Also, basic QM

@acuriousmind I will in time but is there no explanation available to me now, without it? I studied hard what was meant by these definitions in newtonian mechanics back in march. In modern physics and even classical mechanical descriptions of EM, these go out the window it seems.

@0celo7 for a limited subset of meanings of the term scientific community.

@Obliv In classical mechanics, momentum is what changes when you apply a force

Just like kinetic energy is what changes when you do some kind of work. Mass does not need to be present in either of those definitions

@JohnRennie so not even chemists think photons have mass?

Consider EM, where the force is based only on charge and velocity

In that case, momentum changes regardless of what the mass. It is explicitly the Lorentx force

@Obliv Yes. Just wait till you get to quantum mechanics, where most things you learned in classical physics go out the window :)

Lorentz

@0celo7 I'm a (physical) chemist and I don't think photons have a mass. Actually I suspect most chemists don't know and don't care.

Mass is what governs the acceleration of particles in classical mechanics, but you don't need it to have momentum

Yes it does. @william these words, momentum and energy, are associated only with massive objects. Otherwise, you're applying an acceleration to nothing. Energy is defined as a spatial integration of acceleration $\int a dx = \int v dv$ associated with the mass so $m\int a dx = m\int v dv$ momentum is the temporal integration of acceleration $m \int a dt = mv$

@Obliv but photons don't accelerate like matter does

when you do work on a massive particle, it accelerates (unless that energy goes into another degree of freedom like rotation)

@WilliamBulmer Photons don't really "accelerate" at all.

@ACuriousMind they do, and the don't. They change trajectory, though from the perspective of GR, they really aren't doing that either

@ACuriousMind but I am not as well versed on the kinematics of photons as I should be, so I will defer to others who know more

@WilliamBulmer I'm not sure that photons are a good way to describe the behaviour of a light ray, GR or no GR.

Light rays can be deflected and that deflection is associated with a force.

@JohnRennie we might want to avoid appealing to photons to explain momentum without mass in the first place, though

I think that the Poynting vector offers a good enough explanation

@JohnRennie Also, photons do not couple to curvature like light rays do.

I'm getting nowhere..

@Obliv Let's try a different approach. Do you understand how an electric field can have energy?

Err an electromagnetic field

define energy

haha

Sorry, I laugh because that's not a straightforward thing to do

I am not laughing at you

One way to look at energy, at least from a simplistic viewpoint, is as a "battery" that stores all of your ability to do work

and so, of course, you could then ask, "what is work?"

In the case of massive particles, it has a very specific definition

which is the integral of force dotted with the change in position, or variants thereof

If you have no energy, you can't do work.

Though that's a bit simplistic

But it works for now

Now, a field like the electric field does work on charged particles

define what you mean by define @Obliv

So that's what is meant by EM fields having momentum, then? The ability to change the momentum of a particle affected by the field?

by conservation of energy, the E field, acting on particles has energy "stolen" or "taken" from it

I'd say that's a fair thing to start with

yes

@Skillpatrol which makes me wonder, is there a definition of the word define?

Look it up in a dictionary :P

In a lot of ways, energy and momentum should be thought of as fundamental (read axiomatic) quantities.

@william Sure, this makes sense.

So, when you ask what momentum and energy are, it's almost like asking what a number is.

It's more like they are the fundamental entities of physical theory, and the important part is understanding how they can change

^

one of the ways they can change is through interaction of massive particles with fields

I disagree. Momentum & energy have different definitions in modern physics than in newtonian. If they were the same axiomatic definitions, photons wouldn't have energy or momentum.

No, because we changed the axiomatic definitions based off of our observations

duh

In other news, Deep Impact turns 18 this year

We found, for example, that when photons knock electrons out of orbitals, they did so with energy equivalent to hf

@ACuriousMind What is the homology of a "pair"?

@William you didn't give any motivation to change this definition or to what this definition was changed to. If you can at least give the latter I'll be satisfied. Just observing the photoelectric effect doesn't give me any reason to say photons have energy associated to their frequency. Since, if photons are massless, this modifies a previous definition of energy.

@0celo7 Probably what's more often called "relative homology".

@EmilioPisanty Has Deep Impact some cultural significance I'm not aware of?

@ACuriousMind Lol, my prof's response to "Milnor is too hard" is to tell me to read Hirsch, which is coming at me with relative homology

@Obliv well, if energy is not carried by the photons, where is it?

There is nothing massive to carry that energy

And so SOMETHING needs to carry it by conservation of energy

@0celo7 You might want to ask him to assign you books he's more recently read :P

Well you're assuming you have to associate the word energy with the ability to free electrons from their orbitals.

@ACuriousMind I'd honestly be surprised if he knows what relative homology is any more

He's an analyst, not a topologist

@Obliv but that energy has to come from somewhere

is it because electrons, which are massive, now have an increased kinetic energy? @william

by being freed

Looks like I'm...accepting that theorem without...proof

it seems unlikely that it was magically vanish upon leaving the sun, only reappear 8 minutes later once an electron gets knocked out

It hurts...

@Obliv you're right. It does modify the previous definition of energy

but that's my point. We changed the axiom

@0celo7 if you saw my notes you'd probably have an aneurysm. When it tells me to prove things I just jot down ideas and when I understand why it is true, I just move on :D

Actually, I would say we changed the axiom long before the photoelectric effect

We changed it, because we saw that doing so explained physics better than not changing it

@Obliv That's fine if you can fill in the details

We didn't have to call it energy if we didn

't want to

we could have called it "light juice"

However, I cannot fill in the details and I don't think ACM or Balarka understood them either

But what would have been the point

?

I'm just wondering if the photoelectric effect demonstrates photons giving energy to electrons @william

if it does, then the definition of energy most likely should be re-defined to accommodate for this phenomena.

It was

actually, it had already been with classical EM

We already knew fields could carry energy

cries over homology

@ACuriousMind halp

It's just that the photoelectric effect introduced the notion that it could be discretized

why is the betti number function subadditive?

in packets called photons

@william mm.. from the definition of field you gave earlier it doesn't immediately follow that photons/light fits this definition. Can you explain this part too? :D

@ACuriousMind We have the portion of the long exact sequence $\cdots \to H_\lambda(Y,Z)\to H_\lambda(X,Z)\to H_\lambda(X,Y)\to\cdots$

@Obliv fields are, again, axiomatic entites

That we posited by experiment

the homology has field coefficients

and somehow the betti number is subadditive from this?

physical things that didn't act like matter, which, nonetheless, changed the trajectories of matter

and exherted forces

and did work

@0celo7 "subadditive" with respect to what notion of addition?

@ACuriousMind it would be easier for you to check page 28 of Milnor

@William Oh i see what you meant. If photons/light acts as an EM field, doesn't that mean it doesn't interact with neutrally charged objects?

So if you had a wall made out of neutrinos, it would be invisible because photons would pass right through it.

@ACuriousMind Not particularly, just 1998 feels like a lot more recent than eighteen years ago. For feeling old, take your pick of Saving Private Ryan, You've Got Mail, Something About Mary, Godzilla, Patch Adams, Blues Brothers 2000, The Man in the Iron Mask, The Mask of Zorro, etc

@Obliv Well, we are nearing the limits of my understanding, but that's not quite true

@Obliv indeed, you can have a classical electric dipole

Also on the unfortunate pop-culture facts

None of those yield anything interesting much =|.

that is electrically neutral and yet, an EM wave passing through it will cause it to expand and contract

uhh wait this assumes photons can act on electrons from a distance, though, right?

@0celo7 I see. Think about what exactness of the sequence would imply if the dimension of $H(X,Z)$ was larger than that of $H(Y,Z)$ and $H(X,Y)$ together (it implies a contradiction).

also is the energy of an electromagnetic field, then, given as $E = hf$ or where does this equation come from? @William

Or do it without contradiction. In any case, exactness and a minimum of knowledge of vector spaces is enough to get the subadditivity.

@Obliv I don't think so. I am pretty sure the basic idea (which is never exactly right) is that a photon gets locally absorbed by an electron, which then moves to a new energy state

absorbed and moving to a new energy state need to be defined :\ is this a characteristic of fields? We're defining a photon as an EM field , now , right?

@Obliv Don't do that.

@ACuriousMind so is actually additive

huh

A photon is inherently a quantum mechanical object. Don't conflate it with the classical EM field.

or can it be smaller

@Obliv If I remember correctly, E=hf comes from the experiments photoelectric effect. Specifically, in analyzing something called the work function

@0celo7 While the Euler characteristic is additive (that's another linear algebra exercise about vector spaces/exact sequences), the individual dimensions are merely subadditive.

@ACuriousMind Yeah, but they didn't really know that at the time when they formulated E=hf, did they?

@WilliamBulmer So we need to repeat their error?

I am trying to give motivation to @Obliv for why we consider them to have energy

Trying to retrace the discovery of quantum mechanics instead of presenting the modern picture first is responsible for a lot of horrible confusion, imo

And why we associate them with the EM interaction

I understand the motivation now, but I want to understand the standard definition now, in terms that I can comprehend. SO I guess that's impossible as @acuriousmind said earlier

@ACuriousMind what do you mean that the euler char is additive?

how does that work if the betti numbers themselves aren't

@0celo7 Because they are summed with alternating signs.

@ACuriousMind The modern picture is too confusing to people who are not familiar with the steps taken to get there

@ACuriousMind no, how do you actually show that?

does it require any homology or just linear algebr

It's like I told you a story by saying, "And then Aslan came back to life and helped to vanquish the Ice Queen. The end."

There's no context

@WilliamBulmer I disagree. I only understood quantum mechanics when people finally stopped talking about "wave-particle duality" or the photoelectric effect and all those things and started actually teaching me the formalism.

@ACuriousMind You are a mathematician, however.

Sure you should tell people how it was discovered. But I really think that using those old muddled notions of the beginnings of quantum mechanics as an introduction teaches students a lot of false beliefs that they later have trouble getting rid of.

@ACuriousMind It's probably a matter of personal taste, though. Also, people suck at explaining what wave particle duality is

Paul Davies cites QM as the reason he believes the world is non-deterministic.

@0celo7 It's linear algebra + knowing how homology is calculated.

(I am not Paul Davies)

Part of the problem with it, is we've only relatively recently begun to understand how large a role decoherence plays in that epiphenomenon

But I never found myself confused by it, because it never entered explicitly into the problems I was asked to do

@ACuriousMind do you have a reference?

google found something

I noticed from pretty early on that the only thing that matters in QM is the Hamiltonian, and the operators. Anything such as wave function collapse never really entered into the technical discourse

@0celo7 I only have this problem set from my second year where I proved it myself :P

that seems to be a PSE question

Fixed the link

Algebra?

jesus @acuriousmind your handwriting is worse than mine

Also jesus second year

ACM is an AI

I know, I think this is an inf. Hilbert space in that handwriting....

hmm maybe not.. mine is pretty bad too

@ACuriousMind I personally can't stand the whole Bourbaki approach to formalism

I hate it when you begin a subject like, "Define X such that..."

Nobody else talks that way

I don't start conversations with people why walking up to them and going, "Consider a certain person named William Bulmer, such that he is standing next to you..."

@William that's how most math books go. I don't see any problem with it. It's precise which is really important when it comes to conveying points and ideas.

No, it's the WORST way to convey ideas to most people

most people learn by a example. Starting with the most general definition, and developing theory is not providing examples

Well that's just flat out wrong because there are plenty of worse ways to convey ideas to people.

Look at what the literature says, though

Students don't learn by you talking at them

Of course, not all students are this way

I mam talking in a general sense

But, students learn by actually working through examples until they get a good sense of what the general case is

That's the idea of inquiry in education

You don't just tell a student, "A group is a set, together with an operation, *, with the following properties..."

it's preference. I actually prefer learning general definitions in math before applying them to specific problems.

@WilliamBulmer I'm not sure what that has to do with anything. There are plenty of nice examples one can consider in quantum mechanics without sacrificing the formalism. But things like "defining a photon as an EM field", which was what I objected to, are certainly not necessary.

But that's what a group is..

@ACuriousMind well, I was not defining it as an EM field

I was relating it to the EM field

which it is

brb food

My point was that we had already removed mass from energy long before photons

by considering the Poynting vector

and E^2 + B^2

@WilliamBulmer I have no issue with that. But I had an issue with obliv thinking that you're defining the photon as an EM field, and I pointed out that that's not what one should do.

@ACuriousMind Ahh

@0celo7 hm, actually, what I did in what I linked is a bit different. But anyway, the (sub)additivity follows from linear algebra (rank-nullity) + knowing what the kernels and images in an exact sequence are.

"knowing what the kernels and images in an exact sequence are"?

I don't see how you can get additivity from that

Actually, I still don't have a very good technical sense of what a photon is. In other words, if you asked me to mathematically model a photon, I could not. Then again, I am not famila

r with 2nd quantization, either

If you asked me to model an electron, it might not be entirely precise, but I might start with the hamilton

Hamiltonian

@WilliamBulmer : hi William. I know about this stuff. See this question and note the Feynman quote. Also see this.

"An interesting example of energy flow is the energy flowing around an electron. An electron has a charge as well as a magnetic moment. It therefore has a magnetic field. The cross product of the electric field E and magnetic field H gives the Poynting vector S = ExH which is the energy flow per unit volume. See Figure 4 below:"

the feynman quote :D

@JohnDuffield Don't you have a book to write

Yep. But I just popped in for a break, and thought William needed a little encouragement.

@JohnDuffield Ooh, that looks like that might dovetail with my "original" research. Let me take a look

thanks

@JohnDuffield we should make a reading group for Hatcher

William: pursue your ideas. The photon has a non-zero inertial mass. Inertial mass is a measure of energy. It has no rest mass because it's never at rest. However it's effectively at rest when you trap it in a mirror-box. See arxiv.org/abs/1508.06478 . Oh and note that the Poynting vector for a photon isn't something that "just circulates around and around".

Gotta go I'm afraid. Bye.

@JohnDuffield Thanks

Ugh, I need to actually do my job

Physics is so much more interesting than debugging

do I really want to learn about the lattice of subgroups of groups...

kill me @0celo7 i don't understand how u learn pure math all the time

@Obliv I'm not good at it

ask ACM

@ACuriousMind Reading about the path space now

Is the path space really an infinite dimensional manifold?

A Hilbert manifold maybe?

Well

You can add two paths

You can multiply them by scalars

what?

No?

What do you mean by "path" here

are you high

@0celo7 What do you mean? Do you not believe what whatever you're reading is telling you?

@ACuriousMind He puts it in quotes

Getting other people through their exams is harder than taking your own... Who would've guessed?

I'm wondering if it can be made formal

@0celo7 Probably because he didn't want to bother to fix a notion of "infinite-dimensional manifold".

yes

so, can it be made formal?

No, they don't exist and physicists made infinite-dimensional Lie algebras up ;)

@Danu You mean, as a TA?

@ACuriousMind No, as a friend

@Danu ...that would not be surprising to me :P

No doubt the two most stressful days of my exam period behind me

@Danu what?

I know there are infinite-dimensional manifolds

I'm asking if this particular construction is one

@ACuriousMind Come on, there has been a Fields medal for vertex algebras ;D

@0celo7 I guess? Google away, I don't know much about the path space

Loop space is much nicer anyways

Google?

Google: search for information about (someone or something) on the Internet using the search engine Google.

@0celo7 A recent invention that allows you to search the vastness of the internet instead of just my mind for answers.

...but why is it necessary when you exist?

@0celo7 Because it neither sleeps nor gets annoyed.

You're a vampire and a saint, @ACuriousMind, neither do you

I'm a vampire and I'm holy? That sounds like a painful mode of existence

He feigns sleeping in order to fit into society. He exists as an agent from an organization sent from the future to learn the ways of humans. how did u know @0celo7 :O!!

@ACuriousMind No touching yourself

(the ultimate punishment)

lol

↑ shiny

what is that

a pretty funky light beam combination

Some wonky orbital angular momentum stuff

viergruppe sounds german. does it mean anything to you @0celo7 or @acuriousmind

Why would you ask me?

aren't you german?

Uh, no?

Ich bin ein American

@Obliv just means "four-group"

@emilio you're german too?

@0celo7 Du meinst Amerikaner

or speak it

@Obliv Not particularly

@EmilioPisanty Ich weiß genau was ich meine ;)

But I speak some German

Commensurate to the requirements of my current city

@0celo7 you said you had german war machine blood or something along those lines one day, didn't you?

and the fondness of weapons because you were also american

Sounds like something stupid I'd say, yeah

Glorious Riemannian geometry

This is truly the best math

I think Riemannian geometry is the most physical math

I think I'm just gonna put a hold on algebra for now and start learning geometry in my free time instead :\ this stuff is too boring for an aspiring physicist @0celo7 do u have any beginner geometry books to recommend

u said zee was good right

@Obliv I've been meaning to reread Zee

If you read it I would be very happy to help

It's a physics book, not math

you said it requires multi-var calc though, right?

i'm learning it in the fall but I suppose getting a start on it now wouldn't hurt

what was the text called

Einstein Gravity

@Obliv Amen. I think about it this way. Pure math actually is quite surprising and beautiful, but you wouldn't know it from the way that mathematical textbooks describe it.

Not to mention that much of it turns out to be quite useful

@ACuriousMind Milnor skips a lot of details in his Riem geo...

@0celo7 Well, duh. It's called "a rapid course", not "a thorough introduction".

@ACuriousMind No, even later

Like on the top of page 76

Again, that's not surprising, he skips details also in the topology and analysis parts, as you already discovered, no? :P

That equation needs a lot of justification, I think he's just invoking torsion-freeness though

It's like HE in that regard

@ACuriousMind Why do people like this book so much?

It doesn't seem suited for a close reading unless you know what's in it already

I'm quite familiar with all of this geometry, so I just flew through the past 40 pages

@0celo7 How should I know?

I guess precisely because it doesn't turn into an 800-page monstrosity of strictly necessary but utterly boring details

I would understand a 300 page detailed text

@WilliamBulmer That's not true at all---you must've been reading the wrong books. For something truly inspiring, check out Thurston's book on 3-manifolds.

@Danu I'll take a look. Almost without fail, the books that I have read follow the theorem-definition-proof format. Even the ones that purport to be introductory for undergrads. I remember asking an Abstract Algebra prof. once why we were studying solvable groups (I knew the answer, but wanted to see what he was saying.), and his reply was, we're studying them to understand them better

Theorem-definition-proof is not a bad thing.

It's for precision.

Should it not be definition-theorem-proof?

You don't need precision when you are learning a subject

you need the gist

precision can come later once you work through enough examples

That's your opinion, and I thoroughly disagree.

As I said, it isn't just my opinion

that's how humans tend to learn

It is, because I completely disagree.

@WilliamBulmer Ah, the physicist's characteristic call.

And I study mathematics

and physics, too

But to really know mathematics, you need details

Which is precisely why Milnor isn't great

It's not great as a first book

Because it has no details

Have you read it?

@Danu Of course, you do, but to to learn the details, you need a way of absorbing htme efficiently

htme = them

@0celo7 You're talking about which Milnor book? Morse theory, or difftop?

Morse.

Top can be a first book if you're dedicated and have Lee as a backup ;)

I didn't read any of it

"have Lee as a backup" = it's not a good first intro, which Lee is ;)

Lee is good even when you know the material

@Danu and to absorb details efficiently, you need to understand fundamental aspects

It's incredibly precise most of the time

If you need a result, you can get it there without too much fuss

@WilliamBulmer Or you just need to get into mathematics more before you make sweeping generalization based on your personal experience ;)

Very good organization, etc

Lee is great; but it's too tedious to really read in full right now for me

Oh, I would not read it in full

There's so much stuff there

I prefer slightly less detail

@Danu Have you tried Warner, Lang, or Spivak?

I'm getting Spivak Vol 1 from a prof who's retiring.

My advisor got the full set from him yesterday

Spivak, yikes, no fun :P

@Danu My sweeping generalizations are based off of all the textbooks I ever attempted to read. I have a whole book case full of number theory./topology/galois theory/analysis, and so on

Too fucking much text haha

@WilliamBulmer It's still just one person.

@Danu Kobayashi-Nomizu?

Now that book is a terrible introduction.

@0celo7 I think that's more of a reference work than a textbook

Literally zero pictures, a connection is just some distribution, etc.

Bundles are a set of axioms.

Oh but the connection as distribution is nice

reading the preface of Zee makes me really eager to start learning this lol @0celo7 the guy seems so cool :D

@Danu Right, but he does not motivate it at all

@Danu Can you give me an example of "Getting into mathematics more"?

@WilliamBulmer Neh, never mind it.

I don't feel like this is a productive thing to talk about right now.

Jeff Lee's geo book talks about connections as distributions and has lots of pictures

@Danu I really do enjoy math. I just hate how it's presented

@Danu It isn't a textbook

But neither is HE, but people teach courses out of it.

@Obliv start today

I have time to answer questions

i can study it for approximately an hour 10 minutes straight then i have to head home. I'll probably continue then if you're still around.

where are you?

wolframalpha.com/input/?i=apples+vs+oranges

apples have fat??

@EmilioPisanty Are you procrastinating? :P

@0celo7 American apples

@Danu ... maybe?

Mathematica is taking aaaaaages on some contour plots for no good reason at all

@0celo7 school library. closes around 5ish

@Danu I'm German, this does not offend me

@Obliv weird time

summer hours i guess

our summer hours are midnight instead of 24/7

@0celo7 You wish

@Danu What's your problem

@Danu was this what you're referring to? library.msri.org/books/gt3m

@ACuriousMind Why can the solutions of the Jacobi equation be extended along the geodesic?

Existence and uniqueness of ODE solutions is local

@WilliamBulmer No, this. But they're related.

page 77 of Milnor

maybe it works on an interval...