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4:31 AM
@Mr.Feynman if you've used $\approx$ symbol atleast thrice in your life
4:49 AM
@nickbros123 $\pi=e$
5:13 AM
anyone know about the alexander quandle
@Obliv I studied neuro and evolutionary bio for 3 years :) scrolling through this paper, it seems you'd just need to know something akin to an organismal systems/evolutionary bio class to give you background. the details you could look up as reading the paper -- usually for bio research, you dont cover everything in class, it just gives you a foundation that can help make papers more accessible, but details still need to be investigated.
maybe smth like this but im not sure bc i didnt skim through it so closely ocw.mit.edu/courses/8-591j-systems-biology-fall-2014
@Mr.Feynman the answer to the question "what is a harmonic oscillator to you"
@naturallyInconsistent BAHHHHHH
5:29 AM
oh, it is time for squacking?
i wish grading could be automated
5:52 AM
That was what MCQ are meant to be for.
I had always looked down at MCQ for the fact that they do not allow students a free-form answer space to tell us much more about how they are misunderstanding a subject, but a colleague of mine pointed out an advantage, that MCQ are quite democratic, since it would not rely upon the student to memorise keywords and tax on their language and handwriting skills more than what is necessary to understand the question.
@naturallyInconsistent bad thing is that if u get super close to the answer but mess up on smth small u get 0 points :(
i like the idea of oral tests but they seem quite taxing on the grader/prof
i also like project based assessments
oral exams would be awesome
om nom oral exam
6:25 AM
@Relativisticcucumber you mean MCQ or free-form? Because if a question is created smartly, whether MCQ or not, it should test concepts and not minor computational mistakes. The wrong answers in MCQ should be suggestive of conceptual errors rather than of algebraic mistakes.
6:42 AM
@SillyGoose in India whatever exam you write, it all boils down finally to an oral exam :)
7:02 AM
@nickbros123 I am sorry for all the dentists I have inconvenienced
@nickbros123 I'm more a $\sim$ or $\simeq$ guy
@SillyGoose all my master's exams are oral :O
7:18 AM
Atomism wasn't really universally accepted until like the... 20's I think?
Although by the 20's only old fuddy duddy really opposed it I think
Hello, if we say that momentum creation operators commute then the state they act on is symmetric, its my understanding that we say these particles are bosons. in the free field theory, it's also my understanding that we are dealing specifically with spin zero states since we can show that acting with the momentum operator on a one particle state with no momentum gives us zero. however, at some point we need fermions in our fock space, right? how is this resolved?
@naturallyInconsistent true i do agree with that but i think my qualm is with giving zero points when not 0% is understood
Although atomism was popular with chemists early on iirc
@Relativisticcucumber This is where I'm particularly happy with giving extra marks for stuff not on the rubric
@naturallyInconsistent kind soul
@Relativisticcucumber What do you mean by "we need fermions in our fock space"?
If you're looking at a theory of scalar fields, there ain't gonna be any fermions
7:29 AM
@ACuriousMind right so is it just that when we deal with non scalar field we will have CRs that allow for both particle types?
for context this is what i was looking at:
@Relativisticcucumber Due to spin-statistics theorem, all spin-integer types will have commutation relations just like spin-0 case, whereas all the spin-half-integer types will obey anti-commutation relations. Boson v.s. fermion automatically comes from that.
i think i might see my confusion. so in qm we have a general hilbert space regardless of potential, but here we have specific fock spaces for individual theories? and fock space needn't contain all particle solutions ?
so when we consider fermions we will have a new theory and a different fock space?
one difference between QM and QFT is that in QM the Stone-von Neumann theorem forces all theories to be essentially on the same Hilbert space of wavefunctions, while in QFT Haag's theorem forces all theories to be on different spaces
to be precise, the SvN theorem forces all quantum theories with the same finite number of CCR to be on the same hilbert space. this means the interaction term of the Hamiltonian doesnt matter in fixing the Hilbert space
this means that both the free and the interaction hamiltonian r defined on the same hilbert space
7:53 AM
@Relativisticcucumber Note that in QM the (skew-)symmetrisation of the wavefunction is an additional postulate that is experimentally motivated but theoretically just randomly tacked onto the mathematical structure. There is no mathematical connections from it to anywhere else in QM. Whereas in QFT the Fock space concept of having (skew-)commutation relations on the field operators automatically make it impossible to excite the quantum field to have the wrong statistics.
The symmetrization is because of the interchange symmetry
A very mathematical fact
iirc it's one of those superselection rume business
in classical mechanics, it seems like a rare occasion to be able to identify the Hamiltonian function with the total energy of the system. However, in quantum mechanics, we (seemingly) blindly identify the Hamiltonian operator with the total energy of the system in the sense that its eigenvalues correspond to definite total energy values of the system. Why is this so?
What else would you mean by total energy
@SillyGoose why would it be rare?
if we take as definition that energy is the thing that is conserved with respect to time evolution, then why is the Hamiltonian ever not identified with total energy
7:58 AM
name a single system where the Hamiltonian is not the energy
(there are such systems, I'm just curious why you claim they're so common)
That's not even true in classical mechanics
it is just my impression from the constraints which must be met to identify the Hamiltonian with the total energy of the system. 1) the generalized coordinates are time-independent and 2) all forces acting on the system are conservative (derivable from a conservative potential). [these constraints maybe do not give an iff statement about when H is total energy. they come from Goldstein's text]
@SillyGoose and how many classical mechanics systems have you met that don't fulfill both of those requirements?
I have not met any. But I have only studied classical mechanics from an introductory textbook which would only present the most tractable problems
"we neglect friction" is probably the most common sentence physicists say
so your second point is usually tautologically true
8:01 AM
Well any system with dissipation would make condition 2) not met
I mean if you only consider time independent systems, the Hamiltonian is conserved, since [H,H] = 0 by definition
@Slereah which statement?
@SillyGoose Systems with dissipation usually don't have a Hamiltonian description at all!
like, you don't even get to the point where you can wonder whether the Hamiltonian "is the energy" because you don't get to the Hamiltonian
the whole construct of Lagrangian/Hamiltonian mechanics is not made for dissipative systems
Well if the claim being made here is that almost every physically relevant system is a situation in which the Hamiltonian is total energy, then I can perhaps accept this and see it through later experience.
there are specific cases that still admit such a description, but that's very much not what the formalism is made for
8:04 AM
@Slereah i think there r cases like the damped Harmonic oscillator where the Hamiltonian is conserved but it's still not the total energy
this ocillator admits a time independent Hamiltonian
@SillyGoose when you get to the cases where the Hamiltonian isn't total energy you will notice because everyone will treat it as noteworthy :P
there isn't any problem here at all, just stop expecting the texts to somehow treat the most general cases with all possible pathologies up-front - it's hard enough to wrap one's head around the simple quantum mechanical systems at first!
Either that or just learn physics directly from here : arxiv.org/abs/1310.7930
what we need to note here is what even allows us to define the energy of dissipative systems with no Hamiltonian formulation. the answer is just that we r defining the energy in terms of the Hamiltonian of a larger system!!
so energy is always Hamiltonian
it has no other. meaning
well this notion disrupts my current framework for defining observables :P
an observable is a self-adjoint operator on Hilbert space
what kind of "framework" did you cook up :P
8:09 AM
since now it seems like defining energy to be the quantity that is invariant under time-evolution is a little bit incorrect
we just take the Hamiltonian of a larger non dissipative system and call one of its terms the total energy of the dissipative system
i should say physical quantities or something (is what I mean by observable)
this is what allows u to differentiate between the energy and the Hamiltonian of a damped oscillator
@Slereah Lol
@SillyGoose sure - there are cases where the notion of "energy" doesn't really make sense at all!
8:12 AM
im just saying that energy of a system is always defined in terms of some Hamiltonian. it can be the Hamiltonian of a larger non dissipative system
for instance we get like one question a month about how the expansion of the universe is incompatible with energy conservation, and the answer is always "sure it is, energy conservation isn't actually a universal law" :P
this isn't a defect in the physical theory, it's just the students who have to let go of this deep reification of energy as something "real" we teach in all the intro courses :P
hm so then how should one state the law of conservation of energy?
@SillyGoose why do you want to state it?
to be able to correctly state it in the future :P
It's the Noether current of time translation
8:15 AM
yes but what do you need that law for?
Energy isn't even a quantity you can directly observe
@ACuriousMind well say solving a problem in an introductory course in kinematics and newton's laws and so on
@SillyGoose Energy in kinematics is just $\frac{1}{2}mv^2$
possibly with some potential terms, depending on how narrow your definition of kinematics is
what's the problem with that?
well i guess my question is
@Slereah this is almost correct but needs to be modified slightly. energy is still defined for systems without time translational symmetry. we just take a larger system with time translational symmetry and define one of its constituents to be the energy of the smaller system
8:18 AM
one states the conservation of energy as "the total (expectation value of/appropriate generalization of) energy of an isolated system is conserved with respect to time evolution". is this "expanding universe" not an isolated system, or is the conservation of energy statement incorrectly stated?
also, the Neother current defines energy only upto a constant. GR defines energy more perfectly
@SillyGoose The statement just doesn't make any sense in the full cosmological context without a lot of added clarifications
You don't need to have symmetry to define the current
hm okay
The symmetry implies conservation of the current
8:19 AM
it's like how people think "mass" just means something in arbitrary geometries (like: what is the mass of a black hole?) but...it doesn't
some concepts just don't straightforwardly generalize or have multiple conflicting generalizations
@Slereah oh. in that case, is the current just the generator of time translations?
and energy is, to me, one of these concepts that's either completely clear in a given context or it's useless
The generator of the time translation is the generator of time translation
if you find it hard to articulate what energy means in a given context, maybe just...don't?
Independent of the Lagrangian
You need to apply it to your system
8:21 AM
@SillyGoose How 'bout this?---"Energy" is always locally conserved in a closed time-independent system.
@Slereah I wasn't going to fixate on this definition because then we have to talk about what even defines a notion of "time" for a system where you're apparently confused about what energy means
Although the Hamiltonian is the generator of the time translation in the phzse space
i think u need to consider a larger system to be able to define the energy of a damped oscillator as $x(t)^2 + KE (t)^2$
this is the energy corresponding to the oscillator if u considered it as part of a larger system.
this still neglects the interaction energy with the environment, but we dont count that as "energy of the oscillator" according to convention
@Slereah the problem is if we're already being this concerned about pathological cases, that's not even really true (remember generally covariant systems have zero Hamiltonian)
like, yes, in ordinary classical mechanics for most systems the Hamiltonian will "generate time translations" but at some point we just have to admit that the formalism is so general that a completely generic interpretation of "the Hamiltonian" or "time" isn't actually possible
gosh i feel i have no framework to hold on to now :P
8:27 AM
Hi everyone. Can you help me with this? The idea of the space lift dates back to the 19th century, and has been exploited by numerous authors of science fiction. Since the 1960s, it has also been considered scientifically. It would consist of a cable of length h and cross-section σ anchored to the earth's surface and held tensioned by a counterweight of mass m at the other end, in equatorial geosynchronous rotation, on which shuttles can be loaded to facilitate the most expensive initial part of space travel.
defining energy in terms of Einstein field eqn has to be the most precise definition. it removes even the addition of a constant
@ACuriousMind Hehe...I remembered gauge theory and put some quotes around the word energy when writing the above text :p
What exactly would L be in this problem?
@ACuriousMind what is the most general definition of classical theories?
8:31 AM
@ACuriousMind Is a sieve on an object the set of its subsets and inclusion maps [assuming it's concrete] or is it more generally all the objects that map to it
@SillyGoose It's a bit odd that I as a notorious lover of abstraction am the one saying this repeatedly, but physics is still an experimental science - many physical interpretations of specific mathematical objects are dependent on what exactly the system is that we're modelling, you can't get physical meaning just from staring very hard at abstract formulae
Maybe you just didn't stare long enough
My impression from "sieves are basically like covers" is that they would be subsets but from the definition it's not so clear
@Slereah I don't really know a lot about sieves
Unless that's what "closed under precomposition" means
i agree with that. but i would ideally hope that one can identify a mathematics object with a physical object. and after a series of judicious identifications, produce a wholly consistent theory in which abstract manipulations of the mathematics lead to correct physical predictions for every manipulation possible (with respect to the definitions of the mathematical objects)
8:36 AM
@ACuriousMind I believe they were invented by Erathostene
@SillyGoose That hope has kind of been crushed since the late 19th century I fear
@Slereah as a result of what?
we would like to think that to be true, but physics involve subtleties regarding measuements, finite precision of measurements and if measurement of the thing u computed is even possible
Would no one like to help me?
@SillyGoose I won't deny that's the shining goal of mathematical physics, but cold reality is that reality is a bit more complicated than our beautiful abstractions of it :P
@SillyGoose The various failures of making theories where the world is made of tiny marbles or secretly made of water
8:39 AM
all these subtleties r brushed aside in the mathematics
Also you have to remember that technically speaking, it is possible to have a physical theory where we remove all the theoretical terms
Ramsey sentences are formal logical reconstructions of theoretical propositions attempting to draw a line between science and metaphysics. A Ramsey sentence aims at rendering propositions containing non-observable theoretical terms (terms employed by a theoretical language) clear by substituting them with observational terms (terms employed by an observation language, also called empirical language). Ramsey sentences were introduced by the logical empiricist philosopher Rudolf Carnap. However, they should not be confused with Carnap sentences, which are neutral on whether there exists anything...
^using this abomination
You could define all modern physics without ever using energy or wavefunctions if you wanted
We don't because that would be awful
@SillyGoose I think you are one of those rare people who want to overgeneralise on what your professors have oversimplified, and that you might benefit a lot from a professor who is willing to engage your kind separately. But first of all, you should agree that an introduction to a topic should NOT cover all the crazy potential headaches. If this is not something you agree to, then you should raise this to meow asap
As for the particular headache you are having over this, it is nice to have a tiny beginner's introduction to chaos theory, because there are standard and nice little examples of "Hamiltonians not equal to total energy". In fact, it is wonderful to just consider the time-independent conserved Hamiltonian that is not the non-conserved total energy case, and immediately learn a lot from it.
@naturallyInconsistent I would agree that an introduction to a topic should not cover everything. But the line between reasonable simplification and being misleading seems very thin in certain cases
(That is classical mechanics, btw.)
I saw the wiki page on Birkhoff theorem. It says that Israel's theorem (loosely which says static+asymptotically flat implies spherical symmetry which implies Schw soln via EFE) is not valid in Newtonian gravity.
I saw the cited articles which says something interesting: Coulomb's potential (interpreted to be an analogue of staticity here I guess) can arise from a distribution which is spherically asymmetric.
8:46 AM
@naturallyInconsistent Hm I shall take a look at this over the coming break
I saw both the proof of Israel's theorem and the thing I just mentioned. Both of them uses stuff related to conformal transformations---but superficially atleast the methods are entirely different! Is there any intuitive reason as to why Israel's theorem do not hold in Newtonian gravity but in GR?
@SillyGoose But this is one of the situations whereby being misleading is probably more than warranted: While in classical mechanics we are often interested in dissipative systems, in quantum mechanics we can get away with Hamiltonian = energy even at the research level. Why should students be forced to see a generalisation that is not necessarily academically-career-helpful to them?
I would guess it's because Newtonian gravity is linear?
Why would that allow more freedom for sources given a potential than GR?
@naturallyInconsistent well I can only answer for myself in saying: seeing the more general statements of things allows you to more clearly see why the particulars are the way they are. in gaining this knowledge, you are now better equipped to recognize a connection between a particular and a more general statement in a novel situation
which is what one aims to do in research (to my understanding)
I guess I see it as: even if the content is not relevant or cared for (for instance, I am not that interested in classical mechanics but all my recent questions have been about it...), everything that goes into formulating and thinking through the content may still be of immense use in areas one is interested in
8:51 AM
let's see the sources on tihs
It's a book
Can't check em at work
@SillyGoose Why not just "barring cosmological corrections" and be done with it? Why must people solve all the most difficult problems at once? In fact, there are plenty of big name physicists, Feynman being one of them, who actively look down on people who try to do that. A good theoretical physicist should always be ok with rules that hold only for specialised cases, generalising when (s)he sees a way to, but content with special rules for special cases if that does not work out. Like Ohm's Law.
I'd say the universe is a pretty good case of an isolated system
I hope so anyway
@SillyGoose None of what we are saying is in disagreement with this. When pedagogy tells us that it makes sense to present things the way you want to, people sometimes do choose to do things that way. But it is often pedagogically insane to try to do that, and then people overwhelmingly won't. And then there are cases whereby no hooman alive knows how to generalise it, then what are they supposed to do for poor students? Not even teach a successful proto-theory?
@Bml In the question as stated, nothing important, just a placeholder name for a specific value of h that is desired by the question.
8:59 AM
I shared some snapshots...
I saw the proof of Israel theorem from Heusler's lecture notes on black hole uniqueness theorems...
@SillyGoose Would you require uniqueness? Or finality?
@SillyGoose Notice that I was referring to an extremely well-known (to people who learnt it; virtually unknown outside of the tiny sub-category of physicists) situation whereby the total energy is extremely obviously well-understood and well-defined and non-conserved and we also know why and how it is non-conserved, compared to a conserved Hamiltonian conjugate to time-translation symmetry.
This is a direct contradiction to the noisy yapper. But let's not attack more. Instead, it is interesting to note that this implies that we can have a much more nuanced view of all these issues. Sadly, it supports ACM's repeated exhortations that physics is an experimental science, and we can only know what is happening by actually going out and observing a lot of stuff, and then pattern match. If you wanted to be maximal generalisation, then you might end up with nothing at all.
In particular, the actual physics seems to be extremely inconsistent, defying simplistic categorisation at all.
@naturallyInconsistent OK, but how to recognize when T is minimum? I've thought that the length h = L for which the tension is minimum is the length that corresponds to the geostationary orbit, where the angular velocity of the cable matches the angular velocity of the Earth. This is because at this point, the centrifugal force balances the gravitational force, and the cable is in equilibrium. What do you think?
@SillyGoose This is correct but it's only one side of the coin: The other side of the coin is knowing which level of abstraction is appropriate for which questions. Some notions simply are no longer appropriate beyond a certain level of abstraction - notice for instance how central the idea of "force" is to Newtonian mechanics but as soon as we discover the principle of least action and Lagrangian/Hamiltonian mechanics, we stop talking about "forces" almost entirely.
energy is one such concept: It's so fundamental to a lot of "ordinary" systems but when you get to the very general notion of generalized coordinates and the evolution parameter of the Hamiltonian not necessarily even being "time" in a fixed sense (due to relativity or otherwise), it gets a bit murky. There is no universal rule that all concepts have to be able to be generalized - some just stop being useful at some rungs in the ladder of abstraction
9:15 AM
@Bml I think you are correct, but you should simply work out the general solution for all lengths h+R > R and then find the specific L that has the minimum tension. You should even check that it is a minimum, and not the boundary between pulling and pushing. Anyway, I am not about to start computing that.
Oh yay today I'm harvesting stars again~
@Bml I'm used to seeing this exercise without the counterweight: as you make the cable longer, the centrifugal force pulling a section of the cable at the end outward grows while the gravitational force on it gets weaker, so there's a point where it's actually pulled outward by the centrifugal force rather than down by gravity. The length of least tension is when the outward centrifugal pull exactly cancels the downward inward pull
A lot of "fundamental" physical notions you will see pop up in history and then completely fall out of favor and never see again
The time span in which energy was considered as a fundamental idea isn't even that long really
Same; the cable ought to have some defined mass density, or else it is a bit difficult to envision how it is supposed to be used.
Although it is still commonly said to be so today
I wonder for how long schools taught that EM fields were just movement in the ether
@Slereah did schools even teach EM? :P
Maxwell only gave us a unified theory of electromagnetism in 1860, and special relativity arrived around 1905
9:22 AM
If we have to abandon bedrock of all science, we will do it. But for now, conservation of energy and momentum and angular momentum, and 2nd law of thermo, are the greatest organising principles of the vast majority of physics phenomena, followed by quantum postulates and a few other basic ingredients, so we postulate them and use them to explain stuff.
that's only 45 years for schools to pick up "EM fields" in a form we would recognize and us realizing there's no aether
@ACuriousMind I have some old ass textbooks from the 19th century
They did teach like electricity and magnetism although not in so many details
Also I don't have like higher education textbooks from the era so idk what university students would be learning
Plus the description of EM as ether motion is pretty old
like uuuuh Descartes old
The EM field can be called the ether in modern physics
@naturallyInconsistent You're absolutely right, but I have not done so so far simply because the next question is: (b) Assuming now that the length of the cable slightly exceeds L, h = L + l with l << L, calculate the cable tension T(l).
"Ether" in the general context is pretty close to meaningless really
9:26 AM
@Bml There is no reason why you cannot solve (b) before solving (a)
The only general meaning I can divine from it is that it's just something that is everywhere and """behaves like a fluid""", with some pretty wide notion of what that means
i mean that EM waves are waves in the field. it's just that velocities dont add according to v'=c+v
It's just hoping that all is secretly water
As per Thales
@Slereah It's all just careless ontology :P
it is accepted today that the EM field is everywhere, and even is non zero everywhere according to QFT
the only difference is that people expected this field to be non relativistic fluid with a rest frame
if u accept lorentz transforms instead, the ether idea is good. the waves r in a field with no preferred rest frame
9:30 AM
@naturallyInconsistent So you recommend finding the general solution for L+l>l and then finding L for which the tension is minimal? Wouldn't that lead to a dependence on l in point (a)?
As William Thomson said it is the jelly
the jelly of reality I guess
^the Aether
the EM waves are "made of the same thing" as the field, i.e. of electric and magnetic fields. the waves r the field itself
> When Zeus had heard the prophecies from his father,
he swallowed the revered one [or phallus], who [or which] sprang forth
first into the aither [or who first ejaculated aither].
@ACuriousMind "The length of least tension is when the outward centrifugal pull exactly cancels the downward inward pull": does this mean that "the centrifugal force balances the gravitational force, and the cable is in equilibrium", or am I missing something?
Talk about fluids
9:34 AM
@Bml that's what I meant, yes (but that also refers to the situation where the cable has mass)
> sublime Ether, best cosmic element,
radiant, luminous, starlit offspring,
I call upon you and I beseech you to be temperate and clear.
It is apparently the best cosmic element
Hence its enduring popularity
@ACuriousMind When the cable has mass is the last question, but I have no idea how to approach it.
some people say that spacetime is the ether for gravitational waves. these ideas r extremely modified versions of the original ether concept
the original concept had a fluid with a preferred rest frame like water does
it was good as a hypothesis at the time. people also suspected particles to be ripples in fluids
how would these ancient people react to modern physics universe ontology
In varying ways considering their very diverse opinions
9:49 AM
some would be thrilled to know that all this is experimentally tested and hence the truth
they were extremely interested in this. and today, we casually know all this
I'm not sure that was that much of an important position for most
@Bml I said h+R > R, i.e. for all general h > 0, and then find out what specific h=L would have minimal tension
Empiricism was mostly a medical school of thought and it was only vaguely used by pyrrhonists
And pyrrhonists were big on dismissing any epistemology anyway
But if they are going to dismiss epistemology, then how would they claim to know that their treatments work at all?
Trying to learn about the human body by looking at what happens instead of winging it
@naturallyInconsistent Pyrrhonists weren't doctors, they were philosophers
Although some were both
In which case you can look up how Sextus Empiricus claims to do anything by doubting everything :p
9:56 AM
But "looking at what happens" is kinda a form of epistemology too...
Sounds very much like winging everything...
yes, as I said it was more of a doctor's thing
@naturallyInconsistent OK, do you have any hints on how to find the general solution of tension?
Well, to be incredibly fair to them, medical "science" was not yet a science until somewhat late of last century. Quite a lot of the prior knowledge was worse than useless, e.g. bloodletting being the most likely cause of death of Washington. If they were more into winging it and epistemology of any sensible kind, it would not have been that bad.
@Bml did we not talk about computing the forces? That is the general solution already. Your rope isn't even having mass and thus won't need integration, it should be really doable?
it took a little while
Although still plenty of good observations back then
Galen was doing a lot of good work considering the era
like still many people today most ancient greeks just thought the fundamental rules of the universe would be obvious if you thought really hard about it
Counterexamples didn't even need to be tested if it passed the vibe check
10:15 AM
some like Zeno, Plato, Newton, Lebnitz would be thrilled to know modern physics. someone like Aristotle would ignore it
I doubt it
and I doubt you read any of them to make such claims
@naturallyInconsistent I am having difficulty understanding the physical situation. I came across this paper (users.wpi.edu/~paravind/Publications/PKASpace/…), but I just can't understand all the variables involved, there are so many of them and I am slightly confused. Can you try and have a look at this?
@naturallyInconsistent users.wpi.edu/~paravind/Publications/PKASpace%20Elevators.pdf this is the correct one
@Slereah I feel this recurrence of non-empirical thought is a bit of a post-modern feature - at the beginning of the 20th century we were assaulted by empirical observations that didn't make sense and forced us to develop new "fundamental" theories, while nowadays new "fundamental" theories should be desperately looking for observations to explain but mostly argue in circles about naturalness and other aesthetics (if they argue about their relation to reality at all)
I dunno, I think that's just a natural way people think about such things
Especially students
also I think it was still pretty common for people to have preconceived notions of how physics work even back then
Kuhn et al would probably say so :p
Like how Einstein's ideas apparently didn't even come from the Michelson-Morley experiment
And how people stuck with GR even when there were experimental difficulties with it in the 1920's
GR was just too neat an idea to drop
10:44 AM
it is also because the new fundamental theories today cannot be tested right now
so people focus more on beauty which is a proven good method
11:30 AM
@ACuriousMind I would also be voting on the side of Slereah on this; people have always asserted, whatever worldview that they hold, that their way of seeing things is beautiful. It is a bit like all the people claiming that god is so efficient in his work that there is a least action principle, only to suddenly shut up when Feynman showed that it is actually maximally wasteful.
@Bml I am not going to be free any time soon, but what even is the problem? It might be full of maths and symbols and is an in-depth treatment, but a cursory glance seems to show that it is having the considerations we were considering and not too conceptually complicated.
@naturallyInconsistent y is it maximally wasteful?
@naturallyInconsistent One issue is the following: if I equalise T(h) to 0, I find the same value that would be found by equalising centrifugal force and gravitational force, i.e. h=(GM/\omega^2)^1/3, a value already known. However, to find the minimum, shouldn't one take the derivative of T and set it to 0? In this case I would find a solution that is negative and differs from the previous one by 2^(1/3). So?
@RyderRude that is a very well-treated part, why are you not already knowing it?
BRST is such a Brst-rrd...
@Bml You are asserting that T=0 in the first place. That is not what you should be doing. You should be expressing T=T(h) for all possible h, and the differentiating to find its minimum.
Also, stop tagging me. I literally just said that I won't be free.
11:39 AM
@naturallyInconsistent i havent read it. pls share it
I literally said I won't be free
@Slereah I think this is very much an intellectual paradigm in the Kuhnian sense rather than "natural" - you're right that this likely stems originally from the Ancient Greek way of doing philosophy, but I don't think that's natural, it's just what the Greeks did :P
I mean, the appeal of this is clear - "My mind is so powerful it can understand the world by just thinking" is much more enticing than "The world's a mess and all our models are preliminary and faulty" - but I don't think it's inevitable
idk is that any different in any other part of the world
From what I can see at all times and place that's the most common way to do things instead of empirically
Empiricism is something you have to force yourself into
We all fundamentally think that it's tiny gnomes and marbles all the way down
I mean for people who think about such things at all anyway
@Slereah Do we? Or do we think that because we're taught that?
i can confirm this. i thought matter was continuous becuz u cud cut it in half forever
11:51 AM
Maybe idk
little children do physics more akin to philosophy
But I'm not sure I see the "let's only assume the minimum we can" idea a whole lot historically
Maybe because it's not that fertile of an idea
The problem with empiricism is that if you're radical about it you start questioning not only the "silly" truths like "everything is made of water" but also the social truths, and almost every society strongly conditions its members against questioning its own foundations
also it's not something that attracts converts even if you aren't persecuted; it's not an appealing idea compared to just believing in what you want to believe in :P
Oh it has had its trends
But it's pretty tough of an idea to apply in full I guess
some people do question society's foundations
12:52 PM
Okay I submit, Polchinski's BRST section is pretty good
The bosonic string theory book?
what happens if u dont have gravitons in string theory? does it become inconsistent
It's like taking the states $|p>$ out of qft
so it is part of the bare definition of the theory, rather than an add-on particle content
y r closed loops more special than other strings
1:04 PM
idk what you mean by more special
Open strings are very special too
i mean the theory cant work without gravitons
QFT doesnt become inconsistent if u remove or add some fields
or maybe string theory just allows any topology branes to be the incoming and the outgoing particles. so no strings can be artificially removed from the theory
since particles r now identified with the topology unlike in QFT where the particle content can be manually tweaked
is this correct
nothing of what you just said has anything to do with how string theory actually works
@ACuriousMind This might be too hard, but from $S = \int d \tau[p_I \dot{q}^I - u^a \phi_a + i b_c (\dot{c}^c + i f_{ab}^c c^a u^b)] = \int d \tau [p_I \dot{q}^I + i b_a \dot{c}^a + u^a (\phi_a + i f_{ab}^c c^b b_c)]$ we find the constraints $\tilde{\phi}_a = \phi_a + i f_{ab}^c c^a c^b b_c$, thus, since $H_T = u^a \tilde{\phi}_a$, shouldn't the $\tilde{\phi}_a$ generate gauge transformations?
i.e. shouldn't $\epsilon^a \tilde{\phi}_a = i \theta c^a \tilde{\phi}_a = i \theta (c^a \phi_a + i f_{ab}^c c^a c^b b_c) =^? i \theta \hat{Q}$ generate gauge transformations? The right answer is that $Q = c^a \phi_a + \frac{1}{2} i f_{ab}^c c^a c^b b_c$ generates them, and $H_t = \{u^b b_b,Q\}$,
but if we believe Dirac shouldn't we directly find the $\epsilon \tilde{\phi}_a$ as the generators, e.g. like we do when restricted to $p$ and $q$ in $S = \int d \tau [p \dot{q} - e \phi]$ for the point particle
(Might be signs and i's off)
@bolbteppa What exactly generates gauge transformations is a somewhat subtle question. You didn't tell me what theory you're looking at, but that there are fewer generators of "actual" gauge transformations than constraints is a common pitfall (this happens because people aren't careful what exactly they mean by "generating a gauge transformation"), see this answer of mine for the case of electromagnetism
1:25 PM
Right, yes it's an arbitrary point particle gauge theory with an arbitrary number of constraints, and great your answer gets the $\delta u^a$ transformation right (in $\delta \lambda^i$ notation) I was hoping that would make sense, in this case the gauge transformations are reparametrizations so the $H_c = 0$ and $H_t = u^a \phi_a$ initially, then I added the FP factor and you see I'm getting this new $H_T = u^a \tilde{\phi}_a$, these generalized constraints are correct
the only issue is that Dirac's logic implies they generate gauge transformations, so the $\tilde{Q}$ I wrote should be the generator, but the $1/2$ is missing, if I write $H_T = \{u^b b_b,Q\}$ and use this new $Q$ that's not what Dirac says at all
@bolbteppa No, a careful investigation of Dirac's argument only establishes that primary first-class constraints generate gauge transformations
For example, for the rel point particle, $\phi = (p^2 + m^2)/2$, and $\delta x = \{x, i \theta c \phi \}$ gives the right answer as expected, as for $\delta p$, so you'd think that using $\tilde{\phi}_a$ in the general case would work, but it is off by factors of $2$ in $\delta b_a$, and only $Q$ gives the right factors
you cannot just add arbitrary first-class constraints to a theory and hope that the generate gauge transformations - the assumption that all first-class constraints generate gauge transformations is called the Dirac conjecture and it's rather easy to construct counterexamples to it
it's an annoying fact that you do need to know more about the constraints - in particular their character as "primary" or "secondary, tertiary, etc." - to establish the Dirac conjecture
and the point of my answer is that even then, your notion of "gauge transformation" may only involve the symmetries of the canonical action, which are a proper subset of all the gauge transformations generated by the first-class constraints for the extended action
but maybe you're actually talking about a third thing - is the notation with the $c$ and $Q$ here supposed to imply you've already added ghosts to your theory and $Q$ is the BRST charge?
Dirac's argument is just that the $u^a$ in $dF = \{F,H_c\} dt + (dt u^a ) \{F,\phi_a \}$ are arbitrary,
$$dF = \{F,H_c\} dt + (dt u'^a ) \{F,\phi_a\} +dt (u^a - u'^a ) \{F,\phi_a\}$$
is just a gauge transformed version of $dF$ and physically describes the same dynamics, at least this is how basic books set it up and I see no issue with it at all but maybe the controversy is worth looking into, the point is I can easily use the $u^a \tilde{\phi}_a$ and conclude $\tilde{\phi}_a$ generate gauge t's
physics.stackexchange.com/a/307315 @ACuriousMind this answer of urs only talks about a closed string. does this mean that the Hilbert space u've constructed comprises of different states of the same closed loop type particle?
1:37 PM
@bolbteppa the subtlety is exactly the primary/non-primary and canonical/extended action distinction, I allude to this in point 4 here, but really if you want to know all the nasty details of constrained quantization there's no way around reading Quantization of Gauge Systems :P
@RyderRude I have no idea what a "closed loop type particle" is
Okay right, I am assuming they are all first class primary constraints, and the $\tilde{\phi}_a$ are also first class primary constraints, so let's assume that. I'm saying that the $\tilde{\phi}_a$ are 1st class primary, so they should generate gauge transforms, thus $\epsilon^a \tilde{\phi}_a = i \theta \hat{Q}$ should, but this is wrong I need to use $i \theta Q$, which contradicts Dirac
is the particle species identified by the topology of the string or its vibrational mode
@bolbteppa Again, is your $Q$ supposed to be a gauge generator or the BRST charge?
because the 1/2 looks to me like the typical BRST charge
which is not, exactly, the same as the statement about first-class constraints generating gauge transformations
it seems like ur answer is from the conformal field theory perspective. u've taken a field $X^{\mu} (\sigma, \tau)$, decomposed it into mode variables, and promoted it to operaroes
so this Hilbert space is different states of the same particle species
$Q$ is the BRST charge, e.g. $Q = b \frac{1}{2} (p^2+m^2)$ in the rel point particle case and it leads to $\delta x = \{x,i \theta Q \} = i \theta p$ and $\delta b = \{b,i \theta Q \} = - \frac{1}{2} \theta (p^2 + m^2)$.
1:49 PM
@RyderRude the particle species are associated with different combinations of the mode operators
@bolbteppa Ah, but the BRST charge is much more than just a combination of first-class constraints - the BRST charge only exists after we have introduced ghosts etc.
Such a thing is usually in the first chapter of any string theory book
you're getting confused because you're trying to map the Dirac procedure for constraints directly onto the BRST charge, but there's literally about half a book between those two things :P
thanks. i will read about it
My point is, before we even mention brst, in the rel point particle, I know $\delta x = \epsilon p$ is a gauge transformation, and I know that $\delta x = \{x, \epsilon \phi \}$ is an equivalent way to derive this gauge transformation, where $\phi = (p^2+m^2)/2$ is a constraint, simply by invoking Dirac.
To go to BRST, in the rel point particle case BRST I simply need to set $\epsilon = i \theta c$. In the general case, I already introduced ghosts above, and on doing this I get the new $\tilde{\phi}_a$ constraints, and they are first class primary too, so you'd immediately think Dirac tells you to use these and form $\delta x^I = \{x^I, \epsilon^a \tilde{\phi}_a \} = \{x^I, i \theta c^a \tilde{\phi}_a \} = \{x^I, i \theta \tilde{Q} \}$, but this fails it has to be $Q$ to get the right factors
(literally just a factor of 2)
@bolbteppa the thing is that "what Dirac tells you" is in a context without ghosts
there are no ghosts in the Dirac procedure
In some sense, it's the purpose of the ghosts to "reduce" the multidimensional symmetries generated by the $\phi^\alpha$ into a single symmetry generated by the BRST charge
1:57 PM
Yes, but the constraints are still bosonic and they are first class primary, the presence of ghosts does not seem to affect anything, and everything works up to a factor of 2, so that may be the problem but I can't see it
it's why they appear from "gauge fixing" in the Faddeev-Popov trick: The ghosts are very much related to the idea of losing a substantial part of the gauge symmetry
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