BenTheMan

From the arXiv: http://arxiv.org/abs/0708.1721 Are the spectra of geometrical operators in Loop Quantum Gravity really discrete? One of the celebrated results of Loop Quantum Gravity (LQG) is the discreteness of the spectrum of geometrical operators such as length, area and volume operators. This is an indication that Planck scale geometry in LQG is discontinuous rather than smooth. However, there is no rigorous proof thereof at present, because the afore mentioned operators are not gauge invariant, they do not commute with the quantum constraints. The relational formalism in the incarnation of Rovelli's partial and complete observables provides a possible mechanism for turning a non gauge invariant operator into a gauge invariant one. In this paper we investigate whether the spectrum of such a physical, that is gauge invariant, observable can be predicted from the spectrum of the corresponding gauge variant observables. We will not do this in full LQG but rather consider much simpler examples where field theoretical complications are absent. We find, even in those simpler cases, that kinematical discreteness of the spectrum does not necessarily survive at the gauge invariant level. Whether or not this happens depends crucially on how the gauge invariant completion is performed. This indicates that ``fundamental discreteness at Planck scale in LQG'' is an empty statement. To prove it, one must provide the detailed construction of gauge invariant versions of geometrical operators. =============== The big realization seems to be in this paragraph: In other words, the analogue of gauge invariance in GR is space-time diffeomorphisms (coordinate transformations). If one has a discrete spectrum for an area or length or volume operator, the eigenvalues are not gauge invariant---they depend on the choice of coordinates. Because observables must be gauge invariant, the discreteness of these operators is not an observable.

[math]e^{0} = 1[/math] [math]e^{\pi i} =-1[/math] [math]e^{2\pi i} = 1[/math]

We're getting pretty far afield of actual physics here. This argument isn't very good---presumably you'd be smart enough as a time traveler to realize that any thing you influenced in the past could have terrible reprecussions in the future.

Cocaine is a hell of a drug.

High energy physics, mostly phenomenology, some theory. Building heterotic string models that reseble the real world. Trying not to piss my advisor off.

I'm not sure if I agree with some of the things in the post. For example, ``where does the menu come from?'' I am not sure what this means. Physics gives us a very good idea of why things should be as they are. In fact, I can't think of any examples of quantum systems that we don't understand in the sense which I think you think we don't understand them. So perhaps you can clarify what you mean by this. It seems like you are talking of some sort of coherence---many atoms together in an ``environment'' will behave the same way. I think that it is dangerous to think this way. Particles in a quantum system don't necessarily know that they're in a quantum system. Take photons in a laser beam, for example. The photon doesn't know that it's surrounded by tons of OTHER photons. It just knows that it has a very specific frequency and direction. Another example is blackbody radiation---there, a large number of non-interacting particles behave the same way. How can one say that a photon obeys any sort of ``learning rule'' when it is quite clear that this cannot be the case? The correct interpretation of macroscopic phenomena is simply a numbers game, as I said earlier. Again, it is not clear that this is true at all. At least at the sub-Planck scale, we expect the space-time formations to be completely random, as there are (in principle) no symmetries for the space-time to obey. Quantum Gravity doesn't respect ANY symmetries. So I'm not sure how you can claim this. We are mixing vocabularies here. Either way, could you clarify this? Fredrik---I'd point you to my previous posts. If everything is based on probability, then the fact that large lumps of atoms behave in a predictable way was EXACTLY the point I tried to make. If you believe in Statistical Mechanics, then this is trivially true. Take an ideal gas, for example. Quantum mechanically, one cannot define, say, the temperature of a single atom. But if one takes a large collection of such atoms, and writes down a partition function (which is defined in terms of the hamiltonian of all of the individual atoms), then macroscopic phenomena emerge. Now one can take derivatives of the partition function to define temperature, pressure, entropy, etc. This is exactly what foodchain is talking about. At the quantum level, we can't know what one atom of a gas is doing from one instant to the next. Statistically, however, we have a very good idea about what is going on.

Well, a quantum system can't ``remember'' what state it WAS in. So I don't think it would be a problem.

This is an important point to remember: Spin is not a classical quantity! It Take the classical electron radius, which is the Compton wavelength of the electron: [math]r_e=\frac{1}{4 \pi \epsilon_0}\frac{e^2}{m_ec^2}[/math], and the ``spin'' angular momentum: [math]m_ev_0r_e = \frac{\hbar}{2}[/math]. No calculate [math]v_0[/math], which is the speed at the ``equator'' of an electron. [math]v_0=\frac{\hbar}{2m_e r_e}[/math] [math]\Rightarrow v_0 = \frac{4\pi\epsilon_0\hbar c^2}{2e^2}.[/math] But we can define the fine structure constant [math]\alpha[/math], so that [math]\Rightarrow v_0 = \frac{c}{2\alpha}[/math]. We know very well that [math]\alpha = \frac{1}{137}[/math], which tells us that the electron spins 260 (ish) times faster than the speed of light. Here I treated the electron as a sphere of charge, but as someone already pointed out, the electron is a point particle. Because a point is zero dimensional, there is no concept of it spinning.

But the system isn't just a ``random number''... It's a weighted random number, just like my sock example. Certain outcomes are more likely than others, and over very large numbers, the classical outcomes are the most likely ones. If your ``million sided die'' had 900,000 sides that said `1', then if you threw that die enough times, you would expect to see a 1 come up most of the time. You must have heard that the quantum world is ``random'', which I think you have misinterpretted. We are choosing from a range of options---for example, when you make a random choice at a resturaunt, it is from food that is actually on the menu.

I think you're missing the point... Sure, there is a finite chance that one molecule of carbon dioxide will pop apart into carbon and oxygen (ignoring the violation of the second law of thermodynamics, of course). But when was the last time you saw just one molecule of carbon dioxide? The point of having all of the carbon dioxide atoms in, say, your body spontaneously dissociate would be the same as drawing a white sock from the drawer 10^23 times in a row. (Actually, it's even smaller.) The point is, the quantum world is inherintly random, and inherintly governed by the laws of probability, just like your spontatneously dissociating carbon dioxide molecule. But when you are dealing with very large numbers, the effect of one molecule matters very little in the grand scheme of things (like people back home in Texas who voted for Kerry).

I think you're a bit confused. First, I wouldn't say that there is a lot of ``chaos'' in a quantum system. The thing to remember is that quantum mechanics says that you cannot know the outcome of a single experiment, but if you preform many experiments, you know very well what should happen. Think of it like socks in drawers---suppose you have ten blue socks and one white sock in your drawer. Now suppose you draw one sock from the drawer. Will it be white or blue? There's no way to tell before you preform the experiment. Now suppose you repeat the experiment many times. You would (rightly) predict that MOST of the time (91%), you would draw a blue sock. This is the way I understand quantum mechanics in terms of macroscopic phenomena. While INDIVIDUAL molecules will ``arbitrarily dissociate for unknown reasons'', chances are that you'll never notice, because you are made from 10^25 ish individual molecules, and 10^25 - 1 = 10^25 (check on your calculator). So don't worry. You're safe.

I guess that this gets to the heart of the matter---like I said, all we ever measure is an effective theory. As physicists, we cannot talk about what is ``true'' and what is ``right''. We can never know these things. All we can ever talk about is ``agree with experiments''. QM predicts experimental results, which is all we can ever hope for. Because our knowledge of the subatomic world rests with results of experiments, then we can never do any better than QM, UNLESS you can show me an experiment in which QM fails. Also, I don't know what things like this mean : In QM, all one does is use the uncertainty principle, promote classical observables to hermitian operators, and everything else follows. The thing that is really fundamental to QM is the uncertainty principle, and it seems to me that unless you get rid of that, then you are stuck with QM. Finally, Tell me what you mean by this. I think that you're wrong, but I want to know what you mean by ``probability space'' before I say so

Hmm. I don't know how much physics you know, so excuse the vernacular. All we can ever measure is the effective theory. An example would be the standard model---if we only find a single higgs at the LHC and nothing else, then that's it. The natural scale for new physics is too high for us to ever hope to measure---we will know that there is some more fundamental structure, but we'll never be able to test it. And any attempts to describe our universe beyond what we can actually test is not phsyics at all. The same can be said for quantum mechanics. You'll never be able to remove the physical act of measuring from QM. This means you'll ALWAYS be stuck with the uncertainty principle, which pretty much dictates how we see QM. So you can do philosophy, I'll do physics:) We should be in the business of measuring things. If we aren't, then we might as well talk about many universes and landscapes and such. Untill you come up with an experiment which shows some fault with QM, then the question of ``Is QM "first principles" enough'' isn't a question about phsyics at all.

Yeah I met him on another forum, polluting the internet and purporting to ``explain'' things he didn't even understand himself. 100% bullshit (or ``bollocks'' for the EUers). You have to make up your own mind about these things, of course. The trouble is, Farsight has presented himself as an expert, which is what pisses me off...he spouts off a bunch of bad science, and people who don't know any better take him seriously. It's like getting medical advice from a bar tender. Time travel is technically possible, as there are solutions to Einstein's equations which allow it (a fact pointed out by Goedel, and which Einstein was loathe to admit). There are also wormhole solutions to Einstein's equaitons, a fact which no learned physicist will dispute. As I mentioned earlier in the thread, however, I think that (quite generally) wormholes are hidden by horizons (I haven't done the calculation myself, so don't trust me). This means that you could fall into a wormhole, but never get out again. So what happens after you cross the horizon is between you and Allah Warning: This is long and detailed. I hope that someone will read it so that I haven't spent this hour in vain Let me dispute a few of Farsight's claims: I think I stated earlier that this is not true in general. It is only true for an observer infinitely far away. This means that I can watch a star collapse, and I can watch a black hole form, as long as I am not watching it happen from infinitely far away. This is just wrong, and I can show you why with equations (in stead of waving hands, as Farsight does). All you have to do is imagine a black hole that is massive enough. For a sufficiently large black hole, the gravitational pull at the event horizon could be even less than the gravitational pull that you are experiencing sitting in your chair right now. I can show you equations, if you REALLY want to see them, or I could walk you through the calculation so YOU could calculate it. This next one is in response to the idea that space and time are inseparable (post #10): Farsight screws up one of the most fundamental concepts to theoretical physics---the idea of Lorentz Invariance. In my mind, Lorentz Invariance is the most fundamental concept in all of physics. It tells you why you have particles with integer and half integer spins, it tells you why lengths get contracted, but most of all, it tells you that you can NEVER separate space and time. Farsight has yet to understand this. This could be cut and paste from any crackpot rant anywhere on the internet Einstein was brilliant for a few years in his career, then he stopped doing anything useful. Even the graduate students at Princeton in Einstein's later years thought that he was more or less useless---he didn't believe in quantum mechanics and he basically ignored half of the physics that was happening in his latter years in favor of his own brain wanderings. I hope you realize what a tremendous claim this is, besides. Farsight is claiming (on the internet, no less) to understand things that very smart people have dedicated their lives to understanding for over 100 years. For some reason, this really gets on my nerves... The Mobius strip is understood quite well, at a topological level. In fact, it's typically the third example that people talk about when they start learning topology (the circle and torus are first and second). If you're confused about this, you're not alone. I've never heard of elastodynamics, and I have been studying physics for a long time. Again, farsight is claiming to be the only one who understands how the universe works. While this certainly isn't impossible, he also makes statements like Again, I hope you realize what you're dealing with This is amusing in and of itself. What Schwarzchild thought doesn't matter---he is dead (I think). What YOU should know is that the Schwarzchild metric is used to describe the gravitational field of our sun when calculating the perhilion of Mercury, which is generally considered to be one of General Relativity's greatest successes. So, fatty---make up your own mind. Read things and think for yourself. Listen to me, and listen to Farsight. Ask questions---this is what science is about. I can tell you that I think Farsight is an idiot, and I can tell you about my degrees and publications, but at the end of the day, that doesn't matter much because this is the internet, the ultimate democracy. Anyway, I hope this hasn't been for naught If you'll notice, Farsight has ignored me thusfar. He will probably start bitching about me not actually reading his ``... Exlained'' posts---he is right. I read untill I found the first mistake, then stopped and asked him about it. He never answered, so I never felt obliged to continue reading.

don't you need to turn it into a contour integral or something?

hmmm 1 + 1 = 0 mod 1 or 1 + 1 = 0 mod 2. There. There's two cases.

I've always seen quantum mechanics as more of a paradigm for understanding physics which we wouldn't have otherwise understood. One could argue that classical mechanics wasn't fully understood untill Newton taught us how to attack the problem analytically---quantum mechanics may be the same sort of ill-guided attempt at understanding the world that people practiced before Newton. All of this probability stuff may be just a way for us to deal with the measurements we make in the lab. Either way it doesn't matter. We'll probably never be able to do any better. QM gives the right answer, so what difference does it make if it makes any sense? Unless we can find an experiment to preform which disproves QM, then we are stuck with Hilbert spaces, wavefunctions, and uncertainty principles. Humans live in a world that is deterministic, and electrons live in a world ruled by probabilities. When we try to understand electrons, things don't make sense because we evolved in a world that is deterministic. So, in the words of Feynman, ``Shut up and calculate.''

Very much to his credit that he answered such a stupid question

Basically you hit things really hard and see what happens. If it breaks, then it's not fundamental, and if it doesn't break, then either you didn't hit it hard enough, or the thing is actually fundamental. You are correct in that there's no formal proof, but the experiments do place a lower bound on ``compositeness'', as it were. I'll try to remember to come back and post something about how you can use dimensional analysis to look at all of these things when I have more time.

I remember asking a question about the conformal transformations for pure gauge Yang Mills theory in four dimensions on another forum---I had worked out an answer that couldn't be correct, and this guy (Garrett) answered with just one line: [math]F_{\mu\nu}F^{\mu\nu} = F_{\mu\nu}F_{\alpha\beta}g^{\alpha\mu}g^{\beta\nu}[/math]. I felt pretty foolish because this statement should be completely obvious.

I'll save Farsight the trouble: No. No.

I think this thread has wandered a long way from science.

Aparently you are quite confused about what ``REAL NUMBER'' means. Plus your comments are wrong by the first postulate of special relativity, namely that things can't travel faster than the speed of light. Plus tachyons violate Lorentz Invariance, which is the centerpiece of special relativity. So you'll have to do better...

You should find this guy Farsight and talk to him. You two would totally get along.

K--- One doesn't have to look too hard to find alternatives to Dark Matter and Dark Energy. Most of the alternatives (frankly) suck. The best explanation for Dark Matter has to do with supersymmetry, which will probably be discovered at CERN within the decade. As for where dark matter lives, it lives everywhere! There are dark matter particles streaming through you at this moment---they don't interract very strongly with matter (except gravitationally), so you of course won't notice anything. As far as where dark matter lives, it has to live at the outsides of the galaxies---this is how it was first discovered. The normal models of spiral galaxies don't work at the very outer reaches of those galaxies. The models predict that many of the galaxies we see shouldn't exist. However, when one postulates dark matter living in the outer regions of the spiral arms, the models match the data exactly.