title: Has hep-th/0312070 forgotten to fix $s_{0} = 1/2$ for the fermionic states in the second table on page 52?

text: Link to the original paper: The Gauge/String Correspondence Towards Realistic Gauge Theories (arXiv paper) On page 52 we see that, for a theory of Dp-branes placed at an orbifold (orbifold = $C_{2}$/$Z2$) fixed point, we have a certain spectrum of open string NS and R states. The first thing to note is that the orbifold action has the effect of making the spinor labels behave as follows: $s_{3} = s_{4}$ or $s_{3} = -s_{4}$. This means the 3rd and 4th components are no longer independent. Since $s_{i}$ equals $1/2$ or $-1/2$, this means there are now 16 different combinations of the state $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$ corresponding to the different 16 permutations of the values of $s_{0}, s_{1}, s_{2},s_{3} = -1/2,+1/2$. We would have had 32 different permutations corresponding to $s_{0}, s_{1}, s_{2},s_{3}, s_{4} = -1/2,+1/2$ if $s_{4}$ wasn't fixed by the value of $s_{3}$. However, this isn't the end of it. We must also take a GSO projection which projects out half the states. This leaves 8 states (8 permutations) for each $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$. So far so good; we now have 2 different types of Chan-Paton factor in front of our $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$. Thus, we must consider two types of $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$. This is why each row of the second table on page 52 has 2 states with different Pauli matrices in front. Each of these 2 states contributes 8 permutations so we have 16 fermions for each row of the table. But this is where I threw a spanner into the problem. For the _physical_ Ramond sector states, don't we normally fix $s_{0}$ to equal 1/2. This is the case for flat spacetime and is explained on page 23 of Notes on D-branes. So now we have $s_{0}$ and $s_{4}$ fixed leaving permutations of $s_{1}, s_{2}, s_{3} = 1/2, -1/2$. This would give us 8 states per $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$ and after GSO projecting half the states out we'd be left with 4 per $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$. The result: I'd expect 8 fermions per row of the 2nd table on page 52 of our original source. My interpretation seems to be supported by page 30 of UV/IR Mixing in Noncommutative Field Theories and Open Closed String Duality  which says we have 4 copies of each spinor. Note: It ignores $s_{0}$. I'm guessing they do this as it is fixed by the physicality condition. Sorry for the length of my description and thanks in advance.


title: Deriving the Lagrangian for a free particle

text: I'm a newbie in physics. Sorry, if the following questions are dumb. I began reading "Mechanics" by Landau and Lifshitz recently and hit a few roadblocks right away.   1. **Proving that a free particle moves with a constant velocity in an inertial frame of reference** ($\S$3. Galileo's relativity principle). The proof begins with explaining that the Lagrangian must only depend on the speed of the particle ($v^2={\bf v}^2$): $$L=L(v^2).$$ Hence the Lagrance's equations will be $$\frac{d}{dt}\left(\frac{\partial L}{\partial {\bf v}}\right)=0,$$ so $$\frac{\partial L}{\partial {\bf v}}=\text{constant}.$$ And this is where the authors say > Since $\partial L/\partial \bf v$ is a function of the velocity only, it > follows that $${\bf v}=\text{constant}.$$ **Why so?** I can put $L=\|{\bf v}\|=\sqrt{v^2_x+v^2_y+v^2_z}$. Then $$\frac{\partial L}{\partial {\bf v}}=\frac{2}{\sqrt{v^2_x+v^2_y+v^2_z}}\begin{pmatrix} v_x \\\ v_y \\\ v_z \end{pmatrix},$$ which will remain a constant vector $\begin{pmatrix} 2 \\\ 0 \\\ 0 \end{pmatrix}$ as the particle moves with an arbitrary non-constant positive $v_x$ and $v_y=v_z=0$. Where am I wrong here? If I am, how does one prove the quoted statement?   2. **Proving that** $L=\frac{m v^2}2$ ($\S$4. The Lagrangian for a free particle). The authors consider an inertial frame of reference $K$ moving with a velocity ${\bf\epsilon}$ relative to another frame of reference $K'$, so ${\bf v'=v+\epsilon}$. Here is what troubles me: > Since the equations of motion must have **same form** in every frame, the > Lagrangian $L(v^2)$ must be converted by this transformation into a function > $L'$ which differs from $L(v^2)$, if at all, **only by the total time > derivative of a function of coordinates and time** (see the end of $\S$2). First of all, what does **same form** mean? I think the equations should be the same, but if I'm right, why wouldn't the authors write so? Second, it was shown in $\S$2 that adding a total derivative will not change the equations. There was nothing about total derivatives of time and coordinates being **the only functions** , adding which does not change the equations (or their _form_ , whatever it means). Where am I wrong now? If I'm not, how does one prove the quoted statement and why haven't the authors done it? * * * P. S. Could you recommend any textbooks on analytical mechanics? I'm not very excited with this one. Seems to hard for me.


title: Landau's argument for dependence of Lagrangian on magnitude of velocity

text: In chapter 1, of Landau-Lifshitz Mechanics' book, Landau through isotropy and homogeneity of space and homogeneity of time proves that the Lagrangian must depend of magnitude of velocity of the particle. This seemed fine. But he didn't give a reason as to why it must depend of $|v|^2$ and not on some $|v|^n$ where $n\neq2$.


title: How does isotropy of free space imply $L(v^2)$ for a free particle?

text: From Mechanics; Landau and Lifshitz, it's stated on page 5: > Since space is isotropic, the Lagrangian must also be indpendent of the > direction of $ \mathbf{v}$, and is therfore a function only of its > magnitudie, i.e. of $ \mathbf{v} \cdot \mathbf{v}=v^2$: $L = L(v^2)$ How is he able to exlude $L = L(| \mathbf{v}|)$?


title: What if the binding energy becomes larger than the rest mass?

text: Looking at the equation for binding energy and mass defect,   $$ B = m_{\text{free}} - m_{\text{bound}} \\\ \Rightarrow m_{\text{bound}} = m_{\text{free}} - B, $$ my question is the following. Suppose that the binding energy $B$ becomes larger than the free rest mass $m_{\text{free}}$. It would seem that the mass of the bounded system $m_{\text{bound}}$ can become negative. As this is impossible there must be something wrong in my assumptions but I don't see it. Take for example the hydrogen atom, the binding energy for the ground state is $13.6 \text{ eV}$. What would happen to the mass of the hydrogen atom if hypothetically the binding energy would become larger than $ \approx 938.783 \text{MeV} $ (which is just the sum of the rest masses of the proton and the electron) by, for example, increasing the EM-coupling constant.


title: How much energy is needed to curve the space so the Sun will come 1 meter closer to our planet?

text: How much energy/work will it take if our planet is:               5.9736×1024 kg      or               3.0×10−6 Suns      to move Sun and Earth 1 meter closer to each other?


title: Is sun revolving around certain mass in space with reference to some other galaxy?

text: Every planet and satellite revolves around their mother planet and about their own axis as we know. So they are compacted as a system. Then why not sun?


What does the height refer to in the Bernoulli equation? I'm trying to determine the pressure on a surface where the length is short enough that the velocity should be constant across it.               Fan -> -8_________________                     ^-- surface      So, the idea is that there is air forced over the top of the surface and the air below it is not moving. How do I calculate this? I've seen this equation: $$ P_1 + {1 \over 2} \rho v_1^2 + \rho g y_1 = P_2 + {1 \over 2} \rho v_2^2 + \rho g y_2 $$ where: $P$ is the pressure of the fluid, $\rho$ is the fluid density, $g$ is the local gravitational acceleration, $v$ is the fluid's speed and $y$ is the height of the fluid, which seems to be what I am looking for, but I'm not sure what height relates to.


title: What really makes a rainbow happen?

text: Yes, a rainbow is formed when a raindrop refracts light, but why then does the whole sky not become a huge rainbow when it rains? Would the light not be dispersed into ordinary 'white' light? What forces it to look as if each end is nearly touching the ground?


title: Commutators involving functions

text: I am looking for the commutator: $$[e^{aq},p]$$ My approach is to Taylor expand the function: $$[\sum_n \frac{1}{n!}(aq)^n,p]$$ I know that $[q^n,p]=ni\hbar q^{n-1}$ So how do I account for $n$ commutators?


title: Are Stokes' theorem and Gauss's theorem examples of the Holographic Principle?

text: Before I write this question, I'd want to say that I've read this question , and Lubos Motl's answer to it (I found it through the "Questions that may already have your answer"). My question isn't exactly that. I'm asking whether Stokes' theorem and Gauss's theorem are _Examples_ of the Holographic principle . My impression is that it is, since Stokes' theorem, for example, in it's all-intiuitive most general sense, tells us that: \begin{equation} \int\limits_{\partial\Omega}\omega = \int\limits_{\Omega}\mathrm{d}\omega. \end{equation} In other words, it relates something (the RHS) on the region to something (the LHS) on its boundary. So, I had written a blog post about that to summarise my thoughts on Holography and AdS/CFT. However, Mitchell Porter corrected me saying that it really isn't. So, I just need to confirm whether it is at least an example (of courese _not_ the basis) for Holography ? .


title: Why does light not refract when incidented perpendicularly?

text: I had read that light does slow down in glass because photons interact with atoms in glass. They are absorbed and re-emitted and during this phenomenon it's speed decreases. See also this and this Phys.SE post and links therein. But then why does it bend? It has confused me a lot! Furthermore i have very little knowledge of Optics or Physics as a whole.


title: Plano-convex lenses

text: When the spherical side of a planoconvex lens is towards the source we will see reduced aberrations normally we use it for the beam collimation. When we use two planoconvex lenses configuration why we dont use back to back oriented like a single planoconvex? why do we use them as in the second picture below. What will happen if we will use as in the first picture ? Is it only to reduce aberrations ? In practice where do we use the configuration in second picture ? ![enter image description here](http://i.stack.imgur.com/T1xX0.png)


title: Layer thickness measurement

text: I am interested to measure the thickness of the coating on a mirror (e.g. in the picture below). What are the available and best way of measuring it? What will be the solution in case of multiple layer coating on the same mirror how we could measure the thickness of all layers? ![enter image description here](http://i.stack.imgur.com/lKvzR.jpg)


title: What are the restrictions on the Hamiltonian in QM?

text: In quantum mechanics, we usually write the Hamiltonian as: $$\hat{H}=\hat{T}+\hat{V}$$ But in classical mechanics, there are several reasons why it **would not have this form** :   * We've chosen some particular coordinate system where there is no conservation of energy (we're neglecting the source of external forces).   * We can modify the Hamiltonian without changing the equations of motion: The Lagrangian can be written as: $L=\frac{1}{3}T^2+2TV-V^2$. The Hamiltonian would be: $\hat{H}=p\left[\frac{\left(\sqrt{9p^2m^8 +32m^9V^3}+3pm^4\right)^{1/3}}{\sqrt[3]{2}m^2}-\frac{2\sqrt[3]{2}mV}{\left(\sqrt{9p^2m^8 +32m^9V^3}+3pm^4\right)}\right]-L=(T+V)^2$ The eigenvalues of this operator are $E^2$, not the energy. **Do we consider this possibilities in quantum mechanics?** In the negative case, why do we discard them? And (besides the necessity to be self-adjoint and bounded from below) does the Hamiltonian have some restriction on the operators it includes (position, angular momentum,...)?


title: What happens when two high-speed objects going opposite directions crash into a black hole?

text: Imagine the following:               A->  []  <-B      where [] is a black hole, and A and B are two objects of same mass, traveling with the same speed right at the black hole. A and B have immense kinetic energy. When they crash into the black hole, the black hole does not feel any net force; thus it will not accelerate to bring away the kinetic energy. Nor would it heat up, due to the no-hair theorem dictating that black holes don't have a "temperature". Neither would angular momentum change since we assume A and B go absolutely straight towards the black hole. Where does the energy go? Does it get converted to mass and added into the black hole mass? If so, what is the exact mechanism? Obviously "conservation of energy", but in each case there is a more low-level description (say, when A and B crash into a solid object, the heating comes from the kinetic energy of the objects agitating the atoms making up the object).


title: Is Information a potential or kinetic kind of energy?

text: It is said that _the law of least action is that nature tries to convert potential energy into kinetic one as fast as possible._ Information can't be thought without a physical realisation, see here. It may be thought as a form of entropy. So information is physical and therefore it's connected to a certain kind of energy. But is it a potential or a kinetic kind of energy?


title: How to treat differentials and infinitesimals?

text: In my Calculus class, my math teacher said that differentials such as $dx$ are not numbers, and should not be treated as such. In my physics class, it seems like we treat differentials exactly like numbers, and my physics teacher even said that they are in essence very small numbers. Can someone give me an explanation which satisfies both classes, or do I just have to accept that the differentials are treated differently in different courses? P.S. I took Calculus 2 so please try to keep the answers around that level. P.S.S. Feel free to edit the tags if you think it is appropriate.


title: Ionizing an atom with a proton

text: What happens when a proton collide with an atom? How does the ionization occurs, assuming the proton has enough kinetic energy? How much energy is required for the proton to enter the atomic nucleus, resulting a higher chemical element ion?


title: Born rule and Feynman propagators

text: Let us assume that we want to describe the full process of photon emission by electron A and absorption by electron B. Therefore electron B must be on the forward lightcone of electron A. In the normal forwards in time description a virtual photon propagates from A to B depositing a certain amount of energy and momentum onto electron B. But does this process alone also describe the recoil of electron A? Should one also include the backwards-in-time virtual photon which propagates from B to A depositing an equal amount of negative energy and momentum back onto electron A? Between measurements no-one is "looking" at a quantum system so in the spirit of "what isn't prohibited is compulsory" maybe it evolves both forwards and backwards in time. One could say that the product of the forward-in-time and backward-in-time propagators give the full probability of photon emission by A and absorption by B. Is this where the Born rule comes from (as in John Cramer's TIQM)?


title: Inflation problem for exponential potential

text: I'm trying to solve the inflation problem for exponential potential. $$v(\phi) = v_0 \exp(-\alpha \phi)$$ (it's known as barrow or power law inflation) we have two main equations: $$H^2 = 8π G / 3 (1/2 (\dot{\phi})^2 + v(\phi))$$ $$\ddot{\phi} + 3H \dot{\phi} + v(\phi)'=0$$ I must solve these two equations and find $\phi(t)$ & H(Hubble). in the book of cosmology by Weinberg has written,it is easy.many articles have mentioned it;but I can't do it.it has exact solution.answer is: $$\phi(t)=\phi_0 \ln t/ \ln t_0$$ and $$R(t)=R_0 (t/t_0)^b$$ with suitable constant b,$\phi_0$,$R_0$ . Should I use slow-roll condition for solve that? ($\dot{\phi}^2 \ll v(\phi)$ and $\ddot{\phi} \ll v(\phi)'$) or without that I can?


title: Is it possible to express "free"-ness of a time-like world line without referring to "tangent space" (but only directly to causal relations )?

text: I don't know much about _tangent spaces_ , or _tangent vectors_ , "as such"; nor about _affine parametrization_ (which seems to be closely related to the notion of _tangent vectors_ , as far as I understand for instance MTW, Box 10.2, section B). Is it possible to explain/express whether some particular _identifiable point_ (cmp. MTW, Box. 13.1) had been " _free_ " (or had " _moved freely_ "; or was " _in free fall_ "; or was represented by a " _geodesic_ "; etc.) without explicitly using the notion of _tangent vector_ or _affine parametrization_? I'd be especially looking for such a description being given explicitly and exclusively in terms of particular _identifiable points_ and coincidence events in which they took part (or also, which _identifiable points_ didn't take part in some particular coincidence event of other participants); or (equivalently, as far as I understand) in terms of whether coindicence events under consideration are _time-like_ or _space-like_ related to each other (or neither, i.e. _light-like_ ). (I have already put forth some related attempts here or there. But perhaps this rephrased question helps to focus the effort ...)


title: Mass exchange between stars

text: I'm curious, are there specific conditions for a transfer of mass between two stars as they pass near each other? Its for my final project in computing, I'm writing a gravity simulation and thought it'd be a nice touch.


title: Smallest possible object made of atoms that is spherical to a given degree

text: I recently saw a video, World's Roundest Object, which discussed a sphere made of silicon-28. The researcher in the video stated, if the object was scaled to the size of the Earth, the highest and lowest points would only differ by 14 meters in height. **How small can an object be while still being round?** If we need to define _roundness_ , lets compare it to the silicon-28 sphere when both objects are scaled to the size of the Earth. A single atom is boring, so lets exclude that. Two atoms doesn't seem round. Maybe 16? 300? Which atom(s)?


title: Liouville's theorem on integrable Hamiltonian systems is a let down

text: I read a proof of Liouville’s theorem on integrable Hamiltonian systems. According to the theorem, an autonomous Hamiltonian system can be integrated in quadratures, given $n$ involutive first integrals $F_j(q_1\dots q_n,p_1\dots p_n)=a_j,\quad j=1\dots n$, if the Jacobian $\frac{D(F_1,\dots,F_n)}{D(p_1,\dots,p_n)}$ is non zero. Since $\frac{D(F_1,\dots,F_n)}{D(p_1,\dots,p_n)}\not=0$, we can solve the system $F_j(q_1\dots q_n,p_1\dots p_n)=a_j,\quad j=1\dots n$ for $p$. That is $p_j=f_j(q_1,\dots,q_n,a_1,\dots,a_n)$. According to the proof $\sum_{j=1}^nf_jdq_j$ is an exact form and its potential function is Hamilton's characteristic function for the Hamiltonian system. If we wanted to show existence of the solution, we may have referred to the general formula for existence of solutions to ordinary differential equations. It is understandable then why I perceive Liouville's theorem as a bit of a let down, for the proof is still not constructive. It only reduces the problem to integrating the above differential form. What are the theorem's implications? How is it useful? Can we always find a potential function?


title: Quantization of strings, string Fock space and transition to QFT

text: I am not an expert of string theory and am quite uncertain about the basic ideas of string theory that I am going to ask about. I would appreciate some hints of more experienced physicists. What I am trying to understand is how string theory can describe particles as quantum fields as in quantum field theory. This should be possible since, after all, it is said that QFT can be seen as a low energy limit of string theory. What I would like to do is to compare second quantization in QFT to string theory. In QFT, if the wave function of a quantum mechanical particle can be described as a superposition of a complete set of wave functions $\\{\phi_i(x)\\}$, then we can define creation (annihilation) $a_i^+$($a_i$) operators which generate symmetrized/antisymmetrized products of corresponding one-particles states. So, by this construction we have for example $|100\dots\rangle = a_1^+|0\rangle = |\phi_1\rangle$ where again $\langle x|\phi_1\rangle= \phi_1(x)$. The commutation relations for $a_i^+$ and $a_i$ follow by construction, this is then called canonical quantization. Finally, changing the basis from $\\{|\phi_i\rangle\\}$ to $\\{|x\rangle\\}$ we obtain the field operators $\psi(x) = \sum_i\phi_i(x)a_i$ and $\psi(x)^+ = \sum_i\phi_i^*(x)a_i^+$, such that $\psi(x)^+|0\rangle = |x\rangle = \sum_i\phi_i^*(x)|\phi_i\rangle$. So, to sum up, field operators $\psi(x)^+$ create superposition states with a probability distribution which is equal to a delta function: $$ \langle x'|x\rangle = \sum_i\psi_i(x')\psi_i^*(x) = \delta(x-x')$$ And again, commutation relations for field operators follow by definition. In string theory, the coordinates $(\sigma,\tau)$ on the world sheet which parametrize the embedding $X^\mu(\sigma,\tau)$ of the string in spacetime play the role of spacetime coordinates $(x,t)$ in QFT, and the embedded string $X^\mu(\sigma,\tau)$ plays the same role as field operators $\psi(x)$, and with these identifications the quantization is done along the same lines. But with this construction it is not clear to me, what these operators $X^\mu$ really represent. If in QFT $\psi(x)|0\rangle$ was a localized wave function of a one-particle state, what is $X^\mu(\sigma,\tau)|0\rangle$? A wave- function localized on the world-sheet? How can we then identify something that lives on some fictitions two-dimensional parameter space $(\sigma,\tau)$ with particles in 4-dimensional space-time? And what is the Fock space and the states therein in string theory? If someone tells me that something like $\alpha^\mu|0\rangle$ ($\alpha^\mu$ being the modes of $X^\mu$) can for example be seen as a photon state, the only thing that I see in common with the photon is the vector index $\mu$. Photons that I know are bosons described by a probability distribution over the space-time. In this sense, how can I make the identification with $\alpha^\mu|0\rangle$? I would very much appreciate any help to disentangle these ideas!


title: The inertia tensor is "global", how do I make it local?

text: A solid cuboid has the following moment of inertia: $$I= \begin{pmatrix} \frac{1}{12}h^2+d^2 & 0 & 0 \\\ 0 & \frac{1}{12}w^2+d^2 & 0 \\\ 0 & 0 & \frac{1}{12}w^2+h^2 \end{pmatrix} $$ When integrating the torque to the angular velocity, as far as I know, this holds: $$v_a=\int I^{-1} T dt$$ Where $T$ is the torque and $t$ is the time. However, this seems to only hold when the orientation is the identity orientation. How do I extend this onto an arbitrary orientation? Do I need to make the inertia tensor local in some way? Note that the angular velocity $v_a$ is "global" (!)


title: Why can't we see things swallowed by black holes?

text: Apologies in advance, I'm a layman with only a school-level education in physics. If an object approaching the event horizon of a black hole has its light cone progressively bent towards the black hole's singularity, why can all observers in the future not see the light bouncing off of it? In a recent UK TV documentary _The Science of Doctor Who_ , Professor Brian Cox talked about the above and explained that the future light cone of an object would get bent towards a black hole's singularity. In the example he gave, he suggested that an object venturing towards the event horizon would leave a permanent image. If that's the case, why aren't all black holes masked by photos bouncing off of 'swallowed' objects to all possible future moments? My mental model (which I'm sure is wrong) see this a bit like a laser being shined at a mirror rotating from a perpendicular angle through 90 degrees to an orthogonal one - as the mirror moves in a continuous fashion, the laser should be bounced back to all points in the future. This assumes that the flow of photos in the laser is also continuous, and not made up of discrete packets very close together such that they seem continuous. Hopefully that made sense, and many thanks to anyone kind enough to take the time to answer.


title: Experimentally Verifying a Clock's Accuracy

text: So recently one of my professors went off on a tangent, and we ended up discussing atomic clocks and how they work, which is something I've always been fascinated with and thoroughly enjoyed. But it left me with a question he couldn't answer. After you've built an atomic clock (or any clock for that matter), how do you verify that it is, in fact, working at a specified level of accuracy? Let me be clear here. I understand that by knowing the properties of the atoms and the lasers and all the other parts, you can calculate what frequency it _should_ work at, but once it is working, how do you verify that it is working at that level, **_without_** needing a more accurate clock? Once again, to be clear:               I'm NOT asking about how they build/design the clock.     I'm NOT asking about how they calculate the theoretical values.      I want to know how you can experimentally prove that a clock is working at its theoretical accuracy (or within an acceptable level of accuracy).


title: Torque required to rotate panel on one axis

text: I am working on a project that requires solar panel to rotate around its axis, but I am struck with deciding the specs of the Servo Motor that is capable of doing this. Though, the panel hangs on its axis with help of two axle rods on each side attached almost in the mid of the length, and the servo is to be attached on one of the sides. The specification of the panel is LBH - 1665 x 1000 x 50 mm and weighs around 24kgs. I suppose it should not require that much toruqe to rotate the same, but can someone help me deciding the minimum torque needed to do that. Please consider providing an equation so that I can use it for deciding on panels of different dimensions.


title: Dealing with experimental data

text: I have some experimental data about a value $n$, now, I am supposed to give, in the ending, a single value with an error: $n=a\pm b $. I have originally 6 values of $n$, each one comes as an indirect measurement from direct measurement, each one with it's systematic errors, so in the ending I have those 6 values, each one with an error. So what I guess I have to do is to mix the systematic error with the random error the way I've been taught $(E_{sys}^2+E_{rand}^2)^{1/2}$. The systematic is already calculated, what do I use for the error? the mean of systematic errors? Another question is that those values, which are by the way moles of a quantity of a gas, have been got from different ways (basically from calculating different isothermic curves and getting the $n$ value that best fits each of them), so they're actually not from the same kind of measurement, but from different ones. This makes me doubt about how this would affect the calculation of the final error, if it does, or if I can just do it the way I said above.


title: Modeling a two-mass, spring, damper system

text: I'm trying to model a system with two masses, two springs, two dampers, and one applied force using transfer functions. I'll then be inputting it into Simulink. The system looks like this but there is a force applied to the right edge of ${ m }_{ 2 }$ pointing towards the right. I already found the two differential equations of the system. Then I found one transfer function from each differential equation and ended up with these: $\frac { { U }_{ 1 }(s) }{ { U }_{ 2 }(s) }$ and $\frac { { U }_{ 1 }(s){ U }_{ 2 }(s) }{ F(s) } $ I'm a little unsure of how to proceed from here. The end result should be the response of ${ U }_{ 1 }$ and ${ U }_{ 2 }$. I haven't included the exact equations because I don't really want an exact answer (I'll never learn that way!). If I've left out any information please let me know. Thanks for any help! EDIT: I guess the general question here is, how do you model this type of system using transfer functions? EDIT 2: Here are my differential equations. Note that my variables differ slightly from the linked image. ${ \ddot { x } }_{ 1 }=\frac { { k }_{ 12 } }{ { m }_{ 1 } } \left( { x }_{ 2 }-{ x }_{ 1 } \right) +\frac { { d }_{ 12 } }{ { m }_{ 1 } } \left( { \dot { x } }_{ 2 }-{ \dot { x } }_{ 1 } \right) -\frac { { k }_{ 1 } }{ { m }_{ 1 } } { x }_{ 1 }-\frac { { d }_{ 1 } }{ { m }_{ 1 } } { \dot { x } }_{ 1 }$ ${ \ddot { x } }_{ 2 }=\frac { f(t) }{ { m }_{ 2 } } -\frac { { k }_{ 12 } }{ { m }_{ 2 } } \left( { x }_{ 2 }-{ x }_{ 1 } \right) -\frac { { d }_{ 12 } }{ { m }_{ 2 } } \left( { \dot { x } }_{ 2 }-{ \dot { x } }_{ 1 } \right) $


title: Is there an equivalent of a Galton box for a converging probability?

text: This is a question about probability. The Galton box (or quincunx) uses the physical process of shot moving down a pin-board, to demonstrate central limit theorem, eg: ![quincunx](http://i.stack.imgur.com/OQQXg.png) So I am interested in events with converging probabilities (like a coin toss @ 1/2, or card-guessing @ 1/5) and have found this matrix from a paper by E.G. Boring (Boring, E. G. (1941). Statistical frequencies as dynamic equilibria. Psychological Review, 48(4), 279): ![matrix](http://i.stack.imgur.com/VD2Nk.png) And have a stackoverflow question plotting a related graphic in R: ![heatmap](http://i.stack.imgur.com/up6E5.png) If we think of the heatmap as depicting the shape of a gradient, it seems to me that it is possible to imagine a physical board that contains the same gradient, such that shot put on it would follow the same path. **So my question is, is there an equivalent of the Galton-box, that would allow a physical demonstration of this (and similar) Boring matrices?** Update: I have found this graphic of a quincunx which reminds me very much of the heatmap: ![quincunx](http://i.stack.imgur.com/2ccnt.jpg)


title: Explanation of energy levels in molecules, atoms, nuclei and their relationship

text: Why are the energy levels of molecules, the atoms that form them and the nuclei inside the atoms considered separately? Or phrased in a different way- what is it that makes their energy levels so different? What is the relationship between the energy levels of molecules, atoms and nuclei? I'd interested to learn different interpretations on this question.


title: Superspace Uncertainty Principle

text: Do the "operator for translations in superspace" and the "position in superspace operator" follow an uncertainty principle? How "real" is superspace? Aside from being weird (and possibly just a "mathematical trick") is superspace really just an extension of space on the same footing as real space (with off-shell degrees of freedom)?


title: The System and the Measuring Gadget

text: In Quantum Mechanics the value of an observable results from the interaction between the "system" with the "Measuring gadget". But when the experimenter[or the technologist concerned] is devising/constructing the gadget he has a view the "system" with its stand alone properties/attributes.It seems to be so. But theory itself cannot ascribe the value of some observable to the system alone. How does one explain this?


title: Is 't Hooft's Determinism based on the holographic principle?

text: Does 't Hooft's determinism work need the holographic principle in order to work or is it just an extension of his work?


title: Could we build a super computer out of wires and switches instead of a microchip?

text: I thought of this question; would too much wiring make a computer burn down? Or can you build an actual full-speed supercomputer computer using switches and wires? In other words, could simply adding wires and switches into a computer already composed of switch and wire, can you achieve a calculation speed which is limited only by the speed of light, or perhaps 10% of this?


title: Calculating gravity with inclometer?

text: How would one calculate the gravity vector for [X,Y,Z] by using an inclinometer which gives Pitch, Yaw and Roll. A little explanation of the goal: I have a accelerometer I was planning to use to map movement. But as gravity being what it is, it affects everything. So the Accelerometer reports [aX,aY,aZ] = a, and a contains actually two components gravity's [gX,gY,gZ] = g and movements [mX,mY,mZ] = m meaning that we have g+m=x so to do movement detection is more or less impossible unless I could keep the meter leveled at all times, and then i could remove the gravity component. But since this is a moving system I thought that gyrometer could be used, but it has horrible drift and don't work. Next step is naturally to use a Inclometer and then calculate what gravity is supposed to be and subtract that. The inclometer outputs like this in a [Pitch,Yaw,Roll] vector. ![inclometer decriptor](http://i.stack.imgur.com/TptN4.png) I expect the calculated gravity [cX,cY,Cz]=c and that c - g ≈ 0 Making is useful to say movement accel m + g - c ≈ m


title: What are torque parameters?

text: Torque is just the turning force on an object, i.e the force multiplied by the perpendicular distance from the force to the pivot. But when torque parameters are mentioned in mechanical engineering papers, what are they referring to? It's the plural "parameters" thats confusing me. Is it the collection of properties such as maximum torque, starting torque, etc? If so, can somebody please provide me with a list of the all, or some of the "torque parameters", along with what each means? Finally, if looking at the torque parameters of a crankshaft, what would be the important ones to pay attention to, and what are some typical values?


title: Is there a list of all atomic electron state transitions and the corresponding radiation emitted?

text: Here's a quote from Wikipedia: > As an example, the ground state configuration of the sodium atom is > 1s22s22p63s, as deduced from the Aufbau principle (see below). The first > excited state is obtained by promoting a 3s electron to the 3p orbital, to > obtain the 1s22s22p63p configuration, abbreviated as the 3p level. Atoms can > move from one configuration to another by absorbing or emitting energy. In a > sodium-vapor lamp for example, sodium atoms are excited to the 3p level by > an electrical discharge, and return to the ground state by emitting yellow > light of wavelength 589 nm. > > Usually the excitation of valence electrons (such as 3s for sodium) involves > energies corresponding to photons of visible or ultraviolet light. The > excitation of core electrons is possible, but requires much higher energies > generally corresponding to x-ray photons. This would be the case for example > to excite a 2p electron to the 3s level and form the excited 1s22s22p53s2 > configuration. The questions this prompted for me are this: 1) Does each atom have a finite number of possible state transitions, or could any electron in any atom theoretically be excited to some absurdly high orbital, if the photon it absorbed was energetic enough? 2) Does a given transition (e.g. the 3p to 3s sodium transition) always emit a photon of the exact same energy for a given atom (e.g. the 589 nm example above)? Does the same transition in a different atom emit the same photon (i.e. does a 3p to 3s transition always emit a photon of 589 nm)?


title: Definition of phase transitions in statistical mechanics

text: Phase transitions in statistical mechanics are usually taught by working through a bunch of examples. I decided to try and think about them from a more "fundamental" point of view - but I've run into a weird little snag, in that I've come up with a system that seems to satisfy the most common definition of a (first order) phase transition, yet it's too trivially simple to really be interesting. Because of this, I'm wondering whether there's a way to define phase transitions in such a way that they include interesting examples such as the two dimensional Ising model, yet exclude the one I present below. My reason for asking is not that I want to be finicky about definitions for their own sake, but rather because I want to get a handle on what special feature(s) less trivial models have that mine doesn't, which cause them to exhibit interesting effects such as scale free behaviour etc. around the transition point. My trivial model is as follows. Consider a two-state (classical) system, which can either be in state 0 with energy $E_0 = 0$, or state 1 with energy $E_1 = \lambda \varepsilon $. Here $\varepsilon$ is a fixed value (with dimensions of energy) and $\lambda$ is a dimensionless parameter that I will vary later, representing the size of the system. We assume the system is in a Boltzmann distribution with inverse temperature $\beta$. The probabilities $p_i$ of the system being in state $i$ are then as follows: $$ p_0 = 1/Z \quad\text{and}\quad p_1 = e^{-\beta \varepsilon \lambda}/Z, $$ with the normalisation factor ("partition function") being given by $Z=1+e^{-\beta \varepsilon\lambda}$. We now consider changing the "scale" of the system, $\lambda$. The expected energy density in the system is given by $$u(\beta) = \frac{\langle E \rangle}{\lambda} = \frac{\varepsilon e^{-\beta\varepsilon\lambda}}{1+e^{-\beta\varepsilon\lambda}}.$$ It is not hard to see that in the limit of infinite $\lambda$ this becomes a step function, with $u=0$ if $\beta>0$ and $u=\varepsilon$ if $\beta<0$. Similarly, the density of the log partition function (or dimensionless "free energy"), given by $$ \frac{\log Z}{\lambda} = \frac{\log(1+e^{-\beta \varepsilon \lambda})}{\lambda}, $$ becomes piecewise linear in the limit of large $\lambda$, exhibiting a discontinuity in its first derivative at $\beta=0$. In the limit, $\frac{\log Z}{\lambda} = -\beta \epsilon$ if $\beta<0$, and $0$ otherwise. One can also note that at the transition point, $\beta=0$, the variance of $E$ diverges. So this system seems to exhibit the same discontinuities as nontrivial models of phase transitions --- yet it's just a simple two-state system. Intuitively this phenomenon is too simple to be called a phase change, so I would like to understand what the defining difference is between this type of behaviour and the non-trivial phase changes that can take place in Ising type models and in physical systems. Alternatively, I suppose it's possible that some of the "interesting" features of phase transitions _are_ present in my example, but it just takes some clever interpretation to get at them. If this is the case I would greatly appreciate an explanation. **Addendum.** The model as described above has the advantage of being the simplest I could come up with, but it has the disadvantage that the transition point is at $\beta=0$, which corresponds to infinite temperature, since $\beta=1/k_BT$. This makes the model seem somewhat pathological. However, a small change to the model can make the transition take place at a finite, positive temperature. Making this change turns out to be quite instructive. To achieve this we give the higher energy level some degeneracy. That is, there is still only one state with $E_0=0$, but there are now $g$ states with $E_i=\varepsilon$. (If the single state with $E=0$ is aesthetically displeasing, one can also give it a degeneracy; I have chosen not to do this for simplicity.) In this case, as before, $p_0=1/Z$ and $p_i = e^{-\beta\varepsilon\lambda}/Z$ for $i>0$, but now $Z=1+ge^{-\beta\varepsilon\lambda}$. In order to get a transition at a non- zero temperature we must make the additional assumption that the degeneracy scales exponentially with $\lambda$, i.e. $g=e^{a\lambda}$ for some dimensionless constant parameter $a>0$. In this case (as gatsu deduced in his answer), $$ u(\beta) = \frac{\varepsilon e^{(a-\beta\varepsilon)\lambda}}{1+e^{(a-\beta\varepsilon)\lambda}}. $$ In the limit of infinite $\lambda$ this has a transition point at $\beta = a/\varepsilon$, with $u$ being zero if $\beta$ is greater than this value, and $\varepsilon$ if it is below it. the "free energy" $\log Z(\beta)$ changes in a similar way, becoming piecewise linear with a discontinuity in its first derivative at $\beta = a/\varepsilon$. As well as being less pathologically-behaved, this model also feels a bit less trivial, since it has many microstates rather than just two and, as gatsu points out, the transition now arises from a competition between the low energy of state $0$ and the entropy of the degenerate excited states. But nevertheless, as far as I can see, this model lacks such features as power law behaviour and symmetry breaking that are associated with phase transitions in other models. I'm interested in the essential difference between my model and those ones, which gives rise to such non-trivial behaviour around the critical point.


title: Does the rate of expansion of the universe exceed the rate at which the local group is being pulled toward the Great Attractor?

text: I understand that the expansion rate of the universe driven dark energy exceeds the rate at which the local group is being pulled toward The Great Attractor (Hydra-Centaurus Supercluster). Does this mean that the local group will be isolated from the Great Attractor and all other galaxies in 40 billion plus years.


title: Gauge theory in classical electromagnetism

text: I understand gauge theory as the theory of continuous transformation group which keeps Lagrangian (or dynamics) invariant. So some integral invariants could be found. In terms of classical electromagnetism, we choose gauge so that obtain different vector potentials $A$. My question is, since now different potentials doesn't affect dynamics, is that meaningless to choose a gauge? I just have no idea how gauge theory play a role.


title: Is the equivalence principle in General Relativity an approximation?

text: I read in web that Einstein used the principle of equivalence to explain General Relativity but we know the gravitation is approximately equal in all of rested frame in gravitional field. In contrary, we can only create uniform acceleration inside a frame if we move it in a direction (The same $a$ will be gotten in each point of the frame ) . Is it possible to create exact equivalence between accelerated frame and rested frame in gravitational field? How can the general relativity explain the difference?


title: Experimental tests of Cluster Decmposition

text: How tight are experimental and astrophysical tests on whether Cluster Decomposition is satisfied at various space-like separations? Is there a review paper or a standard reference on the question? I note that Cluster Decomposition may be related to the existence of a unique vacuum state and to spontaneous symmetry breaking.


title: Can I perceive the photoelectric effect as "ionizing with EM force"

text: The photoelectric effect demonstrates that EM radiation has particle-wave duality and it can be used to generate electricity. I am still a novice in physics and my understanding of this phenomenon is that when the energy in photon is high enough, it can remove electrons from an atom, and with higher energy it can increase the kinetic energy of the emitted electrons or with higher intensity it can increase the number of emitted electrons. So can I perceive the photoelectric effect as "ionizing with EM force"? Is there any connection between this effect and the ionization triggered by EM radiation like the gamma ray? What are the main differences between this effect and the ionization by EM radiation?


title: How exactly does a resistance reduce current?

text: I've heard that resistors are used to decrease current to a particular appliance, such as in the regulator of a fan. However, I've also heard that the total current in a circuit is always the same- in other words, all the current leaving the positive terminal of the battery reaches as it is to the negative terminal. How can this be? The two statements are contrary. How can the total current be the same if a resistor is reducing current at some point in the circuit?


title: Electrostatic induction

text: Is this a correct definition of electrostatic induction, The **production** of opposite charges on a neutral body when a charged body is brought near to it? I think it's not the production but redistribution.


title: How to write a noisy state as a separable operator

text: Bipartite operators close enough to the identity are separable. But how does one compute the product operator terms of the separable expansion? In particular, if $\left| \Phi \right> = m^{-1/2} \sum_{i=1}^n \left| i \right> \otimes \left| i \right>$ is the $n$-dimensional Bell state then $I \otimes I - \left| \Phi \right>\left< \Phi \right|$ is separable. But how do I write this as a sum of positive product operators $\sum A_i \otimes B_i$?


title: Calculating the the kernel using path integrals for quadratic lagrangians

text: I am reading Feynman and Hibbs on Path Integrals. In section 3.5, they show that the kernel for a lagrangian of the form $L=a(t)\dot{x}^2+b(t)\dot{x}x+c(t)x^2+d(t)\dot{x}+e(t)x+f(t)$ is $K(b,a)=e^{\frac{i}{\hbar}S_{cl}[b,a]}F(t_a,t_b)$. In general, how do I calculate the factor $F(t_a,t_b)$. In the problems after the section, I have calculated the classical action, for the particle in a magnetic field, and the forced harmonic oscillator. But I don't know how to calculate the prefactors. For e.g. this is the problem 3-11 from Feynman and Hibbs asks you to calculate the kernel of the harmonic oscillator driven bby an external force $f(t)$. The Lagrangian is $L=\frac{m}{2}\dot{x}^2-\frac{m\omega^2}{2}x^2+f(t)x$. The answer is $$K=\sqrt{\frac{m \omega}{2 \pi i \hbar \sin{\omega T}}}e^{\frac{i}{\hbar}S_{cl}}$$ where $T=t_f-t_i$ and $S_{cl}$ is the classical action. How can I see that the above is the factor multiplying the exponent directly or via a calculation.


title: fast quantum random number generator + limited decoherence rate => Schrödinger cat state?

text: Suppose that fast quantum random number generator (QRNG, https://qrng.physik.hu-berlin.de/) is placed in a subsystem which has limited interaction with the rest of the world. What would happen if decoherence rate for this subsystem were not fast enough to extract all the random numbers being created by QRNG? It seems that then the subsystem has to evolve into a state with macroscopic coherent properties (Schrödinger cat state). At least this occurs if one simulate QRNG using measurement of a rotated qubit and copying a result in a vector of qubits representing a (partly) closed subsystem. If so, why the question about feasibility of macroscopic coherent states is still not settled?


title: Does the slip-stick phenomenon have any application?

text: The slip-stick phenomenon is present all around us, be it the noise of car breaks or in earthquakes. But does it have any real-life application?


title: What to cite for MSX6 survey?

text: Here is my current dilemma: I used the data from MSX6 Survey and want to put a citation or some kind of acknowledgment but I can't seem to find any concrete paper or reference text to put in the paper. Any of you would know what paper to cite?


title: Direct rocky planet formation without protostellar disk?

text: Inspired by this question, which sort of asked what I'm asking, but emphasized a different point (and the answers accordingly addressed that). Can a rocky planet form directly from the collapse of an interstellar gas cloud? The picture I have of rocky planet formation is that dust and rocky material in a protostellar disk accumulates into a rocky planet. There must be some mechanism to prevent the accretion of significant amounts of (hydrogen/helium) gas, I'm guessing this is something to do with the gas either being unable to cool, reheated if it does cool, or outright removed from some region around the protostar as the stellar wind and/or radiation pressure kick in. My intuition is that even the most metal enriched (astronomer metal, i.e. everything heavier than $\rm{He}$) cloud still by far hydrogen dominated, so that a cloud in isolation must _always_ form a primary star/brown dwarf/gas giant. But has anyone ever heard of a mechanism where a cloud in the right environment (some radiation field that efficiently heats the gas but not the dust/metals perhaps?) could form a rocky body directly, with no associated star? This seems far fetched, but I'm curious to know whether anyone has ever looked at this; literature references appreciated.


title: A pedestrian explanation of Renormalization Groups - from QED to classical field theories

text: shortly after the invention of quantum electrodynamics, one discovered that the theory had some very bad properties. It took twenty years to discover that certain **infinities** could be overcome by a process called renormalization. One might state the physical reason behind this that we are only aware of **effective theories** which are reliable on certain **scales** given by more or less fundamental constants. Renormalization tells us how to deal with this situation and only consider effects of a specific range. The **technique** to perform the calculations is called renormalization group. It is a powerful tool and it is no wonder it is under heavy development since nothing can be calculated without it in quantum field theories. Per-se, this procedure is not limited to its root and one might ask the **question** : ## How can we use the renormalization group to find effective theories for classical field theories? I suppose, an **example** where this has been done very recently can be found in Renormalization Group Analysis of Turbulent Hydrodynamics. I would be thankful for any insight, examples etc.   Sincerely Robert PS.: The question is naturally linked to How to calculate the properties of Photon-Quasiparticles and in a loose line with A pedestrian explanation of conformal blocks.   Since I am no expert in the field please advice me if something is not clear or simply wrong.


title: Energy conservation in General Relativity

text: I understand that energy conservation is not a rule in general relativity, but I'd like to know under what circumstances it can still be possible. In other words, when is it possible to associate a potential energy to the gravitational field, so that the energy is constant in the evolution of the system? Here are some examples, is there a convenient way to define energy in these scenarios?   * Just a system of gravitational waves.   * A point mass moving in a static (but otherwise arbitrary) space-time. Equivalent (if I'm not mistaken) to a test mass moving in the field of a second much larger mass, the larger mass wouldn't move.   * Two rotating bodies of similar mass. Overall, I'm trying to understand what keeps us from associating a potential energy to the metric. When we break the time translation symmetry of a system by introducing an electromagnetic field, we can still conserve energy by defining an electromagnetic potential energy. **Why can't we do the same when we break TT symmetry by making space-time curved?**


title: Is it possible to obtain higher order corrections to the ideal gas law when one allows realistic phenomena to make their way into the equations?

text: I had an interesting thought today that caused me to ask whether it'd be possible to make corrections to the ideal gas law via introducing terms derived from more realistic phenomena to make their ways into the equation.


title: Calculating coefficient of friction

text: > Consider a body attached to a horizontal spring and resting on a surface, > inclined at an angle $\theta$ from the ground.![Set- > up](http://i.stack.imgur.com/F2G97.png) > > The spring constant is $k$. Initially the spring was kept in its natural > length while the body was held still by some external agent. When the > external agent was removed, the body slid $x$ units down the inclined plane > to achieve equilibrium.The coeffecient of (kinetic) frictional force acting > on the body is $\mu$ To solve this problem, I can use two methods: 1.Work-Energy Theorem: This yields $$0=mg(x\sin\theta)-\frac12kx^2-\mu (mgx\cos\theta)$$ 2.Equating forces at equilibrium: This yields $$kx+\mu mg\cos\theta- mg\sin\theta=0$$(along the inclined plane) On solving the equations the two methods give different answers, while they should give the same. Is there anything I am missing out here? Please help me solve this problem.


title: What are Killing spinors?

text: What are Killing spinors? How can they be motivated? Are they directly related to Killing vectors and Killing tensors and is there an overarching motivation for all three objects? Any answer is greatly appreciated but a less formal one would be preferred.


title: Discussion of the Rovelli's paper on the black hole entropy in Loop Quantum Gravity

text: In a recent discussion about black holes, space_cadet provided me with the following paper of Rovelli: Black Hole Entropy from Loop Quantum Gravity which claims to derive the Bekenstein-Hawking formula for the entropy of the black hole. Parts of his derivation seem strange to me, so I hope someone will able to clarify them. All of the computation hangs on the notion of _distinguishable_ (by an outside observer) states. It's not clear to me how does one decide which of the states are distinguishable and which are not. Indeed, Rovelli mentions a different paper that assumes different condition and derives an incorrect formula. It seems to me that the concept of Rovelli's distinctness was arrived at either accidentally or a posteriori to derive the correct entropy formula. > Is the concept of distinguishable states discussed somewhere more carefully? After this assumption is taken, the argument proceeds to count number of ordered partitions of a given number (representing the area of the black hole) and this can easily be seen exponential by combinatorial arguments, leading to the proportionality of the area and entropy. But it turns out that the constant of proportionality is wrong (roughly 12 times smaller than the correct B-H constant). Rovelli says that this is because number of issues were not addressed. The correct computation of area would also need to take the effect of nodes intersecting the horizon. It's not clear to me that addressing this would not spoils the proportionality even further (instead of correcting it). > Has a more proper derivation of the black hole entropy been carried out?


title: Passing a Car With known speeds vs. Time

text: While driving, it takes 22.0 s to pass a car. If my speed was 64.0 mph when I started to pass, and was 80.0 mph as I finished, how far (in feet) did I travel while passing?


title: spin conservation in exchange polarization process

text: Exchange polarization is the process by which spin is transferred between an electron beam and a system of polarized atoms (with a single valence spin). The process occurs as a result of the Pauli Exclusion principle. See Burke and Schey Phys. Rev. 126 163 (1962). The type of scattering processes that can occur at |up up> $\rightarrow$ |up up>, |up down> $\rightarrow$ |up down>, |up down> $\rightarrow$ |down up> etc. where up and down refer to the spin of the electron in the beam or the atom. In any case, you can see that the spin is preserved in each instance. However, when one goes through the mathematics using the density matrix formalism for general initial polarizations of the electron beam and atoms, the resulting polarization $P'_e + P'_a \neq P_e + P_a$ which to me is impossible given that each scattering channel conserves spin. This can be see in the paper cited above (n + p in their notation) and also in the Mott & Massey monograph (Vol. 2 p. 542) and the book Polarized Electrons by Kessler. None of the authors mention any problem with spin conservation and I am confounded at the moment.


title: What is a covalent bond?

text: What is a covalent bond, quantum mechanically? How does it hold the two atoms together, and at one point can you qualify the electron as being shated between two atoms, versus feeling attractive forces?


title: Didn't anybody see an apple falling before Isaac Newton?

text: We all know that Isaac Newton developed the gravitational theory (as is often told) when an apple fell on his head. But my question is, didn't anyone before him notice it?


title: Covalent Bonds, Varities and Limits!

text: Related:http://chemistry.stackexchange.com/q/9459/ I was wondering, covalent bonds tend to share two electrons, apparently rarely more than three, and normally between two electrons. Can someone give something in the way of a proof that a a two electron two body pairing wouls be most stable? Or better, that as you go higher in electrons or atoms in the "bond" that things become less stable?


title: group velocity in non dispersive medium

text: In a non-dispersion medium, phase velocity is equal to group velocity of waves. Consider a free quantum particle moving in one dimension say $\text{x}$ axis. There would be matter waves associated with the particle. According to derivations available in books, the phase velocity is half the velocity of particle while group velocity is equal to velocity of particle. My question is that in non-dispersion medium, will the group velocity of matter waves remain to be equal to particle's velocity or will it become half of it? If group velocity remains the velocity of particle then phase velocity will also be the same as that of particle's velocity but that not what mathematical derivations show.


title: Quantity to measure order in crystals

text: Imagine a cubic lattice for simplicity and put some atomic configuration inside of each unit cell. Lattice is of course a highly ordered entity (long range order). Please, how could I quantitatively describe this via some correlator function? I assume no up/down spins are present on the lattice. This is most probably a very basic question, but I come from a different field of physics and could not find the right references.


title: What is the difference between dot and cross product?

text: 1. What is the difference between dot product and cross product?    2. Why do we use cross product to find torque, why can't we use dot product?   3. Also we use dot product to find work done and not cross product?


title: Transformer ratios - 1:2 vs 50:100

text: I am only the equivalent of a high school student, so please, if possible, don't answer this question with anything too complex or really advanced university level. I am very happy to research new concepts anyone mentions, but can you please keep it reasonably simple. With transformer windings, primary vs secondary, is there any difference between or advantage to using a 50:100 ratio rather than a 1:2 ratio? With a 1:2 ratio it would seem easier to make the wires thicker and allow for larger currents, whereas with a 50:100 ratio to give the same cross sectional area of wire, the coil would have to be much longer, larger, and involve more metal in production. So there are obvious disadvantages to using 50:100, but are there any advantages? Thank you very much.


title: Constructing the $\cal{N}=2$ supersymmetric non-Abelian Chern-Simon's theory

text: This is related to this earlier question I had asked. I am using the so called ``Majorana" representation of gamma matrices in $2+1$ dimensions in which everything is real. After doing the dimensional reduction of the $\cal{N}=1$ supersymmetry transformations of the components of the vector superfield in $3+1$ dimensions the supersymmetry transformations of the resulting $\cal{N}=2$ components of the vector superfield in $2+1$ dimensions are, $$\delta F_A = i \bar{\alpha}^a\lambda_{Aa}$$ $$\delta D_A = i \bar{\alpha}^a\gamma_3^\mu D_\mu \lambda_{Aa}$$ $$\delta V_{A\mu} = i \bar{\alpha}^a\gamma_{3\mu}\lambda_{Aa}$$ $$\delta \lambda_{Aa} = -\frac{1}{2}f^3_{A\mu \nu}\gamma_3^{\mu \nu}\alpha_a + D_A\alpha^a + \gamma_3^\mu D_\mu F_A\alpha ^a$$ where $A,B,..$ are the gauge group indices, $f^3_{A\mu \nu}$ is the non- Abelian field strength and $\alpha$ is a spinor parameter whose components are raised and lowered as, $\alpha^1 = \alpha_2$ and $\alpha^2 = -\alpha_1$. Using the above one can derive the following transformations for the possible terms in the intended super-Chern-Simons' theory, $$\delta(Tr[FD]) = Tr[t_At_B] \\{ i\bar{\alpha}^a \lambda _{Aa}D_B - i \bar{\alpha}^a \gamma_3^\mu\lambda_{Ba}\partial_\mu F_A + i\bar{\alpha}^a \gamma_3^\mu \lambda_{Ca} C_{BB'C}V_{B'\mu}F_A\\}$$ $$\delta(Tr[\bar{\lambda}_a\lambda_a]) = 2Tr[t_A t_B]\\{ \frac{1}{2}\bar{\alpha}_a\gamma_{3\rho}\lambda_{Ba}f^3_{A\mu\nu}\epsilon ^{\mu \nu \rho} + \bar{\alpha}^a\lambda_{Ba}D_A - \bar{\alpha}^a\gamma_3^\mu \lambda_{Ba}\partial_\mu F_A$$ $$- \bar{\alpha}^a\gamma_3^\mu \lambda_{Ba}C_{AB'C}V_{B'\mu}F_C \\}$$ $$\delta(Tr[\epsilon^{\mu \nu \rho}(V_\mu \partial_\nu V_\rho)]) = -i\epsilon^{\mu \nu \rho}Tr[t_At_B]\bar{\alpha}^a\gamma_{3\rho}\lambda_{Aa}(\partial_\mu V_{B\nu} - \partial_\nu V_{B\mu})$$ $$\delta(Tr[\epsilon ^{\mu \nu \rho}V_\mu V_\nu V_\rho]) = -\frac{3}{2}\epsilon^{\mu \nu \rho}Tr[t_At_D]C_{DBC}\bar{\alpha}^a\gamma_{3\mu}\lambda_{Aa}V_{B\nu}V_{C\rho}$$ (where $t_A$ are a chosen basis in the lie algebra of the gauge group such that the structure constants are defined as, $[t_A,t_B]=iC_{DAB}t_D$) It is clear that by choosing a coefficient of $-2$ for the $Tr[FD]$ and $i$ for the $Tr[\bar{\lambda}_a\lambda_a]$ some of the terms the variation of the auxiliary fields can be cancelled and some of the remaining terms of the variation of the fermionic term totally cancel the supersymmetric variation of the kinetic term of the gauge fields. What remains are, $$\delta(Tr[\epsilon^{\mu \nu \rho}V_\mu\partial _ \nu V_\rho + i \bar{\lambda}_a\lambda_a - 2FD]) = Tr[t_At_B]\\{i\bar{\alpha}_a\gamma_{3\rho}\lambda_{Aa}C_{BCD}V_{C\mu}V_{D\nu}\epsilon^{\mu \nu \rho}$$ $$ - 2i\bar{\alpha}^a\gamma_3^\mu \lambda_{Ba}C_{AB'C}V_{B'\mu}F_C - 2i\bar{\alpha}^a\gamma_3^\mu \lambda_{Ca}C_{BB'C}V_{B'\mu}F_A\\}$$ and $$\delta(Tr[\epsilon ^{\mu \nu \rho}V_\mu V_\nu V_\rho]) = -\frac{3}{2}Tr[t_At_B]\bar{\alpha}^a\gamma_{3\mu}\lambda_{Aa}C_{BCD}V_{C\nu}V_{D\rho}\epsilon^{\mu \nu \rho}$$   * Its not clear that a coefficient can be chosen for the last term so that the supersymmetric variation of the sum of the LHSs go to zero.  The above terms seem to be structurally very different and hence its not clear how they will cancel. Like the variation of the fermionic self-coupling term produces a coupling of the fermionic component, gauge field and auxiliary field. Such a term is not produced by the variation of the gauge field cubed term! One expects the lagrangian should look something like, $$Tr[\epsilon^{\mu \nu \rho}(V_\mu\partial _\nu V_\rho -i\frac{2}{3}V_\mu V_\nu V_\rho ) + i \bar{\lambda}_a\lambda_a - 2FD ]$$ I would like to get some help in establishing the above!   * One progress would be if the two terms with the structure constant actually cancel i.e, if $$Tr[t_At_B]\\{C_{AB'C}\lambda_{Ba} V_{B'\mu}F_C + C_{BB'C}\lambda_{Ca}V_{B'\mu}F_A\\} = 0$$ But the above is not clear! NB. My structure constants are defined as $[t_A,t_B]=iC_{DAB}t_D$


title: What am I doing wrong finding the eigenvectors of this rotation matrix?

text: This question is from Principles of Quantum Mechanics by R. Shankar. > Given the operator (matrix) $\Omega$ with eigenvalues $e^{i\theta}$ and > $e^{-i\theta}$ , I am told to find the corresponding eigenvectors. I think these equations are relevant: $$\begin{align}\Omega &= \begin{bmatrix}\cos{\theta} & \sin{\theta} \\\ -\sin{\theta} & \cos{\theta} \end{bmatrix} \\\ \Omega \begin{bmatrix} x_1 \\\ x_2 \end{bmatrix} &= \begin{bmatrix} x_1 \cos{\theta} + x_2 \sin{\theta} \\\ -x_1 \sin{\theta} + x_2 \cos{\theta} \end{bmatrix} \\\ e^{i\theta} \begin{bmatrix} x_1 \\\ x_2 \end{bmatrix} &= \begin{bmatrix} x_1 \cos{\theta} + x_1 i\sin{\theta} \\\ x_2 \cos{\theta} + x_2 i\sin{\theta} \end{bmatrix}\end{align}$$ I let the matrix operate on the generic vector $(x_1, x_2)^T$ and demand that the resulting vector is equal to $(e^{i\theta}x_1, e^{i\theta}x_2)^T$ . From this i get the condition that $x_2 = ix_2$ and $x_1 = -ix_2$ , which implies that $x_1 = x_2 = 0$ . Am i missing something crucial?


title: Can a full image be made in mirror if we change the distance between object and mirror?

text: To see the full height of man, the height of mirror should be half of the height of the man. What happens when we decrease the distance between mirror and man?


title: What will happen if a gamma-ray burst hits the Earth?

text: Is there any probability that a gamma-ray burst (GRB) could affect the Earth?


You throw a baseball directly upward at time t = 0 at an initial speed of 12.1 m/s. What is the maximum height the ball reaches above where it leaves your hand? At what times does the ball pass through half the maximum height? Ignore air resistance and take g = 9.80 m/s2. Find the Earlier time at half maximum height and Later time at half maximum height. For the maximum height, I got 7.4699 m. I do not know what I do next to find the earlier time at half the height or the later time.


title: Quantum numbers and the band structure of solids

text: I got a question concerning the band strucutre of solids. The reference I'm using is the book on solid state physics by Ashcroft/Mermin. My problem is that I don't completely understand the reason why there exists a band-index n. On the one hand Ashcroft gives a quite plausible explanation saying that the general solution of the given eigenvalue problem H $\psi_k(r)$ = $E_{k}$ $\psi_k(r)$ can be decomposed into $\psi_k(r) = e^{i*k*r} u_{k}(r)$. Plugging this ansatz into the Schroedinger-equation and applying H to the $e^{i*k*r}$ first we obtain a new Hamiltonian $H_{k}$ that depends on the "wavevector" k and a new eigenvalue-problem $H_{k} u_{k}(r) = E_{k} u_{k}(r)$. Now we fix k. The operator $H_{k}$ acts on r and therefor produces a certain number of solution u_{n,k}(r). These solution can be counted using the index n. So far so good. Then Ashcroft states that the second proof of Bloch's theorem shows that $u_{k}(r)=\sum_{G} c_{k-G} e^{i G r}$. The sum is taken over all reciprocal lattice vectors G. He says that from this way of writing the function $u_{k}$ it is obvious that multiple solution do exists. I don't really understand why that is the case. First i thought that you may think of the n-th solution to be $u_{k,n}(r)=c_{k-G_{n}} e^{i G_{n} r}$ where $G_{n}$ denotes the n-th reciprocal lattice vector. However this doesn't seem plausible to me, since you need to have the sum to proof that $\psi_{k}$ = $\psi_{k+G}$ and therefor E(k)=E(k+G). As you can see I'm quite confused about all this. Actually I'm also a bit confused why the reduction to the first brillpuin zone doesn't produce more than one Energy value for a given k. Anyway, I'd be more than happy if someone could help me. Thanks in advance!! See you.


title: Difference between thermodynamics and statistical mechanics

text: I've just finished a class a few weeks ago which taught thermo and stat mech, and I still don't know the exact difference between the two. Can someone help clear this up for me? (Yeah it's sad, and even sadder that I got an A- in the course)


title: "Space" in General Relativity and "vacuum" in Standard Model, is it the same thing?

text: And expansion of space is equal to expansion of vacuum?


title: Is a superconductor really a super conductor?

text: It is known that a superconductor is a material with electrical resistance zero. My question is, it is exactly zero, a theoretical zero, or for practical realistic reasons it is effectively zero?


title: The equipartition theorem in momentum space

text: Motivated by the answers to this question on turbulence, I'm interested in an explanation and/or derivation/reference of the equipartition theorem in momentum space. To formulate it as a question: > If one considers a physical configuration, which admits a description in a > dual/momentum space in the framework of statistical mechanics (basically > like fields do), what is the realization of the equipartition theorem in > these $k$-space terms. I.e. how does it affect the distribution of energy and what consequences do different dispersion relations have on the evolution, or even outcome of that partition? How do we understand statistical mechanics, not in the inital degrees of freedom, which are usually well motivated by the model building process, but in abstract $k$-space terms and what do we typically expect?


title: Negative energies and a partition function

text: I'm writing down the partition function for a system, for which I know the dispersion relation $$E \left( \mathbf{k} \right) = \sqrt{ \left| \mathbf{k} \right|^2 + m^2 + \cdots }$$ The exact form is not important, what matters is that technically, as the dispersion relation is the solution of a 2nd degree polynomial equation, there should be a $\pm$ sign before the square root. When I write the partition function I get: $$ Z = \sum_{\mathbf{k}} \exp \left( - \frac{E \left( \mathbf{k} \right)}{\beta} \right) $$ but if I wanted to include the negative energies too, I would get: $$ Z = \sum_{\mathbf{k}} \exp \left( - \frac{E \left( \mathbf{k} \right)}{\beta} + \frac{E \left( \mathbf{k} \right)}{\beta} \right) = \sum_{\mathbf{k}} 1$$ which is clearly absurd: my system has dynamics! ;-) Now my question is: are the negative really unphysical? Wouldn't it be more correct to keep track of the two-branch dispersion relation with something along these lines: $$ Z = \sum_{\mathbf{k}} \exp \left( - 2 \frac{E \left( \mathbf{k} \right)}{\beta} \right) $$


title: Phenomena which are incorrectly declared as resonance phenomena?

text: In standard college physics text books, high-school books and popular level physics books, the collapse of the Tacoma Narrows Bridge is often taken as an example of _resonance_. However a more detailed analysis shows that the collapse was not due to simple resonance (see the article by Billah and Scanlan and Wikipedia). Now my question is, whether there are further examples from mechanics, which are often incorrectly or oversimplified declared as examples of resonance phenomena (for example in physics text books)? **Edit:** I should add, that the definition usually given in textbooks is the following: Consider a mechanical system which can be described by a differential equation like $\frac{d^2}{d t^2}x(t) + 2 \rho \frac{d}{d t} x(t) + \omega_0^2 x(t) = f(t) $ where $f(t) = f_0 \cos(\omega t)$. Then for example the amplitude or the energy of the system as function of $\omega$ has a maximum near $\omega_0$. In this case one speaks of _resonance_ of the system.


title: Merge different wavelength rays

text: Let's say an array of rays of light is given.   Each ray has a specific wavelength (in the range of visible light). Example:               ray1 = 470 nm     ray2 = 640 nm     ray3 = 510 nm      Can these rays be merged into a single ray of "white light" which preserve the information of it's components?


title: stability of hypothetical lunar atmosphere

text: Assume that by some means, the moon could be given an atmosphere, of the same density and pressure at the surface as the earth's. Obviously in a stable atmosphere there are temperature variations from pole to equator, day to night, and a lot of interesting dynamics follow from that, but for argument's sake assume an average of something like 0 to 5 C. How long would it last? Clearly there's no air there now (unless you count traces of potassium, or whatever it has), so such at artificial atmosphere would not be permanent, but presumably the moon could hang onto some air, for some time. With the moon's lower gravity, the density or pressure vs altitude curve will be different, and I'd guess that a significantly larger fraction of the mass would be at higher altitudes than on earth, and there's no protection from the solar wind. Someone with the right equations could probably work out how long it would take the pressure at the surface to drop by half.


title: Significance of 'chiral' form for a quibit?

text: Say I have a qubit with probability amplitude divided evenly among $|0\rangle$ and $|1\rangle$ $$\frac{1}{\sqrt 2}|0\rangle + \frac{1}{\sqrt 2}|1\rangle$$ So it seems that we have a, loosely speaking, 'symmetric' probability amplitude for the states. However I have read that this is not the case and $$\frac{1}{\sqrt 2}|0\rangle + \frac{i}{\sqrt 2}|1\rangle$$ is actually a 'better' representation of a states with a symmetric probability amplitude. Can someone explain why this is so? The explanation can be very handwavey, I'm looking for some intuition on it why this is the case.


title: Does anybody actually do astrometry in the 21st century?

text: I mean the nitty-gritty measurements of positions of stars and planets using transit circles. The transit instruments I've seen are mostly in national observatories and are essentially museum pieces: in Greenwich, U.S. Naval Observatory in Washington, Ottawa, Sidney, Paris, etc. I've never read of anybody in the present time actually logging stars as they cross the meridian. If nobody is actually making these observations regularly, how do we know for sure the rotation period of the Earth and the exact positions of the planets? Modern astronomers study distant galaxies, but is anyone actually measuring the rotation of the Earth the old-fashioned way?


title: Black Holes and String states

text: We seem to know that black hole energy, and apparently event horizon geometry and topology, are functions of Mass, Charge and Angular Momentum. Area is a function of geometry and topology as well, which apparently controls entropy. Strominger, et al, can show that the apparent agreement of string states to event horizon in extremal cases. What is the study of diffeomorphisms from extremal black holes solutions to schwarzschild solutions called? What are routes to show string states describe schwarzschild black holes as charge and angular momentum are taking to limit zero?


title: What are the reasons to expect that gravity should be quantized?

text: What I am interested to see are specific examples/reasons why gravity should be quantized. Something more than "well, everything else is, so why not gravity too". For example, isn't it possible that a quantum field theory on curved space-time would be the way treat QFT and gravity in questions where the effects of neither can be ignored?


title: Why are cgs units the norm in astrophysics?

text: Other physics communities, e.g. the particle physics one, have their own set of units, custom-tailored to their own needs. Now, the astrophysics community is somewhat similar, in that a lot of measurements are often expressed in terms of corresponding numbers related to the Sun (solar mass, solar luminosity, solar radius, et cetera). However, when it comes to using down and dirty numbers, I've seen cgs units used way more often than SI units. Why is that? Is it just a matter of tradition or is there something more subtle at work?


title: Observable which dependes on the cutoff

text: In arXiv:0710.4330v1 Balitsky calculate the eikonal scattering of dipole composed of quark anti-quark, $Tr(U_{x}U^{\dagger}_{y})$, to NLO accuracy. The result he found is: ![enter image description here](http://i.stack.imgur.com/EGe1e.png) Where $\mu$ is the normalization point in the MS bar scheme. I have already read the discussion Why do we expect our theories to be independent of cutoffs? and understood the observable cannot depends on the cutoff. While $Tr(U_{x}U^{\dagger}_{y})$ is observable, taking the derivative according to $\eta$ should produce an equation which is independent of the cutoff (at least naively). Why is it still there? any idea?


title: Advice sought on choosing a camera for a Meade LX10

text: I have a Meade LX10 and have never done any astrophotography (except with some computerised SBIG ST9s on my university's LX200s). I would like to get started at home, but could do with some advice on which low budget camera would be suitable, obtainable in the UK and preferably under £100. Do I need a DSLR? I've seen people posting pics of Jupiter from their webcam, but I would like to target a wider selection of targets (starscapes, nebulae, galaxies, HH objects, etc.) How do I attach it to the back of the 'scope? Can I computer- control the motor from a MacBook Pro? Any other advice would also be welcomed.


title: Calculating (Glass) thermal conductivity

text: I read that glass has a thermal coefficient of 0.8-1 W/mK. Given that my window window thickness is around $5\ mm$, then I would calculate my heat loss per area being $160-200\ Wm^{-2}K^{-1}$. Yet standard glass has a U value of $5.6\ Wm^{-2}K^{-1}$. What is going on here? I'm missing something huge...


title: Question about units of mass, $M = (L^{3})(T^{-2})$?

text: In section 5 of the "Preliminary: On the measurement of quantities" chapter (page 3) in "A treatise on electricity and magnetism" Maxwell uses, total length, $s=mt^{2}/{2r^{2}}$to show that $m=2sr^{2}/{t^2}$ is in units of Length cubed over Time squared. This is the first I have encountered this way of looking at mass and am wondering:   1. Is this just per-relativity nonsense?   2. Does mass being "length cubed over time squared" have any meaning?


title: Optics alignment of a confocal scanning microscope

text: I am facing a challenge in my project regarding optical alignment. See the figure: ![enter image description here](http://i.stack.imgur.com/q3KTF.png) The challenge is with the vertical optical system alignment. I considered placing a mirror and check back if the image and source coincide. But since the light is too low on power (less bright). How can I align it so well that everything is absolutely vertical and good?


title: When to use Cosine or Sine?

text: Three blocks of masses $M_A=15.0 \; \text{kg}$, $M_B=12.0 \; \text{kg}$, and $M_C=8.0 \; \text{kg}$ sit next to each other on a frictionless surface. A force $F=54.0 \; \text{N}$ acts on Block $A$ at an angle of $\theta=25°$ below the horizontal. Calculate the magnitude and direction of all forces acting on each of the blocks. Normal force of $A$: $$\text{Weight of }A + F \sin \theta = 147+54 \sin(25)=170$$ This is just an example of what I'm trying to figure out. When and why do you use sin or cos when calculating magnitudes?


title: jumping into water

text: Two questions:   1. Assuming you dive head first or fall straight with your legs first, what is the maximal height you can jump into water from and not get hurt?    In other words, an H meter fall into water is equivalent to how many meters concrete-pavement fall, force wise? (I'm assuming the damage caused will be mainly due to amount of force and not the duration)   2. Assume you jump head first and hold a sharp and strong long object that cuts the water before you arrive, will that make the entrance to the water more smooth and protect you?![enter image description here](http://i.stack.imgur.com/RGzj6.png)


title: Is it possible to create a solar cell in home?

text: I'm not really sure that this is the place to ask this question, but anyway. I'm curious is it possible to create a solar cell (not panel with 2$ cells from ebay) in home. If not why and if yes how. Thanks in in advance.


title: How can I understand counterintuitive units like $\text{s}^2$?

text: One of the things I never understood, but was too afraid to ask is this: how should I think of things like $\frac{\text{kg}}{\text{s}^2}$. What exactly is a square second? Square foot makes sense to me because I can see it, but square second? I've always assumed it's just one of those things you're supposed to deal with abstractly and not try to get an intuition for, the same way you deal with 4-dimensional space. Should I just keep doing that or is there some bit of wisdom I've been deprived of all these years?


title: Causality and Determinism

text: If one has a deterministic model of physics, why is causality so important? Let's work in a fixed frame. Suppose that event A in the future causes event B, which happens before event A. Now, given the conditions at some time $t_0$ before both events (or at the time of event B), one can predict what will happen in the future, hence one already knows that event A will happen. Therefore, there's no event that happens after B that causes B and that we didn't already know about at the time of event B. Rather, we could say that because event A is caused by the conditions at time $t_0$, event B is in essence caused by the conditions at time $t_0$ and not by event A (in the sense that it can be explained completely in terms of what happens at time $t_0$). **Therefore, if we have a theory that says that event A can cause event B, we could re-explain the exact same theory by considering the mechanism that allows A to cause B as simply part of the laws which allow us to determine all future events from the initial conditions at time $t_0$.** Maybe, there's some mathematical aspect of those laws, some term in the equations, which we would philosophically like to think of as 'event A causing event B.' But physically, causality is already in this sense satisfied under a deterministic model. I came to wonder about this as a result of classical electrodynamics (without any other forces or gravity). If we know the electromagnetic field at a given point in time in addition to its time derivative, as well as the position and velocity of all charge and mass, we can use Maxwell's equations and the Lorentz force law to find out what will happen at all future events. The implication of this is that in special relativity, we shouldn't really have to worry so much about information traveling faster than the speed of light (something we impose on a system because otherwise it would violate the law of causality after applying the Lorentz transformations). It does so happen that the current classical special theory of relativity works nicely mathematically and also satisfies that principle, so we have no reason to reject it, but I don't see why it's so important for information not to travel faster than the speed of light. One further idea: We are assuming that we can choose some time $t_0$ before all events caused by A, but what if A causes events arbitrarily far into the past. One idea is that we could in some sense consider the limit of the state of the universe as $t$ approaches $-\infty$ as our initial conditions. Maybe the independence of causality and determinism could make sense in a universe with 'divine intervention' of sorts, where we could suddenly decide to intervene at any point and break the deterministic model. For example, suppose we could suddenly give an electron in the universe a mysterious push, or we could even magically make an electron appear or disappear. Then if the laws of physics allowed that divine push to influence what happened in the past, that would violate causality. But that situation would not be deterministic.


title: Quantum field theory alternatives

text: Quantum field theory arises from the requirement that the S-matrix is lorentz scalar and obeys the cluster decomposition principle. I want to know if there are _other ways to build invariant S-Matrices_ that obey the cluster decomposition principle without using quantum field theory. QFT is the most natural way to do it but I'm interested in other possible approaches.


title: Question about SL(2,Z) duality of string theory/N=4 SYM

text: $\mathcal N=4$ SYM has an $\mathrm{SL}(2,\mathbb Z)$ duality group. This can be thought of in two ways: 1\. This theory can be obtained by compactifying the 6D $\mathcal N=(2,0)$ theory on a torus, and this group comes from the group of large diffeomorphisms of the torus.   1. The theory can also be obtained as the low energy theory of D3 branes in type IIB string theory. This theory has an $\mathrm{SL} (2,\mathbb Z)$ duality exchanging, eg, strings and D1 branes, but preserving D3 branes, and so this descends to an action on their low energy theory. If both ways of thinking are valid, I'm confused about what exactly this $\mathrm{SL}(2,\mathbb Z)$ symmetry of type IIB is. It seems to be related to torii, but isn't it present even if there are no torii around? How are the two descriptions related?


title: Maximum range of a projectile launched from elevation "dumbed down"

text: I am trying to conceptually understand why the angle which produces the greatest range for a projectile launched with an elevation is not 45 degrees. I have exausted all other options, and I hope that someone here can explain it to me conceptually. That is, I am currently not at the mathematical level to explain this feature of projectiles using derivatives- but I am required to understand why this is the case. The real source of my confusion is this app http://phet.colorado.edu/sims/projectile-motion/projectile-motion_en.html The projectile in this application is launched from an elevation, and yet 45 degrees still produces the maximum range for the projectile! From my understanding the maximum range of projectile depends on configuring the launch to maximum amount of "hang-time." Is this application flawed - or am I going crazy.


title: What is the theoretical limit for farthest we can see back in time and distance?

text: 13.2 billion years ago the universe was rather small, having started only half a billion years ago. Today, with the help of Hubble Space Telescope, we are able to capture the light of galaxies emitted at that time. The point at which Earth exists now must have been quite close to those galaxies back then. If so, why is it that it is only now, after 13.2 billion years later the light from those galaxies has reached us? Or in other words, are we sure that the light we are seeing from those galaxies indeed travelled for 13.2 billion years? It looks like as if there was a race between our point running away from those galaxies (with the expansion of universe and space) and the light that was emitted at that time. And only now that light has reached and overtaken us. But if that is so, then wouldn't it put a limit on the oldest light we can see, no matter how powerful the telescope is (even it is more powerful than the James Webb Space Telescope)? This should be expected, because at the time just after the Big Bang, the light emitted by all objects must have already overtaken all other objects, including the location of earth. Therefore we will never see the light that old (close to the time of Big Bang) no matter how powerful the telescope. If this is so, what is the theoretical limit we can see far back in the past?


I am researching low temperature, near absolute zero, and in particular Bose Einstein Condensate. There is a lot of research information, but it is confusing, and not explained. Technically a BEC is formed from a Gas or liquid at somewhere below 1K. The only three elements that could still be in a liquid state at <1k are Helium, Hydrogen and Neon. Hydrogen and Neon would need to be at a pressure, below atmospheric. So, these are the only three elements that can become BEC's in their natural state. Most of the experiments using other elements, are not in their natural state, but are carried out, using a non natural vapour, of the element's atoms. It appears to me, that Helium is the only element which naturally becomes a BEC,at atmospheric pressure, but even Wiki., when it talks about Helium's strange properties, below its Lambda point, does not mention, it becoming a BEC. My questions are: 1\. Does Helium just naturally display BEC properties at <1K, or does it become a BEC? 2, Are you aware of anyone lowering Hydrogen and/or Neon to a temperature where they enter a BEC state. IE. Can Hydrogen and Neon become BEC's. 3\. Am I assuming correctly, that H,He and Ne are the only three elements that can enter a BEC state naturally. IE. Other elements at <1K are in a solid state, and can not, become a BEC. Thank you Clive Ballard


title: Do particles keep on emitting energy?

text: Accelerated electrons emit electromagnetic waves, but those at rest do not. So accelerated electrons emit energy and those at rest do not. (At rest hypothetically, I know it is not permissible according to uncertainity principle) But lets assume that we keep an electron at rest and start accelerating ourselves pretty fast, now even though we do not move the electron, it seems to move and produce EM waves, we should get these waves, and if we do get these waves then the electron must have released these waves or energy. However someone standing near the electron should not see these and hence must not say that electron is emitting energy. Why are the two observations differing ? Although energy is not conserved across frames, what else can be the source of light in one frame other than the electron which seems to move. If my thought experiment that electron does emit energy is correct, then why doesnt the standing observer see it ? And even if he does see it in some other form, does this imply that particles continuously lose energy ?


title: Where can I find open-source weather forecast software?

text: Are there any software packages/frameworks for running weather forecast numeric models on a single PC? I've found at least two instances, but I'm interested to know whether there are more and where I may find it. My preference is for   1. open-source software (but other licensing schemes can be considered), to be used for personal use (i.e. not professional/enterprise use) and study;   2. synoptic weather forecast (assuming I would be providing the input weather data in some way, as well possibly other frontier-conditions such as orography, etc), but will also consider software more oriented for mesoscale, climatology, etc;   3. software that can run on single PCs, but I'd also like to know about software which runs on a multi-node networked environment (e.g. dedicated "farm"). If you'd also care to provide, I would also be interested to know about what input format is used for the setup, what numeric models it uses (hydrostatic, ...), and any other information you'd care to provide; or just point me to the page which describes that. :-) So far I've only found MM5 (which the authors affirm to be of public domain) and WRF as candidates, but I've neither tried downloading let alone setting it up to see how it works (in other words, I'm still just idly browsing, not actively persuing attempts to install, run, etc). So I'm asking if these are the only ones available more or less freely, or if there are others. Other software frameworks appear to be available under paid licenses (e.g. some things from the ECMWF) and I'm not too sure whether they source code is provided. So I'm basically asking to know what's out there.


title: Can spatial coherence be maintained in fiber optic cables over time?

text: I am doing research with a double slit experiment, using a beam splitter and 2 lengths of fiber optic cable, whose ends brought close together form the effective double slit. I notice that the interference pattern shifts over time and suspect thermal expnsion/contraction to be changing the relative path lengths and causing the shifts or drift that I see. Are there methods for maintaining spatial coherence in single mode fibers? I am scanning a line with a detector at low signal so I need a steady interference pattern over long periods of time. Also, I am using first order interference to characterize a QM second order interference pattern from downconverted light. Is this drift a concern for the position of the interfence pattern in coincidence counts/quantum eraser type experiments?


title: Can the axis of rotation of a celestial body point in any arbitrary direction?

text: I am developing a small computer program that involves moderately simple simulation of elliptical Kepler orbits for fictional, generated star systems. I'm doing this without much prior knowledge of orbits (apart from some basic equations) or astrophysics in general. I'm attempting to create loosely realistic orbits in the sense that they are not all in one plane and that they are elliptical with the parent body at one focus. **The program assumes there are only interactions between a body and the one it orbits.** Planets do not affect other planets' orbits. All other forces are explicitly ignored with orbits following only Kepler's laws. The axis of rotation of each body in this simulation will be static (ie. without precession) and the axial tilt will be pseudorandomly generated. I wish to align an orbit with zero inclination with the equatorial plane of the parent body. The real question, then, is the following: are there are any important constraints to the _direction_ of the axis of rotation of an arbitrary hypothetical body orbiting another hypothetical body of significantly greater mass that I should take into consideration when determining the axis and axial tilt? Which is to say, can the axis of rotation of, say, a planet, "point" in any arbitrary direction? (The axis can be assumed to be a vector in the direction of the north pole. The north pole is here simply the pole that is "above" the orbital plane when axial tilt is zero.)


title: Why is compressible flow near the choke point so efficient?

text: Imagine a steady state, one-dimensional, compressible flow in a horizontal pipe of constant cross sectional area. This flow can be isothermal, adiabatic (Fanno), or diabatic (Rayleigh). As an example, the relevant macroscopic energy balance for Fanno flow is: $$\Delta h=-\Delta (KE)$$   In other words, Fanno flow converts enthalpy into kinetic energy. Below are Fanno lines for a real gas (steam) for different flow rates at the same pipe ID and inlet conditions. ![enter image description here](http://i.stack.imgur.com/IFw8A.jpg) The lines clearly indicate the "efficiency" with which enthalpy is converted to kinetic energy. At low velocity (left upper portion of lines), almost no enthalpy is converted while the entropy generation is large (the line is nearly horizontal). Near Mach 1 (the rightmost maximum entropy point for each curve) the enthalpy conversion is high and entropy generation is nearly zero (the line is nearly vertical). This happens whether you are originally subsonic (the upper branch) or supersonic (the lower branch). Another way of saying no entropy generation occurs is that the viscous losses (friction) are zero. This would imply that the molecules no longer impart (as much?) force upon each other near the choke point. This seems absurd. In isothermal flow, you also find that as you near the choke point (interestingly, below Mach 1!), the flow approaches frictionless. As I understand it, these examples are physically realizable flow regimes (if obviously very short in length scale). Why is compressible flow so efficient near the choke point? Is something at the molecular scale really going on or is it simply a poor mathematical model at these velocities? UPDATE: Obviously, in the supersonic regime (lower branch), the reverse process takes place, i.e. kinetic energy is converted into enthalpy. Still, my question holds since this conversion is nearly perfect as you approach Mach 1.


title: Toward the establishment of non-equilibrium (quasi-equilibrium) magnon BEC theory

text: In 2006, Demokritov et al have reported that they have achieved the observation of quasi-equilibrium magnon Bose-Einstein condensation (BEC) in YIG at finite (room) temperature by using the method called "microwave pumping" http://www.nature.com/nature/journal/v443/n7110/abs/nature05117.html. I know there are various (sometimes controversial) discussion on their experiments and the understanding of the microscopic mechanism; about spontaneous coherence (spontaneous symmetry breaking), spontaneous condensation[A] or driven condensation, thermodynamics properties, the novel dispersion relation of YIG, and thermalization effects. From the view point of theoretical issues (to the best of my knowledge), Prof. Bunkov and Volovik http://arxiv.org/abs/1003.4889 has generalized (extended) the idea of magnon condensation and have proposed a non-equilibrium magnon BEC theory[B], in which they have viewed the magnon annihilation operator associating with (you call) the off-diagonal long-range order, $a_0=\sqrt{N_0} \exp(i\mu t-i\alpha)$, as the macroscopic condensate order paramater. Taking the experimental result by Demokritov et al into account, we can easily expect that the magnitude of the condensate order parameter decreases due to thermalization effects[C] (i.e. The dissipation into environments, phonons) and it takes the following form; $a_0= \sqrt{N_0} \exp(i\mu t-i\alpha -t/\tau)$, [D] where the variable $\tau$ reflects the dissipation. Thus although we can understand an aspect of the above experiment by Demokritov et al, I wonder whether we can regard it as Quasi-equilibrium or non-equilibrium magnon BEC (until $t<\tau$) also in the case where the magnitude of the condensate order paramater is time-dependent; $a_0= \sqrt{N_0} \exp(i\mu t-i\alpha-t/\tau)$ ? [A] In the standard or traditional equilibrium magnon BEC theory treating only equilibrium situations, the idea of "the spontaneous $U(1)$-symmetry breaking" plays the key role. [B] Magnon BEC corresponds to coherent precessions in the language of original spins. [C] By tracing out the enviromental degrees of freedoms, the time-development of the system we focus on becomes a non-unitary form. As an eaxample, please see the famous Caldeira-Leggett model or quantum master equation. [D] When we neglect the thermalization effects or the dissipation, we can easily show and see that the magnitude of the macroscopic condensation order parameter takes a time-independent constant value after switching off microwaves because the $U(1)$-symmetry of the system is recovered. I guess although this theoretically means the stable magnon BEC, it does not reflect the experimental result by Demokritov et al.