Quantum gravity
We prove spectral optical identities in quantum field theories of physical particles (defined by the Feynman $i\epsilon $ prescription) and purely virtual particles (defined by the fakeon prescription). The identities are derived by means of purely algebraic operations and hold for every (multi)threshold separately and for arbitrary frequencies. Their major significance is that they offer a deeper understanding on the problem of unitarity in quantum field theory. In particular, they apply to “skeleton” diagrams, before integrating on the space components of the loop momenta and the phase spaces. In turn, the skeleton diagrams obey a spectral optical theorem, which gives the usual optical theorem for amplitudes, once the integrals on the space components of the loop momenta and the phase spaces are restored. The fakeon
prescription/projection is implemented by dropping the thresholds that involve fakeon frequencies. We give examples at one loop (bubble, triangle, box, pentagon and hexagon), two loops (triangle with “diagonal”, box with diagonal) and arbitrarily many loops. We also derive formulas for the loop integrals with fakeons and relate them to the known formulas for the loop integrals with physical particles.
J. High Energy Phys. 11 (2021) 030 | DOI: https://doi.org/10.1007/JHEP11(2021)030
We study primordial cosmology with two scalar fields that participate in inflation at the same time, by coupling quantum gravity (i.e., the theory $R+R^2+C^2$ with the fakeon prescription/projection for $C^2$) to a scalar field with a quadratic potential. We show that there exists a perturbative regime that can be described by an asymptotically de Sitter, cosmic RG flow in two couplings. Since the two scalar degrees of freedom mix in nontrivial ways, the adiabatic and isocurvature perturbations are not RG invariant on superhorizon scales. It is possible to identify the correct perturbations by using RG invariance as a guiding principle. We work out the resulting power spectra of the tensor and scalar perturbations to the NNLL and NLL orders, respectively. An unexpected consequence of RG invariance is that the theory remains predictive. Indeed, the scalar mixing affects only the subleading corrections, so the predictions of quantum gravity with single-field inflation are confirmed to the leading order.
J. Cosmol. Astropart. Phys. 07 (2021) 037 | DOI: 10.1088/1475-7516/2021/07/037
We study inflation as a “cosmic” renormalization-group flow. The flow, which encodes the dependence on the background metric, is described by a running coupling $\alpha $, which parametrizes the slow roll, a de Sitter free, analytic beta function and perturbation spectra that are RG invariant in the superhorizon limit. Using RG invariance as a guiding principle, we classify the main types of flows according to the properties of their spectra, without referring to their origins from specific actions or models. Novel features include spectra with essential singularities in $\alpha $ and violations of the relation $r+8n_{\text{t}}=0$ to the leading order. Various classes of potentials studied in the literature can be described by means of the RG approach, even when the action includes a Weyl-squared term, while others are left out. In known cases, the classification helps identify the models that are ruled out by data. The RG approach is also able to generate spectra that cannot be derived from standard Lagrangian formulations.
Class. Quantum Grav. 38 (2021) 225011 | DOI: 10.1088/1361-6382/ac2b07
Online talk for the Kavli Institute for Cosmology, Institute of Astronomy, University of Cambridge
Abstract: I introduce the concept of purely virtual particle, or fakeon, and show how to use to make sense of quantum gravity as a quantum field theory. Then I discuss novel features of the classical limit and derive predictions in inflationary cosmology that could be tested in the forthcoming years, paying special attention to the effects of fakeons on perturbation spectra
Online talk for the Max-Planck-Institut für Kernphysik, (MPIK), Heidelberg, Germany
Abstract: I introduce the concept of purely virtual particle, or fakeon, and show how to use to make sense of quantum gravity as a quantum field theory. Then I discuss novel features of the classical limit and derive predictions in inflationary cosmology that could be tested in the forthcoming years, paying special attention to the effects of fakeons on perturbation spectra
Talk at NICPB, Tallinn, Estonia, Oct 14th, 2020
Abstract: I introduce the concept of purely virtual particle, or fakeon, and show how to use to make sense of quantum gravity as a quantum field theory. Then I discuss novel features of the classical limit and derive predictions in inflationary cosmology that could be tested in the forthcoming years, paying special attention to the effects of fakeons on perturbation spectra
We compute the inflationary perturbation spectra and the quantity $r+8n_{T}$ to the next-to-next-to-leading log order in quantum gravity with purely virtual particles (which means the theory $R+R^{2}+C^{2}$ with the fakeon prescription/projection for $C^{2}$). The spectra are functions of the inflationary running coupling $\alpha (1/k)$ and satisfy the cosmic renormalization-group flow equations, which determine the tilts and the running coefficients. The tensor fluctuations receive contributions from the spin-2 fakeon $\chi _{\mu \nu }$ at every order of the expansion in powers of $\alpha \sim 1/115$. The dependence of the scalar spectrum on the $\chi
_{\mu \nu }$ mass $m_{\chi }$, on the other hand, starts from the $\alpha^{2}$ corrections, which are handled perturbatively in the ratio $m_{\phi}/m_{\chi }$, where $m_{\phi }$ is the inflaton mass. The predictions have theoretical errors ranging from $\alpha ^{4}\sim 10^{-8}$ to $\alpha^{3}\sim 10^{-6}$. Nontrivial issues concerning the fakeon projection at higher orders are addressed.
J. Cosmol. Astropart. Phys. 02 (2021) 029 | DOI: 10.1088/1475-7516/2021/02/029
We study the running of power spectra in inflationary cosmology as a renormalization-group flow from the de Sitter fixed point. The beta function is provided by the equations of the background metric. The spectra of the scalar and tensor fluctuations obey RG evolution equations with vanishing anomalous dimensions in the superhorizon limit. By organizing the perturbative expansion in terms of leading and subleading logs, we calculate the spectral indices, their runnings, the runnings of the runnings, etc., to the next-to-leading log order in quantum gravity with fakeons (i.e., the theory $R+R^2+C^2$ with the fakeon prescription/projection for $C^2$). We show that these quantities are related to the spectra in a universal way. We also compute the first correction to the relation $r=−8n_T$ and provide a number of quantum gravity predictions that can be hopefully tested in the forthcoming future.
J. Cosmol. Astropart. Phys. 01 (2021) 048 | DOI: 10.1088/1475-7516/2021/01/048
Testable predictions of quantum gravity with fakeons on the spectra of the CMB radiation
Based on the paper 20A3 Renorm (arXiv: 2006.01163 [hep-th])
Talk given online by M. Piva for the Tokyo Institute of Technology, on Jun 16th, 2020
Talk given online by Marco Piva for the University of Sussex, on July 20th, 2020
Testable predictions of quantum gravity with fakeons on the spectra of the CMB radiation
Talk given by M. Piva at the National Institute of Chemical Physics and Biophysics, Tallinn, Estonia, on Jun 9th, 2020
Based on the paper 20A3 Renorm (arXiv: 2006.01163 [hep-th])
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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
The pdf file of the 2015 Edition is available here: PDF
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