In Relativistic Quantum Mechanics, Bjorken and Drell state that expanding the square root in the equation $$-\hbar^2\frac{\partial^2\psi}{\partial t^2}=\sqrt{-\hbar^2c^2\boldsymbol{\nabla}^2+m^2c^4}\psi$$ would result in a non-local wave equation. What exactly do they mean by this? One could argue that in order to compute $\psi$'s spatial derivatives at a certain point in space-time up to arbitrary order, one has to know the behavior of $\psi$ in points arbitrarily far from the point of interest, even if there is no world line linking those points. But if we were to solve a local equation such as the Dirac equation, we could do that nonetheless. Accordingly, I think there must be a deeper reason for the non-locality that Bjorken and Drell warn for.
The Wikipedia  has a derivation of the Klein-Gordon equation.  It gets to this step:  $$\sqrt{\textbf{p}^2 c^2 + m^2 c^4} = E$$  and inserts the QM operators to get  $$\left( \sqrt{ (-i \hbar \nabla)^2 c^2 + m^2 c^4 } \right) \psi = i\hbar\frac{\partial}{\partial t} \psi$$  The article then says     This, however, is a cumbersome expression to work with because the differential operator cannot be evaluated while under the square root sign. In addition, this equation, as it stands, is nonlocal.   To fix this, the first equation is squared instead to get  $$\textbf{p}^2 c^2 + m^2 c^4 = E^2$$  after which the QM operators are inserted and the expression is simplified to get  $$-\hbar^2 c^2 \nabla^2 \psi + m^2 c^4 \psi = -\hbar^2 \frac{\partial^2}{\partial t^2} \psi$$  A couple things I don't understand.  First, are the solutions to this differential equation not exactly the same as the solutions to the first differential equation?  Both sides of the starting equation were squared, so it seems to me that regardless of the particular form of the resulting differential equation, both of them should have the exact same set of solutions.  Secondly, why is the first differential equation cumbersome to work with?  It seems like it would in fact be easier to work with, since the operator under the square root could be expanded in terms of a Taylor series and then you have an equation that is first order in time.  And finally, can someone elaborate on what nonlocal means? The linked article on the Wikipedia page didn't entirely help me understand it.
Why won't my villagers breed I'm trying to run multiple Nodejs applications on Nginx server running on CentOS 7. I noticed that when I run a Nodejs app on some ports I get an 502 Bad Gateway error in the browser so I checked the error logs:  [notice] 12806#0: signal process started [crit] 12807#0: *13 connect() to 127.0.0.1:7777 failed (13: Permission denied) while connecting to upstream, client: **.**.99.58, server: myapp.com, request: "GET / HTTP/1.1", upstream: "http://127.0.0.1:7777/", host: "myapp.com" [crit] 12807#0: *13 connect() to [::1]:7777 failed (13: Permission denied) while connecting to upstream, client: **.**.99.58, server: myapp.com, request: "GET / HTTP/1.1", upstream: "http://[::1]:7777/", host: "myapp.com"   when I change the app to listen to 8008 for example everything is working fine. I checked permissions and if the process is running as root and everything seems ok. I played with the timeouts as well but no result. Can anyone help? What is the difference between "What are you?" and "Who are you?"? I am often asked "Who are you?" and "What are you?", but I don't know the difference between these two questions.  Please explain it to me. How to enable TRIM? I know that the Linux kernel supports TRIM as of version 2.6.33, so there should be TRIM support in ubuntu.  Is TRIM enabled by default or do I need to change some settings or install additional software for it to work? If so, how? 80s: the "Eighties"   90s: the "Nineties"   00s: the ???   For that matter, what is the second decade called? The "tens" just doesn't sound right. While rendering a mesh with smooth shading on, I get a dark region/shadow on certain faces. The effect is more evident when smooth shading is applied to a mesh containing ngons, with the ngons turning black when rendered.   The blackening effect is visible in the viewport too.  How can I solve this problem? Given two increasing continuous functions $f,g$ prove that $(b-a) \int^b_a f(x)g(x) dx > \int^b_a f(x) dx \int^b_a g(x) dx$ Given two monotonically increasing continuous functions $f,g$ prove that  $$(b-a) \int^b_a f(x)g(x) dx &gt; \int^b_a f(x) dx \int^b_a g(x) dx,\; b&gt;a$$  what I have tried:  Let $$h(x) = (x-a)\int^x_a f(t)g(t) dt - \int^x_a f(t)dt\int^x_a g(t)dt $$ and trying to figure out that $h'&gt;0$ using MVT to reform $h'(x)$ as   $$h'(x) = (x-a)(-f(x)g(\xi_1) - g(x)f(\xi_2) + f(x)g(x) + f(\xi_3)g(\xi_3))$$ where $\xi_{1,2,3} \in (a,x)$ (That method does not seem to work) What is $\Delta t$ in the time-energy uncertainty principle? In non-relativistic QM, the $\Delta E$ in the time-energy uncertainty principle is the limiting standard deviation of the set of energy measurements of $n$ identically prepared systems as $n$ goes to infinity. What does the $\Delta t$ mean, since $t$ is not even an observable? How do I eliminate the vertical realignment when using arrayrulecolor with cmidrule? I'm using consecutive \cmidrule commands in conjunction with \arrayrulecolor to alternately hide and unhide horizontal rules in a table.  However, when \arrayrulecolor is used with \cmidrule, some odd vertical spacing is introduced. I'd like to eliminate it.  (I'm using longtable, so a colorbox, minipage, or anything else tied to a page won't work.)  Below is an image that introduces the challenge. The first block shows a \toprule.  My goal is to create a shaded border around the table, so I've added a dummy column.  The second rule shows a \cmidrule that skips the first column.  The third block shows how I can use a \cmidrule to fill in the whitespace with a shorter \cmidrule.  Finally, I want to recolor this short \cmidrule to the background color and restore the color later.  Note the jagged vertical skipping in case 4 compared to case 3.  Why would \arrayrulecolor change vertical alignment?  How can I eliminate the jagged edges?    Here is the code:   \documentclass[11pt]{article}  \usepackage{longtable} \usepackage{booktabs} \usepackage{color} \usepackage{xcolor} \usepackage{colortbl}  \setlength{\aboverulesep}{0pt} \setlength{\belowrulesep}{0pt} \setlength{\extrarowheight}{0.25ex}  \newcolumntype{L}{&gt;{\columncolor[gray]{0.8}}l} \newcolumntype{C}{&gt;{\columncolor[gray]{0.8}}c} \newcolumntype{R}{&gt;{\columncolor[gray]{0.8}}r}  \begin{document}  \textsf{ \begin{longtable}{LLLLL}  &amp; \multicolumn{3}{@{}L}{\small {\tablename\ \thetable{}} - demonstration of odd vertical alignment with arrayrulecolor} &amp; \\%  &amp; \multicolumn{3}{@{}L}{\small {case 1: the line below is a toprule}} &amp; \\% \toprule  &amp; \multicolumn{3}{@{}L}{\small {case 2: the line below is a cmidrule that spans the middle 3 columns.  Note the whitespace!}} &amp; \\%  &amp; \multicolumn{3}{@{}L}{\small {The challenge: hide the widespace giving the appearance of a spacing around a longtable.}} &amp; \\% \cmidrule[\heavyrulewidth]{2-4}  &amp; \multicolumn{3}{@{}L}{\small {case 3: the line below is 3 cmidrules, 1 each for the first and last columns and 1 for the middle}} &amp; \\%  &amp; \multicolumn{3}{@{}L}{\small {note how the cmidrules align vertically ... perfect!}} &amp; \\% \cmidrule[\heavyrulewidth]{1-1}\cmidrule[\heavyrulewidth]{2-4}\cmidrule[\heavyrulewidth]{5-5}  &amp; \multicolumn{3}{@{}L}{\small {case 4: Now, lets try recoloring the first and last cmidrules}} &amp; \\%  &amp; \multicolumn{3}{@{}L}{\small {how do I eliminate the odd vertical alignments?}} &amp; \\% \arrayrulecolor[gray]{0.80}\cmidrule[\heavyrulewidth]{1-1}\arrayrulecolor[gray]{0.0}\cmidrule[\heavyrulewidth]{2-4}\arrayrulecolor[gray]{0.80}\cmidrule[\heavyrulewidth]{5-5}\arrayrulecolor[gray]{0.00} \endfirsthead  &amp;  Arizona State University &amp;  &amp; ASU &amp;\\   &amp;   Baylor University &amp;  &amp; BAYLOR &amp;\\   &amp;   Boston University &amp; 164988 &amp; BU &amp;\\   &amp;   Brown University &amp; 217156 &amp; BROWN &amp;\\   &amp;   California Institute of Technology &amp; 110404 &amp; CALTECH &amp;\\   &amp;   Carnegie Mellon University &amp; 211440 &amp; CMU &amp;\\   &amp;   Case Western Reserve University &amp; 201645 &amp; CASE &amp;\\   &amp;   Columbia University &amp; 190150 &amp; COLUMBIA &amp;\\   &amp;   Cornell University &amp; 190415 &amp; CORNELL &amp;\\      &amp; Yale University &amp; 130794 &amp; YALE &amp; % \label{tab:schools} \end{longtable} }  \end{document} Find the value of $\int_{-\infty}^{\infty} \frac{e^{-x^2}}{1+x^2} dx$ . Find the value of $\int_{-\infty}^{\infty} \frac{e^{-x^2}}{1+x^2} dx$     My attempt  First, I know the existence of this integral since $$ \mid \int_{-a}^{a} \frac{e^{-x^2}}{1+x^2} dx \mid \;\; \leq\;\;  \mid \int_{-a}^{a} \frac{1}{1+x^2} dx \mid$$  Next, let $R &gt;0 $ be an arbitrary and $\gamma_1(t) = -R(1-t) + Rt , \;\; 0 \leq t \leq1$ and $\gamma_2(t)= Re^{i\pi (t-1)}, \;\; 1 \leq t \leq 2 $.  So, take $f(z) = \frac{e^{-z^2}}{1+z^2}$.   I tried to find the value or upper bound of norm of $$\int_{\gamma_2} \frac{e^{-z^2}}{1+z^2} dz$$.  However, $$ \lvert\int_{\gamma_2} \frac{e^{-z^2}}{1+z^2} dz\rvert \leq  \frac{\text{max}\lvert e^{-z^2}\rvert}{R^2-1}\pi R = \frac{e^{R^2}}{R^2-1}\pi R $$ So, I can't proceed this.  May I ask you how to solve this? What is some good free software for doing physics calculations?  I'm mainly interested in symbolic computation (something like Mathematica, but free). How to get the current time in Python Can a US citizen leave and enter the US using a foreign passport? I am a dual citizen. I have both a US passport and a non-US passport. I live in the US. My US passport is currently being processed for renewal. I am unsure whether I will receive my passport in time for a trip that I have scheduled. Could I use my non-US passport to leave the US and return? I have never entered the US with my non-US passport. There is no immigration leaving the US but I still need to scan my passport for ticketing. Will the airline refuse me a ticket? I know that in this situation returning with a non-US passport can be problematic as well, but will I be able to leave at all? Note that I am traveling to a country which neither of my passports are from. $\sum_{n=-\infty}^{\infty}\frac{1}{(n+\alpha)^2}=\frac{\pi^2}{(\sin \pi \alpha)^2}$? While I was reading Enderton's "A mathematical introduction to Logic", I came across the proof of the following sentence: "The set of all finite sequences of members of the countable set A is also countable".   Proof: The set S of all such finite sequences can be characterized by the equation  $$S=\bigcup_{n \in N} A^{n+1}$$ Since A is countable, we have a function f mapping A one-to-one into N. The basic idea is to map S one-to-one into N by assigning to $(a_0,a_1,...,a_m)$ the number $2^{f(a_0)+1}3^{f(a_1)+1}\cdot ... \cdot p_m^{f(a_m)+1}$, where $p_m$ is the $(m+1)$st prime. This suffers from the defect that this assignment might not be well-defined. For conceivably there could be $(a_0,a_1,...,a_m)=(b_0,b_1,...,b_n)$, with $a_i$ and $b_j$ in A but with $m\neq n$. But this is not serious; just assign to each member of S the smallest number obtainable in the above fashion. This gives us a well-defined map; it is easy to see that it is one-to-one.  Note: P is a finite sequence of members of A iff for some positive integer $n$, we have $P=(x_1,...,x_n)$, where each $x_i \in A$.  First of all, I cannot understand why the former assignment might not be well-defined and the latter assignment is well-defined. Secondly, I cannot understand what Enderton means by "just assign to each member of S the smallest number obtainable in the above fashion". By the way, is $(a,b,c,d) = ((a,b),(c,d))$ true? Also, in which cases can I omit/add parentheses in a tuple so as to have an equal tuple? Is it posible to upload file in Lists Attachments in SharePoint in custom way High Voltage Powerlines - clarification of energy loss I'm trying to get my head around mutable vs immutable objects. Using mutable objects gets a lot of bad press (e.g. returning an array of strings from a method) but I'm having trouble understanding what the negative impacts are of this. What are the best practices around using mutable objects? Should you avoid them whenever possible? How do I handle elastic contacts in a simulation with friction I'm trying to simulate a wheel as it hits the ground.  Problem 1  Suppose a disc is dropped from a height. It has initial velocity of $-x,-y$ caused by throwing and gravity. It has no initial angular velocity. When it hits the ground it should have some rotation resulting from the collision.   How to calculate force that caused torque for that rotation?   Problem 2  The same disc is dropped from height. It doesn't have velocity on the side direction. It already spinning fast. When it hits the ground the spinning should cause some translation.   How to calculate force that caused this linear acceleration? Some rotation must be lost in the collision, how to calculate that?   To simplify assume static $friction = 1$ and $g = -10$. Proving that for a smooth function if $f(\frac 1 k)=0 :\forall k\in \mathbb N$ then $f(x)=0 :\forall x\in[-1,1]$