This paper contributes to a burgeoning area of investigation, the ambiguity inherent in mathematics and the implications for physics of this ambiguity. To display the mathematical form of equations of quantum theory used to describe experiments, we make explicit the knobs by which the devices of an experiment are arranged and adjusted. A quantum description comes in two parts: (1) a statement of results of an experiment, expressed by probabilities of detections as functions of knob settings, and (2) an explanation of how we think these results come about, expressed by linear operators, also as functions of knob settings. Because quantum mechanics separates the two parts of any description, it is known that between the statements of results and the explanations lurks a logical gap: given any statement of results one has a choice of explanations. Here we work out some consequences of this openness to choice. We show how quantum theory as mathematical language in which to describe experiments necessarily involves multiple descriptions: multiple explanations of a given result, as well as multiple statements of results and multiple arrangement of knobs. Appreciating these multiplicities resolves what otherwise is a confusion in the concept of invariance. Implications of multiplicity of description for the security of quantum key distribution are noted.
This paper contributes to a burgeoning area of investigation, the ambiguity inherent in mathematics and the implications for physics of this ambiguity. To display the mathematical form of equations of quantum theory used to describe experiments, we make explicit the knobs by which the devices of an experiment are arranged and adjusted. A quantum description comes in two parts: (1) a statement of results of an experiment, expressed by probabilities of detections as functions of knob settings, and (2) an explanation of how we think these results come about, expressed by linear operators, also as functions of knob settings. Because quantum mechanics separates the two parts of any description, it is known that between the statements of results and the explanations lurks a logical gap: given any statement of results one has a choice of explanations. Here we work out some consequences of this openness to choice. We show how quantum theory as mathematical language in which to describe experiments necessarily involves multiple descriptions: multiple explanations of a given result, as well as multiple statements of results and multiple arrangement of knobs. Appreciating these multiplicities resolves what otherwise is a confusion in the concept of invariance. Implications of multiplicity of description for the security of quantum key distribution are noted.
We note the separation of a quantum description of an experiment into a statement of results (as probabilities) and an explanation of these results (in terms of linear operators). The inverse problem of choosing an explanation to fit given results is analyzed, leading to the conclusion that any quantum description comes as an element of a family of related descriptions, entailing multiple statements of results and multiple explanations. Facing this multiplicity opens avenues for exploration and consequences that are only beginning to be explored. Among the consequences are these: (1) statements of results impose topologies on control parameters, without resort to any quantum explanation; (2) an endless source of distinct explanations forces an open cycle of exploration and description bringing more and more control parameters into play, and (3) ambiguity of description is essential to the concept of invariance in physics.
Everybody agrees that quantum physics is strange, and that the world view it implies is elusive. However, it is rarely considered that the theory might be opaque because the mathematical language it employs is inarticulate. Perhaps, if a mathematical language were constructed specifically to handle the theory's subject matter, the theory itself would be clarified. This article explores that possibility. It presents a simple but rigorous language for the description of dynamics, experiments, and experimental probabilities. This language is then used to answer a compelling question: What is the set of allowed experiments? If an experiment is allowed, then the sum of the probabilities of its outcomes must equal 1. If probabilities are non-additive, there are necessarily sets of outcomes whose total probability is not equal to 1. Such experiments are therefore not allowed. That being the case, in quantum physics, which experiments are allowed, and why are the rest disallowed? What prevents scientists from performing the disallowed experiments? By phrasing these questions within our mathematical language, we will uncover answers that are complete, conceptually simple, and clearly correct. This entails no magic or sleight of hand. To write a rigorous mathematical language, all unnecessary assumptions must be shed. In this way, the thicket of ad hoc assumptions that surrounds quantum physics will be cleared. Further, in developing the theory, the logical consequences of the necessary assumptions will be laid bare. Therefore, when a question can be phrased in such a language, one can reasonably expect a clear, simple answer. In this way we will dispel much of the mystery surrounding quantum measurements, and begin to understand why quantum probabilities have their peculiar representation as products of Hilbert space projection operators.
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.
Introduction Chapter 1 dealt with one of the deepest mysteries in modern science: how to explain consciousness. This Part deals with another mystery: how to make sense of quantum mechanics. Both involve a problem of reconciliation with the classical worldview, which is one hint that the two mysteries might be related. A second is that, unlike classical physics, which makes no reference to or allowance for consciousness, quantum theory raises the issue of consciousness and its relationship to the physical world in a very direct way. These hints do not necessarily mean there is a connection between the two mysteries, but by the end of this Part I hope to have shown why it would be natural to look for one. My more immediate aim here is to give the social scientific reader an introductory understanding of the experimental findings of quantum theory, its key concepts, and the debates about its interpretation. The discussion assumes no prior knowledge of quantum physics and makes no use of mathematics. Quantum theory without equations might seem like an oxymoron, and certainly reading this Part will not enable anyone to use quantum theory. But to a perhaps surprising degree it is possible to understand quantum theory without being able to use it. Its findings can be communicated in ordinary language, and the same goes for its main concepts and interpretive debates. One consequence of this has been the emergence of a virtual cottage industry of books on quantum theory for a lay audience. Most are written by physicists, and many are quite good. However, it is not just popular treatments that do without equations: much of the professional philosophical literature does as well. (I draw on both kinds of literature below.) Philosophers of physics are trained to understand the math, of course, which is necessary for a full appreciation of the issues.
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
While it is widely acknowledged that quantum theory is strange, and that the world view it implies is elusive, there is considerably less consensus as to why this is the case. This article will explore the possibility that the theory may be opaque because the mathematical language it employs is inarticulate. To address this possibility, three interconnected mathematical theories will be developed, each specifically designed to handle aspects of quantum theory's subject matter. The first will describe dynamic systems in terms that are general enough to be applicable to any kind dynamics, including both classical and quantum. That description will then be used to construct a theory of experiments, with the aim of understanding what types of experiments scientific methodology allows. Finally, the theory of experiments will found a theory of experimental probabilities applicable to any collection of experiments, so long as their probabilities are internally consistent. The resulting over-arching mathematical language will then be applied to quantum theory. In the initial application, it will be used to explain the core nature and structure of quantum measurement and probability. Following that, it will be used to explore quantum theory's models. Like any theory written in a mathematical language, quantum theory derives its meaning from its class of models. To understand the class quantum models, two models of central importance will be described. The first will be a conceptually simple construction that follows almost immediately from the above theory of dynamics. It fully satisfies both quantum theory and the above mentioned theory of experimentation; additionally, its macroscopic objects obey classical mechanics. The second will utilize a minimal amount of mathematical structure to satisfy the same conditions.