In this work, a definition of agreement between a physical theory and experiment, proposed in earlier work, is extended to be relative to τ where τ is Zermelo-Fraenkel set theory. The main aim of this work is to show that this definition, unlike that of earlier work, is sufficiently powerful to include relations between limit properties of empirical outcome sequences and expectation values obtained from the physical theory. We also extend, to the more powerful τ, some earlier results on randomness and the empirical determinability of the probability measure which a physical theory assigns to the outcome set of an infinite sequence of experiments.
In this work, a definition of agreement between a physical theory and experiment, proposed in earlier work, is extended to be relative to τ where τ is Zermelo-Fraenkel set theory. The main aim of this work is to show that this definition, unlike that of earlier work, is sufficiently powerful to include relations between limit properties of empirical outcome sequences and expectation values obtained from the physical theory. We also extend, to the more powerful τ, some earlier results on randomness and the empirical determinability of the probability measure which a physical theory assigns to the outcome set of an infinite sequence of experiments.
Quantum mechanics has several deficiencies as a complete theoretical description of the measurement process. Among them is the fact that the quantum mechanical description of correlations between the single measurements of a sequence is quite problematic. A single measurement is defined to be a preparation followed by an observation. In particular, one feels that an infinite sequence of such single measurements which corresponds to the measurement of a question O on a state η where η does not lie entirely in an eigenspace of O should generate a random output sequence. However, quantum mechanics seems to say nothing about this. In this paper, physical theories are defined in such a manner that correlations between single measurements are explicitly included. In particular, a physical theory is considered to be a mapping U, with domain in the set [Qsτ] of infinite instruction strings for carrying out infinite sequences of single measurements and range in the set of probability measures defined on A, the usual σ algebra of subsets of Ω. Ω is the set of all infinite sequences of natural numbers. A fundamental property which any valid physical theory must satisfy is that it agrees with experiment. It is proposed and discussed here that much of the intuitive meaning of agreement between a theory U and experiment with respect to H is given by the statement ∀Qsτ [U(Qsτ) defined⇒E(H,U(Qsτ),ψQsτ)],where U(Qsτ) is the probability measure U associates with the infinite instruction string Qsτ and ψQsτ is the outcome sequence obtained by carrying out Qsτ. E(H, U(Qsτ),ψQsτ) is the statement that all formulas in H with one free sequence variable which are true on Ω almost everywhere with respect to U(Qsτ) are true for ψQsτ. H is a subclass of the class of all formulas in a formal language L. A theorem is proved which states that, if U(Qsτ) corresponds to a nontrivial product probability measure and U H-agrees with experiment, then the outcome sequence ψQsτ is H-random. H-randomness is defined here in terms of the statement E(H, P, φ). Another property of a valid physical theory, which is defined here, is that, for some Qsτ, U(Qsτ) must be determinable on much of AH from ψQsτ. Sufficient conditions for this property to hold are given. AH is the class of all H definable subsets of Ω. Some properties of the statement E(H, P, φ) are given. Among other things, it is proved that, if E(H, P, φ) holds and P is a nontrivial probability measure on A, then φ is not definable in H.
In previous technical work ([1] and [2]) on which his present paper [3] draws, Benioff has presented results conforming with the following argument-scheme: First, if we construe Quantum Mechanics as making claims to the effect that infinite outcome sequences (generated by repeated measurements on a system for a given observable in a given state) be random; and second, if a strong definition of “random” is adopted in this construal, then certain models of Zermelo-Fraenkel set theory (ZF) cannot be “carriers for the mathematics of physics”. How interesting is this? One can approach the matter on two levels, the level of the specific technical results, and the level of more general implications of this pattern of argument for understanding connections between mathematics and physics. The second more general level should be our primary focus, but it will pay at the outset to look briefly at the technical level.
The results of an earlier paper on finite and infinite sequences of measurements are here extended to include decision procedures. It is shown that with each decision procedure Q there is uniquely associated a probability operator measure OQ, which gives the statistical properties of Q. None, some, or all of the paths of Q can be infinitely long. A result of this association is that there are two methods of measuring the probability that carrying out Q on a system in state ρ gives an outcome sequence in some set F. A remarkable aspect of this equivalence is that the purely physical operation of one method is equivalent to, or can replace, the physical operation and mathematical decision procedure of the other method.
This paper is a first attempt to explore the relationship between physics and mathematics ’’in the large.’’ In particular, the use of different Zermelo Frankel model universes of sets (ZFC models) as carriers for the mathematics of quantum mechanics is discussed. It is proved that given a standard transitive ZFC model M, if, inside M, B (ℋM) is the algebra of all bounded linear operators over a Hilbert space ℋM, there exists, outside M, a Hilbert space ℋ and an algebra B (ℋ), along with isometric monomorphisms UM and VM from ℋM into ℋ and from B (ℋM) into B (ℋ). UM and VM are used to relate quantum mechanics based on M to quantum mechanics based on the usual ZFC model. It is then shown that, contrary to what one would expect, all ZFC models may not be equivalent as carriers for the mathematics of physics. In particular, it is proved that if one requires that an outcome sequence, associated with an infinite repetition of measuring a question observable on a system prepared in some state, be random, and if a strong definition of randomness is used, then the minimal standard ZFC model cannot be a carrier for the mathematics of quantum mechanics.
This paper uses the sample probability-space description to review, under general conditions, the measurement process whereby an empirical expectation value is obtained for comparison either with other values or with values calculated from a theory. The emphasis here is on some of the conditions which must be satisfied by general sequences of single measurements, about which an observer may have relatively little knowledge, in order that such sequences yield suitable expectation values. In particular, sequences are considered for which the requirements that the single measurements of the sequence be independent and made of the same physical quantity on ensembles of identically prepared systems are not necessarily satisfied. The differences between sequences which satisfy these requirements and those which do not are discussed in terms of the implications or meaning of the resultant expectation value as a point of contact between theory and experiment.
Identifying the property of the world that enforces the Born rule is a longstanding problem in physics. We prove that in any physical theory that assigns probabilities to the outcomes of ideal measurements, the maximal set of probability assignments for each graph of exclusivity is the one that satisfies the Born rule. Therefore, the agreement between quantum theory and experiments follows from a one-to-one correspondence between the logical possibilities and the physical possibilities and, in particular, implies that the outcomes of quantum measurements are not constrained by any physical reason.