Hardy's axiomatic approach to quantum theory revealed that just one axiom

distinguishes quantum theory from classical probability theory: there should

be continuous reversible transformations between any pair of pure states. It

is the single word `continuous' that gives rise to quantum theory. This

raises the question: Does there exist a finite theory of quantum physics

(FTQP) which can replicate the tested predictions of quantum theory to

experimental accuracy? Here we show that an FTQP based on complex Hilbert

vectors with rational squared amplitudes and rational phase angles is

possible providing the metric of state space is based on p-adic rather than

Euclidean distance. A key number-theoretic result that accounts for the

Uncertainty Principle in this FTQP is the general incommensurateness between

rational $\phi$ and rational $\cos \phi$. As such, what is often referred to

as quantum `weirdness' is simply a manifestation of such number-theoretic

incommensurateness. By contrast, we mostly perceive the world as classical

because such incommensurateness plays no role in day-to-day physics, and

hence we can treat $\phi$ (and hence $\cos \phi$) as if it were a continuum

variable. As such, in this FTQP there are two incommensurate Schr\"{o}dinger

equations based on the rational differential calculus: one for rational

$\phi$ and one for rational $\cos \phi$. Each of these individually has a

simple probabilistic interpretation - it is their merger into one equation

on the complex continuum that has led to such problems over the years. Based

on this splitting of the Schr\"{o}dinger equation, the measurement problem

is trivially solved in terms of a nonlinear clustering of states on $I_U$.

Overall these results suggest we should consider the universe as a causal

deterministic system evolving on a finite fractal-like invariant set $I_U$

in state space, and that the laws of physics in space-time derive from the

geometry of $I_U$. It is claimed that such a deterministic causal FTQP will

be much easier to synthesise with general relativity theory than is quantum

theory.