Reconstructing Quantum Theory: From Complementarity to Quantum Properties Operationally

Quantum theory has always carried an uncomfortable tension at its core. On the one hand, physics as a discipline rests on the promise that if we know the state of the world precisely now, we can predict its future perfectly. On the other, quantum mechanics appears to deny that we can ever know the present state of a system completely. A new paper published in Foundations of Physics by Philip Goyal of the University at Albany confronts this tension head-on, offering an explicit model of what quantum properties actually are—a model that transforms Niels Bohr’s famously obscure principle of complementarity from a philosophical slogan into a precise, operationally grounded statement about the structure of reality.

The research, titled “From Complementarity to Quantum Properties: An Operational Reconstructive Approach,” builds on a recent operational reconstruction of Richard Feynman’s formulation of quantum theory. The reconstruction begins not with the abstract mathematical machinery of Hilbert spaces and operators, but with concrete questions about what measurements can be performed and what outcomes they can yield. From these operational primitives, Goyal shows that a particular form of complementarity—what he calls Complementarity I, the version Bohr formulated in his landmark 1928 Nature paper—emulates itself naturally out of the mathematics. This is the formulation Bohr expressed when he wrote that the space-time coordination of a system and the claim of causality, “the union of which characterizes the classical theories,” must be regarded as “complementary but exclusive features of the description.”

Goyal distinguishes carefully between two versions of complementarity that Bohr developed over roughly three decades. Complementarity I concerns different conceptual perspectives on a single physical situation—such as the wave-particle duality or the tension between specifying where a system is and predicting where it will be. Complementarity II, which Bohr emphasized increasingly after his 1935 engagement with the Einstein-Podolsky-Rosen paradox, concerns what can be learned about the same microscopic object using different experimental arrangements. Goyal argues that Complementarity I is the far more profound formulation, and his reconstruction demonstrates that a precise version of it can be extracted directly from the operational framework—one that mirrors a mathematical tension known from Heisenberg’s uncertainty relations.

The technical heart of the paper is a model of quantum properties that makes explicit what it means for a system to possess a property like position at a given moment. In classical physics, a particle possesses a definite position and a definite velocity simultaneously, and these properties jointly determine its future trajectory through deterministic laws of motion. In the quantum model developed here, the situation is subtler. Measurements in the framework are assumed to be repeatable: if a measurement at time t yields outcome m, then repeating that measurement an infinitesimally short time later yields m with probability approaching unity. This repeatability condition, rigorously stated for both discrete and continuous outcome spaces, ensures that measurement outcomes are not fleeting artifacts but reflect something stable about the measured system at the moment of measurement. It is this stability that grounds the ascription of genuine properties.

To articulate what kind of properties these are, Goyal draws on the ancient philosophical distinction between actuality and potentiality—a distinction stretching back to Aristotle and revived in modern quantum foundations by thinkers such as Henry Margenau, Abner Shimony, and Diederik Aerts. In this framework, some properties of a quantum system are actual: they are possessed in a determinate, definite way, and their values can be measured repeatably. Others remain potential: they are real possibilities encoded in the quantum state, with well-defined probabilities of becoming actual under suitable experimental conditions, but they are not possessed in the classical sense at the present moment. The quantum state, on this reading, is neither a catalogue of hidden classical facts nor a mere bookkeeping device for measurement statistics, but a representation of the full network of actual and potential properties of a system.

This model yields striking dividends when applied to long-standing conceptual puzzles. Perhaps the most celebrated is Zeno’s paradox of motion, the ancient argument that a flying arrow cannot move because at any instant it occupies a space equal to itself and is therefore at rest. Classically, this paradox exposes a genuine puzzle about the very concept of instantaneous velocity—whether a particle can truly be said to have a velocity at a single moment when velocity is defined as a ratio over an interval. The quantum property model resolves the paradox elegantly: at any instant, a system possesses an actual position but its velocity exists only as a potential property, a real propensity governing how position will evolve. There is no instant at which the arrow must be “at rest,” because the property needed to make that claim—simultaneous actual position and actual velocity—is precisely what quantum theory denies.

The model also illuminates phenomena closer to experimental practice. Electron diffraction, in which a beam of electrons produces interference patterns that seem to require the electron to behave as both particle and wave, becomes intelligible when position is understood as actual only upon measurement while the wave-like propagation of potentialities determines the probability structure of the eventual positions. Similarly, Goyal argues the framework provides reliable intuitions about the non-local behaviour of entangled states of non-identical particles—systems in which measurements performed on one particle seem to influence what can be said about another, distant particle. Rather than invoking spooky action at a distance, the property model describes such situations as involving a single web of actual and potential properties that does not factorize into independent properties for each particle, reflecting the deep nonseparability that philosophers such as Vassilios Karakostas have identified as characteristic of quantum objects.

Methodologically, the paper is notable for integrating three standpoints that are usually pursued separately. The operational standpoint, exemplified by work in the tradition of reconstructing quantum theory from information-processing principles—such as Jonathan Barrett’s analysis of generalized probabilistic theories and Christopher Fuchs’s quantum informational approach—treats measurements and their statistical structure as the primitive ingredients of physical theory. The reconstructive standpoint, defended by theorists such as Alexei Grinbaum and Philipp Berghofer, holds that interpretation of quantum theory should proceed by first deriving the theory’s formal structure from minimal operational assumptions, so that interpretive questions are answered by the reconstruction itself rather than grafted on afterward. The metaphysical standpoint, finally, supplies the conceptual vocabulary—actuality, potentiality, property, causality—needed to say what kind of world the reconstructed formalism describes. Goyal’s achievement is to weave these three threads into a single coherent fabric in which each constrains and supports the others.

The timing of the work is significant. Quantum theory passed its centennial in the mid-2020s, and despite a century of extraordinary predictive success, the interpretation of its formalism remains contested. Goyal argues that the proliferation of competing interpretations is partly a symptom of attempting to interpret quantum theory without first understanding why its particular mathematical structure—the complex amplitudes, the Born rule, the symmetrization postulate for identical particles—holds at all. His own research programme, which includes earlier derivations of the origin of complex quantum amplitudes and of Feynman’s rules from elementary considerations of measurement, aims to close precisely this gap. The new property model represents the interpretive payoff: once the formalism is derived operationally, the metaphysics of actuality and potentiality can be read off from the structure rather than imposed upon it.

The paper also engages carefully with classical puzzles about the mathematics of motion itself, including the debate over whether instantaneous velocities are physically meaningful—a question examined by philosophers such as Frank Arntzenius, Marc Lange, and Craig McCoy. By insisting that property ascriptions in the quantum model are always property-valued—there is never a system ascribed position without a corresponding value—and by grounding repeatability in precise probabilistic terms, the framework avoids ambiguities that have plagued earlier discussions. Goyal notes, too, that ideal position measurements yielding real-valued outcomes assume a continuum of spacetime; if spacetime were fundamentally discrete, this assumption would need revision, a caveat he states explicitly.

Whether the model will win broad acceptance remains to be seen, but it represents one of the most systematic attempts to date to make Bohr’s complementarity precise through reconstruction. What emerges is a vision of the quantum world in which exact knowability of the present and perfect predictability of the future are not both available, not because of experimental limitations or human ignorance, but because reality itself is woven from a fabric of actualities and potentialities in which the two desiderata trade off against each other at the deepest structural level. In that sense, the century-old insight of Bohr and Heisenberg is not merely preserved but finally given the rigorous foundation it has long been said to lack.