I am wondering if the masslessness of photons is due to the local gauge invariance of $u(1)$-gauge fields. The reason why I consider about this question is that I remember that the  is not locally $u(1)$-invariant. How do I prove this explicitly in field theory?
Why can't photons have a mass? Could you explain this to me in a short and mathematical way?
How to Remove URLs from Google Search Engine I received a table template, which I'm intended to generate dynamically with data using Java. But when the table becomes to large, the compilation with LaTeX fails. The specific about this template is, that it stretches the table width into the maximal page width and has same column widths for all data columns (except the first column).  Here is a minimal version of the full template, that still works fine:  \documentclass[final,authoryear,3p,twocolumn]{elsarticle}  \usepackage{amssymb,amsmath} \usepackage{calc} \usepackage{array} \newcolumntype{L}[1]{&gt;{\raggedright\arraybackslash}p{#1}} \newcolumntype{R}[1]{&gt;{\raggedleft\arraybackslash}p{#1}} \usepackage{eurosym} \usepackage{microtype} \usepackage{needspace} \usepackage{fp}  \usepackage{graphicx} \DeclareGraphicsExtensions{.pdf} \input{pgfexternal}  \newcommand\tabt{\rule{0pt}{2.6ex}} \newcommand\tabb{\rule[-1.2ex]{0pt}{0pt}}  \begin{document}  \begin{table*}[t]     \footnotesize     \begin{tabular*}{\linewidth}{@{}L{2em}@{}*9{R{(\linewidth-2em)/9}@{}}}\hline\tabt $i$ &amp; $1$ &amp; $2$ &amp; $3$ &amp; $4$ &amp; $5$ &amp; $6$ &amp; $7$ &amp; $8$ &amp; $9$\tabb\tabularnewline\hline\tabt $1$ &amp;  &amp; $48.87$ &amp; $0.01$ &amp; $0.05$ &amp; $1.23$ &amp; $0.00$ &amp; $0.20$ &amp; $5.67$ &amp; $2.35$\tabularnewline     \end{tabular*} \end{table*}  \end{document}   But when extending this example by another column, so that the table would look like the following example, it fails:  \begin{table*}[t]     \footnotesize     \begin{tabular*}{\linewidth}{@{}L{2em}@{}*10{R{(\linewidth-2em)/10}@{}}}\hline\tabt $i$ &amp; $1$ &amp; $2$ &amp; $3$ &amp; $4$ &amp; $5$ &amp; $6$ &amp; $7$ &amp; $8$ &amp; $9$ &amp; a\tabb\tabularnewline\hline\tabt $1$ &amp;  &amp; $48.87$ &amp; $0.01$ &amp; $0.05$ &amp; $1.23$ &amp; $0.00$ &amp; $0.20$ &amp; $5.67$ &amp; $2.35$ &amp; b\tabularnewline     \end{tabular*} \end{table*}   This fails, no matter how many columns I add. So far, it seems for me to be a problem with the division by 10. May it be, that LaTeX isn't able to handle that properly by default? How can I solve this problem and make even larger tables work using this template. Or is that to any other problem I don't see yet? Thanks for any help! How can we print line numbers to the log in java How to print line numbers to the log. Say when outputting some information to the log, I also want to print the line number where that output is in the source code. As we can see in the stack trace, it displays the line number where the exception has occurred. Stack trace is available on the exception object.  Other alternative could be like manually including the line number when printing to the log. Is there any other way? How can a point-particle have properties? I have trouble imagining how two point-particles can have different properties.  And how can finite mass, and finite information (ie spin, electric charge etc.) be stored in 0 volume?  Not only that, but it can also detect all fields without having any structure. Maybe it can check curvature of spacetime to account for gravity, but how can a point contain the information of what the other fields-vectors are? This seems to mean that also the information/volume in space is infinite.  Mathematically, a point cant have any intrinsic structure, so how does physics which is a mathematical theory explain this? How to enable read/write for windows partitions? With 11.04 I could read and write my partitions that was created in Windows however after upgrading to 11.10 somehow I can only read those partitions. How do I enable this ability in 11.10? Say you take a dice and you roll it twice so that you have a pair (X,Y) where X represents the first roll, Y represents the second.   When you have a distribution like max(X,Y), what type of distribution is that? It's not uniform, obviously, because the outcomes of max(X,Y) are clearly more favorable towards the higher numbers. I'm being asked to "describe" the distribution. I know that we can look at each distinct outcome, like:  max(1,1) = 1  max(1,2) = 2  max(1,3) = 3  max(1,4) = 4  max(1,5) = 5  max(1,6) = 6  max(2,1) = 2  max(2,2) = 2  max(2,3) = 3  max(2,4) = 4  max(2,5) = 5  max(2,6) = 6  max(3,1) = 3  max(3,2) = 3  max(3,3) = 3  max(3,4) = 4  max(3,5) = 5  max(3,6) = 6  max(4,1) = 4  max(4,2) = 4  max(4,3) = 4  max(4,4) = 4  max(4,5) = 5  max(4,6) = 6  max(5,1) = 5  max(5,2) = 5  max(5,3) = 5  max(5,4) = 5  max(5,5) = 5  max(5,6) = 6  max(6,1) = 6  max(6,2) = 6  max(6,3) = 6  max(6,4) = 6  max(6,5) = 6  max(6,6) = 6  But is that the most effective way to look at a distribution like this? Is there a faster and more concise way to describe this distribution other than saying the probability of each outcome?  i.e.,  P(max = 1) = 1/36  P(max = 2) = 3/36 = 1/12  P(max = 3) = 5/36  P(max = 4) = 7/36  P(max = 5) = 9/36 = 1/4  P(max = 6) = 11/36  So, obviously, I understand how to LOOK at a discrete distribution and see what the probabilities are, but not until I look at each possible outcome. But what does it mean to "describe" a distribution? This isn't uniform or any sort of discrete distribution we're taught. Can someone provide clarity? Moving set of polygons in ArcGIS Pro? Whenever I want to use the pipe operator in the terminal, I get a weird kind of "bug":    As is shown in the image above, I can type the exact same command twice and it will sometimes register it or sometimes include some spacing in the command, as can be seen in the error message. The only difference between the two commands was the speed I typed them, with fast typing the char sequence "| " (without the "") leading to the error. I've checked in the terminal settings if this sequence somehow triggers a shortcut, but I did not find anything.   In case it matters, I'm using a swiss keyboard. The layout is as follows:    To produce the "|" character, I press "alt + 7". Does anybody know what the cause of this problem is? I've somehow lost my id_rsa.pub file. I still have my id_rsa file. How do I recover the public key from the private key?  (This question came up after ssh-copy-id was reporting "ERROR: No identities found" while ssh-keygen reported "/root/.ssh/id_rsa already exists". ls ~/.ssh pointed out the problem was having id_rsa but not id_rsa.pub. I found the solution, but it was buried at the bottom of a long post on an Ubuntu support forum.) "did shoot" vs "shot" Can I run Ubuntu OS on a Android 4.0 tablet? What does one second after big bang mean? Consider the following statement:     Hadron Epoch, from $10^{-6}$ seconds to $1$ second: The temperature of the   universe cools to about a trillion degrees, cool enough to allow   quarks to combine to form hadrons (like protons and neutrons).   What does it mean to say "from $10^{-6}$ seconds to $1$ second"?   How is time being measured?  One particle might feel just $10^{-20}\ \mathrm s$ having passed and another could feel $10^{-10}\ \mathrm s$ having passed.   Is saying "1 second after the big bang" a meaningful statement? What does it take to understand Maxwell's equations? Assume I want to learn math and physics enough to reach a level where I understand Maxwell's equations (The terms and reasoning in the equations I.e. why they "work"). What would I have to learn in order to have the tools I need to make sense out of it?  I'm kindof looking for a road map which I can use to get started and know what to focus on. The fields are pretty big so some pruning would be very helpful (if possible) I recall in at least two episodes of Star Trek ( from the original series and  from The Next Generation) that there is reference to and contact with species from external galaxies.  By the 29th century, we see that timeships exist.  My question is, is there any reference to intergalactic relations in Star Trek in the future (say, by the 29th Century), or does the Star Trek Universe stay in the Milky Way Galaxy? Is there any way to generalise this problem? Why are useful questions on StackOverflow being closed as non-constructive? Useful threads are being closed as "not constructive:"     This question is not a good fit to our Q&amp;A format. We expect answers to generally involve facts, references, or specific expertise; this question will likely solicit opinion, debate, arguments, polling, or extended discussion. See the .   For example,  was very useful for me since I am new to obfuscation and the question gave me a start. Also that question has lots of votes and bookmarks - these alone are the  markers of usefulness. The question is closed anyway. No idea why.  Same happened to my question, , which received one bookmark immediately.  I was waiting for an answer with the two products downloaded and installed.  I desired that experienced people would share their thoughts and point me to the better one. No. Closed. I.e. I must go to some other forum and ask there or I must search on Google and write down the pros and cons of each myself. I am not asking questions just because I have tons of time and want to chat; I need an answer, that's why I am asking. I am asking because I think that SO has the most professional people to answer and because I didn't find a good answer in Google.  I saw other questions closed which I myself found to be useful and I bookmarked some of the closed questions.  UPDATE  If I want people who have an experience which I do not have with some software products and I want to see which product is better in the terms of usability and reliability, not just better in terms of the price or has a better official website or better described on the official website, how do I ask this question? Or is this a question which I cannot ask on SO? I.e. this sort of information is not available on SO?  Is this one breaking the rules as well:  ? The tilde operator in Python In-universe explanation for single line of planetary entry/departure in Star Wars Why are planetary blockades confined to a single point of a planet instead of being sphere shell shaped around the planet? (I know, it would take an awful lot of spaceships.)  Also, when a rebel ship wants to escape from a planet and there is a single star destroyer hovering in the atmosphere (or some more of them further up, like in the Empire Strikes Back) how come it's not a viable maneuver to fly low altitude for a couple of hundred kilometers horizontally (they would be out of the single-point blockade), then make a vertical dash upwards? My equation is not showing big bracket and not showing in next line What are app domains used for?