It is known that in QFT the Euler-Lagrange equations are used to obtain the equations of the quantum fields. Nevertheless, from the path integral's point of view (where you integrate over all $\it{paths}$/possible field configurations, even if they don't satisfy Euler-Lagrange) the use of this equations makes only sense if you are looking for the classical field, since this one satisfies the least action principle.  So, why do we use these equations to obtain the ones for the quantum field?
To quantize a scalar field theory with the action:  $$S=\int \mathrm d^Dx\mathscr{L}(\phi,\partial_\mu\phi)=\int \mathrm dx^0L(\phi,\partial_0\phi)$$  we promote $\phi(\vec{x})$ and $\pi=\frac{\delta L}{\delta(\partial_0\phi)}=\pi(\vec{x})$ to be field operator at fixed time $x^0$ (with the canonical commutation relation $[\phi(\vec{x}),\pi(\vec{y})]=\delta^{D-1}(\vec{x}-\vec{y})$), then we use the classical equation of motion to find $\phi$ and $\pi$ at all time.  What principles lie behind this? Why can we assume that the field operators satisfy the classical equation of motion?    By example if we have a classical theory with $S=\int \,\mathrm dtL(\phi,\partial_t\phi)$, momentum $\pi=\frac{\delta L}{\delta(\partial_t\phi)}$ hamiltonian $H=\pi\phi-L$, the classical equation of motion is:  $$\frac{\mathrm d\phi}{\mathrm dt}=\frac{\delta L}{\delta(\partial_t\phi)}$$  When we do the canonical quantization, we promote $[\phi,\pi]=\delta(\vec{x}-\vec{y})$ and the equation of motion for $\phi$ is $$\frac{\mathrm d\phi}{\mathrm dt}=i[H,\phi]$$  Why should $i[H,\phi]$ has the same form as $\frac{\delta L}{\delta(\partial_t\phi)}$? This fact comes from where?
407 Proxy Authentication Required When I try to install a software using Ubuntu Software center I get:  Failed to download repository information Check Your Internet connection   When I try to do a apt-get install something, I get:  407  Proxy Authentication Required   I use a proxy server that requires a user-name and a password. I have set my systems proxy manually, by plugging in the required numbers in the Networks proxy and applied it system wide. I guess the problem now is plugging in my user-name and password.  When I use INTERNET via Mozilla, it specifically asks me for my user-name and password. Here's an image from another question:    I get what the pull down resistor is there to do, but I don't really understand exactly how it works.  When there is no voltage applied through the base wire, why does any induced current then go through R2, but when a voltage is applied, it goes to the transistor? My understanding is that voltage always follows the path of least resistance, so if applying say, 5v turns the transistor on, that means that the transistor has a lower resistance than R2, so I would also assume any induced current would also flow through the resistor - so why does it then go through R2? Ethernet: how important is the magnetics topology? I have some Ethernet issues with a set of boards (6/10 are bad). They're possibly just assembly house mistakes... but I'm very concerned about my magnetics jack.  I could have sworn I saw a example schematic with my jack's topology used with my PHY before, but I can't find it. The PHY's datasheet and suggested magnetics jacks are all identical in topology, and very different than what I have.  I'm using a SMSC  PHY, with a Bel  magnetics jack.  The transformer ratios on my jack vs. a suggested jack for this PHY like the  are the same (1:1).  The inductance is the same, 350uH. However, the choke on mine is on the circuit side, not network side. Also, while the TLA has four 75 ohm resistors from taps tied to a single 1nF/2kV cap, mine has an extra set of 1nF caps between the taps and the 75 ohm resistors.  The main difference is that my jack is a POE jack, and I thought I found an example that used one exactly like it. So, the topology is obviously different just because of that feature.  At the moment, I have 4 good boards, 1 'sometimes' board, and 5 boards that just blink a LED at 11.68Hz representing some error that no datasheet explains.  How important is this? Of course there's no other jack with that footprint.  EDIT  I added some detail on the magnetics jacks above. I was just recommended the SI-60152-F, so I'll compare to that one. Of course like all the other recommendations... it also has no stock at anyone on FindChips.  LATE EDIT  It turns out my problem was the crystal layout that fed the Ethernet PLL.  On the respin, I went with a more standard non-POE jack. How do I install iTunes on Ubuntu? I've tried different ways to install iTunes, but to no avail. I tried with Wine, but it didn't work. Is it possible to have iTunes on Ubuntu? Conditional typesetting / build Let's say I have .tex file which is a set of problems and solutions, of which I want to make two pdf files, one which includes the solutions and one which doesn't. so I need to build the .tex file under two different modes, and specify that parts of the document be ignored under one of the modes.  This is something common in programming context. e.g. in C++ i can code  #ifdef DEBUG   // do this additional stuff ... #endif   How can I do something equivalent in latex? Suppose that $A, B, C$, and $D$ are sets with the cardinalities related as  $|A|=|C|$ and $|B|=|D|$. Prove that the cardinality of $A\times B$ is equal to the cardinality of $D\times C$.  I know that I must prove the bijection of the function $f: A \times B \to C \times D$, but I am not sure how to say this? Can anyone help me? "Z-level" in TikZ How to validate an email address in JavaScript Is there a regular expression to validate an email address in JavaScript? Can you combine multiple lists with LINQ? Say I have two lists:  var list1 = new int[] {1, 2, 3}; var list2 = new string[] {"a", "b", "c"};   Is it possible to write a LINQ statement that will generate the following list:  var result = new []{      new {i = 1, s = "a"},     new {i = 1, s = "b"},     new {i = 1, s = "c"},     new {i = 2, s = "a"},     new {i = 2, s = "b"},     new {i = 2, s = "c"},     new {i = 3, s = "a"},     new {i = 3, s = "b"},     new {i = 3, s = "c"} };   ?  Edit: I forgot to mention I didn't want it in query syntax. Anyway, based on preetsangha's answer I've got the following:  var result = list1.SelectMany(i =&gt;  list2.Select(s =&gt; new {i = i, s = s})); I am a history teacher and I like to make my own maps (I can never find exactly what I want).  I am using QGIS 1.8 and .  I want to reproject their cross-blended hypso rasters into projections that are more familiar looking to my students, like Lambert Conic Conformal.  I have tried changing the crs for the whole project, and the warp and assign functions from the raster menu, and changing the crs for the raster layer itself.  None of these seem to work properly - the resulting maps are unrecognizable and extremely large (file size).  I feel like I am making some kind of basic mistake but I don't know where to begin with this.  Any help would be appreciated. List of Ubuntu Versions with Corresponding Linux Kernel Version Example 4.21 in Baby Rudin: Why is the inverse of this function not continuous at this point? I'm trying to amass a list of physics books with open-source licenses, like Creative Commons, GPL, etc. The books can be about a particular field in physics or about physics  in general.  What are some freely available great physics books on the Internet? edit: I'm aware that there are tons of freely available lecture notes online. Still, it'd be nice to be able to know the best available free resources around.  As a starter:         Table of contents sorted by field (in alphabetical order): I understand that unlocking the bootloader will wipe my Android phone, but have been looking around for why.  Seems to be by design, but what is the reasoning for that design?  Is it some security concern, some obscure technical reason, or just for lulz?  I'm looking for something solid to chew on here, something more than because "that's how it is". What is Renly's basis for removing Joffrey? Related question:   I get that he considers himself a better king than Stannis, but why isn't he ?  Did Renly know about Joffrey being illegitimate? I always thought he didn't and was distrustful that Cersei would accept Ned as Regent.  I think in both the books and the series, Renly finds out Joffrey is illegitimate from Stannis' letters, but rewatching that clips makes me think Renly was with Ned, Varys and LF in the plan to arrest Joffrey and Cersei.  [looks up something]  Oh also . So Renly did know? This is a follow up to   I am using a SHA-256 HMAC function on a single-word input: sha256hmac(privatekey,word) = output. The private key length is 128 bits, it does not change. The "word" is literally one short word.  Assuming an attacker knows "word" and "output" for multiple words (e.g., dog = "dsfwrw3r3r323211", cat = "3erwerwetwewer"):   Is it significantly easier for an attacker to identify the private key by brute force with the above word/output pairs, as compared to a situation where they do not have those pairs? Is this brute force vulnerability (with word/output pair knowledge) an actual security concern? How far into space does one have to travel to see the entire sphere of earth? Virgin Galactic will take passengers aboard SpaceShipTwo  above the surface of the earth. But from this altitude, passengers will only be able to see a certain segment of the curvature of the earth through .  How much further into space would SpaceShipTwo have to travel to give passengers a view of the entire sphere of earth through one of these windows? Suppose we rotate the graph of $y = f(x)$ about the $x$-axis from $a$ to $b$.  Then (using the disk method) the volume is $$\int_a^b \pi f(x)^2 dx$$ since we approximate a little piece as a cylinder.  However, if we want to find the surface area, then we approximate it as part of a cone and the formula is $$\int_a^b 2\pi f(x)\sqrt{1+f'(x)^2} dx.$$  But if approximated it by a circle with thickness $dx$ we would get $$\int_a^b 2\pi f(x) dx.$$  So my question is how come for volume we can make the cruder approximation of a disk but for surface area we can't. How can I resize an ext root partition at runtime? How can I increase the size of the root partition of a system at runtime?   I have a partition that is not allocated after the root partition (which is also ext4), how can I add that unallocated space to the space allocated to the root partition without having to shutdown the server? Assuming I have a Czech license and break a minor law in Germany which incurs me a certain number of "violation points" (e.g. speeding), would those points be eventually transferred to my Czech license? I would obviously have to pay the fine (either on the spot or by mail), but it's not clear if EU countries exchange data on traffic points.  I would limit the scope to EU/EEA license holders driving within the EU/EEA, to avoid an overly broad question. Also assume that the person in question is resident in his home country and is only visiting other EU/EEA countries as a tourist.