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Some consequences of the strengthened interpretative rules of quantum mechanics
1.P. A. Benioff, Phys. Rev. D, 7, 3603 (1973).
2.H. Ekstein, Phys. Rev. 184, 1315 (1969).
3.P. A. Benioff, J. Math. Phys. 11, 2553 (1970);
3.P. A. Benioff, 12, 361 (1971); , J. Math. Phys.
3.P. A. Benioff, J. Symbol. Logic, 36, 377 (1971).
4.P. A. Benioff, Found. Phys. 3, 359 (1973).
5.The correct condition is that include operators with nonzero continuous parts. To see this let be the discrete spectrum of A and define the projection operator by is the eigenprojector of A for eigenvalue r. If then A has a nonzero continuous part and The reason this condition is correct rather than the one in the main text is that there exist self‐adjoint operators with nonempty continuous spectra but for which For an example see Ref. 1, footnote 4.
6.M. H. Stone, Linear Transformations in Hilbert Space, American Mathematical Society Colloquium Publ. 1932, Vol. XV, Chap. V. Sec. 5.
7.N. Dunford and J. Schwartz, Linear Operators (Wiley‐Interscience, New York, 1963), Vol. II, Chap. X, p. 902.
8.M. Loeve, Probability Theory (Van Nostrand, Princeton, New Jersey. 1963), 3rd ed., Secs. 30–32.
9.By the Stone‐Weierstrass theorem any continuous where is a polynomial. The difficulty is to prove where is the mean of the first m elements of ψ.
10.Let be a sequence of polynomials for which uniform on For each n let be a sequence of polynomials with rational coefficients such that for each for all r on Then the diagonal sequence is such that uniform on
11.The usual definition (Ref. 8, pp. 102–110) defines the generating set of all functions f of the form if if with a real number and any finite set of pairwise disjoint Borel sets. Now [and ] are countably generated [K. R. Parthasarathy, Probability Measures on Metric Spaces (Academic, New York, 1967), Chap. I, Sec. 2 and Chap. V, Sec. 2] from the set of intervals (a,b) with rational endpoints, and any real number is the limit of a sequence of rationals. Thus the set of defined as above except that is rational and with and rational is a set of τ‐definable functions which is also a countable generating set for the Borel functions.
12.Let φ be a 0, sequence such that if k odd, if and for some even [odd] n. Let θ be a 0, sequence such that if k odd, if and for some even [odd] n. Define ψ by Then if rh and are four rational numbers such that but then exists and equals but for none of the four rational numbers does the limit relative frequency of finding it in ψ exist.
13.This proof depends on there being a “sufficient” supply of preparation procedures available. Possible definitions of sufficiency are that of normal states on a or that is dense in some topology on a.
14.One first shows that for any commuting self‐adjoint operators A and B. the spectral measure for is the convolution of the spectral measures of A and B, or for each E in with From this one shows easily that for each E in From this it follows that if any set F in is τ‐definable from it is τ‐definable from as is τ‐definable. The τ‐correctness of for follows from this and the τ‐correctness of for
15.The author is indebted to Hans Ekstein for suggesting these examples.
16.N. Dunford and J. Schwartz, Linear Operators, (Wiley‐Interscience, New York, 1958), Chap. III, Sec. 5.
17.P. A. Benioff, unpublished work.
18.W. Mackey, Mathematical Foundations of Quantum Mechanics (Benjamin, New York, 1963), pp. 63, 64.
19.R. Giles, J. Math. Phys. 11, 2139 (1970).
20.If ψ is such that exists and equals then the limit mean measure defined by Eq. (20). T is the one sided shift and is the set transformation generated by T. The ergodic theorem (Ref. 8) and the fact that is a product measure give (Ref. 8, p. 230) the result that for almost all ψ. Equation (21) then follows from the τ‐correctness of
21.A. Wald, Mathematischen Kolloquiums (Vienna, 1973), Vol. 8
21.[Selected Papers in Statistics and Probability of A. Wald (McGraw‐Hill, 1955), pp. 79–99].
22.One must show that if E in is τ‐definable from and then Much of the proof consists in showing that if φ is τ‐random and E is τ‐definable from and then The proof of this is done by constructing a measure μ from and in a standard way on and defining for each E in a Borel function by From one has that is true μ almost everywhere. Thus if is τ‐definable from φ, which occurs only if and are τ‐definable, the τ‐randomeness of with gives
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