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.

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: 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

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.

We prove a set of polynomial identities for complex numbers associated with Feynman diagrams. The equations are at the core of perturbative unitarity in quantum field theory.

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.

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

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

We study some properties of the canonical transformations in classical mechanics and quantum field theory and give a number of practical formulas concerning their generating functions. First, we give a diagrammatic formula for the perturbative expansion of the composition law around the identity map. Then, we propose a standard way to express the generating function of a canonical transformation by means of a certain “componential” map, which obeys the Baker-Campbell-Hausdorff formula. We derive the diagrammatic interpretation of the componential map, work out its relation with the solution of the Hamilton-Jacobi equation and derive its time-ordered version. Finally, we generalize the results to the Batalin-Vilkovisky formalism, where the conjugate variables may have both bosonic and fermionic statistics, and describe applications to quantum field theory.

Eur. Phys. J. C 76 (2016) 49 | DOI: 10.1140/epjc/s10052-015-3874-y