algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
In quantum field theory (and string theory) the scattering amplitudes (string scattering amplitudes) “where nothing external scatters,” hence for no incoming and no outgoing states, are called vacuum amplitudes.
As functions of source fields, the vacuum amplitudes, or rather the vacuum energy, serve as the generating functionals for all correlators/n-point functions (e.g. Scrucca, 1.6).
The 1-loop vacuum amplitudes are regularized traces over Feynman propagators/Dirac propagators. These are the incarnations of special values of zeta functions, L-functions and eta functions in physics:
In the presence of supersymmetry 1-loop vacuum amplitudes are typically supposed to vanish.
For the type II superstring, see e.g. (Palti). For the heterotic superstring see e.g. Han 89.
In view of the above relation of 1-loop vacuum amplitudes to special values of L-functions such vanishing reminds one of the Riemann hypothesis. See (ACER 11).
Discussions for particles includes
Claudio Scrucca, Advanced quantum field theory pdf
Ori Yudilevich, Calculating Massive One-Loop Amplitudes in QCD, Utrecht 2009 (pdf)
Robbert Rietkerk, One-loop amplitudes in perturbative quantum field theory, Utrecht 2012 (pdf)
Lecture notes for 1-loop vacuum amplitudes for strings (vacuum string scattering amplitudes) include
José Edelstein, Lecture 8: 1-loop closed string vacuum amplitude, 2013 (pdf)
Eran Palti, The IIA/B superstring one-loop vacuum amplitude (pdf)
Seung Kee Han, Vanishing vacuum amplitude of four-dimensional heterotic string theory compactified on N=2 superconformal field theory, Phys. Rev. D 39, 2322 – Published 15 April 1989 (web)
And with relation to the Riemann hypothesis:
Last revised on July 6, 2023 at 10:09:31. See the history of this page for a list of all contributions to it.