Retarded field of a uniformly accelerated source in non-local scalar field theory
RRetarded field of a uniformly accelerated source in non-local scalar field theory
Ivan Kol´aˇr ∗ and Jens Boos † Van Swinderen Institute, University of Groningen, 9747 AG, Groningen, The Netherlands High Energy Theory Group, Department of Physics,William & Mary, Williamsburg, VA 23187-8795, United States (Dated: 17/02/2021)We study the retarded field sourced by a uniformly accelerated particle in a non-local scalar fieldtheory. While the presence of non-locality regularizes the field at the location of the source, wealso show that Lorentz-invariant non-local field theories are particularly sensitive to the somewhatunphysical assumption of uniform acceleration, leading to logarithmic divergences on the accelerationhorizon. Analytic properties of the non-local retarded Green function indicate that the divergencescan be removed by placing appropriate sources on the acceleration horizon in the asymptotic past.
I. INTRODUCTION
Locality is deeply woven into our notion of physics:from classical mechanics to general relativity and quan-tum field theory, locality has been an undergirding prin-ciple across disciplines. However, there are notable ex-ceptions from that rule. Quantum entanglement is a non-local phenomenon, effective actions in quantum field the-ory typically contain non-local factors, and it has provendifficult if not outright impossible to define local observ-ables in quantum gravity [1]. Therein, the role of non-locality may also play a major role in possible resolutionsof the black hole information loss problem [2].The recent years have seen a flurry of activity witha particular focus on the class of ghost-free infinite-derivative theories [3–5]. These theories propose a fun-damental non-locality by means of non-local form fac-tors f ( (cid:3) ), and have been remarkably successful in al-leviating curvature singularities [6–14] in the context ofweak-field gravity. Some exact non-singular solutions ofinfinite-derivative gravity theories have been constructedin the context of gravitational waves [15, 16] and cos-mology [17, 18]. Implications of such non-local mod-ifications have also been investigated in quantum the-ory [19–21], quantum field theory [22–26], quantum fieldtheory in curved spacetime [27], Hamiltonian mechanics[28, 29], and other aspects of gravitational theory [30–32]. Non-local Green functions have proven a particularlyuseful tool in such studies [33], even though most sce-narios considered in the literature so far are either time-independent or space-independent, implying that the fullspacetime notion of non-local Green functions is not yetvery well understood.This paper aims towards closing that gap by studyingthe retarded non-local scalar field of a uniformly acceler-ated source in flat spacetime. The study of the retardedfield for uniformly and arbitrarily accelerated point parti-cles has a long history, but, to the best of our knowledge,has so far been focused on local field theories. ∗ [email protected] † [email protected] In 1909, Born studied the field of two charges undergo-ing uniform acceleration in opposite directions [34]. Thefollowing decades saw substantial activity in this field,and while much progress was made in analyzing the ra-diation content of such a field configuration—see e.g. theintroduction in Fulton and Rohrlich [35] for a brief histor-ical overview—Bondi and Gold [36] emphasized that thebehavior of the field on the acceleration horizons was sin-gular. Boulware [37] and Das [38] considered physicallymeaningful limiting procedures towards the unphysicalassumption of uniform acceleration, and Bondi [39] usedtheir approach to re-derive the original Bondi–Gold so-lution. While Ginzburg has deemed the problem of theradiation of uniformly accelerated charges solved [40–42],the field is still active, focusing on the influence of gravi-tation [43], studying scalar theory [44], or extending thestudies to de Sitter spacetime [45, 46].These considerations have provided much insight onthe causal structure of fields propagating in Minkowskispacetime, the spacetime properties of retarded Greenfunctions, and have brought to light some unphysical con-sequences of assuming uniform acceleration. This paperpresents a first step towards extending many of these con-siderations from local field theory to a class of non-local field theories.In order to focus our discussion somewhat we shallconsider a simple toy model of a scalar field theory infour-dimensional Minkowski spacetime with the metricd s = g µν d X µ d X ν = − d t + d x + d y + d z , (1)expressed in Cartesian coordinates X µ = ( t, x ) wherewe denoted x = ( x, y, z ) for simplicity. The scalar fieldequation takes the simple form D φ = j , (2)where j is an external source, and D is a differential oper-ator. The local theory is specified by the choice D = (cid:3) , We use the letter j to denote the external source term, but re-call that a scalar field theory couples to a density and not to aconserved current. a r X i v : . [ h e p - t h ] F e b where (cid:3) is the d’Alembert operator, and one recovers themassless Klein–Gordon equation. Suppose now that theexternal source has the following form: j ( X ) = 2 µαδ (2) ( − t + z − α ) δ ( x ) δ ( y ) × θ ( z + t ) θ ( z − t ) , (3)which describes a uniformly accelerated particle of mass µ > α such that the con-stant acceleration of the particle is µ/α , and the particleis located on the positive part of the z -axis. The retardedfield created by such a source may be calculated via theretarded Green function G R ( X (cid:48) , X ) = 12 π δ (2) (cid:2) ( X (cid:48) − X ) (cid:3) θ ( t (cid:48) − t ) , ( X (cid:48) − X ) = − ( t (cid:48) − t ) + ( x (cid:48) − x ) , (4)such that the retarded solution for φ takes the well knownform [43–46] φ ( X ) = ˆ d X (cid:48) G R ( X, X (cid:48) ) j ( X (cid:48) )= − µα π θ ( z + t ) (cid:112) ( X + α ) − α ( z − t ) . (5)This retarded field of a uniformly accelerated source hasseveral remarkable properties.First, this expression diverges when − t + z = α and x = y = 0, that is, at the location of the uniformly accel-erated source. Second, this expression is non-zero only inthe future and right Rindler wedges, while being finite onall horizons. And third, across the past acceleration hori-zon located at u ≡ z + t = 0, the retarded field exhibitsa discontinuity:∆ φ u =0 ≡ φ ( u = 0 + ) − φ ( u = 0 − ) = − µ πα . (6)These three properties are intimately connected to theproperties of the retarded Green function of the localscalar theory.In the remainder of this paper it is our objective tounderstand how the presence of non-locality affects theproperties of the retarded field of a uniformly acceleratedparticle. Our model of non-locality utilizes the followingdifferential operator, D = exp (cid:104) ( − (cid:96) (cid:3) ) N (cid:105) (cid:3) , N = 1 , , . . . , (cid:96) > . (7)This expression is to be understood via a formal expan-sion. N is an integer, and (cid:96) > scale of non-locality , and this class of non-local theories is also re-ferred to as GF N . Here, “GF” stands for “ghost-free”since the inverse of the non-local differential operator inFourier space has no additional poles and is thereby de-void of spurious ghost-like particles typically encounteredin higher-derivative theories. In the local limit (cid:96) → N theories manifestly regularize the field of station-ary sources, but in the time-dependent case only evenvalues for N are permissible, since odd N lead to time-dependent instabilities and divergences in the classicaltheory [26, 47].Moreover, in a true spacetime sense it is impossibleto define “small” Lorentz-invariant spacetime volumes byrelations of the form − ( t (cid:48) − t ) + ( x (cid:48) − x ) < (cid:96) since theyare always hyperbolic in nature. While in many purelyspatial problems the question of time-dependence can beneglected and non-locality truly acts on a small scale, inthe present paper this interpretation is not possible. Forthis reason we will place particular focus and emphasison non-local effects close to the light cone.This paper is organized as follows. In Sec. II we willbriefly introduce some useful coordinate systems and thenotion of Fourier transforms in those curvilinear coordi-nate systems. In Sec. III we will derive an integral ex-pression for the retarded field of a uniformly acceleratedsource in the non-local theory and discuss its propertiesin detail. And last, in Sec. IV, we will summarize ourfindings and outline possible future research directions. II. MINKOWSKI SPACETIME
In what follows it will be useful to work in Rindlercoordinates, so let us briefly fix our notation to encom-pass different coordinate choices both in real space andFourier space.
A. Various coordinates
In this paper we exclusively consider flat Minkowskispacetime, but it is convenient to introduce several co-ordinates. We start with the standard
Cartesian coordi-nates { t, x, y, z } , where the flat metric takes the formd s = − d t + d x + d y + d z . (8)It is useful to transform to null coordinates { u, v } via u = z + t , v = z − t . (9)Finally, let us define the the real Rindler coordinates { τ, ζ, x, y } that are adapted to the boost Killing vector z∂ t − t∂ z such that τ = log | u/v | = artanh [( t/z ) σ u σ v ] ,ζ = (cid:112) | uv | = (cid:112) | − t + z | , (10) Note that in the literature one also finds the alternative defini-tions ˇ u = t − z and ˇ v = t + z . We choose the present conventionsuch that u > v > where σ u = sign( u ) and σ v = sign( v ). The inverse trans-formations are given by u = σ u ζe τ , v = σ v ζe − τ , (11) t = ζ ( σ u e τ − σ v e − τ ) , z = ζ ( σ u e τ + σ v e − τ ) . (12)Introducing the subscript W ∈ { R , L , F , P } we may labelindividual regions of Minkowski spacetime as M W ; seeFig. 1. The metric in Rindler coordinates isd s = σ u σ v (cid:0) − ζ d τ + d ζ (cid:1) + d ρ + ρ d ϕ , (13)where we also introduced the polar cylindrical versiongiven by the standard relations ( x, y ) = ( ρ cos ϕ, ρ sin ϕ ).Denoting the four-dimensional spacetime volume elementby g / = (cid:112) | Det g | , we can write g / = d t d x d y d z = d u d v d x d y = ζρ d τ d ζ d ρ d ϕ . (14)The norm of a position vector X in these coordinatesreads X ≡ X · X ≡ g µν X µ X ν = − t + x + y + z = uv + x + y = σ u σ v ζ + ρ , (15)where the dot denotes the scalar product. The differenceof two such vectors X and ˜ X has the norm( X − ˜ X ) = − ( t − ˜ t ) + ( x − ˜ x ) + ( y − ˜ y ) + ( z − ˜ z ) = e − τ − ˜ τ (cid:0) σ u e τ ζ − σ ˜ u e ˜ τ ˜ ζ (cid:1)(cid:0) σ v e ˜ τ ζ − σ ˜ v e τ ˜ ζ (cid:1) + ρ + ˜ ρ − ρ ˜ ρ cos( ϕ − ˜ ϕ ) . (16) B. Fourier transform
Due to the translational invariance of Minkowskispacetime M it is convenient to employ Fourier trans-form methods. Because Minkowski spacetime is an affinespace, after fixing an arbitrary origin one may freely con-vert coordinate positions into vectors with respect to thatorigin. We denote the Fourier transform of a function f ( X ) as f ¯ X , and in four spacetime dimensions their in-terrelations are given by the formulas f ¯ X = 14 π ˆ M g / ( X ) e + i ¯ X · X f ( X ) , (17) f ( X ) = 14 π ˆ M g / ( ¯ X ) e − i ¯ X · X f ¯ X . (18)In our terminology, the coordinate space vector X as wellas momentum space vector ¯ X live in the same vectorspace. This definition is useful because now several coor-dinate systems can be used both for the Fourier transformand its inverse. The contraction between the momentumspace and coordinate space vectors is given by¯ X · X = − ¯ tt + ¯ xx + ¯ yy + ¯ zz (19)= ¯ ζζ ( σ ¯ u σ v e ¯ τ − τ + σ ¯ v σ u e − ¯ τ + τ ) + ¯ ρρ cos ( ¯ ϕ − ϕ ) . v = u = t z “F”“L” “R”“P” u < , v < u > , v > u > , v < u < , v > j Figure 1. The split of Minkowski spacetime into the fourregions “L,” “R,” “F,” and “P,” here displayed for x = y = 0.The dash-dotted line represents the spacetime location of theuniformly accelerated particle, and the dashed line v = 0( u = 0) represent the future (past) acceleration horizon. Let us emphasize here that the above notation presentsa departure from the common notation where one wouldwrite X µ = ( t, x ) and ¯ X µ = ( ω, k ). Hence, in whatfollows, a barred quantity is the conjugate Fourier mo-mentum to the unbarred real-space variable. For Eu-clidean coordinates this procedure is somewhat odd, butits notational advantage becomes apparent when per-forming Fourier transforms in curvilinear coordinates—as we shall see below—since in that case one does notneed to invent new coordinate symbols for Fourier space.For similar methods in Lorentz-invariant Fourier trans-forms we refer to the insightful paper by DeWitt-Morette et al. [48]. III. NON-LOCAL SOLUTION FOR ANACCELERATED PARTICLEA. Non-local theory
With the brief reminder on Lorentz-invariant Fouriertransforms out of the way, let us discuss our non-localtoy model. In what follows we shall consider a scalarfield theory described by the equation of motion a ( (cid:3) ) (cid:3) φ = j . (20)Here, a ( (cid:3) ) is an analytic operator a ( (cid:3) ) = ∞ (cid:88) k =0 a k (cid:3) k , (21)and j is an external source. In this paper, for simplicity,we focus on so-called GF N theories defined by a ( (cid:3) ) = exp (cid:104) ( − (cid:96) (cid:3) ) N (cid:105) , (22)which reduces to the local case a ( (cid:3) ) = 1 in the local limit (cid:96) → a ( (cid:3) ) is called form factor , and it satisfies twoimportant properties: it is non-vanishing when acting onfunctions, and it satisfies a (0) = 1. B. Retarded solution
Consider a particle with mass µ that uniformly ac-celerates in the direction of the positive z -axis with theconstant acceleration µ/α . The corresponding source islocalized in the region M R and can be parametrized as j ( X ) = 2 µαδ (2) ( − t + z − α ) δ ( x ) δ ( y ) θ ( u ) θ ( v )= µδ ( ζ − α ) δ ( x ) δ ( y ) θ ( u ) θ ( v ) . (23)In order to find the response of the non-local theory tothis source, we employ the Green function method suchthat the retarded solution is given by the integral φ ( X ) = 14 π ˆ M g / ( ¯ X ) e − i ¯ X · X G R¯ X j ¯ X . (24)Real-space expressions for G R are known and can be givenin terms of Meijer-G functions, and we derive an ex-plicit expression in Appendix A, where we also prove thatthey satisfy DeWitt’s asymptotic causality criterion [49].However, their form is rather complicated and hence im-practical for calculational purposes. As we will demon-strate now it is much simpler to perform the calculationsin momentum space.A momentum space Green function for the differentialoperator (cid:3) a ( (cid:3) ) is given by the expression G ¯ X = 1 − ¯ X a ( − ¯ X ) , (25)and we may rewrite it as a sum of two terms, G ¯ X = G ¯ X + ∆ G ¯ X ,G ¯ X = 1 − ¯ X , ∆ G ¯ X = a − ( − ¯ X ) − − ¯ X . (26)Here, G ¯ X denotes the Green function of the (cid:3) -operator.This quantity is a Green function for the local theory anddoes not depend on the presence of non-locality. It hastwo poles in complex Fourier space, and needs to be reg-ulated, typically via a suitable i(cid:15) -prescription. As is wellknown, the choice of i(cid:15) -regularization gives rise to dis-tinct causal properties. The quantity ∆ G ¯ X , on the otherhand, encapsulates the non-local modification of the localtheory: in the limiting case of (cid:96) → a → a (0) = 1, the quantity ∆ G ¯ X isdevoid of any poles in the complex plane and hence ana-lytic. This implies that non-locality, as described in non-local infinite-derivative theories, modifies all local Greenfunctions equally, irrespective of their causal properties.Concretely, making use of Eq. (22), the Green functionfor our scalar non-local theory takes the form G ¯ X = e − ( (cid:96) ¯ X ) N − ¯ X . (27)Because the non-local modification does not change thestructure of the poles in the complex momentum plane,one might be tempted to perform a similar i(cid:15) -prescriptionand contour integration in analogy to the local case.This, however, is impossible, since contour integrationassumes a fall-off behavior of the momentum space rep-resentation of the Green function which is not satisfied inour non-local infinite-derivative model due to the expo-nential factor. Incidentally, this problem is well known inthe non-local literature and lies at the heart of unitarityissues of non-local theories [22–25].At this point we note that it is possible to avoid thenotion of contour integration by following the approachproposed in Ref. [47]. Using the Sokhotski–Plemelj the-orem for continuous functions it is shown that one mayderive non-local Green functions with the correct causalproperties by performing a one-dimensional line integralalong the real axis. To obtain the retarded Green func-tion we shift the poles by an infinitesimal quantity − i(cid:15) in accordance to the local theory, and define G R¯ X ≡ e − ( (cid:96) ¯ X ) N − ¯ X (cid:12)(cid:12) − i(cid:15) ≡ e − [ (cid:96) ( − ¯ t +¯ x +¯ y +¯ z )] N − [ − (¯ t − i(cid:15) ) + ¯ x + ¯ y + ¯ z ]= e − [ (cid:96) ( σ ¯ u σ ¯ v ¯ ζ +¯ ρ )] N − (cid:2) σ ¯ u σ ¯ v ¯ ζ + ¯ ρ + i ¯ ζ (cid:0) σ ¯ u e ¯ τ − σ ¯ v e − ¯ τ ) (cid:15) (cid:3) , (28)where in the second line we employed Rindler coordinatesthat are ideally suited for analytical calculations withuniformly accelerated sources.To that end, the momentum space description of thesource j ¯ X takes the following form in Rindler coordinates: j ¯ X = µα π ˆ M d t d x d y d z e i ( − ¯ tt +¯ xx +¯ yy +¯ zz ) × δ (2) ( − t + z − α ) δ ( x ) δ ( y ) θ ( u ) θ ( v )= µα π ˆ M R d τ d ζ ζ exp (cid:104) i ¯ ζζ ( σ ¯ u e ¯ τ − τ + σ ¯ v e − ¯ τ + τ ) (cid:105) × exp [ i (¯ xx + ¯ yy )] δ (2) ( ζ − α )= µα π ˆ R d τ exp (cid:104) i α ¯ ζ (cid:0) σ ¯ u e ¯ τ − τ + σ ¯ v e − ¯ τ + τ (cid:1)(cid:105) . (29)With the expressions for both G R¯ X and j ¯ X known in mo-mentum space we may now utilize Eq. (24) to arrive atthe real-space expression of the retarded field φ . Sincewe shall employ Rindler coordinates, this step involvesthe integration over four distinct patches of momentumspace, which we refer to as M ¯W (with ¯W = ¯R , ¯L , ¯F , ¯P inanalogy to the real-space covering of Minkowski space-time). For this reason the integration can be split in fourintegrals I ¯W over the regions M ¯W . The retarded solutionfor φ is then given by four contributions, φ ( X ) = (cid:88) ¯W I ¯W ( X ) , (30) I ¯W ( X ) = 14 π ˆ M ¯W g / ( ¯ X ) e − i ¯ X · X G R¯ X j ¯ X . (31) Then, employing the integral expression for the source asper Eq. (29), I ¯W takes the following rather lengthy form: I ¯W ( X ) = µα π ˆ R d¯ τ ∞ ˆ d¯ ζ ˆ R d˜ τ ¯ ζ exp (cid:110) i ¯ ζ (cid:2) σ ¯ u ( αe − ˜ τ − σ v ζe − τ ) e ¯ τ + σ ¯ v ( αe ˜ τ − σ u ζe τ ) e − ¯ τ (cid:3)(cid:111) × (cid:20) ∞ d¯ ρ ¯ ρJ ( ρ ¯ ρ ) e − [ (cid:96) ( σ ¯ u σ ¯ v ¯ ζ +¯ ρ )] N − ( σ ¯ u σ ¯ v ¯ ζ + ¯ ρ ) + iπ ( σ ¯ u − σ ¯ v )2 ¯ ζJ ( ρ ¯ ζ ) ∞ ˆ d¯ ρ δ (2) ( σ ¯ u σ ¯ v ¯ ζ + ¯ ρ ) (cid:21) (32)= µα π ˆ R d¯ τ ∞ ˆ d¯ ζ ˆ R d˜ τ ¯ ζ exp (cid:110) i ¯ ζ (cid:2) σ ¯ u ( αe − ˜ τ − σ v ζe ¯ τ )+ σ ¯ v ( αe ˜ τ − σ u ζe − ¯ τ ) (cid:3)(cid:111) × (cid:20) ∞ d¯ ρ ¯ ρJ ( ρ ¯ ρ ) e − [ (cid:96) ( σ ¯ u σ ¯ v ¯ ζ +¯ ρ )] N − ( σ ¯ u σ ¯ v ¯ ζ + ¯ ρ ) + iπ ( σ ¯ u − σ ¯ v )4 J ( ρ ¯ ζ ) (cid:21) . (33)In the above we first integrated out the angles, π ˆ d ¯ ϕ exp [ − i ¯ ρρ cos ( ¯ ϕ − ϕ )] = 2 πJ (¯ ρρ ) , (34)where J ( x ) denotes the Bessel function of the first kind[50]. Then we made use of the Sokhotski–Plemelj theo-rem to rewrite the regulated expression f (¯ ρ ) − ¯ X (cid:12)(cid:12) − i(cid:15) = p.v. ¯ ρ f (¯ ρ ) − ¯ X + iπ ( σ ¯ u − σ ¯ v )2 f (¯ ζ ) δ (2) ( ¯ X ) , (35)where f (¯ ρ ) is a continuous function. Due to the centralimportance for the causal properties of the solution pre-sented in this paper, we prove the above relation in detailin Appendix B.In the above, the symbol p.v. ¯ ρ denotes the Cauchyprincipal value with respect to the variable ¯ ρ with othercoordinates held fixed. The symbol ffl denotes that theintegration is to be performed with the standard prescrip-tion for the Cauchy principal value. Note that the last term of (35), including the δ -distribution, has supportonly in M ¯F ∪ M ¯P as there are no poles in M ¯R ∪ M ¯L . Con-sequently, in the regions M ¯R ∪ M ¯L the Cauchy principalvalue integral reduces to the standard integral and (35)yields the identity, as it must.Then, in the second equality of Eq. (32), we integratedout the δ -distribution and shifted the variables ˜ τ and ¯ τ .In order to obtain the final expression for the retardedfield φ we now need to sum the contributions I ¯ W , and itis useful to first sum the integrals corresponding to theopposite regions. We arrive at the compact expressions I ± ( X ) ≡ (cid:40) I ¯R ( X ) + I ¯L ( X ) I ¯F ( X ) + I ¯P ( X )= µα π ∞ ˆ d¯ ζ ¯ ζ (cid:20) C ± W ( ζ, ¯ ζ ) ∞ d¯ ρ ¯ ρJ ( ρ ¯ ρ ) e − [ (cid:96) ( ± ¯ ζ +¯ ρ )] N − ( ± ¯ ζ + ¯ ρ ) − π θ ∓ S W ( ζ, ¯ ζ ) J ( ρ ¯ ζ ) (cid:21) , (36)where we defined θ + = 1, θ − = 0, and C ± W ( ζ, ¯ ζ ) and S W ( ζ, ¯ ζ ) denote the following cosine and sine integrals: C ± W ( ζ, ¯ ζ ) ≡ ˆ R d¯ τ ˆ R d˜ τ cos (cid:104) ¯ ζ ( αe − ˜ τ − σ v ζe ¯ τ ) ± ¯ ζ ( αe ˜ τ − σ u ζe − ¯ τ ) (cid:105) ,S W ( ζ, ¯ ζ ) ≡ ˆ R d¯ τ ˆ R d˜ τ sin (cid:104) ¯ ζ ( αe − ˜ τ − σ v ζe ¯ τ ) − ¯ ζ ( αe ˜ τ − σ u ζe − ¯ τ ) (cid:105) . (37)These double integrals can be separated into products ofintegrals (see Eq. (3.868), (1)–(4) in Ref. [51]) and takethe following form in the various regions of Minkowskispacetime: C ± R ( ζ, ¯ ζ ) = (cid:40) π (cid:2) J ( α ¯ ζ ) J ( ζ ¯ ζ ) + Y ( α ¯ ζ ) Y ( ζ ¯ ζ ) (cid:3) , K ( α ¯ ζ ) K ( ζ ¯ ζ ) ,C ± L ( ζ, ¯ ζ ) = (cid:40) π (cid:2) − J ( α ¯ ζ ) J ( ζ ¯ ζ ) + Y ( α ¯ ζ ) Y ( ζ ¯ ζ ) (cid:3) , K ( α ¯ ζ ) K ( ζ ¯ ζ ) , (38) C ± F ( ζ, ¯ ζ ) = C ± P ( ζ, ¯ ζ ) = (cid:40) − πY ( α ¯ ζ ) K ( ζ ¯ ζ ) , − πK ( α ¯ ζ ) Y ( ζ ¯ ζ ) ,S R ( ζ, ¯ ζ ) = S L ( ζ, ¯ ζ ) = 0 ,S F ( ζ, ¯ ζ ) = − S P ( ζ, ¯ ζ ) = 2 πK ( α ¯ ζ ) J ( ζ ¯ ζ ) . Then, the final solution for φ is given by the sum φ ( X ) = I + ( X ) + I − ( X ) . (39)We were not able to find a closed-form analytic expressionfor φ , which is why we refrain from giving an explicitexpression at this point. C. Local case (cid:96) = 0
As a simple consistency check let us recover the knownlocal solution for (cid:96) →
0. Employing Eq. (36) we find φ ( X ) = µα π ∞ ˆ d¯ ζ ¯ ζ (cid:20) − K ( ρ ¯ ζ ) C +W ( ζ, ¯ ζ ) (40)+ π Y ( ρ ¯ ζ ) C − W ( ζ, ¯ ζ ) − π J ( ρ ¯ ζ ) S W ( ζ, ¯ ζ ) (cid:21) , where we used the following principal value integral ex-pressions (for ρ (cid:54) = 0): ∞ d¯ ρ ¯ ρJ ( ρ ¯ ρ ) − ( ± ¯ ζ + ¯ ρ ) = (cid:40) − K ( ρ ¯ ζ ) , π Y ( ρ ¯ ζ ) . (41) Numerical integration of (40) perfectly matches theknown analytic result for the retarded solution [43–46], φ ( X ) = − µα π θ ( u ) (cid:112) ( σ v ζ + ρ + α ) / − σ v α ζ . (42)Note that this field is non-zero in M R ∪ M F and vanishesin M L ∪ M P . Despite the discontinuity across the surface u = 0, it fully satisfies the field equations with distribu-tional source (23). The advanced solution φ A can befound by formally reversing the time direction, t → − t ,which is equivalent to the exchange u ↔ v , φ A ( X ) = − µα π θ ( v ) (cid:112) ( σ u ζ + ρ + α ) / − σ u α ζ . (43)As already pointed out in the Introduction, this localsolution is singular at the location of the source, thatis, in the plane ζ = α whenever ρ = 0. On the futurehorizon t = z , however, the retarded field is regular. Fora more detailed discussion of this local solution, includingquantum radiation, we refer to Ren and Weinberg [44]. D. Non-local case (cid:96) > Let us now study the non-local case (cid:96) >
0. In general,for ρ >
0, we were not able to proceed analytically withthe integral expressions for the non-local retarded fieldvia Eqs. (32), (36), and (39). Restricting ourselves tothe plane ρ = 0, however, the solution reduces to φ ( X ) ≡ φ ( X ) | ρ =0 = µα π ∞ ˆ d¯ ζ ¯ ζ (cid:20) Ei (cid:0) − (cid:96) N ¯ ζ N (cid:1) N C +W ( ζ, ¯ ζ ) (44)+ Ei (cid:0) − ( − (cid:96) ) N ¯ ζ N (cid:1) N C − W ( ζ, ¯ ζ ) − π S W ( ζ, ¯ ζ ) (cid:21) , where we used the principal value integral expression ∞ d¯ ρ ¯ ρ e − [ (cid:96) ( σ ¯ u σ ¯ v ¯ ζ +¯ ρ ) ] N − (cid:0) ± ¯ ζ + ¯ ρ (cid:1) = Ei (cid:0) − ( ± (cid:96) ) N ¯ ζ N (cid:1) N , (45)and Ei( x ) denotes the exponential integral [50]. Inspect-ing (44) one immediately notices that the cases of evenand odd N are quite different. Indeed, the integrals forodd values of N do not converge. This can be shown sim-ply for X ∈ M R ∪ M L . In this case, the first and thirdterm in the integrand of (44) are suppressed for largevalues of ¯ ζ , but the second term grows to infinity. For X ∈ M F ∪ M P , the second term is also unbounded be-cause it oscillates with growing amplitude. On the otherhand, the integral converges for even values of N . Thisbehaviour for even/odd non-local theories seems to be inagreement with Ref. [26, 47].For numerical analysis it is useful to introduce dimen-sionless quantities. Since we assume the scale of non-locality (cid:96) to be fundamental, we choose to normalize thephysical parameters of distance and acceleration with re-spect to that length scale and introduce the quantityˆ α ≡ α(cid:96) . (46)The scalar field is proportional to the mass of the particle µ . Since that constant does not appear anywhere else wedefine the dimensionless scalar field ˆ φ asˆ φ ≡ φ(cid:96)µ . (47)Now the only free parameter is the dimensionless acceler-ation parameter ˆ α , which measures inverse accelerationper unit mass.For the remainder of this paper let us focus on thesimplest case of N = 2, which we refer to as GF theory.Then one findsˆ φ ( ˆ X ) = ˆ α π ∞ ˆ d¯ ζ ¯ ζ (cid:20) Ei (cid:0) − ¯ ζ (cid:1) C +W (ˆ ζ, ¯ ζ ) (48)+ Ei (cid:0) − ¯ ζ (cid:1) C − W (ˆ ζ, ¯ ζ ) − π S W (ˆ ζ, ¯ ζ ) (cid:21) , where we introduced the dimensionless distance ˆ ζ ≡ ζ/(cid:96) .The integration can be performed numerically for eachRindler wedge, and we plot a graphical representationin Fig. 2. For convenience we combined the numericalexpressions for the right and future wedge by artificiallyplotting φ as a function of σ u σ v ζ , and, similarly, in theleft and past Rindler wedge.The retarded field has several noteworthy properties:1. For large timelike and spacelike distances, ζ (cid:29) (cid:96) ,one recovers the local result discussed in theprevious section.2. The non-local field is regular at the location of thesource, ζ = α , in contrast to the local field.3. The non-local field is non-vanishing in the left andpast Rindler wedge, unlike the local field.4. The behavior of the non-local solution around thehorizon appears singular. Closer inspection, as weshall discuss below, reveals that this is an artefactof the unphysical assumption of uniform accelera-tion.Let us now discuss these properties of the retarded non-local field in more detail.
1. Asymptotic timelike and spacelike behavior
As discussed in Sec. III C, in the local limit (cid:96) → ζ ≡ ζ/(cid:96) → ∞ , which corresponds to the limit of (cid:96) → ζ , or to the large-distance limitin the case of finite (cid:96) >
0. From the graphical repre-sentation in Fig. 2 it is clear that the non-local retardedfield approaches the values of the local theory at largespacelike and timelike distances,ˆ φ (ˆ ζ (cid:29)
1) = − ˆ α (1 + σ u )4 π ˆ ζ . (49)This is a non-trivial consistency check since it impliesthat for large timelike and spacelike distances the effectsof non-locality are heavily suppressed.
2. Regularity at the location of the source
In stark contrast to the local solution (42), the non-local field is finite at the location of the particle, ζ = α .It is possible to calculate this value analytically,ˆ φ (ˆ ζ ≈ ˆ α ) = 164 π / ˆ α (cid:20) − π ˆ α F (cid:18) , ; , , , ; ˆ α (cid:19) + π / ˆ α F (cid:18) , ; , , ,
2; ˆ α (cid:19) − √ π ˆ α A (ˆ α ) − √ B (ˆ α ) (cid:105) + O (ˆ ζ − ˆ α ) , (50) A (ˆ α ) ≡ G (cid:32) , ; , , , , , , , , ; − , , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ α (cid:33) , (51) B (ˆ α ) ≡ G (cid:32) , , , , , , ,
1; 0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ α (cid:33) , (52)where G mnpq denote Meijer G-functions [50]. One mayshow that ˆ φ (ˆ ζ ≈ ˆ α ) is finite, smooth, and negative forpositive values of ˆ α , whereas it vanishes for ˆ α →
0. Thismanifestly finite behavior at the location of the sourcematches our expectation that non-locality regularizes thefield of localized sources and, perhaps more importantly,presents a concrete extension from previous static andstationary results known in the literature to the full,time-dependent case.A closer inspection reveals that the linear term O (ˆ ζ − ˆ α ) does not vanish. This corresponds to the fact thatthe minimum of the non-local potential is not located at ζ = α but, rather, is shifted towards smaller values of ζ .This behavior can also be seen in Fig. 2.
3. Causal properties
Recall that the local solution (42) is proportional tothe step function θ ( u ), implying that the local retardedfield is strictly zero in the left and past Rindler wedges. In Figure 2. The local (dashed) and non-local (solid) retarded dimensionless field in the four Rindler wedges plotted as a functionof σ u σ v ζ , in the plane ρ = 0 for a dimensionless acceleration of ˆ α = 7, with a dimensionless step size of 0 .
05. The verticaldashed line in the right wedge indicates the position of the particle at ˆ ζ = ˆ α . the non-local case one might expect that this is no longerthe case. And indeed, Fig. 2 confirms this suspicion:the non-local retarded field is non-zero in the left andpast Rindler wedges. While we were unable to find acomplete analytical description, it is again possible tofind the value of the field analytically at the somewhatad hoc location ζ = α . In the left Rindler wedge we findˆ φ (ˆ ζ ≈ ˆ α ) = − √ π / ˆ α (cid:2) π ˆ α A (ˆ α ) + 2 B (ˆ α ) (cid:3) + O (ˆ ζ − ˆ α ) , (53)whereas for the past Rindler wedge one hasˆ φ (ˆ ζ ≈ ˆ α ) = 18 π ˆ α − ˆ α π / G (cid:32) , ;0 , , − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ α (cid:33) + O (ˆ ζ − ˆ α ) , (54)Let us emphasize that these non-zero values arise solelydue to the presence of non-locality, (cid:96) >
0. In the limit of vanishing non-locality and finite acceleration parameter α one has ˆ α = α/(cid:96) → ∞ , and one may show that in thislimit the above terms vanish identically.In linearized non-local theories it is common wisdomthat “non-locality smears out sharp sources” [33, 52], andone might be tempted to interpret the above expressionsas the result of a smeared out step function similar tothe expression ∼ e (cid:96) (cid:3) θ ( u ). However, due to the lack ofconcrete analytical expressions for the retarded field forarbitrary values of ζ it is not possible to test this ideafurther.
4. Singular behavior in vicinity of acceleration horizons
From our numerical plot in Fig. 2 it is obvious thatthe retarded non-local field behaves somewhat singularlyin proximity to the acceleration horizons. Expanding theintegrand (44) for small values of ˆ ζ one finds the followinglogarithmic behavior:ˆ φ (ˆ ζ (cid:28)
1) = c (ˆ α ) + c (ˆ α ) log ˆ ζ + O (ˆ ζ ) , (55) c (ˆ α ) = ˆ R d¯ ζ ˆ α ¯ ζ π (cid:110) π Ei( − ¯ ζ ) (cid:104) π ( σ u + σ v ) J (ˆ α ¯ ζ )+ 4 Y (ˆ α ¯ ζ ) (cid:16) γ + log ¯ ζ (cid:17) (cid:105) − K (ˆ α ¯ ζ ) (cid:104) π ( σ u − σ v )+ 2Ei( − ¯ ζ ) (cid:16) γ + log ¯ ζ (cid:17) (cid:105)(cid:111) , (56) c (ˆ α ) = 14 π ˆ α G (cid:18) , , , ,
1; 0 (cid:12)(cid:12)(cid:12)(cid:12) ˆ α (cid:19) − π ˆ α G (cid:32) , , , , ,
1; 0 , , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ α (cid:33) . (57)Apparently, the retarded field diverges logarithmically asone approaches the acceleration horizon. Note that c de-pending on σ u and σ v leads to different values in differentRindler wedges. However, the constant c that multipliesthe diverging logarithmic term is universal.This logarithmic divergence arises due to non-localityand is pathological as the retarded field of the local the-ory does not exhibit any singular behavior, except for adiscontinuity on the past horizon, which we shall addressin the next subsection. In what follows we will demon-strate that the pathological logarithmic divergence arisessolely due to the unphysical assumption of a uniformlyaccelerated massive particle. This acceleration would re-quire an infinite amount of energy and result in a massiveparticle moving asymptotically at the speed of light.In order to gain some qualitative understanding of thedivergences, let us consider a simpler setting of a singlepoint-like source located at ( u , v ), j test ( X ) = κδ ( u − u ) δ ( v − v ) δ ( x ) δ ( y ) , (58)where κ is a dimensionless prefactor. Focusing our con-siderations to the plane ρ = 0 the resulting field is thensimply φ test ( X ) = κ G ( u, v ; u , v ) . (59)To study the effects of non-locality it is sufficient to con-sider the non-local modification of the Green function.Since we are interested in a source that becomes asymp-totically null we need to check two cases:(a) Past horizon: Set u = 0 and consider the resultingfield in the limit v → ∞ , evaluated on the pasthorizon ( u = 0).(b) Future horizon: Set v = 0 and consider the result-ing field in the limit u → −∞ , evaluated on thefuture horizon ( v = 0).For a visualization we refer to Fig. 3. The non-localGreen function modification—see Appendix A—can be v = u = t z “F”“L” “R”“P” u < , v < u > , v > u > , v < u < , v > v →∞ u → − ∞ κ κ Figure 3. Test setup to understand the emergence of singularbehavior on the past and future acceleration horizon due tonon-locality. Consider two sources of magnitude κ locatedat ( u ,
0) and (0 , v ) and then take the limit u → −∞ and v → ∞ , shifting the sources into the asymptotic past. written as follows:∆ G = 14 π / s ∞ ˆ d ye − y / sin (cid:18) s y(cid:96) (cid:19) (60)= − sgn [( u − u )( v − v )]4 π / (cid:96) ∞ ˆ d y sin (cid:18) y (cid:19) × exp (cid:2) − y ( u − u ) ( v − v ) / (4 (cid:96) ) (cid:3) , (61)where s = ( u − u )( v − v ). In order to probe the diver-gence on the past horizon we set u = 0. If u (cid:54) = 0 then∆ G ≡ v → ∞ . If u = 0, then the integral diverges logarith-mically close to the past horizon. For the future horizonthe analysis goes through identically, mutatis mutandis .An analytic representation of the non-local Green func-tion modification confirms this behavior:∆ G ( s ) = | s | π (cid:96) G (cid:18) − , − ; − (cid:12)(cid:12)(cid:12)(cid:12) s (cid:96) (cid:19) , (62)where G denotes the Meijer G-function [50]. For aderivation of this expression we refer to Appendix A. Thisfunction has the following asymptotics:∆ G ( | s | (cid:28)
1) = 132 π / (cid:96) (cid:20) − γ − log (cid:18) s (cid:96) (cid:19)(cid:21) , ∆ G ( | s | (cid:29)
1) = 12 √ π | s | sin (cid:32) √ s / · / (cid:96) / (cid:33) (63) × exp (cid:18) − s / · / (cid:96) / (cid:19) . u, v ) and the location of a source at ( u , v ) arenull separated. This is precisely what happens on theacceleration horizons.Extracting the prefactor of the logarithmic divergencecreated by the presence of the test source (58), Z ≡ − κ π / (cid:96) , (64)we may now equate it to the negative of the near-horizonconstant c (ˆ α ) of Eq. (57) while simultaneously restoringa dimensional φ -field, resulting in an expression for thedimensionless constant κ , κ = − √ π µ(cid:96) α (cid:34) G (cid:32) , , , , ,
1; 0 , , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α (cid:96) (cid:33) − π G (cid:18) , , , ,
1; 0 (cid:12)(cid:12)(cid:12)(cid:12) α (cid:96) (cid:19)(cid:21) . (65)This implies that it is possible to regularize the loga-rithmic divergence by adding a counterterm-like sourcewith the above prefactor on the past horizons at both( u → −∞ , v = 0) and ( u = 0 , v → ∞ ); see alsoFig. 3. It is clear that this procedure is necessitatedsolely due to the presence of non-locality, since in thelimit (cid:96) → κ →
0, as expected.Hence, just as in the local case, the singular behav-ior arises due to the unphysical assumption of uniformacceleration: in order to accelerate a particle of mass µ to the speed of light we would require infinite amountof energy. In the local case, due to the simplicity of thelocal Green function, it is possible to consider instead asource which is initially at rest and then starts acceler-ating: see Bondi and Gold [36] and Boulware [37] for theelectromagnetic case, and Ren and Weinberg [44] for thescalar case. Boosting such a source to a finite speed, andthen taking the ultrarelativistic limit, one recovers theunphysical discontinuities on the past acceleration hori-zon that are otherwise absent.Unfortunately, due to the complicated analytical formof the non-local Green function, such a construction isnot feasible in this case. However, based on the abovediscussion we may argue that if the source never reachesthe future light cone (or has never emanated from thepast light cone) then there would be no such singularbehavior. Alternatively, one may place the κ -sources onthe past horizons as a regularization prescription.These considerations confirm our hypothesis that theunphysical assumption of uniform acceleration leads tothe pathological behavior on the past and future hori-zons, and any physically well-behaved source should bedevoid of such artefacts. Non-local theories, such as theGF theory studied in the present work, appear to bemore sensitive to the physicality of sources.
5. “Principal values” across acceleration horizons
While the field is logarithmically divergent on the hori-zon, it is possible to show that the difference of the fieldacross both the past acceleration horizon ( u = 0) as wellas the future acceleration horizon ( v = 0) is finite. Sincethis difference is taken between two diverging expressionswe shall refer to it as a “principal value.” This principalvalue is known from the local case, (cid:96) = 0. In the localtheory the field is manifestly finite on all horizons, andhence the principal value becomes a mere discontinuity.Moreover, this discontinuity only appears across the pasthorizon, and not on the future horizon.In this subsection we will determine the principal val-ues across the acceleration horizons analytically (for ρ =0). Since φ depends only on the coordinate ζ = (cid:112) | uv | ,the near-horizon expressions for the functions in the inte-grand of Eq. (44) can be found by inserting ζ = pq (with p > q >
0) and expanding around q = 0, C +W ( ζ, ¯ ζ ) ≈ π (cid:8) Y ( α ¯ ζ ) (cid:2) log (cid:0) pq ¯ ζ/ (cid:1) + γ (cid:3) + π ( σ u + σ v ) J ( α ¯ ζ ) (cid:9) , (66) C − W ( ζ, ¯ ζ ) ≈ − K ( α ¯ ζ ) (cid:2) log (cid:0) pq ¯ ζ/ (cid:1) + γ (cid:3) , (67) S W ( ζ, ¯ ζ ) ≈ π ( σ u − σ v ) K ( α ¯ ζ ) . (68)It turns out that the difference of these integrals betweeneither side of the horizon is independent of the positionon the horizon p as well as the near-distance coordinate q . As a consequence, the the jumps across u = 0 and v = 0 reduce to the finite expressions∆ φ u =00 = + µα π ∞ ˆ d¯ ζ ¯ ζ (cid:2) Ei (cid:0) − (cid:96) N ¯ ζ N (cid:1) J ( α ¯ ζ ) − K ( α ¯ ζ ) (cid:3) , (69)∆ φ v =00 = − µα π ∞ ˆ d¯ ζ ¯ ζ (cid:2) Ei (cid:0) − (cid:96) N ¯ ζ N (cid:1) J ( α ¯ ζ )+ 4 K ( α ¯ ζ ) (cid:3) . (70)For N = 2 one finds the analytic expressions∆ φ u =00 = − µ πα (cid:20) − Q (ˆ α ) (cid:21) , (71)∆ φ v =00 = − µ πα Q (ˆ α ) , (72) Q (ˆ α ) ≡ F (cid:18) ; 12 ,
12 ; ˆ α (cid:19) − √ π ˆ α F (cid:18) ; 1 ,
32 ; ˆ α (cid:19) , (73)where Q (ˆ α ) captures the influence of non-locality. Let usemphasize that the logarithmic divergence encounteredon the horizon in GF theory precisely cancels out ofthis symmetric limit from both sides of the horizons. In1 Figure 4. A plot of the dimensionless function Q (ˆ α ) and κ (ˆ α )in arbitrary units. They both undergo non-periodic oscilla-tions, and their zeroes do not coincide. the cases of small and large values for the dimensionlessacceleration parameter ˆ α one findslim ˆ α → Q = 1 , lim ˆ α →∞ Q = 0 , (74)The latter equation shows that in the limiting case ofvanishing non-locality, (cid:96) → α → ∞ ), theprincipal value across the future horizon ( v = 0) vanishes.Hence, the principal value across the future horizonis solely related to the presence of non-locality, and theprincipal value across the past horizon is modified by non-locality—in the local theory it is merely a discontinuitysince there are no divergences. Let us also note that thecontributions due to non-locality across these horizonsare equal in magnitude but opposite in sign.It is conceivable that these non-trivial principal val-ues remain present in the non-local retarded field evenafter the κ -subtraction presented in the previous subsec-tion. This is because the logarithmically divergent term,as per Eq. (57), does not depend on the Rindler wedgeand hence cancels out of the symmetric principal valueprescription presented in this subsection. The constantterm, however, as per Eq. (56), differs across the Rindlerwedges, giving rise to the non-trivial principal value.The function Q exhibits damped oscillatory behaviorwith an infinite number of non-periodic zeroes, the firstfew taking place at ˆ α = { . , . , . , . } . Forthese values the principal value vanishes across v = 0. Onthe other hand, the quantity κ viewed as a function ofˆ α also undergoes damped non-periodic oscillations, withthe first zeroes at ˆ α = { , . , . , . } . For thosedistinct values there are no divergences on the horizons,but the solution is discontinuous due to the non-vanishingprincipal value. A graph of the functions Q (ˆ α ) and κ (ˆ α )can be seen in Fig. 4. Their zeroes do not coincide, whichmeans that for select values of dimensionless accelerationˆ α = α/(cid:96) one may have either no principal value or a finitefield at the acceleration horizon. E. A non-local Born-type solution
Before concluding, let us briefly comment on a possi-ble extension of the retarded solution discussed so far.Namely, we would like to construct a non-local gener-alization of the Born solution [34] and comment on itsfeatures in relation to the previously discussed logarith-mic divergences and principal values.Formally, the Born solution may be regarded as thefield resulting of the retarded response of a uniformly ac-celerated particle in the right Rindler wedge superposedwith the advanced field of a uniformly accelerated particlein the left Rindler wedge [45, 46]. Instead of re-derivingEqs. (32), (36), and (39) for that particular case, let usobserve that we can transform the retarded field in theright Rindler wedge into the advanced field in the leftRindler wedge by mapping the null coordinates u → − u and v → − v , which amounts to identifying M R → M L aswell as M F → M P . Let us call the retarded solution φ R and the advanced solution φ A . The Born solution is φ B ≡ φ R + φ A . (75)The Born field in, say, the right Rindler wedge is then thesuperposition of the retarded field in the right Rindlerwedge and the advanced field of the left Rindler wedge,and similar for all other wedges. For this reason thedependence on the factors σ u and σ v , as encounteredin Eqs. (32), (36), and (39), drops out entirely. Thisimmediately implies that the principal values across thehorizons vanish identically for the Born solution.The logarithmic divergences on the horizons, however,as can be seen from Eqs. (55)–(57), do not depend onthe Rindler wedge, and hence are still present in theBorn solution and need to be removed via a suitable κ -subtraction. IV. CONCLUSIONS
In this paper we constructed the retarded field of auniformly accelerated point particle in a non-local scalarfield theory: we employed the Sokhotski–Plemelj theo-rem to construct a non-local causal Green function inmomentum space and found an integral representationfor the resulting field. We then proved that the pres-ence of non-locality regularizes the field at the locationof the source , while—for large timelike and spacelike dis-tances away from the hyperbolically accelerated source—approaching the expression for the retarded field found inthe local theory, in accordance with DeWitt’s notion ofasymptotic causality encountered in non-local theories.On the acceleration horizons of the source, however,the retarded field is mildly logarithmically divergent dueto the presence of non-locality. Using a pair of testsources on a null cone we proved analytically that suchsources indeed give rise to logarithmic divergences in thisparticular non-local theory. We believe that this diver-gence is similar to those artefacts encountered in local2theories, arising due to the unphysical assumption of uni-form acceleration. Our considerations prove that if thesource is never to become asymptotically null (either inits past or future) then there are no such divergencespresent, consistent with the regular field of null sourcesin other non-local theories [13, 15, 16, 53]. Moreover, wedevise a prescription that involves test sources placed inthe asymptotic past of the acceleration horizon which iscapable of removing these spurious divergences. It re-mains to be seen if and how these additional sources arerelated to modified boundary conditions that one mayencounter in non-local field theories. We shall leave thisquestion for future research.Last, we found that the difference of the retarded fieldacross acceleration horizons is finite, even without a reg-ularization procedure, and we demonstrated that for anon-local generalization of the Born solution these prin-cipal values vanish identically. If combined with the reg-ularization procedure of sources in the asymptotic pastone then arrives at a solution that is completely regularon the horizons.It is a natural question to ask how the radiation of aretarded non-local source behaves, but since energy mo-mentum tensors of non-local fields are notoriously hardto compute, see e.g. Ref. [27] for a concrete example ofGF theory, this point deserves further study. Anotheravenue would be the study of non-local electrodynamics,where recently ultrarelativistic objects have been stud-ied by one of the authors [53]. Then, it would also behighly interesting to study implications for the presenceof radiation vis-`a-vis the equivalence principle in Lorentz-invariant non-local theories.Let us emphasize that the results derived in this paperpresent only one step towards improving our understand-ing of the spacetime structure of non-locality. Due to theintrinsic Lorentz invariance that lies at the very heart ofthis class of non-local field theories, modifications of theGreen function can only be a function of the dimension-less 4-distance,∆ G ( t (cid:48) , x (cid:48) ; t, x ) = ∆ G (cid:18) − ( t (cid:48) − t ) + ( x (cid:48) − x ) (cid:96) (cid:19) . (76)Naively speaking, Lorentz-invariant non-local field theo-ries cannot seem to tell whether two points in spacetimeare coincident or null-separated. Whether this presents abug or a feature of this class of non-local theories remainsto be seen. ACKNOWLEDGEMENTS
We would like to thank Valeri Frolov (Edmonton) forhelpful comments on a previous draft of this paper. I.K.was supported by Netherlands Organization for ScientificResearch (NWO) grant no. 680-91-119. J.B. acknowl-edges support by the National Science Foundation undergrant PHY-181957, and is grateful for a Vanier Canada Graduate Scholarship administered by the Natural Sci-ences and Engineering Research Council of Canada aswell as for the Golden Bell Jar Graduate Scholarship inPhysics by the University of Alberta during the earlierstages of this work.
Appendix A: Real-space expression for the non-localmodification of the scalar Green function
The free scalar retarded Green function G R ( X (cid:48) , X ),due to the translational isometry of Minkowski space,depends only on the difference of its arguments, G R ( X (cid:48) , X ) = G R ( X (cid:48) − X ). Moreover, writing X µ =( t, x ), one can further decompose the argument struc-ture as G R ( X (cid:48) , X ) = G R ( t (cid:48) − t ; x (cid:48) − x ). A Green functionin GF theory is a solution of (cid:3) e − (cid:96) (cid:3) G R ( t (cid:48) − t, x (cid:48) − x ) = − δ ( t (cid:48) − t ) × δ (3) ( x (cid:48) − x ) , (A1)and clearly it is sensitive to the existence of non-locality (cid:96) >
0. We may decompose it as G R ( t (cid:48) − t, x (cid:48) − x ) = G R ( t (cid:48) − t, x (cid:48) − x )+ ∆ G ( t (cid:48) − t, x (cid:48) − x ) , (A2)where ∆ G ( t (cid:48) − t, x (cid:48) − x ) is a non-local modification termand G R ( t (cid:48) − t, x (cid:48) − x ) is the local retarded Green functionthat solves (cid:3) G R ( t (cid:48) − t, x (cid:48) − x ) = − δ ( t (cid:48) − t ) δ (3) ( x (cid:48) − x ) , (A3)subject to the retarded constraint G R ( t (cid:48) − t, x (cid:48) − x ) = 0if t (cid:48) < t . From now on we shall denote t (cid:48) − t simply as t and x (cid:48) − x as x . For the local piece one may calculate G R ( t, x ) = 12 π δ (2) ( − t + x ) θ ( t ) , (A4)which, by construction, is only non-vanishing on the fu-ture light cone. Inside the future light cone, as well asanywhere outside of it, it vanishes identically. The non-local part can be calculated as follows:∆ G ( t, x ) = ∞ ˆ −∞ d ω π ˆ R d k (2 π ) e + iωt − i k · x − e − (cid:96) ( ω − k ) ω − k = (cid:96) π / x ∞ ˆ d ω cos ωt ∞ ˆ k d k sin kx × ∞ ˆ e − y / y ˆ d z sin (cid:2) (cid:96) ( ω − k ) z (cid:3) (A5)= (cid:96) π / x ∞ ˆ d ye − y / y ˆ d z × [ I ( t, z ) I ( x, z ) − I ( t, z ) I ( x, z )] , k ≡ | k | as well as x ≡ | x | , and I , I , I , and I denote the following regularized integrals: I ( t, z ) = lim α → ∞ ˆ d ωe − αω cos ωt sin ω (cid:96) z (A6)= (cid:114) π z(cid:96) (cid:20) cos (cid:18) t z(cid:96) (cid:19) − sin (cid:18) t z(cid:96) (cid:19)(cid:21) ,I ( t, z ) = lim α → ∞ ˆ d ωe − αω cos ωt cos ω (cid:96) z (A7)= (cid:114) π z(cid:96) (cid:20) cos (cid:18) t z(cid:96) (cid:19) + sin (cid:18) t z(cid:96) (cid:19)(cid:21) ,I ( x, z ) = lim α → ∞ ˆ k d ke − αk sin kx cos k (cid:96) z (A8)= (cid:114) π z (cid:96) x (cid:20) sin (cid:18) x z(cid:96) (cid:19) − cos (cid:18) x z(cid:96) (cid:19)(cid:21) ,I ( x, z ) = lim α → ∞ ˆ k d ke − αk sin kx sin k (cid:96) z (A9)= (cid:114) π z (cid:96) x (cid:20) sin (cid:18) x z(cid:96) (cid:19) + cos (cid:18) x z(cid:96) (cid:19)(cid:21) . Then one can further regulate ( s ≡ − t + x )∆ G ( t, x ) = − π / (cid:96) ∞ ˆ d ye − y / y ˆ d zz cos (cid:18) s z(cid:96) (cid:19) = − π / (cid:96) ∞ ˆ d ye − y / × lim α → ∞ ˆ /y d ze − αz cos (cid:18) s (cid:96) z (cid:19) (A10)= 14 π / s ∞ ˆ d ye − y / sin (cid:18) s y(cid:96) (cid:19) = | s | π (cid:96) G (cid:18) − , − , − (cid:12)(cid:12)(cid:12)(cid:12) s (cid:96) (cid:19) , where G denotes a Meijer G-function [50]. It is clearthat this function is invariant under s → − s , mean-ing that it does not distinguish between timelike andspacelike distances, consistent with the putative acausal-ity typically encountered in non-local theories.For small and large arguments s one finds the follow- Figure 5. The dimensionless non-local modification∆ G ( | s | ) × (cid:96) plotted as a function of dimensionless 4-distance | s | /(cid:96) , together with its null expansion ( | s | (cid:28)
1) as well aslarge-distance expansion | s | (cid:29) ing asymptotic behavior:∆ G ( | s | (cid:28)
1) = 132 π / (cid:96) (cid:20) − γ − log (cid:18) s (cid:96) (cid:19)(cid:21) , ∆ G ( | s | (cid:29)
1) = 12 √ π | s | sin (cid:32) √ s / · / (cid:96) / (cid:33) (A11) × exp (cid:18) − s / · / (cid:96) / (cid:19) . The non-local modification is logarithmically divergenton the light cone and decreases exponentially fast forlarge spacelike and timelike distances. We plot the func-tion ∆ G ( s ) as well as its asymptotics in Fig. 5. Theexponential suppression happens in accordance with De-Witt’s asymptotic causality criterion [49] which statesthat any causal Green function must satisfylim t (cid:48) − t →−∞ G ( t (cid:48) − t ; x (cid:48) − x ) = 0 , (A12)that is, if the effect precedes the cause arbitrarily, anycausal Green function must vanish. Since local causalGreen functions satisfy DeWitt’s criterion identically—since they are proportional to θ ( t (cid:48) − t )—we only need toverify that the non-local modification satisfies condition(A12), which it does, as can be seen from Eq. (A11). Appendix B: Proof of Eq. (35) using theSokhotski–Plemelj theorem
The Sokhotski–Plemelj theorem may be stated as fol-lows:
For any continuous function f ( x ) one has f ( x − x ) x − x ± i(cid:15) = p.v. x f ( x − x ) x − x ∓ iπδ ( x − x ) . (B1)4These expressions are understood under the integral sign,lim (cid:15) → b ˆ a d x f ( x − x ) x − x ± i(cid:15) = b a d x f ( x − x ) x − x ∓ iπf ( x ) . (B2)First, let us consider ¯ X ∈ M R ∪ M L . Then the function − f (¯ ρ ) / ¯ X has no poles since ¯ X (cid:54) = 0 in that region.This means that (35) is satisfied trivially: the Cauchyprincipal value integral reduces to the standard integral,and the δ -term does not contribute since the momen-tum is spacelike. In other words, in this domain the i(cid:15) -prescription is not necessary and we may simply set (cid:15) = 0.If ¯ X ∈ M ¯F ∪ M ¯P , we define σ ≡ ( σ ¯ u − σ ¯ v ) / σ = 1 in M ¯F and σ = − M ¯P . Then one can show − f (¯ ρ ) − (¯ t − i(cid:15) ) + ¯ x + ¯ y + ¯ z ≈ − f (¯ ρ ) σ ¯ u σ ¯ v ¯ ζ + ¯ ρ + i ¯ ζ (cid:0) σ ¯ u e ¯ τ − σ ¯ v e − ¯ τ ) (cid:15) ≈ − f (¯ ρ ) (cid:104) ¯ ρ − (cid:112) ¯ ζ − iσ ¯ ζ(cid:15) (cid:105)(cid:104) ¯ ρ + (cid:112) ¯ ζ − iσ ¯ ζ(cid:15) (cid:105) ≈ − f (¯ ρ ) (cid:0) ¯ ρ − ¯ ζ + iσε (cid:1)(cid:0) ¯ ρ + ¯ ζ − iσε (cid:1) = 12¯ ζ (cid:34) − f (¯ ρ )¯ ρ − ¯ ζ + iσε − − f (¯ ρ )¯ ρ + ¯ ζ − iσε (cid:35) = p.v. ¯ ρ − f (¯ ρ )¯ ρ − ¯ ζ + iσπf (¯ ζ ) δ (2) (¯ ρ − ¯ ζ ) , (B3)where in several lines we have rescaled (cid:15) by a positiveconstant. Utilizing this relation in Eq. (24) and the fol-lowing steps, one readily obtains Eq. (35) as written inthe main body of the paper. [1] C. Kiefer, Quantum Gravity , Vol. 3 (Oxford UniversityPress, Oxford, 2014).[2] S. B. Giddings, “Nonviolent nonlocality,” Phys. Rev. D , 064023 (2013), arXiv:1211.7070 [hep-th].[3] E. T. 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