Hi Stephen, Sorry for the slow reply, I have been working on various things and also catching up on the many conversations (and naming conventions) on this board. And thanks for your interest! -- I think I have discovered a giant "low hanging fruit", which had previously gone unnoticed since it is rather nonintuitive in nature (in addition to being a subject that many smart people shy away from thinking about...).

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Ok, let me address the Faddeev-Popov, "gauge-invariant information" issue first. I'll start with the final conclusion reduced to its most basic essence, and give more concrete examples later. First, note that any one "structure" can have many different "descriptions". When counting among different structures thus it is crucial to choose only one description per structure, as including redundant descriptions will spoil the calculation. In other words, one only counts over the gauge-invariant information structures. A very important lemma to this is that all of the random noise is also removed when the redundant descriptions are cut, as the random noise doesn't encode any invariant structure. Thus, for instance, I agree with COMP, but I disagree that white rabbits are therefore a problem... The vast majority of the output of a universal dovetailer (which I call A in my paper) is random noise which doesn't actually describe anything (despite "optical illusions" to the contrary...) and can therefore be zapped, leaving the union of nontrivial, invariant structures in U (which I then argue is dominated by the observer class O due to combinatorics). Phrasing it differently, if "anything goes" then there is actually nothing there! This is despite the "optical illusion" of there being a vast number of different possibilities as afforded by the "nonrestrictive policy" of "anything goes". One thus needs *constraints* so that "only some things go and not others" in order to generate nontrivial structures -- such as the constraints introduced by existing within our physical universe (i.e. a complex, nontrivial mathematical structure). Ok, time for examples. I'll start with one so simple that it is borderline silly... Say professor X is tracking a species of squirrel which comes in two populations: a short haired and a long haired version (let's say the long hair version stems from a dominant allele). In one square kilometer of forest he counts 202 short haired squirrels and 277 long haired. 2 years later, after 2 colder than average winters, he sends out his 3 grad students to count the populations again. They all write down that there are 184 short haired squirrels, but student 1 writes that there are 298 long haired squirrels, student 2 writes down that there are 296 group B squirrels, and student 3 records the existence of 301 shaggy squirrels. In a hurry prof X gathers the data, and seeing in the notes that the group B and shaggy populations both have long hair, he adds them all up for a total of 895 long-haired squirrels vs 184 short-haired -- a huge change instead of a mild selection! He rushes off his groundbreaking paper to Science... Anyways, one could concoct a more realistic example (perhaps using more abstract labeling), but the main point holds -- it doesn't matter which description is used (long-haired, group B, or shaggy) but it does need to be consistent to avoid over- counting and getting the wrong answer... A silly example, but things become much, much more subtle when considering different mathematical structures which can have various mathematical descriptions! Consider the case in general relativity. Here the structure of spacetime can take different forms - you can have flat, empty Minkowski space, the "dimpled" spacetime resulting from a single star, or a double helix of dimples due to 2 stars orbiting each other, or a wormhole geometry for a black hole and so forth... Each one of these spacetime structures can then be described by many different coordinate systems and associated metrics. For instance, flat space can be mapped out by a Cartesian coordinate system, or cylindrical coordinates, or spherical coordinates, or an infinite number of alternatives, most of which completely obscure the simple Minkowski spacetime structure. Likewise for the black hole - one can use Schwarzschild, or Eddington-Finkelstein, or Kruskal- Szekeres or an infinite number of other variations... Thus, consider a complex metric which describes some highly warped spacetime geometry as expressed in an intricate coordinate system. It is natural to wonder what components of the metric are directly informing on the exotic shape of the spacetime, and which parts are just artifacts of the peculiar coordinate system that has been chosen. The answer is: it's not clear in general! Relativists thus depend heavily on scalar invariants (The Ricci scalar, Kretschmann scalar and so forth) which are the same in all coordinate systems for a given geometry, and on asymptotic quantities defined at large distances where the spacetime is nearly flat (one can get the total mass, spin, and charge this way) in order to understand complex spacetimes structures. Let's move on to QFT. Say we have the function f(a,x,y) = e^(- ax^2), and we want to integrate this over all values of x and y, thus producing Z(a), which is just a function of a (we will then want to compute things like d(log(Z))/da ...). Ok, fine, we have: / +infinity / +infinity | | - ax*x Z(a)= | | e dx dy | | / -infinity / -infinity But the final result is infinite: Z(a) = (pi/a)^(1/2) * infinity = infinity, so at first it looks like we can't work with this theory... But of course this is also a bit of a silly example -- the original function f(a,x,y) never depended on y: y is a redundant variable, and this redundancy is responsible for the failure of the integration to produce a sensible result. The solution is obvious - just get rid of y and integrate f(a,x) over all values of x. This is essentially what is going on in the Faddeev Popov case, where f (i.e. the Lagrangian) depends on gauge fields (as in electromagnetism, or the strong nuclear force), and f is a constant for some variations of the gauge fields. However in this case the extraction of the "redundant y variable" is much more subtle than in the above example - it is this subtlety which necessitates all of the intricate computational machinery as described in the wikipedia article. So yeah, to get back to your question on the nature of Faddeev-Popov ghosts, I would go with: 1) a computational tool that does not have a “real” physical expression since they violate some key requirements of “reality” This is certainly the consensus view at least. The rich nature of these mathematical tools is in some sense a reflection of the real structures which they assist in the analysis of... That said, it is fun to wonder if some more complex "theory of everything" could make them real (string theory does not as far as I can tell...) - perhaps they could be useful in a star-ship engine in a science fiction story... So, right, one should only consider the real, nontrivial, description-invariant structures when counting among all forms of information. Let's apply this to the white rabbit problem that I have read about on this board (my name for these is "reality discontinuities"...). First note that white rabbits are not completely forbidden in our universe, but rather just exceedingly unlikely - not just exponentially suppressed, but rather doubly- exponentially suppressed. The easiest one to think about is the probability that all the air molecules in a room could spontaneously shift to the left half of the room. The probability for this will be about 1/2^N, where N is the number of air molecules - itself something like 10^27 (in the case that the density is low enough that the mean free path is larger than the dimensions of the room, this should be an accurate estimate...). This should be the general pattern for macroscopic violations of the 2nd law and macroscopic quantum- tunneling - the probability will be something like 1/(10^(10^N)) for some 2 digit N - i.e. fantastically unlikely, if not completely forbidden. But then, if one can simulate our universe in a computer, then one could also simulate what I call an "if-then" type universe - essentially a second program that runs the universe simulation and watches it, and "if" certain conditions are met it "then" pauses the simulation and goes in and changes things, perhaps inserting a "white rabbit" of some sort - thus making the white rabbits quite common in that particular "if-then" program instead of incredibly unlikely. Furthermore there will be an incredible number of different "if-then" type programs with different "if" and "then" conditions, as opposed to just one "unmolested" universe... I don't think this is a problem however! All of these "if-then" universes are essentially an optical illusion - there actually isn't any nontrivial structure there. Note for instance that one is free to pick any "if" condition and any "then" condition - they are both completely arbitrary and contentless. I believe at one point Bruno said that if a physical universe was capable of doing an infinite number of computations then it necessarily would run a universal dovetailer and thus spend the vast majority of its time simulating these random "if-then" simulations and thus causing a white rabbit problem. I disagree with this! In a physical universe where ever more powerful computers are being built (which may be the case in our universe) the vast majority of the time they would be programmed to work on ever more complex nontrivial problems, with very rare excursions out into the "random wilderness". The "if-then" type programs could thus only be a problem in Platonia... And I don't think they are a problem there either! Like I have mentioned earlier, the apparent plethora of white rabbit type universes is essentially an "optical illusion", and the Platonic ensemble is itself composed of nonrandom, nontrivial structures... Let me copy and paste an example I gave in a previous post: "Say ETs show up one day - the solution to the Fermi paradox is just that they like to take long naps. As a present they offer us the choice of 2 USB drives. USB A) contains a large number of mathematical theorems - some that we have derived, others that we haven't (perhaps including an amazing solution of the Collatz conjecture). For concreteness say that all the thereoms are less than N bits long as the USB drive has some finite capacity. In contrast, USB B) contains all possible statements that are N bits long or less. One should therefore choose B) because it has everything on A), plus a lot more stuff! But of course by "filling in the gaps" we have not only not added any more information, but have also erased the information that was on A): the entire content of B) can be compactified to the program: "print all sequences N bits long or less". " I am saying that the universal dovetailer is the dynamical equivalent of USB drive B)! By filling in all the gaps between the nontrivial structures with random noise we end up with no structure what-so-ever -- as evidenced by how trivial it is to generate the content of drive B: "print all sequences N bits long or less". The same is true for a universal dovetailer - it is just slightly obscured as it is now dynamical, but the end result is the same. Ok, this post is getting long, so I'll talk about absorption later, but perhaps it won't be too surprising if I say that it needs to be nonrandom in character: i.e. photons can bounce off of a fig and hit both a monkey and a rock but only one is absorbing information in a nontrivial fashion -- the other is merely being warmed slightly... Sincerely, Travis -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. 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