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Finite and infinite measurement sequences in quantum mechanics and randomness: The Everett interpretation
1.B. DeWitt, in The Many Worlds Interpretation of Quantum Mechanics, edited by B. DeWitt and N. Graham (Princeton U.P., Princeton, New Jersey, 1973), pp. 167–218;
1.in Battelle Recontres, edited by C. DeWitt and J. Wheeler (Benjamin, New York, 1968), pp. 318–32.
2.P. Martin‐Löf, Inf. Control 9, 603 (1966);
2.“Algorithms and Random Sequences,” Lectures at Erlangen University (1966), translated by W. A. Beyer, Los Alamos, New Mexico, 1976, LA‐TR‐67‐105.
3.H. Everett III, Rev. Mod. Phys. 29, 454 (1957);
3.J. A. Wheeler, Rev. Mod. Phys. 29, 463 (1957).
4.J. Von Neumann, “On Infinite Direct Products” in Collected Works, Vol. III, edited by A. H. Taub (Pergamon, New York, 1963), pp. 323–99;
4.M. A. Guichardet, Ann. Ec. Norm. Super. 83, 1 (1966).
5.If is considered as a state evolved from by the action of a unitary time evolution operator then the time evolution of as describing not‐yet‐interacting systems should be included. However this is quite inessential for our purposes here and is thus suppressed.
6.Hartley Rogers, Jr., Theory of Recursive Functions and Effective Computability (McGraw‐Hill, New York, 1967), Chaps. 1–5.
7.Terrence Fine, Theories of Probability (Academic, New York, 1973), Chap. V.
8.P. Martin‐Löf, On the Notion of Randomness, Proceedings of Summer Institute on Proof Theory and Intuitionism, State University of Buffalo, New York, 1968, edited by J. Myhill, A. Kino, and R. Vesley (North‐Holland, Amsterdam, 1970).
9.R. Solovay, Ann. Math. 92, 1 (1970).
10.P. A. Benioff, Phys. Rev. D 7, 3603 (1973);
10.A. H. Kruse, Z. Math. Logik Grundl. Math. 13, 299 (1967).
11.The randomness definitions of Refs. 8–10 are given in terms of sets of measure zero rather than in terms of sequential tests. However one can show that the two types of definitions are equivalent for the definitions of Refs. 8 and 9. For that of Ref. 10 the definition given in terms of sets of measure zero is at least as strong as that given in terms of sequential tests.
12.For these stronger definitions a universal test V is not in general a test in C. Thus a definition of randomness in terms of a universal test V for C is at least as strong a definition given in terms of the tests in C.
13.P. A. Benioff, J. Math. Phys. 17, 618, 629 (1976).
14.To see this let and the object state ψ be such that (This choice is made for convenience only.) Let for each 0–1 sequence θ length n. Then for each . So for odd no sequence θ of length n satisfies and is the zero vector for these n.
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