Abstract
In the six papers devoted to the foundations of quantum mechanics four subjects are treated. They are distributed over the papers as follows:
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Quantum logic
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The Copenhagen interpretation and Bohm
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Hidden parameters and Bell’s theorem
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EPR-situation and Bell’s inequality In the following introduction [27], being merely an account, is touched on only occasionally.
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References
First published as Scheibe 1985.
Scheibe 1964
v. Neumann 1932, Ch.III.5; Birkhoff and v. Neumann 1936
Putnam 1969, p. 235
Beehner 1980
In the discussion Prof, van Praassen and Prof. Kalmbach informed me about the following result recently obtained by Goldblatt and related to the problem posed in the main text: The class of orthomodular frames is not elementary, indeed not even Δ-elementary. See Goldblatt 1973 and 1984. See also Kalmbach 1983, p.348, problem 29
The anthology Hooker 1975 and 1979 may be taken as representative for the usage in question.
See Monk 1976, p. 160, Theor. 9.60, and Sikorski 1969, p. 44, Theor. 14.4
Sikorski 1969, § 8.
For infinitary classical propositional logic see Karp 1964
Kamber 1965, p. 167, Satz III
Scheibe 1964, Ch. II.1
ibid. p. 90. Here fe(t, A) means that the result A has been obtained at time tby a suitable measurement. UT is an arbitrary element of the dynamical group, and EA is the projection operator associated with the result A.
ibid. p. 113, formula (25)
First published as Scheibe 1986a.
Bell 1964 and 1971.
v. Neumann 1932, III.2 and IV 2; Bohm 1952
Bell 1966; Bohm and Bub 1966; Gudder 1970; Belinfante 1973; Jammer 1974, Ch. 7.
An overview concerning these proofs is given in Scheibe 1981.
G. Süßmann 1958, III.6.
Bell 1966, §V; Ochs 1970, §3; 1971, §1.
Bohm 1951, Chs. 6, 8 and 22.
Cf. Mackie 1963, p. 67. Mind that a σ-additive truth-value function in the sense of (5a) is not necessarily a homomorphism of L into the boolean lattice (0, 1). Cf. Kamber 1965. §5, note 6.
Gleason 1957. The application of Gleason’s theorem to two-valued σ-additive measures on an infinite-dimensional Hilbert space is somewhat trivialized by the fact that in this case L has already boolean sublattices not admitting any two-valued σ-additive measures. A recent stronger result in Krips 1977 allows the following modification of our argument: For a quantum mechanical L let Ob be any set of observables, containing all maximal ones. Let Lo = UOb and replace (5) by the weaker assumption that every s ∈ S induces a unique function v: Lo ↣ {1,0} in an obvious manner. Then (5a) follows for these functions. But it follows from Krips’ theorem that even for the smallest Lo possible, namely the set of all 1-dimensional subspaces, no such truth-value functions exist.
Mixtures of product states ϑ ⊗ ψ are cases in point, cf. Selleri and Tarozzi 1981, §5.
Cf. Sikorski 1969, §13.
In the discussion Prof. Suppes raised doubts as to the validity of Theorem 4, this theorem being at variance with his ‘Corollary on Hidden Variables’ in Suppes and Zanotti 1981, p. 198. According to the Corollary a pair of Bell-type inequalities concerning three random variables is a necessary (and sufficient) condition for the existence of ‘a hidden variable... with respect to which the three given random variables are conditionally independent.’ I do not think that the literal incompatibility of this result with theorem 4 is also one in fact. Suppes’ approach to the problem of hidden variables is quite different from the present one. Indeed, it is quite different from any other conception of hidden variables that I know of. This suggests a thoroughgoing comparison that I shall entertain in the near future, and I am sure that this will clear up also the seeming incompatibility in question. For the moment I only want to point out that no theory of hidden variables in my sense would satisfy equation (12), and it is for this reason that Bell-type inequalities are of no harm: they simply cannot be transmitted to the given quantum mechanical theory.
First published as Scheibe 1990a, translated by Hans-Jakob Wilhelm
Bohm 1957; Rosenfeld 1958
Bell 1987; here pp. 160 and 167
Pinch 1977
Bohr 1935, p. 701; emphasis mine
Bohr 1958, p. 39; emphasis mine
Ballentine 1970, p. 360
Concerning the contributions by Bohr and Heisenberg, see Scheibe 1989b; the contribution by Pauli is appreciated in detail in Laurikainen 1988
For the following see Scheibe 1964
Laurikainen 1988, p. 161
Bohr 1958, p. 72
Bohr 1963, p. 60
Bohr 1934, p. 68
Bohr 1935, p. 699
Compare the detailed presentation in Heisenberg 1959a, pp. 27 ff
See no. 15 and Heisenberg 1959b, p. 140
See Laurikainen 1988, pp. 57ff, 144f., 176f
The following presentations may be mentioned: Bell 1966; Ca-passo/Fortunato/Selleri 1970; Belinfante 1973; Jammer 1974, Ch. 7
An overview over this tradition is given in Scheibe 1981a
Bohm 1952; Bohm 1953; Bohm/Vigier 1954; this period together with related attempts is presented in Freistadt 1957
Bohm/ Bub 1966b; Bubl968; Bub 1969
Bohm/ Hiley 1975; Bohm/Hiley 1984; Bohm/Hiley/Kaloyerou 1987
Bohm 1980
Einstein/ Podolsky/ Rosen 1935
v. Neumann 1932 (1955), III.2 and IV.2
An overview of definite results is given in Kruszinsky 1984
See v. Neumann 1932, p. 171
Philippidis et al. 1979
Critical comments by members of the Copenhagen school are: Pauli 1955; Rosenfeld 1955; Heisenberg 1959a, pp. 119 ff; informative is also the controversy between Bohm and the Jauch school: Jauch/Piron 1963; Bohm/Bub 1966a; Jauch/Piron 1968; Gudder 1968; Bohm/Bub 1968
Bohm/ Bub 1966b, p. 465
For the following, see Scheibe 1986a (this vol. eh. VI.26)
Cf. Einstein/ Podolsky/ Rosen 1935
Bohm is clear on this, cf. Bohm/ Bub 1966b, p. 467.
First published as Scheibe 1991d, translated by Hans-Jakob Wilhelm.
v. Neumann 1932 (1955), III.2 and IV.2
For the following, see Scheibe 1986a (this vol. VI.26)
An overview is given in Scheibe 1981a; see also Kruszyiiski 1984
See Kamber 1965, sect.6, Example 3), and sect. 13.2
Besides Kamber 1965 see also Kamber 1964 and Dombrowski/Horneffer 1964
First publication in Bell 1964
In this form the inequality occurs first in Bell 1971; see also Selleri 1972
The origin of this inequality is unknown to me. The following proof I owe to a suggestion of Prof. Stamatescu (Heidelberg). For a proof under more general assumptions see Summers/ Werner 1987, p. 2442
For the cases 3) and 4), see Selleri 1988, pp. 28 f
Einstein/ Podolsky/ Rosen 1935
Scheibe 1991e (this vol. VI.29)
Scheibe 1981a
See, for instance, de Broglie 1957, p. 26
Mind that these existence statements are stronger than those which result from negating Bell’s inequality in general. For we are here confronted with the special ‘product situation’. In the 2-dimensional case, it is true, the violation of Bell’s inequality has been proven for this case first, see Bell 1964
For attempts at greater precision see Hellman 1982 and Rédei 1991
First published as Scheibe 1991e.
Bell 1964
Einstein et al. 1935.
See, for instance, the review papers Clauser/ Shimony 1978 and Selleri 1988.
This is essentially the famous wording of the criterion in the original EPR-paper Einstein et al. 1935.
Equation (3a) is, then, the version of Bell’s inequality that appears in Bell 1971. A proof of (3b) under very general assumptions is given in Summers/Werner 1987, p. 2442.
Besides the papers mentioned in no. 2 see also Selleri/ Tarozzi 1981.
For precursors of Bell’s inequality see the references in Pitowsky 1989, p. 49.
Selleri 1988, pp. 28f.
Einstein 1949, p. 85.
Bell_1964, p. 196; see also ibid. no. 2, p. 200.
For a systematic presentation of this tradition see Scheibe 1981a
Belinfante 1973, Bell 1966, and many others.
The belittling of Von Neumann’s result in Belinfante 1979, Ch. I, is unjustified in view of this function.
Einstein 1953, p. 14.
Bell 1966, p. 451.
For details of this part of the paper see Scheibe 1986a (this vol. VI.26).
Bohr 1935, p. 700.
See Bohm’s own emphasis in Bohm/ Hiley 1975.
This is admitted in Belinfante 1973, p. 94.
First published as Scheibe 1993b
Bell 1964
Einstein/ Podolsky/ Rosen 1935
Scheibe 1991e (this vol. VI.29)
Einstein 1948; Einstein 1949a
Bell 1964, p. 200; Einstein 1949a, p. 85; italics mine
Bell 1964, p. 196
For a closer analysis see Scheibe 1986a, reprinted in this volume ch. VI.26
Scheibe 1991d (this vol. ch. VI.28)
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Scheibe, E. (2001). Foundations of Quantum Mechanics. In: Falkenburg, B. (eds) Between Rationalism and Empiricism. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0183-7_6
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