S-matrix theory

<h2 id="history">History</h2>
<p><i>S</i>-matrix theory was proposed as a principle of particle interactions by <a href="/facts/Werner_Heisenberg/n4qzYvj4">Werner Heisenberg</a> in 1943,<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> following <a href="/facts/John_Archibald_Wheeler/I1A5t0Fd">John Archibald Wheeler</a>'s 1937 introduction of the <i>S</i>-matrix.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p><p>It was developed heavily by <a href="/facts/Geoffrey_Chew/5xVg50pU">Geoffrey Chew</a>, <a href="/facts/Steven_Frautschi/Hh1fDHDp">Steven Frautschi</a>, <a href="/facts/Stanley_Mandelstam/MtfqcXPS">Stanley Mandelstam</a>, <a href="/facts/Vladimir_Gribov/ve2QwzW8">Vladimir Gribov</a>, and <a href="/facts/Tullio_Regge/5EpyGSJM">Tullio Regge</a>. Some aspects of the theory were promoted by <a href="/facts/Lev_Landau/lNXQTC49">Lev Landau</a> in the Soviet Union, and by <a href="/facts/Murray_Gell-Mann/7yg75mjR">Murray Gell-Mann</a> in the United States.
</p>
<h2 id="basic-principles">Basic principles</h2>
<p>The basic principles are:
</p>
<ol><li>Relativity: The <i>S</i>-matrix is a representation of the <a href="/facts/Poincar%25C3%25A9_group/3IerRYG1">Poincaré group</a>;</li>
<li><a href="/facts/Unitarity/x2gXDAsS">Unitarity</a>: 
  
    
      
        S
        
          S
          
            †
          
        
        =
        1
      
    
    {\displaystyle SS^{\dagger }=1}
  
;</li>
<li>Analyticity: integral relations and singularity conditions.</li></ol>
<p>The basic analyticity principles were also called <i>analyticity of the first kind</i>, and they were never fully enumerated, but they include
</p>
<ol><li><a href="/facts/Crossing_(physics)/UNyX4EXs">Crossing</a>: The amplitudes for antiparticle scattering are the <a href="/facts/Analytic_continuation/aBCuEgvt">analytic continuation</a> of particle scattering amplitudes.</li>
<li><a href="/facts/Kramers%25E2%2580%2593Kronig_relations/S2UemwVo">Dispersion relations</a>: the values of the <i>S</i>-matrix can be calculated by integrals over internal energy variables of the imaginary part of the same values.</li>
<li>Causality conditions: the singularities of the <i>S</i>-matrix can only occur in ways that don't allow the future to influence the past (motivated by <a href="/facts/Kramers%25E2%2580%2593Kronig_relations/S2UemwVo">Kramers–Kronig relations</a>)</li>
<li>Landau principle: Any singularity of the <i>S</i>-matrix corresponds to production thresholds of physical particles.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li></ol>
<p>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.
</p>
<h2 id="bootstrap-models">Bootstrap models</h2>
<p class="note">Main article: <a href="/facts/Bootstrap_model/PSDWkyee">Bootstrap model</a></p>
<p>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.
</p><p>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.
</p><p>This path was mostly abandoned because the resulting equations, devoid of any space-time interpretation, were very difficult to understand and solve.
</p>
<h2 id="regge-theory">Regge theory</h2>
<p class="note">Main article: <a href="/facts/Regge_theory/zR8Vr2s3">Regge theory</a></p>
<p>The principle behind the Regge theory hypothesis (also called <i>analyticity of the second kind</i> or the <i>bootstrap principle</i>) is that all strongly interacting particles lie on <a href="/facts/Regge_trajectories/zR8Vr2s3">Regge trajectories</a>. This was considered the definitive sign that all the hadrons are composite particles, but within <i>S</i>-matrix theory, they are not thought of as being made up of elementary constituents.
</p><p>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.
</p><p>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 <a href="/facts/Path_integral_formulation/6lt2MHkW">path integral formulation</a> of field theory also had little need for local fields, since Feynman derived the propagators and interaction rules largely using Lorentz invariance and unitarity.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Landau_pole/kbW2kr6f">Landau pole</a></li>
<li><a href="/facts/Regge_trajectory/zR8Vr2s3">Regge trajectory</a></li>
<li><a href="/facts/Bootstrap_model/PSDWkyee">Bootstrap model</a></li>
<li><a href="/facts/Pomeron/dw4AokqV">Pomeron</a></li>
<li><a href="/facts/Dual_resonance_model/7Uaxw5jV">Dual resonance model</a></li>
<li><a href="/facts/History_of_string_theory/nf21oaah">History of string theory</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Steven_Frautschi/Hh1fDHDp">Steven Frautschi</a>, <i>Regge Poles and </i>S<i>-matrix Theory</i>, New York: W. A. Benjamin, Inc., 1963.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>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. <a href="/wiki/Steven_Giddings" target="_blank">/wiki/Steven_Giddings</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>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. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>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. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>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. <a href="/wiki/Lev_Landau" target="_blank">/wiki/Lev_Landau</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Yuri V. Kovchegov, Eugene Levin, Quantum Chromodynamics at High Energy, Cambridge University Press, 2012, p. 313. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

S-matrix theory open-in-new

S-matrix theory