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.
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.
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.
The topic of this session is “physical randomness”. It might be doubted whether such a subject exists, for definitions of randomness have hitherto almost all been mathematical in nature. The only exceptions of which I am aware are the preceding paper by Benioff and a paper by Wesley Salmon. These attempts to inject some empirical content into randomness are highly desirable. But anyone attempting to formulate a physically based definition of randomness should at some point make clear what the connection is (if any) with a more traditional notion of disorder - that of indeterminism. Repeated reference to quantum mechanical examples whenever physical randomness is discussed indicates that a primary motivation for considering physical randomness to be important is because of the current belief that data sequences associated with quantum mechanical experiments are irreducibly random. (As an indication of this, in any situation in which physical randomness is discussed, translate the remarks about quantum phenomena into remarks about coin tossing, and they will lose much of their interest.)
Abstract Quantum mechanics (QM) predicts probabilities on the fundamentallevel which are, via Born probability law, connected to the formal randomnessof infinite sequences of QM outcomes. Recently it has been shown thatQM is algorithmic 1-random in the sense of Martin-L¨of. We extend this resultand demonstrate that QM is algorithmic ω-random and generic, precisely asdescribed by the ’miniaturisation’ of the Solovay forcing to arithmetic. Thisis extended further to the result that QM becomes Zermelo–Fraenkel Solovayrandom on infinite-dimensional Hilbert spaces. Moreover, it is more likely thatthere exists a standard transitive ZFC model M, where QM is expressed in reality,than in the universe V of sets. Then every generic quantum measurementadds to M the infinite sequence, i.e. random real r ∈ 2ω, and the model undergoesrandom forcing extensions M[r]. The entire process of forcing becomesthe structural ingredient of QM and parallels similar constructions applied tospacetime in the quantum limit, therefore showing the structural resemblanceof both in this limit. We discuss several questions regarding measurability andpossible practical applications of the extended Solovay randomness of QM.The method applied is the formalization based on models of ZFC; however,this is particularly well-suited technique to recognising randomness questionsof QM. When one works in a constant model of ZFC or in axiomatic ZFCitself, the issues considered here remain hidden to a great extent.
This paper continues the study of the use of different models of ZF set theory as carriers for the mathematics of quantum mechanics. The basic tool used here is the construction of Cohen extensions of ZFC models by use of Boolean valued ZFC models [C=axiom of choice]. Let M be a standard transitive ZFC model. Inside M, B (ℋM) is the algebra of all bounded linear operators over some Hilbert space ℋM. It is shown that with each state ρ in B (ℋM) and projection operator ο in B (ℋM) one can associate a unique Boolean valued ZFC model Mℬρο. ℬρο is the algebra of all Borel subsets of {0,1}ω, the set of all infinite 0–1 sequences, modulo sets of Pρο =⊗pρο measure zero with pρο({1}) =Trρο in M. Let ΨM and ΦM be respective maps from the sets of state preparation and question measuring procedures into B (ℋM). Let M=M0, the minimal standard transitive ZFC model. It is then shown that with each state preparation procedure s - Dom(ΨM0) and each question measuring procedure q - Dom(ΦM0) and with each infinite repetition (tsq) of doing s and q at times t (0), t (1),..., if the definition of randomness is sufficiently strong, one can associate the Cohen extension M0[ψtsq] of M0 by ψtsq. ψtsq is the random outcome sequence associated with (tsq). A third condition, in addition to the two given in the previous paper, is then given which must be satisfied if a ZFC model M is to serve as a carrier for the mathematics of quantum mechanics. In essence it says that for each pair (tsq) and (wuk) of distinct infinite repetitions of doing s and q and of doing u and k with s, u - Dom(ΨM) and q, k - Dom(ΦM), the two outcome sequences ψtsq and ψwuk are mutually statistically independent. It is then shown that for a strong definition of independence, corresponding to the definition of randomness used previously, no Cohen extension M0[ψtsq] of M0 can serve as the carrier for the mathematics of quantum mechanics.
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.
When investigating various interpretations of quantum mechanics one notices that each interpretation contains an element which escapes a complete and full description. This element is always associated with the stochasticity of the individual event in the quantum measurement process. It appears that the implications of this limit to any description of the world has not been sufficiently appreciated with notable exceptions of, for example, Heisenberg, Pauli and Wheeler. If we assume that a deeper foundation of quantum mechanics is possible, the question arises which features such a philosophical foundation might have. It is suggested that the objective randomness of the individual quantum event is a necessity of a description of the world in view of the significant influence the observer in quantum mechanics has. It is also suggested that the austerity of the Copenhagen interpretation should serve as a guiding principle in a search for deeper understanding. In recent years, a significant increase of interest in the foundations of quantum mechanics can be observed. This increase is certainly related to the immense progress made during the last two decades in experiments with individual quantum systems. This technological progress has made it possible to really perform more and more of the famous gedanken experiments which were so important in the early days of the theory. Not only did these experiments confirm the theory in every detail, they also opened up avenues for new experimental directions which might even lead some day to new technology. While such applications are certainly still far away, areas like quantum cryptography[1], quantum computation[2] and interaction-free measurement[3], just to name a few examples, certainly deserve the attention they receive. All this activity has also drawn more attention to the problems of understanding and interpreting quantum mechanics. It seems that there, a definitive consensus has not yet been reached. As witnesses quote authorities like Feynman as saying[4]: I think can safely say that nobody today understands quantum mechanics, whereby he apparently included himself, or like Roger Penrose[5], who describes his opinion remarking that while the theory agrees incredibly well with experiment and while it is of profound mathematical beauty, it makes absolutely no sense. Why did even physicists who have contributed so significantly to quantum theory Feynman was even rewarded the Nobel Prize for one of its mathematical formulations choose such emphatic and strong formulations? Why is it that the average physicist hears very little about such problems during his/her education? Why is the comprehension of the theory very often focused on formalism while questions that probe for deeper meaning are usually not tackled? To take a closer look at the situation, let us turn to a brief analysis of the interpretation of quantum mechanics. We note that there are at least two levels of interpreting a formalism, a physical theory. The first, basic, level supplies the rules that determine which element of the formalism corresponds to which measurable quantity or to which observable fact in a concrete experimental situation. Theses rules are a large, mostly not explicated but only implicit, set of instructions. These instructions concern the manner of working with the statements of the theory in order to obtain predictions for future experiments or events, and they concern the instructions on how to proceed in experiment in order to demonstrate or test a theoretical