The h Bar

naturallyInconsistent

01:27

@Amit jokeeee but really

Silly Goose

does $SU(N)$ have an infinite number of Cartan subalgebras? just that they're all isomorphic

naturallyInconsistent

@Mad You would then know that the eigenvalue corresponding to $\left|1\right>$ is half, and that is the only thing you may know for now.

@Mad Especially because we have just shown that it cannot be holding in general, if it is a doable homework question at all, we need the context. You need to provide us with the unalterated full question before we can see if we can help you, not small snippets like what you have been giving us. We cannot know if you are giving us a correct picture of what the question actually is.

@Pygmalion no longer visible?

@Mad Aren't mathematicians supposed to be the ones lecturing us physicists on the uses of counterexamples? The result you wish to hold true does not even hold in the 2x2 case, why do you think it will hold in higher dimensions? The 2x2 counterexample is necessarily and even easily generalised to any higher dimension by just including it in a small part of the bigger matrix, easily by the block matrix form.

Silly Goose

01:46

wow i think i discovered an interesting fractal like behavior in such a mundane linear algebra computation :D

naturallyInconsistent

02:18

@SillyGoose fractally complicated?

PM 2Ring

03:32

@SillyGoose You can get chaos from very simple non-linear maps, eg en.wikipedia.org/wiki/Logistic_map

nickbros123

05:29

Thinking of writing my notes like a theorem → proof→ corollary kind of account on EnM

Silly Goose

i tried to do something like that but got awfully bored :P

my attempts always turn into one pagers with the maxwells and definitions to do with dielectrics and magnetized materials

nickbros123

Mathematical preliminaries section might be larger than entire electrostats + potentials+ dielectrics sections lol

I guess unless someone is writing a textbook or making a course it doesn't make sense to write a whole lot of notes explicitly. I just scribble down stuff on the side of the textbook pages itself. Or use sticky notes. Somewhat sufficient

naturallyInconsistent

06:24

I think it is more about teaching myself, that I would want to write it like a textbook. Too many stupidities to iron out, and so would need to go full details.

123

Hello Everyone...

Hi @JohnRennie Sir

Silly Goose

07:30

i think i finally proved this result :D

naturallyInconsistent

@SillyGoose what result? H O N K

Silly Goose

it is classifying all the global minima of a certain cost function

which is why i have been trying to ask about maximal tori lately :P

the solutions are a maximal torus of $SU(N)$

how convenient :D

the result was there for a few days now...but proving it was quite an ordeal :P

Silly Goose

08:25

i think this is a problem in notation...

naturallyInconsistent

08:37

hahahaha yeah

nickbros123

09:51

I was told here that Taylor expansion of dirac deltas aren't actually explicitly Taylor expanding the delta, but the function it's acting on, what abt a direct differentiation of an expression containing dirac delta. Zangwill differentiates $\rho= \sum q_{i} \delta(r-r_{i}'(t))$ so what is he doing here

He even uses chain rule on the dirac delta, internally i died

ACuriousMind

@nickbros123 What's the problem? While the $\delta$ is not a function, one can define derivatives for distributions in general, and of course all the usual properties of derivatives like product rule and chain rule will hold

Slereah

Product of two dirac distributions

ACuriousMind

@Slereah product rule of course only in the cases where the product is well-defined :P

naturallyInconsistent

@nickbros123 games like these on the Dirac delta distribution is how we physicists write down Lorentz invariant integrals in many places.

ACuriousMind

in general, being dogmatic about $\delta$s will only bring you unnecessary pain, see physics.stackexchange.com/q/726654/50583 for longer discussions of this

nickbros123

10:03

@ACuriousMind I don't know about the theory of distribution. Is it defined like those limits on those exponential functions, and hence we can first differentiate then apply the limit?

@naturallyInconsistent I love the dirac delta under the integrals sign where it belongs. Outside not so much

ACuriousMind

10:21

@nickbros123 not really, but once you have defined what distributions are you can properly show that these "nascent $\delta$s" actually converge towards the distribution in a rigorous sense

Relativisticcucumber

10:46

@bolbteppa well i dont really have a definition of them so i can't say if i get them. i have seen the arguments about spinors being projections onto the complex plan so i guess these are spinor "states?" of SU(2)? i have also heard the statement that spinors have the property that they have a double angle relation between "visible space" and "state space". [...]

[...] sorry if these terms are strange, i dont really know what to call these terms, but what i am referring to is that, for instance, a spatial rotation of a spinor by $2\pi$ puts on a negative sign, so you actually need to do a $4\pi$ rotation in physical space to return to start.

But i think theres a lot being shoved under the rug here so im trying to unpack it a bit

it seems like this might be related to a question that i had before which is how you prove that the poincare group has 10 elements and the lorentz group 6, so i was hoping to discover more about that along the way. though i have seen this can also come from that fact that the matrix representation of these elements are maximally symmetric? im still parsing through these arguments though

ACuriousMind

@Relativisticcucumber if you really want to understand what's going on you'll not get around learning about the concept of projective representations

Relativisticcucumber

@ACuriousMind that statement seems loaded XD

is it neutral XD

ACuriousMind

due to the low dimension there's lots of "geometric" ways of talking about spinors in 3d or 4d but the actual reason these things become relevant in physics is because of how the nature of quantum states as rays necessarily forces us to consider such projective representations rather than "normal" representations

and then specifically for Dirac/Weyl spinors you need to learn about Clifford algebras

@Relativisticcucumber well, arguably I'm just trying to get you to read my popular Q&A on the topic :P

Relativisticcucumber

looking now! random question: what's the point of putting a bounty on your own question?

*your own q&a

ACuriousMind

11:03

@Relativisticcucumber I mean, the general point of bounties is that you want to see better answers to the question or reward an existing answer. Why would that change just because it's your own question or you've already answered the question yourself?

Harshdeep Chhabra

11:18

I heard somewhere that research in superconductivity is sort of stagnant currently

wanted to know the validity

of the statement

Slereah

@ACuriousMind How does one classify all the possible scalars you can build from the metric tensor

I vaguely remember that it may be done

But I can't remember what the process is

ACuriousMind

I mean it depends on what exactly you mean by "building", but generally such classifications are essentially finding the multiplicity of the trivial representation in various tensor products

Slereah

I guess the basis of all the scalars built from derivatives of the metric tensor

Might be in Lovelock's theorem mb

Slereah

11:37

The technical term seems to be concomitant

Scalars concomitant to the metric

ACuriousMind

11:52

never heard of that, actually

seems to be a somewhat old-timey term

nickbros123

@ACuriousMind I'll just take your word for it for now

Slereah

It's a 19th century term I am told

From old algebraic theory

Ryder Rude

according to modern philosophical thought, dualism is not very sophisticated

it is an old fashioned idea that a soul is responsible for the feeling of sentience

several old religions like Buddhism push these ideas hard. it cant be completely discreted because Buddha was one of the best philosopher

i wonder on that what basis he hypothesized a re-icarnating soul. maybe his heart just felt that way

Slereah

Boy learning GR just never ends

Apparently every different version of the weighed metric tensor $g \mathrm{det}(g)^{w}$ is concomitant to the metric tensor $g$, and vice versa, except for the conformal case $w = 2/n$

"the conformal metric densities have only the trivial scalar concomitant −1."

There is a whole program of them, if I am to trust the grandiose terminology : philsci-archive.pitt.edu/3005/1/PSA2006.pdf

Although how grandiose can it be if I never heard of it

though Geroch was involved in it

Relativisticcucumber

12:18

rare picture of snowball not eating a textbook

she is getting fatter i think im feeding her too much

but when she sees me with her food she meows so excitedly that i cannot possibly deny her

Slereah

you must remain strong

Relativisticcucumber

plus i fear what will happen to wald and carroll if i take her real food from her

Slereah

Sometimes you must deny the cat

ACuriousMind

@Relativisticcucumber but we can still see a single tooth peeking out ready to tear into any book

Relativisticcucumber

@ACuriousMind exactly this is why i must feed her whenever she asks

or else who knows what she is capable of

Ryder Rude

12:24

there is some new finding that earth is at the center of the universe

i cant find it on google

it was a non trivial observation. was confunding scientists

Slereah

Are you referring to this perhaps

Axis of evil (cosmology)

The "axis of evil" is a name given to the apparent correlation between the plane of the Solar System and aspects of the cosmic microwave background (CMB). It gives the plane of the Solar System and hence the location of Earth a greater significance than might be expected by chance – a result which has been claimed to be evidence of a departure from the Copernican principle as assumed in the concordance model. == Overview == The cosmic microwave background (CMB) radiation signature presents a direct large-scale view of the universe that can be used to identify whether our position or movement has...

Ryder Rude

yes!

Relativisticcucumber

if the relation $V^a_b = v^i\sigma_{ib}^a$ can be interpreted as mapping between a vector in a vector space to a spinor pair in spinor space, then how should i interpret $g_{ij} \sigma^{ia}_b \sigma^{jc}_d = g^{ac}_{bd}$ physically?

ACuriousMind

what's a "spinor pair"

Slereah

Mapping of a tensor to a tensor of double the rank?

ACuriousMind

12:33

what's "spinor space"

sometimes I hate physics and its steadfast refusal to do things properly :P

Relativisticcucumber

@ACuriousMind er the "factors" of a pauli vector? but i might be wrong

Slereah

It's spinors, it's just gonna be $\mathbb{C}^{2n}$

ACuriousMind

@Relativisticcucumber factors?

Relativisticcucumber

yeah like we can decompose a pauli vector into the product of a row and column vector and these are spinors in dual and nondual space?

am i vastly misunderstanding oh no

ACuriousMind

oh god, that approach

Relativisticcucumber

12:36

oh no

ACuriousMind

I have intense personal dislike of that way of approaching spinors :P

Ryder Rude

i think vectors r suposd to be outer products of spinors

Relativisticcucumber

ldfjsdlkfjlsdkjlds

Slereah

@ACuriousMind It's just the square root of a vector

You know it

Ryder Rude

not the product of column and row spinor (this wud just give a number)

@Slereah u r suposd to take the tensor product

Relativisticcucumber

12:38

@ACuriousMind i found your post to be slightly above my current level :,(

or maybe largely above who knows

Ryder Rude

it's like how a rank 2 tensor can sometimes be factored as $v^a v^b$. This is an outer product

ACuriousMind

@Relativisticcucumber Okay, so we're looking at spinors in 4d, right, and you're talking about writing some vector $V^\mu$ as $V^\mu = \xi^\dagger\sigma^\mu \psi$ where $\xi$ and $\psi$ are 2-component spinors (=Weyl spinors)?

Ryder Rude

similarly a rank 1 vector can sometimes be factored

Relativisticcucumber

@ACuriousMind yes i think so

ACuriousMind

@Relativisticcucumber now you want to look at something strange - your indices on the $g$ translate to something like $g_{\mu\nu}\sigma^\mu\otimes \sigma^\nu = \sigma^\mu \otimes \sigma_\mu$, which I don't really remember seeing anywhere else - why are we looking at this expression?

Slereah

12:43

is that the metric tensor in the spinor rep

I've occasionally seen it

Relativisticcucumber

i swear i saw this in wald but i dont have the book on me right now to be more specific

ACuriousMind

ahhh

yes, Slereah, I think that's it

shouldn't some of these indices be dotted or something?

Slereah

Ideally yes

but we live in a fallen world

ACuriousMind

really indices are cool and all but when spinors are involved I need to think in index-free notation or I lose track of what's going on completely

Slereah

just imagine that it's abstract index notation

ACuriousMind

12:46

arrrgh

really the worst of both worlds

pretending to be abstract but completely isomorphic to the "more concrete" notation

Slereah

I think so far the greatest aspect of my book is helping me to keep track of papers on various topics

I just dump a bunch of them in the bibliography and pepper them at the appropriate chapters

I should include some ViXra paper to spice things up

Let's see the coolest GR paper on vixra

Ryder Rude

i like the abstract notation

Slereah

" Einstein had determined the speed of light for the speed limit and even offered a reductive formula. For the relative speed definition, the speed of light limit applies. In this study, the "hypothetical relativistic" speeds are defined and the limit of this speed is shown to be 2c, and a false notion in cosmic analyses and CERN is clarified. "

Ryder Rude

abstract notation makes u feel that u r not manipulating components

when we say that the speed limit of information transfer is c, it makes sense for particles. how do we interpret a limit to the speed of information transfer in a theory of just fields?

Slereah

13:03

Various ways

Ryder Rude

tell

Slereah

Velocity of a compactly supported field is an example

Ryder Rude

so the geometric optics limit

Slereah

Not quite no

Ryder Rude

i mean a wave packet's group velocity

Slereah

13:05

It is neither the group nor phase velocity here

It is what's called the front velocity

Ryder Rude

oh

Slereah

You can also use the group and phase velocity, although with the usual issues that it presents

ie fake FTL signals

Ryder Rude

i thought only phase velocity did that

Slereah

Both of them can

that is why people had to come up with new notions of field velocity

Ryder Rude

is it bothersome that the Green's function does not vanish outside of the light cone? this cud b interpreted to mean that if i change the field strength at one point, its effect propagates outside the light cone

Slereah

13:07

the Green's function of what

Ryder Rude

of say Maxwell's equation

Slereah

It does vanish outside the lightcone

Ryder Rude

the Green's function falls exponentially outside the light cone

but it non zero

Slereah

In most context the Green's function is very much exclusively on the light cone

You may be thinking of the two point correlation function of QFT, possibly

Ryder Rude

yes. but that's the same as Green's function

Slereah

13:09

Do I have to get the diagram out

Ryder Rude

so if i change th current density at a point, the Green's function propagates its effect on the EM field to outside the light cone?

@Slereah i will hav to look up front velocity

@Slereah pls do

Slereah

there are many contours for the green's function

Ryder Rude

idk what this diagram means but Green's function is the same as two point function

naturallyInconsistent

@Slereah So many Green's functions! takes notes on the notation

Slereah

Mr. DeWitt shows that the symmetric Green's function of EM is indeed zero for spacelike intervals

Symmetric Green's function depends on the Feynman one and some other one which I don't remember the name

The QFT one is usually the Feynmann Green function

Ryder Rude

13:22

yes. but the propagator is the unadulterated Green's function

Slereah

but what does the propagator mean

Ryder Rude

this symmetrization argument is also used in vanishing spacelike commutators in QFT ,i think

Slereah

Also idk what "unadulterated Green's function" mean

Why do you think this specific contour is the real one

Ryder Rude

@Slereah it's the Green's function u wud use in solving the diff equation for the 4-potential

i think the eqn is something like $\partial _{\mu} \partial ^{\mu} A^{\nu}= J^{\nu}$ iirc

so if u change $J^{\nu}$ at a point, the effect wud propagate according to the unadulterated Green's function

it's the one that doesnt vanish outside the light cone

naturallyInconsistent

@Slereah you know he does not know enough to begin having this conversation with you, right?

Slereah

13:27

but then why does he

Ryder Rude

by unadulterated Green's function, I mean the inverse of the differential operator

Slereah

All of these are the inverse of the differential operator

Ryder Rude

it looks like $\frac{1}{p^2-m^2}$ in momentum space

naturallyInconsistent

@Slereah That is not much of a mystery. Instead, what is a mystery is why do so many of you smart folk fall for it?

123

Hello Everyone...

I didn't learned Calculus of Variations intensely . You all learned it deep. Pls suggest what you think is it possible to show calculus of variations geometrically?

Slereah

13:32

If you mean geometrically in the sense of bundles and such then yes

Ryder Rude

@123 what do u mean by "showing it geometrically"?

Slereah

there are big old books on the topic

ACuriousMind

@Slereah I highly doubt they mean that :P

123

@RyderRude can we plot graph or curves of the topics of calculus of variations.

Slereah

Well if you mean like Euclid and such, then also kinda I guess?

Like that is probably how the original calculus of variation was done, IIRC

123

13:34

@Slereah Yes in euclidean plane or space

Ryder Rude

yes. when u first learn the principle of least action, they would plot a graph of the trajectory

Slereah

With Euler and Lagrange

123

@Slereah yes

Slereah

I'm not sure that's a super common way to do it these days, I haven't really seen much of that type

Ryder Rude

minimising the action is like taking a dot product to zero. but it is a dot product of infinite dimensional vectors

Slereah

13:35

But IIRC there are for instance geometric proofs of a straight line being the smallest length between two points

Ryder Rude

so i suppose that could be thought of in terms of infinite dimensional manifolds

Slereah

Which is a geometric proof of a variation problem

ACuriousMind

@123 Why do you think calculus of variations has something to do with geometry?

(not saying it necessarily doesn't, but what are you thinking about here?)

Ryder Rude

i mean each trajectory could be a point on that manifold, and the action could be a scalar function on that manifold

ACuriousMind

I guarantee you that infinite-dimensional manifolds (which are indeed one possible setting for this) are not what they mean

123

13:39

@ACuriousMind I think if we can plot functionals between two points. and the shortest path between these two points can be defined as geometrically. What do you say?

ACuriousMind

I don't know what that means

A functional takes another function (e.g. a path) and assigns a number to it

I don't know what it means to "plot functionals between two points"

Ryder Rude

@ACuriousMind yes. i think there is some subtlety with infinite dimensional manifolds. we discussed it before

ACuriousMind

functionals don't operate on points

123

@ACuriousMind Is it possible to create two different euclidean planes like x-y axes for function and x'-y' for functionals. Does this way work?

Ryder Rude

plus here we seem to need an uncountably infinite dimensional manifold

ACuriousMind

13:40

@123 I don't know what that means either

Ryder Rude

unless we are discretizing the trajectory...in which case we just need a finite dimensional manifold

Slereah

The OG variational problem from Herr Euler :

hal.science/hal-01484262/document

123

@ACuriousMind Pls clear me at my level of understanding, what is the problem defining it geometrically. You all are high level of knowledge but i don't.

ACuriousMind

@123 I cannot explain anything here because I don't understand what your idea is at all

Slereah

Or if you can read Latin it's Institutionum calculi integralis volumen primum for the original

123

13:42

@ACuriousMind I asked this question here. Because if there is any possibility to do this. I will definitely create simulation on calculus of variations.

Because geometry is power tool in understanding the phenomenon. showing changes in parameter real time.

ACuriousMind

The spaces functionals act on are infinite-dimensional (the space of functions of a single real variable is infinite-dimensional), you cannot draw them in any meaningful sense. If you just draw a "finite-dimensional part" of what's happening you'll find you're just drawing functions

like, if you want to draw what the "stationary point" of a functional is, you have to draw something finite-dimensional, and that'll just look like the stationary point of a function (e.g. the extremum of $x^{2n}$)

123

@ACuriousMind Is it possible to draw a curve of any simplest functional. $\int{F(x , y , y')}dx$

Ryder Rude

tbf simulations are discretized anyway. but even then, u wud probably need more than 3 dimensions which is impossible to visualise like usual calculus

ACuriousMind

@123 only if you have infinite-dimensional paper

remember, for any function $f(x)$ where $x$ comes from a n-dimensional space, the graph has to be drawn in a n+1 dimensional space

123

@ACuriousMind :-(

ACuriousMind

13:46

since functionals take arguments from infinite-dimensional spaces, their graphs are infinite-dimensional, too

Ryder Rude

if u discretize and ur end points are not at infinity, u still need too many dimensions than is possible to visualize

123

@RyderRude If is it 4 dimensional we can visualize as we do in imaginary functions using 3D space and 4th variable as color. otherwise we can create two euclidean planes.

Ryder Rude

u can make it 3 too if u want. it depends on what approximates ur trajectory

123

@ACuriousMind It means for calculus of variations we only depend on methods and numbers. we can not create graph of it by any means

Ryder Rude

perhaps u can simulate 3-point discretized approximation of calculus of variations if u feel that wud help u understand it

u can use color for the action then

the point is that u can only visualize crude approximations of calculus of variations

ACuriousMind

13:49

@123 if you want to visualize it, just think of ordinary functions

but as I said you will never be able to "draw" the difference between the calculus of functions and the calculus of functionals

123

@ACuriousMind Ookay. It means calculus of variations is totally non geometrical. we just rely on method we know from calculus of functions to find extremes?

ACuriousMind

I mean, I wouldn't say it's "non-geometrical"

but the notions of "geometry" involved are more general than "stuff I can draw on paper"

what we call geometry in math today hasn't been about stuff you can necessarily draw for a long time

Slereah

You can kind of do it in a geometric way, but only specifically for geometric variational problems

ie if the variation problem you're doing is about lengths, volumes, etc

123

Yes i want to draw geometry or use calculus of variations for physics problems only

ACuriousMind

sure, if your functional is something that computes lengths of paths you can draw how the extremal path and its neighbouring paths look; but unfortunately the main physics functional - the action - doesn't have such a geometric meaning in general

Ryder Rude

14:02

i personally dont feel the need to draw anything to intuit calculus of variations, because it's practically just usual calculus in a very high dimension

@123 If u r comfortable with differentiating >3 variable functions without being able to visualize it, then u wouldnt have problems with calculus of variations

123

@ACuriousMind What is the main purpose of introducing calculus of variations in physics?

Ryder Rude

i havnt seen the calculus of variations used much other than to derive Euler Lagrange equation

Slereah

Also used in continuous optics

123

@RyderRude Ookay. Calculus with more than 3 variables just base on basics idea of one dimensional. We just hypothesize the idea in more than 3D.

@RyderRude You are saying COV only used in euler-lagrange equation.

Ryder Rude

@123 there must be other uses too like Slereah mentioned. but it's not used all the time

123

14:09

@RyderRude It means main use is in E-L equation.

Ryder Rude

yes

and also path integrals in QM get counted under calculus of variations i guess

123

@RyderRude If i don't read Lagrangian and Hamiltonian mechanics. Is there any thing important which i will miss in understanding physics?

ACuriousMind

I mean you won't be able to understand most of modern physics :P

Ryder Rude

@123 They are great for transition to advanced physics like QM and Relativity. other than that, they are great for understanding the role of symmetry in physics (Noether's theorem)

and Lagrangian is great for problem solving too like in constrained systems

123

@ACuriousMind Ooooh.. LM and HM don't support geometry. And i found geometry powerful tool. I learned a lot from geometry.

Ryder Rude

14:14

@123 yes. an action is a function from a path between two fixed points to numbers. now if u discretize the path into a finite bunch of points, the action just becomes a usual function in a larger number of variables. u r just differentiating this function in deriving the Euler Lagrange eqn

123

Is it necessary to learn COV separately with mathematics books for used in physics. Or COV in physics book enough to understand just physics topics

ACuriousMind

learning calculus of variations from the mathematicians won't help you understand what the physicists are doing :P

123

@ACuriousMind Ooooh.. It means physics book is enough to understand LM and HM

Slereah

If you find a good one

I remember seeing a cool geometric proof once that the circle is the best perimeter to area ratio of any shape

123

Pls suggest the books which have good understanding of LM and HM start from elementary level.

nickbros123

14:21

@123 damn, you are saying my investment into gelfand and fomin is a waste?

Meant to @ACuriousMind

bolbteppa

@Relativisticcucumber As ACM said, 'projective representations' will really help with this, the start of the Wald spinor chapter sets up projective representations. Projective representations become important for groups that are connected but not simply connected, which is the case for the rotation group and Lorentz group (component connected to the identity)

Try skim the start of the Wald chapter and look out for that stuff and comments about connectivity

@123 Most people have enormous trouble with Lagrangian mechanics because they didn't study calculus of variations independently first, Gelfand is the best book on this

Relativisticcucumber

okay i will try that. thanks! @bolbteppa

123

@bolbteppa It means first i need to study COV book as u suggested Gelfand

bolbteppa

I don't think Wald justifies things fully e.g. the link between SL(2,C) and the Lorentz group he sort of states it iirc, then justifies it after the fact with that link between vectors and matrices which is a bit messy but most books do it this way

@123 You don't have to, but it's similar to trying to do Newtonian mechanics while knowing bits and pieces of calculus without having a full course in it

123

Relativisticcucumber

14:25

@bolbteppa i think my issue with that chapter at first glance was that is was just a lot to take in and hard to follow but i think ill try again with some of the concepts having been exposed to me now

bolbteppa

Elsgolts is very good too

123

I have these books on COV. Pls suggest the good one. Which start from elementary and goes step by step in rigorous. Similar as i love K&K in physics.

bolbteppa

Do Elsgolts and Gelfand together, you really only need to do maybe 2 or 3 chapters and you have enough for Lagrangian mechanics, the multivariable chapters will help with lagrangian electrodynamics and classical field theory, but again most people just wing it with these subjects so only do it if it will help

123

@bolbteppa Thanks for suggestion.

bolbteppa

I think the NPTEL COV videos might follow Elsgolts or be close to it if you want to check

naturallyInconsistent

14:52

What is more common in what I see is that, just by me writing and presenting the physics notion of CoV in very high detail, with enough rigour that the student does not get confused of what is being done, and writing down the central insanity that is differentiating w.r.t. a derivative, they can understand how E-L equations drop out of it. Yes, I do draw the extremal paths and the perturbations too. For simplicity, I use the "scalar parameter multiple of perturbing path" so it is old calculus

Relativisticcucumber

15:32

Ok schwartz spinors section seems promising

i pray

@naturallyInconsistent meow

naturallyInconsistent

@Relativisticcucumber jigglypuuufffff

Relativisticcucumber

16:21

in the statement "a particle can be defined as a set of states that mix only among themselves under poincare transformations", what does "mix" mean?

ACuriousMind

@Relativisticcucumber it's saying the set of states forms an irreducible representation

Relativisticcucumber

@ACuriousMind ok and it says this means that no subset of states transform only among themselves but i am also not sure what this means. because the definition for representation here is that a set of objects that mix under a transformation group is called a representation of the group but this is not really in line with the math definition i thought i knew

about linearizing action of group elements?

Slereah

Consider the case of two particles for instance

Relativisticcucumber

or more specifically a homomorphism between a group and the general linear group on a vector space

Slereah

(I'm using the classical case because it doesn't change much)

ACuriousMind

16:29

@Relativisticcucumber the vector space is the space of states

Slereah

Your physical system is something like R^6

ACuriousMind

there's a representation of the Poincaré group on it

the claim is that an irrep under the Poincaré group is exactly the (sub)space of states of a single particle

Slereah

But the transformation of those particles under rotation splits between those two particles, your system is like $R(x_1 \oplus x_2) = (R x_1, Rx_2)$

ACuriousMind

the physical intuition there is that the number and kind of particles that are around doesn't change under Poincaré transformations

Slereah

The result isn't gonna involve mixing elements of $x_1$ and $x_2$

If you try such a split with say, the elements of a vector, that's not gonna work

You could try to write your vector as three particles with values in R, but they will mix under rotation

Relativisticcucumber

16:40

Oh no

ACuriousMind

oh yes!

Relativisticcucumber

@ACuriousMind i dont understand this. bah. when you say subspace of states of a single particle i dont quite get what this means

sorry for big confusion

do you mean subspace of states that a single particle can take on?

ACuriousMind

@Relativisticcucumber I mean if you're looking at this kind of definition you're reading about QFT, right?

Relativisticcucumber

this is my first page of qft so im new to it but yes

ACuriousMind

did you stop reading after the first page? this is one of these cases where I'd expect the text to explain what it means exactly in the rest of the text/chapter

Relativisticcucumber

16:44

i read the section but i felt i had too many concepts muddled and they were compounding D:

ACuriousMind

I mean that's normal?

Relativisticcucumber

oh no

ACuriousMind

if you felt you had understood QFT after reading the first chapter of a QFT book I'd be more worried :P

Relativisticcucumber

LOL

okay ill press on a bit and see how the chapter goes and then return

bolbteppa

17:35

Schwartz talks about projective representations somewhere, it may be later in the book

nickbros123

17:53

@naturallyInconsistent are you talking about the $ \epsilon \delta y(x) $ variational function controlled by the $\epsilon$ parameter?

I stopped reading Lagrangian mechanics after I saw this. Very simple thing: the dude was trying to show me that the path of least distance is indeed straight line, thru variation of the length functional. But it was when he wrote down this variation in path did I totally lose track and decide lagrangian stuff is for later

Soumik Mukherjee

18:08

Hello, is anyone here with a knowledge of magneto hydrodynamics?

bolbteppa

Given a path $y(x)$, you want to vary it into some nearby path $y^*(x) = y(x) + \delta y(x)$, in order to take a limit explicitly it makes sense to include a parameter $\epsilon$ in there, $y^*(x) = y(x) + \epsilon \delta y(x)$, now in $S = \int L(x,y,y')dx$ we can work out $\delta S = \int [L(x,y^*,y'^*) - L(x,y,y')]dx$ and you'll get the EL equations. If $L(x,y,y') = \sqrt{1+y'^2}$ you you have the length functional and the EL equations reduce to $y'' = 0$ giving $y = ax + b$

nickbros123

18:26

Well when I started learning this I was skeptical as to how one could write down a small variation in the function as some $epsilon$ times another function that obeyed some boundary conditions. Also, it was a time in which I knew not a single word in multivariable calculus (not even divergence). Also, I didn't understand (still don't) about phase space and conjugate variables. The 2nd page of any lagrangian mech book turned me off and showed me my place

Pygmalion

@naturallyInconsistent The question was closed, so I deleted it.

Slereah