Classes are over for the semester, and I’ve put together the lecture notes for my undergraduate “Quantum Mechanics for Mathematicians” course, which are available here.

The idea for the course was to try and explain the basics of quantum mechanics, from the point of view of unitary representations of Lie groups. While this is a rather advanced topic, I made an effort to do things quite concretely and start at the most basic level (the only prerequisite for the course was calculus and linear algebra). I hope the notes will be useful both to mathematicians trying to learn something about quantum mechanics as well as to physicists who would like to better understand the mathematics behind the way symmetry principles get used in the subject.

More to come next semester. The initial plan is to start with the fermionic oscillator, move on to path integrals, then relativity, the Dirac equation, and U(1) gauge theory (E and M), ending up with some very basic quantum field theory (non-interacting fields). We’ll see how that turns out and at what point I run out of energy and stop writing.

Any corrections, comments or suggestions about how to improve these notes are most welcome.

**Update**: Thanks to all for comments, I’m quite pleased to see how many people have been looking at these notes (6600 downloads and counting!). They’ve also made an appearance in surprising places, including here.

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Hermano, esto es justo lo que buscaba, soy ingeniero de 62 años al borde del retirement y fisico frustrado llegue hasta la path integral y deseo comprender mejor la mecanica cuantica, gracias, un abrazo desde calama, chile

de nada!

Dear Peter,

I have really enjoyed the quick review of the Lecture Notes above. I was looking for such an accurately aimed document about QM from the aspect suitable for veterans of informatics. I do not yet plan to drop from the computing systems and applied informatics, and I feel a need to catch up for quantum computing, which I think is the only fundamental advance in the field of computing for a long time ago.

So thank you for posting this, and I know I will be really happy when your second semester Lecture Notes will be available.

Please continue with remarks to applied QM like quantum computing as you did in the current document.

Best regards,

SuhesZ from Subotica, Serbia

quantum computing, which I think is the only fundamental advance in the field of computing for a long timeReally! This is clearly off-topic but one might as well complain that fundamental physics hasn’t seen much progress because there are still only three colors for the strong charge.

Even disregarding the enormous practical advances and explosion of engineering approaches, principles and philosophies that the field has seen in the last 40 years, and considering only advances in theoretical computer science and complexity theory, it certainly looks to me that there were serious advances indeed.

As for quantum computing, one could do worse than check out the complexity class BQP, then become a reader of Scott Aaronson’s blog, where many surprises await. In particular, his attempt at a timeline of computer science, which probably needs review.

And does anyone know what Schoonship is?

I fear those looking to me for a text on quantum computing will be sorely disappointed…

Dr. Woit, will you be teaching this class again next year (for the Fall 2013 – Spring 2014 terms)?

Dear Peter,

do not burden yourself, your current work is enough to correctly recognize the place of the “qubit black-box” within the QM. For the text on quantum computing someone would look elsewhere. Your notes are as good for students of informatics as you have it aimed for mathematicians.

Of course I have found and now following this Web site driven by my skepticism toward the string theory and SUSY (it would be too long to explain why I’m involved in this) not by my before expressed efforts to understand quantum computing.

Dear Peter,

On page 9, there is strangely truncated sentence

So something like “tion groups” is missing.

Just to say: thank you. I downloaded it and I much look forward to finding time to read it.

Another thanks. Section 12 is brilliant.

Thank you,

Looking forward to going through this over the holidays.

I have one small quibble with the last part of 1.3: for the equivalence between isometric linear operators (preserving the the inner product) and unitary linear operators, the vector space must be strictly finitely dimensional. In infinite dimensional vector spaces isometric linear maps are not necessarily invertible linear maps.

For example consider the space of square integrable functions on the positive real line (modulo measure zero functions) ; the positive translation operator is a linear isometry, yet is not invertible.

While seemingly arcane, such considerations could be important when considering translations of a wave function along a geodesic that intersect with a boundary or singularity.

M Uppal,

Probably will teach something else next year, but there’s no definite plan for 2013-4 classes yet.

Blablon,

Thanks! fixed.

Aaron,

Thanks. I should perhaps make it more clear that for the first half or so of the class, the state spaces under consideration are all finite-dimensional.

And, when infinite dimensional spaces do come into the game, there should be a huge warning label on the text that I’m no analyst and the standard being aimed for is not precise statements, but statements that are morally true and could be made precise without a lot of effort (unless it is made clear that the statement is highly non-trivial and requires serious work to prove).

How did you manage to compose 180 pages of rigorous material in just a few months?

What make of coffee do you drink and where can I get some?

Dear Peter,

I’m curious to know how many complaints you get because students think QM is illogical or not well founded? What’s would you guess the drop out rate is? Are students stunned by the complexity without reason? Or has everyone pretty much accepted that this is the way it is? How needed is a better conceptual foundation for QM? Thanks.

I sat in on the first few lectures of Peter’s (excellent) course, but then appeared to drop out. I took a similar course 30 years ago and completely failed to understand it. Once Peter explained the connection of representations to QM, I was satisfied that I now “get it”, and continued to follow from his online notes and Shankar’s book.

Peter, thank you for clearing up a decades-old confusion!

Hi Peter — in section 2.1, you should either require your representations to be unitary, or your group to be compact, or define “irreducible” to mean simply that there is no proper subrepresentation. The 2-dimensional representation of the additive group C given by t |–>

[ 1 t ]

[ 0 1 ]

is irreducible in the sense you define but clearly violates Schur’s lemma.

Thanks Michael,

I was trying to avoid getting into the indecomposable vs. irreducible business, but you’re right, I need to say something there.

The only example later on where this turns up is when I mention the 3d Heisenberg group (thought of as upper triangular matrices) action on 3d vectors, and that it’s not unitary so not relevant in QM.

Superposition is a more complicated concept than most physicists realize. If f & g are wave functions, then g*e^(i*c) represents the same state as g; but f+g*e^(i*c) defines a one-parameter family of different states! This obvious truism-which physicists all know-is conveniently forgotten when they try to explain Q.T. Even Feynman’s axioms for his path theoretic approach ignores this point. Just like one should start from pseudo-Riemannian geometry in systematically developing General Relativity, one should start from von Neumann’s quantum logics in systematically developing Quantum Theory.

Would it be possible for someone in his 2nd year course (of Bach. in Physics) to follow these lectures?

Anant,

I’ve tried to write the notes so that only linear algebra and calculus are required, no physics beyond the most elementary. Physics students may find the mathematical abstraction hard to follow, especially for the first part of the notes, where I’m trying to explain the crucial concept of a representation of a Lie group.

Dear Peter,

Being a mathematician with ( ugh..) not very good physical background, I truly enjoyed when I took a quick look at these lecture notes.

My suggestion or to say more correctly my wish is that you include an explanation of spin manifold from the physical point of view if you find it relevant, as somehow I can’t find it in the literature.

Newcomer,

These notes are all in Euclidean or Minkowski space, not trying to do things on manifolds, so spin manifolds don’t come into it.

However, there is a great deal in the notes about the relation of Spin(3)=SU(2) and SO(3), and the necessity of using Spin(3) rather than SO(3) to describe one of the most fundamental things we know about: spin-half particles.

These are really excellent notes. Any inclination towards publishing them?

Thanks Zathras,

Maybe someday, but I’d want to make sure there will always be a version available on the web. For now, I’ll see how much more material I can write over the next semester, then see what this looks like after that.

In an ideal world, at some point while teaching the class again I’d go through the whole thing and rewrite it, based on what I learned by doing this the first time. Seems likely to me though that I’d never have the energy for that.

I’ve almost finished going through notes one last time for typos, inconsistencies, etc. There’s a short list of things I’d like to add, sometime soon I’ll stop making changes on those notes and concentrate on thinking about next semester.

So how did the undergrads do in your opinion? Were they able to follow the material satisfactorily?

Zathras,

It was a small class, so I don’t think I have any idea how this would work with a large group of typical students. I think they did well and learned a lot, it’s quite challenging material. Especially difficult was the way they course started off, throwing them into very unfamiliar material: Lie groups, algebras and representations. It’s the kind of thing you can only really start to understand once you’ve seen the crucial examples, so very confusing at first. Maybe some more effort should have been put into motivating this at the beginning.

Would there be any complementarity with Fulton and Harris’s (1991) Representation Theory: A First Course? I found a used copy and it seems very example-oriented.

srp,

Fulton and Harris is also about Lie groups and representations, but mostly covers different material. They’re aiming for the general case of semi-simple Lie groups and their representations (for example, the group SU(n) for arbitrary n). I’m sticking to the specific low dimensional Lie groups that are basic for physics: U(1), SU(2), SO(3), and the Heisenberg group. Some of this overlaps with Fulton and Harris, especially the first half of what I was doing, but the second half is quite different than what they do.

Thank you for uploading your material. Would it work as a preliminary reading to Hatfield’s Quantum Field Theory of Point Particle and Strings?

Michel,

That’s really a QFT and string theory book, you certainly should have a good understanding of quantum mechanics itself before studying those subjects. I think my notes are best used in conjunction with a standard QM book, they emphasize aspects of QM that aren’t emphasized in the standard treatments, don’t go into much detail on issues that are well covered in the usual textbooks.

Dear Peter,

When trying to see where your lecture notes fit in the general scheme of things, I noticed something.

On the very first page, you write upfront: “In quantum mechanics, the state of a system is best thought of as a different sort of mathematical object: a vector in a complex vector space.” No other formulations of quantum mechanics are mentioned. For an inventory of various formulations of quantum mechanics, see e.g. Daniel F. Styer et al., Nine formulations of quantum mechanics, Am. J. Phys. 70, p. 288-297, March 2002.

Apparently all states you deal with are pure ones, but it is not explicitly stated, mixed states and density matrices are not mentioned.

One of the best expositions of quantum mechanics for mathematicians that naturally comes to mind is Mackey’s Mathematical Foundations of Quantum Mechanics, which is missing from the bibliography. Mackey offers a very general axiomatic definition of quantum states that encompasses both mixed and pure states which seems to be a better way to proceed when lecturing to mathematicians.

mo,

Thanks, that’s an excellent comment. I should probably add this somewhere in the notes to make clear what I’m trying to do, but the decision to not discuss mixed states and density matrices was intentional. Those are important if you want to construct a framework which includes both classical mechanics and quantum mechanics, and/or understand measurement theory and see how classical states emerge as certain mixed states.

My point of view is that such material makes things much more complicated and doesn’t belong in the foundations of the subject, but should be put off until one addresses the difficult issue of measurement theory and the relation to classical mechanics. Bringing these issues in at the beginning to my mind obscures what are the fundamental mathematical structures involved.

Peter, do you have a suggestion for a text that covers the fundamentals of QFT from a point of view that would be pleasant for a mathematician? Essentially something that would come right after what you’re planning for the next semester. I have forgotten most of what I learned in haphazard fashion 20 years ago. I would want something more mathematically oriented than most of the standard physics texts, but not so abstract and high-level as the IAS QFT/String 2-volume set.

David,

Of the things I’ve looked at, Folland’s “QFT: A tourist guide for mathematicians” is the best I’ve seen along these lines. I’ll try and cover some of the basics next semester, which would be a good background for reading Folland.

Folland doesn’t get to many of the things that would most interest mathematicians: non-abelian gauge field theories, 2-d conformal field theories and affine Lie algebras, and TQFTs. Some of those topics are in the IAS volumes.

Thanks Peter, that looks like exactly what I’m after, and I know Folland usually writes well. I’ll take a look at it.

Page 43, the argument that the adjoint representation is a real representation could possibly be made clearer (unless it is in the exercises, which I haven’t yet looked at).

Anonyrat,

Thanks, that is a confusing point. I did go over it in more detail in class (perhaps confusing the students even more). I’ll think about what can be done to improve that.

There are actually two confusing things going on here: the first is that, even when dealing with a group defined by complex matrices (like SU(2)), the Lie algebra I’m talking about is a real vector space (real linear combinations of i times Pauli matrices). All complex linear combinations of Pauli matrices is the Lie algebra of SL(2,C), not SU(2). The second confusing thing is that to get raising and lowering operators, you want to complexify and go to the Lie algebra of SL(2,C) anyway.