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I have been reading Quantum Mechanics: A Modern Development by L. Ballentine. I like the way everything is deduced starting from symmetry principles. I was wondering if anyone familiar with the book knows any equally elegant presentation for quantum field theory. Weinberg's books start off nice with the irreducible representations of the Poincare group/algebra but the later chapters lose me with the notation. Also, most books I've read on QFT (Srednicki, Peskin and Schroeder, Mandl and Shaw) make a valiant initial attempt at a nice consistent framework but end up being a big collection of mathematical recipes and intuitive insights that seem to work but the overall structure of the theory seems to be sewn up. The relativistic equations crop out of rather flimsy arguments, the canonical anti/commutation relations are imposed out of density indeterminate air, functional methods are developed cause we know no better, infinities come about with renormalization theory to the rescue but it seems very alien from the initial context. Is there any approach that ties all these things together in an elegant mathematical framework which accounts for all the patch up work that is needed? I am not talking about axiomatization just a global point of view that encompasses all the issues.

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Robin Ticciati's Quantum Field Theory for Mathematicians might be what you are after! –  Alex Nelson Nov 19 '13 at 17:51

5 Answers 5

I have not read it myself, but did you take a look at Zee's book? I heard that while it is not very technical, it is great on a conceptual level.

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Judging by the content the book tries to cover a lot of ground so, as you said, it probably doesn't go deeper in the mathematics. I'll have a look. Thanks. –  Sebby Jan 10 '13 at 7:28
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It's an excellent book to get an overall feel for QFT, - it uses the functional approach quite extensively –  twistor59 Jan 10 '13 at 9:33

I wish I could point to a single book that does what you want. The best I can offer is this suggestion: Read (or rather, skim) a few books on axiomatic field theory. Just get a sense for what the physics being described by the axioms is. Draw cartoons. Don't get too hung up on the topology.

I'd start with Haag's book, Local Quantum Physics. Learn about nets of algebras, which are a way of encoding the idea that physical observables actually are located somewhere in spacetime. Also does a nice job of explaining the idea that there's some sort of algebra of observables, and that states -- both pure states and density matrices -- are linear functionals on this algebra. Notice how crucial the fact that spacelike separated observables commute is to matching up the product of operators with the product of observables.

The thing really missing in Haag's book, if you're reading axiomatics as a way of organizing you're thinking about the material in some particle physics book, is a notion of field. The algebras in Haag's book really ought to be generated by 'local observables'. There should be operator-valued functions -- or rather, distributions -- on spacetime, and you should get the algebras in Haag's book by smearing these operators with test functions and then multiplying them. There is a language for this, which is nicely explained in Streater & Wightman's PCT, Spin, Statistics, and All That. This book also explains a critical reconstruction theorem: If you can write down all of the correlation functions of a QFT, then you can recover the Hilbert space & operators directly from these correlation functions. Thinking of the basic local fields as generators of the algebra of observables is a good idea, but it's more complicated than you might think. The product of two fields at a point is in general singular. Equivalently, many of the correlation functions diverge as you approach the diagonals. You can define non-trivial products by subtracting away these divergences.

It's worth reading Hollands & Wald's paper Axiomatic Quantum Field Theory in Curved Spacetime, at about this point. They axiomatize QFT on globally hyperbolic spacetimes in terms of operator product expansions. Hollands and Wald's point of view is very close to the modern Wilsonian point of view on QFT, which says a) that a QFT is a deformation of a conformal field theory, and b) that a CFT is a collection of local observables obeying an OPE which satisfies the bootstrap condition. You could probably alter their diffeomorphism language to something involving conformal maps and get a definition.

It's also worth spending more time thinking about OPEs and CFTs. Spend some time with diFrancesco et al's Conformal Field Theory or maybe one of the math books on vertex algebras (Kac, Vertex Algebras for Beginners or Frenkel-Ben-Zvi Vertex Algebras & Algebraic Curves)?

Then at last, the path integral. It's not the most general definition of a QFT, but it's very useful. Basically, it gives you a way of writing down the correlation functions, and from the correlation functions, you can read off the OPEs, recover the Hilbert space, etc. The basic idea is best learned first in lattice QFT; I like Montvay & Munster's Quantum Fields on the Lattice. This is also a good time to learn a bit about gauge fields.

The biggest annoyance of the path integral approach is that the funcrtions on field space which are integrable with respect to the continuum measure -- the ones that represent observables in whatever vacuum you're playing in -- can be hard to express in terms of variables that people typically use to construct the measure. This is what renormalization is all about; it gives you a way of using stupid variables and then isolating the main contributions to the correlation functions you're after.

The standard mathematical reference on path integrals is Glimm & Jaffe, Quantum Physics: A Functional Integral Point of View. Most of what you'll want is in Chapter 3,6, or Appendix A. They spend most of the book grinding out a detailed construction of the correlation functions of 2d massive scalar field theory. Figuring out how their constructions are related to standard renormalization language is a worthwhile exercise.

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I enjoyed Brian Hatfield's "Quantum Field Theory of Point Particles and Strings." Part 1 of the book is great IMO, Part 2 is useless. Also make sure to get the newest edition, there are many errors in the older editions, but the work is still good.

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I like that part 1 of the book is rather concise and technical enough to build up a more mathematical view. But again, it seems to follow to same algorithm in writing QFT books as the ones I mention. For introductory purposes this is fine I guess but I'm not necessarily looking for an introduction, rather a restructuring under a small set of neatly justified ideas with associated mathematical objects that fit nicely together. –  Sebby Jan 10 '13 at 7:40
    
+1 one of the few books that deal with the functional Schroedinger picture –  twistor59 Jan 10 '13 at 9:39

I know you're probably aiming for a textbook, but I can recommend giving Coleman's original lectures (made available for all on the Harvard website). While he advances rather slowly, and some of the stuff is fairly outdated, he gives many delightful insights, and I find his manner of introducing Quantum Fields one of the most illuminating I have come across so far. Give them a try if you can find the time. In fact, there are accompanying lecture notes made available on the arXiv, linked to on the same page, if you find the actual lectures are moving a bit slowly.

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If you are looking for a book with an "elegant mathematical framework which accounts for all the patch up work that is needed" to build a realistic perturbatively renormalizable quantum field theory like the recently completed and validated Standard Model, its title could be : Noncommutative Geometry, Quantum Fields and Motives. It is written by Alain Connes and Mathilde Marcolli two mathematicians whose work is strongly inspired by quantum field theories .

The first chapter addresses for instance the conceptual understanding of the algebra (with their representations for particles) required to describe the gauge symmetries (and quantum anomalies cancellations) in the Standard Model as well as a "clear mathematical interpretation to the renormalization procedure used by physicists to extract finite values from the divergent expressions obtained from the evaluations of the integrals associated to Feynman diagrams". Unfortunately this book does not attempt to be instructional and it is clearly focused on a mathematical audience!

Nevertheless non commutative geometry is definitely a deep approach that ties important archipelagos of contemporary quantum field knowledge and could bring new insight for physicists but it is definitely a work still in progress just like M-theory I guess. On the opposite quantum mechanics appears to be built on old firm foundations (linear algebra, Hilbert spaces) elaborated mostly by mathematicians from the XIX century who distilled the models of heat transport and electromagnetic wave propagation imagined by Fourier, Maxwell and others...

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