Nonlocal Scalar Field Research Articles

A closed universe with hybrid nonlocality

In this paper, we explore some cosmological solutions of the Friedmann–Lemaître–Robertson–Walker (FLRW) closed universe with nonlocal de Sitter gravity [Formula: see text] and nonlocal scalar field which has its origin in [Formula: see text]-adic string theory. In this case, we have that both geometrical and matter sectors of equations of motion (EoM) are nonlocal. The cosmological constant [Formula: see text] plays a role of dark energy (DE) and is related to the mass of scalar field as [Formula: see text]. We found that the cosmological constant [Formula: see text] depends on prime number [Formula: see text] as a discrete parameter and we conjectured that [Formula: see text] is not constant during the universe evolution.

A non-local scalar field problem: existence, multiplicity and asymptotic behavior

The main purpose of this work is to study a non-local scalar field equation in bounded domains. More precisely we consider the semi-linear fractional elliptic problem: 0 \\hbox{ in } \\Omega, \\hbox{ and } u= 0 \\hbox{ in }\\mathbb{R}^N\\setminus \\Omega,\\] ]]> ( − Δ ) s u = λ u p − 1 − u m − 1 in Ω , u > 0 in Ω , and u = 0 in R N ∖ Ω , where Ω is a regular bounded domain of R N , m>p>2 and 0 $ ]]> λ > 0 . We discuss the existence, non-existence, and multiplicity of solutions, for the largest possible range of the parameters λ , p , m . Optimal results are obtained in the sub-critical and critical regimes, and partial results are obtained in the super-critical case. Furthermore, a careful asymptotic analysis when m tends to ∞ will be provided and we identify the limiting profiles as solutions to a free boundary type problem with a nonlinear right-hand side.

Nonlocal Fractional Quantum Field Theory and Converging Perturbation Series

The main purpose of this paper is to derive a new perturbation theory (PT) that has converging series. Such series arise in the nonlocal scalar quantum field theory (QFT) with fractional power potential. We construct a PT for the generating functional (GF) of complete Green functions (including disconnected parts of functions) Zj as well as for the GF of connected Green functions Gj=lnZj in powers of coupling constant g. It has infrared (IR)-finite terms. We prove that the obtained series, which has the form of a grand canonical partition function (GCPF), is dominated by a convergent series—in other words, has majorant, which allows for expansion beyond the weak coupling g limit. Vacuum energy density in second order in g is calculated and researched for different types of Gaussian part S0[ϕ] of the action S[ϕ]. Further in the paper, using the polynomial expansion, the general calculable series for Gj is derived. We provide, compare and research simplifications in cases of second-degree polynomial and hard-sphere gas (HSG) approximations. The developed formalism allows us to research the physical properties of the considered system across the entire range of coupling constant g, in particular, the vacuum energy density.

Existence and multiplicity of positive solutions of certain nonlocal scalar field equations

AbstractWe study existence and multiplicity of positive solutions of the following class of nonlocal scalar field equations: where , , , and is a nonnegative functional, that is, whenever u is a nonnegative function in . We establish Palais–Smale decomposition of the functional associated with the above equation. Using the decomposition, we establish existence of three positive solutions to (), under the condition that with as and is small enough (but ). Further, we prove that () admits at least two positive solutions when , as and is small enough (but ). Finally, we prove the existence of a positive solution when under certain asymptotic behavior on the function a.

Ground state solution for nonlocal scalar field equations involving an integro-differential operator

Ground state solution for nonlocal scalar field equations involving an integro-differential operator

Initial value problem in string-inspired nonlocal field theory

We consider a nonlocal scalar field theory inspired by the tachyon action in open string field theory. The Lorentz-covariant action is characterized by a parameter ξ2 that quantifies the amount of nonlocality. Restricting to purely time-dependent configurations, we show that a field redefinition perturbative in ξ2 reduces the action to a local two-derivative theory with a ξ2-dependent potential. This picture is supported by evidence that the redefinition maps the wildly oscillating rolling tachyon solutions of the nonlocal theory to conventional rolling in the new scalar potential. For general field configurations we exhibit an obstruction to a local Lorentz-covariant formulation, but we can still achieve a formulation local in time, as well as a light-cone formulation. These constructions provide an initial value formulation and a Hamiltonian. Their causality is consistent with a lack of superluminal behavior in the nonlocal theory.

Non-unitarity of Minkowskian non-local quantum field theories

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.

Retarded field of a uniformly accelerated source in nonlocal scalar field theory

We study the retarded field sourced by a uniformly accelerated particle in a non-local scalar field theory. While the presence of non-locality regularizes the field at the location of the source, we also show that Lorentz-invariant non-local field theories are particularly sensitive to the somewhat unphysical assumption of uniform acceleration, leading to logarithmic divergences on the acceleration horizon. Analytic properties of the non-local retarded Green function indicate that the divergences can be removed by placing appropriate sources on the acceleration horizon in the asymptotic past.

Nonlocal scalar field in an external potential: The WKB approximation

We consider a nonlocal theory of a scalar massive field in a flat spacetime background in the presence of an external potential and construct WKB solutions for this theory. We use a model in which the kinetic part of the scalar field action is modified by changing $\ensuremath{\square}$ to $\ensuremath{\square}f(\ensuremath{\square})$ operator. We discuss conditions when the corresponding form factor $f$ is chosen so that the theory does not contain new unphysical degrees of freedom. We applied the obtained WKB solutions for study energy levels of the field trapped by a one-dimensional potential and the probability of the barrier penetration. This allows us to illustrate how the effects of the nonlocality change the known results obtained for the local field theory.

Wilson-Fisher fixed points for any dimension

The critical behavior of a non-local scalar field theory is studied. This theory has a non-local quartic interaction term which involves a real power -\beta of the Laplacian. The parameter \beta can be tuned so as to make that interaction marginal for any dimension. The lowest order Feynman diagrams corresponding to coupling constant renormalization, mass renormalization, and field renormalization are computed. In all cases a non-trivial IR fixed point is obtained. Remarkably, for dimensions different from 4, field renormalization is required at the one-loop level. For d=4, the theory reduces to the usual local \phi^{4} field theory and field renormalization is required starting at the the two-loop level. The critical exponents \nu and \eta are computed for dimensions 2,3,4 and 5. For dimensions greater than four, the critical exponent \eta turns out to be negative for \epsilon>0, which indicates a violation of the unitarity bounds.

Non-locality effect on the entanglement entropy in de Sitter

We investigate the effect of infrared non-locality on the entanglement between two causally separated open-charts in the de Sitter space-time. Inspired by the work of Maldacena and Pimentel who gave a precise methodology for the computation of the long-range entanglement for the local massive scalar field theory on de Sitter space-time, we aim to investigate the change in behaviour of this long-range entanglement due to the presence of infrared non-locality in the theory. By considering a non-local scalar field theory where the non-locality becomes important in the infrared, we follow the footsteps of Maldacena and Pimentel to compute the entanglement entropy of the free non-local scalar field theory in the Bunch-Davies vacuum. It is found that the presence of infrared non-locality will have strong effect on the long-range entanglement. In some case it is noted that if the strength of non-locality is large then it will tend to decrease the long-range entanglement in the infrared. We also consider the behaviour of R'enyi entropy, where a strong role of non-locality on the entropy is noticed.

Cutkosky rules and perturbative unitarity in Euclidean nonlocal quantum field theories

We 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 Cutkowsky 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.

Non-local scalar field on deSitter and its infrared behaviour

We investigate free non-local massless and massive scalar field on deSitter (dS) space-time. We compute the propagator for the non-local scalar field for the corresponding theories on flat and deSitter space-times. It is seen that for the non-local theory, the massless limit of massive propagator is smooth for both flat and deSitter. Moreover, this limit matches exactly with the massless propagator of the non-local scalar field for both flat and deSitter space-time. The propagator is seen to respect dS invariance. Furthermore, investigations of the non-local Green's function on deSitter for large time-like separation shows that the propagator has no infrared divergences. The dangerous infrared log-divergent contributions which arise is local massless theories are absent in the corresponding non-local version. Lack of infrared divergences in the propagator hints at the strong role non-localities may play in the dS infrared physics. This study suggests that non-locality can cure IR issues in deSitter.

S-Matrix of Nonlocal Scalar Quantum Field Theory in Basis Functions Representation

Nonlocal quantum theory of a one-component scalar field in D-dimensional Euclidean spacetime is studied in representations of S -matrix theory for both polynomial and nonpolynomial interaction Lagrangians. The theory is formulated on coupling constant g in the form of an infrared smooth function of argument x for space without boundary. Nonlocality is given by the evolution of a Gaussian propagator for the local free theory with ultraviolet form factors depending on ultraviolet length parameter l. By representation of the S -matrix in terms of abstract functional integral over a primary scalar field, the S form of a grand canonical partition function is found. By expression of S -matrix in terms of the partition function, representation for S in terms of basis functions is obtained. Derivations are given for a discrete case where basis functions are Hermite functions, and for a continuous case where basis functions are trigonometric functions. The obtained expressions for the S -matrix are investigated within the framework of variational principle based on Jensen inequality. Through the latter, the majorant of S (more precisely, of − ln S ) is constructed. Equations with separable kernels satisfied by variational function q are found and solved, yielding results for both polynomial theory φ 4 (with suggestions for φ 6 ) and nonpolynomial sine-Gordon theory. A new definition of the S -matrix is proposed to solve additional divergences which arise in application of Jensen inequality for the continuous case. Analytical results are obtained and numerically illustrated, with plots of variational functions q and corresponding majorants for the S -matrices of the theory. For simplicity of numerical calculation, the D = 1 case is considered, and propagator for free theory G is in the form of Gaussian function typically in the Virton–Quark model, although the obtained analytical inferences are not, in principle, limited to these particular choices. Formulation for nonlocal QFT in momentum k space of extra dimensions with subsequent compactification into physical spacetime is discussed, alongside the compactification process.

Nonlocal scalar field equations: Qualitative properties, asymptotic profiles and local uniqueness of solutions

We study the nonlocal scalar field equation with a vanishing parameter:(Pϵ){(−Δ)su+ϵu=|u|p−2u−|u|q−2uinRNu∈Hs(RN), where s∈(0,1), N>2s, q>p>2 are fixed parameters and ϵ>0 is a vanishing parameter. For ϵ small, we prove the existence and qualitative properties of positive solutions. Next, we study the asymptotic behavior of ground state solutions when p is subcritical, supercritical or critical Sobolev exponent 2⁎=2NN−2s. For p<2⁎, the ground state solution asymptotically coincides with unique positive ground state solution of (−Δ)su+u=up, whereas for p=2⁎ the asymptotic behavior of the solutions is given by the unique positive solution of the nonlocal critical Emden–Fowler type equation. For p>2⁎, the solution asymptotically coincides with a ground-state solution of (−Δ)su=up−uq. Furthermore, using these asymptotic profile of positive solutions, we establish the local uniqueness of positive solution.

Critical behavior of a nonlocalϕ4field theory and asymptotic freedom

The critical behavior of a nonlocal scalar field theory is studied. This theory has a nonlocal kinetic term which involves a real power $1\ensuremath{-}2\ensuremath{\alpha}$ of the Laplacian. The interaction term is the usual local ${\ensuremath{\phi}}^{4}$ interaction. The lowest order Feynman diagrams corresponding to coupling constant renormalization, mass renormalization, and field renormalization are computed. Particular features appearing in the renormalization of this nonlocal theory that differ from the case of local theories are studied. The previous calculations lead to the perturbative computation of the coupling constant beta function and critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$. In four dimensions for $\ensuremath{\alpha}&lt;0$ this beta function presents asymptotic freedom in the UV. This is remarkable since no non-Abelian vector fields are included. However, this comes at the expense of losing reflection positivity.

Linear spreading speeds from nonlinear resonant interaction**GF received support from the project NONLOCAL (ANR-14-CE25-0013) funded by the French National Research Agency. MH was partially supported by the National Science Foundation through grant NSF-DMS-1516155. AS was partially supported by the National Science Foundation through grant NSF-DMS-1311740 and through a DAAD Fellowship.

We identify a new mechanism for propagation into unstable states in spatially extended systems, that is based on resonant interaction in the leading edge of invasion fronts. Such resonant invasion speeds can be determined solely based on the complex linear dispersion relation at the unstable equilibrium, but rely on the presence of a nonlinear term that facilitates the resonant coupling. We prove that these resonant speeds give the correct invasion speed in a simple example, we show that fronts with speeds slower than the resonant speed are unstable, and corroborate our speed criterion numerically in a variety of model equations, including a nonlocal scalar neural field model.

Concentration-compactness principle for nonlocal scalar field equations with critical growth

The aim of this paper is to study a concentration-compactness principle for homogeneous fractional Sobolev space Ds,2(RN) for 0<s<min⁡{1,N/2}. As an application we establish Palais–Smale compactness for the Lagrangian associated to the fractional scalar field equation (−Δ)su=f(x,u) for 0<s<1. Moreover, using an analytic framework based on Ds,2(RN), we obtain the existence of ground state solutions for a wide class of nonlinearities in the critical growth range.

Ground states of nonlocal scalar field equations with Trudinger-Moser critical nonlinearity

We investigate the existence of ground state solutions for a class of nonlinear scalar field equations defined on the whole real line, involving a fractional Laplacian and nonlinearities with Trudinger-Moser critical growth. We handle the lack of compactness of the associated energy functional due to the unboundedness of the domain and the presence of a limiting case embedding.

One-loop effective potential in nonlocal scalar field models

In this paper we apply the usual perturbative methodology to evaluate the one-loop effective potential in a nonlocal scalar field theory. We find that the effect induced by the nonlocaliity of the theory is always very small and we discuss the consequences of this result. In particular we argue that, looking at one-loop corrections for matter fields, it is not possible to find signals of the nonlocality of the theory in cosmological observables since, even during inflation when energies are very high, nonlocality-induced corrections are expected to be very small.