David Tong argues that quantum mechanics is ultimately continuous, not discrete.
In other words, integers are not inputs of the theory, as Bohr thought. They are outputs. The integers are an example of what physicists call an emergent quantity. In this view, the term “quantum mechanics” is a misnomer. Deep down, the theory is not quantum. In systems such as the hydrogen atom, the processes described by the theory mold discreteness from underlying continuity. … The building blocks of our theories are not particles but fields: continuous, fluid-like objects spread throughout space. … The objects we call fundamental particles are not fundamental. Instead they are ripples of continuous fields.
Source: The Unquantum Quantum, Scientific American, December 2012.