Perturbation Lagrangian Theory for Dirac Fields-Ward-Takahashi Identity and Current Algebra

In a class of Lagrangian field theories for Dirac spin-½ particles, the Bogoliubov-Parasiuk-Hepp renormalization scheme provides a proof of the operator forms of Euler-Lagrange equations of motion, Noether's theorem, and Ward-Takahashi identities. Time-ordered products for some derivatives of Dirac fields can only be defined with special care in terms of Feynman graphs. Current-algebra Ward-Takahashi identities are obtained if ${\mathcal{L}}^{\mathrm{derivative}}$ is invariant under the algebra; however, Schwinger terms are absent from these identities.