Reinhard Oehme - Work - Dispersion Relations, GMO Sum Rule, and Edge of The Wedge Theorem

Dispersion Relations, GMO Sum Rule, and Edge of The Wedge Theorem

In 1954 in Chicago, Oehme studied the analytic properties of forward Scattering amplitudes in quantum field theories. He found that particle-particle and antiparticle-particle amplitudes are connected by analytic continuation in the complex energy plane. These results led to the paper by him with Marvin L. Goldberger and Hironari Miyazawa on the dispersion relations for pion-nucleon scattering, which also contains the Goldberger-Miyazawa-Oehme Sum Rule. There is good agreement with the experimental results of the Fermi Group at Chicago, the Lindenbaum Group at Brookhaven and others. The GMO Sum Rule is often used in the analysis of the pion-nucleon system. Oehme published a proper derivation of hadronic forward dispersion relations on the basis of local quantum field theory in an article published in Il Nuovo Cimento. His proof remains valid in gauge theories with confinement. The analytic connection Oehme found between particle and antiparticle amplitudes is the first example of a fundamental feature of local quantum field theory: the crossing property. It is proven here, in a non-perturbative setting, on the basis of the analytic properties of amplitudes which are a consequence of locality and spectrum, like the dispersion relations. For generalizations, one still relies mostly on perturbation theory. For the purpose of using the powerful methods of the theory of functions of several complex variables for the proof of non-forward dispersion relations, and for analytic properties of other Greens functions, Oehme formulated and proved a fundamental theorem which he called the “Edge of the Wedge Theorem” (“Keilkanten Theorem”). This work was done mainly in the Fall of 1956 at the Institute for Advanced Study in collaboration with Hans-Joachim Bremermann and John G. Taylor. Using microscopic causality and spectral properties, the BOT theorem provides an initial region of analyticity, which can be enlarged by "Analytic Completion". Oehme first presented these results at the Princeton University Colloquium during the winter semester 1956/57. Independently, a different and elaborate proof of non-forward dispersion relations has been published by Nikolay Bogoliubov and collaborators. The Edge of the Wedge Theorem of BOT has many other applications. For example, it can be used to show that, in the presence of (spontaneous) violations of Lorentz invariance, micro-causality (locality), together with positivity of the energy, implies Lorentz invariance of the energy- momentum spectrum. Together with Marvin L. Goldberger and Yoichiro Nambu, Oehme also has formulated dispersion relations for nucleon-nucleon scattering.

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