What is the Quantum Limit to Measuring Spacetime Geometry?

[marcus]

in case there's interest:

http://arxiv.org/abs/quant-ph/0505064
Quantum limits to the measurement of spacetime geometry
Seth Lloyd
11 pages
"This letter analyzes the limits that quantum mechanics imposes on the accuracy to which spacetime geometry can be measured. By applying the physics of computation to ensembles of clocks, as in GPS, we present a covariant version of the quantum geometric limit, which states that the total number of ticks of clocks and clicks of detectors that can be contained in a four volume of spacetime of radius r and temporal extent t is less than or equal to
rt/(pi x_P t_P),
where x_P, t_P are the Planck length and time. The quantum geometric bound limits the number of events or `ops' that can take place in a four-volume of spacetime and is consistent with and complementary to the holographic bound which limits the number of bits that can exist within a three-volume of spacetime."

$$\frac{rt}{\pi x_P t_P}$$

 

[Chronos]

Thanks for the link, marcus. Seth Lloyd always makes my brain hurt.

 

[marcus]

... Seth Lloyd always makes my brain hurt.

Kea said it right on a similar occasion. She said: "Where is setAI when we need him?"

but actually this Seth Lloyd paper is not so wild and it might not give you a headache.

the formula for the maximum number of spacetime measurement "events" per unit spacetime volume which one can have without forming an horizon seems rather nice, or at least simple

it is not exactly my cup of tea either but I thought it was the sort of thing that might appeal to several here at PF

 

[setAI]

nice find Marcus!

there is a very tantilizing puzzle in that it seems that the quantum geometric limit/ the Beckenstein information bound/ and the covariant entropy bound are all complementarily interelated- that they are different views of the same fundamental causal structure of spacetime- [and that this structure is equivalent to the wiring-diagram of quantum logic gates in a quantum computer]

 

[marcus]

nice find Marcus!
...

thanks, setAI. I am glad you showed up since you are most apt to know what to do with this.

In case you might be interested, I used Lloyd's formula to calculate the maximum density of ticks/clicks, or as he says "ops", in a cubic meter second.

You might wish to check my calculation, in case I made a careless mistake (and the battery on my calculator seems to be getting low)

I got that in the spacetime volume represented by a cubic meter lasting for one second there can be at most

8.7 E76 ops.

The way I reckoned this is to set r = 1 meter and t = 1 second, so that we are talking about a spherical volume lasting for one second (with a spacetime volume of 4pi/3.

and then I evaluated his formula for #, the number of ops that could exist in that spacetime volume. It came out 3.65 E77.
Then I divided by 4pi/3 and got 8.7 E76.


See if you get the same number.

So then we can imagine the densest possible swarm of clocks all ticking as rapidly as possible. Like a dense swarm of gnats where we have a very sensitive doctors stethoscope and can count the individual heartbeats of each gnat. And in a cubic meter, in one second, we listen and count all the heartbeats of all the gnats during that second. And it comes out 8.7 E76


In that case watch out because it means that the cloud of gnats is so dense that it is right on the verge of forming a black hole. and that is why you can't ever get any more "ops", according to Lloyd, than 8.7 E76 in one cubic meter during one second.