title: Isn't Newton's Second "Law" merely a definition of force?

text: Is it just me, or is Newton's Second "Law" in fact no law at all, but a mere definition of force? The concept of force does not exist prior to the statement of N2, and thus it would seem that the law is entirely analytic and does not in itself improve our knowledge about nature. Things were different if the concept of force had been introduced separatedly, and if N2 then postulated a _proportionality_ between $\mathbf F$ and $d\mathbf p/dt$. The fact that N2 is a statement of _equality_ now has the consequence that no better unit can be given for force than the traditional Newton = metre kilograms per second squared. As I see it, the only synthetic statements that could be considered part of N2 are: 1) all behaviours of and interactions between bodies may be described in terms of forces, which are applied _to_ bodies _from_ other bodies (note that this fact is not at all evident in itself); and 2) the values of the individual forces are independent of any other simultaneous forces acting from other objects. 1) now allows us to state _real_ laws about concrete interactions, like gravitation. 2) gives rise to the _superposition principle_ , while 1) and 2) together allow us to extend the concept of force from N2, which only defines the _net force_ acting on a body. The _force_ applied from a body on another body is then defined as what _would_ be the net force on the body if all other objects did not exist. In fact, it is not even evident that the concept of force is necessary for the physical science and its ability to describe the nature around us. The use of force is probably a consequence of its great practical applicability, which is in turn the result of a desire to describe nature in terms of linear differential equations. For entirely mathematical reasons, the study field of linear differential equations is rich, and thus its use in physics gives us invaluable tools that allow us to easily analyse physical systems. Work and energy are examples of concepts with properties that are again direct consequences of the mathematical framework (at least I think most of them are). However, were it not for those rich mathematical systems, it is not obvious that the mathematical framework for classical physics is even the most natural one. For example, a concept like _velocity_ (central to the force- oriented mathematical framework) seems less natural in a relativistic context, where _rapidity_ in some ways seems more in accordance with the physical system itself. So any thoughts about whether Newton's Second Law actually contains synthetic information about the physical world?


title: Operator norm of creation and annihilation operators

text: Are the creation and the annihilation operators $a(f)$ and $a^{\dagger}(f)$ for the bosonic Fock space bounded? What is their norm? So far I did not have found any note about this in the linked Wikipedia article.


title: Olympic games and the local g

text: As you probably know, the two next Olympic games will take place in London and Rio. Randall, author of xkcd, did an interesting observation (that I had never thought of) in http://xkcd.com/852/ , arguing that the difference in latitude may influence comparisons between different world records in pole vaulting. As I started considering other results that might be affected by a small change in the local g, I could not find an interesting model for swimming. Does anyone know a model to determinate if the swimming results would be better or worst in a gravity-increased scenario? i.e., would it be easier to swim on Jupiter or on the moon?


title: History of the elementary particles

text: The current model for the elementary particles is the standard model. I understand its basics It is for me however, difficult to wrap my mind around it without having some kind of knowledge of the predecessors of this model. Just like first KLMN shells are taught first instead of SPDL, I feel the need of following this model from the beginning. I found the Gell-Mann's much simpler to understand, no matter how incomplete it was as compared to today's model. So, could you please list a list of models of elementary particles right from the beginning? To study them that way would be much easier. Thanks!


title: Spinor inner products

text: The spinor inner product in particle physics is given by $\overline{\psi} \psi = \psi^{\dagger} \gamma_0 \psi $, where I take the convention that the zeroth gamma matrix is hermitian while the rest are anti-hermitian. This is invariant under spin group transformations, $\psi \rightarrow e^{\omega_{ab} S^{ab} }\psi$, with $\omega_{ab}$ real parameters and $S^{ab} = \frac{1}{4}[\gamma^a, \gamma^b]$. However, there is a second invariant inner product given by $\psi^{\dagger} \gamma_0 \gamma_5 \psi$, with $\gamma_5 = i\gamma_0 \gamma_1 \gamma_2 \gamma_3$. My question is, why not the other one? Is it down to the fact that there are observable differences between the two, and one is favoured by experiment?


title: Why can't I see far when I look in a mirror?

text: I'm myopic. It's a fact. I understand exactly how it works because Internet told me light rays encounter themeselves too soon in my eyes... that is why I can't see far objects even if I see near objects very good. Yesterday, I was looking myself in a mirror (it seems that I'm a very narcissistic person :) ) and I discovered that I can't see far objects in a mirror exactly like if I was looking to them directly. My question : light rays bouncing on the mirror to my eyes have the exact same length (I guess it works like that but I suppose I'm wrong), the length mirror -> my eyes. So why can't I see far objects good when I look at them using a mirror ? Thank you


title: What is the appropriate scaling for a Fermi-liquid fixed point?

text: I am puzzled with the RG description of Fermi liquids. Consider non- interacting spinless fermions in 2D that form a circular Fermi surface, with an energy cutoff $\Lambda\ll E_F$ ($E_F\equiv |\mathbf{k}_F|^2$ is the Fermi energy). Now, what is the appropriate transformation under which the observables (e.g. $\mathcal{G}(\mathbf{k},\omega)\,=\,\langle \psi_{\mathbf{k},\omega}\bar{\psi}_{\mathbf{k},\omega} \rangle$) remain invariant? Following Polchinski's and Shankar's RG approach, one is tempted to say, scaling $\mathbf{k}-\mathbf{k}_F$, i.e., scaling momenta measured from the Fermi surface. In the present case of a circular Fermi surface, this means scaling $r-r_F$, where $r$ is the radial coordinate in the momentum space ($(k_x,k_y)\to (r,\theta)$) and $r_F=|\mathbf{k}_F|$. I seem to recall that Shankar mentions in his paper that the fact that there is a scale, namely $k_F$, does not mean that Fermi-liquid fixed-points are not scale invariant. Strictly speaking, the scaling that maintains the Fermi surface is a non-linear scaling. Does this mean that for systems that possess a Fermi surface (not necessarily Fermi liquids) we should always cook up a non-linear scaling transformation? Is there a general recipe to deal with cases where the "zero-energy manifold" is not an isolated point?


title: Does the spin precession change sign when the angular momentum does?

text: Say you have a charged particle moving circularly in an electromagnetic field. Basic quantum mechanics tell us that its spin will precess with a certain frequency. If the same particle were traveling circularly in the opposite direction (e.g. clockwise instead of counterclockwise or vice versa) would the spin precess in the opposite direction as well, or would it stay the same? EDIT: I'm assuming that in the second case, it's traveling in the opposite direction in the same field. This may mean that the field has the be purely electric, and not magnetic, for this scenario to be possible. EDIT 2: If the field is purely electric, then classically the spin shouldn't precess at all. However, in the special relativistic case, the electric and magnetic fields are part of one tensor, and they both affect the precession. So if you're looking at this using a quantum mechanical Hamiltonian you may want to keep this in mind and use the Dirac equation instead of the Schrodinger equation.


title: magnetic field outside a solenoid

text: I couldn't picture the idea behind the zero magnetic flux outside the solenoid properly.My teacher explained it as"Components of the magnetic field in other directions are cancelled by opposing fields from neighbouring coils. Outside the solenoid the field is also very weak due to this cancellation effect and for a solenoid which is long in comparison to its diameter, the field is very close to zero. "But I didn't understand anything at all.First he said about the components of magnetic field in other direction being cancelled by magnetic field, but didnt further explain and said nothing to support his arguement.If somebody has some simple,inituitive way of understanding this phenomenon then please explain it...


title: Question about equation 2.27 from Pachos's Introduction to topological quantum computing

text: http://quince.leeds.ac.uk/~phyjkp/Files/IntroTQC.pdf above is the PDF that is hosted on his website. The equation is on page 22 (pg 30 in the pdf). In chapter 2. It is the second equation of the subsection "derivation of the holonomy". On equation 2.27 he splits up the integral from the exponential of a time evolution operator into finite steps (that are infinitely small). Because they are within an exponential the infinite series becomes an infinite product. I get all of that. WHat I don't get is how the two Unitary matrices (the $U$ and the $U^{\dagger}$) come out of the exponential. This has been vexing me for a while. This is as he goes from the center series to the right-hand side product. I'm wondering if this is a mistake or not. I'm not asking for a complete solution since I want to work it out but a hint would be nice.


title: Second fundamental form

text: How do I calculate the integral of the trace of the second fundamental form on a surface? The formula used in the Gibbons, Hawking, York paper _Action integrals and partition functions in quantum gravity,_ how do I derive it? Is it a universal or does it have any assumptions about the kind of space time we are considering? The formula is, $$\int K d\Sigma = \frac{\partial}{\partial n}\int d\Sigma$$ $K$ is the trace of the second fundamental form, $d\Sigma$ is the area element and the derivative is along the outward normal to the surface.


title: Naive unification of scalar QFT and GR is possible?

text: I am thinking on the Klein-Gordon equation with curved (non-diagonal) metrics. Is it possible? Doesn't have it some inherent contradiction? If yes, what? If no, what is this combined formula?


title: Are the fast axes on Thorlabs quarter-waveplates mislabeled?

text: Some members of my lab are performing a polarization-sensitive experiment where they need to use a quarter-waveplate (QWP) with the fast axis in a specific direction. In the process of carefully checking all the polarization optics, they discovered that for QWPs from Special Optics and Thorlabs, different optical axes are labeled as the fast axis. They have done a number of tests (1) and came to the conclusion that the ones from Thorlabs are mislabeled, and what is labeled as the fast-axis is actually the slow axis. For most experiments this wouldn't matter or be easily noticeable, but in our case (and many others) it can cause the sign of the measurement to be incorrect. After a lot of back-and-forth, Thorlabs now agrees that their QWPs are incorrectly labeled, but would like to confirm this with another research group or industrial lab. Does anyone have a QWP from either of these companies and the ability to determine if the fast axis is correctly labeled? If so, could you please answer here and let us know the model number of the QWP and how you determined the actual fast axis? Thanks! 1) Including looking at the magnetoluminescence from CdTe and the phase shift induced from total internal reflection from a glass prism. Edit: Just to clarify, this is not any sort of interesting conceptual question about waveplates, but rather a practical matter that may be relevant to a large number of scientists performing polarization-sensitive optics experiments. Edit2: We would also expect the thicknesses of the two components to be slightly different. We took some high-magnification images today and it does look like the marked edge component is slightly thicker, which also suggests that they are mislabeled. ![Thor QWP out of mount from the edge](http://i.stack.imgur.com/VeW0L.jpg) ![High res image of the edge](http://i.stack.imgur.com/x7Auc.jpg)


title: Lorentz force, multiple conductors?

text: 3 conductors are placed in a magnetic field $B$, that have equal current $I$ flowing, is the Lorentz force totaled as the sum of all those conductors? Assuming that all of them have the same direction of current.


title: A two-dimensional event horizon?

text: Today, I came across an article which says that the Universe began with the collapse of a 4-dimensional [spatial] Universe. Well, I don't know if there's any way to confirm these like it is with all other theories which deal with before the Big Bang. The strange thing was that the article mentioned about 2-dimensional event horizons. Quoting from it : > In our three-dimensional [spatial] universe, black holes have two- > dimensional event horizons -- that is, they are surrounded by a two- > dimensional boundary that marks the "point of no return." In the case of a > four-dimensional [spatial] universe, a black hole would have a three- > dimensional event horizon. Why does the article say that black holes in 3 [spatial] dimensions have 2-dimensional event horizon. I will reach the "point of no return" no matter from what direction I approach, right? Is something missing here?


title: Prove scalar product is distributive

text: The scalar product is defined as r*s = the sum of all r*s. Using this definition, prove that r*(u+v) = r*u + r*v. Also, if r and s are vectors that depend on time, prove that the product rule for differentiation applies to r*s. Ok, so I'm new to proofs and I literally do not know where to even start. Any suggestions?


title: 4-vector velocity to Newtonian?

text: The four-vector condition for a particle free of forces is: $\frac{du}{dτ} = 0$ and the equivalence of this to the statement of newton's first law follows from the expression for four-velocity: ![enter image description here](http://i.stack.imgur.com/DtUxu.png) Then I came across this part in my lecture notes: ![enter image description here](http://i.stack.imgur.com/01vnu.png) Now this got me curious. How exactly does one get to the Newtonian statement i.e. what algebra steps are involved? Thank you to anyone who can enlighten me~


title: Is momentum an extensive quantity?

text: Volume, energy of a system are examples of an extensive quantity, in contrast to intensive properties such as temperature, density etc. In a similar sense, is momentum an extensive quantity in classical mechanics? The above examples are all scalar quantities and pose no problem in being associated with either of the categories, Though momentum has mass as a factor which makes it an extensive quantity, the vectorial nature of momentum which demands parallelogram law of addition for momenta, creates peculiar problems. Again, the velocity becomes an intesive property, but it adds just as momentum does, leading to confusion. The question naturally arises whether the concept of intensive and extensive properties applies to vector quantities? Wikipedia gives momentum as an example of extensive quantity and magnetic field as an example of intensive property. Any clarification on the above will be helpful to understnd the nature of these quantities better.


title: Is there Johnson noise in superconductor?

text: For conductor, the Johnson Noise is $v_n = \sqrt { 4 k_B T R \Delta f }$. It is clear that the noise depends on $R$. I'm curious whether this noise will appear in supercondutor? That is for conductor the leading order of the noise is $v_n = \sqrt { 4 k_B T R \Delta f }$, and for superconductor the leading order is zero but it has subleading order term.


title: Gauge symmetry for p-forms

text: It is well known that the Lorentz invariance of the S-matrix implies Gauge redundancy for 1-forms,'photons'. Does this argument go through to p-forms? That is does lorentz invariance of s-matrix of these fields imply that the following is a symmetry of the action? $A_{a_1 \dots a_p} \to A_{a_1 \dots a_p} + \partial_{[a_1}\lambda_{a_2 \dots a_p]} $ where $\lambda$ is a (p-1) form.


title: Kinematics Physics on Basketball

text: A basketball is thrown straight upwards so that it reaches the bottom of the rim (2.00 m above the launch point) 0.50 s after it is thrown. What is the maximum height above the launch point which the ball reaches? Now, I use the motion equations but the answer is not as desired. It results for me a negative value for displacement. What I did was assumed initial velocity = 0, got acceleration as 16 m/s^2. And then put these in the third motion equation. But I didn't quite get a right displacement value. Any help appreciated.


title: Is there a magnetic field that satisfies $\nabla \times B = 0$?

text: In electrostatics Maxwell's equations for the magnetic field are $\nabla \cdot B = 0$ and $\nabla \times B = \mu_0 J$ Now, take $B = xi-yj$, where $i$ and $j$ are the usual unit vectors, then one can show that $\nabla \times B = 0$ which consequently means that $J=0$. But in Maxwell's equations, isn't that $J$ supposed to be the source of the magnetic field B? Then how come $J$, the source of the magnetic field, is zero yet $B$ is not zero? What am I missing?


title: Can an arrangement of particles be duplicated precisly?

text: Is it possible to teleport or clone someone or something? After watching this TED talk by Max Tegmark - https://www.youtube.com/watch?v=GzCvlFRISIM I find myself wondering if it is then possible to clone someone, and if you had cloned someone how could you tell who the clone is? Would this all be impossible due to the Heisenberg Uncertainty Principle? **Edit:** Let me rephrase a bit and say if we were to clone an insect or animal; would they be identical down to the atomic level? If I were to be atomically cloned would my clone be conscious as I am? How could I tell which is the real me? One article said - "Biological clones are nothing like these idealized versions. The statistical distributions of particles in an organism and its clone are inevitably very different - so physicists would not accept them as true clones at all." http://www.nature.com/news/2002/020521/full/news020520-1.html I also found this to be interesting - http://www.sciencedaily.com/releases/2011/11/111106150759.htm Another interesting bit of information from BioWizard of Sciencechat Forum - "Now looking back at elementary particles and atoms, their properties are inherent to their physics. It can be argued that the properties of atoms are due to the arrangement of protons, neutrons, electrons, quarks, etc within them, but there is no blue print you can copy out and move to another atom to force it to adopt this trait or that. It is not complex enough. As such, there is no such thing as cloning atoms." http://www.sciencechatforum.com/viewtopic.php?f=2&t=3165 **If someone returns to this page I thought I should also add this here: http://dare.ubvu.vu.nl/bitstream/handle/1871/10137/L59.pdf?sequence=1 As though it would seem there is a no-cloning theorem for classical cloning as well, so both are impossible. Although teleportation is possible with the destruction of the original.


title: Why does magnetic field lines go from plus to minus?

text: My question is why the magnetic field lines goes from plus to minus, if there was two charge. Is it true or isn't it true.


title: How to find an old Soviet physics paper?

text: I am interested in an old physics paper by Viktor Hambardzumyan and Dmitrii Ivanenko: Doklady Akademii Nauk SSSR, ser. A, No. 6, p.153 (1930). It concerns the structure of the atomic nucleus and accurately predicted the neutron's existence. I have been unable to locate it online.


title: Is frequency or Bayesian interpretation used in quantum mechanics?

text: In quantum mechanics, we discussed about probability. There are two kinds of interpretations: frequency and Bayesian. Which one is actually used in quantum mechanics? My impression is, it doesn't matter. However, I would like to know if my understanding is correct. (I am not sure if this is opinion-based question)


title: Experimentally measuring coherence length of laser

text: So I am trying to figure out the coherence length of my laser source. I am using a mach zehnder interferomter (image attached). I have theoretically calculated $L_c$ to be somewhere between 800-1200 $\mu m$ (equation on wiki page). So anyways. When I scan the movable mirror through a large range (50 mm), I observe interference fringes everywhere. I thought that outside of the coherence length, there should be no interference. Am I doing something incorrect?![Interferometer](http://i.stack.imgur.com/ULNEk.jpg) NOte: The top part of the MZI interferomter is an adjustable mirror. NOT TO SCALE.


title: Calculate Euler equations of fluid dynamics without division?

text: I'm working on the calculation of the Euler equations with the finite volume method. Unfortunately I'm not allowed to do a division. So I'm wondering if there's a form which does not need a division. At the moment the Euler equations look like this: $$ \frac{\partial}{\partial t} \begin{pmatrix} \rho \\\ \rho v_1 \\\ \rho v_2 \\\ \rho v_3 \\\ \rho E \end{pmatrix} = -\mathrm{div} \begin{pmatrix} \rho v_1 & \rho v_2 & \rho v_3 \\\ \rho v_1^2 + p & \rho v_1 v_2 & \rho v_1 v_3 \\\ \rho v_2 v_1 & \rho v_2^2 + p & \rho v_2 v_3 \\\ \rho v_3 v_1 & \rho v_3 v_2 & \rho v_3^2 + p \\\ (\rho E + p) v_1 & (\rho E + p) v_2 & (\rho E + p) v_3 \end{pmatrix} $$ As you can see, I first need to calculate $\frac{\rho v_1}{\rho}$ to get $v_1$ so I can calculate e.g. $\rho v_1^2$


title: Boundary conditions for Laplace's equation

text: Given a grounded conducting sphere, $V=0$ and $radius = R$, centered at the origin with a pure electric dipole (dipole moment $\vec p$) situated at the origin and pointing along the positive $z$ axis, I should be able to solve Laplace's equation in spherical coordinates to find the potential everywhere in the sphere. I can separate the differential equations and use Legendre polynomials, but I'm having trouble defining and using my boundary conditions. What I (think I) know so far: $$V(r,\theta)=\sum_{n=0}^\infty (A_n r^n + \frac{B_n}{r^{n+1}}) P_n(\cos{\theta})$$ $$r\to R \Rightarrow V\to 0$$ $$\theta \to \frac{\pi}{2} \Rightarrow V\to 0$$ $$\theta \to 0 \space or \space \pi \Rightarrow V\to \frac{\hat{r}\cdot \vec{p}}{4\pi\varepsilon_0r^2}$$ At least I think those boundary conditions work. I need to define a condition as well for the case of $$r \to 0$$ but that would seem to make the potential explode. How can I use these boundary conditions to solve for the potential everywhere inside the sphere?


title: strength calculations - scooter

text: I need some help to compute strength calculations of my homemade scooter. the plate (where you stay on) are 20mm x 500mm x 10mm In the back there are 1 wheel, and in the front there are 2 wheels connected to the control rod. The plate and wheels are aluminium


title: Calculating the force of a gyroscope attached to an armature

text: I stumbled across an illustration of a simple legged armature stabilized using a gyroscope, and I'm attempting to understand the math behind it. I've simplified the model to this diagram: ![enter image description here](http://i.stack.imgur.com/hZe4y.png) Given the radius, orientation and spin of the gyroscope with respect to the ground, how would I calculate the force `f` applied against weight `w0`? How would I determine the amount of spin necessary to stop the armature from toppling over and dropping the weight?


title: Derivation of Landau diamagnetism

text: In deriving the magnetic susceptibility of free electrons, we need to calculate $$\chi = \left( \frac{\partial M}{\partial H} \right)_N = - \left( \frac{\partial^2 F}{\partial H^2} \right)_N.$$ Here, $F = E- TS $ is the free energy. It should be emphasized that the particle number $N$ is held fixed. However, in the statistical mechanics book by Landau, he calculated $ - \left( \frac{\partial^2 \Omega}{\partial H^2} \right)_\mu $, where $\Omega = F - \mu N $ is the grand potential. See his equation 59.11 on page 174. The reason seems to be that it is much easier to calculate $\Omega$ as a function of $T, H, \mu$ than $F$ as a function of $T, H, N$. But, is the calculated quantity the one we really want?


title: Is the half-life of Primordial Nuclide determine only by the age of the earth?

text: Is the half-life of Primordial Nuclide determined only by the age of the earth? Or are their other age defining considerations? The point of the question is to know that the only reason that we have a determination of the half-life of Primordial Nuclide is by its comparison solely to the lifespan of the earth, or if their is any other foundation for determining the half-life. I want an alternate method of determining the duration half-life of a Nuclide. Specifically Primordial Nuclide. The only method that I know about right now is through visual measurement.


title: How do electrons choose their path?

text: Wire A                       -----R-----                          /           \     ------I--->-[p]--             ---------I-->                      \           /                       -----R-----                             Wire B      Wire A and B both have resistance $R$.   So current $\frac{I}{2}$ flows through each of the wires. How come electrons know which wire path to choose? What I am thinking is if we consider electrons at point P are in state $\frac{|A\rangle + |B\rangle}{\sqrt{2}}$ and at the junction some kind of measurement occurs which leaves half of the electrons in state $|A\rangle$ and other half in $|B\rangle$. Please explain how exactly this works. Please point out why argument above is right or wrong (or at all relevant) in this situation. **EDIT:** How can one derive basic laws for series and parallel ckts from more basic principles? Thanks! :-)


title: Home experiment to estimate Avogadro's number?

text: How to get an approximation of Avogadro or Boltzmann constant through experimental means accessible by an hobbyist ?


title: Quantum death like heat death possible?

text: Quantum decoherence is an irreversible process which is the result of interaction of the system with its environment. It prevents interference due to lack of coherence. Environment acts just like a heat bath. Now my question is, is the different branches of the wave function of the universe becoming gradually more and more decoherent so that in a far future no trace of interference can occur? In other words I would like to know whether there will be any quantum death (like heat death of thermodynamics) of the universe if one waits for a sufficiently long time?


title: How to feel or sense magnetic fields from earth?

text: I know that earth has magnetic fields/forces, but why we don't feel them? So if I hold a magnet and earth's magnetic field is positive and my magnet is from the positive side, then why the magnet does still fall down? Shouldn't be there a pullsive force since they will be of the same polarity?


title: How stellar aberration is measured?

text: A simple calculation shows that stellar aberration due orbital motion of earth is roughly 20 arcseconds. My questions are:   1. Practically how this small value is measured?    2. Does this value is in the range of accuracy of a 11 inch reflective telescope with a camera?    3. And how this measurement originally was done by Bradley in the 18th century?


title: absorbing energy from explosion

text: The other day in physics, we went to visit a workgroup that was developing a wall that would absorb energy from explosions (it was a joint project with the military academy). The first prototype they developed which they deemed unsuitable worked a bit like this: [ (solid layer) (cilinders) (to protect wall) ] The idea was that localized pressure changes would be spread evenly by the solid layer and the cilinders would work a bit like pistons. But, a real explosions creates two rapid changes in pressure. First you have a shockwave with a rapid increase in pressure and then there comes a different shockwave with a rapid decrease in pressure. Their protoype would first absorb the first shockwave by crushing the cilinders but the second shockwave would move their solid layer back to the place it first came from and the cilinders would expand again. They then proceeded to say that if you looked at that system before and after the explosion, the solid layer would be at the same position so it didn't absorb any energy and all energy was absorbed by the wall you were supposed to protect. Isn't that reasoning completely falacious? From my point of view it looked as if that wall absorbed energy two times, once in the first shockwave and once in the second shockwave making it pretty effective. I couldn't ask my question at the moment we were given the presentation as any colaboration was not really appreciated by the prof.


title: Basic Free Fall Problem

text: So I am trying to remember my childhood doing some Physics problems, but seems that I forgot almost everything. But it's not a big thing... that's why I'm training! My problem is pretty basic although I remember just a little... let me ennounce it: > A 5 kg block of ice falls away from the edge of the roof of a block of > flats, at a height of 26 m above the ground. Ignoring air drag, find out: > > a) how long it takes for the ice block to hit the ground. b) what is the > speed at which the ice block hits the ground. c) how much energy it > transfers to the surroundings when it comes to a stop and break into pieces. > (g = 9.8 m/s^2) For a), I tried: t = d/r, and gives me 2.653 secs. For b), I tried: 9.8 * 2.653, and this results in 25.9994 m/s For c) I tried nothing since I don't remember what should I do. Can someone take a look at this problem and tell me if I am doing it right, or just messing all the things? Thank you!


title: Notation in the book Symmetry by Hermann Weyl

text: I'm having troubles understanding a notation of the symmetry groups in a book "Symmetry" by Hermann Weyl. On the page 80 of the 1952 Princeton University Press edition of the book, Weyl lists the three-dimensional point groups. Can somebody, who has read this book and has knowledge of these groups (I am at the very beginning of studying group theory), translate his notation to Schönflies notation which I am more familiar with?


title: SpinFET Transistor

text: When a gate voltage is applied to a current of spin polarized electrons, a spin precession will occur. If this spin preccesion is enough to make the bulk electron spin polarization anti-parallel to a ferromagnet, why is it that the electrons will not flow through the ferromagnet? Basically my question is: Why won't electrons spin polarized in a direction anti-parallel to a ferromagnet's spin polarization flow through this material? Assuming everything is ideal...


title: Regarding Non-renormalizatibility of GR

text: I've been doing some reading trying to get to a better understanding of some renormalization issues with the Einstein-Hilbert action. But, something odd came into mind that I'm hoping some users may have the background to comment on it. Specifically, most of the sources I have seen are based on expanding the Einstein Hilbert action about a flat background to achieve a Lagrangian of the form $(\partial h)^2+{1\over{k}}(\partial h)^2h+{1\over k^2}(\partial h)^2h^2+...$with terms that become more and more divergent the further out one goes. But, for some reason I find I'm a little pre-occupied with the notion that perhaps the particular expansion itself has issues with convergence that may or may not fundamentally have to do with the E-H action in its non expanded form. We commonly entertain the notion of a cut-off energy range to deal with the renormalization issues in the 'effective field theory' speak. How do we know that the expansion of the E-H itself isn't only valid as a low energy approximation though? Do we know that there isn't an expansion of an alternate form which has convergent terms for a range of energies where the first runs into problems?


title: How to thoroughly distinguish a coordinate singularity and a physical singularity

text: In a course on general relativity I am following at the moment, it was shown that the singularity $r=2M$ in the Schwarzschild solution is a consequence of the choice of coordinates. Introducing Kruskal-Szekeres coordinates $(u,v)$ resolves this problem: the singularity at $r=2M$ disappears, but if one draws a $(u,v)$ graphs with light cones and such, one still recognizes the event horizon at $r=2M$. The singularity at $r=0$ remains and is said to really by an essential singularity. So in general: if you can find a coordinate transformation to get rid of a divergence in your metric, it is not a true singularity. However, it struck me that the Kruskal-Szekers coordinates were only discovered in 1960 (44 years after the Schwarzschild solution). This leaves me to wonder: is there a more systematic way of distinguishing physical vs. 'fake' singularities? In Carroll's book, I've read something about contractions of curvature quantities diverging at real singularities: E.g. $R^{\alpha \beta \gamma \delta}R_{\alpha\beta\gamma\delta}\propto r^{-6}$ such that $r=0$ is a real singularity (and $r=2M$ not). Could anyone make this ad-hoc rule more quantitative?


title: Frequency difference of absorbed and emitted photons?

text: In a transparent medium, how does the frequency of incoming light compare to the exiting light? I thought it was the same, but doesn't there have to be an energy difference in two electron states in the atoms of the medium that is exactly equal to the energy of the photon? I feel like an exact match would be unlikely. Is there a litlle energy lost in the process that lowers the exit frequency by a slight amount?


title: Recommendations for time-line and road map in graduate school towards specializing in Maldacena's conjecture

text: This question was asked on Theoretical Physics Stackexchange and was grossly misread and closed. I am again posting the question here hoping to get some valuable insights. Also some people were unhappy with the term Maldacena conjecture since there is so much evidence in favour of "AdS/CFT". I guess still the point remains that there is no true "proof". Isn't it true that the Gopakumar-Vafa invariants were discovered in an attempt to "prove" AdS/CFT? To put one of the central questions first - I guess the canonical resource to learn about Maldacena's conjecture is the famous "MAGOO" review or may be the Hoker and Freedman's review. Any other suggestions? The "problem" I see is that standard QFT courses don't teach about N=4 SYM or about N=8 Supergravity and absolute fluency with such stuff seems to be utterly necessary for getting into this field. How does one develop this and from where and how much time should it require? Most bizarrely someone read this question as being asking as to what is the "least" one can study and "become" another Jacob Bourjaily! (whatever that means!) Thats farthest from the intention. There is no denying of the fact that some people have intrinsic IQ edges over others but that is hopefully not a reason for others to stop trying to be the best! (..I am being hopelessly optimistic that non-exceptional people also can produce cutting edge string theory results!..) On the contrary the point of this question is to want to know as to at what rate do the best students work and study. Its more like wanting to know what do speeds of study of "most of the best" students look like. Surely there is not one-size-fits-all answer to this but surely there are generic patterns to how the cutting edge string and field theorists of today did their graduate school. As to what courses did they take at what stage and how much of what they learn by when and from where. I would like to know as to how or at what rate does the academic life or a good graduate student progress in a theoretical physics grad school if he/she is aiming to specialize in topics like say the Maldacena conjecture (or "AdS/CFT" as it is often said), or issues of integrability emerging from it and such related questions. I have in my mind "role models" of certain brilliant students who have recently completed their PhDs in such stuff like Jacob Bourjaily, Tudor Dimofte, Silviu Pufu etc. Assume that a grad student starts learning QFT and related mathematics from the first day of grad school (which I guess is already too late!) I guess some of these recent very successful PhDs were way ahead of such scenarios! Then, how much and how far into QFT should someone know before he/she can get into cutting edge of Maldacena's conjecture? To split up the question,   * How long should it take to learn enough of QFT (what is enough?) ?   * At what stage and after knowing how much of QFT should one be able to start learning String Theory?   * How long and how much of String Theory should one know before being able to get through the literature in the topics mentioned above?   * Can one start reading the papers (or even working?) in these topics along with learning QFT?   * Can one start learning String Theory along with learning QFT or do they have to come in a strict order? For all of these above questions I would love to know of the characteristic time-line in terms of months into graduate school when each of these milestones should be covered. I guess this will help know what is the right pace to work at - which I guess was the speed in which some of these people mentioned earlier worked at.


title: Bullet trajectory at different angles

text: I've read from different sources (can't find them online) that firing a bullet at different angles from horizontal will result in differing amount of bullet drop. For example, a gun is fired at a target 2000 yards away horizontally. The bullet drops _X_ inches. Again the gun is fired at a target 2000 yards away with a 30 degree angle upward. The bullet drops _Y_ inches. _X_ does not equal _Y_. Same scenario, only 30 degrees downward angle. Bullet drops _Z_. _Z_ does not equal _X_ or _Y_. Why? What is happening? The bullet is fired at the same speed at the same distance, the bullet should be in the air for the same time. Acceleration due to gravity is a function of time, so the bullet should drop equally in all cases. Right? The only change in each case is the angle gravity acts on the bullet. My guess is that when the bullet is fired upwards, gravity is pulling it downward, thus slowing the bullet down causing it to be in the air longer and increasing the drop.


title: How do the LASERs in LIGO realize that space has expanded as a gravitational wave passes by?

text: I read an article on LIGO, and I heard it mentioned that it is a nontrivial argument to say that the effect can be measured by interferometry. What happens to space as the wave passes? Does the light change wavelength due to the space that it is traveling through expanding? or does it keep its original wavelength according to a reference frame unaffected by the wave?


title: How can airplanes fly upside down?

text: I've read many times, including here on this very site that the commonly known explanation of flight is wrong, and that airplanes can fly because the shape of their wings deflects air down. This makes sense, but as far as I can tell it doesn't explain upside down flight or symmetric wings. The images I've seen show an inclined wing, which forces the air to go downwards. But how can planes fly upside down then?


title: Can you fuse Anti-matter atoms?

text: I know that this is a simple question, but I could not for the life of me find a source that explains fusion for Anti-matter atoms. intuitively, I can say that there shouldn't be any barrier for fusion of anti-matter atoms since they have the same structure as atoms. Given a large enough anti-hydrogen gas cloud, secluded in a region of space without the interference of matter, an Anti-matter star should be able to form, fuse all the way down to Anti-iron, build up a large enough core, go supernova and keep fusing into heavier elements. I just need a simple yes or no answer or a source explaining this. Thank you


title: Triality and charge

text: I have a few questions about triality for the representations of $SU(3)$. (I have seen the wikipedia page, but it does not make the connection with physics.)   * What is triality, how can you compute it from the dynkin labels?   * What has triality to do with the fractional charges of quarks?   * Why does the fact that $\overline{\underline{6}}$ has the same triality as $\underline{3}$ implies that we cannot have quarks in the irreps of the subgroup $SU(3)\times G_2$ of $E_6$? These questions arises from the follows paragraph in the text "Group theory for unified model building" by Slanksy (on page 98): > The $\underline{27}$ of $E_6$ branches to $(\underline{3},\underline{7}) + > (\underline{\overline{6}},\underline{1})$ of its maximal subgroup > $SU(3)\times G_2$. If $SU(3)$ were color with $U^\text{em}(1)$ in $G_2$, > then the $(\overline{\underline{6}}^c,1)$ states would be neutral. Since > $\underline{6}^c$ has the same triality as $\underline{3}^c$, it is > impossible to maintain the usual quark charges.


title: Distribution of star colours in a galaxy

text: I'm trying to randomly generate a representation of a galaxy. I have some Idea on the spatial distribution of stars within a galaxy, and I can find plenty of material on the colour of stars, but nothing that seems to give either the distribution of numbers of stars by colour or (if any exists) spatial distribution by colour. Are there any models for distribution of the colour of stars in a galaxy either by frequency or by location within a galaxy? What do these models depend on? galaxy age? type of shape of galaxy?


title: Forces involved in a climbing jet aircraft

text: I have a problem with which I need help. The question is: A jet aircraft is climbing at an angle of 45° above the horizontal and is accelerating at 4.5m/s2. What is the total force that the cockpit seat exerts on the 75kg pilot? It says that the y-component of the force is: $$F\sin(a)–mg\cos(45^\circ) = 0.$$ I don't understand how this component is working at all. I was just wondering if anybody could help me understand why this is the y-component of the force exerted by the seat.


title: On conservative forces

text: We say if we calculate the work done by a force in going from 1 to 2 following a path say A is equal to -1 times the work done by the same force in getting from 2 to 1 following a path say B then work done by the force along any path to get to 2 from 1 is the same and the work done is hence path independent while the force is conservative. But how can we prove this rigorously that if a force is conservative then it is path independent?


title: Static Electricity and Grounding Confusion

text: 1. In an electroscope, you take metallic balls, and rub them with either a comb or a glass. Touching the balls with your finger is said to undo the effect of rubbing them with the comb or glass, since your finger is connected to ground, and ground is so enormous that it absorbs or releases the electrons to neutralize the metallic ball.   2. When you rub your shoes against a carpet and then touch a metallic doorknob, a shock results, since the build-up of electrons that was built up by rubbing against the carpet is released via the doorknob to ground. What I don't understand in #2 is how the buildup of electrons occurs in the first place. We've already seen in #1 that you are connected to ground, so in #2, why don't any excess electrons released from the carpet at the time of the rubbing immediately go back to ground? (I guess I'm asking a larger question: how can any rubbing of anything create a surplus or deficit of electrons if the thing being rubbed is connected to ground -- why doesn't that connection immediately discharge any imbalance of electrons created by the rubbing?)


title: Spectrum of CMB vs. duration of last scattering

text: The epoch of last scattering took over 100,000 years. The Visibility Function has maximum at about 370,000 years after the Big Bang when the temperature was about 5,600 K. 50,000 years earlier the temperature was around 6,000 K and 50,000 years after the maximum it was approximately 5,200 K (I used this tool to do the calculations). The temperature difference at the beginning and the end of the last scattering epoch is about 800 K. At 6,000 K the spectrum peaks at 483 nm while at 5,200 K the peak is 557 nm. Some of the CMB photons we see today come from the earlier stage of last scattering, some of them from the later stage, so the resulting spectrum would be a convolution of spectra from various times. However the spectrum is not distorted: it looks perfectly like a blackbody spectrum. Does it mean the CMB photons were cooling at the same rate as the plasma? So all photons which decoupled earlier cooled down (due to expansion of the space) just right to perfectly aligh their wavelengths with the photons that decoupled later on?


title: Can the Lorentz force stabilize the Hydrogen atom?

text: I've recently been working on relative equilibria for some systems of particles. (ie. studying equilibrium solutions in a rotating frame. Saturn's rings for example.) This has evolved into some classical notions of atomic physics and questions of stability for Coulomb interactions there. Entertain for a few moments a classical(planetary) model for the atom. Take an isolated hydrogen atom for example. It seems to me that the $q\left(v\times B\right)$ term of the Lorentz force provides an intrinsic stabilization for the two particle system. Naively play the right hand rule game with the system and you'll see. If we assume that the electron is in a classical orbit and not radiating, then could we establish the stability of atoms with the Lorentz force? The non-radiating electron is a topic of its own and appears to be far from solved. For example, if you would like to make the statement that accelerating charges radiate, therefore...... Try to go one step further and show why this must be true in general and you imediately run into difficulties. Gyro-stabilization might be a relevant topic here. Order of Magnitude investigation: \begin{align} &\text{Proton Charge:}\qquad&q &\approx 1.602\times 10^{-19}\: C\\\\[2mm] &\text{Permeability of free space:}\qquad &\mu_0 &\approx 4\pi\times10^{-7}\frac{volt\cdot s}{amp\cdot m}\\\\[2mm] &\text{Permitivity of free space:}\qquad &\epsilon_0 &\approx 8.854\times10^{-12}\frac{farad}{m}\\\\[2mm] &\text{Electron Speed: }\qquad &v &= \alpha c\approx c/137 \approx 2.19\times10^6\frac{m}{s}\\\\[2mm] &\text{Electron Radius:(Bohr) }\qquad &r & = \frac{\hbar}{m_ec\alpha}\approx 5.29\times10^{-11}m\\\\[2mm] \end{align} Maximum Fields(Im thinking of the fields generated by the electron in a linear approximation for an infinitesimal section of the circular orbit: \begin{align} \text{Biot - Savart}\qquad{\bf B}_{\text{max}} &= \frac{\mu_0 q v}{4\pi r^2}&\approx& 12.537\:\text{T}\\\\[2mm] \text{Coloumb}\qquad{\bf E}_{\text{max}} &= \frac{q}{4\pi\epsilon_0 r^2}&\approx& 5.145\times 10^{11}\frac{N}{C} \end{align} With Force ratio: \begin{align} \frac{q(v\times{\bf B})_{max}}{q{\bf E}_{max}} =\frac{\frac{\mu_0 q^2 v^2}{4\pi r^2}}{\frac{q^2}{4\pi\epsilon_0 r^2}} = v^2\mu_0\epsilon_0 = \frac{v^2}{c^2}\approx\frac{(2.19\times10^6)^2}{(3\times10^8)^2}\approx 5.329\times10^{-5} \end{align} This shows the magnetic force is 5 orders of magnitude smaller than the electrostatic force for this crude model of the ground state hydrogen. Not sure what this conclusively says about the internal dynamics but at least I think I can foresee some problems with this force stabilizing the system. A colleague of mine who does work with NMR(nuclear magnetic resonance) informed me that the magnetic fields they work with in the lab are on the order of 3$T$ and that they do not observe the large fields I am envisoning in practice. The concept he quoted as the reasoning behind this was "orbital quenching". This appears to be a sort of averaging to zero of this field. However, it was also stated that the magnetic field that I am interested in can be measured in beam type experiments.


title: Atomic emissions and energy time uncertainty principle

text: Am I right, according the time-energy uncertainty principle, to say that an excited hydrogen atom in free space could emit photons with energies different from those possible by Bohr's calculations? As you may have noticed, there is no need to mention the fine/hyper-fine corrections and like things.


title: Electric field of half a wire

text: ![Drawing](http://i.stack.imgur.com/1Zoec.jpg) I am trying to calculate the electric field of a half wire in polar coordinates. At $y=R$ is a wire going from $-\infty$ to $0$ with charge density $\rho$. I want to calculate the electric field at $(0,0)$ $$ \vec{E} = \frac{1}{4\pi\epsilon_0} \int_{0}^{-\infty} dx \; \frac{\rho}{r^2} \hat{r} $$ My idea was to integrate over the angle $\theta$ from $0$ to $\pi/2$. Then I get $\vec{r} = \frac{R}{\cos \theta} \hat{\theta}$ and my integral is $$ \vec{E} = \frac{1}{4\pi\epsilon_0} \int_{0}^{\pi/2} d\theta \; \frac{\rho \cos^2 \theta}{R^2} \hat{\theta} $$ How do I solve this integral? I don't know what to do with $\hat{\theta}$ and what the resulting direction of the field is. Any suggestions?


title: Why does the coriolis effect dissapear at the equator?

text: I'm studying from the book "Classical Mechanics" by Goldstein and from a coursebook my Professor provided me. In the coursebook, it says that "the Coriolis effect disappears at the equator (Where the rotation vector $\vec\omega$ of the Earth is horizontal)" Now here's my reasoning: Take the Coriolis term: > $-2m(\vec\omega\times\vec v_r)$ Nearby the north pole, this would cause a force pointing towards the equator if you were moving from west to east. The closer you get to the equator, the more this force starts pointing "upwards" if you were to describe the vector from the surface of the Earth.   At the equator, this vector is perpendicular to the tangent of the equator and pointing outwards.   Intuitively, this would mean that if you move from west to east on the equator you would be accelerating away from Earth. Now if my reasoning is correct, this doesn't mean the Coriolis effect disappears, only it turns into some sort of centrifugal force, but this seems weird because it's described by another term. Where did I go wrong?


title: What is the difference between primary and volume currents in electromagnetism?

text: What is the basic difference between the two currents and how each is contributing to the magnetic field at source and the surface?


title: Ring Resonators

text: If optical field will propagate into the following ring resonator. Optical field will propagate into the ring and propagate to the coupling region. I didn't get where and how the interference will take place and what we will get from the output at the wave guide ?![enter image description here](http://i.stack.imgur.com/K1ocU.png)


title: Is it useful to geometrically represent conservation laws?

text: In my physics class, we just studied collisions. We learned that momentum is conserved in collisions. I decided to examine a one dimensional collision in which two objects collide - object 1 and object 2. We can put the masses of the colliding objects are put in to a vector $\vec{m} = \langle m_1,m_2\rangle $. We can also put the velocities of the colliding objects in to a vector $\vec{v} = \langle v_1,v_2\rangle $, where the directions of $v_1$ and $v_2$ are given by their signs. Then, the momentum of the system is given by: $p = \vec{m}\cdot \vec{v}$, where the direction of the momentum is given by the sign of $p$. Now we can express the conservation of momentum in this way: $$ p = \vec{m}\cdot \vec{v}= \|\vec{m}\|\|\vec{v}\|\cos(\theta) = \text{constant} $$ where $\theta$ is the angle between $\vec{v}$ and $\vec{m}$. So, if the masses are constant, then if $\|\vec{v}\|\cos(\theta) $ is constant, momentum is conserved. If the mass and velocity vectors are based at the origin and the mass vector stays constant, then momentum is conserved as long as the projection of the velocity vector on to the mass vector remains constant. This defines a line along which the tip of the velocity vector can slide while conserving momentum. Through this process the conservation of momentum has been geometrically represented, by a line in this case. With a bit of work, I think this could be done for other conservation laws and with more interacting objects. For instance, another curve could be drawn that corresponds to the conservation of kinetic energy. My question is: Is representing conservation laws in this way useful? That is, does it make understanding any situations easier or allow one to solve certain problems in a better way? Also, if this technique is useful for anything, what sorts of topics could I research in order to learn more about it? Edit: Kyle Kanos made some good points below. I should clarify how this process results in the geometric expression of a conservation law. The ordered pair of masses $(m_1,m_2)$ and the ordered pair of velocities $(v_1,v_2)$ can be plotted in some two dimensional space as directed line segments based at the origin (which I was referring to as vectors above, but I will avoid doing so from now on). For the ordered pair of masses, the directed line segment would go from the point $(0,0)$ to the point $(m_1,m_2)$. Similarly, for the ordered pair of velocities, the directed line segment would go from the point $(0,0)$ to the point $(v_1,v_2)$. So, we have two directed line segments which represent the system in a certain configuration. I would call this a "geometric representation" of the system. Now, as shown above, as long as the projection of the velocity related directed line segment on to the mass related directed line segment remains constant, then the momentum of the system remains constant. So there is a set consisting of an infinite number of velocity related directed line segments for which the momentum of the system is the same. All of the terminal endpoints of these directed line segments fall upon a line. As long as the velocity related directed line segment has its terminal endpoint along this line, momentum is constant. So both before and after the collision, the velocity related directed line segment's terminal endpoint must be somewhere along this line. The conservation law restricts the terminal endpoint of the velocity related line segment to fall upon a line. This is a geometric expression of a conservation law.


title: Bullet: Piercing vs. Deflection

text: I am trying to create a simulation where a projectile (imagine a bullet) is fired at a surface of some kind. When it collides with the surface, I need to figure out what happens between the bullet and surface. > Example: > > Bullet fired at extremely sturdy object, bounces off the object. > > Bullet fired at very soft object, pierces into object. > > Bullet fired at thin sturdy object, dents the object but does not pierce it. 1) Does what I'm looking for have a name? I am having problems finding any type of equations that may relate to my examples. 2) Can somebody please explain to me what the forces are in this example and how they interact? 2b) What are the variables that I need to consider?


title: Coleman-Weinberg potential: resum at 2 loops?

text: Say we want to compute the Coleman-Weinberg potential at 2 loops. The general strategy as we know is to expand the field $\phi$ around some background classical field $\phi \rightarrow \phi_b + \phi$, and do a path integral over the quantum part of the field, $\phi$. We can retrieve the effective action by doing a path integral, something like eq.42 in this reference. There are 2 ways to do this at 1 loop, we can either evaluate a functional determinant or do the classic Coleman-Weinberg thing where we sum up all diagrams we get by inserting any number of background fields $\phi_b^2$ into the loop integral. This is eq. (56) of that same reference again. My question is, why do we not need to do this resummation over background field insertions at 2 loops? For example, in this (quite standard) reference, as well as in chapter 11 in Peskin and Schroeder, the authors seem to claim that the 2 loop contribution to the path integral are simply the "rising sun" and "figure 8" vacuum diagrams, and no summing over classical field insertions is even mentioned. What am I missing? EDIT: To give some more details, in perturbation theory, each diagram contributing to the path integral is spacial integral of some functional derivative acting on the free field path integral with a source: the loop diagram with n insertions of external field $\phi_b$ is the term: $$\left( \phi_b^2 \int dx \left( \frac {\delta}{\delta J(x)}\right)^2 \right)^n Z_0[J]$$ The 2 loop figure 8 is $$\int dx \left( \frac {\delta}{\delta J(x)}\right)^4 Z_0[J]$$ The 2 loop diagrams that it seems like the papers cited above are excluding are contributions like $$\left( \phi_b^2 \int dx \left( \frac {\delta}{\delta J(x)}\right)^2 \right)^n\int dx \left( \frac {\delta}{\delta J(x)}\right)^4 Z_0[J]$$ It seems to me that these terms will indeed arise in the exponential expansion of the interacting lagrangian, so it seems that a resummation over $n$, as in the 1 loop case, is still necessary. Where is my error?


title: Physical interpretation of Parseval's theorem

text: I have read that Parseval's theorem, relating the norm of a function $f$ and the norm of its Fourier transform $g(k)$: \begin{equation} \int |f(x)|^2 dx=\int|g(k)|^2 dk \end{equation} has the simple physical interpretation of "conservation of energy". I just don't see this, so can you suggest me a way to think about it?


title: Why does blowing on hot coffee cool it down?

text: And will it cool off faster if you blow across the top of the cup or directly into the coffee? Does it have to do with the fact that when you blow across the top of the cup the velocity of the air increases which causes an area of low pressure above the cup, resulting in steam from the coffee to be forced up and out of the cup? Or is that not it?


title: Describing Objects in the First Dimension

text: The "third" dimension is the one we experience day-to-day. An example of the "second" dimension is the shadow of a three-dimensional object. How do we describe the "first" dimension? What would be an example of matter in the first dimension?


title: The influence of the expanding universe on the wave function

text: I was puzzled by a question on a physics fora about a possible connection between the expansion of the universe and the wave function. So, a photon's red shift tells us something about the expansion, but also implies a loss of Energy. Where does this Energy goes to, and is there a direct effect on such a particles wavefunction? Over time, shouldn't there be like a correction factor in the wavefunction compensating this expansion phenomena...?


title: What is the status of Everett's Thought Equation?

text: Here is Hugh Everett's PhD Thesis: http://philosophyfaculty.ucsd.edu/faculty/wuthrich/PhilPhys/EverettHugh1957PhDThesis_BarrettComments.pdf On page 9 he says: > "We have the task of making deductions about the appearance of phenomena to > observers which are considered as purely physical systems and are treated > within the theory." If I got this straight, Everett proposed something kind of like a thought experiment. But it wasn't exactly an experiment you did with thoughts. It was an equation you made with thoughts. A thought equation. With quantum mechanics, you can write a wave equation to describe a particle. Or perhaps a group of particles. The more particles you describe, the more complex the equation gets. Everett proposed something bold. A wave equation that contains all the particles of object to be measured. But he didn't stop there. If you're going to measure the object with a particle detector, or a ruler, or a clock, those things are made out of particles themselves, and in the equation they go. Now you've got an object made of particles as well as measuring tools made of particles all in the wave equation. But that's not all. An observer, like you or me, is also made of particles. In the equation we go. Or at least something like us: "an automatically functioning machine" with senses and memory. This idea of Everett's is now that the entire wave equation can be set in motion as a purely physical system. The wave never needs to collapse, as it does in other versions of quantum mechanics, because as the object and the observer interact, the observer's memory will contain the outcome of its measurements. Everett called the measurements encoded in the modeled observer's memory "relative state". The wave equation itself, the "universal wave function" as he put called it, would presumably be the absolute state. Everett's thought equation could, in theory, be turned it into real mathematics. But a wave function describing all the particles needed to make an object, and measuring tools, and an observer would take an unreasonable amount of paper and time to write down. **My question is...** Aren't those limitations gone in the information age? Shouldn't we be able to turn Everett's wave equation that contains an observer into a computer model? I know there is a Human Brain project working on simulating the human brain for medical purposes. What's the status on physicists modeling an observer? **Example of what I'm asking about** We have lots of models of reality. But not one like Everett's thought equation. When we want to make a model of the solar system, we make a model of the solar system. When we want to make a model of a particle of light, we make a model of the particle of light. What we don't do is make a model of an observer made of particles interacting with the particle of light performing a measurement. And that is precisely what is happening in Everett's thought equation. So, if we are to implement Everett's idea with software, the primary requirement is:   1. Make a model of an observer performing a measurement We have plenty of models of things we measure, but we don't have any models of the measurement actually being made by an observer. Now there are some stipulations on how this can be achieved. Everett stresses repeatedly that the observer is a purely physical system, interacting with the object being measured in a purely physical way. This can be translated into the following software requirements:   1. There are no special rules that give an object in the model the power of observation or measurement. The only rules of the model govern the mechanical interactions of the objects within it.   2. Through the mechanical interactions of the objects in the model, the objects arrange into compounds that act similarly to atoms and molecules.   3. The simulated atoms and molecules form the necessary components for an automatically functioning machine with senses (such as cameras or eyeballs) and a memory (such as a computer or a brain)   4. The simulated machine with the sensor and memory interacts mechanically with the object and the measuring tools, and a measurement is recorded in the memory of the observer **Why I'm asking** Software that fulfills these requirements would present two different sets of information to anyone examining it. First, the algorithm and its variables and their values. This could be exposed to the researcher explicitly, or accessed through a debugger. This is how the program looks to the programmer as it's running. This is the program's absolute state. The second set of information in the software cannot be found in the variables of the model directly. Some of the variables in the model form an observer with a memory. You have to look at that memory to find the second set of information. An abstract set of relational data. This is how the program looks to something that exists inside in it. This is the program's relative state. The relative states are the unique feature of software that meets these requirements. With other models, you only have one set of states. By simulating measurement within the model, we have two states, absolute and relative. The idea is to then take the relative states of the model, which are the observations made by a simulated observer, and compare them to our observations of reality. This leads to some very interesting questions and answers. For example, let's ask the simulated observer to measure the position and momentum of a particle simultaneously. How will the observer do? The observer's first measurement will disturb the particle being measured and affect the second measurement. Other models don't have a measurement happening as a mechanical process inside it. So they need to comply with the uncertainty principle in other ways. But the absolute/relative model has a measurement actually occurring inside it, and it seems intuitive the model is going to implicitly produce measurements in accordance with the uncertainty principle. Just as Einstein had to remind us that the space and time we measure and discuss in physics are relative, here is Everett telling us that the matter we measure is also a relative state of reality. Perhaps the absolute states of the program are indeed compliant with the uncertainty principle. But they don't have to be. If the predictions of the model come from the relative states, only the relative states have to be consistent with empirical reality. Nothing in the software requirements dictates anything about the true nature of absolute matter, except that they form an observer whose measurements are consistent with ours. Hopefully that is somewhat clear. **Are there research programs trying to implement Everett's Thought Equation?** How far along are they?


title: Capacitance and energy

text: So I've been reading about capacitance (Specifically in capacitors) and I'm wondering if this reasoning is sound, and why/why not: C = Q/U U = Q/C We also know that: U = E/Q So: E/Q = Q/C Q^2 = EC E = Q^2/C However, if I try to use this it conflicts with some of the answers in my textbook, but I don't see where I made any mistake.


title: What could I test for in a guillotine?

text: I'm doing a school assignment project and I need to investigate something of my choice. I was thinking of doing something about how the height and angle the blade of a guillotine is dropped from affects how accurately and/or how deep it cuts. And also, it wouldn't hurt if anyone knew the directions to building a mini guillotine. So does anyone know the basic Physics of the guillotine, per se? There's not much about it on the internet and I might have to go and search the library for more info... But if anyone could help, I'd soo appreciate it! thanks all!


title: Does using batteries in a series raise the overall Wh's of the batteries used? Why/Why Not?

text: If one has 2 batteries of 6 Volts each with 500 mAh each, then separately the batteries should provide 6 Volts * 500 mAh = 3 Wh each or 6 Wh total. However would it be the case that if someone where to connect the two batteries in a series that they would be able to double the Wh, as now the equation would be 12 Volts * 1000 mAh = 12 Wh. Why/Why not?


title: An introductory resource for learning AdS space

text: Can someone please point me to introductory resources about the geometry of Anti DeSitter Space ? What are some examples of other spaces used in theoretical physics ?.I'm learning Differential Geometry right now , I want to apply what I learn on simple spaces of importance in theoretical physics. Like , I want to investigate the nature of these spaces , calculate their metrics , curvature and other things.


title: Can a very thin sheet of any material float inside a liquid?

text: When a body is immersed in a liquid,buoyancy is the net force of all the forces acting on it. Now the forces are equivalent to those which will act on the same volume of liquid.Rightly so,but considering the chaotic motion of the molecules of the liquid,for every force acting on that particular volume of liquid,there will be another force in the opposite direction but of different magnitude.The magnitude differs because of the difference in height. Now let's consider a very very thin sheet immersed in the liquid.The opposite forces now will be of the same magnitude as difference in height is negligible and hence will cancel each other!so the only force acting will be its weight. So the thin sheet will never be balanced!So does that imply that no thin sheet will ever float inside a liquid??


title: Waves and information

text: I'm looking to transfer very simple information using audio waves. One of the approaches I'm looking into is using different frequencies. For example, "command 1" will be transmitted using 500Hz wave, where "command 2" using 700Hz wave. On the other side, the decoder will look if we have a wave with that frequency and then interpret the command. The challenge appears when I'd like to transmit 2 (or more) commands simultaneously - I get a new wave form. If I remember right from my physics studies - it's a "Wave package", so I get new frequencies, additional to the original ones. My question is - Is there a way to receive this "wave package" and interpret the original commands from it? Many thanks!


title: Ehrenfest's theorem on Gaussians

text: Considering the free evolution of a Gaussian wave packet, is it possible to use Ehrenfest's theorem to determine the average value of momentum given that of position? And I imply the simplified version of the theorem, namely $\frac{\mathrm d}{\mathrm d t}\langle x \rangle = \frac{1}{m}\langle p \rangle$, where $x$ and $p$ are the position and momentum, respectively.


title: induction for massive objects

text: We know that when two charged bodies are kept near one another, they attract the charges and induce some charge near the surface. This happens even when only lne body is charged as the charges are pulled to one side. My question is that why doesnt a similar phenomenon takes for mass. As we know that massive particles are attracted by massive particles, then why doesnt the mass of a body get concentrated at one side when placed in gravitational field of another body ?


title: Is it possible to mechanically isomerize an sp3 hybridized carbon center?

text: Imagine I have an sp3 hybridized carbon attached to four separate polyethylene chains. By pulling on the polyethylene chains in some manner, is it possible for me to mechanically isomerize the chiral center prior to breaking any carbon-carbon bonds? Perhaps a more realistic scenario would involve shining a laser on a compound similar to 2,2-dimethylpropane (i.e. a fully saturated carbon compound where four carbons are attached to an sp3 hybridized carbon), but with asymmetric functionalizations on the bonded carbons to allow for detection of isomerization. Just as in the previous scenario, can mechanical isomerization occur prior to breaking any carbon-carbon bonds? * * * Quoting from Georg's answer: "In "practice" there is a big problem, You would need two Laplacian demons (with two hands each) to do the experiment..." It's not clear to me that this is an impossible experiment. For example, you could attach one polyethylene chain to a surface and then try to make a covalent bond between the other relevant chain and, say, an AFM tip. If this is successful, one would then proceed in a manner similar to that reported in "How Strong is a Covalent Bond?" by Michel Grandbois et. al. (http://www.sciencemag.org/content/283/5408/1727.short), which estimates a ~4.0 nanonewton rupture force for a C-C bond (See Fig. 4) and directly measures the rupture of a single Si-C bond to be ~2.0 +/- 0.3 nanonewtons. In the case where one wants to look for mechanical isomerization of the sp3-hybridized carbon, one would look for an earlier sub-4.0 nN peak in the force vs. extension curve (or whatever critical force is experimentally measured for breaking a C-C bond), perhaps allow relaxation at some critical force to complete the isomerization process, and then compare the observed displacement with a geometrical model. Sounds ridiculously hard, but I see no fundamental reason it couldn't be done.


title: Simplest interferometer

text: I want to build simplest interferometer which should be able to measure movements down to fraction of wavelength. What is the simplest scheme for that, and what are the requirements for a laser? I have a bunch of laser diode-based ones, and I guess they might be not coherent enough... Are green DPSS ones any better?


title: N particles, will there be any rotation after a period of time or everything will collapse

text: This is in context of classical Newtonian physics. Consider a system of n different point mass particles. Initially all are spread around on one plane. No particle possess any velocity to begin with. There is no external force acting on this system. Each particle has some mass. The only force to consider here is gravitation between any two particles. When two particles collide, lets say they form a lump with mass equal to sum of the two. This now behaves like one single particle. My first assumption is, the centre of mass of this system, throughout time won't move. Right? Given a sufficient amount of time, will everything collapse into one single particle at the centre of mass? Or it's possible to see a system where few particles are rotating in one direction and other in other direction, so that angular momentum is conserved and net angular momentum is still zero? If yes, can you give one simple example with points, mass and initial position which will achieve this kind of state. EDIT: On similar note, can we have say 3 particles which are attracted to each other but never collide, something like a simple harmonic motion?


title: If a credit card won't read when swiped, does wrapping it in plastic actually help and if so, why?

text: I've on occasion seen when a cashier has trouble getting a credit card to read, will wrap it in a thin plastic grocery bag, and try swiping it again. Much to my surprise, it seemed to work. Since the card uses a magnetic strip, I can't see how the plastic would help. Is there any reason it should, or is this a case where it worked, but not because of the plastic?


title: What is this form of notation called?

text: $$^{14}_6C \rightarrow ^{14}_7N + E^{-} + \bar{\nu}_e$$ Just curious!


title: Why does the spiral of a positron have a larger radius than that of an electron in this picture in a bubble chamber?

text: The smaller spiral is caused by an electron The bigger spiral is caused by a positron However, they have the same mass and magnitude of charge. So, during this pair production, why does the positron follow a spiral that has a larger radius? ![enter image description here](http://i.stack.imgur.com/9CKRA.jpg)


title: Optimal tunnel shape for travelling inside the earth

text: Say you were to travel from Paris to Tokyo by digging a tunnel between both cities. If the tunnel is straight, one can easily compute that the time for travelling from one city to the other (independently from where both cities are actually) is a constant (about 42 minutes). It goes without saying : only gravitationnal potential and no friction. But let's upgrade on the idea. Fix the two cities : **what is the optimal shape of the tunnel** , so that the time it takes for going from one city to the other is shortest ? I assume a planar movement, where the plane contains both cities and the center of the earth. With Gauss' law, it is easy to compute that the gravitationnal potential on a mass m inside the earth at any distance $r$ from the center : $V(r) = m\alpha r^2$ where $\alpha$ is a constant. I parameterize the tunnel by a polar function $r(\theta)$ where the origin is taken at Earth's center. The problem reduces to minimizing the following functional $\mathcal{T}(\dot{r},r,\theta) = \int \limits_{\theta_1}^{\theta_2} \frac{ds}{|v|} = \int \limits_{\theta_1}^{\theta_2} d\theta \frac{\sqrt{\dot{r}^2 + r^2}}{|v|} = \sqrt{\frac{m}{2}}\int \limits_{\theta_1}^{\theta_2} d\theta \frac{\sqrt{\dot{r}^2 + r^2}}{\sqrt{E - m\alpha r^2}} \equiv \sqrt{\frac{m}{2}}\int \limits_{\theta_1}^{\theta_2} d\theta G(\dot{r},r,\theta)$ $\theta_1$ and $\theta_2$ are the angular coordinates of both cities. $\dot{r}$ is the derivative of $r$ with respect to $\theta$. With this parametrisation, the 'Hamiltonian' is conserved : $\frac{\partial G}{\partial \dot{r}}\dot{r} - G = c$ Which leads to $\frac{-r^2}{\sqrt{E-2\alpha r^2}\sqrt{\dot{r}^2+r^2}} = c$ And here I get stuck. Even if I try to play with special initial conditions, I cannot solve this equation. The easiest way is to perform separation of variables, but the integralcannot be carried out. Any ideas on how to solve this problem ? I am sure the solution is a nice well known symmetric function. I just don't know how to tackle this problem. Would you tink it'd be easier to solve this in cartesian coordinates ?


title: How do I write the Hamiltonian for a 3-level system?

text: I came across following types of three-level systems like V-system, Λ-system and 2-photon absorption ![enter image description here](http://i.stack.imgur.com/Xq1Lo.png) It seems that their Hamiltonians can be written intuitively by checking out the coupled levels and the gap between levels both in general and in the dressed picture (rotating frame approximation). Do V and Λ systems have the same Hamiltonian due to the symmetry in the level structure ? By following the analogy, I designed the Hamiltonian for 2-photon absorption process to be same as the other 2 cases but it is not so :- What are the coupled levels in the 2-photon absorption process and how can the Hamiltonian for this process be written ? How do I extend the design to N-level system -lets say for N-photon absorption in N-level system ?


title: Why does the Alcubierre drive require negative energy?

text: The Alcubierre drive is an idea for a faster-than-light spaceship. It works by contracting space-time in front of the ship, and expanding it behind the ship. Physicists say that this requires the use of negative energy or exotic matter. But why is that? I mean, positive mass and normal gravitational fields _do_ bend space-time. If we could get a hold on creating and manipulating large gravitational fields, wouldn't we be able to create this drive?


title: Large gauge transformations for higher p-form gauge fields

text: **Question: What is the large gauge transformations for higher p-form gauge field on a spatial d-dimensional torus $T^d$ or a generic (compact) manifold $M$?** for p=1,2,3, etc or any other integers. Is there a homotopy group to label distinct classes of large gauge transformations for p-form gauge field on $d$-dimensional torus $T^d$ or any $M$ manifold ? (shall we assume the theory is a topological field theory, or not necessary?) **References are welcome.** * * * **Background:** Large gauge transformation has been of certain interests. The Wiki introduces it as > Given a topological space M, a topological group G and a principal G-bundle > over M, a global section of that principal bundle is a gauge fixing and the > process of replacing one section by another is a gauge transformation. If a > gauge transformation isn't homotopic to the identity, it is called a large > gauge transformation. In theoretical physics, M often is a manifold and G is > a Lie group. **1-form:** The well-known example is a connection $A$ as Lie algebra value 1-form. We have the finite gauge transformation. $$ A \to g(A+d)g^{-1} $$ An example of a large gauge transformation of a Schwarz-type Chern-Simons theory, $\int A \wedge dA$, on 2-dimensional $T^2$ torus of the size $L_1 \times L_2$ with spatial coordinates $(x_1,x_2)$ can be $g=\exp[i 2\pi(\frac{n_1 x_1}{L_1}+\frac{n_2 x_2}{L_2})]$. This way, for the constant gauge profile $(a_1(t),a_2(t))$ (constant respect to the space, satisfying EOM $dA=0$), the large gauge transformation identifies: $$ (a_1,a_2)\to (a_1,a_2)+2\pi (\frac{n_1}{L_1},\frac{n_2 }{L_2}) $$ This seems the two $\mathbb{Z}^2$ integer indices $(n_1,n_2)$ remind me the homotopy group: $\pi_1(T^2)=\pi_1(S^1\times S^1)=\mathbb{Z}^2$. **2-form:** If we consider a 2-form $B$ field for a Schwarz-type TQFT, do we have the identification by $\pi_2(M)$ on the $M$ as the based manifold? (Note that $\pi_2(T^d)=0$ - ps. from the math fact that $\pi_2(G)=0$ for any compact connected Lie group $G$.) Is this the correct homotopy group description? How does large gauge transformation work on $T^d$ or $M$? **3-form:** is there a homotopy group description on large gauge transformation? How does its large gauge transform on $T^d$ or $M$?


title: Do we have magnetic monopole? Is charge, according to yesterdays paper on Nature by Ray-Roukokoski-Kandel-Möttönen-Hall (30.01.14) quantized?

text: As everybody here knows, Maxwell's equation would look more beautiful if a magnetig charge were present. Beyond the aesthetics question, if a single magnetic Dirac monopole would be found, the consequence is that the electric charge would be quantized. While, I guess, everybody should be happy with the news, my experimentalist background doesn't allow me to understand the experimental setting. I therefore ask for the following advice -- a rather theoretically oriented question. A recent paper, > _Observation of Dirac monopoles in a synthetic magnetic field._ M. W. Ray, > E. Ruokokoski, S. Kandel, M. Möttönen and D. S. Hall. _Nature_ **505** , pp. > 657–660 (30 January 2014). claims to have observed such monopoles.   * Can somebody shed light on the theoretical implicacions of this work? Concretely, should we now take care before saying that it's been proven that the electric charge is quantized?


title: Physical meaning of Maxwell's equations and origin of EM waves

text: Is it possible to describe the physical meaning of Maxwell's equations and show how they lead to electromagnetic wave, with little involvement of mathematics ?


title: Work done from force-extension graph

text: ![enter image description here](http://i.stack.imgur.com/BDD2l.png) The answer is C, but I got B... F at 3cm = 10, F at 6cm = 20 $W=F\cdot s$ $(20-10)\times 3=30$ I must have made a mistake or stupid assumption then, how am I meant to get 45 N cm?


title: Can you get a playground to swing from stationary

text: I feel this might be a FAQ but I would love a definitive answer. Imagine a frictionless stationary idealised child's playground swing. If you are sitting on the seat of the swing, is it possible in principle to set it into motion by simply moving your body and pulling on the chains of the swing? If so, is there a simple explanation for how you can change your center of mass without an external force acting on you?


title: extracting energy from cosmological expansion

text: This question is a more concrete reincarnation of an old question about energy conservation in GR. Are there mechanisms to extract energy from the cosmic rate of expansion? putting some extremely long ropes across the universe and extract work from their increase of separation? assuming this is possible ( i don't see why it would not ), what form would take the back-reaction force? in all physics, every time you extract work from some system by some force, this system will exert a force back. Can we slow down cosmic expansion by variations of these mechanisms?


title: Quick question on Perturbation theory - How do I evaluate this probability?

text: We know that hamiltonian for interaction between an electron and external field along $z$ is: $$\hat H = \frac{e\hbar B}{2m}\hat \sigma_z = \frac{\hbar \omega}{2} \hat \sigma_z $$ This has energy eigenvalues of $\pm \frac{\hbar \omega}{2} $ Time evolution operator is given by: $$ exp(i\frac{\omega}{2}\sigma_z) = \left( \begin{array}{ccc} e^{-\frac{i\omega t}{2}} & 0 \\\ 0 & e^{\frac{i\omega t}{2}} \end{array} \right) $$ At time $t=0$, the electron is in state $ \left( \begin{array}{ccc} a_0 \\\ b_0 \end{array} \right) $ and at $t>0$, the state of the electron is: $$ |\psi_{(t)}\rangle = \left( \begin{array}{ccc} a_0 e^{-\frac{i\omega t}{2}} \\\ b_0 e^{\frac{i\omega t}{2}} \end{array} \right) $$ Suppose now we have a small perturbation field in the x-y plane: $$B_1= B'\left( \begin{array}{ccc} cos (\omega t) \\\ sin (\omega t) \\\ 0 \end{array} \right) $$ Thus the perturbation is: $$V_1 = \frac{e\hbar}{m}\vec B_1 \cdot \vec S = \frac{e\hbar}{2m}B' \left( \begin{array}{ccc} 0 & e^{-i\omega t} \\\ e^{i\omega t} & 0 \end{array} \right) $$ I now want to find the probability the electron transits from the lower energy state to the higher energy state. By first order perturbation theory, $$\frac{\partial a_m^{(1)}(t)}{\partial t} = e^{\frac{i\Delta E}{\hbar}t}\frac{1}{i\hbar} \sum_n a_n^{(0)} \tilde V_{mn} \space dt $$ . Thus for this case, it is represed by this matrix: $$i\hbar \frac{\partial}{\partial t} \left( \begin{array}{ccc} a^{(1)} \\\ b^{(1)} \end{array} \right) = \left( \begin{array}{ccc} \langle-|V_1|-\rangle & \langle -|V_1|+\rangle \\\ \langle+|V_1|-\rangle & \langle+|V_1|+\rangle \end{array} \right) \left( \begin{array}{ccc} a_0 \\\ b_0 \end{array} \right) $$ Since the particle is originally in the lower state, and we want to find the chance electron jumps to higher (+) state, we want to find first order perurbation $|a_{+}^{(1)}|^2 = |b^{(1)}|^2 $ with $b_{(0)} = 0 $ $$ i\hbar \frac{\partial}{\partial t} \left( \begin{array}{ccc} a^{(1)} \\\ b^{(1)} \end{array} \right) = \frac{e\hbar B'}{2m} \left( \begin{array}{ccc} 0 & e^{-i\omega t} \\\ e^{i\omega t} & 0 \end{array} \right) \left( \begin{array}{ccc} a_0 \\\ 0 \end{array} \right) $$ Thus $b^{(1)} = \frac{e\hbar B'}{2mi\omega}a_0e^{i\omega t} $ and thus probability of jumping from lower to higher state is: $$ (b^{(1)})^2 = \left(\frac{e\hbar B'}{2m\omega}a_0\right)^2 cos^2 (\omega t) $$ Is it okay to simply take the real value? Because if I don't, it will be $|bb^*| = const.$ which is wrong, as it should oscillate with time.


title: Does physics have some division schema which divide physical amounts into these two classes?

text: Does physics have some division schema which divide amounts into these two classes? :   * A) amounts which can be counted by natural numbers (for example many units can be counted by number of electrons, photons per second etc.)   * B) amounts which cannot be counted by natural numbers (time, speed, length) It looks like we understand much more things from class A and the history of physics can be described as an evolving possibility of counting more and more things with natural numbers. What does this mean? Why are natural numbers so special? And why can't some things be counted with natural numbers?


title: Proof for a time-ordering equation in Negele & Orland (1998)

text: Let $T$ be the time-ordering operator which orders operators $A_1(t_1), A_2(t_2), \ldots$ such that the time parameter decreases from left to right: $$T[A_1(t_1) A_2(t_2)] = A_2(t_2) A_1(t_1) \text{ if } t_2 > t_1 \text{ and }= A_1(t_1)A_2(t_2) \text{ otherwise. } $$ The time $t_i$ does not have to be a physical time, it can also be an imaginary time, etc. > **Question:** I would like to know why the following equation holds: for > $t_i \leq t_1, t_2 \leq t_f$ it holds that $$T\left[A_1(t_1) A_2(t_2) > \exp\left(-i\int_{t_i}^{t_f}H(t) dt\right)\right] \\\ = > T\left[\exp\left(-i\int_{t_{\pi_1}}^{t_f}H(t) dt\right)\right] > A_{\pi_1}(t_{\pi_1}) \cdot > T\left[\exp\left(-i\int_{t_{\pi_2}}^{t_{\pi_1}}H(t) dt\right)\right] > A_{\pi_2}(t_{\pi_2}) \cdot T\left[\exp\left(-i\int_{t_i}^{t_{\pi_2}}H(t) > dt\right)\right] ,$$ where $\pi$ is a permutation such that the times are > ordered. I encountered this equation in Negele & Orland (1998) in eq. (2.49) on p. 63 and in eq. (2.67b) on p. 70, where they split the integral $$\int_{t_i}^{t_f} dt = \int_{t_i}^{t_{\pi_2}} dt + \int_{t_{\pi_2}}^{t_{\pi_1}} dt + \int_{t_{\pi_1}}^{t_f} dt$$ and used the time-ordering. It appears in calculations of greens functions respectively correlation functions. I tried to prove this equation in an elementary way by using $$ T\left[\exp\left(-i\int_{t_i}^{t_f}H(t) dt\right)\right] = 1 + \sum_{n=1}^\infty \frac{(-i)^n}{n!} \int_{t_i}^{t_f} d\tau_1 \ldots \int_{t_i}^{t_f} d\tau_n T \left[ H(\tau_1) \ldots H(\tau_n) \right]$$ [cf. eq. (2.10) on p. 50] and applying the $T$-operator on the expression, but I did not succeed yet. If someone can show me a valid proof or point out some literature where it is proven, I'd be thankful.


title: A change in the mass of the moon

text: Would a change in the mass or radius of the moon have any effect on its speed? When asked this question in an assignment, I stated that it did not. My reasoning being that a change in the mass or radius of the moon would in no way change the acceleration or velocity of the moon. The moon would travel along the same path, the only difference being that the gravitational force being exerted by/on the Earth and Moon would be of a lesser magnitude. As the mass of the Moon decreases by ½, does the gravitational force. F=ma ½F=½m*a The original gravitational force being exerted is double that of the new gravitational force 2(G MiMii/r2)=G Mi(1/2Mii)/r2 I got this answer wrong and I am unsure why. Was I wrong in my approach? Are there other forces that I should have taken into account when coming to my conclusion? Edit:The mass of the moon simply vanishes.


title: What is the meaning of Wheeler's delayed choice experiment?

text: Wheeler's delayed choice experiment is a variant of the classic double slit experiment for photons in which the detecting screen may or may not be removed after the photons had passed through the slits. If removed, there are lenses behind the screen refocusing the optics to reveal which slit the photon passed through sharply. How must this experiment be interpreted?   * Does the photon only acquire wave/particle properties only at the moment of measurement, no matter how delayed it is?   * Can measurements affect the past retrocausally?   * What was the history of the photon before measurement?   * What are the beables before the decision was made?


title: How does a state vector be projected onto an eigenspace after measurement

text: In http://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics#Degenerate_spectra, it is said that > _If there are multiple eigenstates with the same eigenvalue (called > degeneracies),..., The probability of measuring a particular eigenvalue is > the squared component of the state vector in the corresponding eigenspace, > and the new state after measurement is the projection of the original state > vector into the appropriate eigenspace._ My question: Is the state vector after measurement when the eigenspace is degenerated a pure state or mixed state? And what is the mathematical formulation of the mentioned "projection" onto the eigenspace?


(Theoretically) Are there any differences between photons emited /absorbed by antimatter atoms to photon in usual atoms? for example, is it impossible to tell the difference between a photon emmited by an atom and it's anti matter counter part? Assuming the same orbital transitions for their electorn and positrons? Are the absorptions lines the same as well? or are there absoption lines for anti hydrogen are expected to be different than the hydrogen? What is convention for anti elements? e.g is anti matter counter part of He is just denoted as -He or $\bar{He}$?


title: Why is it hard to solve the Ising-model in 3D?

text: The Ising model is a well-known and well-studied model of magnetism. Ising solved the model in one dimension in 1925. In 1944, Onsager obtained the exact free energy of the two-dimensional (2D) model in zero field and, in 1952, Yang presented a computation of the spontaneous magnetization. But, the three- dimensional (3D) model has withstood challenges and remains, to this date, an outstanding unsolved problem.


title: QCD corrections in W boson production

text: The book "Collider Physics", by barger and Phillips, deals with hadronic W and Z productions, and includes the $K$ factor which represents QCD corrections. However, when dealing with the hadronic process $$\bar{u} d \to W \to e \nu$$ and then later with the inclusive $$AB \to e \nu X,$$ (pg. 248, 255) it does not include the $K$ factor. It does, however, again includes the $K$ factor on pg. 260, when dealing with transverse mass. Is this simply an error or is there a reason for this?


title: How can Kaluza-Klein particles be told apart from winding modes at the LHC?

text: I`ve already asked this in the comments below this article http://profmattstrassler.com/articles-and-posts/some-speculative-theoretical- ideas-for-the-lhc/extra-dimensions/how-to-look-for-signs-of-extra-dimensions/ from Prof. Strassler but did not get a satisfactory answer... In the article Prof. Strassler explains that a possible sign of extra dimensions at the LHC could be the detection of Kaluza-Klein particles. The energy (or mass squared) spectrum of the corresponding particle tower would have a step size proportional to 1/r (where r is the radius of a compact extra dimension). Now I`m wondering how such a (hypothetical) KK particle candidate can be told apart from a winding mode particle (with the step size of the energy spectrum proportional to r) which is probably more easily produced depending on the size r of the extra dimension ? By the way what's the actual state of these searches for extra dimensions considering the data already available ? **Clarification** Considering an extra dimension that is much larger than the particle, the KK modes with the 1/r spectrum arise from the quantized angular momentum a particle can have due to it' movement in the extra dimension. If the extra dimension is smaller the particle can wind around the extra dimension leading to the winding mode particles. For these the energy spectrum is determined by the potential energy of stretching. The spectra of the KK and winding modes are T-dual to each other.


title: Is there a non-perturbative remormalization? If so, how does it work?

text: Is there a method to renormalize a theory without using perturbative expansions for the divergences? For example, is there a method to get masses and other renormalized quantities without using expansions and counterterms? O have heard about Lattice Gauge theory but what other solutions of examples of non perturbative renormalization (numerical or analytical ) are there?


title: Topological band theory

text: Why topological insulators were discovered so late? While the band theory was known long time ago! I mean why the topological properties of electronic bands were not noticed in the past?


title: Acceleration and Circular Motion

text: Lets assume that there is a force that makes our body moves in circular motion. We know that the acceleration of a body that moves in circular motion is Velocity ^ 2 / Radius . How is it possible that if we grant velocity to a body with mass M , Its acceleration is not dependent on M ?


title: Alternative interpretation of Off-shell internal QFT propagators?

text: In Quantum Field Theory in a (1, D - 1) space-time, to calculate transition amplitudes, we are using Feynman diagrams, where internal lines (internal propagators) corresponds to momenta which are said to be off-shell : $p^2 \neq 0$. Is it posssible to interpret these momenta as on-shell, but within (2) space- time(s), with a supplementary dimension ? * * * For instance, consider a massless (1,3) space-time Quantum Field Theory, like pure Yang-Mills theory (gluons only). External legs corresponds to on-shell momenta $p^2 = (p_o)^2 - (p_1)^2 - (p_2)^2 - (p_3)^2 = 0$ Internal lines correspond to (1, 3) off-shell momenta, but there are 2 possibilites, $p^2 >= 0$ and $p^2 <= 0$. For $p^2 >= 0$, a (1, 4) on-shell condition could be written : $$ (p_o)^2 - (p_1)^2 - (p_2)^2 - (p_3)^2 - (p_4)^2 = 0$$ For $p^2 <= 0$, a (2, 3) on-shell condition could be written : $$ (p_{-1})^2 + (p_o)^2 - (p_1)^2 - (p_2)^2 - (p_3)^2 = 0$$ So, we could consider than free particles (on-shell external legs) live in a (1, 3) space-time, but than "interactive" particles (off-shell internal lines), live in 2 "interaction" space-times, which have signatures (1, 4) and (2, 3). So we could say, that in fact, everything is on-shell, but interaction make one supplementary dimension appears (depending on the value of $p^2$, this is a time-like or space-like dimension). Another way to say it, is that free particles could live on a space-time which could be the intersection of 2 "interaction" space-times. A even more interesting model would be that free particles live in a space- time which would be the boundary of some "interaction" space-time.


title: How sharp is the direction of magnetic field of a cone magnet?

text: I heard recently that cone magnets can be used to focus and direct the field sharply at one particular point in a particular direction. I jus want to kno how accurate and sharp this field is? And if its so sharp that it does not influence the area surrounding that point but only straight above it, how small can the cones be fabricated and whaether we can fabricated cone electromagnets?


title: Difference between $\Delta$, $d$ and $\delta$

text: I have read the thread regarding 'the difference between the operators between $\delta$ and $d$', but it does not answer my question. I am confused about the notation for **change** in Physics. In Mathematics, $\delta$ and $\Delta$ essentially refer to the same thing, i.e., change. This means that $\Delta x = x_1 - x_2 = \delta x$. The difference between $\delta$ and $d$ is also clear and distinct in differential calculus. We know that $\frac{dy}{dx}$ is always an operator and not a fraction, whereas $\frac{\delta y}{\delta x}$ is an infinitesimal change. In Physics, however, the distinction is not as clear. Can anyone offer a clearer picture?


title: Is dark matter repulsive to dark matter? Why?

text: I think I saw in a video that if dark matter wasn't repulsive to dark matter, it would have formed dense massive objects or even black holes which we should have detected. So, could dark matter be repulsive to dark matter? If so, what are the reasons? Could it be like the opposite pole of gravity that attracts ordinary matter and repulses dark matter?