On the Consistency between Lagrangian and Hamiltonian Formalisms in Quantum Mechanics. III

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A consistent quantization procedure is proposed for the non-linear Lagrangian L = (1/8) ×g^<-1/4>{g_<ik>, q^^・^k}g^<1/2>g^<ij>{g_<jm>, q^^・^m}g^<-1/4>-υ(q). We extend the previous formalism, which was formulated by means of a q-number variation principle for cases with special g_<ij>, so that it is applicable to the general case with arbitrary g_<ij>. The commutation relations of q-number variations δq^i and δq^i with q^iand q^^・^j are taken as unknown functions in deriving the Euler-Lagrange equation. However, in the final expression of the Euler-Lagrange equation and the Hamiltonian, these unknown functions can be eliminated so that the equation of motion and the Hamiltonian are uniquely given by g_<ij> and their derivatives. In general the unknown functions are not well defined in the overall region of q^i, contrary to the special cases given in previous papers. For illustration, a simple example of two-dimensional space is presented. The Hamiltonian as the time generator of the system is given by H=L+2ν-(h^2/4)R where R is the curvature scalar constructed from g_<ij>. With this H the canonical equation of motion is in accord with the Euler-Lagrange equation.
理論物理学刊行会の論文
1972-10-25