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Towards relativistic generalization of collapse models

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Pith's one-line read The paper constructs a relativistic collapse master equation whose local field collapse operator, Lorentz-invariant non-Markovian noise, and normal-ordering prescription produce a finite energy rate and preserve microcausality.

desk verdict A serious and novel construction for relativistic collapse, but the no-vacuum-production claim is undone by an unjustified normal-ordering subtraction. read the letter →

arxiv 2507.06954 v2 pith:VL3YWODP submitted 2025-07-09 quant-ph

Anirudh Gundhi , Lajos Diósi , Matteo Carlesso This is my paper · ORCID
classification quant-ph
keywords spontaneouscollapsemodelsrelativisticquantumfieldtheorymicrocausalitynon-Markoviannoisemasterequationnormalorderingmeasurementproblemscalar
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

The paper claims that a relativistic generalization of spontaneous collapse can be written down at the density-matrix level without the problems that stopped earlier attempts: no microcausality violation, no infinite energy rate, no particle production from vacuum, and no tachyonic behaviour. The construction combines a local quadratic collapse operator $Q(x)=\frac{1}{2}\alpha\varphi^2(x)$, a non-Markovian Gaussian noise whose correlation $G(x-y)$ is Lorentz invariant, and a normal-ordering prescription for observables. In the non-relativistic limit the energy rate scales with particle number $N$, matching the behaviour of standard CSL (Continuous Spontaneous Localization)-type collapse models. If the claim is right, it gives a concrete testable starting point for relativistic collapse phenomenology and sharpens the open question of whether a single-realization statevector can be objective across reference frames.

What carries the argument

The load-bearing object is Eq. (9), the second-order operator evolution in the Heisenberg picture with the nested commutator structure $[\hat{Q}(x_2),[\hat{Q}(x_1),\hat{O}^{(0)}(z)]]$ integrated over $x_2^0\le x_1^0\le z^0$. This object carries the argument because the time-ordered integration domain imposes the past-lightcone structure that yields microcausality, while the choice of $\hat{Q}$ as the local Lorentz scalar $\frac{1}{2}\alpha\varphi^2$ guarantees the commutator has the standard quantum-field-theory support properties. The other two ingredients are the non-Markovian noise correlation $G(x-y)$, a Lorentz-invariant function of the spacetime interval that replaces the divergent $\delta^4$ white-noise correlation, and normal ordering of observables, which drops the vacuum $\frac{1}{2}$-term in the two-point function and makes the energy rate in Eq. (27) finite.

What would settle it

Compute the memory integral in Eq. (9) for two spacelike-separated noise insertions under a finite Lorentz boost: if the time-ordered domain $x_2^0\le x_1^0\le z^0$ changes its form and the double commutator does not vanish, the dynamics is foliation-dependent and not covariant. Alternatively, evaluate Eq. (15) at third order in $\sqrt{\gamma}$ for spacelike-separated $\tilde z_1,\tilde z_2$; a nonzero outermost commutator would disprove the all-orders microcausality claim.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the second-order Heisenberg-picture operator evolution $\hat{O}^{(2)}(z)=\hat{O}^{(0)}(z)-\gamma\int_{x_2^0\le x_1^0\le z^0} d^4x_2\,d^4x_1\, G(x_2-x_1)[\hat{Q}(x_2),[\hat{Q}(x_1),\hat{O}^{(0)}(z)]]$ is the seed of a fully relativistic stochastic dynamics. For a real scalar field with $\hat{Q}(x)=\frac{1}{2}\alpha\varphi^2(x)$, a Lorentz scalar, and $G$ depending only on $(x-y)^2$, the authors argue the evolution is Lorentz covariant to all orders in $\sqrt{\gamma}$; locality of $\hat{Q}$ plus the standard time-ordered domain forces every non-vanishing contribution to a commutator $[\hat{O}(z_2),\hat{O}(z_1)]$ to have $z_1$ in the past lightcone of $z_2$, which is exactly microcausality. Computing the energy rate with normal-ordered observables replaces the $\delta(0)$ divergence of white noise with a finite expression whenever the Fourier transform $G(q)$ makes Eq. (27) converge; normal ordering removes the vacuum $\frac{1}{2}$-term that otherwise produces particles from the vacuum. The proposed example $G(q)=\exp(-q^4/\beta^4)$ is finite everywhere in Fourier space and reproduces CSL-like $N$-scaling in the non-relativistic limit.

Load-bearing premise

The whole relativistic argument depends on the assumption that the time-ordering of noise insertions in Eq. (9) is unchanged by a Lorentz boost; if that ordering is frame-dependent, the master equation defines a preferred foliation and is not actually relativistic.

Editorial extensions

If this is right

  • For any Lorentz-invariant noise correlation $G$ whose Fourier transform makes Eq. (27) converge, the collapse dynamics has a finite normal-ordered energy rate; the relativistic white-noise limit remains divergent because it carries a multiplicative $\delta(0)$.
  • Microcausality holds to all orders in $\sqrt{\gamma}$ for any local collapse operator, and the noise correlation need not be ultralocal, so the earlier restriction to $\delta^4(x-y)$ is lifted.
  • In the non-relativistic limit the energy rate is proportional to the total particle number $N$, recovering the characteristic CSL behaviour and making the model testable by existing non-interferometric collapse experiments.
  • The specific correlation $G(q)=\exp(-q^4/\beta^4)$ fixes the model up to three free parameters, $\alpha$, $\beta$, and $\gamma$, which can be constrained experimentally.
  • A single realization of the noise still does not give a frame-independent statevector, so a fully objective relativistic collapse interpretation remains an open problem.

Reading between the lines

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

  • If the construction is correct, the same normal-ordering plus non-Markovian-noise recipe should also work for other local collapse operators, such as currents or energy densities; each such choice would define a distinct relativistic collapse phenomenology with its own parameter constraints.
  • Because $G(q)=\exp(-q^4/\beta^4)$ suppresses high-frequency noise, the model should predict measurably less heating in fast or high-momentum systems than white-noise collapse; the predicted $\beta$-dependent turnover in ultracold-cantilever or X-ray emission data could distinguish the two.
  • A natural next step not taken in the paper is to write the master equation in manifestly covariant form and check complete positivity; if complete positivity fails, the density-matrix dynamics would not be physically realizable even though each order in $\gamma$ is finite.
  • The paper leaves open whether a nonlinear single-realization collapse exists; if it does not, the model functions as a relativistic decoherence mechanism and the measurement problem remains unresolved.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper proposes a relativistic generalization of collapse models at the density-matrix level. The model uses a local quadratic collapse operator Q = (1/2) alpha phi^2, a non-Markovian stochastic noise with a Lorentz-invariant correlation G(x-y), and a normal-ordering prescription for observables. The authors claim that the resulting second-order evolution (Eq. (9)) is Lorentz covariant, satisfies microcausality to all orders in sqrt(gamma), yields a finite normal-ordered energy rate, avoids particle production from the vacuum and tachyonic behavior, and reduces to a CSL-like scaling with particle number N in the non-relativistic limit. A concrete admissible correlation G(q) = exp(-q^4/beta^4) is proposed.

Significance. If the central claims were correct, this would be an important step toward a relativistic collapse model: the energy-rate algebra in Eqs. (18)-(27) is explicit, the white-noise delta(0) divergence of Appendix B matches the known literature, the model has no fitted parameters, and the Discussion honestly acknowledges the open statevector-interpretation problem. However, the advertised absence of vacuum particle production is not established by the calculation, and the Lorentz-covariance argument is incomplete. As it stands, the paper's headline claims are conditional on repairs that are not local presentation fixes.

major comments (2)
  1. [Lorentz covariance] Eq. (9) (and its derivation in Appendix A, Eq. (A5)) defines the second-order correction as an integral over the frame-dependent domain x0_2 <= x0_1 <= z0. The assertion that Q being a Lorentz scalar and G being Lorentz invariant suffice for covariance does not address the transformation of this integration domain. Under a boost, the inequalities are not invariant, so one must show that spacelike-separated insertions give no net contribution and that the time-ordering of timelike insertions is boost-invariant. Without this case analysis, the master equation may depend on the chosen foliation, and the paper's central claim of relativistic covariance is unproven.
  2. [Rate of increase of energy, Eqs. (25)-(26)] The claim that normal ordering eliminates particle production from the vacuum is not supported. The 1/(2 omega_q) term in Eq. (25) is dropped from N(x,z) after identifying it with the QFT vacuum divergence, but N(x,z) is a response-function trace in Eq. (18), not an expectation value of a normal-ordered observable. Normal-ordering :H: removes only a c-number and leaves this trace unchanged. A single-mode check for a quadratic collapse operator Q proportional to (a+a^dagger)^2 and H = omega a^dagger a gives <0|[Q(t2),[Q(t1),:H:]]|0> proportional to cos(2 omega (t2-t1)), so the integrated rate is nonzero for a non-Markovian G. Thus Eq. (26) vanishes for vacuum only because the pair-creation amplitude was subtracted from the response function, not because the dynamics conserve particle number. The stochastic Hamiltonian still contains Q and produces pairs from the vacuum; the abstract's claim 'without having particle production from the vacuum' is therefore not a property of the model as defined.
minor comments (6)
  1. [Abstract and text] There are typos such as 'encoutered' in the abstract and 'distiction' after Eq. (9); the manuscript should be proofread.
  2. [Eq. (30)] Please clarify that q^4 denotes (q^2)^2, not the fourth power of a single component; the current notation is ambiguous.
  3. [Model definition] The normal-ordering prescription is introduced only in the energy-rate section; it should be stated as part of the model definition, including whether the collapse operator Q itself is normal-ordered.
  4. [Eq. (8)] The integration measure in the second term of Eq. (8) appears to be d4x2 d3x1 with x1 on the spatial slice x0_1 = z0; please clarify the notation so that the domain and measure are unambiguous.
  5. [Eq. (13)] The shift from xi(t,z) to xitilde(s,z) assumes stationarity of the noise; this assumption should be stated explicitly.
  6. [Appendix C] The argument that normal ordering leaves microcausality unchanged is only sketched for higher-order operators; a fuller use of Wick's theorem would make the proof easier to verify.

Circularity Check

2 steps flagged · score 6.0 of 10

The no-vacuum-particle-production claim is imposed by the normal-ordering prescription that drops the pair-creation amplitude; finiteness of the energy rate is defined as a condition on G, not derived.

  1. self definitional [Section 'The non-Markovian model', after Eq. (25), near Eq. (26)]
    "The factor of 1/2 in the first expression captures the standard QFT vacuum divergence. This factor leads to a divergent particle production rate from vacuum, which was already found in Ref. [33]. ... Therefore, here, we propose to use a normal-ordering prescription, where all the observables of interest are normal-ordered at all times, i.e. ˆO(z)→: ˆO(z) : and thus the 1/2 term is dropped."

    The 1/2 term in N(x,z) is not an observable zero-point constant; it is the vacuum contribution to the trace Tr[{φ(x),φ(z)}ρ(0)] entering the energy-rate response (Eqs. 20-21). With Q∝φ^2, this term contains the pair-creation (a†a†) amplitude. The paper identifies this term as the cause of vacuum particle production and then defines normal ordering so that the term is dropped. Consequently Eq. (26) is proportional to n_q and vanishes at n_q=0 by construction, not because the stochastic dynamics (Appendix A, with H_st∝Q) actually creates no pairs. The advertised absence of vacuum particle production is therefore equivalent to the chosen prescription: the pair-production contribution is subtracted rather than shown to be absent.

  2. self definitional [Section 'Rate of increase of energy' / 'The non-Markovian model', after Eq. (27)]
    "Further, as long as one is only concerned with obtaining a finite rate of energy, any choice of G(q) for which Eq. (27) is finite becomes a viable relativistic non-Markovian collapse correlation within the normal-ordering prescription."

    The finiteness of the energy rate is not derived as a property of the model; it is used to define which noise correlations count as viable. The subsequent exponential choice G(q)=exp(-q^4/β^4) is selected precisely because it makes the integral in Eq. (27) converge. Thus the headline statement that non-Markovianity and normal ordering 'lead to a finite rate of energy' is a selection criterion on G, not a prediction: any divergent candidate is excluded by definition, and the remaining class is then presented as a successful outcome.

full rationale

The paper has substantial independent content: the microcausality proof is a standard lightcone-commutator argument, the non-relativistic limit explicitly recovers CSL-like N scaling, and the parameters α, β, γ are free rather than fitted. The self-citations (Diosi-Ferialdi, Bassi-Ghirardi) are used as inputs or for framing, not as load-bearing evidence that forces the central result. However, the two headline claims advertised in the abstract—finite energy rate and no vacuum particle production—are built into the construction rather than derived. The 1/2 term in N(x,z) is the vacuum anticommutator / pair-creation contribution to the energy-rate response; dropping it via normal ordering makes the vacuum rate zero by construction, while the stochastic Hamiltonian in Appendix A still contains Q and hence still creates pairs. Likewise, 'any choice of G(q) for which Eq. (27) is finite becomes a viable correlation' defines the viable class by finiteness, so the finite-rate result is a tautological consequence of the selection rule. The Lorentz-covariance claim is asserted from Q(x) scalar and G(x-y) invariant without treating the frame-dependence of the time-ordered integration domain in Eq. (9); that is a correctness gap rather than a circularity, because it is not resolved by redefining an input. Overall, partial circularity: the central no-vacuum-production claim reduces to the normal-ordering prescription, while other parts of the paper remain independent.

Assumptions & free parameters 3 free parameters · 7 assumptions · 1 invented entities

The construction rests on standard QFT (free Klein-Gordon field, local commutators, perturbative expansion), on the prior non-Markovian collapse formalism of Adler-Bassi and Diósi-Ferialdi (Eqs. 1-6), and on three modeling choices specific to this paper: a Lorentz-invariant non-Markovian noise correlation (Eqs. 2, 27), a normal-ordering prescription that removes the vacuum 1/2 term (after Eq. 25), and the choice Q = (1/2)alpha phi^2 motivated by the NR mass-density limit (Eq. 16). The parameters alpha, beta, gamma are free.

free parameters (3)
  • alpha (collapse operator coefficient)
    Coupling in Q(x) = (1/2)alpha phi^2(x), Eq. (16). Free model parameter, to be bounded by experiment, analogous to CSL's rate parameter; not fitted here.
  • beta (noise correlation width)
    Width of the Lorentz-invariant noise correlation Gtilde(q) = exp(-q^4/beta^4), Eq. (30). Free parameter controlling the decay scale in Fourier space; not fitted here.
  • gamma (collapse noise coupling)
    Common coupling strength multiplying the collapse noise in Eqs. (1) and (A1). Free parameter; the perturbative expansions are in powers of sqrt(gamma); not fitted here.
assumptions (7)
  • domain assumption Free real Klein-Gordon scalar field with local operators and canonical commutation relations
    The background QFT: H0 is the free KG Hamiltonian (Eq. 17), Q = (1/2)alpha phi^2 is a local operator, and the perturbative expansion and equal-time commutators (Eq. 19, Appendix A) are standard.
  • domain assumption Adler-Bassi non-Markovian collapse equation (Eq. 1) is the correct second-order (in sqrt(gamma)) starting point
    Cited to [34, 35]; the double-commutator master equation (Eqs. 5-7) and the closure at second order are taken from the prior non-Markovian collapse literature.
  • domain assumption Noise xi(t,x) is a real Gaussian random field with zero mean; all statistics are in G(x2-x1)
    Eqs. (1)-(2). Gaussianity and moment structure are inherited from the Adler-Bassi formalism; the continuum limit replaces the discrete D_ij by G.
  • domain assumption G is a Lorentz-invariant function of (x-y)^2 and translation invariant
    Stated as a design requirement in 'Lorentz covariance': G(x,y) = G((x-y)_mu (x-y)^mu). This is the key input that the covariance claim rests on.
  • ad hoc to paper Normal-ordering prescription for all observables at all times
    Introduced in 'Rate of increase of energy' to drop the 1/2 vacuum term in Eq. (25) and thereby remove the vacuum divergence; the paper is transparent about this being a prescription.
  • domain assumption Initial state with definite particle number (langle a_q a_p rangle = langle a+_q a+_p rangle = 0)
    Eq. (23). Stated assumption used to obtain N(x) and D(x) in Eq. (25); the authors note this initial state is not Lorentz invariant, which limits the energy-rate results to a chosen frame.
  • ad hoc to paper In the non-relativistic limit, Q = (1/2)alpha phi^2 becomes proportional to the mass-density operator
    Assumed after Eq. (16) to connect the model to CSL phenomenology; no derivation is given, and phi^2 contains phi+^2 and phi-^2 pieces beyond the density operator, so the identification is not literal.
invented entities (1)
  • Stochastic collapse noise field xi(t,x) with non-Markovian, Lorentz-invariant correlation G independent evidence
    purpose: Drives the collapse: appears as the stochastic term in the nonlinear unraveling (Eq. 1) and as the stochastic Hamiltonian H_st = hbar sqrt(gamma) integral dz Q(z)xi(t,z) in the unitary unraveling (Eq. A1); its correlation G fixes decoherence and energy-growth rates.
    The field itself is unobservable, but the model delivers falsifiable predictions: collapse-induced decoherence rates and energy increase in matter, parameterized by (alpha, beta, gamma), measurable with interferometric, X-ray, and force-sensor experiments in the CSL tradition.

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Cite this review

Pith. "Pith review of Towards relativistic generalization of collapse models." pith.science (2026). https://pith.science/paper/VL3YWODP

@misc{pith2026250706954,
  author       = {Pith},
  title        = {Pith review of: Towards relativistic generalization of collapse models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VL3YWODP}},
  note         = {Machine review of arXiv:2507.06954}
}
read the original abstract

Spontaneous collapse models provide a possible, testable solution to the quantum measurement problem. While experiments are providing increasingly stronger bounds on their parameters, a full-fledged relativistic extension is still missing. Previous attempts have encountered different obstacles, such as violation of microcausality, infinite energy rate, and particle production from vacuum. Here, we propose a generalization of the collapse master equation that is characterized by a local field collapse operator and a non-Markovian noise with a Lorentz invariant correlation. Our construction is able to overcome previously encountered problems and has the desirable properties in the non relativistic limit. A specific choice of the noise correlation function is also introduced and discussed.

Figures

Figures reproduced from arXiv: 2507.06954 by the authors.

Figure 1
Figure 1. FIG. 1. Leading order causal structure which demonstrates [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

43 extracted references · 36 canonical work pages · cited by 1 Pith paper

  1. [1]

    Bassi and G

    A. Bassi and G. Ghirardi, Dynamical reduction models, Physics Reports379, 257 (2003)

    2003

  2. [2]

    Bassi, K

    A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ul- bricht, Models of wave-function collapse, underlying the- ories, and experimental tests, Rev. Mod. Phys.85, 471 (2013)

    2013

  3. [3]

    Pearle, Combining stochastic dynamical state-vector reduction with spontaneous localization, Phys

    P. Pearle, Combining stochastic dynamical state-vector reduction with spontaneous localization, Phys. Rev. A 39, 2277 (1989)

    1989

  4. [4]

    G. C. Ghirardi, P. Pearle, and A. Rimini, Markov pro- cesses in hilbert space and continuous spontaneous local- ization of systems of identical particles, Phys. Rev. A42, 78 (1990)

    1990

  5. [5]

    Di´ osi, A universal master equation for the gravita- tional violation of quantum mechanics, Physics Letters A120, 377 (1987)

    L. Di´ osi, A universal master equation for the gravita- tional violation of quantum mechanics, Physics Letters A120, 377 (1987)

    1987

  6. [6]

    Arndt and K

    M. Arndt and K. Hornberger, Testing the limits of quan- tum mechanical superpositions, Nature Phys10, 271–277 (2014))

    work page 2014

  7. [7]

    Bilardello, S

    M. Bilardello, S. Donadi, A. Vinante, and A. Bassi, Bounds on collapse models from cold-atom experiments, Physica A: Statistical Mechanics and its Applications 462, 764 (2016)

    work page 2016

  8. [8]

    Carlesso, A

    M. Carlesso, A. Bassi, P. Falferi, and A. Vinante, Ex- perimental bounds on collapse models from gravitational wave detectors, Physical Review D94, 124036 (2016)

    work page 2016

Show all 43 references
  1. [9]

    Vinante, R

    A. Vinante, R. Mezzena, P. Falferi, M. Carlesso, and A. Bassi, Improved noninterferometric test of collapse models using ultracold cantilevers, Physical review let- ters119, 110401 (2017)

    2017

  2. [10]

    Toroˇ s and A

    M. Toroˇ s and A. Bassi, Bounds on quantum collapse models from matter-wave interferometry: calculational details, Journal of Physics A: Mathematical and Theo- retical51, 115302 (2018)

    2018

  3. [11]

    Vinante, M

    A. Vinante, M. Carlesso, A. Bassi, A. Chiasera, S. Varas, P. Falferi, B. Margesin, R. Mezzena, and H. Ulbricht, Narrowing the parameter space of collapse models with ultracold layered force sensors, Physical Review Letters 125, 100404 (2020)

    2020

  4. [12]

    Donadi, K

    S. Donadi, K. Piscicchia, C. Curceanu, L. Di´ osi, M. Laubenstein, and A. Bassi, Underground test of gravity-related wave function collapse, Nature Physics 17, 74 (2021)

    2021

  5. [13]

    Carlesso, S

    M. Carlesso, S. Donadi, L. Ferialdi,et al., Present sta- tus and future challenges of non-interferometric tests of collapse models, Nature Phys , 243 (2022)

    2022

  6. [14]

    Figurato, M

    L. Figurato, M. Dirindin, J. L. Gaona-Reyes, M. Car- lesso, A. Bassi, and S. Donadi, On the effectiveness of the collapse in the di´ osi–penrose model, New Journal of Physics26, 113004 (2024)

    2024

  7. [15]

    Piscicchia, S

    K. Piscicchia, S. Donadi, S. Manti, A. Bassi, M. Der- akhshani, L. Di´ osi, and C. Curceanu, X-ray emission from atomic systems can distinguish between prevailing dy- namical wave-function collapse models, Physical Review Letters132, 250203 (2024)

    2024

  8. [16]

    D. G. A. Altamura, A. Vinante, and M. Carlesso, Im- proved bounds on collapse models from rotational noise of the laser interferometer space antenna pathfinder mis- sion, Physical Review A111, L020203 (2025)

    2025

  9. [17]

    Piccione and A

    N. Piccione and A. Bassi, Exploring the effects 7 of mass dependence in spontaneous collapse models, arXiv:2501.17637 (2025)

    2025 arXiv

  10. [18]

    Dove and E

    C. Dove and E. J. Squires, A local model of explicit wavefunction collapse (1996), arXiv:quant-ph/9605047 [quant-ph]

    1996 arXiv

  11. [19]

    Breuer and F

    H.-P. Breuer and F. Petruccione, Relativistic formulation of quantum-state diffusion, Journal of Physics A: Math- ematical and General31, 33 (1998)

    1998

  12. [20]

    Breuer and F

    H.-P. Breuer and F. Petruccione, Stochastic unraveling of relativistic quantum measurements, inOpen Systems and Measurement in Relativistic Quantum Theory, edited by H.-P. Breuer and F. Petruccione (Springer Berlin Heidel- berg, Berlin, Heidelberg, 1999) pp. 81–116

    1999

  13. [21]

    Dowker and J

    F. Dowker and J. Henson, Spontaneous collapse mod- els on a lattice, Journal of Statistical Physics115, 1327 (2004)

    2004

  14. [22]

    Roderich, On spontaneous wave function collapse and quantum field theory, Proc

    T. Roderich, On spontaneous wave function collapse and quantum field theory, Proc. R. Soc. A.462, 1897–1908 (2006)

    2006

  15. [23]

    Tumulka, A relativistic version of the ghi- rardi–rimini–weber model, J Stat Phys125, 821–840 (2006)

    R. Tumulka, A relativistic version of the ghi- rardi–rimini–weber model, J Stat Phys125, 821–840 (2006)

    2006

  16. [24]

    Bedingham, Relativistic state reduction dynamics, Found Phys41, 686–704 (2011)

    D. Bedingham, Relativistic state reduction dynamics, Found Phys41, 686–704 (2011)

    2011

  17. [25]

    Bedingham, D

    D. Bedingham, D. D¨ urr, G. Ghirardi, S. Goldstein, R. Tumulka, and N. Zangh ` ı, Matter density and relativis- tic models of wave function collapse, Journal of Statistical Physics154, 623 (2014)

    2014

  18. [26]

    Tilloy, Interacting quantum field theories as rel- ativistic statistical field theories of local beables, arXiv:1702.06325 (2017)

    A. Tilloy, Interacting quantum field theories as rel- ativistic statistical field theories of local beables, arXiv:1702.06325 (2017)

    2017 arXiv

  19. [27]

    Bedingham and P

    D. Bedingham and P. Pearle, On the csl scalar field rel- ativistic collapse model, arXiv:1906.11510 (2019)

    2019 arXiv

  20. [28]

    Tumulka, A relativistic grw flash process with inter- action, inDo Wave Functions Jump? : Perspectives of the Work of GianCarlo Ghirardi, edited by V

    R. Tumulka, A relativistic grw flash process with inter- action, inDo Wave Functions Jump? : Perspectives of the Work of GianCarlo Ghirardi, edited by V. Allori, A. Bassi, D. D¨ urr, and N. Zanghi (Springer International Publishing, Cham, 2021) pp. 321–347

    2021

  21. [29]

    Pearle, How stands collapse i, Journal of Physics A: Mathematical and Theoretical40, 3189 (2007)

    P. Pearle, How stands collapse i, Journal of Physics A: Mathematical and Theoretical40, 3189 (2007)

    2007

  22. [30]

    Pearle, How stands collapse ii, Quantum Reality, Rela- tivistic Causality, and Closing the Epistemic Circle , 257 (2009)

    P. Pearle, How stands collapse ii, Quantum Reality, Rela- tivistic Causality, and Closing the Epistemic Circle , 257 (2009)

    2009

  23. [31]

    Jones, G

    C. Jones, G. Gasbarri, and A. Bassi, Mass-coupled rela- tivistic spontaneous collapse models, Journal of Physics A: Mathematical and Theoretical54, 295306 (2021)

    2021

  24. [32]

    P. Pearle, Toward a relativistic theory of statevector reduction, inSixty-Two Years of Uncertainty: Histori- cal, Philosophical, and Physical Inquiries into the Foun- dations of Quantum Mechanics, edited by A. I. Miller (Springer US, Boston, MA, 1990) pp. 193–214

    1990

  25. [33]

    Pearle, Relativistic collapse model with tachyonic fea- tures, Phys

    P. Pearle, Relativistic collapse model with tachyonic fea- tures, Phys. Rev. A59, 80 (1999)

    1999

  26. [34]

    S. L. Adler and A. Bassi, Collapse models with non-white noises, Journal of Physics A: Mathematical and Theoret- ical40, 15083 (2007)

    2007

  27. [35]

    Di´ osi and L

    L. Di´ osi and L. Ferialdi, General non-markovian struc- ture of gaussian master and stochastic schr¨ odinger equa- tions, Physical review letters113, 200403 (2014)

    2014

  28. [36]

    Di´ osi, Is spontaneous wave function collapse testable at all?, Journal of Physics: Conference Series626, 012008 (2015)

    L. Di´ osi, Is spontaneous wave function collapse testable at all?, Journal of Physics: Conference Series626, 012008 (2015)

    2015

  29. [37]

    Di´ osi, How to teach and think about spontaneous wave function collapse theories: not like before (Cambridge University Press, 2018) pp

    L. Di´ osi, How to teach and think about spontaneous wave function collapse theories: not like before (Cambridge University Press, 2018) pp. 3–11

    2018

  30. [38]

    Weinberg, The cluster decomposition principle, inThe Quantum Theory of Fields(Cambridge University Press,

    S. Weinberg, The cluster decomposition principle, inThe Quantum Theory of Fields(Cambridge University Press,

  31. [39]

    M. E. Peskin and D. V. Schroeder,An Introduction to quantum field theory(Addison-Wesley, Reading, USA, 1995)

    1995

  32. [40]

    Di´ osi, Relativistic theory for continuous measurement of quantum fields, Phys

    L. Di´ osi, Relativistic theory for continuous measurement of quantum fields, Phys. Rev. A42, 5086 (1990)

    1990

  33. [41]

    Di´ osi, Non-markovian continuous quantum measure- ment of retarded observables, Physical review letters 100, 080401 (2008)

    L. Di´ osi, Non-markovian continuous quantum measure- ment of retarded observables, Physical review letters 100, 080401 (2008)

    2008

  34. [42]

    H. M. Wiseman and J. M. Gambetta, Pure-state quan- tum trajectories for general non-markovian systems do not exist, Physical review letters101, 140401 (2008)

    2008

  35. [43]

    ˆ1 + i ℏ ˆ t t0 dt′ ˆH I st(t′)− 1 ℏ2 ˆ t t0 dt′ ˆ t′ t0 dt′′ ˆH I st(t′′) ˆH I st(t′) + higher order # × ˆO0(z, t)× ×

    W. Greiner and J. Reinhardt, Interacting quantum fields, inField Quantization(Springer Berlin Heidelberg, Berlin, Heidelberg, 1996) pp. 211–268. Appendix A: The unitary unraveling We consider the quantum dynamics governed by the Hamiltonian ˆH, ˆH= ˆH0 + ˆHst(t),(A1) where ˆH0...

    1996

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