⚛️ quantum physics

The final version of a recent approach towards quantum foundation

This paper simplifies a previous approach to quantum foundations by removing the assumption of an inaccessible variable, demonstrating instead that the entire Hilbert space formalism can be derived solely from the existence of two complementary maximal accessible variables.

Original authors: Inge S. Helland

Published 2026-04-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Inge S. Helland

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to rebuild the entire rules of the universe (Quantum Mechanics) from scratch, but instead of starting with complex math equations or mysterious particles, you start with questions and answers.

This paper by Inge S. Helland is like a master architect saying, "We don't need to assume the universe is made of weird, invisible magic. We just need to assume that asking a question and getting an answer is the most fundamental thing that happens."

Here is the breakdown of his theory using simple analogies:

1. The Core Idea: The "Question" Variable

In the old versions of this theory, the author assumed there was a giant, invisible "Master Variable" (like a hidden script of the universe) that determined everything. But he realized this was too hard to prove.

In this final version, he drops that heavy assumption. Instead, he starts with a simple idea:

Theoretical Variables: These are just "things we can ask about."
Accessible Variables: These are questions we can actually answer.
Maximal Variables: These are the "ultimate questions." You can't ask anything more detailed about the system without changing the system itself.

The Analogy: Imagine you are looking at a spinning coin.

You can ask, "Is it Heads or Tails?" (This is an accessible variable).
But you can't ask, "Is it Heads, Tails, and spinning at exactly 4.32 degrees per second?" because the act of measuring the spin stops the coin from being a simple Heads/Tails question.
The "Heads/Tails" question is a Maximal Accessible Variable. It's the most you can know at once.

2. The Big Discovery: Two Different "Maximal" Questions

The paper's "magic trick" relies on one specific assumption: In any given situation, there are at least two different "ultimate questions" that cannot be answered at the same time.

The Analogy: Think of a die.

Question A: "What number is on the top face?" (1, 2, 3, 4, 5, or 6).
Question B: "What color is the face pointing North?" (Red, Blue, Green, etc., if we painted the die).

If you look at the die to see the number, you can't simultaneously know the exact orientation of the North face in a way that gives you a single, sharp answer for both. They are Complementary.

Helland says: "If you have two different, equally important questions that you can't answer together, the math of Quantum Mechanics automatically appears."

3. The Result: The "Hilbert Space" (The Playground)

When you have these two complementary questions, the math forces the universe into a specific shape called a Hilbert Space.

What is a Hilbert Space? Imagine a giant, multi-dimensional playground where every possible answer to every possible question lives.
The Operators: In this playground, asking a question (like "Is it Heads?") is like a machine that sorts the balls in the playground. If the ball is a "Heads" ball, the machine lets it pass. If it's "Tails," it stops it.
The Proof: Helland shows that if you start with just the idea of "Two Complementary Questions," you can mathematically prove that this playground must exist, and the machines (operators) must behave exactly like the equations in quantum physics.

4. What About the "Weird" Stuff? (Schrödinger's Cat)

Standard quantum mechanics says a cat can be both dead and alive at the same time (superposition). Helland's theory offers a different, more "human" view called the Epistemic Interpretation.

The Analogy: Imagine you are playing a guessing game with a friend. You don't know if the friend is holding a red or blue card. To you, the card is "Red and Blue" (a superposition of possibilities).
The Twist: The card isn't actually both colors. It's just that your knowledge is a mix.
Helland's View: Quantum states (the math describing the cat) aren't describing the cat itself; they are describing what a specific person (or group of people) knows about the cat.
- If you know the answer to Question A, your "state" is clear.
- If you try to switch to Question B, your state changes instantly.
- This solves the "Schrödinger's Cat" paradox: The cat is never actually dead and alive. It's just that you don't know which one it is until you ask the right question.

5. Why Does This Matter?

This isn't just about physics. The author suggests this "Question and Answer" framework can be used for:

Psychology: How people make decisions when they are confused between two options.
Statistics: How to handle data when you don't have enough information to be sure.
Quantum Decision Theory: Why humans sometimes make "irrational" choices that look like quantum jumps.

Summary

Think of the universe not as a giant clockwork machine, but as a giant conversation.

We can ask questions (Variables).
Some questions are the "best" we can ask (Maximal).
Sometimes, asking one "best" question makes it impossible to ask another "best" question at the same time (Complementarity).
The Magic: If you accept that these two questions exist, the strange, magical math of Quantum Mechanics falls out naturally, like a flower blooming from a seed.

Helland has stripped away the complicated "hidden variables" and left us with a simple, elegant foundation: Reality is defined by what we can ask, and what we can know.

1. Problem Statement

The paper addresses the foundational reconstruction of quantum mechanics (QM). Previous approaches by the author relied on a set of postulates, including a specific, difficult-to-motivate assumption: the existence of a fundamental inaccessible variable $\phi$ from which all accessible variables are derived as functions. This assumption created a metaphysical burden and lacked clear physical justification.

The core problem is to derive the Hilbert space formalism of quantum mechanics from a simpler, more intuitive set of mathematical assumptions, specifically eliminating the need for a pre-existing inaccessible variable $\phi$ . The author aims to establish a purely mathematical theory based on theoretical variables and accessibility, which can then be interpreted physically to recover QM, Quantum Decision Theory, and statistical links.

2. Methodology

The author employs a rigorous mathematical framework based on group theory, measure theory, and functional analysis. The methodology proceeds as follows:

Definitions:
- Theoretical Variables: Undefined entities (real scalars, vectors, or matrices) that may be accessible or inaccessible.
- Partial Ordering: A variable $\theta$ is "less than or equal to" $\lambda$ ( $\theta \le \lambda$ ) if $\theta = f(\lambda)$ for a Borel-measurable function $f$ . If $f$ is bijective, they are equivalent ( $\theta \sim \lambda$ ).
- Maximal Accessible Variables: Variables that cannot be further refined by accessible functions within a given context.
Core Assumption (The "Complementarity" Postulate):
- The theory assumes the existence of a context containing two non-equivalent maximal accessible variables, $\theta$ and $\eta$ , with similar ranges ( $\Omega_\theta \cong \Omega_\eta$ ).
- A transitive group $G$ acts on the range $\Omega_\theta$ with a trivial isotropy group and a left-invariant measure $\mu$ .
- This setup mirrors Niels Bohr's concept of complementary variables.
Mathematical Construction:
- The author constructs a Hilbert space $H = L^2(\Omega_\theta, \mu)$ .
- A regular representation $U_L$ of the group $G$ is defined: $U_L(g)f(\theta) = f(g^{-1}\theta)$ .
- A crucial step involves finding a function $f_0 \in L^2(\Omega_\theta, \mu)$ that is bijective with respect to $\theta$ . The paper proves that while a real-valued $f_0$ may not always exist (specifically when the range is unbounded in both directions), a complex-valued $f_0$ always exists. This necessitates a complex Hilbert space for general cases.
- A group $N$ is constructed acting on the pair $\phi = (\theta, \eta)$ , generated by $G$ , an independent copy $H$ , and a swap element $j$ (swapping $\theta$ and $\eta$ ).
- An irreducible representation $W$ of $N$ is constructed on the Hilbert space.

3. Key Contributions

The paper makes several significant theoretical contributions:

Simplification of Foundations: It successfully drops the assumption of a fundamental inaccessible variable $\phi$ . The entire formalism is derived solely from the existence of two complementary maximal accessible variables.
Derivation of Hilbert Space Formalism: The paper proves that the existence of two complementary variables necessitates a Hilbert space structure and symmetric (self-adjoint) operators corresponding to physical variables.
Resolution of Real vs. Complex Hilbert Spaces: Theorem 4 provides a precise condition for when the Hilbert space must be complex. If the range of the variable is unbounded in both directions (e.g., $-\infty$ to $+\infty$ ), a real-valued bijective function $f_0$ cannot exist in $L^2$ , forcing the use of complex numbers. This offers a mathematical derivation for why QM requires complex numbers.
Linking Operators to Variables: It establishes a one-to-one correspondence between accessible variables and self-adjoint operators, and between maximal variables and non-degenerate eigenvalues.

4. Key Results (Theorems)

Theorem 1: Given two non-equivalent maximal accessible variables $\theta$ and $\eta$ with similar ranges and a transitive group action, there exists a Hilbert space $H$ and two symmetric operators $A_\theta$ and $A_\eta$ corresponding to these variables.
Theorem 2: There exists a unitary operator $S$ such that $A_\eta = S^{-1} A_\theta S$ . This establishes the unitary equivalence between operators of complementary variables.
Theorem 3 (Finite Case):
- The eigenvalues of an operator $A_\zeta$ correspond exactly to the possible values of the variable $\zeta$ .
- A variable is maximal if and only if all eigenvalues of its corresponding operator are non-degenerate.
- Eigenvectors correspond to "questions" (e.g., "What is $\zeta$ ?") with sharp answers.
Theorem 4 (Hilbert Space Construction):
- A complex-valued bijective function $f_0$ always exists, allowing the construction of the Hilbert space.
- If the variable's range is bounded on one side, a real Hilbert space suffices.
- If the range is unbounded in both directions, the Hilbert space must be complex.
Proposition 1: Any accessible variable $\zeta$ (which is a function of a maximal variable) corresponds to a self-adjoint operator derived via the spectral theorem.

5. Significance and Applications

Epistemic Interpretation: The theory supports a general epistemic interpretation of quantum mechanics. Quantum states are not descriptions of an objective reality but represent the knowledge of an observer (or a communicating group) regarding theoretical variables. This aligns with QBism (Quantum Bayesianism) but generalizes it.
- Implication: This approach resolves paradoxes like Schrödinger's cat by restricting state vectors to eigenvectors of meaningful physical operators, effectively breaking the general superposition principle for macroscopic objects unless they are eigenstates of a specific operator.
Quantum Decision Theory (QDT): The framework applies to decision-making where a decision-maker faces two different maximal decision processes (complementary variables). This provides a foundation for QDT, extending beyond finite action sets to infinite ones.
Statistical Theory: The approach links to statistical parameter reduction. If two experiments focus on different maximal sub-parameters, the prior for the second experiment should be treated as a quantum probability.
Unification: The author suggests links to Quantum Field Theory and General Relativity, proposing that the concept of theoretical variables is a unifying thread across physics and statistics.

Conclusion

Helland's paper presents a "final version" of a reconstruction of quantum mechanics that relies on minimal, intuitive assumptions: the existence of complementary maximal accessible variables. By rigorously proving that these assumptions necessitate the Hilbert space formalism (including the necessity of complex numbers in unbounded cases), the author provides a mathematically self-contained foundation for quantum theory that avoids the metaphysical pitfalls of hidden variables while offering a coherent epistemic interpretation.