1314:
bad, it is rather the tale of the unrelenting pursuit of a bad idea. ... Many, many philosophers and physicists have become convinced that a change of logic (and most dramatically, the rejection of classical logic) will somehow help in understanding quantum theory, or is somehow suggested or forced on us by quantum theory. But quantum logic, even through its many incarnations and variations, both in technical form and in interpretation, has never yielded the goods.
5269:
5571:
3109:
Although many treatments of quantum logic assume that the underlying lattice must be orthomodular, such logics cannot handle multiple interacting quantum systems. In an example due to Foulis and
Randall, there are orthomodular propositions with finite-dimensional Hilbert models whose pairing admits
925:
A number of other logics have also been proposed to analyze quantum-mechanical phenomena, unfortunately also under the name of "quantum logic(s)". They are not the subject of this article. For discussion of the similarities and differences between quantum logic and some of these competitors, see
1313:
The horse of quantum logic has been so thrashed, whipped and pummeled, and is so thoroughly deceased that...the question is not whether the horse will rise again, it is: how in the world did this horse get here in the first place? The tale of quantum logic is not the tale of a promising idea gone
2973:
1280:
presents severe difficulties for this goal. Later, Putnam retracted his views, albeit with much less fanfare, but the damage had been done. While
Birkhoff and von Neumann's original work only attempted to organize the calculations associated with the
2781:
1328:
Quantum logic remains in limited use among logicians as an extremely pathological counterexample (Dalla Chiara and
Giuntini: "Why quantum logics? Simply because 'quantum logics are there!'"). Although the central insight to quantum logic remains
2174:. Although much of the development of quantum logic has been motivated by the standard semantics, it is not the characterized by the latter; there are additional properties satisfied by that lattice that need not hold in quantum logic.
3098:. Indeed, modern logics for the analysis of quantum computation often begin with quantum logic, and attempt to graft desirable features of an extension of classical logic thereonto; the results then necessarily embed quantum logic.
1285:
of quantum mechanics, a school of researchers had now sprung up, either hoping that quantum logic would provide a viable hidden-variable theory, or obviate the need for one. Their work proved fruitless, and now lies in poor repute.
1780:
1915:
106:
3457:, p. 94: "Quantum logics are, without any doubt, logics. As we have seen, they satisfy all the canonical conditions that the present community of logicians require in order to call a given abstract object a logic."
3751:, Addison-Wesley Series in Advanced Physics; Addison-Wesley, 1968, this property cannot be used to deduce a vector space structure, because it is not peculiar to (pre-)Hilbert spaces. An analogous claim holds in most
4303:
3288:, vol. 7, 1998. p. 882ff: " differs from the standard sentential calculus....The most notable difference is that the distributive laws fail, being replaced by a weaker law known as orthomodularity."
1259:
popularized Mackey's work in two papers in 1968 and 1975, in which he attributed the idea that anomalies associated to quantum measurements originate with a failure of logic itself to his coauthor, physicist
3068:
2867:
1297:, in the sense of describing a process of reasoning, as opposed to a particularly convenient language to summarize the measurements performed by quantum apparatuses. In particular, modern
1203:
an observer might ask about the state of a physical system, questions that could be settled by some measurement. Principles for manipulating these quantum propositions were then called
2311:, such as position and momentum. This occurs because measurement affects the system, and measurement of whether a disjunction holds does not measure which of the disjuncts is true.
2170:
This semantics has the nice property that the pre-Hilbert space is complete (i.e., Hilbert) if and only if the propositions satisfy the orthomodular law, a result known as the
2679:
1701:
1842:
1301:
argue that quantum logic attempts to substitute metaphysical difficulties for unsolved problems in physics, rather than properly solving the physics problems.
4059:
Richard Blute, Alessio
Guglielmi, Ivan T. Ivanov, Prakash Panangaden, Lutz StraĂburger, "A Logical Basis for Quantum Evolution and Entanglement" in
3861:, Boston Studies in the Philosophy of Science 13, ed. Robert S. Cohen and Marx W. Wartofsky; D. Riedel, 1974. pp. 92-122. DOI:
1963:
This suggests the following quantum mechanical replacement for the orthocomplemented lattice of propositions in classical mechanics, essentially Mackey's
2279:
Expressions in quantum logic describe observables using a syntax that resembles classical logic. However, unlike classical logic, the distributive law
4498:
3101:
The orthocomplemented lattice of any set of quantum propositions can be embedded into a
Boolean algebra, which is then amenable to classical logic.
3020:
2186:
immediately points to a difference with the partial order structure of a classical proposition system. In the classical case, given a proposition
36:
4100:, trans. Robert T. Beyer, ed. Nicholas A. Wheeler; Princeton University Press, 2018 (original 1932). pp. 160-164.
5305:
5066:
1149:(they each have uncertainty 1/3, which is less than the allowed minimum of 1/2). So there are no states that can support either proposition, and
3787:, University of Western Ontario Series in Philosophy of Science 5a, ed. C. A. Hooker; D. Riedel, c. 1975-1979. pp. 123-133.
3537:
4699:
3183:
2220:. In the case of the lattice of projections there are infinitely many solutions to the above equations (any closed, algebraic complement of
1183:
372:
3001:
is the lattice of closed subspaces of a separable
Hilbert space of complex dimension at least 3. Then for any quantum probability measure
4129:
3252:. They are presented here because they are simple enough to enable intuition, and can be considered as limiting cases of operators that
873:
5237:
4432:, vol. 1: "Derivation of Hilbert Space Structure", trans. Leo F. Boron, ed. Karl Just; Springer, 1985. DOI:
1610:. It follows easily from this characterization of propositions in classical systems that the corresponding logic is identical to the
4220:
1671:
are respectively set intersection and set union. The orthocomplementation operation is set complement. Moreover, this lattice is
4548:
1634:
947:
status of the rules of propositional logic. Modern philosophers reject quantum logic as a basis for reasoning, because it lacks a
124:
5249:
1559:
on the state space. Examples of observables are position, momentum or energy of a particle. For classical systems, the value
580:
3756:
1225:
in terms of quantum propositions. Although Mackey's presentation still assumed that the orthocomplemented lattice is the
4933:
4491:
353:
2968:{\displaystyle \operatorname {P} \!\left(\bigvee _{i=1}^{\infty }E_{i}\right)=\sum _{i=1}^{\infty }\operatorname {P} (E_{i}).}
4867:
4441:
3870:
1047:
To illustrate why the distributive law fails, consider a particle moving on a line and (using some system of units where the
536:
935:
Quantum logic has been proposed as the correct logic for propositional inference generally, most notably by the philosopher
4790:
3284:
2308:
459:
4007:
Luc Bouten; Ramon van Handel; Matthew R. James (2009). "A discrete invitation to quantum filtering and feedback control".
1793:
formulation of quantum mechanics as presented by von Neumann, a physical observable is represented by some (possibly
5298:
4286:
4267:
4061:
Categories and Types in Logic, Language, and
Physics: Essays Dedicated to Jim Lambek on the Occasion of His 90th Birthday
1630:
1344:, which has engendered a proliferation of new logics for formal analysis of quantum protocols and algorithms (see also
4421:
1659:
We summarize these remarks as follows: The proposition system of a classical system is a lattice with a distinguished
4521:
4484:
4253:
3133:. Consequently, quantum logic struggles to represent the passage of time. One possible workaround is the theory of
119:
2812:, then there is a bijective correspondence between Mackey observables and densely-defined self-adjoint operators on
2425:). Moreover: if the relevant Hilbert space for the particle's dynamics only admits momenta no greater than 1, then
5602:
4631:
208:
5204:
4913:
4908:
4616:
3596:
N.J. Cutland; P.F. Gibbins (Sep 1982). "A regular sequent calculus for
Quantum Logic in which ⚠and ⧠are dual".
866:
315:
295:
163:
4174:, Boston Studies in the Philosophy of Science V, ed. Robert S. Cohen and Marx W. Wartofsky, 1969.
5291:
5216:
4888:
4196:, in Current Issues in Quantum Logic, Plenum Press, New York, ed. E. Beltrametti et al., 1981, pp. 333-340
2987:
555:
285:
5575:
5194:
4971:
4893:
4765:
4661:
4531:
2829:
595:
333:
233:
1244:, GĂŒnther Ludwig and others later developed axiomatizations that do not assume an underlying Hilbert space.
5607:
4928:
4862:
4857:
4828:
4526:
3249:
1145:" each assert tighter restrictions on simultaneous values of position and momentum than are allowed by the
531:
526:
497:
348:
129:
5597:
5537:
4996:
4903:
4237:
3130:
1611:
1485:
919:
565:
310:
300:
2776:{\displaystyle \varphi \left(\bigcup _{i=1}^{\infty }S_{i}\right)=\sum _{i=1}^{\infty }\varphi (S_{i}).}
939:, at least at one point in his career. This thesis was an important ingredient in Putnam's 1968 paper "
922:, but the structure of experimental tests in quantum mechanics forms a much more complicated structure.
5612:
5552:
5352:
5272:
5034:
4842:
4813:
4242:
3126:
2624:
2366:
859:
511:
482:
5382:
5367:
5056:
4923:
4847:
4808:
4745:
4724:
4681:
4574:
2787:
1817:
1282:
1218:
516:
477:
430:
405:
328:
188:
2446:
be the momentum functions (Fourier transforms) for the projections of the particle wave function to
5525:
5520:
5449:
5424:
5409:
5404:
5399:
5123:
5103:
5093:
5083:
5039:
4599:
4335:
3726:
1513:
1192:
600:
3708:
4818:
4729:
1971:
The propositions of a quantum mechanical system correspond to the lattice of closed subspaces of
1932:
1048:
975:
769:
487:
395:
5232:
4780:
4355:
2365:
and hence does not have non-isolated zeroes. Therefore, there is no wave function that is both
1356:
Quantum logic can be axiomatized as the theory of propositions modulo the following identities:
445:
343:
109:
5542:
5464:
5143:
4918:
4898:
4823:
4676:
4408:, Springer-Verlag, 1989. Elementary and well-illustrated; suitable for advanced undergraduates.
3896:
3721:
3716:. 8th Int. Conf. on Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX).
3357:
1298:
1265:
774:
492:
320:
290:
253:
4719:
4323:
400:
5515:
5489:
5314:
5168:
4666:
4646:
3203:
2978:
Every quantum probability measure on the closed subspaces of a
Hilbert space is induced by a
2156:
2104:
1798:
1525:
1509:
1330:
1269:
1146:
1123:
243:
228:
4154:
3223:
5454:
5377:
5199:
5128:
5073:
4803:
4611:
4569:
4362:, vol. 21, V. F. Hendricks and J. Malinowski (eds.), Springer, 2003. arXiv
4227:, ed. K. Engesser, D. M. Gabbay, and D. Lehmann; Elsevier, 2009. pp. 49-78.
4026:
3911:
3671:
3634:
3208:
3142:
3118:
2825:
2238:
to {â„,â€} must be
Boolean. A standard workaround is to study maximal partial homomorphisms
2235:
1553:
1541:
1493:
1430:
1277:
948:
940:
560:
472:
198:
155:
804:
8:
5547:
5505:
5474:
5434:
5419:
5394:
5184:
5153:
5098:
5078:
4986:
4943:
4798:
4709:
4636:
4626:
4538:
3213:
3193:
3138:
2422:
2066:
1951:. That subspace can be interpreted as the quantum analogue of the classical proposition
1626:
1622:
1607:
1533:
1529:
1306:
1273:
991:
895:
659:
467:
385:
213:
193:
145:
4030:
3915:
3675:
3638:
3110:
no orthomodular model. Likewise, quantum logic with the orthomodular law falsifies the
5439:
5329:
5244:
5113:
5011:
4704:
4651:
4543:
4383:
4109:
4042:
4016:
3987:
3969:
3927:
3857:
3687:
3650:
3605:
3188:
3122:
1991:
of quantum propositions is also sequentially complete: any pairwise-disjoint sequence {
1775:{\displaystyle \operatorname {LUB} (\{E_{i}\})=\bigcup _{i=1}^{\infty }E_{i}{\text{.}}}
1252:
887:
380:
305:
238:
150:
4346:, vol. 6, D. Gabbay and F. Guenthner (eds.), Kluwer, 2002. arXiv
3960:
V. P. Belavkin (1992). "Quantum stochastic calculus and quantum nonlinear filtering".
3815:
5484:
5372:
5342:
5337:
5254:
5163:
5133:
5061:
5024:
5019:
5001:
4966:
4956:
4760:
4656:
4621:
4604:
4507:
4437:
4249:
3983:
3931:
3866:
3691:
3654:
3198:
3134:
3111:
2983:
2354:
2350:
2228:
2077:
1695:
1477:
1341:
1261:
907:
814:
789:
729:
724:
624:
590:
570:
168:
27:
4046:
2603:. Such interference is key to the richness of quantum logic and quantum mechanics.
1910:{\displaystyle f(A)=\int _{\mathbb {R} }f(\lambda )\,d\operatorname {E} (\lambda ).}
5389:
4981:
4976:
4833:
4714:
4465:
4125:
4121:
4113:
4034:
3991:
3979:
3919:
3679:
3642:
3218:
2801:
2231:
has unusual properties in quantum logic. An orthocomplemented lattice admitting a
1813:
1794:
1641:
1481:
1415:
1334:
1248:
1241:
1188:
994:
959:
915:
911:
819:
809:
799:
699:
679:
664:
634:
502:
390:
4277:
4246:
4136:, series II, vol. 37, issue 4, pp. 823â843, 1936. JSTOR
3855:
Jeffery Bub and William Demopoulos, "The Interpretation of Quantum Mechanics," in
2171:
5459:
5444:
5414:
5362:
5357:
5211:
5138:
5118:
5088:
5051:
5046:
4951:
4775:
4281:
4201:
4190:
4179:
4159:
4064:
3752:
3547:
3358:
Attempt of an Axiomatic Foundation of Quantum Mechanics and More General Theories
2617:
2362:
2358:
2314:
For example, consider a simple one-dimensional particle with position denoted by
2070:
1619:
1505:
1290:
1237:
1233:
1226:
987:
963:
844:
714:
694:
440:
280:
3248:
Due to technical reasons, it is not possible to represent these propositions as
1199:
can be viewed as propositions about physical observables; that is, as potential
5479:
5469:
5189:
5158:
5148:
4770:
4750:
4579:
4433:
4339:
4311:
4137:
3862:
3780:
3772:
3146:
2979:
2232:
1825:
974:
can be defined in terms of functions on or to the lattice, giving an alternate
779:
739:
719:
689:
669:
619:
585:
435:
425:
218:
4451:
2065:
The standard semantics of quantum logic is that quantum logic is the logic of
1586:
concerning a classical system are generated from basic statements of the form
5591:
5530:
4961:
4755:
4671:
4641:
4594:
4167:
4149:
3172:
2073:
1790:
1256:
1210:
1196:
971:
936:
839:
834:
764:
734:
704:
575:
248:
223:
101:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
4416:(in German), Springer, 1954. The definitive work. Released in English as:
4105:
3662:
Hirokazu Nishimura (Jul 1994). "Proof theory for minimal quantum logic II".
3517:
3515:
3513:
3511:
1340:
Quantum logic's best chance at revival is through the recent development of
5510:
4991:
4589:
4584:
3818:, which cites G.D. Crown, "On some orthomodular posets of vector bundles,"
3625:
Hirokazu Nishimura (Jan 1994). "Proof theory for minimal quantum logic I".
3228:
3178:
3154:
3150:
3088:
1615:
1512:
on a Hilbert space. However, the main ideas can be understood in the
1489:
952:
944:
829:
824:
759:
744:
709:
203:
4371:
4006:
2421:, which is certainly not false (there are states for which it is a viable
918:, that the structure of experimental tests in classical mechanics forms a
5347:
5029:
4363:
4347:
4232:
3947:
V. P. Belavkin (1978). "Optimal quantum filtration of Makovian signals".
3717:
3508:
3167:
3092:
3010:
2349:
Now, position and momentum are Fourier transforms of each other, and the
1629:) with the set operations of union and intersection corresponding to the
1583:
1423:
1374:
1370:
1302:
910:. The formal system takes as its starting point an observation of
903:
794:
749:
684:
639:
5283:
3609:
1348:). The logic may also find application in (computational) linguistics.
3923:
3800:
3683:
3646:
3341:
3303:
2093:
2085:
2054:
1784:
1537:
1419:
1230:
967:
784:
754:
674:
649:
644:
629:
4038:
1504:
The remainder of this article assumes the reader is familiar with the
4883:
4564:
4322:
A. Baltag, J. Bergfeld, K. Kishida, J. Sack, S. Smets and S. Zhong, "
4141:
4068:
4021:
3974:
3837:
3533:
1470:
275:
4476:
4425:, vol. 1, trans. Carl A. Hein; Springer-Verlag, 1983.
4295:
3129:
in a certain technical sense reduces the class of propositions to a
4391:
4375:
4315:
1059:
654:
4262:
2216:
have exactly one solution, namely the set-theoretic complement of
1289:
Most philosophers find quantum logic an unappealing competitor to
3538:
Venn and Euler type diagrams for vector spaces and abelian groups
891:
4310:, vol. 16, issue 3, pp. 491-525, 2006. DOI
1217:), attempted to axiomatize quantum logic as the structure of an
4225:
Handbook of Quantum Logic and Quantum Structures: Quantum Logic
3779:, vol. 6, no. 4, 1957. pp. 885-893. DOI:
3291:
1293:. It is far from evident (albeit true) that quantum logic is a
1126:
of momenta between 0 and +1/6 and positions between â1 and +3.
16:
Theory of logic to account for observations from quantum theory
4185:, Z. Logik und Grundl. Math., vol. 20, 1974, pp. 395-406.
3257:
1476:
Alternative formulations include propositions derivable via a
4354:
M. L. Dalla Chiara, R. Giuntini, and R. Leporini, "
4101:
3063:{\displaystyle \operatorname {P} (E)=\operatorname {Tr} (SE)}
1466:
1122:
in other words, that the state of the particle is a weighted
958:
Mathematically, quantum logic is formulated by weakening the
4353:
4245:"Contemporary Philosophy in Focus" series, 2005. DOI:
3553:
3438:
3436:
1499:
4455:
4324:
PLQP & Company: Decidable Logics for Quantum Algorithms
4263:"Quantum Logic in Historical and Philosophical Perspective"
3858:
Logical and Epistemological Studies in Contemporary Physics
3460:
3153:
that is very close to quantum logic, can handle arbitrary
4321:
3843:
3775:, "Measures on the Closed Subspaces of a Hilbert Space",
3433:
4330:, vol. 53, issue 10, pp. 3628-3647, 2014.
3805:
Workshop on Physics and Computation (PhysComp '92)
3595:
3484:
1923:
is the indicator function of an interval , the operator
3825:
3763:, vol. 348, no. 5, 1996. pp. 1839-1862.
2627:
from the orthocomplemented lattice of Borel subsets of
1544:. In the simplest case of a single particle moving in
3496:
2606:
2517:
differently so as to make such states possible; also,
2111:(since for those states, the probability of observing
1640:
In fact, a stronger claim is true: they must obey the
1594:
yields a value in the interval for some real numbers
986:
The most notable difference between quantum logic and
4384:
Reasoning Formally About Quantum Systems: An Overview
3807:
proceedings. See also the discussion at
3564:
3562:
3381:
3379:
3377:
3023:
2870:
2682:
1845:
1704:
1264:. Putnam hoped to develop a possible alternative to
1221:, and recognized that a physical observable could be
39:
4334:
4212:
3876:
3740:
3706:
3521:
3454:
3429:, vol. 14, no. 5, 1984. pp. 409-430.
1931:) is a self-adjoint projection onto the subspace of
1785:
Propositional lattice of a quantum mechanical system
1433:, which additionally satisfy the orthomodular law:
4260:
4158:, W. A. Benjamin, 1963. HathiTrust
4120:
3744:
3309:
2842:
with values in such that P("â„)=0, P(â€)=1 and if {
2177:
1606:through the conventional arithmetic operations and
4406:An Introduction to Hilbert Space and Quantum Logic
3785:The Logico-Algebraic Approach to Quantum Mechanics
3559:
3374:
3062:
2967:
2775:
2669:are pairwise-orthogonal propositions (elements of
1909:
1774:
1207:by von Neumann and Birkhoff in a 1936 paper.
100:
4296:Mathematical study and computational applications
3747:. Despite suggestions otherwise in Josef Jauch,
3472:
3391:
3315:
2874:
2857:is a sequence of pairwise-orthogonal elements of
2635:. In symbols, this means that for any sequence {
1548:, the state space is the positionâmomentum space
1345:
928:
5589:
3258:§ Quantum logic as the logic of observables
2224:solves it; it need not be the orthocomplement).
1618:of the state space. They thus obey the laws of
4063:; Springer, 2014. pp. 90-107. DOI:
3822:, vol. 15 issue 1-2: pp. 11â25, 1975.
3554:Dalla Chiara, Giuntini & Leporini 2003
3408:
3406:
3336:C. Piron, "Axiomatique quantique" (in French),
3082:
2819:
2010:has a least upper bound. Here disjointness of
1519:
1176:
3801:Linear logic for generalized quantum mechanics
2808:is the lattice of closed subspaces of Hilbert
5299:
4492:
4304:LQP: The Dynamic Logic of Quantum Information
4261:de Ronde, C.; Domenech, G.; Freytes, H.
4218:
4194:Orthomodular Logic as a Hilbert Type Calculus
4155:Mathematical Foundations of Quantum Mechanics
4098:Mathematical Foundations of Quantum Mechanics
3442:
3184:Mathematical formulation of quantum mechanics
1575:, is obtained by a process of measurement of
1215:Mathematical Foundations of Quantum Mechanics
1184:Mathematical Foundations of Quantum Mechanics
867:
4328:International Journal of Theoretical Physics
3664:International Journal of Theoretical Physics
3627:International Journal of Theoretical Physics
3403:
2472:to momenta that are (in absolute value) â„1.
2369:in momentum space and vanishes on precisely
2274:
1727:
1714:
1337:, discussions rarely mention quantum logic.
1309:by making the problem impossible to state."
902:is a set of rules for manipulation of
95:
69:
4308:Mathematical Structures in Computer Science
4301:
4095:
3894:
3831:
3297:
1414:("¬" is the traditional notation for "
5570:
5306:
5292:
4499:
4485:
4000:
3949:Problems of Control and Information Theory
1488:system. Despite the relatively developed
874:
860:
5313:
4020:
3973:
3795:
3793:
3725:
3137:developed in the late 1970s and 1980s by
1885:
1867:
1500:Quantum logic as the logic of observables
4430:An Axiomatic Basis for Quantum Mechanics
4200:
4189:
4178:
3882:
3584:
3580:
1698:, specifically the set-theoretic union:
1305:writes that quantum "logic 'solves' the
955:, of which quantum logic is a fragment.
951:; a common alternative is the system of
4390:, 36(3), 2005. pp. 51â66. arXiv
4231:
3502:
3466:
3385:
3364:, vol. 4, 1967. pp. 331-348.
3175:(An approach to temporal quantum logic)
2786:Equivalently, a Mackey observable is a
1319:
962:for a Boolean algebra, resulting in an
5590:
4411:
4398:
4369:
4356:Quantum Computational Logics: A Survey
4278:"Quantum Logic and Probability Theory"
4166:
4148:
3790:
3777:Indiana University Mathematics Journal
3568:
3425:T. A. Brody, "On Quantum Logic",
3397:
3367:
3321:
2650:of pairwise-disjoint Borel subsets of
1663:operation: The lattice operations of
1633:and subset inclusion corresponding to
1469:and ""â„" the traditional notation for
1380:There is a maximal element â€, and †=
1351:
5287:
4506:
4480:
4464:
3745:de Ronde, Domenech & Freytes
3710:Gentzen-like Methods in Quantum Logic
3522:Dalla Chiara & Giuntini 2002
3490:
3478:
3455:Dalla Chiara & Giuntini 2002
3424:
3347:
3141:. It is known, however, that System
2611:
2060:
1465:("â€" is the traditional notation for
1129:On the other hand, the propositions "
3310:Birkhoff & von Neumann 1936
3285:Routledge Encyclopedia of Philosophy
4287:Stanford Encyclopedia of Philosophy
4268:Internet Encyclopedia of Philosophy
4082:
3117:Quantum logic admits no reasonable
2607:Relationship to quantum measurement
2513:= 0 (this holds even if we defined
1492:, quantum logic is not known to be
1422:", and "∧" the notation for "
1346:§ Relationship to other logics
1092:= "the particle is in the interval
1082:= "the particle is in the interval
929:§ Relationship to other logics
13:
4472:, W. A. Benjamin, 1976.
4414:Der Grundlagen der Quantenmechanik
3282:Peter Forrest, "Quantum logic" in
3024:
2940:
2935:
2896:
2871:
2746:
2707:
1889:
1824:. In particular, for any bounded
1820:E defined on the Borel subsets of
1752:
1675:, in the sense that any sequence {
406:Sum-over-histories (path integral)
92:
66:
22:Part of a series of articles about
14:
5624:
4275:
4235:, "The Tale of Quantum Logic" in
4213:Modern philosophical perspectives
3412:
2595:might interfere to produce zero |
1694:of elements of the lattice has a
1571:for some particular system state
43:
5569:
5268:
5267:
4450:
4422:Foundations of Quantum Mechanics
3962:Journal of Multivariate Analysis
3820:Journ. of Natural Sci. and Math.
3808:
3749:Foundations of Quantum Mechanics
3741:Dalla Chiara & Giuntini 2002
3070:for any self-adjoint projection
2178:Differences with classical logic
1975:; the negation of a proposition
1959:yields a value in the interval .
1213:, in his 1963 book (also called
4344:Handbook of Philosophical Logic
4053:
3938:
3888:
3849:
3766:
3757:Decompositions in Quantum Logic
3734:
3707:Uwe Egly; Hans Tompits (1999).
3700:
3616:
3589:
3574:
3527:
3448:
3418:
3340:vol. 37, 1964. DOI:
2625:countably additive homomorphism
1418:", "∨" the notation for "
1219:orthocomplemented lattice
981:
964:orthocomplemented lattice
5217:Relativistic quantum mechanics
4470:Foundations of Quantum Physics
4207:, Academic Press, London, 1983
4130:The Logic of Quantum Mechanics
3895:RomĂĄn, L.; Rumbos, B. (1991).
3327:
3276:
3242:
3104:
3057:
3048:
3036:
3030:
2959:
2946:
2767:
2754:
2491:) corresponds to states with |
1901:
1895:
1882:
1876:
1855:
1849:
1730:
1711:
556:Relativistic quantum mechanics
88:
81:
62:
1:
5195:Quantum statistical mechanics
4972:Quantum differential calculus
4894:Delayed-choice quantum eraser
4662:Symmetry in quantum mechanics
2830:Quantum statistical mechanics
2623:, a Mackey observable Ï is a
1979:is the orthogonal complement
1835:, the following extension of
1181:In his classic 1932 treatise
1044:are propositional variables.
906:inspired by the structure of
596:Quantum statistical mechanics
4336:M. L. Dalla Chiara
4247:10.1017/CBO9780511614187.006
3984:10.1016/0047-259X(92)90042-E
3270:
3250:quantum-mechanical operators
3083:Relationship to other logics
2820:Quantum probability measures
2547:corresponds to states with |
2038:. The least upper bound of {
1520:Logic of classical mechanics
1177:History and modern criticism
7:
4997:Quantum stochastic calculus
4987:Quantum measurement problem
4909:MachâZehnder interferometer
4065:10.1007/978-3-642-54789-8_6
3160:
2838:is a function P defined on
2836:quantum probability measure
2242:with a filtering property:
943:" in which he analysed the
888:mathematical study of logic
566:Quantum information science
10:
5629:
5553:Thermoacoustic heat engine
4243:Cambridge University Press
4077:
3087:Quantum logic embeds into
2823:
2322:, and define observables:
2307:) fails when dealing with
1839:to operators can be made:
1251:'s then-recent defence of
978:for quantum computations.
5565:
5538:Immersive virtual reality
5498:
5328:
5321:
5263:
5225:
5177:
5057:Quantum complexity theory
5035:Quantum cellular automata
5010:
4942:
4876:
4789:
4738:
4725:Path integral formulation
4692:
4557:
4514:
4434:10.1007/978-3-642-70029-3
4312:10.1017/S0960129506005299
4302:A. Baltag and S. Smets, "
3897:"Quantum logic revisited"
3863:10.1007/978-94-010-2656-7
3781:10.1512/iumj.1957.6.56050
2788:projection-valued measure
2618:orthocomplemented lattice
2275:Failure of distributivity
1818:projection-valued measure
1429:Some authors restrict to
1333:as an intuition pump for
1283:Copenhagen interpretation
5521:Digital scent technology
5124:Quantum machine learning
5104:Quantum key distribution
5094:Quantum image processing
5084:Quantum error correction
4934:Wheeler's delayed choice
4089:Arranged chronologically
3235:
2454:â„ 0 respectively. Let |
2432:To understand more, let
2357:nonzero function with a
2309:noncommuting observables
2155:) refers to states that
1933:generalized eigenvectors
1567:), that is the value of
1552:. An observable is some
1532:have three ingredients:
601:Quantum machine learning
354:Wheeler's delayed-choice
5603:Systems of formal logic
5040:Quantum finite automata
4338:and R. Giuntini, "
4160:2027/mdp.39015001329567
3832:Baltag & Smets 2006
3761:Transactions of the AMS
2229:propositional valuation
2129:is the intersection of
2084:is associated with the
2053:is the closed internal
1299:philosophers of science
1100:We might observe that:
1049:reduced Planck constant
966:. Quantum-mechanical
311:LeggettâGarg inequality
5543:Magnetic refrigeration
5144:Quantum neural network
4372:Quantum Logic Explorer
4219:Guido Bacciagaluppi, "
3904:Foundations of Physics
3427:Foundations of Physics
3338:Helvetica Physica Acta
3064:
3009:there exists a unique
2969:
2939:
2900:
2777:
2750:
2711:
2465:be the restriction of
2405:) is false. However,
2080:, where an observable
1911:
1814:spectral decomposition
1776:
1756:
1510:self-adjoint operators
1326:
1323:, pp. 184â185
990:is the failure of the
102:
5516:Cloak of invisibility
5315:Emerging technologies
5169:Quantum teleportation
4682:Waveâparticle duality
4205:Orthomodular Lattices
4134:Annals of Mathematics
4096:J. von Neumann,
3755:; see John Harding, "
3298:von Neumann 1932
3204:Quantum contextuality
3065:
2970:
2919:
2880:
2778:
2730:
2691:
2555:= 0. As an operator,
2333:| †1 (in some units)
2105:orthogonal complement
2086:set of quantum states
1912:
1799:self-adjoint operator
1777:
1736:
1673:sequentially complete
1627:de Morgan's laws
1431:orthomodular lattices
1331:mathematical folklore
1317:â Maudlin,
1311:
1270:wavefunction collapse
1189:John von Neumann
1147:uncertainty principle
916:John von Neumann
296:Elitzur–Vaidman
286:Davisson–Germer
103:
5200:Quantum field theory
5129:Quantum metamaterial
5074:Quantum cryptography
4804:Consistent histories
3342:10.5169/seals-113494
3209:Quantum field theory
3119:material conditional
3021:
2984:nonnegative operator
2868:
2680:
2236:lattice homomorphism
2092:(when measured) has
2067:projection operators
1843:
1702:
1661:orthocomplementation
1635:material implication
1631:Boolean conjunctives
1554:real-valued function
1437:If †= ¬(¬
1398:∨¬(¬
1058:= "the particle has
949:material conditional
561:Quantum field theory
473:Consistent histories
110:Schrödinger equation
37:
5608:Non-classical logic
5548:Phased-array optics
5506:Acoustic levitation
5185:Quantum fluctuation
5154:Quantum programming
5114:Quantum logic gates
5099:Quantum information
5079:Quantum electronics
4539:Classical mechanics
4399:Quantum foundations
4221:Is Logic Empirical?
4172:Is Logic Empirical?
4126:J. von Neumann
4031:2009SIAMR..51..239B
3916:1991FoPh...21..727R
3676:1994IJTP...33.1427N
3639:1994IJTP...33..103N
3493:, pp. 428â429.
3362:Commun. Math. Phys.
3214:Quantum probability
3194:Quantum Bayesianism
3155:discrete spacetimes
3135:quantum filtrations
2423:measurement outcome
1939:with eigenvalue in
1804:on a Hilbert space
1623:propositional logic
1530:classical mechanics
1352:Algebraic structure
1274:quantum measurement
1201:yes-or-no questions
941:Is Logic Empirical?
896:quantum foundations
349:Stern–Gerlach
146:Classical mechanics
5598:Mathematical logic
5238:in popular culture
5020:Quantum algorithms
4868:Von NeumannâWigner
4848:Objective collapse
4544:Old quantum theory
4382:N. Papanikolaou, "
4276:Wilce, Alexander.
4183:Orthomodular Logic
3924:10.1007/BF00733278
3844:Baltag et al. 2014
3684:10.1007/bf00670687
3647:10.1007/BF00671616
3600:. Nouvelle SĂ©rie.
3598:Logique et Analyse
3469:, p. 159-161.
3443:Bacciagaluppi 2009
3189:Multi-valued logic
3060:
2965:
2773:
2612:Mackey observables
2061:Standard semantics
1907:
1797:) densely defined
1772:
1514:finite-dimensional
1272:in the problem of
1255:, the philosopher
1253:general relativity
1032:where the symbols
537:Von NeumannâWigner
517:Objective-collapse
316:Mach–Zehnder
306:Leggett inequality
301:Franck–Hertz
151:Old quantum theory
98:
5613:Quantum mechanics
5585:
5584:
5561:
5560:
5368:complexity theory
5353:cellular automata
5281:
5280:
5255:Quantum mysticism
5233:Schrödinger's cat
5164:Quantum simulator
5134:Quantum metrology
5062:Quantum computing
5025:Quantum amplifier
5002:Quantum spacetime
4967:Quantum cosmology
4957:Quantum chemistry
4657:Scattering theory
4605:Zero-point energy
4600:Degenerate levels
4508:Quantum mechanics
4442:978-3-642-70029-3
4112:available at the
4039:10.1137/060671504
3871:978-94-010-2656-7
3356:GĂŒnther Ludwig, "
3199:Quantum cognition
3112:deduction theorem
2826:Gleason's theorem
2355:square-integrable
2351:Fourier transform
2182:The structure of
2078:pre-Hilbert space
2031:is a subspace of
1770:
1696:least upper bound
1490:proof theory
1478:natural deduction
1342:quantum computing
1278:Gleason's theorem
1262:David Finkelstein
884:
883:
591:Scattering theory
571:Quantum computing
344:Schrödinger's cat
276:Bell's inequality
84:
59:
28:Quantum mechanics
5620:
5573:
5572:
5450:machine learning
5425:key distribution
5410:image processing
5400:error correction
5326:
5325:
5308:
5301:
5294:
5285:
5284:
5271:
5270:
4982:Quantum geometry
4977:Quantum dynamics
4834:Superdeterminism
4766:RaritaâSchwinger
4715:Matrix mechanics
4570:Braâket notation
4501:
4494:
4487:
4478:
4477:
4473:
4461:
4447:
4428:GĂŒnther Ludwig,
4419:GĂŒnther Ludwig,
4412:GĂŒnther Ludwig,
4379:
4366:
4364:quant-ph/0305029
4350:
4348:quant-ph/0101028
4331:
4318:
4291:
4282:Zalta, Edward N.
4272:
4257:
4228:
4208:
4197:
4186:
4175:
4163:
4145:
4117:
4114:Internet Archive
4083:Historical works
4072:
4057:
4051:
4050:
4024:
4004:
3998:
3995:
3977:
3956:
3942:
3936:
3935:
3901:
3892:
3886:
3880:
3874:
3853:
3847:
3841:
3835:
3829:
3823:
3799:Vaughan Pratt, "
3797:
3788:
3783:. Reprinted in
3770:
3764:
3738:
3732:
3731:
3729:
3715:
3704:
3698:
3695:
3670:(7): 1427â1443.
3658:
3620:
3614:
3613:
3593:
3587:
3578:
3572:
3566:
3557:
3551:
3545:
3531:
3525:
3519:
3506:
3500:
3494:
3488:
3482:
3476:
3470:
3464:
3458:
3452:
3446:
3440:
3431:
3430:
3422:
3416:
3410:
3401:
3395:
3389:
3383:
3372:
3331:
3325:
3319:
3313:
3307:
3301:
3295:
3289:
3280:
3264:
3246:
3219:Quasi-set theory
3069:
3067:
3066:
3061:
2982: — a
2974:
2972:
2971:
2966:
2958:
2957:
2938:
2933:
2915:
2911:
2910:
2909:
2899:
2894:
2802:Spectral theorem
2782:
2780:
2779:
2774:
2766:
2765:
2749:
2744:
2726:
2722:
2721:
2720:
2710:
2705:
2543:=0). Meanwhile,
2525:corresponds to |
2318:and momentum by
2227:More generally,
2190:, the equations
2096:1. From there,
1950:
1916:
1914:
1913:
1908:
1872:
1871:
1870:
1781:
1779:
1778:
1773:
1771:
1768:
1766:
1765:
1755:
1750:
1726:
1725:
1655:
1642:infinitary logic
1608:pointwise limits
1590:"Measurement of
1528:formulations of
1482:sequent calculus
1335:categorification
1324:
1266:hidden variables
1249:Hans Reichenbach
1242:Constantin Piron
1234:linear subspaces
1095:
1085:
1075:
1073:
1072:
1068:
1062:in the interval
995:distributive law
960:distributive law
912:Garrett Birkhoff
876:
869:
862:
503:Superdeterminism
156:Braâket notation
107:
105:
104:
99:
91:
86:
85:
77:
65:
60:
58:
47:
19:
18:
5628:
5627:
5623:
5622:
5621:
5619:
5618:
5617:
5588:
5587:
5586:
5581:
5557:
5494:
5405:finite automata
5317:
5312:
5282:
5277:
5259:
5245:Wigner's friend
5221:
5212:Quantum gravity
5173:
5159:Quantum sensing
5139:Quantum network
5119:Quantum machine
5089:Quantum imaging
5052:Quantum circuit
5047:Quantum channel
5006:
4952:Quantum biology
4938:
4914:ElitzurâVaidman
4889:DavissonâGermer
4872:
4824:Hidden-variable
4814:de BroglieâBohm
4791:Interpretations
4785:
4734:
4688:
4575:Complementarity
4553:
4510:
4505:
4401:
4388:ACM SIGACT News
4370:Norman Megill,
4360:Trends in Logic
4298:
4215:
4142:10.2307/1968621
4085:
4080:
4075:
4058:
4054:
4005:
4001:
3959:
3946:
3943:
3939:
3899:
3893:
3889:
3881:
3877:
3854:
3850:
3842:
3838:
3830:
3826:
3798:
3791:
3771:
3767:
3739:
3735:
3713:
3705:
3701:
3661:
3624:
3621:
3617:
3604:(99): 221â248.
3594:
3590:
3579:
3575:
3567:
3560:
3552:
3548:
3532:
3528:
3520:
3509:
3501:
3497:
3489:
3485:
3477:
3473:
3465:
3461:
3453:
3449:
3441:
3434:
3423:
3419:
3411:
3404:
3396:
3392:
3384:
3375:
3332:
3328:
3320:
3316:
3308:
3304:
3296:
3292:
3281:
3277:
3273:
3268:
3267:
3256:feasible. See
3247:
3243:
3238:
3233:
3224:SolĂšr's theorem
3163:
3131:Boolean algebra
3107:
3085:
3022:
3019:
3018:
2953:
2949:
2934:
2923:
2905:
2901:
2895:
2884:
2879:
2875:
2869:
2866:
2865:
2856:
2850:
2832:
2824:Main articles:
2822:
2761:
2757:
2745:
2734:
2716:
2712:
2706:
2695:
2690:
2686:
2681:
2678:
2677:
2668:
2662:
2649:
2643:
2614:
2609:
2602:
2594:
2590:
2583:
2579:
2573:, and nonzero |
2572:
2565:
2554:
2542:
2535:
2531:
2512:
2508:
2501:
2497:
2471:
2464:
2460:
2445:
2438:
2389:are false, so (
2359:compact support
2277:
2180:
2063:
2052:
2046:
2037:
2030:
2023:
2016:
2006:of elements of
2005:
1999:
1955:Measurement of
1940:
1866:
1865:
1861:
1844:
1841:
1840:
1787:
1767:
1761:
1757:
1751:
1740:
1721:
1717:
1703:
1700:
1699:
1693:
1683:
1654:
1652:
1644:
1612:Boolean algebra
1522:
1506:spectral theory
1502:
1445:)∨¬(
1354:
1325:
1316:
1315:
1291:classical logic
1240:Hilbert space,
1179:
1093:
1084:[â1, 1]
1083:
1070:
1066:
1065:
1063:
988:classical logic
984:
945:epistemological
920:Boolean algebra
880:
851:
850:
849:
614:
606:
605:
551:
550:Advanced topics
543:
542:
541:
493:Hidden-variable
483:de BroglieâBohm
462:
460:Interpretations
452:
451:
450:
420:
412:
411:
410:
368:
360:
359:
358:
325:
281:CHSH inequality
270:
262:
261:
260:
189:Complementarity
183:
175:
174:
173:
141:
112:
87:
76:
75:
61:
51:
46:
38:
35:
34:
17:
12:
11:
5:
5626:
5616:
5615:
5610:
5605:
5600:
5583:
5582:
5580:
5579:
5566:
5563:
5562:
5559:
5558:
5556:
5555:
5550:
5545:
5540:
5535:
5534:
5533:
5523:
5518:
5513:
5508:
5502:
5500:
5496:
5495:
5493:
5492:
5487:
5482:
5477:
5472:
5467:
5465:neural network
5462:
5457:
5452:
5447:
5442:
5437:
5432:
5427:
5422:
5417:
5412:
5407:
5402:
5397:
5392:
5387:
5386:
5385:
5375:
5370:
5365:
5360:
5355:
5350:
5345:
5340:
5334:
5332:
5323:
5319:
5318:
5311:
5310:
5303:
5296:
5288:
5279:
5278:
5276:
5275:
5264:
5261:
5260:
5258:
5257:
5252:
5247:
5242:
5241:
5240:
5229:
5227:
5223:
5222:
5220:
5219:
5214:
5209:
5208:
5207:
5197:
5192:
5190:Casimir effect
5187:
5181:
5179:
5175:
5174:
5172:
5171:
5166:
5161:
5156:
5151:
5149:Quantum optics
5146:
5141:
5136:
5131:
5126:
5121:
5116:
5111:
5106:
5101:
5096:
5091:
5086:
5081:
5076:
5071:
5070:
5069:
5059:
5054:
5049:
5044:
5043:
5042:
5032:
5027:
5022:
5016:
5014:
5008:
5007:
5005:
5004:
4999:
4994:
4989:
4984:
4979:
4974:
4969:
4964:
4959:
4954:
4948:
4946:
4940:
4939:
4937:
4936:
4931:
4926:
4924:Quantum eraser
4921:
4916:
4911:
4906:
4901:
4896:
4891:
4886:
4880:
4878:
4874:
4873:
4871:
4870:
4865:
4860:
4855:
4850:
4845:
4840:
4839:
4838:
4837:
4836:
4821:
4816:
4811:
4806:
4801:
4795:
4793:
4787:
4786:
4784:
4783:
4778:
4773:
4768:
4763:
4758:
4753:
4748:
4742:
4740:
4736:
4735:
4733:
4732:
4727:
4722:
4717:
4712:
4707:
4702:
4696:
4694:
4690:
4689:
4687:
4686:
4685:
4684:
4679:
4669:
4664:
4659:
4654:
4649:
4644:
4639:
4634:
4629:
4624:
4619:
4614:
4609:
4608:
4607:
4602:
4597:
4592:
4582:
4580:Density matrix
4577:
4572:
4567:
4561:
4559:
4555:
4554:
4552:
4551:
4546:
4541:
4536:
4535:
4534:
4524:
4518:
4516:
4512:
4511:
4504:
4503:
4496:
4489:
4481:
4475:
4474:
4462:
4448:
4446:
4445:
4426:
4409:
4400:
4397:
4396:
4395:
4380:
4367:
4351:
4340:Quantum Logics
4332:
4319:
4297:
4294:
4293:
4292:
4273:
4258:
4229:
4214:
4211:
4210:
4209:
4198:
4187:
4176:
4164:
4146:
4118:
4092:
4091:
4084:
4081:
4079:
4076:
4074:
4073:
4052:
4015:(2): 239â316.
3999:
3997:
3996:
3968:(2): 171â201.
3957:
3951:(in Russian).
3937:
3910:(6): 727â734.
3887:
3875:
3848:
3836:
3824:
3789:
3765:
3733:
3727:10.1.1.88.9045
3699:
3697:
3696:
3659:
3633:(1): 103â113.
3615:
3588:
3573:
3558:
3546:
3526:
3507:
3505:, p. 174.
3495:
3483:
3471:
3459:
3447:
3432:
3417:
3402:
3390:
3373:
3371:
3370:
3365:
3352:
3351:
3345:
3326:
3314:
3302:
3290:
3274:
3272:
3269:
3266:
3265:
3240:
3239:
3237:
3234:
3232:
3231:
3226:
3221:
3216:
3211:
3206:
3201:
3196:
3191:
3186:
3181:
3176:
3170:
3164:
3162:
3159:
3147:deep inference
3106:
3103:
3084:
3081:
3080:
3079:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
2990:1. Formally,
2980:density matrix
2976:
2975:
2964:
2961:
2956:
2952:
2948:
2945:
2942:
2937:
2932:
2929:
2926:
2922:
2918:
2914:
2908:
2904:
2898:
2893:
2890:
2887:
2883:
2878:
2873:
2852:
2846:
2821:
2818:
2784:
2783:
2772:
2769:
2764:
2760:
2756:
2753:
2748:
2743:
2740:
2737:
2733:
2729:
2725:
2719:
2715:
2709:
2704:
2701:
2698:
2694:
2689:
2685:
2664:
2658:
2645:
2639:
2613:
2610:
2608:
2605:
2600:
2592:
2588:
2581:
2577:
2570:
2563:
2552:
2540:
2533:
2529:
2510:
2506:
2499:
2495:
2469:
2462:
2458:
2443:
2436:
2381:and similarly
2347:
2346:
2340:
2334:
2276:
2273:
2272:
2271:
2214:
2213:
2203:
2179:
2176:
2168:
2167:
2138:
2120:
2062:
2059:
2048:
2042:
2035:
2028:
2021:
2014:
2001:
1995:
1985:
1984:
1961:
1960:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1884:
1881:
1878:
1875:
1869:
1864:
1860:
1857:
1854:
1851:
1848:
1826:Borel function
1786:
1783:
1764:
1760:
1754:
1749:
1746:
1743:
1739:
1735:
1732:
1729:
1724:
1720:
1716:
1713:
1710:
1707:
1685:
1679:
1650:
1648:
1604:
1603:
1521:
1518:
1501:
1498:
1463:
1462:
1412:
1411:
1393:
1378:
1367:
1363:= ¬¬
1353:
1350:
1312:
1178:
1175:
1174:
1173:
1120:
1119:
1098:
1097:
1094:[1, 3]
1087:
1077:
1030:
1029:
983:
980:
908:quantum theory
882:
881:
879:
878:
871:
864:
856:
853:
852:
848:
847:
842:
837:
832:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
752:
747:
742:
737:
732:
727:
722:
717:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
647:
642:
637:
632:
627:
622:
616:
615:
612:
611:
608:
607:
604:
603:
598:
593:
588:
586:Density matrix
583:
578:
573:
568:
563:
558:
552:
549:
548:
545:
544:
540:
539:
534:
529:
524:
519:
514:
509:
508:
507:
506:
505:
490:
485:
480:
475:
470:
464:
463:
458:
457:
454:
453:
449:
448:
443:
438:
433:
428:
422:
421:
418:
417:
414:
413:
409:
408:
403:
398:
393:
388:
383:
377:
376:
375:
369:
366:
365:
362:
361:
357:
356:
351:
346:
340:
339:
338:
337:
336:
334:Delayed-choice
329:Quantum eraser
324:
323:
318:
313:
308:
303:
298:
293:
288:
283:
278:
272:
271:
268:
267:
264:
263:
259:
258:
257:
256:
246:
241:
236:
231:
226:
221:
219:Quantum number
216:
211:
206:
201:
196:
191:
185:
184:
181:
180:
177:
176:
172:
171:
166:
160:
159:
158:
153:
148:
142:
139:
138:
135:
134:
133:
132:
127:
122:
114:
113:
108:
97:
94:
90:
83:
80:
74:
71:
68:
64:
57:
54:
50:
45:
42:
31:
30:
24:
23:
15:
9:
6:
4:
3:
2:
5625:
5614:
5611:
5609:
5606:
5604:
5601:
5599:
5596:
5595:
5593:
5578:
5577:
5568:
5567:
5564:
5554:
5551:
5549:
5546:
5544:
5541:
5539:
5536:
5532:
5531:Plasma window
5529:
5528:
5527:
5524:
5522:
5519:
5517:
5514:
5512:
5509:
5507:
5504:
5503:
5501:
5497:
5491:
5490:teleportation
5488:
5486:
5483:
5481:
5478:
5476:
5473:
5471:
5468:
5466:
5463:
5461:
5458:
5456:
5453:
5451:
5448:
5446:
5443:
5441:
5438:
5436:
5433:
5431:
5428:
5426:
5423:
5421:
5418:
5416:
5413:
5411:
5408:
5406:
5403:
5401:
5398:
5396:
5393:
5391:
5388:
5384:
5381:
5380:
5379:
5376:
5374:
5371:
5369:
5366:
5364:
5361:
5359:
5356:
5354:
5351:
5349:
5346:
5344:
5341:
5339:
5336:
5335:
5333:
5331:
5327:
5324:
5320:
5316:
5309:
5304:
5302:
5297:
5295:
5290:
5289:
5286:
5274:
5266:
5265:
5262:
5256:
5253:
5251:
5248:
5246:
5243:
5239:
5236:
5235:
5234:
5231:
5230:
5228:
5224:
5218:
5215:
5213:
5210:
5206:
5203:
5202:
5201:
5198:
5196:
5193:
5191:
5188:
5186:
5183:
5182:
5180:
5176:
5170:
5167:
5165:
5162:
5160:
5157:
5155:
5152:
5150:
5147:
5145:
5142:
5140:
5137:
5135:
5132:
5130:
5127:
5125:
5122:
5120:
5117:
5115:
5112:
5110:
5109:Quantum logic
5107:
5105:
5102:
5100:
5097:
5095:
5092:
5090:
5087:
5085:
5082:
5080:
5077:
5075:
5072:
5068:
5065:
5064:
5063:
5060:
5058:
5055:
5053:
5050:
5048:
5045:
5041:
5038:
5037:
5036:
5033:
5031:
5028:
5026:
5023:
5021:
5018:
5017:
5015:
5013:
5009:
5003:
5000:
4998:
4995:
4993:
4990:
4988:
4985:
4983:
4980:
4978:
4975:
4973:
4970:
4968:
4965:
4963:
4962:Quantum chaos
4960:
4958:
4955:
4953:
4950:
4949:
4947:
4945:
4941:
4935:
4932:
4930:
4929:SternâGerlach
4927:
4925:
4922:
4920:
4917:
4915:
4912:
4910:
4907:
4905:
4902:
4900:
4897:
4895:
4892:
4890:
4887:
4885:
4882:
4881:
4879:
4875:
4869:
4866:
4864:
4863:Transactional
4861:
4859:
4856:
4854:
4853:Quantum logic
4851:
4849:
4846:
4844:
4841:
4835:
4832:
4831:
4830:
4827:
4826:
4825:
4822:
4820:
4817:
4815:
4812:
4810:
4807:
4805:
4802:
4800:
4797:
4796:
4794:
4792:
4788:
4782:
4779:
4777:
4774:
4772:
4769:
4767:
4764:
4762:
4759:
4757:
4754:
4752:
4749:
4747:
4744:
4743:
4741:
4737:
4731:
4728:
4726:
4723:
4721:
4718:
4716:
4713:
4711:
4708:
4706:
4703:
4701:
4698:
4697:
4695:
4691:
4683:
4680:
4678:
4675:
4674:
4673:
4672:Wave function
4670:
4668:
4665:
4663:
4660:
4658:
4655:
4653:
4650:
4648:
4647:Superposition
4645:
4643:
4642:Quantum state
4640:
4638:
4635:
4633:
4630:
4628:
4625:
4623:
4620:
4618:
4615:
4613:
4610:
4606:
4603:
4601:
4598:
4596:
4595:Excited state
4593:
4591:
4588:
4587:
4586:
4583:
4581:
4578:
4576:
4573:
4571:
4568:
4566:
4563:
4562:
4560:
4556:
4550:
4547:
4545:
4542:
4540:
4537:
4533:
4530:
4529:
4528:
4525:
4523:
4520:
4519:
4517:
4513:
4509:
4502:
4497:
4495:
4490:
4488:
4483:
4482:
4479:
4471:
4467:
4463:
4460:
4458:
4453:
4452:Quantum Logic
4449:
4443:
4439:
4435:
4431:
4427:
4424:
4423:
4418:
4417:
4415:
4410:
4407:
4403:
4402:
4393:
4389:
4385:
4381:
4377:
4373:
4368:
4365:
4361:
4357:
4352:
4349:
4345:
4341:
4337:
4333:
4329:
4325:
4320:
4317:
4313:
4309:
4305:
4300:
4299:
4289:
4288:
4283:
4279:
4274:
4270:
4269:
4264:
4259:
4255:
4254:9780521012546
4251:
4248:
4244:
4240:
4239:
4238:Hilary Putnam
4234:
4230:
4226:
4222:
4217:
4216:
4206:
4203:
4199:
4195:
4192:
4188:
4184:
4181:
4177:
4173:
4169:
4165:
4161:
4157:
4156:
4151:
4147:
4143:
4139:
4135:
4131:
4127:
4123:
4119:
4115:
4111:
4107:
4103:
4099:
4094:
4093:
4090:
4087:
4086:
4070:
4066:
4062:
4056:
4048:
4044:
4040:
4036:
4032:
4028:
4023:
4018:
4014:
4010:
4003:
3993:
3989:
3985:
3981:
3976:
3971:
3967:
3963:
3958:
3955:(5): 345â360.
3954:
3950:
3945:
3944:
3941:
3933:
3929:
3925:
3921:
3917:
3913:
3909:
3905:
3898:
3891:
3884:
3883:Kalmbach 1981
3879:
3872:
3868:
3864:
3860:
3859:
3852:
3845:
3840:
3833:
3828:
3821:
3817:
3813:
3811:
3806:
3802:
3796:
3794:
3786:
3782:
3778:
3774:
3769:
3762:
3758:
3754:
3750:
3746:
3742:
3737:
3728:
3723:
3719:
3712:
3711:
3703:
3693:
3689:
3685:
3681:
3677:
3673:
3669:
3665:
3660:
3656:
3652:
3648:
3644:
3640:
3636:
3632:
3628:
3623:
3622:
3619:
3611:
3607:
3603:
3599:
3592:
3586:
3585:Kalmbach 1983
3582:
3581:Kalmbach 1974
3577:
3570:
3565:
3563:
3555:
3550:
3544:(blog), 2021.
3543:
3539:
3535:
3530:
3523:
3518:
3516:
3514:
3512:
3504:
3499:
3492:
3487:
3480:
3475:
3468:
3463:
3456:
3451:
3444:
3439:
3437:
3428:
3421:
3414:
3409:
3407:
3399:
3394:
3387:
3382:
3380:
3378:
3369:
3366:
3363:
3359:
3355:
3354:
3349:
3346:
3343:
3339:
3335:
3334:
3330:
3323:
3318:
3311:
3306:
3299:
3294:
3287:
3286:
3279:
3275:
3262:
3259:
3255:
3251:
3245:
3241:
3230:
3227:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3205:
3202:
3200:
3197:
3195:
3192:
3190:
3187:
3185:
3182:
3180:
3177:
3174:
3173:HPO formalism
3171:
3169:
3166:
3165:
3158:
3156:
3152:
3148:
3144:
3140:
3136:
3132:
3128:
3124:
3120:
3115:
3113:
3102:
3099:
3097:
3094:
3090:
3077:
3073:
3054:
3051:
3045:
3042:
3039:
3033:
3027:
3016:
3012:
3008:
3004:
3000:
2996:
2993:
2992:
2991:
2989:
2985:
2981:
2962:
2954:
2950:
2943:
2930:
2927:
2924:
2920:
2916:
2912:
2906:
2902:
2891:
2888:
2885:
2881:
2876:
2864:
2863:
2862:
2860:
2855:
2849:
2845:
2841:
2837:
2831:
2827:
2817:
2815:
2811:
2807:
2803:
2799:
2795:
2793:
2789:
2770:
2762:
2758:
2751:
2741:
2738:
2735:
2731:
2727:
2723:
2717:
2713:
2702:
2699:
2696:
2692:
2687:
2683:
2676:
2675:
2674:
2672:
2667:
2661:
2657:
2653:
2648:
2642:
2638:
2634:
2630:
2626:
2622:
2619:
2604:
2598:
2587:
2576:
2569:
2562:
2558:
2550:
2546:
2539:
2528:
2524:
2520:
2516:
2505:
2494:
2490:
2486:
2482:
2478:
2473:
2468:
2457:
2453:
2449:
2442:
2435:
2430:
2428:
2424:
2420:
2416:
2412:
2408:
2404:
2400:
2396:
2392:
2388:
2384:
2380:
2376:
2372:
2368:
2364:
2360:
2356:
2352:
2344:
2341:
2338:
2335:
2332:
2328:
2325:
2324:
2323:
2321:
2317:
2312:
2310:
2306:
2302:
2298:
2294:
2290:
2286:
2282:
2269:
2265:
2261:
2257:
2253:
2249:
2245:
2244:
2243:
2241:
2237:
2234:
2230:
2225:
2223:
2219:
2212:
2208:
2204:
2201:
2197:
2193:
2192:
2191:
2189:
2185:
2175:
2173:
2172:SolĂšr theorem
2165:
2161:
2158:
2154:
2150:
2146:
2142:
2139:
2136:
2132:
2128:
2124:
2121:
2118:
2114:
2110:
2106:
2102:
2099:
2098:
2097:
2095:
2091:
2087:
2083:
2079:
2075:
2072:
2068:
2058:
2056:
2051:
2045:
2041:
2034:
2027:
2020:
2013:
2009:
2004:
1998:
1994:
1990:
1982:
1978:
1974:
1970:
1969:
1968:
1966:
1958:
1954:
1953:
1952:
1948:
1944:
1938:
1934:
1930:
1926:
1922:
1917:
1904:
1898:
1892:
1886:
1879:
1873:
1862:
1858:
1852:
1846:
1838:
1834:
1830:
1827:
1823:
1819:
1816:, which is a
1815:
1811:
1807:
1803:
1800:
1796:
1792:
1791:Hilbert space
1782:
1762:
1758:
1747:
1744:
1741:
1737:
1733:
1722:
1718:
1708:
1705:
1697:
1692:
1688:
1682:
1678:
1674:
1670:
1666:
1662:
1657:
1647:
1643:
1638:
1636:
1632:
1628:
1624:
1621:
1617:
1616:Borel subsets
1613:
1609:
1601:
1597:
1593:
1589:
1588:
1587:
1585:
1580:
1578:
1574:
1570:
1566:
1562:
1558:
1555:
1551:
1547:
1543:
1539:
1535:
1531:
1527:
1517:
1515:
1511:
1507:
1497:
1495:
1491:
1487:
1483:
1479:
1474:
1472:
1468:
1460:
1456:
1452:
1448:
1444:
1441:∨¬
1440:
1436:
1435:
1434:
1432:
1427:
1425:
1421:
1417:
1409:
1405:
1401:
1397:
1394:
1391:
1387:
1384:∨¬
1383:
1379:
1376:
1372:
1368:
1366:
1362:
1359:
1358:
1357:
1349:
1347:
1343:
1338:
1336:
1332:
1322:
1321:
1320:Hilary Putnam
1310:
1308:
1304:
1300:
1296:
1292:
1287:
1284:
1279:
1275:
1271:
1267:
1263:
1258:
1257:Hilary Putnam
1254:
1250:
1245:
1243:
1239:
1235:
1232:
1228:
1224:
1220:
1216:
1212:
1211:George Mackey
1208:
1206:
1205:quantum logic
1202:
1198:
1197:Hilbert space
1194:
1190:
1186:
1185:
1172:
1168:
1164:
1160:
1156:
1152:
1151:
1150:
1148:
1144:
1140:
1136:
1132:
1127:
1125:
1124:superposition
1118:
1114:
1110:
1106:
1103:
1102:
1101:
1091:
1088:
1081:
1078:
1061:
1057:
1054:
1053:
1052:
1050:
1045:
1043:
1039:
1035:
1027:
1023:
1019:
1015:
1011:
1007:
1003:
1000:
999:
998:
996:
993:
992:propositional
989:
979:
977:
973:
969:
965:
961:
956:
954:
950:
946:
942:
938:
937:Hilary Putnam
933:
931:
930:
923:
921:
917:
913:
909:
905:
901:
900:quantum logic
897:
893:
889:
877:
872:
870:
865:
863:
858:
857:
855:
854:
846:
843:
841:
838:
836:
833:
831:
828:
826:
823:
821:
818:
816:
813:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
786:
783:
781:
778:
776:
773:
771:
768:
766:
763:
761:
758:
756:
753:
751:
748:
746:
743:
741:
738:
736:
733:
731:
728:
726:
723:
721:
718:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
617:
610:
609:
602:
599:
597:
594:
592:
589:
587:
584:
582:
579:
577:
576:Quantum chaos
574:
572:
569:
567:
564:
562:
559:
557:
554:
553:
547:
546:
538:
535:
533:
532:Transactional
530:
528:
525:
523:
522:Quantum logic
520:
518:
515:
513:
510:
504:
501:
500:
499:
496:
495:
494:
491:
489:
486:
484:
481:
479:
476:
474:
471:
469:
466:
465:
461:
456:
455:
447:
444:
442:
439:
437:
434:
432:
429:
427:
424:
423:
416:
415:
407:
404:
402:
399:
397:
394:
392:
389:
387:
384:
382:
379:
378:
374:
371:
370:
364:
363:
355:
352:
350:
347:
345:
342:
341:
335:
332:
331:
330:
327:
326:
322:
319:
317:
314:
312:
309:
307:
304:
302:
299:
297:
294:
292:
289:
287:
284:
282:
279:
277:
274:
273:
266:
265:
255:
252:
251:
250:
249:Wave function
247:
245:
242:
240:
237:
235:
232:
230:
229:Superposition
227:
225:
222:
220:
217:
215:
212:
210:
207:
205:
202:
200:
197:
195:
192:
190:
187:
186:
179:
178:
170:
167:
165:
162:
161:
157:
154:
152:
149:
147:
144:
143:
137:
136:
131:
128:
126:
123:
121:
118:
117:
116:
115:
111:
78:
72:
55:
52:
48:
40:
33:
32:
29:
26:
25:
21:
20:
5574:
5511:Anti-gravity
5455:metamaterial
5429:
5383:post-quantum
5378:cryptography
5108:
4992:Quantum mind
4904:FranckâHertz
4852:
4746:KleinâGordon
4700:Formulations
4693:Formulations
4622:Interference
4612:Entanglement
4590:Ground state
4585:Energy level
4558:Fundamentals
4522:Introduction
4469:
4456:
4429:
4420:
4413:
4405:
4387:
4359:
4343:
4327:
4307:
4285:
4266:
4236:
4224:
4204:
4193:
4182:
4171:
4153:
4133:
4110:1955 edition
4106:j.ctt1wq8zhp
4097:
4088:
4067:. HAL
4060:
4055:
4022:math/0606118
4012:
4008:
4002:
3975:math/0512362
3965:
3961:
3952:
3948:
3940:
3907:
3903:
3890:
3878:
3856:
3851:
3839:
3827:
3819:
3809:
3804:
3784:
3776:
3768:
3760:
3748:
3736:
3709:
3702:
3667:
3663:
3630:
3626:
3618:
3601:
3597:
3591:
3576:
3549:
3541:
3529:
3503:Maudlin 2005
3498:
3486:
3474:
3467:Maudlin 2005
3462:
3450:
3426:
3420:
3393:
3386:Maudlin 2005
3361:
3337:
3329:
3317:
3305:
3293:
3283:
3278:
3263:for details.
3260:
3253:
3244:
3229:Vector logic
3179:Linear logic
3151:linear logic
3149:fragment of
3116:
3108:
3100:
3095:
3089:linear logic
3086:
3075:
3071:
3014:
3006:
3002:
2998:
2994:
2977:
2858:
2853:
2847:
2843:
2839:
2835:
2833:
2813:
2809:
2805:
2797:
2796:
2791:
2785:
2670:
2665:
2659:
2655:
2651:
2646:
2640:
2636:
2632:
2628:
2620:
2615:
2596:
2585:
2574:
2567:
2560:
2556:
2548:
2544:
2537:
2526:
2522:
2518:
2514:
2503:
2492:
2488:
2484:
2480:
2476:
2474:
2466:
2455:
2451:
2447:
2440:
2433:
2431:
2426:
2418:
2414:
2410:
2406:
2402:
2398:
2394:
2390:
2386:
2382:
2378:
2374:
2373:â„ 0. Thus,
2370:
2367:normalizable
2348:
2342:
2336:
2330:
2326:
2319:
2315:
2313:
2304:
2300:
2296:
2292:
2288:
2284:
2280:
2278:
2267:
2263:
2262:) = â€, then
2259:
2255:
2251:
2247:
2239:
2226:
2221:
2217:
2215:
2210:
2206:
2199:
2195:
2187:
2183:
2181:
2169:
2163:
2159:
2152:
2148:
2144:
2140:
2134:
2130:
2126:
2122:
2116:
2112:
2108:
2100:
2089:
2081:
2064:
2049:
2043:
2039:
2032:
2025:
2018:
2011:
2007:
2002:
1996:
1992:
1988:
1986:
1980:
1976:
1972:
1964:
1962:
1956:
1946:
1942:
1936:
1928:
1924:
1920:
1918:
1836:
1832:
1828:
1821:
1809:
1805:
1801:
1788:
1690:
1686:
1680:
1676:
1672:
1668:
1664:
1660:
1658:
1645:
1639:
1605:
1599:
1595:
1591:
1584:propositions
1581:
1576:
1572:
1568:
1564:
1560:
1556:
1549:
1545:
1523:
1503:
1475:
1464:
1458:
1454:
1450:
1446:
1442:
1438:
1428:
1413:
1407:
1403:
1399:
1395:
1389:
1385:
1381:
1364:
1360:
1355:
1339:
1327:
1318:
1294:
1288:
1247:Inspired by
1246:
1222:
1214:
1209:
1204:
1200:
1182:
1180:
1170:
1166:
1162:
1158:
1154:
1142:
1138:
1134:
1130:
1128:
1121:
1116:
1112:
1108:
1104:
1099:
1089:
1079:
1055:
1046:
1041:
1037:
1033:
1031:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
985:
982:Introduction
957:
953:linear logic
934:
927:
924:
904:propositions
899:
894:analysis of
885:
521:
431:KleinâGordon
367:Formulations
204:Energy level
199:Entanglement
182:Fundamentals
169:Interference
120:Introduction
5526:Force field
5475:programming
5435:logic clock
5420:information
5395:electronics
5250:EPR paradox
5030:Quantum bus
4899:Double-slit
4877:Experiments
4843:Many-worlds
4781:Schrödinger
4730:Phase space
4720:Schrödinger
4710:Interaction
4667:Uncertainty
4637:Nonlocality
4632:Measurement
4627:Decoherence
4617:Hamiltonian
4314:arXiv
4233:Tim Maudlin
4202:G. Kalmbach
4191:G. Kalmbach
4180:G. Kalmbach
4122:G. Birkhoff
4009:SIAM Review
3816:Revision 42
3718:SUNY Albany
3569:Megill 2019
3398:Putnam 1969
3368:Ludwig 1954
3360:" II,
3322:Mackey 1963
3168:Fuzzy logic
3105:Limitations
3093:modal logic
3011:trace class
1538:observables
1526:Hamiltonian
1375:associative
1371:commutative
1369:∨ is
1303:Tim Maudlin
1193:projections
1191:noted that
968:observables
820:von Neumann
805:Schrödinger
581:EPR paradox
512:Many-worlds
446:Schrödinger
401:Schrödinger
396:Phase-space
386:Interaction
291:Double-slit
269:Experiments
244:Uncertainty
214:Nonlocality
209:Measurement
194:Decoherence
164:Hamiltonian
5592:Categories
5440:logic gate
5338:algorithms
5178:Extensions
5012:Technology
4858:Relational
4809:Copenhagen
4705:Heisenberg
4652:Tunnelling
4515:Background
4404:D. Cohen,
4392:cs/0508005
4316:2110.01361
3773:A. Gleason
3753:categories
3542:What's New
3491:Brody 1984
3479:Brody 1984
3348:Piron 1976
3123:connective
3017:such that
2997:. Suppose
2094:eigenvalue
2088:for which
2055:direct sum
1987:The space
1051:is 1) let
815:Sommerfeld
730:Heisenberg
725:Gutzwiller
665:de Broglie
613:Scientists
527:Relational
478:Copenhagen
381:Heisenberg
239:Tunnelling
140:Background
5485:simulator
5373:computing
5343:amplifier
4884:Bell test
4739:Equations
4565:Born rule
4168:H. Putnam
4150:G. Mackey
3932:123383431
3722:CiteSeerX
3692:189850106
3655:123183879
3534:Terry Tao
3271:Citations
3046:
3028:
3013:operator
2944:
2936:∞
2921:∑
2897:∞
2882:⋁
2752:φ
2747:∞
2732:∑
2708:∞
2693:⋃
2684:φ
2429:is true.
2417:) equals
2157:superpose
2071:separable
1965:Axiom VII
1899:λ
1893:
1880:λ
1863:∫
1795:unbounded
1753:∞
1738:⋃
1709:
1625:(such as
1620:classical
1494:decidable
1238:separable
1064:[0, +
976:formalism
845:Zeilinger
690:Ehrenfest
419:Equations
96:⟩
93:Ψ
82:^
70:⟩
67:Ψ
44:ℏ
5390:dynamics
5273:Category
5067:Timeline
4819:Ensemble
4799:Bayesian
4761:Majorana
4677:Collapse
4549:Glossary
4532:Timeline
4466:C. Piron
4376:Metamath
4069:01092279
4047:10435983
3610:44084050
3353:Ludwig:
3161:See also
3139:Belavkin
3127:monotone
3125:that is
3091:and the
2616:Given a
2450:†0 and
1919:In case
1542:dynamics
1486:tableaux
1388:for any
1060:momentum
892:physical
890:and the
770:Millikan
695:Einstein
680:Davisson
635:Blackett
620:Aharonov
488:Ensemble
468:Bayesian
373:Overview
254:Collapse
234:Symmetry
125:Glossary
5480:sensing
5460:network
5445:machine
5415:imaging
5363:circuit
5358:channel
5330:Quantum
5226:Related
5205:History
4944:Science
4776:Rydberg
4527:History
4454:at the
4378:, 2019.
4284:(ed.).
4140:. DOI
4138:1968621
4078:Sources
4027:Bibcode
3992:3909067
3912:Bibcode
3672:Bibcode
3635:Bibcode
3333:Piron:
3261:et seq.
2995:Theorem
2798:Theorem
2536:=0 and
2345:â x â„ 0
2339:â x †0
2209:∧
2198:∨
2119:) = 0),
2103:is the
2074:Hilbert
1789:In the
1653:,ω
1471:falsity
1453:) then
1449:∨
1402:∨
1307:problem
1227:lattice
1223:defined
1137:" and "
1069:⁄
886:In the
810:Simmons
800:Rydberg
765:Moseley
745:Kramers
735:Hilbert
720:Glauber
715:Feynman
700:Everett
670:Compton
441:Rydberg
130:History
5470:optics
5322:Fields
4919:Popper
4440:
4358:", in
4342:", in
4252:
4223:", in
4104:
4045:
3990:
3930:
3869:
3803:," in
3724:
3690:
3653:
3608:
3121:; any
2804:). If
2673:) and
2363:entire
2270:) = â€.
2024:means
1812:has a
1649:ω
1534:states
1516:case.
1276:, but
1231:closed
1161:) or (
1020:) or (
972:states
840:Zeeman
835:Wigner
785:Planck
755:Landau
740:Jordan
391:Matrix
321:Popper
5499:Other
5430:logic
4829:Local
4771:Pauli
4751:Dirac
4280:. In
4102:JSTOR
4043:S2CID
4017:arXiv
3988:S2CID
3970:arXiv
3928:S2CID
3900:(PDF)
3714:(PDF)
3688:S2CID
3651:S2CID
3606:JSTOR
3540:" on
3413:Wilce
3236:Notes
2988:trace
2861:then
2654:, {Ï(
2584:and |
2483:) âš (
2397:) âš (
2353:of a
2299:) âš (
2291:) = (
2233:total
2147:= ÂŹ(ÂŹ
2137:, and
2069:in a
1949:]
1941:[
1467:truth
1295:logic
1236:of a
1195:on a
1171:false
1107:and (
1074:]
1012:) = (
1004:and (
795:Raman
780:Pauli
775:Onnes
710:Fermi
685:Debye
675:Dirac
640:Bloch
630:Bethe
498:Local
436:Pauli
426:Dirac
224:State
5576:List
4756:Weyl
4438:ISBN
4250:ISBN
4124:and
3867:ISBN
3743:and
3583:and
3145:, a
2828:and
2439:and
2254:and
2205:â„ =
2194:†=
2162:and
2133:and
2115:, P(
2017:and
1669:join
1667:and
1665:meet
1582:The
1540:and
1524:The
1426:".)
1406:) =
1373:and
1169:) =
1165:and
1157:and
1141:and
1133:and
1117:true
1115:) =
1040:and
1024:and
1016:and
970:and
914:and
830:Wien
825:Weyl
790:Rabi
760:Laue
750:Lamb
705:Fock
660:Bose
655:Born
650:Bohr
645:Bohm
625:Bell
5348:bus
4459:Lab
4436:.
4386:",
4374:at
4326:",
4306:",
4144:.
4132:,"
4128:, "
4108:.
4035:doi
3980:doi
3920:doi
3865:.
3812:Lab
3759:,"
3680:doi
3643:doi
3536:, "
3254:are
3074:in
3005:on
2986:of
2790:on
2631:to
2502:= |
2409:⧠(
2361:is
2329:â |
2283:⧠(
2246:if
2202:and
2107:of
2076:or
1935:of
1831:on
1808:.
1706:LUB
1614:of
1508:of
1484:or
1473:.)
1424:and
1416:not
1268:or
1229:of
1111:or
1008:or
997::
5594::
4468:,
4265:.
4241:;
4170:,
4152:,
4041:.
4033:.
4025:.
4013:51
4011:.
3986:.
3978:.
3966:42
3964:.
3926:.
3918:.
3908:21
3906:.
3902:.
3814:,
3792:^
3720:.
3686:.
3678:.
3668:33
3666:.
3649:.
3641:.
3631:33
3629:.
3602:25
3561:^
3510:^
3435:^
3405:^
3376:^
3157:.
3143:BV
3114:.
3043:Tr
2834:A
2816:.
2794:.
2663:)}
2601:â„1
2599:|âŸ
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2582:â„1
2580:|âŸ
2566:+
2559:=
2553:â„1
2551:|âŸ
2534:â„1
2532:|âŸ
2521:â§
2511:â„1
2509:|âŸ
2500:â„1
2498:|âŸ
2487:â§
2479:â§
2463:â„1
2461:|âŸ
2413:âš
2401:â§
2393:â§
2385:â§
2377:â§
2303:â§
2295:â§
2287:âš
2151:â§ÂŹ
2101:ÂŹp
2057:.
1967::
1656:.
1637:.
1602:."
1598:,
1579:.
1536:,
1496:.
1480:,
1457:=
1420:or
1187:,
1036:,
1028:),
932:.
898:,
5307:e
5300:t
5293:v
4500:e
4493:t
4486:v
4457:n
4444:.
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