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A Deflationary Account of Quantum Theory and its Implications for the Complex Numbers

This paper proposes a deflationary interpretation of quantum theory where systems are viewed as indivisible stochastic processes in configuration space, arguing that complex numbers are necessary specifically to ensure the Hilbert-space formalism functions as a valid Markovian embedding.

Original authors: Jacob A. Barandes

Published 2026-02-03

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Jacob A. Barandes

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

The Big Question: Why Do We Need "Imaginary" Numbers?

For nearly a century, physicists have been puzzled by one thing: Why does the math of quantum mechanics (the rules governing atoms and particles) rely so heavily on complex numbers? These are numbers that include "i" (the square root of -1), which don't exist on the regular number line.

Standard textbooks treat these complex numbers as fundamental building blocks of reality. This paper argues the opposite: Complex numbers are not the "real" stuff of the universe; they are just a clever mathematical trick.

The Core Idea: The "Map vs. Territory" Analogy

Imagine you are trying to describe a very bumpy, winding hiking trail.

The Territory (Reality): The actual trail is messy. To know where you will be in 10 minutes, you need to know where you were 5 minutes ago, 10 minutes ago, and maybe even 20 minutes ago. The path depends on your entire history. In physics, this is called a non-Markovian process (history matters).
The Map (The Trick): To make the math easier, you could invent a new way of describing the trail. Instead of tracking your position on the ground, you track your position and your momentum (speed and direction) together as a single, giant "state." Suddenly, the trail looks smooth and predictable. You only need to know your current "state" to predict the future. This is called a Markovian embedding.

The Paper's Claim: Quantum theory (with its wave functions and complex numbers) is just the "Map." It is a simplified, smooth mathematical representation of a much messier, history-dependent reality underneath.

The "Indivisible" Reality

The author suggests that the "real" underlying reality is a type of stochastic process (a random process, like rolling dice) that is "indivisible."

What does "indivisible" mean? Imagine a movie. In a normal movie, you can pause it, look at frame 10, and then look at frame 20, and the story flows logically from 10 to 20.
In an indivisible process, you cannot break the story down like that. Even if you know the state at time A and time B, you cannot simply multiply the probabilities to get the state at time C. The connection between A and C is "glued" together in a way that doesn't allow for simple, step-by-step calculation.
The Analogy: Think of a complex knot. If you try to untie it by looking at just one loop at a time, it makes no sense. You have to see the whole knot as a single, unbreakable unit to understand how it works. The "indivisible" process is that knot.

So, Where Do Complex Numbers Come From?

If the real world is just a messy, random knot of probabilities (using only normal, real numbers), why do we need "imaginary" numbers to describe it?

The paper argues that complex numbers are the price we pay for turning that messy knot into a smooth, easy-to-solve equation.

The Transformation: When you take that messy, history-dependent "knot" and force it into a smooth, first-order mathematical system (like the Schrödinger equation), the math demands a new kind of number to make the equations work.
The Matrix Trick: The author shows that you can represent these complex numbers using simple 2x2 grids of real numbers (matrices). It's like realizing that "i" isn't a magical ghost number; it's just a specific way of rotating a grid.
The Conclusion: We don't need complex numbers because the universe is "imaginary." We need them because they are the most efficient tool to translate the messy, indivisible reality into a clean, solvable math problem.

The "Strocchi-Heslot" Connection

The paper points to a specific mathematical discovery (by Strocchi and Heslot) that acts as a Rosetta Stone. They showed that a quantum system (which looks like a wave) is mathematically identical to a giant collection of coupled springs (classical harmonic oscillators).

The Spring Analogy: Imagine a room full of springs connected to each other. If you pull one, they all wiggle.
The Insight: The quantum "wave function" is just a fancy way of describing the position and speed of all these springs at once.
The Catch: For this to work, the "room" of springs has to be infinitely large, even for a single tiny particle (like an electron). This suggests that the quantum world is actually a massive, complex machine of springs, and the "wave" is just the shadow it casts.

The "Indivisible Interpretation"

The paper proposes a new way to look at quantum theory, called the "Indivisible Interpretation." Here is what it changes:

No "Spooky" Superposition: In standard quantum theory, a particle is often described as being in two places at once (superposition). In this new view, the particle is just in one place, but the probability of finding it there is part of a complex, indivisible knot. It's not "in two places"; it's just that the rules connecting past and future are too tangled to be broken down simply.
Wave Functions are Not Real: The wave function (the math symbol $\Psi$ ) is not a physical object floating in space. It's like a map legend or a recipe. It tells you how to calculate probabilities, but it isn't the food itself.
No Measurement Problem: The famous "Schrödinger's Cat" paradox (is the cat dead or alive?) disappears. The cat is always either dead or alive in the underlying reality. The confusion only arises because we are looking at the "Map" (the wave function) instead of the "Territory" (the indivisible process).

Summary

Think of the universe as a giant, complex puzzle where every piece is connected to every other piece in a way that depends on the entire history of the puzzle.

Old View: We think the puzzle pieces are made of "magic" (complex numbers) and that the picture is blurry until we look at it.
New View (This Paper): The puzzle pieces are just normal, everyday things (probabilities). The "magic" (complex numbers) is just the special language we invented to describe the puzzle quickly. The "blurry picture" (wave function) is just a shadow of the puzzle, not the puzzle itself.

By accepting this, the author argues we can strip away the "exotic" and "mysterious" parts of quantum theory and see it as a straightforward, albeit very complex, system of probabilities.

Technical Summary: A Deflationary Account of Quantum Theory and its Implications for the Complex Numbers

Problem Statement
The paper addresses a foundational mystery in quantum theory: the apparent necessity of complex numbers within the standard Dirac-von Neumann axioms. While previous literature has attempted to establish the necessity of complex numbers through specific measurement scenarios (e.g., Renou et al. 2021), this work adopts a more foundational approach. It questions whether the Hilbert-space formalism, with its reliance on complex numbers, is a fundamental description of reality or merely a mathematical artifact of a specific representation. The central inquiry is whether the complex numbers are ontologically required or if they arise as a tool to render a non-Markovian process Markovian.

Methodology
The paper employs a structural analysis of dynamical systems, specifically focusing on the concept of Markovian embeddings. The methodology proceeds through the following logical steps:

Review of Complex Numbers and Algebra: The paper establishes complex numbers ( $\mathbb{C}$ ) as a unique two-dimensional normed division algebra, noting their isomorphism to specific $2 \times 2$ real matrix structures (linear complex structures). It introduces the complex conjugation operator $K$ and its relation to pseudo-quaternions.
Markovian Embeddings: The author demonstrates how a non-Markovian system (where dynamics depend on past configurations) can be transformed into a Markovian system (depending only on the present state) by expanding the configuration space (ontology) to include additional variables. This "nomology-ontology trade-off" simplifies the dynamical laws (making them first-order) at the cost of a more complex state space. The paper illustrates this with discrete and continuous examples, showing how second-order laws can be rewritten as first-order systems using complex variables to unify the equations.
The Strocchi-Heslot Formulation: The paper reviews the reformulation of quantum mechanics as a system of coupled classical harmonic oscillators (Strocchi 1966; Heslot 1985). By decomposing the Schrödinger equation into real and imaginary parts, the Hilbert-space evolution is shown to be isomorphic to classical Hamiltonian dynamics. The paper argues that the infinite size of the Strocchi-Heslot state space (even for a qubit) suggests that the Hilbert-space formalism is an embedding of an underlying, extremely non-Markovian process.
Indivisible Stochastic Processes: The core theoretical innovation is the introduction of "indivisible stochastic processes." These are defined as equivalence classes of non-Markovian processes that agree on a sparse set of first-order conditional probabilities but fail the "iterative" property (i.e., $\Gamma(t \leftarrow t_0) \neq \Gamma(t \leftarrow t')\Gamma(t' \leftarrow t_0)$ ).
Stochastic-Quantum Correspondence: The paper establishes a bidirectional mapping:
- Stochastic-to-Quantum: Starting with an indivisible stochastic process, the non-negativity of transition probabilities is satisfied by introducing complex-valued "potential" matrices. For $N > 2$ , the requirement that these matrices be unitary (to preserve probability) necessitates the introduction of complex numbers, as real-valued orthogonal matrices are insufficient (the distinction between unistochastic and orthostochastic matrices).
- Quantum-to-Stochastic: Starting with a unitarily evolving quantum system, one can define an indivisible stochastic process by taking the squared moduli of the unitary evolution matrix elements ( $\Gamma_{ij} = |U_{ij}|^2$ ).

Key Contributions and Results

The Indivisible Interpretation: The paper proposes the "indivisible interpretation" of quantum theory. In this view, a quantum system is fundamentally an "indivisible" stochastic process unfolding in a classical configuration space. Wave functions and the Hilbert-space formalism are demoted from ontological status; they are secondary, derived mathematical tools used to describe the process.
Deflationary Account of Complex Numbers: The paper argues that complex numbers are not fundamental to the ontology of the universe. Instead, they are a necessary mathematical resource to ensure that the Hilbert-space formalism functions as a valid Markovian embedding of the underlying indivisible stochastic process. Specifically, they are required to represent the transition matrices as unitary operators when the system dimension $N > 2$ .
Resolution of the Measurement Problem (Claimed): By treating the measuring device as a subsystem within the overarching indivisible stochastic process, the paper claims the framework naturally yields the correct measurement-outcome probabilities (Born rule) without requiring a collapse postulate or treating superposition as a literal physical state of multiple configurations.
Linearity of the Schrödinger Equation: The linearity of the Schrödinger equation is derived as a consequence of the linearity of the law of total probability in the underlying stochastic process.
Ontological Status: The paper posits that configurations (the "hidden variables" in this context) are the primary ontological furniture, corresponding to what is observed in experiments. Wave functions are described as encoding a blend of epistemic and nomological information, rather than being physical entities.

Significance and Claims
The paper claims to provide a "deflationary" account of quantum theory that simplifies its axioms. By grounding quantum mechanics in ordinary probability theory and classical configuration spaces, the author argues that:

The "exotic" nature of quantum phenomena is an artifact of the Hilbert-space representation.
The theory avoids the measurement problem by reifying the stochastic process rather than the wave function.
The complex numbers are revealed as a mathematical convenience for handling the "indivisibility" of the underlying process, rather than a fundamental feature of nature.
The framework suggests a path toward a causally local microphysical theory of quantum influences.

The paper concludes that the Hilbert-space formalism serves as a powerful "analytical mechanics" for highly non-Markovian stochastic systems, providing a systematic method for calculating predictions and studying symmetries, but it is not the fundamental description of reality. Future research directions suggested include applications to dynamical systems, quantum causal models, statistical mechanics, and quantum gravity, as well as generalizations of quantum theory based on the axioms of indivisible stochastic processes.