### Course

19R1 D. Anselmi
Theories of gravitation

Program

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### Recent papers and theorems

19R2 Damiano Anselmi
Fakeons, quantum gravity and the classical limit

Talk given at the Conference "Scale invariance in particle physics and cosmology", CERN, on January 29th, 2019 A new quantization prescription is able to endow quantum field theory with a new type of "particle", the fakeon (fake particle), which mediates interactions, but cannot be observed. A massive fakeon of spin 2 ... [more]

19A1 Damiano Anselmi
Fakeons and the classicization of quantum gravity: the FLRW metric

Under certain assumptions, it is possible to make sense of higher derivative theories by quantizing the unwanted degrees of freedom as fakeons, which are later projected away. Then the true classical limit is obtained by classicizing the quantum theory. Since quantum field theory is formulated perturbatively, the classicization is also ... [more]

19R1 Damiano Anselmi
Theories of gravitation

Program Differential geometry topological spaces, manifolds, differential manifolds, derivations, vector fields, differential forms, tangent space, cotangent space, tangent bundle, cotangent bundle, Lie derivative, metric, covariant derivative, torsion, curvature, metric compatibility, tetrad, spin connection, Christoffel symbols, Riemann tensor, Ricci tensor, Ricci scalar, Bianchi identities. General relativity Hilbert action, Palatini formulation, field equations. Coupling of gravity to ... [more]

18R2 Damiano Anselmi
From fakeons to quantum gravity

Talk given at the Department of Physics and Astronomy of Southampton University, UK, on Nov 16th, 2018 I introduce the concept of fake particle and study how it is used to formulate a consistent theory of quantum gravity. Fakeons arise from a new quantization prescription, alternative to the Feynman one, for ... [more]

18A7 Damiano Anselmi
On the nature of the Higgs boson

Several particles are not observed directly, but only through their decay products. We consider the possibility that they might be fakeons, i.e. fake particles, which mediate interactions but are not asymptotic states. A crucial role to determine the true nature of a particle is played by the imaginary parts of ... [more]

18A6 Damiano Anselmi
Let the dice play God

We define life as the amplification of quantum uncertainty up to macroscopic scales. A living being is any amplifier that achieves this goal. We argue that everything we know about life can be explained from this idea. We study a ladder mechanism to estimate the probability that the amplification occurs ... [more]

18R1 Damiano Anselmi
The correspondence principle in quantum field theory and quantum gravity

Talk given at the conference Progress and Visions in Quantum Theory in View of Gravity: Bridging foundations of physics and mathematics Max Planck Institute for Mathematics in the Sciences, Leipzig October 04, 2018 I claim that the best correspondence principle for quantum field theory and quantum gravity is made of unitarity, locality and proper ... [more]

18A5 Damiano Anselmi
The correspondence principle in quantum field theory and quantum gravity

We discuss the fate of the correspondence principle beyond quantum mechanics, specifically in quantum field theory and quantum gravity, in connection with the intrinsic limitations of the human ability to observe the external world. We conclude that the best correspondence principle is made of unitarity, locality, proper renormalizability (a refinement ... [more]

18A4 Damiano Anselmi
Fakeons, microcausality and the classical limit of quantum gravity

We elaborate on the idea of fake particle and study its physical consequences. When a theory contains fakeons, the true classical limit is determined by the quantization and a subsequent process of "classicization". One of the major predictions due to the fake particles is the violation of microcausality, which survives ... [more]

18A3 Damiano Anselmi and Marco Piva
Quantum gravity, fakeons and microcausality

We investigate the properties of fakeons in quantum gravity at one loop. The theory is described by a graviton multiplet, which contains the fluctuation $h_{\mu \nu }$ of the metric, a massive scalar $\phi$ and the spin-2 fakeon $\chi _{\mu \nu }$. The fields $\phi$ and $\chi _{\mu ... [more] 18A2 Damiano Anselmi and Marco Piva The ultraviolet behavior of quantum gravity A theory of quantum gravity has been recently proposed by means of a novel quantization prescription, which is able to turn the poles of the free propagators that are due to the higher derivatives into fakeons. The classical Lagrangian contains the cosmological term, the Hilbert term,$ \sqrt{-g}R_{\mu \nu }R^{\mu \nu ... [more]

18A1 Damiano Anselmi
Fakeons and Lee-Wick models

The "fakeon" is a fake degree of freedom, i.e. a degree of freedom that does not belong to the physical spectrum, but propagates inside the Feynman diagrams. Fakeons can be used to make higher-derivative theories unitary. Moreover, they help us clarify how the Lee-Wick models work. In this paper we ... [more]

17A3 Damiano Anselmi
On the quantum field theory of the gravitational interactions

We study the main options for a unitary and renormalizable, local quantum field theory of the gravitational interactions. The first model is a Lee-Wick superrenormalizable higher-derivative gravity, formulated as a nonanalytically Wick rotated Euclidean theory. We show that, under certain conditions, the $S$ matrix is unitary when the cosmological constant ... [more]

17A2 Damiano Anselmi and Marco Piva
Perturbative unitarity of Lee-Wick quantum field theory

We study the perturbative unitarity of the Lee-Wick models, formulated as nonanalytically Wick rotated Euclidean theories. The complex energy plane is divided into disconnected regions and the values of a loop integral in the various regions are related to one another by a nonanalytic procedure. We show that the one-loop ... [more]

17A1 Damiano Anselmi and Marco Piva
A new formulation of Lee-Wick quantum field theory

The Lee-Wick models are higher-derivative theories that are claimed to be unitary thanks to a peculiar cancelation mechanism. In this paper, we provide a new formulation of the models, to clarify several aspects that have remained quite mysterious, so far. Specifically, we define them as nonanalytically Wick rotated Euclidean theories. ... [more]

A theory of quantum gravity has been recently proposed by means of a novel quantization prescription, which is able to turn the poles of the free propagators that are due to the higher derivatives into fakeons. The classical Lagrangian contains the cosmological term, the Hilbert term, $\sqrt{-g}R_{\mu \nu }R^{\mu \nu }$ and $\sqrt{-g}R^{2}$. In this paper, we compute the one-loop renormalization of the theory and the absorptive part of the graviton self energy. The results illustrate the mechanism that makes renormalizability compatible with unitarity. The fakeons disentangle the real part of the self energy from the imaginary part. The former obeys a renormalizable power counting, while the latter obeys the nonrenormalizable power counting of the low energy expansion and is consistent with unitarity in the limit of vanishing cosmological constant. The value of the absorptive part is related to the central charge $c$ of the matter fields coupled to gravity.

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J. High Energ. Phys. 05 (2018) 27 | DOI: 10.1007/JHEP05(2018)027

arXiv: 1803.07777 [hep-th]

The “fakeon” is a fake degree of freedom, i.e. a degree of freedom that does not belong to the physical spectrum, but propagates inside the Feynman diagrams. Fakeons can be used to make higher-derivative theories unitary. Moreover, they help us clarify how the Lee-Wick models work. In this paper we study the fakeon models, that is to say the theories that contain fake and physical degrees of freedom. We formulate them by (nonanalytically) Wick rotating their Euclidean versions. We investigate the properties of arbitrary Feynman diagrams and, among other things, prove that the fakeon models are perturbatively unitary to all orders. If standard power counting constraints are fulfilled, the models are also renormalizable. The S matrix is regionwise analytic. The amplitudes can be continued from the Euclidean region to the other regions by means of an unambiguous, but nonanalytic, operation, called average continuation. We compute the average continuation of typical amplitudes in four, three and two dimensions and show that its predictions agree with those of the nonanalytic Wick rotation. By reconciling renormalizability and unitarity in higher-derivative theories, the fakeon models are good candidates to explain quantum gravity.

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J. High Energy Phys. 02 (2018) 141 | DOI: 10.1007/JHEP02(2018)141

arXiv: 1801.00915 [hep-th]

hal-01900285

We study the main options for a unitary and renormalizable, local quantum field theory of the gravitational interactions. The first model is a Lee-Wick superrenormalizable higher-derivative gravity, formulated as a nonanalytically Wick rotated Euclidean theory. We show that, under certain conditions, the $S$ matrix is unitary when the cosmological constant vanishes. The model is the simplest of its class. However, infinitely many similar options are allowed, which raises the issue of uniqueness. To deal with this problem, we propose a new quantization prescription, by doubling the unphysical poles of the higher-derivative propagators and turning them into Lee-Wick poles. The Lagrangian of the simplest theory of quantum gravity based on this idea is the linear combination of $R$, $R_{\mu \nu}R^{\mu \nu }$, $R^{2}$ and the cosmological term. Only the graviton propagates in the cutting equations and, when the cosmological constant vanishes, the $S$ matrix is unitary. The theory satisfies the locality of counterterms and is renormalizable by power counting. It is unique in the sense that it is the only one with a dimensionless gauge coupling.

J. High Energy Phys. 06 (2017) 086 | DOI: 10.1007/JHEP06(2017)086

arXiv: 1704.07728 [hep-th]

hal-01900209

We study the perturbative unitarity of the Lee-Wick models, formulated as nonanalytically Wick rotated Euclidean theories. The complex energy plane is divided into disconnected regions and the values of a loop integral in the various regions are related to one another by a nonanalytic procedure. We show that the one-loop diagrams satisfy the expected, unitary cutting equations in each region: only the physical degrees of freedom propagate through the cuts. The goal can be achieved by working in suitable subsets of each region and proving that the cutting equations can be analytically continued as a whole. We make explicit calculations in the cases of the bubble and triangle diagrams and address the generality of our approach. We also show that the same higher-derivative models violate unitarity if they are formulated directly in Minkowski spacetime.

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Phys. Rev. D 96 (2017) 045009 | DOI: 10.1103/PhysRevD.96.045009

arXiv: 1703.05563 [hep-th]

The Lee-Wick models are higher-derivative theories that are claimed to be unitary thanks to a peculiar cancelation mechanism. In this paper, we provide a new formulation of the models, to clarify several aspects that have remained quite mysterious, so far. Specifically, we define them as nonanalytically Wick rotated Euclidean theories. The complex energy plane is divided into disconnected regions, which can be related to one another by a well defined, albeit nonanalytic procedure. Working in a generic Lorentz frame, the models are intrinsically equipped with the right recipe to treat the pinchings of the Lee-Wick poles, with no need of external ad hoc prescriptions. We describe these features in detail by calculating the one-loop bubble diagram and explaining how the key properties generalize to more complicated diagrams. The physical results of our formulation are different from those of the previous ones. The unusual behaviors of the physical amplitudes lead to interesting phenomenological predictions.

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J. High Energy Phys. 06 (2017) 066 | DOI: 10.1007/JHEP06(2017)066

arXiv: 1703.04584 [hep-th]

It is normally believed that viewing time as time, that is to say a real coordinate $t$ with a Minkowski metric, is equivalent to viewing it as a space coordinate $x^4$ (with a Euclidean metric) that is turned into imaginary values by means of the Wick rotation. Indeed, most quantum field theories, including the standard model, can be equivalently formulated directly in Minkowski spacetime or by Wick rotating their Euclidean versions.

However, in a recent paper it was shown that the two formulations are not always equivalent. In particular, they are not equivalent in a wide realm of quantum field theories that is relevant for the search of quantum gravity.

The two formulations differ so much that one of the two, the Minkowski one, is mathematically inconsistent, because it leads to nonlocal divergences that cannot be subtracted away. The only viable formulation of quantum field theory is thus the Wick rotation of a Euclidean theory.

This observation could have very broad consequences. Ultimately, it tells us that the environment of quantum field theory is not Minkowski spacetime, but a different kind of spacetime, which we may call Wick spacetime, that is to say the Wick rotated Euclidean space.

If we believe that quantum field theory is the correct framework to describe nature, as all experimental evidence suggests so far, the conclusion extends from quantum field theory to nature itself, i.e.

the universe does not live in Minkowski spacetime, but in Wick spacetime.

Said differently,

time is not time, but an imaginary space.

The cutting equations are diagrammatic identities that are used to prove perturbative unitarity in quantum field theory. In this paper, we derive algebraic, upgraded versions of them. Differently from the diagrammatic versions, the algebraic identities also holds for propagators with arbitrary, nonvanishing widths. In particular, the cut propagators do not need to vanish off shell. The new approach provides a framework to address unsolved problems of perturbative quantum field theory and a tool to investigate perturbative unitarity in higher-derivative theories that are relevant to the problem of quantum gravity, such as the Lee-Wick models and the fakeon models.

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Ann. Phys. 394 (2018) 294 | DOI: 10.1016/j.aop.2018.04.034

arXiv: 1612.07148 [hep-th]

We show that Minkowski higher-derivative quantum field theories are generically inconsistent, because they generate nonlocal, non-Hermitian ultraviolet divergences, which cannot be removed by means of standard renormalization procedures. By “Minkowski theories” we mean theories that are defined directly in Minkowski spacetime. The problems occur when the propagators have complex poles, so that the correlation functions cannot be obtained as the analytic continuations of their Euclidean versions. The usual power counting rules fail and are replaced by much weaker ones. Self-energies generate complex divergences proportional to inverse powers of D’Alembertians. Three-point functions give more involved nonlocal divergences, which couple to infrared effects. The violations of the locality and Hermiticity of counterterms are illustrated by means of explicit computations in scalar models and higher-derivative gravity.

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Eur. Phys. J. C 77 (2017) 84 | DOI: 10.1140/epjc/s10052-017-4646-7

arXiv: 1612.06510 [hep-th]

We reconsider perturbative unitarity in quantum field theory and upgrade several arguments and results. The minimum assumptions that lead to the largest time equation, the cutting equations and the unitarity equation are identified. Using this knowledge and a special gauge, we give a new, simpler proof of perturbative unitarity in gauge theories and generalize it to quantum gravity, in four and higher dimensions. The special gauge interpolates between the Feynman gauge and the Coulomb gauge without double poles. When the Coulomb limit is approached, the unphysical particles drop out of the cuts and the cutting equations are consistently projected onto the physical subspace. The proof does not extend to nonlocal quantum field theories of gauge fields and gravity, whose unitarity remains uncertain.

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Phys. Rev. D 94 (2016) 025028 | DOI: 10.1103/PhysRevD.94.025028

arXiv: 1606.06348 [hep-th]

Using the background field method and the Batalin-Vilkovisky formalism, we prove a key theorem on the cohomology of perturbatively local functionals of arbitrary ghost numbers, in renormalizable and nonrenormalizable quantum field theories whose gauge symmetries are general covariance, local Lorentz symmetry, non-Abelian Yang-Mills symmetries and Abelian gauge symmetries. Interpolating between the background field approach and the usual, nonbackground approach by means of a canonical transformation, we take advantage of the properties of both approaches and prove that a closed functional is the sum of an exact functional plus a functional that depends only on the physical fields and possibly the ghosts. The assumptions of the theorem are the mathematical versions of general properties that characterize the counterterms and the local contributions to the potential anomalies. This makes the outcome a theorem on the cohomology of renormalization, rather than the whole local cohomology. The result supersedes numerous involved arguments that are available in the literature.

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Phys. Rev. D 93 (2016) 065034 | DOI: 10.1103/PhysRevD.93.065034

arXiv: 1511.01244 [hep-th]

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### Book

14B1 D. Anselmi
Renormalization

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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.)