[Paper Review] On the diversity and similarity of mathematical models in science

Inge S. Helland|arXiv (Cornell University)|May 2, 2022

Quantum Mechanics and Applications4 citations

TL;DR

This paper proposes a unified conceptual framework for mathematical models in science by linking statistical inference and quantum theory through the notion of maximally accessible conceptual variables. It introduces a decision-theoretic model grounded in observer-based measurements, showing how quantum mechanics can be reinterpreted using probabilistic states derived from statistical principles, offering a more intuitive foundation than traditional Hilbert space formalism.

ABSTRACT

In this article, the notion of a mathematical model in science is attempted to be enlightened from several points of view. In particular, it is shown that mathematical models are introduced differently and used differently in different areas of science. In the present article the use of models in statistics is taken as a basis, but links are given to several other areas. Two particular such links are described in some detail here: First the link connected to the chemometrical Partial Least Squares algorithm, a link that now has been generalized to the more sophisticated envelope model. Secondly, statistics, as is well known, relies heavily on making decisions, and it may also in certain cases be connected to a process of taking measurements. A mathematical model for these two activities, connected to the mind of an observer, is introduced. This model is taken further, and shown to be important in a new proposal for a foundation of quantum theory. Quantum mechanics, as seen from this point of view, is described in some detail. The discussion here is close to the discussion in a recent book and in recently published articles by the author.

Motivation & Objective

• To clarify the role of mathematical models across diverse scientific disciplines, especially in statistics and quantum mechanics.
• To bridge the cultural and conceptual gap between statistics and quantum theory by identifying shared foundational structures.
• To propose a new, interpretable foundation for quantum mechanics based on decision-making and measurement processes in the observer's mind.
• To demonstrate how statistical concepts like confidence distributions and decision models can underlie quantum states and Born's rule.
• To advocate for a common language of mathematical modeling that transcends disciplinary boundaries in science.

Proposed method

• Models conceptual variables θ as maximally accessible through measurement, with outcomes defined by questions like 'What will θ be if we measure it?'
• Introduces a state vector |ψ⟩ corresponding to a sharp value θ = u, analogous to a pure quantum state.
• Uses projection operators Pi onto eigenspaces of observable operators A^θ to represent statements like η = vi, with orthogonal projections summing to identity.
• Defines mixed states ρ as convex combinations of pure state projectors: ρ = ∑i pi Pi, where pi are probabilities (Bayesian, frequentist, or fiducial).
• Applies Born’s formula to compute probabilities from the density operator ρ, derived from foundational assumptions in statistical inference.
• Generalizes the Partial Least Squares (PLS) and envelope models in statistics to illustrate how statistical modeling can inform quantum foundations.

Experimental results

Research questions

• RQ1 How can mathematical models in science be unified across disciplines like statistics and quantum mechanics?
• RQ2 What is the role of the observer’s mind and measurement in defining quantum states?
• RQ3 Can statistical decision models and confidence distributions serve as a foundation for quantum theory?
• RQ4 How do concepts like causality, decisions, and maximally accessible variables connect statistical inference and quantum mechanics?
• RQ5 Can a common language of mathematical modeling improve interdisciplinary communication in science?

Key findings

• The paper shows that quantum states can be interpreted as statements of the form 'θ = u' where θ is a maximally accessible conceptual variable, offering a more intuitive alternative to abstract Hilbert space vectors.
• Mixed quantum states are represented as density operators ρ = ∑i pi Pi, where probabilities pi can be interpreted as Bayesian priors, posteriors, or fiducial distributions.
• The derivation of Born’s rule is grounded in statistical principles, suggesting that quantum probability emerges from decision-theoretic modeling of measurement uncertainty.
• The Partial Least Squares (PLS) and envelope models in statistics are shown to be structurally analogous to quantum models, supporting a deeper link between statistical and quantum modeling.
• The framework provides a new, interpretable foundation for quantum mechanics that avoids the conceptual difficulties of traditional formulations, particularly the measurement problem.
• The approach suggests that scientific modeling across disciplines—especially in statistics and quantum physics—can be unified through shared principles of measurement, decision, and accessible variables.