Objective-collapse theory

Last updated

Objective-collapse theories, also known spontaneous collapse models [1] or dynamical reduction models, [2] are proposed solutions to the measurement problem in quantum mechanics. [3] As with other interpretations of quantum mechanics, they are possible explanations of why and how quantum measurements always give definite outcomes, not a superposition of them as predicted by the Schrödinger equation, and more generally how the classical world emerges from quantum theory. The fundamental idea is that the unitary evolution of the wave function describing the state of a quantum system is approximate. It works well for microscopic systems, but progressively loses its validity when the mass / complexity of the system increases.

Contents

In collapse theories, the Schrödinger equation is supplemented with additional nonlinear and stochastic terms (spontaneous collapses) which localize the wave function in space. The resulting dynamics is such that for microscopic isolated systems, the new terms have a negligible effect; therefore, the usual quantum properties are recovered, apart from very tiny deviations. Such deviations can potentially be detected in dedicated experiments, and efforts are increasing worldwide towards testing them.

An inbuilt amplification mechanism makes sure that for macroscopic systems consisting of many particles, the collapse becomes stronger than the quantum dynamics. Then their wave function is always well-localized in space, so well-localized that it behaves, for all practical purposes, like a point moving in space according to Newton's laws.

In this sense, collapse models provide a unified description of microscopic and macroscopic systems, avoiding the conceptual problems associated to measurements in quantum theory.

The most well-known examples of such theories are:

Collapse theories stand in opposition to many-worlds interpretation theories, in that they hold that a process of wave function collapse curtails the branching of the wave function and removes unobserved behaviour.

History of collapse theories

Philip Pearle's 1976 paper pioneered the quantum nonlinear stochastic equations to model the collapse of the wave function in a dynamical way; [4] : 477  [5] [6] [7] this formalism was later used for the CSL model. However, these models lacked the character of “universality” of the dynamics, i.e. its applicability to an arbitrary physical system (at least at the non-relativistic level), a necessary condition for any model to become a viable option.

The next major advance came in 1986, when Ghirardi, Rimini and Weber published the paper with the meaningful title “Unified dynamics for microscopic and macroscopic systems”, [4] [8] where they presented what is now known as the GRW model, after the initials of the authors. The model has two guiding principles: [4]

  1. The position basis states are used in the dynamic state reduction (the "preferred basis" is position);
  2. The modification must reduce superpositions for macroscopic objects without altering the microscopic predictions.

In 1990 the efforts for the GRW group on one side, and of P. Pearle on the other side, were brought together in formulating the Continuous Spontaneous Localization (CSL) model, [9] [10] where the Schrödinger dynamics and a randomly fluctuating classical field produce collapse into spatially localized eigentstates. [4] : 478 

In the late 1980s and 1990s, Diosi [11] [12] and Penrose [13] [14] and others [4] : 508  independently formulated the idea that the wave function collapse is related to gravity. The dynamical equation is structurally similar to the CSL equation.

Three models are most widely discussed in the literature:

The Quantum Mechanics with Universal Position Localization (QMUPL) model [12] should also be mentioned; an extension of the GRW model for identical particles formulated by Tumulka, [15] which proves several important mathematical results regarding the collapse equations. [16]

In all models listed so far, the noise responsible for the collapse is Markovian (memoryless): either a Poisson process in the discrete GRW model, or a white noise in the continuous models. The models can be generalized to include arbitrary (colored) noises, possibly with a frequency cutoff: the CSL model has been extended to its colored version [17] [18] (cCSL), as well as the QMUPL model [19] [20] (cQMUPL). In these new models the collapse properties remain basically unaltered, but specific physical predictions can change significantly.

In all collapse models, the noise effect must prevent quantum mechanical linearity and unitarity and thus cannot be described within quantum-mechanics. [21] : 423  Because the noise responsible for the collapse induces Brownian motion on each constituent of a physical system, energy is not conserved. The kinetic energy increases at a constant rate. Such a feature can be modified, without altering the collapse properties, by including appropriate dissipative effects in the dynamics. This is achieved for the GRW, CSL, QMUPL and DP models, obtaining their dissipative counterparts (dGRW, [22] dCSL, [23] [24] dQMUPL, [25] DP [26] [24] ). The QMUPL model has been further generalized to include both colored noise as well as dissipative effects [27] [28] (dcQMUPL model).

Tests of collapse models

Collapse models modify the Schrödinger equation; therefore, they make predictions that differ from standard quantum mechanical predictions. Although the deviations are difficult to detect, there is a growing number of experiments searching for spontaneous collapse effects. They can be classified in two groups:

Problems and criticisms to collapse theories

Violation of the principle of the conservation of energy

According to collapse theories, energy is not conserved, also for isolated particles. More precisely, in the GRW, CSL and DP models the kinetic energy increases at a constant rate, which is small but non-zero.

This is often presented as an unavoidable consequence of Heisenberg's uncertainty principle: the collapse in position causes a larger uncertainty in momentum. This explanation is wrong; in collapse theories the collapse in position also determines a localization in momentum, driving the wave function to an almost minimum uncertainty state both in position and in momentum, [16] compatibly with Heisenberg's principle. The reason the energy increases is that the collapse noise diffuses the particle, thus accelerating it.

This is the same situation as in classical Brownian motion, and similarly this increase can be stopped by adding dissipative effects. Dissipative versions of the QMUPL, GRW, CSL and DP models exist, [22] [23] [25] [24] where the collapse properties are left unaltered with respect to the original models, while the energy thermalizes to a finite value (therefore it can even decrease, depending on its initial value).

Still, in the dissipative model the energy is not strictly conserved. A resolution to this situation might come by considering also the noise a dynamical variable with its own energy, which is exchanged with the quantum system in such a way that the energy of the total system and noise together is conserved.[ citation needed ]

Relativistic collapse models

One of the biggest challenges in collapse theories is to make them compatible with relativistic requirements. The GRW, CSL and DP models are not. The biggest difficulty is how to combine the nonlocal character of the collapse, which is necessary in order to make it compatible with the experimentally verified violation of Bell inequalities, with the relativistic principle of locality. Models exist [30] [31] that attempt to generalize in a relativistic sense the GRW and CSL models, but their status as relativistic theories is still unclear. The formulation of a proper Lorentz-covariant theory of continuous objective collapse is still a matter of research.

Tails problem

In all collapse theories, the wave function is never fully contained within one (small) region of space, because the Schrödinger term of the dynamics will always spread it outside. Therefore, wave functions always contain tails stretching out to infinity, although their “weight” is smaller in larger systems. Critics of collapse theories argue that it is not clear how to interpret these tails. Two distinct problems have been discussed in the literature. The first is the “bare” tails problem: it is not clear how to interpret these tails because they amount to the system never being really fully localized in space. A special case of this problem is known as the “counting anomaly”. [32] [33] Supporters of collapse theories mostly dismiss this criticism as a misunderstanding of the theory, [34] [35] as in the context of dynamical collapse theories, the absolute square of the wave function is interpreted as an actual matter density. In this case, the tails merely represent an immeasurably small amount of smeared-out matter. This leads into the second problem, however, the so-called “structured tails problem”: it is not clear how to interpret these tails because even though their “amount of matter” is small, that matter is structured like a perfectly legitimate world. Thus, after the box is opened and Schroedinger's cat has collapsed to the “alive” state, there still exists a tail of the wavefunction containing “low matter” entity structured like a dead cat. Collapse theorists have offered a range of possible solutions to the structured tails problem, but it remains an open problem. [36]

See also

Related Research Articles

An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics might correspond to experienced reality. Quantum mechanics has held up to rigorous and extremely precise tests in an extraordinarily broad range of experiments. However, there exist a number of contending schools of thought over their interpretation. These views on interpretation differ on such fundamental questions as whether quantum mechanics is deterministic or stochastic, local or non-local, which elements of quantum mechanics can be considered real, and what the nature of measurement is, among other matters.

In quantum mechanics, wave function collapse, also called reduction of the state vector, occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an observation and is the essence of a measurement in quantum mechanics, which connects the wave function with classical observables such as position and momentum. Collapse is one of the two processes by which quantum systems evolve in time; the other is the continuous evolution governed by the Schrödinger equation.

<span class="mw-page-title-main">Big Bounce</span> Model for the origin of the universe

The Big Bounce hypothesis is a cosmological model for the origin of the known universe. It was originally suggested as a phase of the cyclic model or oscillatory universe interpretation of the Big Bang, where the first cosmological event was the result of the collapse of a previous universe. It receded from serious consideration in the early 1980s after inflation theory emerged as a solution to the horizon problem, which had arisen from advances in observations revealing the large-scale structure of the universe.

<span class="mw-page-title-main">Quantum Zeno effect</span> Quantum measurement phenomenon

The quantum Zeno effect is a feature of quantum-mechanical systems allowing a particle's time evolution to be slowed down by measuring it frequently enough with respect to some chosen measurement setting.

In quantum mechanics, the measurement problem is the problem of definite outcomes: quantum systems have superpositions but quantum measurements only give one definite result.

<span class="mw-page-title-main">Black hole information paradox</span> Puzzle of disappearance of information in a black hole

The black hole information paradox is a paradox that appears when the predictions of quantum mechanics and general relativity are combined. The theory of general relativity predicts the existence of black holes that are regions of spacetime from which nothing—not even light—can escape. In the 1970s, Stephen Hawking applied the semiclassical approach of quantum field theory in curved spacetime to such systems and found that an isolated black hole would emit a form of radiation. He also argued that the detailed form of the radiation would be independent of the initial state of the black hole, and depend only on its mass, electric charge and angular momentum.

The Schrödinger–Newton equation, sometimes referred to as the Newton–Schrödinger or Schrödinger–Poisson equation, is a nonlinear modification of the Schrödinger equation with a Newtonian gravitational potential, where the gravitational potential emerges from the treatment of the wave function as a mass density, including a term that represents interaction of a particle with its own gravitational field. The inclusion of a self-interaction term represents a fundamental alteration of quantum mechanics. It can be written either as a single integro-differential equation or as a coupled system of a Schrödinger and a Poisson equation. In the latter case it is also referred to in the plural form.

<i>Shadows of the Mind</i> Book by Roger Penrose

Shadows of the Mind: A Search for the Missing Science of Consciousness is a 1994 book by mathematical physicist Roger Penrose that serves as a followup to his 1989 book The Emperor's New Mind: Concerning Computers, Minds and The Laws of Physics.

The Ghirardi–Rimini–Weber theory (GRW) is a spontaneous collapse theory in quantum mechanics, proposed in 1986 by Giancarlo Ghirardi, Alberto Rimini, and Tullio Weber.

Some interpretations of quantum mechanics posit a central role for an observer of a quantum phenomenon. The quantum mechanical observer is tied to the issue of observer effect, where a measurement necessarily requires interacting with the physical object being measured, affecting its properties through the interaction. The term "observable" has gained a technical meaning, denoting a Hermitian operator that represents a measurement.

The SP formula for the dephasing rate of a particle that moves in a fluctuating environment unifies various results that have been obtained, notably in condensed matter physics, with regard to the motion of electrons in a metal. The general case requires to take into account not only the temporal correlations but also the spatial correlations of the environmental fluctuations. These can be characterized by the spectral form factor , while the motion of the particle is characterized by its power spectrum . Consequently, at finite temperature the expression for the dephasing rate takes the following form that involves S and P functions:

In theoretical physics, the logarithmic Schrödinger equation is one of the nonlinear modifications of Schrödinger's equation, first proposed by Gerald H. Rosen in its relativistic version in 1969. It is a classical wave equation with applications to extensions of quantum mechanics, quantum optics, nuclear physics, transport and diffusion phenomena, open quantum systems and information theory, effective quantum gravity and physical vacuum models and theory of superfluidity and Bose–Einstein condensation. It is an example of an integrable model.

A dynamical reduction theory (DRT) is an extension of quantum mechanics (QM) that attempts to account for the collapse of the wave function. It is necessary because QM does not account for the specific measurements of observable quantities or events, in the familiar realm of Newtonian or classical physics, that we make in QM experiments.

<span class="mw-page-title-main">Time crystal</span> Structure that repeats in time; a novel type or phase of non-equilibrium matter

In condensed matter physics, a time crystal is a quantum system of particles whose lowest-energy state is one in which the particles are in repetitive motion. The system cannot lose energy to the environment and come to rest because it is already in its quantum ground state. Time crystals were first proposed theoretically by Frank Wilczek in 2012 as a time-based analogue to common crystals – whereas the atoms in crystals are arranged periodically in space, the atoms in a time crystal are arranged periodically in both space and time. Several different groups have demonstrated matter with stable periodic evolution in systems that are periodically driven. In terms of practical use, time crystals may one day be used as quantum computer memory.

Giancarlo Ghirardi was an Italian physicist and emeritus professor of theoretical physics at the University of Trieste.

The quantum jump method, also known as the Monte Carlo wave function (MCWF) is a technique in computational physics used for simulating open quantum systems and quantum dissipation. The quantum jump method was developed by Dalibard, Castin and Mølmer at a similar time to the similar method known as Quantum Trajectory Theory developed by Carmichael. Other contemporaneous works on wave-function-based Monte Carlo approaches to open quantum systems include those of Dum, Zoller and Ritsch and Hegerfeldt and Wilser.

In quantum probability, the Belavkin equation, also known as Belavkin-Schrödinger equation, quantum filtering equation, stochastic master equation, is a quantum stochastic differential equation describing the dynamics of a quantum system undergoing observation in continuous time. It was derived and henceforth studied by Viacheslav Belavkin in 1988.

Quantum foundations is a discipline of science that seeks to understand the most counter-intuitive aspects of quantum theory, reformulate it and even propose new generalizations thereof. Contrary to other physical theories, such as general relativity, the defining axioms of quantum theory are quite ad hoc, with no obvious physical intuition. While they lead to the right experimental predictions, they do not come with a mental picture of the world where they fit.

The continuous spontaneous localization (CSL) model is a spontaneous collapse model in quantum mechanics, proposed in 1989 by Philip Pearle. and finalized in 1990 Gian Carlo Ghirardi, Philip Pearle and Alberto Rimini.

The Diósi–Penrose model was introduced as a possible solution to the measurement problem, where the wave function collapse is related to gravity. The model was first suggested by Lajos Diósi when studying how possible gravitational fluctuations may affect the dynamics of quantum systems. Later, following a different line of reasoning, Roger Penrose arrived at an estimation for the collapse time of a superposition due to gravitational effects, which is the same as that found by Diósi, hence the name Diósi–Penrose model. However, it should be pointed out that while Diósi gave a precise dynamical equation for the collapse, Penrose took a more conservative approach, estimating only the collapse time of a superposition.

References

  1. ↑ Bassi, Angelo; Lochan, Kinjalk; Satin, Seema; Singh, Tejinder P.; Ulbricht, Hendrik (2013). "Models of wave-function collapse, underlying theories, and experimental tests". Reviews of Modern Physics. 85 (2): 471–527. arXiv: 1204.4325 . Bibcode:2013RvMP...85..471B. doi:10.1103/RevModPhys.85.471. ISSN   0034-6861. S2CID   119261020.
  2. ↑ Bassi, Angelo; Ghirardi, GianCarlo (2003). "Dynamical reduction models". Physics Reports. 379 (5–6): 257–426. arXiv: quant-ph/0302164 . Bibcode:2003PhR...379..257B. doi:10.1016/S0370-1573(03)00103-0. S2CID   119076099.
  3. ↑ Bell, J. S. (2004). Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy (2 ed.). Cambridge University Press. doi:10.1017/cbo9780511815676. ISBN   978-0-521-52338-7.
  4. 1 2 3 4 5 Bassi, Angelo; Lochan, Kinjalk; Satin, Seema; Singh, Tejinder P.; Ulbricht, Hendrik (2013-04-02). "Models of wave-function collapse, underlying theories, and experimental tests". Reviews of Modern Physics. 85 (2): 471–527. arXiv: 1204.4325 . Bibcode:2013RvMP...85..471B. doi:10.1103/RevModPhys.85.471.
  5. ↑ Pearle, Philip (1976). "Reduction of the state vector by a nonlinear Schr\"odinger equation". Physical Review D. 13 (4): 857–868. Bibcode:1976PhRvD..13..857P. doi:10.1103/PhysRevD.13.857.
  6. ↑ Pearle, Philip (1979). "Toward explaining why events occur". International Journal of Theoretical Physics. 18 (7): 489–518. Bibcode:1979IJTP...18..489P. doi:10.1007/BF00670504. ISSN   0020-7748. S2CID   119407617.
  7. ↑ Pearle, Philip (1984). "Experimental tests of dynamical state-vector reduction". Physical Review D. 29 (2): 235–240. Bibcode:1984PhRvD..29..235P. doi:10.1103/PhysRevD.29.235.
  8. 1 2 Ghirardi, G. C.; Rimini, A.; Weber, T. (1986). "Unified dynamics for microscopic and macroscopic systems". Physical Review D. 34 (2): 470–491. Bibcode:1986PhRvD..34..470G. doi:10.1103/PhysRevD.34.470. PMID   9957165.
  9. ↑ Pearle, Philip (1989). "Combining stochastic dynamical state-vector reduction with spontaneous localization". Physical Review A. 39 (5): 2277–2289. Bibcode:1989PhRvA..39.2277P. doi:10.1103/PhysRevA.39.2277. PMID   9901493.
  10. 1 2 Ghirardi, Gian Carlo; Pearle, Philip; Rimini, Alberto (1990). "Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles". Physical Review A. 42 (1): 78–89. Bibcode:1990PhRvA..42...78G. doi:10.1103/PhysRevA.42.78. PMID   9903779.
  11. ↑ Diósi, L. (1987). "A universal master equation for the gravitational violation of quantum mechanics". Physics Letters A. 120 (8): 377–381. Bibcode:1987PhLA..120..377D. doi:10.1016/0375-9601(87)90681-5.
  12. 1 2 3 Diósi, L. (1989). "Models for universal reduction of macroscopic quantum fluctuations". Physical Review A. 40 (3): 1165–1174. Bibcode:1989PhRvA..40.1165D. doi:10.1103/PhysRevA.40.1165. ISSN   0556-2791. PMID   9902248.
  13. 1 2 Penrose, Roger (1996). "On Gravity's role in Quantum State Reduction". General Relativity and Gravitation. 28 (5): 581–600. Bibcode:1996GReGr..28..581P. doi:10.1007/BF02105068. ISSN   0001-7701. S2CID   44038399.
  14. ↑ Penrose, Roger (2014). "On the Gravitization of Quantum Mechanics 1: Quantum State Reduction". Foundations of Physics. 44 (5): 557–575. Bibcode:2014FoPh...44..557P. doi: 10.1007/s10701-013-9770-0 . ISSN   0015-9018.
  15. ↑ Tumulka, Roderich (2006). "On spontaneous wave function collapse and quantum field theory". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 462 (2070): 1897–1908. arXiv: quant-ph/0508230 . Bibcode:2006RSPSA.462.1897T. doi:10.1098/rspa.2005.1636. ISSN   1364-5021. S2CID   16123332.
  16. 1 2 Bassi, Angelo (2005). "Collapse models: analysis of the free particle dynamics". Journal of Physics A: Mathematical and General. 38 (14): 3173–3192. arXiv: quant-ph/0410222 . doi:10.1088/0305-4470/38/14/008. ISSN   0305-4470. S2CID   37142667.
  17. ↑ Adler, Stephen L; Bassi, Angelo (2007). "Collapse models with non-white noises". Journal of Physics A: Mathematical and Theoretical. 40 (50): 15083–15098. arXiv: 0708.3624 . Bibcode:2007JPhA...4015083A. doi:10.1088/1751-8113/40/50/012. ISSN   1751-8113. S2CID   118366772.
  18. ↑ Adler, Stephen L; Bassi, Angelo (2008). "Collapse models with non-white noises: II. Particle-density coupled noises". Journal of Physics A: Mathematical and Theoretical. 41 (39): 395308. arXiv: 0807.2846 . Bibcode:2008JPhA...41M5308A. doi:10.1088/1751-8113/41/39/395308. ISSN   1751-8113. S2CID   118551622.
  19. ↑ Bassi, Angelo; Ferialdi, Luca (2009). "Non-Markovian dynamics for a free quantum particle subject to spontaneous collapse in space: General solution and main properties". Physical Review A. 80 (1): 012116. arXiv: 0901.1254 . Bibcode:2009PhRvA..80a2116B. doi:10.1103/PhysRevA.80.012116. ISSN   1050-2947. S2CID   119297164.
  20. ↑ Bassi, Angelo; Ferialdi, Luca (2009). "Non-Markovian Quantum Trajectories: An Exact Result". Physical Review Letters. 103 (5): 050403. arXiv: 0907.1615 . Bibcode:2009PhRvL.103e0403B. doi:10.1103/PhysRevLett.103.050403. ISSN   0031-9007. PMID   19792469. S2CID   25021141.
  21. ↑ Leggett, A J (2002-04-22). "Testing the limits of quantum mechanics: motivation, state of play, prospects". Journal of Physics: Condensed Matter. 14 (15): R415–R451. doi:10.1088/0953-8984/14/15/201. ISSN   0953-8984.
  22. 1 2 Smirne, Andrea; Vacchini, Bassano; Bassi, Angelo (2014). "Dissipative extension of the Ghirardi-Rimini-Weber model". Physical Review A. 90 (6): 062135. arXiv: 1408.6115 . Bibcode:2014PhRvA..90f2135S. doi:10.1103/PhysRevA.90.062135. ISSN   1050-2947. S2CID   52232273.
  23. 1 2 Smirne, Andrea; Bassi, Angelo (2015). "Dissipative Continuous Spontaneous Localization (CSL) model". Scientific Reports. 5 (1): 12518. arXiv: 1408.6446 . Bibcode:2015NatSR...512518S. doi:10.1038/srep12518. ISSN   2045-2322. PMC   4525142 . PMID   26243034.
  24. 1 2 3 Di Bartolomeo, Giovanni; Carlesso, Matteo; Piscicchia, Kristian; Curceanu, Catalina; Derakhshani, Maaneli; Diósi, Lajos (2023-07-06). "Linear-friction many-body equation for dissipative spontaneous wave-function collapse". Physical Review A. 108 (1): 012202. arXiv: 2301.07661 . doi:10.1103/PhysRevA.108.012202. ISSN   2469-9926.
  25. 1 2 Bassi, Angelo; Ippoliti, Emiliano; Vacchini, Bassano (2005). "On the energy increase in space-collapse models". Journal of Physics A: Mathematical and General. 38 (37): 8017–8038. arXiv: quant-ph/0506083 . Bibcode:2005JPhA...38.8017B. doi:10.1088/0305-4470/38/37/007. ISSN   0305-4470. S2CID   43241594.
  26. ↑ Bahrami, M.; Smirne, A.; Bassi, A. (2014-12-01). "Role of gravity in the collapse of a wave function: A probe into the Diósi-Penrose model". Physical Review A. 90 (6): 062105. arXiv: 1408.6460 . doi:10.1103/PhysRevA.90.062105. ISSN   1050-2947.
  27. ↑ Ferialdi, Luca; Bassi, Angelo (2012). "Dissipative collapse models with nonwhite noises". Physical Review A. 86 (2): 022108. arXiv: 1112.5065 . Bibcode:2012PhRvA..86b2108F. doi:10.1103/PhysRevA.86.022108. ISSN   1050-2947. S2CID   119216571.
  28. ↑ Ferialdi, Luca; Bassi, Angelo (2012). "Exact Solution for a Non-Markovian Dissipative Quantum Dynamics". Physical Review Letters. 108 (17): 170404. arXiv: 1204.4348 . Bibcode:2012PhRvL.108q0404F. doi:10.1103/PhysRevLett.108.170404. ISSN   0031-9007. PMID   22680843. S2CID   16746767.
  29. ↑ Carlesso, Matteo; Donadi, Sandro; Ferialdi, Luca; Paternostro, Mauro; Ulbricht, Hendrik; Bassi, Angelo (February 2022). "Present status and future challenges of non-interferometric tests of collapse models". Nature Physics. 18 (3): 243–250. arXiv: 2203.04231 . Bibcode:2022NatPh..18..243C. doi:10.1038/s41567-021-01489-5. ISSN   1745-2481. S2CID   246949254.
  30. ↑ Ghirardi, G. C.; Grassi, R.; Pearle, P. (1990). "Relativistic dynamical reduction models: General framework and examples". Foundations of Physics. 20 (11): 1271–1316. Bibcode:1990FoPh...20.1271G. doi:10.1007/BF01883487. ISSN   0015-9018. S2CID   123661865.
  31. ↑ Tumulka, Roderich (2006). "A Relativistic Version of the Ghirardi–Rimini–Weber Model". Journal of Statistical Physics. 125 (4): 821–840. arXiv: quant-ph/0406094 . Bibcode:2006JSP...125..821T. doi:10.1007/s10955-006-9227-3. ISSN   0022-4715. S2CID   13923422.
  32. ↑ Lewis, Peter J. (1997). "Quantum Mechanics, Orthogonality, and Counting". The British Journal for the Philosophy of Science. 48 (3): 313–328. doi:10.1093/bjps/48.3.313. ISSN   0007-0882.
  33. ↑ Clifton, R.; Monton, B. (1999). "Discussion. Losing your marbles in wavefunction collapse theories". The British Journal for the Philosophy of Science. 50 (4): 697–717. doi:10.1093/bjps/50.4.697. ISSN   0007-0882.
  34. ↑ Ghirardi, G. C.; Bassi, A. (1999). "Do dynamical reduction models imply that arithmetic does not apply to ordinary macroscopic objects?". The British Journal for the Philosophy of Science. 50 (1): 49–64. arXiv: quant-ph/9810041 . doi:10.1093/bjps/50.1.49. ISSN   0007-0882.
  35. ↑ Bassi, A.; Ghirardi, G.-C. (1999). "Discussion. More about dynamical reduction and the enumeration principle". The British Journal for the Philosophy of Science. 50 (4): 719–734. doi:10.1093/bjps/50.4.719. ISSN   0007-0882.
  36. ↑ McQueen, Kelvin J. (2015). "Four Tails Problems for Dynamical Collapse Theories". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 49: 10–18. arXiv: 1501.05778 . Bibcode:2015SHPMP..49...10M. doi:10.1016/j.shpsb.2014.12.001. ISSN   1355-2198. S2CID   55718585.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.