The h Bar

vzn

18:00

think there is enough energy seen in the chat room to maintain a blog. there are so many great topics in the chat room. just writing some of these up in brief blogs would be regardable as sufficient. but think/ have noticed that something like perfectionism shows up and interferes & becomes an obstacle.

ACuriousMind

@0celo7 Yes, somewhat. I can explain it in the $C^*$-algebra approach to QM, where it basically arises from the tensor-hom adjunction. Yet, e.g. Wikipedia claims (without source) that the tensor product is also the categorial product of projective Hilbert spaces, which a discussion of mine with ThomasKlimpel showed to be false at least naively.

I don't really know how to work with (projective) Hilbert spaces alone to get that the tensor product is what we should be looking at

Though you can find an intuitive explanation when you consider that the tensor product is essentially what you get when you remove the restriction that every state of your combined system should yield a unique state of the subsystems

In a sense, the tensor product is the "free" space on the tupels of $H_1 \times H_2$.

0celo7

@ACuriousMind What is the best way to learn mathematical quantum mechanics? I've done quite a bit of studying concerning rigorous GR (Hakwing & Ellis, Straumann, Jost, Lee, etc.), but I don't know how to approach the same goal for QM.

Is Zeidler any good?

ACuriousMind

@vzn Yes, I perfectionism can be a curse :)

vzn

re math blog, halfway into it the mods got very involved (not initially saying they would) & then set a high bar.

ACuriousMind

@0celo7 I'm very bad at recommending books because I have never even read a textbook cover to cover. What I know comes mostly from lectures.

vzn

18:05

they seemed to apply nearly the same criteria to se answers as they did to the blog, treating it as a sort of long-form format for se answers. now this is understandable, but also questionable.

0celo7

@ACuriousMind Grad school is still 4 years out. My "lectures" consist of spoon-feeding AP test material.

ACuriousMind

@0celo7 I'm not saying you shouldn't read those books, I'm saying I cannot recommend any

0celo7

@ACuriousMind: The problem is that when I buy I book, I have to drop $100 on it. I can't buy a book and then find out it's unreadable or I'm in way over my head.

vzn

in theory all books are readable ;)

0celo7

Can I make a thread concerning a specific author or book?

ACuriousMind

18:10

I understand the problem. Have you searched for comprehensive lecture notes, or otherwise freely accessible material?

0celo7

I have a bunch of PDFs, but I want a self-contained resource.

vzn

0ce you mean on the main site? se generally dislikes stuff like that wrt policy. in chat, ask about anything

0celo7

Yeah I mean on the site. But if people who frequent the chat haven't read his books, then it doesn't make sense to ask about it in the chat.

ACuriousMind

@0celo7 Asking "Is this book/author good?" is off-topic. The allowed resource-recommendation questions are more of the type "What books are there on X?", and then the answers will give some titles along with their assessment/summary of them.

vzn

0ce there are a lot of user lists of "good"/ "preferred" books on amazon. try browsing those.

ACuriousMind

18:13

9

Q: A book on quantum mechanics supported by the high-level mathematics

I'm interested in quantum mechanics book that uses high level mathematics (not only the usual functional analysis and the theory of generalised functions but the theory of pseudodifferential operators etc, certainly the modern mathematics). If there isn't something similar please give me a refere...

Seems that would be essentially your question

0celo7

I'm interested in Zeidler specifically because of what I saw in the table of contents. (Knot theory, Lebesgue, Homotopy theory, etc.) I'm just worried about the prereqs.

Thats the thread that turned me on to Zeidler!

ACuriousMind

Personally, I think the Reed&Simon emphasis on functional analysis is not worth it. Neumaier's summary of Zeidler sounds as if it would be great

0celo7

Should I just comment and ask about the prereqs?

ACuriousMind

@0celo7 That's certainly not forbidden ;)

alarge

You usually can find the first chapter or so available through the publisher or Google Books. It is also quite easy to obtain the full PDFs, which you could check to see if the book is worth the buy. I regularly buy books off AbeBooks, where I typically either go for used books or international editions (often meant for resale in India) which are a lot cheaper (also, Dover reprints of old books are also cheap). The price typically drops quite a bit if you're fine with an older edition.

This all depends on the book in question, though.

ACuriousMind

18:20

@0celo7 Reading the TOC, it seems to me real analysis and a bit of topology are required, but not much else, though the complex analysis is dealt with rather fast.

0celo7

@alarge: The first chapter is the "historical introduction" and was quite enjoyable, but didn't use any mathematics beyond undergrad, which I have decently covered.

@ACuriousMind: How much topology do you think? I have Lee.

I know enough topology for Wald and can struggle though Hawking & Ellis.

ACuriousMind

@0celo7 The more the better, topology is a beautiful thing ;) But for the book (again, I haven't read it) the TOC seems to me as if you'll be fine if you know the basic ideas of open/closed, continuity, and the separation properties (e.g. Hausdorff).

0celo7

@ACuriousMind That may be, but I have to look up the definitions every time I solve a problem in topology! I totally feel Hitler youtube.com/watch?v=SyD4p8_y8Kw

I have no idea what clopen is.

ACuriousMind

@0celo7 A clopen set is one that is both closed and open

0celo7

Not that. How do I understand it intuitively?

ACuriousMind

18:33

@0celo7 The existence of a non-trival clopen set means that the space is disconnected - a clopen set is something like $[0,1]$ in $[0,1] \cup [2,3]$.

The clopen sets are essentially the connected components of the space

0celo7

Isnt that intersection empty?

ACuriousMind

It's a union, not an intersection

0celo7

I need to install MathJax. I read that as /cap

ACuriousMind

I just took the idea that, if you have two topological spaces $X,Y$, then the disjoint union $X \cup Y$ is disconneted and $X$ and $Y$ are both clopen

In most "nice" settings, the spaces are connected, and clopen sets do not occur.

0celo7

Do I have to click on "start ChatJax" in the bookmarks bar to start it?

ACuriousMind

18:39

@0celo7 Yes

It should start to render, then

Congrats, by the way, not many people find the ChatJaX link on their own ;)

0celo7

Hmm, the ChatJax link doesn't want to get into the bookmarks bar. The render MathJax one works though.

Wait. $[0,1]$ is by definition closed so why is it clopen in $[0,1]\cup[2,3]$?

David Z

@0celo7 "render MathJax" doesn't automatically render math posted in new messages

0celo7

@DavidZ: Chrome is dumb. It works now.

David Z

cool

0celo7

@ACuriousMind Are you aware of a proof of "if you have two topological spaces $X$,$Y$, then the disjoint union $X\cup Y$ is disconnected and $X$ and $Y$ are both clopen"

ACuriousMind

18:46

@0celo7 It follows directly from the fact that $X$ and $Y$ are both open, and complements of each other

(They are both open because the topology on the union is just the union of the topologies)

0celo7

@ACuriousMind But in the above example they were both closed. How is a closed set also clopen?

Oh they're clopen in the topology of the union?

ACuriousMind

Closed and open are not opposites

In my $[0,1] \cup [2,3]$ example, it is indeed easier to say that they are both closed and complements of each other

0celo7

Is [0,1] clopen in the topology of $\mathbb{R}^n$?

ACuriousMind

No, $\mathbb{R}^n$ is connected and has no non-trivial clopen sets

Yeah, sorry, the intervals were probably a bit confusing

0celo7

Exactly, so how is [0,1] clopen? Is it clopen in the topology of $[0,1]\cup[2,3]$?

ACuriousMind

18:50

@0celo7 Yes, it is clopen in the union of the intervals with the subspace topology inherited from $\mathbb{R}$.

0celo7

I forget -- what is topological definition of complement?

@ACuriousMind: That makes more sense, thank you.

ACuriousMind

@0celo7 Just the space without it - the complement of $Y\subset X$ is just $X - Y$ (or $X \text{backslash} Y$, you see both notations)

Dammit, I forgot how to escape a \ in TeX.

HDE 226868

Hey, is this flaggable?

physics.stackexchange.com/review/first-posts/73316

0celo7

\setminus

HDE 226868

It's non-mainstream, which isn't a valid reason for flagging, but I don't think it addresses the question.

0celo7

18:54

@ACuriousMind I'm confused by "It follows directly from the fact that X and Y are both open, and complements of each other"

What do you mean by that last bit?

complements of each other?

ACuriousMind

@0celo7 Well, $X\cup Y - X = Y$, right?

(recall the union was disjoint)

And the complement of an open set is closed by definition

David Z

@HDE226868 if it doesn't answer the question, flag it as not an answer

HDE 226868

@DavidZ Just to clarify - the non-mainstream flagging idea is dead, gone and buried, right?

3

Q: Adding "Non-mainstream physics" to the answer deletion/flag reasons

Okay, so we have the question VTC reason of "we deal with mainstream physics..." yadda yadda yadda. And everyone is grateful for it. But what about answers? Surely everyone would agree that we don't want to see someone answer a valid question with non-mainstream physics and/or unpublished (crackp...

feature-request scope answers review-tools flagging

ACuriousMind

@DavidZ It's a bit weird because the non-mainstream nonsense there says there is no dark matter, so it would answer the question, if it were right, by showing that the premise of the question "there is dark matter" is flawed

HDE 226868

I just want to make sure I don't flag future answers incorrectly.

ACuriousMind

18:57

For the record: I downvoted it but did not flag it

HDE 226868

I just caught it in the first post queue but I haven't acted on it yet.

David Z

@HDE226868 theoretically it should be, but people still cast flags for that reason from time to time

@ACuriousMind hm, I see what you're saying.

...

0celo7

@ACuriousMind Back the interval example. You said that $X$ and $Y$ are open and thus their disjoint union is open, right? But the intervals are closed. I'm derping.

David Z

Well, I think our judgment of whether the answer is an answer shouldn't be influenced by whether it's non-mainstream - that is, assuming it's not a complete layperson espousing their unfounded beliefs about physics

ACuriousMind

@0celo7 Ah, but the intervals are open in their own subspace topology! (The space itself is always open)

(That's why intervals were probably a bad example to pick)

@DavidZ So, this particular case is an answer, just a really bad one.

0celo7

19:02

@ACuriousMind This might be a bad example, but one I understand it I will understand the good examples.

Hopefully.

David Z

(what I mean by that last phrase is that, in some cases, I'll treat answers which are completely wrong and clearly given by someone with no relevant knowledge as non-answers, even if they do constitute an attempt by the poster to answer the question. Like "not a legitimate answer" in a sense.)

@ACuriousMind I'm leaning that way, yes.

Though it doesn't seem like an obvious decision.

0celo7

@ACuriousMind So $X$ is trivially open in the indiscrete topology of $X$, right? How about the discrete topology? Is $X$ considered to be a subset of itself?

ACuriousMind

@0celo7 No matter what topology a set $X$ carries, $X$ is always open in it.

And yes, a set is a subset of itself

(Though there may be fields/books where the convention is that sets aren't subsets of themselves. But for topology, set are subsets of themselves)

0celo7

@ACuriousMind Let me try to summarize everything. Let $X=[0,1]$ and $Y=[2,3]$. Then $X$ and $Y$ are both trivially open in their subset topologies. We form the disjoint union $X\cup Y$ which is also trivially open in its topology.

@ACuriousMind Then we write $X\cup Y-Y=X$ and similarly for $Y$, which shows that $X$ and $Y$ are closed. So how do we conclude that $X$ and $Y$ are clopen? They are open in their subset topologies, so how to we know that they are open in $X\cup Y$? Additionally, since $X$ and $Y$ are deemed open in distinct subset topologies (or are they isomorphic?), how do we conclude that the union is open?

Wait no the union is trivially open in its own topology.

So how do we conclude that $X$ and $Y$ are open in the union topology?

ACuriousMind

@0celo7 Ah, well, if you want to fully formalize this, you first have to formalize what a "disjoint union" is, topologically. And given $(X,T_x)$ and $(Y,T_y)$ as topological spaces with their respective topologies, we define the disjoint union to be $(X\cup Y, T_{xy})$ where $T_{xy} := \{U \subset X\cup Y \vert \exists U_x \in T_x \wedge U_y \in T_y : U = U_x \cup U_y\}$.

If you know what a basis for a topology is, you just take the basis of the union topology as the union of the basis of the components.

Essentially, this boils down to $X,Y$ being open in the union by definition

Which is why clopen sets do not occur in connected spaces - the occurence of a clopen set implies the space can be written as the disjoint union of two open sets

0celo7

19:20

@ACuriousMind What's the wedge in your definition of $T_{xy}$?

ACuriousMind

@0celo7 logical and

I always forget the semantically correct TeX for it

The generalisation of the disjoint union of sets/topological spaces to arbitrary objects is called the coproduct, by the way.

@0celo7 My definition of $T_{xy}$ says in natural language that the open sets of the union are exactly those which can be written as the union of an open set from X and an open set from Y

0celo7

@ACuriousMind Let's resume this later. I'm sure I still have questions that I can't think of ATM.

Thanks for all your help

ACuriousMind

19:36

@0celo7 No problem :)

user54412

20:07

@JamalS He did. I think Panda Inn is his favorite restaurant.

user54412

And wow that blog makes Caltech sound like a cheery place where everyone's always smiling and having fun.

user54412

@Danu Be careful. \d is defined by latex and various other packages. I personally use \dd for \mathrm{d}, since it has fewer conflicts.

user54412

(and similarly I use \ee and \ii)

0celo7

@ACuriousMind you still here?

ACuriousMind

@0celo7 Yep

0celo7

20:24

@ACuriousMind Alright, so to show that $X$ is open in the union topology, we have to establish that $X$ is a subset of $T_{xy}$, is that right?

user54412

@ACuriousMind Whereas I have read a number of books cover to cover, and I still can't recommend many for being good. I probably should have paid more attention in lectures instead.

ACuriousMind

@0celo7 An element, not a subset.

0celo7

Ok, the topology is a set of sets, right?

ACuriousMind

The topology is a subset of the powerset of the topological space, so subsets of the space are elements of the topology (if they're open)

@ChrisWhite Depends on whether your lecturers were good or not, I think

0celo7

So if we can find a $U$ such that $U=X$, then by your definition of the union topology $X$ is open? If so, this seems trivial. Choose $U_x=X$ and $U_y=\emptyset$, right?

ACuriousMind

20:27

@0celo7 And yes, the topology is a set of sets - a subset of the powerset, the set of all subsets of the set

@0celo7 Yes, correct.

0celo7

Glorious. I think I understand your bad example.

Lol

I have to write this down now.

ACuriousMind

If I write set a few more times, it will become totally meaningless

user54412

Is there a name for that phenomenon?

ACuriousMind

@ChrisWhite Semantic satiation is a thing.

user54412

In English or German?

user54412

20:30

Semantische Sättigung

Semantische Sättigung (auch verbale Sättigung) ist ein psychologisches Phänomen, bei dem die mehrfache Wiederholung eines Wortes zu einem temporären Bedeutungswandel oder -verlust führt. Bei vollständiger Sättigung wird das Wort nur noch als bedeutungslose Aneinanderreihung von Tönen ähnlich unbekannter Wörter aus Fremdsprachen empfunden. Das Phänomen kann damit erklärt werden, dass die verbale Wiederholung ein spezifisches neurales Muster in der Großhirnrinde anregt, das für die Bedeutung des Wortes steht. Die wiederholte Erregung der Nervenzellen führt demnach zur neuronalen Inhibition, wodurch…

user54412

I was kinda hoping it would be a single word

0celo7

I haven't spoken German (intensively) in 4 years.

I grew up there though. It's amazing how much vocabulary deteriorates.

You need two or three words to express a thought which you used to be able to use one word for.

ACuriousMind

@ChrisWhite Heh. We're not always using single words, we do have adjectives.

user54412

20:45

So as part of a pet project of mine (I'll probably post results on meta at some point), I'm going through all my old answers and assigning them a subjective quality score.

user54412

I've noticed a number of times where I say something like "I thought there was a good answer here, but it's gone."

user54412

Now I see those deleted answers. Often they're @dmckee answers that I upvoted before he deleted them.

dmckee

@ChrisWhite Generally a sign that I've been convinced some other answer to that question is much better than my own.

I suppose I could edit to fess up, but it is easier just to consign them to 10k land.

Danu

@ChrisWhite Of course, I redefined the old meaning (which I never use). Moreover, \dd is already in use (double derivative...)

David Z

I've seen \od or sometimes \nd for derivatives

and \pd for partials

Thomas Klimpel

20:49

@ACuriousMind I would like to ask you two questions. The first is: "How much have you thought about category theory and its relation to QM. Have you seen or browsed John Baez' Physics, Topology, Logic and Computation: A Rosetta Stone at least once?"

David Z

In practice I just use the physics package

2

Danu

Oh, what does that do?

David Z

ALL THE THINGS

Danu

(although I'm typing/editing a script on differential geometry atm so it may not be too useful)

Why did I never hear of this yet lol

David Z

but seriously: differentials and derivatives, trig and hyperbolic trig functions, matrix operators, auto-sized brackets, Dirac notation, etc.

It's a reasonably new package, only since late 2012

0celo7

20:51

given a package, how does one get a list of the operations?

David Z

Look at the documentation

0celo7

documentation?

Danu

I'll have a look at it

just google it, Ocelot

The documentation is usually among the first few hits

also thanks @DavidZ I think I'll stick with \od for now

and \odd I guess

user54412

And the documentation is usually good, since people who make tex packages are the types who enjoy documenting their work

Danu

Sounds... odd

(har-har-har)

@ChrisWhite Undeniably true

It's just a little tedious at times

0celo7

20:53

by documentation do you mean this get-software.net/macros/latex/contrib/physics/physics.pdf

or is it something else?

David Z

What else would it be?

0celo7

I don't know. Perhaps it was something in the distro.

Danu

@DavidZ yikes, it has so many things that I already made custom definitions for... hmm

user54412

@0celo7 Do you know about CTAN?

user54412

For instance ctan.org/pkg/physics

user54412

20:56

It makes it clear what the documentation is

0celo7

@ChrisWhite Thanks.

I did not know about that.

ACuriousMind

@ThomasKlimpel I have read the physics part and skimmed the logic part of the paper by Baez you link, and I have a reasonable grasp of the functorial formulation of a TQFT as a functor on cobordism categories, I think. I've not really thought much about category theory as it relates to "ordinary QM", though I also recall reading something from Baez that casts the action and path integral in functorial terms.

DanielSank

@Danu: 100% interested in blog idea.

Danu

Btw, what do you guys think about distinguishing the differential vs. the exterior derivative ( $d$ vs. $\mathrm{d}$)?

user54412

@0celo7 I've never encountered a public tex package that was not on that site

Danu

20:57

@DanielSank Heh, knew you'd be!

DanielSank

@Danu It is am important distinction, but is audience dependent.

Danu

Hmm?

user54412

@Danu What is the differential?

David Z

@Danu in my experience it has been distinguished by setting the exterior derivative in bold

DanielSank

In Munkres's Analysis on Manifolds he points out that "the differential" is actually a pretty poorly formed notion.

Danu

20:58

@DavidZ Bold? Really?

DanielSank

@DavidZ I've never seen that.

Danu

@ChrisWhite The induced map on tangent spaces

@DanielSank I don't agree with that... I think?

DanielSank

I've seen a Roman $\mathrm{d}$ for the "differential".

David Z

but anyway, the semantically proper way to do it would be to have separate commands like \d and \extd, or \dif and \extderiv if you want to be more verbose, or so on. Then you can easily adjust the formatting as desired

Danu

@DavidZ Urgh, you're right. But I'm LAZY

user54412

20:59

@Danu Trying to interpret that statement...

DanielSank

@DavidZ Regarding that: I've started putting a bunch of \newcommand's at the start my of SE answers. Makes writing them a lot easier.

Danu

I'll go for \d and \dif

user54412

What is inducing a map on which tangent spaces?

DanielSank

...and mathjax is nice about it.

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