I finally have finished a draft version of the book that I’ve been working on for the past four years or so. This version will remain freely available on my website here. The plan is to get professional illustrations done and have the book published by Springer, presumably appearing in print sometime next year. By now it’s too late for any significant changes, but comments, especially corrections and typos, are welcome.

At this point I’m very happy with how the book has turned out, since I think it provides a valuable point of view on the relation between quantum mechanics and mathematics, and contains significant amounts of material not well-explained elsewhere. I’m simultaneously rather unhappy with it, very much aware of a long list of ways in which it could be improved. Any of these though would require putting more time into the project, and right now I’m thoroughly sick of it, desperately wanting to think about other things. So, this is pretty much it.

I’ve learned a huge amount by writing this, and I hope to apply some of this in work on several different new projects. As I work on these, perhaps I’ll do some more writing that would partially take the form of new chapters extending what’s in the book. We’ll see…

“on the the behavior” page 141

congratulations! I’ve been quietly following this as you’ve worked on it over the last few years and am delighted it’s now ‘complete enough’ and you can focus on other things. Also, thanks for making it available before it’s hidden behind a paywall. I’m looking forward to having the time to get into it..

Peter, congrats for finishing the book

Let me add my own congratulations–and also hearty thanks, since I’ve learned so much from working through the early chapters of your draft. You’ve managed to convey the mathematical structure of quantum mechanics in a way that’s refreshingly elegant and clear.

page 547, “include a a connection”

I downloaded the file and skimmed through sections of it, and seen that it is very much mathematics, while I am very much into physics now. The only thing that jumps out at me is the so-far hand-drawn illustrations and your stated desire to “get professional illustrations done”; having authored my own book, with well over 200 color illustrations I designed and made myself, I would say the illustrations in your book are not hard to do yourself, using only the “Paint” program in Windows for example, and you could take real pride in doing so, and also have no one else to blame if you are unhappy with the “professional” versions (choose your own fonts and line thicknesses, for example, and extend axes to your liking). (Most of my illustrations came from the interactive graphical software I developed myself, about 18 years ago, to do my unprecedented research — into what turned out to be the single objective origin of all of the world’s “ancient mysteries” — and so were churned out “ready-made”–with that software cookie cutter, as it were–but a fair number were done from scratch, or just finished, using Windows Paint, as yours could be.) I am my own “professional”, or craftsman, in every aspect of my book, and the result is complete satisfaction.

Peter: What an accomplishment! Congratulations. And give that mind of yours a little time off!

While I haven’t read the whole book, I’ve read parts of earlier drafts quite closely. I’ve been amazed at how much there is to recommend this contribution. In many texts one reads the same lines (almost verbatim) that one has read in earlier works. This means that if the original phraseology confused you, you have the illusion that you can’t learn the material because the same phraseology occurs repeatedly and seemingly independently.

As a famous example of this kind of phenomena, many years ago Atiyah and Singer had to explain what an anomaly was mathematically to those of us geometry minded folks who had swallowed the mystical physicists description “An anomaly is what prevents a classical symmetry from surviving quantization.” Many physicists claimed this work of A&S was “nothing new”, but it showed me in an instant how important a change in pedagogy can be to those of us who cannot understand the standard descriptions.

This book excels at this most important task in my opinion. People may claim that it breaks no new ground, but this will be incorrect. It is a pedagogical innovation to be able to explain things from a new viewpoint so that those who were mystified by the original descriptions can find a new path to understanding. This book has repaid close reading every time I have invested in resolving a long-standing confusion.

Well done.

Thanks to all for the kind words. Especially to Eric, since he’s praising what I would like to think is the best feature of the book, not different results, but a somewhat different viewpoint than the usual one. While teaching the course, I did try and rethink for myself the subject from the ground up rather than following closely other texts.

Anonymous pedants,

Thanks!!

Technical comment about harmonic oscillator: “If we start off with a state |0> that is a non-zero eigenvector of N with eigenvalue 0, we see that the eigenvalues of N will be the non-negative integers, etc”

Actually one must argue that this MUST be the case (see Dirac book): if we start off with an eigenstate |c> with fractional eigenvalue c, then by repeated applications of the annihilation operator a we obtain a sequence of eigenstates of N with eigenvalues c-1, c-2, … which will eventually take negative values. This is impossible (as proved earlier in the text), hence the sequence MUST terminate with a state “x” such that a|x> = 0. It follows that the eigenvalue of N is zero for this state, and we may label the state |0>. Then by repeated applications of the creation operator, we derive that the eigenvalues of N are the non-negative integers.

Page 15 “…2 by two matrices…”

amz review,

An earlier version had that argument, but I decided it best to leave it out. The problem is that if you try and make general statements about ALL representations of the algebra of annihilation and creation operators you have to worry about various possible pathologies, whether operators are defined, etc. In general the philosophy of the book is to avoid trying to prove general theorems about what MUST happen, since getting those exactly right typically requires making explicit a complicated set of assumptions. Better to not get involved if it’s not something I need for what I want to explain.

It looks great; I like the way it starts at an elementary level, gradually increasing in difficulty. The important thing now is to not worry too much about it being perfect, but that the number of mistakes are minimized, laying the foundations for a later second edition if successful.

It has the potential to be a classic like ‘Tensor Geometry: The Geometric Viewpoint and its Uses’ which all the reviews rave about it laying the foundations clearly:

https://www.amazon.com/Tensor-Geometry-Geometric-Viewpoint-Mathematics/dp/354052018X

You write (page 100) that CP^1 is C^2 quotiented out by C^*. Actually it is C^2 minus the origin which must be quotiented out by C^* : no point in CP^1 has homogeneous coordinates (0:0).

Congratulations, and thank you for making this final draft freely available.

In the 2nd ed., perhaps Chap. 41 could also touch upon the (less well-known) isomorphism between SO+(3,1) [the standard matrix representation of the orthochronous proper Lorentz group] and SO(3,C) [the group of 3×3 complex orthogonal matrices with unit determinant]. The latter representation is convenient for (e.g.) transforming the Faraday tensor F between orthonormal frames, or transforming the Weyl tensor W.

[It goes something like this: As F is a bivector, it can be expressed as a complex 3-vector, and as the Riemann tensor R is (in some sense) the symmetric product of two bivectors, it can be expressed as a complex 3×3 matrix. The Weyl tensor W is a symmetric and trace-free 3×3 complex matrix. Under Lorentz transformations, F’ = A F and W’ = A W A^t, where A is the appropriate matrix in SO(3,C)].

struwwel,

Thanks. Fixed.

Shouldn’t the expression “Planck’s constant” show up somewhere in Section 1.2.1?

Richard,

A good point. I seem to have discussed this constant without ever using its name, will fix that.

The topological reason for this this is that

The relation between between the quadratic

angular momentum of an object spinning about about some axis

If you have a Knuth style $1 per typo policy, I’ll write some more regex and find others 😉

regex,

Thanks! This inspired me to run a full check for repeated words on my .tex file and found a couple more. Fixed.

No, no bounty for typos, I seem to be doing very well getting them found for free!

On the other hand, if people can find sign errors in equations, I might think about it. Current plan is to check all the equations one last time once this thing is in proofs, and I’m not looking forward to that…

If Springer doesn’t have any TeX-savvy manuscript editors, I’m always looking for freelance work (just looking at chapter 1, that $\langle\cdot ,\cdot\rangle$ construction could use some touching up, and just above section 1.3.3, “i.e. ” should be “i.e.,\ ” [or “i.e.\ ” if you insist] along with a comma after the equation). Whether they outsource, I don’t know.

^ Whoops, “i.e., “. Probably not the best advertisement.

Anyway, “Choosing local coordinates on $G$, $\pi$ will given by” (between the definition of a representation and an irreducible representation) doesn’t really read right. I’m also unclear on the meaning of the subscript $|W$ introduced just below.

Near the end of page 325 it should be “One thus sees that on C^n, as in the symplectic case, up to change of basis there is only one non-degenerate

symmetricbilinear form.”.Mateusz,

Thanks! Fixed.

The letter font for some of the SU( , SO( and Spin( like in the title of 21.2 or the last paragraph of page 246 is different from the rest. Is that correct?

@Anonymous: Fraktur denotes the algebra, italics the group.

The $\frac{1}{2}$ instances on page 246 “should be” \tfrac, but I’m doing my level best to STFU about such things.

But, seriously, don’t do, e.g., $\dot{p_j}$. Springer’s never going to catch the horizontal misplacement (even directly compared with $\dot{q_j}$). Use ${\dot{p}_j}$, etc.

Congratulations on finishing the book! Just a small comment: I was surprised not to see a mention of or reference to Jean-Marie Souriau, who was (pretty much simultaneously with Kostant) the first to introduce the general notion of momentum map as we now know it, as well as introducing geometric quantization (again, independently but essentially simultaneously with Kostant). His book “Structure des systèmes dynamiques” (published in English as “Structure of Dynamical Systems – A Symplectic View of Physics” by Birkhauser in ’97) is still a worthy read.

Johan,

Thanks! Souriau was one of the places I first started learning about symplectic geometry. In general my references are just a selection of expository things I happen to have looked at in recent years. I try not to get into references related to who originated ideas, since that gets very complicated very fast. The Kostant reference is an anomaly.