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added to S-matrix a useful historical comment by Ron Maimon (see there for citation)
Sounds interesting, but what’s the key point? String theory ought to be considered as a S-matrix theory to make best sense of it?
To me a key point is that the ways of the scientific community can be convoluted (reminds me of Nolte’s account of the history of the notion of “phase space”) and that one should trust in genuine understanding and not just the in the current folk lore, as that may be confused in subtle ways over decades. In particular it’s important to know what is actually known, what is just expected.
Yes, perturbative string theory is an S-matrix theory. This is so by definition and by design. The point here is that it is a curious irony of history: first S-matrix theory is overthrown and abandoned in favor of quantum field theory, then later some people start to say that quantum field theory needs to be refined by string theory – which in turn is an S-matrix theory that contradicts the claim that every sensible S-matrix is that of a field theory.
One statement of Maimon that particularly resonated with me is the “…the main S-matrix theory, string theory, is not properly explained and remains completely enigmatic even to most physicists.” This is a feeling that have all the time: so much discussion of string theory in the public domain, and so much confusion about what it really is. It may be all wrong, but not for the alleged reasons most commonly voiced about it.
So I think it’s good to be aware of the history here and beware of common lore without thorough scrutinization.
added a quote from a talk by Weinberg in 2009 to the History-section at S-matrix (search for “Weinberg” to find it)
I have tweaked the Idea-section at S-matrix a little and collected the historical comments in a single History-section and then moved that further towards the end of the entry. Then I copied over the technical discussion of the S-matrix in quantum mechanics and perturbative quantum field theory from the entry interaction picture.
(This means that now these two entries have a considerable overlap. But that seems better than one of the two lacking this basic information. Eventually the entries will grow in different directions.)
If
$\vert \psi(t)\rangle_I \coloneqq \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \vert \psi(t)\rangle_S$shouldn’t
$\vert \psi(t) \rangle_S = \exp\left({\tfrac{t}{i \hbar} H}\right) \vert \psi(t) \rangle_I$have $H_{free}$?
Thanks for catching this. Absolutely, as the surrounding text explained, the point is that we have $H_{free}$ here. I have fixed it now.
I have also added another paragraph of text below the definition of the state in the interaction picture, here.
Since that section had been copied (#5) from Dirac interaction picture, those same changes are needed there.
Yes, done now. I have also done a bunch of other edits to this section.
Just in case you are watching the logs and wondering:
I am working on bringing in content on the S-matrix in field theory at S-matrix.
But I am in the middle of it. Don’t have a look unless you are serious about joining in non-trivial editing. I hope to have something coherent, consistent and readable later this week.
I have been further working on S-matrix.
There is now a complete proof spelled out that causal perturbation theory indeed does yield a local net of quantum observables. It culminates in the sub-section Causal locality and Quantum observables The full proof of this crucial statement of pAQFT is somewhat hidden in the literature, it needs the computation of the support of the “generating” retarded products in section 8.1 of the old Epstein-Glaser article, now reproduced as this prop in the entry.
I have also spelled out now in detail how the S-matrix is expressed by the Feynman pertubation series over Feynman diagrams “away from the diagonal” (this section). What remains is discussion of how these products of Feynman propagator distributions are extended to the diagonal, but that discussion should probably be kept only at renormalization, not to overburden the entry on the S-matrix. I’ll get to that.
Also, the first sections on spacetime causality and on free field algebras are not yet expository at the moment, but just terse collection of what is needed later on. I’ll eventually expand on that to make it more readable.
I have added further details to the subsection Perturbative S-matrix and Time-ordered products, such as to complete the full details of the proof of the statement (this prop) that the causal factorization axiom on the time-ordered products implies the causal additivity axiom on the S-matrix.
I have a student interested in the story of the paths of the S-Matrix and QFT approaches. Is there anything to say in retrospect about the structures involved as to why we should have this strange history, leaving aside all the sad history of the psychology drives of the participants. Do we now see why it isn’t so surprising that Veneziano wrote out that amplitude, when apparently thinking about the strong force?
Or to put it another way, is there a “rational reconstruction” which has these two approaches find what each needs in the other, allowing a friendly merger?
There’s been a similar reworking of the phlogiston/oxygen theories of combustion where the best of each survives and hastens the development of chemistry.
On the one hand, Ron Maimon’s comment is spot on in pointing out that even though people eventually realized that most S-matrices come from local Lagrangian field theory, examples from string theory show that not all do. On the other hand, it seems to have remained underappreciated that the only rigorous construction of these QFTs is (or was, until Collini 16) by axiomatization of their S-matrices, namely via causal perturbation theory.
This causal perturbation theory is really what unifies the two perspectives: On the one hand it is just an axiomatization of the S-matrix, on the other hand the key axiom (“causal additivity”) relates the S-matrix to spacetime causality.
But apart from this I feel like the situation remains very interesting in the sense of still requiring deeper understanding. For instance the surprising story of Veneziano’s amplitude that you mention by and large remains just that: endlessly surprising. It feels like despite all these developments, we are still just at the beginning of understanding the full picture.
By the way, you may enjoy reading Geoffrey Chew’s physico-philosophical essays, such as “Quark or Bootstrap”
Thanks! Your second paragraph should be of great interest.
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