# Irrational Distances in Quantised Geometry

1. Apr 16, 2006

### dx

Since distances have to be multiples of the quantum of length, how can there be irrational distances? Please provide a non-technical explanation if possible, or correct me if my assumption is wrong.

2. Apr 16, 2006

### Hurkyl

Staff Emeritus
I don't know of any theories that talk about distance, but I can mention a similar situation for area in LQG:

In LQG, area is given to a surface based on how many edges of a spin network pass through it.

So if all the edges are in their fundamental state, then the area of a surface is just some quantum of area multiplied by the number of edges.

But if an edge is more energetic, it will contribute a greater amount of area... which is simply not an integer multiple of its fundamental value.

But anyways, the point is that "quantized" does not mean "integer multiples of some fundamental value".

3. Apr 17, 2006

### rtharbaugh1

Hi dx.

Perhaps "distance" is not the accepted word. Still, I think I get the sense of your question. It is a very old question, going back to the Pythagoreans, a group of Greek mystics. They wanted all measurements to be rational, and were infuriated by the incontrovertible fact of square root of two as a sensible number. It doesn't end, it goes on out the decimal line forever, so it cannot be a multiple of any base integer. How irritating. So if you draw a square, one by one, this rotten, fuzzy cross-section square root of two emerges, real as it can be, but not nice. You can plot it on a piece of paper but it's position remains forever in doubt, unfinished. Enough to make poor Zeno mad. You can point to it, but in no rational sense is it at the position to which you point. Be as precise as you will and yet you can never put an arrow through that target. It recedes from sensibility indefinitely. The only thing we can do is make a mark, and call it good enough, put the fidgity hare to sleep and let the turtle overtake the goal. Not good enough to bet your money on, but the only way to win.

Anyway, we set the quantum length and call it good. This does not remove the onerous burden of irrationality. We can still never fix the separation of two points across the square. It is in unconscious recognition of this fact that early American towns were commonly plotted around a town square in the center of which stood a courthouse topped with a dome. The mystery of Pi, the scales and blindfold of justice, the rule of law blocked the unbalenced crossing. If you want to get from the drugstore to the barber shop, you have to go around.

The Planck length is the center of balance, unitary, like the speed of light and the horizon. The sword cuts, but the two sides are never equal. So the black hole contains more space than all the rest of the multiverse combined!

Just take it as a mystery and be glad, for it is the source of your free will, your right of being, and death forever to tyranny. You can make a definition, you can impose it upon others, but you can never change the fact that every definition is a lie. This is a wisdom gained at terrible expense, and yet is it yours for free, easy as plucking the apple from the famous tree. Taste it. Will the world ever be the same? Was it, in fact, ever the same before? No. And that is affirmation.

Richard.