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In a recent paper (arXiv:2107.04761), Sen critiques a superdeterministic model of quantum physics, Invariant Set Theory, proposed by one of the authors. He concludes that superdeterminism is `unlikely to solve the puzzle posed by the Bell correlations. He also claims that the model is neither local nor $psi$-epistemic. We here detail multiple problems with Sens argument.
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Relying on some auxiliary assumptions, usually considered mild, Bells theorem proves that no local theory can reproduce all the predictions of quantum mechanics. In this work, we introduce a fully local, superdeterministic model that, by explicitly violating settings independence--one of these auxiliary assumptions, requiring statistical independence between measurement settings and systems to be measured--is able to reproduce all the predictions of quantum mechanics. Moreover, we show that, contrary to widespread expectations, our model can break settings independence without an initial state that is too complex to handle, without visibly losing all explanatory power and without outright nullifying all of experimental science. Still, we argue that our model is unnecessarily complicated and does not offer true advantages over its non-local competitors. We conclude that, while our model does not appear to be a viable contender to their non-local counterparts, it provides the ideal framework to advance the debate over violations of statistical independence via the superdeterministic route.
One implication of Bells theorem is that there cannot in general be hidden variable models for quantum mechanics that both are noncontextual and retain the structure of a classical probability space. Thus, some hidden variable programs aim to retain noncontextuality at the cost of using a generalization of the Kolmogorov probability axioms. We generalize a theorem of Feintzeig (2015) to show that such programs are committed to the existence of a finite null cover for some quantum mechanical experiments, i.e., a finite collection of probability zero events whose disjunction exhausts the space of experimental possibilities.
In this comment we critically review an argument against the existence of objective physical outcomes, recently proposed by R. Healey [Foundations of Physics, 48(11), 1568-1589]. We show that his gedankenexperiment, based on a combination of Wigners friend scenarios and Bells inequalities, suffers from the main criticism, that the computed correlation functions entering the Bells inequality are in principle experimentally inaccessible, and hence the authors claim is not verifiable. We discuss perspectives for fixing that by adapting the proposed protocol and show that this, however, makes Healeys argument virtually equivalent to other previous, similar proposals that he explicitly criticises.
Invariant Set Theory (IST) is a realistic, locally causal theory of fundamental physics which assumes a much stronger synergy between cosmology and quantum physics than exists in contemporary theory. In IST the (quasi-cyclic) universe $U$ is treated as a deterministic dynamical system evolving precisely on a measure-zero fractal invariant subset $I_U$ of its state space. In this approach, the geometry of $I_U$, and not a set of differential evolution equations in space-time $mathcal M_U$, provides the most primitive description of the laws of physics. As such, IST is non-classical. The geometry of $I_U$ is based on Cantor sets of space-time trajectories in state space, homeomorphic to the algebraic set of $p$-adic integers, for large but finite $p$. In IST, the non-commutativity of position and momentum observables arises from number theory - in particular the non-commensurateness of $phi$ and $cos phi$. The complex Hilbert Space and the relativistic Dirac Equation respectively are shown to describe $I_U$, and evolution on $I_U$, in the singular limit of IST at $p=infty$; particle properties such as de Broglie relationships arise from the helical geometry of trajectories on $I_U$ in the neighbourhood of $mathcal M_U$. With the p-adic metric as a fundamental measure of distance on $I_U$, certain key perturbations which seem conspiratorially small relative to the more traditional Euclidean metric, take points away from $I_U$ and are therefore unphysically large. This allows (the $psi$-epistemic) IST to evade the Bell and Pusey et al theorems without fine tuning or other objections. In IST, the problem of quantum gravity becomes one of combining the pseudo-Riemannian metric of $mathcal M_U$ with the p-adic metric of $I_U$. A generalisation of the field equations of general relativity which can achieve this is proposed.
We identify points of difference between Invariant Set Theory and standard quantum theory, and evaluate if these would lead to noticeable differences in predictions between the two theories. From this evaluation, we design a number of experiments, which, if undertaken, would allow us to investigate whether standard quantum theory or invariant set theory best describes reality.
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