Discretised Hilbert Space and Superdeterminism

Abstract: In computational physics it is standard to approximate continuum systems with discretised representations. Here we consider a specific discretisation of the continuum complex Hilbert space of quantum mechanics - a discretisation where squared amplitudes and complex phases are rational numbers. The fineness of this discretisation is determined by a finite (prime-number) parameter $p$. As $p \rightarrow \infty$, unlike standard discretised representations in computational physics, this model does not tend smoothly to the continuum limit. Instead, the state space of quantum mechanics is a singular limit of the discretised model at $p=\infty$. Using number theoretic properties of trigonometric functions, it is shown that for large enough values of $p$, discretised Hilbert space accurately describes ensemble representations of quantum systems within an inherently superdeterministic framework, one where the Statistical Independence assumption in Bell's theorem is formally violated. In this sense, the discretised model can explain the violation of Bell inequalities without appealing to nonlocality or indefinite reality. It is shown that this discretised framework is not fine tuned (and hence not conspiratorial) with respect to its natural state-space $p$-adic metric. As described by Michael Berry, old theories of physics are typically the singular limits of new theories as a parameter of the new theory is set equal to zero or infinity. Using this, we can answer the challenge posed by Scott Aaronson, critic of superderminism: to explain when a great theory in physics (here quantum mechanics) has ever been grudgingly accommodated' rather thangloriously explained' by its candidate successor theory (here a superdeterministic theory of quantum physics based on discretised Hilbert space).