Apr 122015

Take something, say a cake, and break it in half. Take one piece and break it again. Keep breaking, until a small piece becomes a tiny crumb, then a barely visible speck, then an invisible molecule and an atom and a subatomic particle. Then what? However imperceptibly minuscule in size, an elementary particle is still something. As such, it can itself be broken in half – conceptually at least, if not in practice. So can each resulting piece, and so on. But we know that this cannot go on forever. At some point – at a size of about 10-35 metres, called the Planck length – an amazing phenomenon occurs: something becomes nothing. Where nothing is not an approximation to a really tiny something: it is nothing – nothing at all.

This is very odd, but indisputably true. We know it from Zeno’s Dichotomy paradox, a funny version of which is as follows: Zeno and Epicurus see a beautiful woman from a distance, who tells them: “Walk half the way between us, every ten seconds. When you get here, you can have me”. “Impossible!” – says Zeno and storms away. “Here I come” – says Epicurus (who, never mind, was born 150 years after Zeno) – “in a couple of minutes I’ll be close enough for all practical purposes”.

The distance between Epicurus and the lady – or, similarly, between Achilles and the tortoise – goes from something to nothing. Let’s say it starts at 100 metres. After 10 seconds, it is 50 metres. After 20 seconds, 25 metres. After N∙10 seconds, it is 100∙(1/2)N metres: 2.4 centimetres in two minutes. Epicurus is right. However, Zeno, the ascetic stickler, is not having it: 2.4 centimetres is still some distance. In ten seconds, it will be 1.2 centimetres, then 6 millimetres, 3, 1.5 and so on: it will never go to zero. But he is wrong. N such that 100∙(1/2)N is less than the Planck length P=1.616199∙10-35 is Log(P/100)/Log(0.5), which is about 122.2: it takes 1222 seconds – 20 minutes and 22 seconds – to go from 100 metres to nothing – nothing at all.

Notice that, in mathematical terms, Zeno is right. The infinite series S=x+x2+x3+… equals x/(1-x) for x<1. So, for x=1/2, S=1. S converges to 1, but it does so only ad infinitum. S has many more than 122 terms: it has an infinite number of them, each one a non-zero addendum to the series. In mathematics, numbers can be smaller – infinitely smaller – than the Planck length. But in reality nothing – no thing – can be. Some thing with a Planck length breaks into no thing. That’s why Epicurus is happy and Achilles beats the tortoise.

A Planck length is really, really small: 10-20 the diameter of a proton – far too small for direct measurement with currently available instruments. But it is also the shortest measurable length: the lower limit to how small things can be. Nothing can be smaller than P metres. Infinitesimal calculus, which contemplates numbers smaller than P, is valid and useful, but it does not reflect the reality of small things, where addenda in S beyond N>122 are exactly zero. Nothing is infinitesimally small.

Something turning into nothing is weird. But that’s just one of the many oddities we encounter in the quantum world. Here is an excellent introduction by Brian Greene:

In Greene’s last words:

As strange as quantum mechanics may be, what’s now clear is that there is no boundary between the worlds of the tiny and the big. Instead, these laws apply everywhere, and it’s just that the weird features are most apparent when things are small. And so the discovery of quantum mechanics has revealed a reality – our reality – that’s both shocking and thrilling, bringing us that much closer to fully understanding the fabric of the cosmos.

Indeed. Quantum mechanics shows that spacetime is inherently granular, as implied by Loop Quantum Gravity, one of the most promising attempts – put forward by Lee Smolin – at unifying quantum mechanics and general relativity in a so-called Theory of Everything. At a more basic level, however, granularity is a simple consequence of Zeno’s paradox. That’s why it is a paradox.

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