In my course on particle physics the KG and Dirac equations were explained in an historical way. However, it was more confusing than clarifying. The problems with the KG equation were   The probability density can take negative values. Energy doesn't have a lower bound.   To solve this the Dirac equation was introduced. However, it only solves the first problem but not the second. Then the antiparticle interpretation is presented to solve the negative energies problem and everybody is happy. Also, the probability density problem of the KG equation is then solved saying that it is actually a charge current so it can take negative values...  So, if density is actually charge density and negative energies are antiparticles, what was the problem with the KG equation in the first place? How can we interpret the probability density of the Dirac equation as charge density if its always positive? Does QFT solve the negative energy problem without the need of postulating weird stuff like particles going backwards in time?
Why can't the Klein-Gordon equation with a Couloumb potential describe the hydrogen atom?  Why can the first order Dirac equation explain it? What are the failures?
Is there a reason the British omit the article when they "go to hospital"? Can a PC with the Alert feat alert his friends? Assume the following situation: The party has been surprised by an ambush. Every PC is surprised except PC Bob because he has the Alert feat. Thanks to his high initiative bonus Bob gets to go first. He shouts "look out" to alert his party members of the imminent threat. The idea is that his party members are no longer surprised because he warned them.   While I think that this makes sense from a storytelling perspective, I am afraid this would be too strong. as it allows one PC with the Alert feat to give the "not surprised" benefit to other members of his party pretty much for free.  Is this allowed? Are there rules or an official ruling or similar on this?  If not, do you have any experience with this situation, and how did it play out in your games? Homeomorphism that maps given points to given points How do I install iTunes? I have an iPhone and run Ubuntu on all of my personal computers. Since I did not want to keep a separate partition with Windows on it for the sole purpose of running iTunes, I attempted to install It using Wine.   I installed Wine 1.4 from the Software Center and installed iTunes 10.6.3. When I tried to run it I got a slew of error messages. I hopped over to google where it was suggested that I install it through PlayOnLinux. I did so with the same result. Further googling revealed that iTunes 10.6.x is confirmed to work with Wine 1.5.1 and up.   I installed Wine 1.5.1 following the instructions I found and was unable to get it to open. I did the same with 1.5.9 with the same results. I opened the Package Manager and installed the Wine 1.5.9 packages through it, and it appears to have installed properly.   When trying to install iTunes I got he error "This iTunes installer requires Windows Vista 64 bit or later". Realizing that Wine uses XP as a default I ran winecfg and changed it to Windows 7. This changed nothing and I tried changing it through winetricks to no avail. I even changed it to Vista with the same results.   Does anyone know what is going wrong here and how to fix it? Thanks Why the middle mouse button in Blender 2.80 changes the orthographic view to perspective? If I choose an orthographic view by using the numpad and then try to orbit the geometry by clicking the middle mouse button Blender goes from orthographic to perspective. Why is that the default now? Wouldn't it be more logical if instead of that Blender kept the orthographic mode? In other words: Why Auto Perspective is turned on by default? This is not a duplicate of this .  The linked question says that it suffices to show that if $(x_n)\to x$ then $(\frac{x_1+\cdots+x_n}{n})\to x$ to prove my question, but how so? I tried using the same strategy as how one proves that if $(x_n)\to x$ then $(\frac{x_1+\cdots+x_n}{n})\to x$, by "splitting" the product in th $N$th term:  $$\sqrt[n]{x_1x_2\cdots x_n}=\sqrt[n]{x_1x_2\cdots x_Nx_{N+1}\cdots x_n}=\sqrt[n]{x_1x_2\cdots x_N} \sqrt[n]{x_{N+1}\cdots x_n}$$  but it seems I can't use for now the definition of convergence of $(x_n)$ because of the $n$th root. I also tried to use a result: if $(x_n)\to x$ then $(\frac{x_n}{n})\to 1$ but don't know if this is true. Sadly I've had no real progress. Any help will be greatly appreciated, thanks in advance! How does accepting an answer work? What does it mean to accept an answer? How do I accept an answer, and what are the rules? Can I accept my own self-answer? What are the reputation benefits? Which answer should I accept? Should I upvote the accepted answer? What should I do if none of the answers are suitable?      For more information, see "" in the . I have no idea how to do this. Any help would be appreciated. The chapter I'm on is about differentiation and the mean value theorem.  Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b &gt; 0,\; 0 &lt; \alpha &lt; 1$ Cardinality of set of real continuous functions In order to animate a character so that it grabs an object an throws it, I added a Child-Of constrained and keyframed its influence. In the F-curve Editor I can't set the curve which controls the influence to interpolation-mode Constant without changing the other curves (Which should be linear or bezier).   How can the interpolation mode of a curve changed independently? Syntax error on print with Python 3 How did the writers not notice that the mailbox of Leela's parents says "Turanga," and not "Leela"?  Turanga Leela is Leela's proper name, with Leela being her surname, so shouldn't it follow that her folks would be the same? How could it possible to factorise $x^8-1$ in product of irreducibles in the rings $(\mathbb{Z}/2\mathbb{Z})[x]$ and $(\mathbb{Z}/3\mathbb{Z})[x]$? How could it possible to factorize $x^8-1$ in product of irreducibles in the rings $(\mathbb{Z}/2\mathbb{Z})[x]$ and $(\mathbb{Z}/3\mathbb{Z})[x]$?  I'm having a hard time starting the problem. Could anyone help me at this point? Why are there $736$ matrices $M\in \mathcal M_2(\mathbb{Z}_{26})$ for which it holds that $M=M^{-1}$? I'm currently trying to introduce myself to cryptography.  I'm reading about the Hill Cipher currently in the book Applied Abstract Algebra. The Hill Cipher uses an invertible matrix $M$ for encryption and the inverse matrix $M^{-1}$ for decryption. The book recommends to use a matrix $M$ for which it holds that $M=M^{-1}$, since we then have the same key for encryption and decryption.  In the book, they use a $2 \times 2$ matrix $M$ over $\mathbb{Z}_{26}$ as example, and state that for there are 736 $2 \times 2$ matrices for which it hold that $M=M^{-1}$.  I'm trying to pick up on as much as possible when reading things, since I find it counter-productive for learning to skip something, when you don't get the theory behind it. Can someone enlighten to me, as to why it is that there are 736 possible $2 \times 2$ $M=M^{-1}$ matrices and how to find them? Proof/Derivation of Closed form of Binomial Expression $\sum\limits_{k=0}^{2n}(-1)^k\binom{2n}{k}^2$ The binomial expression given as follows: $$\sum_{k=0}^{2n}\left(-1\right)^{k}\binom{2n}{k}^{2}$$ results nicely into the following closed form: $$(-1)^{n}\binom{2n}{n}$$ I wish to know how exactly is it being done? I haven't been able to make much progress in solving it. My approach: \begin{align} \sum_{k=0}^{2n}(-1)^{k}\binom{2n}{k}^{2} = \binom{2n}{0}^{2} - \binom{2n}{1}^{2} + \binom{2n}{2}^{2} - ... -\binom{2n}{2n-1}^{2} + \binom{2n}{2n}^{2} \\ = \binom{2n}{0}.\binom{2n}{0} - \binom{2n}{1}.\binom{2n}{1} + \binom{2n}{2}.\binom{2n}{2} - ... -\binom{2n}{2n-1}.\binom{2n}{2n-1} + \binom{2n}{2n}.\binom{2n}{2n} \\ \text{By Symmetry of binomial coefficients} \\ = \binom{2n}{2n}.\binom{2n}{0} - \binom{2n}{2n -1}.\binom{2n}{1} + \binom{2n}{2n-2}.\binom{2n}{2} - ... -\binom{2n}{1}.\binom{2n}{2n-1} + \binom{2n}{0}.\binom{2n}{2n} \\ = \binom{2n}{2n}.\binom{2n}{0} + \binom{2n}{2n -1}.\binom{2n}{1} + \binom{2n}{2n-2}.\binom{2n}{2} + ... +\binom{2n}{1}.\binom{2n}{2n-1} + \binom{2n}{0}.\binom{2n}{2n} - 2.\left(\binom{2n}{2n -1}.\binom{2n}{1} + \binom{2n}{2n -3}.\binom{2n}{3} + ... + \binom{2n}{1}.\binom{2n}{2n-1}\right) \\ \text{By Vandermond's identity, first component, i.e.not enclosed within -2.(...) evaluates to C(4n, 2n)} \\ \binom{4n}{2n} - 2.(\binom{2n}{2n -1}.\binom{2n}{1} + \binom{2n}{2n -3}.\binom{2n}{3} + ... + \binom{2n}{1}.\binom{2n}{2n-1}) \end{align} I'm lost beyond this point. It will be extremely helpful if someone can direct me in the right direction or provide the answer to this perplexing and challenging problem. Thank you. Do internal forces cause change in velocity? Connect Java to a MySQL database Does $\lim_{n\to\infty}\underset{n}{\underbrace{\cos(\cos(...\cos x))}}$ exist? Possible Duplicate:         Does the following limit exist?  $$\lim_{n\to\infty}\underset{n}{\underbrace{\cos(\cos(...\cos x))}}$$  If yes, find the limit.  If no, please explain why the limit doesn't exist.  I think, the limit exists.  So, I tried to use Squeeze theorem but didn't work. Confidence interval interpretation in linear regression when errors are not normally distributed Should I upgrade my body or lens first?