Gauge invariance and the degrees of freedom 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 ?
Counting the number of propagating degrees of freedom in Lorenz Gauge Electrodynamics 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).
$S^2 \times S^2$ is diffeomophic to $G_2(\mathbb{R}^4)$ $G_2(\mathbb{R}^4)$ is the Grassmannian manifold of two-dimensional subspaces of $\mathbb{R}^4$. I would like a detailed proof. Can it be done explicitly? I mean, showing the map and checking its differentiability using charts? Is there something like "Firebug for IE" (for debugging JavaScript)? I'm trying to fix some JavaScript bugs. Firebug makes debugging these issues a lot easier when working in Firefox, but what do you do when the code works fine on Firefox but IE is complaining? I execute quite a lot of sudo commands.  I noticed that if I execute one and then another one within few minutes, the 2nd time I don't get the message:     [sudo] password for my_username:   but I do get it when there is more time between the execution of the two commands.  What's the period of time in which the message is not appearing? How can we check it / update it? Recently, I accidentally deleted a partition off of my external hard disk. I had a copy of Minitool Power Data Recovery left over when the same drive had become RAW about a year ago. When I selected lost partition recovery, it immediately identified my deleted partition and allowed me to access and recover the files that were there.  My question is this. Is there any way to change the deleted partition back to NTFS without having to have a drive that you need to recover the files to and then move them back after formatting, as I don’t have any space left on either the computer or on my other external hard drive, so that I don’t need to buy a new one. The files on the disk were really important.  I am using Windows 7 Home Premium.  I appreciate any help in solving this crisis. Create default home directory for existing user in terminal How do I do this? I created a user without a home directory and now I want to create one. Not just a folder called new_user, I want the default home directory with all it's folders and hidden files, etc. How do you parse and process HTML/XML in PHP? Windows 10 - Users already logged in at boot Following on from this question:   It seems all feats carry over to your new form if your new form is capable of using them (so Mounted Combatant and Inspiring leader would usually be ruled as unusable in most circumstances). But what about Tough?     Your hit point maximum increases by an amount equal to twice your   level when you gain this feat. Whenever you gain a level thereafter,   your hit point maximum increases by an additional 2 hit points.   Would this increase your maximum HP by the same amount when in the form of a rat, or brown bear? How to clip an Openlayers OSM Background to a Vector Boundary? I'm new to QGIS. In my project i have two layers: one is my vector (city boundaries) and the other is the OSM (obtained with the plugin). How can I cut the OSM along the edges of the vector? I want to obtain only a region of the map, to save it as an image. Calculate sum of squares of first $n$ odd numbers Is there an analytical expression for the summation $$1^2+3^2+5^2+\cdots+(2n-1)^2,$$ and how do you derive it? What are the blue dots next to App names in iOS and macOS? They disappear once you open the apps and then return to the home screen, but I can't tell what they indicate.  Are they apps that haven't been opened since an app update? Or that it thinks have never been opened (not accurate in my case, but that could be a function of a "restore" confusing the phone). This is a reformulation of a deleted question:  If  $a_1 &gt; 0$ and $a_2 &gt; 0$ and $2a_{n+2} \le a_{n+1}+a_n$,  show that  $\lim \sup a_n \le \frac23 a_2 + \frac13 a_1$.  My proof involves showing that $a_{n+2} \le u_{n} a_{n+1} + (1-u_n)a_n $ where $u_n \to \frac23$, and I wondered if there is a simpler proof.  I will post my proof in a couple of days if no one does. Deductible expenses paid with credit card: In which tax year would they fall? If I paid my business expenses with a credit card during the month of December, which year can I deduct them from my taxes? 2011 (when the purchase was made) or 2012 (when I paid off the credit card)?  IRS pub reference greatly appreciated In which cases should I use the subjunctive mood?     I suggest that every applicant fill out the form carefully.        If she were rich, she would live on Long Island. Let $A,B \in \mathbb{M}_n (\mathbb{C})$. If $A^2B + BA^2 = 2ABA$ then exist $k \in \mathbb{N}$ where $(AB-BA)^k = 0$.  I tried solve with minimal polynomial, but I did not have much effect. My machine is continously making udp dns traffic request. what i need to know is the PID of the process generating this traffic.  The normal way in TCP connection is to use netstat/lsof and get the process associated at the pid.  Is UDP the connection is stateles, so, when i call netastat/lsof I can see it only if the UDP socket is opened and it's sending traffic.  I have tried with lsof -i UDP and with nestat -anpue, but I can't find wich process is doing that request because i need to call lsof/netstat exactly when the udp traffic is sended, if i call lsof/netstat before/after the udp datagram is sended is impossible to view the opened UDP socket.  Call netstat/lsof exactly when 3/4 udp packet is sended is IMPOSSIBLE.  How I can identify the infamous process? I have already inspected the traffic to try to identify the sended PID from the content of the packet, but is not possible to identify it from the contect of the traffic.  Anyone can help me ?  I'm root on this machine FEDORA 12 Linux noise.company.lan 2.6.32.16-141.fc12.x86_64 #1 SMP Wed Jul 7 04:49:59 UTC 2010 x86_64 x86_64 x86_64 GNU/Linux I have the following file:  abcde kwakwa &lt;0x1A&gt; line3 linllll   Where &lt;0x1A&gt; represents a byte with the hex value of 0x1A. When attempting to read this file in Python as:  for line in open('t.txt'):     print line,   It only reads the first two lines, and exits the loop.  The solution seems to be to open the file in binary (or universal newline mode) - 'rb' or 'rU'. Can you explain this behavior ? I'm supposed to calculate: $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using WolframAlpha, I might guess that the limit is $\frac{1}{2}$, which is a pretty interesting and nice result. I wonder in which ways we may approach it. I would like to install Ubuntu  for almost 120 users. But it's very difficult in using USB and drives.  Any suggestions? How can I copy from Tmux terminal and paste it other application in my OS? I am using tmux on ubuntu. I read this  for copy and paste inside tmux. But I would like to know how can I copy from TMUX terminal and paste it to other application e.g browser in my OS?