Citation? I was under the impression that this wasn't a solved problem, and both discrete and continuous models can be useful in different situations.
http://arstechnica.com/science/2011/11/the-insanely-weird-qu...
Totally agree with how unsettling our model of physics is if we grant that space-time is in fact discrete. I do believe, however, it is not so much about making the math easier as it is about making the universe conform to our perception of beauty. Continuous is just sexier than discrete. :-)
Obviously? It might be obvious to you but it isn't to me.
[1] http://en.wikipedia.org/wiki/Planck_length [2] http://en.wikipedia.org/wiki/Planck_units
1. http://en.wikipedia.org/wiki/Quark#Electric_charge
2. http://en.wikipedia.org/wiki/Fractional_quantum_Hall_effect
3. http://physicsworld.com/cws/article/news/1997/oct/24/fractio...
As for the Fractional quantum Hall effect, based on a quick glance, it seems that thus far only quasiparticees with charge e/3 have been discovered. So, while the theory could be correct, we will have wait and see if the other predicted fractional charges are detected.
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From the Quark wikipedia page you cited: What is the quantum of charge? All known elementary particles, including quarks, have charges that are integer multiples of 1⁄3 e. Therefore, one can say that the "quantum of charge" is 1⁄3 e. In this case, one says that the "elementary charge" is three times as large as the "quantum of charge". On the other hand, all isolatable particles have charges that are integer multiples of e. (Quarks cannot be isolated, except in combinations like protons that have total charges that are integer multiples of e.) Therefore, one can say that the "quantum of charge" is e, with the proviso that quarks are not to be included. In this case, "elementary charge" would be synonymous with the "quantum of charge".
"According to the generalized uncertainty principle, the Planck length is in principle, within a factor of order unity, the shortest measurable length - and no improvements in measurement instruments could change that."
There is no claim that it is the smallest length possible, and there is certainly no claim that every length has to be an integral multiple of the Planck length.
Some pertinent quotes:
http://todayinsci.com/QuotationsCategories/M_Cat/Measurement...
On careful examination the physicist finds that in the sense in which he uses language no meaning at all can be attached to a physical concept which cannot ultimately be described in terms of some sort of measurement. A body has position only in so far as its position can be measured; if a position cannot in principle be measured, the concept of position applied to the body is meaningless, or in other words, a position of the body does not exist. Hence if both the position and velocity of electron cannot in principle be measured, the electron cannot have the same position and velocity; position and velocity as expressions of properties which an electron can simultaneously have are meaningless. — Percy W. Bridgman Reflections of a Physicist (1950), 90.
The strength and weakness of physicists is that we believe in what we can measure. And if we can't measure it, then we say it probably doesn't exist. And that closes us off to an enormous amount of phenomena that we may not be able to measure because they only happened once. For example, the Big Bang. ... That's one reason why they scoffed at higher dimensions for so many years. Now we realize that there's no alternative... — Michio Kaku Quoted in Nina L. Diamond, Voices of Truth (2000), 333-334.
p2 doesn't follow from p1.
i.e. Planck lengths is the shortest measurable length and thus you can't measure anything of smaller length, and so the smallest length any object can have is the Planck length.