joigus

Oh. Got you. Yes, you're right. It should be what you say. Classic books in QFT tend to be rather fast-and-loose with the indices. \( \partial_{\nu}\mathcal{L}=\partial_{\mu}\left(g_{\mu\nu}\mathcal{L}\right) \) is not a tensor equation. \( \partial_{\nu}\mathcal{L}=\partial_{\mu}\left(\left.\delta^{\mu}\right._{\nu}\mathcal{L}\right) \) is. Although I should say there is no fundamental difference between \( g \) and \( \delta \) really. \( \delta \) is just \( g \) (viewed as just another garden-variety tensor) with an index raised (by using itself). Bogoliubov is similarly cavalier with the indices if I remember correctly. Is it from the 50's?

No, no typo. It is actually a theorem (or lemma, etc) of tensor calculus that the gradient wrt contravariant coordinates is itself covariant. It is just a fortunate notational coincidence that the "sub" position in the derivative symbol seems to suggest that. Proof:

It would. !!! The 1/2 factor doesn't change Lorentz invariance of the metric measure, but it's quite essential to the formalism that comes later. One would think we're done with negative energies/frequencies, and such. But no. They keep biting our buttocks later with the Fourier transform. That's where the Stueckelberg-Feynman prescription for antiparticles comes in.

You want energies to be positive. As k0 (the zeroth component) of the 4-momentum is the energy component, all states must be decreed to have zero amplitude for that choice. That's achieved by the step function trick. You missed a well-known trick for delta "functions"... The delta function satisfies, \[ \delta\left(f\left(x\right)\right)=\sum_{x_{k}\in\textrm{zeroes of }f}\frac{f\left(x-x_{k}\right)}{\left|f'\left(x_{k}\right)\right|} \] for any continuous variable \( x \) and "any" well-behaved function \( f \) of such variable. Taking as your corresponding function and variable both \( k^{0} \) and, \[ f\left(k^{0}\right)=\left(k^{0}\right)^{2}-\left(\omega_{\mathbf{k}}\right)^{2} \] you get, And that's why you need the step function: to kill the un-physical \( k^0 \)'s. Negative energies do appear again in the expansion of the space of states, but they're dealt with in a different manner. This is just to define the measure for the integrals. All kinds of bad things would happen if we let those frequencies stay. I'm sure there are better explanations out there. But the delta identity is crucial to see the point.

Absolutely. It's an erratum. The potential is quadratic, so it's the force that's linear. Close to equilibrium the Taylor expansion of the potential must be quadratic, as at the equilibrium position, the gradient (the force) must be zero. So the next-to-zeroth-order term for the force is proportional to V''(x0). V(x)=V(x0)+(1/2)V''(x0)(x-x0)2+...

Reminds me of this priceless piece of comedy: Sigh

Ahem

Fifty billion flies...

Yes, very much so. I'm re-reading what I said as well as your comment that motivated it. I was kinda losing track of what I was trying to say, and thinking 'why the hell did I mention QFT?' And (after re-reading) I see it's because you said, The reason I mentioned QFT (or the SM as a particular case) is because I wanted to point out that sometimes, even though force and mass are not pillars of the theory, you still have to do a lot of work with this mass, so it's very far from disappearing from most considerations. But it's not like the theory is telling you what this parameter actually is or does.

SM is but one particular QFT. Are you sure you're not splitting hairs here? SM cannot be in addition to QFT the same way the statistical mechanics of an Ising magnet wouldn't be in addition to statistical mechanics. It's given by a particular choice of Hamiltonian within the general procedures of quantum statistical mechanics. SM is QFT under a particular choice of Lagrangian, including gauge groups, global gauge groups, and Higgs multiplets. Unless I overlooked an essential point you made, which is certainly possible, especially of late.

Ultimately it's a major SM issue, I think. But there are very general arguments in QFT in which Yang-Mills pretty much appears as the only interesting generalisation for gauge invariance. So what I mean I suppose is that from QFT to SM there's "just" (ahem) a choice of symmetry groups, generations, and mixing parameters. A very wise expert in QFT nothing fundamentally different from the general principles of QFT conveniently generalised.

That's certainly what happens in GR. In QFT I think the process is much more painful. The theory is not force-based either, but we must start with mass being a parameter that discriminates between different types of fields (massless vs massive). But the physical mass (inertia) becomes more of a dynamical attribute that depends on the state and has to be calculated perturbatively. And there is no explanation for the spectrum of masses.

I should have said Newton's 2nd law, obviously. I have this kind of dyslexic-like glitch that makes me do that very much like @studiot's problem with the typing. Yes, that's exactly what I meant. Now, if all forces of Nature were like that, I wouldn't find it surprising at all. After all, the word "surprise" has to do with contrast in comparison to previous experience, or inference from that. Electricity is not like that, nor any other interaction. Thereby the word "amazed". True. In fact Newton used Kepler's third law to guess his inverse-square law. Textbooks generally point out that the power law is implied. But the equivalence principle is too. The thing is, because the mass on the receiving end of the gravitational interaction (not the mass as a source) disappears from all the physics, it is almost inescapable that the distorsion that a source introduces around it can be described in some geometric way as a distorsion of space-time itself. I think this is amazing even after one learns about GR.

Exactly (my highlight in bold red). This is at no detriment to the use that @Janus and @Genady here in particular (and most physicists elsewhere as well) have given to Newton's third law as an equation. Any definition, identity, or formula can be postulated as an equation the moment one gives numerical values to any of the terms involved, or values in terms of further parameters. So, for example: sin2a+cos2a=1 is an identity. It says something obvious. A further substitution, eg sin2a=1/2 makes it an equation.

They are not defined separately. Not operationally at least. They are inextricably linked, and hidden assumptions operate between both concepts, as I will try to show. Consider, F = ma = (2m)(a/2) = (3m)(a/3) = ... for different test masses m, 2m, 3m, etc we measure to good approximation impinged accelerations a, a/2, a/3, etc for a fixed spring of given elastic constant. This measures F. There is a hidden assumption here. Namely: masses are additive, and so I will be able to stick together identical test objects and assume they operate in Newton's law as twice the mass, three times the mass, etc. Something by no means obvious. In fact we know that to be false from relativity. OTOH, for a fixed m and a fixed direction, we apply different sets of springs by hooking them together (in parallel, so that the spring constants are additive) and measure to good approximation that m = F/a = 2F/2a = 3F/3a =... Mind you, this also assumes something about connecting springs together. This measures m. I don't think this is what one does to measure either mass or force, but it's introduced in books of mechanics from the 50's to the 60's or so. I would think methods based on displacement from an equilibrium position would be more accurate. But I'm not sure. Even in that case, hidden assumptions about mass and force are operating there that boil down to additivity, I'm sure. In fact, the whole lot of Newton's mechanics can be put more simply in terms of F=ma plus a principle of additivity or external transitivity, if you like; and sub-divisibility or internal transitivity, if you like. Then it becomes just one law, instead of three, plus this principle that Newton's laws are to be applied to any level of integrating sub-parts or conglomerates of parts: F=ma (first and only law) 1) System is free (F=0) => a=0 => v=constant (first law) 2) System is free F=0 but it's made of sub-parts 1 and 2. F=F12+F21 => F21=-F12 (third law) And the principle (not so hidden, but explicitly stated throughout history) that certain "magical" frames of reference exist (inertial frames) where the conglomerate of all the parts can be looked upon as free, and ultimately all the dynamics can be analysed in terms of internal forces. Laboratory -> Earth -> Solar System -> etc. The traditional expositions that there are 3 laws, inertial systems, inertia, and the whole shebang are very good to get started, but only obscure these very simple principles IMO, of F=ma (very, very strong assumption that we can isolate interactions into action on something, F, and reaction to that action, -ma, and ultimately bound to be false, as we know); and applicability both to sub-parts and super-parts.

(My emphasis.) There it is. There's your standard of force if you want to make the definition operational. The fixed spring is your standard of force. It's actually inescapable that one needs the other, as F=ma involves both and neither F, nor m, is a primitive concept with a direct observational interpretation, like time or space have. I see no a priori reason to rule out a more complicated mathematical dependence like, say, F=(m+C2m2+C3m3+...)a with m being the additive parameter representing the "amount of stuff", F being our standard spring, C2, C3, etc, very small coefficients under a wide range of dynamical conditions, and the other force laws that we know and love later accomodating this complicated dependence. I'd challenge anybody to provide a robust argument why it cannot be that way. Newton's choice is very sound and very natural, and harmonises wonderfully with symmetries, known behaviours, etc, but I see no a priori reason why it should be that way, and not some other. I don't essentially disagree with anything Janus has said, by the way.

Oh I get it. Is it 'beautiful', 'praised', 'commendable'? I've read different translations.

Well, good point. It's a bit subtler than what I said. I do remember a similar operational definition to what you say in Mechanics by Keith R. Symon. But even for your operational definition of force, you need to set a unit of mass. So it's kind of circular. Mass helps you define force, while you need a standard of force to define mass. They're tied to each other, really. Let me put it in my words: If you think it makes sense to fix a standard of force independent on anything it acts on, to that extent, you can define mass. If you think it makes sense to fix a standard of mass independent of the force that acts on it, then you can define force. It could be more complicated. It could be that there is no way to abstract the 'push or pull' that you exert on a body from the parameters that define the body. Maybe it's something closer to what I called a formula (mathematicians use that distinction, I know). What it is not is an equation, unless you use the formula to plug in numbers and solve for the unknown, of course.

Those were words by Kepler. It's Latin for 'wherever there is matter, there is geometry'. It goes to prove that the idea that geometry held the key to understanding physics has been around for a long time. The factoring out of one of the masses from the equation of motion (or the fact that you could talk about gravitation without without one of the masses not really being there, and the other being replaced by energy, as Genady suggested) is a subtle clue that geometry is at the core of gravity. As to push or pull, I think you mean something about attractive vs repulsive forces perhaps? But then it's not about F=ma vs F=GmM/r². F=ma is the definition of force. It's a definition, rather than an equation really. F=GmM/r² is a law of force, and it has a very different content. It's when we equate both, as @Janus illustrated, ma=GmM/r² that we do have an equation, ie, an equality to be solved. The mere F=ma cannot be solved. Definitions cannot be 'solved'. Not all equalities are equations. This is a common misconception. There are definitions, identities, formulas and equations. Definition: velocity=space/time Identity: x²-y²=(x+y)(x-y) Equation: x²-2x+1=0 Formula: (c1)²+(c2)²=h², where c1 and c2 are the catheti of a right rectangle and h is the hypotenuse of the same triangle A definition is just a labeling, an identity is an algebraic equivalence that's always true, an equation is the expression of a hypothesis to be solved from its statement in the algebraic language, and a formula is an algebraic statement involving ideas that can be abstract, geometrical, etc. There is a long tradition of calling physical laws, definitions (and perhaps formulas) all 'equations', which might be at the root of your confusion.

Ubi materia ibi geometria Einstein took it to the next level. 👍 push/pull is not a physical distinction. It's anthropomorphic. 'Work' is another anthropomorfic term, although extremely useful. But push vs pull is not useful at all in terms of physics.

We should all be amazed that this little m disappears from the equation. It goes very deep in the nature of gravitation.

Not true. That depends on how you prepare the initial situation and what you want to measure. Some outcomes you cannot predict. All the outcomes compatible with the eigenstate you have prepared (AKA 'knowing the initial situation') can be predicted with 100% accuracy. But I have the feeling that something, very much related to information, is becoming quickly unavailable on this thread.

This is wrong. As @exchemist said, entropy does not represent the total information, but only the part that your description "cannot see". Say you have 2 grams of hydrogen. This is about 6x10²³ molecules. The number of coordinates in phase space (positions and momenta) of this system is about 3.6x10²⁴. All of this is information, and this information is never lost. In theoretical physics that's called microscopic entropy, or also volume of phase space. The fact that it is a constant is called Liouville's theorem or "conservation of phase-space volume", or "conservation of information", or "conservation of the number of distinctions". Now, usually you want to describe that sample of matter, not in terms of all the molecular coordinates, but in terms of a reduced number of parameters: pressure, volume, temperature, internal energy, and so on. It is because you can do that for all relevant purposes that this entropy that we use in thermodynamics becomes relevant. It is this entropy the one that always increases. In the case of a BH, something similar happens. The BH is described by just (M, Q, J) --mass, electric charge, and angular momentum. Yet, it must have some microscopic degrees of freedom that account for this thermality. It's not so clear to me what you mean when you say,

Oh well. That probably changes everything then.

This did come across as threatening. I take it that's not what you meant, but it did sound like that. Ok. Yes. That's my personal idea. You say you have no academic background and a prestigious journal asks you to referee for them? Seriously? It's like when a very hot young lady asks me to contact her on FB. Do you seriously think life is just that kind? I'm just trying to be helpful here. And Bufofrog probably was too, although by using humour.