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2 Renormalizing the Non-Renormalizable

The modern view on renormalization has been shaped by Kadanoff and Wilson. The more familiar perturbative notion of renormalizability is neither sufficient (e.g. 4 φ theory in d = 4) nor necessary (e.g. Gross–Neveu model in d = 3) for renormalizability in the Kadanoff–Wilson sense. For the convenience of the reader we summarize the main ideas in Appendix A, which also serves to introduce the terminology. The title of this section is borrowed from a paper by Gawedzki and Kupiainen [88].

In the present context the relevance of a Kadanoff–Wilson view on renormalization is two-fold: First it allows one to formulate the notion of renormalizability without reference to perturbation theory, and second it allows one to treat to a certain extent renormalizable and non-renormalizable on the same footing. The mismatch between the perturbative non-renormalizability of the Einstein–Hilbert action and the presumed asymptotic safety of a functional measure constructed by other means can thus be systematically explored.

In a gravitational context also the significance of renormalizabilty is less clear cut, and one should presumably go back to the even more fundamental property for which renormalizability is believed to be instrumental, namely the existence of a genuine continuum limit, roughly in the sense outlined in Section 1.3. Since rigorous results based on controlled approximations are unlikely to be obtained in the near future, we describe in the following criteria for the plausible existence of a genuine continuum limit based on two uncontrolled approximations: renormalized perturbation theory and the functional renormalization group approach. Such criteria are ‘implicit wisdom’ and are hardly ever spelled out. In the context of Quantum Gravidynamics, however, the absence of an obvious counterpart of the correlation length and non-renormalizability of the Einstein–Hilbert action makes things more subtle. In Sections 2.1 and 2.2 we therefore try to make the implicit explicit and to formulate critera for the existence of a genuine continuum limit which are applicable to Quantum Gravidynamics as well.

In Section 2.3 we describe the renormalization problem for Quantum Gravidynamics and in Section 2.4 the dimensional reduction phenomenon outlined before.

For a summary of basic renormalization group concepts we refer to Appendix A and for a review of the renormalization group for the effective average action to Appendix C.

 2.1 Perturbation theory and continuum limit
 2.2 Functional flow equations and UV renormalization
 2.3 Towards Quantum Gravidynamics
  2.3.1 The role of Newton’s constant
  2.3.2 Perturbation theory and higher derivative theories
  2.3.3 Kinematical measure
  2.3.4 Effective action and states
  2.3.5 Towards physical quantities
 2.4 Dimensional reduction of residual interactions in UV
  2.4.1 Scaling of fixed point action
  2.4.2 Anomalous dimension at non-Gaussian fixed point
  2.4.3 Strict renormalizability and 4 1∕p propagators
  2.4.4 Spectral dimension and scaling of fixed point action

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