@imbAF I know it's probably not right (I was never that great at QFT) but maybe try treating it as just a complex scalar field? That looks to be what it turns out to get defined as in the end anyway

There's only ever a 1% chance I know what I'm talking about with real qft stuff but that's what I'd try; at the very least it gives you an answer and you can check afterwards if that's what it menat

@controlgroup The line right after Equation (1) explicitly stated that H is a 3-component complex scalar field.

hi

@imbAF You should be able to make an educated guess from the fact that they explicitly say $j,k,l = 1,2,3$ and then write things like $H_k$...

and particularly after the "complex scalar field $H_j(x)$" right after eq. (3) I'm not really sure what's unclear here

that is, sure, you can complain that they didn't define the phrase "N-component complex scalar field", but there is really only exactly one way to interpret what's going on here if you read the entirety of the exercise first

morning

the task of converting a physics theory to experimental data is called phenomenology en.wikipedia.org/wiki/Phenomenology_(physics)

i think, to do phenomenology, one would have to know the physics theory and the physical equipment theory and statistics theory

it sounds like a nightmare to do this. there could be all sorts of errors

theoretical physics is much easier than this. phenomenologists take one for the team

they are like the protectors of science who have the back of all theoretical physicists, making sure their work isn't worthless

@ACuriousMind I did read them and I have a result. Do you mind if you can confirm whether what I got is the accurate result?

For the euler lagrange eq.

@ACuriousMind I can understand the fact that because of the j index you have 3 of such terms H_j. But since it says that the field is a complex scalar field, then does that mean that the field also contains H_j* ?

@imbAF What do you mean by "the field" "containing $H_j^\ast$"?

The statement is that H is a three-component complex scalar field

Normally when we talk about complex scalar field

we say that it contains two dof

$\psi$ and $\psi*$

But I don't think the dof are the same thing as saying "three-component"

I would not say that the complex field "contains the two d.o.f. $\psi$ and $\psi^\ast$", no

$\psi$ is the field, it's not "contained" in it

And what would you say?

what's true is that $\psi$ and $\psi^\ast$ represent independent d.o.f., but I don't know what this "contains" language is supposed to do here - what's the container?

the container would be

the expression that is the field theory

I don't know what that means at all

That expression contains psi and psi*

the Lagrangian?

in the Lagrangian ,we vary these two as independent variables, yes

yes the Lagrangian density

why would you say "the expression that is the field theory" instead of "the Lagrangian" ????

cuz I just woke up

i think it should depend on the Lagrangian tbh whether or not both of these show up

usual, they do show up. i haven't seen Lagrangian without these two

but u can always write an alternative Lagrangian in terms of the real and imaginary parts instead of psi and psi*

in any case, you're deeply confused about the nature of these terms: When someone says something like "$\psi$ is a complex scalar field", they mean that (classically), $\psi$ is a function $\mathbb{R}^{1,3}\to\mathbb{C}$

It's not a statement about any Lagrangian or any specific physics

The statement that $\psi$ contains two real d.o.f. is likewise independent of the physics, since $\psi = \psi_1 + \mathrm{i}\psi_2$ in terms of two real scalar fields $\psi_i : \mathbb{R}^{1,3}\to\mathbb{R}$

Yes I assumed such and with that in mind

I got for Euler lagrange for H_j

I assume that one can have a euler langrange for $H_j*$ as well

But because I am asked specifically for H_j I, of course, ignore the expression I would get for H_j*

It should be accurate what I get.

I'm afraid that's wrong in at least two ways

You didn't apply the E-L equations correctly/did the differentiations wrong

What have I done wrong?

I'm not solving your homework for you :P

Ok

One thing

Why is that the first and the last term in the lagnrangian are in eistein notation while the 2nd is not ?

there is no significance to that at all, you could just as well write $H^\ast_j H_j$ instead of $H^\dagger H$.

$\frac{\\partial L}{\partial H_j}$

This term, from my understanding give ((D_\nu H)_j^\dagger)^\Dagger)

Because it is multiplied with the covariant derivative $(D_\nu H)_j)$ which, as the exercise has contains H_k. I assume that I made the mistake of not writing the terms with which H_k is also multiplied

That would be one of the two wrong ways

@M.N.Raia hi

Hi Ryder

@M.N.Raia are you interested in the measurement problem

So, I'm having a bit of a hard time trying to understand the Blandford-Znajek mechanism

yeah. i read your message. i am not familiar with this

oh....ok =(

@Slereah might be your best bet for that among the chat regulars

@M.N.Raia are you more into GR or QFT or cond matter?

In the exercise that i gave, the gauge field A_\mu has an index a, which goes from 1 to 8. Isn't A the field whose excitations are photons, and it has just four components. What is the a for ?

In the sense that two quarks of different type cannot exchange photons together?

@imbAF There are not 8 quarks, and the $A$ here is not the photon field.

what is it then?

I refuse to believe you could have been given this exercise without having discussed non-Abelian gauge fields in the lecture.

I just want to clarify that this exercise has NOTHING to do with lecture and lecture notes. We haven't done strong interaction, gell man matrices, and Higgs mechanism in strong interaction. So i have no clue

And we haven't discussed non-Abeligian fields

And I can easily prove it by giving the lecture notes

We just mentioned that the Higgs mechanism can be used in such cases

Actually

Is better to just show things, right?

The red underlined part is all we have regarding non-abelian gauge fields :)

You can argue that I can read about it on my own. I am doing that, in order to solve this exercise, for my own personal gains. But as I am reading through different notes that I am finding, there is a multitude of things that I don't understand

very strange pedagogical choice

We only did the Higgs mechanism for the complex scalar field, which from what I get, is an abelian gauge theory

so what you're telling me is that the "gauge field of SU(3)" in that exercise text is the the first time you've seen a gauge field that's not just $A_\mu$ but $A_\mu^a$?

Ever

As I just said

the higgs mechanism was introduced for the complex scalar field

Why would I lie

very strange, but in that case you're not expected to understand anything about what the fields in this exercise represent, I guess :P

This is what we got about the Higgs mechanism. And I also read QFT from mandl and shaw. The corresponding chapter

@ACuriousMind Yes but I would like to understand. That is why I want, at least someone to help me through point a)

@imbAF What I mean is that you shouldn't try any physical interpretation like associating $A$ with photons or whatever.

I want to understand the physical picture (what is the scenario described) and the mathematics. While, I don't know what non-abelia etc ist. I believe I have enough knowledge that will help me understand

You don't need that kind of interpretation to just plug stuff into the E-L equations, which is just what a) wants you to do

I don't need to but I want to. Yes

I can turn off my brain and solve the math (failing at that)

But that is not what I want. I'd like to also have an understanding of the physics.

I re calculated the eom

May I describe what I did, without you giving me an aswer

about what the actual result of EL is

@imbAF I'm not really sure what you mean by "an understanding of the physics". The Lagrangian in your exercise is not part of the Standard Model, it is not part of any theory we would use to describe the real world, it's just a toy model.

But i read that it describes interaction of a

complex scalar field with itself and photon field

and I assume that quarks are excitations of complex scalar fields

Just like in classical mechanics you can be expected to solve the equations of motion for any given potential $V(x)$ without necessarily having a physical description of where or why such potentials occur, you can be expected in QFT to solve such exercises with no additional picture. It's just about mastery of the mathematical tools.

But that is what I read, I don't want to derail from 1a)

@ACuriousMind I see

ok

@imbAF well, you're wrong, quarks are not scalars

the Higgs boson is the only scalar in the Standard Model

the only scalar boson?

Because the covariant derivative contains a term of the form H_k

and because I am considering the EL of H_j and not H_j*

I only focus on (D_^\nu H)_j in the first term

which contains H_k, which multiplies $(D_^\nu H)_j^\dagger$

@ACuriousMind " it's just a toy model." How can you tell just by looking at a lagrangian whether it's a toy model or a thing that occurs in nature? Is it because of experience?

Or are you familiar with all the field equations, and if you are given one, you say that you have never encountered that in nature and hence a toy model ?

@imbAF yes (but generally unless an exercise explicitly tells you otherwise, I would always assume the things you see in exercises are toy models)

The more I read from these conversations the more I wonder about the structure of your lectures

Well luckily the semester is at end, and the first thing I will do is read some suggestions ACM did. Re collect my thoughts and perhaps his suggestions + the books

I will have a less fragmented understanding of stuff

Right now I just want to solve 1a

So i can have an anchor point

a reference

of wtf is taking place

@ACuriousMind Kinda makes sense to go like this

ACM can you tell whether A^\mu which is not a spin 1 massless vector field (w/e other short name there is) is complex or real?

it is a spin-1 massless vector field

Each of the $A_\mu^a(x)$ is real.

@ACuriousMind but its excitations are not photons?

no

ok

@ACuriousMind how can you tell?

He has inside info

because I know what the phrase "gauge field of SU(3)" means :P

Good day evryone

@ACuriousMind I mean, isn't it the same as gauge field U(1), with the only difference the transformation that takes place?

@imbAF I don't know what you mean.

You claimed you haven't done non-Abelian gauge fields

same as = transformation that doesn't change the coordinates but the field value

so how could you possibly know anything about SU(N) gauge symmetry?

which is what U(1) local gauge transformation does

@ACuriousMind well because it's a gauge symmetry

@User198 No

first of all you would have to fix a definition of "Lie group" that also encompasses infinite-dimensional versions for that question to be meaningful

Than I am curious, what is the appropriate Lie group of the Poisson algebra.

@User198 Infinite-dimensional Lie algebras do not necessarily possess corresponding Lie groups

Ah

Ok thanks

a Lie group is a manifold and the theory of infinite dimensional manifolds is ill defined @User198

@User198 but u can consider a subset of Poisson algebra generators like rotations + translations+boosts. these form a finite dimensional Lie Group

u can also consider the Heisenberg algebra

@RyderRude Does that group have a name? That finite dimensional Lie group?

@User198 I wasn't finished ;) In any case, once you fix the definitions so that everything makes sense, some people call the corresponding group the quantomorphism group but it's not talked about a lot

@RyderRude What is its corresponding Lie group?

it is called Galilean group in Newtonian mech and Poincaire group in relativity @User198

these two are different groups

@ACuriousMind Thanks. Sounds cool.

the latter group is extremely important in QFT in flat spacetime

@RyderRude You're wrong, quite famously there is no Galilean group or algebra on phase space.

i mean the Poincaire group

@ACuriousMind i meant the extended phase space like the time dependent phase space

but I was hesitating to include boosts there, yea

without the time dependent phase space, u don't have boosts @User198

u r left with translations + rotationa

on the usual phase space

Or does it just mean that the lie algebra approximates linearly the things happening on te Lie group?

note that a time dependent phase space is a different notion from "phase space with time evolution"

@User198 a lie group is a manifold. u can visualise a manifold maybe

@User198 yes

@User198 please just learn lie groups

a simple example is the group called SU(2). U can visualise this as a three-sphere @User198

u also have SO(3), which is like a 3D ball with antipodal points identified

I can visualise a smooth ball and a plate toutching.

ooh and the simplest one is just a circle @User198

@RyderRude I read that yes

@RyderRude That's not what I mean, and you're wrong again. It's well-known (among those that care about Hamiltonian mechanics) that the Galilean algebra gets centrally extended to the Bargmann algebra when we want to talk about its action on the phase space, see e.g. this post by Qmechanic and links therein.

I.e. there is no Galilean algebra on phase space, but the Bargmann algebra

this is the rotation group. the tangent to a circle is 1 dimensional. So it's Lie algebra is 1D vector space @User198

But I was trying to put that in the context of classical mech, and trying to see if a poisson algebra has its group and maybe visualise that.

@User198 see this answer of mine for that exact question

Ok thanks

@ACuriousMind ooh. i am familiar with this idea too, but I wasn't careful

but I am not deeply familiar with it

@ACuriousMind but u still need the time dependent phase space for this idea, right? u can't have this on the usual phase space

i think the generator here is mX-Pt

it does not exactly satisfy the Galilean algebra commutation relations, which is why it is formalised using extension stuff

@RyderRude No.

but the generator is time dependent?

mX-Pt

yes. The dimension being referred to is the dimension of the vector space of functions on the phase space

@User198 Yes. Because the space of functions $\mathbb{R}^2\to\mathbb{R}$ is already infinite-dimensional.

Ah okok. Thanks.

@ACuriousMind please elaborate

as far as I know, the boost generator is always time dependent, in both Newtonian and relativity

@User198 what subjects are you interested in

@ACuriousMind i went to this link but it doesn't mention the phase space representation of the Bargmann algebra

i think the representation is time dependent

QMechanic also lists this post physics.stackexchange.com/a/104881/156987 . It has the representation

@RyderRude Hamiltonian mechanics, but since I discovered the Poisson brackets I want to study Lie algebra and Lie groups now.

the representation is time dependent. but nevertheless, the answer never mentions the term "time dependent phase space"

@User198 oh

How does the Lie derivative come into play? What is its relationship to the Lie algebra and Lie group?

A Lie derivative is a concept in differential geometry. A Lie bracket is a special case of a Lie derivative

that is one way to look at a Lie bracket

in the usual formalism, a Lie bracket is just a bracket on a vector space which satisfies 3 axioms

@RyderRude Yes, the Lie derivative of the Hamiltonian vector field can be seen as the time evolution of the system.

@User198 u want to say : dA/dt =[A,H] Poisson bracket, right?

yes

this is related to the Lie derivative. i think the right hand side is the Lie derivative of A wrt the Hamiltonian vector field of H

@User198 please see a differential geometry book for Lie derivatives

i think Carroll's or Wald's book on general relativity

i think you are learning things in a top-down approach

you encounter a concept and you go off learning it

$ df/dt = \{f,H\}= {\displaystyle {\mathcal {L}}_{X_{H}}f}$

@User198 The Lie derivative of one vector field w.r.t. another is their Lie bracket (which, on a phase space, is the Poisson bracket of their corresponding functions if they are Hamiltonian vector fields). Vector fields generate transformations/maps of the space they live on (their flows). This is the reason Lie algebras/groups are so interesting: They are abstractions of the relationship between vector fields and flows.

All these things are called "Lie" because Lie considered them first in this geometrical setting.

@User198 Yes!!

But is that connected to the notion of say linear translation that keeps the Hamiltonian (or lagrangian) invariant. Also a symetry right

And that symetry manifests as the conservation of linear momentum.

I'm not really sure what the question is

not all canonical transformations/symplectomorphisms are symmetries

@ACuriousMind Me neither... xD

Ok thanks. That is a lot of info in a short time.

Will examine it all. Thanks for all the references and answers.

@User198 cya

@User198 are you interested in the measurement problem from quantum mechanics and interpretations

Not in particular.

oh

Do you know what is that famous picture representing

which one

What kind of Lie group is that trying to convey?

And where all are Lie groups used in physics?

i haven't seen this picture before but it looks really deep

@User198 groups are used extensively in QFT

@RyderRude hahah, it is litteraly the picture that pops out when you search Lie group

but other fields may use them too

@User198 oh

@RyderRude Can you state a concrete example

What is the manifold in that case?

With respect to what is it symetrical?

relativistic QFT has the Poincaire group for starters. i cant visualise the manifold of this cuz it is high dimensional

u mostly don't bother with the manifold. u just care about the lie algebra

@User198 the action of QFT

@RyderRude Ok thanks

apart from this group, the standard model group is called U(1)xSU(2)xSU(3)

@RyderRude The action of QFT

That is the same action that is in classical mech, at least conceptually?

@User198 there are many quantum field theories. each have a different action. but yes

@RyderRude Ah true!

@User198 yea. we usually try to quantise a classical theory to get a quantum theory

@RyderRude And the same goes in classical mech, we usually don't bother with the manifold, we work with the Poisson algebra?

@User198 a lot of Grand Unified Field Theory approaches are about searching for a group which describes all the forces

@User198 You can go to the Wikimedia page of the image to see it's a graphical representation of the Lie group $E_8$. Mostly it's supposed to look pretty :P

like people investigate groups like SU(5) and E8

@User198 yes

@RyderRude And in this group U1SU2Su3

That is also a manifold, ok. What is its Lie algebra?

@User198 we do care about some global properties of the manifold like compactness and stuff. but we never care about visualising the manifold

Ok ok

@ACuriousMind Yeah xD

@User198 this is the Cartesian product of three groups. the first is the circle group. The second is the 3-sphere i told u about. The third is an 8 dimensional Lie group

Ah ok. And each group has it appropriate algebra

the circle group and the SU(2) group represent electroweak theory

and the SU(3) group represents the QCD

@User198 yes

grand unified field theories are about looking for a pretty group which has all the forces

Einstein was working on this stuff but i doubt he was doing group theory

i mean he was working on a grand unified field theory

@User198 What do you mean by "the manifold"? You seem to be entering mechanics from the wrong end - we don't start with Lie groups or Lie algebras or anything like that, we build mechanics and along the way recognize that a lot of information can be usefully organized in Lie-theoretic terms.

If you want you can introduce Lie theory when you do basic Newtonian mechanics and rotations

if you want you can almost infinitely delay it and never speak of it even while doing non-Abelian quantum gauge theories :P

(usually you do something in-between)

Is seems that there is a switch, from purely analysis and differential equations, to more algebraic approach in physics.

Around what time did that happen?

Mid 1900?

physics is mostly still diff eqns

True

@User198 I'm not sure you really mean the right thing with this, but the "invasion" of group-theoretic methods into physics was pioneered by Wigner and Weyl, among others, and derisively called the gruppenpest

algebra just came long as another thing that is powerful in physics. i think the guy who tried to bring group theory in physics lived in the 1900s and he faced too much resistance

Group theory was called a plague

Hah cool.

note, however, that this was mostly specific to quantum mechanics, which due to its linear structure lends itself to "pure" algebraic methods (for some value of "pure" :P) much more than classical mechanics

It is the same as Poisson, Liouville and the rest of the crew invented math for QM before QM.

Commutators and so.

yes vector spaces were formalised a few decades before QM

Grothendieck was born in 28, after QM

if vector spaces and linear algebra hadn't been there, people would be thinking of Schrodinger eqn as a wave equation

which people did for a few years before Dirac cleared that up

@User198 oh

@User198 it was someone else then

We've talked about this before, the "father" of modern linear algebra is Graßmann, with Peano formalizing it, see chat.stackexchange.com/transcript/71?m=66997691#66997691

yes. I think i confused Grassman with Grothendieck

which is why I deleted it. i wasn't sure

@User198 are you interested in the hard problem of consciousness

@RyderRude Yes I am

@User198 what ideas do you subscribe to rn?( if you can summarise it...)

What is rn?

right now

@User198 they are related. i will try to find links

@RyderRude Than I don't understand your question. To what Ides do I subscribe to? Regarding what?

@User198 this post physics.stackexchange.com/a/51504

assuming you mean path integral

@User198 about a solution to the hard problem

there are some philosophies like panpsychism and illusionism

No. I am more prone to this:

I was reading Wittgenstein, and I don't think we can use philosophy to come to any real conclusions.

nvm It is you saying that, not Wittgenstein

Yes

My interpretation xD

Also I like fallabilism

lol. Wittgenstein also said something similar iirc

like, philosophy is all word games

Yes

@User198 i agree with this. my philosophy is that one can never have a 100% belief in anything

because a Cartesian demon could be tricking u, for starters

Agreed

But I also like this quote from Schopenhauer:

It is about philosophical scepticism

"it is impregnable but its garrison does not pose any threat since it never sets foot outside the fortress."

i have been thinking about some purpose to the universe. like the universe intends for things to happen. and the initial conditions of the universe are just the reflection of the universe's intention.

i think Schelling also expresses these ideas

Meh, idk. Every theory that a person could construct can be further examined, and so on we go into a perpetuum loop of questioning. So I think that we will never know anything.

@User198 what does this mean

The so called "theory of anything" if such thing exists, if it would be constructed, one could question it and so on.

I talked about that in this post

:

@User198 yeah. you can never get behind reasoning. Cuz there is always meta reasoning.

some people say "u can't get behind consciousness". i say " u can't get behind reasoning"

@User198 i am also on philosophy SE. Lemme see

@User198 yes. Feynman's onion is also a relevant idea here but I find his formulation quite naive

@RyderRude Yes, heard of it.

he says reality could be like an onion, with deeper theories always hidden in the layers

Yes

@User198 this is also related to Agrippas trilemma

Yes, that is exactly it

i came across this trilemma a while back. it says explanations can either be infinite regress or circular or dead end

it bothered me immensely back then

It also bothered me

But from than, since I figured out that you can question everything, and there is no "absolute truth".

I just see all maths, physics, philosophy, as a form of art.

my current philosophy is kind of like Kant. We understand nature in a form that our experience allows

there is nothing objective about our understanding of nature

except for some things

I agree

@RyderRude Like what?

I like this one also.

@User198 i like to believe in something like a correspondence theory of truth. that when I make a statement like "this chair is flying", it is true out there in some sense, even tho my understanding of it and my formulation of it is entirely a product of my experience

So kind of like Platonism?

now, i say "true" in some sense but I don't mean true in the sense of classical logic. because classical logic is itself a product of what we are as human minds

I also heard that there are many various notions of logic. And that they are not all compatible with eachother.

Sound interesting, but I didn't look much into that

@User198 nah. i just means that there is an outside world that we are studying. So my theories and my ideas about it are maybe not all a product of my own

@User198 wow

@User198 my idea is more like there are facts (but what they are is incomprehensible to us). They are not facts in the sense of classical logic

but the way we understand anything is entirely interpretational

@User198 yes. There is also intuitionistic logic which is a three valued logic

@RyderRude I agree.

Quantum mechanics does present some resistance to the correspondence theory of truth. and I have been thinking about it

i have a post about this

this one philosophy.stackexchange.com/q/121408/27700

i think the fine tuning is an interesting problem. but we can't apply probability theory to the parameters of the universe, cuz the universe is non repeatable (as far as we know)

so the premise that the parameters are unlikely is invalid

if we assume the universe to be a repeatable experiment, then probabilities do apply. but then we are begging the question

Interesting view.

Didn't occur to me

Cool

i got this from a god debate on youtube :P

All in all

I know that I know nothing. xD

i instead like to think about the problem of the initial conditions of the universe. and I have a formulation of that problem which doesn't use probability theory

Nice chatting with you

I am going to eat

@User198 :)

@User198 thanks for discussing :)

cya

hi