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Stability of Classical Solutions of Nonlocal Gravity
No full textIn this section we discuss the stability of classical solutions of nonlocal gravity. We consider a special class of nonlocal theories that admits all the maximally symmetric and Ricci-flat vacuum solutions of general relativity with and without cosmological constant, and show that such solutions are as stable as in Einstein's theory. This implies the stability of the Minkowski spacetime, black hole solutions and gravitational waves
Unitarity and Cutkosky Rules in Nonlocal Quantum Field Theory
No full textIn this section we discuss the unitarity of nonlocal quantum field theories. For simplicity, we focus our considerations on the case of a nonlocal scalar field, which encodes all the features of the nonlocality. We show that, for a specific class of nonlocal form factors, one can give a prescription for defining the scattering amplitudes, such that the theory is unitary. This prescription consists in defining the amplitudes in Euclidean signature, assuming that all the external and loop energies are purely imaginary. Then, physics is recovered by analytic continuation of such amplitudes to real and positive values of the external energies. We prove that the imaginary part of the analytically continued scattering amplitudes is given by the Cutkosky rules, which imply the unitarity of the theory. We conclude by showing that, if the nonlocal theory is defined directly in Minkowskian signature, assuming real external and loop energies, unitary is flawed
A System of 2 Nonlinearly Coupled ODEs Which is Explicitly Solvable and Possibly Isochronous Provided Its Coefficients are Suitably Restricted
No full textIn this paper we discuss some remarkable properties of the autonomous system of 2 first-order ordinary differential equations (ODEs), which equates the derivatives x ̇n(t) (n=1,2) of the 2 dependent variables xn(t) to the ratios of polynomials (with constant coefficients) in the 2 variables xn(t): each of the 2 (a priori different) polynomialsP3(n)(x1,x2) in the 2 numerators is of degree 3; the 2 denominators are instead given by the same polynomial P1(x1,x2) of degree 1. Hence this system features 23 a prioriarbitrary input numbers, namely the 23 coefficients defining these 3 polynomials. Our main finding is to show that if these 23 coefficients are given by 23 (explicitly provided) formulas in terms of 15 a prioriarbitrary parameters, then the initial values problem (with arbitrary initial data xn(0)) for this dynamical system can be explicitly solved. We also show that it is possible (with the help of Mathematica) to identify 12 explicitconstraints on these 23 coefficients, which are sufficient to guarantee that this system belongs to the class of systems we are focusing on. Several such explicitly solvable systems of ODEs are treated (including the subcase with P1(x1,x2)=1, implying that the right-hand sides of the ODEs are just cubic polynomials: no denominators!). Examples of the solutions of several of these systems are reported and displayed, including cases in which the solutions are isochronous
Non-unitarity of Minkowskian non-local quantum field theories
Full text linkAbstract We show that Minkowskian non-local quantum field theories are not unitary. We consider a simple one loop diagram for a scalar non-local field and show that the imaginary part of the corresponding complex amplitude is not given by Cutkosky rules, indeed this diagram violates the unitarity condition. We compare this result with the case of an Euclidean non-local scalar field, that has been shown to satisfy the Cutkosky rules, and we clearly identify the reason of the breaking of unitarity of the Minkowskian theory
On the occurrence of gauge-dependent secularities in nonlinear gravitational waves
Full text linkWe study the plane (not necessarily monochromatic) gravitational waves at nonlinear quadratic order on a flat background in vacuum. We show that, in the harmonic gauge, the nonlinear waves are unstable. We argue that, at this order, this instability can not be eliminated by means of a multiscale approach, i.e. introducing suitable long variables, as is often the case when secularities appear in a perturbative scheme. However, this is a non-physical and gauge-dependent effect that disappears in a suitable system of coordinates. In fact, we show that in a specific gauge such instability does not occur, and that it is possible to solve exactly the second order nonlinear equations of gravitational waves. Incidentally, we note that this gauge coincides with the one used by Belinski and Zakharov to find exact solitonic solutions of Einstein's equations, that is to an exactly integrable case, and this fact makes our second order nonlinear solutions less interesting. However, the important warning is that one must be aware of the existence of the instability reported in this paper, when studying nonlinear gravitational waves in the harmonic gauge
Cutkosky rules and perturbative unitarity in Euclidean nonlocal quantum field theories
No full textWe prove the unitarity of the Euclidean nonlocal scalar field theory to all perturbative orders in the loop expansion. The amplitudes in the Euclidean space are calculated assuming that all the particles have purely imaginary energies, and afterwards they are analytically continued to real energies. We show that such amplitudes satisfy the Cutkosky rules and that only the cut diagrams corresponding to normal thresholds contribute to their imaginary part. This implies that the theory is unitary. This analysis is then exported to nonlocal gauge and gravity theories by means of Becchi-Rouet-Stora-Tyutin, or diffeomorphism invariance, and Ward identities
Nonlinear stability of Minkowski spacetime in nonlocal gravity
No full textWe prove that the Minkowski spacetime is stable at nonlinear level and to all perturbative orders in the gravitational perturbation in a general class of nonlocal gravitational theories that are unitary and finite at quantum level
Conjectures analogous to the Collatz conjecture
Full text linkIn this paper we introduce some conjectures analogous to the well-known Collatz conjecture
Form factors, spectral and Källén-Lehmann representation in nonlocal quantum gravity
Full text linkWe discuss the conical region of convergence of exponential and asymptotically polynomial form factors and their integral representations. Then, we calculate the spectral representation of the propagator of nonlocal theories with entire form factors, in particular, of the above type. The spectral density is positive-definite and exhibits the same spectrum as the local theory. We also find that the piece of the propagator corresponding to the time-ordered two-point correlation function admits a generalization of the K & auml;ll & eacute;n-Lehmann representation with a standard momentum dependence and a spectral density differing from the local one only in the presence of interactions. These results are in agreement with what already known about the free theory after a field redefinition and about perturbative unitarity of the interacting theory. The spectral and K & auml;ll & eacute;n-Lehmann representations have the same standard local limit, which is recovered smoothly when sending the fundamental length length scale l* in the form factor to zero
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