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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, 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 Although later superseded by quantum chromodynamics, the ideas of S-matrix theory remain important, especially in modern approaches to quantum gravity and in connections with the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.^[1]

History

S-matrix theory was proposed by Werner Heisenberg in 1943,^[2] building on earlier work by John Archibald Wheeler, who introduced the concept of the S-matrix in 1937.^[3]

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".^[4]

Basic principles

The theory is based on a set of general principles:

Relativity: the S-matrix forms a representation of the Poincaré group.
Unitarity: $S S^{†} = 1$ , ensuring probability conservation.
Analyticity: scattering amplitudes are analytic functions with well-defined singularities.

Additional analyticity conditions include:

Crossing symmetry, relating particle and antiparticle processes.
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.^[5]

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 models. 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 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.

Table of contents (138 articles)

Index

Core theory

Foundations
Conceptual and interpretations
Mathematical structure and systems
Atomic and spectroscopy
Wavefunctions and modes
Quantum dynamics and evolution
Measurement and information
Quantum information and computing

Applications and extensions

Quantum optics and experiments
Open quantum systems
Quantum field theory
Statistical mechanics and kinetic theory
Condensed matter and solid-state physics
Plasma and fusion physics
Timeline
Advanced and frontier topics

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Physics:Quantum S-matrix

History

Basic principles

Bootstrap models

Regge theory

Scattering interpretation

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