According to the chapter II.1 in "Quantum Field Theory in a Nutshell" by A. Zee, Dirac was trying to write down the relativistic wave equation linear in spacetime derivative.  The author stated that     The equation is supposed to have the form "some linear combination of $\partial_\mu$ acting on some field $\psi$ is equal to some constant times the field." Denote the linear combination by $c^\mu \partial_\mu$. If the $c^\mu$'s are four ordinary number, then the four-vector $c^\mu$ defines some direction and the equation cannot be Lorentz invariant   What I do not understand is the last part of this statement. If $a_\mu$ and $b_\mu$ are four-vectors, it is obvious that $a_\mu b^\mu$ must be Lorentz invariant. Why can it not be true in this case since both $c_\mu$ and $\partial_\mu$ are also four-vectors?
I have a question about the following passage on pg. 89 of Zee's QFT in a nutshell:     At first sight, what Dirac wanted does not make sense. The equation is supposed to have the form "some linear combination of $\partial_\mu$ acting on some field $\psi$ is equal to some constant times the field." Denote the linear combination by $c^\mu\partial_\mu$. If the $c^\mu$'s are four ordinary numbers, then the four-vector $c^\mu$ defines some direction and the equation cannot be Lorentz invariant.   The simplest equation I can construct that he is referring to is  $$(c^{\mu}\partial_{\mu}-m)\phi(x)=0, $$  for $m&gt;0$. Taking $\phi(x)\rightarrow\phi(\Lambda^{-1}x)$, and $c^{\mu}\rightarrow\Lambda^{\mu}_{\ \ \nu}c^{\nu}$. Then, the left term transforms like   \begin{align*} \Lambda^{\mu}_{\ \ \nu}c^{\nu}(\Lambda^{-1})^{\sigma}_{\ \ \mu}\partial_{\sigma}\phi(\Lambda^{-1}x) &amp;= \delta^{\sigma}_{\ \ \nu}c^{\nu}\partial_{\sigma}\phi(\Lambda^{-1}x)\\ &amp;=c^{\sigma}\partial_{\sigma}\phi(\Lambda^{-1}x). \end{align*}  Then, we have that   $$ (c^{\sigma}\partial_{\sigma}-m)\phi(\Lambda^{-1}x)=0.$$  But doesn't this show that the equation is Lorentz-invariant?
Can you trade from Pokemon Red on GBA to Pokemon gold on virtual console? I have a Game Boy Advance SP and Pokemon Red GEN 1, and I'm curious if it is possible for me to trade a Pokemon from Red to the Virtual Console release of Pokemon Gold on the 3DS Minecraft Failed To Bind Port I made a server and used my IPv4 address for the IP. I decided to scratch that server and make a Bukkit server so I can use plugins. I tried to use my IPv4 address which I used for my other server but I keep getting the message [17:31:15 WARN]: ****FAILED TO BIND TO PORT!  Is there any thing I can do to get a new address to use or is there a way to delete the old server? Do I need to use a different computer? Please help! I've don't know how to fix this problem. If you have any suggestions please tell me and I'll give it a try. Why do birds sitting on electric wires not get shocked? "Compared with" vs "Compared to"—which is used when? Is only one of them correct? Are they used in different situations? Or are they interchangeable? Does anyone know if there are any video formats (besides the HUGE uncompressed avis) that you can render in that maintain the alpha channel? (i.e. for after FX etc.)   I used to use quicktime, but that's no longer an option.  I know you can render in png. or other image formats, but video is significantly smaller, and I often work on machines with less space &amp; resources. Getting video dimension / resolution / width x height from ffmpeg Odd behavior in a QGIS plugin: my function is triggered twice I have a plugin with a function (triggered by a button) that selects a specific layer by name, then shows its attribute table. When I close the plugin and open it again, the attribute table pops up twice, and repeating this keeps adding another table popup. What's wrong?  Code:  def showTable(self):     for lyrs in QgsMapLayerRegistry.instance().mapLayers().values():         if lyrs.name() == "TestFeatures":             lyrTest = lyrs     self.iface.showAttributeTable(lyrTest) Question Prove that for every integer k≥1, exists a self-complementary graph with 4k vertices half of which are of degree 2k-1 and the other half of degree 2k.  My approach So, I think the easiest way to prove what is asked is by induction. I can prove the base case for k=1, but as far as the inductive step is concerned, I cannot seem to reach a conclusion as to why if the statement holds for n, then it holds for n+1.  Any help would be appreciated. Proof of "well-founded iff there is no sequence $x_n$ s.t. $(x_{n+1}, x_n) \in R$" I tried to do the following exercise from a book I'm reading, can you tell me if my solution is correct? Thanks!  Theorem 2: Let $R$ be a binary relation on a set $X$. Then $R$ is strictly well-founded iff there is no sequence $(x_n)_{n \in \mathbb N}$ of elements of $X$ such that $(x_{n+1}, x_n) \in R$ for all $n \in \mathbb N$.  Exercise:  (i) Formulate an analogue of Theorem 2 for  well-founded relations.   (ii) Prove Theorem 2.    (i) Theorem 2': Let $R$ be a binary relation on a set $X$. Then $R$ is well-founded iff there is no sequence $(x_n)_{n \in \mathbb N}$ of elements of $X$ such that $(x_{n+1}, x_n) \in R$ and $x_n \neq x_{n+1}$ for all $n \in \mathbb N$.  (ii)   $\implies$: Assume there is a sequence $x_n$ with $(x_{n+1},x_n) \in R$ for all $n \in \mathbb N$. Then the set $Y = \{x_n \mid n \in \mathbb N\}$ and hence $R$ is not well-founded since for every element $x_n$ there is $x_{n+1} \in Y$ with $(x_{n+1},x_n) \in R$.  $\Longleftarrow$: Let $R$ be strictly well-founded. Assume we have a sequence $x_n$ with $(x_{n+1},x_n) \in R$ for all $n \in \mathbb N$. Then by the same argument as above $R$ is not well-founded, hence there cannot be such a sequence.   I must be missing something but I can't seem to figure out what. I have no idea how to prove it by using only what the book has talked about so far. Can anyone help? The proof shouldn't be using set theory as set theory is only mentioned in the following chapter. The proof should only make use of the addition of natural numbers, order properties of natural numbers, the trichotomy of order for natural numbers and principle of induction.  Proposition 2.2.14 (Strong principle of induction). Let $m_0$ be a natural number, and let $P(m)$ be a property pertaining to an arbitrary natural number $m$. Suppose that for each $m ≥ m_0$, we have the following implication: if $P(m')$ is true for all natural numbers $m_0 ≤ m' &lt; m$, then $P(m)$ is also true. (In particular, this means that $P(m_0)$ is true, since in this case the hypothesis is vacuous.) Then we can conclude that $P(m)$ is true for all natural numbers $m ≥ m_0$. Exercise 2.2.5. Prove Proposition 2.2.14. (Hint: define $Q(n)$ to be the property that $P(m)$ is true for all $m_0 ≤ m &lt; n$; note that $Q(n)$ is vacuously true when $n &lt; m_0$.)  Thanks! Why "hide" Luke from his father somewhere Anakin knows? In Episode II Anakin goes to the Lars homestead and meets his extended family. How would sending Luke there be "hidden" to Vader in any way? At least Leia had a random planet and family. Show $\operatorname{Aut}(C_2 \times C_2)$ is isomorphic to $S_3=D_6$ MATLAB superclass can not be found on the MATLAB's search path There are a few "automatically protect questions" requests out there, but I'd like to add another twist.  I'd like to request that questions that have three or more deleted answers by users with &lt;50 rep be automatically protected.  The reasoning behind this is that we have lots and lots of "not an answer" flags in the mod queue.  But we cannot protect a question from the mod queue, so we have to open each answer, evaluate if the question needs protecting, then protect it.    This process takes a long time, and can definitely be automated, as the only thing you really need to check is how many new users are posting non-answers or are spamming their awful products. Seeing that Defense Against the Dark Arts is a crucial subject, both in Harry's life as well as a part of the school-going wizard's curriculum, it must be in the hands of a good professor.   Is it known who Minerva McGonagall appointed as DADA prof. once she was made Headmistress. Was it someone we know or was it just a random powerful wizard. Is there any information about this available? In windows we have an option to view recently opened media files as my recent documents, I was  wondering does there exists something like that in Ubuntu?  via terminal or via GUI? How do I force TeX to completely use the 1st column? How do I force TeX to completely use the 1st column before writing to the second column?  Right now I dont have enough content to fill one whole page. So the left and right columns are equally filled, leaving 25% symmetrical empty page at bottom. How do I fill the left column first, and then the right?  \documentclass{article}  \usepackage{multicol} \usepackage[scale=0.9]{geometry} \usepackage{amssymb} \renewcommand{\labelitemi}{\tiny$\blacksquare$} \begin{document} \begin{multicols}{2}      %lots of text goes here  \end{multicols} \end{document} Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate? Your favourite application of the Baire Category Theorem When drawing a line between two nodes in TikZ, you can specify the anchors on the beginning and end nodes in order to control how the line is drawn.  I would like to specify the anchor on the beginning node, and the angle at which the line is drawn, and let TikZ figure out where it will intercept the end node.  The usual use case is when I'm connecting to a "big node" and I would like the lines to be vertical.  I can cheat by placing a coordinate behind the big node, drawing a line to the coordinate instead, and then drawing the big node so that it covers things up.  However this won't work if I need to put an arrow on the end.  I could try to calculate the intercept coordinate on the edge of the big node manually, but that seems against the TikZ spirit.  |- doesn't work in this case.  \documentclass[10pt]{minimal} \usepackage[active,tightpage,pdftex]{preview} \usepackage{tikz}  \begin{document} \begin{preview}  \tikzstyle{RectObject}=[rectangle,fill=white,draw,line width=0.5mm] \tikzstyle{line}=[draw] \tikzstyle{arrow}=[draw, -latex]   \begin{tikzpicture}   %%%%%%%%%%%%%%%%%%%%%%%%   %%  LEFT SIDE   %%%%%%%%%%%%%%%%%%%%%%%%   \draw (1,0) node[RectObject] (Small1A) {A};   \draw (3,0) node[RectObject] (Small1B) {B};   \coordinate (Cheat1A) at (1,2);   \coordinate (Cheat1B) at (3,2);   \path [line] (Small1A.north) -- (Cheat1A);   \path [line] (Small1B.north) -- (Cheat1B);    % need to draw Big Node last to cover up our cheating   \draw (2,2) node[RectObject, inner xsep=1cm] (Big1) {Big Node};    \draw (2,4) node[text width=4cm] (text1) {L: Cheating to control line orientation. Arrows would be hard.};    %%%%%%%%%%%%%%%%%%%%%%%%   %%  RIGHT SIDE   %%%%%%%%%%%%%%%%%%%%%%%%   \draw (7,2) node[RectObject, inner xsep=1cm] (Big2) {Big Node};   \draw (6,0) node[RectObject] (Small2A) {A};   \draw (8,0) node[RectObject] (Small2B) {B};   \path [arrow] (Small2A) -- (Big2);   \path [arrow] (Small2B) -- (Big2);    \draw (7,4) node[text width=4cm] (text2) {R: No cheating. Arrows easy. Can't control line orientation.}; \end{tikzpicture} \end{preview} \end{document}   Here's a side-by-side example of "cheating", which shows what I want to be able to do, and "not cheating", which does not do what I want.    The left side of this example shows what I want to be able to do, except I want to do it without cheating, and I'd like to put arrows on the end of the lines.