Narrow Results

We derive the one-loop effective action for scalar, pseudoscalar, and electromagnetic fields coupled to a Dirac fermion in an extension of QED with Yukawa couplings. Using the Schwinger proper-time formalism and zeta-function regularization, we calculate the full nonperturbative effective action to one loop in the constant background field approximation. Our result is nonperturbative in the external fields, and goes beyond existing results in the literature which treat only the first nontrivial order involving the pseudoscalar. The result has an even and odd part, which are related to the modulus and phase of the fermion functional determinant. The even contribution to the effective action involves the modulus of the effective Yukawa couplings and is invariant under global chiral transformations while the odd contribution is proportional to the angle between the scalar and pseudoscalar couplings. In different limits the effective action reduces either to the Euler–Heisenberg effective action or the Coleman–Weinberg potential. We also comment on the relationship between the odd part of the effective action and the chiral anomaly in QED.

The conformal anomaly provides a useful tool for deriving vacuum effective action. Such contributions to the gravitational action lead to the exponential inflation, similar to the cosmological constant term. In the presence of a background electromagnetic field, there is no exponential solution, including the case when the the anomaly-induced correction in the electromagnetic sector is taken into account. In this contribution we explore the role of the anomaly-induced terms in the form of "equation of state" in both gravitational and radiation sectors. Furthermore, we derive the scaling laws for these terms for the dS, radiation and matter-dominated cases.

On this 75th anniversary of the publication of the Heisenberg-Euler paper on the full non-perturbative one-loop effective action for quantum electrodynamics I review their paper and discuss some of the impact it has had on quantum field theory.

The equations for the QED effective action derived in Ref. 3 are considered using singular perturbation theory. The effective action is divided into regular and singular (in coupling constant) parts. It is shown that expression for the regular part coincides with usual Feynman perturbation series over coupling constant, while the remainder has essential singularity at the vanishing coupling constant: $e\to 0$. This means that in the frame of quantum field theory it is impossible “to switch off” electromagnetic interaction in general and pass on to “free electron”.

The explicit expressions for the one-loop non-perturbative corrections to the gravitational effective action induced by a scalar field on a stationary gravitational background are obtained both at zero and finite temperatures. The perturbative and nonperturbative contributions to the one-loop effective action are explicitly separated. The nonperturbative part of the renormalized one-loop effective action at zero temperature is proved to depend explicitly on the Killing vector defining the vacuum state of quantum fields. This part cannot be expressed in a covariant way through the metric and its derivatives alone.