S-matrix theory
Encyclopedia
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 QCD 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 the best accepted 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 are 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
.
in 1941, 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.
The basic analyticity principles were also called analyticity of the first kind, and they were never fully enumerated, but they include
These principles were to replace the notion of microscopic causality in field theory, the idea that field operators exist at each space time point, and that spacelike separated operators commute with one another.
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.
The Regge theory hypotheses 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.
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
as the basic principle of elementary particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
.
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
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, which was plagued with the zero interaction phenomenon
Landau pole
In physics, the Landau pole is the momentum scale at which the coupling constant of a quantum field theory becomes infinite...
at strong coupling. Applied to the strong interaction, it led to the development of string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
.
S-matrix theory was largely abandoned by physicists in the 1970s, as QCD 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 the best accepted approach to the problem of quantum gravity.
The S-matrix theory is related to the holographic principle
Holographic principle
The holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region—preferably a light-like boundary like a gravitational horizon...
and the AdS/CFT correspondence
AdS/CFT correspondence
In physics, the AdS/CFT correspondence , sometimes called the Maldacena duality, is the conjectured equivalence between a string theory and gravity defined on one space, and a quantum field theory without gravity defined on the conformal boundary of this space, whose dimension is lower by one or more...
by a flat space limit. The analog of the S-matrix relations in AdS space are the boundary conformal theory.
The most lasting legacy of the theory is string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
. Other notable achievements are the Froissart bound, and the prediction of the pomeron
Pomeron
In physics, the pomeron is a Regge trajectory, a family of particles with increasing spin, postulated in 1961 to explain the slowly rising cross section of hadronic collisions at high energies.-Overview:...
.
History
S-matrix theory was proposed as a principle of particle interactions by Werner HeisenbergWerner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
in 1941, following John Archibald Wheeler
John Archibald Wheeler
John Archibald Wheeler was an American theoretical physicist who was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr in explaining the basic principles behind nuclear fission...
's 1937 introduction of the S-matrix.
It was developed heavily by Geoffrey Chew
Geoffrey Chew
Geoffrey F. Chew is an American theoretical physicist.He has worked as a professor of physics at the UC Berkeley since 1957 and has been an emeritus since 1991. Chew holds a PhD in theoretical particle physics from the University of Chicago. Between 1950 and 1956, he was a physics faculty member...
, Steven Frautschi
Steven Frautschi
Steven Frautschi is an American theoretical physicist, Professor of Physics at the California Institute of Technology. He is known for his contributions to the bootstrap theory of the strong interactions....
, Stanley Mandelstam
Stanley Mandelstam
Stanley Mandelstam is a South African-born theoretical physicist. He introduced the relativistically invariant Mandelstam variables into particle physics in 1958 as a convenient coordinate system for formulating his double dispersion relations...
, Vladimir Gribov
Vladimir Gribov
Vladimir Naumovich Gribov was a prominent Russian theoretical physicist, who worked on high-energy physics, quantum field theory and the Regge theory of the strong interactions.His best known contributions are the pomeron, the DGLAP equations, and the Gribov copies.-Life:Gribov completed his...
, and Tullio Regge
Tullio Regge
Tullio Regge is an Italian theoretical physicist. He obtained a degree in physics from the University of Turin in 1952, and a Ph.D. in physics from the University of Rochester in 1957 under the direction of Robert Marshak. From 1958 to 1959 Regge held a post at the Max Planck Institute for...
. Some aspects of the theory were promoted by Lev Landau
Lev Landau
Lev Davidovich Landau was a prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics...
in the Soviet Union, and by Murray Gell-Mann
Murray Gell-Mann
Murray Gell-Mann is an American physicist and linguist who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles...
in the United States.
Basic Principles
The basic principles are:- Relativity : The S-matrix is a representation of the Poincaré group.
- Unitarity :
- 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 dispersion relation)
- 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 space time 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
also called analyticity of the Second Kind, or the bootstrap principle. The 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 hypotheses 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
- Froissart bound
- Double dispersion relations
- Landau singularities
- Regge trajectory
- Bootstrap modelBootstrap modelIn physics, the term bootstrap model is used for a class of theories that use very general consistency criteria to determine the form of a quantum theory from some assumptions on the spectrum of particles...
- PomeronPomeronIn physics, the pomeron is a Regge trajectory, a family of particles with increasing spin, postulated in 1961 to explain the slowly rising cross section of hadronic collisions at high energies.-Overview:...
- N/D equations
- Dual resonance modelDual resonance modelIn theoretical physics, a dual resonance model arose the early investigation of string theory as an S-matrix theory of the strong interaction....
- History of string theoryHistory of string theoryThe history of string theory is probably more relevant to its core science than histories of other physical sciences. String theory is presently, and essentially, a non physically testable science, and thus arguably not Physics, yet its derivation parallels testable physics...