## 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

689 | Quantum Theory: Concepts and Methods - Peres - 1993 |

581 | Can Quantum-Mechanical Description of Physical Reality - Einstein, Podolsky, et al. - 1935 |

551 |
Speakable and Unspeakable in Quantum Mechanics
- Bell
- 1987
(Show Context)
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... |

274 | Topology, a First Course - Munkres - 1975 |

264 |
The logic of quantum mechanics
- Birkhoff, Neumann
- 1936
(Show Context)
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... |

222 |
The Feynman lectures on physics
- Feynman
- 1963
(Show Context)
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 ... |

220 | The problem of hidden variables in quantum mechanics - Kochen, Specker - 1967 |

155 | Measures on the closed subspaces of a Hilbert space - Gleason - 1957 |

153 | The Theory of Representations for Boolean Algebras - Stone - 1936 |

150 | Quantum Theory and Measurement - Wheeler, Zurek - 1988 |

145 | On the problem of hidden variables in quantum mechanics - Bell - 1966 |

143 |
Foundations of Quantum Physics
- Piron
- 1976
(Show Context)
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... |

136 | The Philosophy of Quantum Mechanics - Jammer - 1974 |

128 | Die gegenwärtige Situation in der Quantenmechanik.” Naturwissenschaften - Schrödinger - 1935 |

117 | Foundations of quantum mechanics - Jauch - 1968 |

117 | Orthomodular Lattices - Kalmbach - 1983 |

79 | Automaton logic - Schaller, Svozil - 1996 |

68 |
Orthomodular Structures as Quantum Logics
- Pták, Pulmannová
(Show Context)
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... |

55 | The Conceptual Development of Quantum Mechanics - Jammer - 1967 |

53 | The logic of quantum mechanics - Neumann, Birkhoff - 1936 |

36 | Die Logik nicht gleichzeitig entscheidbarer Aussagen. Dialectica - Specker - 1960 |

34 | Two simple proofs of the Kochen-Specker theorem - Peres - 1991 |

29 | Hidden variables and the two theorems of john bell - Mermin - 1993 |

24 | The Conceptual Development of Quantum Mechanics, McGraw-Hill - Jammer - 1966 |

24 |
Greechie diagrams, nonexistence of measures in quantum logics and Kochen–Specker type constructions
- Svozil, Tkadlec
- 1996
(Show Context)
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... |

22 | Multiparticle interferometry and the superposition principle - Greenberger, Horne, et al. - 1993 |

21 |
and Revs. Modern Phys
- Rev
- 1939
(Show Context)
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 ... |

18 | Realism: A Prolegomenon to the Philosophy of Quantum Mechanics - Redhead, Incompleteness - 1987 |

18 |
Haunted” measurements in quantum theory
- Greenberger, YaSin
- 1989
(Show Context)
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... |

18 | The present situation in quantum mechanics: a translation of Schrödinger’s “cat paradox - Trimmer |

17 | Law without law - Wheeler - 1983 |

15 |
Complementarity and the quantum eraser
- Herzog, Kwiat, et al.
- 1995
(Show Context)
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 ... |

15 |
Automaton partition logic versus quantum logic
- Schaller, Svozil
- 1995
(Show Context)
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}},{... |

15 |
Boolean embeddings of orthomodular sets and quantum logic
- Zierler, Schlessinger
- 1965
(Show Context)
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... |

14 |
An Elementay Proof of Gleason’s Theorem
- Cooke, M, et al.
- 1984
(Show Context)
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... |

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

12 | Density conditions for quantum propositions - Havlicek, Svozil - 1996 |

10 | Partition logics of automata. Il Nuovo Cimento 109 B - Schaller, Svozil - 1994 |

9 | Philosophical Papers. 1. The Methodology of Scientific Research Programmes - Lakatos - 1978 |

9 | Getting contextual and non-local elements-of-reality the easy way Am - Clifton - 1993 |

9 |
Quantum mechanics and Hilbert space
- Mackey
- 1957
(Show Context)
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... |

8 |
Measures and Hilbert Lattices
- Kalmbach
- 1986
(Show Context)
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 |

5 | Nature 371 - Bennett - 1994 |

5 |
Schrödinger :‘Die gegenwärtige situation in der quantenmechanik’, Naturwiss
- unknown authors
- 1935
(Show Context)
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 |
QuasiBoolean Algebras and Simultaneously Definite
- Bell, Clifton
- 1995
(Show Context)
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 |
Orthomodular lattices whose MacNeille completions are not orthomodular. Order 8
- HARDING
- 1991
(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
- Mayet, Navara
- 1995
(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... |