title: Shouldn't the fourth dimension be energy/mass, and not time?

text: First of all, I am in 8th grade. That doesn't mean you have to make your answer simple and easy to understand, but I ask you not to be too critical of my query/theory, because I don't consider myself an expert in this in any sense of the word. So, imagine gravity. What is the Newtonian equation for gravity? The gravitational constant times the sum of the masses of the two objects (gravitated and gravitator) divided by the square of the radius of the gravitator squared. The gravitational constant times the sum of the masses of the two objects This illustrates the relation between density and gravitational strength. Now, if we consider that mass and energy are, in a sense, parallel expressions of one another (i.e. parallel lines with the same slope, but one with a y-intercept of 1, the other with a y-intercept of whatever the speed of light squared is), than we come to the conclusion that they are, in their parallelity, almost interchangeable (of course, they must be converted into each other). So, really, mass is concrete energy and energy is abstract mass. Mass is a measure of matter, correct? If so, dense objects are objects with small volumes and great amounts of mass/energy. Now, as the first three dimensions are dimensions of space, and therefore incompressible, as objects merely are in space, but not actually "using" or literally "taking up" space,(imagine it this way...if you draw a cube on a piece of paper, along with just it's length, width, and height, there is no possible way to calculate the cube's mass, weight, etc.) then space is stagnant without the introduction of a fourth dimension to bind, change, and affect space (this is how we know there is a fourth dimension, because otherwise, change, movement, and other principles would be inconceivable). So, what should this dimension be? Well, it must be a duality. It must be concrete, to influence concrete objects in concrete ways, but it must be abstract, to influence the abstract (concepts, things, objects?) in abstract ways. Well, if we need such a duality, we can try the concept of energy/mass. Does this work? There are a few ways to check.   1. Does this "dimension" affect concrete objects? We use energy/mass to move and influence objects.    2. Does this "dimension" influence abstractions? Energy/mass is crucial to many abstractions, as we know.    3. Is there a "real-world" proof of this new "dimension"? To make sure this is valid, we apply our new "dimension" to a proven concept/theory/law. And so, we return to gravity. We already decided that density is a direct correlation between mass/energy and space. In Newtonian terms, the gravitational constant times the sum of the masses of the two objects (gravitated and gravitator) divided by the square of the radius of the gravitator squared. In more planar, abstractual terms, imagine the three dimensions of space, incompressible by just "space-taking" objects that only expand in those three dimensions (remember...if you draw a cube on a piece of paper, along with just it's length, width, and height, there is no possible way to calculate the cube's mass, weight, etc.). Now, how does mass/energy fit into this? Well, if mass/energy is also a planar value, it has some different properties than space (mass/energy would not be invariant). Because that you now have mass/energy, different objects have different masses and amounts of energy in different amounts of space. This means that mass/energy must be compressed differently into different spaces, right? So, the plane of mass/energy has been distorted, warped, compressed. Because or new "dimension" has now created "space-energy/mass" as a four-dimensional, universal fabric, space has also been distorted, warped, compressed as well. Now objects and abstractions traveling through the universe naturally take the path of least resistance and follow the now-curved and distorted planes of "space-mass/energy", creating the gravitational effect. The more space/mass is distorted, the more space is distorted, and thus the more powerful gravity is. There is more to this theory, but, until I receive validation of this theory, I will wait to digress more. Please contact me as soon as possible.


title: Why does (potential) fluid flow bend around a solid surface in the flow?

text: Potential flow obeys Laplace's equation with certain boundary conditions (i.e. no fluid penetrates the solid body in flow, and far away from the body, the flow is uniform with a given velocity and pressure). So let's condider the potential flow around a cylinder. After the fluid moves to the "top" of the cylinder (above the "+" in the image) it then bends around the surface instead of continuing in a straight line. ![Cylinder flow](http://www.thermopedia.com/content/5637/TUBES_CROSSFLOW_OVER_FIG1.gif) Why does it bend and not continue in a straight line from the point before the bending? Since the flow is potential (no viscosity) the only thing causing the fluid to bend around can be the pressure field, but im not sure how this works. EDIT: Thanks to the first comment I do realize that at the "top" of the cylinder surface there is a pressure gradient normal to the flow (i.e. lower pressure nearer the surface) which would cause flow turning at that point. But how is this pressure gradient established in the first place? It seems like one is arguing that the flow bends because of the pressure gradient, but there exists a pressure gradient only because the flow bends... Am I missing something here?


title: Why do real gases deviate from ideal behavior above the Boyle temperature?

text: My textbook says that Boyle temperature is the temperature at which a real gas shows maximum ideal gas behavior. Below the Boyle temperature, molecules come too close and intermolecular forces skew off its behavior. But what about above the Boyle temperature? Why do gases deviate from the ideal gas law above it? I mean, rise in temperature can only mean less of intermolecular attraction, right?


title: Rubber bands and Hooke's law

text: Why does increasing the width of a rubber band increase its elastic constant($k$)? And why are the two variables directly proportional? Explain it in terms of the structure of the band, if that is relevant.


title: Why Does Germanium Spontaneously Extrude Long Screw Dislocations?

text: > Pure germanium is known to spontaneously extrude very long screw > dislocations. They are one of the primary reasons for the failure of older > diodes and transistors made from germanium; depending on what they > eventually touch, they may lead to an electrical short. -Wikipedia ![enter image description here](http://i.stack.imgur.com/Tz9gp.jpg) I'm not sure what is going on in the below figure: the caption is confusing. ![enter image description here](http://i.stack.imgur.com/6KhlF.jpg) > FIG. 7. Point contact between tungsten-molybdenum whisker and germanium > surface: (a) Before forming; (b) After forming; (field of view measures 50 > μm from left to right; Smith 1956). Silicon Carbide also form screw dislocations which are more pipe-like. Pictures of SiC screw dislocation. > A team led by University of Wisconsin-Madison chemist Song Jin, writing this > week (April 23, 2010) in the journal Science, shows that a simple crystal > defect known as a "screw dislocation" drives the growth of hollow zinc oxide > nanotubes just a few millionths of a centimeter thick. > > Dislocations are fundamental to the growth and characteristics of all > crystalline materials. As their name implies, these defects prompt the > creation of spiral steps on an otherwise flawless crystal face. As atoms > alight on the crystal surface, they form a structure strikingly similar in > appearance to the spiral ramps of multistory parking structures. In earlier > work, Jin and his research group showed that screw dislocations drive the > growth of one-dimensional nanowire structures that looked like tiny pine > trees. That, says Jin, was a critical clue to understanding the kinetics of > spontaneous nanotube growth. > > It turns out that " **making the structure hollow and making it twist are > two good ways of relieving such strain and stress** ," Jin explains. "In > some cases, the large screw dislocation strain energy contained within the > nanomaterial dictates that the material hollow out its center around the > dislocation, thus resulting in the spontaneous formation of nanotubes." Is it stress that causes screw dislocations in Germanium? What causes the stress?


title: What is the body force in the Navier Stokes Equations?

text: The incompressible Navier Stokes equations are: $\rho(\frac{\partial v_i}{\partial t} + v_j\frac{\partial v_i}{\partial x_j}) = -\frac{\partial p}{\partial x_i} + \mu\frac{\partial^2 u_i}{\partial x_j \partial x_j} + f_i$ for $i =1,2,3$ Reading around I have gathered that the force $f_i$ is a body force which can be that due to gravity, or some other force on fluid due to the presence of a body. My question is, isn't this force explained by pressure gradients? I.e. if there is a body in the flow (like a wing say) then there will be change in pressure of the flow as it comes close to the wing, which is effectively the influence of the wing on the flow. So why do we include this force term in the equations (assuming we dont care about gravity or any other external forces apart from those due to surfaces in the flow?)


title: Lightbulbs in circuits. Why does the current go UP?

text: > **Possible Duplicate:**   >  Why does the light bulb's brightness decrease? ![enter image description here](http://i.stack.imgur.com/uZeX6.jpg) So I have a circuit in the picture with three resistors. If I were to remove/disconnect $R_2$ from the circuit, what would happen to the overall brightness of $R_3$ and $R_1$? Now the answer I already have is that $R_1$'s brightness will go down and $R_3$ will go up. I understand that $R_1$'s brightness goes down because by removing the $R_2$, the net resistance goes up (destroying a parallel combination) But I don't understand why $R_3$ would go up? Conceptually this makes no sense to me. How do I know that the new current would always be larger than when the current splits at the junction? I would like a conceptual **physics** answer and not equations comparing inequalities.


title: Why will two bubbles floating on water surface attract each other?

text: Two identical bubbles floating on water surface will form clumps, according to the "cheerio effect". But what's the detail about the force? It's necessary to calculate the shape of water surface, in order to find the force?


title: Reissner-Nordström Black Holes

text: The Reissner-Nordström black holes are described by the metric, \begin{align} ds^2 = -\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)dt^2 + \frac{1}{1-\frac{2M}{r}+\frac{Q^2}{r^2}}+r^2d\Omega^2 \end{align} where $M$ is the mass of the black-hole and $Q$ is the charge of the black hole. I am using the convention of $(-,+,+,+)$ here. The minimum mass horizon, which one can calculate, is given by $M_{min} = Q^2$. Part of the derivation is at: http://aesop.phys.utk.edu/ads-cft/L3.pdf. (Notes on AdS/CFT Correspondence from University of Tennessee, notes 3). Suppose we approach the horizon, the metric begins to approach the $AdS_2 \times S^2$ metric, given by, \begin{align} ds^2 = R_1^2\frac{-dt^2}{z^2}+R_1^2 \frac{dz^2}{z^2} +R_2^2 z^2 d\Omega^2. \end{align} Note that in the link I provided, there is a sign switch (since the note writes for a Euclidean AdS). I was wondering, what would be the overall generators of the group? Is it just the generators of $AdS_2$ and the generators of $S^2$ (which are from $SO(3)$)? Are there not some more complicated generators that mix the two spaces?


title: Is brane world scenario in contrast with inflation theory?

text: Is brane world scenario in contrast in some way with inflation theory ?


title: How to deal with the notation of a function $f$ vs its value $f(x)$ in Physics?

text: This doubt is very silly, but anyway, I think it's worth asking. The problem is: when we work with mathematics, in many situations we want to consider sets $A$ and $B$ and functions $f : A \to B$. When we specify a function, the letter we use is irrelevant, so that $f : \Bbb R \to \Bbb R$ given by $f(x)=x^2$ is the same thing as $f(y)=y^2$ or $f(E)=E^2$. Also, we try to make a sharp distinction between the function $f$ and its value $f(a)$ for the point $a$. Now, in Physics I'm a little confused. For instance, when we think about momentum, we have $p = mv$, but wait, there are some details that appear to be not clear at first. If we are to differentiate, integrate, or whatever, this thing _must_ be a function. But, there are many ways in which this can be a function:   1. We can think of $\mathbf{p} : \mathbb{R}^3 \to \mathbb{R}^3$ as a function of velocity: $\mathbf{p}(\mathbf{v})=m\mathbf{v}$ and think of $m \in \mathbb{R}$ fixed.   2. We can think of $\mathbf{p} : \mathbb{R} \to \mathbb{R}^3$ as a function of mass: $\mathbf{p}(m)=m\mathbf{v}$ for $\mathbf{v}$ fixed.   3. We can think of $\mathbf{p} : \mathbb{R}\times \mathbb{R}^3 \to \mathbb{R}^3$ as a function of both mass and velocity: $\mathbf{p}(m,\mathbf{v})=m\mathbf{v}$.   4. We can think of $\mathbf{p} : \mathbb{R} \to \mathbb{R}^3$ as a function of time: $\mathbf{p}(t)=m\mathbf{v}(t)$ where we think of $\mathbf{v}$ as a function of time.   5. We can think of $\mathbf{p} : \mathbb{R} \to \mathbb{R}^3$ as a funciton of time: $\mathbf{p}(t)=m(t)\mathbf{v}(t)$ where both mass and velocity are time dependent. And these are only some of possible ways to think about it. Indeed, if we allow mass and velocity vary with even more parameters, then we can get $p$ as function as much more things. In that case, it's not clear at first what function $p = mv$ defines. There are tons of situations in physics where this kind of thing occur, and it becomes a little messy to deal with it this way. The books don't use the notation of of function, so we end up without knowing well what is happening. So, how do we deal with this kind of situation? How to remove these ambiguities?


title: Acceptance probability 2D Ising Model

text: Disclaimer: I just found a possible solution - eventhough i don't really understand, whats wrong with my prior approach. Edit: I just tried to calculate it from scratch and found the following: $E = -J*s\sum_r s_r$ and the new energy $E_n = -J*s_n \sum_r s_r$ where $s_n = -s$. So the difference is: $E_n - E = -J*s_n \sum_r s_r - -J*s\sum_r s_r = (-J\sum_r s_r)(s_n -s) = (-J\sum_r s_r)(-s -s) = (-J\sum_r s_r)(-2s)$. So there's a factor 2 instead of a factor 4 in the energy difference. My prior thought was: If i change the spin from -1 to 1 from a point surrounded by +1 spins the energy of that point changes from +4J to -4J and all the surrounding points energy get reduced by 2, which adds up to the total difference $\delta E = (-4J-4J)+4*(-2J) = -16J Why is that wrong? The old problem, somewhat solved. Question- name still applies. I'm currently writing a simulation of the 2D Ising model and there's something strange with my program. I calculate my energy difference before a spinflip (Metropolis Hastings algorithm) with               dE = -4*s*S      where `s` is the spin value of the (soon to be flipped) lattice site and `S` is the sum of the nearest neighbours. Usually there would be a coupling constant--I assumed it's just 1--and there is no field. Then I calculate the probability with:               Math.exp(-dE* Beta) //Beta == 1/T      Here are the calculated acceptance prbpabilities, the first value is the temperature ($kT/J = kT$), the second value is the energy difference (if it's zero or below zero the probability is 100% so i didn't include those). The third value is the probability.               0.5 16.0 1.2664165549094176E-14     0.5 8.0 1.1253517471925912E-7          1.0 16.0 1.1253517471925912E-7     1.0 8.0 3.3546262790251185E-4          2.0 16.0 3.3546262790251185E-4     2.0 8.0 0.01831563888873418          2.2669 16.0 8.604140030289146E-4     2.2669 8.0 0.029332814440979144          3.0 16.0 0.004827949226449516     3.0 8.0 0.06948344570075318          4.0 16.0 0.01831563888873418     4.0 8.0 0.1353352832366127      This is a screen shot of the simulation at $T=T_c = 2.2669$ which differs from what I expect: http://i.stack.imgur.com/Dmpi3.png Can someone confirm my probabilities or suggest what is wrong with them?


title: Accelerated electromagnetic system leads to divergent calculation?

text: ![paradox](http://i.stack.imgur.com/aVhRr.jpg) Consider the above system comprising a pair of oppositely charged parallel plates, connected by a rigid rod of length $r$, constrained to move in the x-direction. The forces on the plates from the Coulomb electric fields and the magnetic forces are all in the vertical direction so that they do not affect the motion of the system. The only fields that do affect the system are the horizontal radiation fields due to the acceleration of the charged plates. Let us assume that we initially apply a force $F_0$ so that we have: $$F_0 = m a_0$$ As charge $+q$ has an acceleration $a_0$ then it will produce a horizontal electric field at the negative plate given by: $$E_+ = \frac{-qa_0}{4 \pi \epsilon_0 c^2 r}$$ This will lead to a horizontal force $f_0$ on the negative plate, pointing towards the right, given by: $$f_0 = \frac{q^2a_0}{4 \pi \epsilon_0 c^2 r}$$ Likewise as charge $-q$ has an acceleration $a_0$ then it will produce a horizontal electric field at the positive plate given by: $$E_- = \frac{qa_0}{4 \pi \epsilon_0 c^2 r}$$ Again this will lead to a horizontal force $f_0$ on the positive plate, pointing towards the right, given by: $$f_0 = \frac{q^2a_0}{4 \pi \epsilon_0 c^2 r}$$ Thus the total force on the system is now given by: $$F_1 = F_0 + 2f_0$$ Therefore the system has an increased acceleration given by: $$a_1 = \frac{F_1}{m}$$ $$a_1 = a_0 \left(1+\frac{q^2}{2\pi\epsilon_0c^2rm}\right)$$ The argument can be repeated indefinitely leading to the calculated acceleration diverging exponentially. What's gone wrong? I think the best way to imagine the experiment is to simply drop the system in a gravitational field. The induced electric forces cause the system to accelerate faster than gravity which of course is forbidden by the equivalence principle. There are no $\dot{a}$ terms in the calculation so I don't think the Abraham- Lorentz reaction force can save the day. A positive feedback involving forces and accelerations seems to preclude a consistent initial value for the acceleration. Here's my (rather radical) solution: perhaps the mass has to change to ensure that the acceleration stays constant. Perhaps we have: $$F_0 = m_0 a_0$$ $$F_1 = m_1 a_0$$ Therefore: $$\frac{m_1}{m_0} = \frac{F_1}{F_0}$$ $$ \frac{m_1}{m_0} = \frac{1}{m_0a_0}\left(m_0a_0+\frac{q^2a_0}{2\pi\epsilon_0c^2r}\right)$$ $$m_1 = m_0 + \frac{q^2}{2\pi\epsilon_0c^2r}$$


title: Particle-hole symmetry of Bogoliubov - de Gennes equation

text: Bogoliubov - de Gennes equation reads, $$\left( \begin{array}{cc} H_{0} - E_{F} & -i\sigma_{y}\Delta \\\ i\sigma_{y}\Delta^{*} & E_{F} - H_{0}^{*} \end{array}\right) \left( \begin{array}{c} \psi_{e}^{\uparrow} \\\ \psi_{e}^{\downarrow} \\\ \psi_{h}^{\uparrow} \\\ \psi_{h}^{\downarrow} \end{array}\right) = \mathcal{E} \left( \begin{array}{c} \psi_{e}^{\uparrow} \\\ \psi_{e}^{\downarrow} \\\ \psi_{h}^{\uparrow} \\\ \psi_{h}^{\downarrow} \end{array}\right) $$ The above Hamiltonian which is $4$x$4$ obeys particle-hole symmetry, $$H = -\mathcal{C}H\mathcal{C}^{-1}$$ where $\mathcal{C} = \tau_{x}\mathcal{K}$, with $\tau_{x} = \sigma_{x} \oplus \sigma_{x}$ and complex conjugation operator $\mathcal{K}$. This symmetry implies that if You have solution $\Psi$ with energy $\mathcal{E}$, then You also have solution $\mathcal{C}\Psi$ with energy $-\mathcal{E}$. I always thought that if Hamiltonian posseses some symmetry then You have solution with the same energy. I am confused, cause I also read that holes are time-reversed electrons, $$\psi_{h} = \mathcal{T}\psi_{e}$$ with $\mathcal{T} = i\sigma_{y}\mathcal{K}$, but I can't show this from above BdG equation.


title: Angular Momentum commuting with Hamiltonian

text: I've been given an assignment where I have to prove that the angular momentum operators $L_j = \varepsilon_{jkl}q_{k}p_{l}$ commute with the Hamiltonian, given as $H = \frac{p^2}{2m} + V(r)$. Now, I can prove that $[L_j, \frac{p^2}{2m}] = 0$, and we've been given that $V(r) = \sum_{i\in Z}C_i r^n$, which basically means that I just have to show $[L_j, r] = 0 = [L_j, r^{-1}]$. Now, using the fact that I can get the following from $[r^2, p_j]$, and then subsequently show that $[L_j, r] = 0$; $$[r^2, p_j] = [q_{i}q_{i} , p_j] = 2i q_j$$ $$=> [r^2, p_j] = 2r[r, p_j] = 2i q_j$$ $$=> [r, p_j] = \frac{i q_j}{r}$$ $$=> [p_j, r] = \frac{-i q_j}{r}$$ Then, $$[L_j, r] = [\varepsilon_{jkl} q_k q_l, r] = \varepsilon{jkl}(q_k[p_l, r] + [q_k, r]p_l) = \varepsilon_{jkl}(q_k \frac{-i q_j}{r}) = \frac{-i \varepsilon_{jkl} q_j q_k}{r} = \frac{-i \varepsilon{kjl} q_k q_j}{r} = \frac{i \varepsilon_{jkl} q_k q_j}{r}$$ $$=> [L_j, r] = 0$$ However, I am completely lost/struggling to prove $[L_j, r^{-1}] = 0$. Any help would be fantastic!!


title: How Does Hubble's Expansion Affect Two Rope-Tied Galaxies?

text: Suppose we have two galaxies that are sufficiently far apart so that the distance between them increases due to Hubble's expansion. If I were to connect these two galaxies with a rope, would there be tension in the rope? Would the tension increase with time? Is the origin of the tension some sort of drag between the expanding space and matter?


title: Is the observable universe enclosed by an infinitely dense shell?

text: In modern cosmology, it is commonly stated to the public that as we look deeper and deeper into space, we are seeing further into the past. CMB is the relic of the big bang after photon decoupling following recombination. However, if we pushed further back in our conceptualization, eventually we get to the period of inflation, and then to a period before that where energy density of the universe was some large value (possibly infinite). So why would it be wrong to conceptualize the universe as being enclosed by an infinitely dense shell? Update: For clarification, wikipedia has a good definition of observable universe, and answers in the affirmative on how every observer will see a different observable universe. This question is whether there are valid approaches in theories to avoid issues of big bang singularites. Update2: I scaled back the question and accepted an answer, I think I was being overly vague in the write up and plan on asking the question again.


title: Continuum theory from lattice theory

text: I am looking for references on how to obtain continuum theories from lattice theories. There are basically a few questions that I am interested in, but any references are welcome. For example, you can obtain the Ising chiral CFT from a lattice theory. How does this work exactly? Intuitively is is clear that one should do something like taking the lattice spacing to zero. Is this worked out somewhere in detail for this example? One can also image, say, quantum spin models with sites on the edges of some graph, such that the interactions do not depend on the distance between the sites. One can imagine subdividing this graph further, to obtain an inclusion of the associated algebras of observables. This leads to an increasing sequence of algebras, and one can take the direct limit of this. Can one in this way obtain a continuum theory? I suppose that one might have to impose some conditions on the dynamics of the system at each step. Is something like this done in the literature? I'm mainly interested in a mathematical treatment of these topics.


title: Information geometry of 1D Ising model in complex magnetic field regime

text: Consider the one-dimensional Ising model with constant magnetic field and node-dependent interaction on a finite lattice, given by $$H(\sigma) = -\sum_{i = 1}^N J_i\sigma_i\sigma_{i + 1} - h\sum_{i = 1}^N\sigma_i$$ where $\sigma = \\{\sigma_i\\}_{i = 1,\dots, N}\in\Omega := \\{\pm 1\\}^N$, $\\{J_i\\}_{i = 1,\dots, N}$ are nearest neighbor interaction strength couplings, and $h \in \mathbb{R}$ is the magnetic field. Let's consider the ferromagnetic case, that is, $J_i \geq 0$ for $i = 1, \dots, N$, and for the sake of simplicity (though this doesn't matter in the thermodynamic limit), take periodic boundary conditions. Neither in the finite volume, nor in the thermodynamic limit does this model exhibit critical behavior for finite temperatures. On the other hand, as soon as we allow $h$ to be complex (and fix the temperature), even in the finite volume $N$, the partition function has zeros as a function of $h$. In the thermodynamic limit these zeros accumulate on some set on the unit circle in the complex plane (Lee-Yang circle theorem). Now the question: let's consider information geometry of the Ising model, as described above, when $h$ is real. In this case the induced metric is defined and the curvature does not develop singularities (obviously, since there are no phase transitions). Now, what about information geometry of the Ising model when $h$ is complex? This is a bit puzzling to me, since then the partition function attains zeros in the complex plane, so that the logarithm of the partition function is not defined everywhere on the complex plane, and the definition of metric doesn't extend directly to this case (the metric involves the log of the partition function), let alone curvature. Is anyone aware of any literature in this direction? I thought it would be a good idea to ask before I try to develop suitable methods from scratch. One could of course try to define the metric and curvature first for real $h$ and only then extend the final formulas to complex $h$. This seems a bit unnatural to me, and even dangerous. EDIT: Allow me to elaborate on what I mean by "information geometry." Let us consider for simplicity the finite volume model (i.e., $N < \infty$ above). The Gibbs state (i.e., the probability distribution on $\Omega$ of maximal entropy), given by $$P(\sigma) = \frac{e^{\beta H(\sigma)}}{\sum_{\sigma\in\Omega} e^{\beta H(\sigma)}}$$ where $\beta$ is inverse temperature, obviously depends on the temperature and the magnetic field. So it is convenient to write, for example, $P := P_{(\beta, h)}$ to make this dependence explicit. Now, the Gibbs states can be identified with points in the parameter space $$M := \\{(\beta, h): \beta, h\in (0,\infty)\\}$$ On this space one can define a metric, the so-called Fisher information metric, that naturally measures the distance between two Gibbs states $P_{(\beta, h)}$ and $P_{(\beta', h')}$. The definition of this metric (as you may have guessed!) involves the partition function. This metric then induces a geometry on the parameter space $M$, the so-called statistical manifold (see, for example, http://en.wikipedia.org/wiki/Information_geometry for more details). The curvature (induced by the metric) is an interesting quantity to study. As it turns out, the curvature develops singularities at phase transitions (also something you may have guessed, since the metric involves the partition function, and hence so does the curvature tensor). Actually, the _definition_ of the metric involves the logarithm of the partition function. All is fine so far, since $h$ is real (assume nonzero), and everything is well-defined. However, as soon as we pass to complex $h$, the partition function admits zeros, and it is no longer clear (at least to me) how the above constructions should generalize. See, for example, **Section 4** in http://eprints.nuim.ie/268/1/0207180.pdf).


title: Superconformal Multiplet Calculus in 6D

text: A convenient method for dealing with off-shell formulations of supergravity theories is provided by the superconformal multiplet calculus. This calculus was originally constructed for 4d ${\cal N}=2$ supergravity by de Wit et al. For 5d supergravity, conformal supergravity approach was developed relatively recently by several groups of Bergshoeff et al, Ohashi et al,Hanaki et al. http://xxx.lanl.gov/abs/hep-th/0104113 , http://xxx.lanl.gov/abs/hep- th/0104130 , http://xxx.lanl.gov/abs/hep-th/0611329 . The main idea is that the Poincare algebra is extended to the superconformal algebra to obtain an off-shell version of Poincare supergravity. It turns out that extending conformal supergravity to a gauge theory of superconformal algebra provides an irreducible off-shell realization of the gravity and matter multiplets. Then, by imposing constraints, the gauge theory is identified as a gravity theory. Upon gauge fixing the extra superconformal symmetries, one obtains the Poincare supergravity. In this formalism, one has a systematic way to construct invariant actions since the transformation laws are simpler and completely fixed by the superconformal algebra. Following this approach, one gets an off-shell formulation of supergravity coupled to vector multiplets. As far as I know, there is no off-shell formulation of 6d supergravity by superconformal multiplet calculas. Why is there no conformal supergravity in 6D? Is there any obstruction to the formulation?


title: Cosmic bubble collision

text: I'm reading this review right now. The claim seems to be that when you have an expanding "false vacuum", finite size bubbles form due to phase transition (cause by finite action instantons) (which, in itself might then evolve, etc). My questions is, if I imagine these bubbles to be moving away from each other due to exponential expansion of the false vacuum, then how will they come near each other and collide?


title: Energy difference in General Relativity

text: Why exactly are absolute energies important in General Relativity, unlike for example EM where only energy differences matter?


title: Internal observers in a mathematical model

text: In this abstract the idea of a mathematical model having an internal observer is raised. A similar suggestion was made to me in a comment just recently. Is there a name for this methodology of employing an internal observer of the mathematics? Is it widely used and known?


title: Why complex functions for explaining wave particle duality?

text: I have this very bad habit of going to the scratch, discarding all the developments of a theory and worldly knowledge, and ask some fundamental (mostly stupid and naive, as some may say) questions as to why why we needed so and so assumption, why we had to consider this way, could we assume $X$ instead of $Y$ and get a different theory and so on and so forth. As a part of this, is the following question : Way back in the early twentieth century, physicists struggled to explain certain phenomenon like photo electric effect which needed light waves to behave a particles (photons), and the interference effect of electrons (diffraction) which needed them to behave as waves. So they said, hey, consider a wave (don't ask me know what it is, but just consider it)... ok $$\Psi(x,t) = e^{i(kx-\omega t)},$$ now without asking what is $\Psi$, we can explain the interference of electrons and also the photo electric effect, basically the wave/particle duality, if we make an analogy between wave and particle nature as $p = \hbar k$, $E = \hbar \omega$. My seemingly blunt question is, if you want to explain wave nature and interference by considering a wave function $\Psi$, why the heck do we need complex numbers, why not just real function? Cant we consider something like $\Psi(x,t) : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ or something like that. Hold on, Water waves are successfully explained by real wave function for example $\Psi(x,t) = \cos(kx-\omega t)$, so why the heck we need complex waves for making an anology for wave-particle duality? What if we just unlearn all the QM and start with a real function for waves, what is going to happen? Sorry for being a bit presumptuous, but some gentlemen will start talking about probability amplitude, uncertainty principle, and so on and so forth, but Gentlemen, wait, I haven't gone that far! I don't have Born interpretation and I don't have uncertainty principle or for that matter the Schrodinger equation yet, So your logic will lead to circular arguments! After all our goal is to explain physical phenomenon...what if we venture into this jungle of real functions and come up with a totally different theory which explains physical phenomenon. ((If you ask me to do research in theoretical physics, I'll throw all the QM books in garbage (no disregard though) and start thinking from this point of view...Thats my style of working!) My expected answer is in this spirit, "Hey if you go in that direction, you are bound to end up in a quick sand, for so and so reason"


title: Calculating correlation functions of exponentials of fields

text: In their book Condensed Matter Field Theory, Altland and Simons often use the following formula for calculating thermal expectation values of exponentials of a real field $\theta$: $$ \langle e^{i(\theta(x,\tau)-\theta(0,0))} \rangle = e^{-\frac12 \langle (\theta(x,\tau)-\theta(0,0))^2 \rangle} $$ An example can be found in chapter 4.5, problem "Boson-fermion duality", part c). (This refers to the second edition of the book, page 185.) In other words, expectation values of exponentials can be cast as exponentials of expectation values under certain conditions. Unfortunately, I seem to be unable to find an explanation of why this can be done and what the conditions on the Lagrangian of $\theta$ are. Hence, my question is: > How to derive the above formula? What do we need to know about $\theta$ for > it to be valid in the first place? Ideally, I am looking for a derivation using the path integral formalism. (I managed to rederive a very special case in terms of operators and the Baker- Campbell-Hausdorff formula, but I'd like to gain a more thorough understanding.)


title: RG flow from a UV scale invariant field theory to a gapped phase in the IR

text: On the section 3 of http://arxiv.org/abs/1309.2921 the authors consider the RG flow from a scale invariant field theory in the UV to a gapped theory in the IR. The theory is couple to a background metric $g_{\mu\nu}$. They state that since the theory in the IR is gapped the low energy effective action is a local functional of the background metric. How can I prove this statement? They say in a footnote that their main argument could be extended to the case when the IR is a CFT. In that case I wouldn't have a gapped theory in the IR, right?! So what is the importance of the IR being in a gapped phase?


title: on quantum steering

text: I have become interested in quantum steering after listening a talk and tried to read more about it. I think I am more confused now. My understanding is as follows: > Sharing a (entangled) state, Bob wants to make a series of measurements and > Alice wants to make a guess of the measurement outcome. After which I am totally at loss, as a few papers which I have seen refers to a steering inequality. However it is heavily state dependent, and not like Bell (or Bell type) inequalities. Can someone point out some basic references or clear up my confusion. I am sorry for not writing it more clearly. May be by the questions, I shall be able to understand. Advanced thanks for any help.


title: Difficult to visualize Franck-Condon overlap

text: The franc condon principle says about the vertical transitions, means the coordinates of the nuclei remains unchanged during an electronic transition. But, what is the physical meaning or how to visualize 1) Franck Condon overlap or Franck Condon Factor? 2) I specifically came across the vibrationally mediated photodissociation (VMP). Here it says VMP is effective when there is a favorable Franck Condon overlap between vibrationally excited states of the ground electronic states and those of the electronically excited dissociative or predissociative states. Does this mean the potential energy surfaces of the said states really overlap? if anyone can give me a pictorial explanation, that would be nice.


title: What does this observation of instantaneous velocity in Brownian particles mean?

text: I read this artice: Physicists Prove Einstein Wrong with Observation of Instantaneous Velocity in Brownian Particles > “We’ve now observed the instantaneous velocity of a Brownian particle,” says > Raizen. “In some sense, we’re closing a door on this problem in physics. But > we are actually opening a much larger door for future tests of the > equipartition theorem at the quantum level.”   * **What did Einstein propose and what did they actually prove?**   * **What door does this experiment open?**


title: $\langle TT\rangle$ correlator of the boundary CFT from metric fluctuations in the bulk gravity

text: * Is there a reference which explains how the $\langle TT\rangle $ correlation of the boundary conformal field theory (CFT) can be holographically calculated from the bulk gravity? (..I am often getting referred to some paper by Skenderis et. al but I would like to see some more explicit explanation of the idea..)  In the following we defined $C_T$ as the coefficient of the leading singularity of the $\langle TT\rangle $ correlation. * * * Roughly that I understand the following possible algorithm works -   * first expand the gravity Lagrangian about an AdS background making a fluctuation (say $h$ ) along two orthogonal spatial directions of the boundary. Like say add to the metric two off-diagonal components of the type, $g_{xy} = g_{yx} = \epsilon h (r,z)$ where $x,y,z$ are 3 randomly chosen orthogonal spatial directions on the boundary and $r$ is the bulk direction orthogonal to the boundary.    * Now pick out all terms in the Lagrangian which are quadratic in $h$ and focus on all the terms which have a $\partial_r$ and rewrite those terms by pulling out a total derivative in $r$. (...I don't know how to make sense of this if one has terms of the kind $(\partial _r h)(\partial_r ^2 h)$ (and such terms do occur in examples I have tried)    * Now in this "boundary Lagrangian" which is formed by the total derivatives in $r$ and quadratic in $h$ terms substitute a Fourier expansion for $h$ as $h(r,z) = \sum_p e^{ipz}H_p(r)$.    * Now apparently there is (a universal? irrespective of the gravity Lagrangian?) value for this $H$ function which makes the fluctuation go on-shell and for Gauss-Bonnet gravity that apparently is, $H_p(r) = c BesselK[ d/2, (L^2p)/(\sqrt{f_\infty} )]$. And this $c$ is determined by demanding that $H_p (r \rightarrow \infty) = 1$ (where $r=\infty$ is the boundary) [...here its not clear to me as to how will this $f_\infty$ be defined from just the GB Lagrangian - AFAIK this $f_\infty$ is a parameter that is defined only when one is looking for asymptotically $AdS$ blackholes in GB gravity)    * Now expand in a power series in $r$ about $r=\infty$ this boundary Lagrangian and then (for any of its Fourier mode $k$?) for a d-dimensional boundary if the $d$ is even then this $C_T$ is given by the coefficient of the $k^dlog k$ terms and for $d$ odd this is given by the $k^d$ term.  * * *   * I would like to know if I understand the above algorithm right and if so then what is the reference for its derivation.


title: What is the spin state of a spin-1/2 particle when it comes out of a Stern-Gerlach apparatus?

text: Having a particle entering the apparatus with spin state $|+\rangle$, for which $\hat S_x|+\rangle=+\frac\hbar 2|+\rangle$, I have a question about how to express the spin state when it comes out. I mean: will the spin state be an eigenstate of the spin $z$ component? or would it be a superposition of the $z$ component's eigenstate ?


title: Help needed to interpret question - Spin States of electron pair in Helium?

text: ![enter image description here](http://i.stack.imgur.com/nCg1K.png) For the last part, I'm not sure what they mean by "explain how to form eigenstates of the total spin $\hat S^2$ and $S^z = S_1^z + S_2^z$. Are they simply referring to the spin singlet and tripplet states? I know for tripplet states total spin is 1, the states are $|S^2,m_{s1} + m_{s2} \rangle$: |1,1>,|1,-1> and |1,0> and for the singlet states total spin is 0, the only state is |0,0>.


title: Deriving Cartan formula

text: I have trouble deriving Cartan formula of the form: $$ \mathrm{d} \omega (X,Y) = X[\omega(Y)] - Y[\omega(X)] - \omega([X,Y]) \tag{1} $$ where $\mathrm{d}$ is the exterior derivative, $\omega$ is a one-form, $X$ and $Y$ are tangent vectors in the coordinate basis $\\{e_\mu \\}$, and $[ \cdot,\cdot]$ denotes the Lie bracket. I can show that the right-hand side equals: \begin{equation} X[\omega(Y)] - Y[\omega(X)] - \omega([X,Y]) = (X^\nu Y^\mu - Y^\nu X^\mu ) \partial_\nu \omega_\mu \tag{2} \end{equation} but I have difficulties evaluating the left-hand side of equation $(1)$. I have tried this: $$ \begin{aligned} \omega(X,Y) & = \omega_\mu \mathrm{d} x^\mu X^\nu e_\nu Y^\lambda e_\lambda \\\& = \omega_\mu \delta^\mu_\lambda X^\nu Y^\lambda e_\nu \\\& = \omega_\mu X^\nu Y^\mu e_\nu \end{aligned} $$ This is not any kind of form anymore, so I'm not sure if the exterior derivative is well-defined to act on it. If I would try it, then I get: $$ \begin{aligned} \mathrm{d} \omega & = \left( \partial_\lambda \omega_\mu X^\nu Y^\mu e_\nu \right) \mathrm{d} x^{\lambda} \\\& = \partial_\lambda \omega_\mu X^\nu Y^\mu \delta_\nu^\lambda \\\& = \partial_\nu \omega_\mu X^\nu Y^\mu \end{aligned} $$ which does not equal the right-hand side equation $(2)$. And so it does not obey equation $(1)$. Does anybody where I went wrong in my derivation?


title: Energy dependent cross sections for neutrons

text: Do you know where I can find a table with energy dependent cross sections for neutrons? Thanks!


title: Pendulum (highschool)

text: I have a question concerning pendulum. On our physics lessons we built simple pendulum consisting of weight and a thread ($\ell=0.31$ m). We tried to calculate $g$ from the well known formula $$T_m = 2\pi\sqrt{\frac{\ell}{g}}$$ which leads to $$g = \frac{4\pi^2 \ell}{T_m^2}$$ Of course, this formula comes from conclusion that $\sin(\alpha)-\tan(\alpha)$ is neglectable for "small" $\alpha$. Hence our target was to find formula, which works for larger $\alpha$ as well. According to this two sources, this should do the trick: $$T_m = 2\pi\sqrt{\frac{\ell}{g}}(1+\frac{1}{4}\sin^2(\frac{\alpha}{2}))$$ from which I came to this conclusion: $$g=\frac{\ell\pi^2(\cos(\alpha)-9)^2}{16T_m^2}$$ Here is my problem: After using this formula and our measured data, I got $\overline g = 8.9085$, which is obviously not very accurate (for ~48°N). Any ideas where I've made a mistake?


title: Can you gauge a $U(1)_L$ symmetry?

text: I recently calculating the one loop correction for the propagator of a gauge boson, $\hspace{5cm}$![enter image description here](http://i.stack.imgur.com/gk0dJ.png) I assumed arbitrary left and right couplings, $ g _L $ and $ g _R $. I found that the one loop correction was, \begin{equation} = \frac{ - 4i }{ ( 4\pi ) ^{ d / 2}} \int dx \frac{ \Gamma ( 2 - d / 2 ) }{ \Delta ^{ 2 - d /2 }} \left[ ( g _L ^2 + g _R ^2 ) x ( 1 - x ) \left( g _{ \mu \nu } - \frac{ p _\mu p _\nu }{ p ^2 } \right) + g _{ \mu \nu } ( g _L ^2 - g _R ^2 ) m ^2 \right] \end{equation} Now we also know that for a $ U(1) $ invariant theory we should have the Ward identity and hence this one loop correction would be of the form, \begin{equation} \Pi _{ \mu \nu } = \Pi ( p ^2 ) \left( g _{ \mu \nu } - \frac{ p _\mu p _\nu }{ p ^2 } \right) \end{equation} So it appears that in order for this propagator to arise from a $U(1)$ invariant theory, we must have $ g _L = g _R $ (unless I made a mistake). I found this very strange. Is it not possible to gauge a $ U(1) _L $ symmetry and if so why? As a working example I invented the theory below which seems like it could be gauged: \begin{equation} {\cal L} = i \psi ^\dagger \bar{\sigma} ^\mu \partial _\mu \psi + i \chi \sigma ^\mu \partial _\mu \chi + \phi ^0 \chi \chi + \phi ^{+ + } \psi \psi + \phi \mbox{ terms} \end{equation} where $ \psi $ is a left chiral spinor and $ \chi $ is a right chiral spinor. The particles transform under the symmetry as, \begin{equation} \psi \rightarrow e ^{ i \alpha } \quad \chi \rightarrow \chi \quad \phi ^0 \rightarrow \phi ^0 \quad \phi ^{ + + } \rightarrow e ^{ - 2i \alpha } \phi ^{ + + } \end{equation} This seems like a perfectly fine theory, whose symmetry can be gauged but if my conclusion above is correct for some reason I shouldn't be able to gauge this $ U(1 ) _L $ symmetry. Did I make a mistake or is there some deep reason this can't be done?


title: Model to convert radiation data from one tilted surface to a different one?

text: I have measuring data of global radiation on a tilted surface (a solar collector surface). Now I need to do 2 things:   1. split the tilted global radiation into its diffuse and beam parts    2. calculate the same terms (diffuse + beam radiation) on a different surface (different tilt and/or azimuth angle)* I was able to solve **problem (1)** by inverting a tilted radiation model (HDKR), and combining it with the Erbs model on the horizontal plane. In this way I get quite good results for diffuse and beam parts on the tilted surface. Problems near sunrise / sunset and numeric problems are all under control. For **problem(2)** however, I failed. I mention problem (1) because I intended to use its solution as a basis for problem (2): I intended to take the diffuse + beam parts on the horizontal (an intermediate result of problem (1)), feed it into HDKR to calculate diffuse + beam on an arbitrary surface. Now, these steps do not work. From step (1), I do not get diffuse + beam horizontal at times where I need it for step 2. This is because due to the different orientation, sunrise / sunset / sun behind collector happen at different times compared to the original surface. This is why my approach to step (2) fails. I do not know whether this is a very unusual problem? For us it would be really useful to solve problem (2). It would allow us to use radiation data from one solar energy installation to evaluate a nearby plant with possibly different orientation. Are there any special models treating with this or a similar problem? Any help is greatly appreciated! Thank you all!


title: Amplitude of motion for a cart on an inclined plane attached to a spring

text: A very practical question. So, let's imagine that we have a small cart attached to a spring lying on an inclined plane. If I move my cart by 10 cm (measured as if there were a ruler lying along the inclined plane) such that the spring is compressed, and then release it, will my cart oscillate in simple harmonic motion with an amplitude of 10cm measured along the plane? I thought about this for some time now. But, I think the answer is simply no. Because there will always be the impact of a component of the force of gravity. So, the cart should still move in SHM, but not with an amplitude of 10 cm, right? If I am wrong, please explain with detail.


title: Why Green's function will diverge at the same spacetime point?

text: In $d+1$ dimensional quantum field theory, the 2-point Green's function will diverge at the same spacetime point when $d\geq1$. When $d=0$, $\phi(t)=q(t)$, that is the case of QM, and 2-point Green's function at the same spacetime point $\langle\Omega|T(q(t)q(t))|\Omega\rangle$ is well-defined. While $d \geq1$, the 2-point Green's function at the same spacetime point $\langle\Omega|T(\phi(x)\phi(x))|\Omega\rangle$ will diverge. So what's the physical or mathematical essence of this diverge. I especially want to know the physical picture in the path integral. Why the randomly walking of a particle will be different from that of a string?


title: Order of Monte Carlo integration and frequency summation

text: I am currently trying to calculate an integration formula of a linear response function by Monte Carlo method. It is a multiple integration over three 3D vectors, i.e., nine dimensions in all. And because of green function, there will also be an infinite discete frequency summation. This can be done either inside the integral, i.e., when evaluating the integrand, or outside the integral, i.e., summing up the integral result for each discete frequency point. My question is that whether this order issue matters or not.


title: Solid-body rotation of fluid in polar coordinates: How to compute the stress tensor

text: In a course on continuum mechanics, we are given an exercise concerning solid- body rotation of a fluid in polar coordinates. In the first parts (feel free to correct any errors here) we are tasked with finding the velocity field (which we have simply stated to be $\vec v=\begin{bmatrix}v_r \\\ v_\theta \end{bmatrix}=\begin{bmatrix}0 \\\\\omega_0 r \end{bmatrix}$ where $\omega_0$ is the angular velocity), and find out if the solid-body rotation features shear strain (no, since the strain rate tensor is zero), expansion (no, same reason) or rotation (yes, this is given by the antisymmetric part of $\nabla\vec v$ which in this case is identically equal to $\nabla\vec v$). The next part of the exercise would have us compute the stress tensor for this flow, and this is where we're stuck. How could we go about doing this? We are not given any specific details about the fluid in question, so we don't know if we could just use the constitutive relations for e.g. a Newtonian fluid, $\sigma_{ij}=-p\delta_{ij}+2\mu d_{ij}$. (The last part of the exercise asks if this flow satisfies the Navier-Stokes or Euler equations, and we're also stuck here, feel free to help out.)


title: Schwarzschild Metric coordinate sign change in $0\leq r \leq 2GM$

text: In the event horizon of the Schwarzschild-metric not only the time coordinate but also the radial space coordinate seems to change sign: $$ds^2=\left(1-\frac{2M}{r}\right)\mathrm{d}t^2-\left(1-\frac{2M}{r}\right)^{-1}\mathrm{d}r^2-r^2\mathrm{d}\theta^2-r^2\sin^2\theta \, \mathrm{d}\phi^2$$ (in units wherein $c=G=1$). As we can see, for $0<r<2M$, $t$ be negative (just like the space coordinates) and $r$ will be positive (just like the time coordinate), although $r$ has a singular point at $r=2M$. Does this sign change have some physical meaning? **Reacting comments**   1. Under "normal" circumstances (not stepping through the event horizon of a Schw BH), a such change like this don't happen, this is because I have the question, what about physical meaning could have a such change.   2. As I understand the Kruskal-Szekeres metric, it has multiple sections. It effectively handles the problem, what happens if somebody steps through to event horizon. But stepping through the event horizon is impossible only for an external observer, the object falling in the Schw BH _can_ watch its fall into the BH. What will it experience [as a theoretical possibility], as its radial spatial coordinate transforms to a temporal and vice versa?


title: Electric field near surface of a conductor?

text: > If the electric field on a positive charge is non-zero, then the charge > accelerates in the direction of the field. The field at the surface of a conductor is perpendicular to the surface. Why don't the surface charges accelerate away from the surface of the conductor then?


title: Why does rubbing feet against sheets create a spark?

text: Sometimes when I'm in bed at night, I rub my feet against the sheets and light is created. What exactly is going on? I'm familiar with static electricity, and the triboelectric effect (creating a charge by rubbing 2 insulators together), and electrostatic discharge (releasing charge by touching a grounded conductor). I believe that all of that is what is at work here, but I'm slightly confused since in other cases, the triboelectric effect and electrostatic discharge seem to occur as two separate steps, with different substances involved, and here, the two effects seem to be occurring simultaneously, with the same substances. Can someone help me with a step-by-step explanation of the dynamics of what is going on?


title: Harmonic Oscillator (Quantum Mechanics)

text: Griffiths uses an algebraic "brute force" technique to solve the harmonic oscillator. I'm somewhat confused regarding a few parts. $$\frac{1}{2m}[p^2 + (m \omega x)^2] \psi = E \psi$$ $H = \frac{1}{2m}[p^2 + (m \omega x)^2]$ We are about to factor $H$, noting that the numerical equavilant is $u^2 + v^2 = (iu+v)(-iu+v)$. We now define $a_{\pm} = \frac{1}{\sqrt{2 \hbar m \omega }}(\pm ip+m \omega x)$ Skipping through the commutator, I don't understand how we can say: $a_+ a_ - = \frac{1}{\hbar \omega}H + \frac{1}{2}$ Looking at the original equation, we factored $[p^2 + (m \omega x)^2]$, so we can replace this with $a_+ a_ - $. Under this, couldn't we just say that $H = \frac{1}{2m} (a_+ a_ -) $ My second question has to do with the statement (this is a direct quote): **Now, here comes the crucial step: I claim that if $\psi$ satisfies the Schrödinger equation with energy $E$, (that is: $H \psi = E \psi$), then $a_+ \psi$ satisfies the Schrödinger equation with energy $(E+ \hbar \omega)$** I don't understand why we're multiplying $\psi$ by a piece of Hamiltonian, and the by the Hamiltonian again.


title: Cluster Expansion vs Cluster Decomposition

text: Are the cluster expansion (which we encounter in Statistical Physics), and cluster decomposition (in Quantum Field Theory) related to each other? (I have a reason to believe they are)


title: A question about a complex integration in Peskin's QFT textbook

text: In page 27 (2.52), the integration is $$\int_{-\infty}^{\infty}dp \frac{p e^{ipr}}{\sqrt{p^2+m^2}}$$ He says that there are two branch cuts starting from $\pm im$ ![enter image description here](http://i.stack.imgur.com/pyO89.png) But I learn in complex analysis that $\sqrt{z^2+m^2}$ has only one branch cut from $-im$ to $im$, because point going around $im$ or $-im$ only will gets a minus, but point going around $\infty$ only will keep the sign. Therefore $\infty$ point is not the branch point and the branch cut is from $-im$ to $im$. So who is wrong?


title: Brillouin Zones in a nanowire

text: My professor told me something I didn't understand the other day: I was reading a paper on a crystalline nanowire (NW), and in the paper they look at how the band structure changes (from that of the unconfined bulk) as you make the diameter smaller (essentially, the confinement makes a valence and conduction band at a certain point of the Brillouin Zone (BZ) split apart). Well, he essentially said that it's kind of useless to look at the bulk band structure when talking about a nanowire. I didn't really understand his reasoning for it, so that's what my question is about. Now, here's what I thought (which must be incorrect in some part): I thought that the wave vector determines direction of travel of an electron and the Brillouin Zone gives all the directions you can travel in the crystal with respect to its crystal structure. And, the band structure diagram tells you the energy at and between the major points of the BZ, like so. So, I thought, if you're confined to travel in one crystal direction like in the NW, if you want to find the dispersion relation $E(\vec k)$, you basically take a "slice" from the band structure diagram. For example, if the crystal orientation of your nanowire only let you travel along the $\Gamma -X$ direction in this BZ: ![BZ](http://i.stack.imgur.com/7Cbhn.png) in the band structure diagram in the link above (assuming they went together, which they probably don't; I just took a couple random ones to illustrate my point), you'd only look at the region between $\Gamma$ ($\vec k = 0$) and $X$ ($\vec k = \vec b_1$). Essentially, that a NW is just a little piece of a bulk material, so it has the same band structure, but only part of it. But apparently this isn't true, according to him (and I trust him, I just didn't understand him). Can anyone tell me why? Thanks!


title: Has any permanent magnet motor been proven to run?

text: I have read lots of articles about permanent magnet motors, some of which claim the possibility and other which refute it. Is it possible to have a permanent magnet motor that runs on the magnetic force of permanent magnets?


title: criteria for quantaization of excitations in matter

text: We know that elementary excitations in plasma, like the free electron gas in a metal, can be quantized. There are known as plasmons. We also know that the elemntary vibrations in the lattice of solids can also be quantized as phonons. My question is two fold.   1. What is the critera for quantaiztion?    2. These so called quasi-particels do not really exist, but these excitations can be treated, at least mathematically, as particles. What is the property or feature that makes the physics or math work.


title: Triple point temperature and freezing point

text: Why does the triple point temperature have very similar values to the freezing point, in most substances?


title: Information loss in a black hole

text: How does the Holographic Principle help to establish the fact that all the information is not lost in a black hole?


title: References for circular restricted 3 body problem?

text: Does anyone know of any good references for the CR3BP -- the circular restricted 3 body problem? Emphasizing on real-life applications, and interpretation of the numerical solutions? Thank you. There are quite a few hits when I googled the topic, but I am wondering if anyone knows of any "Bible" for the subject.


title: Non-Hermiticity when Fourier transforming onto a finite lattice

text: I'm doing numerical simulations. I have the Haldane model in a honeycomb lattice where $$ H = \sum \limits_{<ij>}a^\dagger_i b_j + h.c $$ Where $i$ belongs to sublattice $A$, and $j$ to sublattice $B$, which comes from the fact that there are two atoms per unit cell. $a$ is the annihilator on sublattice $A$ and $b$ on sublattice $B$. This operator is Hermitean. I can use its Fourier representation $$ H \sim \sum \limits_k H(k) e^{ikr} $$ Where $r$ denotes lattice vectors and $k$ the corresponding lattice momentum. The amplitudes $H(k)$ are not Hermitean but they satisfy $H(k)^\dagger = H(-k)$ obviously, so the above sum remains Hermitean if I take out pairs of $(k,-k)$. I cannot figure if by discretizing the elementary cell in $k$-space, thus effectively taking out some of the $k$'s in the above sum, the Hermiticity of this truncated Hamiltonian would get lost.


title: "S-duality" between confinement and the Higgs mechanism?

text: I feel picked by the second to last sentence in this answer to a question about what would happen if EM and QCD were spontaneously broken, which says _"In fact, there is a sense in theoretical physics in which confinement is complementary to Higgsing – it's the same thing in different ("opposite", "S-dual") variables"_ to explain further why a broken $SU(3)$ gauge group could not be confining. I'd like to see a further explanation of this. For example which of the weak/strong regime would correspond to the higgsing/confinement regime? Is it as I naively expect from the cited comment, the strong coupling constant that is used to establish the S-duality, or does it work via a more unified higher energy scale coupling constant? Heck, I'd just like to know in some more detail than just this comment, how it works.


title: Young's Double slit experiment (Finding the value for slit separation)

text: The question illustrates mono-chromatic red light passing through 2 slits and forming an interference pattern on the screen. The question then asks to suggest a suitable value for the slit separation. How do I find a value for this or what basis should I approximate my answer?


title: Potential Difference Between Capacitors in Series

text: I am struggling to find an answer to this, hopefully relatively simple, question. I had a search on stackexchange but couldn't find anything helpful. We are learning about capacitors in Physics and I understand that when capacitors are connected in series, the charge stored in each is equal. My only issue with this is that, when the capacitors (lets assume there are two) have different capacitance, the potential difference across each will be different according to the formula $ V = \frac{q}{C} $. If this is true, and the potential difference across each is different, then why is there no potential difference between the two capacitors, as otherwise, charge would flow from one to the other and the resulting stored charges would not be equal. One possible explanation for this behaviour that I can come up with is that since the magnitude of the charge on each face of the each capacitor is equal, the attractive force between the charge on the positive face of one capacitor and the negative charge on the face of the adjacent capacitor is balanced by the attractive force between that charge and the negative charge on the other face of the same capacitor. Is that correct? I assume I must be misunderstanding something. **Edit 1** As I (clearly incorrectly) understand it, the electrons 'fill up' (as James Ngai Chun Tat put it) one plate (plate A) and as a result electrons are repelled away from the other plate (plate B) and so there is a difference in charge across the plates, resulting in a potential difference. As electrons are repelled from the positive plate (plate B), they move to the negative plate of the adjacent capacitor (plate C) and fill it up. As electrons collect on this plate, it becomes negatively charged. I would have thought that as plate B is positively charged and plate C is negatively charged, there would be a potential difference between the two (causing electrons to flow in the opposite direction). What is wrong with the model I have come to understand? **Edit 2** We were taught that when charging a resistor, as charge flows to the plate from the battery, negative charge builds up on the plate beside the battery. As a result, the potential difference between that plate and the negative terminal on the battery falls, resulting in an increasingly low current until eventually charge stops flowing altogether (when the potential difference across all of the capacitors is equal to that across the power supply). Is that right?


title: Super High Frequency Electromagnetic Radiation - String Theory

text: I am a serious high school student with one year of physics class experience, so please point out if there are any flaws in my question or reasoning. Thanks! Gamma ray radiation possesses a wavelength below 10 picometers ($10^{-11}\: \mathrm{m}$) and is capable of ionizing atoms. If a photon was endowed with sufficient energy ($~1.24 \times 10^{10}\: \mathrm{J}$) to yield a wavelength on the scale of a Planck length ($~1.62 \times 10^{-35}\: \mathrm{m}$), the theorized length of strings, would it be capable of producing interference or some other influence on strings, if they exist? From what I have learned, string theory postulates that all matter and energy is composed of tiny vibrating strings that require 11 dimensions (please correct me if I am wrong). Would EM radiation with a wavelength on the scale of a string's length then be able to cause some sort of interference or influence on the string's properties or vibrational pattern? Considering that an electron is so much larger than a string, such an experiment seems like trying to cause a ripple through a floating one inch piece of string by rapidly shifting an adjacent mountain one inch back and forth. At such high energy levels, would an electron even be able to remain stable, or would it dissociate into its constituent particles? If so, would there be some amount of sufficient energy that would dissociate even those particles? Here is the equation that I used to approximate the energy required to endow an electron with a wavelength of $~1.6 \times 10^{-35}\: \mathrm{m}$: $$E=\frac{hc}{\lambda} = \frac{6.63 \times 10^{-34}\: \mathrm{Js} \cdot 3.00 \times 10^8\: \mathrm{m/s}}{1.62 \times 10^{-35}\: \mathrm{m}}= 1.23 \times 10^{10}\: \mathrm{J}$$


title: Is a particle subject to dissipation proportional to its velocity a Hamiltonian system?

text: Why or why not? I'm pretty sure that this isn't a Hamiltonian system because it involves a dissipation term, but using the Hamiltonian flow it gives me that the system is Hamiltonian.


title: What is the total kinetic energy per second of the particles accelerated by the LHC

text: Related: how many protons per second can the LHC accelerate?


title: Determining the path of a particle in a velocity selector

text: I am wondering how you can determine the path of a particle in a velocity selector provided its velocity doesn't equal $ \frac{E}{B} $. I understand that the equation is derived from this force equation, $\sum \vec{F} = qvB- qE = ma$, where $a = 0$, but I cannot show analytically how its path would differ should its velocity be greater than or less than $ \frac{E}{B} $. Any help would be much appreciated.


title: How to reconcile angular momentum measurements?

text: Imagine a particle tracing a counter-clockwise circular path on a flat table with a certain speed. The particle is tied with a massless string of length $R$ to a point $P$ at the center of the circular path. Will the particle rotate about $P$ forever at constant speed in the absence of any external force? Consider using different origins to measure the physical quantities. **Measurement Case 1 (origin $O$ at $P$):** If I choose the origin $O$ of a Cartesian coordinate system to be at $P$, the flat table is the xy-plane and the particle rotates about the point $O$ as described. Specifically, the angular velocity $\vec{\omega}$ of the particle points in the direction of the positive z-axis, the position of the particle at any time is specified by the position vector $\vec{r}$, the angle between $\vec{\omega}$ and $\vec{r}$ is always $\frac{\pi}{2}$, and the angular momentum $\vec{L}$ is as follows: $\vec{L} = \vec{r}\times\vec{p}\;\ldots\text{ definition} \\\ \hphantom{\vec{L}} = m\,(\vec{r}\times\vec{v})\;\ldots\text{ definition of linear momentum }\vec{p} \\\ \hphantom{\vec{L}} = m\,(\vec{r}\times(\vec{\omega}\times\vec{r}))\;\ldots\text{ definition of linear velocity in terms of angular velocity} \\\ \hphantom{\vec{L}} = m\,(\vec{\omega}\,(\vec{r}\cdot\vec{r}) + \vec{r}\,(\vec{r}\cdot\vec{\omega}))\;\ldots\text{ scalar triple product}\\\ \hphantom{\vec{L}} = m\,(\vec{\omega}\,(r^2) + \vec{r}\,(0))\;\ldots\;\vec{r} \perp\vec{\omega}\\\ \hphantom{\vec{L}} = (m\,r^2)\,\vec{\omega}$ That is, $\vec{L}$ and $\vec{\omega}$ have the same direction, and $\vec{L}$ does not change direction and magnitude while the particle is rotation about $P$. Therefore, the particle will rotate about $P$ forever at constant speed in the absence of any external force because the angular momentum $\vec{L}$ is conserved (the presence of a centripetal force does not affect the angular momentum $\vec{L}$ in any way because a centripetal force is always parallel to the moment arm, and therefore, does not give rise to a torque $\vec{\tau}$). **Measurement Case 2 (origin $O'$ vertically directly beneath $P$):** Now if I choose to describe the same problem by choosing an origin $O'$ of a Cartesian coordinate system to be vertically directly beneath $P$, the particle's position vector $\vec{r}'$ makes an angle $\phi$ with the z-axis and the particle's angular momentum $\vec{L}'$ also makes a $\phi$ angle with the z-axis. As the particle rotates, however, $\vec{r}'$ also rotates about the z-axis, and therefore, the angular momentum $\vec{L}'$ keeps changing direction as shown below: $\vec{L}' = \vec{r}'\times\vec{p}'\;\ldots\text{ definition} \\\ \hphantom{\vec{L}'} = m\,(\vec{r}'\times\vec{v}')\;\ldots\text{ definition of linear momentum }\vec{p}' \\\ \hphantom{\vec{L}'} = m\,(\vec{r}'\times(\vec{\omega}\times\vec{r}'))\;\ldots\text{ definition of linear velocity in terms of angular velocity} \\\ \hphantom{\vec{L}'} = m\,(\vec{\omega}\,(\vec{r}'\cdot\vec{r}') + \vec{r}'\,(\vec{r}'\cdot\vec{\omega}))\;\ldots\text{ scalar triple product}\\\ \hphantom{\vec{L}'} = m\,(\vec{\omega}\,(r'\,^2) + \vec{r}'\,(r'\,\omega\,\cos(\angle(\vec{r}', \vec{\omega})))) \\\ \hphantom{\vec{L}'} = (m\,r'\,^2)\,\vec{\omega} + (r'\,\omega\,\cos\phi)\,\vec{r}'$ Because the angular momentum $\vec{L}'$ is not conserved due to changing direction, and a changing $\vec{L}'$ requires the presence of an external force to give rise to a net torque $\vec{\tau}'$ about the point that is used to measure $\vec{L}'$, which is the origin $O'$, the particle will not rotate about $P$ forever in the absence of any external force. But then, a contradiction arises: **the same phenomenon has a different outcome depending on the choice of origin $O$ or $O'$!** That is _unacceptable_ in Physics! How to resolve this contradiction?


title: the nature of the big bang

text: If space-time expanded together with matter then why do physicist bother extrapolating backwards the expansion back to a point in time? I mean does that really tell us anything? I mean if the speed of the passage of time (measured by atomic vibrations) is dependant on the local space-time geometry, and the measurement of space is dependant on having 'stuff' (light, matter) in it to measure the distances, then it seems to me that the concept of space, time, its interacting fields, and the resulting quantum actualities are all dependant on each other for their survival as a concept. What explanatory power have we really added to the discussion on the origins of reality if we talk about a finite time periods back to the infinite? How can a finite time period escape from the infinite? Therefore what are we really learning by saying the universe is 14 billion years old? In my ignorance, its like we are applying logic based on newtonian physics to post Einstein physics and walking away happy that we said something sensible when we just got ourselves (and me especially ;) really confused (I'm sure the reality is different, thats why I'm asking the question :)). Furthermore, if everything (within the fabric of nothing) expanded together then how could we prove anything expanded at all? The red-shift must be showing that matter expanded FASTER than the space-time right? Otherwise there would be no measurable difference. How could we ever measure that if space- time is expanding with it? I guess what I'm asking is 3 questions (that are really all the same question); why do we talk about space-time expanding when "expansion" is a term that needs space to make sense. Wouldn't the permission to use that term require our space-time existing in a greater spacial reality? Why do we talk about matter expanding when if the space-time expanded with it then it really didn't expand at all. Why do we talk about a finite time period when the maths shows the passage of time retreats to infinity and therefore we haven't really explained anything.


title: Origin of Elasticity

text: Why is it that not all bodies possess Elastic behavior? What is the origin of elasticity or plasticity? I mean, it's a physical property. So, how does it relate to atoms or molecules in different phases? (It should have some relation with atoms)


title: Wannier functions on a ring

text: Let's say I have a single particle hamiltonian in a periodic potential. For example a 1D lattice such that: $$H = -\frac{\partial_x^2}{2m} + V(x) $$ with $ V(x+a) = V(x)$ where $a$ is the lattice spacing between the atoms or sites. It is known by Bloch's theorem that a solution to such a system will have the form $$\psi_{k}(x)=e^{ikx}u_k(x)$$ where $u(x+a)=u(x)$. My questions is about the boundary conditions. If we take $$\psi(x+Na) = \psi(x)$$ we get, if $N$ is large enough, a lot of different values for $k$ in the Brillouin zone: $$-\frac{\pi}{a}<k<\frac{\pi}{a},$$ so we get a band of possible states. In this case we can define Wannier functions which using Fourier over the wave-functions: $$\phi_k(x-R) = \sum_k e^{-ik R} \psi_k(x)$$ And the summation if over all the $k$'s in the first Brillouin zone. But if I take the B.C $$\psi(x+a) = \psi(x)$$ I get a single value for the momentum in each Brillouin zone $$k = 0, \pm 2\pi , \pm 4\pi,...$$ Is it still possible to define a Wannier function for such a state? I mean what will the Fourier be like if we have a single possible value of $k$??


title: Is anti-gravity possible in theoretical physics?

text: Is anti-gravity possible in string theory? I have read some articles about scientists making assumptions about the existence of anti-gravity, but is it possible in string theory?


title: Solving the BCS Hamiltonian via the Bogoliubov Transformation

text: I was doing a calculation in Giamarchi's Introduction to Many Body Physics, chapter 3, on BCS theory and second quantization, and ran into some confusion with the BCS Hamiltonian. The pdf is here for your reference: http://dpmc.unige.ch/gr_giamarchi/Solides/Files/many-body.pdf The main confusion comes with eqn. 3.154. Here, the BCS Hamiltonian is given by $$ H=\sum_k \left( A(k)(\beta_k^\dagger \beta_k-\alpha_k^\dagger \alpha_k)+\Delta(\alpha_k^\dagger \beta_k+\beta_k^\dagger \alpha_k)\right)+ \sum_k\xi(k) $$ Where $\xi(k)$ and $A(k)$ are functions of $k$ and $\alpha_k$ and $\beta_k$ are fermionic operators. Now, I know that the tight-binding Hamiltonian with a periodic potential is given by eqn. 3.128: $$ H=\sum_{k}\bigg( A(k)(\beta_k^\dagger \beta_k-\alpha_k^\dagger \alpha_k)+V(\alpha_k^\dagger \beta_k+\beta_k^\dagger \alpha_k)\bigg) $$ The solution to this is easy to solve with the Bogoliubov transformation, and is given by $$ E(k)=\sqrt{\xi(k)^2+V^2} $$ I was able to derive this without any problems. However, my question is this: would the solution of the BCS Hamiltonian then be $$ E_{BCS}(k)=\sqrt{\xi(k)^2+V^2}+\xi(k) $$ Or would it be identical to the tight-binding Hamiltonian? Would the eigenvectors as given by the Bogoliubov transformation change as well?


title: Quantum Field Theory cross sections integrals

text: Where can I find some examples of cross sections calculations in QFT done step-by-step? Those integrals are a little horror. For example - a simple scalar+scalar -> scalar+scalar at the tree level in a theory scalar $\phi^4$ ?


title: How electromagnetic fields travel through vacuum?

text: How electromagnetic fields travel through vacuum is it right to say that they propagate or travel? do they alternately form each other by charged particles?


title: 4th equation of motion

text: Ive been given the 4 equations of motions The fourth being: $$s=ut+\frac{1}{2}at^2$$ If rearranged it forms the quadratic equation $$at^2+2ut-2s=0$$ But that means that t has 2 values. Will one of them always be negative. So only one value is realistically possible? And how could you rearrange it to get t on it's own? Would that be using the quadratic formula?


title: How does porous anti-reflection coating work?

text: We know ideally that if we stick a coating of thickness $d = \frac{\lambda}{4}$ with refractive index $n_{coat} = \sqrt{n_{air}n_{glass}} \approx 1.2$ there would be no reflected wave. But it's hard to manufacture these coatings witn $n < 1.4$. Instead, porous anti-reflection coatings are used. How do they ensure low reflection?


title: Notation of plane waves

text: Consider a monochromatic plane wave (I am using bold to represent vectors) $$ \mathbf{E}(\mathbf{r},t) = \mathbf{E}_0(\mathbf{r})e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}, $$ $$ \mathbf{B}(\mathbf{r},t) = \mathbf{B}_0(\mathbf{r})e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}. $$ There are a few ways to simplify this notation. We can use the _complex field_ $$ \tilde{\mathbf{E}}(\mathbf{r},t) = \tilde{\mathbf{E}}_0 e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)} $$ to represent both the electric and magnetic field, where the real part is the electric and the imaginary part is proportional to the magnetic. Often it is useful to just deal with the _complex amplitude_ ($\tilde{\mathbf{E}}_0$) when adding or manipulating fields. However, when you want to coherently add two waves with the same frequency but different propagation directions, you need to take the spatial variation into account, although you can still leave off the time variation. So you are dealing with this quantity: $$ \tilde{\mathbf{E}}_0 e^{i\mathbf{k} \cdot \mathbf{r}} $$ My question is, what is this quantity called? I've been thinking _time- averaged complex field_ , but then again, it's not really time-averaged, is it? _Time-independent_? Also, what is its notation? $\langle\tilde{\mathbf{E}}\rangle$?


title: What keeps electrons in an atom from flying away or falling into the nucleus?

text: In atoms, what force or charge, etc. keeps electrons from flying away or into their nucleus? is there a kind of weak-force at work on the atomic scale? Note I am aware the electron positions are only abstract variables and can be referred to as the electron field and the like. This is not the question. What is the reason an electron is bound to that nucleus to the point it can sustain an "orbit" or variable probability path, and not fly away or into it's nucleus?


title: Can one obtain free energy from the vacuum?

text: It is known that from the vacuum of a quantum field theory, virtual particle pairs are created and destroyed; is it possible to capture these particles thus obtaining free energy from the vacuum?


title: Dimensional Analysis to Determine a Formula

text: The kinetic energy of a particle confined to a spherical region with a uniform internal potential depends on its mass, the radius of the sphere, and the Planck constant. An electron, confined to such a region of radius $1$ nm, has kinetic energy $0.38$ eV. Use dimensional analysis to find the kinetic energy of a proton confined to a region the size of an atomic nucleus (radius ≈ $5 × 10^{−15}$ m). I cannot produce a formula which results in $0.38$ eV as in the example.


title: Commutator summation notation

text: I have the relation $ e^L M e^{-L}=\sum_{n=0}^\infty \frac 1{n!} [L,M]_{(n)}$ where $L$ and $M$ are operators. What does the subscript $n$ after the commutator bracket denote?


title: Spin eigenvalues and eigenvectors problem. Is this the correct way to solve it?

text: An electron is described by the Hamiltonian $ H=\frac{e}{mc}\bar{S}\cdot\bar{B} $ where $\bar{S} =(S_x,S_y,S_z)$ is the spin operator and $\bar{B}$ the magnetic field. For $t>0$ $\bar{B}=B_0\hat{x}$ and for $t=0$ the electron is in the state $|\psi\rangle=\frac{1}{\sqrt3}|+\rangle+\frac{\sqrt2}{\sqrt3}|-\rangle$, with $|+\rangle$ and $|-\rangle$ eigenvectors of the operator $S_z$ (with eigenvalues $\pm \frac{\hbar}{2}$). I want to determine the time evolution of the state. Since $B$ is along x, the dot product gives me $S_xB_0$ and I know that the operator $S_x$ is rapresented by the matrix $ \frac{\hbar}{2}\bigl(\begin{smallmatrix} 0 & 1 \\\1 & 0\end{smallmatrix}\bigr)$. For convenience $\frac{e\hbar B_0}{2mc}=\epsilon$. Now, avoiding all the calculations (hoping I did them right), what I did is to find the eigenvalues and eigenvectors of my operator by setting the equation: $ H(\alpha|+\rangle+\beta|-\rangle)=E(\alpha|+\rangle+\beta|-\rangle) $ From which the eigenvalues are $\epsilon$ and $-\epsilon$, and the respective eigenvectors (determining the relation between $\alpha$ and $\beta$ and normalizing) are: $|\psi_1\rangle=\frac{1}{\sqrt2}|+\rangle+\frac{1}{\sqrt2}|-\rangle$ $|\psi_2\rangle=\frac{1}{\sqrt2}|+\rangle-\frac{1}{\sqrt2}|-\rangle$ Now I need to express the state $|\psi\rangle$ as a combination of the two eigenvectors: $ |\psi\rangle=\frac{\sqrt2+2}{2\sqrt3}|\psi_1\rangle+\frac{\sqrt2-2}{2\sqrt3}|\psi_2\rangle $ And the time evolution is: $ |\psi(t)\rangle=\frac{\sqrt2+2}{2\sqrt3}|\psi_1\rangle \exp[{-i\frac{\epsilon}{\hbar}t}]+\frac{\sqrt2-2}{2\sqrt3}|\psi_2\rangle \exp[{i\frac{\epsilon}{\hbar}t}] $ This is how I solved the problem but since I'm pretty new to the quantum mechanics I'd like to have some opinions. Is this a reasonable procedure? Did I made some terrible mistakes or non-sense considerations?


title: How to calculate speed difference between objects close to the speed of light?

text: If two different objects (for example two rockets) move in opposite direction at close to the speed of light (for example 0.8c and 0.9c), how do I calculate the difference in speed between the two (which in classical physics would be 0.8c + 0.9c)?


title: (Iran's nuclear timetable) How IAEA knows that iran is able to make nuclear bombs in four months?

text: 'Iran could have enough material for bomb within 4 months' actually its within 3 months now. I feel, iran will be able to produce nuclear weapons, given the weakness of Western. because West has no real policy to deal with this application. And they are preparing their selves for the bang


title: Extending mixed states to pure state

text: Let us consider any pure state $|\psi\rangle\in\mathbb{C^n\otimes C^n\otimes C^n}$. Its reduced bipartite density matrix represent a pure state or mixed state depending on whether $|\psi\rangle$ is entangled or not (exactly how it is entangled, on which system we take the partial trace, etc). My question is given any arbitrary (mixed) state $\rho\in\mathcal{B}(\mathbb{C^n\otimes C^n})$, can we find out a pure state $|\psi\rangle\in\mathbb{C^n\otimes C^n\otimes C^n}$ (or in some suitable higher dimension which needs to be determined) such that $\rho$ is the reduced density matrix of $|\psi\rangle$. In particular, I do not want only an existential result, I also want to an algorithmic method to determine such $|\psi\rangle$. Obviously such state will not be unique. Advanced thanks for any help.


title: What is the minimum physical distance between two electrons?

text: If electrons repel, then it seems to me that energy is required to decrease the distance between two electrons. Since energy and mass are equivalent, adding energy to a system also adds some mass. If you confine enough mass/energy in a small enough region, the gravitational force begins to dominate. As two electrons were brought arbitrarily close together, the amount of energy to halve their remaining distance would increase drastically, furthermore, the space within which that energy must be confined would decrease commensurately. It seems obvious to me that this could not continue forever. Eventually the two electrons and the energy needed to bring them into such close proximity would be confined within the corresponding Schwarzschild radius. Even if electrons are not to be considered as point-particles, wouldn't you eventually bump up against pressure from the Pauli exclusion principle, to the same effect? I understand that the uncertainty principle could come to bear quite strongly in a case like this, but I'm really looking for an answer rather than being told my question is wrong, so if you have to, consider it a cloud of some greater number of electrons, to cut to the heart of the question, which I guess is: At what distance would it become impossible to bring N electrons any closer together without producing a gravitational singularity? I would prefer an answer where N=2, despite the fact that such a scenario may not be meaningful, it will help me visualize the general principle behind it. For example, one could treat the electrons as point particles, ignore the uncertainty principle, and just plot a graph of the amount of energy needed for a given separation, and then find where that line intersects with the Schwarzschild radius - which I acknowledge would only answer the question - "Despite it not being possible to bring two electrons to a mutual distance of 10^-49 m, if you were to naively project the math involved, that is the point where escape velocity would equal c." Please forgive my limited understanding, I engage in physics thought experiments recreationally only and have no formal training or background. Please help me enhance my intuition with a concrete solution to the distance at which 2 electrons would need to be, for the energy required to bring them there to exceed the Schwarzschild radius of that energy.


title: While falling into a black hole, what color would the flashlight have?

text: If, hypothetically, me and my rocket powered flashlight were falling straight toward the center of a black hole. The flashlight is a few kilometers behind me in our travels toward the center of the black hole, but since it is rocket powered, it manages to maintain the exact distance to me for a while. The point is; The distance between me and my flashlight is constant as long as I am observing it. The photons coming from the flashlight would obviously not be rocket powered - and they would be affected by the black holes gravitation. Would the light I see from the flashlight be shifted towards red or blue, even though the distance between me and my dear flashlight is maintained? If so; switching the positions of me and my flashlight, would it change the color I'd observe? If we turn off the rocket on the flashlight, I assume it would be redshifted regardless of which were closer to the singularity, and the magnitude of redshift would appear to accelerate?


title: Do photons see shorter distances due to space contraction?

text: I may be wrong, but this is just what I've heard. People often talk about what would happen if we, humans, were able to make a spaceship travel at the speed of light. If I'm not mistaken, then we should not be able to make it out of our own galaxy, since it's millions of light years wide. But, I've heard, that due to the contraction effect, at the speed of light, the distance traveled will actually be much less than what is actually the width of the galaxy, so we should be able to reach the end of the galaxy, and even beyond in a much smaller time span than you would think. Is this right ? And, if that is actually true, does this hold for light as well ? We've heard that light is about 8 minutes to travel from the sun to the earth, but that is not when taking contraction into consideration, is it ? Or is it different with photons that with ordinary objects at the speed of light ? Yeah, a lot of questions, and I might just have misread everything I've ever heard. In that case, sorry :)


title: Find the points where potential is null

text: Let's say we have two charges called $q_1$ and $q_2$, respectively $20 \, C$ and $-40\,C$, at a distance $d=1\,m$ We want to find all the points where electric potential is null. I solved the equation $$\frac{q_1}{4\pi\epsilon_0r_1} + \frac{q_2}{4\pi\epsilon_0(d-r_1)}=0$$ For $r_1$ (distance from $q_1$), and found $r_1=\frac13\,m$ However this is not the only solution: there is another point not in-between the charges, but $1\,m$ left from $q_1$ ($r_1=-1\,m$) How can I set up an equation giving me both the solutions?


title: Representation of phase in quantum mechanics

text: _[Note: My discussion of the three answers can be found just after the question.]_ Imagine three points in space that differ only by a phase angle of "something" (what doesn't really matter). One way to represent the difference in phase angle between the three points is assigning each one a specific angle designation, e.g. $0^{\circ}$, $90^{\circ}$, and $180^{\circ}$, or in the equivalent complex number forms, $1$, $i$, and $-1$. [1] However, assigning phase angles assumes that the starting angle of 0 degrees is a universal constant across the entire space in which such points could exist. For many phenomena, especially in quantum mechanics, it turns out that how you select this universal or global $0^{\circ}$ setting doesn't really matter. The ability to select the $0^{\circ}$ point arbitrarily -- or equivalently, the ability to add any arbitrary angle or phase to each point in the space -- is typically described as a symmetry of physics. If you don't mind exploding the number of mathematical quantities involved, you can instead express the relationships without assuming any global constants to exist. More specifically, you can use a Mach-like approach in which phase angles only exist between _pairs_ of locations in the space. With this approach and assuming transitive phase relationships, you can use $n$ _phase vectors_ (that's a phrase I made up, so don't bother looking it up) to specify all phase relationships between all particles, where $n$ is the number of points (or particles) in space that have phase angles: > $p_1 \rightarrow p_2 = i = 90^{\circ}$ > > $p_1 \rightarrow p_3 = -1 = 180^{\circ}$ Notice that in the above example, setting all phase vectors to start at the same location provides a close equivalent to assuming a universal $0^{\circ}$. The only difference is that here you have _explicitly_ stated which point will serve as the universal $0^{\circ}$ value by referencing all other particles to it. The above two phase vectors fully define the phase relationships of the three points I described earlier, but there are many ways to capture the phase relationships. The full expansion of phase vectors implied by $\\{(p_1=1), (p_2=i), (p_3=-1)\\}$ is: > $\\{(p_1 \rightarrow p_2 = i), (p_2 \rightarrow p_1 = -i),$ > > $(p_2 \rightarrow p_3 = i), (p_3 \rightarrow p_2 = -i),$ > > $(p_1 \rightarrow p_3 = -1), (p_3 \rightarrow p_1 = -1)\\}$ Now, at last, my questions. Feel free to answer any or all, or create your own. Answers of the type "Terry you dunce this is just [...] phrased badly!" are not only welcome but hugely appreciated, if accurate.   1. Is phase symmetry real, or is it just an artifact of using mathematical methods that assume the existence of a universal $0^{\circ}$ value that has no real physical meaning, since phase can only be measured by using _pairs_ of locations?   2. Phase arrows have their own symmetry, since any set of $n-1$ phase arrows [2] that fully connects $n$ particles can be used to represent their phase relationships. So, do phase arrows just obscure phase symmetry by representing it in an oddly particle-centric way?   3. Or is it the other way around: Is the equivalence of all fully connected sets of phase arrows the deeper symmetry, since it reflects an entanglement-like preservation of phase across measurements of many, many particles?   4. If phase symmetry is just an artifact of assuming a global $0^{\circ}$, what does that imply for expressing electromagnetism as a phase symmetry? What would change? * * * **2012-09-02 - Assessment of the three answers** All three answers were useful and as best I can discern, accurate. I have selected @NickKidman's as "the" answer mostly due to the nice link to the Wikipedia page on heaps, which most succinctly addressed my underlying question of why it seems possible to represent such issues in an affine-like way. But the @Kostya answer was also very nice from a somewhat different mathematical perspective, and I appreciated the answering of each of the four sub-questions. Finally, I think @RonMaimon best caught the important difference in just _how_ phase symmetry is used in gauge theory. So, my recommendation to anyone coming across this question is that you should go carefully through all three of the answers provides as of 2012-09-02. Each adds value. My thanks to all for such excellent answers, and my apologies that I forgot about not closing this one out earlier. * * * [1] This note is for any readers who may not familiar with complex-plane phase angles. You may be wondering why @Kostya's complex-plane angle notation looks so different from my assertion that $1$, $i$, and $-1$ represent "angles." We really are talking about the same thing, I assure you! Angles are very often represented and calculated in physics disciplines using a compact exponential notation with remarkable computational properties. That notation works like this: If you raise the mathematical constant $e$ ($\approx 2.71828$) to an _imaginary_ exponent -- and yes, once upon a time and many years ago, a lot of _deep_ mathematical thinking went into figuring out what it even means to raise a number to an imaginary exponent -- then a most curious relationship emerges. The first part of what happens is the magnitude $\phi$ of the imaginary exponent -- that is, the real number by which $i$ is multiplied in the exponent -- ends up specifying a complex value that is always one unit away from the origin ($0$) of the complex plane. Next, if you interpret the origin $0$ of the complex plane as the vertex of an angle, and the line from $0$ to $1$ as one side of that angle, then the line from $0$ to the unit-length complex value generated by $e^{i\phi}$ will define the other side of that angle. To make that work correctly, the angle $\phi$ must be expressed in units of radians rather than degrees, where $180^\circ = \pi$ (radians). (Side note: The truly amazing relationship between $e$, $i$, $\pi$, and $-1$ that this relationship produces is known as Euler's Identity.) So, if you start out with a _real_ angle $\phi=\pi/2=90^\circ$, then what the function $e^{i\phi}$ does is "translate" that somewhat abstract real angle into a tangible and easily visualized angle formed by the complex-plane vertex $0$, the unit value $1$ (which defines the "always assumed" horizontal side of the angle), and the output of $e^{i\phi}$, which in this case would be $i$. Since $i$ is straight up from the vertex $0$, the resulting angle is $90^\circ$. So, I sort of cut to the chase and spoke only of a few carefully selected examples of how $1=0^\circ$, $i=90^\circ$, $-1=180^\circ$, and $-i=270^\circ$ show the _potential_ for complex numbers to represent angles. That potential is realized in the $e^{i\phi}$, which enables unit-circle complex angles to be represented simply and elegantly. Still, using complex numbers alone for angles has advantages too. In particular, simply _multiplying_ two unit complex numbers is the same as _adding_ two angles -- a neat trick indeed, and a lot faster than doing a pile of trig functions! It also provides a different way of understanding why $i^2=-1$. If those numbers are instead interpreted as angles, you are simply adding two $90^\circ$ angles (represented by the two $i$ values) to produce a new angle of $180^\circ$ (the -1 result). [2] Corrected from $n$; thanks @NickKidman!


title: Is there some connection between the Virial theorem and a least action principle?

text: Both involve some 'averaging' over energies (kinetic and potential) and make some prediction about their mean values. As far as the least action principles, one could think of them as saying that the actual path is one that makes an equipartition between the two kinds of energies...


title: Can I study Quantum Computing or Quantum Mechanics with an Engineering background?

text: I am currently studying Electrical & Electronic Engineering. I wish to pursue Quantum Mechanics or Quantum Computing as my research subject. Is it possible for me to do my M.Tech. and then pursue my research subject? What are the prerequisites for studying these subjects? I would be grateful if you could help me.


title: Why is a 1mW laser dangerous?

text: In our Physics lab we have a 1 milliwatt (0.001W) helium neon laser. Despite the low power, we were cautioned not to even look at reflections of the beam as it could cause permanent eye damage - why is such a low power level hazardous? I have a 3W LED which I can look directly into which is more than 3,000x more powerful.


title: Maximising entropy when energy is shared between systems

text: ![enter image description here](http://i.stack.imgur.com/9kYAR.png) This is a problem to do with statistical physics, and the exchange of energy when we have two microcanonical ensemble. I don't understand why there should be a minus sign in the middle, if Energy* is the energy of system 1 , Energy total - E* is the energy of another system, Two system were put together and interact, I was told this equation can maximise entropy, but I can't really see why? I thought S = (S1(E*))+S(E total - E*) is the total entropy. therefore it should be a positive sign when you differentiate.. Just to clarify, S(E*) and S(E total - E*) , they are just labels, not function of E* and E total - E*


title: In the Wheeler-de Witt formalism, do big bangs act as sources and future singularities as sinks?

text: In the Wheeler-De Witt formalism of quantum gravity, do big bangs act as sources and future singularities as sinks? A finite time ago, existence emerged from the big bang, and existence will fade in a finite time for future singularities. Evolving WDW solutions forward in time leads to no change for parts of the wavefunctional away from singularities, but does it act as a sink for future singularities leading to a _reduction_ in the norm of the wavefunctional? Do big bangs act as a wavefunctional source increasing its norm over time? Then the best we can hope for WDW solutions is a steady state under extended hamiltonian time evolution with "wavefunctional probability density" created at the big bang and ending at future singularities?


title: Exact energies of spherical harmonic oscillator in Dirac equation

text: The potential is given by: $$ V(r) = {1\over 2} \omega^2 r^2 $$ and we are solving the radial Dirac equation (in atomic units): $$ c{d P(r)\over d r} + c {\kappa\over r} P(r) + Q(r) (V(r)-2mc^2) = E Q(r) $$ $$ -c{d Q(r)\over d r} + c {\kappa\over r} Q(r) + P(r) V(r) = E P(r) $$ What is the analytic expression for the eigenvalues $E$ in atomic units? It is ok to provide source code (any language) to obtain it if one needs to solve some simple analytic equation. Here are the (I think correct) energies from my numerical code that I would like to compare against analytic solution (for $c = 137.03599907$ and $\omega=1$):                 n  l  k kappa        E            1  0  0 -1       1.49999501       2  0  0 -1       3.49989517       2  1  0 -2       2.49997504       2  1  1  1       2.49993510       3  0  0 -1       5.49971547       3  1  0 -2       4.49983527       3  1  1  1       4.49979534       3  2  0 -3       3.49994176       3  2  1  2       3.49987520       4  0  0 -1       7.49945592       4  1  0 -2       6.49961564       4  1  1  1       6.49957571       4  2  0 -3       5.49976206      If one only solves the radial Schroedinger equation, then the analytic formula is $$E_{nl} = \omega (2n - l - {1\over 2})$$ I am looking for the relativistic version. I found for example the paper: Qiang Wen-Chao: Bound states of the Klein- Gordon and Dirac equations for scalar and vector harmonic oscillator potentials. Vol 11, No 8, 2002, Chin. Phys. Soc., but it only shows a formula for nonzero scalar and vector potentials in Dirac equation (above we only have the scalar potential, the vector potential is zero).


title: Angular momentum and EM wave

text: Is there any sense in saying that circularly polarized EM waves have angular momentum?