title: More questions on string theory and the standard model

text: This is a followup question to How does string theory reduce to the standard model? Ron Maimon's answer there clarified to some extent what can be expected from string theory, but left details open that in my opinion are needed to justify his narrative. Let me state what his answer suggested to me: Based on some indirect evidence, it is conjectured that   (i) one day, string theory produces in one of the many possible low energy approximations a theory equivalent to the standard model or a slight variation of it.   (ii) string theory cannot predict that the universe must be in this low energy approximation, as the actual low energy approximation is a random result of a stochastic dynamics, and hence as unpredictable as the masses and distances of sun and planets in the solar system.   (iii) In particular, the phenomenological input needed (roughly corresponding to boundary conditions in a classical theory) includes ''the qualitative structure of the standard model, plus the SUSY, plus say 2-decimal place data on 20 parameters''. From this, it is conjectured that infinitely accurate values of all parameters in the standard model are determined by string theory. Support for these conjectures is a matter of belief. To me, none of the three statements is so far plausible:   (i) can be supported only by finding such an approximation.   _What other finding could support this conclusion?_ (ii) is quite different from what we see at lower energies. For example, while we cannot predict the shape of a particular bunch of water, we can derive completely all macroscopic (i.e., low energy) properties of water from microscopic principle and a few constants defining the microscopic laws (masses of H, O, and the electron, and the fine structure constant). And the parameters of the standard model define in principle the properties of all nuclei (though we haven't yet good enough numerical methods).   _What makes string theory unique in that not only the particular low energy states but also the low energy dynamical laws are random results?_ (iii) could only be substantiated if it could be shown at least for some of the many low energy approximations that knowing some numbers to low pecision fixes all coupling constants in theory to arbitrary precision.   _Is there such a case, so that one could study the science behind this claim?_ Also (iii) is somewhat disappointing, as it means that most of the potential insight an underlying theory could provide must already be assumed: Quite unlike Newton's theory, which is indispensible to derive from a few dozen constants the complete motion of the planets, or quantum chemistry and statistical mechanics, which are indispensible to derive from a few dozen constants all properties of macroscopic materials, string theory does not provide a similar service for elementary particle theory, as quantum field theory already contains all machinery needed to draw the empirical consequences from 'the qualitative structure of the standard model, plus the SUSY, plus'' say 4-decimal place data on 33 parameters.


title: Index raising and lowering - how does it work?

text: In the context of four-dimensional spacetime, how does the metric turn a tangent vector into a gradient, and vice versa? By this I mean that I know the metric can be used to raise and lower indices: $$g^{\mu\nu}V_{\nu}=V^{\mu}$$ and $$g_{\mu\nu}V^{\nu}=V_{\mu}.$$ But my understanding is that, at a point in spacetime, the components of a contravariant vector can be thought of as a tangent vector to a parameterised curve $$V^{\nu}=\frac{dx^{\nu}}{d\lambda}$$ where $\lambda$ is the parameter along the curve. And the components of a covariant vector can be thought of as the gradient of a scalar field $$V_{\nu}=\frac{\partial\phi\left(x\right)}{x^{\nu}}.$$ How does the metric flip $\frac{dx^{\nu}}{d\lambda}$ to $\frac{\partial\phi\left(x\right)}{x^{\nu}}$ or vice versa? To my (non- mathematical) eyes these two things (a tangent to a parameterised curve and a scalar field) look very different to each other.


title: Does the second law of thermodynamics tell me how the entropy changes?

text: In thermodynamics I can e.g. compute the properties of ideal gases with certain energies $U_1,U_2$ in boxes with certain volumes $V_1$ and $V_2$. Say I have two such boxes and they have some specific parameter values, like say I know their temperatures $T_1,T_2$. Now I _put them together_ , the gases can interact and I thereby construct a new box with volume $V_1+V_2$ and the energy is $U_1+U_2$. Using the laws of thermodynamics, I can compute everything else again now. Say I find a new temperature $T_{\text{new}}$. One says that the temperatures of the gases in the systems _changed_ by putting the boxes together. However, on the computational side what I did was just considering a system of new specifications. To get the new result, I didn't have to enter the real world staring conditions except for the variables which also were necessary to compute their respecive properties - effectively, the theory didn't have to tell me how the system changed, just what the restrictions are - and I basically entered into a new system with these values. For example, it's not relevant which gas had with specifications before. I just did a little trivial algebra and computed what has to be. When then talk about a _change_ of entropy and how the gas behaves, but that seems to be only decorative. If I say that the second law of thermodynamics tells me that heat flows from a system with high temperatures to a system of low temperature I'll be able to derive rules for the entropy say, how the combined system has to look like if I state their variables and insist on extensivity of certain variables and so on - but it's not about change in the sense that I compute how the system developes from one point to the other in details. Rather I just compute how the end configuration has to be. My question is > Is the starting configuration of a thermodynamics system ever relavant? And secondly, > In thermodynamics, if I compute "the change a system" in the sense of the > above example, do I always induce the necessarity for doing such a thing by > stating "now we bring system one and system two in contact"?


title: When one can make telescope that can distuingish 1 mas?

text: When one can make telescope that can distuingish(angular resolution) 1 mas object by angular diameter? If already possible, tell the name of telescope/observatory. How to calculate angular resolution $\theta$ of Telescope from the diameter $D$ of its mirror? Just in case I'm understood wrong; is it through Rayleigh equation; $\theta \approx 1,22 \lambda/D$?


title: Divergence of One and Two Graviton Exchanges

text: At the bottom of pg. 3, Kiritsis states the following > To appreciate the difficulties with the quantization of Einstein gravity, we > look at a single-graviton exchange between two particles (Fig. 1.1a)). Then, > the amplitude is proportional to $E^2/M_P^2,$ where $E$ is the energy of the > process and $M_P$ is the Planck mass, $M_p\sim 10^{19}\, \mathrm{GeV}$. . . > . Therefore, the graviational interaction is irrelevant in the IR ($E\ll > M_P$) but strongly relevant in the UV. In particular, this implies that two- > graviton exchange diagram (Fig. 1.1b)) is proportional to the dimensionsless > ratio $$ \frac{E^2}{M_P^4}\int _0^\Lambda d\tilde{E}\tilde{E}\sim > \frac{\Lambda ^2E^2}{M_P^4}. $$ My question is: How does he obtain the $E^2/M_P^2$ and $\Lambda ^2E^2/M_P^4$ with so little effort? What is this physicist black-magic? I would have to at least:   1. Write down the interaction terms in the Lagrangian.   2. Write down the kinetic term for the graviton in the Lagrangian.   3. Deduce the graviton propagator (my attempt at this physicist black-magic is telling me that it is just proportional to $1/k^2$?).   4. Deduce the interaction Feynman rule.   5. Write down the amplitude.   6. Determine the superficial degree of divergence.   7. Write down the result. And even upon trying to do this in my head, I still didn't get the right $\Lambda$ dependence. (In the two-graviton case, there is $1$ loop, $2$ bosonic propagators, and $2$ fermionic propagators, so the superficial degree of divergence should be $4\cdot 1-2\cdot 2-2\cdot 1=-2$, no? The only thing I can think of is that the scalar interaction term would contain derivatives, increasing the superficial degree of divergence, but I don't see how this would happen with fermions.)


title: Why does large curvature of spacetime imply high temperature?

text: I`ve just stumbled about a sentence which says that high curvature of spacetime implies that **any** matter present is at high temperature. This somehow confuses me, so my probably dumb question(s) are: 1) How is this general (?) relationship between the temperature of matter and curvature of spacetime derived or explained? Just looking at Einsteins field equations, I dont see why there cant be some cold mass just "sitting there" leading to large curvature ... 2) Is this relationship generally valid or is its domain of applicability restricted somehow? For example has the curvature of spactime to be so large that quantum effects kick in and it has something to do with the uncertainty principle?


title: Interference, photon's phase, and the Hilbert space

text: Dirac said that a photon can only interfere with itself. This is consistent with the tensor product of two photon spaces representation. On the other hand, it is known that there is interference between distinct sources, between two photons, between two photons again, and between two electrons. This is related to this SE question, and this one. Let $\frac 1 {\sqrt 2}\left(|1\rangle|2\rangle + |2\rangle|1\rangle\right)$ be a two-photon state. Let's consider interference between the two photons. If one "rotates" one of the wavefunctions with a phase factor, one expects that this will affect the interference, and the result will depend on the phase factor. So, the interference between $|1\rangle$ and $|2\rangle$ should be different than the interference between $|1\rangle$ and $e^{i\varphi}|2\rangle$, and that between $e^{i\varphi}|1\rangle$ and $|2\rangle$ (where $\varphi$ is an arbitrary overall phase). This can be checked experimentally by using a phase shifter. Suppose that there is an algorithm to calculate the interference between the two photons, and that this algorithm receives as input the two-photon state. The state $\frac 1 {\sqrt 2}\left(|1\rangle|2\rangle + |2\rangle|1\rangle\right)$ is equivalent to $\frac 1 {\sqrt 2}\left(|1\rangle|2'\rangle + |2'\rangle|1\rangle\right)$, where $|2'\rangle=e^{i\varphi}|2\rangle$. If the algorithm which gives us the interference depends only on the two-photon state, then it apparently gives the same result, even when we phase-shift the second photon. Apparently, the interference between photons depends on the way we decompose the two-photon state in two photons: in terms of $|1\rangle,|2\rangle$ we obtain different interference than in terms of $|1\rangle,e^{i\varphi}|2\rangle$, respectively $e^{i\varphi}|1\rangle,|2\rangle$. So, either the Hilbert space does not contain all the needed information, or I am missing something (which is by far more likely). Could you please explain me where I am wrong, and what is the correct description of interference using the two-photon state? _P.S. Let me emphasize that I am fully aware that QM has been tested successfully by so many experiments, and that most likely I am missing something. I just don't know what._


title: How often does a cosmic ray hit our eyes when we are on Earth?

text: Astronauts often see flashes when they are in space. How often these particles make their ways down to the Earth and hit us? What's the chance they hit our eyes and see flashes?


title: Einsteins equation : Black hole solution

text: Let Einstein's equations satisfy $ R_{\mu \nu } = 0 $. Suppose we solve it numerically with the aid of a computer. Can we know from the numerical solution if there is a black HOle in the solutions? For example, how can you know when you solve Einstein's equation if your solution will be a black hole or other particular non-smooth solution?


title: Will Pluto be in the ecliptic plane when New Horizons passes by? Was this deliberate?

text: From the illustrations at the New Horizons website it seems that Pluto may be in or near the ecliptic plane when New Horizons passes by. Is this intentional? Of course Pluto was to move to that position whether the Earthlings send a probe or not, but was the probe sent at a time to coincide with Pluto's pass through the ecliptic plane? My line of thought is that it would require less energy to get to ~40 AU in the ecliptic plane as the launchpoint (Earth) already has some velocity in that plane. Also, while travelling in the ecliptic plane the probe might happen upon other interesting things on the way (asteroids).


title: Numerical renormalization

text: is there a numerical algorithm (Numerical methods) to get 'renormalization' ?? i mean you have the action $ S[ \phi] = \int d^{4}x L (\phi , \partial _{\mu} \phi ) $ for a given theory and you want to get it renormalized without using perturbation theory .. can this be made with numerical methods ?? or on the other side let us suppose we have a divergent integral in perturbative renormalization and you want to evaluate it with numerical methods to extract some finite part of it.


title: Are there solid materials with controllable porosity?

text: In analogy to piezoelectric materials, where the application of an electrical field creates mechanical deformation in the material, I have the following question. Are there solid materials whose porosity can be altered or controlled? Of course, I have googled this. This lead me to a lot of websites where _controllable porosity_ means that the porosity of a material is very-well defined when it is produced. This is not what I am looking for. I am looking for a material that can alter its porosity according to an external control quantity, such as electric voltage.


title: Collision between Neutron stars and Black holes

text: When two neutron stars or black-holes come closer and closer, their angular velocity increases with decrease of distance and they start to revolve around a constant center with high relativistic speeds due to the influence of gravity... At last,   * Neutron-stars _collide_ to produce a giant explosion...   * Black-holes _collide_ to form super-massive black-holes... (Even two neutron stars merge to form black-holes, based on the answer) So, What are observed during and after this collision..? Magnetic fields & short GRBs only..?


title: Would Titan moon blow up if you light a match?

text: Given that the atmosphere of Titan is mostly methane, and from a comment on this TED talk... Would it blow up from a lighted match-stick? Seems like it's at _least_ plausible. But, wouldn't the lack of oxygen hinder this event? Or perhaps I'm confusing the _blow up_ with **combustion**?


title: Is strictly harmonic 2D lattice made of Hooke springs possible?

text: If we connect a set of point masses in a 1D lattice with Hooke springs and consider longitudinal oscillations, we'll have a strictly harmonic system, for which there exist eigenmodes and the frequency of oscillations doesn't depend on amplitude. If we choose transverse oscillations, they cease to be harmonic: the potential for small oscillations appears to be $\sim \Delta x^4$. Now I'd like to generalize the first case, where the oscillations are linear, to 2 dimensions. However, it appears that if we make, e.g. a rectangular lattice with nearest neighbours connected by Hooke springs, then any displacement of a lattice site leads to longitudinal deformation of two springs _and_ transverse deformation of two others, thus making the returning force nonlinear function of displacement. Arranging the masses into a triangular lattice doesn't help to compensate the nonlinearity. So, the question is: is it possible to make a _strictly harmonic_ 2D lattice using just point masses connected by Hooke springs? What would be an example of such an arrangement?


title: Effective field theory for fermion gas

text: Reading about fermion gas in a paper they used the following Lagrangian, which describes an effective field theory for nonrelativistic fermions (I neglect the four point interaction term). $$ L = \psi^{\dagger}(\partial_{\tau} - \frac{\nabla^2}{2m})\psi. $$ They work in two-dimensional Euclidean space-time, where $\psi_{\sigma}(\tau,x)$ is a four component Grassmann-valued spinor with spin components labelled by the index $\sigma$. Somehow it looks very familiar, but I am not able to deduce this from something I know. Since this is an Effective Field Theory, means this that it can be deduced with an expansion of the relativistic Lagrangian?


title: What does this quote about the four dimensional divergence of an antisymmetric tensor mean?

text: > _In the beginning, God said that the four dimensional divergence of an > antisymmetric second rank tensor equals zero and there was light._ Can someone explain what is the meaning of this quote by Michio Kaku?


title: Are irrelevant terms in the Kahler potential always irrelevant, even at strong coupling?

text: I've been reading about the duality cascade in Strassler's TASI '03 lectures (hep-th/0505153). He reminds us of the non-renormalization theorem theorem for the superpotential so that the beta functions for superpotential couplings go like their classical scaling dimension plus anomalous dimension terms from wavefunction renormalization. This allows us to make statements about the behavior of these terms even when a theory become strongly coupled, e.g. when the anomalous dimension becomes large. For example, we can say if an operator becomes 'dangerous irrelevant': it is irrelevant classically, but becomes relevant in the deep IR. In light of the existence of such behavior at strong coupling, I am confused why we can disregard higher order terms that may be generated in the Kahler potential. The Kahler potential has no protection against renormalization, so I would expect higher order classically irrelevant terms to be generated---how do I know that these do not become 'dangerous irrelevant' in the strongly coupled deep IR regime? More importantly, how do I know that these higher order Kahler potential terms (if they exist) do not destabilize the fixed points of the superpotential couplings that were important for analyzing the duality cascade? Thanks!


title: Eikonal approximation for wave optics. Why follow the unit vector parallel to the Pointing vector?

text: The description of the passage from wave optics to geometrical optics claims that light rays are the integral curves of a certain vector field (the Pointing vector direction, normalized to 1). Here are the details, could you fill the blanks: The wavelength $\lambda$ is much smaller than all other characteristic lengths.   1. The setting is a "nice" medium (with spatially varying refraction index $n(x)$) through which an "almost plane" wave is propagating. The wave (if linearly polarized) turns out to be representable by $\vec{E}{(x,t)} = \vec{E}_0\exp(i(\chi{(x)}-\omega t))$ and $\vec{B}{(x,t)} = \vec{B}_0\exp(i(\chi{(x)}-\omega t))$ with constant $\vec{E}_0$ and $\vec{B}_0$.   2. Maxwell's equations imply $(\vec{\nabla}\chi)^2=\frac{n^2\omega^2}{c^2}$ and a time-averaged Pointing vector $\vec{S}=\frac{c}{n}\vec{s}$, where $\vec{s}$ is the unit vector $\vec{s}=\frac{\vec{\nabla}\chi}{n\omega/c}$.   3. The integral curves of the field of unit vectors $\vec{s}$ are the light rays.   4. Working through the equations results for this for a ray trajectory $X(\tau)$ (where $\tau$ is just a parameter): $\frac{d}{d\tau}(n\frac{d\vec{X}}{d\tau})=\vec{\nabla} n$ How does one prove the jump from 2 to 3. Why follow the unit vectors and not the Pointing vectors themselves? Or even, why should the light rays be tangential to the Pointing vectors at all (besides intuition like "light rays should transport energy")? Could someone give me the proof of 3. or point me to a reference?


title: Can the zeroth-component of a 4-velocity be negative?

text: Is it allowed to have the zeroth-component of a four-velocity be negative? I presume the answer is yes, but I just want to make sure. Many thanks. * * * This is referring to $V^0$ for a curved space metric with signature $-+++$.


title: Is there any case in classical mechanics where Newton's (strong) third law doesn't hold?

text: Is there any case in classical (non relativistic) mechanics where the strong form of Newton's third law does not hold (that is, reaction forces are not collinear)? For example, if we consider a system of two point particles in equilibrium with each other upon which a constraint acts so that the reaction forces are directed in a direction that is not collinear. Is such a situation possible?


title: Can mass dimension of a field be viewed as another 'quantum number'?

text: While studying SUSY in 4D, I noticed the dynamical chiral superfield has dimension [GeV], whereas the dynamical vector superfield (for gauge theories) is unitless. Because I was introduced to the chiral and vector superfields as being irreducible components of a more general superfield, it seemed strange to me that they could have different mass dimensions. But then I realized that this is not alien to me: in the Standard Model of electroweak interactions, the Dirac spinors which have left-chiral and right- chiral irreducible components carry different weak charges! So is it reasonable to think of the mass dimension of a field to be yet another quantum number? For example, the chiral superfield is charge +1 and vector superfield is charge 0. Could I write down a transformation law? Can I do something as perverse as make it _local_?


title: Virtual differentials approach to Euler-Lagrange equation - necessary?

text: I'm currently teaching myself intermediate mechanics & am really struggling with the d'Alembert-based virtual differentials derivation for the Euler- Lagrange equation. The whole notion of, and justification for, using 'pretend' differentials over a time interval of zero just isn't sinking in with me. I also notice that not all textbook authors invoke it, so I'm wondering how necessary it is, given that Hamilton's principle gets us to E-L just fine. So, will I ever need this virtual displacement/work approach for something other than a route to E-L, or can I safely wave bye-bye to it?


title: Could a spatial filter improve a heterodyne signal?

text: Consider two beams of light at slightly difference frequencies that are interfered at a detector. The signal of interest is contained in the phase of the observed signal. As the beams travel around they pass through a variety of optics, which slightly distort the beams' wavefronts. As I understand it, a spatial filter is supposed to clean up the wavefronts. What would be the effect of the spatial filter on the observed signal? Would it give a less noisy measurement of the signal phase?


title: Energy conversions, on terrestrial scale, before and during a massive Earthquake

text: Ethan Siegel is an astrophysicist who is now making a living as science blogger. There is a post titled: Where does an Earthquake's energy come from? Ethan Siegel points out that massive earthquake events are accompanied by a measurable change in length-of-day, in the order of microseconds. (A very large event was the 2004 Sumatra quake, causing a 6.8 microsecond shortening of length-of-day.) Obviously, to speed up the Earth must contract. That is, what must be happening is that the rearrangement of tectonic plates during the quake allows the Earth to shrink a bit. As I interpret Ethan Siegel's post he is arguing that the energy of the seismic waves of such a quake come _directly_ from the contraction of the Earth during the event. In my own words: as I understand Ethan Siegel he is arguing that the Earth contraction during a quake is effectively an _implosion event_ , causing seismic waves directly. Quoting Ethan Siegel: > Plate tectonics tells you _where_ earthquakes are most likely to occur, and > the geophysics of the Earth’s crust tells you the different types of faults > that cause these quakes, but neither of these tells you where the _energy_ > for these quakes comes from. Please read Ethan Siegel's original post to see if you agree with my interpretation of his words. In comment #5 on that blog post I argued against Ethan's theory. I submitted: > when a rotating system contracts its kinetic energy increases (Angular > momentum is *conserved*; rotational kinetic energy *increases*) So: during > an Earth-rotation-rate-changing-earthquake the gravitational contraction of > the Earth is doing work.   >   >  Here's my point: I argue that all of that work goes to the increase of > Earth rotation rate, not to the quake!   >  I argue that *during the quake* the energy of the seismic waves comes > entirely from the release of elastic potential energy that has been building > up in the tectonic plates. Elaborating on the above:   I think that the contraction of the Earth-as-a-whole is a consequence and not a cause. I think: when tectonic plates slip relative to each other their rearrangement _allows_ a contraction of the Earth to occur, and following that there is an increase in Earth angular velocity. My question: which one of the above described opposite points of view is likely to be the correct one?


title: Number of the Generators of Poincare Group

text: It is said that the Poincare group, $P(3,1)$ has $10$ generators. $6$ of them are the generators of the Lorentz group, $O(3,1)$ and the other $4$ generators are the generators of $4D$ translational group. But Brian C Hall in his book named Lie Groups, Lie Algebras,and Representations puts that the Lie algebra of $P(n;1)$ is the space of all $(n+2) \times (n+2)$ real matrices of the form \begin{align} \begin{pmatrix} Y & y \\\ 0 & 0 \\\ \end{pmatrix} \end{align} with $Y \in $ **so** $(n,1)$ and $y \in \mathbb{R}^{n+1}$ is given by \begin{align} y = \begin{pmatrix} y_1 \\\ y_2 \\\ \vdots\\\ y_{n+1} \end{pmatrix} \end{align} If I use $n=3$ then $Y \in $ **so** $(3,1)$. Now there are $6$ generators in **so** $(3,1)$. For them I will get in total $6$ generators of the Poincare group, $P(3,1)$. But from where I will get the other $4$ generators according to the scheme given in Hall's book?


title: Is the black hole surface area actually zero?

text: From quantum mechanics we know that entropy of a part can be smaller than the entropy of the whole. I wonder whether a similar rule works for GR as well. Quantum mechanics predicts that the entropy of the whole universe is zero. At the same time we know that all astronomical objects put far enough take greater solid angle than a similar object near us. Far enough objects even in opposite directions have smaller distance between them than from them to us. For example, a tennis ball put sufficiently close to the cosmic horizon will span over sufficient part of the sky. This ultimately means that we are surrounded by a point-like singularity. And entropy of such singularity (proportional to the area) is zero. A similar consideration is true for a black hole. Any object approaching it will cover the greater share of the surface as it approaches. Even a planck- scale partile will cover the whole BH surface when approaching close enough. This may mean that the true area of the horizon is actually zero, with finite area being only an illusion appearing in the flat coordinates. This gives total entropy of a black hole to be zero which is in accordance with the fact that all of the BH information is located outside the horizon (to avoid the information paradox). If we subtract from the horizon's area(as seen by an observer at infinity) the total entropy of objects visible outside the horizon, we will arrive at zero. Similarly if we subtract from the supposed mass of the BH the total mass of matter visible outside the horizon, we also will arrive at zero: all mass of the BH is in the matter immediately surrounding the horizon. Is this correct?


title: What causes fluorescent light bulb to light up when it is near a plasma globe? Can this phenomenon be used to generate electricity in a circuit?

text: When a fluorescent light bulb is placed near a plasma globe, despite the fact there isn't a closed circuit, a part of the light blulb would light up. > ![plasma globe](http://i.stack.imgur.com/Cdesy.jpg) I tried measuring the voltage of the air around it with the probes of a multimeter, and discovered that the shorter the distance between these probes, the lower the voltage the reading shows. And when one of the probes is at a large distance away from the globe, the reading on the multimeter would increase as the other probe gets closer to the globe. The reading can increase up to 300 volts. What does this imply? Am I conducting a wrong measurement? Does the plasma globe ionized the air around it and the electrons in the ionized air excite the mercury vapor in the light blub and cause the phosphor to fluoresce? Are all plasmas (cold & hot) kept in glass container capable of ionizing the air around it? If yes, is it possible to generate electricity current that flows in a circuit from this phenomenon? (By _this phenomenon_ I'm talking about the phenomenon that causes the light bulb to light up. I'm not talking about plasma itself where nuclear fusion can take place.)


title: Other application of Liouville's theorem besides thermodynamics

text: Are there any other important practical and theoretical consequences of Liouville's theorem on the conservation of phase space volume besides the calculation of the microcanonical potential in Thermodynamics?


title: do relativistic velocities change the apparent entropy content of the moving object?

text: Imagine a box of hot gas. It has a certain (large) amount of entropy, which we can relate to the amount of information needed to completely specify the position and velocity of every gas particle in the box. Attach a clock and rocket motor to the box and accellerate it to close to the speed of light. As the rocket zooms past us, we notice that the hands on the clock it carries are seemingly frozen, and with some special apparatus we notice that the gas particles themselves do not seem to be moving either. Furthermore, the length of the box in the direction of travel has apparently vanished, rendering the box two-dimensional. The amount of information needed to completely specify the contents of the box is reduced, because 1) the box is missing one degree of freedom it had while at rest (relative to us) and 2) every one of the gas particles now has exactly the same velocity, say .99999c in the direction of travel, and their positions within the box itself are no longer changing with time. It would seem then that the box as observed from a nonmoving reference frame now contains less entropy than it did while at rest in that frame. (An observer attached to the box would detect no such changes in it.) What is wrong with my understanding of this?


title: Is there a field equation which can reduce into all three flavors of spin (zero, one, one half)?

text: Is there a known particle field equation of a similar form $$ \begin{equation} (\Gamma^n \pi_n)^2 \Psi = (mc)^2 \Psi \tag{1} \end{equation} $$ such that by reducing the number of degrees of freedom for the spinor $\Psi$ into a spinor of lesser degrees of freedom, such as a scalar $\psi_0$, two three-vectors $\boldsymbol{\psi}_\pm$ or two two-vectors $\boldsymbol{\phi}_\pm$, it reduces Eq. 1 into either ...   * a spin zero field equation $$ \begin{equation} \pi^n \pi_n \psi_0 = (mc)^2 \psi_0, \tag{2} \end{equation} $$   * a spin one field equation $$ \begin{equation} (I\pi_0\pm i \boldsymbol{\pi} \times) (I\pi_0\mp i \boldsymbol{\pi} \times) \boldsymbol{\psi}_ \pm = (mc)^2 \boldsymbol{\psi}_ \pm \tag{3} \end{equation} $$   * or a spin 1/2 field equation $$ \begin{equation} (I\pi_0\pm\boldsymbol{\sigma}\cdot\boldsymbol{\pi}) (I\pi_0\mp\boldsymbol{\sigma}\cdot\boldsymbol{\pi}) \boldsymbol{\phi}_\pm = (mc)^2 \boldsymbol{\phi}_\pm? \tag{4} \end{equation} $$ In these expressions $\pi_n$ is the four-component momentum operator which includes the electromagnetic four-potential interaction $A_n$ with the particle's charge $q$ written as $$ \begin{equation} \pi_n = i\hbar \partial_n - q A_n , \tag{5} \end{equation} $$ and $$ \begin{equation} \boldsymbol{\pi} = -i\hbar \boldsymbol{\nabla} - q \boldsymbol{A} \tag{6} \end{equation} $$ uses bold to indicate a euclidean vector, specific to 3-components. The three two- by-two matrices $\boldsymbol{\sigma}$ in Eq. 4 are the Pauli Spin Matrices.


title: What Is Energy? Where did it come from?

text: The simplistic undergrad explanation aside. I've never really understood what energy really is. I've been told that it's _something_ when converted from one kind of something to another kind does some "work", as defined by us, but what is that something? Moreover, if the total amount of energy in the universe is finite and we cannot create energy. Then, where did it come from? I've learnt from thermodynamics where it _goes_ , but where does it come from? I know this sounds as something trivially simple, but there is so much going on over here and I just can't grasp what it is. Of course, I lack the mathematical understanding to grasp the subtle things the universe is doing, but what is it doing and how do I get to point of understanding what it's doing? ( **Note:** What prompted me to ask this was this answer. I'm afraid that it just puzzled me further and I sat there staring at the screen for a good 10 minutes.)


title: What is energy?

text: What exactly is energy (in terms a non-physicist/non-mathematician can understand)? I've been researching this topic, and it seems that energy is not a real physical state, i.e. it is only a scalar that is assigned to the state of physical systems for mathematical abstraction. In other words, energy appears to be a trait of mathematics used to describe traits of matter and non-matter fields. How accurate is this assumption? If energy is "nothing," how can the breaking of a chemical bond catalyze another reaction (e.g. formation of a bond)? That energy that is transferred, what exactly is it?


title: Why is electric charge conserved?

text: We have long been taught that electric charges are neither created nor destroyed. But somehow it is okay to destroy two oppositely charged particles at once ! Why is that so? Let's just take a look at electron-positron annihilation; Two equally oppositely charged particle are bombarded into each other, both are instantly destroyed (lose their existence) and gamma ray photons emerge. It is certainly convenient to say that at $t$ second the net charge in system was 0 and at $t+1$ second it is also zero. But certainly it also raises the question what if you observe only a positron or an electron, in observed system certainly charge will not be conserved however overall it maybe. It jumps out and say that you cannot destroy individual charges, but if you get 2 equally and oppositely charged particles we can destroy both ! (kind of Romeo-Juliet) Then why is the fundamental postulate of conservation of charge that /charge can neither be created nor destroyed"? Since, clearly it can be destroyed, just in pairs (also created in pairs)


title: Why does a force field leave the momentum operator unchanged in the Schrödinger equation?

text: The reasoning leading to the Schrödinger equation goes as follows: A plane wave in empty space has the following form: $$\psi = e^{i(kx-\omega t)}$$ Einstein had previously explained the photoelectric effect, i.e. the emission of electrons from a metal surface through light, by suggesting that light is made from photons containing the momentum: $$p_\text{Photon}=\hbar k$$ and energy: $$E_\text{Photon}=\hbar\omega$$ This was proven to be correct by experiment. De Broglie then suggested that the same relations hold for electrons, so that, given the momentum and energy of an electron, one could find the wavelength and frequency of an electron's plane wave: $$\begin{gather} k_\text{Electron}=\frac{p_\text{Electron}}{\hbar}\\\ \omega_\text{Electron}=\frac{E_\text{Electron}}{\hbar} \end{gather}$$ Given the plane-wave function of an electron, one can then use certain operators to extract momentum and energy from it: $$\begin{gather} \hat{p}\psi = -i\hbar\partial_x\psi = \hbar k \psi\\\ \hat{E}\psi = i\hbar\partial_t\psi = \hbar\omega\psi \end{gather}$$ But for a free electron outside of a force field, the relationship between energy and momentum is given by: $$E = \frac{p^2}{2m}$$ The Schrödinger Equation for a free electron can be derived from this equation by replacing energy and momentum by the extraction operators: $$\hat{E}\psi = i\hbar\partial_t\psi = \frac{\hat{p}^2}{2m}\psi=-\frac{ \hbar^2\partial^2_x}{2m}\psi$$ Schrödinger now argued that placing the free electron in a Potential would modify the equation simply through the addition of the Potential Energy: $$i\hbar\partial_t\psi =\biggl(-\frac{\hbar^2\partial^2_x}{2m}+V(x)\biggr)\psi$$ Now my question is: The introduction of a potential is going to mess up the plane waves very seriously, transforming them into something quite different, so the extraction operators for Energy and Momentum - which only work for plane waves, for which they were designed - are no longer going to work! How can the Schrödinger equation still hold up?


title: A question about the asymptotic series in perturbative expansion in QFT

text: Related post I heard about the argument that the perturbative expansion in QFT must be asymptotic, such as http://ncatlab.org/nlab/show/perturbation+theory#DivergenceConvergence > Roughly this can be understood as follows: since the pertrubation is in the > coupling constant about vanishing coupling, a non-zero radius of convergence > would imply that the theory is finite also for negative coupling (where > “things fly apart”), which will not happen in realistic theories. My question is, why the negative coupling causing "things fly apart" will lead to asymptotic series? Suppose we have an electron and positron, if they fly apart, it corresponds to electron and electron, and vice versa.


title: Maximum rail adhesion utlilization during vehicle towing

text: Ok Physics Stackexchange, I've got a question for you regarding maximum adhesion utilization regarding railway vehicles (a EMU, with distributed motorized axles to be more specific) during a special, although simplified, case (towing), which I can't seem to work out. I am a fairly new engineer in the railway industry and have asked my colleagues to look at my calculations, but they are mildly interested, since the time spent wont produce much of value. This is not for solving any particular problem, but rather for my own curiosity and understanding. Consider this: Two train units, one with no passenger load, and one with full passenger load. The train with full passenger load is faulty (can not provide tractive effort and is thus essentially a trailer vehicle) and needs to be towed by the empty train. The empty train can only provide a tractive force $F$ (e.g. no load compensation available for this train). My problem is that I'm having trouble understanding why it appears that the level of maximum adhesion utilization increases when the gradient and/or total train-set weight increases (towing consist is considered as two rigid bodies connected), even though the adhesion weight and available tractive effort ($F = M/r$, where M is motor torque and r is wheel radius for the empty train) remains constant (i.e. the empty train adhesion weight does not change, and neither does the maximum available tractive effort). The adhesion utilization level is the ratio between force transferred to the rail over adhesion weight: $$ \alpha = \frac { F_{\alpha}} {(m_{\alpha}*g)} = \frac {F - m_{eq,dr}*a_{max}} {m_{\alpha}*g} $$ where $\alpha$ is the maximum adhesion utilization level, $m_{\alpha}$ is the adhesion weight (weight on driven axles), $m_{eq,dr}$ is the equivalent mass of the driven axles, and $a_{max}$ is the maximum achievable level of acceleration for a given train mass and maximum tractive effort. On a side note: should the acceleration due to gravity be considered an external force trying to turn all wheels in the opposite direction than desired, and if so, will this force me to treat all axles as driven? $$ \alpha = \frac {F - m_{eq,dr}*a_{max} - m_{eq,drUndr}*g*\sin(\arctan(\frac{grad}{1000}))} {m_{\alpha}*g*\cos(\arctan(\frac{grad}{1000}))} $$ Furthermore, $a_{max}$ can be calculated according to: $$ a_{max} = \frac{F-D} {m_{eq,tot}} $$ where $D = A + Bv + Cv^{2}+ G$ (Davis' equation), but for this purpose the running resistance is assumed to be speed-independent (i.e. $Bv$ and $Cv^{2}$ are zero, due to relatively low speeds during towing). The running resistance can then be expressed as: $$ D = A + G = A + m_{tot}*g*\sin(\arctan(\frac{grad}{1000})) $$ where $grad$ is the track gradient in per mille. Calculating the adhesion utilization for a single train (e.g. the empty train) on a level track is straightforward. However, when another train (fully loaded, undriven) is added, the adhesion utilization appears to go up, even though the adhesion weight and available traction force is the same. Why is this? The above model/set of equations exhibit the same behaviour if the track slope is increased (gradient resitance increase -> maximum achievable acceleration decrease -> maximum adhesion utilization increases). My colleagues tell me that the maximum adhesion utilization should remain the same and that I should not worry about such issues as the above, but I want to understand how to calculate this in a correct way. Assume the following data for the train: $m_{empty} = 150 ton$, $m_{full} = 200 ton$, $F = 150 kN$, $A= 5 kN$, $m_{eq,dr} = 10 ton$, $m_{eq,undr} = 8 ton$,


title: Is relative velocity invariant under special relativity?

text: If a metre stick passes an observer at speed $v$, would all observers in any inertial frame of reference say the speed of the meter stick relative to the observer is exactly $v$? If so what is it also true for non inertial frames of reference?


title: Understanding the four fundamental forces of the standard model - are they magic

text: Don't misunderstand the question, my purpose is exploration and understanding of what defines "mainstream physics". It is not asked idly, or with ill purpose.... My understanding of current mainstream physics, is that there are four fundamental forces. All four fundamental forces reach through vacuums, to attract or repel other objects, with no transport mechanism whatsoever. And to me that sounds like magic. [Mirriam Webster - Magic: an extraordinary power or influence seemingly from a supernatural source (among other definitions)] Is there a difference between the nature of the fundamental forces of nature, and nature of magic as we define it? If so, what is that difference?? Are any of the alternate models, designed to replace empty space with a transport mechanism? Do all of the currently entertained mainstream physics model alternatives, depend necessarily on the continued belief in non-mechanical interactions at the sub-atomic level?? ![ einstein/potter blur ](http://i.stack.imgur.com/v2Sak.jpg) My understanding of the fundamental forces as currently accepted would include believing the contents of this picture to be accurate enough to support the nature of my question. ![ fundamental forces of nature ](http://i.stack.imgur.com/PEZhn.jpg) My understanding of the quantum theory as currently accepted would include believing the contents of this picture to be accurate enough to support the nature of my question. ![enter image description here](http://i.stack.imgur.com/qQrGU.png) Thank you for your patience with my question. :-)


title: existing bounds on maximum density achieved by a Bose condensate

text: As we know, fermions are subject to exchange interactions that limit the densities they can achieve. However bosons (simple or composite) are not constrained by this, which implies physical phenomena like the Bose-Einstein condensate, where the main component is Helium-4 Question: Are there physical limits to the density that can be achieved with Bose-Einstein condensates made up from composite bosons? my suspicion is that there is a critical density above which the fermion character of the fermion components will play up, but i couldn't find any mention about this anywhere


title: Please clarify how entropy increases when matter gravitationally coalesces

text: On John Baez's website, http://math.ucr.edu/home/baez/entropy.html, he discusses the problem of how entropy increases when a cloud of ideal gas collapses gravitationally (no black holes - keeping it simple). But he doesn't give away the answer - instead he implies that, I presume, either gravitons or photons are released during collapse and these increase the overall system entropy. Can someone please answer which it is, gravitons or photons? If gravitons (or gravity waves), then does this mean that pre-20th century physicists, with no notion of gravity waves, would have concluded that gravitational attraction violates the 2nd law? And if photons, then what about if the gas particles were totally neutral - wouldn't they not emit any EM radiation in that case? Or am I wrong that gravitons or photons are the answer? Thanks!


title: The age of the universe

text: Many times I have read statements like, "the age of the universe is 14 billion years" . For example this wikipedia page Big Bang. Now, my question is, which observers' are these time intervals? According to whom 14 billion years?


title: Trace of an operator matrix (Quantum computation and quantum information)

text: I'm reading the book Quantum computation and quantum information by Mike & Ike and I'm stuck at 2.60/2.61. There, the author says that, given the operator $A|ψ⟩⟨ψ|$, its trace is: $${\rm tr}(A|\psi\rangle\langle\psi|) = \sum\limits_i\langle i|A|\psi\rangle\langle\psi|i\rangle$$ Why would that be true? Why can we rearrange the bras and kets like that?


title: Derivation of f(R) field equations, problem with integration by parts

text: I am following the derivation of the field equations on the the Wikipedia page for $f(R)$ gravity. But I do not understand the following step: $$ \delta S = \int \frac{1}{2\kappa} \sqrt{-g} \left(\frac{\partial f}{\partial R} (R_{\mu\nu} \delta g^{\mu\nu}+g_{\mu\nu}\Box \delta g^{\mu\nu}-\nabla_\mu \nabla_\nu \delta g^{\mu\nu}) -\frac{1}{2} g_{\mu\nu} \delta g^{\mu\nu} f(R) \right) $$ the wiki article says, the next step is to integrate the second and third terms by parts to yield: $$ \delta S = \int \frac{1}{2\kappa} \sqrt{-g}\delta g^{\mu\nu} \left(\frac{\partial f}{\partial R} R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} f(R)+[g_{\mu\nu}\Box -\nabla_\mu \nabla_\nu] \frac{\partial f}{\partial R} \right)\, \mathrm{d}^4x $$ In other words, integrating by parts should yield: $$ \int \sqrt{-g} \left(\frac{\partial f}{\partial R} (g_{\mu\nu}\Box \delta g^{\mu\nu}-\nabla_\mu \nabla_\nu \delta g^{\mu\nu}) \right)\, d^4x = \int \sqrt{-g}\delta g^{\mu\nu} \left([g_{\mu\nu}\Box -\nabla_\mu \nabla_\nu] \frac{\partial f}{\partial R} \right) \mathrm{d}^4x $$ From there getting the usual f(R) field equations is trivial. What I'm confused by is how to integrate by parts to get that. I have tried many different ways the one I think is most correct is: assuming $g_{\mu \nu} \Box$ and $\nabla_\mu \nabla_\nu$ are differential operators then $u' = g_{\mu \nu} \Box \delta g^{\mu\nu}$ and $v = f'$, similarly with the $\nabla_\mu \nabla_\nu$ so using the formula for integration by parts: $$ \int u'v = uv -\int uv' $$ I get: $$ \int \sqrt{-g} \left(f' (g_{\mu\nu}\Box \delta g^{\mu\nu}-\nabla_\mu \nabla_\nu \delta g^{\mu\nu}) \right)\, d^4x = -\int \sqrt{-g}\delta g^{\mu\nu} \left([g_{\mu\nu}\Box -\nabla_\mu \nabla_\nu] f' \right) \mathrm{d}^4x $$ because the $uv$ term will disappear. So can any one explain to me why I have the minus sign and Wikipedia doesn't? Is it ok to use $g_{\mu \nu} \Box$ as a differential operator? I have tried other ways such as writing $\Box$ explicitly and using integration by parts twice but I also couldn't get the correct answer as i end up with terms such as $\nabla_\nu \nabla_\mu$ which cant be correct. There is a similar post on physics forums on this step but it does not answer my question and is now closed.


title: calculate force between two magnets

text: I have used a couple of online calculators to estimate the attractive force between two cylindrical bar magnets. Using d=1"; t=1"; x=0.1"; Neo35 (Br=12,300 Gauss): source 1 gives 63.5 lbf   source 2 gives 29.6 lbf source 2 explains data is based on measured data, so this accounts for the difference from source 1. But in general, I think this is the right ballpark number. I then tried to reproduce these numbers myself, using a formula I found on Wikipedia. I could not get close to the right answer - my answer is about 4000 lbf. **Could someone please explain what I did wrong in my calculation?** ![formula](http://i.stack.imgur.com/boNho.png) ![enter image description here](http://i.stack.imgur.com/WUd0K.png) **EDIT** Tough crowd. Let me try and be more specific. The problem itself is pretty straightforward. I did not really mean to imply I need someone to check the numbers. I mean more on if I am applying the right approach to the problem:   1. Is the formula above valid when trying to replicate results in the calculators I linked? (i.e. "Calculate the Pull and Repulsion Forces Between Two Disc Magnets")   2. Perhaps "real life" is significantly different than theory for this application? In which case, the engineering calculators would apply a de-rating factor that is not represented in the Wikipedia formula. I did find any references to this, but perhaps someone with more knowledge could advise?   3. Assuming the formula is valid here, I think Magnetic Flux Density (B_0) is the most likely source of my error. It is my understanding that this is a physical property of the material. Is that correct? The value I found for Neo35 is Br = 12,300 Gauss (CGS units). I converted this to SI 1.23 Tesla using a 10^-4 conversion factor. Is this the correct interpretation of B_0?


title: Simulation of fluid flow using Euler equation

text: I have been looking on Euler's equations for a while and can't grasp one thing. Suppose we have initial system state with volumes of fluid "hanging" in air (time is frozen and equal to zero), each of them has its initial (x, y) coordinates and velocity vector (Vx, Vy). The space is divided in stationary volumes in which measurement takes place (Euler measurement model). ![2 dimension representation of segmented space and restrictions](http://i.stack.imgur.com/AQBYW.jpg) As I understand Euler's equations impose mutual restrictions between functions of velocity ( u(x, y, t) ), mass density ( m(x, y, t) ), and pressure ( p(x, y, t) ) at any instance of time for stationary volume of space with coordinates (x, y) that contains volume of fluid (in that instance). So why each function has time parameter when equations must be solved for any instance of time (iteration process) ? We must have new set of these functions for any moment of time that satisfy equations. And it's unclear whether volume of space contains volume of fluid or not, how do we take this in consideration ? And the second question is how state of each volume of fluid (its coordinates and velocity vector) is related to these functions ? I need a system that has some initial state (array of fluid volumes with their coordinates and velocities) and uses Euler's equations to compute next state, which becomes initial for next iteration and so on. Something like this: ![scheme](http://i.stack.imgur.com/DRe1Q.jpg) I don't need math details, I just want to grasp basic idea in context of computer simulation (how method of simulation can be bounded to standard analytic equations). Most papers are very complicated, with additional physical factors and user interaction, and they don't cover this "link". I want to consider the easiest case and how to start iteration process (just idea and how it is usually done). Sorry if some things sound ridiculous, please adjust my statements if they are wrong. If my model of understanding is completely wrong just point me in right direction, thanks.


title: In what ratio does the charge distribute if a charge and uncharged body touch each other?

text: Lets say a large charged body has $n$ elementary charges and is brought in contact with a small uncharged body. When the charge distributes, does it distribute in the ratio of number of atoms or each body gets half of the charge?


title: Tetrad formalism: getting back to coordinate basis

text: Let $\omega^{\hat{a}}$ be an orthonormal basis, and $\theta^{\hat{a}}_{\hat{b}}$ be the associated connections. From Cartan's second structure equation, we may compute the curvature 2-form, i.e. $$R^{\hat{a}}_{\hat{b}} = \mathrm{d}\theta^{\hat{a}}_{\hat{b}} + \theta^{\hat{a}}_{\hat{c}} \wedge \theta^{\hat{c}}_{\hat{b}}.$$ We may also relate the curvature 2-form to the Riemann curvature, via the relation, $$R^{\hat{a}}_{\hat{b}} = \frac{1}{2}R^{\hat{a}}_{\hat{b} \hat{c} \hat{d}} \omega^{\hat{c}} \wedge \omega^{\hat{d}}$$ from which we may 'read off' the terms. However, both the Ricci and Riemann curvature are in an **orthonormal basis.** How can I 'convert' $R^{\hat{a}_{\hat{b}}}$ and $R^{\hat{a}}_{\hat{b}\hat{c}\hat{d}}$ back to an ordinary **coordinate basis**?


title: Potential difference (PD) and electromotive force (EMF) in terms of electrons?

text: ![enter image description here](http://i.stack.imgur.com/Zg4Me.png) ![enter image description here](http://i.stack.imgur.com/NNsSz.png) ![enter image description here](http://i.stack.imgur.com/qAfRc.png) I am trying to understand potential difference and electromotive force in terms of electrons. In the above 3 circuit diagrams the red shows (what I think to be) the distribution of electrons around the circuit. (on reflection I do not think there would be more electrons at B, F and L as indicated on the diagram but rather an even distribution of electrons between B and D, F and H and J, and L and P respectively). I have several parts to by question:   1. Is emf simply the potential difference between A and B (just focusing on the first diagram for now) i.e. if you sum all the $\frac{e}{4\pi \varepsilon_0 r}$ from all the electrons for position A and then for B and subtract them.   2. On the same principle is the pd across the resistor the same as above expect done for C and D.   3. Do these diagrams seem to be write or have I got totally the wrong idea.


title: Applications of the Feynman-Vernon Influence Functional

text: I am looking for a reference where the Feynman-Vernon influence functional was defined and used in the context of relativistic quantum field theory. This functional is one method to describe non-equilibrium dynamics for open systems (e.g. coupled to noise) which seems (naively, as an outsider) to be particularly well-suited for field theories where path integral methods are more intuitive. As a consolation prize, I'd be also interested in applications to other areas of physics (such as dissipative quantum systems), or for more effective or popular methods to describe non-equilibrium dynamics (of open or closed systems) in the context of relativistic quantum field theory (preferably in the path integral language).


title: Holographic Renormalization in non-AdS/non-CFT

text: In AdS/CFT, the story of renormalization has an elegant gravity dual. Regularizing the theory is done by putting a cutoff near the conformal boundary of AdS space, and renormalization is done by adding counterterms on that surface. Mathematically this is also interesting, since this utilizes the Lorentzian generalization of the Graham-Fefferman expansion. But, in the spirit of “effective holography”, one ought to be able to do that in spacetimes which do not admit a conformal boundary. I am wondering if anyone has ever seen an attempt to systematically define holographic renormalization in such spaces, for example for p-branes ($p \neq 3$), the NS fivebrane, or the Sakai-Sugimoto model, etc. In such cases one can still take a cutoff surface at the UV of the theory, take the fields to be essentially non-fluctuating, but one does not have a conformal boundary and all the associated machinery.


title: Analyticity and Causality in Relativity

text: A few weeks ago at a conference a speaker I was listening to made a comment to the effect that a function (let's say scalar) cannot be analytic because otherwise it would violate causality. He didn't make this precise as it was a side comment, and although it intrigued me (I had never heard that said before) I didn't give it much thought until this morning. Now that I think about it, it actually seems quite obvious. Just look in $1+1$ Minkowski space: suppose $f$ is analytic around some point $(t_0, x_0)$, then I can find some ball about this point such that at another point $(t,x)$ in the ball but outside the light cone of $(t_0,x_0)$ we have that $f(t,x)$ is completely determined by the value of the function and its derivatives at $(t_0,x_0)$. This seems to be against the spirit of causality. If the above is correct, does anyone know when this was first discussed? I imagine that it would have been quite a long time ago. It's interesting because until this conference I had never heard it said before. Perhaps it is deemed to be uninteresting/obvious?


title: Connection: Work on dipole <-> optical absorption

text: If I consider a two-level system with a ground and an excited state, I can compute the instantaneous rate at which work is being done on the dipole $\mu(t)$ by an electric field $E(t)$: $\dot{W} = E(t)\tfrac{\mathrm{d}}{\mathrm{d}t}\mu(t)$ If I integrate over time, I would end up with with the work that is being done on the dipole by the field. What I am interested in is, how can I connect this with the optical absorption due to the dipole transition?


title: Is the universe expanding on a preferred axis, or asymmetrically?

text: I remember back in 2011 i found this paper on the research a team of Chinese physicist completed about the universe may having a prefered axis when expanding. On ARXIV I haven't seen any other papers released on the same topic and i'm wondering if any new evidence or papers have been released since then. Or if anyone has some information on it that would be great.


title: Optical interferometers resolution

text: I found this interesting list of optical interferometers: http://en.wikipedia.org/wiki/List_of_astronomical_interferometers_at_visible_and_infrared_wavelengths But I can't understand which is the smallest size of a visible object in Earth orbit (let's suppose 300 km high). Or at least it would be enough to know the resolution in radiants or degrees.


title: Degrees of freedom in the early Universe

text: I am reading Dodelson's textbook on cosmology. On page 66 we find equation 3.26: $$\rho = \frac{\pi^2}{30}T^4\biggl[\sum_{i=\text{bosons}}g_i+\frac{7}{8}\sum_{i=\text{fermions}}g_i\biggr]\equiv g_\star \frac{\pi^2}{30}T^4 $$ On the next page, there is the following statement: > By the time decays become important, electrons and positrons have > annihlated, so $g_\star$ in Eq. (3.26) is 3.36 I don't understand where 3.36 comes from. If you take the above equation, the bosons are the photons and so they have $g=2$ and the fermions are the 3 generations of neutrinos and their anti-particles and so they have $g=6$ (these numbers are also given in the text). This then gives $g_\star=2+{7\over 8}\times 6=7.25$. Any help in understanding why $g_\star=3.36$ rather than 7.25 would be greatly appreciated.


title: Distribution of energy in an electric and magnetic field

text: Are the energy of the electric field and the energy of the magnetic field concentrated on their sources OR are they scattered in the environment where the fields arent zero? What is the formula that supports your answer?


title: Can we observe changes in the fine-structure constant?

text: The _fine structure constant_ is a number of constants rolled into one equation. Brian Cox mentioned in the April edition of Focus magazine that it is possible that the speed of light was once faster, say, in the earlier universe - hence I would conjecture this constant must then have been different. If any or each of the constants that make up the fine structure constant are said to be changing, then is there anywhere in the universe where we can reliably observe this; and if so, what are the consequences in such a case? What of our known _laws of physics_ in such cases, those that are used so broadly to garner great results in the field of astronomy to this day, do they become questionable or blatantly break down? For reference, this is an excerpt from the magazine concerning what was said regarding observations of the speed of light: > What the astronomers are seeing in their study of the distant gas clouds is > the last cosmic moments of that decline. He further mentions gravity with hints of possibilities of change: > Others look for variations in big G as this could be used to develop a new > understanding of gravity. These are fitting statements for the source of them, but seemingly fanciful without further detail.


title: Is the length of the day increasing?

text: In _Frontiers of Astronomy_ , Fred Hoyle advanced an idea from E.E.R.Holmberg that although the Earth's day was originally much shorter than it is now, and has lengthened owing to tidal friction, that further increases in the length of the day would not occur because of resonance effects in the atmosphere caused by the gravitational field of the sun. I haven't heard that suggested anywhere else, though, and wondered whether it was really the case, or is the length of the day increasing?


title: Transition integral from 1-D cartesian into 3-D polar coordinate system

text: Lets say we have an electron which can be in two states. Its wavefunction for two states is then $\Psi=A\Psi_n + B\Psi_m$, where $\Psi$ is time dependent wave function. I know that the **transition integral** can be written in 1D cartesian coordinates like this - $\psi$ marks the time independent wavefunction: $$\int\limits_{-\infty}^{\infty}x\,\overline\psi_{m}\psi_{n} dx.$$ How do I write the selection rule for the 3D polar coordinate system? I have been thinking about writing something like this: $$\int\limits_{V}r\,\overline\psi_{m}\psi_{n} \,dV=\int\limits_{0}^{\infty}r^3\overline\psi_{m}\psi_{n} \,dr\int\limits_{0}^{\pi}\sin\theta \,d\theta\int\limits_{0}^{2\pi}d\phi.$$ Is this correct? Am I missing anything? I used the volume differential $dV=dr\cdot r\sin\theta d\theta \cdot rd\phi$. I am not sure if it is correct to just replace the $x$ with $r$. Should I instead replace $x$ with its polar analog which is $r\sin\theta\cos\phi$?


title: Magnetic moment of electron

text: Since magnetic moment come from the circulation of charge, what is the origination of the electron's magnetic moment? Because spin of electron is not the classical spin of particle. Can we say that corresponding to every angular momentum there is a magnetic moment and since spin angular momentum is conformed from Stern-Gerlach experiment so there is a magnetic moment.


title: What's the altitude of the flat base of Cumulus clouds?

text: Is there a formula that describes the altitude of clouds with respect to the local temperature? I think of the _flat bases_ of Cumulus clouds. Thanks...


title: Effective masses for different direction

text: Assume we have an indirect semiconductor where the effective mass becomes anisotropic in different directions. Usually, one talks about a mass in parallel and perpendicular direction referring to momentum (or k-)space. So if I have a semiconductor sample for which I know the growth direction is lets say the parallel, I know which mass I have to consider when I look along this axes in real space. But what mass acts along the perpendicular direction in real space? Since real and k-space are connected via a Fourier transform, the mass perpendicular to the parallel mass in real space is not the one perpendicular to the parallel mass in k-space. So how can I find the mass acting perpendicular to the growth direction in real space when I know the mass along the growth direction and the mass perpendicular to that but I know it in k-space?


title: What is phenomenological equation and phenomenological model?

text: I come across these terms in some papers. My understanding is that it is an equation or model describing a phenomenon. Usually, the equations are given and claimed to be true with only some explanations and justification, but not derived from the first principle. The reason is often a bit obscure to follow, on what base they can do that with high confidence? How can they derive it in the first place? If the only purpose is to generate the expected phenomenon, how can they justify it is really the equation governing the phenomenon? Is there good definition and explanations of these two terms? I expect few solid physics examples should make it easy to understand.


title: Intensity of light that produces largest current in Photoelectric Effect + Stopping Voltage Question

text: ![For the Sodium metal, the Current produced by the cell is the highest when shined on by 196nm light - Not the highest, why is that? What determines the intensity of the light that creates the largest current?](http://i.stack.imgur.com/87Dje.png) ![enter image description here](http://i.stack.imgur.com/dg4YQ.png) For the first image: For the Sodium metal, the Current produced by the cell is the highest when shined on by 196nm light - Not the highest, why is that? What determines the intensity of the light that creates the largest current? For the second image: When I change the voltage of the battery the current stops increasing when the voltage reaches 0V. Even when I increase the voltage even higher the current does not increase. Any idea why? Thanks in advance!


title: Tensor decomposition of $\partial_\mu A_\nu$

text: In the decomposition of a rank-2 Minkowski tensor into irreducible representations, I expect the 16 components of the tensor product $M_\mu N_\nu$ to reduce to the sum of a scalar (1), a rank-2 anti-symmetric tensor (6), and a rank-2 traceless symmetric tensor (9). Applying this to $\partial_\mu A_\nu$, where $A_\nu$ is the electromagnetic 4-potential, I find the usual anti-symmetric field tensor $F_{\mu\nu}$, and the the 4-divergence $\partial\cdot A$ (which vanishes in Lorentz gauge). But I don't recognize the traceless symmetric tensor. Have I missed something in the decomposition, messed up my indices, or forgotten something from E & M? thanks!


title: Addition according to significant digits

text: I have been studying about significant digits and addition rules and I can't quite digest the rules of addition completely. It states that in the answer number of decimal places will be equal to the least number of decimal places in the terms. $//$What my teacher has taught me and what my book says This makes sense first : $1000.1+1.15=1001.2$ If we had went for number of significant digits rule we would have retained only $3$ significant digits. But consider this case : $1. 10^3+1.0=1001$ The number of significant digits in both of initial terms were $1$ and $2$ respectively, but in the final answer they are $4$. There are more significant digits in the answer. Isn't it wrong as last three digits of $1.10^3$ are insignificant Please clear my doubt or whether the rule has an extension. **EDIT** $1000$ changed to $1.10^3$. I don't think anyone understands what am I asking. I $know$ what the rules are and how to apply them but I want to know that $1.10^6+1.0=1000001$ . Don't $you$ think it is wrong as we are not sure of second last digit of $1.10^6$ but we are of $1000001$


title: Sterile Neutrinos and Supersymmetry

text: Just read a Scientific American review of a recent article which suggests evidence for sterile neutrinos: http://www.nature.com/news/cosmic-mismatch-hints-at-the-existence-of-a- sterile-neutrino-1.14752 The article is a little unclear about exactly what evidence, but my question is more general. A sterile neutrino would certainly not necessarily imply a forth generation to the standard model - if memory serves there is some nice anomaly cancellation which would be ruined if we had four generations. I am mostly interested in what a sterile neutrinos would mean for supersymmetry - are they part of the MSSM? What is the minimal number of SUSY particles which must come along with such a neutrino? Do they HAVE to be SUSY, or can they be part of an extended SM?


title: Significant figures vs. absolute error

text: On NIST Avogadro's Number, $N_A = 6.022\;141\;29 \times 10^{23} \text{ mol}^{-1}$ has 9 significant figures and a standard uncertainty of $0.000\;000\;27 \times 10^{23} \text{ mol}^{-1}$. First, can it be written as the following? $$ N_A = 6.022\;141\;29 \times 10^{23} \pm 0.000\;000\;27 \times 10^{23} \text{ mol}^{-1} $$ If yes, consider the following quote from Rice's lab guide on error analysis (emphasis added): > Whenever you make a measurement, the number of meaningful digits that you > write down implies the error in the measurement. For example if you say that > the length of an object is 0.428 m, you imply an uncertainty of about 0.001 > m. To record this measurement as either 0.4 or 0.42819667 would imply that > you only know it to 0.1 m in the first case or to 0.00000001 m in the > second. You should only report as many significant figures as are consistent > with the estimated error. The quantity 0.428 m is said to have three > significant figures, that is, three digits that make sense in terms of the > measurement. Notice that this has nothing to do with the "number of decimal > places". The same measurement in centimeters would be 42.8 cm and still be a > three significant figure number. **The accepted convention is that only one > uncertain digit is to be reported for a measurement. In the example if the > estimated error is 0.02 m you would report a result of 0.43 ± 0.02 m, not > 0.428 ± 0.02 m.** and the following quote from A Project of Joliet Junior College (IL), Lee College (TX) and the National Science Foundation (emphasis added): > The absolute error places bounds telling you that the actual value is > expected to fall between x - D x and x + D x. The absolute error is an easy > way to report the error and makes it easy to check that you are not > reporting too many significant figures. **For example, suppose you had a > measurement that was reported to be 10.02 ± 0.3 m. Because the reported > value is uncertain in the tenths place, it is wrong to report it to the > hundredths place. It should be reported to the same decimal place as the > absolute error: 10.0 ± 0.3 m.** Another point is that the absolute error and > the reported value must have the same units. If there is such a convention (cf. Rice's lab guide) and it is wrong to report more significant figures than is warranted by the absolute error (cf. the joint project's guidelines), and NIST should have adhered to such convention and striven to be correct, is it true that NIST adheres and is correct because the standard uncertainty has two least significant digits (i.e., $27$)? Had the standard uncertainty been $0.000\;000\;2\text{ mol}^{-1}$, would NIST have reported Avogadro's Number as $N_A = 6.022\;141\;3\times 10^{23}$ with 8 significant figures?


title: Rotation of angular momentum eigenfunctions?

text: I am struggling to understand this apparently obvious example in my group theory notes: ![enter image description here](http://i.stack.imgur.com/0YfRl.jpg) Where do the $e^{i\phi} $ and $e^{-i\phi} $ factors come from? I know that the $m_l$ = -1,0 and +1 angular momentum eigenstates have the $\phi$ dependence incorporated into a complex exponential factor of the very same form... But should the $\phi$ there be a variable? As opposed to a fixed rotation angle?


title: Why does burning magnesium explode when sprinkled with water?

text: Magnesium powder burns extremely well and reaches temperatures of 2500°C. However, attempts to extinguish such a magnesium fire with conventional water (e.g. from a garden hose) only make it worse: the flame grows astronomically and the whole thing gets even hotter. Why is this?


title: Time dilation - why the observers see each other the slow one but then one of them is older or younger?

text: I'm in trouble with time dilation: Suppose that there's two people on the Earth (A,B), they are twins and each other has a clock. (So they are at the same reference frame). B travels in a spaceship and is orbiting around the Earth, as B's speed has increased there's some time dilation. So it's supposed that if A "looks" to B, B's clock will run slower than A's, and viceversa. And if one day B decides come back to the Earth, he will be younger than his twin and the hour of his clock would be different (earlier than A's clock) because of time dilation. My first question is, why each person see the other's clock as the slow one? (In fact, I found some information that may solved it but I'm not sure if it's the right answer, and if it is, I don't understand it. -> It's because everyone thinks about himself as the reference frame. In other words, that for B, he is the stationary and who moves is A, so he sees A's clock run slower as B is "stopped" and the same for A. B is who is moving and A the stationary.) The other question I have is that if it's true that each one sees the other as the slow one there's something that i missed out. We all agree that B (after going to the space and came back to the Earth) he is younger than his brother. So, why B don't see A moving and aging very fast if A see his brother and his things going very slow? Let's imagine they were looking to each other all the time, if both of them see the other going slowly once they meet and "discover" B is the young one... how is that possible? [I think that in one episode of Cosmos of Carl Sagan (though may be I'm mistaken) he said that a neutrino "borned" in the Big Bang could have seen the creation of the Universe until today in few seconds due to its high speed, here starts my doubts: or i misunderstood something or there's contradictory information] (I don't know anything about physics, only what is taught at high school as I'm a teenager so it'd be better to answer with no kind of calculation as I'd not understand it.)


title: Covariant derivative and Leibniz rule

text: I read the Wikipedia page about the covariant derivative, my main problem is in this part: http://en.wikipedia.org/wiki/Covariant_derivative#Coordinate_description Some of the formulas seem to lead to contradictions, I assume I'm making some mistakes. Here are some formulas from that page. They define the Covariant derivative in the direction $\mathbf e_j$, denoted $\nabla_{\mathbf e_j}$ or $\nabla_j$ so that: $\nabla_{\mathbf e_j} \mathbf e _i = \nabla_j \mathbf e _i = \Gamma^k_{\ \ ij}\mathbf e_k$ And define it so it obeys Leibniz' rule. They then go on to show that ![Covariant Derivative](http://i.stack.imgur.com/G4YQF.png) Where it seems they used $\nabla_{\mathbf e_i} u^j = \frac {\partial u^j}{\partial x^i}$ But then later they define here: http://en.wikipedia.org/wiki/Covariant_derivative#Notation $\nabla_{\mathbf e_i} u^j = \frac {\partial u^j}{\partial x^i} + u^k \Gamma^j_{\ \ ki}$ 1) Is this a misunderstanding of mine or a problem in Wikipedia? Also instead of the definition: $\nabla_j \mathbf e _i = \Gamma^k_{\ \ ij}\mathbf e_k$ I saw in other places the Christoffel symbols defined so $\partial_j \mathbf e _i = \Gamma^k_{\ \ ij}\mathbf e_k$ 2) Is the covariant derivative of basis vectors the same as the regular derivative of a basis vector?or are these just two different definition of the Christoffel symbols? Another contradiction I saw is that they write the following formula: ![](http://upload.wikimedia.org/wikipedia/en/math/6/c/e/6ced9c5ef44f7b4e2406d9a0ae1f5ee2.png) ![](http://upload.wikimedia.org/wikipedia/en/math/1/1/5/11550e62cbbaf4b08343c173de229d1d.png) in the end of the section "Coordinate Description" where you add here a Gamma for each upper index and subtract a Gamma for each lower index according to the rule written there. According to this it seems to me that: $\nabla_j \mathbf e _i = \partial_j \mathbf e _i - \Gamma^k_{\ \ ij}\mathbf e_k$ Which is also inconsistent with how they defined the covariant derivatove 3) Is this a contradiction or a confusion of mine? Thank you very much, sorry it's so long If it's a problem I can break the question up into two questions or something


title: A question about conformal transformation in Polchinski's string theory

text: I have one more stupid question in Polchinski's string theory book. P. 46, it is said > _It is convenient to take a basis of local operators that are eigenstates > under rigid transformation (2.4.9)_ $$\mathcal{A}'(z',\bar{z}')=\zeta^{-h} > \bar{\zeta}^{-\bar{h}} \mathcal{A}(z,\bar{z}). \tag{2.4.13} $$ Eq. (2.4.9) is $$z'=\zeta z.\tag{2.4.9}$$ I simply don't understand the sentence "...basis of local operators that are eigenstates under...". Which "local operator", $\mathcal{A}$ or $\mathcal{A}'$? How Eq. (2.4.13) is obtained? How to see later that the derivative $\partial_z$ increases $h$ by one, the derivative $\partial_{\bar{z}}$ increases $\tilde{h}$ by one?


title: Pedagogic reference for calculation of 2-loop anomalous dimension (supersymmetric)

text: I want to know of pedagogic references which teach how to compute anomalous dimensions (..wave-function renormalization..) at lets say 2-loops. I guess there might be specialized techniques for supersymmetric theories and especially may be adapted for $2+1$ dimensions.   * I have in mind being able to understand calculations like say in this paper in Appendix D starting on page 39.  I found the treatment of this subject in books like Peskin and Schroeder very sketchy and not explanatory enough. I would like to know of any more detailed "modern" reference which will help lets say with the calculation linked above.


title: Impact Force versus Non-impact force why are they so different?

text: A mass that has velocity and a force f1, can create a sound and even break a wall upon impact. However if the same force f1 is applied to the wall without any motion (velocity) then the wall will not break and no sound can be heard. Why is this? I suspect that since the mass is in motion it is traveling as a wave and upon impact this wave is what generates a vibration and sound. I also suspect that the vibration caused by this impact causes the molecules around the impact area to vibrate at its harmonic frequency (causing the molecules to break apart). Is any of this true? Or am I off base? What equations exists that describes the vibrations upon impact?? Thanks!!


title: Has anyone measure the strength of the force of gravity in relation to the molecular bonding?

text: It's clear to me that gravity it a function of mass. It is also clear to me that gasses are less affected by gravity. So I'm thinking that there exists a measurable minimum binding distance $d$ between molecules that determines whether the molecules will collectively experience a gravitational pull. In other words, for the same element molecules $M_1$ and $M_2$ the gravitational force is normally represented by the addition of both masses ($M_1 + M_2$). But when $M_1$ and $M_2$ exceed a distance $d$ of separation, the mass used to determine the gravitational force is just $M_1$ or $M_2$. Then if there were a molecular bond that separates the molecules by a distance greater than $d$, gravity could be nullified. Is this possible or am I totally off base?


title: What is the smallest increment of space and the smallest increment of time?

text: I believe that every point in space is a surface point. And if it was possible to break through or tear a point in space, it will create a black hole. I believe that's why a black hole can exist anywhere in space. Since every point in space is a surface edge then punching through a point in space will causes matter to get suck into a very powerful and vast vacuum (outside of our universe). Does any of this hold water? And what about the smallest increment of time and space? Where can I find supporting Theory and equations, if any?


title: Transmission Line Theory: Coaxial Cable Per Unit Length Series Resistance Derivation

text: **Brief Background** A coaxial cable is an example of a transmission line. A transmission line can be analyzed by using a distributed element model in which there are a repeating set of per unit length parameters including   * A series resistance $R'$      * In a coaxial cable, this is the component representing the losses the electromagnetic wave propagating down the coaxial cable incurs as it encounters a nonideal conductor.     * There are two nonideal conductors (outer and inner) in a real coaxial cable.   * A shunt conductance $G'$      * In a coaxial cable, this is due to the nonideal dielectric that causes some current to flow from one conductor to the other.   * A series inductance $L'$      * In a coaxial cable, there are two parts to series inductance -- an internal inductance and an external inductance.   * A shunt capacitance $C'$      * In a coaxial cable, the presence of two conductors separated by a dielectric can be modeled as a capacitance. where the prime notation is there to indicate it is a per unit length parameter. **Question** How can I derive the series resistance per unit length $R'$? From electrostatics I know that resistance can be computed from $$R = \frac{V_{ab}}{I} = \frac{-\int_a^b \vec{E} \cdot d\vec{l}}{\int_S \sigma \vec{E} \cdot d\vec{S}}$$ where $S$ is the cross-sectional surface through which current $I$ flows. But does this apply in the case of electrodynamics as well? I am aware of how current flows through the bulk of a conductor at DC, but due to induced eddy currents within the conductor at AC, especially at higher frequencies, the current passing through a conductor gets concentrated at the surface. **Possible Solution** I have seen people derive this series resistance per unit length $R'$ by considering how much power is lost due to Ohmic losses in the two conductors. It appears that it is assumed that current flows along the surface of these conductors. Consider http://www.dannex.se/theory/4.html starting around Equation 71. It may be that the inspiration for that website's derivations were found on pages 376 and 387 in Cheng's Field and Wave Electromagnetics in sections related to general transmission lines. Can this be done with a more explicit application of Maxwell's equations? I'd also like to add as an aside that from my attempts to find a solution it appears as though the derivation of the other three parameters is very much more commonly seen.


title: Ideal 2D Unicycle Kinematics

text: A particle is connected to a massive wheel by a rigid rod. The wheel can roll without slipping on a horizontal surface. The particle is free to rotate around the centre of the wheel. I believe the system has two degrees of freedom: The centre of the wheel and the particle each have x- and y- positions, and the wheel has an angle of rotation. The constraints are:   * The wheel is on a horizontal surface, fixing its y-coord   * The wheel cannot slip, so its x-coord is directly proportional to its angle of rotation   * The particle is a fixed distance from the centre of the wheel Is this correct? This leaves two generalised coordinates, which I have taken to be the angle of rotation of the wheel and the angle between the rod and the y-axis. After struggling (and failing) with a Newtonian approach I constructed a Lagrangian for the system and applied the Euler-Lagrange equation(s), using the angles as generalised coordinates. After much algebra out popped two second-order non-linear differential equations. To my surprise, I could eliminate the wheel rotation completely from one equation, leaving it concerning only the rod angle and its derivatives. What, if at all, is the significance of this? And finally, I would like to simulate the situation computationally. Is there a general way of simulating rigid constraints acting on rigid bodies/particles, or must one find and solve (numerically) the differential equations governing the system?


What is probability current in quantum mechanics? Why define such a thing? I mean the meaning of probability current. I know the formula for it but I just don't get the idea of a flow of probability over time. Does it imply that the probability of finding a particle along dx changes over time such that it is not fixed. In other words, at t=0, the probability can be 0, then t=1, probability can be 0.5. I am quite new to quantum mechanics, so I am still trying to understand this new world.


title: Mixing generators of different dimensionality

text: Reading a paper about compactified manifolds used in Kaluza Klein theories the author discusses in which ways you can get $SU(2)\times{}U(1)\times{}U(1)$ as a subgroup of $SU(3)\times{}SU(2)\times{}U(1)$. At a certain point, it starts considering ways of getting $U(1)$. It says you can do it using the $\lambda_8$ Gell-Mann's matrix as a generator of $U(1)$. The original $U(1)$ group as well can be, of course, used, whose generator he calls $Y$, and finally any of the $U(1)$ generators of $SU(2)$, which he calls $T_3$. Then he combines the three generators in a linear combination $p\lambda_8+qT_3+rY$ Which is supposed to be a generator (via complex exponentiation) of a $U(1)$ group. How is this supposed to be understood? Being all three generators matrices of different dimensionality, how do we add them?


title: Inflation and the Meaning of Time

text: I'm not quite sure how to ask this so that it can be answered in layman's terms, but I have lately seen, in several places, that with cosmological inflation, there was a point where the universe expanded faster than the speed of light. In this explanation, the following is stated:   * Gravity itself would have split off at the Planck time, 1e-43 of a second   * The strong nuclear force by about 1e-35 of a second    * Within about 1e-32 of a second, the scalar fields would have done their work, doubling the size of the Universe at least once every 1e-34 of a second (some versions of inflation suggest even more rapid expansion than this). It goes on to say that this rapid expansion is enough to take a quantum fluctuation 1e-20 times smaller than a proton and inflate it to a sphere about 10 cm across in about 15 x 1e-33 seconds. The conclusion is that this expansion occurred at faster than light speed. My main question is about the role of time in all of this, and I'm only indirectly interested in this apparent violation of the cosmic speed limit. I have watched at least one interview (and I can't find a link to it right now) where a well-known physicist sums up the meaning of time as a big unknown, not being intuitively "fundamental" the way space is. However, in the very least, and perhaps most abstract way, time is an important part to the mathematics when it comes to explaining how things work. My own understanding of time is that it is a way of gauging change relative to some known standard that is changing. (And acknowledging that it can change, depending on one's frame of reference.) That standard might be the swinging of a pendulum or the oscillations of some subatomic particle. But in the earliest part of the universe's lifetime, before and during inflation, what was the standard for measuring time? Why would it be acceptable to say that the time duration of the inflationary period was so short and thus violated an accepted principle. Couldn't an alternative view be that the inflationary period lasted longer and nothing travelled faster than the speed of light? How was it decided that the duration of time for the inflationary period was so short?


title: Is linear momentum conserved in a system with open boundary conditions?

text: In a one dimensional lattice system with periodic boundary conditions, in which the last and the first site of the lattice are the same site, momentum is conserved modulo a vector of the reciprocal lattice. My question is if for a system with open boundary conditions, in which the single particle wave functions vanish in the two extremes of the chain, momentum or quasi-momentum is conserved. In principle, there is no discrete (nor continuous) translation symmetry, so momentum shouldn't be conserved. However, I wonder if momentum conservation is completely lost or if there are some remains.


title: Unitary gauge for non-abelian case

text: I'm reading Chapter 19 of Mandle and Shaw's Quantum field theory. In the first section it is explained that one can go with a $SU(2)$ followed by a $U(1)$ transformation from $$\begin{bmatrix}\eta_1(x) + i\eta_2(x) \\\ v + \sigma(x) + \eta_3(x)\end{bmatrix} $$ to the state $$\begin{bmatrix} 0 \\\ v + \sigma(x)\end{bmatrix}. $$ I tried to first make of this first vector a 'down isospin' by multiplying with an generic element of $SU(2)$ $$\begin{bmatrix} \alpha & \beta \\\ -\beta^\star &\alpha^\star\end{bmatrix}, $$ where $|a|^2+|b|^2=1$. Is there maybe a easier representation if $SU(2)$ which leads to the correct needed transformation?


title: Distinguish between instantaneous speed and instantaneous velocity

text: I encountered a line in my text book of physics that: > _Average speed over a finite interval of time is greater or equal to the > magnitude of the average velocity. But instantaneous speed at an instant is > equal to the magnitude of the instantaneous velocity at that instant._ Now the question is how instantaneous speed = | instantaneous velocity |?


title: A question about deriving Eq. (6.2.13) in Polchinski's string theory book volume 1

text: I have a question about deriving Eq. (6.2.13) in Polchinski's string theory book volume I. It is claimed that > Now consider the path integral with a product of tachyon vertex operators, > $$A_{S_{2}}^{n}(k,\sigma)=\left\langle [e^{ik_{1}\cdot > X(\sigma_{1})}]_{r}[e^{ik_{2}\cdot X(\sigma_{2})}]_{r}\cdots[e^{ik_{n}\cdot > X(\sigma_{n})}]_{r}\right\rangle _{S_{2}}\tag{6.2.11}$$ This corresponds to > $$J(\sigma)=\sum_{i=1}^{n}k_{i}\delta^{2}(\sigma-\sigma_{i})\tag{6.2.12}$$ > The amplitude (6.2.6) then becomes > $$A_{S_{2}}^{n}(k,\sigma)=iC_{S_{2}}^{X}(2\pi)^{d}\delta^{d}(\sum_{i}k_{i})\times\exp(-\sum_{i,j=1;i<j}^{n}k_{i}\cdot > k_{j}G'(\sigma_{i},\sigma_{j})-\frac{1}{2}\sum_{i=1}^{n}k_{i}^{2}G_{r}'(\sigma_{i},\sigma_{i}))\tag{6.2.13}$$ > where > $C_{S_{2}}^{X}=X_{0}^{-d}(\det'\frac{-\nabla^{2}}{4\pi^{2}\alpha'})_{S_{2}}^{-d/2}$ > and $G_{r}'(\sigma,\sigma')=G'(\sigma,\sigma')+\frac{\alpha'}{2}\ln > d^{2}(\sigma,\sigma')$ Eq. (6.2.6) is > > $$Z[J]=i(2\pi)^{d}\delta^{d}(J_{0})(\det'\frac{-\nabla^{2}}{4\pi^{2}\alpha'})^{-d/2}\times\exp(-\frac{1}{2}\int > d^{2}\sigma d^{2}\sigma'J(\sigma)\cdot > J(\sigma')G'(\sigma,\sigma'))\tag{6.2.6}$$ My question is: where do $X_0^{-d}$ and $G_r'$ come from in Eq. (6.2.13)? I could try to plug (6.2.12) into (6.2.6) to see all other term appears, but not $X_0^{-d}$ nor $G_r'$.


title: O(N) sigma model at large N

text: I would like to better understand the main principles of large-N expansion in quantum field theory. To this end I decided to consider simple toy-model with lagrangian (from Wikipedia) $ \mathcal{L} = \frac{1}{2}(\partial_{\mu} \phi_a)^2-\frac{m^2}{2}\phi_a^2 - \frac{\lambda}{8N}(\phi_a \, \phi_a)^2 $ My aim was to renormalize this theory in all orders of perturbation theory in leading order of $\frac{1}{N}$. The calculation of counterterms in two loops in leading order of $\frac{1}{N}$ almost coincides with the corresponding calculation in $\phi^4$ theory. In leading order of $1/N$ the counterterms are (using MS-scheme): 1 Loop: $\Delta \mathcal{L}_{\phi^4}^{1} = -\lambda^2 \mu^{2\epsilon} \frac{1}{32 \pi^2 \epsilon} \frac{(\phi_a \, \phi_a)^2}{8N}$ $\Delta \mathcal{L}_{\phi^2}^{1} = - \frac{\lambda}{32 \pi^2 \epsilon} \frac{m^2 \phi_a^2}{2}$ 2 Loops: $\Delta \mathcal{L}_{\phi^4}^{1} = -\lambda^3 \mu^{2\epsilon} (\frac{1}{32 \pi^2 \epsilon})^2 \frac{(\phi_a \, \phi_a)^2}{8N}$ $\Delta \mathcal{L}_{\phi^2}^{1} = - (\frac{\lambda}{32 \pi^2 \epsilon})^2 \frac{m^2 \phi_a^2}{2}$ I am quite sure (though I haven't proven it properly yet) that in $n$ loops the leading contribution to counterterms comes from a chain of "fish" diagrams for 4-point Green's function and chain of bubbles for 2-point Green's function (I think, it's quite easy to imagine): $\Delta \mathcal{L}_{\phi^4}^{1} = -\lambda^{n+1} \mu^{2\epsilon} (\frac{1}{32 \pi^2 \epsilon})^n \frac{(\phi_a \, \phi_a)^2}{8N}+O(\frac{1}{N})$ $\Delta \mathcal{L}_{\phi^2}^{1} = - (\frac{\lambda}{32 \pi^2 \epsilon})^n \frac{m^2 \phi_a^2}{2}+O(\frac{1}{N})$ If this speculation is correct, the summation of perturbation series is quite trivial (we have geometric series). When we do it and then take limit $\epsilon \rightarrow 0$ we will find that $\Delta \mathcal{L}^{\infty}_{\phi^2} = \frac{m^2 \phi_a^2}{2}$ $\Delta \mathcal{L}^{\infty}_{\phi^4} = \frac{(\phi_a \phi_a)^2}{8N}$ and hence the total lagrangian is simply (in the leading order of $1/N$). $\mathcal{L} = \frac{1}{2}(\partial_{\mu} \phi_a)^2$ This result seems to me highly suspicious... Did anybody do similar calculation? I looked all over the Internet and didn't find anything :( I will be very grateful for remarks and links to books or may be articles where similar problem is considered.


title: Supercritical Steam Expansion

text: Ten kg of steam at 500 bar is expanded at constant pressure until its volume increases to seven times its initial value of 0.01 $m^3 $. (a) Calculate the initial and final temperature of the steam (b) Calculate the heat that must be supplied to carry out the process * * * An attempt at solving this problem: $\Delta H$ + $\Delta E_k$ + $\Delta E_p$ = $Q - W_S$ (open system) Taking out unnecessary terms leaves us with $\Delta H = Q$ 500 bar = $5*10^7 Pa$ $\Delta H$ = $H_f - H_i$ $H_i = (5*10^7 Pa)*(0.01 m^3) = 5*10^5 J$ $H_f = (5*10^7 Pa)*(0.07 m^3) = 3.5*10^6 J$ Converting to kJ/kg (10 kg of steam) $\hat{H}_i = 50 kJ/kg$ $\hat{H}_f = 350 kJ/kg$ And that's where I stopped. I tried looking up the values of $\hat{H}_i$ and $\hat{H}_f$ in steam tables to interpolate the temperatures, but the values are too low. Since the pressure is too high, I also know the ideal gas law $(PV=nRT)$ wont apply to this scenario. Can someone point me in the right direction? Thank you for your time.


title: Wilson loops and gauge invariant operators (Part 2)

text: These questions are sort of a continuation of this previous question.   * I would like to know of the proof/reference to the fact that in a pure gauge theory Wilson loops are all the possible gauge invariant operators (... that apparently even the local operators can be gotten from the not-so-obviously-well-defined infinitesimal loop limit of them!..)    * If the pure gauge theory moves into a confining phase then shouldn't there be more observables than just the Wilson loops... like "glue balls" etc? Or are they also somehow captured by Wilson loops?    * If matter is coupled to the gauge theory are the so called ``chiral primary" operators, $Tr[\Phi_{i1}\Phi_{i2}\cdots\Phi_{im}]$ a separate class of observables than either the baryons or mesons (for those field which occur in the fundmanetal and the anti-fundamental of the gauge group) or Wilson loops?..is there a complete classification of all observables in the confined phase? {..like as I was saying last time..isn't the above classification the same thing as what in Geometric Invariant Theory is well studied as asking for all G-invariant polynomials in a polynomial ring (..often mapped to $\mathbb{C}^n$ for some $n$..) for some group $G$?..)   * But can there be gauge invariant (and hence colour singlet?) operators which are exponential in the matter fields?    * In the context of having new colour singlet( or equivalently gauge invariant?) observables in the confining phase I would like to ask the following - if one is working on a compact space-(time?) then does the Gauss's law (equation of motion for $A_0$) somehow enforce the colour-singlet/confinement condition on the matter observables?..hence possibly unlike on flat space-time, here even at zero gauge coupling one still has to keep track of the confinement constraint?


title: Electrial Conductivity of Thin Metal Films

text: What is the best way to find specific/electric conductivity which is dependent of very thin film thickness?


title: What are the 'oscillators' in the Drude Lorentz model?

text: Jackson's Electrodynamics defines the Drude-Lorentz model as a set of harmonic oscillators (running over indices $j$ below), which, if you write out the equations of motion and rearrange a little, gives you the the famous DL equation for the permittivity of a material: $$\epsilon(\omega) = \epsilon_0 + \frac{Ne^2}{m}\sum_j \frac{f_j}{\omega_j^2 - \omega^2 - i\omega \gamma_j}$$ ($N$ = density, $f_j$ = oscillator strength, $\omega_j$ = resonant frequency for oscillator $j$, $\gamma_j$ = damping constant for oscillator $j$.) Now the way he derives it, the electrons are simply oscillating spatially. But someone mentioned to me that the oscillations are actually transitions between different orbitals, which kind of makes sense to me: you could get the transition rates and such from Fermi's Golden Rule, and there's probably some analogy for the damping constant. Is this true? Could it be that the derivation presented by Jackson is just a heuristic derivation which doesn't hold up in all possible cases? Or is it completely rigorous?


title: Is there a simple, accurate formula for calculating transit times from rise and set times?

text: I have access to "rise" and "set" times for astronomical objects, and want to determine the corresponding times of culmination. Is there a reliable and accurate way to do this? It isn't clear to me from the sources of the data I have whether the times I have are for the geometric centers of the objects, or take into account refraction, or the angular extents of the objects, so I'm hoping there is an approach that is insensitive to these factors. In particular, I'm interested in transit times for the Sun.


title: constraint on scaling dimension

text: How can we show that for any scalar operator $\Delta\geq1$ (where $\Delta$ is the scaling dimension)? Where can I find a reference for reading where it comes from?