Is Segal's notion of conformal field theory a quantum field theory in the sense of Wightman axioms?

Question

For context, I heard that the Atiyah-Segal formalism of quantum field theory is equivalent to the traditional Wightman approach (at least within the scope of certain theories). However, I could not find a precise proof of this, only physical arguments.

Answer 1

My understanding is that Segal invented his formalism (which was then adapted by Atiyah) by thinking about the same thing Wightman was thinking about: formalising the theory of local operators. In hindsight, the theory Segal was groping towards was essentially the theory of factorization algebra of Costello and Gwilliam. Segal was unable to come up with the Weiss locality axiom in factorization algebra: his approach with cobordisms was the best locality he could think of. In hindsight, Segal was extremely prescient, as functorial/cobordism field theory has proved more powerful than you would be able to tell from 2D examples. For example, functorial field theory can organically talk about extended operators, whereas factorization algebra, as currently defined, has trouble axiomatizing them.

For comparison, factorization algebras as we currently know them evolved out of Beilinson–Drinfeld's work on conformal field theory chiral algebras, which globalize the vertex operator algebras first defined mathematically by Frenkel–Lepowsky–Meurman and Borcherds. These are, in turn, essentially Wightman QFTs, implemented for 2D Euclidean-signature conformal field theory.

At the technical level, Segal formalism and Wightman formalism are different. They have different locality conditions, different analytic conditions, and different algebraic conditions. But "formalism" is a vague term. One can, and should, consider ways to adapt formalisms to meet examples. Formalisms are as much about worldviews and mindsets as they are about definitions.

As far as I am aware, in spacetime dimension $>2$, there is no real comparison between these formalisms, except perhaps in the topological case where the comparison becomes much more formal. One issue is a lack of understanding of even what either formalism should be in the nontopological case. Bare categories are not a good setting in which to describe the operator products of nontopological extended operators, because categories are inherently "topological" in the sense that their composition laws are associative and do not depend on the geometry of the pasting/composition diagrams. The Wightman axioms as they exist today only describe the composition of local operators, not extended operators, and these are insufficient to completely describe the quantum field theory. On the cobordism side, it is a technically hard challenge to marry geometry with unitality. Some of these challenges were resolved by Grady–Pavlov, but I find their geometric cobordism category not to accommodate the examples I would consider vital. (Not that every formalism must accommodate every example!) Most notably, Grady–Pavlov do still have lots of adjunctibility/dualizability built into their theory, and this is hard to justify on a moral level.

In the special case of 2D conformal field theory, where all the formalisms are under the best analytic and algebraic control (well, 1D is even better: the formalisms are all under complete control and there is nothing formal left to do), one can hope to establish a collection of adjectives to add to both formalisms so that all the examples you want enjoy those adjectives, and so that those adjectives are enough to make the two formalisms equivalent. I believe that Tenner has done the most meaningful work in this direction.

I mentioned parenthetically that the 1D case is fully controlled. This is a very good example to think through carefully. Stated more strongly, it is a very good example for you, gentle reader, whoever you are, to think through carefully. What is a 1D Segal QFT? (What is a 1D modular functor?) What is a 1D Wightman QFT? How do these compare to the textbook pictures with names like "Schrödinger" and "Heisenberg"? What role does functional analysis play? (What role does the Stone–von Neumann theorem play?) What role does Wick rotation play? And so on. Will you learn something new about quantum mechanics from doing this? Probably not, depending on how much you already "know". But still you, and I really mean you, should think through this carefully: your conclusions might not match someone else's (for example, me), and there is a value to having diverse conclusions.