Adiabatic approximation for \(W^*\) quantum systems with particles of mass m, M as \((m/M)^{1/2}\to 0\) (Q1066061)

The \((m/M)^{1/2}\to 0\) limit and the adiabatic approximation for a non- relativistic quantum system, consisting of light and heavy particles of masses m, M, are studied in the standard (Tomita) representations of the relevant \(W^*\)-algebras. The paper introduces the standard representations of N-particle quantum mechanics, and shows the adequacy of such representations in the study of asymptotic limits which may not have any irreducible representations. The Primas-Raggio theorem is derived, which implies that, in the limit \((m/M)^{1/2}\to 0\), the heavy-particle subsystem is a static, classical system in the sense of having a commutative algebra of observables. The adiabatic approximation is derived as the leading correction.