S-matrix theory. It was a thing. A proposal, really, for how to break free from the shackles of local quantum field theory when discussing elementary particle physics. Think of it as a desperate attempt to find a new language when the old one was stammering. Instead of focusing on what happens here and now, it looked at the grand, overarching narrative—the transition from the infinitely distant past to the equally distant future. It treated the S-matrix, this grand operator that maps initial states to final states, as the fundamental building block. No messy intermediate steps, no decomposing reality into infinitesimal time slices. Just a single, elegant, or perhaps just brutally efficient, transformation.

This whole endeavor was quite the buzz in the 1960s. Quantum field theory, bless its heart, was drowning in its own complexities, particularly at strong coupling, leading to this rather inconvenient phenomenon known as the Landau pole. S-matrix theory offered a way out, a plausible alternative for understanding the strong interaction. And, as it turns out, it was the fertile ground from which string theory eventually sprouted.

By the 1970s, though, the initial fervor for S-matrix theory had waned. Quantum chromodynamics had emerged, offering a solution to the strong interaction problems within the familiar framework of field theory. But the ghost of S-matrix theory lingered. In the guise of string theory, it found new life, persisting as a popular avenue for tackling the notoriously difficult problem of quantum gravity.

The connections it forged are still relevant. The S-matrix theory bears a relationship to the holographic principle and the AdS/CFT correspondence through a flat space limit. In the context of anti-de Sitter space, the analog of the S-matrix relations is found in the boundary conformal theory.[1]

Its most enduring legacy, without question, is string theory. But it also gave us other notable achievements, like the Froissart bound, which sets an upper limit on the total cross-section in particle scattering, and the prediction of the pomeron, a theoretical particle that describes the behavior of strong interactions at high energies.

History

The genesis of S-matrix theory can be traced back to 1943, when Werner Heisenberg proposed it as a fundamental principle governing particle interactions.[2] This built upon the earlier work of John Archibald Wheeler, who had introduced the concept of the S-matrix in 1937.[3]

The theory was significantly advanced by a group of physicists including Geoffrey Chew, Steven Frautschi, Stanley Mandelstam, Vladimir Gribov, and Tullio Regge. Aspects of it were also championed by Lev Landau in the Soviet Union and Murray Gell-Mann in the United States.

Later, in 1979, Steven Weinberg established a crucial link between S-matrix theory and effective field theories with his now-famous "folk theorem".[4]

Basic Principles

The core tenets of S-matrix theory were built upon a few fundamental pillars:

• Relativity: The S-matrix was understood as a representation of the Poincaré group, ensuring that the theory respected the principles of special relativity.
• Unitarity: This is expressed by the equation $SS^{\dagger }=1$. It essentially means that the total probability of all possible outcomes of an interaction must sum to one, a fundamental requirement for any probabilistic theory.
• Analyticity: This principle involved integral relations and conditions on the singularities of the S-matrix. These analyticity properties were often referred to as analyticity of the first kind. While not exhaustively enumerated, they encompassed crucial concepts like:
  • Crossing: The scattering amplitudes for antiparticles were seen as the analytic continuation of the scattering amplitudes for particles. This provided a way to relate different types of scattering processes.
  • Dispersion relations: These relations stated that the values of the S-matrix could be determined by integrating the imaginary part of the S-matrix over internal energy variables.
  • Causality conditions: These were designed to prevent the future from influencing the past, a natural consequence of the Kramers–Kronig relations.
  • Landau principle: This principle posited that any singularity in the S-matrix corresponded to the threshold for the production of physical particles.[5][6]

These principles were intended to supplant the notion of microscopic causality found in field theory—the idea that field operators exist at every point in spacetime and that operators separated by spacelike intervals commute.

Bootstrap Models

The fundamental principles of S-matrix theory, while elegant, were too general to directly describe the complexities of the real world. Any self-consistent field theory, after all, would satisfy them. To make progress, additional principles were introduced, leading to the development of what were known as "bootstrap models."

The phenomenological approach involved taking experimental data and using the dispersion relations to predict new scattering limits. This iterative process led to the discovery of certain particles and provided successful parameterizations for the interactions of pions and nucleons.

However, this path was largely abandoned. The equations generated by these models, lacking a clear spacetime interpretation, proved exceptionally difficult to comprehend and solve.

Regge Theory

A significant development within S-matrix theory was Regge theory, sometimes referred to as analyticity of the second kind or the bootstrap principle. The core hypothesis here was that all strongly interacting particles reside on Regge trajectories. This was interpreted as strong evidence that all hadrons were composite, but within the framework of S-matrix theory, they weren't viewed as being built from more fundamental constituents in the traditional sense.

Regge theory provided a crucial foundation for the construction of string theories, which were derived from bootstrap principles. An additional assumption made was the narrow resonance approximation. This started with stable particles on Regge trajectories and then incorporated interactions in a loop-by-loop fashion, akin to a perturbation series.

Later, string theory was given a more concrete interpretation through the Feynman path-integral formulation. In this context, the path integral represented a sum over possible string paths, rather than a sum over field configurations as in traditional quantum field theory. It's worth noting that Feynman's original path integral formulation of field theory itself had a somewhat diminished need for local fields, as Feynman derived propagators and interaction rules primarily through Lorentz invariance and unitarity.

See Also

• Landau pole
• Regge trajectory
• Bootstrap model
• Pomeron
• Dual resonance model
• History of string theory

Notes

• ^ Giddings, Steven B. (1999-10-04). "Boundary S-Matrix and the Anti–de Sitter Space to Conformal Field Theory Dictionary". Physical Review Letters. 83 (14): 2707–2710. arXiv:hep-th/9903048. Bibcode:1999PhRvL..83.2707G. doi:10.1103/physrevlett.83.2707. ISSN 0031-9007.
• ^ Heisenberg, W. (1943). "Die beobachtbaren Größen in der Theorie der Elementarteilchen". Zeitschrift für Physik (in German). 120 (7–10). Springer Science and Business Media LLC: 513–538. Bibcode:1943ZPhy..120..513H. doi:10.1007/bf01329800. ISSN 1434-6001. S2CID 120706757.
• ^ Wheeler, John A. (1937-12-01). "On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure". Physical Review. 52 (11). American Physical Society (APS): 1107–1122. Bibcode:1937PhRv...52.1107W. doi:10.1103/physrev.52.1107. ISSN 0031-899X.
• ^ Cushing, James T. (1990). Theory Construction and Selection in Modern Physics: The S Matrix. Cambridge University Press. ISBN 978-0-521-38181-9.
• ^ Landau, L.D. (1959). "On analytic properties of vertex parts in quantum field theory". Nuclear Physics. 13 (1). Elsevier BV: 181–192. Bibcode:1959NucPh..13..181L. doi:10.1016/0029-5582(59)90154-3. ISSN 0029-5582.
• ^ Yuri V. Kovchegov, Eugene Levin, Quantum Chromodynamics at High Energy, Cambridge University Press, 2012, p. 313.