dhillonv10, if you want I can PM you some related info with very interesting results.
Also what about the time considerations? Could time be affected by such a transformation? The equations of physics are time invariant and we "feel" time through increase in entropy, a concept that Eddington calls the arrow of time. Another speculation, it may be the case that the instantaneous signalling of time is not instantaneous after all, the time is takes for an interaction to occur in the bulk seems instantaneous (this time we are going the other way) but through the holographic mapping, at the boundary it is actually stretched out or more precisely said, the time is dilated as we would experience in Einstein's GR.
atyy: that was a good find, they are discussing some ideas very similar to what we were speculating. Although they seem to be approaching this through the use of guage/gravity duality. Nonetheless thanks for the link.
The wavefunctions on the horizon are the boundary condition on the form of the wavefunctions specifying the probability distributions of the ﬁnite number of mass quanta distributed throughout the featureless background space of a closed universe with a constant vacuum energy density (cosmological constant). The wavefunction for the probability of ﬁnding a mass quantum anywhere in the universe is a solution to the Helmholtz wave equation in the closed universe.
The inverse of this perspective is to think about events happening on the boundary (on the circle, the perimeter of the disk). There might be solitonic waves of different sizes, traveling around the rim. The size of a wave is like the size of a shadow; the longer the wave, the deeper its holographic information reaches into the bulk. A wave that is really really small corresponds to events in the bulk which are only a short distance away from the boundary, but a wave which wraps most of the way around the circle will map to points which are very close to the center.
edit: See figure 1 in this paper for the de Sitter case. And they're saying that the de Sitter case, for real variables ... which ought to be the version of holography relevant for the real world ... is related to the anti de Sitter case, for complex variables! - the "complexified boundary" mentioned above. I'll write more when I understand it.
This reduces the dimensionality of the bulk space of states and makes it possible to ﬁnd a one to one mapping into the boundary states.
But if we suppose that the version of dS/CFT that is relevant for the real world involves past and future boundaries, maybe this doesn't matter, because the boundary theory has no dynamics! In this version of dS/CFT, the time direction is the bulk - space-time holographically emerges from a boundary which is purely spatial.I don't think this would resemble the concrete examples of gauge/gravity holography that have been discovered so far, because the dynamics of the boundary theory should produce entanglement on the boundary in all such cases.
6.2 Bulk Feynman diagrams
In this section we show how the Feynman diagrams associated with a local theory in the bulk can be mapped over to CFT calculations. This will provide yet another way of deriving the CFT operators which are dual to local bulk observables.
edit #2: Also see page 7 here for a boundary-to-bulk map for de Sitter space.
Vikram Dhillon, <Thu, Jun 9, 2011 at 1:30 AM> To: email@example.com Hi Prof. Lowe, I recently came across your paper on A new twist on dS/CFT and on page 7 you mention a mapping from boundary to bulk by promoting the modes on the circle to the modes on the de Sitter. I have a question about that, is it possible to formulate an inverse of this mapping? Can the inverse of this mapping be written down where we have a function mapping bulk-to-boundary in dS/CFT? Thanks for your time. - Vikram David Lowe <firstname.lastname@example.org> Thu, Jun 9, 2011 at 4:36 PM To: Vikram Dhillon Yes, the inverse map is easier -- you just look at the asymptotics of the bulk mode near infinity (I think we had in mind past infinity), and extract the appropriate coefficient of the time dependent piece. If the bulk mode is a positive frequency mode with respect to the Euclidean vacuum, this time dependent factor should be uniquely defined.
So then studying the ds/CFT paper is probably a good idea, I think I will be doing that for the next few days. Originally I had in mind to go study twister theory and its implication that you provided but now in the light of these new developments, mitchell is it a good idea to spend time on twisters? The boundary-to-bulk mapping that is provided in that paper is derived from another paper so I'll probably start there and then come back to this one.
Such a reformulation of dS/CFT is natural from the bulk point of view, since the quantization of a scalar field on ...
The real heart of the correspondence is the re-expression of the bulk correlation functions at past infinity, in terms of CFT operators.
So the holographic correspondence is telling us that, for example, correlations between those bulk fields can actually be calculated from correlators of the corresponding boundary operators (the "tr ABC" products of boundary fields).
5.6 Mapping Type IIB Fields and CFT Operators
Given that we have established that the global symmetry groups on both sides of the
AdS/CFT correspondence coincide, it remains to show that the actual representations of
the supergroup SU(2, 2|4) also coincide on both sides.
So, returning to the paper: their region-on-the-boundary-to-point-in-the-bulk map for anti de Sitter space is an analytic continuation of the region-on-the-past-boundary-to-point-in-the-present-bulk map for de Sitter space. More precisely, they use a formula which makes sense in de Sitter space, because all the space and time distances in the formula are specified in real numbers, and then they transform that into a formula which looks like the anti de Sitter formula by making some quantities imaginary.
I made a diagram for reference (see attachment)... I mentioned that you could analyse a holographic mapping, from bulk to boundary, into two stages. First, you go from the interior of the bulk to the edge of the bulk: for example, from a point in the interior to a region on the boundary. Then, you re-express everything in terms of the boundary theory, so that bulk fields become boundary operators.So does that imply that there is to some extent a mapping established from bulk to the boundary in terms of AdS space?? If so couldn't that be extended out to dS spacetime using what you said earlier: (using the complex numbers approach)