A simple systematic method is presented for the evaluation of the derivative expansion of the functional determinant with covariant differential operators, space-time-dependent background fields, and internal symmetry. The results are directly applicable to the one-loop perturbative effective-action expansion for bosons and fermions. Derivative expansions up to four-derivative terms for Trlnl -Il 2 + U(X)] and the effective action of the SO( N) linear cr model are calculated.

I present a systematic method for the development of the effective-action expansion in perturbation theory. The multiderivative terms beyond the effective potential can be evaluated in a direct and simple manner relying only on the familiar momentum space and Feynman propagator. I have used the self-coupled scalar field to explain the details of this new formulation. The effectiveaction expansion for this model in the one-loop approximation is evaluated up to the terms containing four derivatives. This method can be readily generalized to other models with spins and internal symmetries.It has always been an important problem in physics to extract the dominant contribution of the shortdistance effects on the large-distance behavior. A classical example is the multipole expansion. In the local relativistic quantum field theory the short-distance effects are mostly due to quantum fluctuations. In the low-energy limit, short-distance effects are not explored in detail. Typical examples are the effects of heavy particles, confining particles, and quantum fluctuation of the light particles. It is more convenient io eliminate those degrees of freedom not directly observable and incorporate their effects into an effective action of the observable fields. The expansion of this necessarily nonlocal effective action into an infinite series of local actions in the order of the number of space-time derivatives is known as the effective-action expansion.This effective-action expansion is best formulated in the functional integral method in which the unobserved fields are integrated out. The calculation of the effective potential, which is the leading term of the expansion with no derivative, is well known.

No abstract