{\huge Zeno Proof Paradox}
Another discussion of paradoxes
\includegraphics[width=1.2in]{Zeno_of_Elea.jpg}
Zeno of Elea was a Greek philosopher who lived almost 2400 years ago. He is famous for the creation of paradoxes at the juncture of mathematics and the real world.
Today I want to talk about a type of Zeno paradox.
The paradoxes are claimed to be due to Zeno, although given the time gap, there is some discussion whether they are due to him or others. The central paradox has been claimed to be resolved by modern mathematics by some, and claimed to still be an important paradoxical notion by others.
Zeno's Most Famous Paradox
I assume that you probably know this paradox well: Here is a version from our friends at Wikipedia.
\includegraphics[width=3in]{zeno.png}
In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.
I think this is answered simply in math by ``so what?---the real line has lots of convergent subsequences, but they don't prevent you from graphing objects in motion as far as you like.'' Ken, however, agrees with those who say a satisfying resolution requires quantum mechanics. In a discrete time instant, Achilles either moves some multiple of a discrete unit of space, or he doesn't. With overwhelming probability it's the former, and so in some instant he lurches into a sudden lead and never looks back. There has been argument about further paradoxes that emerge with this view, but Ken believes they are rectifiable in several possible ways, not just in string theory's marriage of relativity and quantum, but also in the relational model of spacetime advanced by Lee Smolin.
The Lamp Paradox
I knew the above paradox well, but was not aware of another variant due to James Thomson, a British philosopher, who created it in 1954. A bit after Zeno. Of course Zeno could not have imagined the following, since he did not have lamps and switches. Again from Wikipedia here is the paradox: Consider a lamp with a toggle switch. Flicking the switch once turns the lamp on. Another flick will turn the lamp off. Now suppose that there is a being able to perform the following task: starting a timer, he turns the lamp on. At the end of one minute, he turns it off. At the end of another half minute, he turns it on again. At the end of another quarter of a minute, he turns it off. At the next eighth of a minute, he turns it on again, and he continues thus, flicking the switch each time after waiting exactly one-half the time he waited before flicking it previously. The sum of this infinite series of time intervals is exactly two minutes. So far no issue. The key question is: Is the lamp switch on or off after exactly two minutes? Good luck.
Zeno's Arrow
Perhaps you have seen the effect of a rapidly flashing strobe light making dancers appear to stay momentarily still. If the flash itself is just an instant, nothing in what you see will suggest motion. So how can the dancer be moving? Always one to go to extremes, Zeno let both the duration of a flash and the interval between flashes approach zero.
Because he didn't have strobe lights either, Zeno imagined a flying arrow rather than a dancer. His logic was: the arrow isn't moving to where it is not, because it doesn't have any time in which to do so, and it's not moving to here because it's here already. So it can't be moving. It sounds like something Yogi Berra would say, and maybe being able to picture a baseball as standing still is why Yogi was such a good hitter.
My Zeno Proof Paradox
My humble observation is about trying to prove hard theorems. The Zeno proof effect is based on my current difficulty. I am trying to solve an open problem---not that one. But it's still a hard open problem, as most open problems are. Else why would they be open?
I have a collection of ideas that all seem to shed light onto the problem at hand. Yet the set of ideas has yet not yielded a proof. But each time I am about to give up and work on something else, I seem to always get another small insight not the problem. The insight seems to take me a bit closer to my goal---but not enough to actually get me there.
Thus, I feel a bit like Achilles trying to reach the Tortoise. Each insight say halves my distance to the goal, the Tortoise. But I am only closer: the goal seems nearer, but it is still not quite here. If I get another insight then I am again a bit closer. Alas, did I say that?, alas I keep getting insights, keeping getting closer. Yet I never have been able to catch the Tortoise, never yet been able to finish the proof.
The worst part is that I have the illusion that I am making progress. Should I quit the ``race'' to find the proof? Or should I keep trying for more and more insights? Will the insights go on for ever? Or will they eventually yield a proof? Very frustrating, indeed.
This reminds me of a quotation from Stanislaw Ulam's book ``Adventures of a Mathematician'':
\includegraphics[width=3in]{ulam.png}
Quantum Zeno Effect
As usual strange things are standard in quantum mechanics, so what is a paradox becomes an effect. The quantum Zeno paradox, now an effect, is analogous to Zeno's arrow. Ken thinks the name is actually unfortunate, because the point is that the effect is felt already when you're close to 0 but not necessarily approaching 0. Indeed it has been seen in experiments. It involves the quadratic difference between amplitude and probability, between $latex {\epsilon}&fg=000000$ and $latex {\epsilon^2}&fg=000000$. The experiments were first performed in the 1970's, but the idea was remarked on by Alan Turing in 1954, and is implicit in the mathematical formulation of quantum mechanics laid out in a text by John von Neumann in 1932.
Here's how it works in a standard situation. Consider a population of unstable particles that follow an exponential decay rule: after $latex {t}&fg=000000$ time units one expects only an $latex {e^{-t}}&fg=000000$ portion to be left in the original state. Near $latex {t = 0}&fg=000000$, both $latex {e^{-t}}&fg=000000$ and its derivative are close to 1 (or -1). Hence the expected number of decays over a short timespan $latex {t = \epsilon}&fg=000000$ according to the exponential decay rule is well approximated by a linear function $latex {L(\epsilon)}&fg=000000$. At least this is what the rule says to expect.
However, the transformation taking a particle from the unstable state $latex {\phi}&fg=000000$ to a stable state $latex {\psi}&fg=000000$ is also linear and evolves over time. For small time intervals $latex {\epsilon}&fg=000000$ again we may approximate the exponential involved in unitary evolution $latex {e^{-iHt}}&fg=000000$ by a simple linear function, so as to describe the system as being in the state
$latex \displaystyle ((1-\epsilon)\phi + \epsilon\psi) \qquad \text{normalized by}\qquad \frac{1}{\sqrt{(1 - \epsilon)^2 + \epsilon^2}}.&fg=000000$
Hence after a measurement, it is found to be in state $latex {\psi}&fg=000000$ with probability only
$latex \displaystyle \frac{\epsilon^2}{(1 - \epsilon)^2 + \epsilon^2} \simeq \epsilon^2 \ll \epsilon.&fg=000000$
(See this for a more proper derivation via unitary evolution.) Thus repeated measurements over tiny time intervals tend to preserve the unstable state $latex {\phi}&fg=000000$ with probability much closer to 1 than the model provides, even though the model is correct for larger times.
This is like saying if our strobe light performs such a measurement and we flash it fast at our decaying particles---regarding them as like a hail of flying arrows---a lot of them will appear to stay still. The ratio of those that do, instead of scaling as $latex {1 - \epsilon}&fg=000000$ as $latex {\epsilon \longrightarrow 0}&fg=000000$, approaches 1 much faster. The argument over this seems to go into murky questions over: what exactly is a quantum measurement?
Open Problems
When you are stuck on a proof, is it more like:
---?