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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? How do I set up my LaTeX document to enable me to cite from my BibTeX bibliography? This may be an overly simplistic question but please bear with me...  I have a BibTeX bibliography containing all of my references, set up using BibDesk. I am now trying to cite these references in my LaTeX documents. Following the advice on the LaTeX wikibooks page, I have included the following lines into my LaTeX document...  \bibliographystyle{plain} \bibliography{/Users/Graeme/Documents/References/BIBliography.bib}   ... where BIBliography is my .bib file, created in BibDesk.  Unfortunately, when I cite something in the document, using \cite{citekey} and typeset the document, the log tells me that there were undefined references. In the .pdf that is produced, there is a [?] where the references should be.  Am I doing something wrong? Do I need to store my .bib for in a particular location? I know there can be some errors that mean you need to typeset the document several times, but I have tried countless times to no avail.  I would be very grateful if someone could help out. Sci-Fi Movie/Show Where a Soul is Captured in a Glass Vessel I remember very little of this, I was quite young -  It was on television in the mid 90's, but seemed older. My wife was telling me about the 21 grams experiment and the memory clicked into place. Setting Inside of a mad scientist sort of lab, with the plasma balls and everything else, a character (Who in my mind resembles Gene Wilder in Young Frankenstein) is conducting an experiment. The room is hazy, lots of hues of purple and red. He has an assistant present. They were conducting some sort of experiment to try to figure out the secrets of immortality (or possibly reanimation). I remember them &quot;trapping&quot; a human soul in a glass vessel to see what it weighed or what kind of volume it took up - I believe they shot someone in this vessel, and because their soul could not &quot;escape&quot;, they did not die. In the first half of Tenet, there is a scene where The Protagonist and Neil break into Freeport and find a glass wall with bullet holes (could only find a good photo with Christopher Nolan in it):  In the next scene, we find out that these bullet holes were caused by an inverted armed man entering the room (exiting the room from their perspective) and reverse-shooting the glass, which removes the bullet holes from the future-facing perspective. This all makes sense, but raises a question: how long were the bullet holes there in the first place? Were they in place as soon as the glass was installed, possibly years ago? Or even back when the glass was manufactured? How to convert an "object" into a function in JavaScript? JavaScript allows functions to be treated as objects--if you first define a variable as a function, you can subsequently add properties to that function.  How do you do the reverse, and add a function to an "object"?  This works:  var foo = function() { return 1; }; foo.baz = "qqqq";   At this point, foo() calls the function, and foo.baz has the value "qqqq".  However, if you do the property assignment part first, how do you subsequently assign a function to the variable?  var bar = { baz: "qqqq" };   What can I do now to arrange for bar.baz to have the value "qqqq" and bar() to call the function? Disable App Update Notifications on OSX? Does the Dueling Fighting Style prevent using a shield? The Dueling fighting style doesn't seem to state anything about shields, stating that the character can only wield one weapon while fighting to gain the benefit. Does a shield count as a weapon or otherwise stop the damage bonus from occurring? Disable Narrator shortcut key in Windows I've install Windows 8 on a BootCamp macbook air The problem is that the command key located differently from the winkey in a normal keyboard, therefor, I keep clicking Winkey+Enter which starts the narrator (which is really annoying!)  Is there a way to disable this shortcut? I want to clear all previous commands from the history of my server. I used history -c and it seems all things are cleared but when I ssh to the server, all the commands are still there.  How can I clear them permanently? An example of a sequence of Riemann integrable functions $(f_n)$ that converges pointwise to a function $f$ that is not Riemann integrable. Power supplies are available in a wide range of voltage and current ratings.  If I have a device that has specific voltage and current ratings, how do those relate to the power ratings I need to specify?  What if I don't know the device's specs, but am replacing a previous power supply with particular ratings?  Is it OK to go lower voltage, or should it always be higher?  What about current?  I don't want a 10 A supply to damage my 1 A device. I have a DSLR (Canon 550D) and since I'm taking my first steps in the world of RAW photography I don't know which would be a good program to use to edit these files. I've tried  and  but wasn't satisfied. I'm sure there must be better software out there... $16$ rooks on a chess board. Place $8$ pairwise non-attacking white rooks and black rooks on a $8\times8$ chess board. If one can swap rows and columns, is it possible for the black rooks to take the initial position of the white rooks and vice versa?  My attempt:  Let the rows be $a_1,a_2,...,a_8$ and columns $b_1,b_2,...,b_8$. If there are white rooks at ($a_i$,$b_j$) and ($a_j$,$b_a$) as well as black rooks at ($a_i$,$b_i$) and ($a_j$,$b_j$), then we can reduce it to a $2\times2$ and a $6\times6$.   I don't know what to do next. Any help appreciated. Proving $\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$ without induction I was looking at: $$\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$$  It's pretty easy proving the above using induction, but I was wondering what is the actual way of getting this equation? How are parentheses interpreted at the command line? I am using an USB port as a power supply. Is there a way I can take the weak 5V of a regular laptop USB and transform if to 100V+ while begin in the smallest package possible. I've heard of boost converters but they are usualy too big. I want something that has the size of like a normal USB key.   What I want to do is to charge up around 5 capacitors at 100+V and then discharge them later. Power efficiency not a concern for me.  It would be helpful to include a schematic of the circuit. Computing $\mathrm{Hom}(\mathbb Z_n,\mathbb Z_m)$ as $\mathbb Z$-module Does the Polyjuice Potion Dark Mark work? I was just reading Harry Potter and the Deathly Hallows, and I was wondering, when Hermione Granger changed into Bellatrix Lestrange, she should've obtained the Dark Mark, temporarily.  Now, my question is: would the Dark Mark work if pressed? Here's a situation that came up during a back lot league softball game a few weeks ago. (Note: While our league has some of its own rules we try to defer to MLB rules in all causes that aren't specifically addressed.)  There is a runner on second (R2) and a runner on first, with no outs. A ground ball is hit to the third baseman (3B) who cleanly fields the ball away from third base. Seeing that R2 is running toward him, 3B does not go to third and instead waits to tag the incoming R2. R2, in an attempt to buy some time for the other runners, stops advancing to third base. As soon as R2 stops, 3B declares R2 is out (without making a tag) and throws the ball to first to make a play.  During the discussion after the play, the 3B claimed that any player advancing on a force cannot stop moving toward their next bag or they are automatically out (the same way a runner leaving the baseline to avoid a tag is considered out). Our rules don't have a specific clause that addresses this situation and I couldn't find anything in official MLB rules (though it's possible I missed something, as 3B claims it is an MLB rule).  So, is 3B’s claim correct? That if a runner who is forced to advance to their next base stops (in particular to avoid/delay a tag) they are automatically out? Use early 2009 20" iMac as external display for 2015 rMBP