Unitarity of quantum field theory
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.
J. High Energy Phys. 02 (2018) 141 | DOI: 10.1007/JHEP02(2018)141
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
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.
Phys. Rev. D 96 (2017) 045009 | DOI: 10.1103/PhysRevD.96.045009
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.
J. High Energy Phys. 06 (2017) 066 | DOI: 10.1007/JHEP06(2017)066
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.
Ann. Phys. 394 (2018) 294 | DOI: 10.1016/j.aop.2018.04.034
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.
Phys. Rev. D 94 (2016) 025028 | DOI: 10.1103/PhysRevD.94.025028
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Quantum Gravity 
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14B1 D. Anselmi
Renormalization
Course on renormalization, taught in 2015.
Last update: September 15th 2023, 242 pages
The final (2023) edition is vaibable on Amazon:
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
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