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تسجيل مستخدم جديدInvariant Set Theory: Violating Measurement Independence without Fine Tuning, Conspiracy, Constraints on Free Will or Retrocausality
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تأليف
Tim Palmer
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Invariant Set (IS) theory is a locally causal ontic theory of physics based on the Cosmological Invariant Set postulate that the universe $U$ can be considered a deterministic dynamical system evolving precisely on a (suitably constructed) fractal dynamically invariant set in $U$s state space. IS theory violates the Bell inequalities by violating Measurement Independence. Despite this, IS theory is not fine tuned, is not conspiratorial, does not constrain experimenter free will and does not invoke retrocausality. The reasons behind these claims are discussed in this paper. These arise from properties not found in conventional ontic models: the invariant set has zero measure in its Euclidean embedding space, has Cantor Set structure homeomorphic to the p-adic integers ($p ggg 0$) and is non-computable. In particular, it is shown that the p-adic metric encapulates the physics of the Cosmological Invariant Set postulate, and provides the technical means to demonstrate no fine tuning or conspiracy. Quantum theory can be viewed as the singular limit of IS theory when when $p$ is set equal to infinity. Since it is based around a top-down constraint from cosmology, IS theory suggests that gravitational and quantum physics will be unified by a gravitational theory of the quantum, rather than a quantum theory of gravity. Some implications arising from such a perspective are discussed.
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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.
Bells theorem is often said to imply that quantum mechanics violates local causality, and that local causality cannot be restored with a hidden-variables theory. This however is only correct if the hidden-variables theory fulfils an assumption called
Statistical Independence. Violations of Statistical Independence are commonly interpreted as correlations between the measurement settings and the hidden variables (which determine the measurement outcomes). Such correlations have been discarded as finetuning or a conspiracy. We here point out that the common interpretation is at best physically ambiguous and at worst incorrect. The problem with the common interpretation is that Statistical Independence might be violated because of a non-trivial measure in state space, a possibility we propose to call supermeasured. We use Invariant Set Theory as an example of a supermeasured theory that violates the Statistical Independence assumption in Bells theorem without requiring correlations between hidden variables and measurement settings.
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, wh
ich, if undertaken, would allow us to investigate whether standard quantum theory or invariant set theory best describes reality.
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In arXiv:1707.08641, Tim Maudlin claims to construct a counterexample to the result of Proc. Roy. Soc. A vol. 473, iss. 2202, 2017 (arXiv:1607.07871), in which it was shown that no realist model satisfying a certain notion of time-symmetry (in additi
on to three other assumptions) can reproduce the predictions of quantum theory without retrocausality (influences travelling backwards in time). In this comment, I explain why Maudlins model is not a counterexample because it does not satisfy our time-symmetry assumption. I also explain why Maudlins claim that one of the Lemmas we used in our proof is incorrect is wrong.
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