Minisuperspace models as infrared contributions

Abstract

A direct correspondence of quantum mechanics as a minisuperspace model for a self-interacting scalar quantum-field theory is established by computing, in several models, the infrared contributions to 1-loop effective potentials of Coleman-Weinberg type. A minisuperspace approximation rather than truncation is thereby obtained. By this approximation, the spatial averaging scale of minisuperspace models is identified with an infrared scale (but not a regulator or cut-off) delimiting the modes included in the minisuperspace model. Some versions of the models studied here have discrete space or modifications of the Hamiltonian expected from proposals of loop quantum gravity. They shed light on the question of how minisuperspace models of quantum cosmology can capture features of full quantum gravity. While it is shown that modifications of the Hamiltonian can well be described by minisuperspace truncations, some related phenomena such as signature change, confirmed and clarified here for modified scalar field theories, require at least a perturbative treatment of inhomogeneity beyond a strict minisuperspace model. The new methods suggest a systematic extension of minisuperspace models by a canonical effective formulation of perturbative inhomogeneity.

Minisuperspace models of discrete systems

“…Our results provide further support for the canonical effective methods proposed for quantum gravity in [36,37], with an extension to the computation of effective potentials in [44].…”

“…The first term is twice the non-interacting contribution from a single spin with the same form as in (39), the second term, −γs 2h2 , is γ Ĵ z h1 Ĵ z h2 for antiparallel alignment in the z-direction, and the last term, −γsh 2 , adds up the two non-zero covariances in (44). This result agrees with the ground-state energy (29)…”

Critical Evaluation of Common Claims in Loop Quantum Cosmology

“…In particular, the prevalent application of minisuperspace models in quantum cosmology can be meaningful, provided one takes into account the correct role played by the averaging region V as an infrared scale (but not an infrared regulator). This role can be inferred from analogous constructions [50] in the much better-understood case of scalar quantum field theory on a flat background space-time, where the Coleman-Weinberg potential [51] provides crucial insights.…”

Testing Nonassociative Quantum Mechanics

“…Instead, we solve equations of motion for fluctuations in an adiabatic approximation (giving stationary moments in a near-coherent state) and saturating uncertainty relations (minimizing fluctuations). For well-understood (associative) systems such as anharmonic oscillators [25] or Coleman-Weinberg potentials in self-interacting scalar field theories [26], the correct results are obtained in this way [22,24]. In our new situation, we minimize fluctuations by saturating the uncertainty relations, in the standard form for q i and p i and for non-commuting momentum components.…”

“…Following the methods of [24], we can compute the relevant state properties without using a wave function. Instead, we solve equations of motion for fluctuations in an adiabatic approximation (giving stationary moments in a near-coherent state) and saturating uncertainty relations (minimizing fluctuations).…”