Approximate representations of SU(2) ordered exponentials in the adiabatic and stochastic limits

Publisher Website
Google Scholar
Abstract
Approximate representations for the SU(2) ordered exponential U(t‖E) =(exp[i∫t0dt′ σ⋅E(t′)])+, written as a functional of its input field E(t), are derived in the adiabatic ( ρ≪1) and stochastic ( ρ≫1) limits, where ρ≡‖dÊ/dt‖/E, Ê=E/E, E=+(E2)1/2. An algorithm is set up for the adiabatic case, and fixed-point equations are obtained for situations of possible convergence. In the stochastic regime, ‘‘averaged’’ functions describing U(t‖E) are derived which reproduce its slowly varying dependence of large magnitude while missing, or approximating, rapid oscillations of small magnitude. Several functional integrals, analytic and machine are carried out over these approximate forms, and their results compared with the same functional integrals over the exact U(t‖E).