The World is Continuous not Discrete?

Double pendulum as Lagrangian mechanics in generalised angular coordinates as two discs modeled as particles of finite size carrying mass. 

The old question if the World on small scales is (i) continuous (fields) or (ii) discrete (particles) is handled by modern physicists by saying that it is both: fields create particles which create fields and so the World is both continuous and discrete. Both fields and particles. Very clever, but is it illuminating and more importantly, can it be true? 

An example of a continuum is the set $\Re$ of real numbers as digital (decimal o binary) expansions without limit on the number of digits. We may say that $\Re$ offers infinite precision, to be compared with finite precision if a limit is set to the number of digits. 

We may compare with the resolution of analog photo with no clear smallest size and a digital photo with a smallest pixel, and we may say that with the resolution of today it is difficult to distinguish between analog and digital.  We know that the Digital World is a world of finite precision, but what about the Real World?  

We know that the physical world has a range of scales from very large cosmological scales to very small atomistic or subatomic scales. It is a natural to believe that there is a biggest scale as the scale of the Universe, but we do not know its size. Atom physics is physics on small scales and it is natural to ask if there is a smallest scale. In quantum mechanics as the physics of small scales, Planck's constant $h$ serves the role of smallest scale of energy and also determines smallest scales in space and time. On the other hand, the mathematical models of quantum mechanics like Schrödinger's and Dirac's equations take the form of differential equations over the continuum of real numbers as continuum mechanics models.  

The Real World consists, roughly speaking, of matter with mass subject to gravitation and light as electromagnetic waves in a vacuum. It is natural to view a vacuum as a continuum without smallest scale and thus allow light as waves to arbitrarily short wave length/high frequency, even if there are limits because power increases quadratically with frequency. 

Concerning matter with mass the situation is less clear and has invited to use the concept of particle as something without extension in space yet capable of carrying mass, then as a concept borrowed from macroscopic continuum mechanics modeled as discrete systems of point masses to allow digital computation with finite precision as in the finite element method used in engineering. Here the continuum model remained as the starting point for discretisation for computation. 

Classical mechanics was perfected in Lagrangian mechanics in generalised coordinates of particle positions and velocities with equations of motion derived from a Principle of Least Action. This was extended to the Hydrogen atom by Both and Rutherford in 2013 with a Hydrogen atom viewed as a little planet system with an electron orbiting around a kernel. But this model could not explain the stability of the ground state and so was replaced in 1925 by Schrödinger's equation for a wave function as a continuum mechanics model. However, in extensions to atoms with more than one electron the particles of Lagrangian mechanics reappeared to serve the statistical interpretation required by the multi-dimensionality of the wave function. 

Atom physics thus took the form of classical Lagrangian particle mechanics rather than continuum mechanics of solid/fluid mechanics and electromagnetics, although Schrödinger's equation is a continuum mechanics model. Planck's constant $h$ then emerged as the smallest possible action.         

We find no compelling reason to insist that atom physics is particle physics, nor that an elastic body is the same as the mass-spring system effectively used in (finite element) computation.  

RealQM offers a continuum model of atoms with the electrons of an atom appearing as non-overlapping continuous charge distributions. See lecture on Structural Mechanics of the Atom.

Summary: Macroscopic particles of finite small size have physical meaning, while microscopic particles of no size does not seem to make much sense. Microscopics as continuum wave mechanics makes sense. There is no fundamental difference between macroscopic continuum wave mechanics and microscopic continuum wave mechanics, which opens to human understanding of microscopics from experience of macroscopics. Like a Hydrogen atom as a cloud of negative charge attracted by a positive kernel with minimal total energy as kinetic and potential energy.  

To be compared with the standard view that quantum mechanics cannot be understood, only used as a black box to predict outcomes of experiments.