Mechanics deduced from observations, no prior mathematical assumptions needed.

The foundations of quantum mechanics, despite their unparalleled predictive power, continue to invite re-examination of their underlying principles. Researchers are increasingly exploring alternative formalisms, not to supplant the existing theory, but to offer fresh perspectives on persistent conceptual challenges. Piotr Szańkowski, from the Institute of Physics, Polish Academy of Sciences, Davide Lonigro of Friedrich-Alexander-Universität Erlangen-Nürnberg, and colleagues present a novel approach in their paper, “Phenomenological quantum mechanics II: deducing the formalism from experimental observations”. They demonstrate a deduction of quantum formalism directly from observed multi-time probability distributions obtained through sequential measurements, deliberately avoiding pre-defined mathematical structures and instead building the theory from purely phenomenological inputs. The resulting formalism, while empirically equivalent to standard quantum mechanics, reveals conceptual distinctions that may offer new avenues for tackling unresolved problems within the field.

Contemporary quantum mechanics, while remarkably successful in predicting experimental outcomes, invites scrutiny and alternative formulations of its foundational structure. Researchers actively explore approaches that reconstruct the theory not from pre-defined mathematical axioms, but directly from observed phenomena, specifically the probabilities derived from sequential measurements of physical observables. This work contributes to this ongoing effort, aiming to deduce the formalism of quantum mechanics solely from these phenomenological inputs, establishing a theory that emerges naturally from observed probabilities.

The investigation focuses on multi-time probability distributions, quantifying the likelihood of obtaining specific measurement results at different points in time, and estimates these distributions from repeated sequential measurements performed on a quantum system. By analysing these probabilities, the authors seek to identify the underlying mathematical framework governing quantum behaviour, effectively ‘reverse-engineering’ the theory from experimental data and contrasting this approach with the traditional axiomatic formulation where mathematical structure is assumed a priori. Establishing this framework allows for detailed analysis and the potential for reassembling components in novel ways, extending predictive power and addressing conceptual challenges within quantum mechanics.

The authors propose a ‘bi-trajectory’ formulation, a mathematical structure intended to capture the evolution of quantum systems based on these observed probabilities, detailed in a preceding publication. This formalism serves as the foundation for the current investigation, which focuses on establishing a connection between this bi-trajectory approach and the more familiar Hilbert space representation, complex vector spaces equipped with an inner product, providing a framework for describing quantum states and their evolution.

This research presents a distinctive approach to quantum mechanics, reconstructing its formalism not from established postulates, but solely from observed probabilities derived from sequential measurements. Rather than beginning with Hilbert spaces and operators, the authors meticulously build the mathematical structure from the ground up, starting with multi-time probability distributions obtained through repeated observations of physical systems, deliberately avoiding preconceived notions about the underlying mathematical framework. The core innovation lies in the development of a ‘bi-trajectory formalism’, which describes the evolution of quantum systems through joint probabilities – bi-probabilities – that account for both the system being measured and the measuring apparatus itself.

This bi-trajectory approach fundamentally alters how quantum measurement is conceptualised, portraying measurement as an evolution governed by probabilities where both the measured system and the measuring device follow intertwined trajectories. These trajectories are not singular paths but rather probability distributions reflecting the inherent uncertainty of quantum systems, naturally leading to decoherence, the process by which quantum superposition gives way to classical definiteness without requiring the introduction of wave function collapse as a separate postulate. Decoherence arises organically from the interaction between the quantum system and the measuring device, as described by the evolving bi-probabilities.

The resulting formalism, while mathematically equivalent to standard quantum mechanics in terms of predicting experimental outcomes, differs substantially in its conceptual underpinnings, offering a fresh perspective on long-standing issues within the theory. This is particularly relevant when considering the measurement problem and paradoxes like Wigner’s friend, challenging the objectivity of quantum measurements, treating the measuring device as an integral part of the quantum system to reconcile different observers’ perspectives and avoid conceptual difficulties. The deliberate construction of the formalism from purely phenomenological inputs represents a significant methodological advancement, creating a framework grounded in observable data and offering a novel lens through which to examine the foundations of quantum mechanics.

The article presents a novel derivation of quantum mechanics, beginning solely from observed multi-time probability distributions obtained through sequential measurements, departing from conventional axiomatic approaches. The authors reconstruct the formalism without presupposing Hilbert spaces or other established mathematical structures, yielding a bi-trajectory formalism, a representation where the state of a quantum system evolves according to paired forward and backward trajectories, rather than a single wavefunction. The mathematical foundation rests upon the Kolmogorov extension theorem, ensuring a consistent probabilistic interpretation throughout, and exhibits a clear relationship to standard quantum mechanics, readily identifying analogues of familiar operators such as projection operators.

The authors demonstrate substantial conceptual differences between the derived formalism and the conventional approach, despite complete agreement in empirically testable predictions, prioritising trajectories and offering an alternative perspective on the quantum world. This reconstruction provides a fresh lens through which to examine long-standing conceptual issues within quantum mechanics, particularly those surrounding the measurement problem, circumventing the need for a separate measurement postulate or wavefunction collapse. Instead, measurement emerges as a process that selects a specific trajectory from an ensemble of possibilities, offering a potentially more intuitive understanding of state reduction.

Furthermore, the formalism provides a consistent framework for describing interactions between quantum and classical systems, functioning as a filter selecting specific trajectories from the quantum system, avoiding conceptual difficulties when reconciling these two domains. This approach offers a potentially more natural way to model hybrid systems and understand the role of observation in quantum mechanics, suggesting that this novel formulation, while empirically equivalent to standard quantum mechanics, may prove more successful in addressing persistent theoretical challenges.

This work presents a formalism derived solely from phenomenological observations, eschewing pre-defined mathematical structures, beginning with multi-time probability distributions obtained from sequential measurements of observables. The authors effectively reconstruct a framework from experimental data rather than imposing a theoretical model, yielding a formalism exhibiting affinities with Hilbert spaces, allowing for the identification of analogues to familiar quantum mechanical operators, such as projection operators. Despite the correspondence with established quantum mechanics regarding empirically testable predictions, the derived formalism diverges conceptually from the standard approach.

The core distinction lies in the construction of the theory, built upon ‘bi-trajectories’ and their associated probabilities, offering a novel perspective on fundamental issues within quantum theory, prioritising observable quantities and avoiding assumptions about underlying reality. This bi-trajectory formalism describes quantum systems through bi-probabilities, probabilities defined over trajectories, providing a detailed account of system evolution and historical information, contrasting with standard quantum mechanics which focuses primarily on probabilities of outcomes. By modelling measurement as a specific type of classical-quantum hybrid system, the authors circumvent traditional conceptual difficulties associated with the measurement problem, suggesting that wave function collapse arises naturally from the interaction between quantum and classical components.

The authors demonstrate the consistency of their approach through the application of the Kolmogorov extension theorem, ensuring well-defined probabilities across extended spaces, connecting to the objectivity of classical stochastic processes, and offering a means to describe classical randomness independently of the observer. This emphasis on objectivity and observable quantities positions the work as a contribution to Phenomenological Quantum Mechanics, a program dedicated to building theory from empirical foundations, and future work will likely focus on applying this formalism to specific physical systems, providing concrete examples of its predictive power and exploring its potential advantages over standard quantum mechanics. Investigating the implications of bi-probabilities for quantum information theory and exploring connections to other approaches to the measurement problem represent further avenues for research, with empirical validation, through the design of experiments capable of distinguishing between predictions of the bi-trajectory formalism and standard quantum mechanics, remaining a crucial step in establishing the validity and utility of this novel approach.