joigus

One of the girls in the last cover looks strangely familiar...

That would be the Lorenz gauge --different Loren*s --. https://en.wikipedia.org/wiki/Ludvig_Lorenz It's first order. It becomes second order when substituted in Maxwell's equations, which I think is probably what you meant, Markus. ----- Maybe a split is in order? I agree with @studiot and @MigL. Although I find the discussion with @Markus Hanke very interesting. It's OK. If somebody had to be in charge of the Inquisition I'd rather it be 21st-Century Canadians. Much more humane, I'm sure. Nobody expects... that joke. 😆

http://www.folksong.org.nz/soon_may_the_wellerman/index.html

Let me get back to you when I get more time to think about this and review some literature. It's correct that radiation is generally described as having "2 DOF" in the sense of having two polarisation states. But the term "degrees of freedom" is a bit ambiguous. Some people use it in the sense of counting spin states. But there are also space-time variables, according to, \[\left|\boldsymbol{k},s\right\rangle\] The \( k_x \), \( k_x \), \( k_x \), and \( s \) would be four variables, more in the sense that I was talking about (DOF as number of variables necessary to describe the states). As I said, let me get back to you. Also, I think you can do the counting on the E's and B's, which are gauge invariant. You can do it on the A's, but AFAIR there's just one gauge fixing condition, because you're in \( U\left(1\right) \)... Still thinking. Apparently I'm outside my comfort zone too. LOL

Ok. There's a couple of things that you're seeing there that are already taken care of in the mainstream formalism. And have been mentioned repeatedly. One of them is that for any physical 4-vector the pseudo-norm with signature (+,-,-,-) must be positive --causality. The other one that I at least forgot to mention --it's possible that either @Ghideon and/or @Markus Hanke have mentioned it and I've missed it--, is that the naught component must be positive. Those are called orthochronous 4-vectors. So you cannot add two arbitrary 4-vectors and hope that that has any physical meaning. When you add two orthochronous 4-vectors, you obtain another orthochronous 4-vector. Also, the orthochronous character ( \( v^0 > 0 \) ) is split in mutually disconnected subsets of the Minkowski space, so the idea that you suggest that there is a "perforated" structure is actually a misguided intuition. The reason is that both the sign of the Minkowski pseudometric and the zero component are continuous and differentiable function of their arguments: \[ f\left(\alpha^{0},\alpha^{1},\alpha^{2},\alpha^{3},\beta^{0},\beta^{1},\beta^{2},\beta^{3}\right)=\alpha\cdot\beta \] \[g\left(\alpha\right)=\alpha^0\] So you can't have either changes in sign of the product, nor changes in the sign of any component for arbitrarily close 4-velocities, as you posit.

(My emphasis.) Your tangents are always very interesting. Of course, you're right. I meant "in a sense" in the sense that it is an object that only has 6 non-zero components in general, which can be relevant when there is charged matter around, although in that case the new degrees of freedom can be attributed to matter densities --see below. Let me check the counting. Pure EM field must have identically null Lagrangian,* \[\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}\left(E^{2}-B^{2}\right)\] as in the vacuum \( \left|\boldsymbol{E}\right|=\left|\boldsymbol{B}\right| \), so the Lagrangian is identically null on solutions, as corresponds to massless particles. That leaves 5. If we now throw in the transversality, \[\boldsymbol{E}\cdot\boldsymbol{B}=0\] that leaves 4 DOF, which are the ones you're talking about. Correct me if I'm wrong. When light goes through matter, we no longer have the \( \boldsymbol{E} \) and the \( \boldsymbol{B} \), but also the \( \boldsymbol{D} \) and the \( \boldsymbol{H} \), so that, \[\mathcal{L}=\frac{1}{2}\left(\boldsymbol{E}\cdot\boldsymbol{D}-\boldsymbol{B}\cdot\boldsymbol{H}\right)\neq0\] with, \[\boldsymbol{D}=\varepsilon\left(x\right)\boldsymbol{E}\] \[\boldsymbol{B}=\mu\left(x\right)\boldsymbol{H}\] so we have two "additional" DOFs, the matter medium, characterised by \( \varepsilon\left(x\right) \) and \( \mu\left(x\right) \) that take us back to 6. But it's a fictional 6. *Edit: \( c=1 \) throughout.

Your premises are wrong, precisely for the reasons that Swansont has pointed out. A proton exerts a force on all the other electrons in the universe, and all the other protons, as well as on any other charged particles. On any charged particle, the electrostatic force is the sum of all the forces exerted on it by all the other charged particles in the universe, according to the superposition principle. You could say that the electric force is 6 dimensional in some sense, if you include the magnetic effects. But 1-dimensional? That certainly is not the case. You also seem to be confusing the number of particles on which a charged particle can act with the number of dimensions. Neither of them is one.

Virtual particles indeed appear in QFT always as a result of perturbative calculations. It could make sense to ask "are they really there?" They're really there in the sense that you must take them into account if you want to calculate the renormalised quantities, like mass, charge, etc. They're also "there" in the sense that, whenever you put in the required energy, they "leap into reality" and new real, detectable particles appear. In the decades after the glorious years of perturbative QFT around the 50s, towards the 80s and 90s, new non-perturbative methods were developed, by people like 't Hooft and Polyakov. Topological field theories are also non-perturbative, but I think we're still not there. I could be wrong, but I don't think we completely understand QFT from a non-perturbative POV. There are new games in town, but it may well be the case that a new synthesis of the formalisms is necessary.

Perturbation techniques is a different idea. It applies to both classical as well as quantum theories. It has to do with studying an interaction that you cannot solve exactly as one that you can solve exactly plus a small deviation from it, or "perturbation," and then expanding the perturbed states as a power series of the small perturbation parameter, and the states of the exactly solvable theory. The closest to what you're saying is what @Halc has explained, AFAIK.

It interacts badly with the round brackets also when you want to show the inline maths tags. It may interact badly with other tags too. I think the safest thing when you want to show code is the code button on the toolbar.

Meanwhile in Nepal... Here's some people paying good money to get a dose of harmful radiation, compounded with hypoxia, frostbite, and perhaps more, all compounded by stress from overcrowding. https://www.bbc.com/news/world-asia-48401491 It is not inconceivable that some of the same people would gladly buy those stickers.

@Anamitra Palit, Maybe you can try some LateX editor, like, https://www.google.com/search?q=wysiwyg+latex+editor+online&oq=wysiwyg+LateX+editor+online and then nest your equations with braces, like this: \[ \textrm{ \[ <LateX code here> \] } \] or with round brackets for inline maths. This would considerably improve the communication of mathematical arguments.

That one's easy, and much less convoluted than the story about the CC. It was Maxwell's equations for electromagnetism. He actually didn't think in terms of 4-dimensional non-Euclidean space for quite a while, until Minkowski came up with his formalism of time as a fourth dimension. For Einstein it was always Maxwell's great unification of electricity and magnetism what was the inspiration, and the experimental corroboration by Hertz. From then on, he always thought the equations must be correct, and must be telling us something very deep about Nature. The equations say the speed of light doesn't depend on the observer, so I'll take it to heart and see what the consequences are. Space and time must be changed to comply with these field equations? It must be it. AAMOF, he conjectured gravitational field equations as kind of an analogue of EM --Maxwell's equations-- for curved spaces. But two corrections about the currently accepted formalism --: Light is not ether. In fact there is no ether, as has been proven to death. There is no lack of an appropriate term: Space-time is the accepted term. It has symmetry properties that explain a great deal of how light behaves. The non-Euclidean character is not a property of the field, but of the space-time continuum. Fields are defined on it, so they inherit this structure.

AKA "overcoming the curse of knowledge." I like to think about it in terms of tying the knot, and untying the knot.

That's nice. Only thing is that you're forced to leave out the braces, which maybe you want to make explicit. But you gave me an idea. Let's see if it works: \[ \textrm{ \[ \frac{a}{b} \] } \] \[ \text{ \[ \frac{a}{b} \] } \] Look at that. It does... You only have to escape the math mode inside the LateX with the \textrm{} or \text{} commands.

Oops! Mmmm. Ok. I have nothing to oppose to that. But, as Swansont has pointed out, I misunderstood. I thought you were talking about arguments in the direction that "something can come from nothing", when you were talking about the argument "anything must come from something". These kind of arguments always give me a headache. Please, carry on.

It must have been Lawrence Krauss: https://en.wikipedia.org/wiki/A_Universe_from_Nothing But it's not a postulate. I would call it a philosophical argument on the periphery of physics.

\[ \frac{a}{b} \] \[\frac{a}{b}\] Oh, so that's how you escape LateX. I thought I'd seen it somewhere. Thank you @Ghideon.

A sphere has no poles. A chart does (a system of coordinates). You can map the sphere in many ways. They all have different poles. So spheres have no special points. Neither does the universe, as far as we know. But it is a mathematical theorem that no matter what the system of coordinates you choose, you must always leave a point out, which would be your pole. But which point it is is up to you. Expanding universes have their problems when cosmology meets quantum mechanics. But that's outside of the scope of what you are proposing here. You seem to be thinking of a classical universe, and we do know already that it's quantum.

I use the backslash + square brackets for maths sections, and the backslash + round brackets for inline maths. I don't know how to escape LateX to show the code. Perhaps a "code" tag... \\[ \\] \\( \\) Simple backslash is a common escape tag. But it doesn't work here.

No. I meant the Euclidean norm of both vectors \( \boldsymbol{\alpha} \) and \( \boldsymbol{\beta} \). That's why I wrote them with boldface characters. If you read carefully, I did introduce this qualification: So I meant different things by plain \( \alpha \) and \( \left| \boldsymbol{\alpha} \right| \). I should have been more explicit --and standard-- though, and written, \[ \left(\alpha^{\mu}\right)=\left(\alpha^{0},\boldsymbol{\alpha}\right) \] I'm sorry about that. Nevertheless, the argument stands.

AFAIK, trans fats are known to be harmful on a number of levels. I'm here really to kindle conversation from the experts and learn more. Take everything I say with a grain of salt. Those studies that I remember mostly showed correlations between trans-fats and atheroschlerosis or cancer. Then I learnt from the MIT courses that part of the problem was the shape of the molecules affecting the melting point. What @CharonY says suggests that more research has further clarified the metabolic connections at the molecular level, which I think is very interesting.

This image from Khan Academy is a bit more realistic: https://www.khanacademy.org/science/biology/properties-of-carbon/hydrocarbon-structures-and-functional-groups/a/hydrocarbon-structures-and-isomers The chain of, eg., -C-C- that is usually represented as a straight line, looks as a gentle zigzag, because the carbon atoms in the main chain are really tetrahedral, not linear. The double bonds make this effect of trans vs cis even more dramatic, as you see in the picture. So they differ considerably from the schematic representation. The effect of both hydrogen atoms being on the same side (cis) amounts to both atoms bending the molecular skeleton to the opposite side. The fatty functional group doesn't affect this feature much, unless they're sitting next to the double bond.

I see. Very interesting. The info that I had comes from an outdated source. Those were lectures from 8+ya --MIT Hazel Sive & also Graham Walker, which are on Youtube. What you tell me suggests that a lot of research has gone into studying them in much more detail. Thanks a lot. It also suggests that TFAs are in part incorporated to the membrane after they're metabolized. Would that be correct?

I don't think they're the same. The static universe is one model; the steady state is a different one. It's explained in the reference you provided. The steady state is better known as an idea by Bondi, Hoyle, and Gold. I didn't know that Einstein had anticipated a steady-state model. He apparently didn't publicize it much. Einstein's hypothesis of the constancy of the speed of light dates back to 1905, while redshift of the galaxies was discovered in 1929. So, no. Edit: Even if you consider Slipher''s observations, as Swansont says, it doesn't fit the timeline. OTOH, so-called recession velocities are actually due related to an expansion factor in the FLRW solution to the Einstein equations, which are quite different from local velocities. So, again, no.