Covariant Differential Operator under General Coordinate Transformation and Structure of Lagrangian Operator

概要

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Quantum-theoretical lagrangian formalism proposed in a previous article is generalized so that we can deal with quasi-linear system for which classical lagrangian is given by L_<c1>=1/2αq^^・^αg_<αβ>q^^・^β. The only necessary modification is to decompose g into a product of three matrices d^†u^^-d. Prescription to extract u^^-, d and real variation δq from generalized Euler equation, which is necessary to get equation of motion, is fixed. It is confirmed that each step of manipulation in our formalism can be performed without referring to any group-theoretical property of the system. So our formalism is proved applicable to any kind of quasi-linear system. When we consider general coordinate transformation in internal symmetry space, we can determine quantum-theoretical covariant differential from (Dq)_α. It is shown that invariant bi-linear form of (Dq)_α is not affected by the ambiguity in the decomposition of g although the form of (Dq)_α itself depends on the ambiguity. Therefore, the ambiguity cannot be utilized to give rise to the scalar curvature term in lagrangian and hamiltonian operators.
理論物理学刊行会の論文
1976-04-25