## Embedding Quantum Universes in Classical Ones (1999)

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Citations: | 3 - 1 self |

### BibTeX

@MISC{Calude99embeddingquantum,

author = {Cristian S. Calude and Peter H. Hertling and Karl Svozil},

title = {Embedding Quantum Universes in Classical Ones},

year = {1999}

}

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### Abstract

this paper; the propositional structure encountered in the quantum mechanics of spin - state measurements of a spin one-half particle along two directions ( mod p) , that is, the modular, orthocomplemented lattice MO 2 drawn in Fig. 1 ( where p 2 = ( p + ) and q 2 = ( q + ) )

### Citations

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Citation Context ... , Of course rÞ 1, since 1¢ = 0 and 0 Î I. Let r Ï I and r¢ Ï I ( 3) J= In ( r) ( 4) where ( r) = {s Î L | s< r} is the principal ideal of r [note that (r) is indeed an ideal]. Then, under assu=-=mption (3)-=-, using ( 1) above, we have that J is an ideal which properly contains I. This contradicts the maximality of I and ends the proof of the assertion ( 2). For claim ( iii) we have to show the relation f... |

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Citation Context ...d logical functions, then it is possible to give a classical meaning to quantum physical statements, thus giving raise to an "understanding" of quantum mechanics. Quantum logic, according to=-= Birkhoff [5]-=-, Mackey [28], Jauch [21], Kalmbach [23], Pulmannov a [37], identifies logical entities with Hilbert space entities. In particular, elementary propositionssp,q, . . . are associated with closed linear... |

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Citation Context ...novel ``counterintuitive’’ phenomena ( see Refs. 12 and 50) even almost a century after its development ( cf. Refs. 19, 20, and 42). Yet it can be safely stated that quantum theory is not understo=-=od. (10)-=- Indeed, it appears that progress is fostered by abandoning long-held beliefs and concepts rather than by attempts to derive it from some classical basis (Refs. 4, 13, and 18). But just how far might ... |

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Citation Context ...quantum universes in classical theories is a progressive or a degenerative case ( compare Ref. 27). APPENDIX A: PROOF OF THE GEOMETRIC LEMMA In this appendix we prove the geometric lemma due to Piron =-=(36) -=-which was formulated in Sec. 2.2. First let us restate it. Consider a point q in the northern hemisphere of the unit sphere S 2 = { p Î R 3 | I pI = 1}. By C( q) we denote the unique great circle whi... |

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Citation Context ...ical meaning to quantum physical statements, thus giving raise to an “understanding” of quantum mechanics. Quantum logic, according to Birkhoff [5], Mackey [28], Jauch [21], Kalmbach [23], Pulmannová =-=[37]-=-, identifies logical entities with Hilbert space entities. In particular, elementary propositions p,q,... are associated with closed linear subspaces of a Hilbert space through the origin (zero vector... |

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Citation Context ... this approach is a result proven for the first time by Kochen and Specker [26] (cf. also Specker [43], Zierler and Schlessinger [52] and John Bell [2]; see reviews by Mermin [32], Svozil and Tkadlec =-=[48], and a forthcoming -=-monograph by Svozil [46]) stating the impossibility to "complete" quantum physics by introducing noncontextual hidden parameter models. Such a possible "completion" had been sugges... |

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Citation Context ... be restated as Q( p¢ ) = I \ Q( p) I Î Q( p¢ ) iff I Ï Q( p) for all I Î I . But this means p¢ Ï I iff p Î I, which follows directly from condition 1 in the definition of an ideal and from as=-=sertion (2).-=- We proceed to claim ( i), which states that Q is injective, i.e., if pÞ q, then Q( p) Þ Q( q). But pÞ q is equivalent to p< q or q < p. Furthermore, if we can show that there is a maximal ideal ... |

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Citation Context ...od (Feynman [10]). Indeed, it appears that progress is fostered by abandoning long--held beliefs and concepts rather than by attempts to derive it from some classical basis, cf. Greenberger and YaSin =-=[13]-=-, Herzog, Kwiat, Weinfurter and Zeilinger [18] and Bennett [4]. But just how far might a classical understanding of quantum mechanics be, in principle, possible ? We shall attempt an answer to this qu... |

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Citation Context ...gs of quantum logics into classical logics. One physical motivation for this approach is a result proven for the first time by Kochen and Specker [26] (cf. also Specker [43], Zierler and Schlessinger =-=[52] and John -=-Bell [2]; see reviews by Mermin [32], Svozil and Tkadlec [48], and a forthcoming monograph by Svozil [46]) stating the impossibility to "complete" quantum physics by introducing noncontextua... |

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Citation Context ...ality very nicely (cf. Svozil and Tkadlec [48], Svozil [46]), we shall give two geometric arguments which are derived from proof methods for Gleason's theorem (see Piron [36], Cooke, Keane, and Moran =-=[7]-=-, 4 and Kalmbach [24]). Let L be the lattice of closed linear subspaces of the three-dimensional real Hilbert space R 3 . A two-valued probability measure or valuation on L is a map v : L # {0,1} whic... |

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Citation Context ...gress is fostered by abandoning long--held beliefs and concepts rather than by attempts to derive it from some classical basis, cf. Greenberger and YaSin [13], Herzog, Kwiat, Weinfurter and Zeilinger =-=[18]-=- and Bennett [4]. But just how far might a classical understanding of quantum mechanics be, in principle, possible ? We shall attempt an answer to this question in terms of mappings of quantum worlds ... |

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Citation Context ...T , but f (B) #= ( f (E)) # . One needs not be afraid of order-preserving embeddings which are no lattice morphisms, after all. Even automaton logics (see Svozil [47, Chapter 11], Schaller and Svozil =-=[39, 40, 41]-=-, and Dvurecenskij, Pulmannova and Svozil [8]) can be embedded in this way. Take again the lattice MO 2 depicted in Figure 1. A partition (automaton) logic realization is, for instance, {{{1},{2,3}},{... |

13 | K.: Partition logics, orthoalgebras and automata - Dvurečenskij, Pulmannová, et al. - 1995 |

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Citation Context ...nctions, then it is possible to give a classical meaning to quantum physical statements, thus giving raise to an "understanding" of quantum mechanics. Quantum logic, according to Birkhoff [5=-=], Mackey [28]-=-, Jauch [21], Kalmbach [23], Pulmannov a [37], identifies logical entities with Hilbert space entities. In particular, elementary propositionssp,q, . . . are associated with closed linear subspaces of... |

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Citation Context ...f. Svozil and Tkadlec [48], Svozil [46]), we shall give two geometric arguments which are derived from proof methods for Gleason's theorem (see Piron [36], Cooke, Keane, and Moran [7], 4 and Kalmbach =-=[24]-=-). Let L be the lattice of closed linear subspaces of the three-dimensional real Hilbert space R 3 . A two-valued probability measure or valuation on L is a map v : L # {0,1} which maps the zerodimens... |

7 | private communication - Harding - 2000 |

6 | Measures and Hilbert Lattices (World Scientific - Kalmbach - 1986 |

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Citation Context ...ry successful theory which appears to predict novel "counterintuitive" phenomena (see Wheeler [50], Greenberger, Horne and Zeilinger [12]) even almost a century after its development, cf. Sc=-=hrodinger [42]-=-, Jammer [19, 20]. Yet, it can be safely stated that quantum theory is not understood (Feynman [10]). Indeed, it appears that progress is fostered by abandoning long--held beliefs and concepts rather ... |

4 | Quantum mechanics and Hilbert - Mackey - 1957 |

4 |
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Citation Context ...ons are quite trivial as well. Another weakening of (iii) is to restrict oneself to particular physical states and study the embeddability of quantum logics under these constraints; see Bell, Clifton =-=[1]-=-. In the following sections we analyze a completely different option: Is it possible to embed quantum logic into a Boolean algebra when one does not demand preservation of all ortholattice operations?... |

4 |
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(Show Context)
Citation Context ...method of embedding an arbitrary partially ordered set into a concrete orthomodular lattice which in turn can be embedded into a Boolean algebra has been used by Kalmbach [22] and extended by Harding =-=[16]-=- and Mayet and Navara [31]. In these Kalmbach embeddings, as 7 they may be called, the meets and joins are preserved but not the complement. The Kalmbach embedding of some bounded lattice L into a con... |

4 |
Classes of logics representable as kernels of measures, inContr. General Algebra 9
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(Show Context)
Citation Context ...itrary partially ordered set into a concrete orthomodular lattice which in turn can be embedded into a Boolean algebra has been used by Kalmbach [22] and extended by Harding [16] and Mayet and Navara =-=[31]-=-. In these Kalmbach embeddings, as 7 they may be called, the meets and joins are preserved but not the complement. The Kalmbach embedding of some bounded lattice L into a concrete orthomodular lattice... |