19S1 D. Anselmi
Theories of gravitation




D. Anselmi
From Physics To Life

A journey to the infinitesimally small and back

In English and Italian

Available on Amazon:
US: book | ebook  (in EN)
IT: book | ebook  (in IT)

Recent Papers

Damiano Anselmi

The physics of fundamental interactions is going through a concerning, prolonged period of stagnation. The incredible success of the standard model of particle physics and the lack of new experimental data have frustrated our hopes in the future. On top of that, the scientific community shattered into a large number of isolated groups. Many mainstreams have consolidated, leaving not much room for the advancement of bright, original proposals. In frontier domains, like quantum gravity, most mainstreams have disavowed the inheritance of the glowing past and embarked on uncertain routes (string theory, loop quantum gravity and many others). It is time to make room for approaches that are really out of the box and can truly trigger a renaissance of particle physics. Yet, they can only be believable if they are solidly rooted in the successes of the past. This ERC project pursues a research line that does stem from the achievements of the past, but is radically new and has the potential to take us out of this dark period. It is based on the notion of purely virtual particle, which upgrades in a crucial way our understanding of fundamental interactions through quantum field theory. One of its key predictions in primordial cosmology could be confirmed experimentally within a decade. Nevertheless, the scientific community cannot afford another decade like the past ones, so it is imperative to act now. The new idea opens the door to unthinkable scenarios and has a huge amount of ramifications and applications to all areas of fundamental physics, with the potential to build bridges between quantum gravity, primordial cosmology and the phenomenology of particle physics beyond the standard model. More key predictions are expected to follow, together with crucial ideas for future colliders. Hopefully, they will trigger the breakthroughs that we need to make a U turn, activate a virtuous circle, reunite the scientific community and lead to the renaissance of particle physics.


The techical part of the application can be viewed here

We review the concept of purely virtual particle and its uses in quantum gravity, primordial cosmology and collider physics. The fake particle, or “fakeon”, which mediates interactions without appearing among the incoming and outgoing states, can be introduced by means of a new diagrammatics. The renormalization coincides with one of the parent Euclidean diagrammatics, while unitarity follows from spectral optical identities, which can be derived by means of algebraic operations. The classical limit of a theory of physical particles and fakeons is described by an ordinary Lagrangian plus Hermitian, micro acausal and micro nonlocal self-interactions. Quantum gravity propagates the graviton, a massive scalar field (the inflaton) and a massive spin-2 fakeon, and leads to a constrained primordial cosmology, which predicts the tensor-to-scalar ratio r in the window 0.4≲1000r≲3.5. The interpretation of inflation as a cosmic RG flow allows us to calculate the perturbation spectra to high orders in the presence of the Weyl squared term. In models of new physics beyond the standard model, fakeons evade various phenomenological bounds, because they are less constrained than normal particles. The resummation of self-energies reveals that it is impossible to get too close to the fakeon peak. The related peak uncertainty, equal to the fakeon width divided by 2, is expected to be observable.


Symmetry 2022, 14(3), 521 | DOI: 10.3390/sym14030521

arXiv: 2203.02516 [hep-th]

We reconsider the Lee-Wick (LW) models and compare their properties to the properties of the models that contain purely virtual particles. We argue against the LW premise that unstable particles can be removed from the sets of incoming and outgoing states in scattering processes. The removal leads to a non-Hermitian classical limit, besides clashing with the observation of the muon. If, on the other hand, all the states are included, the LW models have an unbounded Hamiltonian or negative norms. Purely virtual particles, on the contrary, lead to a Hermitian classical limit and are absent from the sets of incoming and outgoing states without implications on the observation of long-lived unstable particles. We give a vademecum to summarize the properties of most options. We study a method to remove the LW ghosts only partially, by saving the physical particles they contain. Specifically, we replace a LW ghost with a certain superposition of a fakeon and an observable particle, and drop only the former from the sets of the external states. The trick can be used to make the Pauli-Villars fields consistent and observable, without sending their masses to infinity, or build a finite QED, by tweaking the original Lee-Wick construction. However, it has issues with general covariance, so it cannot be applied as is to quantum gravity, where a manifestly covariant decomposition requires the introduction of a massive spin 2 multiplet.


arXiv: 2202.10483 [hep-th]

We study the resummation of self-energy diagrams into dressed propagators in the case of purely virtual particles and compare the results with those obtained for physical particles and ghosts. The three geometric series differ by infinitely many contact terms, which do not admit well-defined sums. The peak region, which is outside the convergence domain, can only be reached in the case of physical particles, thanks to analyticity. In the other cases, nonperturbative effects become important. To clarify the matter, we introduce the energy resolution $\Delta E$ around the peak and argue that a “peak uncertainty” $\Delta E\gtrsim \Delta E_{\text{min}}\simeq \Gamma _{\text{f}}/2$ around energies $E\simeq m_{\text{f}}$ expresses the impossibility to approach the fakeon too closely, $m_{\text{f}}$ being the fakeon mass and $\Gamma _{\text{f}}$ being the fakeon width. The introduction of $\Delta E$ is also crucial to explain the observation of unstable long-lived particles, like the muon. Indeed, by the common energy-time uncertainty relation, such particles are also affected by ill-defined sums at $\Delta E=0$, whenever we separate their observation from the observation of their decay products. We study the regime of large $\Gamma _{\text{f}}$, which applies to collider physics (and situations like the one of the $Z$ boson), and the regime of small $\Gamma _{\text{f}}$, which applies to quantum gravity (and situations like the one of the muon).


to appear in J. High Energy Phys.

arXiv: 2201.00832 [hep-ph]

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:

arXiv: 2109.06889 [hep-th]

Extensions to the Standard Model that use strictly off-shell degrees of freedom – the fakeons – allow for new measurable interactions at energy scales usually precluded by the constraints that target the on-shell propagation of new particles. Here we employ the interactions between a new fake scalar doublet and the muon to explain the recent Fermilab measurement of its anomalous magnetic moment. Remarkably, unlike in the case of usual particles, the experimental result can be matched for fakeon masses below the electroweak scale without contradicting the stringent precision data and collider bounds on new light degrees of freedom. Our analysis, therefore, demonstrates that the fakeon approach offers unexpected viable possibilities to model new physics naturally at low scales.


Phys. Rev. D 104 (2021) 035009 | DOI: 10.1103/PhysRevD.104.035009

arXiv: 2104.03249 [hep-ph]

We introduce a new way of modeling the physics beyond the Standard Model by considering fake, strictly off-shell degrees of freedom: the fakeons. To demonstrate the approach and exemplify its reach, we re-analyze the phenomenology of the Inert Doublet Model under the assumption that the second doublet is a fakeon. Remarkably, the fake doublet avoids the most stringent $Z$-pole constraints regardless of the chosen mass scale, thereby allowing for the presence of new effects well below the electroweak scale. Furthermore, the absence of on-shell propagation prevents fakeons from inducing missing energy signatures in collider experiments. The distinguishing features of the model appear at the loop level, where fakeons modify the Higgs boson $h\rightarrow\gamma\gamma$ decay width and the Higgs trilinear coupling. The running of Standard Model parameters proceeds as in the usual Inert Doublet Model case. Therefore, the fake doublet can also ensure the stability of the Standard Model vacuum. Our work shows that fakeons are a valid alternative to the usual tools of particle physics model building, with the potential to shape a new paradigm, where the significance of the existing experimental constraints towards new physics must necessarily be reconsidered.


J. High Energy Phys. 10 (2021) 132 | DOI: 10.1007/JHEP10(2021)132

arXiv: 2104.02071 [hep-ph]

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

arXiv: 2105.05864 [hep-th]

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

arXiv: 2103.01653 [hep-th]

The Youtube Channel Quantum Gravity has just crossed the threshold of 1100 subscribers. Its URL address is

Quick statistics up to date, at 3 years since creation:

Total number of views: 68.6 k
Total watched time: 10.6 k hours
Number of subscribers: 1103
Number of videos: 130

Most watched videos:

I quanti e la quasi realtà: 3090 views, 756 hours, 82 likes
Quantum gravity from fakeons – Catania 2019: 2910 views, 87.5 hours, 32 likes
Quantum gravity al pub: 2767 views, 651 hours, 45 likes
L’atomo: 2514 views, 238 hours, 106 likes

Theories of Gravitation Lecture 1: 2033 views, 100 hours, 27 likes
La MQ mette a dura prova la ns comprens. della natura: 1835 views, 571 hours, 51 likes
Entanglement. Tagliare l’atomo come il burro: 1533 views, 221 hours, 61 likes
Il principio di indeterminazione: 1485 views, 246 hours, 67 likes
La gravità quantistica: 1482 views, 280 hours, 60 likes

La fisica e la vita – Viaggio nell’infinitamente piccolo, e ritorno: 1417 views, 165 hours, 19 likes

Number of visits to this website since creation (2013): 51.4 k

Download count for the papers posted on archives other than arXiv (which does not allow download count):

18A5 The corr. principle in q. field th. and q. grav.: 1725 downloads (5 archives)
17A3 On the quantum field theory of the grav. interactions: 1033 downloads (2 archives)
18A6 Let the dice play God: 574 downloads (4 archives)
18A7 On The Nature of the Higgs Boson: 427 downloads (4 archives)

Last update: Sept 16, 2021

Search this site

YouTube Channel

Quantum Gravity Youtube Channel Quantum Gravity Quantum Gravity - Youtube Channel


14B1 D. Anselmi

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

Last update: May 9th 2015, 230 pages

Avaibable on Amazon:


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