On Consistency between Lagrangian and Hamiltonian Formalisms in Quantum Mechanics. II

概要

論文の詳細を見る
We examine whether a consistent quantization can be carried out for a non-linear Lagrangian when one cannot resort to the transformation from the given Lagrangian L=(1/8)g^<-1/4>×{g_<ir>, q^^・^r} g^<1/2>g^<ij>{g_<is>, q^^・^s} g^<-1/4>-υ(q) to the standard form L=1/2Q^aQ^a-V(q). In order to obtain the variational equation the commutation relations for δq^i and δq^^・^i are settled so as to be consistent with the Euler-Lagrange equation. Under the premise of δq^^・^i=(d/dt)δq^i, it is shown that consistent Lagrangian and Hamiltonian formalisms exist when a space defined by metrics g_<ij> is of constant curvature. All consequences of a previous paper are derived and the dynamical model reduces to the case of flat space, if the commutability between the Euler equation and δq^i is supposed to hold. As a non-trivial example of the irreducible dynamical models, the case of the three-dimensional space with constant curvature is investigated. It is shown that for this model the explicit form of [q^^・^i, δq^j] and [q^^・^i, δq^^・^j] are all determined and the consistent quantization is carried out.
理論物理学刊行会の論文
1972-03-25