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. Physics handles the infinities of quantum field theory by the mathematical trick of renormalization, that makes the infinities of field theory go away by requiring particles to interact via other particles in Yang-Mills interactions. Dirac described this tactic 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 was even blunter:

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

Renormalization pulls physical reality from the quantum hat although 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 space and time are continuous, rather than discrete…(Barrow, 2007) p57

Quantum realism concludes that space isn’t continuous because a digital reality has no half pixels and time isn’t continuous because it has no half cycles. Processing by definition chooses from a finite set that doesn’t allow infinite values. It 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, so the hare catches the turtle.

Denying the infinitely small avoids the infinitely large. 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.

Our space breaks down at the order of Planck length because it is discontinuous. 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 nothing below the Planck length can be known. Planck length and time are the irreducible limits of our reality.

This predicts what current physics doesn’t, that repeatedly dividing our space gives a pixel that can’t be split and 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 irreducible Planck lengths. If physical reality is a screen image, the Planck limits are its resolution and refresh rate, so the pixel size of physical reality is 10-33 meters and its refresh rate is 1043 times per second.