Transformation Properties of Dynamical Systems at the Quantum Level

Abstract

Based on the generating functional of Green function for a dynamical system, the general equations of transformation properties at the quantum level are derived. In some cases they can be reduced to the quantum Noether theorem. In some other cases they can be reduced to momentum theorem or angular momentum theorem etc. at the quantum level. An example is presented and it shows that the classical conservation laws don't always preserve in quantum theories.KEY WORDS: symmetry and conservation laws; constrained Hamiltonian systems; path integral quantization.

The Transformation Properties at the Quantum Level in Field Theories for a Gauge-Invariant System with a Higher-Order Lagrangian

“…As is well known, in classical theories, those equations of transformation properties corresponding the spatial translation yields the momentum theorem and spatial rotation yields the moment of momentum theorem etc respectively [21,22]. Some results hold true in quantum mechanics for a system with a finite number of degrees of freedom [20]. In this paper we discuss the quantal transformation properties of a gauge-invariant system with a higher-order Lagrangian in field theories and give some applications.…”

“…Now let us consider a more general case in which the action of the gauge-invariant (local invariance) system is variant under the global transformation in configuration space, we shall further study the transformation properties of such a system at the quantum level, the preliminary studies was performed for a system with a finite number of degrees of freedom in the previous works [20]. As is well known, in classical theories, those equations of transformation properties corresponding the spatial translation yields the momentum theorem and spatial rotation yields the moment of momentum theorem etc respectively [21,22].…”