@Relativisticcucumber hugs, but don't worry so much about other people. I've been there, so I know how it feels - but you and @SillyGoose are both motivated and can support each other! and you have hbar! just do your own thing. not everyone shows interest and enthusiasm in the same way, too.
does it make any sense at all to consider the continuum limit of say, d'alembert's principle, or is there an obvious reason it's not mentioned in texts?
@ACuriousMind I guess I was reading qmechanic's answer here physics.stackexchange.com/a/78149 and was thinking I've seen taking the continuum limit for the Lagrangian -> lagrangian density but not seen it done from his "key identity"
Lanczos actually says things like "sometimes d'alembert is more fundamental" (or some paraphrasing of it) but i only read that in a discrete newtonian context
if it's so key or basic it's just funny that it never gets mentioned again
@ACuriousMind for the first part, is the rationale that if the Hamiltonian remains symmetric under exchange, the Hamiltonian and the exchange operator commute. Then, they share a basis, and if nondegenerate, the basis is unique, so eigenstates must be the same?
but the point of the meme is that the two (or more) things/people/whatever pointing at each other share some kind of unexpected or funny similarity, it's not a metaphor for "pointing at each other" :P
the ferromagnetic 2D ising model takes place on a lattice $\Lambda$ with $N := \lvert \Lambda \lvert$ sites. the (classical) configuration space is $X := \{ \sigma : \sigma_i \in \{-1, 1\} \}$. the Hamiltonian is $H: X \to \mathbb{R}$ defined by $H(\sigma) := -J \sum_{\langle i j \rangle} \sigma_i \sigma_j$ where $J$ is a positive constant.
Is there a simple way to compute the partition function $Z(\beta) := \sum_{\sigma \in X} \exp(-\beta H(\sigma))$?
There seems to be Onsager's original method, Kaufman's method, Baxter's method, and (what seems to be attributed to) Feynman's method. But they all seem not simple.
@SillyGoose this is known in the textbooks; which is why they are all omitted from textbook discussions unless the textbook is specific to ising model itself
@DIRAC1930 it just cannot be the case that the pioneers didn't know that they were working towards a dead end, that something down the road they were working on is a showstopping mistake. It must be the case that everybody after the construction of the Standard Model are doing things wrongly. It cannot be the case that the newer guys learnt more than the pioneers.
@ACuriousMind @Relativisticcucumber actually, it is worse than what you wrote. As far the as spatial wavefunction is concerned, one behaves as "repelling bosons", because Coulomb+exchange interaction still pushes the electrons away in correlation, whereas the other behaves as the usual repelling fermions. It is not "no exclusion effect", but rather two extremes in a highly restricted repelling scenario.
@Relativisticcucumber If it were the other two triplet terms, the spin part would respond to applied magnetic field. That is an annoyance that we do not wish to handle, and it is trivial to see that we can always just pick the 0 term, and so that is nice to pick as the definition.
@Relativisticcucumber The statement is just wrong. It is not that "removing the degeneracy leads to symmetric and skewsymmetric states". Those states always existed and could be taken as your basis (or not). The issue is that degeneracy breaking leads to them being of elevated status, namely that the non-symmetric states are no longer Hamiltonian eigenstates, and obviously, working with Hamiltonian eigenstates bring conveniences to the table.
@SillyGoose CDP concerns itself with a general long range decay behaviour of S matrices, Green's functions, or correlation operators. Things like entanglement are built using the results of the CDP, not beforehand. Like, you should be able to find a much more precise definition of CDP that covers your specific nitpick.
@Relativisticcucumber it is not phrased as "eigenstates must be the same", but that if the exchange operator lifts the degeneracy, then by definition of lifting degeneracy, the eigenstates of the Hamiltonian (and exchange operator) will depend upon the exchange operator's eigenvalue and no longer be independent and free for you to choose any linear combination within each originally degenerate subspace. In particular, you can no longer solve each quantum well alone and expect to get correct results.
@handan_toddler watched the book playlist (that is still being populated). There are some parts that I really agree, but other parts where he is straight up pushing an extremely biased narrative that he hasn't yet fleshed out. Some stuff that I'm quite sure are just straight up wrong. I dont think I'll be able to recommend that book even if I'd want to adopt some of his points.
there was once an email exchange which drifted into this point - i didn't even know this was a misconception people had that acceleration and gravity were "completely interchangeable", but I guess I see how it might arise now
I guess I wanted to make what I felt was an important conceptual point. Maybe I overdid it, but I think it does no harm to present that perspective.
To be honest part of my motivation was a reaction to Duffield as at the time he was polluting the site with answers about GR that are far more misleading than anything I have posted.
a stopped clock is right twice a day. I don't know what else he has said, but his particular point that the equivalence principle holds at a point was imo an important one
> The equivalence principle holds at a single point
But I think this is misleading way to state the EP.
Well, there are lots of subtly different EPs, but the core of the EP is the same for all of them and I din't think your statement captures it well.
They thing to understand is that neither the coordinate acceleration nor the Christoffel symbols are tensors, but in the geodesic equation their sum is.
The point of the EP is that changing the coordinate system changes the coordinate acceleration and the Christoffel symbols in complementary ways i.e. coordinate transformations interchange coordinate acceleration and curvature.
In that sense they are interchangeble.
@qwerty But the statements about the EP being true only at a point lead the reader to the misleading view that it's just about the coordinate acceleration.
@qwerty without mentioning you would have to consider (similar to how it is sometimes stated in newtonian mechanics) that the gravitational field is uniform, i.e. only "at a point" or "approximately in a small region"
you don't need "really" coordinates to do GR unless e.g. you're computing observables... in fact purists would love to write everything in a coordinate-free form. i don't see what coordinates have to do with a purely geometric object like a "point".
@qwerty The pop science statement of the EP is that you can't tell the difference between a gravitational acceleration and a non-gravitational acceleration.
Then the relativity deniers say, yes you can because all gravitational accelerations change with distance i.e. there are always tidal forces, and therefore the EP is false.
Or at least trivial since it only applies at a single point.
@JohnRennie I am actually "kind of" ok with this being illustrated by say, an accelerating rocket (which by default is a small lab). what my comment was referring to as being misleading is when you start talking about "the surface of the earth" without caveat - I imagine the entire globe tbh
the other way is to invoke a freely falling elevator or lift
@qwerty Ah, OK, it had not occurred to me that a reader would interpret the surface of the Earth as the whole sphere. I meant the patch of the surface where the object is of course.
@JohnRennie yes - I think we may have been talking past each other :)
I think the popularity of using trains, rockets, lifts etc (or if you're Galileo, then a ship) to talk about these sorts of concepts is not without merit
@Relativisticcucumber Not only is this guy an excuse for a professor/advisor, but he doesn't seem to even understand that being quick with a reply doesn't mean anything, I mean, it's physics, not stand-up comedy. Demolishing ideas is acceptable in science, attacking people personally, whatever the intention.
@qwerty i suspect you've not helped us fight against crackpots a lot. When fighting perpetual motioners, it is not the right time to discuss having minor quantum violations of the 2nd law. When fighting cold fusion, it is not the right time to bring up uncertainties in low energy cross section data. When communicating the climate catastrophe, it is time to cry.
@ACuriousMind does this follow a logarithmic scale of severity?
@ACuriousMind I see that SSB is already tough by itself but I swear the unholy mess I'm reading in SC books makes me wonder if some people just do things badly on purpose :P
@naturallyInconsistent The purpose of modern texts is to supply the reader with a set of working rules to be able to calculate scattering amplitudes used in colider experiments. If one is insterested in lab based experiments and the applications of qed to it (with an interest in exactly why it works and the inner workings of it), one must look elsewhere
@DIRAC1930 this is manifestly nonsense. We continue to do lab work today. We know a lot more about QFT in general today, and how that impacts the instruction of QED, today than 70 years ago.
plus the terminology is not really that uniform - everyone thinks they mean the same when they say stuff like "this symmetry is a gauge symmetry" but the devil is in the details
@DIRAC1930 Wilsonian renormalisation came from cond matt and immediately got reused for HEP. You are clearly so far away from current research, that one would prefer if you would just stop maligning entire communities
For instance, @HerrFeinmann, another "fun" exercise: Reconcile the standard story of how the Higgs mechanism "breaks" a gauge symmetry with Elitzur's theorem (or more generally the idea that only gauge-invariant quantities are meaningful)
There are many quirks of computational chemistry that is coming from, say, numerical simulations, and it is very different from how we want to conceptualise the subject in general, since we usually want to link to experiments.
@ACuriousMind I'll be honest, I've been thinking about it during the last few days because I realized I never had an understanding of what it means to break a gauge symmetry (even at the physics level of rigor). I can follow the equations in the Higgs case but I never understand the conclusion concerning the breaking of the symmetry
@HerrFeinmann The secret is that the standard presentation doesn't really make a lot of sense and they will sometimes outright admit this by showing some charge operators diverge or stuff like that, then shrug it off and just go "I guess it works anyway" :P
what is "really" going on is to some extent still a matter of debate. You can find people who say that what's happening is the spontaneous breaking of the residual global symmetry after the gauge has been fixed (e.g. physics.stackexchange.com/a/203739/50583), people who rephrase everything in terms of gauge-invariant quantities (Fröhlich et el.) and find everything somehow still works out (in terms of practical results) without assuming a non-zero VEV for the gauge-variant Higgs
and people who invert the whole reasoning and claim the Higgs sector is just a necessary component if you want to formulate a consistent theory of a massive vector field (like a ghost field that does not decouple) and I'm not sure that's an exhaustive list
e.g. linkinghub.elsevier.com/retrieve/pii/055032138190448X is part of a series of papers by Fröhlich, Morchio and Strocchi where they insist in formulating all gauge theories entirely in terms of gauge-invariant fields and quantities first
(it's on my pile of things to actually understand some time :P)
the "we need it for consistency" approach I have only seen in CPT/Epstein-Glaser material like arxiv.org/abs/0906.1952
@naturallyInconsistent yes
but if it is at least possible it contradicts the usual folklore that trying to solve the constraints to obtain the actual physical phase space is hopeless
@Allie Misunderstanding what? I am literally quoting textbooks that have the answer that I am trying to understand. If someone gives an answer that goes against what is in the textbook that I am trying to understand fully, then either they are wrong, or the textbook is wrong. I have nothing to do with this.
i am having trouble with what i think should be a simple counting exercise...how do i count the number of distinct pairs in the bulk of a $n \times n$ square lattice. by "in the bulk" I mean ignoring "periodic boundary conditions" as I can compute these pairs as $n+n = 2n$
i believe the correct answer should be $2(n^2 - n) = 2n(n-1)$
"professor, why do you say xyz" "student, try to find a counterexample" "professor, here's a counterexample" "you just happened to think of the one counterexample there is but trust me im right most of the time" x.x
@HerrFeinmann hey, if it works as well in practice as the others I'm not against it
mathematical physicist: "Hm, I don't think step 3 in your derivation is justified without better bounds on that integral" normal physicsist: "Immortality machine go brrrr"
@ACuriousMind I don't trust first order solutions in physics, why would I with my life?! You sure haven't read FMA (which I already know you didn't :P)
(I also know you don't know it stands for Fullmetal Alchemist)
@Relativisticcucumber This sounds strange because the usual argument is the inverse of this - that the finiteness of life make the time matter more, not less
@SillyGoose i mean living is just not fun to me lol so the thought of doing it forever would surely make me end it all. but i can hang in there or like 80 years
@ACuriousMind but its like "worst case scenario this situation goes poorly and i am sad and eventually i die" so its not like theres an eternal impact thats possible
@Relativisticcucumber Why wouldn't I? I think about life like about a really good book: When you're reading it, you always tell yourself "one more page", and then it's done, and you're sad it's done because you need to find a new one. I would never choose for the good book not to have a next page, and likewise I would never choose for life not to have a next day
I mean from December 20/21th to January 6th, that's what we have in schools and university. Workers only have December 25th, 26th, January 1st and January 6th. I think I should go for academia :P
@ACuriousMind Academia was my original project and then it just trailed off because a lot of things about it and about me were not as I expected. Now I have realized that the easiest job I can find with my background (yes)
alas i encounter another spin singlet triplet issue. back to the case of two regions: left and right. if we have two cases where we have two electrons: case 1) one in left and one in right case 2) two in right. this review is saying the spin triplet splitting only occurs in case 2, but i do not see why this is the case?
i expect singlet triplet splitting to occur in both cases, so the q is why its not present for case 1
