Is it indeed the case there are several different S-matrices?
The S-matrix is approached in a quite a few different ways. Here are
seven (or is it six?) examples. I personally bless only the final
three, and only the last of the blessed three is useful in practice.
A. In reslib.com/book/Relativistic_Quantum_Mechanics#88
Bjorken and Drell follow an old approach of Feynman's ("the propagator
approach") which they acknowledge is "less compelling" than using QFT.
B. In reslib.com/book/Quantum_field_theory__Lewis_H_Ryder#175
Ryder may be echoing "the propagator approach". Whatever it is, it
differs from the approaches below, and he attributes it to unpublished
lectures by Veltman.
C. In reslib.com/book/Introduction_to_the_Theory_of_Quantized_Fields#207
Bogolubov and Shirkov follow an old approach (attributed to
Stueckelberg) in which the operator S is a functional of a
spacetime-cutoff function g. (Ick.)
In sections II.3 and II.4 of
Haag describes (or touches on) four approaches:
D. S = lim exp(iH₀t₂) exp(-iH(t₂ - t₁)) exp(-iH₀t₁)
Or some such. If you split the middle exponential into two factors,
you end up with
S = Ω₊ Ω₋
where Ω₊ and Ω₋ are "Moller operators". Haag mentions this only for
comparison. It is not suitable for QFT. But I was taught it in my
first QFT class.
D. Haag-Ruelle collision theory. Now we are getting somewhere. This
is what you would (or should) call a fully-interacting S-matrix. We
start with a Wightman field theory with a field that creates
single-particle states. [We also assume a mass gap. But let's put
infrared issues aside.] By appropriately applying fields to |0> and
taking limits, we obtain "in" and "out" states |α out> and |β in>. α
and β are descriptive of the Fock states of a free field theory.
Indeed, there are free "in" and "out" fields defined on the Hilbert
space along with the interacting field. The S-matrix is then defined
S[α,β] = <α out | β in>
Good, but computationally intractable.
E. Araki-Haag collision theory. I won't go into the details. But we
can now drop the requirement that we have a field that creates
single-particle states. This is important to algebraic QFT-ers,
because fields that create single fermions are unobservable. Again,
F. LSZ collision theory.
Bingo! The starting assumption is that the interacting field weakly
approaches the "in" and "out" fields (mentioned above) in the two
asymptotic limits. Lehmann, Symanzik, and Zimmermann then show how to
S[α,β] = <α out | β in>
from Green functions, ie, from vacuum expectation values of
time-ordered products of fields. And Green functions can be obtained
in perturbation theory using either the Gellman-Low expansion or
Feynman path integrals. (I vote for the latter. In Euclidean space,
followed by Wick rotation.)
To summarize: In practice, we (should) obtain Green functions from
Feynman diagrams (by one of two methods) and then obtain S-matrix
elements from the Green functions à la LSZ. I feel safe in saying
that this is the approach to Feynman diagrams and the S-matrix that is
found in "better" textbooks.
If so, why is it "carefully hidden" in the literature?
Physics went through a difficult phase between 1930 and 1970. Please
read the following quote by Jost:
The recovery may be incomplete.
Is it possible to compute the "fully interacting" S-matrix in
perturbation theory using the Bethe-Salpeter equation to extract the
bound state spectrum?
What you're asking for, if I understand correctly, is some way to get
the infinite sums (that are probed by the B-S equation) into the Green
functions for composite fields that create bound states. Then you
could, ideally, compute proton-proton cross sections.
I'm not aware of such a method. (But I probably wouldn't be aware even
if it existed.)