QUANTUM BASIC INVARIANTS AND CLASSICAL NOETHER THEOREM

Taking the generalized time-dependent harmonic oscillator (GTHO) and singular time-dependent harmonic oscillator (STHO) as illustrative examples, we discuss the connection between the quantum basic invariants and classical Noether's theorem and, for the system, find: (1) The classical basic invariants and general form of classical invariants. (2) The current algebra and corresponding operator algebra. (3) A kind of converse of the Noether's theorem. (4) The connection between the so-called symmetry algebra ({G(k)} algebra) and the classical current algebra through the operator algebra.