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Natural Science Forum / Physics / Research / January 2008A theoretical physics FAQ
The theoretical physics FAQ at http://www.mat.univie.ac.at/~neum/physics-faq.txt contains answers to some more or less frequently asked questions from theoretical physics. Starting in 2004, the topics were edited from my answers to postings to the moderated newsgroup sci.physics.research (or, for some, translated from postings to the unmoderated newsgroup de.sci.physik). Currently, the FAQ contains 140 topics, grouped into 19 Chapters, and filling almost 10000 lines of text, corresponding to a book of about 150 pages. In the last year, the following topics have been added: S1n. The role of the ergodic hypothesis S1o. Does quantum mechanics apply to single systems? S2f. What is a photon? S2g. Particle positions and the position operator S4a. How do atoms and molecules look like? S5i. Why Feynamn diagrams? S6e. Constructive field theory S10k. Time in quantum mechanics S15e. Theoretical physics as a formal model of reality S16k. What is real? S17c. How to respond to critical referee's reports S18d. Research at age 16 Your comments are welcome. If you like the FAQ and/or found it useful, please link to it from your home page to make it more widely known. If you spot errors or have suggestions for improvements, please write me (at Arnold.Neumaier@univie.ac.at). If you have questions, please post them to the moderated newsgroup sci.physics.research (http://www.lns.cornell.edu/spr)! Of course, the FAQ refers only to a tiny part of theoretical physics, namely to what I happened to discuss on sci.physics.research. The answers are only as good as my understanding of the matter. This doesn't mean that they are poor but probably that they are not perfect. This is not a book, so don't expect completeness or comprehensiveness in any sense. On topics where the physics community has not yet reached a consensus, my point of view is of course only one of the possibilities, and not always the mainstream view. In any case, I try to be accurate, consistent, and intelligible. Happy Reading! Arnold Neumaier > The theoretical physics FAQ at > http://www.mat.univie.ac.at/~neum/physics-faq.txt Quite readable stuff on some of the most tricky subjects! One question on the "information in one particle". The FAQ reads: [..] > A pure state of an electron is defined by its wave function > (up to a phase). Thus knowing all about an electron requires in the > traditional interpretation to know all about this wave function - > an infinite amount of information. How does this relate to the entropy formula S = k ln(Omega)? Gerard Gerard Westendorp schrieb: >> The theoretical physics FAQ at >> http://www.mat.univie.ac.at/~neum/physics-faq.txt [quoted text clipped - 11 lines] > > How does this relate to the entropy formula S = k ln(Omega)? Thanks for asking. I added the following to the FAQ entry: How is this notion of information related to information in terms of entropy? Entropy measures the amount of information missing for a complete probabilistic description of a system. The formula for the entropy S found in every statistical mechanics textbook is, for a system in a mixed state described by the density matrix rho, S = <kbar log rho> where <f> = Tr rho f and kbar is Boltzmann's constant. (I use the bar to be free to use k as an index.) In any representation where rho is diagonal, rho = sum_k p_k |k><k|, this gives S = kbar sum_k p_k log p_k; also, since <1>=1 and rho is positive semidefinite, sum_k p_k = 1 , all p_k >= 0. Thus p_k can be consistently interpreted as the probability of the system to occupy state |k>. This probability interpretation depends on the orthonormal basis used to represent rho; which basis to use is a famous and not really solved problem in the foundations of quantum mechanics. For a pure state psi, rho has rank 1, and the sum extends only over the single index k with |k> = psi. Thus in this case, p_k = 1 and S = kbar 1 log 1 = 0, as it should be for a state of maximal information. The amount of missing information is zero. For more along these lines, and in particular for a way to avoid the probabilistic issues indicated above, see Sections 6 and 12 and Appendix A of my paper A. Neumaier, On the foundations of thermodynamics, arXiv:0705.3790 http://lanl.arxiv.org/abs/0706.0155 Arnold Neumaier [..] >>> A pure state of an electron is defined by its wave function (up to a >>> phase). Thus knowing all about an electron requires in the traditional [quoted text clipped - 16 lines] > and kbar is Boltzmann's constant. (I use the bar to be free to use k > as an index.) It seems confusing though that the entropy of let's say a box of 1 mole of Helium, at room temperature, is finite, while the information of just one particle is infinite. The good thing about entropy in the first sense (the box with 10^23 Helium atoms), is that the entropy is actually measurable. The other information quantity, the one with the single particle, is a lot trickier. A positivist probably wouldn't talk about the information in a single particle. I'm not really a positivist, so I'm OK with imagining aspects of a particle that we may not be able to measure. I don't have much trouble imagining a wave function with an infinite amount of information. It is the fact that *measured* entropy is finite that bothers me most. When I look at a box of gas, I find it counterintuitive that there is something finite about its information content. Gerard Gerard Westendorp schrieb: > [..] > [quoted text clipped - 22 lines] > mole of Helium, at room temperature, is finite, while the information > of just one particle is infinite. This is because the concept of information is somewhat fuzzy. Once one realizes that entropy is not information but _missing_ information the picture is much less confusing. More precisely, in the statistical interpretation, the state belongs not to a single particle but to an ensemble of particles. Entropy is the mean number of binary questions that must be asked in an optimal decision strategy to determine the state of a particular realization given the state of the ensemble to which it belongs. See Section 6 and in particular Appendix A of my paper A. Neumaier, On the foundations of thermodynamics, arXiv:0705.3790 http://lanl.arxiv.org/abs/0705.3790 (the link in my previous post was wrong!) Specifying a mixed state _exactly_ provides already an infinite amount of information, since the density matrix rho must be specified to infinite precision. Defining the eigenstates that are of interest in measurement amounts to specifying a Hamiltonian operator H _exactly_, which again provides already an infinite amount of information, since the coefficients of H in an explicit description must be specified to infinite precision. Then only a finite amount of information is missing to determine in which of the eigenstates a particular particle is. Of course in practice one just _postulates_ rho and H, pretending them to be known, while knowing well that one knows them only approximately. > The good thing about entropy in the first sense (the box with 10^23 > Helium atoms), is that the entropy is actually measurable. This is only because of a number of approximations made. One postulates exact equilibrium, hence a grand canonical ensemble, which of course is not exactly valid. Deviations from equilibrium are handled by means of a hydrodynamical approximation, in which entropy is no longer a number but a field - and specifying the entropy density again requires an infinite amount of information. Of course, one also represents this only to some limited accuracy, to keep things tractable. > The other > information quantity, the one with the single particle, is a lot > trickier. A positivist probably wouldn't talk about the information > in a single particle. It is the same theoretical concept, applied to a smaller system. Measuring a system gets of course trickier the smaller the system... > I'm not really a positivist, so I'm OK with imagining aspects of > a particle that we may not be able to measure. I don't have much [quoted text clipped - 3 lines] > counterintuitive that there is something finite about its > information content. As shown above, finiteness is enforced by making simplifying assumptions which are valid only if one doesn't look too closely. Arnold Neumaier [..] > Specifying a mixed state _exactly_ provides already an infinite amount > of information, since the density matrix rho must be specified to > infinite precision. Perhaps an analogy can be made with throwing a slightly biased die. If the die is thrown and temporarily kept hidden (as in some dice games), the observer needs an infinite amount of information to describe his knowledge of the state of the die: A probability distribution consisting of real numbers. On the other hand the state of the die can also be thought of consisting of much less information: Just a choice of 1 from 6. In this case, the exact description of ignorance about a system requires much more information than the system itself contains. Gerard Thus spake Gerard Westendorp <westy31@xs4all.nl> >> The theoretical physics FAQ at >> http://www.mat.univie.ac.at/~neum/physics-faq.txt [quoted text clipped - 11 lines] > >How does this relate to the entropy formula S = k ln(Omega)? I find this sort of thing very worrying in an faq. The wave function is determined, up to phase, by the very finite amount of information which comes from measurement. Regards  SignatureCharles Francis moderator sci.physics.foundations. charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and braces) http://www.teleconnection.info/rqg/MainIndex Oh No schrieb: > Thus spake Gerard Westendorp <westy31@xs4all.nl> >>> The theoretical physics FAQ at [quoted text clipped - 14 lines] > determined, up to phase, by the very finite amount of information which > comes from measurement. You may be thinking of von Neumann projection measurements. Here the wave function is forced into a Procrustes bed where it can only take a very finite amount of possibilities, from which then the resulting (not the original) state is determined. http://www.mythweb.com/teachers/why/basics/procrustes.html ''Procrustes was a host who adjusted his guests to their bed. Procrustes, whose name means "he who stretches", was arguably the most interesting of Theseus's challenges on the way to becoming a hero. He kept a house by the side of the road where he offered hospitality to passing strangers, who were invited in for a pleasant meal and a night's rest in his very special bed. Procrustes described it as having the unique property that its length exactly matched whomsoever lay down upon it.'' But modern measurements are no longer confined to the idealized measurements von Neumann introduced in 1932 which are discussed in typical textbooks. For example, look up quantum tomography in google which allows you to measure the density matrix of a mixed quantum system, and a fortiori the wave function of a pure quantum system. For a N-state system, one needs N^2-1 independent pieces of information. Most systems have infinitely many states, and therefore need an infinite amount of information for the reconstruction of their state to full accuracy. To get high (but finite) accuracy, you still need lots of measurements, not just a very few. Arnold Neumaier Gerard Westendorp wrote {478147e2$0$85782$e4fe514c@news.xs4all.nl} on Sun, 06 Jan 2008 20:44:26 -0500: > Arnold Neumaier wrote: >> The theoretical physics FAQ at [quoted text clipped - 14 lines] > > Gerard General definition of entropy is S = - k Tr{RHO {LN RHO}} for a *isolated* system *at equilibrium* RHO = 1 / OMEGA Substituting S = k Tr{RHO {LN OMEGA}} = k {LN OMEGA} Tr{RHO} = k {LN OMEGA} Textbooks often take this like definition for entropy but the general definition is that of above. An electron at pure state |PSI> is RHO = |PSI><PSI| and S = - k Tr{RHO {LN RHO}} = 0 -- I follow http://canonicalscience.com/guidelines.txt |
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