Higher time derivatives in effective equations of canonical quantum systems

Abstract

Quantum-corrected equations of motion generically contain higher time derivatives, computed here in the setting of canonically quantized systems. The main example in which detailed derivations are presented is a general anharmonic oscillator, but conclusions can be drawn also for systems in quantum gravity and cosmology.

Minisuperspace models as infrared contributions

“…Both assumptions have been shown to lead to correct results for anharmonic oscillators in quantum mechanics [4,7] and, in the context of the Coleman-Weinberg potential, for quartic self-interactions of a scalar field [5]. In the rest of this paper, we work with general potentials W (φ) but assume that approximations as in the preceding brief derivation are valid both in the (full) quantum-field theory and the minisuperspace quantum-mechanics model.…”

“…We begin by providing a more-detailed derivation of the field-theory effective potential with a 1-dimensional spatial manifold, following [5]. We take this opportunity to show more details of the derivation that leads to the integral used in (16), but at the same time give an example which demonstrates the dependence of the infrared contribution on the spatial dimension.…”

“…The effective potential is well-defined if the moments behave adiabatically, corresponding to a derivative expansion in time [7]. To lowest order in an adiabatic expansion, we ignore the time derivatives on the left-hand sides of (10)- (12) and solve the resulting linear equations for the moments.…”

“…In this last form, the effective potential is directly obtained by canonical methods [5], briefly reviewed below. Being canonical, the derivation does not make covariance manifest, and accordingly only the spatial wave vector k appears.…”

Deformed covariance in spherically symmetric vacuum models of loop quantum gravity: Consistency in Euclidean and self-dual gravity

“…This observation suggests that there is room for further explorations of possibly new models. A likely candidate for a generic extension is the inclusion of canonical quantum back-reaction effects [31][32][33], which in an action formulation provide higher-curvature terms with generic higher derivatives. However, quantum back-reaction on its own does not modify the hypersurface-deformation brackets of constraints [34] and is therefore unlikely to change our conclusions about modified space-time structures.…”