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If a mathematician who doesn't know much about the physicist's jargon and conventions had the curiosity to learn how the so called Standard Model (of particle physics, including SUSY) works, where should (s)he have a look to?

References (if they exist!) written for a mathematical target (so can assume e.g. basic differential geometry, basic Lie group theory...) in a "mathematical style" with rigorous definitions, theorems and proofs would be appreciated.

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Google says: Introduction-Elementary-Particles-David-Griffiths – Niyazi Mar 7 '11 at 13:01
@Niyazi: thank you for noting about google. I dont't think the question is offtopic though: we can gather various references for the benefit of the readers. [oh, I should probably communitywikify the question!] – Qfwfq Mar 7 '11 at 13:25
QFT for mathematicians by Ticciati: – Steve Huntsman Mar 7 '11 at 14:38
I find Ticciati's book merely okay. He leaves an awful lot of loose mathematical ends lying around. – Todd Trimble Mar 7 '11 at 21:58
This is similar to this question at the Physics Exchange:… – aleph0 Oct 31 '11 at 19:41

11 Answers 11

For the standard model, and in particular for its representation-theoretic aspects (which are crucial), I would refer you to the excellent recent article by John Baez and John Huerta from the Bulletin of the American Mathematical Society which can be found here:

There are also references to other articles and books here that could lead you further.

If you are interested more generally in quantum field theory and its description for mathematicians (where differential geometry plays a big role, in addition to representation theory), then there is the infamous 2-volume "Quantum Field and Strings: A course for mathematicians" which is written by (mostly) mathematicians. It's not going to necessarily give you the correct physical insight, however. Here are the links:

Volume 1

Volume 2

Other good possibilities are Freed-Uhlenbeck's "Geometry of Quantum Field Theory" from the PCMI (Park City) series, or the gargantuan "Mirror Symmetry" from the Clay Math monographs.

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Presumably these two volumes correspond to the Autumn and Spring semesters of the programme in Quantum Field Theory for mathematicians held at the Institute for Advanced study during the academic year 1996--97. The notes are available at $$ $$ $$ $$ – Chandan Singh Dalawat Mar 7 '11 at 14:52
@Chandan: Yes, they are exactly the lectures (probably greatly expanded) from that IAS program. – Spiro Karigiannis Mar 7 '11 at 15:37
@Spiro: I don't think that the lectures have been greatly expanded, actually. The stuff Chandan has linked to is very close (if not identical) to the books. – José Figueroa-O'Farrill Mar 7 '11 at 17:33
@José: Good to know. I wish I knew that before I bought the books. But still, PDFs onscreen cannot compare to physically holding a book in your hands. – Spiro Karigiannis Mar 7 '11 at 17:37
Sorry, but I find those volumes fairly daunting, dispiriting, almost a cruel joke. A much gentler introduction would be most desirable. – Todd Trimble Mar 7 '11 at 22:01

An excellent introduction for a mathematician without previous exposure to quantum field theory is the book by Gerald Folland: "Quantum field theory, a tourist guide for mathematicians", ISBN: 978-0-8218-4705-3. To understand the standard model, one first needs to learn about quantum field theory, since this is an example of QFT model, although a rather formidable one. I think you will have a hard time finding a more pedagogical introduction to this subject than Folland's book.

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Another book in that vein which just came out is "quantum mechanics and quantum field theory, a mathematical primer" by Jon Dimock at Cambridge U. Press. Check it out! It does not go all the way to the standard model, but gets to QED at least. It is quite short so could be ideal as a first thing to read before going into the other more advanced references mentioned in this post. – Abdelmalek Abdesselam Mar 22 '11 at 15:26

The Folland book mentioned here is quite good. One of the most straight-forward physics references might be Pierre Ramond's "Field Theory: A Modern Primer", but it's still a long ways from mathematical rigor. Some comments about the other books and topics discussed here:

Weinberg's books are very good in their own way, but not really appropriate for mathematicians. The first one develops QFT not so much in terms of fundamental objects, but as a phenomenological framework forced upon us by principles such as special relativity and locality. The second one does gauge theory without using geometry, or coordinate-invariant notation, which is not a great idea for mathematicians. The third one is just about SUSY, concentrating on the parts of the subject not of much mathematical interest (the IAS volumes do the opposite).

About the IAS volumes, one should keep in mind that the main point of that exercise was to try to explain to mathematicians Seiberg-Witten theory as understood by physicists in terms of N=2 supersymmetric QFT. This has nothing to do with the Standard Model, and from what I remember the Standard Model doesn't appear in those volumes. They do contain a truly spectacularly good set of lectures by Witten on QFT (but not written up by him...), aimed at getting to the Seiberg-Witten story. This involves some heavy-duty use of non-perturbative supersymmetric quantum field theory, of the sort that is of mathematical interest in building TQFTs.

Besides not explaining the Standard Model, I don't think the IAS lectures really explain the use of supersymmetry to extend the Standard Model (the MSSM "minimal supersymmetric standard model"). This is a subject that has always been heavily advertised without much explanation of its significant problems, one of which is an extra 120 or so parameters. Initial results from the LHC rule out nearly half the most popular region in parameter space, chosen for simplicity and assuming that supersymmetry can be used to solve certain problems (dark matter particle, anomaly in measurement of muon magnetic moment). This still leaves the other half, as well as a lot of other less popular regions of parameter space. Over the next year or two I believe we'll see increasingly large regions of parameter space ruled out, but there is no way the LHC can rule out all of it. All it can do is change somewhat how physicists evaluate the likelihood of nature being described by conventional supersymmetric extensions of the Standard Model, a process which has started and will continue.

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I'm neither a physicist nor a mathematical physicist but I've taken some recreational interest in learning about the subject and about QFT and the standard model in particular. What follows is the recommendations of a complete QFT novice and is admittedly somewhat off topic, but hopefully will be useful to someone. I found both Feynman's "The strange theory of light and matter" and Griffith's "Introduction to elementary particles" very helpful. These are not math books and Feynman's has essentially no details (or even equations). But the last half of Feynman's book (especially the last lecture) is good for giving an intuitive understanding of what the mathematics is trying to formalize (this is something I found maddening about the many mathematical accounts of QFT I've read). It is also appealing that you can read the book in a couple afternoons (probably no other book on this topic can boast this). Griffith's book fills in a number of blanks in Feynman's book. My main reaction to the mathematical treatments I've seen of QFT is that it is hard to gain intuition as to what the definitions and axioms are really intending to model. Both these books helped a lot in remedying this, at least for me. Read them first and then hunt down the mathematical treatments.

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For a straightforward and quick intro to the standard model, try "Groups and Symmetries: From Finite Groups to Lie Groups" by Kosmann-Schwarzbach. It's rigorous and does a good job motivating the standard model in its later chapters. You'll learn what a quark is from the mathematical point of view.

In addition, Griffith's textbook on elementary particle physics would be a good historical supplement. It took physicists many years and guesses to work out the standard model. The first few chapter of Griffith's book read like a good mystery novel. Plus, you'll be a little more familiar with weird concepts like isospin, strangeness and color.

Finally, for more talk related to particle physics the classic text "Quarks and Leptons" by Halzen and Martin is really in-depth, but does assume a good grasp on physics. It does a good job of explaining concepts in the context of group theory. I would say, try to read the discussions in it rather then get bogged down in the physics.

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+1 for pointing this book out. I didn't know it -- but it looks like a nice undergraduate level book for physicists. – José Figueroa-O'Farrill Mar 7 '11 at 18:49
The title of the book makes me very optimistic: I understand finite groups and their representations rather well and have a halfway decent working knowledge of Lie groups / algebras / representations thereof. Am I really so close to being able to understand modern particle physics? Why did no one tell me sooner?? – Pete L. Clark Mar 7 '11 at 19:27
I don't know, Pete: did you ask? :-) – Todd Trimble Mar 7 '11 at 21:55
@Pete, Todd: Every physical theory (and QFT is no exception) has both "kinematical" and "dynamical" aspects. Kinematics is basically representation theory, but dynamics is not. (The exception to this rule might large chunks of two-dimensional conformal field theory; although not everyone might agree.) Hence with a good knowledge of representation theory, one can understand how to set up the QFT models. To extract physically meaningful results, though, requires studying the dynamics... and therein lies the rub, as they say. – José Figueroa-O'Farrill Mar 7 '11 at 23:08
@Todd: well, not recently, no. I always thought that my "math: yes; physics: huh?" stance was pretty clear. – Pete L. Clark Mar 9 '11 at 6:54

All of the above answers are good, but I think these might be closer to what you're looking for:

Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists (v. 1)

Quantum Field Theory II: Quantum Electrodynamics: A Bridge between Mathematicians and Physicists

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These are not good books to learn from. I find these two to be to long and drawn out. They lack focus. They are missing and overall arc or plot and feel more like an amalgam of thousands of snippets written by different people with little regard to what the others were writing. These books may contain everything, but they also contain everything. On the plus side, I enjoy the historical annotations and stories. – aleph0 Oct 31 '11 at 20:01

I make no pretense at understanding the standard model myself, but would like to mention Connes and Marcolli's "Noncommutative Geometry, Quantum Fields and Motives" (see chapters 9-19. This quote is from the introduction to chapter 9:

"Our main purpose is to show that the full Lagrangian of the Standard Model minimally coupled to gravity, in a version that accounts for neutrino mixing, can be derived entirely from a very simple mathematical input, using the tools of noncommutative geometry. This will hopefully contribute to providing a clearer conceptual understanding of the wealth of information contained in the Standard Model, in a form which is both palatable to mathematicians and that at the same time can be used to derive specific physical predictions and computations."

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I thought that "Mathematical aspects of quantum field theory" by Edson de Faria and Welington de Melo was nicely written.

Summary from the Publisher: "Over the last century quantum field theory has made a significant impact on the formulation and solution of mathematical problems and has inspired powerful advances in pure mathematics. However, most accounts are written by physicists, and mathematicians struggle to find clear definitions and statements of the concepts involved. This graduate-level introduction presents the basic ideas and tools from quantum field theory to a mathematical audience. Topics include classical and quantum mechanics, classical field theory, quantization of classical fields, perturbative quantum field theory, renormalization, and the standard model. The material is also accessible to physicists seeking a better understanding of the mathematical background, providing the necessary tools from differential geometry on such topics as connections and gauge fields, vector and spinor bundles, symmetries, and group representations"

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Yes, although I only had a quick glance at it, it looks quite good. However, I was disappointed by section 8.1 about BPHZ renormalization. It does not even state what the theorem is. Just defining the renormalized amplitude of a diagram by Zimmermann's forest formula and saying that it converges does not foot the bill. – Abdelmalek Abdesselam Mar 9 '11 at 19:36

G. Scharf, Quantum Gauge Theories: Spin One and Two (can be downloaded as a PDF)

A thin book that covers the basics of free and interacting fields in a mathematically rigorous way, at the level of formal power series in the interaction strength. Essentially proves the renomralizability of quantum Yang-Mills theories (a large part of the standard model) and the necessity for a Higgs field. Treats perturbative gravity as well. Since the emphasis is on the methods, not much time is spent exploring features specific to the standard model. But this book is really hard to beat when it comes to mathematical rigor among other books aimed at a similar audience.

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Where did you find this book available as a PDF? A quick Google hunt didn't turn it up. – L Spice Sep 24 '12 at 15:18
@L Spice: Follow the Google Books link. It's not very intuitive, but the direct PDF download link is available under the "gear" drop-down menu. – Igor Khavkine Sep 24 '12 at 16:00
@Igor Khavine, thanks! – L Spice Sep 26 '12 at 22:37

This was just posted to the arXiv today:

M. J. D. Hamilton. "The Higgs boson for mathematicians. Lecture notes on gauge theory and symmetry breaking." arXiv:1512.02632 [math.DG].

"These notes form part of a lecture course on gauge theory. The material covered is standard in the physics literature, but perhaps less well-known to mathematicians. The purpose of these notes is to make spontaneous symmetry breaking and the Higgs mechanism of mass generation for elementary particles more easily accessible to mathematicians interested in theoretical physics. We treat the general case with an arbitrary compact gauge group G and an arbitrary number of Higgs bosons and explain the situation in the classic case of the electroweak interaction where G=SU(2)xU(1). Prerequisites are only a basic knowledge of Lie groups and manifolds. No prior knowledge of gauge theory or bundle theory is assumed."

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It is worth to note that SUSY model now is in doubt, because of the LHC experiments. It is growing consensus that it is probably wrong at least as regard to supersymmetry, although it may not have influence on mathematical apparatus behind of it,which is worth of knowing of course.

I suggest to check Steven Weinberg "The Quantum Theory of Fields" Vol 1,2,3 -

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The LHC is at such an early stage that it has not ruled anything out. You have clearly been reading blogs with an axe to grind with respect to SUSY. I'm not arguing one way or the other (I like SUSY mathematically, but despite being "in the house of Higgs" have a fundamental mistrust of fundamental scalars), but remain agnostic until the data comes in. – José Figueroa-O'Farrill Mar 7 '11 at 17:32
Weinberg is a bad choice, in my opinion - the book is incredibly thick and difficult to understand on first read. Also, I'm not sure if it would appeal to a mathematically-minded person. – Michal Kotowski Mar 7 '11 at 18:26
Weinberg's book I think does a slightly better job with the mathematics than many QFT books written by physicists, such as the one by Peskin and Schroeder. – Todd Trimble Mar 7 '11 at 21:52

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