While reading about pair production, I came across the concept of negative energy. What is negative energy exactly? And can there be negative mass corresponding to this energy?
When Dirac found his equation for the electron $(-i\gamma^\mu\partial_\mu+m)\psi=0$ he famously discovered that it had negative energy solutions. In order to solve the problem of the stability of the ground state of the electron he invoked Pauli's exclusion principle and postulated that negative energy states well already filled by a "sea" of electrons. This allowed him to predict the positron, viewed as a hole in the sea.  This interpretation was ultimately discarded owing to its inapplicability to bosons and difficulties with explaining the invisibility of the infinite charge of the sea.  According to my understanding, the modern argument goes something like this. There is a discrete symmetry of the lagrangian called charge conjugation $\psi \rightarrow \psi^c$ which allows the negative energy solutions to be interpreted as positive energy solutions for a second mode of excitation of the electron field with opposite charge, called positrons. The decay of electrons to positrons is then suppressed by the $U(1)$ gauge symmetry of the lagrangian forcing conservation of electrical charge.  According to this interpretation What Dirac would have missed was the lagrangian formalism. Is this historically and physically correct?
A text says:     The surface tension of a liquid results from an imbalance of intermolecular attractive forces, the cohesive forces between molecules:   A molecule in the bulk liquid experiences cohesive forces with other molecules in all directions.   A molecule at the surface of a liquid experiences only net inward cohesive forces   If this is the case: Why is surface tension considered or experienced as a force parallel to the interface while it is so obvious that it must be perpendicular to it? Is there a standard and accepted method for selecting the number of layers, and the number of nodes in each layer, in a feed-forward neural network? I'm interested in automated ways of building neural networks. Using sets of arguments from a list "I'm right, aren't I?" My dictionary reports that aren't is also an abbreviation for am not, when used in questions.  Is this use of aren't common? If it is common, are there any differences between the different English dialects?     Why aren't I being given a pay raise? The set of convergent points of a sequence of continuous functions is Borel Suppose that $f_n$ are continuous.  Prove that $E=\{x\in \mathbb{R}:f_n(x)\text{ is convergent}\}$ is Borel.   My first instinct is to take balls around $\lim_{n\to\infty}f_n(x)$, for each $x$ in $E$, and then show that there has to be some ball small enough so that everything inside that ball is a member of $E$. But this would imply not only that $E$ is Borel, but moreover that it is open, so I don't think that can be right.  Can someone show me how to do this? I have a friend who is playing with me on a server we've created. Unfortunately, he is on a  Mac whereas I am on a Windows machine. I told him that I should be able to compile and send a .minecraft folder to him so that we can both play. Unfortunately, he can't find his .minecraft folder where it is supposed to be. Is it hidden and how can I unhide it? I need to externally limit a process/session to a certain number of cores.  Are there any other possibilities than CPU affinity (I don't like the need to specify the actual cores) and cgroups (hard to integrate into our project)? I've asked several questions about 4th Edition D&amp;D because I may be able to start a game. However, I'd need to purchase (some) materials. I am aware of the 4 book set (core) and the expected Essentials Red Box.  So now I need to know: presumably buying either the 4 book set or essentials will be enough to get my game on. Would there be something 'lacking' though? Would I be missing character types, for instance, as I've seen that the splatbook strategy is still in effect?  I'm an experienced player and GM and plenty creative, so many holes I'd be able to fill in myself. I just don't want to get sucked into too much financial impact. Ubuntu 13.10 Macbook Air Mavericks Parallels 9 Black Screen I can install Ubuntu 13.10 in parallels ok and then log in.  Problem is Ubuntu does an update and I get black screen / desktop such that I can't see any app windows that I open.  I've tried a few of the other suggestions and they don't work for me. General integral to find resistance My question is: is there a simple and truly general equation for the resistance between two electrical equipotential surfaces?.  Obviously, if so, what is it, and if not, why?  It would be very difficult to solve, granted, but I just want to see a calculus equation that is fully descriptive.  I have two frameworks under which this could be entertained, I'll write those out and then explain the motivation.  To start with, we need propose that the volume separating the two surfaces has a volumetric resistivity, $\rho$ in units of $(\Omega m)$.  Single Volume Framework  We can limit the discussion to a defined volume, then the surfaces reside in that volume or on the surface of it.  This volume may have a constant resistivity $\rho$ while everywhere outside the volume is completely electrically insulating.  Infinite Volume Framework  An alternative to the above approach that might make the task more or less difficult would be to replace a constant resistivity with a spatial dependence $\rho(\vec{r})$ and no longer require a boundary condition.  In that case we only have 3 mathematical inputs to the problem, which is the resistivity defined for all $\vec{r}$ and a definition of the two surfaces, $S_1$ and $S_2$.  Known Algebraic Analogs  The  that I find insufficient is:  $$R = \rho \frac{\ell}{A}$$  Where $l$ is the length of the restive material that is any shape which has translation symmetry over that length, and $A$ is the cross-sectional area.  Obviously, this is a rather simple equation that won't apply to more complicated geometry.  Even more  seem to give equations that fall short of what I'm asking.  For example:  $$R = \rho \int_0^l \frac{1}{A(x)} dx$$  I think it's obvious that an equation such as this is built upon a myriad of assumptions.  For a thought experiment, imagine that the area starts out as very small and then pans out to very large quickly.  Well, accounting for the larger area in the above sense underestimates the resistance, because the charge has to diffuse out perpendicular to the average direction of flow as well as parallel to it.  I have some reasons to suspect this might actually be rather difficult.  A big reason is that all the approaches I'm familiar with require the flow paths to be established beforehand, which can't be done for what I'm asking.  So maybe this will result in two interconnected calculus equations.  Motivation  I had an interest in , and it occurred to me that I can't quickly and simply write down the equation for resistance between two points.  The unique thing about Squishy Circuits is that it calls for two types of dough, one that conducts and one that is mostly insulating.  However, the recipes aren't perfect and because of that, the young children who play with these circuits regularly encounter the limits of conductor and insulator definitions.  If you make your conductor dough too long and/or too thin, you will encounter dimming of the light you connect with it.  Similarly, a thin insulator layer will lead to a lot of leakage current which also dims the light. iPhone screen randomly turns on Installing Guest Additions causing problems Display keyboard without animation Basically asking what were the motives behind each theory. What was it that led physicists toward these ideas? There is no fruit jet pack. I can't get it. How do I get it? I looked at the jetpacks many times. There are none. Select all columns except one in MySQL? I'm trying to use a select statement to get all of the columns from a certain MySQL table except one. Is there a simple way to do this?  EDIT: There are 53 columns in this table (NOT MY DESIGN) Why is there such a big difference between "Size" and "Size on disk"? If $a+\frac1b=b+\frac1c=c+\frac1a$ for distinct $a$, $b$, $c$, how to find the value of $abc$? If $a, b, c$ be distinct reals such that $$a + \frac1b = b + \frac1c = c + \frac1a$$ how do I find the value of $abc$?  The answer says $1$, but I am not sure how to derive it. how to concatenate string variables into a third? What program can be used to package software into one executable? I need to package an application into a single executable file (so that I don't have to install it, so that it will be portable).  Does anybody know what I can use to make a program portable?