub events. Correlated micro sites (e.g., common mode fields, shared geometry) induce non Poisson cluster statistics. Practical tests: (i) variance scaling of inferred T0across controlled subsystems (Fano factor Ffrom Eq. 18 exceeding hypo exponential predictions flags positive correlation); (ii) surrogate shuffle controls that preserve marginals but destroy cross correlations; (iii) AIC/BIC comparison of constant/ageing hazard fits to a simple cluster/Hawkes alternative. •Composite systems. The averaging of hazards of world line vs. world tube must be fixed apriori; report sensitivity. Give the Lorentz Covariant construction appropriate to each instrument geometry. •Deriving micro hazards. Tie λ(0) to specific spin foam amplitudes when feasible; otherwise, use empirically calibrated effective rates and validate using variance / covariance diagnostics. •General C(δ)forms. 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