FQFT and cohomology
Types of quantum field thories
From this physics.SE comment by Ron Maimon:
The history of physics cannot be well understood without appreciating the unbelievable antagonism between the Chew/Mandelstam/Gribov S-matrix camp, and the Weinberg/Glashow/Polyakov Field theory camp. The two sides hated each other, did not hire each other, and did not read each other, at least not in the west. The only people that straddled both camps were older folks and Russians— Gell-Mann more than Landau (who believed the Landau pole implied S-matrix), Gribov and Migdal more than anyone else in the west other than Gell-Mann and Wilson. Wilson did his PhD in S-matrix theory, for example, as did David Gross (under Chew).
In the 1970s, S-matrix theory just plain died. All practitioners jumped ship rapidly in 1974, with the triple-whammy of Wilsonian field theory, the discovery of the Charm quark, and asymptotic freedom. These results killed S-matrix theory for thirty years. Those that jumped ship include all the original string theorists who stayed employed: notably Veneziano, who was convinced that gauge theory was right when t'Hooft showed that large-N gauge fields give the string topological expansion, and Susskind, who didn’t mention Regge theory after the early 1970s. Everybody stopped studying string theory except Scherk and Schwarz, and Schwarz was protected by Gell-Mann, or else he would never have been tenured and funded.
This sorry history means that not a single S-matrix theory course is taught in the curriculum today, nobody studies it except a few theorists of advanced age hidden away in particle accelerators, and the main S-matrix theory, string theory, is not properly explained and remains completely enigmatic even to most physicists. There were some good reasons for this— some S-matrix people said silly things about the consistency of quantum field theory— but to be fair, quantum field theory people said equally silly things about S-matrix theory.
Weinberg came up with these heuristic arguments in the 1960s, which convinced him that S-matrix theory was a dead end, or rather, to show that it was a tautological synonym for quantum field theory. Weinberg was motivated by models of pion-nucleon interactions, which was a hot S-matrix topic in the early 1960s. The solution to the problem is the chiral symmetry breaking models of the pion condensate, and these are effective field theories.
Building on this result, Weinberg became convinced that the only real solution to the S-matrix was a field theory of some particles with spin. He still says this every once in a while, but it is dead wrong. The most charitable interpretation is that every S-matrix has a field theory limit, where all but a finite number of particles decouple, but this is not true either (consider little string theory). String theory exists, and there are non-field theoretic S-matrices, namely all the ones in string theory, including little string theory in (5+1)d, which is non-gravitational.
From (Weinberg 09, p. 11):
I offered this in my 1979 paper as what Arthur Wightman would call a folk theorem: “if one writes down the most general possible Lagrangian, including all terms consistent with assumed symmetry principles, and then calculates matrix elements with this Lagrangian to any given order of perturbation theory, the result will simply be the most general possible S-matrix consistent with perturbative unitarity, analyticity, cluster decomposition, and the assumed symmetry properties.”
There was an interesting irony in this. I had been at Berkeley from 1959 to 1966, when Geoffrey Chew and his collaborators were elaborating a program for calculating S-matrix elements for strong interaction processes by the use of unitarity, analyticity, and Lorentz invariance, without reference to quantum field theory. I found it an attractive philosophy, because it relied only on a minimum of principles, all well established. Unfortunately, the S-matrix theorists were never able to develop a reliable method of calculation, so I worked instead on other things, including current algebra. Now in 1979 I realized that the assumptions of S-matrix theory, supplemented by chiral invariance, were indeed all that are needed at low energy, but the most convenient way of implementing these assumptions in actual calculations was by good old quantum field theory, which the S-matrix theorists had hoped to supplant.
Here a quantum field theory is given by a functor
To a codimension-1 slice of space this assigns a vector space – the (Hilbert) space of quantum states over ;
to a spacetime/worldvolume manifold with boundaries one assigns the quantum propagator which is the linear map that takes incoming states to outgoing states via propagation along the spacetime/worldvolume . This is alternatively known as the the scattering amplitude or S-matrix for propagation from to along a process of shape .
Now for genuine topological field theories all spaces of quantum states are finite dimensional and hence we can equivalently consider the dual vector space (using that finite dimensional vector spaces form a compact closed category). Doing so the propagator map
equivalently becomes a linear map of the form
Notice that such a linear map from the canonical 1-dimensional complex vector space to some other vector space is equivalently just a choice of element in that vector space. It is in this sense that is equivalently a vector in .
Segal’s axioms for FQFT (CFT in his case) were originally explicitly about the propagators/S-matrices, while Atiyah formulated it in terms of the correlators this way. Both perspectives go over into each other under duality as above.
Notice that this kind of discussion is not restricted to topological field theory. For instance already plain quantum mechanics is usefully formulated this way, that’s the point of finite quantum mechanics in terms of dagger-compact categories.
Wikipedia, S-matrix theory