Operator ordering in quantum field theory. Nonlinear Lagrangian of scalar fields

Abstract

Operator ordering in the Lagrangian formalism of the relativistic quantum field theory is investigated for a system corresponding to a classical one described byLc=–1/2∂μϕcaϱab(ϕc)∂μϕcb-v(ϕc). The most general form of the Lagrangian density operator and that of current operator accommodating the freedom of the order of the field operators are taken first, and the conditions on them required for the realization of the symmetry properties in the operator formalism are investigated. The requirement of the Poincare symmetry determines the Lagrangian density operator, the field equation and the current operators up to an unknown function. The unknown function is restricted to a scalar under the transformations of the Poincare and internal symmetries.