Stable and Unstable Modes at a Saddle Point in Micro-Macro Duality (Mathematical Aspects of Quantum Fields and Related Topics)

Abstract

Longitudinal modes associated with local gauge invariance cause such difficulties as breakdown of Lorentz covariance in Coulomb mode and/or that of the probabilistic interpretation owing to "indefinite metric". The latter one can be viewed in the context of dynamics as a saddle point causing bifurcation into stable and unstable modes. In the light of "Cartan involution" characterizing symmetric space structure, natural connections can be found between the problem caused by longitudinal modes in gauge theory and the roles played by heat baths in thermodynamics, from the viewpoint of mutual relations between finite- and infinite-dimensional realizations of noncompact Lie groups in the ligth of indefinite metric. These problems can be treated consistently from the viewpoint of bifurcations into stable and unstable modes at a saddle points in Micro-Macro duality.