Perturbative and non-perturbative renormalization of quantum field theories and gravity

Abstract

This thesis is about the exploration of perturbative and non-perturbative renormalization methods in quantum field theory and gravity. After we introduce and review these main concepts we investigate off-shell perturbative renormalisation of quantum gravity. We show that at each new loop order, the divergences that do not vanish on-shell are constructed from only the total metric, whilst those that vanish on-shell are renormalised by canonical transformations involving the quantum fields. Purely background metric divergences do not separately appear, and the background metric does not get renormalised. We verify these assertions by computing leading off-shell divergences to two loops, exploiting off-shell BRST invariance and the renormalisation group equations. Although some divergences can be absorbed by field redefinitions, we explain why this does not lead to finite beta-functions for the corresponding field.

Afterwards we explore non-perturbative methods applied to d-dimensional scalar field theory in the Local Potential Approximation. Sturm-Liouville methods allow the eigenoperator equation to be cast as a Schrödinger-type equation. Combining solutions in the large field limit with the Wentzel–Kramers–Brillouin approximation, we solve analytically for the scaling dimension of high dimension potential-type operators around a non-trivial fixed point. These results are universal, independent of the choice of cutoff function.

Finally, we review the functional f(R) approximations in the asymptotic safety approach to quantum gravity. It mostly focuses on the application of methods used to study scalar fields. In particular, one can use these methods to establish that there are at most a discrete number of fixed points, that these support a finite number of relevant operators, and that the scaling dimension of high dimension operators is universal up to parametric dependence inherited from the single-metric approximation. Formulations using adaptive cutoffs, are also reviewed, and the main differences are highlighted.

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Identifiers

ORCID for Dalius Stulga: orcid.org/0009-0006-6304-4053

ORCID for Tim Morris: orcid.org/0000-0001-6256-9962

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