@DavidZ yeah, but unlike the usual ones that I see in the Maxwell equations, I dont really get what E and B what they mean physically when defined in relation to the four potential
@ACuriousMind yeah I get what a gauge transformation is mathematically, but I dont really underatand what this tells me about B and E.
A gauge transformation does, in essence, precisely nothing. The presence of a gauge transformation means you have superfluous degrees of freedom in your system - you are using more variables than you need to describe your system
Nevertheless, there are good reasons to do this - maybe you don't know how to reduce the d.o.f., maybe the formalism with the additional d.o.f. has a certain advantage
And in the case of electromagnetism, the advantage is a powerful one: It allows us a completely covariant description of the system
The transformation that would perform the reduction is quite complicated, and the coordinates you end up with are nothing like the usual $(x,y,z)$ Cartesian coordinates we usually deal with.
@Danu : see above. That question came from Bosoneando carping about my answer here about why charged particles move the way that they do. Doubtless somebody will say it's because they're tossing photons back and forth. As if hydrogen atoms twinkle, and magnets shine.
So I read about different gauges in Griffiths E&M book, is the superfluousness of this DOF part of why we can choose different gauges? IIRC, he mentioned something called the Coulomb gauge
Ehhh...it's a field theory, so that's a bit of a difficult question
"One d.o.f." would essentially be the value of the potential at one point, so since we have four values of the potential at every point we can't really count them
I think the $A^3=0$ gauge isn't partial, though, so on the field level you might think of "one extra d.o.f."
But it's more difficult so say "how many" d.o.f. the differential gauge fixing conditions like the Lorenz gauge fix
So is a gauge freedom then just a term to signify a DOF that manifests through a gauge tranaformation, like the case for B and E in relation to the four potential?
@StanShunpike The fact that, if I have more d.o.f. describing my system than I really need, I can do with the superfluous parts of them whatever the hell I want.
@StanShunpike Not exactly. It's saying that given one set of variable describing my system, I may change to a different one without changing the physics. (E.g. going from $\vec A = 0$ to $\vec A = \nabla\chi$ in vacuum)
@JohnDuffield Bosoneando is completely correct in asserting that quantum electrodynamics can distinguish between positive and negative charges regardless of the spin of the charged particle, since the Coulomb law derives just from the form of the bosonic propagator of the photon, and Coulomb+special relativity gives Lorentz force.
@ACuriousMind can u elaborate a bit more? I am still a bit confused. So does this mean gauge freedom is what allows me to use either Coulomb to Lorentz gauge?
@StanShunpike Yes. See, gauge freedom is the statement that, given any $A$, the choice $A+\mathrm{d}\chi$ for any scalar function $\chi$ is completely equivalent.
If you now examine all the gauge choices that are commonly presented to us, you can find that the solutions to the gauge fixing equations are all related to one another by such a $\chi$.
For example, the statement is that I can use Lorenz gauge, $\partial^\mu A_\mu = 0$. This is achieved by taking any $A$ that describes my system and solving the equation $\partial^\mu \partial_\mu \chi = \partial^\mu A_\mu$. The solution $\chi$ then gives $A'^\mu = A^\mu - \partial^\mu\chi$ with $\partial_\mu A'^\mu = 0$.
This also shows why not all gauge fixings are always desirable - the explicit solution for $\chi$ might be hard to find, and if you can't modify your procedure to get the $A$ in the first place to respect the gauge fixing condition so you don't need to search for $\chi$, then the gauge fixing is not particularly useful
@0celo7 Not really. (I also missed the day of that talk, the only one I missed :/) But in the end, I think all it and the Unruh effect tell us is that different observers might disagree on the number of particles in a state, i.e. the notion of particle is simply not an invariant notion in regimes where the Lorentz invariance of special relativity is not enough
But the notion of particle is also not a good notion in strongly interacting theories, so I'm not terribly disturbed by that
Assigning the black hole a temperature and thinking of it as thermal radiation is the way I think of it, and I'm pretty certain the explanation with the "particle pairs forming and one of them falling in" is non-sense.
@JohnDuffield Let me be blunt: I've checked the moderator tools concerning inappropriate voting being directed at you at least once a week for the last month, and tihe evidence isn't there.
What there is, is lots of evidence that many people think you are wrong very often.
@ACuriousMind Basically Unruh and Hawking effect can be approximately translated into each other in "1 to 1 fashion", sending Rindler $\to $ Boulware vacuum (and corresponding states) and Minkowski $\to$ Kruszkal vacuum (and corresponding states)
Stack Exchange sites use crowdsourced evaluation. That makes poorly suited for introducing new ideas, but then introducing new ideas was never what they were designed to do.
The word "wave" in the comment I quoted was not an issue, claiming that it was in particular an electromangnetic wave is a big claim, contradicts successful theories and you would have to back it up with a solution to Maxwell's equations to get it treated seriously.
@0celo7: I think I'll be getting a MBAir after all. Just don't think the retina display of the MB12 can justify the price increase with such a drastic performance drop.
@ACuriousMind in classical mechanics, we focus a lot on position, velocity, and momentum. Why does quantum mechanics just have this vague term called an "observable"?
@StanShunpike Classical Hamiltonian mechanics has the perfectly analogous term of "classical observable"! Those are the smooth functions on the phase space. The phase space is the space of coordinates and momenta, and everything that is classically of interest is a function of those
In fact, the canonical commutation relations $[x,p] = \mathrm{i}\hbar$ come from the classical relation $\{x,p\} = 1$, where $\{\dot{},\dot{}\}$ is the Poisson bracket on the classical observables
So term is not vague at all - the quantum notion of "algebra of observables" is the perfect analogue of "algebra of smooth functions on the phase space"
It's just that, in the classical treatment, you don't have to stress this algebra of functions very much, because position and momentum have simultaneously well-defined values on every possible state, so you can deduce a well-defined value of every classical observable by just knowing its position and momentum
@ACuriousMind indeed there seems to be a few interchangeably used terms for observables in CM, smooth functions, differentiable functions, classical observables, I for one always use just observable :D
@Phonon You need to choose the smooth and not just (n times) differentiable functions because otherwise the Poisson bracket of two observables is not guaranteed to be an observable again.
@Phonon Ah, well, I ran out of things to say very quickly. Or rather, I realized I need to learn about many topics more in depth if I want to write pieces on them I'm not ashamed of when I read them again later on :P
In the Hamiltonian formalism, Noetherian charges are the generators of their corresponding symmetry transformations.
Electric charge is a bit of a bad example, though...it only becomes a proper Noetherian charge when you consider matter fields to be complex fields, which one does not usually do classically
@StanShunpike yes, one way to convince yourself would be to see that it can be written in terms of the canonical momentum, $\mathbf{P} = m\mathbf{v} + q\mathbf{A}$ with q the charge and A a magnetic vector potential
@0celo7 Both bevor and vor are German words. Bevor is solely a "temporal word", akin to the archaic English "ere", while vor is a "spatial word", akin to "in front of".
> Euclid: The Game gamifies the 2300 years old book "The Elements" written by the ancient Greek mathematician Euclid in Alexandria.
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When you throw an electron through a solenoid, it moves helically around the field lines, as per this schoolphysics illustration:
© Keith Gibbs 2013
Then if we were to throw a positron through the solenoid, it would also move helically, but "the other way". One could liken their paths to left...
