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Fra

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Given the recent discussion about a possible formulation of holographic principle, and also what geometric notions suchs distance and area may mean without prior metric I thought I'd throw in this suggestive comparasion out for discussion. I feel the discussion is still too much tied to geometric notions, I think we nee dto be more radical.

Tom said he wante to understand the AdS/CFT beyond AdS, but how about trying to find a formulation that is more indepdenent of geometric notions altogheter, where we might even reconstruct the geometric notions from a deeper principle?

If we for a second, as a simple example consider the information state of an information processing observer to consist of "prior probabilities" of certain events encoded in a finite discrete form, meaning the values take on constrained rational values, not the full [0,1] contnuum, and the this observer wants to know the conditional probability P for "observing" a future sequence of M k-valued events ,the the multinomial distrubition gives.

[tex]P= \left\{M! \frac{ \prod_{i=1..k} (m_{i}/M)^{(m_{i})} }{ \prod_{i=1..k}m_{i}! } \right\} e^{-M S_{KL}}[/tex]

Define S = - ln P as usual then

[tex]S = M S_{KL} + ln w[/tex]

Also define D = maximum S_KL (normally this is infinitiy, but for a finite complexity observer, there is a maximum)

[tex]S \leq MD - ln w[/tex]

ln w -> 0, as the complexity of the observer is large so we can ignore it for the simple argument

[tex]S \leq MD[/tex]

It's simply interesting to compare this to the Bekenstein bound (ignore the constants)

[tex]S \leq ER[/tex]

The interesting thing here is that we get an interpretation of the variables.

S - is the "entropy" of the system beyond the observable horizon, but

E - is the number of events, or the "size" of the future-sequence. The amount of "data", rather than amount of information.

R - is the directed maximum "distance" from the observer measured looselt in terms of bits - how may "bits" do the observer need to flip - to the distinguishable horizon of possible states from the point of view of the inside observer.

This is to me, a pure INSIDE view, a "cosmological" view. Ie. humans looking as cosmo horizon. Now if we instead consider the flip side of this, we humans looking into a microscopic black hole the situation is flipped as the observer is on the "wrong" side of the horizon, so somehow there seems to be then that both the microscopics BH horizon and the cosmo horizon enters the same equations.

Also this is just a toy model, and yes S_KL is no proper metric but that's a more detailed discussion, but I'm curious to hear what progress we can expect in "this direction". Note that the above is just to fuel associations, I'm not making a precise hypothesis here, just noting that there are many extremely suggestive things here that has nothing to do with ST, AdS or anything else; but which might eventually help us understand how and why space is emerges.

Constructive Ideas in this direction?

/Fredrik

Tom said he wante to understand the AdS/CFT beyond AdS, but how about trying to find a formulation that is more indepdenent of geometric notions altogheter, where we might even reconstruct the geometric notions from a deeper principle?

If we for a second, as a simple example consider the information state of an information processing observer to consist of "prior probabilities" of certain events encoded in a finite discrete form, meaning the values take on constrained rational values, not the full [0,1] contnuum, and the this observer wants to know the conditional probability P for "observing" a future sequence of M k-valued events ,the the multinomial distrubition gives.

[tex]P= \left\{M! \frac{ \prod_{i=1..k} (m_{i}/M)^{(m_{i})} }{ \prod_{i=1..k}m_{i}! } \right\} e^{-M S_{KL}}[/tex]

Define S = - ln P as usual then

[tex]S = M S_{KL} + ln w[/tex]

Also define D = maximum S_KL (normally this is infinitiy, but for a finite complexity observer, there is a maximum)

[tex]S \leq MD - ln w[/tex]

ln w -> 0, as the complexity of the observer is large so we can ignore it for the simple argument

[tex]S \leq MD[/tex]

It's simply interesting to compare this to the Bekenstein bound (ignore the constants)

[tex]S \leq ER[/tex]

The interesting thing here is that we get an interpretation of the variables.

S - is the "entropy" of the system beyond the observable horizon, but

**calculated relative to the finite computational systme of the observer**!E - is the number of events, or the "size" of the future-sequence. The amount of "data", rather than amount of information.

R - is the directed maximum "distance" from the observer measured looselt in terms of bits - how may "bits" do the observer need to flip - to the distinguishable horizon of possible states from the point of view of the inside observer.

This is to me, a pure INSIDE view, a "cosmological" view. Ie. humans looking as cosmo horizon. Now if we instead consider the flip side of this, we humans looking into a microscopic black hole the situation is flipped as the observer is on the "wrong" side of the horizon, so somehow there seems to be then that both the microscopics BH horizon and the cosmo horizon enters the same equations.

Also this is just a toy model, and yes S_KL is no proper metric but that's a more detailed discussion, but I'm curious to hear what progress we can expect in "this direction". Note that the above is just to fuel associations, I'm not making a precise hypothesis here, just noting that there are many extremely suggestive things here that has nothing to do with ST, AdS or anything else; but which might eventually help us understand how and why space is emerges.

Constructive Ideas in this direction?

/Fredrik

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