[Paper Review] Quantum theory as inductive inference

Ryszard Paweł Kostecki|arXiv (Cornell University)|Sep 10, 2010

Statistical Mechanics and Entropy77 references4 citations

TL;DR

This paper proposes a foundational reformulation of quantum theory and probability using algebraic integration, information geometry, and maximum relative entropy, replacing Hilbert spaces and measure spaces with abstract $W^*$-algebras and positive linear functionals. It derives quantum kinematics and dynamics from constrained entropy maximization, recovering standard quantum mechanics as a special case while providing a relational, intersubjective interpretation free from ontological commitments.

ABSTRACT

We present the elements of a new approach to the foundations of quantum theory and probability theory which is based on the algebraic approach to integration, information geometry, and maximum relative entropy methods. It enables us to deal with conceptual and mathematical problems of quantum theory without any appeal to frameworks of Hilbert spaces and measure spaces.

Motivation & Objective

• To develop a foundation of quantum theory independent of Hilbert spaces and measure spaces.
• To resolve foundational conflicts in probability theory (e.g., Bayes–Laplace vs. Borel–Kolmogorov) using the Daniell–Stone integration framework.
• To establish quantum kinematics and dynamics through constrained maximization of quantum relative entropy on $W^*$-algebras.
• To provide a relational, intersubjective interpretation of quantum theory that avoids operationalism and realism.
• To recover standard quantum mechanics as a special case of the proposed framework via isomorphisms to $L_2(\mathcal{N})$ spaces.

Proposed method

• Replaces Borel–Kolmogorov probability foundations with integration on abstract commutative and non-commutative $C^*$-algebras.
• Uses the Daniell–Stone theory of integration on vector lattices to resolve inconsistencies in classical probability foundations.
• Equips the space of finite positive integrals with non-symmetric information deviation (negative relative entropy) to define information geometry.
• Introduces a family $D_p$ of relative entropy functionals as quantum Bregman entropies, inducing preferred information geometries on $\mathcal{M}(\mathcal{N})$.
• Defines quantum dynamics via constrained maximization of $D_p$, interpreted as non-linear projections in information geometry.
• Reconstructs standard Hilbert space quantum mechanics via isomorphism of $L_2(\mathcal{N})$ to Haagerup’s standard representation and GNS construction.

Experimental results

Research questions

• RQ1 Can quantum theory be reconstructed without relying on Hilbert spaces or measure-theoretic probability?
• RQ2 How can information geometry and relative entropy provide a unified foundation for both classical and quantum theories?
• RQ3 What is the role of the $D_p$ family of relative entropies in defining preferred quantum information geometries?
• RQ4 How does constrained entropy maximization generate dynamics in the absence of unitary evolution?
• RQ5 Can a relational, intersubjective interpretation of quantum theory be formulated without ontological or operationalist commitments?

Key findings

• The quantum information model $\mathcal{M}(\mathcal{N})$ is defined as a subset of normal positive finite linear functionals on a $W^*$-algebra $\mathcal{N}$, replacing the role of Hilbert space in kinematics.
• The $L_2(\mathcal{N})$ space is unitary isomorphic to both Haagerup’s standard representation and the GNS construction for any faithful state, enabling reconstruction of standard quantum mechanics.
• The mapping $\mathfrak{P}^{D_{\gamma}}_{F(t)}(\omega_0)$, derived from constrained entropy maximization, provides a non-linear projection that models temporal information dynamics.
• The family $D_p$ of quantum relative entropies corresponds to non-commutative $L_p(\mathcal{N})$ spaces and induces a preferred information geometry on $\mathcal{M}(\mathcal{N})$.
• The concept of a 'measured system' is reintroduced only as a tensor product decomposition, with purely epistemic meaning and no foundational role.
• The approach eliminates ontological commitments to terms like 'particles', 'fields', or 'universe', replacing them with intersubjectively verifiable experimental designs.