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I'm a student of mathematics with not much background in physics. I'm interested in learning Quantum field theory from a mathematical point of view.
Are there any good books or other reference material which can help in learning about quantum field theory?
What areas of mathematics should I be familiar with before reading about Quantum field theory?
You can read any standard source, so long as you supplement it with the text below. Here are a few which are good:
There are major flaws with other presentations, these are pretty much the only good ones. I explain the major omission below.
In order for the discussion of the path integral to be complete, one must explain how non-commutativity arises. This is not trivial, because the integration variables in the path integral for bosonic fields or particle paths is over ordinary real valued variables, and these quantities cannot be non-commutative themselves.
The resolution of this non-paradox is that the path integral integrand is on matrix elements of operators, and the integral itself is reproducing the matrix multiplication. So it is only when you integrate over all values at intermediate times that you get a noncommutative order-dependent answer. Importantly, when noncommuting operators appear in the action or in insertions, the order of these operators is dependent on exactly how you discretize them--- whether you put the derivative parts as forward differences or backward differences or centered differences. These ambiguities are all important, and they are discussed only in a handful of places (Negele/Orland Yourgrau/Mandelstam Feynman/Hibbs Polchinski and Wikipedia) and nowhere else.
I will give the classical examples of this, which are sufficient to resolve the general case, assuming you are familiar with simple path integrals like the free particle. Consider the free particle Euclidean action
$$ S= -\int {1\over 2} \dot{x}^2 $$
and consider the evaluation of the noncommuting product $x\dot{x}$. This can be discretized as
$$ x(t) {x(t+\epsilon) - x(t)\over \epsilon} $$
or as
$$ x(t+\epsilon) {x(t+\epsilon) - x(t)\over \epsilon}$$
The first represents $x(t)p(t)$ in this operator order, the second represents $p(t)x(t)$ in the other operator order, since the operator order is the time order. The difference of the second minus the first is
$$ {(x(t+\epsilon) - x(t))^2\over \epsilon} $$
Which, for the fluctuating random walk path integral paths has a fluctuating limit which averages to 1 over any finite length interval, when $\epsilon$ goes to zero. This is the Euclidean canonical commutation relation, the difference in the two operator orders gives 1. For Brownian motion, this relation is called "Ito's lemma", not dX, but the square of dX is proportional to dt. While dX is fluctuating over positive and negative values with no correlation and with a magnitude at any time of approximately $\sqrt{dt}$, dX^2 is fluctuating over positive values only, with an average size of dt and no correlations. This means that the typical Brownian path is continuous but not differentiable (to prove continuity requires knowing that large dX fluctuations are exponentially suppressed--- continuity fails for Levy flights, although dX does scale to 0 with dt).
Although discretization defines the order, not all properties of the discretization matter--- only which way the time derivative goes. You can understand the dependence intuitively as follows: the value of the future position of a random walk is (ever so slightly) correlated with the current (infinite) instantaneous velocity, because if the instantaneous velocity is up, the future value is going to be bigger, if down, smaller. Because the velocity is infinite however, this teensy correlation between the future value and the current velocity gives a finite correlator which turns out to be constant in the continuum limit. Unlike the future value, the past value is completely uncorrelated with the current (forward) velocity, if you generate the random walk in the natural way going forward in time step by step, by a Markov chain.
The time order of the operators is equal to their operator order in the path integral, from the way you slice the time to make the path integral. Forward differences are derivatives displaced infinitesimally toward the future, past differences are displaced slightly toward the past. This is is important in the Lagrangian, when the Lagrangian involves non-commuting quantities. For example, consider a particle in a magnetic field (in the correct Euclidean continuation):
$$ S = - \int {1\over 2} \dot{x}^2 + i e A(x) \cdot \dot{x} $$
The vector potential is a function of x, and it does not commute with the velocity $\dot{x}$. For this reason, Feynman and Hibbs and Negele and Orland carefully discretize this,
$$ S = - \int \dot{x}^2 + i e A(x) \cdot \dot{x}_c $$
Where the subscript c indicates infinitesimal centered difference (the average of the forward and backward difference). In this case, the two orders differ by the commutator, [A,p], which is $\nabla\cdot A$, so that there is an order difference outside of certain gauges. The correct order is given by requiring gauge invariance, so that adding a gradiant $\nabla \alpha$ to A does nothing but a local phase rotation by $\alpha(x)$.
$$ ie \int \nabla\alpha \dot{x}_c = ie \int {d\over dt} \alpha(x(t))$$
Where the centered differnece is picked out because only the centered difference obeys the chain rule. That this is true is familiar from the Heisenberg equation of motion:
$$ {d\over dt} F(x) = i[H,F] = {i\over 2} [p^2,F] = {i/2}(p[p,F] + [p,F]p) = {1\over 2}\dot{x} F'(x) + {1\over2} F'(x) \dot{x}$$
Where the derivative is a sum of both orders. This holds for quadratic Hamiltonians, the ones for which the path integral is most straightforward. The centered difference is the sum of both orders.
The fact that the chain rule only works for the centered difference means that people who do not understand the ordering ambiguities 100% (almost everybody) have a center fetishism, which leads them to use centered differences all the time.
THe centered difference is not appropriate for certain things, like for the Dirac equation discretization, where it leads to "Fermion doubling". The "Wilson Fermions" are a modification of the discretized Dirac action which basically amounts to saying "Don't use centered derivatives, dummy!"
Anyway, the order is important.Any presentation of the path integral which gives the Lagrangian for a particle in a magnetic field without specifying whether the time derivative is a forward difference or a past difference, is no good at all. That's most discussions.
A good formalism for path integrals always thinks of things on a fine lattice, and takes the limit of small lattice spacing at the end. Feynman always secretly thought this way (and often not at all secretly, as in the case above of a particle in a magnetic field), as does everyone else who works with this stuff comfortably. Mathematicians don't like to think this way, because they don't like the idea that the continuum still has got new surprises in the limit. Mathematicians are snobby and wrong.
The other thing that is hardly ever explained properly (except for Negele/Orland, David John Candlin's Neuvo Cimento original article of 1956, and Berezin) is the Fermionic field path integral. This is a separate discussion, so I will refer to these sources for the time being.
Quantum field theory (QFT) in curved spacetime is nowadays a mature set of theories quite technically advanced from the mathematical point of view.
There are several books and reviews one may profitably read depending on his/her own interests. I deal with this research area from a quite mathematical viewpoint, so my suggestions could reflect my attitude (or they are biased in favor of it).
First of all, Birrell and Davies' book is the first attempt to present a complete account of the subject. However the approach is quite old both for ideas and for the the presented mathematical technology, you could have a look at some chapters without sticking to it. Parker and Toms' recent textbook should be put in the same level as the classic by Birrel Davis' book in scope, but more up to date.
Another interesting book is Fulling's one ("Aspects of QFT in curved spacetime"). That book is more advanced and rigorous than BD's textbook from the theoretical viewpoint, but it deals with a considerably smaller variety of topics.
The Physics Report by Kay and Wald on QFT in the presence of bifurcate Killing horizons is a further relevant step towards the modern (especially mathematical) formulation as it profitably takes advantage of the algebraic formulation and presents the first rigorous definition of Hadamard quasifree state.
An account of the interplay of Euclidean and Lorentzian QFT in curved spacetime exploiting zeta-function and heat kernel technologies, with many applications can be found in a book I wrote with other authors ("Analytic Aspects of Quantum Fields" 2003)
A more advanced approach of Lorentzian QFT in curved spacetime can be found in Wald's book on black hole thermodynamics and QFT in curved spacetime. Therein, the microlocal analysis technology is (briefly) mentioned for the first time.
As the last reference I would like to suggest the PhD thesis of T. Hack http://arxiv.org/abs/arXiv:1008.1776 (I was one of the advisors together with K. Fredenhagen and R. Wald). Here, cosmological applications are discussed.
ADDENDUM. I forgot to mention the very nice lecture notes by my colleague Chris Fewster! http://www.science.unitn.it/~moretti/Fewsternotes.pdf
ADDENDUM2. There is now a quick introductory technical paper, by myself and I.Khavkine, on the algebraic formulation of QFT on curved spacetime: http://arxiv.org/abs/1412.5945 which in fact will be a chapter of a book by Springer.
Best Answer
Let me add just a couple of things to what was already mentioned. I do think that the best source for QFT for mathematicians is the the two IAS volumes. But since those are fairly long and some parts are not easy for mathematicians (I participated a little in writing those down, and I know that largely it was written by people who at the time didn't understand well what they were writing about), so if you really want to understand the subject in the mathematical way, I would suggest the following order:
1) Make sure you understand quantum mechanics well (there are many mathematical introductions to quantum mechanics; the one I particularly like is the book by Faddeev and Yakubovsky http://www.amazon.com/Lectures-Mechanics-Mathematics-Students-Mathematical/dp/082184699X)
2) Get some understanding what quantum field theory (mathematically) is about. The source which I like here is the Wightman axioms (as something you might wish for in QFT, but which almost never holds) as presented in the 2nd volume of the book by Reed and Simon on functional analysis; for a little bit more thorough discussion look at Kazhdan's lectures in the IAS volumes.
3) Understand how 2-dimensional conformal field theory works. If you want a more elementary and more analytic (and more "physical") introduction - look at Gawedzki's lectures in the IAS volumes. If you want something more algebraic, look at Gaitsgory's notes in the same place.
4) Study perturbative QFT (Feynmann diagrams): this is well-covered in IAS volumes (for a mathematician; a physicist would need a lot more practice than what is done there), but on the spot I don't remember exactly where (but should be easy to find).
5) Try to understand how super-symmetric quantum field theories work. This subject is the hardest for mathematicians but it is also the source of most applications to mathematics. This is discussed in Witten's lectures in the 2nd IAS volume (there are about 20 of those, I think) and this is really not easy - for example it requires good working knowledge of some aspects of super-differential geometry (also disccussed there), which is a purely mathematical subject but there are very few mathematicians who know it.
There are not many mathematicians who went through all of this, but if you really want to be able to talk to physicists, I think something like the above scheme is necessary (by the way: I didn't include string theory in my list - this is an extra subject; there is a good introduction to it in D'Hoker's lectures in the IAS volumes).
Edit: In addition, if you want a purely mathematical introduction to Topological Field Theory, then you can read Segal's notes http://web.archive.org/web/20000901075112/http://www.cgtp.duke.edu/ITP99/segal/; this is a very accessible (and pleasant) reading! A modern (and technically much harder) mathematical approach to the same subject is developed by Jacob Lurie http://www.math.harvard.edu/~lurie/papers/cobordism.pdf (there is no physical motivation in that paper, but mathematically this is probably the right way to think about topological field theories).