imported>WikiHarold: Replace raw Quantum Collection backlink with B backlink template

2026-05-08T19:05:12Z

Replace raw Quantum Collection backlink with B backlink template

← Older revision	Revision as of 19:05, 8 May 2026
(No difference)

imported>WikiHarold: Replace raw Quantum Collection backlink with B backlink template

2026-05-08T19:05:12Z

Replace raw Quantum Collection backlink with B backlink template

New page

{{Quantum book backlink|Quantum dynamics and evolution}}

'''''S''-matrix theory''' was a proposal for replacing local [[quantum field theory]] as the basic principle of elementary [[particle physics]]. It avoids the notion of space and time by replacing it with abstract mathematical properties of the [[S-matrix|''S''-matrix]], which relates the infinite past to the infinite future in a single step without intermediate time evolution.

The approach was influential in the 1960s as an alternative to quantum field theory, which at the time faced problems such as the [[Landau pole]] at strong coupling. Applied to strong interactions, it led to the development of [[string theory]]. Although later superseded by [[quantum chromodynamics]], the ideas of ''S''-matrix theory remain important, especially in modern approaches to [[quantum gravity]] and in connections with the [[holographic principle]] and the [[AdS/CFT correspondence]].<ref>{{cite journal |author1-link=Steven Giddings| last=Giddings | first=Steven B. | title=Boundary S-Matrix and the Anti–de Sitter Space to Conformal Field Theory Dictionary | journal=Physical Review Letters | volume=83 | issue=14 | date=1999-10-04 | doi=10.1103/physrevlett.83.2707 | pages=2707–2710|arxiv=hep-th/9903048}}</ref>

[[File:Scattering theory illust-y.png|thumb|400px|Illustration of wave scattering: an incoming wave interacts with a medium and produces a scattered outgoing wave.]]

= History =
''S''-matrix theory was proposed by [[Werner Heisenberg]] in 1943,<ref>{{cite journal | last=Heisenberg | first=W. | title=Die beobachtbaren Größen in der Theorie der Elementarteilchen | journal=Zeitschrift für Physik | volume=120 | issue=7–10 | year=1943 | pages=513–538}}</ref> building on earlier work by [[John Archibald Wheeler]], who introduced the concept of the ''S''-matrix in 1937.<ref>{{cite journal | last=Wheeler | first=John A. | title=On the Mathematical Description of Light Nuclei | journal=Physical Review | volume=52 | issue=11 | date=1937-12-01 | pages=1107–1122}}</ref>

The theory was further developed by physicists such as [[Geoffrey Chew]], [[Steven Frautschi]], [[Stanley Mandelstam]], [[Vladimir Gribov]], and [[Tullio Regge]]. Related ideas were also promoted by [[Lev Landau]] and [[Murray Gell-Mann]]. In 1979, [[Steven Weinberg]] connected ''S''-matrix ideas with [[effective field theory]] through his "folk theorem".<ref>{{Cite book |last=Cushing |first=James T. |title=Theory Construction and Selection in Modern Physics: The S Matrix |date=1990 |publisher=Cambridge University Press}}</ref>

== Basic principles ==
The theory is based on a set of general principles:

* '''Relativity''': the ''S''-matrix forms a representation of the [[Poincaré group]].
* '''[[Unitarity]]''': <math>S S^{\dagger} = 1</math>, ensuring probability conservation.
* '''Analyticity''': scattering amplitudes are analytic functions with well-defined singularities.

Additional analyticity conditions include:

* [[Crossing (physics)|Crossing symmetry]], relating particle and antiparticle processes.
* [[Kramers–Kronig relations|Dispersion relations]], connecting real and imaginary parts of amplitudes.
* Causality conditions restricting the allowed singularities.
* The '''Landau principle''', stating that singularities correspond to physical particle production thresholds.<ref>{{cite journal | last=Landau | first=L.D. | title=On analytic properties of vertex parts in quantum field theory | journal=Nuclear Physics | volume=13 | year=1959 | pages=181–192}}</ref>

These principles were intended to replace the notion of local interactions in spacetime used in quantum field theory.

== Bootstrap models ==
Because the general principles were too broad, additional assumptions were introduced in so-called [[bootstrap model]]s. These models attempted to determine particle properties self-consistently using experimental data and dispersion relations.

Although successful in some cases, the resulting equations were mathematically complex and lacked a clear spacetime interpretation, limiting their usefulness.

== Regge theory ==
In [[Regge theory]], strongly interacting particles are organized along [[Regge trajectory|Regge trajectories]]. This suggested that hadrons are related through a unified structure, rather than being fundamental.

These ideas led to the development of early string models, where particle interactions were interpreted as arising from extended one-dimensional objects. This provided a bridge between ''S''-matrix theory and modern [[string theory]].

== Scattering interpretation ==
In quantum mechanics, the ''S''-matrix provides a direct relation between initial and final states of a scattering process. Instead of tracking time evolution step by step, it encodes the full interaction in a single operator.

This approach is especially useful in high-energy physics, where scattering experiments probe the structure of matter by analyzing incoming and outgoing particle states.

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

== References ==
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum S-matrix|1}}

Physics:Quantum S-matrix - Revision history

imported>WikiHarold: Replace raw Quantum Collection backlink with B backlink template

imported>WikiHarold: Replace raw Quantum Collection backlink with B backlink template