arXiv subjects
Publications and source records attributed to Fulvio Sbisà.
We propose a novel definition of Fourier transform, with the property that the transform of a real function is again a real function (without doubling the number of real components). We prove the inversion theorem for the novel definition, and show that it shares the good properties of the usual definition.
We study a regularized version of Van Hove's 1952 model, in which a quantum field interacts linearly with sources of finite width lying at fixed positions. We show that the central result of Van Hove's 1952 paper on the foundations of Quantum Field Theory, the orthogonality between the spaces of state vectors which correspond to different values of the parameters of the theory, disappears when a well-defined model is considered. We comment on the implications of our results for the contemporary relevance of Van Hove's article.
In 1952, Léon van Hove published an article, in French, with the title ``Les difficultés de divergences pour um modèle particulier de champ quantifié''. The article is frequently cited in relation to Haag's theorem and to the issue of the existence of unitarily inequivalent representations of the canonical commutation relations in Quantum Field Theory. Summarizing in brief, it suggests a link between the appearance of divergences in perturbative Quantum Field Theory and the fact that quantum states belonging to an interacting theory do not belong to the same Hilbert space of the free theory. It also suggests that renormalization fails to provide an accurate description of the time evolution of the quantum field, although it correctly accounts for the S matrix. Due to its relevance, and to the difficulty of finding an English translation, the ideas contained in this article are proposed again here, expanded with derivations and accompanied by a discussion aimed at putting the analysis into context. We highlight the main points from the perspective of a contemporary reader, and underline the differences with the standard approach usually taught in curricular courses in Quantum Field Theory.
We address the issue of the existence of inequivalent definitions of gravitational mass in $R^{2}$-gravity. We present several definitions of gravitational mass, and discuss the formal relations between them. We then consider the concrete case of a static and spherically symmetric neutron star, and solve numerically the equations of motion for several values of the free parameter of the model. We compare the features of the mass-radius relations obtained for each definition of gravitational mass, and we comment on their dependence on the free parameter. We then argue that $R^{2}$-gravity is a valuable proxy to discuss the existence of inequivalent definitions of gravitational mass in a generic modified gravity theory, and present some comments on the general case.
We investigate the linear regime of f(R) = R + alpha R^2 gravity for static, spherically symmetric and asymptotically flat configurations of matter. We show that, in vacuum and deep inside the range of the extra scalar degree of freedom, the post-Newtonian parameter gamma is not equal to 1/2, as established in the literature, but it assumes larger values depending on the pressure of the star. We provide an explicit expression for gamma in terms of the mass, of the integrated pressure of the star and of the ratio between the star's radius and the range of the extra degree of freedom. We corroborate our results by providing numerical solutions for the case of a neutron star.
We consider the 6D Cascading DGP model, a braneworld model which is a promising candidate to realize the phenomenon of the degravitation of vacuum energy. Focusing on a recently proposed thin limit description of the model, we study solutions where the induced metric on the codimension-2 brane is of the de Sitter form. While these solutions have already been recovered in the literature imposing by hand the bulk to be flat, we show that it is possible to derive them without making this assumption, by solving a suitably chosen subset of the bulk equations.
A thin limit description of the 6D Cascading DGP model is derived, starting from a configuration where both the codimension-1 and the codimension-2 branes are thick. Postulating that the thicknesses of the two branes obey a hierarchic relation, the thin limit is executed in two steps. First the thin limit of the codimension-1 brane is executed, obtaining a system where a "ribbon" codimension-2 brane is embedded inside a thin codimension-1 brane with induced gravity, and then the thin limit of the ribbon brane is considered. By proposing a geometric ansatz on the limit configuration, the junction conditions which are to hold at the thin codimension-2 brane are derived. The latters are fully non-perturbative and covariant and, together with the Israel junction conditions at the codimension-1 brane and the Einstein equations in the bulk, constitute the looked-for thin limit formulation of the 6D Cascading DGP model. It is commented on how wide is the class of thin source configurations which can be placed on the thin codimension-2 brane.
We investigate the presence of instabilities in the Cascading DGP model. We start by discussing the problem of the cosmological late time acceleration, and we introduce the modified gravity approach. We then focus on brane induced gravity models and in particular on the Cascading DGP model. We consider configurations of the latter model where the source term is given simply by vacuum energy (pure tension), and we study perturbations at first order around these configurations. We perform a four-dimensional scalar-vector-tensor decomposition of the perturbations, and show that, regarding the scalar sector, the dynamics in a suitable limit can be described by a master equation. This master equation contains an energy scale (critical tension) which is related in a non-trivial way to the parameters of the model. We give a geometrical interpretation of why this scale emerges, and explain its relevance for the presence of ghost instabilities in the theory. We comment on the difference between our result and the one present in the literature, and stress its importance regarding the phenomenological viability of the model. We finally provide a numerical check which confirms the validity of our analysis.
The aim of these notes is to provide a self-contained review of why it is generically a problem when a solution of a theory possesses ghost fields among the perturbation modes. We define what a ghost field is and we show that its presence is associated to a classical instability whenever the ghost field interacts with standard fields. We then show that the instability is more severe at quantum level, and that perturbative ghosts can exist only in low energy effective theories. However, if we don't consider very ad-hoc choices, compatibility with observational constraints implies that low energy effective ghosts can exist only at the price of giving up Lorentz-invariance or locality above the cut-off, in which case the cut-off has to be much lower that the energy scales we currently probe in particle colliders. We also comment on the possible role of extra degrees of freedom which break Lorentz-invariance spontaneously.
We study the behaviour of weak gravitational fields in the 6D Cascading DGP model using a bulk-based approach. To deal with the ambiguity in the thin limit of branes of codimension higher than one, we consider a specific regularization of the internal structure of the branes where the 5D brane can be considered thin with respect to the 4D one. We consider the solutions corresponding to pure tension sources on the 4D brane, and study perturbations at first order around these background solutions. We adopt a 4D scalar-vector-tensor decomposition, and focus on the scalar sector of perturbations. We show that, in a suitable 4D limit, the trace part of the 4D metric perturbations obeys a decoupled equation which suggests that it is a ghost for background tensions smaller than a critical tension, while it is a healthy field otherwise. We give a geometrical interpretation of the existence of the critical tension and of the reason why the relevant field is a ghost or not depending on the background tension. We however find a value of the critical tension which is different from the one already found in the literature. Differently from the results in the literature, our analysis implies that, choosing the background tension suitably, we can construct ghost-free models for any value of the free parameters of the theory. We suggest that the difference lies in the procedure used to evaluate the pillbox integration across the codimension-2 brane. We confirm the validity of our analysis by performing numerically the integration in a particular case where the solution inside the thick cod-2 brane is known exactly. We stress that the singular structure of the perturbation fields in the nested branes set-ups is very subtle, and that great care has to be taken when deriving the codimension-2 junction conditions.
The recent observational data in cosmology seem to indicate that the universe is currently expanding in an accelerated way. An intriguing interpretation of these data is that they may just be signalling that Einstein's General Relativity is not the correct description of gravity when we consider distances of the order of the present horizon of the universe. In this thesis we consider two models which modify General Relativity at very large distances, the Cascading DGP and the dRGT massive gravity, and investigate their phenomenological viability. We start with a general introduction to standard cosmology and we introduce the late time acceleration problem and the cosmological constant problem. We then provide a pedagogical introduction to the DGP model, of which the Cascading DGP is an extension, and to the dRGT massive gravity. Concerning the Cascading DGP, we show that the thin limit of the 4D brane inside the (already thin) 5D brane is well defined, at least for the class of configurations that we consider, and confirm that gravity is regularized in these set-ups. We give a geometrical interpretation of the presence of the critical tension, and comment on the difference between the results in the literature and our results, which we support with a numerical calculation. Regarding the dRGT massive gravity, we focus on the branch of solutions in which the Vainshtein mechanism can occur. We determine analytically the number and properties of local solutions which exist asymptotically on large scales (but still below the gravitational Compton wavelength), and of local (inner) solutions which exist on small scales. We characterize exactly the properties of global solutions in every point of the phase space, and characterize precisely in which regions the Vainshtein mechanism takes place. We also provide numerical solutions which confirm our analysis.
We study static, spherically symmetric solutions in a recently proposed ghost-free model of non-linear massive gravity. We focus on a branch of solutions where the helicity-0 mode can be strongly coupled within certain radial regions, giving rise to the Vainshtein effect. We truncate the analysis to scales below the gravitational Compton wavelength, and consider the weak field limit for the gravitational potentials, while keeping all non-linearities of the helicity-0 mode. We determine analytically the number and properties of local solutions which exist asymptotically on large scales, and of local (inner) solutions which exist on small scales. We find two kinds of asymptotic solutions, one of which is asymptotically flat, while the other one is not, and also two types of inner solutions, one of which displays the Vainshtein mechanism, while the other exhibits a self-shielding behaviour of the gravitational field. We analyse in detail in which cases the solutions match in an intermediate region. The asymptotically flat solutions connect only to inner configurations displaying the Vainshtein mechanism, while the non asymptotically flat solutions can connect with both kinds of inner solutions. We show furthermore that there are some regions in the parameter space where global solutions do not exist, and characterise precisely in which regions of the phase space the Vainshtein mechanism takes place.
We study the behaviour of weak gravitational fields in models where a 4D brane is embedded inside a 5D brane equipped with induced gravity, which in turn is embedded in a 6D spacetime. We consider a specific regularization of the branes internal structures where the 5D brane can be considered thin with respect to the 4D one. We find exact solutions corresponding to pure tension source configurations on the thick 4D brane, and study perturbations at first order around these background solutions. To perform the perturbative analysis, we adopt a bulk-based approach and we express the equations in terms of gauge invariant and master variables using a 4D scalar-vector-tensor decomposition. We then propose an ansatz on the behaviour of the perturbation fields when the thickness of the 4D brane goes to zero, which corresponds to configurations where gravity remains finite everywhere in the thin limit of the 4D brane. We study the equations of motion using this ansatz, and show that they give rise to a consistent set of differential equations in the thin limit, from which the details of the internal structure of the 4D brane disappear. We conclude that the thin limit of the "ribbon" 4D brane inside the (already thin) 5D brane is well defined (at least when considering first order perturbations around pure tension configurations), and that the gravitational field on the 4D brane remains finite in the thin limit. We comment on the crucial role of the induced gravity term on the 5D brane.