arXiv subjects

Inge S. Helland

Publications and source records attributed to Inge S. Helland.

At least 19 recordsLinked to original sources

arXiv2026

The final version of a recent approach towards quantum foundation

In several articles, this author has advocated an alternative approach towards quantum foundation based upon a set of postulates, and based upon the notions of theoretical variables and of accessible theoretical variables. It is shown in this article that this basis can be considerably simplified. In particular, the assumption that there exists an inaccessible variable $ϕ$ such that all the accessible ones can be seen as functions of $ϕ$, can be dropped. This assumption has been difficult to motivate in the previous articles. From this, I get a simple basis for the main Theorems.The essential assumption is that there in the given context exist two different maximal accessible variables, what Niels Bohr would have called two complementary variables. From this, the whole Hilbert space formalism may be derived. It is also discussed in some detail how this Hilbert space can be chosen. The resulting theory is a purely mathematical theory, but it leads to quantum mechanics by letting the variables be physical variables. Other applications of the main theory are also considered. The mathematical proofs are mostly deferred to the Appendix.

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arXiv2026

Three ways to find comfort with the Bell proof and the results of the Bell experiments

Bell's theorem states that no description of a Bell experiment can be simultaneously local, realistic in the sense of counterfactual definiteness, and free of conspiracy between settings and hidden state. The recent generation of experiments has confirmed the predicted violation of the CHSH inequality, so one of the assumptions must be abandoned. Which one, and how one reconstructs a coherent worldview after doing so, is a question on which many authors disagree. This paper is written by three such authors. All three reject both counterfactual definiteness and conspiratorial violation of statistical independence of setting choices and state. After a joint exposition of the classical half of Bell's theorem in the language of Pearl-style causal graphs, a joint summary of the loophole-free experiments, and a joint survey of the recent literature, each author states where they have presently arrived. Gill accepts irreducible and non-local quantum randomness and finds the choice between locality and realism a false dichotomy. In his later works, Bell derives counterfactual definiteness from classical local causality, and that is what has to go. The metaphysical concepts "realism", "locality", "causality" need to be reconsidered. Helland reconstructs the Hilbert-space formalism from a theory of accessible variables, and from this theory he concludes that every observer must be limited in a specific sense. Jongejan proposes a geometric hidden-variable construction in which the degree of violation of the CHSH inequality depends on the number of dimensions of space, Tsirelson's bound corresponding to three dimensions. The authors conclude with a discussion.

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arXiv2026

A new foundation of quantum decision theory

Quantum decision theory is introduced here, and new basis for this theory is proposed. It is first based upon the author's general arguments for the Hilbert space formalism in quantum theory, next on arguments for the Born rule, that is, the basis for calculating quantum probabilities. A basic notion behind the quantum theory foundation is that of theoretical variables, that are divided into accessible and inaccessible ones. This is here specialized to decision variables. It is assumed that all accessible variables can be seen as functions of a specific inaccessible variable. Another assumption is that there exist two different maximal accessible theoretical variables in the given situation. Two basic assumptions behind the Born rule are 1) the likelihood principle, 2) the actor in question has motivations that can be modeled by a hypothetical perfectly rational higher being. The theory is illustrated by a medical example. Finally, a brief discussion of decision processes is given.

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arXiv2026

An alternative foundation of quantum theory

The final version of a new approach to quantum theory is formulated in this paper. The basis is taken to be theoretical variables, variables that may be accessible or inaccessible, i.e., it may be possible or impossible for an observer to assign arbitrarily sharp numerical values to them. In an epistemic process, the accessible variables are just ideal observations connected to an observer or to some communicating observers. Group actions are defined on these variables, and group representation theory is the basis for developing the Hilbert space formalism here. Operators corresponding to accessible theoretical variables are derived, and in the finite-dimensional case, it is proved that the possible physical values are the eigenvalues of these operators. The focus of the paper is some mathematical theorems paving the ground for the proposed foundation of quantum theory. It is indicated here that the groups and transformations needed in this approach can be constructed explicitly in the case where the accessible variables are finite-dimensional. In case, this simplifies the theory considerably: To reproduce the Hilbert space formulation, it is enough to assume the existence of two complementary variables. What is new in this version of the article, is that the assumptions given behind the main theorems are simplified considerably, and that all proofs are given explicitly or refered to in other recent papers.

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arXiv2025

Some mathematical issues regarding a new approach towards quantum foundations

In this article, the weakest possible theorem providing a foundation for the Hilbert space formalism of quantum theory is stated. The necessary postulates are formulated, and the mathematics is spelt out in detail. It is argued that, from this approach, a general epistemic interpretation of quantum mechanics is natural. Some applications to the Bell experiment and to decision theory are briefly discussed. The article represents the conclusion of a series of articles and books on quantum foundations.

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arXiv2025

Quantum probability for statisticians; some new ideas

It is argued from several points of view that quantum probabilities might play a role in statistical settings. New approaches toward quantum foundations have postulates that appear to be equally valid in macroscopic settings. One such approach is described here in detail, while one other is briefly sketched. In particular, arguments behind the Born rule, which gives the basis for quantum probabilities, are given. A list of ideas for possible statistical applications of quantum probabilities is provided and discussed. A particular area is machine learning, where there exists substantial literature on links to quantum probability. Here, an idea about model reduction is sketched and is motivated from a quantum probability model. Quantum models can play a role in model reduction, where the partial least squares regression model is a special case. It is shown that for certain experiments, a Bayesian prior given by a quantum probability can be motivated. Quantum decision theory is an emerging discipline that can be motivated by this author's theory of quantum foundations.

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arXiv2025

A new approach towards quantum foundation and some consequences

A general theory based upon 7 postulates is introduced. The basical notions are theoretical variables that are associated with an observer or with a group of communicating observers. These variables may be accessible or inaccessible. From these postulates, the ordinary formalism of quantum theory is derived. The mathematical derivations are not given in this article, but I refer to the recent articles [9, 10]. Three possible applications of the general theory can be given; 1) The variables may decision variables connected to the decisions of a person or of a group of persons. 2) The variables may be statistical parameters or future data, But most importantly here: 3) The variables are physical variables in some context. This last application gives a completely new foundation of quantum mechanics, a foundation which in my opinion is much more easy to understand than the ordinary formalism.The other applications seem also to give interesting consequences of the approach. Socalled paradoxes like that of Schrödinger's cat can be clarified under the theory. Explanations of the outcomes of David Bohm's version of the EPR experiment and of the Bell experiment are provided. Finally, references to links towards relativity theory and to quantum field theory are given. The concluding remarks point at further possible developments.

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arXiv2024

On optimal linear prediction

The main purpose of this article is to prove that, under certain assumptions in a linear prediction setting, optimal methods based upon model reduction and even an optimal predictor can be provided. The optimality is formulated in terms of the expected mean square prediction error. The optimal model reduction turns out, under a certain assumption, to correspond to the statistical model for partial least squares discussed by the author elsewhere, and under a certain specific condition, the partial least squares predictors is proved to be good compared to all other predictors. It is also proved in this article that the situation with two different model reductions can be fit into a quantum mechanical setting. Thus, the article contains a synthesis of three cultures: mathematical statistics as a basis, algorithms introduced by chemometricians and used very much by applied scientists as a background, and finally, notions from quantum foundation as an alternative point of view.

math.ST↗

arXiv2024

On probabilities in quantum mechanics

This is an attempt to clarify certain concepts related to a debate on the interpretation of quantum mechanics, a debate between Andrei Khrennikov on the one side and Blake Stacey and Rüdiger Schack on the other side. Central to this debate is the notion of quantum probabilities. I first take up the probability concept in the QBist school, and then give my own arguments for the Born formula for calculating quantum probabilities. In that connection I also sketch some consequences of my approach towards the foundation and interpretation of quantum theory. I discuss my general views on QBism as a possible alternative interpretation before I give some final remarks.

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arXiv2023

An explanation of the Bell experiment

The Bell experiment is discussed in the light of a new approach to the foundation of quantum mechanics. It is concluded from the basic model that the mind of any observer must be limited in some way: In certain contexts, he is simply not able to keep enough variables in his mind when making decisions. This has consequences for Bell's theorem, but it also seems to have wider consequences.

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arXiv2023

A Unified Scientific Basis for Inference

Every experiment or observational study is made in a context. This context is being explicitly considered in this book. To do so, a conceptual variable is defined as any variable which can be defined by (a group of) researchers in a given setting. Such variables are classified. Sufficiency and ancillarity are defined conditionally on the context. The conditionality principle, the sufficiency principle and the likelihood principle are generalized, and a tentative rule for when one should not condition on an ancillary is motivated by examples. The theory is illustrated by the case where a nuisance parameter is a part of the context, and for this case, model reduction is motivated. Model reduction is discussed in general from the point of view that there exists a mathematical group acting upon the parameter space. It is shown that a natural extension of this discussion also gives a conceptual basis from which essential parts of the formalism of quantum mechanics can be derived. This implies an epistemological basis for quantum theory, a kind of basis that has also been advocated by part of the quantum foundation community in recent years. Born's celebrated formula is shown to follow from a focused version of the likelihood principle together with some reasonable assumptions on rationality connected to experimental evidence. Some statistical consequences of Born's formula are sketched. The questions around Bell's inequality are approached by using the conditionality principle for each observer. The objective aspects of the world are identified with the ideal inference results upon which all observers agree (epistemological objectivity).

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arXiv2023

Towards quantum mechanics from a theory of experiments

Essential elements of quantum theory are derived from an epistemic point of view, i.e., the viewpoint that thetheory has to do with what can be said about nature. This gives a relationship to statistical reasoning and to other areas of modelling and decision making. In particular, a quantum state can be defined from an epistemic point of view to consist of two elements: A (maximal) question about the value of some parameter together with the answer to that question. Quantization itself can be approached from the point of view of model reduction under symmetry.

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arXiv2023

Quantum Mechanics from Focusing and Symmetry

A foundation of quantum mechanics based on the concepts of focusing and symmetry is proposed. Focusing is connected to c-variables - inaccessible conceptually derived variables; several examples of such variables are given. The focus is then on a maximal accessible parameter, a function of the common c-variable. Symmetry is introduced via a group acting on the c-variable. From this, the Hilbert space is constructed and state vectors and operators are given a clear interpretation. The Born formula is proved from weak assumptions, and from this the usual rules of quantum mechanics are derived. Several paradoxes and other issues of quantum theory are discussed.

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arXiv2023

Possible connections between relativity theory and a version of quantum theory based upon theoretical variables

An alternative approach towards quantum theory is described, and tentative attempts to connect his approach to special and general relativity are discussed. Important concepts are gauge groups and information/entropy connected to some physical systems. Some recent results on information in connection to black holes are touched upon, and it is indicated how expected information can be argued to be conserved. This argument only depends on what happens outside the black hole. Everything connected to the interior of the black hole is inaccessible.

physics.hist-ph↗

arXiv2022

The Bell experiment; an epistemological approach

The Nobel prize in physics for 2022 was given for performing Bell experiments with varying degree of sophistication. The interpretation of this experiment is discussed by first recalling Bell's simple argument behind his inequalities, in particular the CHSH inequality. It is argued that any independent observer must have a limitation: He is not able to keep all relevant variables in his mind at the same time when trying to model the experiment. This is contrasted to the solution proposed by Tim Maudlin, where nonlocality is crucial.Maudlin criticizes the Nobel price committee, but this critique is countered. Finally, a related new approach to the foundation of quantum mechanics is briefly sketched, giving references to the relevant literature.

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arXiv2022

A simple quantum model linked to a theory of decisions

This article may be seen as a summary and a final discussion of the work that the author has done in recent years on the foundation of quantum theory. It is shown that quantum mechanics as a model follows under certain specific conditions from a quite different, much simpler model. This model is connected to the mind of an observer, or to the joint minds of a group of communicating observers. The model is based upon conceptual variables, and an important aspect is that an observer (a group of observers) must decide on which variable to measure. The model is then linked more generally to a theory of decisions. The results are discussed from several angles. In particular, macroscopic consequences are treated briefly.

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arXiv2022

On reconstructing parts of quantum theory from two relates maximal conceptual variables

In the book [4] the general problem of reconstructing the Hilbert space formulation in quantum theory is discussed from the point of view of what I called conceptual variables, any variables defined by a person or by a group of persons. These variables may be inaccessible, i.e., impossible to assign numerical value to by experiments or by measurements, or accessible. One basic assumption in [4] and here is that group actions g 2 G are defined on a space where some maximally accessible variable varies, and then accessible functions of these maximal variables are introduced. By using group representation theory the basic Hilbert space formalism is restored under the assumption that the observator or the set of observators has two related maximally accessible variables in his (their) mind(s). The notion of relationship is precisely defined here. Symmetric (self-adjoint) operators are connected to each variable, and in the discrete case the possible values of the variables are given by the eigenvalues of the operators. In this paper the main results from [4] are made more precise and more general. It turns out that the conditions of the main theorem there can be weakened in two essential ways: 1) No measurements need to be assumed, so the result is also applicable to general decision situations; 2) States can have arbitrary phase factors. Some consequences of this approach towards quantum theory are also discussed here.

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arXiv2022

Epistemological and ontological aspects of quantum theory

In this paper, epistemology and ontology of quantum states are discussed based on a completely new way of founding quantum theory. The fundamental notions are conceptual variables in the mind of an observer or in the joint minds of a group of observers. These conceptual variables are very often accessible, that is, it is possible to find values of the variables by doing experiments or by making measurements. An important notion is that of maximal accessibility. It is shown here that this new machinery may facilitate the discussion of when a specific quantum state can be given an ontological interpretation, and also the more speculative question whether all states can be given such an interpretation. The discussion here is general, and has implications for the basic problem of how one should look upon information from experiments and measurement, in particularly the question concerning when this information may reflect properties of the real world.

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