1400:. These are phenomenological parameters, whose values are not fixed by any principle and should be understood as new constants of Nature. Comparison of the model's predictions with experimental data permits bounding of the values of the parameters (see CSL model). The collapse rate should be such that microscopic object are almost never localized, thus effectively recovering standard quantum mechanics. The value originally proposed was
1943:
4094:
90:). The imprecise “observer” and “measurement” that plague the orthodox interpretation are thus avoided because the wave function collapses spontaneously. Furthermore, thanks to a so called “amplification mechanism” (later discussed), collapse theories recover both quantum mechanics for microscopic objects, and classical mechanics for macroscopic ones.
1642:
3789:
3779:
85:
avoid the measurement problem by merging the two dynamical principles of quantum mechanics in a unique dynamical description. The physical idea that underlies collapse theories is that particles undergo spontaneous wave-function collapses, which occur randomly both in time (at a given average rate),
139:, according to which the wave function does not give a complete description of a physical system. The GRW theory differs from standard quantum mechanics for the dynamical principles according to which the wave function evolves. More philosophical issues related to the GRW theory and to
749:
284:
1312:
66:
every time an observer performs a measurement. One thus faces the problem of defining what an “observer” and a “measurement” are. Another issue of quantum mechanics is that it forecasts superpositions of macroscopic objects, which are not observed in nature (see
1938:{\displaystyle \psi (x)={\frac {1}{(\pi \sigma )^{\frac {1}{4}}}}\,e^{-{\frac {(x-a)^{2}}{2\sigma ^{2}}}}\quad \longrightarrow \quad \psi _{a}(x)={\hat {L}}_{x=a}\,\psi (x)={\cal {N}}e^{-{\frac {(x-a)^{2}}{2r_{C}^{2}}}}\,e^{-{\frac {(x-a)^{2}}{2\sigma ^{2}}}}}
1075:
4089:{\displaystyle {\hat {T}}_{\mathrm {CM} }=\left({\frac {1}{\pi r_{C}^{2}}}\right)^{\frac {3}{2}}\int _{\infty }^{\infty }d^{3}x\,e^{-{\frac {({\hat {Q}}-x)^{2}}{2r_{C}^{2}}}}\,\rho _{\mathrm {CM} }(t)\,e^{-{\frac {({\hat {Q}}-x)^{2}}{2r_{C}^{2}}}}}
2980:
3530:
4475:
process according to which the off-diagonal elements of the statistical operator are suppressed exponentially. This is a feature that the GRW theory shares with other collapse theories: those involving white noises are associated to
2114:
2392:
365:
574:
589:
3432:
3263:
4460:
186:
4321:
1137:
4272:
1449:
945:
1504:
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1552:
4468:
GRW is a non relativistic theory, its relativistic extension for non-interacting particles has been investigated by
Tumulka, while interacting models are still under investigation.
3523:
952:
4583:
Ghirardi, Gian Carlo; Pearle, Philip; Rimini, Alberto (1990-07-01). "Markov processes in
Hilbert space and continuous spontaneous localization of systems of identical particles".
4167:
4784:
Bassi, Angelo; Ferialdi, Luca (2009-07-31). "Non-Markovian dynamics for a free quantum particle subject to spontaneous collapse in space: General solution and main properties".
1562:
When the wave function is hit by a sudden jump, the action of the localization operator essentially results in the multiplication of the wave function by the collapse
Gaussian.
406:
3464:
4462:. Although such an energy increase is negligible, this feature of the model is not appealing. For this reason, a dissipative extension of the GRW theory has been investigated.
3016:
2146:
2003:
4203:
3359:
788:
178:
3276:
This is one of the most important features of the GRW theory, because it allows us to recover classical mechanics for macroscopic objects. Let us consider a rigid body of
1110:
3052:
3323:
1970:
1344:
900:
3774:{\displaystyle {\frac {d}{dt}}\rho _{\mathrm {CM} }(t)=-{\frac {i}{\hbar }}-\sum _{i=1}^{N}\lambda _{i}\left(\rho _{\mathrm {CM} }(t)-{\hat {T}}_{\mathrm {CM} }\right)}
127:(or state vector) represents the most accurate possible specification of the state of a physical system. This is a feature that the GRW theory shares with the standard
4566:
4223:
4137:
4117:
1371:
474:
2425:
1583:
4119:
that is the sum of the rates of its constituents: this is the amplification mechanism. If for simplicity one assumes that all particles collapse with the same rate
3491:
1398:
835:
2178:
2418:
2204:
1635:
1609:
874:
5670:
3294:
1130:
808:
494:
446:
426:
4323:. Fast reduction of macroscopic object superpositions is thus guaranteed, and the GRW theory effectively recovers classical mechanics for macroscopic objects.
1554:
for the localization distance. This is a mesoscopic distance, such that microscopic superpositions are left unaltered, while macroscopic ones are collapsed.
2010:
71:). The theory does not tell where the threshold between the microscopic and macroscopic worlds is, that is when quantum mechanics should leave space to
2148:
is another normalization factor. One thus finds that after the sudden jump has occurred, the initially delocalised wave function has become localized.
98:
93:
The GRW is the first spontaneous collapse theory that was devised. In the following years several different models were proposed. Among these are
2209:
292:
4500:
128:
744:{\displaystyle {\hat {L}}_{x}^{i}=\left({\frac {1}{\pi r_{C}^{2}}}\right)^{\frac {3}{4}}e^{-{\frac {({\hat {q}}_{i}-x)^{2}}{2r_{C}^{2}}}}}
499:
114:
the coloured QMUPL model, which is the only collapse model involving coloured stochastic processes for which the exact solution is known.
4465:
The GRW theory does not allow for identical particles. An extension of the theory with identical particles has been proposed by
Tumulka.
5245:
as its practically unique effect, that of inducing a very fast suppression of coherence among macroscopically distinguishable states
5542:
Diósi, L.; Ferialdi, L. (2014-11-12). "General Non-Markovian
Structure of Gaussian Master and Stochastic Schr\"odinger Equations".
3296:
particles whose statistical operator evolves according to the master equation described above. We introduce the center of mass (
279:{\displaystyle |\psi \rangle \rightarrow {\frac {|\psi _{x}^{i}\rangle }{\sqrt {\langle \psi _{x}^{i}|\psi _{x}^{i}\rangle }}}}
3364:
3268:
We thus see that the
Gaussian that is hit by the localization is left unchanged, while the other is exponentially suppressed.
4406:
1307:{\displaystyle {\frac {d}{dt}}\rho (t)=-{\frac {i}{\hbar }}-\sum _{i=1}^{N}\lambda _{i}\left(\rho (t)-{\hat {T}}_{i}\right)}
5310:
Smirne, Andrea; Vacchini, Bassano; Bassi, Angelo (2014-12-31). "Dissipative extension of the
Ghirardi-Rimini-Weber model".
4344:
4277:
3059:
5675:
4228:
1403:
905:
1458:
136:
4350:
105:
140:
82:
59:
28:
4975:
Ferialdi, Luca; Bassi, Angelo (2012-04-26). "Exact
Solution for a Non-Markovian Dissipative Quantum Dynamics".
1509:
1112:
that the system keeps evolving according to the Schrödinger equation. Accordingly, the GRW master equation for
4523:
Ghirardi, G.C., Rimini, A., and Weber, T. (1986). "Unified dynamics for microscopic and macroscopic systems".
1070:{\displaystyle {\hat {T}}_{i}\equiv \int dx\,{\hat {L}}_{x}^{i}|\psi \rangle \langle \psi |{\hat {L}}_{x}^{i}}
1506:(with an uncertainty of two orders of magnitude) is more adequate. There is a general consensus on the value
2151:
Another interesting case is when the initial state is the superposition of two
Gaussian states, centered at
842:
68:
51:
3496:
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132:
63:
55:
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370:
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1975:
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4532:
4175:
1347:
453:
3328:
757:
155:
5163:
4845:
Bassi, Angelo; Ferialdi, Luca (2009-07-28). "Non-Markovian
Quantum Trajectories: An Exact Result".
4490:
4472:
76:
72:
5373:
Tumulka, Roderich (2006-06-08). "On spontaneous wave function collapse and quantum field theory".
5097:
Bassi, Angelo; Lochan, Kinjalk; Satin, Seema; Singh, Tejinder P.; Ulbricht, Hendrik (2013-04-02).
4480:
master equations, while the coloured QMUPL model follows a non-Markovian Gaussian master equation.
1972:
is a normalization factor. Let us further assume that the initial state is delocalised, i.e. that
1083:
5646:
5612:
5585:
5551:
5524:
5469:
5435:
5408:
5382:
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4740:
4713:
3021:
2122:
36:
4914:
Ferialdi, Luca; Bassi, Angelo (2012-08-08). "Dissipative collapse models with nonwhite noises".
3434:. One can show that, when the system Hamiltonian can be split into a center of mass Hamiltonian
3299:
2975:{\displaystyle \psi _{a}(x)={\cal {N}}e^{-{\frac {(x-a)^{2}}{2r_{C}^{2}}}}\left={\cal {N}}\left}
1951:
1320:
879:
5603:
Ferialdi, L. (2016-03-22). "Exact Closed Master Equation for Gaussian Non-Markovian Dynamics".
4343:). This introduces a steady amount of energy in the system, thus leading to a violation of the
5638:
5577:
5516:
5461:
5226:
5136:
5018:
5010:
4949:
4888:
4880:
4819:
4758:
4705:
4662:
4654:
4629:
Diósi, L. (1989-08-01). "Models for universal reduction of macroscopic quantum fluctuations".
4608:
4548:
4333:
4208:
4122:
4102:
3361:) position operators, which allow us to rewrite each particle's position operator as follows:
1356:
580:
459:
47:
32:
5426:
Tumulka, Roderich (2006-11-01). "A Relativistic Version of the Ghirardi–Rimini–Weber Model".
1568:
5630:
5569:
5508:
5453:
5400:
5345:
5337:
5284:
5257:
Romero-Isart, Oriol (2011). "Quantum Superposition of Massive Objects and Collapse Models".
5218:
5128:
5071:
5002:
4941:
4872:
4811:
4750:
4697:
4646:
4600:
4540:
1452:
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1376:
813:
4340:
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2397:
2183:
1614:
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5565:
5504:
5449:
5396:
5333:
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5214:
5124:
5067:
4998:
4937:
4868:
4807:
4693:
4642:
4596:
4536:
4347:
principle. For the GRW model, one can show that energy grows linearly in time with rate
856:
4731:
Bassi, Angelo (2005-04-08). "Collapse models: analysis of the free particle dynamics".
3279:
1115:
850:
793:
479:
431:
411:
5222:
5189:"Lower and upper bounds on CSL parameters from latent image formation and IGM heating"
5075:
4754:
4339:
The collapse noise repeatedly kicks the particles, thus inducing a diffusion process (
111:
the QMUPL model, which proves important mathematical results on collapse theories; and
5664:
5650:
5296:
5148:
5083:
4961:
4831:
124:
5589:
5528:
5473:
5412:
5359:
5030:
4900:
4770:
4717:
5634:
5573:
5375:
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
5006:
4876:
50:
has two fundamentally different dynamical principles: the linear and deterministic
2109:{\displaystyle \psi _{a}(x)\simeq {\cal {N}}'e^{-{\frac {(x-a)^{2}}{2r_{C}^{2}}}}}
841:
In between two localization processes, the wave function evolves according to the
5488:
4477:
5341:
5288:
5099:"Models of wave-function collapse, underlying theories, and experimental tests"
4945:
4815:
1611:, and let us assume that this undergoes a localization process at the position
5457:
5132:
5044:
Bassi, Angelo; Ghirardi, GianCarlo (June 2003). "Dynamical reduction models".
1353:
Two new parameters are introduced by the GRW theory, namely the collapse rate
5520:
5465:
5230:
5140:
5014:
4953:
4884:
4823:
4762:
4709:
4658:
4650:
4544:
4172:
An object that consists of in the order of the Avogadro number of nucleons (
853:
formalism. Since the localization process is Poissonian, in a time interval
87:
5642:
5581:
5404:
5022:
4892:
4680:
Penrose, Roger (May 1996). "On Gravity's role in Quantum State Reduction".
4604:
4666:
4612:
4552:
2387:{\displaystyle \psi (x)={\frac {1}{2(\pi \sigma )^{\frac {1}{4}}}}\,\left}
5440:
5387:
5205:
5058:
4745:
452:
The localization process is random both in space and time. The jumps are
152:
Each particle of a system described by the multi-particle wave function
5512:
5350:
4701:
360:{\displaystyle |\psi _{x}^{i}\rangle ={\hat {L}}_{x}^{i}|\psi \rangle }
5170:(Summer 2020 ed.), Metaphysics Research Lab, Stanford University
180:
independently undergoes a spontaneous localization process (or jump):
5188:
569:{\displaystyle P_{i}(x)=\langle \psi _{x}^{i}|\psi _{x}^{i}\rangle }
5617:
1346:
is the Hamiltonian of the system, and the square brackets denote a
5556:
5324:
5271:
5115:
4989:
4928:
4859:
4798:
849:
These principles can be expressed in a more compact way with the
4099:
One thus sees that the center of mass collapses with a rate
3088:
2640:
2453:
2129:
2039:
1957:
1818:
4205:) collapses almost instantly: GRW's and Adler's values of
476:; the probability density for a jump to occur at position
143:
in general one have been discussed by Ghirardi and Bassi.
4332:
The GRW theory makes different predictions than standard
3427:{\displaystyle {\hat {q}}_{i}={\hat {Q}}+{\hat {r}}_{i}}
4455:{\displaystyle \simeq 10^{-14}\mathrm {erg\,\,s} ^{-1}}
4336:, and as such can be tested against it (see CSL model).
2985:
If one assumes that each Gaussian is localized (
101:, which is formulated in terms of identical particles;
4409:
4353:
4280:
4231:
4211:
4178:
4145:
4125:
4105:
3792:
3533:
3499:
3472:
3440:
3367:
3331:
3302:
3282:
3062:
3024:
3018:) and that the overall superposition is delocalised (
2991:
2428:
2400:
2212:
2186:
2157:
2125:
2013:
1978:
1954:
1645:
1617:
1591:
1571:
1565:
Let us consider a Gaussian wave function with spread
1512:
1461:
1406:
1379:
1359:
1323:
1140:
1118:
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955:
908:
882:
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796:
760:
592:
502:
482:
462:
434:
414:
373:
295:
189:
158:
3525:
evolves according to the following master equation:
108:, which relates the spontaneous collapse to gravity;
5489:"On the generators of quantum dynamical semigroups"
4316:{\displaystyle \Lambda =10^{15}\,\mathrm {s} ^{-1}}
3258:{\displaystyle \psi _{a}(x)\simeq {\cal {N}}'\left}
123:The first assumption of the GRW theory is that the
5193:Journal of Physics A: Mathematical and Theoretical
4471:The master equation of the GRW theory describes a
4454:
4395:
4315:
4267:{\displaystyle \Lambda =10^{7}\,\mathrm {s} ^{-1}}
4266:
4217:
4197:
4161:
4131:
4111:
4088:
3773:
3517:
3485:
3458:
3426:
3353:
3317:
3288:
3257:
3046:
3010:
2974:
2412:
2386:
2198:
2172:
2140:
2108:
1997:
1964:
1937:
1629:
1603:
1577:
1546:
1498:
1444:{\displaystyle \lambda =10^{-16}\mathrm {s} ^{-1}}
1443:
1392:
1365:
1338:
1306:
1124:
1104:
1080:In the same time interval, there is a probability
1069:
940:{\displaystyle \rho =|\psi \rangle \langle \psi |}
939:
894:
868:
829:
802:
782:
743:
568:
488:
468:
440:
420:
400:
359:
278:
172:
4624:
4622:
1499:{\displaystyle \lambda =10^{-8}\mathrm {s} ^{-1}}
902:that a collapse occurs, i.e. that the pure state
4733:Journal of Physics A: Mathematical and General
4396:{\displaystyle \lambda \hbar ^{2}/4mr_{C}^{2}}
16:Objective collapse theory in quantum mechanics
43:Measurement problem and spontaneous collapses
8:
4565:: CS1 maint: multiple names: authors list (
4403:, which for a macroscopic object amounts to
1032:
1029:
947:is transformed into the statistical mixture
926:
923:
563:
525:
354:
316:
270:
232:
227:
198:
167:
5162:Ghirardi, Giancarlo; Bassi, Angelo (2020),
75:. The aforementioned issues constitute the
58:postulate. The orthodox interpretation, or
3493:, the center of mass statistical operator
5616:
5555:
5439:
5386:
5349:
5323:
5270:
5204:
5114:
5057:
4988:
4927:
4858:
4797:
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4443:
4438:
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4408:
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4367:
4361:
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4297:
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4018:
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2529:
2525:
2505:
2500:
2485:
2466:
2462:
2452:
2451:
2433:
2427:
2399:
2394:. If the localization occurs e.g. around
2368:
2353:
2334:
2330:
2312:
2297:
2278:
2274:
2264:
2250:
2228:
2211:
2185:
2156:
2128:
2127:
2124:
2095:
2090:
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2056:
2052:
2038:
2037:
2018:
2012:
1989:
1977:
1956:
1955:
1953:
1924:
1909:
1890:
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1881:
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1827:
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1757:
1737:
1722:
1703:
1699:
1694:
1680:
1661:
1644:
1616:
1590:
1570:
1547:{\displaystyle r_{C}=10^{-7}\mathrm {m} }
1539:
1530:
1517:
1511:
1487:
1482:
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1460:
1432:
1427:
1417:
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5168:The Stanford Encyclopedia of Philosophy
4578:
4576:
4512:
4358:
3584:
1179:
5493:Communications in Mathematical Physics
4558:
3518:{\displaystyle \rho _{\mathrm {CM} }}
7:
5671:Interpretations of quantum mechanics
4518:
4516:
4501:Interpretations of quantum mechanics
4162:{\displaystyle \Lambda =N\,\lambda }
456:distributed in time, with mean rate
129:Interpretations of quantum mechanics
39:, Alberto Rimini, and Tullio Weber.
54:, and the nonlinear and stochastic
4682:General Relativity and Gravitation
4439:
4434:
4431:
4428:
4300:
4281:
4251:
4232:
4146:
4106:
4003:
4000:
3907:
3902:
3829:
3826:
3811:
3808:
3748:
3745:
3730:
3727:
3694:
3691:
3628:
3625:
3610:
3607:
3558:
3555:
3509:
3506:
3450:
3447:
1637:. One thus has (in one dimension)
1540:
1483:
1428:
401:{\displaystyle {\hat {L}}_{x}^{i}}
14:
3459:{\displaystyle H_{\mathrm {CM} }}
428:-th particle around the position
3011:{\displaystyle \sigma \ll r_{C}}
1998:{\displaystyle \sigma \gg r_{C}}
790:is the position operator of the
579:The localization operator has a
367:is the state after the operator
5187:Adler, Stephen L (2007-03-07).
4198:{\displaystyle N\simeq 10^{23}}
1752:
1748:
86:and in space (according to the
62:of quantum mechanics, posits a
5635:10.1103/PhysRevLett.116.120402
5574:10.1103/PhysRevLett.113.200403
5428:Journal of Statistical Physics
5007:10.1103/PhysRevLett.108.170404
4877:10.1103/PhysRevLett.103.050403
4052:
4039:
4030:
4015:
4009:
3959:
3946:
3937:
3844:
3841:
3835:
3817:
3800:
3763:
3760:
3754:
3736:
3719:
3706:
3700:
3643:
3640:
3634:
3599:
3589:
3570:
3564:
3412:
3396:
3375:
3354:{\displaystyle {\hat {r}}_{i}}
3339:
3309:
3221:
3208:
3079:
3073:
2956:
2943:
2601:
2588:
2545:
2532:
2482:
2469:
2445:
2439:
2350:
2337:
2294:
2281:
2247:
2237:
2222:
2216:
2072:
2059:
2030:
2024:
1906:
1893:
1847:
1834:
1810:
1804:
1782:
1769:
1763:
1749:
1719:
1706:
1677:
1667:
1655:
1649:
1373:and the localization distance
1330:
1296:
1290:
1278:
1265:
1259:
1214:
1211:
1205:
1193:
1184:
1165:
1159:
1050:
1039:
1022:
1004:
981:
975:
963:
933:
916:
837:is the localization distance.
783:{\displaystyle {\hat {q}}_{i}}
768:
707:
688:
678:
600:
544:
519:
513:
381:
347:
329:
297:
251:
208:
201:
191:
173:{\displaystyle |\psi \rangle }
160:
1:
5166:, in Zalta, Edward N. (ed.),
5076:10.1016/S0370-1573(03)00103-0
1105:{\displaystyle 1-\lambda dt}
131:, and distinguishes it from
21:Ghirardi–Rimini–Weber theory
5223:10.1088/1751-8113/40/12/s03
4755:10.1088/0305-4470/38/14/008
3466:and a relative Hamiltonian
3047:{\displaystyle 2a\gg r_{C}}
2141:{\displaystyle {\cal {N}}'}
5692:
5342:10.1103/PhysRevA.90.062135
5289:10.1103/PhysRevA.84.052121
4946:10.1103/PhysRevA.86.022108
4816:10.1103/PhysRevA.80.012116
3318:{\displaystyle {\hat {Q}}}
1965:{\displaystyle {\cal {N}}}
1339:{\displaystyle {\hat {H}}}
895:{\displaystyle \lambda dt}
5458:10.1007/s10955-006-9227-3
5133:10.1103/RevModPhys.85.471
5103:Reviews of Modern Physics
69:Schrödinger’s cat paradox
60:Copenhagen interpretation
4651:10.1103/PhysRevA.40.1165
4218:{\displaystyle \lambda }
4132:{\displaystyle \lambda }
4112:{\displaystyle \Lambda }
1455:proposed that the value
1366:{\displaystyle \lambda }
469:{\displaystyle \lambda }
133:hidden variable theories
5605:Physical Review Letters
5544:Physical Review Letters
4977:Physical Review Letters
4847:Physical Review Letters
4545:10.1103/PhysRevD.34.470
3272:Amplification mechanism
2005:. In this case one has
1578:{\displaystyle \sigma }
876:there is a probability
5405:10.1098/rspa.2005.1636
4605:10.1103/PhysRevA.42.78
4496:Penrose interpretation
4456:
4397:
4317:
4268:
4219:
4199:
4163:
4133:
4113:
4090:
3775:
3669:
3519:
3487:
3460:
3428:
3355:
3319:
3290:
3259:
3048:
3012:
2976:
2414:
2388:
2200:
2174:
2142:
2110:
1999:
1966:
1939:
1631:
1605:
1579:
1548:
1500:
1451:, while more recently
1445:
1394:
1367:
1340:
1308:
1240:
1126:
1106:
1071:
941:
896:
870:
831:
804:
784:
745:
570:
490:
470:
442:
422:
402:
361:
280:
174:
137:de Broglie–Bohm theory
79:in quantum mechanics.
64:wave function collapse
35:, proposed in 1986 by
5487:Lindblad, G. (1976).
4457:
4398:
4318:
4269:
4220:
4200:
4164:
4134:
4114:
4091:
3776:
3649:
3520:
3488:
3486:{\displaystyle H_{r}}
3461:
3429:
3356:
3320:
3291:
3260:
3049:
3013:
2977:
2415:
2389:
2201:
2175:
2143:
2111:
2000:
1967:
1940:
1632:
1606:
1580:
1549:
1501:
1446:
1395:
1393:{\displaystyle r_{C}}
1368:
1341:
1309:
1220:
1127:
1107:
1072:
942:
897:
871:
832:
830:{\displaystyle r_{C}}
805:
785:
746:
571:
491:
471:
443:
423:
403:
362:
281:
175:
56:wave packet reduction
4407:
4351:
4278:
4229:
4209:
4176:
4143:
4123:
4103:
3790:
3531:
3497:
3470:
3438:
3365:
3329:
3300:
3280:
3060:
3022:
2989:
2426:
2398:
2210:
2184:
2173:{\displaystyle x=-a}
2155:
2123:
2011:
1976:
1952:
1643:
1615:
1589:
1569:
1510:
1459:
1404:
1377:
1357:
1321:
1138:
1116:
1084:
953:
906:
880:
857:
851:statistical operator
843:Schrödinger equation
814:
794:
758:
590:
500:
480:
460:
432:
412:
371:
293:
187:
156:
52:Schrödinger equation
5676:Quantum measurement
5627:2016PhRvL.116l0402F
5566:2014PhRvL.113t0403D
5505:1976CMaPh..48..119L
5450:2006JSP...125..821T
5397:2006RSPSA.462.1897T
5381:(2070): 1897–1908.
5334:2014PhRvA..90f2135S
5281:2011PhRvA..84e2121R
5215:2007JPhA...40.2935A
5164:"Collapse Theories"
5125:2013RvMP...85..471B
5068:2003PhR...379..257B
4999:2012PhRvL.108q0404F
4938:2012PhRvA..86b2108F
4869:2009PhRvL.103e0403B
4808:2009PhRvA..80a2116B
4694:1996GReGr..28..581P
4643:1989PhRvA..40.1165D
4597:1990PhRvA..42...78G
4537:1986PhRvD..34..470G
4491:Quantum decoherence
4392:
4345:energy conservation
4080:
3987:
3911:
3878:
3189:
2938:
2908:
2861:
2795:
2765:
2718:
2688:
2510:
2413:{\displaystyle x=a}
2199:{\displaystyle x=a}
2100:
1875:
1630:{\displaystyle x=a}
1604:{\displaystyle x=a}
1066:
1020:
735:
648:
616:
562:
542:
397:
345:
315:
269:
249:
226:
106:Diósi–Penrose model
77:measurement problem
73:classical mechanics
27:) is a spontaneous
5513:10.1007/BF01608499
4702:10.1007/BF02105068
4452:
4393:
4378:
4313:
4264:
4225:give respectively
4215:
4195:
4159:
4139:, one simply gets
4129:
4109:
4086:
4066:
3973:
3897:
3864:
3771:
3515:
3483:
3456:
3424:
3351:
3315:
3286:
3255:
3175:
3044:
3008:
2972:
2924:
2894:
2847:
2781:
2751:
2704:
2674:
2496:
2410:
2384:
2196:
2170:
2138:
2106:
2086:
1995:
1962:
1935:
1861:
1627:
1601:
1575:
1544:
1496:
1441:
1390:
1363:
1336:
1304:
1122:
1102:
1067:
1043:
997:
937:
892:
869:{\displaystyle dt}
866:
827:
810:-th particle, and
800:
780:
741:
721:
634:
593:
566:
548:
528:
486:
466:
438:
418:
408:has localized the
398:
374:
357:
322:
301:
276:
255:
235:
212:
170:
147:Working principles
37:Giancarlo Ghirardi
5312:Physical Review A
5199:(12): 2935–2957.
4916:Physical Review A
4786:Physical Review A
4739:(14): 3173–3192.
4631:Physical Review A
4585:Physical Review A
4525:Physical Review D
4334:quantum mechanics
4082:
4042:
3989:
3949:
3894:
3880:
3803:
3722:
3602:
3587:
3547:
3415:
3399:
3378:
3342:
3312:
3289:{\displaystyle N}
3246:
3190:
3153:
2940:
2863:
2797:
2720:
2626:
2570:
2512:
2375:
2319:
2262:
2258:
2102:
1931:
1877:
1785:
1744:
1692:
1688:
1333:
1281:
1196:
1182:
1154:
1125:{\displaystyle N}
1053:
1007:
966:
803:{\displaystyle i}
771:
737:
691:
664:
650:
603:
489:{\displaystyle x}
441:{\displaystyle x}
421:{\displaystyle i}
384:
332:
274:
273:
141:collapse theories
83:Collapse theories
48:Quantum mechanics
33:quantum mechanics
5683:
5655:
5654:
5620:
5600:
5594:
5593:
5559:
5539:
5533:
5532:
5484:
5478:
5477:
5443:
5441:quant-ph/0406094
5423:
5417:
5416:
5390:
5388:quant-ph/0508230
5370:
5364:
5363:
5353:
5327:
5307:
5301:
5300:
5274:
5254:
5248:
5241:
5235:
5234:
5208:
5206:quant-ph/0605072
5184:
5178:
5177:
5176:
5175:
5159:
5153:
5152:
5118:
5094:
5088:
5087:
5061:
5059:quant-ph/0302164
5052:(5–6): 257–426.
5041:
5035:
5034:
4992:
4972:
4966:
4965:
4931:
4911:
4905:
4904:
4862:
4842:
4836:
4835:
4801:
4781:
4775:
4774:
4748:
4746:quant-ph/0410222
4728:
4722:
4721:
4677:
4671:
4670:
4637:(3): 1165–1174.
4626:
4617:
4616:
4580:
4571:
4570:
4564:
4556:
4520:
4461:
4459:
4458:
4453:
4451:
4450:
4442:
4425:
4424:
4402:
4400:
4399:
4394:
4391:
4386:
4371:
4366:
4365:
4322:
4320:
4319:
4314:
4312:
4311:
4303:
4296:
4295:
4273:
4271:
4270:
4265:
4263:
4262:
4254:
4247:
4246:
4224:
4222:
4221:
4216:
4204:
4202:
4201:
4196:
4194:
4193:
4168:
4166:
4165:
4160:
4138:
4136:
4135:
4130:
4118:
4116:
4115:
4110:
4095:
4093:
4092:
4087:
4085:
4084:
4083:
4081:
4079:
4074:
4061:
4060:
4059:
4044:
4043:
4035:
4028:
4008:
4007:
4006:
3992:
3991:
3990:
3988:
3986:
3981:
3968:
3967:
3966:
3951:
3950:
3942:
3935:
3921:
3920:
3910:
3905:
3896:
3895:
3887:
3885:
3881:
3879:
3877:
3872:
3856:
3834:
3833:
3832:
3816:
3815:
3814:
3805:
3804:
3796:
3780:
3778:
3777:
3772:
3770:
3766:
3753:
3752:
3751:
3735:
3734:
3733:
3724:
3723:
3715:
3699:
3698:
3697:
3679:
3678:
3668:
3663:
3633:
3632:
3631:
3615:
3614:
3613:
3604:
3603:
3595:
3588:
3580:
3563:
3562:
3561:
3548:
3546:
3535:
3524:
3522:
3521:
3516:
3514:
3513:
3512:
3492:
3490:
3489:
3484:
3482:
3481:
3465:
3463:
3462:
3457:
3455:
3454:
3453:
3433:
3431:
3430:
3425:
3423:
3422:
3417:
3416:
3408:
3401:
3400:
3392:
3386:
3385:
3380:
3379:
3371:
3360:
3358:
3357:
3352:
3350:
3349:
3344:
3343:
3335:
3325:) and relative (
3324:
3322:
3321:
3316:
3314:
3313:
3305:
3295:
3293:
3292:
3287:
3264:
3262:
3261:
3256:
3254:
3250:
3249:
3248:
3247:
3245:
3244:
3243:
3230:
3229:
3228:
3206:
3193:
3192:
3191:
3188:
3183:
3174:
3173:
3172:
3159:
3154:
3152:
3151:
3150:
3137:
3136:
3131:
3127:
3111:
3096:
3092:
3091:
3072:
3071:
3053:
3051:
3050:
3045:
3043:
3042:
3017:
3015:
3014:
3009:
3007:
3006:
2981:
2979:
2978:
2973:
2971:
2967:
2966:
2965:
2964:
2963:
2941:
2939:
2937:
2932:
2923:
2922:
2909:
2907:
2902:
2890:
2889:
2879:
2866:
2865:
2864:
2862:
2860:
2855:
2843:
2842:
2832:
2831:
2830:
2817:
2812:
2811:
2806:
2802:
2798:
2796:
2794:
2789:
2777:
2776:
2766:
2764:
2759:
2747:
2746:
2736:
2721:
2719:
2717:
2712:
2703:
2702:
2689:
2687:
2682:
2670:
2669:
2659:
2644:
2643:
2634:
2630:
2629:
2628:
2627:
2625:
2624:
2623:
2610:
2609:
2608:
2586:
2573:
2572:
2571:
2569:
2568:
2567:
2554:
2553:
2552:
2530:
2515:
2514:
2513:
2511:
2509:
2504:
2491:
2490:
2489:
2467:
2457:
2456:
2438:
2437:
2419:
2417:
2416:
2411:
2393:
2391:
2390:
2385:
2383:
2379:
2378:
2377:
2376:
2374:
2373:
2372:
2359:
2358:
2357:
2335:
2322:
2321:
2320:
2318:
2317:
2316:
2303:
2302:
2301:
2279:
2263:
2261:
2260:
2259:
2251:
2229:
2205:
2203:
2202:
2197:
2179:
2177:
2176:
2171:
2147:
2145:
2144:
2139:
2137:
2133:
2132:
2115:
2113:
2112:
2107:
2105:
2104:
2103:
2101:
2099:
2094:
2081:
2080:
2079:
2057:
2047:
2043:
2042:
2023:
2022:
2004:
2002:
2001:
1996:
1994:
1993:
1971:
1969:
1968:
1963:
1961:
1960:
1944:
1942:
1941:
1936:
1934:
1933:
1932:
1930:
1929:
1928:
1915:
1914:
1913:
1891:
1880:
1879:
1878:
1876:
1874:
1869:
1856:
1855:
1854:
1832:
1822:
1821:
1799:
1798:
1787:
1786:
1778:
1762:
1761:
1747:
1746:
1745:
1743:
1742:
1741:
1728:
1727:
1726:
1704:
1693:
1691:
1690:
1689:
1681:
1662:
1636:
1634:
1633:
1628:
1610:
1608:
1607:
1602:
1584:
1582:
1581:
1576:
1553:
1551:
1550:
1545:
1543:
1538:
1537:
1522:
1521:
1505:
1503:
1502:
1497:
1495:
1494:
1486:
1480:
1479:
1453:Stephen L. Adler
1450:
1448:
1447:
1442:
1440:
1439:
1431:
1425:
1424:
1399:
1397:
1396:
1391:
1389:
1388:
1372:
1370:
1369:
1364:
1345:
1343:
1342:
1337:
1335:
1334:
1326:
1313:
1311:
1310:
1305:
1303:
1299:
1289:
1288:
1283:
1282:
1274:
1250:
1249:
1239:
1234:
1198:
1197:
1189:
1183:
1175:
1155:
1153:
1142:
1132:particles reads
1131:
1129:
1128:
1123:
1111:
1109:
1108:
1103:
1076:
1074:
1073:
1068:
1065:
1060:
1055:
1054:
1046:
1042:
1025:
1019:
1014:
1009:
1008:
1000:
974:
973:
968:
967:
959:
946:
944:
943:
938:
936:
919:
901:
899:
898:
893:
875:
873:
872:
867:
836:
834:
833:
828:
826:
825:
809:
807:
806:
801:
789:
787:
786:
781:
779:
778:
773:
772:
764:
750:
748:
747:
742:
740:
739:
738:
736:
734:
729:
716:
715:
714:
699:
698:
693:
692:
684:
676:
666:
665:
657:
655:
651:
649:
647:
642:
626:
615:
610:
605:
604:
596:
575:
573:
572:
567:
561:
556:
547:
541:
536:
512:
511:
495:
493:
492:
487:
475:
473:
472:
467:
447:
445:
444:
439:
427:
425:
424:
419:
407:
405:
404:
399:
396:
391:
386:
385:
377:
366:
364:
363:
358:
350:
344:
339:
334:
333:
325:
314:
309:
300:
285:
283:
282:
277:
275:
268:
263:
254:
248:
243:
231:
230:
225:
220:
211:
205:
194:
179:
177:
176:
171:
163:
5691:
5690:
5686:
5685:
5684:
5682:
5681:
5680:
5661:
5660:
5659:
5658:
5602:
5601:
5597:
5541:
5540:
5536:
5486:
5485:
5481:
5425:
5424:
5420:
5372:
5371:
5367:
5309:
5308:
5304:
5256:
5255:
5251:
5242:
5238:
5186:
5185:
5181:
5173:
5171:
5161:
5160:
5156:
5096:
5095:
5091:
5046:Physics Reports
5043:
5042:
5038:
4974:
4973:
4969:
4913:
4912:
4908:
4844:
4843:
4839:
4783:
4782:
4778:
4730:
4729:
4725:
4679:
4678:
4674:
4628:
4627:
4620:
4582:
4581:
4574:
4557:
4522:
4521:
4514:
4509:
4487:
4426:
4413:
4405:
4404:
4357:
4349:
4348:
4341:Brownian motion
4329:
4298:
4287:
4276:
4275:
4249:
4238:
4227:
4226:
4207:
4206:
4185:
4174:
4173:
4141:
4140:
4121:
4120:
4101:
4100:
4062:
4051:
4029:
4019:
3994:
3969:
3958:
3936:
3926:
3912:
3860:
3851:
3850:
3820:
3793:
3788:
3787:
3739:
3712:
3685:
3684:
3680:
3670:
3619:
3592:
3549:
3539:
3529:
3528:
3500:
3495:
3494:
3473:
3468:
3467:
3441:
3436:
3435:
3405:
3368:
3363:
3362:
3332:
3327:
3326:
3298:
3297:
3278:
3277:
3274:
3235:
3231:
3220:
3207:
3197:
3164:
3160:
3142:
3138:
3117:
3113:
3112:
3102:
3101:
3097:
3085:
3063:
3058:
3057:
3034:
3020:
3019:
2998:
2987:
2986:
2955:
2914:
2910:
2881:
2880:
2870:
2834:
2833:
2822:
2818:
2768:
2767:
2738:
2737:
2728:
2724:
2723:
2694:
2690:
2661:
2660:
2650:
2649:
2645:
2615:
2611:
2600:
2587:
2577:
2559:
2555:
2544:
2531:
2521:
2520:
2516:
2492:
2481:
2468:
2458:
2429:
2424:
2423:
2396:
2395:
2364:
2360:
2349:
2336:
2326:
2308:
2304:
2293:
2280:
2270:
2269:
2265:
2246:
2233:
2208:
2207:
2182:
2181:
2153:
2152:
2126:
2121:
2120:
2082:
2071:
2058:
2048:
2036:
2014:
2009:
2008:
1985:
1974:
1973:
1950:
1949:
1920:
1916:
1905:
1892:
1882:
1857:
1846:
1833:
1823:
1775:
1753:
1733:
1729:
1718:
1705:
1695:
1676:
1666:
1641:
1640:
1613:
1612:
1587:
1586:
1567:
1566:
1560:
1526:
1513:
1508:
1507:
1481:
1468:
1457:
1456:
1426:
1413:
1402:
1401:
1380:
1375:
1374:
1355:
1354:
1319:
1318:
1271:
1255:
1251:
1241:
1146:
1136:
1135:
1114:
1113:
1082:
1081:
956:
951:
950:
904:
903:
878:
877:
855:
854:
817:
812:
811:
792:
791:
761:
756:
755:
717:
706:
681:
677:
667:
630:
621:
620:
588:
587:
503:
498:
497:
478:
477:
458:
457:
430:
429:
410:
409:
369:
368:
291:
290:
206:
185:
184:
154:
153:
149:
121:
45:
29:collapse theory
17:
12:
11:
5:
5689:
5687:
5679:
5678:
5673:
5663:
5662:
5657:
5656:
5611:(12): 120402.
5595:
5550:(20): 200403.
5534:
5499:(2): 119–130.
5479:
5434:(4): 821–840.
5418:
5365:
5302:
5249:
5236:
5179:
5154:
5109:(2): 471–527.
5089:
5036:
4983:(17): 170404.
4967:
4906:
4837:
4776:
4723:
4688:(5): 581–600.
4672:
4618:
4572:
4531:(2): 470–491.
4511:
4510:
4508:
4505:
4504:
4503:
4498:
4493:
4486:
4483:
4482:
4481:
4469:
4466:
4463:
4449:
4446:
4441:
4436:
4433:
4430:
4423:
4420:
4416:
4412:
4390:
4385:
4381:
4377:
4374:
4370:
4364:
4360:
4356:
4337:
4328:
4327:Other features
4325:
4310:
4307:
4302:
4294:
4290:
4286:
4283:
4261:
4258:
4253:
4245:
4241:
4237:
4234:
4214:
4192:
4188:
4184:
4181:
4158:
4154:
4151:
4148:
4128:
4108:
4078:
4073:
4069:
4065:
4058:
4054:
4050:
4047:
4041:
4038:
4032:
4026:
4022:
4017:
4014:
4011:
4005:
4002:
3997:
3985:
3980:
3976:
3972:
3965:
3961:
3957:
3954:
3948:
3945:
3939:
3933:
3929:
3924:
3919:
3915:
3909:
3904:
3900:
3893:
3890:
3884:
3876:
3871:
3867:
3863:
3859:
3854:
3849:
3846:
3843:
3840:
3837:
3831:
3828:
3823:
3819:
3813:
3810:
3802:
3799:
3769:
3765:
3762:
3759:
3756:
3750:
3747:
3742:
3738:
3732:
3729:
3721:
3718:
3711:
3708:
3705:
3702:
3696:
3693:
3688:
3683:
3677:
3673:
3667:
3662:
3659:
3656:
3652:
3648:
3645:
3642:
3639:
3636:
3630:
3627:
3622:
3618:
3612:
3609:
3601:
3598:
3591:
3586:
3583:
3578:
3575:
3572:
3569:
3566:
3560:
3557:
3552:
3545:
3542:
3538:
3511:
3508:
3503:
3480:
3476:
3452:
3449:
3444:
3421:
3414:
3411:
3404:
3398:
3395:
3389:
3384:
3377:
3374:
3348:
3341:
3338:
3311:
3308:
3285:
3273:
3270:
3253:
3242:
3238:
3234:
3227:
3223:
3219:
3216:
3213:
3210:
3204:
3200:
3196:
3187:
3182:
3178:
3171:
3167:
3163:
3157:
3149:
3145:
3141:
3135:
3130:
3126:
3123:
3120:
3116:
3109:
3105:
3100:
3095:
3090:
3084:
3081:
3078:
3075:
3070:
3066:
3041:
3037:
3033:
3030:
3027:
3005:
3001:
2997:
2994:
2970:
2962:
2958:
2954:
2951:
2948:
2945:
2936:
2931:
2927:
2921:
2917:
2913:
2906:
2901:
2897:
2893:
2888:
2884:
2877:
2873:
2869:
2859:
2854:
2850:
2846:
2841:
2837:
2829:
2825:
2821:
2815:
2810:
2805:
2801:
2793:
2788:
2784:
2780:
2775:
2771:
2763:
2758:
2754:
2750:
2745:
2741:
2734:
2731:
2727:
2716:
2711:
2707:
2701:
2697:
2693:
2686:
2681:
2677:
2673:
2668:
2664:
2657:
2653:
2648:
2642:
2637:
2633:
2622:
2618:
2614:
2607:
2603:
2599:
2596:
2593:
2590:
2584:
2580:
2576:
2566:
2562:
2558:
2551:
2547:
2543:
2540:
2537:
2534:
2528:
2524:
2519:
2508:
2503:
2499:
2495:
2488:
2484:
2480:
2477:
2474:
2471:
2465:
2461:
2455:
2450:
2447:
2444:
2441:
2436:
2432:
2409:
2406:
2403:
2382:
2371:
2367:
2363:
2356:
2352:
2348:
2345:
2342:
2339:
2333:
2329:
2325:
2315:
2311:
2307:
2300:
2296:
2292:
2289:
2286:
2283:
2277:
2273:
2268:
2257:
2254:
2249:
2245:
2242:
2239:
2236:
2232:
2227:
2224:
2221:
2218:
2215:
2206:respectively:
2195:
2192:
2189:
2169:
2166:
2163:
2160:
2136:
2131:
2098:
2093:
2089:
2085:
2078:
2074:
2070:
2067:
2064:
2061:
2055:
2051:
2046:
2041:
2035:
2032:
2029:
2026:
2021:
2017:
1992:
1988:
1984:
1981:
1959:
1927:
1923:
1919:
1912:
1908:
1904:
1901:
1898:
1895:
1889:
1885:
1873:
1868:
1864:
1860:
1853:
1849:
1845:
1842:
1839:
1836:
1830:
1826:
1820:
1815:
1812:
1809:
1806:
1803:
1797:
1794:
1791:
1784:
1781:
1774:
1771:
1768:
1765:
1760:
1756:
1751:
1740:
1736:
1732:
1725:
1721:
1717:
1714:
1711:
1708:
1702:
1698:
1687:
1684:
1679:
1675:
1672:
1669:
1665:
1660:
1657:
1654:
1651:
1648:
1626:
1623:
1620:
1600:
1597:
1594:
1585:, centered at
1574:
1559:
1556:
1542:
1536:
1533:
1529:
1525:
1520:
1516:
1493:
1490:
1485:
1478:
1475:
1471:
1467:
1464:
1438:
1435:
1430:
1423:
1420:
1416:
1412:
1409:
1387:
1383:
1362:
1332:
1329:
1302:
1298:
1295:
1292:
1287:
1280:
1277:
1270:
1267:
1264:
1261:
1258:
1254:
1248:
1244:
1238:
1233:
1230:
1227:
1223:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1195:
1192:
1186:
1181:
1178:
1173:
1170:
1167:
1164:
1161:
1158:
1152:
1149:
1145:
1121:
1101:
1098:
1095:
1092:
1089:
1064:
1059:
1052:
1049:
1041:
1037:
1034:
1031:
1028:
1024:
1018:
1013:
1006:
1003:
995:
992:
989:
986:
983:
980:
977:
972:
965:
962:
935:
931:
928:
925:
922:
918:
914:
911:
891:
888:
885:
865:
862:
847:
846:
824:
820:
799:
777:
770:
767:
733:
728:
724:
720:
713:
709:
705:
702:
697:
690:
687:
680:
674:
670:
663:
660:
654:
646:
641:
637:
633:
629:
624:
619:
614:
609:
602:
599:
585:
584:
577:
565:
560:
555:
551:
546:
540:
535:
531:
527:
524:
521:
518:
515:
510:
506:
485:
465:
437:
417:
395:
390:
383:
380:
356:
353:
349:
343:
338:
331:
328:
321:
318:
313:
308:
304:
299:
272:
267:
262:
258:
253:
247:
242:
238:
234:
229:
224:
219:
215:
210:
203:
200:
197:
193:
182:
181:
169:
166:
162:
148:
145:
120:
117:
116:
115:
112:
109:
102:
44:
41:
15:
13:
10:
9:
6:
4:
3:
2:
5688:
5677:
5674:
5672:
5669:
5668:
5666:
5652:
5648:
5644:
5640:
5636:
5632:
5628:
5624:
5619:
5614:
5610:
5606:
5599:
5596:
5591:
5587:
5583:
5579:
5575:
5571:
5567:
5563:
5558:
5553:
5549:
5545:
5538:
5535:
5530:
5526:
5522:
5518:
5514:
5510:
5506:
5502:
5498:
5494:
5490:
5483:
5480:
5475:
5471:
5467:
5463:
5459:
5455:
5451:
5447:
5442:
5437:
5433:
5429:
5422:
5419:
5414:
5410:
5406:
5402:
5398:
5394:
5389:
5384:
5380:
5376:
5369:
5366:
5361:
5357:
5352:
5347:
5343:
5339:
5335:
5331:
5326:
5321:
5318:(6): 062135.
5317:
5313:
5306:
5303:
5298:
5294:
5290:
5286:
5282:
5278:
5273:
5268:
5265:(5): 052121.
5264:
5260:
5253:
5250:
5246:
5240:
5237:
5232:
5228:
5224:
5220:
5216:
5212:
5207:
5202:
5198:
5194:
5190:
5183:
5180:
5169:
5165:
5158:
5155:
5150:
5146:
5142:
5138:
5134:
5130:
5126:
5122:
5117:
5112:
5108:
5104:
5100:
5093:
5090:
5085:
5081:
5077:
5073:
5069:
5065:
5060:
5055:
5051:
5047:
5040:
5037:
5032:
5028:
5024:
5020:
5016:
5012:
5008:
5004:
5000:
4996:
4991:
4986:
4982:
4978:
4971:
4968:
4963:
4959:
4955:
4951:
4947:
4943:
4939:
4935:
4930:
4925:
4922:(2): 022108.
4921:
4917:
4910:
4907:
4902:
4898:
4894:
4890:
4886:
4882:
4878:
4874:
4870:
4866:
4861:
4856:
4853:(5): 050403.
4852:
4848:
4841:
4838:
4833:
4829:
4825:
4821:
4817:
4813:
4809:
4805:
4800:
4795:
4792:(1): 012116.
4791:
4787:
4780:
4777:
4772:
4768:
4764:
4760:
4756:
4752:
4747:
4742:
4738:
4734:
4727:
4724:
4719:
4715:
4711:
4707:
4703:
4699:
4695:
4691:
4687:
4683:
4676:
4673:
4668:
4664:
4660:
4656:
4652:
4648:
4644:
4640:
4636:
4632:
4625:
4623:
4619:
4614:
4610:
4606:
4602:
4598:
4594:
4590:
4586:
4579:
4577:
4573:
4568:
4562:
4554:
4550:
4546:
4542:
4538:
4534:
4530:
4526:
4519:
4517:
4513:
4506:
4502:
4499:
4497:
4494:
4492:
4489:
4488:
4484:
4479:
4474:
4470:
4467:
4464:
4447:
4444:
4421:
4418:
4414:
4410:
4388:
4383:
4379:
4375:
4372:
4368:
4362:
4354:
4346:
4342:
4338:
4335:
4331:
4330:
4326:
4324:
4308:
4305:
4292:
4288:
4284:
4259:
4256:
4243:
4239:
4235:
4212:
4190:
4186:
4182:
4179:
4170:
4156:
4152:
4149:
4126:
4097:
4076:
4071:
4067:
4063:
4056:
4048:
4045:
4036:
4024:
4020:
4012:
3995:
3983:
3978:
3974:
3970:
3963:
3955:
3952:
3943:
3931:
3927:
3922:
3917:
3913:
3898:
3891:
3888:
3882:
3874:
3869:
3865:
3861:
3857:
3852:
3847:
3838:
3821:
3797:
3785:
3782:
3767:
3757:
3740:
3716:
3709:
3703:
3686:
3681:
3675:
3671:
3665:
3660:
3657:
3654:
3650:
3646:
3637:
3620:
3616:
3596:
3581:
3576:
3573:
3567:
3550:
3543:
3540:
3536:
3526:
3501:
3478:
3474:
3442:
3419:
3409:
3402:
3393:
3387:
3382:
3372:
3346:
3336:
3306:
3283:
3271:
3269:
3266:
3251:
3240:
3236:
3232:
3225:
3217:
3214:
3211:
3202:
3198:
3194:
3185:
3180:
3176:
3169:
3165:
3161:
3155:
3147:
3143:
3139:
3133:
3128:
3124:
3121:
3118:
3114:
3107:
3103:
3098:
3093:
3082:
3076:
3068:
3064:
3055:
3054:), one finds
3039:
3035:
3031:
3028:
3025:
3003:
2999:
2995:
2992:
2983:
2968:
2960:
2952:
2949:
2946:
2934:
2929:
2925:
2919:
2915:
2911:
2904:
2899:
2895:
2891:
2886:
2882:
2875:
2871:
2867:
2857:
2852:
2848:
2844:
2839:
2835:
2827:
2823:
2819:
2813:
2808:
2803:
2799:
2791:
2786:
2782:
2778:
2773:
2769:
2761:
2756:
2752:
2748:
2743:
2739:
2732:
2729:
2725:
2714:
2709:
2705:
2699:
2695:
2691:
2684:
2679:
2675:
2671:
2666:
2662:
2655:
2651:
2646:
2635:
2631:
2620:
2616:
2612:
2605:
2597:
2594:
2591:
2582:
2578:
2574:
2564:
2560:
2556:
2549:
2541:
2538:
2535:
2526:
2522:
2517:
2506:
2501:
2497:
2493:
2486:
2478:
2475:
2472:
2463:
2459:
2448:
2442:
2434:
2430:
2421:
2407:
2404:
2401:
2380:
2369:
2365:
2361:
2354:
2346:
2343:
2340:
2331:
2327:
2323:
2313:
2309:
2305:
2298:
2290:
2287:
2284:
2275:
2271:
2266:
2255:
2252:
2243:
2240:
2234:
2230:
2225:
2219:
2213:
2193:
2190:
2187:
2167:
2164:
2161:
2158:
2149:
2134:
2117:
2096:
2091:
2087:
2083:
2076:
2068:
2065:
2062:
2053:
2049:
2044:
2033:
2027:
2019:
2015:
2006:
1990:
1986:
1982:
1979:
1946:
1925:
1921:
1917:
1910:
1902:
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119:Description
5665:Categories
5618:1512.07244
5174:2020-05-26
4507:References
1348:commutator
5651:206271698
5557:1408.1273
5521:0010-3616
5466:1572-9613
5325:1408.6115
5297:118401637
5272:1110.4495
5231:1751-8113
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