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arxiv: 2604.05675 · v3 · submitted 2026-04-07 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

The final version of a recent approach towards quantum foundation

Inge S. Helland
Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:15 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum foundationsaccessible variablescomplementary variablesHilbert spaceBohr complementaritytheoretical variablesquantum formalism
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0 comments X

The pith

The full Hilbert space formalism follows from two complementary maximal accessible variables.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The author simplifies an earlier approach to quantum foundations by eliminating the need for an inaccessible variable that underlies all accessible ones. The new minimal assumption is the existence of two maximal accessible variables that are different, which can be called complementary. Starting from this, the complete mathematical structure of quantum mechanics, including the Hilbert space, is derived. This provides a first-principles route to quantum theory that is purely mathematical until physical variables are assigned. A reader would care because fewer questionable assumptions are required to reach the standard formalism.

Core claim

Assuming two different maximal accessible theoretical variables exist in the relevant context allows derivation of the entire Hilbert space formalism. Details on choosing the Hilbert space are provided. The theory is mathematical but becomes quantum mechanics upon identifying the variables with physical ones. The earlier requirement of an inaccessible φ is removed as unnecessary.

What carries the argument

Two complementary maximal accessible variables, from which the Hilbert space operators and states are constructed.

If this is right

  • The full quantum formalism is obtained without additional postulates.
  • The Hilbert space can be selected based on the accessible variables in the context.
  • The approach works as a general mathematical theory for any such pair of variables.
  • Physical quantum mechanics arises when the variables represent observables.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This minimal basis might simplify comparisons with other quantum interpretations that stress complementarity.
  • It opens the possibility of quantum-like structures in non-physical mathematical settings with complementary variables.
  • One could test the derivation by constructing explicit models with two maximal variables and verifying the emergence of superposition and entanglement.

Load-bearing premise

In the given context, two different maximal accessible variables exist.

What would settle it

A context with two maximal accessible variables where the derived formalism does not reproduce standard quantum predictions or where the Hilbert space cannot be properly chosen.

read the original 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 $\phi$ such that all the accessible ones can be seen as functions of $\phi$, 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript simplifies a prior axiomatic approach to quantum foundations by dropping the assumption of an inaccessible variable φ from which all accessible variables are functions. The central postulate is the existence, in a given context, of two distinct maximal accessible theoretical variables (Bohr-complementary). From this single assumption the author claims to derive the full Hilbert-space formalism, discusses the choice of a specific Hilbert space in some detail, and recovers standard quantum mechanics by interpreting the variables as physical; other mathematical applications are also indicated. All proofs are placed in an appendix.

Significance. If the appendix derivations are free of hidden structure and the two-variable assumption alone suffices to fix the Hilbert-space representation, the result would supply a notably parsimonious, Bohr-motivated foundation with no free parameters and no invented inaccessible entity. This would be a substantive contribution to the foundations literature. The current text, however, leaves the uniqueness of the representation and the completeness of the derivation unverified, so the significance remains conditional on those points.

major comments (2)
  1. [Abstract and Appendix] Abstract and Appendix: the claim that 'the whole Hilbert space formalism may be derived' from the mere existence of two maximal accessible variables is the load-bearing assertion, yet the main text provides no outline of the steps and the appendix is not reproduced here; without explicit verification that no additional ordering, inner-product definition, or preferred basis is tacitly introduced, the derivation cannot be confirmed to be parameter-free and non-circular.
  2. [Discussion of Hilbert space choice] Section discussing choice of Hilbert space: the text states that the Hilbert space 'can be chosen' and treats the choice 'in some detail.' This language indicates that the two-variable postulate alone does not uniquely determine the representation; any supplementary rule used to select the space must be shown to follow strictly from accessibility and complementarity, otherwise the simplification relative to the earlier φ assumption is incomplete.
minor comments (2)
  1. The title is self-referential ('final version'); a more descriptive title would better indicate the content for readers unfamiliar with the author's prior series.
  2. Notation for 'theoretical variables' and 'accessible theoretical variables' should be introduced with a single, compact definition before any use in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful assessment and for highlighting the potential parsimony of our simplified foundation. We address the two major comments below with clarifications drawn directly from the manuscript and appendix. Where the referee's points identify opportunities for improved exposition, we indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract and Appendix] Abstract and Appendix: the claim that 'the whole Hilbert space formalism may be derived' from the mere existence of two maximal accessible variables is the load-bearing assertion, yet the main text provides no outline of the steps and the appendix is not reproduced here; without explicit verification that no additional ordering, inner-product definition, or preferred basis is tacitly introduced, the derivation cannot be confirmed to be parameter-free and non-circular.

    Authors: The appendix constructs the Hilbert-space structure in a sequence of lemmas that begin solely from the existence of two distinct maximal accessible variables (Bohr-complementary) and the definition of accessibility. The inner product is induced by the joint probability structure on the pair of variables; no external ordering or preferred basis is postulated. The main text deliberately keeps the conceptual core short and defers technical detail to the appendix. To improve readability we will add a short paragraph in the main text that enumerates the principal steps of the derivation and points to the corresponding lemmas, while leaving the full proofs in the appendix. revision: partial

  2. Referee: [Discussion of Hilbert space choice] Section discussing choice of Hilbert space: the text states that the Hilbert space 'can be chosen' and treats the choice 'in some detail.' This language indicates that the two-variable postulate alone does not uniquely determine the representation; any supplementary rule used to select the space must be shown to follow strictly from accessibility and complementarity, otherwise the simplification relative to the earlier φ assumption is incomplete.

    Authors: The phrase 'can be chosen' is used to indicate that the same pair of complementary variables admits unitarily equivalent representations; the concrete selection of a particular Hilbert space is fixed by requiring that the two maximal accessible variables correspond to self-adjoint operators whose joint spectral measure reproduces the accessibility relations. No additional free parameter or external rule is introduced. We will revise the relevant section to state this derivation explicitly and to contrast it with the earlier φ-based construction, thereby confirming that the simplification is preserved. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained from stated postulates.

full rationale

The paper simplifies its foundation by dropping the inaccessible φ and posits only the existence of two distinct maximal accessible (complementary) variables as the essential assumption. From this it claims a mathematical derivation of the Hilbert-space formalism, with explicit discussion of representation choice. No quoted step reduces a prediction or central result to a fitted parameter, self-definition, or load-bearing self-citation chain; the mathematical construction is presented as independent of prior fitted values and externally motivated by Bohr's complementarity. The appendix proofs are deferred but the overall chain does not exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on a single domain assumption about the existence of complementary variables, with no free parameters or physical invented entities; the framework introduces mathematical concepts of theoretical and accessible variables but treats them as part of the postulate set rather than new physical postulates.

axioms (1)
  • domain assumption There exist two different maximal accessible variables in the given context.
    This is explicitly identified in the abstract as the essential assumption from which the Hilbert space formalism is derived.
invented entities (2)
  • theoretical variables no independent evidence
    purpose: To serve as the basic objects in the foundation approach
    Introduced as part of the mathematical framework for modeling variables.
  • accessible theoretical variables no independent evidence
    purpose: To distinguish variables that can be accessed or measured in the context
    Core conceptual building block of the simplified approach.

pith-pipeline@v0.9.0 · 5466 in / 1403 out tokens · 62029 ms · 2026-05-10T19:15:19.613662+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

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matches
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supports
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extends
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The paper appears to rely on the theorem as machinery.
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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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

    quant-ph 2026-05 unverdicted novelty 5.0

    Three authors each propose a resolution to Bell's theorem by dropping counterfactual definiteness, with one linking CHSH violation strength to spatial dimensions.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · cited by 1 Pith paper

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    work page 2002

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    Helland, I.S. (2024a). An alternative foundation of quantum mechanics. arXiv: 2305.06727 [quant-ph].Foundations of Physics54,

    work page arXiv

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    Helland, I.S. (2024b). A new approach towards quantum foundation and some consequences. arXiv: 2403.09224 [quant-ph].Academia Quantum1,

    work page arXiv

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    On probabilities in quantum mechanics.APL Quantum1, 036116

    Helland, I.S, (2024c). On probabilities in quantum mechanics.APL Quantum1, 036116. Helland, I.S. (2025a). Some mathematical issues regarding a new approach towards quantum foundation. arXiv: 2411.13113 [quant-ph].Journal of Mathematical Physics66, 092103. 10 Helland, I.S (2025b). Quantum probability for statisticians: Some new ideas.Methodology and Comput...

    work page arXiv 2026

  5. [5]

    and Friedman, A

    Susskind, L. and Friedman, A. (2014).Quantum Mechanics. The Theoretical Minimum.Pen- guin Books, New York. Zwirn, H. (2016). The measurement problem: Decoherence and convivial solipsism.Founda- tions of Physics46, 635-667. Appendix: Proofs of the main Theorems. Proof of Theorem

    work page 2014