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The Requirement of (at least) Complex Structure for Quantum Mechanics

This paper argues that quantum mechanics fundamentally requires a complex structure, demonstrating that ostensibly real-valued formulations merely represent complex linear algebra and that complex numbers are strictly necessary for time evolution under the assumptions of state normalization and energy conservation.

Original authors: M. P. Vaughan

Published 2026-09-22

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: M. P. Vaughan

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

At the heart of how we understand the universe lies a peculiar mathematical tool: the complex number. In the standard language of physics, these are not just abstract curiosities but the very fabric of reality, used to describe how particles behave, how they move, and how they interact. For decades, a lingering question has haunted physicists: is this reliance on complex numbers a fundamental necessity of nature, or merely a convenient shortcut? Could the entire edifice of quantum mechanics be rebuilt using only ordinary, real numbers, stripping away the imaginary parts to reveal a simpler, more intuitive foundation? This question matters because if the universe can be described with real numbers alone, it would suggest that the strange, counterintuitive features of quantum theory are artifacts of our mathematical choices rather than intrinsic properties of the physical world.

A recent paper by M.P. Vaughan tackles this enduring puzzle by examining the attempts to construct quantum mechanics using only real numbers. The author argues that many of these so-called "real" formulations are deceptive. While they appear to use only real numbers on the surface, they secretly preserve the complex structure they claim to have discarded. The paper demonstrates that if you try to build a version of quantum mechanics that evolves over time while keeping energy constant, you cannot escape the need for this complex structure. Even if you write everything using real numbers, the rules you must follow to make the system work force those numbers to behave exactly like complex numbers. The study concludes that complex algebra is not just a helpful tool but a minimum requirement for time evolution in quantum systems, meaning that the universe, as we model it, fundamentally relies on this specific mathematical architecture.

To understand why this is a problem for the "real-only" camp, one must first look at how complex numbers are usually handled. A complex number is essentially a pair of real numbers working together, often visualized as a point on a two-dimensional plane. In standard quantum mechanics, these numbers are used to calculate probabilities and describe the state of a particle. Some researchers have tried to replace these complex numbers with pairs of real numbers, arranging them into grids or matrices. On paper, this looks like a successful translation: the imaginary unit, often called "i," is replaced by a specific arrangement of real numbers. However, Vaughan points out that this is a disguise. When you perform calculations with these real-number grids, the rules you must follow to keep the math consistent are identical to the rules for complex numbers. The complex arithmetic is still there, hidden inside the real-number structure, much like a secret code that only reveals its true nature when you try to make the system change over time.

The core of Vaughan's argument rests on two fundamental pillars of physics: the conservation of energy and the requirement that a system's total probability remains constant as it evolves. In any physical theory, if you want to describe how a system changes from one moment to the next, you need a mathematical operator that drives that change. In the real-number versions of quantum mechanics, this operator must be "antisymmetric," a specific type of mathematical arrangement where the numbers flip signs across the diagonal. When the author applies the strict rule that energy must be conserved—meaning the total energy of a closed system does not change over time—a surprising constraint emerges. For the math to work without breaking the laws of physics, the energy levels of the system must come in pairs. Every energy state must have a twin, a duplicate with the exact same energy value.

This requirement for paired energy levels forces the mathematical structure of the system to double in size. If you start with a system that has a certain number of states, the real-number version must have twice as many dimensions to accommodate these pairs. More importantly, the way these paired states interact creates a specific pattern in the math. The author shows that this pattern is mathematically identical to the behavior of complex numbers. The real-number grids that describe the system's evolution end up behaving exactly like the complex numbers they were supposed to replace. The "imaginary" part of the equation is not gone; it has simply been reorganized into a block of real numbers that acts as a perfect substitute for the imaginary unit. The paper argues that this is not a coincidence but a necessity. Without this complex structure, the system cannot evolve in time while keeping energy constant.

The study also addresses the idea that perhaps even more complicated mathematical systems, such as those involving quaternions (a type of four-dimensional number system), could offer a way out. The author concedes that these more complex structures might also satisfy the requirements for time evolution. However, the crucial finding is that you cannot go simpler than complex numbers. You cannot strip away the complex structure and remain with a purely real, simple system. The complex structure is the floor, the minimum requirement. Any successful theory of quantum mechanics, whether it uses complex numbers, quaternions, or some other hyper-complex entity, must include a mathematical framework that mimics the behavior of complex numbers.

This conclusion has significant implications for how we view the nature of reality. It suggests that the debate over whether the universe is "real" or "complex" is somewhat of a false dichotomy. The paper argues that the attempts to create a purely real version of quantum mechanics have not actually removed the complex numbers; they have only hidden them behind a different notation. The complex algebra is still doing the heavy lifting. The author emphasizes that believing these formulations have done away with complex numbers is misleading. It prevents researchers from gaining a deeper insight into the fundamental nature of quantum mechanics. Instead of thinking we have simplified the theory, we should recognize that the mathematics of complex numbers is likely a necessary feature of how the universe operates, even if we choose to represent it using real matrices.

The paper also touches on the challenges of applying these ideas to systems with multiple parts, such as entangled particles. When physicists try to combine two separate real-number systems to describe a larger, joint system, they run into trouble. If they simply multiply the real-number grids together, they create a mathematical space that is too large and includes possibilities that do not exist in the complex version. To fix this, they must impose extra rules to restrict the space, effectively forcing the real-number system to behave like the complex one again. This process of restriction is essentially a way of re-injecting the complex structure that was supposedly removed. The author notes that this is a recurring theme: any attempt to build a real-valued quantum theory that matches the predictions of the complex theory inevitably requires imposing a complex structure on the real numbers.

In the end, the paper serves as a corrective to the notion that complex numbers are merely a convenient fiction. The author argues that the complex structure is not an optional layer of mathematics but a fundamental requirement for describing time evolution in quantum systems. The "real" formulations that claim to bypass this requirement are, in reality, just complex linear algebra wearing a real-number mask. The imaginary unit "i" is not just a symbol; it represents a specific algebraic relationship between two real numbers that is essential for the universe to function as we observe it. To think that we can discard this relationship is to miss a vital insight into the nature of quantum dynamics. The universe may be described using real numbers, but only if those numbers are arranged in a way that preserves the complex structure, proving that the complexity is in the fabric of the theory itself, not just in our choice of symbols.

Technical Summary: The Requirement of (at least) Complex Structure for Quantum Mechanics

Problem Statement
The paper addresses a perennial debate in quantum foundations: whether complex numbers are fundamentally necessary for the formulation of quantum mechanics or if a formulation based solely on real number algebra is viable. While recent experimental work (Renou et al., 2021) suggested that real-valued quantum mechanics could be falsified, subsequent theoretical challenges (Hita et al., 2023; Hoffreumon and Woods, 2023) proposed real-valued frameworks that replicate complex predictions. The author argues that these "real-valued" constructions are often only nominally real. They preserve the underlying complex algebraic structure through specific constraints on operators and states, effectively disguising complex numbers as real matrices rather than eliminating them. The paper seeks to demonstrate that, under the physical requirements of state normalization and strict energy conservation, a complex structure (or a more complex hyper-complex structure) is mathematically required to allow for time evolution in any real Hilbert space formulation.

Methodology
The paper employs a mathematical analysis of operator mappings and time evolution constraints within real Hilbert spaces.

Representation of Complex Numbers: The author establishes the isomorphism between complex numbers and $2 \times 2$ real matrices. A complex number $z = a + ib$ is mapped to the matrix form $a\hat{I}_2 + b\hat{J}_2$ , where $\hat{I}_2$ is the identity and $\hat{J}_2 = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ acts as the imaginary unit ( $\hat{J}_2^2 = -\hat{I}_2$ ).
Construction of Nominally Real Operators: The paper analyzes the mapping of operators from an $N$ -dimensional complex Hilbert space ( $\mathbb{C}^N$ ) to a $2N$ -dimensional real Hilbert space ( $\mathbb{R}^{2N}$ ). By replacing each complex matrix element with its $2 \times 2$ real equivalent, the resulting $2N \times 2N$ matrices retain the complex arithmetic properties of the original operators via the Kronecker product structure ( $\tilde{A} = \hat{A}_r \otimes \hat{I}_2 + \hat{A}_i \otimes \hat{J}_2$ ).
Time Evolution and Normalization: The author derives the time evolution equation for a state $|\Psi(t)\rangle$ in a real Hilbert space. To satisfy the normalization condition $\langle\Psi|\Psi\rangle = 1$ , the generator of time evolution, $\tilde{\Omega}$ , must be an antisymmetric matrix.
Energy Conservation Constraint: The paper imposes the condition that the expectation value of energy must be strictly conserved ( $\frac{d}{dt}\langle E \rangle = 0$ ). This requires the Hamiltonian $\tilde{H}$ and the evolution generator $\tilde{\Omega}$ to commute ( $[\tilde{H}, \tilde{\Omega}] = 0$ ).
Degeneracy Analysis: By analyzing the commutation relation in the energy eigenbasis, the author demonstrates that for non-trivial time evolution to exist without violating energy conservation, the energy eigenvalues must be at least 2-fold degenerate. This forces the Hamiltonian and evolution operator into a block-diagonal form composed of $2 \times 2$ blocks, which inherently reconstructs the complex structure described in Section 2.

Key Contributions and Results

Identification of "Nominally Real" Constructions: The paper argues that many proposed real-valued formulations (including those by Stueckelberg, Hita et al., and Hoffreumon and Woods) do not eliminate complex numbers but rather represent them via $2 \times 2$ real matrices. The complex arithmetic is preserved through the restriction of operators to a specific subspace of the endomorphism of $\mathbb{R}^{2N}$ .
Proof of Minimum Requirement: The central result is a proof that any real Hilbert space construction allowing for time evolution, subject to state normalization and strict energy conservation, must possess a complex structure. Specifically, the energy spectrum must exhibit at least 2-fold degeneracy, and the evolution operator must commute with a universal antisymmetric operator $\tilde{J}$ (analogous to $i$ ) that squares to $-I$ .
Derivation of the Schrödinger Equation: By setting $\Omega_i = -E_i/\hbar$ , the author shows that the derived real-matrix evolution equation $\tilde{H}|\Psi(t)\rangle = \hbar \tilde{J} \frac{d}{dt}|\Psi(t)\rangle$ is structurally identical to the Schrödinger equation, with $\tilde{J}$ serving the role of the imaginary unit.
Extension to Hyper-Complex Structures: The analysis notes that while complex structure is the minimum requirement, the mathematical framework allows for more complex structures, such as quaternions (represented by $4 \times 4$ blocks), which would also satisfy the conservation and evolution requirements.
Multipartite System Consistency: The paper highlights a critical pitfall in constructing real-valued multipartite systems. If one maps individual subsystems to real spaces and then takes the tensor product, the resulting space has incorrect degrees of freedom and may falsely indicate entanglement. The correct procedure requires mapping the tensor product of the complex spaces first, then converting to the real representation, thereby preserving the complex structure across the composite system.

Significance and Claims
The paper claims that the belief that complex numbers can be "done away with" in quantum mechanics is mistaken and misleading. The author asserts that:

Complex Algebra is Necessary: The algebraic properties of complex numbers (specifically the existence of an element squaring to $-1$) are necessary to describe quantum dynamics, even if the underlying representation uses real matrices.
Misleading Terminology: Labeling these formulations as "real" obscures the fundamental nature of quantum mechanics. The "real" representations are merely different encodings of complex linear algebra.
Counter-Productive Research: If the goal of such research is to gain deeper insight into the fundamental nature of quantum mechanics, attempting to remove complex numbers is counter-productive. The correct insight is that quantum mechanics requires complex (or hyper-complex) algebra, which can be supported by real matrices but cannot be reduced to simple real scalar algebra without losing the ability to describe time evolution and energy conservation.

The paper concludes that while real Hilbert spaces can replicate the results of complex quantum mechanics, they do so only by imposing a complex structure upon them. Therefore, the necessity of complex numbers is not a matter of notation but a fundamental requirement for the theory's dynamical consistency.

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