Area Law and accuracy of momentum measurement at plank energies

1. Jun 8, 2012

Prathyush

The general question that I am interested in, is to obtain the number of bits of information one can infer about the horizon by placing measuing apparatus only in a Region of size D.

I Imagine a telescore of diameter D that(or devices of that general nature)
The standard Theory says that*
angular resolutiom θ ~ λ/D
The larger the momentum the more accurate our resolution is.

How does this result change when we approach wavelength near plank length. The quantum mechanics seems to imply that resolution is infinite.
However we expect effects of quantum gravity to step in and limits the information that we can obtain about the horizon.

The number of angular bits at a given momentum is 4pi/θ^2 ~(d^2/λ^2)
(total solid angle/resolution)

The momentum is resolved upto ~h/D
(I am not entirely sure about correctness of this, but i expect some such relation to be true)

then,
The number of bits when when momentum P is measured is 4 pi/θ^2 ~ (d^2/λ^2)

To obtain the total number of bits we must integrate over all λ(d to plank length lp) and divide by resolution of mementum measurement.
no of bits = (D^2/lp^3) / (h/D) ~ D^3/lp^3

Our naive analysis implies that number of bits about the horizon scale as volume.

However, holographic principle says that number of bits must increase as Area
To recover the holographic principle we must posulate that the accuracy of momentum measurement at wavelenghts near plank scale is not h/D but h/lp

And We can recover the area Law for entropy.
D^2/lp^3 / (h/lp) = D^2/lp^2

It appears that the area law implies that, accuracy of momentum measurement near plank energies must be h/lp. Independent of the detection scheme involved.

I am not sure about about the validity of my assumptions and hence conclusions, but the Question of how many bits can one infer about the Horizon by making measurements in a small region of size D is of interest to me. Any comments or further reading will be much appriciated.

2. Jun 8, 2012

Staff: Mentor

If you want to measure areas down to the planck area, I would expect that the momentum resolution there does not depend on the total size of the object, but on the size of the observed region (which indirectly scales with the observation wavelength). This removes the main contribution from the size of the object. It then just gives a lower limit for relevant frequencies, and even that might be just redundancy (as you scan the area with better resolution anyway). And without doing the math, I would expect a logarithmic dependence in this case.