I am currently starting to learn some QFT. And I am very confused by the fact that the freedom to choose a gauge for a theory controls the physical degrees of freedom.  Please correct me if i am wrong with the following because i think i might have internalized by myself undetected misconceptions.  The free electro-magnetic field for example has only two physical degrees of freedom, since electromagnetic waves are exclusively transversal waves so there are no longitudinal degrees of freedom.  The four-potential $A^\mu$ has four entries so with no further information $4$ degrees of freedom, so we need to relate the entries of $A^\mu$.  I do know and understand that the fields described by $F^{\mu\nu}$ are invariant under a transformation $G$ with $$ G:\quad A^\mu \longrightarrow A^{\mu} + \partial^\mu \phi $$ where $\phi$ is some arbitrary scalar field, which can be easily shown by explicitly writing out the field tensor.  Question: Why does the freedom to choose such a gauge cancel out 2 degrees of freedom ?
How do I definitively show that there are only two propagating degrees of freedom in the Lorenz Gauge $\partial_\mu A^\mu=0$ in classical electrodynamics.  I need an clear argument that   involves the equations of motion for just the potentials $A^0$ and $\mathbf{A}$, and not the electric and magnetic fields.   includes sources $\rho$ and $\mathbf{J}$ in the equations of motion.  This is to justify the assertion that there are degrees of freedom that decouple from the rest of system. does not critically rely on quantum field theoretic arguments (although any supplementary remarks are welcome).     To illustrate the level of clarity I expect, I provide an argument in the Coulomb gauge $\nabla\cdot \mathbf{A} = 0$:  Of the four field degrees of freedom, the gauge condition $\nabla\cdot \mathbf{A} = 0$ removes one degree of freedom (the longitudinally polarized EM waves).  To show that among the remaining three degrees of freedom only two are propagating, consider the field equations of motion in the Coulomb gauge:  \begin{align} \nabla^2 A^0 &amp;= -\rho/\epsilon_0,\\ \big[\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2\big]\mathbf{A}&amp;= \mu_0 \mathbf{J}-\nabla \frac{1}{c^2}\frac{\partial}{\partial t}A^0. \end{align}  The first equation is NOT a wave equation for $A^0$, and thus does not propagate.  The final equation IS a wave equation, and describes the propagation of two degrees of freedom (Gauss' law in first eq. can be solved, and then inserted into the second equation to show that $\mathbf{A}$ only couples to the Solenoidal part of the current).
What does $b_i\mid b_{i+1}$ mean in this context? In the computational topology literature, the reduction algorithm for computing the Smith normal form of a boundary matrix uses the notation $b_j &gt; 1 \: \text{ and }\: b_j\mid b_{j+1}$ in the context of the diagonal elements of the Smith matrix. Can anyone give me an idea for what it means? Show that $\mathbb{E}(T) = \sum\limits^\infty_{k=1}\mathbb{P}(T \geq k)$ for $T$ nonnegative integer valued and $E[T] < \infty$ Let T be a non-negative integer-valued random variable with $\mathbb{E}(T) &lt; \infty $. Prove that $\mathbb{E}(T) = \sum^\infty_{k=1}\mathbb{P}(T \geq k)$.  Had a few attempts, haven't really got anywhere. I'm wondering as I'm typing this if proof by induction is a good way to go.  Edit: One major thing I forgot to add, am I correct in thinking that also, $\mathbb{E}(T) = \sum^\infty_{k=1}k\mathbb{P}(T = k)$? When I try to sculpt with dyntopo on a sword it creates really weird results while it works just fine on other meshes. Any idea what causes this? Scenario:   Create SymLink: C:\Data\workspaces --&gt; C:\Users\****\Documents\workspaces Navigate to C:\Data\Workspaces in explorer Click and drag the the path from the address bar onto "Quick Access" in the navigation pane.   Expected Result:   Clicking on the shortcut takes you to C:\Data\workspaces   Actual Result:   Clicking on the shortcut takes you to C:\Users\****\Documents\workspaces   How do you make it so that the shortcut points to the correct path. (the SymLink)  Command Used to create the SymLink:  mklink /D C:\Data\workspaces C:\Users\****\Documents\workspaces   Updated with pictures:  Please note the workspaces symlink inside C:\Data:    When I double-click on the symlink it works correctly. (note the address bar.    When i pin it to the Quick access bar, however, it uses the address that the symlink points to and NOT the symlink address. What is rank deficiency, and how to deal with it? Fitting a logistic regression using  ends with   Error in mer_finalize(ans) : Downdated X'X is not positive definite.    A likely cause of this error is apparently rank deficiency. What is rank deficiency, and how should I address it? Linear regression with fixed point I have a data set to which I want to add a trend line. For my system, there will always be a fixed point on (2, 0). I have the line representing this system, but currently I do not account for the fixed point. Is there functionality to achieve this?  \begin{figure} \begin{tikzpicture} \begin{axis} [     ylabel={Some Time [ms]},     ]     \addplot[only marks, color=red, error bars/.cd,                        y dir=both, y explicit]         table [x=num_of_nodes, y=MyData, y error=stdev_delay_var] {MyData.dat};     \label{plot_one}     \addplot +[mark=none, dashed]          table[y={create col/linear regression={y=MyData}}]         {MyData.dat}; \end{axis} \end{tikzpicture} \caption{Test Results} \label{fig:test_results} \end{figure} nVidia driver reset after each kernel update I am using the official nVidia driver because my card (GeForce 750 Ti) isn't yet supported by the Ubuntu one. Every time the kernel updates, my system breaks.  I've formatted my system two-three times because of kernel updates and today I decided to try to fix it. I tried to fix it with this method and surprisingly it works.   Switch to console (After waiting for the nVidia driver to crash, as it does when you switch consoles :p) Run a complete uninstall  sudo ./&lt;DRIVER&gt;.run -uninstall  Reboot and switch to the on-board card Update  sudo apt-get update sudo apt-get dist-upgrade  Reboot again Reinstall driver  sudo ./&lt;DRIVER&gt;.run  Reboot and switch back to the NVIDIA GPU   Is there a better/easier/faster way to update without going through all this? I have now almost completed my family tree, having each branch going back to approximately the French revolution, discovering dozens of previously unknown ancestors.  I would like to share the family tree with my family, the problem is that I cannot have a tree in a nice format that could be printed on a standard A4 sheet (maybe using both sides or whathever).  I have a SVG tree that use graphics, but it is unconventionally large and if scaled down to a A4 sheet, the text would be so small that it would not be legible. The maximum width is of about 50 ancestors of a generation, then it goes sparse again.  I'd like to know if a format exist where the family tree (with direct ancestors only) would be printable, with names but also other complementary information, and the format should be printable as not all my family users are particularly skilled with computers. When using custom path style in a tikz graph the quotes label isn't printed I see that there a number of questions (, , , , ) on best practices, none of them answer the issue I am going to talk about.  We all need to prepare large documents (or slides) from time-to-time. And once you have reached a certain number of pages, you find that compiling the whole document takes a substantial time, which you need to do to check that the part you have written is error free, specially if it contains a number of images.   To circumvent the situation, I have used the following practices over years to keep compile time at minimum.   Divide the document in to many chapters: The chapters are put in separate files. Only the one I am working one is uncommented.  \documentclass{book}  \begin{document}  %\input {chapterone.tex} \input {chaptertwo.tex} %\input {chapterthree.tex} %\input {chapterfour.tex}  \end{document}    The above technique can also be used for \include.   Keep a preamble replica of the main file in a temporary file (build file): I use this technique when I am preparing a presentation with many slides, say a hundred. Prepare a main file, copy it to another build file. Delete everything between  \begin{document} and \end{document} in the build file. Prepare a slide (or a group of slides) in the build file. Cut and paste the contents between \begin{document} and \end{document} in the build file to relevant location in the main file only when satisfied with the output (image locations, overfull boxes etc.)     We can hear your inputs about the best practices you practice when a you are preparing a document or presentation from scratch.    Now, I have failed to find a good answer in the case when you have finished the whole document or presentation and now you need tweaking the small things, may be edit a text here, add an image there. What will be your best practice?  Perhaps, again keeping only one chapter uncommented will be usable here. But what about large presentations? May be adding a \end{document} just after the point-of-change will be beneficial, but only if you can put that before midpoint of the file.  Commenting a part of the file will also work, but that is error-prone, at least for me. How to safely change OS name in grub boot menu? I have a system where I am dual-booting Ubuntu GNOME 15.10 and Ubuntu GNOME 16.04, however after installing Ubuntu GNOME 16.04, I have found that in the GRUB boot entry list it is called "Ubuntu", whereas the other one is called something like "Ubuntu GNOME 15.10", is there any way I can change the text of the "Ubuntu" one to "Ubuntu GNOME 16.04"? And also change the names of the other options so that instead of "Advanced options for Ubuntu" it says "Advanced options for Ubuntu GNOME 16.04" etc?  I rather do it manually if it is safe to do so, or a piece of software could be recommended, however nothing from a PPA or other untrusted not official source (I will only install software from the Ubuntu official repositories). I just saw now that a new build had rolled out in which the comment upvote and flag options are always visible. Previously, one had to mouse over the area in order to see them.  Why was this just changed? Is it to support touch screens? I'll admit, it was difficult to find the flag button using my touchscreen laptop running Microsoft Edge.  Cross-site duplicate on MSO: Who invented "dooblidoo"? The word  is used by several different youtube channels as a different word for the youtube description bar. I've seen it used by the  and by .  Who was first to use and invent the term? Vertical line only in one part of a table I need some help on  putting a vertical line in the second part of the table, I don't know how to do it since the tabularx I declared does need to have that only in the second part. The code I used is:  \begin{tabularx}{\linewidth}{|XX|} \hline \multicolumn{2}{|c|}{\textbf{title name}}\\ \hline something: &amp; else \\ something: &amp; else \\ something: &amp; else \\ \hline \multicolumn{2}{|c|}{bla bla blah}\\ \hline title text: &amp; description 1 \\ title text: &amp; description 1 \\ \hline \end{tabularx}   And the render of the table looks like this:    I just need a \vline between title text and description (it I use \vline it works only for the first line but the description text and sometimes the title, are few lines long). How to install qt 4.8 from repository? QGIS save postgis table error : no result buffer I've a polygon layer on PostGIS, and when I try to add featureS using QGIS it gets error when I save the edit.   Here is the error notification "no result buffer".     I found a similar case here : . But it doesn't give me the answer to solve it. Any ideas ? Using Inclusion/Exclusion to solve $x_1+x_2+x_3=15$ with $x_1,x_2\leq 5$ and $x_3\leq 7$ for non negative integers $x_1,x_2,x_3$ I want to solve for:     Number of integers solutions to the equation $x_1+x_2+x_3=15$ with $x_1,x_2\leq 5$ and $x_3\leq 7$ for non negative integers $x_1,x_2,x_3$   How can this be done using the principle of inclusion/exclusion? What does the 'static' keyword do in a class? Can you enter UK later than intended date of travel on the visa application? Is there any way in QGIS to apply service area buffers e.g if I have a point on a map I wish to show the service coverage of 1 mile, 2 mile, 3 mile, breaks using  the road network?