Relativistic Collapse Model with Quantised Time Variables

Daniel J. Bedingham; Philip Pearle

REVIEW 3 major objections 5 minor 24 references

Relativistic Collapse Model with Quantised Time Variables

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims a relativistic collapse model in which each particle's time is a quantum operator, producing definite masses and definite spacetime configurations.

desk verdict Novel relativistic CSL construction with quantized time, but the N-particle localisation claim has a load-bearing boost-degeneracy problem. read the letter →

arxiv 2506.07959 v1 pith:IALG5HE4 submitted 2025-06-09 quant-ph

Daniel J. Bedingham , Philip Pearle This is my paper

classification quant-ph

keywords relativisticcollapsemodelscontinuousspontaneouslocalisationquantisedtimeKlein-GordonequationPoincarécovarianceBornrulewavefunctionstochasticSchrödinger

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes a relativistic version of continuous spontaneous localisation in which each particle carries its own time operator alongside its position operator, and the state evolves stochastically in an evolution parameter s rather than in a common background time. The authors claim that the collapse dynamics drives every individual state vector toward a common eigenstate of Poincaré-invariant collapse operators: a single particle ends in a state of definite mass that satisfies the Klein-Gordon equation, and a system of four or more distinguishable particles ends in a state with definite relative spacetime coordinates. Born-rule probabilities emerge from the norm mechanism of the stochastic dynamics, and energy is conserved in stochastic expectation, which would resolve a long-standing difficulty with relativistic collapse models. The appeal of the construction is that it treats position and time symmetrically as quantum variables, so the collapse operators can be Lorentz scalars.

What carries the argument

The load-bearing mechanism is the stochastic differential equation d|ψ,s⟩ = (-iĤ ds - ½(Â - ⟨Â⟩)·Λ·(Â - ⟨Â⟩) ds + (Â - ⟨Â⟩)·dB_s)|ψ,s⟩, a CSL evolution in which the collapse operators Â are Poincaré scalars and B_s is a Brownian motion with (dB_s)² = λ ds. The two families of collapse operators are the mass-squared scalars p̂_i² - Ê_i² and the invariant pair separations A_ij = (x̂_i - x̂_j)² - (t̂_i - t̂_j)². The key identity is the density-matrix solution of Eq. (72): off-diagonal elements in the configuration basis decay like exp[-λ(s - s₀)/2 Σ_{i<j}(A_ij - A'_ij)²], so entanglement between any pair of configurations with different interval sets is suppressed, and the surviving state is a common eigenstate of the interval set. The counting argument of Eq. (75) is what converts this into a claim of definite spacetime configuration for N ≥ 4.

What would settle it

Apply a global Lorentz boost to any N-particle configuration and every pair interval A_ij is unchanged, so a one-parameter family of boosted configurations carries identical collapse eigenvalues. A calculation of the dimension of the boost orbit for the relative coordinates would show that the constraint count in Eqs. (74)-(75) misses this free direction, and the collapse dynamics would then leave a spread over the orbit rather than a unique configuration.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that a CSL-type Itô stochastic Schrödinger equation written with an anti-Hermitian, white-noise Hamiltonian and a Hermitian scalar Hamiltonian can be made Poincaré covariant by choosing the collapse-generating operators to be Lorentz scalars built from the per-particle position-time and energy-momentum operators. With evolution parameter s and joint probability P(x,t) = (1/S)∫ ds |⟨x,t|ψ,s⟩|², the conditional distribution P(x|t) reproduces a relativistic world-line for a Gaussian packet. The mass-squared operator p̂² - Ê² drives one-particle states to satisfy (∂_t² - ∂_x² + μ²)ψ = 0, while the pair-interval operator (Δx̂)² - (Δt̂)² drives multi-particle states toward definite relative configurations; for N ≥ 4 the constraint count in Eq. (75) is taken to show that all 2(N-1) relative coordinates are fixed. The same norm-based mechanism yields Born-rule probabilities, and the Itô calculus shows the quantum expectation of each particle energy is conserved in stochastic expectation in the full model.

Load-bearing premise

The claims about four or more particles depend on the premise that specifying all pair intervals (x_i - x_j)² - (t_i - t_j)² uniquely fixes the particles' relative spacetime coordinates, so that one common eigenstate of the interval operators corresponds to a single configuration rather than to a family of configurations.

Editorial extensions

If this is right

Single-particle collapse yields states of definite mass obeying the Klein-Gordon equation, so the model produces relativistic dispersion as an emergent outcome.
For four or more distinguishable particles, superpositions of different relative spacetime configurations are suppressed, giving localisation in time as well as space.
Outcome probabilities match the Born rule through the norm mechanism of the stochastic dynamics, preserving standard quantum measurement statistics.
Energy is conserved in stochastic expectation for the full model, directly addressing the infinite-energy problem that has blocked earlier relativistic collapse proposals.
Adding a Poincaré-invariant potential V depending on the interval separations would extend the model to interacting particles within the same framework.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because every pair interval is boost-invariant, the counting argument needs an extra step: either additional scalar operators that break the boost degeneracy must be added, or the claimed unique configuration is actually a one-parameter orbit of configurations.
The construction suggests a general design principle for relativistic collapse: use the full set of Poincaré invariants available in the algebra of positions, times, momenta, and energies, rather than localising only spatial positions.
The free collapse strengths λ_i and the scale parameter m are not fixed by the paper; interference and radiation experiments could be used to bound them, though no such bounds are derived here.
Formulating the model in one spatial dimension leaves open whether the configurational localisation survives in 3+1 dimensions, where there are more scalar combinations to choose among.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The reader's take is basically right, and the stress-test lands. This is a serious, original attempt at relativistic collapse using per-particle time operators and Poincaré-scalar collapse generators. The single- and two-particle math is standard CSL and is worked through cleanly. The Klein-Gordon equation emerging for collapsed single-particle states is a definitional consequence of choosing A=p^2-E^2, not an independent prediction, but that is not a fatal objection—the paper does not oversell it as a prediction, and the model's internal consistency is still worth having.

The construction is genuinely new relative to the cited CSL literature: per-particle time operators, the scalar collapse generators A_i=p_i^2-E_i^2 and A_ij=(Δx_ij)^2-(Δt_ij)^2, and the uniform-s spacetime interpretation. The paper is also honest about limitations: one spatial dimension, distinguishable particles, no interactions, and tachyonic mass-squared possibilities are explicitly acknowledged. Energy conservation in expectation is demonstrated. That is real formal work.

The load-bearing flaw is in Section VI. The counting in Eq. (75) omits that every A_ij is invariant under a global boost. Fixing all A_ij eigenvalues leaves a one-parameter family of configurations related by a common rapidity. For N=4, the level set in the 6-dimensional relative-coordinate space is generically 1-dimensional. The master equation (69) commutes with the global boost K, so coherence between boost-related configurations is never damped. Therefore "the particles will be localised in space and in time" does not follow; at best the model localises to a boost-equivalence class. This is a serious, curable but non-cosmetic flaw. The paper should either add operators that break the degeneracy, or revise the claim to localisation up to a global boost and examine what physical meaning remains.

A minor point: the two-particle collapse example in Sec. V relies on Δt being well localised in a frame, which is not boost invariant and is later retracted for N particles. The paper acknowledges the instability, but the argument for spatial localisation is only heuristic. The citation pattern is appropriate; Pearle's own prior work is cited precisely because it set up the obstacle. Self-citation here is not a flaw.

Overall, this is a serious paper with clear thinking and honest engagement. It deserves a serious referee. The current version's central claim needs repair, but the flaw is identifiable and potentially fixable. I would send it out with a clear request to address the boost degeneracy. People working on relativistic collapse and the problem of time will want to engage with it.

Referee Report

3 major / 5 minor

Summary. This paper proposes a Poincaré-covariant collapse model in 1+1 dimensions in which each particle has both a position operator x_i and a time operator t_i, with state evolution governed by a CSL-type Itô equation parametrised by an evolution parameter s. Spacetime probabilities are defined by uniformly averaging |<x,t|ψ,s>|^2 over s. For one particle, the collapse operator A=p^2-E^2 drives states to eigenstates of A and hence to states satisfying the Klein-Gordon equation. For two particles, adding A=(x_1-x_2)^2-(t_1-t_2)^2 collapses superpositions with different invariant separations. For N≥4, the paper claims that the set of all pair-interval operators fixes all relative spacetime coordinates and therefore localises the particles in space and time. The paper also claims that energy is conserved in stochastic expectation.

Significance. The model is original in treating time as a per-particle operator and in attempting to avoid the infinite-energy problem of relativistic collapse by using scalar collapse operators. The single-particle and two-particle calculations in Sections III and V are clean applications of standard CSL mathematics, and the explicit demonstration of energy conservation in stochastic expectation for the chosen operators is a useful technical feature. The paper's central claim, however, is the N-particle spacetime localisation, and that claim is invalid because of a continuous boost degeneracy. The advertised result of 'definite configuration in spacetime' is therefore not achieved. The paper may still be of interest for its formal construction, but its main physical conclusion requires substantial revision.

major comments (3)

[VI, Eqs. (68)-(75)] The counting argument leading to N≥4 is incorrect. Each operator A_ij=(x_i-x_j)^2-(t_i-t_j)^2 is a Poincaré scalar and is invariant under the global boost generated by K=Σ_i(x_i E_i - t_i p_i). Consequently all A_ij are unchanged along the one-parameter family of configurations obtained by boosting every particle by the same rapidity. The master equation (69) and the density-matrix solution (72) therefore commute with K and do not suppress coherence between configurations related by a global boost. In the 2(N-1)-dimensional relative-coordinate space, the gradients of the N(N-1)/2 constraints are not independent: the boost vector field lies in the kernel, so the level set of fixed A_ij eigenvalues is generically one-dimensional for N=4 and remains non-discrete for all N. Hence the conclusion 'The particles will be localised in space and in time' does not follow from Eq. (75). At most the model localises the state to a boost-equivalence class.
[V, Eqs. (49)-(58)] The two-particle analysis already contains the same degeneracy. The demonstration of spatial localisation assumes that Δt is well localised near zero, as stated in the paragraph before Eq. (53). Without that assumption, A=Δx^2-Δt^2 cannot distinguish states that differ by a boost, and the 'no collapse' example in Section V.A is a special case of this degeneracy. The promised resolution via N≥4 in Section VI does not remove the degeneracy, because the global boost direction is in the kernel of every A_ij for any N. The claim that collapse occurs for 'different configurations' should therefore be restricted to configurations not related by a global boost.
[VI, full collapse-operator set] The same boost degeneracy affects the full set of collapse operators, including the mass operators p_i^2-E_i^2, since these are also invariant under the global boost. Adding the mass operators therefore cannot break the degeneracy. The paper should either introduce a collapse mechanism that selects a boost parameter, which would break the stated Poincaré covariance, or explicitly revise the claimed result to localisation up to global boost.

minor comments (5)

[II.A, Eq. (3)] The uniform probability assignment over s is a postulate, not a consequence of the stochastic dynamics; its status should be stated more explicitly, and the normalisation issue noted in footnote 3, where P(t) is not normalised under the stated approximations, deserves a fuller discussion.
[IV, Eq. (33)] Equation (33) defines the multi-particle spacetime density as a product of single-particle marginals. For an entangled state this is not the joint distribution obtained from the state; since the paper says interactions and entanglement are handled at the state level, the physical meaning of this product density should be clarified.
[III, Eq. (26)] The Klein-Gordon result is a direct consequence of choosing A=p^2-E^2 as the collapse operator; the paper should present it as a consistency property rather than as an independent prediction, to avoid the appearance of circularity.
[Throughout] The manuscript contains numerous LaTeX/OCR encoding artifacts, such as 'Schr¨ odinger', 'Itˆ o', and broken square-root expressions; these should be corrected in the published version.
[VI, Eq. (75)] The inequality in Eq. (75) is written with factorials; the simpler expression N(N-1)/2 would be clearer, and the counting should explicitly account for the constraint degeneracy that invalidates the conclusion.

Circularity Check

2 steps flagged · score 6.0 of 10

Single-particle 'definite mass / Klein-Gordon' is loaded into the collapse operator by construction; the N-particle localisation claim rests on an invalid counting of constraints.

self definitional [Section III, Eq. (16) and Eqs. (23)-(26)]
"we choose Â≡p̂²−Ê² ... the asymptotic (collapsed) states are eigenstates of Ê²−p̂² with eigenvalues E²−p²≡µ² ... The dynamics has therefore resulted in collapsed states, each state in the ensemble satisfying the Klein Gordon equation."

The collapse-generating operator is stipulated to be p̂²−Ê², which is exactly the operator whose vanishing eigenvalues define the Klein-Gordon (mass-shell) condition. The master-equation analysis only shows that the state approaches an eigenstate of the chosen operator; calling that eigenstate a KG solution restates the choice of Â. The Klein-Gordon equation is therefore not an independent consequence of the dynamics; the target property was put in when Â was selected.
other [Section VI, Eqs. (74)-(75)]
"Since the number of coordinates in this set is 2(N−1) ... in order to fix all these coordinates we require N!/(2!(N−2)!) ≥ 2(N−1) =⇒ N≥4. This means ... The particles will be localised in space and in time."

This is not a circular reduction but a load-bearing unsupported inference: every A_ij=(x_i−x_j)²−(t_i−t_j)² is invariant under a global Lorentz boost, so the common eigenspaces carry a continuous one-parameter boost degeneracy. Fixing all A_ij fixes only the boost-invariant relative shape; the counting of 2(N−1) relative coordinates omits the boost direction and therefore does not establish a unique spacetime configuration, even for N≥4.

full rationale

The paper is an explicit model construction: the stochastic CSL machinery (Eqs. (17), (45), (69)) and the Born-rule asymptotics are standard and are imported from the CSL literature rather than re-derived from the new assumptions; the energy martingale (63)-(64) is a genuine stochastic-calculus consequence. The main circular element is the single-particle mass/KG claim: Â is chosen as p²−E² and the asymptotic eigenstates are then recognised as KG solutions. That is a design choice restated as a result. The N-particle localisation claim is not circular, but the counting argument at Eq. (75) is invalid because the constraints A_ij are boost-invariant; the state can remain a coherent superposition of boosted configurations, so the claimed localisation is not proven. For this reason the central definite-configuration result is not independently established.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The model's central claims rest on several posited ingredients: the time operators, the uniform-s probability postulate, the scalar collapse generators, and the assumption that negative-energy and tachyon branches can be ignored. The most consequential unstated premise is the independence of the Aij constraints in the N-particle counting.

free parameters (3)

m_i = not fixed
Dimensionful constants in H=(p_i^2-E_i^2)/(2m_i); set the rate of s-to-time flow and wave-packet spreading. Chosen by hand, not derived.
λ_i (collapse strengths) = not fixed
Collapse rates for each scalar operator; determine which A_i are localised and how fast. Free model parameters.
S (evolution interval) = large, finite
Regularization for the uniform s-probability; appears in P(x,t) and requires Sλ>>1 and S<<mσ^2.

assumptions (4)

ad hoc to paper Uniform probability over evolution parameter s: P(x,t)=(1/S)∫ds|<x,t|ψ,s>|^2
This is the only bridge from s-evolution to spacetime probabilities; it is posited without derivation in Section II.A Eq. (3).
domain assumption Positive-energy restriction for Klein-Gordon states: |ψ,S/2> uses only E=ω_p>0
In Eq. (24) only positive-frequency states are integrated; the model itself allows negative-energy and tachyon branches, acknowledged in Section III.
standard math CSL Itô and master equations from Ref. [4]
The unraveling Eq. (17) and density matrix Eq. (20) are taken from the authors' own textbook; they are standard stochastic calculus but are assumed.
domain assumption Hamiltonian must be a Lorentz scalar to be boost invariant
Section II.B.2 demands H'=e^{iθk}He^{-iθk}=H, restricting H to functions of p^2-E^2.

invented entities (1)

per-particle time operator \hat t_i
purpose: Gives each particle its own quantum time, conjugate to energy \hat E_i, enabling quantized-time relativistic collapse.
No experimental handle is provided; it is a new quantum degree of freedom in the model.

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Cite this review

Pith. "Pith review of Relativistic Collapse Model with Quantised Time Variables." pith.science (2026). https://pith.science/paper/IALG5HE4

@misc{pith2026250607959,
  author       = {Pith},
  title        = {Pith review of: Relativistic Collapse Model with Quantised Time Variables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IALG5HE4}},
  note         = {Machine review of arXiv:2506.07959}
}

read the original abstract

A relativistic collapse model for distinguishable particles is presented. Position and time, for each particle, are the fundamental operators of the theory. The Schr\"odinger equation is of the CSL form, with a Hermitian Hamiltonian and an anti-Hermitian, white-noise dependent, Hamiltonian. It generates state vector evolution parametrised by an "evolution parameter". It is shown how this can be interpreted as an evolving state in spacetime with collapses satisfying Born rule probabilities, and how certain choices of collapse generating operators lead to states of definite mass and definite configuration in spacetime. The model is Poincar\'e covariant and conserves energy in expectation.

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Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

We also note the commutation relations [ˆk,ˆx] =iˆt,[ ˆk, ˆt] =iˆx

Poincar´ e Generators The Poincar´ e generators are space-translation ˆp, time-translationˆEand boost ˆk= ˆxˆE− ˆtˆp, whose mutual non-vanishing commutation relations are, as usual, [ˆk,ˆp] =i ˆE,[ ˆk, ˆE] =iˆp. We also note the commutation relations [ˆk,ˆx] =iˆt,[ ˆk, ˆt] =iˆx. Consider first the generation of boosts. The unitary transformation of boosts...

work page

[2] [2]

This condition, that ˆHbe a Lorentz scalar means that it has to be a function of Lorentz scalars

Choice of Hamiltonian Applying the unitary boost operator to the Schr¨ odinger equation, the boosted state vector |ψ, s⟩′ =e iθˆk|ψ, s⟩satisfies d ds |ψ, s⟩′ =−ie iθˆk ˆHe −iθˆk|ψ, s⟩′.(5) For boost invariance, it is necessary that|ψ, s⟩ ′ also be a possible evolution in the original frame, that is, a solution of the Schr¨ odinger equation (2), which requ...

work page

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The same calculation for particle 2 leads to the same result as (42) with 2 in place of 1

e − [ ¯E1(x1−¯x1)−¯p1(t1− ¯t1)]2 2(σ2x1 ¯E2 1 +σ2 t1 ¯p2 1) .(42) This means that the probability density is focussed about a path through spacetime determined by x1 = ¯x1 + ¯p1 ¯E1 (t1 − ¯t1).(43) Particle 1 follows a straight line path through spacetime with velocity ¯p 1/ ¯E1. The same calculation for particle 2 leads to the same result as (42) with 2 ...

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Ghirardi, A

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