2:26 AM
I guess projectors are unitary on their own subspace and not on the rest

2:50 AM
hello
why do physicist treat electric field as fluid
I mean why use analogy of amount of volume of water flowing through an area per unit time
velocity V multiplied by area
and for electric field use $E\cos(\theta)\Delta A$

3:15 AM
it's just an analogy, we don't "treat it as a fluid"

3:40 AM
no
now I understand it is because electron flow on lines
likd how when you put object on river

1 hour later…
5:07 AM
0

This question was recently closed: What is the expression for the single-particle energy of an orbital of a nucleus? It is true that the question has a misunderstanding due to ignorance. I took the trouble to answer it because it seemed to me that the misunderstanding might be common and the answ...

5:49 AM
Good Mornin everyone!

@BannedUser there are analogies to be had, e.g., the mathematical correspondence between 2D electric fields and 2D fluid flow
but that's simply a result of them satisfying similar equations. doesn't mean we think of electric field as having 'substance' in the way that we do a fluid
and in 3D i don't think the analogy works at all. with 3D fluid flow you really don't talk about point sources/sinks
i guess it's not such a big deal to ignore that tho

6:06 AM
Right, even a fluid is itself described by 'something else' too: the fact that an EM field or a fluid are defined everywhere in some region is why they are similar, a continuously infinite number of degrees of freedom, one at each point, where you can attach an arrow at each point giving the direction it's moving in, thus they are both described by vector fields
I have a beyond shameful understanding of fluid mechanics

plus, it's not really correct to say that 'objects move along streamlines'
that's only true if you can ignore the mass of the object
luckily, that is often a good approximation to make

That Stokes' law thing still gives me nightmares

i don't think that by itself is so bad
trying to understand the transitions between various drag laws
noooooo thanks

6:20 AM
Hi All...
Hello @JohnRennie Sir
If we have two different but same looking bodies, the value of moment of Force on first body about point is equal to the value of couples on second body about some point.

"different but same looking"?

Do both bodies have same amount of rotation?
@Semiclassical Here just i wanted to on one body we apply moment of force.
and other body we apply couples on other body.

that's not what i was getting at. what are "different but same looking" bodies?

So that's characteristic of body does not effect my question.
my question related to moment of force and couples.

mmkay

6:27 AM
@Semiclassical Put another way first we apply moment on body initially at rest. Then put body again rest and then apply couple on the same body. But values of moment and couple are equal
Do the body have same amount of rotation. Whether we apply moment or couple if the value is same.

i'll note that there's a language barrier here. i'm not conversant with the moment/couple terminology; in physics we tend to use the language of forces/torques

I am reading mechanics book. Principle of Mechanics Synge Griffith.
In that moment of force and torque is same with two different names. In which single force F apply on body about some reference point O. The perpendicular distance from O and line of action of force is r. Then moment of force M = r x F

so in the second case you apply a set of forces which apply zero net force but some net torque
in the first case you just apply the net torque directly
i can't see how there would be any difference in effect, though
same moment of inertia, same net torque, so same angular acceleration

Couples : Two equal and opposite forces (F, -F) apply on the body about some reference point O. $r_1$ position vector from O to F, $r_2$ position vector O to -F, $r_1 - r_2$ position vector between F & -F.

the wikipedia page for Couples allows a system of forces more generally, in such a way that there's zero net force but still some net torque

6:34 AM
The Couple $G = (r_1 - r_2) \times F = dF \sin\theta$

\times

@Semiclassical Thanks

unless you're one of those weird people who uses $\wedge$

3 hours later…
9:26 AM
morning

10:06 AM
Sometimes I just want to entirely rewrite Synge's book with LaTeX
Terrible notation

10:33 AM
I should read a basic differential geometry book because I'm pretty sure most GR books that talk about the worldfunction do it poorly
It seems to be the distance function in more Riemannian territories

the thing is, in the Riemannian context you can define distance via a minimum/infimum
you can't do that in indefinite signatures, and Synge's definition works only for points connected by unique geodesics, so "generic" Lorentzian geometry won't care about this function because it can't even define it

Yeah but I think some Riemannian books use that function?
$$d(p, q) = g(\exp_p^{-1}(q), \exp_p^{-1}(q))$$

as I said, in a Riemannian context you can define a global distance function without problems
in the pseudo-Riemannian context, you can't, because the indefiniteness means you can't rely on an infimum/supremum of distance existing

Mathematicians do not strike me as the kind of people who will not do something because there is a simpler way to do it

What I'm saying is you won't find this in a "basic" book because it's a construction that doesn't exist on a "basic" Lorentzian manifold

10:43 AM
I am talking the other way around, though
A book on Riemannian geometry that has the "worldfunction"

ah, sure, that you'll find, it's just the Riemannian distance function
but it will not be obvious which of its properties will carry over to the setting of geodesic neighbourhoods on a Lorentzian manifold

I can probably work that out
I don't know why more GR books don't talk about the worldfunction more
It seems like a neat tool
Does MTW even talk about it?
Let's check
Not under the name "worldfunction", anyway

I mean, the most interesting thing about the Riemannian distance function is that it's a proper metric on the space, i.e. it induces the same topology and metric structure as the actual Riemannian metric
the worldfunction only exists locally and doesn't do any of that

It does many other interesting things, though

it does?

10:49 AM
Also "locally" in GR usually means like "A billion kilometers"
@ACuriousMind You can work out distance measurements in GR with it using particle exchange!
Basically using the GR version of the Pythagorean theorem

but do you need to talk about it as a function or could you just as well just talk about the lengths of geodesics between some points?

@ACuriousMind There is some Taylor expansion involved

it's physics, there's always Taylor expansion involved :P

The GR Pythagorean theorem is shockingly similar to the regular one, although as with most such formulas, the last term does a lot of heavy lifting
The famed bounce
The formula is like $$\sigma^2(p, q) = \sigma^2(p, r) + \sigma^2(q,r) - 2 g(d \sigma(p, r), d \sigma(q, r)) + \phi$$
The $\phi$ does a bit of the heavy lifting
It's like $$\phi = \frac13 \int_0^{\bar v} (\bar{v} - v)^3 D_v^4(\sigma) dv$$
Basically a curvature term
In flat space $\phi = 0$ and it's just the Lorentz geometry law
there are also probably some extra unpleasant terms if not all sides of the triangle are geodesics

11:29 AM
I should learn how to do error bound estimation
Because idk what phi is supposed to be in a reasonable case

12:12 PM
Hm, actually I think Lee does discuss the "worldfunction" for Lorentzian manifolds
not in these words but same idea
In section 13.8
technically not the worldfunction but the geodetic interval, although the two are related

12:45 PM
@Slereah You should translate 19'th century French math books
12

I am trying to find an English translation of Camille Jordan's work "Cours D'analyse". Only the French edition is on Amazon, so since this is a somewhat specialized topic, I thought perhaps someone in this forum might know. TIA, Matt

12

I am interested in the details of Elie Cartan's thesis, and, more specifically the explicit construction of the exceptional Lie groups as groups of symmetries of some specific homogeneous polynomials (according to what I have read in many places). I am interested in the details. For instance, wha...

Two huge opportunities just sitting there
@glS Weinberg and Hamermesh are usual references
The latter even does them for finite groups by relating them to linear-fractional/'mobius' transformations

idk what's a cool old timey math book that is only in French
Is Tonnelat's book translated
Or maybe one of those Lichnerowicz or Cartan books
I could start the project rly
Dunno about finishing it but I can start

1:27 PM
@bolbteppa thanks. Referring to "Group Theory and Its Application to Physical Problems" I guess?

Yeah

I still find it somewhat amusing that these are all books for physicists. I kinda try to avoid learning math from phycists. I find all the shortcuts and abuse of notation only make it harder to understand what's going on
I mean it's great to read those after you already know all the theory, as you can get more intuition and useful examples to keep in mind etc, but not to learn the subject itself

I'd say it's worth taking a mixed approach, sometimes it's just too hard to go too general, sometimes generality is simply unavoidable, the reality is it's all hard

probably. But I'll still never understand why should it be in any way useful for physicists to e.g. not clearly distinguish between Lie groups and their algebras, or between what's a group and what's a representation of it

e.g. one doesn't need to first learn the Sylow theorems in finite group theory before using Lie groups in the standard model, yet they are a basic aspect of any pure group theory discussion
With group theory in physics, a lot of it is a harsh reality of having not first studied the Cartan classification and how it applies to real Lie algebras (the latter of which is very difficult) and so trying to deal with things despite that

1:41 PM
was thinking more about not distinguishing between different concept. Another example being the, in my opinion, awful way differential geometry is used in physics.

Like, I absolutely understand not wanting to go through the trouble of formally defining manifolds, charts, tangent spaces and so on, but is it that hard to introduce the intuition associated to those geometric objects and define tensor fields and such accordingly, rather than just saying "tensors are things that behave in such and such when coordinates transform"?
@bolbteppa uhm, probably my ignorance speaking, but being Sylow theorems about classifying finite groups, why do you need them to talk about Lie groups?

@glS you don't, that's the point
but you won't find "math group theory" that doesn't first do finite groups before doing Lie groups

If you want to study Lie groups, logically you'd first study the basics of finite groups, and the Sylow theorems are one of the basic things of a basic group theory course. Indeed classification problems in finite group theory are an 'introduction' to the classification of semi-simple Lie algebras used all the time in physics

well, ok, but I'm not saying physcists should do math like mathematicians do. It's totally fine to focus on some arguments rather than others. I'm saying however that it's counterproductive to confuse some concepts "for the sake of simplicity"

Schottenloher is 100% mathematical in nature, it's got theorems and everything :P
i.e. don't think this is a "physics-style" book just because it says it's about conformal field theory
Weinberg exists in a weird space of his own because he doesn't do stuff like mathematicians do but usually his approaches are very different from what "typical" physicists would do
where I absolutely agree is that you should probably not learn representation theory from something called "group theory for physicists" or anything like that :P

@ACuriousMind I noticed. In fairness what I'm saying doesn't apply that much for this sort of subject. I suppose when the subject becomes "less basic" the distinction between "mathematician's" and "physicist's" approaches tend to converge

1:47 PM
(I don't know Hamermesh but most stuff titled like that absolutely butchers the math)
@glS oh, not necessarily
(2d) CFTs are an interesting example because we actually know how to do it rigorously, so there's a lot of very mathematical literature on it and also a lot of non-rigorous physics

if anything it may be more different

ah, so never mind then, there's no hope =)
the best hope is to have as many of these subjects and discussions covered in a miriad of topical questions on SE!

1 hour later…
3:00 PM
@Slereah Ted Shifrin, a RO of the Mathematics chat, has a link to a popular diff geo textbook he wrote on his profile page.

@PM2Ring Related to that point?
I don't lack books to scour

No idea. I haven't read the book. But people in the Math chat seem to like it. :) And Ted seems to be a rather good maths teacher.

3:13 PM
Does the electric displacement field
represents the electric field inside a dielectric ?

Also I think typically in diff. geom. texts, they don't use bitensors
They use the one-variable equivalent, the radial distance function

What's a bitensor

Tensor but bi

Like a projective representation is to a representation?
Or maybe:

It's just a tensor that depends on two points

3:25 PM
1

I take a look about Theory of Holors: A Generalization of Tensors but I never heard this term "holor" I thought that the generalization of the tensors were the supertensors or hypertensor like that A vector or tensor whose components are themselves tensors is very hard to find a good genera...

"components that are themselves tensors" reminds me of the continuous spin representation, which has spinors as their index
but no a bitensor is just like $$B : M \times M \to T^r_s M$$

Parry Hiram Moon (; February 14, 1898 â€“ March 4, 1988) was an American electrical engineer who, with Domina Eberle Spencer, co-wrote eight scientific books and over 200 papers on subjects including electromagnetic field theory, color harmony, nutrition, aesthetic measure and advanced mathematics. He also developed a theory of holors. == Biography == Moon was born in Beaver Dam, Wisconsin, to Ossian C. and Eleanor F. (Parry) Moon. He received a BSEE from University of Wisconsin in 1922 and an MSEE from MIT in 1924. Unfulfilled with his work in transformer design at Westinghouse, Moon obtained a...

He floats like a butterfly and stings like a bee

"electromagnetic field theory, color harmony, nutrition, aesthetic measure and advanced mathematics"
ambitious

Does the electric displacement field
represents the electric field inside a dielectric ?

4:35 PM
Hm
For the worldfunction, there is the equality $$\sigma(p, q) = \frac{1}{2} \langle \nabla \sigma(p, q), \nabla \sigma(p, q)\rangle$$
Is that due to the Gauss lemma

@glS #notAllPhysicists
@ACuriousMind random matrix theory is also a bit like that, the rigorous stuff is relatively straightforward but it's also ugly and boring :P
Hall's book on Lie groups does representation theory without talking about finite groups first

maybe a cool book?
I know Perlick did a lot of cool papers anyway

5:34 PM
Why is non-relativistic QM not behaving like QFT, anyway
You can describe it as a QFT of one dimension
But as far as I know it's operator-valued functions of time, not operator-valued distributions
Although I guess the distributions in QFT are only spatial

If the divergence of the electric displacement gives us the free charge density and the divergence of the polarisation gives us the bound charge density
what gives us the total charge density in a material?
the divergence of the electric field inside it?

the sum?

Yes but what is the sum in this case

Remember that most of these terms are approximations

if E_0 is the external magnetic field, responsible for the polarisation of the moleculkes
yes but the problem is that idk what expression and what is the E field inside
If E_0 is the external E field

5:39 PM
The "real" theory is just EM in a vacuum, the medium is just an approximation by getting rid of all the charges making the medium
So the "real" total charge is $\mathrm{div}(E)$

and E
inside the medium right?

which is the sum of uuuuh

D = \epsilon_0 E +P
this is the expression for D

yes that's the one

but in this formula it is not said
whether E is the internal E or external E

5:40 PM
In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is said to be polarized. The electric dipole moment induced per unit volume of the dielectric material is called the electric polarization of the dielectric.Polarization density also describes how a material responds to an applied electric field as well as...
^see here

yes I've seen that. That is what I actually am watching
but it doesn't specify whether the E in the above equation is the E inside the dielectric or the E which causes the polarisation, the external E
so is this E filed the external one ?

@Slereah anti-particles
Putting it in second quantized language is the closest one can get, and it's the anti-particle issue that really sticks out as a difference

6:05 PM
@Slereah
I don't think the \div E = \rho _ (total)
because \nabla D = \rho_(free) and since D=\espilon_0 \epsilon E, substituing this 2nd equation to the initial one, shows you that the divergence of the electrice field gives you the free charge density. And the divergence of the polarisation gives you the bound charge density, which begs the question, what give the total charge density? Or is there actually something whose divergence gives us the total charge density

1 hour later…
7:18 PM
In wikipedia, regarding the definition of magnetisation the following is said:
The origin of the magnetic moments responsible for magnetization can be either microscopic electric currents resulting from the motion of electrons in atoms, or the spin of the electrons or the nuclei.
But even in non magnetic materials. you do have electrons rotating around the molecules
why don't we have magnetic moment there?

8:03 PM
We do
there's plenty of magnetic moments all around
but microscopic magnetic moments don't necessarily translate to macroscopic ones
that's down to the magnetic phase of the material

magnetic phase?

8:23 PM
the way magnetic moments are distributed in the material
most materials are paramagnetic
Magnetic moments point in random directions

8:39 PM
Ok
But
one further question
It is also said that
before I ask my main thing
how do you calculate magnetic phase
which I assume is an indicator as to which material can be classified as magnetic and which as not
?

8:52 PM
Probably not the best direction if you're doing EM :p
Just look at the permeability/permissivity of the material
That's good enough

aha
Ok my main question is
We say that an external magnetic field induces the molecular current loops and their field
What do we mean with induces
because that would imply something that wasn't there before the magnetic field caused it
which is not true
since molecular/atomic currents exist in a material all the time, whether a magentic field is present or not
I know that the magnetic field forces a redirection of the magnetic dipol of the particles
what affect does that have?

9:14 PM
@Slereah this is what I mean : "When any material is placed into a magnetic field its atoms acquire an induced magnetic moment pointing in a direction opposite to that of the external field. " This implies that the atomic magnetic dipole moment was induced, and it wasn't there prior to the presence of the external field, which is not the case since , each atom has it's own (randomly) directed magnetic dipole moment. The only thing that the magnetic field does is that it affects the direction