## Quantum gravity

The properties of quantum gravity are reviewed from the point of view of renormalization. Various attempts to overcome the problem of nonrenormalizability are presented, and the reasons why most of them fail for quantum gravity are discussed. Interesting possibilities come from relaxing the locality assumption, which can inspire the investigation of a largely unexplored sector of quantum field theory. Another possibility is to work with infinitely many independent couplings, and search for physical quantities that only depend on a finite subset of them. In this spirit, it is useful to organize the classical action of quantum gravity, determined by renormalization, in a convenient way. Taking advantage of perturbative local field redefinitions, we write the action as the sum of the Hilbert term, the cosmological term, a peculiar scalar that is important only in higher dimensions, plus invariants constructed with at least three Weyl tensors. We show that the FRLW configurations, and many other locally conformally flat metrics, are exact solutions of the field equations in arbitrary dimensions $d>3$. If the metric is expanded around such configurations the quadratic part of the action is free of higher-time derivatives. Other well-known metrics, such as those of black holes, are instead affected in nontrivial ways by the classical corrections of quantum origin.

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Mod. Phys. Lett. A 30 (2015) 1540004 | DOI: 10.1142/S0217732315400040

Course on renormalization, taught in Pisa in 2015. (More chapters will be added later.)

Last update: May 9th 2015, 230 pages

Contents:

Preface

1. Functional integral

• 1.1 Path integral
• Schroedinger equation
• Free particle
• 1.2 Free field theory
• 1.3 Perturbative expansion
• Feynman rules
• 1.4 Generating functionals, Schwinger-Dyson equations
• 1.6 Massive vector fields
• 1.7 Fermions

2. Renormalization

• 2.1 Dimensional regularization
• 2.1.1 Limits and other operations in $D$ dimensions
• 2.1.2 Functional integration measure
• 2.1.3 Dimensional regularization for vectors and fermions
• 2.2 Divergences and counterterms
• 2.3 Renormalization to all orders
• 2.4 Locality of counterterms
• 2.5 Power counting
• 2.6 Renormalizable theories
• 2.7 Composite fields
• 2.8 Maximum poles of diagrams
• 2.9 Subtraction prescription
• 2.10 Regularization prescription
• 2.12 About the series resummation

3. Renormalization group

• 3.1 The Callan-Symanzik equation
• 3.2 Finiteness of the beta function and the anomalous dimensions
• 3.3 Fixed points of the RG flow
• 3.4 Scheme (in)dependence
• 3.5 A deeper look into the renormalization group

4. Gauge symmetry

• 4.1 Abelian gauge symmetry
• 4.2 Gauge fixing
• 4.3 Non-Abelian global symmetry
• 4.4 Non-Abelian gauge symmetry

5. Canonical gauge formalism

• 5.1 General idea behind the canonical gauge formalism
• 5.2 Systematics of the canonical gauge formalism
• 5.3 Canonical transformations
• 5.4 Gauge fixing
• 5.5 Generating functionals
• 5.6 Ward identities

6. Quantum electrodynamics

• 6.1 Ward identities
• 6.2 Renormalizability of QED to all orders

7 Non-Abelian gauge field theories

• 7.1 Renormalizability of non-Abelian gauge theories to all orders
• Raw subtraction

A. Notation and useful formulas

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The classical action of quantum gravity, determined by renormalization, contains infinitely many independent couplings and can be expressed in different perturbatively equivalent ways. We organize it in a convenient form, which is based on invariants constructed with the Weyl tensor. We show that the FLRW metrics are exact solutions of the field equations in arbitrary dimensions, and so are all locally conformally flat solutions of the Einstein equations. Moreover, expanding the metric tensor around locally conformally flat backgrounds the quadratic part of the action is free of higher derivatives. Black-hole solutions of Schwarzschild and Kerr type are modified in a non-trivial way. We work out the first corrections to their metrics and study their properties.

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JHEP 1305 (2013) 028 | DOI: 10.1007/JHEP05(2013)028

arXiv:1302.7100 [gr-qc]

I prove that classical gravity coupled with quantized matter can be renormalized with a finite number of independent couplings, plus field redefinitions, without introducing higher-derivative kinetic terms in the gravitational sector, but adding vertices that couple the matter stress-tensor with the Ricci tensor. The theory is called “acausal gravity”, because it predicts the violation of causality at high energies. Renormalizability is proved by means of a map M that relates acausal gravity with higher-derivative gravity. The causality violations are governed by two parameters, a and b, that are mapped by M into higher-derivative couplings. At the tree level causal prescriptions exist, but they are spoiled by the one-loop corrections. Some ideas are inspired by the usual treatments of the Abraham-Lorentz force in classical electrodynamics.

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JHEP 0701 (2007) 062 | DOI: 10.1088/1126-6708/2007/01/062

arXiv:hep-th/0605205

In flat space, $\gamma_5$ and the epsilon tensor break the dimensionally continued Lorentz symmetry, but propagators have fully Lorentz invariant denominators. When the Standard Model is coupled with quantum gravity $\gamma_5$ breaks the continued local Lorentz symmetry. I show how to deform the Einstein lagrangian and gauge-fix the residual local Lorentz symmetry so that the propagators of the graviton, the ghosts and the BRST auxiliary fields have fully Lorentz invariant denominators. This makes the calculation of Feynman diagrams more efficient.

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Phys.Lett. B596 (2004) 90-95 | DOI: 10.1016/j.physletb.2004.06.089

arXiv:hep-th/0404032

As it stands, quantum gravity coupled with matter in three spacetime dimensions is not finite. In this paper I show that an algorithmic procedure that makes it finite exists, under certain conditions. To achieve this result, gravity is coupled with an interacting conformal field theory $C$. The Newton constant and the marginal parameters of $C$ are taken as independent couplings. The values of the other irrelevant couplings are determined iteratively in the loop- and energy-expansions, imposing that their beta functions vanish. The finiteness equations are solvable thanks to the following properties: the beta functions of the irrelevant couplings have a simple structure; the irrelevant terms made with the Riemann tensor can be reabsorbed by means of field redefinitions; the other irrelevant terms have, generically, non-vanishing anomalous dimensions. The perturbative expansion is governed by an effective Planck mass that takes care of the interactions in the matter sector. As an example, I study gravity coupled with Chern-Simons $U(1)$ gauge theory with massless fermions, solve the finiteness equations and determine the four-fermion couplings to two-loop order. The construction of this paper does not immediately apply to four-dimensional quantum gravity.

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Nucl.Phys. B687 (2004) 124-142 | DOI: 10.1016/j.nuclphysb.2004.03.024

arXiv:hep-th/0309250

In three spacetime dimensions, where no graviton propagates, pure gravity is known to be finite. It is natural to inquire whether finiteness survives the coupling with matter. Standard arguments ensure that there exists a subtraction scheme where no Lorentz-Chern-Simons term is generated by radiative corrections, but are not sufficiently powerful to ensure finiteness. Therefore, it is necessary to perform an explicit (two-loop) computation in a specific model. I consider quantum gravity coupled with Chern-Simons U(1) gauge theory and massless fermions and show that renormalization originates four-fermion divergent vertices at the second loop order. I conclude that quantum gravity coupled with matter, as it stands, is not finite in three spacetime dimensions.

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Nucl.Phys. B687 (2004) 143-160 | DOI: 10.1016/j.nuclphysb.2004.03.023

arXiv:hep-th/0309249

I study some aspects of the renormalization of quantum field theories with infinitely many couplings in arbitrary space-time dimensions. I prove that when the space-time manifold admits a metric of constant curvature the propagator is not affected by terms with higher derivatives. More generally, certain lagrangian terms are not turned on by renormalization, if they are absent at the tree level. This restricts the form of the action of a non-renormalizable theory, and has applications to quantum gravity. The new action contains infinitely many couplings, but not all of the ones that might have been expected. In quantum gravity, the metric of constant curvature is an extremal, but not a minimum, of the complete action. Nonetheless, it appears to be the right perturbative vacuum, at least when the curvature is negative, suggesting that the quantum vacuum has a negative asymptotically constant curvature. The results of this paper give also a set of rules for a more economical use of effective quantum field theories and suggest that it might be possible to give mathematical sense to theories with infinitely many couplings at high energies, to search for physical predictions.

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Class.Quant.Grav. 20 (2003) 2355-2378 | DOI: 10.1088/0264-9381/20/11/326

arXiv:hep-th/0212013

We study a regularization of the Pauli-Villars kind of the one loop gravitational divergences in any dimension. The Pauli-Villars fields are massive particles coupled to gravity in a covariant and nonminimal way, namely one real tensor and one complex vector. The gauge is fixed by means of the unusual gauge-fixing that gives the same effective action as in the context of the background field method. Indeed, with the background field method it is simple to see that the regularization effectively works. On the other hand, we show that in the usual formalism (non background) the regularization cannot work with each gauge-fixing.In particular, it does not work with the usual one. Moreover, we show that, under a suitable choice of the Pauli-Villars coefficients, the terms divergent in the Pauli-Villars masses can be corrected by the Pauli-Villars fields themselves. In dimension four, there is no need to add counterterms quadratic in the curvature tensor to the Einstein action (which would be equivalent to the introduction of new coupling constants). The technique also works when matter is coupled to gravity. We discuss the possible consequences of this approach, in particular the renormalization of Newton’s coupling constant and the appearance of two parameters in the effective action, that seem to have physical implications.

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Phys.Rev. D48 (1993) 5751-5763 | DOI: 10.1103/PhysRevD.48.5751

arXiv:hep-th/9307014

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Last update: May 9th 2015, 230 pages

Contents: Preface | 1. Functional integral | 2. Renormalization | 3. Renormalization group | 4. Gauge symmetry | 5. Canonical formalism | 6. Quantum electrodynamics | 7. Non-Abelian gauge field theories | Notation and useful formulas | References

Course on renormalization, taught in Pisa in 2015. (More chapters will be added later.)