Transmission of Information in Non-Local Field Theories
Alessio Belenchia, Dionigi M. T. Benincasa, Stefano Liberati, Eduardo Martin-Martinez
TTransmission of Information in Non-Local Field Theories
Alessio Belenchia, Dionigi M. T. Benincasa,
2, 3
Stefano Liberati,
2, 3 and Eduardo Mart´ın-Mart´ınez
4, 5, 6 Institute for Quantum Optics and Quantum Information (IQOQI), Boltzmanngasse 3 1090 Vienna, Austria. SISSA - International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy. INFN, Sezione di Trieste, Trieste, Italy. Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo, Ontario, N2L 2Y5, Canada
The signaling between two observers in 3+1 dimensional flat spacetime coupled locally to a non-local field is considered. We show that in the case where two observers are purely timelike related –so that an exchange of on-shell massless quanta cannot occur – signaling is still possible because of aviolation of Huygens’ principle. In particular, we show that the signaling is exponentially suppressedby the non-locality scale. Furthermore, we consider the case in which the two observers are light-likerelated and show that the non-local modification to the local result is polynomially suppressed inthe non-locality scale. This may have implications for phenomenological tests of non-local theories.
I. INTRODUCTION
Assuming standard local dynamics, the geometry ofspacetime fully determines how free particles propagate.In particular, the strong Huygens’ principle (SHP) statesthat for flat spacetimes of even dimensions, d >
2, theGreen function of a free massless theory only has supporton the lightcone, that is, information propagates onlyalong null geodesics. In the presence of curvature how-ever (or for flat spacetimes in an odd number of spacetimedimensions) the SHP for massless particles ceases to holdin general [1–4], so that information can in principle betransmitted inside the light cone, similarly to what hap-pens with free massive particles in d = 4. Violation ofthe SHP implies that two observers, Alice (the sender)and Bob (the receiver), both locally coupled to a mass-less quantum field, are able to communicate informationvia the field without necessarily having Alice send real,on-shell quanta to Bob [5]. In fact, it was further shownin [5] that this timelike information can be accessed byBob only if he is willing to pay for it (by spending en-ergy), and that no energy is directly transmitted fromAlice to Bob, as would be expected given that no on-shell quanta was exchanged. For this reason this processwas dubbed “quantum collect calling”.Nonlocal field theories provide another example of the-ories for which the SHP ceases to hold even in d = 4 flatspacetime. In particular, we have in mind a class of free,massless, nonlocal scalar field theories whose dynamicsare defined by f ( (cid:3) ) φ = 0, where f is a non-analyticfunction with a branch cut for timelike momenta p ≤ f ( (cid:3) )[6, 8–13].In this work we investigate the signaling between twoobservers, Alice (the sender) and Bob (the receiver), liv- ing in a flat 4-dimensional spacetime, locally coupled toa non-local, massless scalar field via two-level Unruh-DeWitt particle detectors. Since the nonlocal field vio-lates the SHP one might expect a non-vanishing signalingcontribution between the two observers even when theyare purely timelike separated, something that would beimpossible (in the absence of reflecting mirrors) if thefield satisfied the usual local dynamics. Furthermore, onemight also expect a modification to the signaling whenthe observers are in light-like contact, since the responseof a single Unruh–DeWitt detector alone is already mod-ified by the non-locality significantly [14]. As we willshow, in the timelike case the signaling is exponentiallysuppressed in the non-locality scale, whereas a polyno-mial suppression is present in the case of light-like con-tact between the detectors. While the latter case opensan interesting window for constraining the non-localityscale, the former indicates that Huygens violations aresensitive to the UV structure of the theory.The paper is organized as follows. In section II we re-view how signaling between two observers is computedin perturbation theory. In section III, we introduce thenon-local field theory under investigation and, in partic-ular, its Pauli–Jordan function (expectation of the fieldcommutator), which is the central object of this work. Insection IV we compute the non-local contribution to thesignaling between Alice and Bob in the cases where theyare timelike related and lightlike related. In section Vwe discuss the findings of our work and in section VI weconclude with a summary and outlook. II. SETUP
We will consider a setup consisting of two partners. Asender (Alice) and a receiver (Bob) operating quantumantennas modeled as Unruh–DeWitt particle detectors[15]. As well as being very simple, these particle detectormodels also have the advantage of capturing the funda-mental features of light-matter interaction in scenarioswhere the exchange of angular momentum does not play a r X i v : . [ qu a n t - ph ] J u l a prominent role [16–18].This signalling setup has been studied in previousworks regarding the exchange of information in localquantum field theory [5, 19–29].The Unruh–DeWitt interaction Hamiltonian couplingan inertial particle detector to a scalar field is given by H I,ν = λ ν χ ν ( t ) m ν ( t ) φ [ x ( t ) , t ] , (1)where ν ∈ { A,B } is a label indicating Alice’s andBob’s detectors, λ ν are coupling constants, χ ν ( t ) are theswitching functions controlling the coupling-decouplingspeed and the duration of the detector-field interac-tion, m ν ( t ) = ( σ + ν e iΩ ν t + σ − ν e − iΩ ν t ) are the detector’smonopole moment (Ω ν is the energy gap between de-tector ν ’s two energy levels), and φ [ x ν ( t ) , t ] is the fieldevaluated along the detector’s trajectories. In the follow-ing we set c = 1.We are going to consider the same signalling setup asin [5, 23, 26, 30]. Let each detector start out in thepure state ρ ,ν = | ψ ,ν (cid:105)(cid:104) ψ ,ν | , where | ψ ,ν (cid:105) = α ν | e ν (cid:105) + β ν | g ν (cid:105) with | g ( e ) ν (cid:105) the ground state and excited state ofdetector ν respectively, and let the field start out in anarbitrary state ρ ,φ . Hence, the initial state of the systemis ρ = ρ , a ⊗ ρ , b ⊗ ρ ,φ . (2)Allowing the system to evolve under the full interactionHamiltonian H i ( t ) = H i , a ( t ) + H i , b ( t ) for a time T resultsin the state ρ T = U ρ U † , where U is the time evolutionoperator U = T exp (cid:20) − i (cid:90) ∞−∞ d tH i ( t ) (cid:21) , (3)and T denotes time-ordering of the exponential. Thefinal state of Bob’s detector is obtained by tracing outthe field and the state of Alice from ρ T : ρ T, B = tr φ, a ( ρ T ) . (4)Consider the situation in which Alice — the sender —turns on her detector for a time T and, subsequently,Bob — the receiver — turns on his own detector . Theprobability that Bob (who is in the causal future of Alice)finds his detector in the excited state after a time T ∗ > T from when he turns it on can be written as P e ( T ∗ ) = | α | + R ( T ∗ ) + S ( T ∗ ) . (5)The first term in (5) stands for the probability of theinitial state to be excited, R ( T ∗ ) is noise due to the cou-pling of Bob’s detector to the field for a finite time, andthe last term is the signaling term that depends on Alicebeing coupled to the field in the past. Here we will consider the case in which Alice and Bob are at restrelative to each other.
At leading order in perturbation theory the signalingterm can be written as S = λ a λ b S + O ( λ ν ) where S isgiven by (see [5]) S =4 λ b λ a (cid:90) dt (cid:90) dt χ a (t ) χ b (t )Re( α ∗ a β a e iΩ a t ) (6) Re ( α ∗ b β b e i Ω b t [ φ ( x a , t ) , φ ( x b , t )]) , the χ s are the detectors’ switching functions and theintegration is over the times t of Alice and Bob re-spectively.The last expression contains the Pauli–Jordan func-tion of the field theory. This guarantees that, if Huy-gens’ principle holds, the signaling between timelike re-lated detectors vanishes. Moreover, in a causal theory ,the signaling between spacelike observers also vanishes aswell so that no faster-than-light signaling is allowed.The signalling term allows for communication of Alicewith Bob, even if they are timelike separated, as long asthe commutator between the spacetime regions where Al-ice and Bob exist is non-vanishing. Lower bounds to thechannel capacity (in bits per use of the channel) havebeen studied by setting up a concrete communicationprotocol in several scenarios in flat [5, 31] and curvedspacetimes [23, 26, 30]. Notice that protocols that op-timize the choice of the initial state of Alice’s detectorand the measured observable on Bob’s detector have alsobeen considered in previous literature [28].In 4D flat spacetime it is not possible to communicatethrough a local massless scalar field, when the partiesare purely timelike related, since the commutator is non-zero only between light-connected events. However non-localities in the theory will induce modifications of thefield commutator that will manifest in two ways: theywill enable some form of timelike communication andthey will modify the channel capacity through lightlikecommunication. III. NON-LOCAL FIELD THEORY:PAULI–JORDAN FUNCTION
We are interested in a massless scalar field with dynam-ics given by a real, retarded, Poincar´e invariant operator, (cid:101) (cid:3) := f ( (cid:3) ). It can be shown [32] that operators of thiskind have Fourier transforms which depend both on k –as one would expect from a Lorentz invariant operator –and sgn( k ); the latter property being a consequence ofthe retarded nature of these operators. That is (cid:101) (cid:3) e i k µ x µ = B (sgn( k ) , k ) e i k µ x µ . (7) We assume supp( χ A ) ∪ supp( χ B ) = ∅ . A theory which respects (micro)-causality, i.e., [ φ ( x ) , φ ( y )] = 0 ∀ x , y spacelike. The dependence on sgn( k ) implies that the function B possesses a branch cut along k ≤ D (+) ( x − y ) = (cid:90) d k(2 π ) (cid:102) W ( k ) e i k · ( x − y ) , (8)with (cid:102) W ( k ) = 2Im( B ) θ ( k ) | B | . (9) (cid:102) W can be rewritten as (cid:102) W ( k ) = 2 π (cid:101) ρ ( − k ) where (cid:101) ρ canbe further split into a divergent part and a finite part: (cid:101) ρ ( µ ) = δ ( µ ) + ρ ( µ ). With this is mind one can seethat D (+) can actually be written as a sum of two parts:one being the standard Wightman function for a localmassless scalar field, D (+)0 , and the other an integral overthe Wightman function of a local massive field, G (+) µ ,weighted by the finite part of the discontinuity function, ρ ( µ ), i.e. D (+) ( x − y ) = (cid:90) d k(2 π ) πθ ( k ) δ ( k ) e i k · ( x − y ) (10)+ (cid:90) ∞ d µ ρ ( µ ) (cid:90) d k(2 π ) πθ (k ) δ (k + µ )e i k · ( x − y ) . For every choice of (cid:101) (cid:3) there corresponds a specific ρ . Inthis paper we are interested in a discontinuity functionof the form ρ ( µ ) = (cid:96) e − α(cid:96) µ . (11)where α is an order one numerical coefficient [33]. Thischoice of ρ is a simple function which captures all thefundamental features of more complex spectral densitiesin previous literature (see [6, 34] and references therein). A. Pauli-Jordan
Generalizing the previous discussion on the Wightmanfunction to the Pauli-Jordan function it should be clearthat the latter will also decompose into a sum of a localmassless Pauli-Jordan function and the integral over allmasses of massive Pauli-Jordan functions weighted by thespectral density, i.e.[ φ ( x ) , φ ( y )] = [ φ ( x ) , φ ( y )] + (cid:90) d µ ρ ( µ )[ φ ( x ) , φ ( y )] µ , (12)where [ φ ( x ) , φ ( y )] µ represents the Pauli-Jordan for a fieldof mass µ and is given by [35][ φ ( x ) , φ ( y )] µ = [ φ ( x ) , φ ( y )] − µ π √− σ Θ( − σ ) J ( µ √− σ ) , (13) with σ = − ∆ t + ∆ x and J is a Bessel function of thefirst kind. Thus, we can rewrite the commutator of thenon-local field as[ φ ( x ) , φ ( y )] = [ φ ( x ) , φ ( y )] + 1 α [ φ ( x ) , φ ( y )] (14)+ (cid:90) d µ ρ ( µ ) (cid:18) − µ π √− σ Θ( − σ )J ( µ √− σ ) (cid:19) , where the 1 /α in the second term is given by the integralover all µ of (11). From now on we will set α = 1 forconvenience of notation. For σ <
0, the last term on theRHS of (14) is given by − π(cid:96) e σ (cid:96) , (15)so that the commutator takes the following form[ φ ( x ) , φ ( y )] = 2[ φ ( x ) , φ ( y )] − π(cid:96) e σ (cid:96) Θ( − σ ) . (16) B. Distributional local limit
At first sight the above expression may seem puzzlingbecause the local result appears to have been modifiedby an (cid:96) -independent term (having chosen α = 1, a factorof two appears in front of the local result). However, thisis just an artefact. Indeed, the quantity − π(cid:96) e σ (cid:96) (17)converges, in a weak sense, to − δ ( σ ) / π for vanishing (cid:96) .This can be checked explicitly by integrating the aboveexpression against test functions and then taking the lo-cal limit — we will see this in the next sections. Thus,in the local limit the Pauli–Jordan function convergesweakly to the local result — the second term in (14) be-ing cancelled by the weak limit of the last one. IV. NON-LOCAL SIGNALING CONTRIBUTION
Given the splitting of the Pauli-Jordan function we canwrite S in (6) as S = S local + S non − local , (18)where S non − local = S local + S ( (cid:96) )2 and the second term isthe one that is dependent on the non-locality scale (cid:96) . Thelocal limit is recovered since S ( (cid:96) )2 converges to − S local inthe (cid:96) → α X and β X are real for X = A, B , and also that Ω a = Ω b = Ω.The (cid:96) dependent term in (18) then becomes S ( (cid:96) )2 = − α b β b α a β a (cid:90) dt χ b (t ) (cid:90) dt χ a (t ) (19) × cos(Ω t ) cos(Ω t ) 18 π(cid:96) e − ∆ t R (cid:96) Θ( − σ ) . We consider different configurations of the two detectorsfor various switching functions.
A. Bob in the lightband of Alice: polynomialsuppression
We start by investigating the case in which Bob is in-side the lightband of Alice, see Fig.1. In this case the twoobservers are in lightlike contact and can exchange realquanta of a massless field. Signaling in this configurationis therefore allowed both in the local and non-local the-ory, and we are interested in how the non-locality modi-fies the non-vanishing S computed from a local theory.As our calculations will show, the nonlocal correctionto the signaling between Alice and Bob in this case ispolynomially suppressed in the nonlocality scale. Fur-thermore, we will argue that the polynomial suppressiondoes not depend on the UV details of the discontinuityfunction.
1. Case 1: delta-switching
Consider the case in which Alice’s detector is suddenlyswitched on at 0 and then suddenly back off at T (notethat while this would introduce divergences in the detec-tor’s response in 3+1D that would have to be regularized,as discussed in [5, 23], the signalling term is devoid of anyUV divergences), whereas Bob’s detector is on only at t = τ , with τ > T . We also assume that R < τ < R + T ,where R is the constant spatial distance between the twoobservers. This is tantamount to assuming Bob is insideAlice’s light band (see the left panel in Fig.1). In thiscase S ( (cid:96) )2 ∝ (cid:90) dt (cid:90) T0 dt cos(Ωt ) cos(Ωt ) 18 π(cid:96) e − (t1 − t2)2+R24 (cid:96) (20) × Θ(( t − t ) − R ) δ ( t − τ )= (cid:90) T dt cos(Ωt ) cos(Ω τ ) 18 π(cid:96) e − (t1 − τ )2+R24 (cid:96) × Θ(( τ − t ) − R )= (cid:90) τ − R dt cos(Ωt ) cos(Ω τ ) 18 π(cid:96) e − (t1 − τ )2+R24 (cid:96) . The integral above can be computed analytically and theresult is given by S ( (cid:96) )2 = − α b β b α a β a κ √ π(cid:96) (cid:104) cos( τ Ω) e R (cid:96) − Ω ( (cid:96) Ω+i τ )(21) × (cid:18) erf (cid:16) τ (cid:96) − i (cid:96) Ω (cid:17) + erfc (cid:18) R (cid:96) − i (cid:96) Ω (cid:19) − e τ Ω (cid:18) erf (cid:16) τ (cid:96) + i (cid:96) Ω (cid:17) − erf (cid:18) R (cid:96) + i (cid:96) Ω (cid:19)(cid:19)(cid:19)(cid:21) , where erf and erfc are the error function and comple-mentary error function respectively. Note here that κ has dimensions of length (needed to dimensionally bal-ance Bob’s delta-function switching) and characterizeshow much of an energy perturbation we introduce withthe “kick”. Consistently with our previous discussion,the local limit of the above expression — obtained bytaking the limit for vanishing (cid:96) — coincides with minus S local , i.e. the signaling in the local case which is givenby S local = 2 kα b β b α a β a π (cid:90) dt (cid:90) T0 dt δ ((t − t ) − R )(22) × cos(Ω t ) cos(Ω t )Θ(( t − t ) − R ) δ ( t − τ )= kα b β b α a β a cos(Ω τ ) cos(Ω τ − Ω R ) πR , where τ > t by assumption. The fact that S ( (cid:96) )2 weaklyconverges to − S local for vanishing (cid:96) — and thus Eq.(18)gives back the local result — is consistent with the non-local theory reducing, in the same limit, to the local one.In order to determine the leading order correction tothe local result when the non-locality scale is small com-pared to every other scale in the problem, we need to ex-pand the whole non-local contribution to the signaling: S local + S ( (cid:96) )2 (see Eq.(18)). The leading order correctionis given by kα b β b α a β a (cid:96) πR [ R Ω (sin(Ω R ) + sin( R Ω − τ Ω)) (23)+ cos(Ω R ) + cos( R Ω − τ Ω)] . This expression shows that the non-local contribution tothe signaling is polynomial in the non-locality scale, afact that resembles the polynomial modification of theresponse of a single Unruh–DeWitt detector coupled toa non-local field [14].
2. Case 2: Bob extended interaction
We now consider the case in which Bob’s switchingprofile is the same as Alice’s, that is, sudden switching onand off for a finite amount of time, so that Bob interactswith the field for a finite period of time (see right panel inFig.1). We will show that the polynomial suppression ofthe non-local signaling contribution persists in this case,as one might have expected.The non-local signaling contribution with these switch-ing functions is given by S ( (cid:96) )2 = − α b β b α a β a (cid:90) ba dt (cid:90) T0 dt cos(Ωt ) cos(Ωt )(24) × π(cid:96) e − ∆ t R (cid:96) Θ( − σ ) . t x Alice Bob τ R 0 T t x
Alice Bob R a b FIG. 1. Left, spacetime diagram sketching the configuration of the two detectors in which Bob is in the “lightband” of Aliceand has a delta-switching detector. Right, same configuration as before with Bob detector “on” for a finite amount of time.
Using the following change of variables − y ≡ − ∆ t + R (25) t = t − (cid:112) R + y, (26) dt = − (cid:112) R + y dy, (27)we can rewrite S ( (cid:96) )2 as S ( (cid:96) )2 =4 α b β b α a β a (cid:90) ba dt cos(Ω b t ) (cid:90) max[0 , (t − T) − R ]t − R (28) × dy2 (cid:112) R + y cos(Ω a ( t − (cid:112) R + y ) 18 π(cid:96) e − y (cid:96) . Note that we have not yet assumed that Bob is insideAlice’s lightband. If the two observers are purely timelikerelated then t > R + T and ( t − T ) − R > ∀ t ∈ [ a, b ],while whenever t ≤ R + T the two detectors will belightlike related. In the latter case the dependence on T drops out of the signaling because we are only requiringthat Bob be inside the lightband of Alice, i.e. b < T + R , so that no information on the position of the innerboundary of Alice’s lightband is needed, since the signalto Bob comes from Alice’s detector from t = 0 to t = b − R .Since here we are interested in the case in which Bobis inside the lightband of Alice, the expression for S ( (cid:96) )2 becomes S ( (cid:96) )2 = − α b β b α a β a (cid:90) ˜ b ˜ a d˜t cos(˜t) (cid:90) − ˜R (29) × d ˜ y (cid:113) ˜ R + ˜ y cos(˜ t − (cid:113) ˜ R + ˜ y ) 18 π ˜ (cid:96) e − ˜ y (cid:96) , where we have introduced dimensionless variables — de-noted by tildes — defined in units of Ω, ˜ a ≥ R and˜ a < ˜ b ≤ R + T . This integral can be computed analyt-ically and the result is given in appendix A. As before,the local limit of S ( (cid:96) )2 coincides with − S local . The leadingcontribution to the signaling coming from S ( (cid:96) )2 + S local isagain polynomial in (cid:96) as is shown in Fig.2. In the par-ticular case in which a = R and b = R + T this correction(after inserting back dimensional quantities) is given by − α b β b α a β a (cid:96) RT sin(Ω R ) + sin(Ω( R + 2 T )) + Ω(3 R + 2 T ) cos(Ω R ) + Ω R cos(Ω( R + 2 T )) − sin(Ω R )2 π Ω R . (30)Finally, in this case it can be also verified (by numericalmeans) that the polynomial nature of the correction tothe local result is independent of the specific UV details In the following we drop the tilde for notational convenience. of the discontinuity function ρ , and indeed holds whenone approximates ρ simply by l , which is exactly whatdetermines the polynomial suppression. ����� ����� ����� ����� ������� - �� �� - � �� - � �� - � FIG. 2. Numerical integration of eq.(29) vs. non-locality scale (cid:96) . Here we neglected the coupling constants λ ν , α ν , β ν as wellas multiplicative numerical factors. We chose R = 7, a = 8, b = 8 .
1, Ω = 1. (cid:96) goes from 10 − to 10 − . The (red) pointsrepresents the numerical values of S whereas the continuum(blue) line is their interpolation with a curve quadratic in (cid:96) . t x Alice Bob
FIG. 3. Spacetime diagram sketching the configuration of thetwo detectors in which Bob is timelike to Alice and has adelta-switching detector.
B. Bob timelike to Alice: violations of the SHP
In the previous section we investigated the case inwhich Alice and Bob are allowed to communicate bothin the non-local and local theory. In this section we con-sider the case in which Bob is purely timelike related toAlice, see Fig.3. In a local massless theory (and in theabsence of any mirrors) Bob would not be able to receiveany information from Alice in this configuration, since noreal quanta can be exchanged by the two. In the contextof a non-local massless theory however, where the SHPfails to hold, one faces the possibility of establishing acommunication channel between the two observers, evenwhen they are timelike related. From a phenomenologi-cal point of view this situation is particularly interestingsince it allows for a binary test to asses the presencenon-local effects. However, for this to be of any practi- cal interest, we first need to determine how the signalingbetween Bob and Alice is suppressed by the non-localityscale.Consider the simple case where Alice’s detector isswitched on for a finite amount of time, from 0 to T ,abruptly, while Bob’s detector is on only for an instantof time at τ > R + T , see Fig.3. In this case S is givenby S ( (cid:96) )2 ∝ (cid:90) dt (cid:90) T0 dt cos(Ωt ) cos(Ωt ) 18 π(cid:96) e − (t1 − t2)2+R24 (cid:96) (31) × Θ(( t − t ) − R ) δ ( t − τ )= (cid:90) T dt cos(Ωt ) cos(Ω τ ) 18 π(cid:96) e − (t1 − τ )2+R24 (cid:96) . The integral can be computed and gives S ( (cid:96) )2 = − α b β b α a β a √ π(cid:96) cos( τ Ω) e R (cid:96) − Ω ( (cid:96) Ω+i τ ) (32) × (cid:18) erf (cid:16) τ (cid:96) − i (cid:96) Ω (cid:17) + erf (cid:18) T − τ (cid:96) + i (cid:96) Ω (cid:19) + e τ Ω (cid:18) erf (cid:16) τ (cid:96) + i (cid:96) Ω (cid:17) + erf (cid:18) T − τ (cid:96) − i (cid:96) Ω (cid:19)(cid:19)(cid:19) The local limit (cid:96) → τ → T + R , equation(32) becomes of order l , implying that the polynomialcorrection to the local result arises from contributionscoming from a future-timelike neighbourhood of Alice’sfuture lightband boundary (in the case where Alice hasa delta switching the future lightband boundary reducesto Alice’s future lightcone).Finally, expanding the result for small (cid:96) shows thatin general the suppression of the signaling is exponen-tial in the non-locality scale. Similar results hold if Bobinteracts with the field for a finite amount of time. Soeven if in principle we have a binary test for such non-local phenomena, the fact that the effect is exponentiallysmall makes it of no practical use, unless some ampli-fication effects can be introduced. Note also that, con-trary to the lightlike case, in this setup the signaling isno longer determined by the IR behaviour of the theoryalone (provided the distance from the future boundaryof the sender’s lightcone is greater than the non-localityscale), since the explicit form of the suppression dependson the full ρ . Crucially, this result holds true in the limit of vanishing (cid:96) only if τ approaches T + R faster than the rate at which (cid:96) vanishes. V. DISCUSSION
We have shown that when Alice and Bob are lightlikerelated (meaning that they have the possibility of ex-changing real quanta of a massless field) the predictionfor the signaling coming from a non-local theory differsfrom the local one by a correction suppressed only poly-nomially in the non-locality scale. When Alice and Bobare timelike related, which precludes signaling in a mass-less, local theory, the correction due to non-locality is ex-ponentially suppressed in (cid:96) , unless Bob is in a neighbour-hood of Alice’s light of cone of size smaller than O ( (cid:96) ).Given the leading order correction to the signalingwhen Alice and Bob are lightike related and both interactwith the field for a finite amount of time, we can com-pute the ratio R = (cid:0) S − S local (cid:1) / S local . From eq.(18)we know that the ratio is given by R = S ( nl )2 S local , (33)which vanishes in the limit of vanishing (cid:96) . The leadingorder contribution to the above expression is suppressedby (cid:96) . It is interesting to note that the oscillating char-acter of the local result (see Appendix A), together withthe fact that the local result and the non-local correction(eq. (30)) have different zeros in general, allows for anamplification of the non-local signal. Of course, such anamplification requires a fine tuning of the parameters ofthe system. Another interesting case is the one of degen-erate detectors, i.e, vanishing Ω. In this case we get fromeq.(30) a particularly simple expression for R , R = 8 (cid:96) R (cid:18) R + TT (cid:19) . (34)In both cases the suppression in the non-locality scaleshows that in order to cast stringent constraints on thenon-locality scale using signaling between light-like re-lated observers a large sample of events is necessary. Thisis similar to what happens in the case where the responseof a single Unruh–DeWitt detector is used to cast con-straints on the non-locality scale, see [14].As already discussed in previous sections, this polyno-mial suppression follows from the IR properties of thespectral function of the non-local theory, i.e. that ρ ≈ (cid:96) for µ →
0, and is independent of the UV details of ρ .To further validate this fact we have numerically com-puted the signalling for the case in which both Bob andAlice are interacting with the field for a finite amount oftime when ρ is set to (cid:96) . This again gives the expectedpolynomial suppression in (cid:96) .Regarding the case of time-like related observers, wehave shown that the signaling is exponentially suppressedin the non-locality scale — as far as Bob is bounded awayfrom a neighborhood of Alice’s light-cone. Let us stressonce again that, the form of the suppression depends onthe specific spectral function chosen and cannot be de-rived just by looking at the IR behaviour of the spec- tral function. In order to retrieve a polynomial suppres-sion consistent with just the IR property of the spectralfunction, Bob’s detector has to be placed (impractically)close to Alice’s lightcone: closer than the non-localityscale we would like to probe. This fact, in conjunctionwith the observation made at the end of section IV A 2,that the polynomial correction is only sensitive to theIR behaviour of ρ , strongly suggests that the polyno-mial correction is not sensitive to the UV also in thisscenario. Indeed, since the signal that arises in the casewhere Bob and Alice are exactly lightlike related doesnot depend on the nonlocality scale, it is hard to imag-ine where else the correction may come from if not froma neighbourhood of Alice’s lightcone. Ideally one wouldwant to check this explicitly by repeating the calculationof section IV B with ρ = l . Unfortunately we have beenunable to perform this computation as of yet, meaningthat a definitive claim on this matter is still out of reach.Finally note that the exponential suppression of S ,when the detectors are timelike related, is not a univer-sal feature but a consequence of the particular form ofthe spectral function Eq. (11). Nevertheless, this de-pendence is interesting since it entails a dependence ofHuygens’ principle violations on the UV structure of thenon-local theory and promotes them to potential probesof it.
VI. CONCLUSIONS AND OUTLOOK
In this work we have investigated how the signalingbetween two observers locally coupled to a scalar field ismodified by a non-local dynamics for the field. Consider-ing the local interaction of two Unruh–DeWitt detectorswith a non-local scalar field, we have shown that when-ever communication is allowed in the local theory, thenon-locality introduces modifications that are polynomi-ally suppressed by the non-locality scale. Furthermore, inthe case where the two observers are timelike related —which does not allow for communication in the local the-ory — the signaling is non-vanishing for a non-local fieldtheory, in accordance with the observation that there areSHP violations.We have explicitly shown that the signaling is expo-nentially suppressed in the non-locality scale making thiseffect phenomenologically irrelevant, unless amplificationprocesses are introduced. However, the exponential sup-pression shows that SHP violations are sensitive to theUV structure of the theory. If this was not the case, weshould have obtained a polynomial suppression dictated It is interesting to note that, in causal set theory — where thesame kind of non-local field operators we are discussing werefirstly derived from first principles — the spectral functions,while not having the same functional form as the simplified oneused in this work, need to decay in the UV faster than any poly-nomial in order for the local limit to be recovered. by the IR behavior. Accordingly, the specific form ofthe suppression is related to the functional form of thespectral function used.The findings of this work lay down the basis for futurephenomenological studies of non-locality exploiting Huy-gens’ principle violations as well as other vacuum effects.The non-local corrections to S , eqs.(23),(30), imply apolynomial in (cid:96) correction to the capacity of the quan-tum communication channel between Alice and Bob —defined in terms of bits per use of the channel in [5].This opens up the possibility to envisage efficient com-munication protocols that, together with a high statisticof events, could allow for stringent constraints on thenon-locality scale.It should be noted that, an analogous polynomial sup-pression of the response of a single Unruh–DeWitt detec-tor was used in [14] to argue for the possibility to castconstraints on the non-locality scale outperforming high-energy experiments. In [14], the effect of non-locality wasshown to be more relevant in the case of spontaneousemission, with respect to excitation due to vacuum fluc-tuations of the field.Following the same logic, it would be interesting toconsider which initial state of Alice and Bob’s detectorsmaximize the signature of non-locality in a more realis-tic communication protocol. A simple optimization oninitial states is not expected to bring about a differentpower dependence on the non-locality scale (as one cananticipate from the results in [28]), but a more involvedcommunication protocol may provide an opportunity toaccumulate signal and, similar to [14], compensate forthe small value of the non-locality scale.Furthermore, the observation that the polynomial sup-pression is due to modes of the field localized in aneighborhood of the sender’s light-cone gives an indi-cation on to where to look for significant deformationsof other effects involving two detectors’ interaction withthe field’s vacuum — in particular, entanglement har-vesting [18, 36, 37].Finally, in order to arrive at a watertight phenomeno-logical study of non-locality with Unruh–DeWitt detec-tors, a crucial step is the extension of the model to anAbelian U (1) gauge theory. This extension could be fromfirst principles, like it was done for the scalar field incausal set quantum gravity [32], or — more conserva-tively — motivated by the requirement to maintain thesame analytic structure for propagators while imposinggauge invariance on the theory. This would result in anon-local electrodynamics and in new effects related tothe vectorial nature of non-local fields in such a model. We leave the exploration of this avenue for future works. ACNOWLEDGEMENTS
AB and DMTB would like to thank Medhi Saravanifor helpful discussion concerning the distributional na-ture of the local limit of the Pauli-Jordan function.DMTB and SL would like to acknowledge financial sup-port from the John Templeton Foundation (JTF), grantNo. 51876. AB wish to acknowledge the support ofthe Austrian Academy of Sciences through Innovations-fonds ”Forschung, Wissenschaft und Gesellschaft“, andthe University of Vienna through the research platformTURIS. E.M-M was partially funded by the NSERC Dis-covery grant programme. E. M-M would like to thankAchim Kempf for his helpful insights.
Appendix A: Calculation inside the lightband
In this appendix we perform the computation ofEq.(28) (that we rewrite here for convenience) S ( (cid:96) )2 = − α b β b α a β a (cid:90) ˜ b ˜ a dt cos( t ) (cid:90) t − ˜ R (A1) d ˜ y (cid:113) ˜ R + ˜ y cos( t − (cid:113) ˜ R + ˜ y ) 18 π ˜ (cid:96) e − ˜ y (cid:96) . In order to compute this double integral we proceed asfollow. Firstly we compute the inner integral, i.e., (cid:90) t − R dy (cid:112) R + y cos( t − (cid:112) R + y ) 18 π(cid:96) e − y (cid:96) (A2)= ie R (cid:96) − (cid:96) (cid:0) (cid:61) (cid:0) e i t erfi (cid:0) (cid:96) − i R (cid:96) (cid:1)(cid:1) + 2i (cid:61) (cid:0) e − i t erfi (cid:0) (cid:96) + i t (cid:96) (cid:1)(cid:1)(cid:1) √ π(cid:96) It should be noted that, in accordance with the discussionin the main text about the distributional local limit of thePauli-Jordan function, in the local limit this expressionreduces to − cos( R − t ) / πR , i.e., to the result of (cid:90) t − R dy (cid:112) R + y cos( t − (cid:112) R + y ) (cid:18) − δ ( y )2 π (cid:19) . (A3)Now, the remaining integral can be performed givingthe final result S ( (cid:96) )2 = − α b β b α a β a (A4) e R l √ π(cid:96) (cid:96)e − a l cos( a ) √ π + e − (cid:96) (cid:18) (cid:60) (cid:18)(cid:0) e ib − e ia (cid:1) erfi (cid:18) (cid:96) − i R (cid:96) (cid:19)(cid:19) − a − b ) (cid:61) (cid:18) erfi (cid:18) (cid:96) + i R (cid:96) (cid:19)(cid:19)(cid:19) + e − (cid:96) (cid:16)(cid:0) (cid:96) − (cid:1) (cid:61) (cid:16) erf (cid:16) a (cid:96) − i (cid:96) (cid:17)(cid:17) − (cid:61) (cid:16) e − ia erf (cid:16) a (cid:96) − i (cid:96) (cid:17)(cid:17) + 4 a (cid:60) (cid:16) erf (cid:16) a (cid:96) − i (cid:96) (cid:17)(cid:17)(cid:17) − le − b l cos( b ) √ π + 2 e − l (cid:18)(cid:0) − l (cid:1) (cid:61) (cid:18) erf (cid:18) b (cid:96) − i (cid:96) (cid:19)(cid:19) + (cid:61) (cid:18) e b erf (cid:18) b (cid:96) − i (cid:96) (cid:19)(cid:19) − b (cid:60) (cid:18) erf (cid:18) b (cid:96) − i (cid:96) (cid:19)(cid:19)(cid:19)(cid:21) . 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