Lorentz covariance and Matthews's theorem for derivative-coupled field theories

Abstract

The canonical quantization of scalar-field Lagrangians involving at most first derivatives of the fields ("first-order" Lagrangian) or second derivatives ("second-order") is discussed. A direct, but necessarily perturbative, quantization procedure for a general first-order Lagrangian is used to show that such theories yield a Lorentz-invariant $S$ matrix to low orders of perturbation theory provided a covariant regularization scheme (e.g., dimensional or Pauli-Villars---but not a high-momentum cutoff) is employed. Matthews's theorem is verified in this context---the naive Feynman rules are valid. Second-order Lagrangians, quadratic in second but arbitrary in first derivatives, are shown to satisfy Matthews's theorem to all orders of perturbation theory and to be equivalent to first-order theories with Pauli-Villars regularization, thereby yielding a proof of Matthews's theorem for an arbitrary Pauli-Villars---regulated first-order theory. It is shown that for spectral reasons second- (and presumably, higher-) order theories are unacceptable physically. Finally, the canonical quantization of a second-order gauge theory is performed explicitly; the results show that (a) the naive Faddeev-Popov prescription remains valid in the presence of higher derivatives, and (b) the spectral pathology of second-order theories persists in gauge theories.