acoustics - What "propagates" a force through the rest of a solid?

So, in typing the title of this question I was recommended this awesome one, which confirmed my guess that this effect "propagates" at the speed of sound (though I just had a feeling, I don't really understand why). I also only have high-school level physics experience.

I feel like I don't know physics well enough to really know what to ask correctly, but I'll try to explain.

So, I rotate my arm, and, even though tendons are just pulling it at a certain location, the rest of it follows suit. Why is that? What is happening at an atomic/molecular scale that ends up "conveying forces" over a distance?

And, something else I don't understand- what is so special about the speed of sound that makes it this "fundamental unit of translation", or something? Maybe the better question is "what processes at an atomic/molecular scale lead to the speed of sound being associated with all of these behaviors", which ties both questions together.

Answer

Solids are just a collection of atoms that are bound together chemically, i.e. through the electromagnetic force. They aren't perfectly rigid. Think about a linear chain of atoms. Pushing on the end atom will cause it to get closer to its neighbour. They will repel each other, so the neighbour will then move away, getting too close to its neighbour on the other side. This process continues all the way along the chain of atoms. By the time the last atom has moved, you have moved the entire chain. Pulling on the end of course just has the opposite effect as pushing on it. The same effect is what happens in solids, but instead of a linear chain you have a 3D lattice. Wikipedia has a nice demonstration of this in 2D.

You see that it isn't an instantaneous process. You correctly identify the speed of the propagation of this effect as the speed of sound, $c$. However, this is just how the speed of sound is defined: the propagation speed of a small deformation in the material. This depends on the stiffness (measured by the bulk modulus $K$) and the density $\rho$ of the material: $$ c = \sqrt{\frac{K}{\rho}} $$ This can vary a lot. The speed of sound in air at SATP is about 343 m/s, whereas in steel it is closer to 6000 m/s.

homework and exercises - Instantaneous angular momentum of a disc

Suppose we have a disk of radius $r$ and mass $m$ travelling at velocity $v$. I want to calculate the instantaneous angular momentum with axis through the edge of the disc (on the circumference).

Angular momentum $= I \omega$. $I = \frac{1}{2}mr^2 + mr^2 = \frac{3}{2}mr^2$ by the parallel axis theorem. $\omega = \frac{v}{r}$. Therefore, angular momentum $= \frac{3mrv}{2}$.

Alternatively, angular momentum $=p\times r= m r \times v = mrv$.

Why do these two methods differ? Which, if any, are correct?

How stupid is this theory of gravity?

As will be evident, I am not a physicist. I've always been interested in physics but my education tapered out with general relativity and basic quantum mechanics, years ago. Several years ago a sort of thought experiment began to nag at me and I've wanted for those more knowledgeable to basically explain to me why this idea must be wrong -- not simply that it is very silly or (obviously) at odds with existing models, but provably wrong. (It is those things.)

The thought experiment:

Imagine a piece of graph paper, with 2 circles of radius 1, centers 10 units apart. Now imagine that the circles are growing at a steady rate of 1 unit/sec. The edges of the circles approach each other and touch after 4 seconds.

Now imagine the same sequence, but the graph paper is also growing at a steady rate, matching that of the circles. The circles are not "tethered" to the expanding graph paper; their centers remain the same distance, hence fewer graph paper units as the paper expands.

Finally, imagine that you, the observer are growing at the same rate as the graph paper and the circles, so that the graph paper appears fixed. What you would perceive would be 2 circles of fixed size accelerating towards each other.

This hypothetical scene is not describing physics, it's just a geometrical construct. It shows that what appears to be constant motion can be seen as acceleration (or vice versa) based on the perceived intertial frame of reference (the graph paper).

We perceive gravity as a force that causes the distance between masses to decrease over time, accelerating just as the two circles described. This assumes the existence of a stable "graph paper" in space, where distance exists independent of the objects. What if we remove that assumption? What if distance in space cannot be separated from distance in time?

Imagine that what appear to be subatomic particles are in fact standing waveforms, emergent from constantly expanding waves. Wave interference gives rise to particles at certain scales with consistent properties, just as fractals give rise to recurring forms at specific scales. These "standing" wave forms are themselves expanding. But, expanding relative to what? Depends on the scale of space/time. At certain scales, wave interactions maintain fixed distances between "particles", e.g. electron valences, atomic distributions. This holds true for the scales of matter we directly experience, where Newtonian laws apply. Everything we perceive to be fixed in size is actually expanding at essentially constant rate, including every atom in our bodies, every electron, every photon. Every unit of measurement we think of as constant is in fact constantly expanding.

Now imagine that this expansive property is not uniform (if it were uniform it would be meaningless), but diminishes with the distance in space/time of the source of the waves. As masses get further apart, their expansion becomes less uniform, causing them to expand towards each other. As the wave sources move further apart, the resulting standing waves (particles) are not expanding in uniform directions (they don't share the same "graph paper"), and so appear to accelerate towards each other.

These gravitational waves are not, therefore, caused by the presence of mass, but rather, are the substrate from which mass originates. The interference of waves at different scales not only causes particles with varying sizes and speeds, but causes complex motions like orbital ellipses, and non-uniform expansion with different arrangements of particles, as with the swelling of the oceans that causes the tides.

Besides being silly, the radical idea here is that everything we assume about the shape of space, and the concept of distance, is distorted by the fact that the stuff we are made of is constantly expanding. Gravity as a force doesn't exist, it's an illusion arising from this distortion of perception.

Please tell me this idea is ludicrous. How would one disprove it, other than citing centuries of accepted models of physical reality? Which parts of established physics is it incompatible with? As silly as it sounds, it has nagged at me for years because it does describe what we perceive as gravity, and explains things like how gravity's effect can be "instantaneous" at a distance (it isn't; it's just an illusion; there is no graph paper). It doesn't seem incompatible with Einstein's observation that mass bends space/time, but rather a different conception of how mass and space/time are related to each other. For a long time, I thought orbital motion was unexplainable with this model, until I realized that just as the fixed scale of imaginary graph paper can be seen to be an illusion, so can the fixed directionality of "constant motion". Motion cannot follow a fixed path relative to non-existent graph paper, it can only be relative to other objects, and wave interaction could conceivably result in orbital motion.

I'm confident this question will either receive a great deal of flak or be closed as inappropriate or such. I am prepared for the flak but hope people will debunk it in earnest.

electrostatics - Weird consequence of Gauss's law

According to Gauss's Law, the electric field at a surface is the function of only the charge enclosed inside it. But that doesn't make sense. I mean, if I put the surface in an electric field, won't the resultant electric field at the surface change?

rotational kinematics - If angular velocity & angular acceleration are vectors, why not angular displacement?

Are angular quantities vector? ... It is not easy to get used to representing angular quantities as vectors. We instinctively expect that something should be moving along the direction of a vector. That is not the case here. Instead, something(the rigid body) is rotating around the direction of the vector. In the world of pure rotation,a vector defines an axis of rotation, not a direction in which something moves.

This is what my book (authored by Halliday, Walker, Resnick) says. But they mentioned with a caution that

Angular displacements cannot be treated as vectors. ....to be represented as a vector, a quantity must also obey the rules of vector addition..... Angular displacements fail this test.

Why do they fail in this test? What is the reason?

Answer

Set your copy of Halliday, Walker & Resnick on the table so the front cover is parallel to the table and visible to you. Now your right hand flat on the book with your thumb and forefinger forming a ninety degree angle. Orient your hand so your thumb is parallel to the spine of the book and pointing toward the top edge. Your forefinger should be parallel to the lines of text on the cover, pointing to the right.

This makes a nice basis for a book-based coordinate system. Your thumb points along the x-hat axis, your forefinger along the y-hat axis. To complete a right-hand system, the z-hat axis points into the book.

Pick the book up and make a +90 degree rotation about the x-hat axis. The spine of the book should be horizontal and facing up. Now make a +90 degree rotation about the y-hat axis (as rotated by that first rotation). You should be looking at the front cover of the book but oriented vertically with text flowing toward the ground.

If you start all over again but reverse the order of operations, +90 degree rotation about the y-hat axis followed by a +90 degree rotation about the x-hat axis, you should be looking at the spine of the book rather than the front cover. The front cover is oriented vertically, but with text running parallel to the ground. You can put your book back in the bookcase.

Rotation in three dimensional space and higher is not commutative (rotation A followed by rotation B is not necessarily the same as rotation B followed by rotation A).

Another hint that there's something different between rotation and translation is the number of parameters needed to describe the two in some N-dimensional space. Translation in N dimensional space obviously needs N parameters. Lines have one degree of freedom, planes, two, three dimensional space three, and so on. Lines don't rotate. There are no rotational degrees of freedom in one dimensional space. Rotation does make sense in two dimensional space, where a single scalar (one rotational degree of freedom) completely describes rotations. Three dimensional space has three degrees of freedom. Four dimensional space? It has six. Three dimensional space is the only space for which the number of rotational degrees of freedom and number of translational degrees of freedom are equal to one another.

This unique characteristic of three dimensional space is why you can treat angular velocity as a vector. The vector cross product (something else that is unique to three dimensional space) means introductory students can be taught about rotations without having to learn about Lie theory or abstract algebras. That wouldn't be the case for students who live in a universe with four spatial dimensions.

general relativity - Real-world evidence that non-massive entities (or even: antiparticles), and their behaviors, are sources of gravity?

The theory of general relativity tells us that non-massive entities, and their behaviors, are possible sources of gravity. Mass isn't needed, the theory says.

What's the real-world evidence that non-massive entities, and their behaviors, are sources of gravity? I'm guessing that even if something responds to curved spacetime, that does not necessarily mean it's a source of curvature, itself--?

Also, I do not see how a correspondence between shapes of present-day gravitating structures and CMB radiation inhomogeneities would be an answer to my question. Even during the universe's radiation-dominated era, there was still some matter, which I imagine wasn't perfectly homogeneous (and weren't production/annihilation events happening anyway, generating mass in random spots even as transparency started to take effect?)? So I wonder if mass inhomogeneities were really the only early gravity wells. Not radiation inhomogeneities, per se.

Likewise, what's the real-world evidence that even massive entities, and their behaviors, are sources of gravity when those entities are antiparticles...forgive me for not yet making this an entirely separate Physics StackExchange question...

Thanks very much for your time.

newtonian mechanics - Mass Dropped on Scale

When a mass is dropped onto something like a bathroom scale, the reading on the scale temporarily exceeds the actual weight of the mass. How do I explain this using forces and a force body diagram? Also, let's say instead of a mass and a scale, its just a person, a ball, and a scale. The person is standing on the scale with the ball in hand and throws it up in the air. When the person catches the ball, should the scale also read a value greater than the weight of the human and the ball combined? Is the reasoning for this the same as the mass and scale example?

Edit: Could the explanation be that at the instantaneous moment when the mass comes in contact with the scale, there is an instantaneous force caused by the impulse?

electromagnetism - Do stars remain electrically neutral?

How electrically neutral do stars remain through their lifetime? As an example, I could imagine processes such as coronal mass ejections leaving the Sun in a slightly charged state. Are there such processes that will leave a star with overall charge?

Answer

Stars are composed of plasmas, which are an ionized gas that exhibit a collective behavior much like a fluid.

There are two important aspects of plasmas to keep in mind. The first is that they act like very highly conductive metals in that the electrons can move very freely in order to cancel out any charge imbalance. The consequence is that they are said to be quasi-neutral over distances larger than the Debye length. By quasi-neutral, I mean: $$ n_{e} = \sum_{s} \ Z_{s} \ n_{s} \tag{1} $$ where $n_{e}$ is the total electron number density, $n_{s}$ is the number density of ion species $s$, and $Z_{s}$ is the charge state of ion species $s$ (e.g., +1 for protons). Generally the Debye length is considered to be as microscopic as one would ever care about in most situations.

The second thing is that unless driven, plasmas tend to satisfy a zero current condition given by: $$ \sum_{s} \ Z_{s} \ n_{s} \ \mathbf{v}_{s} = 0 \tag{2} $$ where $s$ in this case includes electrons and $\mathbf{v}_{s}$ is the bulk flow velocity of species $s$ (i.e., the first velocity moment). This is specifically referring to the flow of plasma out of the sun (i.e., the solar wind), which is derived from the continuity equation for electric charge. There are, of course, large localized currents all through the sun from its interior to its upper atmosphere.

How electrically neutral do stars remain through their lifetime?

Very. If they "charged up" they would produce tremendous Coulomb potentials that would prevent particles of certain charge signs from leaving the surface. Meaning, much like in a conductor, electric fields will do work to get rid of themselves. This is why plasmas are found to be quasi-neutral over distances larger than the Debye length.

The second condition given by Equation 2 results in relative drifts between different particle species in order to maintain a zero net current flow out of the sun (except during active periods like in coronal mass ejections or solar flares). Yet even so, these relative drifts do not act to increase the net charge of the sun.

As an example, I could imagine processes such as coronal mass ejections leaving the Sun in a slightly charged state.

No, generally they do not alter the macroscopic charge state of the sun. They consist of plasmas, which as I said before, are quasi-neutral.

Are there such processes that will leave a star with overall charge?

None of which that I am aware. As I said before, if you suddenly "charged up" a star the resultant electric fields would do work on the system until those electric fields no longer existed.

While I maintain that Equation 1 generally holds, for various reasons it does appear that there is a small net charge to the sun, as mentioned in this answer:
https://physics.stackexchange.com/a/73773/59023.

After further discussion with colleagues and my finally gaining access to the paper in question [i.e., Neslusan, 2001], I have a few comments.

The electric field to which that article refers, now called the Pannekoek and Rosseland (P-R) electric field, is only valid for a stellar corona in hydrostatic equilibrium. This field is not consistent with observations because it only produces a sort of breeze, not a supersonic wind like the one that has been observed since the 1960s. This field would also not remain stable. Meaning, it would eventually accelerate protons away from the star and eliminate any net charging.

The more correct approach, which is now generally accepted, is called an exospheric model [e.g., Zouganelis et al., 2005]. This model, unlike the P-R model, can include the multiple components of the solar wind electron velocity distributions (i.e., core, halo, and strahl). It still includes the mass of the star through a gravitational term but more importantly, it is actually consistent with the observations.

• Neslusan, L. "On the global electrostatic charge of stars," Astron. & Astrophys. 372, pp. 913-915, doi:10.1051/0004-6361:20010533, 2001.


• Zouganelis, I. et al., "Acceleration of Weakly Collisional Solar-Type Winds," Astrophys. J. 626, pp. L117-L120, doi:10.1086/431904, 2005.

quantum mechanics - Wick rotation and the arrow of time

It is well known that we can switch from a statistical system to a quantum mechanical system by a Wick rotation. Has this rotation some implication on the way the time flow? namely, this is an accident or has a deep meaning?

newtonian mechanics - A curious phenomenon with a stick

Recently, I've encountered a really curious thing, namely:

Support in the air a straight stick of length, say, 1 meter, using your index fingers in such a way that the left one is e.g. 30cm from the center of mass and the right one is e.g. 20cm. Now start slowly moving your fingers to each other.

I noticed that the fingers always meet at the point where the center of mass is. What is the physics explanation for this?

quantum field theory - Electromagnetic Unruh/Hawking effect? (Improved argument)

This is an improved version of the argument in Electromagnetic Unruh effect?

In the quantum vacuum particle pairs, with total energy $E_x$, can come into existence provided they annihilate within a time $t$ according to the uncertainty principle $$E_x\ t \sim \hbar.$$ If we let $t=x/c$ then we have $$E_x \sim \frac{\hbar c}{x}$$ where $x$ is the Compton wavelength of the particle pair.

Let us assume that there is a force field present that immediately gives the particles an acceleration $a$ as soon as they appear out of the vacuum.

Approximately, the extra distance, $\Delta x$, that a particle travels before it is annihilated is $$\Delta x \sim a t^2 \sim \frac{ax^2}{c^2}.$$ Therefore the particle pairs have a new Compton wavelength, $X$, given by $$X \sim x + \Delta x \sim x + \frac{ax^2}{c^2}.$$ Accordingly the energy $E_X$ of the particle pairs, after time $t$, is related to their new Compton wavelength $X$ by \begin{eqnarray} E_X &\sim& \frac{\hbar c}{X}\\ &\sim& \frac{\hbar c}{x(1+ax/c^2)}\\ &\sim& \frac{\hbar c}{x}(1-ax/c^2)\\ &\sim& E_x - \frac{\hbar a}{c}. \end{eqnarray} Thus the particle pair energy $E_X$ needed to satisfy the uncertainty principle after time $t$ is less than the energy $E_x$ that was borrowed from the vacuum in the first place. When the particle pair annihilates the excess energy $\Delta E=\hbar a/c$ produces a photon of electromagnetic radiation with temperature $T$ given by $$T \sim \frac{\hbar a}{c k_B}.$$ Thus we have derived an Unruh radiation-like formula for a vacuum that is being accelerated by a field. If the field is the gravitational field then we have derived the Hawking temperature. By the equivalence principle this is the same as the vacuum temperature observed by an accelerating observer. But this formula should be valid for any force field.

Let us assume that the force field is a static electric field $\vec{E}$ and that the particle pair is an electron-positron pair, each with charge $e$ and mass $m_e$. The classical equation of motion for each particle is then $$e\ \vec{E}=m_e\ \vec{a}.$$ Substituting the magnitudes of the electric field and acceleration into the Unruh formula gives $$T \sim \frac{\hbar}{c k_B}\frac{e|\vec{E}|}{m_e}.$$ If we take the electric field strength $|\vec{E}|=1$ MV/m then the electromagnetic Unruh/Hawking temperature is $$T\approx 10^{-2}\ \hbox{K}.$$ If this temperature could be measured then one could experimentally confirm the general Unruh/Hawking effect.

Is there any merit to this admittedly non-rigorous argument or can the Unruh/Hawking effect only be analyzed using quantum field theory?

Answer

You aren't really asking a question but here is my assessment of your argument.

The Unruh effect states that if one were to couple a detector to a quantum field, the detector would detect a thermal excitation as it is being accelerated. More generally, however, this excitation has to do with the thermal character of the vacuum and not necessarily the coupling of a detector. So the acceleration argument is not exactly necessary. In fact (due to an argument by Sciama), the necessary and sufficient condition is that the vacuum be stable and stationary from the perspective of a uniformly accelerated frame.

Your argument is very hand-wavy. There is a confusion of frames, there is no reference to a thermal density matrix, you have not constructed a boost Hamiltonian, you have not addressed the subtleties of the "quantum" equivalence principle, I don't know what metric you're talking about and so on.

quantum field theory - Is electric charge truly conserved for bosonic matter?

Even before quantization, charged bosonic fields exhibit a certain "self-interaction". The body of this post demonstrates this fact, and the last paragraph asks the question.

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Notation/ Lagrangians

Let me first provide the respective Lagrangians and elucidate the notation.

I am talking about complex scalar QED with the Lagrangian $$\mathcal{L} = \frac{1}{2} D_\mu \phi^* D^\mu \phi - \frac{1}{2} m^2 \phi^* \phi - \frac{1}{4} F^{\mu \nu} F_{\mu \nu}$$ Where $D_\mu \phi = (\partial_\mu + ie A_\mu) \phi$, $D_\mu \phi^* = (\partial_\mu - ie A_\mu) \phi^*$ and $F^{\mu \nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$. I am also mentioning usual QED with the Lagrangian $$\mathcal{L} = \bar{\psi}(iD_\mu \gamma^\mu-m) \psi - \frac{1}{4} F^{\mu \nu} F_{\mu \nu}$$ and "vector QED" (U(1) coupling to the Proca field) $$\mathcal{L} = - \frac{1}{4} (D^\mu B^{* \nu} - D^\nu B^{* \mu})(D_\mu B_\nu-D_\nu B_\mu) + \frac{1}{2} m^2 B^{* \nu}B_\nu - \frac{1}{4} F^{\mu \nu} F_{\mu \nu}$$

The four-currents are obtained from Noether's theorem. Natural units $c=\hbar=1$ are used. $\Im$ means imaginary part.

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Noether currents of particles

Consider the Noether current of the complex scalar $\phi$ $$j^\mu = \frac{e}{m} \Im(\phi^* \partial^\mu\phi)$$ Introducing local $U(1)$ gauge we have $\partial_\mu \to D_\mu=\partial_\mu + ie A_\mu$ (with $-ieA_\mu$ for the complex conjugate). The new Noether current is $$\mathcal{J}^\mu = \frac{e}{m} \Im(\phi^* D^\mu\phi) = \frac{e}{m} \Im(\phi^* \partial^\mu\phi) + \frac{e^2}{m} |\phi|^2 A^\mu$$ Similarly for a Proca field $B^\mu$ (massive spin 1 boson) we have $$j^\mu = \frac{e}{m} \Im(B^*_\mu(\partial^\mu B^\nu-\partial^\nu B^\mu))$$ Which by the same procedure leads to $$\mathcal{J}^\mu = \frac{e}{m} \Im(B^*_\mu(\partial^\mu B^\nu-\partial^\nu B^\mu))+ \frac{e^2}{m} |B|^2 A^\mu$$

Similar $e^2$ terms also appear in the Lagrangian itself as $e^2 A^2 |\phi|^2$. On the other hand, for a bispinor $\psi$ (spin 1/2 massive fermion) we have the current $$j^\mu = \mathcal{J}^\mu = e \bar{\psi} \gamma^\mu \psi$$ Since it does not have any $\partial_\mu$ included.

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"Self-charge"

Now consider very slowly moving or even static particles, we have $\partial_0 \phi, \partial_0 B \to \pm im\phi, \pm im B$ and the current is essentially $(\rho,0,0,0)$. For $\phi$ we have thus approximately $$\rho = e (|\phi^+|^2-|\phi^-|^2) + \frac{e^2}{m} (|\phi^+|^2 + |\phi^-|^2) \Phi$$ Where $A^0 = \Phi$ is the electrostatic potential and $\phi^\pm$ are the "positive and negative frequency parts" of $\phi$ defined by $\partial_0 \phi^\pm = \pm im \phi^\pm$. A similar term appears for the Proca field.

For the interpretation let us pass back to SI units, in this case we only get a $1/c^2$ factor. The "extra density" is $$\Delta \rho = e\cdot \frac{e \Phi}{mc^2}\cdot |\phi|^2$$ That is, there is an extra density proportional to the ratio of the energy of the electrostatic field $e \Phi$ and the rest mass of the particle $mc^2$. The sign of this extra density is dependent only on the sign of the electrostatic potential and both frequency parts contribute with the same sign (which is superweird). This would mean that classicaly, the "bare" charge of bosons in strong electromagnetic fields is not conserved, only this generalized charge is.

After all, it seems a bad convention to call $\mathcal{J}^\mu$ the electric charge current. By multiplying it by $m(c^2)/e$ it becomes a matter density current with the extra term corresponding to mass gained by electrostatic energy. However, that does not change the fact that the "bare charge density" $j^0$ seems not to be conserved for bosons.

---

Now to the questions:

• On a theoretical level, is charge conservation at least temporarily or virtually violated for bosons in strong electromagnetic fields? (Charge conservation will quite obviously not be violated in the final S-matrix, and as an $\mathcal{O}(e^2)$ effect it will probably not be reflected in first order processes.) Is there an intuitive physical reason why such a violation is not true for fermions even on a classical level?


• Charged bosons do not have a high abundance in fundamental theories, but they do often appear in effective field theories. Is this "bare charge" non-conservation anyhow reflected in them and does it have associated experimental phenomena?

Mathematical background for Quantum Mechanics

What are some good sources to learn the mathematical background of Quantum Mechanics?

I am talking functional analysis, operator theory etc etc...

popular science - Is there an intuitive way of thinking about the extra dimensions in M-Theory?

Why are 11 dimensions needed in M-Theory? The four I know (three spatial ones plus time) have an intuitive meaning in everyday life. How can I think of the other seven? What is their nature (spatial, temporal, ...) and is there an intuitive picture of what they are needed for?

Answer

This should be a comment since I am not a string theorist but its too big. When Luboš (Luboš correct me if I'm wrong) speaks of the "shape" in his comment:

They're spatial dimensions - new temporal dimensions always lead to at least some problems if not inconsistencies - but otherwise they're the same kind of dimensions as the known ones, just with a different shape. To a creature much smaller than their shapes' size, they're exactly the same as the dimensions we know. The theory implies that the total number of spacetime dimensions is 10 or 11. We don't have "intuition" for higher-dimensional shapes because the extra dimensions are much smaller than the known ones but otherwise they'er intuitively exactly the same as the known spatial dimensions

he means that the higher dimensions are "compactified". A simple example of a compact space is a circle, or a Cartesian product of circles (a torus) or a high dimensional sphere. The crucial idea is that they are topologically compact, meaning roughly that they are finite and closed i.e. they have no boundary just like the torus or sphere have no boundary. So Luboš's little creatures would return to their beginning point if they walked far enough in the same direction.

As I understand it, one of the proposed "shapes" for the compactified dimensions is the Calabi-Yau manifold. Wholly for gazing on beauty's sake, its worth also looking these here at the Wolfram demonstrations site. Be aware that you're looking at a projection, hence the seeming "edges" are not the manifold's boundary. Like the torus and the sphere, these manifolds would let a little creature return to their beginning point eventually by travelling in a constant direction and nowhere would they come across a barrier or boundary.

Actually, it's not out of the question that the three spatial dimensions of our wonted experience are like this too, just that we're talking awfully big distances (10s to 100s of billions of light years) for us to come back to our beginning points if we blasted off into space and kept going in the same direction. As I understand this, this idea is seeming less and less likely since our universe globally is observed to be very flat indeed. See an interesting discussion at MathOverflow on what the fundamental group of the Universe might be like

Update: See also this answer clarifying some of my description of compactified dimensions. If they are big enough (as for our everyday three spatial dimensions, if they are compactified too) even though a constant direction vector can be integrated to a closed loop through space, the fact that the Universe is expanding means that one cannot traverse this loop in a finite time.

Integrating Factor Solution for Plasma Wave Equation

As part of a derivation in Bernstein '58 [1] a linear first-order (eqn. (9) in the image) appears:

But the general solution I would usually take (as appears in Gradshteyn and Ryzhik and checked in Mathematica) here would be

$$F=c_1G+\frac1G\int^\phi_1\frac{G(\phi^\prime)\Phi(\phi^\prime)}\Omega$$ where $\Phi$ is the RHS of eqn. (9). I suppose it depends on the requirement of periodicity and that $s$ has positive complex part (it comes from a Laplace transform) but I can't see how Bernstein's solution is obtained or equivalent, i.e. where the factor of $1/G$ goes. I'm sure I'm missing something silly, but I've wasted a bit of time on this already and can't get it to work out. The same derivation appears in slightly neater form in Montgomery and Tidman[2] but without further insight. Any help justifying this result would be greatly appreciated.

[1] Bernstein, I. (1958). Waves in a plasma in a magnetic field. Physical Review. Retrieved from http://journals.aps.org/pr/abstract/10.1103/PhysRev.109.10

[2] Montgomery D.C., Tidman, D.A. (1964). Plasma Kinetic Theory, McGraw-Hill advanced physics monograph series

homework and exercises - Understanding weight on an inclined plane

I'm trying to solve a problem where I have an object resting on an inclined plane, with the angle of the plan being alpha, and the weight being w. I'm having trouble figuring out how I can calculate the component of the weight parallel to the plane. I also want to find out the weight component perpendicular to the plane.

I don't want an outright answer, more of an explanation to help me understand. Thanks!

homework and exercises - What is the best way to calculate impact time with collisions?

I've been teaching myself physics and I've been wondering about the impact time in collision calculations. The scenario I've been using to learn is an object with a mass of 4000 kilograms colliding with a human being, while travelling at 17m/s. The object has a surface area the size and shape of a human elbow (which I very roughly guesstimated to be around 20cm2.

When calculating the force of this impact I need the momentum and the duration of the impact. The momentum is easy enough to calculate, but how is the duration of impact worked out? I know that it isn't referring to how long the objects are in contact, as this would mean that swords would harmlessly rub against a person if they were swung. I assume then, that the time is referring to how long it takes one object to impart the force of it's momentum into the other object.

How am I supposed to do this? The obvious way is to measure it, but given I'm an art student I can't exactly go around driving cars into people to measure how long it takes them to react to the impact. So far I've just been using .1 seconds, but I feel like this is far too slow.

quantum field theory - Condensed Matter or High Energy examples in which higher-loop $nge 2$ correction can have important physical consequence?

In QFT for high energy or condensed matter, tree level diagram means classical result or mean field. Loop diagram means quantum correction or (thermal or quantum) fluctuation above mean field. In most cases, we only need to compute $1$-loop correction, and in general it seems that higher-loop correction has no other important meaning except increasing the accuracy of numerical value. I even heard some professors say "If your advisor want you to compute higher loop correction, then you need to consider changing advisor." I'm curious about whether in high energy or condensed matter there are some cases in which higher-loop $n\ge 2$ correction has important physical consequences? Or are there some effects that can be found only after taking higher-loop correction into consideration?

Answer

Sure. Consider the following examples:

• 
  

  The anomalous magnetic moment of the electron is known (and needed) to five loops (plus two loops in the weak bosons). It is used to measure the fine-structure constant to a relative standard uncertainty of less than one part per billion. Similarly, the anomalous magnetic moment of the muon has been proposed as a rather clean and quantitative evidence for physics beyond the Standard Model (cf. this PSE post). More importantly (and essentially due to leptonic universality), the one-loop computation of the anomaly is mass-independent (i.e., the same for the three generations). You have to calculate the anomaly to at least two loops to be able to observe a difference (historically, this difference, and its agreement with the calculation, was the most convincing evidence for the fact that the muon is a lepton, that is, a heavy electron; for some time people thought that it could rather be the Yukawa meson).

  
  



• 
  

  Similarly, the muon decay width is the best parameter to measure the weak coupling constant, and the current experimental precision requires a theoretical calculation to several loops. (More generally, several precision tests of the electroweak part of the Standard Model are already measuring two-loops and beyond).

  
  


• 
  

  The fact that massive non-abelian Yang-Mills is non-renormalisable can only be established by computing two loops (cf. this PSE post). In the one-loop approximation, the theory appears to be renormalisable.

  
  


• 
  

  The fact that naïve quantum gravity (in vacuum) is non-renormalisable can only be established by computing two loops (cf. this PSE post). In the one-loop approximation, the theory appears to be renormalisable.

  
  


• 
  
  

  Some objects are in fact one-loop exact (the beta function in supersymmetric Yang-Mills, the axial anomaly, etc.). This can be established non-perturbatively, but one is usually skeptical about these results, because of the usual subtleties inherent to QFT. The explicit two-loop computation of these objects helped convince the community that there is an overall coherent picture behind QFT, even if the details are sometimes not as rigorous as one would like.

  
  


• 
  

  In many cases, the counter-terms that arise in perturbation theory actually vanish to one loop (e.g., the wave-function renormalisation in $\phi^4$ in $d=4$). When this happens, you need to calculate the two-loop contribution in order to obtain the first non-trivial contribution to the beta function and anomalous dimension, so as to be able to tell, for example, whether the theory is IR/UV free.

  
  


• 
  

  In supersymmetric theories, dimensional regularisation breaks supersymmetry (essentially because the number of degrees of freedom of fermions grow differently with $d$ from those of bosons). To one-loop order this only affects the finite part of the counter-terms (which is not a terrible situation), but from two-loops and beyond the violation of SUSY affects the divergent part of the counter-terms as well, which in turns affects the beta functions. (The solution is to use the so-called dimensional reduction scheme).

  
  


• 
  

  In an arbitrary theory, to one-loop order the beta function is independent of the regularisation scheme. From two loops on, the beta function becomes scheme dependent (cf. this PSE post). This has some funny consequences, such as the possibility of introducing the so-called 't Hooft renormalisation scheme (cf. this PSE post), where the beta function is in fact two-loop exact!

  
  
  


• 
  

  It was no-so-long ago suggested that there might be some choices of the gauge parameter $\xi$ that cure all divergences. For example, to one-loop, the Yennie gauge $\xi=3$ eliminates the IR divergence in QED (associated to the massless-ness of the photon), and people pondered on the possibility that this may hold to any loop order. Similarly, the Landau gauge $\xi=0$ does the same thing on the UV divergences. Now we know that in both cases this is just a coincidence, and no such cancellation holds to higher orders. But we only know this because the actual computation was performed to higher loops; otherwise, the possibility that such a cancellation works to any order would still be on the table. And it would definitely be a desirable situation!

  
  


• 
  

  The fact that the vacuum of the Standard Model is unstable if the Higgs mass is $m_h>129.4\ \mathrm{GeV}$ requires a two-loop computation (cf. arXiv:1205.6497). This is fun: why is the bound to close to the measured value? Could higher loops bring the number even closer? Surely that would mean something!

  
  


• 
  

  A meaningful and consistent estimate of the GUT point has been obtained by taking into account two loops (e.g., arXiv:hep-ph/9209232, arXiv:1011.2927, etc.; see also Grand Unified Theories, from the pdg).

  
  



• 
  

  Resummation of divergent series is a very important and trending topic, not only as a matter of practice but also as a matter of principle. It is essential to be able to calculate diagrams to a very high loop order so as to be able to test these resummation methods.

  
  


• 
  

  Historically speaking, the first tests of the renormalisation group equation were performed by comparison to an explicit two-loop computation. Indeed, the RGE allows you to estimate the two-loop large-log contributions given the one-loop calculation. The fact that the explicit two-loop computation agreed with what the RGE predicted helped convince the community that the latter was a correct and useful concept.

  
  


• 
  

  In the same vein, the initial one-loop calculation of the critical exponents (at the Wilson-Fisher fixed point) of certain systems was viewed with a lot of skepticism (after all, it was an expansion in powers of $\epsilon=1$, with $d=4-\epsilon$). The agreement with the experimental result could very well have been a coincidence. Higher loops consolidated the Wilsonian picture of QFT and the whole idea of integrating out irrelevant operators. Nowadays the critical exponents (in $(\boldsymbol \phi^2)^2$ theory) have been computed up to five loops, and the agreement (after Borel resummation) with experiments/simulations is wonderful. And even if the asymptotic series were not numerically accurate, one could argue that the result is still very informative/useful, at least as far as classifying universality classes is concerned.

  
  


• 
  
  

  Generically speaking, loop calculations become much more interesting (and challenging) from two loops on, because of overlapping divergences, the emergence of transcendental integrals (polylogarithms), etc. To one-loop order, naïve power counting arguments are essentially all one needs in order to establish convergence of Feynman integrals. The non-trivial structure of a local QFT can only be seen from two loops on (e.g., the factorisation of the Symanzik polynomials in the Hepp sectors, which is the key to Weinberg's convergence theorem, etc.).

  
  

Some of these examples are admittedly more contrived than others, but I hope they work together to help convince you that higher orders in perturbation theory are not merely a textbook exercise.

electromagnetism - Exciting Surface Plasmon-Polaritons with Grating Coupling

I'm very new the topic of SPPs and have been trying to understand this particular method of exciting surface plasmons using a 1D periodic grating of grooves, with distance $a$ between each groove. If the light incident on the grating is at an angle $\theta$ from the normal and has wavevector ${\bf k}$, then apparently if this condition is met:

$\beta = k \sin\theta \pm\nu g$

where $\beta$ is corresponding SPP wavevector, $g$ is the lattice constant $2 \pi/a$ and $\nu={1,2,3,...}$ then SPP excitation is possible.

I haven't ever really had a formal course in optics, so my question is where this condition comes from. It seems like Fraunhofer diffraction, but only for the light being diffracted at a $90^\circ$ angle to the normal. Most books don't state how they get this result, they just say it's because of the grating "roughness" which really confuses me.

Any help would be greatly appreciated.

Answer

Don't worry, I did research in surface plasmons and even then I was more than a year into it before I truly understood, on an intuitive level, how the light gets a 'kick' from the grating. You are correct that it is diffraction at a 90 degree angle to the normal, but there is an easier way to think about it.

You say you've never taken a formal course in optics so I'll talk a little bit about diffraction gratings in general. You might have come across one before and know that if a beam of light hits it, it is diffracted into several different beams. Transmissive diffraction gratings are what one usually encounters in high school physics so I'll illustrate one below:

The numbers at the end of each beam are known as the order $\nu$ of that beam. The grating equation is $d(\sin\theta_i + \sin\theta_o) = \nu\lambda$, where d is the distance between lines of the grating, $\lambda$ is the wavelength of the light, $\theta_i$ is the angle of incidence, and $\theta_o$ is the angle of the outgoing beam. In the above illustration, $\theta_i$ is zero.

Next we consider a reflective grating (for example a piece of metal with 1D periodic grooves), as in the following illustration:

The same mathematics govern this situation as well. You'll notice the $\nu=+2$ order being very close to grazing the grating surface. Adjusting the angle of incidence a little bit would cause it to do so. In that case, it would have the required wave vector to launch a surface plasmon, which is the phase matching condition that you started out with. You get the $\beta = k\sin\theta\pm\nu g$ when you convert the grating formula to wave vectors (reciprocal space) which I'm too lazy to do right now.

I suppose you could technically say that the light got a momentum 'kick' from the $\nu=-2$ order being launched in the opposite direction, but thinking of it as the light getting a 'kick' is really misleading in my experience.

newtonian mechanics - What is the typical orbital life of an artificial satellite?

The orbit of satellites around Earth eventually decays, or so I read. This is typically caused either by atmospheric drag, or by tides. I would assume most satellites have a limited service life in orbit. Hence the question - What is the typical orbital life of a satellite? How long before it's successor must be launched into orbit ?

EDIT: For instance, a weather satellite

newtonian mechanics - For what reason is the difference of potential energy $Delta U=-W$ equal to the *opposite* of the work done?

In my classical mechanics physics textbook (a translation of the Walker-Halliday-Resnick Fundamentals of Physics) the difference of potential energy is defined as

$$ \Delta U = -W \qquad (1) $$

I have done extensive research (taking me 5+ hours) and I claim to have a reasonable understanding of this model. In particular, I understand that if we throw a solid object in a straight upward direction then the work (i.e., the quantity of kinetic energy conveyed or subtracted from a body) exerted by the Earth's gravitational force is negative because they act on opposite directions: $W = \vec{F} \cdot \vec{d} = F \cdot cos(\phi) \cdot d$, where $cos(\phi) = -1$ due to $\phi$, the angle between the movement and the gravitational force, being $180°$.

However, I couldn't find anywhere an explanation for this. I was demonstrated that for a conservative force $\vec{F}$ doing work along a path $ab$, $W_{ab} = -W_{ba}$, and I also know that we can always associate a potential energy to a conservative force. But I'm still missing a link, and not knowing how the negative work of a force relates to its potential energy gives me brain fog.

Can you please provide an explanation, or an appropriate proof, for $(1)$? Please note that my physics knowledge only extends up to what is taught in university-level Physics I and Physics II courses.

classical mechanics - What makes a space a real space?

By "real space" I mean a space in which physical particles move.

Consider a color sphere and let a bunch of objects "move" on its surface. "Move" means "change colors". Let there be some rules governing the changes of color, maybe giving rise to something like inertness or collisions. .

Under which conditions will we be willing or forced to think of the movements as real movements in real space?

Photoelectric Effect, Why can't two quanta interact with an electron at the same time?

I understand that assuming light is quantized implies that if a lower energy interacts with metal, it is possible that that quanta will not have enough energy to eject the electron. What prevents two quanta from interacting with the electron at the same time, though, and summing to have enough energy to release the particle. Indeed, what if two separate light rays of the same frequency impact the electron, why can't that free it?

Answer

The answer is that electrons will sometimes be emitted when two photons, each of less than the work function energy, hit the surface. But there is no special significance to this so you shouldn't leap out of your bath and run naked down the street just yet.

The electrons in a metal have a natural oscillation frequency called the plasma frequency. Even in total darkness lattice vibrations will randomly transfer energy to the plasma oscillations, and especially at high temperatures random redistributions of this energy may concentrate enough energy in a small volume to eject an electron. This is the phenomenon of thermionic emission, and while the probability of emission is only significant at high temperatures there is in principle a finite probability of electron emission at room temperature.

Incident photons also transfer energy to plasma oscillations, but they transfer a lot of energy to a small volume so it's much more probable the energy can end up ejecting an electron. Even so, the quantum yield for photoemission is typically much less than one because in most cases the energy transferred to the plasma oscillations just leaks away into the bulk. A quick Google suggests typical quantum yields are around $10^{-6}$ i.e. only one photon in a million ends up ejecting an electron.

The point of all this preliminary rambling is that there is a small probability than even a single sub work function photon will eject an electron because you could get a combined photo/thermionic transfer of energy resulting in emission. The probability of this is exceedingly low, but then there are an exceedingly large number of electrons in metals.

Given the above, it should be obvious that there will also be a finite probability of two sub work function photons ejecting an electron. But is this really photoemission? You could argue that the first photon just causes local heating and this increases the chance of the second photon ejecting an electron through combined photo/thermionic emission.

So the pedant in me would have to concede that two sub work function photons can eject an electron. However the experimental scientist in me would regard an unmeasurably small probability as being zero, and would suggest that the pedant should get out more.

newtonian mechanics - How can an object with zero acceleration move?

My physics text has a problem in which it is said that a person moves a block of wood in such a way so that the block moves at a constant velocity. The block, therefore, is in dynamic equilibrium and the vector sum of the forces acting on it is equal to zero:

$\sum{F} = ma = m \times 0 = 0$

This is where I get confused. If the person is moving the block in such a way so that the sum of the forces acting on it is equal to zero, how can he be moving it at all? I realize that

1. even if there is no force acting on an object it can still have velocity (Newton's First Law)


2. force causes acceleration (a change in velocity--not velocity itself)

but for some reason I don't understand how it is possible for the object to move with constant acceleration velocity. Do I have all the pieces? I really feel like I am missing something.

EDIT

The block was at rest to begin with, and the person is moving the block through the air

EDIT: THE ACTUAL PROBLEM

For the sake of clearing up ambiguity, here is the actual problem. (I thought it was wooden blocks at first, sorry)

Two workers must pick up bricks that like on the ground and place them on a worktable. They each pick up the same number of bricks and put them on the same height worktable. They lift the bricks so that the bricks travel upward at a constant velocity. The first gets the job done in one-half the time that the second takes. Did one of the workers do more work than the other? If so, which one? Did one of the workers exert more power than the other? If so, which one?

Basically, the intent of this question is to ask if the above problem is entirely hypothetical. My text does not indicate that it is, which is what I find confusing.

Answer

If the person is moving the block in such a way so that the sum of the forces acting on it is equal to zero, how can he be moving it at all?

Consider a person pushing the block of wood along a surface with friction where the force due to friction (a force proportional to the speed of the block) exactly cancels the pushing force from the person.

The forces add to zero so the block does not accelerate. However, in order for the forces to add to zero, the block must be moving.

---

This addendum addresses the (latest) edited version of the question:

The first gets the job done in one-half the time that the second takes. Did one of the workers do more work than the other?

First let's ignore the accelerations at the beginning and end.

Work is force through distance. A brick lifted with constant speed against the pull of gravity to a given height requires a certain amount of work to be done by the worker regardless of the time spent lifting.

So, comparing the amount of work done while the bricks move with constant speed, there is no difference.

However, there is a difference in the power since power is the rate at which work is done.

If the same amount of work is done in half the time, the associated power is double. One worker does work at twice the rate of the other.

Now, let's look at the accelerations. To start a brick moving requires that the worker do work on the brick such that the brick gains kinetic energy.

But to stop the brick moving requires that the brick do work on the worker such that the brick loses that kinetic energy.

Thus, the work associated with the accelerations (ideally) cancel and don't factor into this calculation.

Let's tie this together with your original question:

How can an object with zero acceleration move?

Clearly, in this problem, each brick is at rest, is then briefly accelerated to some speed, moves at this speed for some distance, and then is briefly decelerated to rest.

During the portion in which a brick moves upward with constant speed, the worker provides a force which cancels the force of gravity. The momentum of the brick is constant since there is zero net force acting on the brick.

But, the brick is moving upward due to the brief acceleration where it gained momentum and kinetic energy.

what breakthrough Physics needs to make quantum computers work?

I read some posts on this forum and some articles which repeatedly state that it is not impossible to build q-comps but to make it successful, physics needs a great breakthrough. I tried finding but most of the material talks about how it works and what are current limitations, but how physics is going to help solving it?

Can somebody please explain precisely 'what is that breakthrough?'.

Answer

It can be argued that there are two primary challenges associated with quantum computing. One is a coding challenge, and the other is a purely mechanical (or in some minds physical) challenge.

As Ron indicated, viable Quantum Error Correction Code (QECC) is probably the largest breakthrough that makes quantum computers possible. Peter Shor was the first to demonstrate a viable Quantum Error Correction Code that can also be characterized as a Decoherence Free Subspace.

At this point, it can be argued that most of the problems associated with Fault Tolerant Quantum Computing are resolvable in terms of theory. However, their is a question of scaling as it relates to the number of qubits that are required to make the system viable, and the physical size of the system that is required to support those qubits. Most arguments that say there is a needed breakthrough in physics can be traced to this question of scalability.

In many cases, the difficulties are related to decoherence times and the speed at which the computers can perform calculations, which would allow the QECC time to operate. At this point those are now considered largely engineering challanges.

As far as physical breakthroughs, it is likely that those conversations are in regards to Topological Quantum Computers, which rely upon the construction of Anyons in order to operate. Currently these are largely viewed as pure mathematical constructions, however, the construction and use of stable anyons, even as quasiparticles, as part of topological quantum computer would be a major physical breakthrough.

Can a parallel computer simulate a quantum computer? Is BQP inside NP?

If you have an infinite memory infinite processor number classical computer, and you can fork arbitrarily many threads to solve a problem, you have what is called a "nondeterministic" machine. This name is misleading, since it isn't probabilistic or quantum, but rather parallel, but it is unfortunately standard in complexity theory circles. I prefer to call it "parallel", which is more standard usage.

Anyway, can a parallel computer simulate a quantum computer? I thought the answer is yes, since you can fork out as many processes as you need to simulate the different branches, but this is not a proof, because you might recohere the branches to solve a PSPACE problem that is not parallel solvable.

Is there a problem strictly in PSPACE, not in NP, which is in BQP? Can a quantum computer solve a problem which cannot be solved by a parallel machine?

Jargon gloss

1. BQP: (Bounded error Quantum Polynomial-time) the class of problems solvable by a quantum computer in a number of steps polynomial in the input length.


2. NP: (Nondeterministic Polynomial-time) the class of problems solvable by a potentially infinitely parallel ("nondeterministic") machine in polynomial time


3. P: (Polynomial-time) the class of problems solvable by a single processor computer in polynomial time


4. PSPACE: The class of problems which can be solved using a polynomial amount of memory, but unlimited running time.

Answer

There is no definitive answer due to the fact that no problem is known to be inside PSPACE and outside P. But recursive Fourier sampling is conjectured to be outside MA (the probabilistic generalization of NP) and has an efficient quantum algorithm. Check page 3 of this survey by Vazirani for more details.

condensed matter - Landau Theory of Phase Transitions

As often stated in books, near a phase transition we may express the free energy density as a power series in the order parameter $\phi(\mathbf{r})$. Up to quartic contributions, we have $$f=f_{0}-h\phi(\mathbf{r})+a_{2}\phi(\mathbf{r})^{2}+\frac{1}{2}|\mathbf{\nabla}\phi(\mathbf{r})|^{2}+a_{3}\phi(\mathbf{r})^{3}+a_{4}\phi(\mathbf{r})^{4}$$ where the second term is a coupling to an external field.

Now, I'm a bit confused about a few things:

1. 
   

   The terms with coefficients $a_{i}$ make sense to me - if we do a Taylor series we get a polynomial. However, why can't we include a term $a_{1}\phi(\mathbf{r})$?

   
   


2. 
   

   The external field term sort of makes sense, but this doesn't really stem from a power series does it? It's just something we put in by hand.

   
   



3. 
   

   Where does the gradient term come from? Is this also put in by hand - I don't see how this could come from a power series... And, why does it have to have coefficient $\frac{1}{2}$? Why can't we use $|\mathbf{\nabla}\phi(\mathbf{r})|$ and higher powers?

   
   

If anyone has any input on these points, or can suggest a good source of explanation, that would be useful :)

Glueball mass in non-abelian Yang Mills theory

How can the glueball mass be calculated in Yang-Mills theory?

Answer

In principle one has to calculate the pole of correlation functions involving gauge invariant operators like $\text{Tr}F_{\mu\nu}F^{\mu\nu}$. The problem is that due to asymptotic freedom, QCD is not solvable perturbatively at low energies. This is why nonperturbative techniques like lattice QCD are used to calculate such spectra. A key achievement in this direction was the calculation of the glueball spectrum by Morningstar and Peardon: http://arxiv.org/abs/hep-lat/9901004.

Another approach would be holographic QCD, where glueballs are mapped from the Yang-Mills theory to a theory of gravity and are represented by graviton modes propagating in space. It is relatively easy to compute their spectra within this formalism, in good agreement with lattice results: http://arxiv.org/abs/hep-th/0003115

As side remark: it is notable that within holographic QCD and in particular the Sakai-Sugimoto model, it is possible to calculate glueball decay to various mesons, which might help with the experimental confirmation of glueballs: http://arxiv.org/abs/arXiv:0709.2208

electromagnetism - 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.

Answer

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.

general relativity - Why do we need coordinate-free descriptions?

I was reading a book on differential geometry in which it said that a problem early physicists such as Einstein faced was coordinates and they realized that physics does not obey man's coordinate systems.

And why not? When I am walking from school to my house, I am walking on a 2D plane the set of $\mathbb{R} \times \mathbb{R}$ reals. The path of a plane on the sky can be characterized in 3D parameters. A point on a ball rotates in spherical coordinates. A current flows through an inductor via cylindrical coordinates.

Why do we need coordinate-free description in the first place? What things that exist can be better described if we didn't have a coordinate system to describe it?

Answer

That's a very good question. While it may seem "natural" that the world is ordered like a vector space (it is the order that we are accustomed to!), it's indeed a completely unnatural requirement for physics that is supposed to be built on local laws only. Why should there be a perfect long range order of space, at all? Why would space extend from here to the end of the visible universe (which is now some 40 billion light years away) as a close to trivial mathematical structure without any identifiable cause for that structure? Wherever we have similar structures, like crystals, there are causative forces that are both local (interaction between atoms) and global (thermodynamics of the ordered phase which has a lower entropy than the possible disordered phases), which are responsible for that long range order. We don't have that causation argument for space (or time), yet.

If one can't find an obvious cause (and so far we haven't), then the assumption that space "has to be ordered like it is" is not natural and all the theory that we build on that assumption is built on a kludge that stems from ignorance.

"Why do we need coordinate free in the first place?"... well, it's not clear that we do. Just because we have been using them, and with quite some success, doesn't mean that they were necessary. It only means that they were convenient for the description of the macroscopic world. That convenience does, unfortunately, stop once we are dealing with quantum theory. Integrating over all possible momentum states in QFT is an incredibly expensive and messy operation that leads to a number of trivial and not so trivial divergences that we have to fight all the time. There are a few hints from nature and theory that it may actually be a fools errand to look at nature in this highly ordered way and that trying to order microscopically causes more problems than it solves. You can listen to Nima Arkani Hamed here giving a very eloquent elaboration of the technical (not just philosophical) problems with our obsession with space-time coordinates: https://www.youtube.com/watch?v=sU0YaAVtjzE. The talk is much better in the beginning when he lays out the problems with coordinate based reasoning and then it descends into the unsolved problem of how to overcome it. If anything, this talk is a wonderful insight into the creative chaos of modern physics theory.

As a final remark I would warn you about the human mind's tendency to adopt things that it has heard from others as "perfectly normal and invented here". Somebody told you about $\mathbb R$ and you have adopted it as if it was the most natural thing in the world that an uncountable infinity of non-existing objects called "numbers" should exist and that they should magically map onto real world objects, which are quite countable and never infinite. Never do that! Not in physics and not in politics.

lagrangian formalism - Energy-momentum tensor of the Dirac field

I'm trying to compute the energy momentum tensor for the dirac field $$\mathcal{L}=\bar\psi(i\gamma_\mu\partial^\mu-m)\psi $$$$T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial(\partial_\mu\psi)}\partial^\nu\psi-\eta^{\mu\nu}\mathcal{L}$$ and I'm not clear on how to treat the term in $\eta^{\mu\nu}\mathcal{L}$: the first term gives $i\bar\psi\gamma^\mu\partial^\nu\psi$ which is the tensor given by Peskin-Schroeder but I don't get how to compute the term in $\eta$

quantum information - Using delayed choice interference experiments as a computing device

I had an idea how to design a "quantum computer":

How about designing interference-experiments where the design of the experiments itself represents algorithmical or mathematical problems that are coded in a way that the solution to that problem is interference = “true” and no interference = “false” as a result?

These results would be acquired instantaneously because the algorithm would not be processed step by step as with classical computers. If the design was changed on the run - thereby representing another problem - the solution would even be acquired faster than the quanta going through the apparatus as predicted by “delayed choice”-experiments.

Are there already such ideas for “quantum computers” of this kind? They would not have the problem of decoherence but on the contrary decoherence vs. no-decoherence would be their solution set of the problems represented by the (flexible) experimental design.

Does this make sense?

EDIT: Perhaps my question is not formulated well - perhaps I shouldn't call it "quantum computer". My point is that any configuration with these double slit experiments - no matter how complicated - instantly shows an interference pattern when there is any chance of determining the way the photons took. My idea is to encode some kind of calculation or logical problem in this configuration and instantly find the answer by looking at the outcome: interference = “yes” and no interference = “no” –

Answer

Here is my understanding of what you are asking (and I believe it is a little different to Lubos's interpretation, so our answers will differ): Can you build a computer that uses the interference effects to perform computation, where you use the presence of light in a particular place to represent 1 and no light to represent 0?

The answer is yes, though with certain caveats. First let me note that there is something already called a quantum computer which exploits quantum effects to outperform normal (classical) computers. Classical computation is a special case of quantum computation, so a quantum computer can do everything a classical computer can do, but the converse is not true.

If you have single photon detectors, you can use the interference effects of a network of beam splitters and phase plates together with the detectors to create a universal quantum computer. This is something called linear-optics quantum computing or LOQC. Perhaps the best known scheme is the KLM proposal, due to Knill, Laflamme and Milburn.

Now the caveats: you need to have a fixed finite number of photons, and you need to adapt the network based on earlier measurement results. This adaptive feed-forward is quite difficult to achieve in practice, though not impossible, and computation with such a setup has been demonstrated.

A further interpretation of the question is whether it is sufficient to use such a linear network, but only make measurements at the end. This is an open question, though there is strong evidence that such a device is not efficiently simulable by a classical computer (see this paper by Scott Aaronson). It is however not yet known whether you can implement universal classical or quantum computing on such a device.

newtonian mechanics - How to calculate wasted energy

Suppose you are pulling a weight along a track at an angle (in the picture 45°).

If the object is dislocated by a distance $D_{45}$ let's assume that the mechanical work done on/energy transmitted to the object is: $$ W_{45} = (\vec{F}\cdot\vec{d} = Fd\cos_{45}) = K J$$, which in the picture is called: working (real) power.

If you had pulled along the track, presumably the distance covered by the object would be $D_0$ = $D_{45} /cos_{45}$ and the work done would be $$W_0 (= E_0) = (\vec{F}\cdot\vec{d} = Fd\cos_{0}) = K/cos_{45} = K * 1.41 J$$, which in the image is called: total (apparent) power

If this is correct, can we legitimately suppose that the same amount of energy/calories were burned in both cases, that is: $E_0 = E_{45} = 1.41 * K$, but, since the work done is less, that is: ($W_{45} < W_0 = 0.7* W_0$) can we therefore conclude that the energy wasted in the first case (calories burned doing no mechanical work) is 41%? That energy was wasted trying to dislocate the track or derail the cart: non working (reactive) power

(If this is not correct, how can we calculate the energy wasted when we apply a force at an angle greater than 0°?)

update

So no mechanical energy is being wasted by pulling at an angle.... However this overlooks the fact that muscles are inefficient things and consume energy even when no mechanical work is being done.

I knew that no mechanical work in excess is done, that is because of the peculiarity of the definition of work. I tried to dodge this ostacle speaking of calories burned: I thought we can desume the exact value by, so to speak, reverse engineering.

The logic is this: we have instruments (I am referring to instruments, so we avoid inefficiency of human muscles) to produce and measure force. If we measure the force something exerts on a blocked object, then remove the block and measure the oucome of that force, can't we deduce that the same force has been applied before and the same amount of energy/calories has been burned withoud doing any work? If this logic is valid, the same logic has been applied to the horse.

A biologist can tell yus the percentage of calories burned in excess with regard to the effective force applied/ transmitted: the inefficiency of the human machine is around 80%. WE burn 4 - 5 times more energy than we put to avail. Efficiency may vary from 18% to 26%. In the case above, energy actually burned would therefore be 41 * 4.5 = ca. 185%

statistical mechanics - Is the Landau free energy scale-invariant at the critical point?

My question is different but based on the same quote from Wikipedia as here. According to Wikipedia,

In statistical mechanics, scale invariance is a feature of phase transitions. The key observation is that near a phase transition or critical point, fluctuations occur at all length scales, and thus one should look for an explicitly scale-invariant theory to describe the phenomena. Such theories are scale-invariant statistical field theories, and are formally very similar to scale-invariant quantum field theories.

Question I understand that at the critical point the correlation length $\xi$ diverges and as a consequence, the correlation functions $\langle\phi(\textbf{x})\phi(\textbf{y})\rangle$ behave as a power law. Power laws are scale-invariant. But for a theory itself to be scale-invariant (as Wikipedia claims) the Landau Free energy functional should have a scale-invariant behaviour at the critical point. But the free energy functional is a polynomial in the order parameter and polynomials are not scale-invariant.

Then how is the claim that the relevant statistical field theory is scale-invariant justified?

Answer

I answered a very similar question here, but in the context of quantum field theory rather than statistical field theory. The point is that it is impossible to have a nontrivial fixed point classically (i.e. without accounting for quantum/thermal fluctuations) for exactly the reason you stated: the dimensionful coefficients will define scales.

We already know that quantum/thermal fluctuations can break scale invariance, e.g. through the phenomenon of dimensional transmutation, where a quantum theory acquires a mass scale which wasn't present classically. And what's going on here is just the same process in reverse: at a nontrivial critical point the classical scale-dependence of the dimensionful coefficients is exactly canceled by quantum/thermal effects. Of course this cancellation is very special, which is why critical points are rare.

Double Slit Experiment: How do scientists ensure that there's only one photon?

Many documentaries regarding the double slit experiment state that they only send a single photon through the slit. How is that achieved and can it really be ensured that it is a single photon?

Answer

In the double slit experiment, if you decrease the amplitude of the output light gradually, you will see a transition from continuous bright and dark fringe on the screen to a single dots at a time. If you can measure the dots very accurately, you always see there is one and only one dots there. It is the proof of the existence of the smallest unit of each measurement which is called single photon: You either get a single bright dot, or not.

So, probably you may ask why it is not a single photon composite of two "sub-photon", each of them passing through the slit separately and then interference with "itself" at the screen so that we only get one dot. However, the same thing occurs for three slits, four slits, etc... but the final results is still a single dot. It means that the photon must be able to split into infinitely many "sub-photon". If you get to this point, then congratulation, you basically discover the path-integral formalism of quantum mechanics.

newtonian mechanics - Wasted energy applying a torque

In another question I asked how find a way to deduce wasted energy, but I did not get a satisfactory answer. I have changed the setting here which brings us more close to the solution. Instead of a horse pulling a cart on a track, let's consider a torque:

If we apply a force tangentially to a rod and exert a force of 100 N we get a torque of 100Nm (J/r), if we apply it at an angle of 60° (30°) we get 86.6 J/r.

Can we say, in this case , that we wasted 13.4 J?

What your drawing shows is an excess force, not wasted energy. Since energy conservation holds, where would that wasted energy have gone in your scenario? If there was no wasted energy, and since the force is in excess of what's needed, how could you modify your calculation in such a way that it becomes consistent? – CuriousOne

In the case of the cart on the track the WE (in the original article is called reactive, non working power) has gone to the track as actually you wer trying to dislocate the track or derail the cart, it has been turned into thermal energy

Same here, if you put a dynamometer under the fulcrum you'll see that it will be compressed by an equivalent amount of energy, WE has cone to compress the spring. Is not that right?

The moment you start using the Joule, it automatically means you are measuring energy, which torque is not.

I hope we can get rid soon of this misunderstanding. I am not saying or implying that torque is energy, although a superficial reading of their units

Newton metres and joules are "dimensionally equivalent" in the sense that they have the same expression in SI base units: $$1 \, \mathrm{N} > \! \cdot \! \mathrm{m} = 1 \frac{\mathrm{kg} \, > \mathrm{m}^2}{\mathrm{s}^2} \quad , \quad 1 \, \mathrm{J} = 1 > \frac{\mathrm{kg} \, \mathrm{m}^2}{\mathrm{s}^2}$$ Again, N⋅m and J are distinguished in order to avoid misunderstandings where a torque is mistaken for an energy or vice-versa.

might lead one to think so. They are fundamentally the same thing with a subtle difference. Torque is measured in joule per radian, the same difference we have between mechanical work $J = W = \vec F \cdot \vec d$ and energy: they are not theoretically the same thing but the are closely related, and work is energy and measured in J's. The difference is mainly that on the left side of the equation we have a scalar and on the right hand a vector.

This hair-splitting distinction becomes irrelevant if, knowing exactly what we are doing, we use all information we have to make deductions. Just to give a more striking example, the same difference can be pinpointed between speed S (distance covered) and velocity V: we know they are not exactly the same 'concept' but if we want to deduce the KE of a body it is risible to state that we cannot rightfully and legitimately deduce $E = m*S^2/2$.

In the case of the torque we have the same problem, instead of d (S) we have r, instead of dot we have cross product, but we know that it is equivalent to, that the outcome of a torque (N*m) is energy (per radian), expressed in J's. Why shouldn't we put this to our advantage to deduce what we like?

quantum field theory - Equivalence of canonical quantization and path integral quantization

Consider the real scalar field $\phi(x,t)$ on 1+1 dimensional space-time with some action, for instance

$$ S[\phi] = \frac{1}{4\pi\nu} \int dx\,dt\, (v(\partial_x \phi)^2 - \partial_x\phi\partial_t \phi), $$

where $v$ is some constant and $1/\nu\in \mathbb Z$. (This example describes massless edge excitations in the fractional quantum Hall effect.)

To obtain the quantum mechanics of this field, there are two possibilities:

1. Perform canonical quantization, i.e. promote the field $\phi$ to an operator and set $[\phi,\Pi] = i\hbar$ where $\Pi$ is the canonically conjugate momentum from the Lagrangian.


2. Use the Feynman path integral to calculate all expectation values of interest, like $\langle \phi(x,t)\phi(0,0) \rangle$, and forget about operators altogether.

My question is

Are the approaches 1. and 2. always equivalent?

It appears to me that the Feynman path integral is a sure way to formulate a quantum field theory from an action, while canonical quantization can sometimes fail.

For instance, the commutation relations for the field $\phi$ in the example above look really weird; it is conjugate to its own derivative $\Pi(x,t) = -\frac{1}{2\pi\nu}\partial_x\phi(x,t)$. The prefactor is already a little off. For this to make sense, we have to switch to Fourier transformation and regard the negative field modes as conjugate momenta, $\Pi_k=\frac{1}{2\pi\nu}(-ik)\phi_{-k}$.

A more serious example: it seems to me that the Feynman integral easily provides a quantum theory of the electromagnetic gauge field $A_\mu$ whereas in canonical quantization, we must first choose an appropriate gauge and hope that the quantization does not depend on our choice.

Could you give a short argument why 2. gives the right quantum theory of the electromagnetic field? (standard action $-\frac1{16\pi} F^{\mu\nu}F_{\mu\nu})$

Answer

This type of problems is often referred to as constrained mechanical system. It was studied by Dirac, who developed the theory of constrained quantization. This theory was formalized and further developed by Marseden and Weinstein to what is called "Symplectic reduction". A particularly illiminating chapter for finite dimensional systems may be found in Marsden and Ratiu's book: "Introduction to mechanics and symmetry".

When the phase space of a dynamical system is a cotangent bundle, one can use the usual methods of canonical quantization, and the corresponding path integral. However, this formalism does not work in general for nonlinear phase spaces. One important example is when the phase space is defined by a nonlinear surface in a larger linear phase space.

Basically, Given a symmetry of a phase space, one can reduce the problem to a smaller phase space in two stages

1. Work on "constant energy surfaces" of the Hamiltonian generating this stymmetry.


2. Consider only "invariant observables" on these surfaces.

This procedure reduces by 2 the dimensions of the phase space, and the reduced dimension remains even. One can prove that if the original phase space is symplectic, so will be the reduced phase space.

May be the most simple example is the elimination of the center of mass motion in a two particle system and working in the reduced dynamics.

There is a theorem by Guillemin and Sternberg for certain types of finite dimensional phase spaces which states that quantization commutes with reduction. That is, one can either quantize the original theory and impose the constraints on the quantum Hilbert space to obtain the "physical" states. Or, on can reduce the classical theory, then quantize. In this case the reduced Hilbert space is automatically obtained. The second case is not trivuial because the reduced phase space becomes a non-linear symplectic manifold and in many cases it is not even a manifold (because the group action is not free).

Most of the physics applications treat however field theories which correspond to infinite dimensional phase spaces and there is no counterpart of the Guillemin-Sternberg theorem. (There are works trying to generalize the theorem to some infinite dimensional spaces by N. P. Landsman). But in general, the commutativity of the reduction and quantization is used in the physics literature, even though a formal proof is still lacking. The most known example is the quantization of the moduli space of flat connections in relation to the Chern-Simons theory.

The most known example of constrained dynamics in infinite dimensional spaces is the Yang-Mills theory where the momentum conjugate to $A_0$ vanishes. It should be mentioned that there is an alternative (and equivalent) approach to treat the constraints and perform the Marsden-Weinstein reduction through BRST, and this is the usual way in which the Yang-Mills theory is treated. In this approach, the phase space is extended to a supermanifold instead of being reduced. The advantage of this approach is that the resulting supermanifold is flat and methods of canonical quantization can be used.

In the mentioned case of the scalar field, the phase space may be considered as an infinite numer of copies of $T^{*}\mathbb{R}$. The relation $\Pi = \partial_x \phi$ is the constraint surface. In a naive dimension count of the reduced phase space dimensions one finds that in every space point the 1+1 dimensions the phase space (The field and its conjugate momentum) are fully reduced by the constraint and its symmetry generator. Thus we are left with a "zero-dimensional" theory. I haven't worked out this example, but I am quite sure that if this case is done carefully we would have been left with a finite number of residual parameters. This is a sign that this theory is topological - which can be seen through the quantization of the global coefficient".

Update:

In response to Greg's comments here are further references and details.

The following review article (Aspects of BRST Quantization) by J.W. van Holten explains the BRST quantization of electrdynamics and Yang-Mills theory (Faddeev-Popov theory) as constrained mechanical systems.The article contains other example from (finite dimensioal phase space) quantum mechanics as well.

The following article by Phillial Oh. (Classical and Quantum Mechanics of Non-Abelian Chern-Simons Particles) describes the quantization of a (finite dimensional) mechanical system performing the symplectic reduction directly without using BRST. Here, the reduced spaces are coadjoint orbits (such as flag manifolds or projective spaces). The beautiful geometry of these spaces is very well known and this is the reason why the reduction can be performed here directly. For most of the reduced phase spaces, such an explicit knowledge of the geometry is lacking. In field theory, problems such as quantization of the two dimensional Yang-Mills theory possess such an explicit description, but for higher dimensional I don't know of an explicit treatment (besides BRST).

The following article by Kostant and Sternberg, describes the equivalence between the BRST theory and the direct symplectic reduction.

Now, concerning the path integral. I think that most of the recent physics achievements were obtained by means of the path integral, even if it has some loose points. I can refer you to the following book by Cartier and Cecile DeWitt-Morette, where they treated path integrals on non-flat symplectic manifolds and in addition, they formulated the oscillatory path integral in terms of Poisson processes.

There is a very readable reference by Orlando Alvarez describing the quantization of the global coefficients of topological terms in, Commun. Math. Phys. 100, 279-309 (1985)(Topological Quantization and Cohomology). I think that the Lagrangian given in the question can be treated by the same methods. basically, the quantization of these terms is due to the same physical reason that the product of electric and magnetic charges of magnetic monopoles should be quantized. This is known as the Dirac quantization condition. In the path integral formulation, it follows from the requirement that a gauge transformation should produce a phase shift of multiples of $2\pi$. In geometric quantization, this condition follows from the requirement that the prequantization line bundle should correspond to an integral symplectic form.

Mixing of Ideal gas - Thermodynamic equilibrium

I always get confused what exactly happens when two ideal gases mix.

Consider the initial situation where two gases are in a box, separated by a rigid and adiabatic wall between them. Now when the wall between them is removed, they come to equilibrium (of course assuming the process is done quasi-statically). Initially the thermodynamic quantities of the gas be

$$ U_i,\: S_i,\:T_i,\:P_i,V_i \:\:\: \:\:\: \:\:\: \:\:\: \:\:\: \:\:\: \:\:\:i=1,2 $$ Now after the wall is removed, the condition for equilibrium is attained by the condition $$ dS = 0 $$ I am not able to apply this and find out exactly the final values of the thermodynamic quantities for an arbitrary ideal gas situation (I am not particularly able convince the situation physicaly)

(for instance if we consider and the two ideal gases to obey equation, $$ P_iV_i = c_iN_iRT_i \:\:\: \:\:\: \:\:\:i=1,2 $$ where $ c_i $ is the degrees of freedom of the gas, like $ c_i = \frac{3}{2},\:\frac{5}{2} $ for monatomic and diatomic respectively.

Answer

The total initial internal energy is $U=U_1 + U_2=\frac{\nu_1}{2}c_1RT_1 + \frac{\nu_2}{2}c_2 RT_2 $ where the last equality comes from Joules' first law for ideal gases and where $c_i$ is the number of moles of species $i$ and $\nu_i$ is the number of degrees of freedom of the molecule (3 for atoms, 5 for diatomic molecules etc..).

Now, once equilibrium is reached everybody should have the same temperature $T$. Since you are dealing with an ideal gas it implies that:

$$U= \frac{(\nu_1c_1+\nu_2c_2)RT}{2}$$ and hence since the whole system is isolated

$$T = \frac{\nu_1c_1T_1 + \nu_2c_2 T_2}{\nu_1c_1+\nu_2c_2}$$

Once the temperature is known, the rest follows easily. The pressure can be gotten straightforwardly as

$$P=\frac{(c_1+c_2)RT}{V_1+V_2}$$

because the ideal gas law is independent of the number of degrees of freedom of the different species.