The Tortoise and the Quantum Zeno Effect

Zeno was a Greek philosopher and mathematician. The most famous of his paradoxes is the paradox of Achilles and the tortoise: Achilles runs twice as fast as the tortoise, which is given a head start of 100 m (a head start of one stadium is probably what has been handed down, and Achilles runs 12 times as fast as the tortoise, but the exact values are irrelevant to the argument).

Achilles can demonstrably never catch up with the tortoise, for when he has caught up with the tortoise's head start, the tortoise is already 50 metres further on, and when he has reached that mark, the tortoise is already 25 metres further on, and so on ad infinitum.

Astonishing! Anyone would probably have thought that after 200 metres Achilles would come darting past the panting tortoise, and now this! Achilles simply cannot catch the tortoise, however much he struggles!

Zeno coined yet another paradox, presumably to defend the thinking of his teacher Parmenides ("There is only the infinitely One, and all motion is only illusion") with his own line of thought:

He said that a flying arrow occupies a certain, exactly delineated place at every moment of its trajectory. At an exactly delineated place the arrow is at rest, for at one place it cannot move. Since the arrow is therefore at rest at every moment, it should as a whole be at rest. Paradox: but we assume that the arrow flies.

Zeno commented on this:

"That which is moved moves neither in the space in which it is, nor in the space in which it is not."

G. W. F. Hegel says of this in "Science of Logic, The Doctrine of Essence":

"Something moves only not in that it is here in this Now and in another Now there, but in that it is in one and the same Now here and not here, in that it is at the same time in this Here and is not. One must concede to the ancient dialecticians the contradictions they point out in motion, but it does not follow from this that motion therefore is not, but rather that motion is the existing contradiction itself."

In contrast to Zeno, who claims that the arrow rests at its place, quantum mechanics states that the arrow at point x has no definable velocity at all! Heisenberg's uncertainty relation states that the more precisely the position of the arrow is determined, the more indeterminate its velocity is.

The closer we therefore approach the infinitely large as well as the infinitely small, the more not only our perception and its rational processing fail, but the laws of nature themselves:

The Planck units mark a limit for the validity of the known laws of physics. One must assume that for distances smaller than the Planck length (about 10 to the power of -35 metres) and times shorter than the Planck time (about 10 to the power of -43 seconds), space and time lose their familiar properties as a continuum. Every object that would be smaller than the Planck length would, on account of the so-called uncertainty relation, have so much energy or mass that it would collapse into a black hole.

The latest theories of quantum gravity now attempt to eliminate the infinities. Since, according to these theories, spacetime is both a field but also possesses granular properties, in Zeno's paradox above spacetime could not be cut into infinitely small pieces, for the quanta of the quantum field cannot be smaller than the Planck unit. Thus there is nothing infinitely small and also no singularity (where everything concentrates into a point without extension). Thus there is also no "Big Bang" as calculated by general relativity, but a "Big Bounce": after the universe has expanded for a certain time, it contracts again, but only concentrates up to the point at which the quanta have reached their minimum size, and then explodes again. The same applies to a black hole. This too accumulates mass only up to a certain degree and not up to the singularity.

Whether this theory is correct or not is beside the point. The fact is, however, that our rational thinking cannot operate beyond the Planck units. So either reality does us the favour of remaining within the range operable for thought, or we will never be able to calculate the nature of reality (and of course far less truly understand reality).

Quantum mechanics and relativity theory tell us that there is nothing truly objective. Everything is directly connected with the observer, i.e. for example that there is no objective absolute time, but all time is dependent on the relation of subject and object.

How our own observation can actually change reality becomes clear in the so-called Quantum Zeno Effect, an effect of quantum theory in which a decay process (e.g. of a radioactive atom) can be prevented by pure observation. Roughly, the argument works as follows:

When an atom decays, according to quantum mechanics it does not do so at a specific point in time, but passes into a superposition of the states "not decayed" and "decayed." If, however, one looks (measures) whether the atom has already decayed, then one finds only either "atom decayed" or "atom not decayed," whereby the probability is given by the proportion of the respective measured state in the superposition.

If now at the beginning the atom is in the state "not decayed," then after a very short time the admixture of the state "decayed" is still very small, namely smaller than one would expect according to the exponential decay law (this follows from continuity considerations, because the probability cannot become greater than 1). If one now looks to see whether the atom has already decayed, it will with very high probability not yet have decayed. Through the observation, however, it in this case passes again into the exact state "not decayed" and the decay begins again from zero.

Altogether one thus obtains, with frequent observation, a decay rate that lies clearly below the "unobserved" decay rate. If one lets the intervals of the observations approach zero (which amounts to continuous observation), then the decay probability also approaches zero, i.e. the continuously observed atom should on account of this observation no longer be able to decay at all.

And this is in fact the case! The Quantum Zeno Effect was experimentally confirmed at the University of Texas at Austin and other laboratories.

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