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Tricky spring on a surface question I have this relative simple-looking question that I haven't been able to solve for hours now, it's one of those questions that just drive you nuts if you don't know how to do it.
This is the scenario:
I have a spring that is on a flat surface, the springs details are like this:
spring constant = 100N/m
height = 0.1m
mass = 0.5kg
g = 10m/s^2
there is nothing attached to the spring.
The initial force exerted on the surface is 5N.
I compress the spring halfway until the force exerted on the surface is double, now 10N and then let it go.
The (simple) oscillation starts, and at one point the force exerted on the surface will be 0N (weightless).
I need to find out how much time has passed after letting it go, and reaching weightlessness.
as in:
10(N)---time--->0(N)
p.s. not homework, read comments.
| Basically at a point on the spring where y is displacement from the equilibrium condition, you'll get a differential equation $d^2/dt^2 (Y \times density) = -d^2Y/dZ^2 \times k$ the spring constant. (sorry I can't use Latex)! If we postulate that solutions look like $e^{ikZ +i\omega t}$, $ 2\pi\omega$ will be the frequency. Plug in $\omega$ or $k$ and you can solve for the other one.
Then at $Z$=height of zero the boundary condition is displacement =0, this will imply that the spatial part of the solution looks like $\sin{(kx)}$. At the top of the spring $dY/dZ$ must be zero, else there would be an unbalanced force, that means that $kZ$ must be an odd multiple of $\pi/2$. The $\omega$ values that satisfy these conditions are your eigenvalues.
Next you need to discover the amplitudes for the infinately number of modes excited. i.E.
your initial displacement as a function of Z is proportional to Z, and you must find Ai such that
$$Z=\sum_{i= odd N} Ai\cos{(ikZ)} (k*.1 = \pi/2)$$.
You may need to lookup Fourier analysis to do this, but you should get a simple formula
for all the Ai. The solution at any future time will be the sum of these (note each term as its own frequency dependence). You should discover that when time is $\pi$ times the lowest eigenvalue, you've reversed the value of F everywhere, so that will be your answer.
Undergrad physics should teach solving the wave equation, and show you how to apply these methods.
| {
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movement of photons In a typical photon experiment the photon is depicted as moving across the page, say from right to left.
Suppose we were actually able to witness such an experiment, from the side (to position of reader to a page).
If the photon is actually moving from left to right can I, standing at 90 degrees to the motion, see the photon?
| The simple answer is no, the eye can only detect photons by their direct interaction with the retina. In this case, the photon is not "visible", since it is not itself incident on the eye, nor emits "secondary" photons that indicate its position.
| {
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Vortex in liquid collects particles in center At xmas, I had a cup of tea with some debris at the bottom from the leaves. With less than an inch of tea left, I'd shake the cup to get a little vortex going, then stop shaking and watch it spin. At first, the particles were dispersed fairly evenly throughout the liquid, but as time went on (and the vortex slowed, although I don't know if it's relevant) the particles would collect in the middle, until, by the time the liquid appeared to almost no longer be turning, all the little bits were collected in this nice neat pile in the center.
What's the physical explanation of the accumulation of particles in the middle?
My guess is that it's something to do with a larger radius costing the particles more work through friction...
| If you check Bernoulli's equation across streamlines (not along), you will see that particles with larger radius have higher static pressure than those with small radius, which actually drives the motion towards the center:
$${P\over\rho} + \int {V^2\over R}dn + gz = {\rm const}$$
When: $R \to {\rm small} \implies \int {V^2\over R}dn \to {\rm big} \implies P \to {\rm small}$
When: $R\to {\rm big} \implies \int {V^2\over R} dn \to {\rm small} \implies P \to {\rm big}$
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Your favorite Physics/Astrophysics blogs? What are the Physics/Astrophysics blogs you regularly read? I'm looking to beef up my RSS feeds to catch up with during my long commutes. I'd like to discover lesser-known gems (e.g. not well known blogs such as Cosmic Variance), possibly that are updated regularly and recently.
Since this is a list, give one entry per answer please. Feel free to post more than one answer.
| Cosmic Variance
Science, Technology, and The Future
NOTE It would be great if someone who knows the blog well would write a few words about it. Just give a little more detail of what it's about. 3 sentences is more than enough. This is community wiki, so most people can edit it freely.
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What would happen if you put your hand in front of the 7 TeV beam at LHC? Some speculation here:
http://www.youtube.com/watch?v=_NMqPT6oKJ8
Is there a possibility it would pass 'undetected' through your hand, or is it certain death?
Can you conclude it to be vital, or only loose your hand?
Would it simply make a small cylindrical hole through your hand, or is there some sort of explosion-effect?
Assume your hand has a cross section of 50cm², and a thickness of 2cm, how much of the beam's energy would be transferred to your hand?
| What Gibbs said (+1), except that because the beam is highly relativistic, the probability of radiation being sent in directions other than down the beam line is very low. You can see this by looking at the problem in the center of mass reference frame for a typical collision. Because special relativity increases the mass of the protons in the beam, the center of mass of the collision products will still be moving at nearly the speed of light.
Basically it will drill a painful hole in your hand, something less than 1mm in diameter, depending on how its focused. The LHC beam dump takes a beam 0.3mm in diameter. For the effects of lower energy proton beams, see http://www.aapm.org/meetings/09PRS/documents/Flanz_AAPM_Final.pdf
By the way, the reason those physicists in the youtube didn't have realistic answers is because they were thinking on their feet. I think this was an ambush question.
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What is the difference between a complex scalar field and two real scalar fields? Consider a complex scalar field $\phi$ with the Lagrangian:
$$L = \partial_\mu\phi^\dagger\partial^\mu\phi - m^2 \phi^\dagger\phi.$$
Consider also two real scalar fields $\phi_1$ and $\phi_2$ with the Lagrangian:
$$L = \frac12\partial_\mu\phi_1\partial^\mu\phi_1 - \frac12m^2 \phi_1^2
+\frac12\partial_\mu\phi_2\partial^\mu\phi_2 - \frac12m^2 \phi_2^2.$$
Are these two systems essentially the same? If not -- what is the difference?
| I think the free Lagrangian alone does not give the physical content. We can also alternatively represent $\phi = \phi_0 \exp(i \theta)$. Then we have
$$ L = { 1 \over 2} \partial^\mu \phi_0 \partial_\mu \phi_0 + m^2 \phi_0^2 + {1 \over 2} \partial^\mu \theta \partial_\mu \theta $$
Here we can also ask whether we have one charged massive field or one massive neutral field and one massless one.
In order to decide the field content, one must couple the scalar field with the vector field or spinor field.
The complex scalar representation can have a coupling with the vector gauge field, while the two real scalar representation does not have one.
Now I have another question: If we want to couple a complex scalar field with a Dirac spinor, how can we choose from the two following
$$ L_1 = \bar \psi (\phi^\dagger + \phi) \psi $$
or alternatively
$$ L_2 = i \bar \psi (\phi^\dagger - \phi) \psi $$
And what is the physical meaning of the above two interactions?
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Can spacetime be non-orientable? This question asks what constraints there are on the global topology of spacetime from the Einstein equations. It seems to me the quotient of any global solution can in turn be a global solution. In particular, there should be non-orientable solutions.
But does quantum physics place any constraint? Because it seems to me that if space by itself is non-orientable then what happens to, say, neutral kaon interactions along two different paths that come back to the same spot with opposite orientations?
So then, is there any reason why time cannot be non-orientable? For example my mental picture (two space dimensions suppressed) is of a disc. The big bang is the centre, time is the radial direction, space is the circumferential direction. A timelike geodesic that avoids black or white holes will start on the big bang, go out to the edge of the disc, continue on the opposite edge with time and orientation (and presumably matter/anti-matter) reversed, and return to the big bang (which is also therefore the same as the big crunch). The "reflection time" of the universe would be large enough that thermodynamic violations are not observed.
| Space could be non-orientable. All spacetimes are locally orientable. To be non-orientable a space has to have some loops that cannot be shrunk to a point. It is only when going round such a loop that non-orientability could reveal itself (For example you don't know that a Mobius strip is non-orientable unless you go right round it.) So you need non-trivial topology before you could even consider a lack of orientability. It could simply be that we have never experimentally explored non trivial regions of space.
Time orientability is more complicated. Mathematically it is simply defined, but experimentally it is not at all clear what an experiment would look like. See my paper for examples. There is an argument that particle antiparticle annihilation is an experimental manifestation of time reversal.
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Two slit experiment: Where does the energy go? In Physics class we were doing the two slit experiment with a helium-neon red laser. We used this to work out the wavelength of the laser light to a high degree of accuracy. On the piece of paper the light shined on there were patterns of interference, both constructive and destructive. My question is, when the part of the paper appeared dark, where did the energy in the light go?
| Energy is indeed conserved. The link below gives a nice explanation...
http://skullsinthestars.com/2010/04/07/wave-interference-where-does-the-energy-go/
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Stopping Distance (frictionless) Assuming I have a body travelling in space at a rate of $1000~\text{m/s}$. Let's also assume my maximum deceleration speed is $10~\text{m/s}^2$. How can I calculate the minimum stopping distance of the body?
All the formulas I can find seem to require either time or distance, but not one or the other.
| The formula you want is
$$v_f^2 = v_i^2 + 2a(x_f - x_i)$$
It's one of the basic kinematic formulas taught in high school (or even middle school) physics classes.
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Will tensile strength keep a cable from snapping indefinitely? Trying to secure a wall hanging using magnets; me and a coworker came up with an interesting question:
When the hanging is hung using 1 magnet, the weight of it causes it to quickly drag the magnet down and the hanging drops. Using n magnets retards this process; causing it to fall more slowly, but does there exist a number of magnets m such that their combined strength will prevent the hanging from slipping, entirely and permanently?
Because this doesn't make for a very good question; we worked at it and arrived at a similar one; but slightly more idealized:
A weight is suspended, perfectly still, from a wire in a frictionless vacuum. If the mass of the weight is too great; it will gradually distend the cable, causing it to snap and release the weight; but will a light enough weight hang there indefinitely, or will the mass of the weight (and indeed the cable) cause the cable to snap sooner or later?
| You slightly misinterpreted your results. They don't just fall more slowly, they accelerate more slowly.
More magnets will cause the acceleration of the object to reduce. Once you have enough magnets to provide enough force to overcome the force on your object due to gravity, then it will stay up.
The same is true of your rope. Let's say atoms in a rope have some attraction to each other, much like a magnet. If the force between these atoms is high enough to overcome the force due to gravity, it will stay together.
As people have pointed out, these explanations work well in a classical world made of spheres in a vacuum, but in the real world, nothing will stay together forever.
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Empty universe in the past, non-empty in the future My question is the following. Are there solutions to the Einstein field equations, which have the property that there is a hypersurface of constant time and to the past of that surface space is empty (Minkowski space-time) and to the future it is not (non-vanishing stress-energy tensor)? At first thought it seems strange to have nothing and suddenly something. On the other hand define $g_{\mu\nu}(x)$ to be the Minkowski metric for the past of the surface and any functions (sufficiently smooth) that match it on the surface and are different than the Minkowski for the future. There are many ways (infinitely many) to do this. Then one can define the stress-energy tensor using Einstein's equations. This would seem to work, but there are certain conditions on the tensor, for example positivity, or other restrictions for physical reasons that I am unaware of, so it may be there are no such functions.
So the questions is: does anyone know of an explicit example or a more convincing argument than "there are so many functions, there has got to be some that work"? Of course a reason why it doesn't work, if that is the case, would be good too.
Thanks.
| If there existed at some point the same amount of matter and antimatter, should there not be a signature in the microwave background radiation? I would expect that the annihilation of electrons and positrons should still be a separate bump?
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Why beauty is a good guide in physics? Dirac once said that he was mainly guided by mathematical beauty more than anything else in his discovery of the famous Dirac equation. Most of the deepest equations of physics are also the most beautiful ones e.g. Maxwell's equations of classical electrodynamics, Einstein's equations of general relativity. Beauty is always considered as an important guide in physics. My question is, can/should anyone trust mathematical aesthetics so much that even without experimental verification, one can be fairly confident of its validity? (Like Einstein once believed to have said - when asked what could have been his reaction if experiments showed GR was wrong - Then I would have felt sorry for the dear Lord)
| Although I agree, beauty can be seductive and may be related to our evolutionary development of "pattern forming" in the brain--we seem inclined to find symmetry beautiful. Effective theories are often ugly as sin---take a look at the Standard Model Lagrangian and tell me it is beautiful :)
http://nuclear.ucdavis.edu/~tgutierr/files/stmL1.html
Download the plain version of the Standard Model Lagrangian Density: [ps][pdf][tex][txt]
Or download a "fun yet soul-crushing exam question" based on it: [ps][pdf][tex][txt]
I was tempted to ask the exam question here to see what response it evoked :)
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Polarization of the gluon I think that, by now, it's understood that the gluon propagator in QCD has a dynamically generated mass. Ok, so my question is the following: where does the extra polarization degree of freedom come from? Or, asking in another way: suppose you try to define an S matrix for QCD, apart from the usual problems for doing so, would it be unitary? How?
In the case of a Higgs mechanism, it is clear that the extra degree of freedom comes from "eating" the Higgs, as they say. But and where does is come from in theories where the mass is dynamically generated?
| I'm not sure this is useful, but I suppose that the problem with an S-matrix for gluons is that the gluons are not free. That is, an S-matrix deals with free particles that interact, perhaps exchange bodily fluids (i.e. charge or whatever), and then escape to infinity. But the real questions with gluons have to do with bound states. So the problem is the absence of free particles at the beginning and end of the interaction.
On the other hand, I want to believe that an S-matrix "formalism" exists even with bound particles. Instead of letting the participants escape to infinity, you assume that they eventually return to their initial state. Then perhaps you can derive some restrictions based on the requirement that the initial state repeating. This is, sort of like a dual to the usual S-matrix.
Here's the wikipedia article on S-matrix theory: http://en.wikipedia.org/wiki/S-matrix
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Determining Maximum Velocity of an object traveling horizontally I'm in the process of working on a physics related game. I'm looking to find the maximum velocity of an object given it's mass and the force acting on it when it is traveling horizontally. I believe there must be a method of calculating this but I'm unaware of it. Is there a formula for this scenario? I'm unaware if this is the same calculation as terminal velocity but I don't believe it is. Thanks for any help!
| Suitable for game physics and BoTE calculations:
*
*Assume a functional form for the frictional resistances (all the apply from rolling, sliding and fluid (wind or water)), and solve for the total resistance equal to the driving force.
*If that does not limit the speed to a reasonable velocity and this even takes place on a planet, compute the orbital velocity at that elevation. Above that speed a downward force will have to be applied to maintain local horizontalness for the motion...
Toward (1):
*
*rolling and sliding friction tend to be independent of speed, except that heat dissipation may cause secondary effects.
*Wind resistance is generally proportional to $v^2$ except in the neighborhood of the speed of sound.
*Water resistance for surface vessels is complicated with several parts proportional to square (viscous) and cubic (wave making) terms at least.
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Does entropy apply to Newton's First Law or does "acted upon" always require an external factor?
First law: Every body remains in a state of rest or uniform motion (constant velocity) unless it is acted upon by an external unbalanced force. This means that in the absence of a non-zero net force, the center of mass of a body either remains at rest, or moves at a constant speed in a straight line.
*
*Wikipedia — Newton's laws of motion
Doesn't the law of increasing entropy affect all objects though, since they are all in the closed system of the universe at large, and therefore they are all subject to slowing down, regardless of the containing medium, given enough time?
I guess what I'm curious is, can there ever be a body that will remain at uniform motion or uniform rest given that entropy must increase?
| If you ignore the microscopic explanation of entropy, entropy is just an internal state of a system, on par with the system's volume, or the number of particles in the system. If you have a gas with a fixed entropy (let's store it in a vacuum tube so it doesn't escape, and let's give the tube infinite insulation so it doesn't leech out any heat) out in deep space, and you throw it, it will just happily trail off in a straight line forever with a constant entropy, volume, and number of particles.
You get changes in entropy and whatnot only when the system in question interacts either with another system or its environment. But it really is best thinking of these things, at least on a macroscopic scale, as internal degrees of freedom of the system.
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How to explain the weak force to a layman? I'm trying to explain in simple terms what the weak interaction does, but I'm having trouble since it doesn't resemble other forces he's familiar with and I haven't been able to come up (or find on the web) with a good, simple visualization for it.
| I would actually emphasize the difference between the forces, rather than the similarity. Although we (as theorists) like to bundle the whole shebang into a "neat" $U(1)\times SU(2) \times SU(3)$ gauge structure (and possible some gauge version of gravity), it doesn't mean that reality has to be that neat (e.g. chirality of electroweak, neutrino masses, etc.)
So:
*
*Electromagnetism is long ranged, and drops off in strength with distance.
*Strong force is actually also long-ranged, but gets stronger with distance! This causes the side effect that trying to separate a pair of opposite charges causes pair creation, and so we always see neutral composite particles.
*Weak force is intrinsically short ranged (order $1/M_W$), and primarily it does not transmit a force --- but transmutates particles. Electrons go to neutrinos, quarks mix, etc.
And for gravity, say whatever your favourite quantum gravity picture say it is :-)
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What would happen if $F=m\dot{a}$? What would happen if instead of $F=m \frac{d^2x}{dt^2}$, we had $F=m \frac{d^3x}{dt^3}$ or higher?
Intuitively, I have always seen a justification for $\sim 1/r^2$ forces as the "forces being divided equally over the area of a sphere of radius $r$".
But why $n=2$ in $F=m\frac{d^nx}{dt^n}$ ?
| Newton second law is known as fundamental law of mechanics, because it is supposed to solve the fundamental problem of mechanics, that is, finding the position of a particle at any given moment in time, i.e., to find
$$
x=f(t)
$$
Plot of $x=f(t)$ can only be a straight line (a special type of curve with curvature=0) or a curve (any curve with curvature <>0). However, any plot just depends on a starting point (current value or initial value) and on how it changes when moving forward or backwards from from that given point. Such a change is represented by curve’s curvature, i.e, from this point on, your choices are, continue straight ahead (curvature=0), go up (curvature>0) or go down (curvature<0). How much you go down or up, dependes on curvature’s magnitude.
It happens that curvature depends only on second and first order derivatives
$$
\textrm{curvature}=\frac{x^{\prime\prime}}{\left(1+{x^\prime}^2\right)^{3/2}}
$$
So, any possible curve for $x=f(t)$ would just be characterized by its first and second derivatives provided that force in $F(x’,x,t)=m \frac{d^2x}{dt^2}$is properly defined.
In an universe with higher order derivative (with respect to us), one could always set that universe’s straight line to be the solution of our $n^{th}-1$ derivative, meaning that in that particular universe Newton’s first law would be a curve with respect to us, but not with respect to themselves, and all they would need to discriminate their natural state of motion (motion in a systems straight line) would be that universe’s second order derivative.
In summary second order derivative is all one needs to differentiate natural states of motion with affected states of motion.
One needs to understand that even though many quantities like, energy, momentum, velocity, acceleration, force, jerk and so on… are (and may be) defined in mechanics being afterwords useful in other branches of science, the ultimate goal of mechanics is to find $x=f(t)$.
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Homework about spinning top I have a top of unknown mass that has a moment of inertia $I=4\times 10^{-7} kg \cdot m^2$. A string is wrapped around the top and pulls it so that its tension is kept at 5.57 N for a distance of .8 m.
Could somebody help me derive some equations to help with this? Or to get me in the right direction? I have been trying to derive some sort of equations from $E=\frac{I \cdot \omega ^{2}}{2}$ but I cant get anywhere without ending up at radius = radius or mass = mass.
I need the final angular velocity.
| This question and ones like it are trying to get at the 'transferability' of energy between different frames of reference. You begin in a linear domain and move to a rotational domain. Whether you are considering the problem in the linear or rotational sense the inherent physics remains the same (at least in elementary examples such as this.), and so you can transfer physical quantities from one to the other.
The point I learned was that it helps to select a regime in which it is easy to calculate some figure of merit and which can be transfered simply to a regime in which the answer exists.
Energy is often useful in this case as it is scalar and invariant in the transform.
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neutrinos by formation of "neutron pairs" Here :
http://www.newscientist.com/article/dn20084-neutron-star-seen-forming-exotic-new-state-of-matter.html
are news on superfluidity in a neutron star. The necessary bosons they say are pairs of neutrons. So far, so good.
But then they postulate the production of neutrinos in the formation of those neutron pairs.
When neutrons pair up to form a superfluid they release neutrinos which should pass easily through the star,...
What kind of reaction is that? n + n => neutrino + "n-n" + "XYZ" ?
| The phenomenon was first predicted in this paper:
"Neutrino pair emission from finite-temperature neutron superfluid and the cooling of young neutron stars"
Flowers E. G., Ruderman M., Sutherland P. G., 1976, ApJ, 205,541
PDF here
The process they are describing is actually $n^*n^*\to\nu\overline\nu$, where $n^*$ is a quasiparticle excitation above the superfluid condensate. It is the annihilation of those quasipartilces that produces those neutrinos.
Of course, the "real" neutrons do not annihilate, but the description of the process in terms of the "real" neutrons is way too complex, because there is a whole medium made of constantly interacting neutrons.
Which is once again shows the usefulness of a concept of a quasiparticle.
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Electromagnetic fields vs electromagnetic radiation As I understand, light is what is more generally called "electromagnetic radiation", right?
The energy radiated by a star, by an antenna, by a light bulb, by your cell phone, etc.. are all the same kind of energy: electromagnetic energy, i.e. photons traveling through space.
So far, so good? (if not please clarify)
On the other hand, there are these things called "electromagnetic fields", for example earth's magnetic field, or the magnetic field around magnets, the electric field around a charge or those fields that coils and capacitors produce.
Now here is my question:
*
*Are these two things (electromagnetic fields and electromagnetic radiation) two forms of the same thing? or they are two completely different phenomena?
*If they are different things, What does light (radiation) have to do with electromagtetism?
I'm not asking for a complex theoretical answer, just an intuitive explanation.
| Electromagnetic radiation consists of waves of electric and magnetic fields, but not all configurations of electric and magnetic fields are described as "radiation." Certainly static fields, like the Earth's magnetic field and the other fields you describe, are not called "radiation."
There is a standard technical definition of electromagnetic radiation, but roughly speaking, we think of a configuration of electromagnetic fields as constituting radiation when it has "detached" from its source and propagates on its own through space. One of Maxwell's equations says, in effect, that a changing magnetic field produces an electric field. Another says that a changing electric field produces a magnetic field. An electromagnetic wave results from these two processes producing a steady flow of radiated energy that persists far from the source.
| {
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What is the usefulness of the Wigner-Eckart theorem? I am doing some self-study in between undergrad and grad school and I came across the beastly Wigner-Eckart theorem in Sakurai's Modern Quantum Mechanics. I was wondering if someone could tell me why it is useful and perhaps just help me understand a bit more about it. I have had two years of undergrad mechanics and I think I have a reasonably firm grasp of the earlier material out of Sakurai, so don't be afraid to get a little technical.
| I will not get into theoretical details -- Luboš ad Marek did that better than I'm able to.
Let me give an example instead: suppose that we need to calculate this integral:
$\int d\Omega (Y_{3m_1})^*Y_{2m_2}Y_{1m_3}$
Here $Y_{lm}$ -- are spherical harmonics and we integrate over the sphere $d\Omega=\sin\theta d\theta d\phi$.
This kind of integrals appear over and over in, say, spectroscopy problems. Let us calculate it for $m_1=m_2=m_3=0$:
$\int d\Omega (Y_{30})^*Y_{20}Y_{10} = \frac{\sqrt{105}}{32\sqrt{\pi^3}}\int d\Omega \cos\theta\,(1-3\cos^2\theta)(3\cos\theta-5\cos^3\theta)=$
$ = \frac{\sqrt{105}}{32\sqrt{\pi^3}}\cdot 2\pi \int d\theta\,\left(3\cos^2\theta\sin\theta-14\cos^4\theta\sin\theta+15\cos^6\theta\sin\theta\right)=\frac{3}{2}\sqrt{\frac{3}{35\pi}}$
Hard work, huh? The problem is that we usually need to evaluate this for all values of $m_i$. That is 7*5*3 = 105 integrals. So instead of doing all of them we got to exploit their symmetry. And that's exactly where the Wigner-Eckart theorem is useful:
$\int d\Omega (Y_{3m_1})^*Y_{2m_2}Y_{1m_3} = \langle l=3,m_1| Y_{2m_2} | l=1,m_3\rangle = C_{m_1m_2m_3}^{3\,2\,1}(3||Y_2||1)$
$C_{m_1m_2m_3}^{j_1j_2j_3}$ -- are the Clebsch-Gordan coefficients
$(3||Y_2||1)$ -- is the reduced matrix element which we can derive from our expression for $m_1=m_2=m_3=0$:
$\frac{3}{2}\sqrt{\frac{3}{35\pi}} = C_{0\,0\,0}^{3\,2\,1}(3||Y_2||1)\quad \Rightarrow \quad (3||Y_2||1)=\frac{1}{2}\sqrt{\frac{3}{\pi}}$
So the final answer for our integral is:
$\int d\Omega(Y_{3m_1})^*Y_{2m_2}Y_{1m_3}=\sqrt{\frac{3}{4\pi}}C_{m_1m_2m_3}^{3\,2\,1}$
It is reduced to calculation of the Clebsch-Gordan coefficient and there are a lot of, tables, programs, reduction and summation formulae to work with them.
| {
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Does Wick rotation work for quantum gravity? Does Wick rotation work for quantum gravity? The Euclidean Einstein-Hilbert action isn't bounded from below.
| Wick rotation might work for backgrounds with an asymptotically timelike Killing vector isometry like asymptotically flat metrics and asymptotically anti de Sitter. It might also work when there's an asymptotic Killing vector which is either timelike to null at some horizon with what lying beyond truncated. Examples are stationary black holes and de Sitter space. On the other hand, Wick rotation can't possibly apply to backgrounds like Friedmann-Robertson-walker. If anyone disagrees with me, please feel free to point out why.
| {
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Colder surface radiates to warmer surface When radiation from a colder source arrives at a warmer surface there is some debate about what happens next. To make the question more concrete lets say that the colder source is at temperature 288K. The warmer surface is at 888K and has emissivity of 1.
3 possibilities
*
*We ignore such radiation because it cannot happen.
*The radiation is subtracted from the much larger radiation of every wavelength leaving the hotter surface.
*The radiation is fully absorbed and its effect is to be re radiated at characteristic temperature of 888K (plus infinitesimally small T increase due to radiation absorption).
I would have thought that 2. and 3 are more plausible than 1.
Both 2 and 3 satisfy the Stephan Boltzmann equation.
3 however seems to imply that the radiation from colder object is transformed into much higher quality radiation and a possible second law of thermodynamics infringement.
| The hotter object absorbs external radiation as well as it absorbs its own radiation inside the body before it reaches its surface.
The heat loss of a hot object (radiated power) is determined with the object temperature but the rate of cooling down (if there is cooling down $dT/dt < 0$) is smaller in case of additional external source of energy in form of radiation coming from exterior.
| {
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Did spacetime start with the Big bang? Did spacetime start with the Big Bang? I mean, was there any presence of this spacetime we are experiencing now before big bang? And could there be a presence/existence of any other space-time before the big bang?
| This relies on an empirical finding as to whether spacetime is finite or infinite. We already know that the Big Bang occured a finite amount of time ago.
If spacetime now is discovered to be finite then the Big Bang started at a point.
If spacetime now is discovered to infinite then spacetime at the Big Bang was also infinite because infinity divided by any finite number is infinite. It also means that the Big Bang didn't occur at a point but everywhere, all at once, in an immense flood of light. I say light, because everything at the Big Bang was massless (the Higgs field, at the Big Bang has a vev that is too low to give any elementary particle, including the Higgs boson, mass) and hence moved at the speed of light and so was akin to light.
It also shows that the density of particles at the Big Bang can't have been infinite either, again because infinity divided by any finite number is still infinite).
Hence, let there be light as a description of the world is pretty accurate.
| {
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Resistance between two points on a conducting surface Suppose we have a cylindrical resistor, with resistance given by $R=\rho\cdot l/(\pi r^2)$
Let $d$ be the distance between two points in the interior of the resistor and let $r\gg d\gg l$.
Ie. it is approximately a 2D-surface (a rather thin disk).
What is the resistance between these two points?
Let $r,l\gg d$, (ie a 3D volume), is the resistance $0$ ?
Clarification:
A voltage difference is applied between two points a distance $d$ apart, inside a material with resistivity $\rho$, and the current is measured, the proportionality constant $V/I$ is called $R$. The material is a cylinder of height $l$ and radius $r$, and the two points are situated close to the center, we can write $R$ as a function of $l$, $r$ and $d$, $R(l,r,d)$, for small $d$.
The questions are then:
What is $$ \lim_{r \rightarrow \infty} \lim_{l \rightarrow \infty} R(l,r,d) $$
What is $$ \lim_{r \rightarrow \infty} \lim_{l \rightarrow 0} R(l,r,d) $$
| The resistance between two electrodes each of radius r0 (not points) separated by a distance of 2S on an infinite plane is given by
arcosh(s/r0)/pi/resistivity
https://www.physicsforums.com/proxy.php?image=http%3A%2F%2Fimg11.hostingpics.net%2Fpics%2F843563pourforum11.jpg&hash=7b69021109a0ada9b50b7ba16cfd1414
See also thermoconductivity Shape Factors
| {
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Should you really lean into a punch? There's a conventional wisdom that the best way to minimize the force impact of a punch to the head is to lean into it, rather than away from it.
Is it true? If so, why?
EDIT: Hard to search for where I got this CW, but heres one, and another. The reason it seems counter-intuitive is that I'd think if you move in the direction that a force is going to collide into you with, the collision would theoretically be softer. You see that when you catch a baseball barehanded; it hurts much more when you move towards the ball, rather than away from the ball, as it hits your hand.
| Possibly because the punch has less force behind it, if you meet it earlier. Can't say I ever heard of this conventional wisdom, but then again, I haven't been into many fights or rings.
| {
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Cooling a cup of coffee with help of a spoon During breakfast with my colleagues, a question popped into my head:
What is the fastest method to cool a cup of coffee, if your only available instrument is a spoon?
A qualitative answer would be nice, but if we could find a mathematical model or even better make the experiment (we don't have the means here:-s) for this it would be great! :-D
So far, the options that we have considered are (any other creative methods are also welcome):
Stir the coffee with the spoon:
Pros:
*
*The whirlpool has a greater surface than the flat coffee, so it is better for heat exchange with the air.
*Due to the difference in speed between the liquid and the surrounding air, the Bernoulli effect should lower the pressure and that would cool it too to keep the atmospheric pressure constant.
Cons:
*
*Joule effect should heat the coffee.
Leave the spoon inside the cup:
As the metal is a good heat conductor (and we are not talking about a wooden spoon!), and there is some part inside the liquid and another outside, it should help with the heat transfer, right?
A side question about this is what is better, to put it like normal or reversed, with the handle inside the cup? (I think it is better reversed, as there is more surface in contact with the air, as in the CPU heat sinks).
Insert and remove the spoon repeatedly:
The reasoning for this is that the spoon cools off faster when it's outside.
(I personally think it doesn't pay off the difference between keeping it always inside, as as it gets cooler, the lesser the temperature gradient and the worse for the heat transfer).
| If you hold a spoon firmly and move it back and forth perpendicular to the face of the spoon at a certain rate you will cause stable cavitation and extreme turbulence that does not splash from the cup. I do this all the time when dissolving sugar in kool-aid but with practice I think it might cool the coffee faster. Might be a close second to the spoonful-lift-pour technique.
| {
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Difference between electric field $\mathbf E$ and electric displacement field $\mathbf D$ $$\mathbf D = \varepsilon \mathbf E$$
I don't understand the difference between $\mathbf D$ and $\mathbf E$.
When I have a plate capacitor, a different medium inside will change $\mathbf D$, right?
$\mathbf E$ is only dependent from the charges right?
| The electrical field $\mathbf E$ is the fundamental one. In principle, you don't need the electrical displacement field $\mathbf D$, everything can be expressed in terms of the field $\mathbf E$ alone.
This works well for the vacuum. However, to describe electromagnetic fields in matter, it is convenient to introduce another field $\mathbf D$. Maxwells original equations are still valid, but in matter, you have to deal with additional charges and currents that are induced by the electric field and that also induce additional electric fields. (More precisely, one usually makes the approximation that the electric field induces tiny dipoles, which are described by the electric polarization $\mathbf P$.) A little bit of calculation shows that you can conveniently hide these additional charges by introducing the electrical displacement field $\mathbf D$, which then fulfills the equation
$$ \nabla· \mathbf D = \rho_\text{free} .$$
The point is that this equation involves only the "external" ("free") charge density $\rho_\text{free}$. Charges that accumulate inside the block of matter have already been taken into account by the introduction of the $\mathbf D$ field.
| {
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How fast does gravity propagate? A thought experiment: Imagine the Sun is suddenly removed. We wouldn't notice a difference for 8 minutes, because that's how long light takes to get from the Sun's surface to Earth.
However, what about the Sun's gravitational effect? If gravity propagates at the speed of light, for 8 minutes the Earth will continue to follow an orbit around nothing. If however, gravity is due to a distortion of spacetime, this distortion will cease to exist as soon as the mass is removed, thus the Earth will leave through the orbit tangent, so we could observe the Sun's disappearance more quickly.
What is the state of the research around such a thought experiment? Can this be inferred from observation?
| Here's a recent science update to this question: Gravity propagates at the speed of light at least to a precision of one part in $10^{15}$. This has been measured directly in 2017 when a Binary Neutron Star Merger happened that was both seen in gravitational waves (GW170817) and in gamma rays (GRB 170817A). In particular, here is Figure 2 from that paper:
While the neutron star merger happened across a cosmological distance of some 100 million light years, the light and gravitational wave signal were seen within a couple seconds. The ratio of these two numbers thus directly gives an upper limit on the difference of the speed of light and gravitational waves. See also this viewpoint for some context of why this measurement is really qualitatively different from anything done before, and in short, pretty cool.
| {
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What does it take to become a top physicist? What does it take to become a top physicist?
Why do so many extremely talented young upstarts totally flop as they move to more advanced physics?
| Do not Despair :) .
What does it take to become a top physicist?
To start with one has to be a physicist. A physicist is one who studies physics because of a burning curiosity about how the material world works and tries to satisfy it by going to graduate school and accumulating knowledge .
Now a "middle of the drawer" physicist has in addition to be either very good in math if aiming in theory, or be very good in experiments if aiming to be an experimentalist, and be able to use the accumulated knowledge in research at the frontier of the unknown.
Top physicist is a bit of luck and a bit of attention and dedication to science politics and university politics etc.
Why do so many extremely talented young upstarts totally flop as they move to more advanced physics?
Define "extremely talented" . If they burn to learn physics and they are good in maths why would they flop? Unless it is the "soft generation syndrome": lack of direction and persistence, giving up at the first hurdle.
| {
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Has every possible interaction between elementary particles been observed? There are some interactions that are forbidden by conservation laws, e.g. an electron cannot turn into a positron by conservation of charge and a photon cannot turn into a positron electron pair by conservation of momentum.
My question is if every interaction (between say up to 3 or 4 particles) that is consistent with all known conservation laws have been observed.
| *
*proton decay is un-observed, and suspected on the basis of various Beyond the Standard Model theories
*Nothing directly involving the Higgs had been published when the question was asked, but ATLAS and CMS have pretty definitive observations of the most easily measured channels. Some of the harder channels are still under invesitgation.
*Neutrinoless double beta decay would be the signature of $\nu + \nu \to \text{::nothing::}$ (off-shell, of course) and would indicate that neutrinos are Majorana particles. There is a report of it, but the significance is limit and it is unconfirmed
*Evidence for any kind of dark matter interactions outside of gravity is pretty sparse on the ground
*No super-symmetric partners have been observed
*I doubt anyone has seen $\nu_\tau + n \to \tau + p$, though this is required by the current electoweak formalism. Or at least, they haven't been able to show that this is what they saw
Basically lots of dark corners. Note that much of this is Beyond the Standard Model, so may or may not represent the real state of physics.
| {
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Do extra-dimensional theories like ADD or Randall-Sundrum require string theory to be true? What I mean is could it turn out that the world is not described by string theory / M-Theory, but that nevertheless some version of one of these extra-dimensional theories is true?
I have no real background in this area. I just read Randall and Sundrum's 1999 paper "A Large Mass Hierarchy from a Small Extra Dimension" (http://arxiv.org/PS_cache/hep-ph/pdf/9905/9905221v1.pdf). Other than the use of the term "brane" and a couple of references to string excitations at TeV scale, I don't see much about string theory, and I notice their theory only requires 1 extra dimension, not 6 or 7.
| Extra-dimensional scenarios may be described as "inspired" by string theory but they are independent hypotheses and they may be true even if string theory is not. However, one has to reduce the ambitions and standards of consistency.
Sociologically, it's surely true that the research of models with extra dimensions has been adopted and pursued by many people who have never take studied proper string theory or taken a course in it.
Despite the academic independence, a confirmation of experimentally accessible extra dimensions - which is extremely unlikely to occur, due to their likely tiny size - would be a huge evidence supporting string theory because it's the only framework in which the extra dimensions actually have a justification (many).
| {
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References about rigorous thermodynamics Can you suggest some references for rigorous treatment of thermodynamics? I want things like reversibility, equilibrium to be clearly defined in terms of the basic assumptions of the framework.
| *
*The pioneer of the rigorous treatment of thermodynamics is Constantin Carathéodory. His article (Carathéodory, C., Untersuchung über die Grundlagen der Thermodynamik, Math. Annalen
67, 355-386) is cited everywhere in this context, but probably you want some newer and more modern things.
*Buchdahl wrote a lot of papers about this subject in the 40's, 50's and 60's. He summarized these in the book:
H.A. Buchdahl, The Concepts of Classical Thermodynamics (Cambridge Monographs on Physics), 1966.
*There was a recent series of articles on this subject by Lieb and Yngavason which became famous. You can find the online version of these here, here, here and here :).
*Finally, I have come across the book T. Matolcsi, "Ordinary Thermodynamics" (since a few friends of mine went to the class of the author), which treats thermodynamics in a mathematically very rigorous way.
I hope some of these will help you.
| {
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Electricity takes the path of least resistance?
Electricity takes the path of least resistance!
Is this statement correct?
If so, why is it the case? If there are two paths available, and one, for example, has a resistor, why would the current run through the other path only, and not both?
| This statement is true and a direct consequence of the 5th Law of Thermodynamics, the Onsager Relations for which Lars Onsager of Yale received the Nobel Prize in 1968.
In an electrical circuit, for DC, current takes the path of least reisitance;
For AC it takes the path of least inductance (impedance).
So a pulse of voltage will cause the current distribution to be determined by path inductance and then finish with the distribution determined by resistance.
On a circuit board, this is critical as the ground plane insures the differences between these two distribution paths are minimal.
| {
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How do you derive Noether's theorem when the action combines chiral, antichiral, and full superspace? How do you derive Noether's theorem when the action combines chiral, antichiral, and full superspace?
| You can think of a generalized innerproduct, where terms get integrated over the "correct" superspace. In my view, the best way of deriving the propagators for chiral fields. It's much cleaner than lifting everything up to full superspace.
An example of this occurs in section 4.8 of Ideas and methods of supersymmetry and supergravity and probably other papers by the authors. In this section only chiral and antichiral spaces are used, but it can be generalized to include a full superspace sector.
Once you've set up your conventions for this type of innerproduct, then everything basically works as you'd expect.
| {
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Is it really possible for water to be held in a "cone shape" for a brief period of time? I just saw this "trick" where a cup of water is turned over onto a table without spilling (using a piece of cardboard. After removing the cardboard from underneath the cup, the person then removes the cup in a particular way (lifts straight up and twists) and lo and behold, the water stays in it's position as if the cup were still there!? (watch the video to fully understand) Is this really possible? If so, and the real question I'm looking to have answered is, how?
After further research it appears that deionized water is needed as well.
| No, It was CG. The video-maker himself said it. Here: http://forums.cgsociety.org/showthread.php?t=957350
| {
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meaning of an integral in the continuity equation This is about continuity equation. What does the last integral mean?
$$\frac{\mathrm{d}Q_V}{\mathrm{d}t}=\iiint_V \mathrm{d}^3x \,\frac{\partial\rho}{\partial t}=-\iiint_V\! \mathrm{d}^3x\,\operatorname{div}\,\mathbf{j}=-\iint\limits_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf j\;\cdot\mathbf n\,{d}S\,,$$
EDIT:
And what does $\partial V$ mean? Why not S like surface?
| The last integral is a surface integral. It is the fluid (current) flux crossing the volume surface integrated over the whole surface. It is how much charge is leaving the volume per second. ($\partial V$ means boundary of V and you sum all local $j_\bot dS$.)
| {
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Magnetic moment of relativistic rotating ring Let's consider rotating charged ring. Theoretically mass of this ring has no limit as rotation speed increases. So what about magnetic moment of the ring? Is it limited by the value of speed of light?
| The magnetic moment of a rotating charged ring is
$$
m=IA=\lambda v A.
$$
Here $I$ is current, $\lambda$ is linear charge density, and $A$ is the enclosed area. This expression is true even relativistically. The quantities $\lambda,A$ don't depend on the rotation speed, so the magnetic moment is limited to $\lambda c A$.
| {
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Physics for mathematicians How and from where does a mathematician learn physics from a mathematical stand point? I am reading the book by Spivak Elementary Mechanics from a mathematicians view point. The first couple of pages of Lecture 1 of the book summarizes what I intend by physics from a mathematical stand point. I wanted to find out what are the other good sources for other branches of physics.
| E. Zeidler, Quantum Field theory I Basics in Mathematics and Physics, Springer 2006. http://www.mis.mpg.de/zeidler/qft.html
is a book I highly recommend. It is the first volume of a sequence, of which not all volumes have been published yet. This volume gives an overview over the main mathematical techniques used in quantum physics, in a way that you cannot find anywhere else.
It is a mix of rigorous mathematics and intuitive explanation, and tries to build ''A bridge between mathematiciands and physicists'' as the subtitle says.
| {
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Do we take gravity = 9.8 m/s² for all heights when solving problems? Why or why not? Do we take gravity = 9.8 m/s² for all heights when solving problems?
| $g$ becomes $ g \approx 9.7 \frac{m}{s^2}$ at a height of about 35km, so it would be ok to use the value $9.81$ for "down to earth" problems.
The relevant wikipedia article has lots of useful information, like for example the following approximation formula for different heights: $$ g_h=g_0\left(\frac{r_e}{r_e+h}\right)^2 $$
Where $g_h$, is the gravity measure at height $h$ above sea level; $r_e$, is the Earth's mean radius and $g_0$, is the standard gravity.
| {
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Arguments for and against Many Worlds? I would like to hear the best arguments for and against the Many Worlds interpretation of QM.
| A good argument against is Occham's razor. Another is the fact that it is not an experimental testable hypothesis. The best argument I think against it is the fact that the only reasons for Many-worlds are based in human language, whose intuition is only developed for a classical setting, as such any human meta-reasoning cannot be expected to apply to non-classical areas, there are no mathemtical or scientific reasoning involved.
| {
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Neutron star references? I'm looking for contemporary reviews on neutron stars. Seems like this area is pretty active, so even reviews from five or ten years ago are somewhat lacking, though certainly not worthless. Does anyone have recommendations? Newer is better.
Books are okay too, but I'd prefer one of those 70-ish page reviews you find in journals.
| I would suggest these from Living Reviews in Relativity,
Rotating Stars in Relativity, by Nikos Stergioulas,
Physics of Neutron Star Crusts, by Nicolas Chamel and Pawel Haensel and
Relativistic Fluid Dynamics: Physics for Many Different Scales, by Nils Andersson and Gregory L. Comer
| {
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List of freely available physics books I'm trying to amass a list of physics books with open-source licenses, like Creative Commons, GPL, etc. The books can be about a particular field in physics or about physics in general.
What are some freely available great physics books on the Internet?
edit: I'm aware that there are tons of freely available lecture notes online. Still, it'd be nice to be able to know the best available free resources around.
As a starter:
http://www.phys.uu.nl/~thooft/theorist.html
jump to list sorted by medium / type
Table of contents sorted by field (in alphabetical order):
*
*Chaos Theory
*Earth System Physics
*Mathematical Physics
*Quantum Field Theory
| Mathematical Tools for Physics, James Nearing
Also available in paperback from Dover. Undergraduate-level math methods book. Clear writing, many problems and exercises (usually without solution). IMHO better than Boas.
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Where should a physicist go to learn chemistry? I took an introductory chemistry course long ago, but the rules seemed arbitrary, and I've forgotten most of what I learned. Now that I have an undergraduate education in physics, I should be able to use physics to learn general chemistry more effectively. What resources, either books or on-line, are useful for physicists to learn the fundamentals of chemistry? I'm not enrolled at a university, so official courses and labs aren't realistic.
Please note that I am not looking for books on specialized advanced topics, but a general introduction to chemistry that takes advantage of thermodynamics, statistical mechanics, and quantum mechanics while requiring little or no prior knowledge of chemistry.
| I think you should have a look at "Understanding Molecules: Lectures on Chemistry for Physicists and Engineers". I haven't read it myself but it looks promising.
| {
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Gamma Ray Bubble at the center of our galaxy seen by Fermi Telescope How could we measure high energy photons, whithout measuring them ?
I can't understand how we can "see" those Gamma Ray Bubbles if they are not reaching here
In this graph from Nasa you can see those "bubbles" are not reaching solar system:
Then how could be measure that Gamma Ray without the Gamma Rays
Thanks for any answer!
| Concerning the 8-shaped bubbles around the galaxy, see
http://motls.blogspot.com/2010/11/fermi-milky-way-cutting-x-ray-infinity.html
http://arxiv.org/abs/arXiv:1005.5480
They're not pictures of photons - X-rays themselves. The infinity symbol is a picture of X-ray sources: we are observing the X-rays that came from those sources here. Note that the whole structure is smaller than 100,000 light years or so - very tiny when compared to the cosmological distances. So if the 8-shaped sources were created 10 million years ago, the time needed for the photons to get here is negligible. They're here "immediately".
It's hard to measure the distance from which an X-ray is approaching us. However, you should understand that the Sun and the Earth are not in the middle of the Milky Way. They're not in the middle of the 8-shaped figure. We're looking at the situation from the "side" (the Solar System is somewhere between the center and the visible edge of our Galaxy) so we literally see something that is 8-shaped in the skies. Assuming that the distribution of the sources is rotationally symmetric - with respect to the Milky Way's axis - one can actually reconstruct the shape of the sources in 3D from the 2D picture we see (because the 3D picture only depends on 2 dimensions, because of the axial symmetry).
| {
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I need help with finding distance traveled How do I find the distance traveled of an object if the speed is not constant?
| In principle, as the others say, you need to calculate the integral of the speed over time to determine the distance traveled.
But a non-constant speed doesn't necessarily mean that the function that describes the speed is complicated. For instance, you may be able to know the average speed simply analyzing the speed function.
Say that the speed increases linearly with time: constant acceleration. Then, you know the starting speed (at A) and the ending speed (at B), and you can easily calculate the average:
$$ v_{avg} = \frac{v_{B} - v_{A}}{t_B - t_A} $$
| {
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Snell's law starting from QED? Can one "interpret" Snell's law in terms of QED and the photon picture? How would one justifiy this interpretation with some degree of mathematical rigour? At the end I would like to have a direct path from QED to Snell's law as an approximation which is mathematically exact to some degree and gives a deeper physical insight (i.e. from a microscopic = qft perspective) to Snell's law.
| This appears to be explained in detail in Feynman's "QED the strange theory of light and matter" in Chapter 2, page 39 to 45, of the 2006 edition, in more or less plain English.
| {
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Are water waves (i.e. on the surface of the ocean) longitudinal or transverse? I'm convinced that water waves for example:
are a combination of longitudinal and transverse. Any references or proofs of this or otherwise?
| Maybe sea waves are longitudinal at sea but when they hit the shallows of the shore they become transverse waves and take the shape of a wheel and roll towards the shore.
Just guessing this from my years of surfing. The waves are up & down out in the deep but turn into tubes when they reach the shallows.
| {
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Why are Saturn's rings so thin? Take a look at this picture (from APOD https://apod.nasa.gov/apod/ap110308.html):
I presume that rocks within rings smash each other. Below the picture there is a note which says that Saturn's rings are about 1 km thick.
Is it an explained phenomenon?
| There seems to be a known explanation. I quote from Composition, Structure, Dynamics, and Evolution of Saturn’s Rings, Larry W. Esposito (Annu. Rev. Earth Planet. Sci. 2010.38:383-410):
[The] rapid collision rate explains why each ring is a nearly flat disk. Starting with a set of particle orbits on eccentric and mutually inclined orbits (e.g., the fragments of a small, shattered moon), collisions between particles dissipate energy but also must conserve the overall angular momentum of the ensemble. Thus, the relative velocity is damped out, and the disk flattens after only a few collisions to a set of nearly coplanar, circular orbits.
I think the key is that particles in a thick ring would not move in parallel planes but would have slanted trajectories, colliding all the time and losing their energy very fast.
| {
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Why is the decibel scale logarithmic? Could someone explain in simple terms (let's say, limited to a high school calculus vocabulary) why decibels are measured on a logarithmic scale?
(This isn't homework, just good old fashioned curiousity.)
| I don't know anything about the history of the Bel and related measures.
Logarithmic scales--whether for audio intensities, Earthquake energies, astronomical brightnesses, etc--have two advantages:
*
*You can look at phenomena over a wide ranges of scales with numbers that remain conveniently human-sized all the time. An earthquake you can barely detect and one that causes a regional disaster both fit between 1 and 10. Likewise the stillness of an audio-dead room and the pain of an amp turned up to 11 fit between 10 and 130.
*Fractional measures are converted into differences which most people find easier to compute quickly. Three decibels reduction is always the same fractional difference; the EEs get a lot of mileage out of this.
These scales may seem very artificial at first, but if you use them they will become second nature.
| {
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Is it possible for one side of the universe to "meet" the other? I've variously heard the shape of the universe being described as multi-dimensional, like a helix or mobius strip, and super string theorem seems to say there are lots of universes all piled up next to each other in vibrating planes.
My question is, can the edges of our universe ever join up? Are they already all joined up? as in is our universe really in the shape of a giant donut (yum). or is this shear lunacy?
I guess I am assuming a model of the universe where the "edge" is not necessarily expanding constantly "out" into infinite amounts of empty vacuum, but rather one where the edges are the boundaries to that expansion (so that it seems from an observable position that the universe is expanding, when really its flexing inside these boundaries). As indicated in the comments no doubt an inaccurate description, but I was imagining it like a balloon.
| The short answer is no!
More about imagination how space of our universe looks (and how expanding) today and in past check here :WMAP
| {
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Coriolis effect on Tsunami The Japanese tsunami, moving at about 700 km/h, affected areas as distant as Chile's coast, 20 hours after the earthquake. How does the Coriolis force affect tsunami? Also, I saw an image of a boat caught within a large whirlpool. Is the whirlpool's rotation due to Coriolis force?
| It does have an effect.
Also see this paper about modelling tsunami propagation.
As noted in the paper, the Coriolis force only becomes important over large distances.
Here's an article on MathWorld including many references.
| {
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estimating ion collisions from Vlasov equations Suppose you have a distribution function $ f_{\alpha}( \vec{r} , \vec{p} , t)$ obtained from Vlasov equations for a certain $\alpha$ species, say some ions.
Is there a rigorous way (in the domain of validity of the model, that is) to estimate a collision density function $ p^{collisions}_{\alpha}( \vec{r} , E_{CM} , t)$ , where $E_{CM}$ is a variable describing collision energy parameters? A link or reference would be great
| I am not sure to understand your question because the vlasov equation is only valid for a collisionless plasma ... The interactions between particles is done through the long range mean electromagnetic field. If you want to include the collision operator you need to work with the Landau's equation or Fokker-Planck one. For an uncharged gas, the Boltzmann's equation gives good results.
One of the best author is probably Radu Balescu. He has written many good books on kinetic theory.
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Uses of the 'Golden Ratio' in Physics What are some physics applications of the golden ratio?
$$\varphi~=~ \frac{1+\sqrt{5}}{2}~\approx~ 1.6180339887\ldots$$
Does it ever function specifically as a constant in any formulas or theorems?
EDIT: Original title said Golden Radio... facepalm. I originally asked this question at math.stackexchange but the answers there were all too abstract or useless for me.
| Googling arxiv comes up with lots of hits. For example:
NewJ.Phys.11:063026 (2009), Adisorn Adulpravitchai, Alexander Blum, Werner Rodejohann, Golden Ratio Prediction for Solar Neutrino Mixing :
It has recently been speculated that the solar neutrino mixing angle is connected to the golden ratio $\phi$. Two such proposals have been made, $\cot(\theta_{12}) = \phi$ and $\cos(\theta_{12}) = \phi/2$. We compare these Ansatze and discuss a model leading to $\cos(\theta_{12}) = \phi/2$ based on the dihedral group $D_{10}$. This symmetry is a natural candidate because the angle in the expression $\cos(\theta_{12} = \phi/2$ is simply $\pi/5$, or 36 degrees. This is the exterior angle of a decagon and $D_{10}$ is its rotational symmetry group. We also estimate radiative corrections to the golden ratio predictions.
| {
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Can a nuclear reactor meltdown be contained with molten lead? If lead can absorb or block radiation, would it be possible to pump molten lead into a reactor core which is melting, so that it would eventually cool and contain the radiation?
Is there something that can be dumped into the core that will both stop the reaction (extremely rapidly) AND will not combine with radioactive material and evaporate into the atmosphere, thus causing a radioactive cloud?
| The GE Mark V containment system used at Fukushima has a design basis that calls for the concrete containment vessel to withstand a complete meltdown. This is typical of reactors in the US. There are many penetrations of the vessel and some of them may be leaking, but that does not necessarily mean the vessel was breached.
Pebble bed reactors can be shut down completely and not melt down. The next reactors in the US will also be able to lose all coolant flow and not melt down.
| {
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Why does the light side of the moon appear not to line up correctly with the evening sun? I live at roughly $(52.4^\circ,-2.1^\circ)$. On sunny evenings I've often looked at the Moon and the Sun and noticed that the light part of the Moon does not appear to line up with the Sun. For example, at about 17:00 GMT on 13 Mar 2011, I noticed the half Moon was facing toward a point roughly $10^\circ-20^\circ$ above where the Sun appeared to be. Why?
| I am puzzled by you question. When one has two points, the sun and the moon, one can always find a line connecting them, by definition of line.
If you mean why the earth is not part of that straight line, it is because the moon has an orbit around the earth, and the angle of the line earth-moon changes. It is the reason the moon has phases. Earth, moon and sun are on the same line during full moon, and the moon rises while the sun sets.
Edit: If as Ted Bun says you mean the bisecting line from the center of the moon, then the drawing given in wikipedia gives an angle because of the motion of the moon around the earth and the motion of the earth around the sun, except at full moon and new moon (if the rays shown are a correct depiction of the sun's direction).
Edit2: If one looks at the drawing In Carl Brannen's answer in combination with the wiki drawing, I think the parallax arises depending on the phase of the moon, because what one sees from the earth is not the total lit up semicircle of the moon. Part of it is hidden from the human observer so the apparent bisector is not the real bisector.
| {
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For an accelerated charge to radiate, is an electromagnetic field as the source necessary? For an accelerated charge to radiate, must an electromagnetic field be the source of the force?
Would it radiate if accelerated by a gravitational field?
| I suspect that two charged objects orbiting one another due to gravitational attraction would radiate, but I can't support that assertion with a citation.
The question of whether or not a charge radiates when it is uniformly accelerated by gravity is an open question; read the link for an excellent discussion of why.
EDIT:
The link I posted isn't inspiring trust, so I searched for peer-reviewed work. I found two relevant papers:
Physical interpretation of the Schott energy of an accelerating point charge and the question of whether a uniformly accelerating charge radiates
The significance of the Schott energy for energy-momentum conservation of a radiating charge obeying the Lorentz–Abraham–Dirac equation
I'm not qualified to comment on the quality of the papers, but both attack the question 'does a falling charge radiate', implying an open question. I didn't find any experimental work on the subject.
This brings up a topic better suited for meta-discussion: I would like to see even more external citations in the answers here. I would also like to see more answers clearly indicate their logical foundation. Is your answer based on...
- Original research?
- Predicted by peer-reviewed theory but unverified?
- Indirectly experimentally verified?
- Directly experimentally verified?
Don't make us guess!
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CPT violation and how could quark masses differ from anti-quark masses? A recent experimental paper measures a difference between the top quark and anti-top quark masses:
Fermilab-Pub-11-062-E, CDF Collaboration, Measurement of the mass difference between $t$ and $\bar{t}$ quarks
We present a direct measurement of the
mass difference between $t$ and
$\bar{t}$ quarks using $t\bar{t}$
candidate events in the lepton+jets
channel, collected with the CDF II
detector at Fermilab's 1.96 TeV
Tevatron $p\bar{p}$ Collider. We make an
event by event estimate of the mass
difference to construct templates for
top quark pair signal events and
background events. The resulting mass
difference distribution of data is
compared to templates of signals and
background using a maximum likelihood
fit. From a sample corresponding to an
integrated luminosity of 1/5.6 fb, we
measure a mass difference,
$\mathrm{M}_{t} -
> \mathrm{M}_{\bar{t}}$ $= -3.3 \pm
> 1.4(\textrm{stat}) \pm 1.0(\textrm{syst})$, approximately two standard deviations away from the CPT
hypothesis of zero mass difference.
This is the most precise measurement
of a mass difference between $t$ and
its $\bar{t}$ partner to date.
http://arxiv.org/abs/1103.2782
This seems to pile on to the recent evidence showing differences between the masses of the neutrinos and anti-neutrinos. But unlike neutrinos, quarks can't be Majorana spinors. So what theoretical explanations for this are possible?
| There is one simple, obvious, and almost certainly correct theoretical explanation: two-sigma effects show up all the time and, like most of them, this one is not real.
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Why does nuclear fuel not form a critical mass in the course of a meltdown? A BWR reactor core may contain up to 146 tons of uranium. Why does it not form a critical mass when molten? Are there any estimates of the critical mass of the resulting zirconium alloy, steel, concrete and uranium oxide mixture?
| Several things are required to cause a nuclear explosion. It is not just about mass... In terms of a melt down, perhaps heat is the real issue, because it expands the gap between atoms and this diminishes the target cross section for a neutron to strike, thus lengthening the mean neutron pathway.
Other issues include isotope purity of the fissile mass... Also impurity from decay products like Xenon 135. In a nuclear melt down like Chernobyl for example there was very high Xenon poisoning before the steam/hydrogen explosion which destroyed containment buildings.
Geometry of a critical mass is a further consideration, but given the two points above once it is too hot and contaminated Geometry is the least consideration.
Compression of the critical mass is another and this is related to heat. If the mass were squashed tight or frozen the chances of explosion would be higher. In a nuclear warhead too much Alpha activity raises the temperature and degrades the bomb. With heat in a reactor melt down there is virtually no compression.
Having an effective reflector would increase the chances of a nuclear explosion, but nuclear reactors are not normally configured to reflect neutrons back into the pile. In any case in a melt down the confinement vessel is usually ruptured.
Even if you somehow overcame all this there still has to be sufficient neutron flux and this is totally unlikely. In terms of Neutron Flux one has to ensure that 35-40% of neutrons do not escape the mass to obtain an explosion.
| {
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What is the most energy efficient way to boil an egg? Starting with a pot of cold tap water, I want to cook a hard-boiled egg using the minimum amount of energy. Is it more energy efficient to bring a pot to boil first and then put the egg in it, or to put the egg in the pot of cold water first and let it heat up with the water?
| I know the answer :).
The most energy efficient way to get a hard boiled egg is to have a pot with a good cover on an electric range.
1)cover the bottom of pot with 1mm water, put eggs in and pot on the range and and turn it on to the maximum.
2)when the cover starts popping, turn off heat completely, leave it on the range, for the residual heat.
3)wait 3 minutes for very soft, 4 to 6 for medium and 8 or more for hard.
Pot with good cover means the cover is not popping once the heat is off.
| {
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Quaternions and 4-vectors I recently realised that quaternions could be used to write intervals or norms of vectors in special relativity:
$$(t,ix,jy,kz)^2 = t^2 + (ix)^2 + (jy)^2 + (kz)^2 = t^2 - x^2 - y^2 - z^2$$
Is it useful? Is it used? Does it bring anything? Or is it just funny?
| Cornelius Lanczos has a chapter on quaternions and special relativity in his "The Variational Principles of Mechanics". So, is has been used. But it seems more straightforward to consider the multivector algebra of spacetime so t,x,y,z really are on the same footing.
| {
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Can electricity transfer radioactivity? If a cable used to power something is exposed to a radioactive source will it over time make the entire cable radioactive?
| A cable cannot transport radioactivity away from the reactor using electrical current as a transport mechanism.
Nevertheless - if a material is exposed to a source of particles of sufficiently high energy (high energy particle beam, or neutron beam, for instance), the material can become activated (meaning that some of the atoms in the material will become radioactive), and may remain activated for a very long time, so use of materials that had been previously used in particle beam applications must be carefully considered.
So, the answer is yes, in the sense that material can become radioactive from exposure to high energy particles as one would find in a reactor/particle beams and that activation can be long lived and could potentially migrate through reuse/re-purposing of components, but no in the sense that the continued activation is dependent on the local presence of such a high energy source.
| {
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Are regular light bulbs better for the eyes than CFLs or "tube lights"? I've heard that regular light bulbs with a filament are better for the eyes. Is the spectrum of one worse than the other? If so, are there any regulations for their use in industrial settings for worker safety?
| It is possible that children growing up under one kind of light would be more likely to become nearsighted (for example) than under another kind of light. But we don't know.
What we do know is that children who spend a lot of time under sunlight grow up with better vision than those who spend more time indoors:
http://www.nytimes.com/2011/06/21/opinion/21wang.html
The people who discovered this fact, guessed (very plausibly) that this is because sunlight is much much brighter than indoor light. But as far as anyone knows, the spectrum could make a difference too.
The spectra of CFLs and incandescents are certainly different. Both are different from sunlight, in different ways. So it's not impossible that they would have a different effect on vision development. Maybe CFLs would cause less nearsightedness than incandescents, maybe more, maybe the same amount. (I'd say "same amount" is the safest bet.) It would be interesting for someone to study this, if they haven't already.
I know virtually nothing about this topic beyond the New York Times article above. Someone can please correct me if I said something wrong. :-)
| {
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What properties do you need for building a tower? When I was a boy I used to daydream about building a tower so tall that the top of it would project into near space.
There would perhaps be a zero gravity area in the penthouse where my friends and I could bounce around and play space versions of various earth-based games and sports in most excellent zero-g conditions.
Much to my continued disappointment and despite all the technological advances of the last thirty or so years, no one has built such a structure.
Can anyone explain the physical limitations/constraints that are preventing someone from realising my fantasy of a 'Space Tower'?
UPDATE: This Kickstarter Project seems to be pretty confident ...
| Towers supported from the bottom are a bit tricky.
Buckling
limits how tall a column can be. One needs to additional lateral stiffness to overcome this, usually by putting up guy wires. Even so there are going to be real limits, as Anonymous Coward has mentioned above, solids obtain their stiffness from chemical interactions between molecules and atoms, and the strength to weight ratio is limited. There are some plans for some structures up to about a kilometer, but the cost per unit volume of building goes up for tall buildings. We could probably go a lot higher by the use of carbon nanotubes, but we are years away from being about to construct practical guy wires from them.
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A Basic Question about Gravity, Inertia or Momentum or something along those lines Why is it that if I'm sitting on a seat on a bus or train and its moving quite fast, I am able to throw something in the air and easily catch it? Why is it that I haven't moved 'past' the thing during the time its travelling up and down?
| Momentum is conserved. If you are on a frame (the bus) moving at a velocity that is constant, then everything else is as well. The momentum of every object is $p~=~mv$. This is whether or not there is something holding to the frame. In the absence of some force a body maintains a constant momentum.
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Do cosmological and Doppler redshift produce different patterns? For a given black body radiation curve, would the changes to the spectrum resulting from cosmological expansion and those from Doppler effects be distinguishable on the basis of the shapes of the resulting curves alone? Or, put another way, starting from the same spectrum, can both processes produce the same observation (for suitably chosen magnitudes of expansion or velocity)?
| The redshift due to cosmological expansion is identical to a Doppler shift in its effect on the spectrum of any source. To be specific, both phenomena "stretch" all wavelengths by the same factor.
There's a very good reason for this: in a suitable coordinate system, the cosmological redshift is a Doppler shift.
You'll find statements in some textbooks saying that this isn't true, but a weak version of this statement, which is nonetheless strong enough to explain why the effects on the spectra are identical, is uncontroversially true. To be specific, the redshift of a distant galaxy can be thought of as the accumulation of many infinitesimal Doppler shifts along the line of sight. (Each member of a family of comoving observers is in motion relative to her neighbor, and each can "watch" the redshift build up gradually due to these relative velocities.)
One perspective on this subject (mine, to be precise) can be found in this paper. Even if you don't like our point of view in this paper, our description of and references to other treatments may be of interest.
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Will Earth Hour do damage to power supply system? There is always a debate around Earth Hour every year, and the opposite side of Earth Hour usually claims that
The (sudden) decrease and increase of the power usage in the start and end of Earth Hour will cause much more power loss (than the save of power), and even do damage to the power supply system.
Is this statement true? To what extent?
Thank you very much.
| It is unlikely that the Earth Hour will cause substantial damage to the power supply system. However, it will require a lot of energy and manpower to adjust the power plants to the changes.
First of all, generators are automatically taken off the grid if it is detected that the power supply is higher than the current demand.
When the Earth Hour ends, power generation companies will have to turn the generators on (which also consumes time and energy) is such a way that they can supply energy to all those who turn their lights on. But they can't do this too rapidly, or the generators will automatically shut down again.
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What future technologies does particle physics and string theory promise? What practical application can we expect from particle physics a century or two from now? What use can we make of quark-gluon plasmas or strange quarks? How can we harness W- and Z-bosons or the Higgs boson? Nuclear physics has given us nuclear plants and the promise of fusion power in the near future. What about particle physics? If we extend our timeframe, what promise does string theory give us? Can we make use of black holes?
| Quantum Chromodynamics, the electroweak theory, or general theory of relativity - or quantum gravity and string theory - are not methods to obtain new devices; they're theories meant to understand the truth about the Universe.
I find it unlikely that any of those things will become practically useful. It may still hypothetically happen, but if it will, no one can predict how this could happen at present - but even if those things happen to have practical applications sometime in the future, that's not why they've been and why they are being studied.
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Is it true that the angular momentum of electromagnetic waves in an anisotropic medium is an integral of motion? Extending my previous question Angular moment and EM wave, does it make sense to talk about the angular momentum of electromagnetic waves in an anisotropic medium? It is not obvious that the angular momentum is conserved in this case. However, if the anisotropy is introduced by the external magnetic field (eg, magnetized plasmas), the projection of the angular momentum of a wave packet on the direction of the magnetic field might be conserved.
| As was pointed out in an answer to your previous question, the polarization of your beam can be thought of as the "spin" angular momentum being carried by it. Circularly polarized light carries angular momentum, linearly polarized light does not (not counting the "orbital" angular momentum carried by the spatial profile of the beam).
Viewed in this light, I would say that angular momentum is not conserved when light propagates through an arbitrary anisotropic medium. A simple example is a half-waveplate, a device which can convert right-hand circular polarized into left-circular. Waveplates are very simple -- they are just a small plate of anisotropic material, cut to a certain width and with their "fast axis" oriented at some set angle relative to the input polarization.
Summing up, the angular momentum of light propagating in an anisotropic medium is not necessarily conserved. The excess angular momentum is, presumably, taken up by the medium.
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Introductory texts for functionals and calculus of variation I am going to learn some math about functionALs (like functional derivative, functional integration, functional Fourier transform) and calculus of variation. Just looking forward to any good introductory text for this topic. Any idea will be appreciated.
| The standard encyclopedic treatise of nonlinear functional analysis is the 5 volume opus of Eberhard Zeidler, "Nonlinear Functional Analysis and Its Applications". It covers a lot of material about variational calculus, for example, in volume III "Variational Methods and Optimization". The applications are usually applications from physics.
If that is too much material, there is also a two volume version including some topics of linear functional analysis as well, "Applied functional analysis. Main principles and their applications." and "Applied functional analysis. Applications to mathematical physics."
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Why does Venus spin in the opposite direction? Given: Law of Conservation of Angular Momentum.
*
*Reverse spinning with dense atmosphere (92 times > Earth & CO2 dominant sulphur based).
*Surface same degree of aging all over.
*Hypothetical large impact is not a sufficient answer.
Assuming any object large enough to alter a planets rotation or even orbit would likely destroy most of its shape, yet Venus has retained a spherical property with a seemingly flat, even terrain indicating no volcanoes,and few if any visible meteor impacts. It would be fragmented and dispersed for billions of years. Even the question of what meteor, comet, asteroid composition could survive traveling that close to the sun's temperature, radiation, electromagnetic energy, solar flares, or gravity to equal a mass reactionary change as to alter it's spin.
| Well, I Binged and found some references.
Seems that a collision is most probable, if it happened at a time when the whole system was malleable. But there is no solid explanation.
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Hawking's alternative to Higgs Boson I have seen in popular media, claims that Hawking does not believe the Higgs boson exists due to microscopic black holes and even made a bet against it. This is based on something published in journal Physical Review D.
I don't have access to journal Physical Review D, and I can't find a clear detailed explanation what his claim is, and what his proposed alternative is. Can someone explain it for this curious layman?
| There is a book review which mentions this:
Famously, Stephen Hawking made his bet for $100 with Gordy Kane that neither LEP nor any other particle collider would ever see the Higgs boson because of virtual black holes. Perhaps unsurprisingly, Peter Higgs was very doubtful of the calculations that lead Hawking to this conclusion (and Peter certainly wasn’t alone) and this lead to a little public drama between the two scientists in the media.
Maybe the book has details.
See also Lubos Motl's blog on the subject where also the reference to the paper of Hawking is discussed, rather unfavorably. :;
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Massless charged particles Are there any massless (zero invariant mass) particles carrying electric charge?
If not, why not? Do we expect to see any or are they a theoretical impossibility?
| There are no massless particles with no electric charge. All the fermions have mass and the leptons that are not neutrinos have electric charge. The quarks also have electric charge. The Bosons, W+ and W- have mass and are charged. So as far as we know all particles that have charge have a reasonable amount of mass. However the particle with a charge and has the lowest mass, is the electron(and the positron).
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Does the foam on top of boiling maple sap affect the rate of evaporation? This is a serious question from someone engaged in evaporating large quantities of water to turn sap into syrup at this time of year.
Probably some background will help. When sap boils vigorously it creates quite a bit of foam, which will overflow the evaporator (incidentally filling the building with a pleasing maple caramel smell as it burns on the side of the evaporator). When the foam gets too high we touch it with a bit of lard and the foam level drops (surface tension - I know). However, it is tempting for me to give a good swipe so that the foam almost disappears (instead of just dropping). The old-timers however contend that I should just reduce the foam to the point where it isn't overflowing any more. They say that it will take longer to boil away the water if I eliminate the foam.
I fail to see how the foam will improve evaporation (although it seems to me that it might slow it down).
Edit: by request ( @georg ) , a link to the evaporator in question https://sites.google.com/site/lindsayssugarbush/_/rsrc/1240515239201/Home/2005-03-30--12-25-21.jpg
| It could go either way. If you are heating the liquid to the boiling point, then the foam will not limit boiling (unless it raises the pressure), but will limit convection/advection of air near the surface. Note that latent heat of water vapor is not the only method of heat loss from your pot. If air advects/convects over the surface, you are also heating air molecules. Also some heat is being lost by the surface via thermal radiation (probably roughly a kilowatt per meter squared). So the bubbles provide insulation, so that the heat loses other than into latent heat of water vapor are reduced..
But, if it is not actually boiling, but the temperature is controlled to be some value below boiling, then it loses water via evaporation, and that requires fluid to flow to and away from the surface, and the foam would seriously inhibit that.
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Collision time of Brownian particles Let's assume two spherical particles $p_1$ and $p_2$ of finite radius $r_1$ and $r_2$, which are at locations $(\pm\frac{d}{2},0,0)$ a distance $d$ apart at initial time $t$. These particles diffuse with coefficients $D_1$ and $D_2$, respectively. How can I obtain the probability distribution of the collision time (that is, if the time at which they collide is $t + \Delta t$, I would like to know the probability density function of the random variable $\Delta t$), or at least some of its moments, in these two cases:
*
*when the domain is infinite, e.g., $\mathbb{R}^3$, and
*when it is bounded (inside a sphere with surface area $A$ and the origin $(0,0,0)$ as its center)?
I tried to look up for an answer in Van Kampen's book, but I couldn't really come up with an answer.
| Marek suggested I post my comment (which doesn't completely answer the question) as an answer. Here it is:
Suppose you have two Brownian motions with diffusion coefficients $D_1$ and $D_2$, which start at the same point at $t=0$. Let $x_i$ be the average displacement vector for particle $i$ after time $t$. Then, $\langle x_i^2 \rangle = 2D_it$, where $x_i^2$ means the square of the length of vector $x_i$. Now, consider the relative position of these two random walks after time $t$ (assuming they start out at the same place). The expected value of the relative displacement will be $$\langle (x_1 - x_2)^2 \rangle = \langle x_1^2 \rangle + \langle x_2^2 \rangle = 2 (D_1 + D_2)t,$$ where you can ignore the cross terms since the expected inner product of the two displacement vectors is 0.
Hopefully, this argument convinces you that the relative position of two points undergoing Brownian motion is still Brownian. You can make it completely rigorous if you want.
So your question boils down to: what is the probability distribution for the time it takes Brownian motion starting at distance $d$ from the origin to first reach a disc of radius $r=r_1+r_2$ around the origin. It's straightforward to find the probability that Brownian motion is in that disc after time $t$, but computing the probability distribution for the first time it's reached is harder. I am sure this has been done, but right now I don't have time to do a literature search for this.
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Measuring the spin of a single electron Is it possible to measure the spin of a single electron? What papers have been published on answering this question? Would the measurement require a super sensitive SQUID, Superconductive Quantum Interference Device?
| The ion trap experiments by Hans Dehmelt might be of interest. Though the scientific focus was the precision measurement of the g factor, you can't get far with that without first knowing that your trapped electron has spin 1/2 - or if you don't know that, you'll find out pretty quick when theory doesn't match experiment even to first order.
You might find this a good read: Stern-Gerlach experiments: past, present, and future
Jean-Francois Van Huele and Jared Stenson - link to PDF is at http://www.physics.byu.edu/Research/theory/paps.aspx
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Why are mirror images flipped horizontally but not vertically? Why is it that when you look in the mirror left and right directions appear flipped, but not the up and down?
| Take a picture and look at it. Now turn the picture to face the mirror. Question one: who flipped the picture? Answer: you did. Now, face the picture back to you, and walk to the nearest refrigerator. Turn the picture to face the refrigerator. Wow! Refrigerators flip images too! Don't believe me? Take your flipped page and hold it up to a bright light. The image is flipped; no mirror required.
Now, most people will turn a page around the vertial axis when they want to face it away from themselves. However, you could flip the page around any axis you choose, as long as it's in the plane of the page. You could easily, for example, flip the page around the horizontal axis. If you still believe that mirrors flip images, you'll notice that you've now tricked the mirror into flipping the image top-to-bottom, not left-to-right. Flip the page around a diagonal axis, and you'll get a very different result.
Bottom line: mirrors don't flip images; people do.
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Are the basic postulates of QM the only set of postulates that can give rise to a sensible semi-probabilistic physical theory? Are the basic postulates of QM, such as complex Hilbert space, unitary evolution, Hermitian operator observables, projection hypothesis etc., the unique and only set of postulates that gives rise to a semi-deterministic and semi-probabilistic theory, in which the time evolution is non-degenerate? By non-degenerate, I mean different initial states never produce the same final state probabilities, which in QM is guaranteed by unitarity.
Phrased in another way, is it possible to prove from some general principles, such as semi-determinism, semi-reversibility (not for collapse), causality, existence of non-compatible observables etc., that a physical theory must satisfy these postulates? In particular, is it possible to prove that complex numbers, or a mathematical equivalent, must be fundamental to the theory?
I haven't studied anything about foundational issues of QM, so feel free to point out if I'm being a crackpot. I suppose this question may be similar to something like "can you prove that gravity must be a metric theory entirely from the equivalence principle?", whose answer is no, but I'll be glad if it turns out to be otherwise.
| There are some recent efforts in trying to derive the mathematical structure of quantum mechanics from some reasonable and/or operational axioms. You may want to give a look, for example, at http://arxiv.org/abs/1011.6451 and references therein.
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What are the conditions to be satisfied by a theory in order to be a quantum theory? This is in continuation to my previous question. It is not a duplicate of the previous one. This question arises because of the answers and discussions in that question.
Can we call a theory, quantum theory, if it is consistent with HUP? For example, suppose there is a finite and self consistent theory of gravity which incorporates the uncertainty principle. Can we at once call this theory a quantum theory of gravity or does it have to satisfy other conditions too?
This question may be too basic but it is intriguing my mind.
| Apparently, No. To quote SEP
A second point is the question whether the theoretical structure or the quantitative laws of quantum theory can indeed be derived on the basis of the uncertainty principle, as Heisenberg wished. Serious attempts to build up quantum theory as a full-fledged Theory of Principle on the basis of the uncertainty principle have never been carried out. Indeed, the most Heisenberg could and did claim in this respect was that the uncertainty relations created "room" (Heisenberg 1927, p. 180) or "freedom" (Heisenberg, 1931, p. 43) for the introduction of some non-classical mode of description of experimental data, not that they uniquely lead to the formalism of quantum mechanics. A serious proposal to construe quantum mechanics as a theory of principle was provided only recently by Bub (2000). But, remarkably, this proposal does not use the uncertainty relation as one of its fundamental principles.
All the mystique surrounding the Heisenberg Uncertainty principle vanishes if you take into account the deBroglie hypothesis. I remember from the Feynman lectures, '...what follows is something that is familiar to anyone who works with waves'. It might be worthwhile also to look at Terry Tao's post.
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Are there examples in classical mechanics where D'Alembert's principle fails? D'Alembert's principle suggests that the work done by the internal forces for a virtual displacement of a mechanical system in harmony with the constraints is zero.
This is obviously true for the constraint of a rigid body where all the particles maintain a constant distance from one another. It's also true for constraining force where the virtual displacement is normal to it.
Can anyone think of a case where the virtual displacements are in harmony with the constraints of a mechanical system, yet the total work done by the internal forces is non-zero, making D'Alembert's principle false?
| I have a interesting example:
Consider two blocks moving in a line, and an electric intelligent rod connects them. Everything is frictionless. The rod can make measurements of the coordinates of the two blocks, and change length to always makes sure that $x_2 = 2x_1$. Then we assume the mass of the rod is negligible, so that the forces it gives to the two blocks are exactly opposite. Now, we have an equation of constraint, and whenever the rod changes length and applies a non-zero force, D'Alembert's principle fails.
The way to fix Lagrange equation for this kind of constraint is to add the generalized force $Q_i^{(c)}$ created by the constraint ($0$ if D'Alembert's principle holds) to the right:
$$
\frac{\mathrm{d}}{\mathrm{d}{t}} \frac{\partial L}{\partial \dot q_i} - \frac{\partial L}{\partial q_i} = Q_i + Q_i^{(c)}
$$
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Why do all the planets of the solar system orbit in roughly the same 2D plane?
*
*Most images you see of the solar system are 2D and all planets orbit in the same plane. In a 3D view, are really all planets orbiting in similar planes? Is there a reason for this? I'd expect that the orbits distributed all around the sun, in 3D.
*Has an object made by man (a probe) ever left the Solar System?
| Nic and Approximist's answers hit the main points, but it's worth adding an additional word on the reason the orbits lie roughly in the same plane: Conservation of angular momentum.
The Solar System began as a large cloud of stuff, many times larger than its current size. It had some very slight initial angular momentum -- that is, it was, on average, rotating about a certain axis. (Why? Maybe just randomly! All of the constituents were flying around, and if you add up those random motions, there'll generically be some nonzero angular momentum.) Because angular momentum is conserved, as the cloud collapsed the rotation rate sped up (the usual example being the figure skater who pulls in her arms as she spins, and speeds up accordingly).
Further collapse in the direction perpendicular to the plane of rotation doesn't change the angular momentum, but collapse in the other directions would change it. So the collapse turns the initial cloud, whatever its shape, into a pancake. The planets formed out of that pancake.
By the way, you can see the signs of that initial angular momentum in other things too: not only are all of the planets orbiting in roughly the same plane, but so are most of their moons, and most of the planets' rotations about their axes as well.
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How to measure the spin of a neutral particle? If a charged particle with charge $q$ and mass $m$ has spin $s \neq 0$ we can measure an intrinsic magnetic moment $\mu = g \frac{q}{2m}\hbar \sqrt{s(s+1)}$. This is how spin was discovered in the first place in the Stern-Gerlach Experiment.
But for a neutral particle $\mu = 0$, so we cannot measure the spin of the particle in the same manner. But it is said, that e.g. the Neutron or the Neutrino both have a spin $s=1/2$. How was or can this be measured?
| Conservation of angular momentum is invoked for the neutrinos because beams of neutrinos cannot be collimated for an experimental measurement. Neutron spin can be measured in a Stern Gerlach setup.
The interactions and decays were carefully examined in various experiments and the only consistent spin values are the ones assigned.
Edit: I see that the question should be formulated as : why the neutron has a Dirac magnetic moment, although it is neutral, which is the formula that is displayed above, and does the neutrino have a Dirac magnetic moment?
The neutron, and other baryons, has a magnetic moment because the quarks that compose it have a Dirac magnetic moment. See for example Perkins, Introduction to High Energy Physics, section: baryon magnetic moments for the derivation.
Whether the neutrino has a magnetic moment due to higher order loop diagrams is a research question.
So, though spin in charged point like particles is connected to magnetic moment with the formula above, analogous to classical charges circulating in a loop having a magnetic moment, , charge is not necessary for spin to appear. There is intrinsic spin which for the neutrino comes from the angular momentum balance in the interactions where it appears. The neutrino is a spinor in the Dirac formalism.
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Can a disk like object (like UFO's) really fly? UFOs as shown in movies are shown as disk like objects with raised centers that emit some sort of light from bottom. Can such a thing fly?
My very limited knowledge in physics tell me that a disk like object may not be able to maneuver unless it has thrusters on sides and simple light can not be enough to make any object go up in the air.
Is it possible?
| The light below a UFO is actually ionized air. A light beam would not be visible by itself. According to some witnesses mentioned by Feindt the ionized air below is rotating, creating a Taylor column, which could explain why the heavy flying saucer shaped vehicle doesn't fall down. According to Hill pulsating forces can be felt near the UFO. One can imagine that the forces mainly propagates in the weak plasma below the saucer. Hill writes that the saucer type of UFOs tilt to turn.
Electromagnetic radiation beam powered propulsion can not be the case since it would loose too much energy to the ground than can be observed. Light and other radiation might be used to affect the plasma though.
Good reads on the physical aspects of UFOs are
Paul R. Hill - Unconventional Flying Objects: A Former NASA Scientist Explains How UFOs Really Work
Carl W. Feindt - UFOs and Water: Physical Effects of UFOs on Water Through Accounts by Eyewitnesses
A credible UFO witness in regard to physical (as well as other) aspects is Billy Meier. His stories, photos and drawings are revealing but also very demanding to understand.
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Making a "heavier-than-air" craft float How big would a hollow rigid object need to be to float, (not in water but in air) if all of the air was vacuumed out and the container sealed?
| Another way to look at this is to solve for what weight/area you are allowed to use. Assume a spherical craft. Surface area goes up in proportion to r^2. Volume (which will generate your lift) is proportional to r^3. If you do the math, you find that you are allowed to use (r x 0.4)kg/m^2, or about (r x 0.88) lbs/m^2. So the problem gets easier to solve as the craft gets bigger.
The force experienced by a totally evacuated body at sea level, would be about 14.7 psi, which is about 11.4 ~TONS~ per m^2. (Don't you just love mixing metric and English units? I do...)
At r = 1m, you can only use 0.88 lbs of material for skin and structure to hold back that massive force (good luck with that).
At r = 10m, you get 8.8 lbs/m^2. That equates to a (1/17) inch thick aluminum skin and no internal supporting structure. ---> Still a no-go.
At r = 100m, you are allowed 88 lbs/m^2. ... O.K., now we are talking. 1/8 inch aluminum plate weighs about 19lb./m^2. That leaves 69 lbs./m^2 for ridigization of the skin panels and internal structure. Something that approximated a monocoque structure (like an egg shell: able to take tremendous pressure with no deep internal bracing) might work. As an engineer, I'd be willing to take on that challenge; but... that is a H-U-G-E craft! OVER two football field lengths in diameter, and weighing in at about 5 million kg before evacuation!
Maybe someone would be able to do it for r = 50m, which would be a budget of 44lbs./m^3. My guess is that technical feasibility lies somewhere in the design area of 50m < r < 100m.
| {
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Have CMB photons "cooled" or been "stretched"? Introductory texts and popular accounts of why we see the "once hot" CMB as microwaves nearly always say something about the photons "cooling" since the Big Bang. But isn't that misleading? Don't those photons have long ("cool") wavelengths because space expanded since they were emitted. There's no separate "cooling" process, is there?
| I think the best way to think about it is that the sentence "the photons have cooled" is simply describing a fact, not explaining that fact. At early times, the photons at any given location had a thermal (blackbody) distribution corresponding to a high temperature (as measured by observers at rest in the natural, comoving reference frame). At later times, the photons at any given location had a thermal distribution corresponding to a lower temperature. That's what we mean when we say that they "cooled."
Of course, it's then very natural to ask why they cooled. That's where the "stretching of space" explanation comes in. I think that that explanation is problematic, as I wrote at great length here, but others disagree.
| {
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How many bits are needed to simulate the universe? This is not the same as: How many bytes can the observable universe store?
The Bekenstein bound tells us how many bits of data can be stored in a space. Using this value, we can determine the number of unique states this space can be in.
Imagine now, we are to simulate this space by enumerating each state along with which states it can transition to with a probability for each transition.
How much information is needed to encode the number of legal transitions and the probabilities? Does this question even make sense? If so, is there any indication that any of these probabilities are or are not Computable numbers?
Edit
Here's a thought experiment.
*
*Select your piece of space and start recording all the different states you see.
*If the Bekenstein bound tells us we can store n bits in our space, wait until you see 2^n different states. Now we've seen all the states our space can be in (otherwise we can violate the Bekenstein bound).
*For any state, record any other state that the space can legally transition to without violating any physical laws.
To simulate this portion of space, take it's state and transition it to a legal state. Repeat.
We have only used a finite number of bits and we have modeled a section of space.
| Multiply the area of the cosmological horizon by 4 - you'll get the needed information quantity in nats. Convert into bits by dividing by $\ln 2$. You'll get the needed value.
| {
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Supergravity calculation using computer algebra system in early days I was having a look at the original paper on supergravity by Ferrara, Freedman and van Nieuwenhuizen available here. The abstract has an interesting line saying that
Added note: This term has now been shown to vanish by a computer calculation, so that the action presented here does possess full local supersymmetry.
But the paper was written in 1976! Do you have any info what kind of computer and computer algebra system did they use? Is it documented anywhere?
| Van Nieuwenhuizen's PhD advisor, Matinus Veltman, was arguably the first person to develop a computer algebra system in the early 1960s, and the program was used in the proof of renormalization of gauge theories.
| {
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Does Quantum Mechanics assume space and time are continuous? I was confused when I was listening to a Quantum Mechanics lecture online. Are space and time assumed to be continuous or discrete in Quantum Mechanics?
I can see the question is vague, but this is so since I am confused.
|
...(the idea) "that space is continuous is, I believe, wrong."
— Professor Richard Feynman
The Messenger Series: Seeking New Laws
A discontinuous space | continuous meta-space model fits observed facts, and provides refreshing insights into many aspects of life.
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More on matter and anti-matter
*
*Does every particle that has rest mass also have an anti-particle with which it would annihilate?
*Does annihilation only occur between like particles? For example what happens if a antineutron (anti u, anti d, anti d) collides with a proton (uud)? What happens if a positron collides with a proton?
*Since the Tevatron accelerates antiprotons is this more difficult to handle and dump?
*I've read about WIMP annihilation detection. Why would one assume there is any different proportion of WIMPS to anti-WIMPS than as is for non-WIMP matter (where there is far more matter than anti-matter).
| *
*Yes, for charged particles and some neutral particles. But there are so-called Majorana neutral particles that have no antiparticles (neutral pion or eta-meson, for example). In other words, a Majorana particle is its own antiparticle.
*At low energies a proton annihilates with an antineutron like a proton with an antiproton - mainly into neutral and charged pions (charge should be conserved).
A positron does not annihilate with a proton. They have the same charge.
| {
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What is the Physical Meaning of Commutation of Two Operators? I understand the mathematics of commutation relations and anti-commutation relations, but what does it physically mean for an observable (self-adjoint operator) to commute with another observable (self-adjoint operator) in quantum mechanics?
E.g. an operator $A$ with the Hamiltonian $H$?
| When two qm operators do not commute, it means that we are missing stuff in Nature. That is quantum mechanics is a theory of measurement but not of Nature because of non-commutation. Hence this means that the stuff we miss cannot be described by quantum mechanics, and this leads to the conclusion that qm is not a complete description of Nature.
| {
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Why is it hard to solve the Ising-model in 3D? The Ising model is a well-known and well-studied model of magnetism. Ising solved the model in one dimension in 1925. In 1944, Onsager obtained the exact free energy of the two-dimensional (2D) model in zero field and, in 1952, Yang presented a computation of the spontaneous magnetization. But, the three-dimensional (3D) model has withstood challenges and remains, to this date, an outstanding unsolved problem.
| I solved the Ising model:
I. A. Stepanov. Exact Solutions of the One-Dimensional, Two-Dimensional, and Three-Dimensional Ising Models. – Nano Science and Nano Technology: An Indian Journal. 2012. Vol. 6. No 3. 118 - 122. (The paper is on the Journal site with a free access)
| {
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When water is about to boil Have ever noticed? When water is about to boil, no matters the kettle, there is some sound I have no idea where it comes from, sometimes long before it boils.
Is there any explanation for this phenomena?
| I don't know how good this explanation is, but it is certainly plausible. The sound must be from bubble formation/popping. It seems the difference between the noise in the beginning and when the water is really boiling well is because in the initial phase, bubbles don't reach the surface.
| {
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Anti-matter repelled by gravity - is it a serious hypothesis?
Possible Duplicate:
Why would Antimatter behave differently via Gravity?
Regarding the following statement in this article:
Most important of these is whether ordinary gravity attracts or repels antimatter. In other words, does antihydrogen fall up or down?
Is this a seriously considered hypothesis? What would be the consequences on general relativity?
If this is seriously studied, can you point to some not-too-cryptic studies on the (anti ;-)matter?
| I can't guarantee the authenticity of the article. But it seems to me quite bizarre since I fail to see how something (even antimatter) can behave differently than matter in a gravitational field without violating the equivalence principle.
A positron for example is a hole in the Dirac sea and it has the same mass as of the electron and behaves exactly similar to an electron in a gravitational field.
The only repulsive gravity that exists in GR is the cosmological constant in the Einstein's equations which is in no way related to antimatter.
| {
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In what order should the subjects be studied in order to get to String Theory I know:
*
*Quantum Mechanics (Griffiths Level, currently doing Sakurai Level)
*Mechanics (Newtonian+ Lagrangian/Hamiltonian but at level lower than Goldstein/Landau)
*Classical Electrodynamics (Griffiths + electro/magnetostatics from Jackson)
*Statistical Physics (Pathria)
I know very little relativity from undergrad mechanics class. Nothing about General Relativity, nothing about QFT, etc. I wish to reach string theory in a proper way without leaving any gaping holes in my education. What subjects should be studied in what order?
| You can try reading Zwiebach 'A first course in String theory' which is roughly at your level right now. Its very handwavy, but well thats the best you can hope for at this level.
Otherwise, to really learn the subject you will absolutely need
1) Grad level GR
2) Quantum II, + 2 semester long courses in QFT
And then you can start thinking about it.
Personally I find the above level a little loose, so to make it more comprehensive and less opaque i'd recommend in addition to the above, to have some experience in Semiclassical gravity (Wald or Birrel and Davies), Conformal field theory (Di Francesco) and Supersymmetry (Weinberg or Wess and Bagger)
| {
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Laplace's equation: Why is $\ell$ to be a non-negative integer? I have got some mathematical difficulties in the following exercise:
Calculate the potential of the polarized sphere along the $z$-axis. There are no free charges.
For this, we need to solve Laplace's equation, by using the method of separation of variables.
$$\phi (r, \theta, \phi) = R(r) \Theta(\theta) $$
We obtain a partial differential equation on $r$ and $\theta$:
$$\frac{1}{R} \frac{\partial}{\partial r} r^2 \frac{\partial}{\partial r} R = \ell(\ell+1) $$
$$\frac{1}{\Theta} \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \sin \theta \frac{\partial}{\partial \theta} \Theta = - \ell(\ell+1). $$
My question is : "Why is $\ell$ to be a non-negative integer?" The solution I got in some course takes it for granted, but I do not know why.
| Since this question has a physics context, reasoning by its underlying physics can give you an answer. First of all, it is easy to see the radial equation has two solutions: $\frac{1}{r^l}$ and $r^{l+1}$. Since the field outside of the sphere must contain multipole components decaying as $\frac{1}{r}, \frac{1}{r^2}, ...$. This shows if the physics is correct then $l$ must be integers. The choices of $l$ being $0,1,2,...$ or $-1,-2,-3,...$ are actually equivalent since $l(l+1)$ will produce the same set of values, but $0,1,2,...$ is obvious the better choice here.
A more mathematical explanation is that the angular part is the Legendre equation if we use variable $x = \cos\theta$ instead. Its only eigenvalues are $l(l+1)$ with $l=0,1,...$ if one requires its solutions are bounded at $x=\pm 1$ (again, for physical reasons), as explained in this question. The other two answers are restatements of this mathematical fact in the context of quantum mechanics.
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Neutrino Speed in Supernova I've read that neutrinos in supernova can be affected by "neutrino refraction." Is this analagous to the refraction of light, and if so, is the speed of these neutrinos similarly reduced from their near c speeds via this index of refraction?
| First, check this reference on Wikipedia.
Now, it is generally true that the "speed" (or, more accurately, the dispersion relation) of any particle is affected by a medium, where it travels. Well, of course, if the particle interacts with the medium.
For neutrinos the "slowing down" itself is absolutely negligible even in very dense media. What is important is that interaction of the electron neutrino with ordinary matter is much stronger, so it affects the patterns of neutrino oscillations -- the effect is known as MSW effect.
Finally, this is not particularly related to supernovae. The idea behind interest in supernovae is that during an explosion there are a lot of neutrinos so one must account for the "neutrino matter" and its effect on oscillations as well.
| {
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$cm^3/g$ as a unit of adsorption I recently saw $cm^3/g$ as a unit for amount adsorbed. Usually, you see either $\mathrm{kg_{adsorbate}/kg_{adsorbent}}$ or $\mathrm{mole_{adsorbate}/kg_{adsorbent}}$. Does anyone know the meaning of this unit?
| I've only ever seen those units described as "specific surface area". It does have implications for adsorption, but I don't think it is the be all and end all.
| {
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Why is $\frac{dx}{dt}=0$ in this average momentum calculation? In the following excerpt from S. Gasiorowicz's Quantum Physics, he derives an expression for the average momentum of a free particle. $\psi(x,t)$ is the wave function of a free particle, $\psi^*$ denotes its complex conjugate.
We try the following: since classically,
$$ p = mv = m\frac{dx}{dt} $$
we shall write
$$ <p> = m\frac{d}{dt}<x> = m\frac{d}{dt}\int{dx \psi^*(x,t) x \psi(x,t)} $$
This yields
$$ <p> = m\int_{-\infty}^\infty{dx\left( \frac{\partial\psi^*}{\partial t} x \psi + \psi^* x \frac{\partial\psi}{\partial t} \right)} $$
Note that there is no $dx/dt$ under the integral sign. The only quantity that varies with time is $\psi(x,t)$, and it is this variation that gives rise to a change in $x$ with time.
I seem to have trouble understanding the difference between the position $x$ and the average position $<x>$. Why can it be assumed that $\frac{dx}{dt}=0$? What is x?
| The confusion seems to stem from a) not understanding what kind of objects you are dealing with and b) usual custom of not writing (all) arguments of functions when they are understood.
To clarify a) note that the position operator $\hat x$ does not depend on time, and so also its kernel $\left< x \right | \hat{x} \left | x' \right> = x\delta(x-x')$ with respect to "position vectors" $\left | x \right >$ also doesn't depend on time. This $x$ is the one that is present in your integral. So in particular ${{\rm d} x \over {\rm d} t} = 0$.
On the other hand, the average of the operator $\hat A$ in the state $\psi$ (which depends on time) obviously depends on the state $\psi$: $\left< \hat {A} \right> := \left< \psi \right | \hat {A} \left | \psi \right>$ and so if you perform averages on a family of vectors $\psi(t)$ so also the average will depend on time. In your case, this should be written $\left< \hat{x} \right > (t)$ to make it obvious that one is dealing with a function of time. But this dependence is usually understood and omitted.
| {
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What is "pure energy" in matter-antimatter annihilation made of? I used to read the term "pure energy" in the context of matter-antimatter annihilation. Is the "pure energy" spoken of photons? Is it some form of heat? Some kind of particles with mass?
Basically, what does "pure energy" in the context of matter-antimatter annihilation refer to?
| Energy is the ability of a system to perform work on another system. In other words, it's a property of a physical system. The annihilation of an electron and a positron performs work on the electromagnetic field by creating two photons. We usually do not talk about these processes in the same language that we use in classical mechanics, but there is absolutely nothing wrong with thinking about it in exactly the same way. We merely have to replace the names of the systems. In classical mechanics we may have been elevating a mass "above the floor" with a pulley by performing mechanical work on it. In the case of fields we are creating field excitations that are "above" the ground state.
| {
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Are all superalgebras Clifford algebras? I believe the answer to be yes, but I realize that sometimes physicists place additional constraints that might not be obvious. If superalgebras are Clifford algebras, why make a literary distinction?
| The short answer is no.
The subject of super algebras is a large one as its a fairly new subject and the terminology hasn't settled down to a form that is clear and transparent to both physicists and mathematicians.
A super algebra, in its most widest sense, is an algebra that has an odd-even grading, usually called a $\mathbb{Z}_2$ grading. Standard examples of these are the exterior algebra and the Clifford algebra. Another example are Weyl algebras. Whilst Clifford algebras can be viewed as deformations of exterior algebras, Weyl algebras are deformations of the symmetric algebra.
Now, in both these examples we begin with an ordinary vector space before constructing the Clifford or Weyl algebra over it. If we begin instead with a super vector space then we can unify the construction such that in odd degree we get a Weyl algebra and in even degree, we get a Clifford algebra.
Its this unification that Lubos Motl has written about implicitly - ie in bosonic and ferminionic degrees - but not explicitly, as he doesn't seem to know of the appropriate mathematical terminology.
Moreover, his description of a Clifford algebra is wrong. He is describing a Clifford algebra with a choice of basis. Moreover, he describes them representationally, that is through matrices. Whilst this is concrete, it isn't neccesary.
| {
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