# Comparison of Different Types of QFT

As far as I can tell, there are a number of major types of quantum field theory. For example,

Constructive QFT, which has two major branches (Algebraic/Axiomatic QFT and Functorial QFT).

Topological QFT, which has three major subsets (Cohomological QFT, Homotopy QFT, and Topological Conformal Field Theory).

Conformal Field Theory, which is the 2d statistical mechanical interpretation of QFT.

Now, what are the major distinctions between these and how do they relate to each other? Are all mathematically rigorous? Which are not dependent on supersymmetry? Are there any other major types of QFT?

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@CarloBeenakker Thanks. – Jimbo Jun 23 '14 at 17:59
That answer doesn't explain if Topological QFTs are rigorous... can they be easily shown to follow the Wightman or Haag-Kastler Axioms? – Jimbo Jun 23 '14 at 18:25
Jimbo, it sounds like you would like someone to write a survey essay on this topic, which I think goes against the customs here in MO. I'll just mention two things. First, your list is missing the kind of QFTs you find described in physics textbooks. Second, your question about 'mathematical rigor' is ill posed. There is not one definition QFT that any mathematical object has to satisfy to get those letters in its name. Everyone has their favorite definitions and works with examples that satisfy it. Nothing non-rigorous about it. – Igor Khavkine Jun 24 '14 at 9:08

There is quite a bit of confusion in the statement "Constructive QFT, which has two major branches (Algebraic/Axiomatic QFT and Functorial QFT)."

Axiomatic QFT provides mathematical definitions for the concept of a QFT, for instance the Garding-Wightman axioms or the Haag-Kastler axioms. Here the word "axiom" can be misleading. This is not about stating self-evident facts that one takes for granted and from which one builds the rest of mathematics like the ZFC axioms. Rather these are similar to the axioms which make up the definition of a group or a vector space (associativity, distributive property etc.). Then, starting from these definitions, Axiomatic QFT studies the consequences which can be derived from them like for instance the CPT or spin-statistics Theorems.

Now this would be a vacuous theory if one did not have specific nontrivial examples of objects satisfying these axioms/definitions. This is where Constructive QFT comes in. It produces (after lots of hard analysis work, mostly about the probability theory of generalized stochastic processes) such nontrivial examples. Typically, the axioms are not the hypotheses but rather the conclusions of a theorem in Constructive QFT.

Algebraic QFT may be seen as a continuation of Axiomatic QFT which emphasizes the operator algebra approach and the Haag-Kastler formulation instead of the Garding-Whiteman axioms. Note however that some recent work (for instance by Grosse and Lechner) in Algebraic QFT also constructs examples of nontrivial QFTs and can therefore be viewed as Constructive QFT also, albeit relying on a different set of techniques. So the boundaries between these complementary areas of research are not sharp.

"Conformal Field Theory, which is the 2d statistical mechanical interpretation of QFT." is also incorrect. Given a QFT say characterized by Euclidean correlators $\langle \phi(x_1)\cdots\phi(x_n)\rangle$ one can consider the limits $\lambda\rightarrow\infty$ and $\lambda \rightarrow 0$ of $\langle \phi(\lambda x_1)\cdots\phi(\lambda x_n)\rangle$ times a suitable power of $\lambda$. If these limits exist then they can be interpreted as the correlators of scale invariant theories ($\lambda\rightarrow\infty$ is the IR or large distance scaling limit and $\lambda\rightarrow 0$ is the UV or short distance scaling limit). Typically the scale invariance of these theories upgrades to the much richer conformal invariance and the primary goal of Conformal QFT is to study such particular examples of QFTs which tell us about the IR or UV behavior of more general (non-scale invariant) QFTs. Conformal QFTs also correspond to renormalization group fixed points and it is important to study them in order to map out the space of more general QFTs.

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