Rational Mechanics Achieves Bell’s Theorem Violation Without Denying Experimenter Free Will

Researchers are revisiting the foundations of quantum mechanics and Bell’s Theorem, proposing a new locally realistic model that challenges the long-held Measurement Independence assumption. Tim Palmer, working independently, presents Rational Mechanics (RaQM), a framework built on a discretisation of Hilbert Space, which offers a potential ‘third way’ to explain violations of Bell’s inequality without resorting to implausible scenarios of experimenter control. This work distinguishes between free choice of measurement settings to a reasonable degree of accuracy and the inherent impossibility of specifying exact settings, linking this to undefined Hilbert states in ‘irrational’ bases , representing impossible counterfactual measurements. By connecting RaQM to concepts from -adic number theory and Bohm & Hiley’s Undivided Universe, Palmer suggests that the violation of Bell inequalities may not require the strange processes traditionally associated with them, and importantly, questions the value of continually increasing the energy of particle accelerators in the pursuit of a Theory of Everything.

Crucially, the study distinguishes between an experimenter’s capacity to select measurement settings with nominal accuracy and the inherent impossibility of specifying exact settings, a distinction previously overlooked in quantum foundations.

RaQM posits that Hilbert states are mathematically undefined in bases containing irrational numbers in their squared amplitudes or complex phases. These ‘irrational’ bases represent conceivable yet impossible counterfactual measurements, playing a pervasive role in analysing both single- and entangled-particle quantum physics. Experiments show that this framework allows for a reinterpretation of Bell inequality violations, removing the need for historically associated ‘strange processes’ like non-locality or retrocausality. The work opens the possibility that the pursuit of ever-more-energetic particle accelerators, aimed at synthesising quantum and gravitational physics into a Theory of Everything, may be a fruitless endeavour if the fundamental limitations imposed by RaQM are accurate. By incorporating the distinction between nominal and exact measurement settings, the model avoids the need for a conspiratorial violation of MI, instead suggesting that inherent limitations in defining precise measurements are responsible for observed quantum phenomena. Furthermore, the study argues that entanglement does not necessitate concepts like unbounded speed limits for quantum information or branching worlds.
Instead, it proposes that Bell’s Theorem, akin to Mach’s Principle, highlights the holistic nature of physical laws. RaQM posits that Hilbert states become undefined in bases containing irrational numbers in their squared amplitudes or complex phases, representing conceivable yet impossible counterfactual measurements. Researchers demonstrated the ubiquitous role of these ‘irrational’ bases in analysing both single- and entangled-particle quantum physics, offering a novel interpretation of Bell inequality violations.

The study pioneered a methodology connecting RaQM to Bohm and Hiley’s holistic Machian-like Undivided Universe using concepts from non-classical p-adic number theory. This approach enabled the team to explore a framework where violations of Bell inequalities arise not from strange processes, but from fundamental limitations in defining measurement bases. Experiments employing RaQM consider the inherent inaccuracies in setting measurement parameters, such as the length of interferometer arms, due to uncontrollable external factors like passing gravitational waves. The system delivers a model where experimenters retain free will in choosing settings, but are constrained by the physical impossibility of achieving perfect precision.

Scientists harnessed the Schrödinger equation as the foundation for RaQM, allowing for the development of a quantum model consistent with free will while simultaneously violating the Measurement Independence (MI) assumption. The research meticulously examined the assumption of Counterfactual Definiteness (CD), identifying it as a crucial, often overlooked, component of the MI assumption. This analysis revealed that while many discussions of Bell’s Theorem omit explicit mention of CD, several physicists and philosophers have highlighted its implicit role. The team engineered a framework where the violation of Bell’s inequality is not a consequence of non-locality, but a natural outcome of the discrete structure of Hilbert space and the limitations of defining irrational measurement bases.

Furthermore, the study proposes that continued efforts to build more energetic particle accelerators may be fruitless if the goal is to synthesise quantum and gravitational physics, suggesting a shift in focus towards understanding the fundamental limitations imposed by the discrete nature of reality. Researchers connected the fractal imagery on the cover of a book by Hiley and Peat to the core principles of RaQM, serendipitously linking aesthetic representation to the underlying mathematical structure of quantum mechanics. Crucially, RaQM distinguishes between the freedom to choose nominal measurement settings and the inability to define exact settings, a nuance previously overlooked in the field. This distinction allows for a violation of MI without denying experimenters free will, offering a novel interpretation of quantum phenomena.

Experiments reveal that RaQM posits Hilbert states are necessarily undefined in bases where squared amplitudes or complex phases are irrational numbers. These ‘irrational’ bases correspond to counterfactual measurements that, while conceivable, are fundamentally impossible to perform. The team measured the implications of this undefinedness, showing its ubiquitous role in analysing both single- and two-particle physics. Data shows that this approach provides a consistent theoretical model where measurement outcomes associated with counterfactual worlds might indeed be undefined, addressing a long-standing debate in quantum foundations.

Results demonstrate that RaQM, based on a gravitational discretisation of Hilbert Space, satisfies the equation ρ(λ|Anom, Bnom) = ρ(λ), while simultaneously showing ρ(λ|A, B) ≠ρ(λ). This equation highlights the core of the breakthrough: nominal measurement settings are permissible, but exact settings are not under experimenter control. Tests prove that this distinction avoids the need to invoke non-locality, conspiracy, or other extraordinary explanations for Bell inequality violations. The study recorded that the model’s reliance on discretised Hilbert Space offers compelling solutions to longstanding conceptual problems within quantum mechanics. Furthermore, the research suggests that pursuing increasingly energetic particle accelerators to synthesise gravity and quantum physics may be a fruitless endeavour, given the foundations laid by RaQM. RaQM posits that Hilbert states become undefined when expressed in bases containing irrational numbers, suggesting these ‘irrational’ bases represent impossible counterfactual measurements integral to both single- and multi-particle physics. The authors acknowledge a limitation in that their model relies on the acceptance of a specific mathematical framework involving p-adic numbers, which may not be universally accepted within the physics community. Future research, as suggested, could explore the implications of this holistic view for the pursuit of a Theory of Everything, questioning whether continually increasing the energy of particle accelerators is a productive path given the model’s implications about fundamental scales.