Non-perturbative corrections to the one-loop free energy induced by a massive scalar field on a stationary slowly varying in space gravitational background

Abstract

Abstract: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. It is proved that, after a suitable renormalization, the perturbative part of the effective action at zero temperature can be expressed in a covariant form solely in terms of the metric and its derivatives. This part coincides with the known large mass expansion of the one-loop effective action. The non-perturbative 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. The implications of this result for the structure and symmetries of the effective action for gravity are discussed.

High-temperature expansion of the one-loop effective action induced by scalar and Dirac particles

“…This expression can be used to obtain the number of particle-antiparticle pairs in the system [19,33] in the high-temperature regime. The leading order terms coincide with that known in the literature [33].In [17,18], we developed a general procedure of how to obtain the complete high-temperature expansion when the spectral problem is reduced to the solution of the KG type equation with a self-adjoint operator. The background fields are supposed to be of a general form, in particular, A 0 = 0 and the metric is stationary.…”

“…for some ǫ > 0. Then, the one-loop Ω-potential takes the form [18] Ω = − ∞ 0 dω ζ + (0, ω) e β(ω−µ) ± 1 + ζ + (0, −ω) e β(ω+µ) ± 1 .…”

“…The first term in this expression is a contribution from particles, and the second one corresponds to antiparticles. The theorem 1 allows us to give a simple proof of the general formula for the high-temperature expansion of the Ω-potential [17,18]. Recall the main assumptions made about the zeta-function:…”

“…6, and so we do not dwell on it here.As is known [9,10,[26][27][28], the high-temperature expansion of the one-loop Ω-potential without the vacuum contribution can be used to obtain the energy of vacuum fluctuations at zero temperature. Therefore, we shall derive a nonperturbative representation of the vacuum energy for the Dirac fermions in a stationary electromagnetic field of a general configuration without resorting to the Wick rotation prescription (as for bosons, see [17][18][19]). This vacuum energy is real-valued for not too wild fields and corresponds to the standard definition of the vacuum of quantum fields on stationary backgrounds (see, e.g., [29]).…”

“…In [17,18], we developed a general procedure of how to obtain the complete high-temperature expansion when the spectral problem is reduced to the solution of the KG type equation with a self-adjoint operator. The background fields are supposed to be of a general form, in particular, A 0 = 0 and the metric is stationary.…”