I am taking a course this semester on QFT, which deals much with constructive quantum field theory. Some of its topics so far involve relationships between non-Gaussian probability measures,Feynman path integrals and so on. While I am totally new to the subject of QFT (I only have some basic ideas of classical mechanics, classical field theory and quantum mechanics in terms of physics, with knowledge in geometry, topology and analysis for a 2nd year graduate student), I found myself totally lost in the study of that course. On the other hand, my professor tends to talk about everything in a rather hand-waving way with neither strict definitions nor proofs, which made my struggling even worse. Now I have to come for help: are there any references on constructive quantum field theory that would save me out of these? I expect to see clear definitions and basic introductions so that I can enter this field without the trouble of getting myself familiar with the most fundamental stuff firstly. Thank you!
Constructive Quantum Field Theory – Reference Request
mp.mathematical-physicspr.probabilityquantum-field-theoryreference-request
Related Solutions
If I read your updated question correctly, you are asking whether people have considered non-linear modifications of quantum mechanics in order to accommodate interacting QFTs. I'm sure someone, somewhere has, but that's certainly not mainstream thought in QFT research, either on the mathematics or theoretical physics sides. Consider the analogous question in the quantum mechanics of particles: do non-linear equations of motion require a non-linear modification of quantum mechanics? The answer is most certainly No.
Without going into generalities, the Hydrogen atom and the double-well potential are prominent examples of systems with non-linear (Heisenberg) equations of motion that live perfectly well within the standard quantum formalism (states form a linear Hilbert space, observables are linear operators on states, time evolution is unitary on states in the Schroedinger picture and conjugation by unitary operators in the Heisenberg picture). When going from particle mechanics to field theory, what changes is the number of space-time dimensions, not the type of non-linearities in the equations of motion. So there is no mathematical reason to expect a non-linear modification of quantum mechanics in the transition.
Now, a few words about your intuition regarding states as solutions to the equations of motion. Unfortunately, it is somewhat off the mark. As you should be aware, relativistic QFT is usually discussed in the Heisenberg picture. This means that it is the field operators $\hat{\phi}(t,x)$ that obey the possibly non-linear equations of motion. For example, $\square\hat{\phi}(t,x) - \lambda{:}\hat{\phi}^3(t,x){:}=0$, where $\square$ is the wave operator and the colons denote normal ordering. On the other hand, states are just elements $|\Psi\rangle$ of an abstract Hilbert space (with the vacuum state $|0\rangle$ singled out by Poincaré invariance), entirely independent of spacetime coordinates. At this point, it should be clear why states have nothing to do with the equations of motion.
Your intuition is not entirely without basis, though. Spelling it out, also shows how the standard formalism of QFT (Wightman or any related one) already accommodates non-linear interactions. One can define the following hierarchy of $n$-point functions (sometimes called Wightman functions): \begin{align} W^0_\Psi &= \langle 0|\Psi\rangle \\ W^1_\Psi(t_1,x_1) &= \langle 0|\hat{\phi}(t_1,x_1)|\Psi\rangle \\ W^2_\Psi(t_1,x_1;t_2,x_2) &= \langle 0|\hat{\phi}(t_1,x_1)\hat{\phi}(t_2,x_2)|\Psi\rangle \\ & \cdots \end{align} It is a fundamental result in QFT (known under different names, such as the Wightman reconstruction theorem, multiparticle representation of states, or simply second quantization) that knowledge of all the $W^n_\Psi$ is completely equivalent to the knowledge of $|\Psi\rangle$.
These Wightman functions, by virtue of the Heisenberg equations of motion, satisfy the following infinite dimensional hierarchical system of equations \begin{align} \square_{t,x} W^1_\Psi(t,x) &= \lambda W^3_\Psi(t,x;t,x;t,x) + \text{(n-ord)} \\ \square_{t,x} W^2_\Psi(t,x;t_1,x_1) &= \lambda W^4_\Psi(t,x;t,x;t,x;t_1,x_1) + \text{(n-ord)} \\ \square_{t,x} W^2_\Psi(t_1,x_1;t,x) &= \lambda W^4_\Psi(t_1,x_1;t,x;t,x;t,x) + \text{(n-ord)} \\ & \cdots \end{align} I'm being a bit sloppy with coincidence limits here. The Wightman functions are singular if any two spacetime points in their arguments coincide, the terms labeled (n-ord) represent the necessary regulating subtractions to make this limit finite. This necessary regularization also explains why the non-linear terms in the equations of motion needed normal ordering.
If $\lambda=0$, the theory is non-interacting, then each of the above equations for the $W^n_\Psi$ becomes self-contained (independent of $n$-point functions of different order) and identical to the now linear equations of motion. At this point it should be clear how your intuition does in fact apply to the states of a non-interacting QFT. States $|\Psi\rangle$ can be put into correspondence with multiparticle "wave functions" solving the linear equations of motion (which are actually the Wightman functions $W^n_\Psi$).
Finally, when it comes to trying to construct models of QFT, people usually just concentrate on the Wightman functions associated to the vacuum state, $W^n_0 = \langle 0|\cdots|0\rangle$, which are sufficient to reconstruct the corresponding $n$-point functions for all other states. In short, the standard approaches to constructive QFT already incorporate non-linear interactions in a natural way. And non-linear modifications to the quantum mechanical formalism are simply a whole different, independent topic.
Modern constructive field theory is based on rigorous implementations of the renormalization group (RG) approach. To get an idea of what this is about see this short introductory paper. The RG is an infinite dimensional dynamical system and constructing a QFT essentially means constructing an orbit which typically joins two fixed points. So first you need a fixed point (for instance the massless Gaussian field) and you need it to have an unstable manifold which is not entirely made of Gaussian measures (trivial QFTs). In 4d the only fixed point we have at our disposal is the Gaussian one and at least at the level of perturbation theory one has strong indications that for models like phi-four and even much more complicated generalizations, the corresponding unstable manifold is Gaussian. The only models in 4d known not to suffer from this problem are non-Abelian gauge theories and their construction (in infinite volume) is a difficult question (one of the 7 Clay Millennium Problems).
The main technical obstacles for having good candidates to even consider constructing are stability (being in the region of positive coupling constant) and Osterwalder-Schrader positivity. In 4d one should be able to construct a phi-four model with fractional propagator $1/p^{\alpha}$ with $\alpha$ slightly bigger than 2 (the standard propagator). There are partial rigorous results in this direction by Brydges, Dimock and Hurd: "A non-Gaussian fixed point for $\varphi^4$ in $4−\varepsilon$ dimensions". Unfortunately, such a model would most likely not satisfy OS positivity.
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Best Answer
The standard reference for constructive QFT is the classic book by J. Glimm and A. Jaffe, Quantum Physics: a Functional Integral Point of View (2nd. ed., Springer-Verlag, 1988). It is certainly more than satisfactory from the viewpoint of mathematical rigor, it has a lot of background material (specially the second edition linked above) and parts of it can also be read by theoretical physicists with benefit, since it collects and derives many useful formulae. I have a friend who works on string theory and wanted to have this book badly for this reason.
Other books which deal with more restricted questions and / or methods in constructive QFT include:
R. Fernández, J. Fröhlich, A. D. Sokal, Random Walks, Critical Phenomena and Triviality in Quantum Field Theory (Springer-Verlag, 1992). It has an excellent discussion of renormalization group ideas and continuum limits. Its main goal are the famous triviality results in constructive QFT;
V. Rivasseau, From Perturbative to Constructive Renormalization (Princeton University Press, 1991). It discusses the transition mentioned in the title with more detail than Glimm-Jaffe, albeit in a slightly more informal way. It has a last chapter on the construction of finite-volume Yang-Mills theory in 4 dimensions, but it lacks the later results which comprise Rivasseau's landmark paper together with Magnen and Sénéor on this hard problem;
G. Battle, Wavelets and Renormalization (World Scientific, 1999). This book focuses on multiscale cluster expansion methods, just as Rivasseau's book above, but Battle's approach (developed together with P. Federbush) is based on certain families of wavelets he proposed together with P. G. Lemarié in the 80's;
V. Mastropietro, Non-Perturbative Renormalization (World Scientific, 2008).
See also this related MO question for more references.