A paradox of the Planck-world?

by Jürgen Buchmüller, March 2007

(Deutsch)

The designation Planck-world is the name for a physical model. The Planck-world is based on the discovery of the German physicist Max Planck, who justified the quantum theory in the year 1900. Planck defined the quantum energy designated after him and the Planck-world describes, among other things, a smallest length (the Planck-length) a shortest time (the Planck time) and the maximum mass of a conceivable elementary particle (the Planck mass).
[Alpha Centauri, BR, Professor Harald Lesch (German): What is the Planck-world?]

A well trained horse including bridle: Physics

The limits of the Planck-world do not seem surmountable. They would be comparable perhaps with the limited optical resolution of a certain microscope, however the Planck values are absolute, and they can not go below or above those limits. They can not be improved in any way.

The Wikipedia article about the Planck constant describes the connections between the Planck quantized energy, the gravitational constant and the speed of light in vacuum. The formulæ of the Planck units, which are based on the three natural constants for mass, length, and time, describe the limiting resolution of the spatial and temporal dimensions, as well as that of the mass. The equations for the Planck temperature and the Planck charge then introduce further natural constants, such as the Boltzmann constant and the permittivity of free space. A whole lot of derived units then follows from these new fundamental values.

A meaningful application of the conversion from the SI unit system with its kilograms, meters and seconds, to the system of the Planck units, is the simplification of equations in the field of quantum physics, the general relativity theory and cosmology. Some particular units or combinations of the three basic values, Dirac's constant, gravitational constant, or speed of light, are set to equal unity to achieve this kind of simplification. The combined substitution of all Planck units at once defines the natural system of units of quantum gravitation.

If you followed professor Harald Leschs lecture to the edge of reality, you probably noticed the critical point at which our physical conception of the world loses its ability to explain the universe. It is this the point at which the Heisenberg uncertainty of a particle's location is exactly equal to the Schwarzschild radius, which is the radius of a certain mass when it becomes a black hole. This Planck length amounts to about 1.61624 · 10-35m, and no lengths smaller than the Planck length are physically relevant, says Lesch. Now the Planck-world describes the beginning of the universe with its non-singular starting point at Big Bang time, which is 5.39121 · 10-44s, or the Planck time. The physical meaningfulness of the Big Bang thus did not start off with every value being zero, not with the singularity, but at those minimum and maximum values of the Planck-world. The search of physicists for an obvious effect of quantum gravitation, which is based on these values, meanwhile continues. Like Lesch briefly mentioned, physicists were — for example — endeavored to measure an effect of a quantized length in quantum space, with the minimum space cube's side lengths being just the Planck-length, on the propagation of light from far distances. Yet, until today, nobody was able to prove the existence of quantum gravitation this way.

Same nag bridled from the rear: Mathematics

A theologian was driving to his workplace at school at the other end of his city on each and every day. He started in point A and drove around the city — a city, which consisted of obviously very square houses and of very straight roads. He drove all the way from A up to point C, then turned his car right by 90 degress, and finally arrived at the school in point B. The distance from A to C amounted to 10 kilometers, just as the distance from C to B. In each case he drove 20 kilometers to and fro work, and 40 kilometers in total. Now he wanted to shorten the route and drove through the city, instead of around its outside. In the evening he looked at his odometer, and he was seriously disappointed, because he saw he drove exactly the same distance of 40 kilometers. Since he could not explain for himself why there was no saving in the distance, he made yet another attempt, and now turned his car left and right at each and every crossing, riding on a zig-zag course through the city. Yet, the length of his way remained constant. He went 20 kilometers to his workplace, and 20 kilometers back. Now the theologian was eager to learn what happened. He searched his cellar and stack, and found the collection of mathematical formulæ from highschool time, and he ascertained himself whether perhaps he thought wrongly. But no, there was this formula of Pythagoras which he remembered, and which of course remains valid to this very day: a2 + b2 = c2. He determined by means of his pocket calculator that the square root of 102km + 102km, i.e. 200km, should be about 14.14 kilometers. Why then, for heavens sake, did it always take 20 kilometers to school and another 20 kilometers back? Surely his zigzag course was not fine grained enough. It was not sufficiently similar to the shortest diagonal connection, which a helicopter in a direct flight path from point A to point B would go. It was not clear to him, though, why the distance did not even become just a little bit shorter. If he didn't reach the optimistic *click* *click* *click* 70.7 percent of the diagonal distance with his zigzag course yet, when would a measurable difference appear on his odometer?

He described his problem to a friend, who was a mathematician, and he asked him whether a shorter way, driving to his workplace with his car, would be possible, if he would drive on an arbitrarily fine grained zigzag course. He expressed his opinion that this course should certainly approximate the diagonal flight path of a helicopter, and that the length of this flight path should be 14.14 kilometers.

The friend now tried to explain to him the problem like this:

We must first specify what arbitrarily fine grained means, and whether a route on an arbitrarily fine grained raster is different from that on an infinitely fine grained raster.

If one drives on an arbitrarily fine grained zigzag course from A to B, then this does not change the very fact that he still always drives short vertical and horizontal sections, relating perpendicularly one to another, of the overall length of his path. Whether he drives half of these sections into one direction first, and the other half in the other direction afterwards, or whether he arbitrarily frequently changes the direction, neither choice does make a difference for the overall length of the path!

This is because he will never actually go on the shortest diagonal connection from A to B, which - as you already recognized - the Pythagoras formula tells you, would be equal to the square root of the sum of the distances a² plus b², with a being the distance from A to C, and b being the distance from C to B.

The reason why the way to your workplace doesn't become any shorter lies in the simple fact that the graphs of an arbitrarly fine grained zigzag curve, and that of the diagonal, do merely approximate — and they even approximate so far that these two graphs are alike in the border crossing to an infinitely fine grained raster — however, the path of the two graphs is not the same, because you always go on a detour by driving this zigzag course. Using a mathematically exact phrase this means: the direction vectors of the two graphs do not converge.

Our theologian's mind was now boggled. The explanation itself seemed so simple, and it was nevertheless against his intuitive expectation. Going a path of even infinitely fine grained sections, which are perpendicular one to another, is different from going the real diagonal path that otherwise looks exactly the same.

Warning! Questions.

A spectator of this double-ended horse bridling might now ask: "Where will this nag run to?"

And after the horse is truely bridled from both ends, I would like to leave the discussion of the question, or even the answer, if you dare, to You, the reader.

• Is the concept of a smallest length, the Planck length, mathematically and at the same time physically at all conceivable?
• How could Pythagoras' formula have ever been proven, or ever become provable by instrumentation, and thus physically?
• Does the acceptance of a Planck length not, at the same time, prevent the acceptance in reality of the mathematically defined direction vectors, if there is a smallest possible surface, or a smallest possible space in three dimensions, where anything — be it a ray of light as waves or photons, or any matter — would have to go or extend through diagonally?
• Is this question in itself senseless, because the physical, and also the mathematical explanation, are beyond their respective limits of meaningfulness?
• Is the question senseless, because energy — be it as a probability wave or as a photon flux — does not move in our macroscopic sense through the Planck-world, and does thus move neither diagonally, nor on a zigzag course?
• Then perhaps mathematics supplies meaningful statements for longer (strictly speaking: for shorter lengths) than physics?
• Did the physicists, who proceeded with the search for the effects of quantum gravitation, do this under the impossible assumption that a zigzag course of the light should, over a sufficiently long distance, yield a measurable deviation in the order of magnitude of the Planck length?
• And if physicists really expected this effect, could the same physicist then, at the same time, assume to be able to prove the formula of Pythagoras by any experiment that makes use of light or matter?
• Is Pythagoras (no longer) valid in the Planck-world?