2 Einstein established mass-energy equivalence: E=mc It is clear from relationships between entropy, energy, and information, that something similar can be said there, but what is it? Consider Heisenber's Uncertainty principle, conventionally written DxDp > h where Dx is uncertainty in x position and Dp is uncertainty in momentum. Now, momentum is mass*velocity, so this can equally validly be written DxDvm > h where m is mass. Solving for m gives h m > ---- DxDv But uncertainty is just the inverse of information: As our information about x grows, our uncertainty about its position decreases in direct proportion: They are merely two different ways of expressing the same thing. So we can rewrite as m > hIxIv where Ix is information on x position and Iv is information on x veloctity. Next, note that the inequality only came from consideration of experimental inadequacies: We do not always get as much information about x and v about a particle as we potentially could. The particle, however, contains the same amount of information whether or not we happen to measure it well: If our interest is fundamental physics rather than the current state of experimental technique, we may validly rewrite as m = hIxIv Finally, the above expresses the fact that the finite information content of a particle may slosh back and forth between x and v representations. We may legitimately choose to ignore this detail and instead write m = hI Which is in fact the information/mass equivalency counterpart to Einstein's mass/energy equivalence equation: Mass and information are the same thing, with Plank's constant h as the constant of proportionality. Returning for the moment to Einstein's formula, if 2 E = mc then E --- = m 2 c but m = hI so E --- = hI 2 c or E 2 - = hc I or 2 E = hc I Now if (as I am told) h has units of "action" == energy*time, then hc**2 has units of energy*area/sec. This seems to imply I has units sec/area. So bits==sec/area? I find that toally baffling, which presumably means it is either ridiculous or profound. :) Going back to the inverse representation, it means the units of uncertainty are area/second? something/second is at least a more intuitive notion, a rate. Is there any intuition to be had here? Volume/second is something a moving 2-D wavefront sweeps out: area/second is something a moving line sweeps out. Is this by any extreme chance a link to string theory?? Suppose we imagine a closed loop of string-theory style string somehow sweeping its way along. To start out simple, we might suppose it to be a photon, constrained to travel at lightspeed. If the wavelength of the photon is proportional to the length of the string, than as the wavelength grows, the area/sec swept out grows in direct proportion -- and of course the uncertainty also grows in direct proportion. Is this how we make sense of uncertainty ~~ area/sec? If so, is there more intuition to be gotten from information ~~ sec/area than just deriving it as the inverse of the previous? As the hypothetical string gets shorter, its information increases, and it takes longer to sweep out a given area: That gives a more direct visualization of area/sec being proportional to information, but not a very satisfyingly direct relationship. What if the Universe is in some interesting sense one-dimensional? And what if our string-theory string is now considered to be open instead of a closed loop, and to be occupying some subset of the full universal length? Now we can clearly play something closely akin to the standard information-theoretic bitcounting game (I think it counter- productive to think -too- carefully about the details of this argument at this point...): * If a string occupies the entire universal interval, it takes zero bits of information to specify it. * If a string occupies half the universal interval, it takes one bit to specify it: Which half of the universal string? * If a string occupies one quarter of this universe, it requires two bits to specify it. And so forth: As a string gets shorter, its position carries more information. We can also turn the relation back the other way: If we can measure the information content of a string, and independently its length, we can infer the length of the universe. 2 Does this by any chance mean that E=mc and m=hI plus standard lab measurements are in principle sufficient to establish the size of the universe? This isn't clear to me. :) (What about particles that travel at less than lightspeed? I've thought for years that all particles travel at lightspeed -- it is just that some do so in a more or less straight line, and some are doing something like travelling partly or wholly around one of the rolled-up dimensions of string theory (say) instead of directly along one of our familiar three unrolled dimensions. You may take this, if you like, as a explanation for why we "need" those extra rolled-up dimensions: They provide the only way for matter to be (effectively) slow-moving enough to form molecules, and hence intelligences -- the Anthropic Principle strikes again *wrygrin*.) --- This above all has some interesting consequences to think about, actually. If E=hI holds, and energy is conserved, then obviously information is also conserved, in the sense intended at least, that of total information content. (A particular datum, such as the angle of some particular leaf at some particular time in the Jurassic, may of course not be conserved. True quantum randomness together with fixed total information content imply that specific data are constantly being lost.) Or consider a hydrogen electron dropping from a "high" to a "low" orbital. The bigger the energy difference, the more energetic will be the photon given off: That's the conventional interpretation. What is the information interpretation? It would appear perhaps to be that when an electron drops from a very large, extended orbital down to a small, confined orbital, its position carries less information relative to the nucleus than before: Only information sufficient to locate it to the same "absolute" precision within a much smaller volume is now needed. Information being conserved, the excess capacity must go somewhere, so it is carried away in the form of increased precision in the position of the emitted photon, which is to say, a shorter photon wavelength. If this is correct, we should be able to compute the energy level of an orbital as a fairly simple integral of the volume times the electron probability density at each point in the volume, no? I wonder if this is significantly different from, or simpler than, the standard way of computing orbital energy levels, whatever it is. (I don't recall ever seeing such a computation done, or even described.)