The final version of a recent approach towards quantum foundation

Abstract: 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 should be chosen. The resulting theory is a purely mathematical theory, but it leads to qunantum 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.

• The paper presents a rigorous axiomatic derivation of Hilbert space formalism from complementary accessible variables, reducing the need for global hidden variables.
• It employs group actions, spectral theorem, and Zorn-type maximality arguments to establish operator self-adjointness and unitary equivalence.
• The results, validated in finite spectrum cases, extend to interdisciplinary applications in quantum cognition, decision theory, and statistical inference.

A Finalized Axiomatic Approach to Quantum Foundations via Complementary Accessible Variables

Theoretical Framework and Motivation

This work presents a formalized mathematical structure for quantum foundations centering on the notion of theoretical variables which are characterized as either accessible or inaccessible. The technical core dispenses with earlier reliance on the existence of a global inaccessible variable $\phi$ encoding all accessible ones as functions, streamlining the set of axioms. Instead, the focal postulate becomes the existence of two non-equivalent, maximal accessible variables with similar ranges—a situation matching Bohr's concept of complementarity. This shift reduces irreducible assumptions and aligns the derivation of the Hilbert space formalism with more operationally transparent conditions.

The theoretical variables are only required to be real scalars, vectors, or matrices, with measurability under Borel functions. Partial ordering is defined through functional dependence, and maximal accessible variables are defined with standard Zorn-type maximality arguments. The existence of a transitive group action with a left-invariant measure on the range of a maximal accessible variable enables construction of the left regular representation on $L^2$ spaces, supporting Hilbert space constructions foundational for quantum mechanics.

Main Results: Reconstruction of Quantum Formalism

The essential technical advance is encapsulated in Theorem 1: Two non-equivalent maximal accessible variables with similar range and appropriate group action yield a Hilbert space $\mathcal{H} = L^2(\Omega_\theta, \mu)$ and two symmetric (often self-adjoint) operators corresponding to these variables. The identification of self-adjointness under weak conditions enables the spectral theorem to be imported directly, establishing correspondence between theoretical variables and observables.

Proposition 1 extends the operator correspondence to all accessible variables, preserving self-adjointness. Theorem 2 establishes unitary equivalence (similarity) of the operators associated to the two complementary variables, implementing the action of the symmetry group and providing operational and physical symmetry (e.g., position-momentum, spin components).

Theorem 3 specializes this structure to finite spectrum cases (e.g., spin/qubit systems), yielding direct interpretation of spectra as possible variable values, and a correspondence of eigenvectors to maximal sharp questions. Maximality is characterized by nondegeneracy of the spectrum.

A granular analysis of Hilbert space construction (Theorem 4) demonstrates that, depending on the topological and measure-theoretic properties of the range, the Hilbert space may be forced to be complex—excluding a general construction on the real field except for bounded ranges. This connects to known physical distinctions between real and complex quantum mechanics and circumscribes the mathematical landscape for potential generalizations.

Role and Implications of Inaccessible Variables

Although the existence of a global inaccessible variable $\phi$ is not required for the main derivation, the structure identifies important physical examples where such a variable is naturally defined (e.g., the full Bloch sphere state underlying spin projections). When assumed, the formalism neatly relates pairs of maximal accessible variables through transformations on $\phi$ and establishes that operator similarity captures this relational structure (Theorem 5).

This treatment clarifies how redundancy in the space of theoretical variables can be leveraged for additional structure but is not foundationally required for the core quantum mechanical machinery. The relational interpretation of accessibility aligns with epistemic interpretations of quantum states.

Interpretational and Applied Implications

A salient feature of the formalism is its compatibility with epistemic interpretations of quantum mechanics. The framework supports the view that quantum state vectors pertain to knowledge—localized to agents or agent groups—and not to direct statements about underlying reality. The derived restriction to eigenvectors of physically meaningful operators eschews unconstrained superposition, potentially sidestepping classical paradoxes such as Schrödinger's cat—a notable, explicit claim.

Further, the structure is shown to subsume quantum decision theory: Choice variables in decision making can be cast as complementary accessible variables. The general theory thus provides a route to quantum-like modeling in cognitive and macroscopic domains, aligning with the program of quantum cognition and quantum-like models in non-physics contexts.

Connections to statistical theory are established via parameter reduction strategies and Bayesian updates, with prior probability for maximal variables acquiring the formal status of quantum probabilities. This signals implications for quantum statistical inference and foundations of statistical learning in high-dimensional or ill-posed settings.

Future Perspectives

The formal structure raises several lines of inquiry. The required existence of complementary maximal accessible variables is operationally motivated but invites scrutiny concerning its physical realization in quantum field theory and theories of quantum gravity. The dichotomy between complex and real Hilbert spaces also links to open problems regarding the field of scalars in quantum theory's ultimate formulation.

The approach provides a platform for deriving the Born rule and quantum probabilities from elementary operational assumptions, as developed in related works by the author. Further generalizations could explore the extension of this foundation to infinite-dimensional systems, open quantum systems, or quantum information scenarios involving generalized measurements.

Conclusion

This work achieves a streamlined, rigorous derivation of the Hilbert space and operator formalism underlying quantum theory, based on accessible, operationally interpretable postulates. By elevating the role of complementary accessible variables and reducing reliance on global hidden variables, the structure supports an epistemic reading of quantum mechanics and naturally extends to interdisciplinary applications. The implications for the interpretation of quantum states, the structure of quantum theory, and its generalizations to decision theory and statistics signal fertile ground for further foundational and applied research in quantum foundations and quantum-like frameworks.