QR2.2.1 Continuum Problems

Continuum problems have plagued physics since Zeno’s paradoxes two thousand years ago (Mazur, 2008):

1. If a tortoise running from a hare sequentially occupies infinite points of space, how can the hare catch it? Every time it gets to where the tortoise was, the tortoise has moved a little further on; OR

2. If space-time is not infinitely divisible, there must be an instant when the arrow from a bow is in a fixed unmoving position. If so, how can many such instants beget movement?

To deny the first paradox exposes one to the second and vice-versa. Zeno’s paradoxes resurface today as infinities in field equations, e.g. an electron as a dimensionless point has infinite mass and charge density unless one assumes other dimensions as string theory does. The infinities of quantum field theory were resolved by the mathematical trick of renormalization that makes the infinities of field theory go away by requiring particles to interact via other particles in what are called Yang-Mills interactions. Dirac described it as follows:

Sensible mathematics involves neglecting a quantity when it turns out to be small – not neglecting it just because it is infinitely great and you do not want it!

Feynman said the same even more bluntly:

“No matter how clever the word, it is what I call a dippy process! … I suspect that renormalization is not mathematically legitimate.”

Renormalization is the mathematical trick that pulls physical reality from the quantum hat but digitizing physical reality achieves the same effect in a simpler way. It is often forgotten that continuity is a mathematical convenience not a proven empirical reality:

… although we habitually assume that there is a continuum of points of space and time this is just an assumption that is … convenient … There is no deep reason to believe that that space and time are continuous, rather than discrete…(Barrow, 2007) p57

In contrast, a digital reality can’t be continuous because there are no “half pixels” or “half cycles”. Quantum realism then answers Zeno’s questions as follows:

There is indeed an instant when the arrow is in a fixed, unmoving position but there is still movement as another quantum cycle generates the next physical state. Equally the hare cannot get closer to the turtle forever as there is a minimum pixel distance that can’t be divided further, so the hare catches the turtle.

 Denying the infinitely small avoids the infinitely large, so a digital world of irreducible pixels and indivisible ticks makes the infinities of continuity disappear, like ghosts in the day. Reality as a series of frames strung together, as in a movie, resolves the paradoxes that continuity cannot. Processing as a choice from a finite set by definition doesn’t allow infinite values.

Our space is discontinuous because it breaks down at the order of Planck length. To study the very small needs short wavelength light that is high energy light but putting too much energy into a space gives a black hole that hides information from us. If you probe the black hole with more energy it expands its horizon to reveal no more, so what is below the Planck length cannot be known. In the same way one can never observe below Planck time. Planck length and time are the irreducible limits of our digital reality.

Quantum realism predicts what current physics does not, that repeatedly dividing our space gives a pixel that can’t be split and that repeatedly dividing our time gives a cycle that can’t be paused. Just as closely inspecting a TV screen reveals only irreducible dots, closely inspecting our space reveals only irreducible pixels. If physical reality is an image on a screen, the screen’s resolution and refresh rate are defined by the Planck limits, where the pixel size is a Planck length of 10-33 meters and the refresh rate Planck time implies is 1043 times per second.

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