What is the largest wavelength of electromagnetic radiation that can produce quantum effects that we can detect? Radio, microwave?  What are the constraints to measure photons of larger wavelengths?  Edit: I'm not looking for radiation you can measure with an antenna. But measured with an system that may measure a single photon at a time.
says that a photon of  to $10^{-62}$ kg has been detected:     10×10−62 kg   Mass equivalent of the energy of the lightest photon   detected [citation needed]   This equates ($mc^2=E=hf=h\frac{c}{\lambda}$) to a wavelength of $\lambda=2.2\times 10^{20}$ m (roughly a million ly)! Can anyone provide a citation for this measurement? And how can such a large wavelength/low energy photon be detected?  Edit  Apart from the quotation (true/false un/reliable un/documented ....), I'd like to learn about the lightest photon detected, or about the instrumental limit: what is the lower limit of energy that can be registerd by an intrument (or the greatest wavelength), and how it is done? For example, can you detect a radiowave of 1 Hz?
Localization of a UFD that is a PID Why exactly do sometimes universal covers, and sometimes central extensions feature in the application of a symmetry group to quantum physics? There seem to be two different things one must consider when representing a symmetry group in quantum mechanics:   The universal cover: For instance, when representing the rotation group $\mathrm{SO}(3)$, it turns out that one must allow also $\mathrm{SU}(2)$ representations, since the negative sign a "$2\pi$ rotation" induces in $\mathrm{SU}(2)$ is an overall phase that doesn't change the physics. Equivalently, all representations of the Lie algebra are what we seek. ($\mathfrak{so}(3) = \mathfrak{su}(2)$, but although every representation of the algebra is one of the universal cover, not every representation of the algebra is one of $\mathrm{SO}(3)$.) Central extensions: In conformal field theory, one has classically the  of infinitesimal conformal transformations. From the universal cover treatment one is used to in most other cases, one would expect nothing changes in the quantum case, since we are already seeking only representation of an algebra. Nevertheless, in the quantization process, a  appears, which is often interpreted to arise as an "ordering constant" for the now no longer commuting fields, and we have to consider the  instead.   The question is: What is going on here? Is there a way to explain both the appearence of universal covers and central extensions in a unified way? Aliens like giant grasshoppers invade earth, give black people power over whites Would a fast interstellar spaceship benefit from an aerodynamic shape? Some (generous) assumptions:   We have a spaceship that can reach a reasonable fraction of light speed. The ship is able to withstand the high energies of matter impacting at that speed.   Given the amount of matter in inter-stellar space, at high speed, would it encounter enough of it and frequently enough that an aerodynamic shape would significantly reduce its drag (and thus save fuel)? Status of B1/B2 visa after citizenship change? I have a valid B1/B2 visa valid to US till March 2019 on Indian passport, I changed my citizenship to Australian. Is my B1/B2 visa on Indian passport valid? Where can I most efficiently buy all items for the "Shut up and take my money" achievement? Having migrated to Linux from Windows, I would like to find an alternative software to Winmerge or rather learn command line tools to compare and sync two folders on Linux. I would be grateful if you could tell me how to do the following tasks on the command line... (I have studied diff and rsync, but I still need some help.)  We have two folders: "/home/user/A" and "/home/user/B"  Folder A is the place where regular files and folders are saved and folder B is a backup folder that serves as a complete mirror of folder A. (Nothing is directly saved or modified by the user in folder B.)  My questions are:   How to list files that exist only in folder B? (E.g. the ones deleted from folder A since the last synchronization.) How to copy files that exist in only folder B back into folder A? How to list files that exist in both folders but have different timestamps or sizes? (The ones that have been modified in folder A since last synronization. I would like to avoid using checksums, because there are tens of thousands of files and it'd make the process too slow.) How to make an exact copy of folder A into folder B? I mean, copy everything from folder A into folder B that exists only in folder A and delete everything from folder B that exists only in folder B, but without touching the files that are the same in both folders. Did spacetime start with the Big bang? Did spacetime start with the Big Bang?  I mean, was there any presence of this spacetime we are experiencing now before big bang? And could there be a presence/existence of any other space-time before the big bang? Way to list "Contact Us" email address on web site, yet reduce likelihood of spam? What's a good way to list a "Contact Us" email address on a web site, while reducing the likelihood it will get spammed?  Is putting the email address in an image the best technique, or are there others? Proving question on sets I am unable to understand this question. I have to say whether its true of false and then prove it but I can't proceed with the question unless I understand it.     Let $S$ be the set of stars in our galaxy. There is exactly the same   number of subsets of stars in our galaxy as there are functions   $f:S\rightarrow\{\alpha,\beta\}$ Can I use Mage Hand to fly by Polymorphing myself and picking myself up with the hand? Could I first cast Mage Hand, then Polymorph into a Tiny creature less than 10 pounds then use the Mage Hand to pick myself up and fly at 30 Ft per turn?  Polymorph says:     The new form can be any beast ... The target's game statistics,   including mental ability scores, are replaced by the statistics of the   chosen beast. It retains its alignment and personality ... The   creature is limited in the actions it can perform by the nature of its   new form and it can't speak, cast spells, or take any other action   that requires hands or speech.   This keeps me from casting spells, which keeps me from just casting Mage Hand to fly as long as I want to over and over. But for that initial duration of the Mage Hand cantrip, could I use the Mage Hand to fly as a Tiny polymorphed creature until the spell runs out? As you are all most likely aware already all domain registrars ask for your personal information.   Such personal information includes:   Forename and surname Address information Email address Telephone number   Personal information via WHOIS  Depending on the domain type such as  or  this information may become immediately available after the registration of the domain via a simple whois. If you do not want that (and in most cases you certainly don't) they offer a service where you can remove this information from the whois service but of course hiding your personal information adds yearly costs to the domain  purchase.  What happens if I use fake details when registering a domain? For any symbol, how can I get the full context-qualified name of the symbol as a string? Divisibility Rule for 9 I'm working through an elementary number theory course right now and I think I've come up with a proof here but wanted some feedback on my logic.    Question:  If the sum of the digits in base 10 is divisible by 9, then the number itself is divisible by 9.    Proof: Suppose that $9|d_1+d_2+...+d_n$ then $d_1+d_2+...+d_n=0\mod9$  Now consider $d_1(10^{n-1})+d_2(10^{n-2})+...+d_{(n-1)}(10^1)+d_n(10^0)$ Each power of $10$ is equivalent to $1\mod9$   therefor  $d_1(10^{n-1})+d_2(10^{n-2})+...+d_{(n-1)}(10^1)+d_n(10^0)=(1\mod9)(d_1+d_2+...d_n)$  $9|(d_1+d_2+...d_n)$ by our assumption, thus $9|(1\mod9)(d_1+d_2+...d_n)$  Thus we have shown that if 9 divides the sum of the digits in base 10, 9 divides the number itself.    The only question I really have is whether I'm jumping the gun on my assumption concerning the powers of 10 being $1\mod 9$.  I think this is fair game here but not 100% confident.  Thanks. How do I install fonts? Question says it all. I need them to be available in gimp. Construct a Borel set on R such that it intersect every open interval with non-zero non-"full" measure I want to convert a std::string to lowercase. I am aware of the function tolower(). However, in the past I have had issues with this function and it is hardly ideal anyway as using it with a std::string would require iterating over each character. Is there an alternative which works 100% of the time? Relationship between Binomial and Beta distributions I'm more of a programmer than a statistician, so I hope this question isn't too naive.  It happens in sampling program executions at random times. If I take N=10 random-time samples of the program's state, I could see function Foo being executed on, for example, I=3 of those samples. I'm interested in what that tells me about the actual fraction of time F that Foo is in execution.  I understand that I is binomially distributed with mean F*N. I also know that, given I and N, F follows a beta distribution. In fact I've verified by program the relationship between those two distributions, which is  cdfBeta(I, N-I+1, F) + cdfBinomial(N, F, I-1) = 1   The problem is I don't have an intuitive feel for the relationship. I can't "picture" why it works.  EDIT: All the answers were challenging, especially @whuber's, which I still need to grok, but bringing in order statistics was very helpful. Nevertheless I've realized I should have asked a more basic question: Given I and N, what is the distribution for F? Everyone has pointed out that it's Beta, which I knew. I finally figured out from Wikipedia () that it appears to be Beta(I+1, N-I+1). After exploring it with a program, it appears to be the right answer. So, I would like to know if I'm wrong. And, I'm still confused about the relationship between the two cdfs shown above, why they sum to 1, and if they even have anything to do with what I really wanted to know. Order of items in a context menu How does MySQL process ORDER BY and LIMIT in a query?