S-matrix theory was a proposal for replacing
local quantum field theory as the basic principle
of elementary particle physics.
It avoided the notion of space and time by
replacing it with abstract mathematical properties
of the S-matrix. In S-matrix theory, the S-matrix
relates the infinite past to the infinite
future in one step, without being decomposable
into intermediate steps corresponding to time-slices.
This program was very influential in the 1960s,
because it was a plausible substitute for
quantum field theory, which was plagued with
the zero interaction phenomenon at strong
coupling. Applied to the strong interaction,
it led to the development of string theory.
S-matrix theory was largely abandoned by physicists
in the 1970s, as quantum chromodynamics was
recognized to solve the problems of strong
interactions within the framework of field
theory. But in the guise of string theory,
S-matrix theory is still a popular approach
to the problem of quantum gravity.
The S-matrix theory is related to the holographic
principle and the AdS/CFT correspondence by
a flat space limit. The analog of the S-matrix
relations in AdS space is the boundary conformal
theory.The most lasting legacy of the theory
is string theory. Other notable achievements
are the Froissart bound, and the prediction
of the pomeron.
== History ==
S-matrix theory was proposed as a principle
of particle interactions by Werner Heisenberg
in 1943, following John Archibald Wheeler's
1937 introduction of the S-matrix.It was developed
heavily by Geoffrey Chew, Steven Frautschi,
Stanley Mandelstam, Vladimir Gribov, and Tullio
Regge. Some aspects of the theory were promoted
by Lev Landau in the Soviet Union, and by
Murray Gell-Mann in the United States.
== Basic principles ==
The basic principles are:
Relativity: The S-matrix is a representation
of the Poincaré group;
Unitarity:
S
S
†
=
1
{\displaystyle SS^{\dagger }=1}
;
Analyticity: integral relations and singularity
conditions.The basic analyticity principles
were also called analyticity of the first
kind, and they were never fully enumerated,
but they include
Crossing: The amplitudes for antiparticle
scattering are the analytic continuation of
particle scattering amplitudes.
Dispersion relations: the values of the S-matrix
can be calculated by integrals over internal
energy variables of the imaginary part of
the same values.
Causality conditions: the singularities of
the S-matrix can only occur in ways that don't
allow the future to influence the past (motivated
by Kramers–Kronig relations)
Landau principle: Any singularity of the S-matrix
corresponds to production thresholds of physical
particles.These principles were to replace
the notion of microscopic causality in field
theory, the idea that field operators exist
at each spacetime point, and that spacelike
separated operators commute with one another.
== Bootstrap models ==
The basic principles were too general to apply
directly, because they are satisfied automatically
by any field theory. So to apply to the real
world, additional principles were added.
The phenomenological way in which this was
done was by taking experimental data and using
the dispersion relations to compute new limits.
This led to the discovery of some particles,
and to successful parameterizations of the
interactions of pions and nucleons.
This path was mostly abandoned, because the
resulting equations, devoid of any space-time
interpretation, were very difficult to understand
and solve.
== Regge theory ==
The 
principle behind the Regge theory hypothesis
(also called analyticity of the second kind
or the bootstrap principle) is that all strongly
interacting particles lie on Regge trajectories.
This was considered the definitive sign that
all the hadrons are composite particles, but
within S-matrix theory, they are not thought
of as being made up of elementary constituents.
The Regge theory hypothesis allowed for the
construction of string theories, based on
bootstrap principles. The additional assumption
was the narrow resonance approximation, which
started with stable particles on Regge trajectories,
and added interaction loop by loop in a perturbation
series.
String theory was given a Feynman path-integral
interpretation a little while later. The path
integral in this case is the analog of a sum
over particle paths, not of a sum over field
configurations. Feynman's original path integral
formulation of field theory also had little
need for local fields, since Feynman derived
the propagators and interaction rules largely
using Lorentz invariance and unitarity.
== See also ==
Landau pole
Regge trajectory
Bootstrap model
Pomeron
Dual resonance model
History of string theory
== Notes
