The h Bar

bolbteppa

4:00 PM

I wonder how Bohm considers path integrals if they axiomatically introduce wave functions

Semiclassical

Not sure. It's worth emphasizing that Bohmian trajectories != Feynman paths though

heather

@BalarkaSen I see, okay. I found a copy of Gromov's book, so I'll try to look at that section. Perhaps I'll be able to plow my way through =)

Novalium Company

Guys, I'm looking forward to getting into chemistry. I know chemistry is a very broad and big subject and that's why I am looking for a simple book to introduce me to the main concepts etc... What book should I get? (that's available in a pdf format as well)

enumaris

oh my memory on path integrals is a bit fuzzy... XD

Semiclassical

there's a lot more freedom to feynman paths than there is to Bohmian trajectories

Anonymous

4:00 PM

@NovaliumCompany At what level?

bolbteppa

@Semiclassical what do you mean re 1-D quantum system

heather

@NovaliumCompany i mean, try just going to a used book store and finding their book on "general chemistry" or something along those lines. There's a lot of books on it, like there are for "intro" physics.

Semiclassical

Like a particle in a confining 1D potential

Novalium Company

Well, not too basic, I know some stuff, but not to advanced too. I need it to be with simple grammar if possible.

Semiclassical

e.g. a harmonic oscillator

though probably for the present context that's not the best one to consider. a gaussian wavepacket will do

bolbteppa

4:02 PM

How does this relate to the topic?

Semiclassical

It's my way of getting to QEH.

bolbteppa

I'm trying to get at something more fundamental, i.e. why are you even allowed to use the word quantum in the first place, why can't you just use classical mechanics on a theoretical level

Novalium Company

@Blue Any suggestions? Something easy to read but still teaches you a lot. I'm not really interested in how electrons behave yet... but everything else is good. I feel like electron behaviour is too complicated for me yet.

bolbteppa

If Bohm does what usual QM does once you get the basic 'theory' established that's fine, I really suspect Bohm is just stealing QM without any theory behind it and saying 'experiment!' to justify it the same way one could say 'angels told us to use wave functions!', which normal quantum mechanics doesn't do

Balarka Sen

@heather I don't recommend trying to read Gromov. He's very dense and impossible to read. Usually if people can make through Gromov they'll get fantastic research materials.

heather

4:05 PM

@BalarkaSen what is your "vague idea of how it could relate" then?

Anonymous

@NovaliumCompany ncert is good

heather

to get an idea of the approach.

Anonymous

If you can get the pdf versions

Semiclassical

well, let's be precise: The quantum version of the Hamilton-Jacobi equation in the second-order formulation is:

Anonymous

I don't know what textbooks they use in Bulgarian schools, so can't say much

Novalium Company

4:05 PM

@Blue ncert.nic.in/ncerts/l/kech101.pdf ?

Anonymous

@NovaliumCompany Yep

Constantine Black

@Danu OK, thanks. I'd say I'm more into diff. geom. and algebra and some topology(based also to what little I have read). The program of Hamburg (https://www.math.uni-hamburg.de/master/mphys/courses/index.html) on theoretical physics seems interesting to me considering mathematics and physics together. If I can do such mathematics I would be okay I think.
The only question I have then is this: are the mathematics one learns in a pure math program applicable in some sense to theoretical physics, or the individual will feel like having two separate knowledges that he won't be able to communic…

Novalium Company

@Blue Ok then, thanks :)

Anonymous

That's just one chapter

Anonymous

If you search a bit you'll get the full book

heather

4:06 PM

i suppose, if you have a PDE, you could probably also define a PDE that solved produces the "inverse", so then you could maybe generate a PDR for that that defines your solution. Or, perhaps, you define a PDE for the inverse operator itself, i.e., what action it has, and then solve that using the PDR method...

something along the lines of inverse(x'(t)) = x(t)

Semiclassical

$$-\frac{\partial S}{\partial t} =\frac{(\nabla S)^2}{2m} + V + \frac{\hbar^2}{2m}\frac{\nabla^2 R}{R}$$

where $\Psi=Re^{i S}$

vzn

@bolbteppa am (sometimes) trying to relate feynman path dynamics to fluid dynamics & finding some scattered refs. Tumulka has been cited/ discussed in here re bohmian mechanics + feynman path integrals arxiv.org/abs/quant-ph/0501167

Semiclassical

If you ignore that third term on the right, that's the classical hamilton-jacobi equationo

subject to the usual identification in CM that $\vec{p}=\nabla S$

But that third term is decidedly weird classically.

For completeness, lemme see if I can find the Newtonian version of that...

bolbteppa

"Bohmian mechanics as presented here is a first-order theory, in which it is the velocity, the rate of change of position, that is fundamental. It is this quantity, given by the guiding equation, that the theory specifies directly and simply. The second-order (Newtonian) concepts of acceleration and force, work and energy do not play any fundamental role.

Bohm, however, did not regard his theory in this way. He regarded it, fundamentally, as a second-order theory, describing particles moving under the influence of forces, among which, however, is a force stemming from a “quantum potential”."

hmm

Semiclassical

Yeah

bolbteppa

4:11 PM

That really sync's up with what I was saying about not knowing the force laws

Semiclassical

I'm not a fan of the second-order theory

I am much more sympathetic to the first-order one

Balarka Sen

@heather The idea, I think, is that if you have a PDE $\Phi(f, \partial_i f, \partial^2_{ij} f, \cdots, \partial^r_{i_1 i_2 \cdots i_r} f) = 0$ (shorthand), then you can extract a differential operator out of this: Namely, $\Delta = \Phi(-, \partial_i, \partial^2_{ij}, \cdots, \partial^r_{i_1 i_2 \cdots i_r})$. How is this an operator? Well, it eats a function $F$ and spits the function $\Delta F = \Phi(F, \partial_i F, \partial^2_{ij} F, \cdots, \partial^r_{i_1 i_2 \cdots i_r} F)$.

Just by considering the derivatives to be operators with blank slots which you feed the functions to.

Semiclassical

mostly because I like that, in the first order theory, extracting the trajectories from the wavefunction is simple

heather

hmm, interesting

Semiclassical

You extract the probability density $\rho=|\Psi|^2$ and the probability current $\vec{j}=\dfrac{\hbar}{2mi}(\Psi^* \nabla \Psi-\Psi \nabla \Psi^*)$ from knowledge of the wavefunction

Balarka Sen

4:14 PM

Then this differential operator can be thought as a map $\Delta : C(\Omega) \to C(\Omega)$ where $C(\Omega)$ is the "space of functions on $\Omega$"

This $C(\Omega)$ is an infinite dimensional space.

Semiclassical

and then you note that $\vec{j}/\rho$ is, dimensionally speaking, a velocity field

hence you can write down flow lines for this velocity field

bolbteppa

"Moreover, the connection between classical mechanics and Bohmian mechanics that the quantum potential suggests is rather misleading. Bohmian mechanics is not simply classical mechanics with an additional force term. In Bohmian mechanics the velocities are not independent of positions, as they are classically, but are constrained by the guiding equation."

Balarka Sen

A version of Nash inverse function theorem should be applicable to conclude that $\Delta$ is, locally, invertible (if it satisfies some infinitisimal conditions akin to determinant of a matrix being nonzero).

bolbteppa

That's what I was looking for, that's why it's not simply trivially just saying F = ma is wrong

Balarka Sen

If the PDR corresponding to $\Delta$ has a solution, then one should be able to patch these local solutions to $\Delta$ to get a solution to the PDE, I suspect is the key idea.

Semiclassical

4:17 PM

well, I did repeatedly say that initial position and initial velocity weren't independent for Bohm

bolbteppa

Sure, but I still think this might be an inherent flaw in the whole theory, need to think about it

I really tried to break normal QM when learning the basics (still tons to learn), tried to find flaws with it etc

Semiclassical

It's definitely a property of the theory

Emilio Pisanty

@Blue No. Not at all. That's the whole point of tensor products.

Semiclassical

It's why I say that one can choose to like this story or not, but it is an internally consistent story

bolbteppa

Alright that's progress with Bohm at least

Semiclassical

4:20 PM

@bolbteppa huh, is that from the paper "What is Bohmian Mechanics?"

I ended up at that same paragraph by coincidence without realizing it

Emilio Pisanty

@DanielSank yeah, that name was used around the time the bonkers washboard thing got introduced

bolbteppa

plato.stanford.edu/entries/qm-bohm

Emilio Pisanty

I still think it's bonkers

but then again I don't use them directly so I imagine they're useful

enumaris

bonkers I tell ya

Secret

(makes a bohmian joke: Perhaps what you said is already encoded by the guiding equation)

heather

4:21 PM

@enumaris pure madness

Semiclassical

ahh

bolbteppa

"it is not very satisfying to think of the quantum revolution as amounting to the insight that nature is classical after all, except that there is in nature what appears to be a rather ad hoc additional force term, the one arising from the quantum potential."

Semiclassical

well, check out page 9 here: cds.cern.ch/record/529132/files/0112008.pdf

bolbteppa

At least I'm on the right track :p

heather

@BalarkaSen thank you for the explanation. I'm guessing to understand any of the books on the subject I'll have to read up on topology first, at minimum.

Semiclassical

4:22 PM

@bolbteppa hence why I'm not a fan of the second-order formulation :P

Anonymous

@EmilioPisanty So, physically it doesn't matter which qubit has the phase factor? We can just shift it around on any qubit ?

Anonymous

I still can't get the intuition for it :/

Secret

I wonder... can we actually control the quantum potential term via external means?

e.g. magnetic fields, light etc

Emilio Pisanty

@DanielSank well, I notice that the eigenvectors that correspond to those eigenvalues are precisely the complex mode amplitudes $\Phi \pm i Q$ (ish), i.e. the creation and annihilation operators

Semiclassical

Inasmuch as we can prepare different initial wavefunctions and how they evolve, yes

Secret

4:23 PM

right

Emilio Pisanty

@Blue that is precisely right

Balarka Sen

@heather Yup. But there are approaches to learn topology through an analytic lens if that's your knack.

Semiclassical

@bolbteppa oh, and that "page 9" link above was intended for you

Anonymous

@EmilioPisanty But why exactly? It would be helpful if you could provide some intuition for it :)

Semiclassical

the paragraph at the bottom is more or less verbatim the one in the Stanford entry, lol

heather

4:25 PM

@BalarkaSen do you have a book recommendation for that?

(or generic resource recommendation)

Balarka Sen

Hmm, let me think.

Secret

Picture an array of 4 balls printed with arrows, if you rotate them all by 90 deg, all the arrows are rotated by 90 deg, but you can then switch your view and noticed that the system is not changed because all arrows stay in the same oritnetation relative to each other

Balarka Sen

How much multivariable calculus do you know?

Secret

An overall phase factor does not change the physics

Emilio Pisanty

@Blue here's one way to think about it. Whatever the state $|\Psi\rangle$ of your two-particle system is, the most general object you can use to speak about its physics is the matrix element $\langle \Phi | \hat A |\Psi \rangle$, where $|\Phi\rangle $ and $\hat A$ are arbitrary (but independent of $|\Psi\rangle$

Semiclassical

4:27 PM

@bolbteppa but, the sentence which shows up in the paper right after that paragraph (but not in the Stanford entry) is pertinent: "In Bohmian mechanics the second-order (Newtonian) concepts of acceleration and force, work and energy play no fundamental role. Rather, they are fundamental to the theory to which Bohmian mechanics converges in the classical limit, namely Newtonian mechanics. "

Emilio Pisanty

if $|\Psi\rangle$ lives in a tensor product state, then that matrix element is actually a bilinear function over the two tensor factors

and, basically, for any bilinear function $F(\psi,\varphi)$, you have $$ F(a\psi,\varphi) = aF(\psi,\varphi) = F(\psi,a\varphi).$$

heather

@BalarkaSen not much. i haven't actually taken any calculus classes, but i can handle single-variable calculus generally - not in the sense of "solve any problem" but "able to follow what's going on and solve some problems". i know a bit about partial derivatives.

Semiclassical

Hence why I say that, inasmuch as the quantum potential makes it easier to compute stuff, I'm fine with it

heather

that's about it.

Semiclassical

but I dislike treating it as fundamental, since it gives such a misleading impression

$d\vec{Q}/dt = \vec{j}/\rho$ is my preference

bolbteppa

4:31 PM

My sense throughout all this is that if you're allowed to even define equation (2) of that pdf then one could actually prove Bohm has to be just classical mechanics with that additional force, that's what I was trying to get at yesterday, normal QM bans you writing $dQ/dt$ and $Q$, my guess is they can't justify that Bohm is not just classical mechanics + a force, because they say they take (2) and (3) as axiomatic, so they are the most fundamental, as if that sidesteps the problem

Secret

In other news, counterfactuality is still something I am trying to wrap my brain around

Anonymous

@EmilioPisanty Mathematically that makes complete sense, yeah :P But what I was worried about is whether the fact we only use the "matrix elements" $\langle \Phi | \hat A |\Psi \rangle$ to talk about the physics is just a technical limitation or a fundamental limitation imposed by nature

Semiclassical

given that the theory doesn't permit you to write down two Bohmian trajectories with the same position and different velocities---no, I'll again insist that it's not just classical mechanics

classical mechanics doesn't give a damn about trajectories crossing in configuration space

it doesn't allow them to cross in phase space, but in configuration space it doesn't care at all

But the fact that the guidance equation exists means that that's not true in Bohmian mechanics

Danu

@ConstantineBlack Saying you're into differential geometry and algebra and topology makes it sound like perhaps you're maybe not too familiar with a lot of mathematics yet :P You'll have to specialize a lot more than that.

Emilio Pisanty

@Blue the latter

Semiclassical

4:35 PM

the moment you specify (1) what the wavefunction is throughout a given region of spacetime, and (2) the presence of the particle at a given spacetime point in that region, the guidance equation specifies what the velocity of the particle at that point will be

Danu

The theoretical & mathematical physics master's here in Hamburg is decent, but there are not that many courses. I personally think that, for precisely this reason, the master's in Munich (same topic) is a better choice.

Semiclassical

you simply do not have the freedom to create trajectories which cross in configuration space

Danu

one reason one might prefer Hamburg, though, is that you can get into research stuff quicker (at the cost of not knowing much about other topics)

Anonymous

@EmilioPisanty Umm, and why so? (Sorry for my long chain of follow up questions :/)

Emilio Pisanty

@Blue that's just how QM is built

Balarka Sen

4:37 PM

@heather Unfortunately there are not too many good books on multivariable calculus. I know two large volumes by Duistermaat and Kolk, but that has a loooot of stuff.

Emilio Pisanty

@BalarkaSen what?

Balarka Sen

Ted has a book but it's not available for free.

Emilio Pisanty

Div Grad Curl And All That

Anonymous

@EmilioPisanty I mean there must be some motivation behind it (?)

Danu

How about the standard books used in the German system, like Forster's? Those are pretty good.

But they're really analysis, not calculus

Emilio Pisanty

4:37 PM

@Blue yes

> QM is linear

basically

but that's it

it's just the single core mystery

@BalarkaSen also Buck's Advanced Calculus

Semiclassical

That's just a basic property of Bohmian trajectories: regardless of what the wavefunction happens to be, the guidance equation implies that the resulting trajectories cannot cross in configuration space. Full stop

Danu

@ConstantineBlack Almost all the mathematics one learns, even in a pure math program, has applications in GR, QFT or string theory (especially the latter, haha).

Anonymous

I don't know how : "We use $\langle \Phi | \hat A |\Psi \rangle$ to talk about the physics" follows from "QM is linear though" :P

Balarka Sen

@EmilioPisanty I haven't heard of that book but I am immediately skeptic it has the mathematical preliminaries.

I have heard of Buck's book.

Danu

But that does not mean that one can converse with physicists; the two communities are quite separate.

bolbteppa

4:39 PM

My point is, the very idea of a trajectory sneaks in classical mechanics

Semiclassical

And my point is that it doesn't

Emilio Pisanty

@BalarkaSen what do you mean by preliminaries? as in, does it do introductory single-variable calculus?

Semiclassical

classical mechanics has position and velocity as independent quantities

Anonymous

@heather For multivariable calculus Khan academy is good

Semiclassical

bohmian mechanics does not.

Anonymous

4:39 PM

If you're looking for intuition

Secret

Quantum is linear

4

Q: Are there examples of history dependent quantum dynamics that evolve like biological life?

There are examples of time evolution of quantum dynamics with history dependence, such as these quantum random walk examples which make use of a memory parameter to influence the distribution of the random walk. I am wondering whether the rules of quantum mechanics allow the construction of a ve...

Hence why you cannot make it to behave life like

Semiclassical

whether or not that's satisfactory is another question. but to say that the Bohmian velocity is independent of the Bohmian position is simply not a valid characterization of the theory

Balarka Sen

@EmilioPisanty A good multivariable text has to have three things: (0) Jacobian, directional derivatives, and the multivariable derivative done right (1) Implicit and inverse function theorem (2) Multivariable second derivative test (3) Integration done right, and a proof of the multivariable change of variables formula

Emilio Pisanty

@Blue yeah, that's just a theorem though

Balarka Sen

(0 does not count because if some book doesn't have that immediately throw it out of the window)

Do those book pass those criterions?

Emilio Pisanty

4:41 PM

@BalarkaSen that's a pretty steep list for an intro book

seriously, have a good look at Schey before you knock it

Anonymous

@EmilioPisanty Which one? QM is linear or "we use matrix elements"?

Emilio Pisanty

@Blue the latter as a consequence of the former

Semiclassical

I'll draw a distinction here between "multivariable calculus" and "multivariable analysis"

Balarka Sen

0, 1, 2, 3 are very very essential for anything mathematical you want to do with multivariable calculus.

Secret

difference?

Balarka Sen

4:42 PM

Multivariable calculus, to me, is an intro to differential topology and differential geometry for $\Bbb R^n$.

Anonymous

@EmilioPisanty Could you give me any reference material for that deduction? It's not immediately obvious to me

enumaris

relegating multivariable calculus as just an intro to another topic eh...

Semiclassical

For the former, you don't worry about proving stuff like the Jacobian formula so much as worrying about how to use it to actually calculate multiple integrals

An E&M class uses multivariable calculus, but it largely doesn't give a damn about multivariable analysis

ACuriousMind

@DanielSank Neat parlor trick ;) However, the fact that the eigenvalues are frequencies seems to be a consequence of your specific choice of dimensions for the coordinates. Non-dimensionalize your coordinates and the eigenvalues of the matrix that appears in the classical Hamiltonian e.o.m. are always energies! I'm not sure this really points at anything except "the Hamiltonian has dimensions of energy".

Balarka Sen

Engineering multivariable calculus is terrible.

DanielSank

4:43 PM

@EmilioPisanty That's true.

@ACuriousMind Well, there being a matrix in the classical case where the eigenvalues are energies and the eigenvectors correspond to the raising and lowering operators seems rather not a coincidence.

heather

@BalarkaSen, okay, but could I manage with a less rigorous text?

Semiclassical

If I'm going to be able to read Jackson's Classical Electrodynamics, knowing multivariable calculus is far more essential than multivariable analysis

Balarka Sen

@heather For sure, depending on what you want to do.

heather

@BalarkaSen well in this case, topology, though multivariable calculus is pretty generally useful =)

Balarka Sen

"Topology" is very broad

bolbteppa

4:46 PM

@Semiclassical I think I would like Bohm more if it was simply claiming what those quotes above are saying it's not :p Will think about how position and velocity are connected and how it's not just F = ma with extra quantum potential forces

Balarka Sen

What kind?

ACuriousMind

@DanielSank Surely not a coincidence - the algebra of ladder operators exists classically, after all, otherwise it could not (at least not in naive quantization) appear quantumly.

heather

@BalarkaSen "But there are approaches to learn topology through an analytic lens " and you were recommending books for that...

because I was looking at topology for the h-principle for inverse differential operators, which now that I look back on it is a rather convoluted chain =P

Anonymous

Multivariable calculus isn't analysis though

Semiclassical

@bolbteppa again, you're talking to someone who is sympathetic to the first-order formulation far more than the second-order one

Balarka Sen

4:47 PM

I was thinking of differential topology when I said that, for which multivariable calculus - a proper course on it - is essential.

ACuriousMind

When you seek "eigenvectors" for the classical Hamiltonian, you are essentially solving the equation $\{H, f\} = c f$ for a phase space function $f$. After quantization, that means that $f$ is a ladder operator for $H$, since the Poisson bracket becomes the commutator.

heather

@BalarkaSen, oh, gotcha. maybe i should just stick with the point-set kind then.

Balarka Sen

That'll be far from $h$-principles though :)

Sir Cumference

Did Godel shoot logicism dead?

enumaris

Godel shot first

heather

4:49 PM

@BalarkaSen oh. well dang it =P

Semiclassical

I will say (following my example from yesterday) it's certainly possible to get situations in classical mechanics in which, for all practical purposes, the position and velocity are not independent

ACuriousMind

@SirCumference That entirely depends on what exactly you mean by "logicism" :P

Sir Cumference

@ACuriousMind I guess the idea that math is a branch of logic

ACuriousMind

I don't know what that means.

enumaris

When you guys use the words "classical mechanics" do you include SR and GR in that

Semiclassical

4:50 PM

and the FAPP model you get in that manner is certainly well-defined and consistent unto itself

@enumaris hell no

bolbteppa

Not in this case

Sir Cumference

@ACuriousMind Well, assuming that the axioms are true statements, and seeing what you can derive from those

bolbteppa

Non-rel class mech

ACuriousMind

@SirCumference That's what math is.

enumaris

Would you call SR and GR classical theories?

Semiclassical

4:51 PM

No.

bolbteppa

Yes definitely

ACuriousMind

Gödel just shows you that what is true by the axioms is not necessarily derivable

Semiclassical

I wouldn't call them quantum, but to call them classical seems abusive

enumaris

^Seems to be a difference of opinion here

Sir Cumference

@ACuriousMind Doesn't it state that no finite set of axioms exists?

Semiclassical

4:52 PM

though tbf when people talk about 'semiclassical gravity' they do mean it in the way bolbteppa does

enumaris

hmmm

Semiclassical

in that respect the word 'semiclassical' is a bit ill-defined

enumaris

I think we need some new terminology

bolbteppa

Landau volume 2 on SR, EM and GR is "classical theory" of fields

Secret

Classical, quantum, <insert suitable regime> ?

Semiclassical

4:52 PM

'semiquantum' be better

heather

@BalarkaSen would Milnor (that's the book that keeps popping up in google searches) be a good book to start with?

Semiclassical

@bolbteppa that's fair

Balarka Sen

You need multivariable calculus to read Milnor :)

heather

but that is the book?

bolbteppa

Milnor is crazy

Even though it's tiny it's still scary

Semiclassical

4:53 PM

semi-newtonian :)

heather

@bolbteppa well that's encouraging =)

Anonymous

@heather What are you trying to learn topology for?

bolbteppa

The 'usual' way to get to Milnor would be reading Spivak's Calculus, Spivak's Calculus on Manifolds, then maybe Milnor

Balarka Sen

Milnor's little book (Topology from a Differentiable Point of View) is indeed the correct text.

bolbteppa

Or the one by those guys, G&P Guillemin and Pollack

Balarka Sen

4:55 PM

But you do need background for that.

ACuriousMind

@SirCumference No. Gödel just states that sufficiently powerful axiomatic systems can formulate statements they can neither prove nor disprove.

heather

@Blue because Balarka got me interested in the h principle and its use in finding inverses of differential operators.

dmckee

I've seen "classical" used to mean both "not quantum" and "neither quantum nor Einsteinian" in various places.

My sense is that the former usage dominates.

Danu

@heather I think that probably Guillemin & Pollack's book is a lot nicer than Milnor's.

Don't you think, @BalarkaSen?

enumaris

Physicists need to get their terminology together

heather

4:56 PM

@Danu what background does that require - similar to Milnor?

enumaris

and unified

Danu

yeah, similar to milnor

calculus on manifolds, nothing else really---technically they explain that too but it's brief

bunch of linear algebra

point-set topology (connectedness, compactness, etc.)

ACuriousMind

@enumaris I think in context there is rarely real confusion about whether classical means "not relativistic" or "not quantum".

heather

@Danu so I do need to learn point-set topology first.

ACuriousMind

But I agree that when there is the danger of confusion, people should just use the "not"-variants.

Danu

4:57 PM

@heather That's usually covered in books on analysis, though.

enumaris

yeah

heather

@Danu so...like baby rudin?

Danu

I don't really know. I never took a proper course on analysis.

I'm learning the basic theorems now by teaching them :D

Balarka Sen

I think @heather is finding that the mathematical rabbithole goes way too deep. I think this isn't the right approach.

Danu

It's rarely a good idea to try to work only in order to achieve a certain goal

better enjoy the journey

Anonymous

4:59 PM

@heather I don't know h-principles but if you were looking forward to learning GR, I found Schuller's lectures good (he gives an easy introduction to topology, differential geometry and multilinear algebra in a coherent fashion)....and I'm following that currently. I don't know how much physics you'll learn from there but for the math, it is good :P

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