Relativistically Covariant Formulation in Non-Linear Lagrangian Theories and Factor Ordering Problems

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Abstract
Factor ordering problems in non-linear Lagrangian theories are discussed. It is shown that symmetrization of the Hamiltonian is necessary in order to get the covariant Dyson formula by considering the factor ordering. As a result, there appear terms proportional to ℏ2(δ(0))2 in the effective Hamiltonian density. In the evaluation of divergent integrals appearing in non-linear theories it is also necessary to take care of the order of integrations. Following the prescriptions thus obtained, Charap's discussions of the soft-pion theorems in chiral dynamics are reexamined in each order of the perturbation expansion. It is concluded that the soft-pion theorems are maintained, order by order, in perturbation theory also under the condition that the factor ordering is completely taken into account.

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