Comment by bollu | Hacker News Reader

Comment by bollu | Hacker News Reader

I like the list! It's very "physics person" oriented. Unfortunately, I have faced an insane amount of frustration trying to learn from the same sources because they "skip" on mathematical detail, often leading to more confusion down the line. I personally enjoyed the books I write down below to learn "rigorous physics":

- Classical mehhanics by VI Arnold: We first abstract define what "space" is, analyze its properties, then move on to build Lagrangian and Hamiltonian formalisms. Is a great book to understand formally (i) what we mean by "observer" in classical mechanics, (ii) variational problems, (iii) the precise relationships between the Lagrangian and the Hamiltonian

- Wald + Misner Thorne Wheeler for general relativity. MTW is the "bible", with great pictorial explanations for everything. Wald prefers to write everything in terms of differential geometric language which makes for an entertaining read. For example, he does not write "the particle is at rest". Rather, he writes "the particle is invariant under time-translational symmetry". It was eye-opening to learn how to "think like a geometer" about these sorts of ideas.

There is also Leonard Susskind's video lectures called as "the theoretical minimum" where he explains, rigorously, classical/quantum/GR/QM/stat-mech: https://theoreticalminimum.com/courses. I've personally watched the GR lectures and found them very informative.

Finally, there is Landau and and Lipschitz: These are dense, terse, often difficult, but honestly, written with extreme clarity. This was the only undergrad stat-mech book I could find which actually formally defines the phase space, the averaging procedure we use, etc.

For a more casual reader interested in the mathematics and practical applications of GR, I'd recommend Peter Wald's book. It still provides a rigorous introduction, but has a slower pacing while building up the mathematical foundation of differential geometry. It provides an enjoyable ~150 page read which can be worked through in a few days/weeks if one is comfortable with Calculus.

- A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity

https://www.amazon.com/Most-Incomprehensible-Thing-Introduct...

Peter Collier?

I think all undergrad STEM programs should put A LOT MORE emphasis on mathematics and other fundamentals before bothering with heavy specializations.

The Arnold book is great, but it presumes a level of mathematical sophistication that isn't practical to achieve (at least in the US) until well into graduate school, and realistically?, only for Physics theory students or some Math students that are motivated to dive into Physics.

There are major theorems in the famous "Calculus on Manifolds" by Spivak that are exercises (and not end-of-chapter or starred exercises either) in Arnold.

Is it “a lot more emphasis” or that the math education of STEM students should be rebalanced?

I’m an EE and I think I had a math class every quarter for two years if I remember right. I tend to agree with Strang’s too much calculus essay.

http://web.mit.edu/18.06/www/Essays/too-much-calculus.pdf

You need the bread around the sandwich too, i.e. if you learnt statistical mechanics solely through Landau I think you'd miss some of the intuition in terms of heat and work and so forth (It's in the book but blurred behind the russian-ness). Bra-ket notation really helps me focus in big equations when battling attention span (certainly not present in my library edition of L&L)

Jackson's book is surprisingly practical considering its reputation (I wasn't expected any discussion of Galerkins method for example)

MTW is great although a little odd by today's standards (how many books talk about tensors as machines with slots and holes)

I am glad you brought up bra-ket notation. I have wasted a lot of time trying to understand how to formalize bra-ket notation. I kept seeing the words "C* algebra" and "rigged hilbert space" being thrown around. I recently learnt the full story:

- Von Neumann's paper "mathematical foundations of quantum mechanics"[http://alpha.math.uga.edu/~davide/The_Mathematical_Foundatio...] builds up hilbert space theory and solves problems without ever using dirac deltas. So if one wants to use bra-ket without dirac deltas, this is the formalism.

- To make dirac deltas themselves precise, forgetting the bra-ket context, one uses the language of distributions. So we view the dirac delta distribution as a linear functional which takes a function `f` and spits out the value `f(0)`, and all that jazz.

- To join distribution theory with Hilbert space theory, we construct rigged Hilbert spaces. This is the theory that carefully delineates from which subset of Hilbert space we can pick kets, and from which larger space of distribution we can pick bras, to allow the bra-ket formalism to continue to work.

I found the reference "The role of rigged Hilbert spaces in QM"[https://arxiv.org/pdf/quant-ph/0502053.pdf] incredibly valuable.

This is the sort of thing that drives me nuts. I bought and tried to read Shankar, because I was told it is "rigorous". It's not. It casually uses dirac deltas and all sorts of "punning" with bra-kets with zero formalism.

Do you have a good recommendation for learning this heat and work perspective? I always found this confusing [eg. "adiabatic work", "infinitely slowly" and all that]. FWIW, I wanted to learn stat-mech for a closer look at entropy and information, which I felt I got with some LL.

Isn't doing physics in a mathematically rigorous way very hard or close to impossible? I think Quantum Field Theory and the Standard Model remain non-rigorous because of their dependence on path integrals.

Yes, doing QFT rigorously is hard because of path integrals. It's possible to do QM rigorously, as the "rigged Hilbert spaces" formalism suffices.

Ballentine's book is a modern textbook that relies on that (as opposed to Shankar which you mentioned)

Nice, thank you, this was exactly the kind of resource I was looking for :)

I don't have the mathematical maturity to comment on the rigour of the bras and kets, but as to heat and work I was referring to thermodynamics in the sense of Carnot and co.

I picked Zemansky and Dittman off the shelf in the library and it's honestly really nice little book - the aim here is for intuition (particularly for experiment) while also making connections to the more theoretical perspective found in L&L (in my case this is because I can't be bothered to read 300 pages of formalism to get to the applications although YMMV). It is quite handwavey in the ways that you describe but I think that is the jazz that the great physicists played in order to get the result in the first place.

I learned stat mech from L&L, and had the same problem that I never really understood thermodynamics.

I've been working through Bohren and Albrecht's 'Atmospheric Thermodynamics' recently, which I've been really enjoying. They are much more concrete, and very opinionated about ditching notation and concepts that are unclear, such as the differentials that get bandied about in most other books.

I think the key problem is that most thermodynamics books try to develop the axiomatic theory in the abstract, as opposed to introducing real materials and building physical intuition with them first.

Carroll as an intro to general relativity is a great recommendation from the OP. I have been learning it (albeit with preexisting differential geometry knowledge) from Carroll and MTW.

An unmentioned benefit of MTW is that, because it is so heavy, it is perfect for placing on my forehead while lying down with a migraine.

> Leonard Susskind's video lectures called as "the theoretical minimum"

This is apparently a book too, co-authored by Susskind and Hrabovsky [1].

[1]. https://en.wikipedia.org/wiki/The_Theoretical_Minimum

It's one of a series, even. There is The Theoretical Minimum, Special Relativity and Classical Field Theory: The Theoretical Minimum, and Quantum Mechanics: The Theoretical Minimum.

I own the series. Combine them with video's and you have a pretty good combination. What you don't understand from one, you can pick up from the other.

Susskind deliberately set out to create something an amateur could use to understand physics. Accordingly he doesn't just teach the physics, he teaches the math too as you need both, of course. It seems so obvious, but he's the only one I've seen that does both.

>ecause they "skip" on mathematical detail, often leading to more confusion down the line. I personally enjoyed the books I write down below to learn "rigorous physics":

That is possibly because it would require too much space to explain those concepts in greater detail. The books are already very long even when skimping on the details.

Is this a joke? He wrote that he liked Arnold's and Wald's books. I have tried to study from typical physics undergrad books and was also missing some rigour which I would assume most people coming from a math background would appreciate more in introductory physics books. Arnold's book or also Spivak's book are much better in that regard.

Hilariously wrong reply to OP. Reading Wald means you’re comfortable with differential geometry which an undergrad in math encounters several years after calculus. Not to mention that other texts mentioned cover Lagrangian and Hamiltonian mechanics which rely on calculus of variations and differential equations.

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