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

Why does quantum theory need the complex numbers? With a view toward answering this question, this paper argues that the usual Hilbert-space formalism is a special case of the general method of Markovian embeddings. This paper then describes the indivisible interpretation of quantum theory, according to which a quantum system can be regarded as an indivisible stochastic process unfolding in an old-fashioned configuration space, with wave functions and other exotic Hilbert-space ingredients demoted from having an ontological status. The complex numbers end up being necessary to ensure that the Hilbert-space formalism is indeed a Markovian embedding.

💡 Research Summary

The paper tackles the long‑standing question of why quantum theory appears to require complex numbers. It argues that the usual Hilbert‑space formalism is not a primitive structure but rather a special case of a more general mathematical technique called a Markovian embedding. In a Markovian embedding, a non‑Markovian dynamics—one that depends on past configurations as well as the present—is recast as a first‑order Markov process by enlarging the state space. The author illustrates this with simple discrete and continuous examples: a second‑order deterministic rule x(t+1)=F(x(t),x(t‑1)) becomes Markovian when the state is taken to be the pair (x(t),x(t‑1)). The same trick works for stochastic dynamics and for systems with arbitrarily many degrees of freedom.

A key observation is that once the state space is enlarged, the dynamics can be expressed compactly using a complex variable z(t)=x(t)+i y(t). The complex conjugation operator K then implements time‑reversal transformations (z→K z(−t)), showing that the introduction of i is mathematically convenient but not ontologically essential. The complex number thus serves as a bookkeeping device that bundles two real degrees of freedom into one algebraic entity, simplifying the description of the embedded Markov process.

The paper then revisits the Strocchi‑Heslot formulation, which rewrites a quantum system with an N‑dimensional Hilbert space as a classical Hamiltonian system of coupled harmonic oscillators. By expanding the state vector |Ψ⟩ into real and imaginary components (q_i, p_i) and separating the Hamiltonian into real symmetric and imaginary antisymmetric parts, the Schrödinger equation becomes a set of real first‑order equations identical to Hamilton’s equations. In this picture the factor i in the Schrödinger equation is nothing more than the structure that couples q and p; it does not represent a mysterious physical entity.

Building on these observations, the author proposes the “indivisible interpretation.” In this view a quantum system is regarded as an indivisible stochastic process defined on an ordinary configuration space. Different non‑Markovian processes that share the same sparse set of first‑order conditional probabilities (i.e., the same Markovian embedding) belong to the same equivalence class. Each class corresponds to a Hilbert‑space description, and the complex numbers are required precisely to guarantee that the Hilbert‑space formalism faithfully reproduces the embedded Markovian dynamics. Without the complex structure the embedding would fail, and the original non‑Markovian dynamics could not be recovered.

Thus the necessity of complex numbers is re‑interpreted as a structural requirement for the Markovian embedding rather than an intrinsic feature of quantum reality. The paper concludes that this perspective opens new avenues: it clarifies the relationship between quantum and classical stochastic theories, suggests alternative formulations of quantum information protocols that avoid unnecessary complex‑valued objects, and provides a fresh framework for studying highly non‑Markovian open quantum systems. Future work is suggested on extending the embedding to infinite‑dimensional systems, exploring the role of other normed division algebras, and investigating experimental signatures that could distinguish the indivisible stochastic picture from standard quantum mechanics.