Why does quantum theory need the complex numbers? With a view toward answering this question, I argue that the usual Hilbert-space formalism is a special case of the general method of Markovian embeddings. I then describe 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.
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The paper asks why quantum theory uses complex numbers and proposes interpreting its Hilbert-space formalism as a Markovian embedding of an underlying “indivisible” stochastic process. Such a process specifies configuration probabilities and transition matrices without requiring stochastic factorization through every intermediate time. Wave functions are treated as mathematical encodings rather than physical ontology.
The argument combines examples of Markovian embeddings with the Strocchi–Heslot reformulation of Schrödinger dynamics as classical Hamiltonian evolution of $N$ coupled oscillators with $2N$ real variables. For the stochastic-to-quantum construction, transition probabilities are written as $\Gamma_{ij}=|\Theta_{ij}|^2$. If $\Gamma$ is unistochastic, $\Theta$ can be unitary; otherwise, a Kraus construction and Stinespring dilation yield an enlarged unitary representation, with stated dimension bound $N'\leq N^3$. A sufficiently smooth unitary family gives a Schrödinger equation through $H(t)=i\dot U(t)U^\dagger(t)$. The proposed role of complex numbers follows from the fact that, for $N>2$, unistochastic matrices need not be orthostochastic.
Conversely, the paper assigns a stochastic transition matrix $\Gamma_{ij}(t\leftarrow0)=|U_{ij}(t\leftarrow0)|^2$ to unitary quantum evolution in a chosen basis. It argues that measurement probabilities are retained when apparatus and environments are included and the basis captures pointer configurations. Detailed measurement analysis and broader correspondence results are deferred to other work. The construction requires enlargement for general stochastic matrices and smoothness to obtain differential dynamics; claims concerning the measurement problem and causal locality are interpretive conclusions, with the latter supported by separate work.
No sign of AI-generated textOverclaimsUnclear noveltyStrong assumptionsPossible error
Central claim may not hold: The claimed necessity of complex numbers is not established: every column-stochastic matrix has a real orthogonal dilation reproducing its transition probabilities after auxiliary configurations are marginalized, so fixed-dimension nonorthostochasticity is insufficient.
The proposed embedding represents transitions from one selected conditioning time but does not establish simultaneous reproduction of all conditioning-time transition maps allowed by the stated stochastic-process axioms.
The displayed Strocchi–Heslot Hamiltonian has the wrong sign on its imaginary-Hamiltonian cross term and therefore does not reproduce the stated Schrödinger equation when B is nonzero.
Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.
Why does quantum theory need the complex numbers? With a view toward answering this question, I argue that the usual Hilbert-space formalism is a special case of the general method of Markovian embeddings. I then describe 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.
The paper asks why quantum theory uses complex numbers and proposes interpreting its Hilbert-space formalism as a Markovian embedding of an underlying “indivisible” stochastic process. Such a process specifies configuration probabilities and transition matrices without requiring stochastic factorization through every intermediate time. Wave functions are treated as mathematical encodings rather than physical ontology.
The argument combines examples of Markovian embeddings with the Strocchi–Heslot reformulation of Schrödinger dynamics as classical Hamiltonian evolution of N coupled oscillators with 2N real variables. For the stochastic-to-quantum construction, transition probabilities are written as Γij=∣Θij∣2. If Γ is unistochastic, Θ can be unitary; otherwise, a Kraus construction and Stinespring dilation yield an enlarged unitary representation, with stated dimension bound N′≤N3. A sufficiently smooth unitary family gives a Schrödinger equation through H(t)=iU˙(t)U†(t). The proposed role of complex numbers follows from the fact that, for N>2, unistochastic matrices need not be orthostochastic.
Conversely, the paper assigns a stochastic transition matrix Γij(t←0)=∣Uij(t←0)∣2 to unitary quantum evolution in a chosen basis. It argues that measurement probabilities are retained when apparatus and environments are included and the basis captures pointer configurations. Detailed measurement analysis and broader correspondence results are deferred to other work. The construction requires enlargement for general stochastic matrices and smoothness to obtain differential dynamics; claims concerning the measurement problem and causal locality are interpretive conclusions, with the latter supported by separate work.
This paper argues that every quantum system can be understood as a sufficiently general kind of stochastic process unfolding in an old-fashioned configuration space according to ordinary notions of probability. This argument is based on an exact correspondence between the class of `indivisible' stochastic processes and quantum theory. This new stochastic-quantum correspondence demotes the wave function from a primary ontological ingredient to a secondary mathematical tool, and yields a deflationary account of exotic quantum phenomena, such as interference, decoherence, entanglement, noncommutative observables, and wave-function collapse. At a more practical level, the stochastic-quantum correspondence leads to a novel reconstruction of quantum theory, alongside the Hilbert-space, path-integral, and quasiprobability representations, and also provides a framework for using Hilbert-space methods to formulate highly generic, non-Markovian types of stochastic dynamics, with potential applications throughout the sciences.
We introduce a realist, unextravagant interpretation of quantum theory that builds on the existing physical structure of the theory and allows experiments to have definite outcomes, but leaves the theory's basic dynamical content essentially intact. Much as classical systems have specific states that evolve along definite trajectories through configuration spaces, the traditional formulation of quantum theory asserts that closed quantum systems have specific states that evolve unitarily along definite trajectories through Hilbert spaces, and our interpretation extends this intuitive picture of states and Hilbert-space trajectories to the case of open quantum systems as well. We provide independent justification for the partial-trace operation for density matrices, reformulate wave-function collapse in terms of an underlying interpolating dynamics, derive the Born rule from deeper principles, resolve several open questions regarding ontological stability and dynamics, address a number of familiar no-go theorems, and argue that our interpretation is ultimately compatible with Lorentz invariance. Along the way, we also investigate a number of unexplored features of quantum theory, including an interesting geometrical structure---which we call subsystem space---that we believe merits further study. We include an appendix that briefly reviews the traditional Copenhagen interpretation and the measurement problem of quantum theory, as well as the instrumentalist approach and a collection of foundational theorems not otherwise discussed in the main text.
This paper introduces several new classes of mathematical structures that have close connections with physics and with the theory of dynamical systems. The most general of these structures, called indivisible stochastic processes, collectively encompass many important kinds of stochastic processes, including Markov chains and random dynamical systems. This paper then states and proves a new theorem that establishes a precise correspondence between any indivisible stochastic process and a unitarily evolving quantum system. This theorem therefore leads to a new formulation of quantum theory, alongside the Hilbert-space, path-integral, and quasi-probability formulations. The theorem also provides a first-principles explanation for why quantum systems are based on the complex numbers, Hilbert spaces, linear-unitary time evolution, and the Born rule. In addition, the theorem suggests that by selecting a suitable Hilbert space, together with an appropriate choice of unitary evolution, one can simulate any indivisible stochastic process on a quantum computer, thereby potentially opening up an extensive set of novel applications for quantum computing.
Standard quantum theory was formulated with complex-valued Schrödinger equations, wave functions, operators, and Hilbert spaces. Previous work attempted to simulate quantum systems using only real numbers by exploiting an enlarged Hilbert space. A fundamental question arises: are the complex numbers really necessary in the standard formalism of quantum theory? To answer this question, a quantum game has been developed to distinguish standard quantum theory from its real-number analog, by revealing a contradiction between a high-fidelity multiqubit quantum experiment and players using only real-number quantum theory. Here, using superconducting qubits, we faithfully realize the quantum game based on deterministic entanglement swapping with a state-of-the-art fidelity of 0.952. Our experimental results violate the real-number bound of 7.66 by 43 standard deviations. Our results disprove the real-number formulation and establish the indispensable role of complex numbers in the standard quantum theory.
We summarize a new realist interpretation of quantum theory that builds on the existing physical structure of the theory and allows experiments to have definite outcomes, but leaves the theory's basic dynamical content essentially intact. Much as classical systems have specific states that evolve along definite trajectories through configuration spaces, the traditional formulation of quantum theory asserts that closed quantum systems have specific states that evolve unitarily along definite trajectories through Hilbert spaces, and our interpretation extends this intuitive picture of states and Hilbert-space trajectories to the case of open quantum systems as well. Our interpretation---which we claim is ultimately compatible with Lorentz invariance---reformulates wave-function collapse in terms of an underlying interpolating dynamics, makes it possible to derive the Born rule from deeper principles, and resolves several open questions regarding ontological stability and dynamics.
The received view in the philosophy of physics is that a quantum theory is given its mathematical formulation in a so-called Hilbert space, with the states of a quantum system given by the elements of this space and observables—measurable quantities of the system—given by so-called operators on this space. But the status of this fundamental postulate becomes threatened when one moves to complex systems, like the quantum fields underlying fundamental particle physics or statistical systems composed of infinite collections of quantum particles. In these more complex cases, one finds many inequivalent Hilbert space representations that appear to be competing quantum theories. In looking for a mathematical structure of the theory to interpret as representing the physical world, one is faced with a choice: either arbitrarily pick one Hilbert space representation (out of the many competing representations) as a Hilbert Space Conservative, or else look for the abstract algebraic structure that all of these competing representations have in common as an Algebraic Imperialist. This dissertation provides an extended argument for the second route.