On the Quantization in Non-Linear Theories

概要

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The canonical quantization of non-linear Lagrangians is discussed. When one can find a Cartesian coordinate system O__〜 (reducible case), the covariant Hamiltonian can be obtained by starting with the Lagrangian L=(1/8)g^<-1/4>{g_<im>, q^^・^m} g^<1/2>g^<ij>{g_<jn>, q^^・^n}g^<-1/4>-υ(q). On the other hand, when no overall flat coordinate system can be found, a local Euclidean frame is attached at each point in q-coordinate space and auxiliary variables are introduced in order to obtain the Hamiltonian. It is shown that the procedure is not covariant under a local rotation of n-legs. In a reducible case, the quantum mechanical transformation operator from O__〜 to q^i is found and the unitarity of the operator is verified in quantum mechanics. Extension to non-linear field theories is carried out, and the transformation operator is shown to be an abnormal operator in particle number representations though it is formally unitary.
理論物理学刊行会の論文
1971-10-25