Ideally/theoretically, all of the above is required in any final, fundamental, and true/ultimate theory (or description) of physics.

It’s a bit technical. Why don’t you send me an email at my university address and I’ll answer this offline. I’m assuming you can find my address so I don’t have to publish it here.

Tom

I have a number of questions which possibly I could answer for myself if I studied your papers closely enough. I’m frustratingly short of time, and I want to have this discussion, so I will just state the questions or issues, and you can comment if you like. (Or perhaps just say where the question is answered.)

1) I don’t understand the relationship between the overlap formalism and the matrix formalism. How do you understand Matrix Theory in terms of overlapping causal diamonds? How do you derive this picture of one dS horizon volume per diagonal, from the overlap formalism for dS? Can such a derivation be performed for a specific model, such as the DBHF?

2) How is that you consider yourself to belong to the “one horizon volume” school of thought, when in the next comment you talk about different horizon volumes? Do you just mean that physical calculations must always take place in the context of a particular horizon volume?

3) There is apparently something called Holographic Eternal Inflation that can be described using the overlap formalism. What is the relationship between HEI and the Eternal Inflation that you were criticizing in this post?

It’s something like this: You think of the different diagonals as different horizons. The Hamiltonian of an observer in one horizon has a piece that just gives you free particle propagation in each horizon. Then it has a correction which thermalizes all the degrees of freedom. Now you can ask what the initial state is . For the observer in one horizon volume it’s natural to take it to be a decoupled state of ITS particles from the rest, with a maximally uncertain density matrix for the rest. The physics of that initial state is the usual thermal physics of dS space, described approximately by QFT. The particles in other horizon volumes look like a thermal bath on the horizon of one. There are other, much more special states, in which the particle degrees of freedom in all the different horizons have unentangled wave functions. But for the more generic state I described, the only degrees of freedom that are behaving like approximately free particles are those in the single horizon.

I don’t understand how I should view these other “copies”. Should I think of myself (and all other observers that I can communicate with) as living on one diagonal, and the other diagonals as something like other horizon volumes?

I’m afraid Willy and I belong to the one horizon volume camp. Something that might make you feel better is the observation we made in our “Recurrent Nightmares” paper:

If you count localizable particle degrees of freedom, with the constraint that you don’t form a black hole as big as the horizon volume, then you find the maximum entropy in the single volume scales like N^{3/2} , where N is the radius in Planck units. This means that the N^2 degrees of freedom in a single horizon volume can be thought of as N^{1/2} copies of the local degrees of freedom in a single horizon. When the Planck length goes to zero N goes to infinity and you get back your picture with an infinite number of copies of local d.o.f.

I disagree with you that the attempt to base dS holography on observer dependent horizons is a mistake, and I think it’s the only sensible proposal if one looks at all the evidence. In HST we do have independent copies but if they never have an overlap (remain forever out of causal contact) then you can throw them away without changing any measurement any observer can make. So they are “gauge copies”.

That’s been my informal opinion about the subject, to be examined more closely at a future date when I think about it properly. I had noticed that in HST there are multiple coexisting horizon volumes, and from my perspective that’s a plus, it indicated a more objective conception of the regions beyond our horizon. But the thought that eternal inflation is making a mistake by supposing the existence of infinitely many volumes with independent degrees of freedom is new to me. It makes HST seem like a middle way, between the subjectivism of observer-dependent dS holography (only one volume’s worth of independent degrees of freedom) and the locality of eternal inflation (infinitely many independent volumes).

I recall from Perimeter your point about actually performing measurements on the timescales of CDL transitions, and I think this is an interesting and potentially important notion. This would be even more salient if we knew “where we are” in the spacetime, since then things far away wouldn’t ever matter to us. But since we don’t know “where we are,” I’m not yet sure what to conclude from these observations. (This is related to my hesitancy to draw too strong of conclusions from the existence of the dS horizon, vis-a-vis the classical continuation of spacetime beyond it.)

Thanks again. You’ve shown incredible patience dealing with everyone’s comments.

In those cases where CDL is, IMHO, a possible description of a real quantum theory of gravity (these are the cases “above the Great Divide” in the jargon of my paper with Aguirre and Johnson), the interpretation of the CDL transition OUT of the lowest dS minimum is a transition to a low entropy state. It’s analogous to all of the air in a room collecting in a 1 cm^3 corner. In the case of transitions to a higher dS vacuum, this is seen quite explicitly in the CDL calculation and the “detailed balance at infinite temperature” ratio of the two transition rates is a piece of evidence for dS having a finite number of quantum states. In the case of a transition to a negative c.c. crunch (I really object to using the word AdS to describe this. AdS is a maximally symmetric space. the real solution does not approach it. in fact, the field doesn’t even stay in the basin of attraction of the negative c.c. minimum. CDL said this a thousand years ago. Why do people persist in using misleading language?), we can’t describe the backward transition semi-classically, because the crunch is a place where the semi-classical approximation breaks down. However, the covariant entropy bound strongly suggests that the crunching region is one of small finite entropy. The most conservative interpretation is that the crunch makes a rather rapid transition back to the lowest dS minimum (I know, everyone finds this hard to believe, but if you throw away your semi-classical crutches you will find enlightenment) and that the system has a finite number of states, most of which resemble the lowest dS vacuum.

I agree with you that the mathematics of the theory will predict that this transition happens an infinite number of times. But, if you’ll look at the “Recurrent Nightmare ” paper I wrote with Willy and Sonia Paban, you’ll see an analysis which suggests that , above the Great Divide, nothing that happens on even one CDL tunneling time for the crunch , can be measured. The most robust “classical” measuring device one can construct in dS space, will suffer an exponentially large number of huge quantum fluctuations of its pointers before one such transition occurs. What this means in practice is that we can write a huge number of alternative Hamiltonians for this finite system, each of which gives the same predictions for everything that’s actually measurable, up to the maximal precision one can get to, but will give radically different predictions for things that happen on the CDL tunneling time scale. IMHO, this means that talking about even one CDL transition in this system, let alone an infinite number, really corresponds to taking our mathematical models of physics a bit too seriously (this is roughly the same point I was making about BBs, though the details are different).

IM(Not So)HO, I think our task as theoretical physicists is to find models that fit all the data we can possibly imagine being measured. If the theory itself tells us that certain aspects of its predictions cannot be accessed, even in principle, then I don’t think we should take those predictions seriously.

I want to make one thing clear about this. CDL is supposed to be an approximate description of a mathematical model of quantum gravity. I believe that Above the Great Divide, ABJ gave a completely coherent explanation of every computable aspect of CDL transitions in terms of a model with a finite number of states. That model then makes a mathematical statement that the reverse transition from the crunch to dS space will take place at a certain rate. I think this is a correct statement about mathematics. It also predicts that the transition will occur an infinite number of times in the future. BUT IT ALSO PREDICTS THAT NO MEASURING DEVICE IN THAT SYSTEM WILL EVER BE ABLE TO REGISTER THAT TRANSITION MORE THAN ONCE. THE PROBABILITY OF REGISTERING IT EVEN ONCE IS INCREDIBLY TINY.

I’m trying to figure out where exactly we disagree, and how strongly.

I view the CDL instanton as a transition amplitude from a Hubble volume of false vacuum to a Hubble volume containing a bubble of lower-energy vacuum. I don’t have a clear idea of what this actually means in terms of quantum gravity, but I think there is sufficient evidence to make it worthwhile to investigate the consequences, considering the transition to be localized (in space and time) and the subsequent evolution to be classical, the geometry given by the CDL analysis, up to small perturbations.

Without such a transition, I’d say positive vacuum energy has eternal inflation: the static patch has infinite spacetime volume. With just one such transition, if the bubble has positive vacuum energy, there is again infinite spacetime volume. These are statements about the classical geometry and I have not yet read your HST proposal, but here and elsewhere all I am asserting is plausibility (in particular, further investigation is worthwhile). Your essay expresses skepticism about the validity of more than one CDL transition but I haven’t gotten there yet. So perhaps you have objections to the “semi-classical” approach? Or maybe you mean to conflate “eternal inflation” with a global picture of spacetime or something else in addition to what I said?

However I also don’t understand your comments about many bubble nucleations. Considering the attractor evolution of inflation, and the infinite time in the causal patch, it is hard for me to see what prevents a second CDL transition long after an earlier one (assuming positive vacuum energy). I want to view the CDL instanton as describing a local transition, local in space and time, and I’m happy to restrict attention to a causal patch. I’m not understanding why I should worry about tails here any more than in say the QFT transitions; I’m inclined to think that, within the causal patch, 100 Hubble times after a CDL transition, a Hubble volume looks the same as if there had not been a CDL transition (as compared to starting in the second vacuum, of course).

In this perspective, eventually there will be a transition to AdS (assuming appropriate landscape), and a big crunch. Perhaps we disagree about semantics, but I would *still* say there is eternal inflation, even though the total spacetime in the causal patch is finite. I’ll try to explain this below, but I think if we agree up to this point it is just an issue of jargon (not to say jargon shouldn’t be misleading).

The above picture is my understanding of how the causal patch measure sees eternal inflation. It depends on initial conditions. Maybe this is the way the world is. But it begs for a theory of initial conditions, since it seems to me we can pretty much get whatever cosmology we want, by appropriately contrived choice of initial conditions. Among the ideas one might pursue for the theory of initial conditions is to look at the *global* attractor indicated by the (semi-)classical “continuation” of the spacetime within the causal patch. I admit this is a fuzzy and super-speculative suggestion; my only assertion is that it’s worth exploring, to develop more evidence for or against it. I have in mind that a pure-state description of spacetime might restrict to the causal patch, but considering an ensemble of initial conditions converts this into a density matrix, and the density matrix may encode “classical” uncertainties for which the global picture might be relevant. If information about beyond the horizon is somehow encoded in Hawking radiation from the horizon, this might lend a different form of credence to this possibility. It’s also not clear to me whether the full quantum calculations should be performed from the perspective of single worldline observer, or giving equal weight to all worldline observers in the causal patch defined by some worldline, or something else. I have stuck my neck out to mention some fuzzy, speculative thoughts because they are the questions that to me determine what is the correct “measure” of “eternal inflation.” It is because of these questions, and the implications of different answers, that I prefer to think of the picture attained at the end of the last paragraph as “eternal inflation,” even though at that point we were focused on a single causal patch with finite spacetime volume.

I apologize that this is a rambling comment. The last paragraph was more just to defend myself; I don’t think it would be productive to further discuss those comments in this venue. However I would like to know to what extent we agree or disagree on the previous paragraphs. Again, I appreciate that there are some assumptions and unanswered questions here; my interest is in evidence that this picture is really *wrong*, so much that we shouldn’t spend more time thinking about it and its consequences. I don’t really see what in your essay speaks so strongly against this.

e.

Thanks so much for taking the time to respond. Just to clarify for my own understanding, based on your numbers then, there is an enormous gap (on the order of 10^20 – 10^30) between the information generated by all of humanity and the available bits on the “screen” given our current best calculated values of the size/acceleration of our observable universe. Stated another way: We (humanity) have no quantifiable impact on the c.c.

e.

The holographic bound on information, assuming we have a positive c.c. as the explanation for the accelerated expansion of the universe, is about 10^{124} . By contrast, the amount of information in the atomic structure of a human being is of order 10^{2X} (twenty something in the exponent, where the X respresents my ignorance about atomic physics and human beings). Neglecting the difference between 2 and the base of the natural logarithm, this says that if we measure the information content of the universe in bits, then it’s a one with 124 zeroes after it. If we measure it in units of the information content of human beings it’s a 1 with close to a hundred zeroes in it. The number of bits that can actually be processed by our brain is much smaller than the total information content in all the states of the atoms in our bodies (most of which are states in which we’re dead). So there’s no problem of having complex information processing in a universe obeying the information bound with the observed value of the c.c. . In fact, the total information in all the matter and radiation in the universe (excluding black holes) is the exponent of ten to the eighty something.