As a graduate student of physics in the early 2000's, our particle physics classes started with quantum field theory, since QCD had long been established as a good model of the nuclear strong force. We sometimes talked about the S-matrix, but as a part of quantum field theory, not an alternative to it: we evaluated Feynman diagrams to compute S-matrix elements.

Recently, I've been reading a lot about the history of the field. In the 1960's, the S-matrix theory of Wheeler and Heisenberg was advocated as an alternative to quantum field theory, one that didn't make assumptions about space-time being a meaningful concept in the intermediate states of a quantum interaction. The troubles they had understanding renormalization and their lack of a field theory model for the strong force would have made it tempting for mid-century physicists to weaken their assumptions about space-time in nuclear interactions, but since the QCD model was discovered and verified (from the observation of gluon jets to the precision of today's lattice QCD calculations) and renormalization is much better understood today (RGEs, etc), those weakened assumptions no longer seem to be necessary. We can be bold in asserting that space-time is a classical manifold on nuclear scales (~10⁻¹⁵ m), just as it is on human scales (~1 m).

Agnosticism about what happens between the incoming free particles and the outgoing free particles may still be relevant at quantum gravity scales (~10⁻³⁵ m) and I hear that S-matrix theory is an important part of modern string/M theory. Frankly, it makes more sense for the notion of a classical space-time manifold to break down at the Plank scale than at the nuclear scale, anyway.

Am I correct in understanding S-matrix theory as a weakened form or subset of quantum field theory—that is, quantum field theory is S-matrix theory plus additional assumptions/claims? I'm asking if the "S-matrix theory" of the 1960's might have also made assumptions/claims that quantum field theory refutes, which would not make it a strict subset.

If it didn't, then S-matrix theory isn't really wrong, is it? It's just incomplete.

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    $\begingroup$ This feels like asking whether Taylor expansion is a subset of Calculus. Yes, sure, but Calculus is not just TE plust additional assumptions. For one thing, Calculus also deals with functions that do not admit a TE at all. On the other hand, as long as you restrict your attention to functions that do admit a TE, it is perfectly reasonable to "bootstrap" Analysis as the study of TE's. One can argue that old-school mathematicians sometimes implicitly assumed that all functions were analytic in their theorems, which they claimed to be valid for all functions. $\endgroup$ Sep 5, 2020 at 14:47
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    $\begingroup$ Furthermore, TE's in a sense go beyond the scope of calculus, because you can also define formal power series in abstract variables. So Calculus and TE's have a very large overlap, but neither contains the other. Similarly, generic QFTs and S-matrix theory tend to deal with mostly the same issues, and for a large number of problems they are both good approaches, but there are QFTs that do not have an S-matrix, and non-QFTs that can be described in the S-matrix spirit (e.g., String Theory). $\endgroup$ Sep 5, 2020 at 14:49
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    $\begingroup$ The modern amplitudes programme - which has replaced the S matrix theory of the 60's - is largely separate from QFT. They are closely related, with some things having a formulation in terms of a path integral and a set of amplitudes, but then there are things only defined on one side, as @AccidentalFourierTransform mentioned. What goes beyond this are things like the amplituhedron, where concepts hard-coded into QFT could be emergent, e.g. locality and unitarity. In that sense, it may be that QFT (+ gravity?) emerges from the S matrix... $\endgroup$
    – Akoben
    Sep 5, 2020 at 15:17
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    $\begingroup$ @AccidentalFourierTransform I think you probably intended to just make a comment, but that's an answer. $\endgroup$
    – user1504
    Sep 5, 2020 at 18:39
  • $\begingroup$ If it's an answer, it's not an answer to the question I was looking for. I know that the use of S-matrices is a technique in QFT (and also potentially beyond it), just as a Taylor expansion is a technique in calculus, but for several decades, "S-matrix theory" was seen as a competitor to QFT. It's that "S-matrix theory" that I'm asking about, not the S-matrix that we use in QFT today. This is a historical question about how physicists in the 1960's to 1970's thought about the two approaches. When they switched to QCD, did they think of it as a stronger statement about nature in a strict sense? $\endgroup$ Sep 6, 2020 at 15:41


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